diff --git a/.nojekyll b/.nojekyll index 53f99e67..68a1edf7 100644 --- a/.nojekyll +++ b/.nojekyll @@ -1 +1 @@ -ef04abae \ No newline at end of file +a822e0bc \ No newline at end of file diff --git a/Key-Notes-in-Plasma-Physics.pdf b/Key-Notes-in-Plasma-Physics.pdf index b38718d0..6d4cbc89 100644 Binary files a/Key-Notes-in-Plasma-Physics.pdf and b/Key-Notes-in-Plasma-Physics.pdf differ diff --git a/contents/shock.html b/contents/shock.html index 31683bd7..24c79559 100644 --- a/contents/shock.html +++ b/contents/shock.html @@ -523,11 +523,11 @@

where \[ -\mathbf{S} = \rho\,\mathbf{V}\,\mathbf{V} + \left(p+ \frac{B^2}{2\mu_0}\right)\mathbf{I}- \frac{\mathbf{B}\mathbf{B}}{\mu_0} +\mathbf{S} = \rho\,\mathbf{U}\,\mathbf{U} + \left(p+ \frac{B^2}{2\mu_0}\right)\mathbf{I}- \frac{\mathbf{B}\mathbf{B}}{\mu_0} \tag{21.2}\] is the total (i.e., including electromagnetic, as well as plasma, contributions) stress tensor, \(\mathbf{I}\) the identity tensor, \[ -K = \frac{1}{2}\rho V^2 + \frac{p}{\gamma-1} + \frac{B^2}{2\mu_0} +K = \frac{1}{2}\rho U^2 + \frac{p}{\gamma-1} + \frac{B^2}{2\mu_0} \tag{21.3}\] the total energy density, and \[ -\mathbf{w} = \left(\frac{1}{2}\rho V^2+ \frac{\gamma}{\gamma -1}p\right)\mathbf{V} + \frac{\mathbf{B}\times (\mathbf{V}\times \mathbf{B})}{\mu_0} +\mathbf{w} = \left(\frac{1}{2}\rho U^2+ \frac{\gamma}{\gamma -1}p\right)\mathbf{U} + \frac{\mathbf{B}\times (\mathbf{U}\times \mathbf{B})}{\mu_0} \tag{21.4}\] the total energy flux density.

Let us move into the rest frame of the shock. For a 1D shock, suppose that the shock front coincides with the \(y\)-\(z\) plane. Furthermore, let the regions of the plasma upstream and downstream of the shock, which are termed regions 1 and 2, respectively, be spatially uniform and time-static, i.e. \(\partial/\partial t = \partial/\partial y = \partial/\partial z = 0\). Moreover, \(\partial/\partial x=0\), except in the immediate vicinity of the shock. Finally, let the velocity and magnetic fields upstream and downstream of the shock all lie in the x-y plane. The situation under discussion is illustrated in the figure below.

@@ -1142,6 +1142,33 @@

21.4.4 Shock Structure

+

Figure 21.9 shows observations from one of the first unambiguous satellite crossings of a quasi-perpendicular supercritical (magnetosonic Mach number \(M_\text{ms} \sim 4.2\)) shock at the Earth’s bow shock.

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+Figure 21.9: Time profiles of plasma and magnetic field parameters across a real quasi-perpendicular shock that had been crossed by the ISEE 1 and 2 spacecraft on November 7, 1977 in near-Earth space (Sckopke et al. 1983). \(N_E\) is the electron density, \(N_i\) the reflected energetic ion density, both in \(\text{cm}^{-3}\), \(T_P\), \(T_E\) are proton and electron in K. \(V_P\) is the proton (plasma) bulk velocity in \(\text{km}\,\text{s}^{-1}\), \(P_e\) electron pressure in \(10^{-9}\,\text{N}\,\text{m}^{-2}\), B the magnitude of the magnetic field in nT, and \(\theta_{Bn}\). The vertical lines mark the first appearance of reflected ion, the outer edge of the foot in the magnetic profile, and the ramp in the field magnitude, respectively. The abscissa is the Universal Time UT referring to the measurements. The upper block are observations from ISEE 1, the lower block observations from ISEE 2. +
+
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+
+

21.4.4.1 Observational evidence

+

The crossing occurred on an inbound path of the two spacecraft ISEE 1 (upper block of the figure) and ISEE 2 (lower block of the figure) from upstream to downstream in short sequence only minutes apart. In spite of some differences occurring on the short time scale the two shock crossings are about identical, identifying the main shock transition as a spatial and not as a temporal structure. Temporal variations are nevertheless visible on the scale of a fraction of a minute.

+

In this case, \(\theta_{Bn}\) is close to \(90^\circ\) prior to shock crossing (in the average \(\theta_{Bn} \sim 85^\circ\)), and fluctuates afterwards around \(90^\circ\) identifying the shock as quasi-perpendicular. Accordingly, the shock develops a foot in front of the shock ramp as can be seen from the slightly enhanced magnetic field after 22:51 UT in ISEE 1 and similar in ISEE 2, and most interestingly also in the electron pressure. At the same time the bulk flow velocity starts decreasing already, as the result of interaction and retardation in the shock foot region. The foot is also visible in the electron density which increases throughout the foot region, indicating the presence of electrons which, as is suggested by the increase in pressure, must have been heated or accelerated.

+

The best indication of the presence of the foot is, however, the measurement of energetic ions (second panel from top). These ions are observed first some distance away from the shock but increase drastically in intensity when entering the foot. These are the shock-reflected ions which have been accelerated in the convection electric field in front of the shock ramp. Their occurrence before entrance into the foot is understood when realising that the shock is not perfectly perpendicular. Rather it is quasi-perpendicular such that part of the reflected ions having sufficiently large parallel upstream velocities can escape along the magnetic field a distance larger than the average upstream extension of the foot. For nearly perpendicular shocks, this percentage is small.

+

The shock ramp in Figure 21.9 is a steep wall in B and \(P_E\), respectively. The electron temperature \(T_E\) increases only moderately across the shock while the ion temperature \(T_P\) jumps up by more than one magnitude, exceeding \(T_E\) downstream behind the shock. This behaviour is due to the accelerated returning foot-ions which pass the shock. \(P_E\), \(B\), and \(N_E\) exhibit overshoots behind the shock ramp proper. Farther away from the shock they merge into the highly fluctuating state of lesser density, pressure, and magnetic field that can be described as some kind of turbulence. Clearly, this region is strongly affected by the presence of the shock which forms one of its boundaries, the other boundary being the obstacle (Earth) which is the main responsible for the formation of the shock.

+

The evidence provided by the described measurements suggests that the quasi-perpendicular shock is a quasi-stationary entity. This should, however, not been taken apodictic. Stationarity depends on the spatial scales as well as the time scales. A shock is a very inhomogeneous subject containing all kinds of spatial scales; being stationary on one scale does not imply that it is stationary on another scale. For a shock like the Earth’s bow shock considered over times of days, weeks or years the shock is of course a stationary subject. However on shorter time scales of the order of flow transition times this may not be the case. A subcritical shock may well be stationary on long and short time scales. However, for a supercritical shock the conditions for forming a stationary state are quite subtle. From a single spacecraft passage like that described above it cannot be concluded to what extent, i.e. on which time scale and on which spatial scale and under which external conditions (Mach number, angle, shock potential, plasma-β,…) the observed shock can be considered to be stationary (a discussion of the various scales has been given, e.g, by Galeev et al, 1988). Comparison between the two ISEE spacecraft already shows that the small-scale details as have been detected by both spacecraft are very different. This suggests that – in this case – on time scales less than a minute variations in the shock structure must be expected.

+

Generally spoken, one must be prepared to consider the shock locally (on the ion gyroscale) and temporarily (on the ion-cyclotron frequency scale) as a non-stationary phenomenon (this has been realised first by Morse et al, 1972) which depends on many competing processes and, most important though only secondarily related to non-stationarity, a shock as a whole is not in thermal equilibrium. It needs to be driven by some energy source external to the shock in order to be maintained. It will thus be very sensitive to small changes in the external parameters and will permanently try to escape the non-equilibrium state and to approach equilibrium. Since its non-equilibrium is maintained by the conditions in the flow, it is these conditions which determine the time scales over which a shock evolves, re-evolves and changes its state. Real supercritical shocks, whether quasi-perpendicular or quasi-parallel, are in a permanently evolving state and thus are intrinsically nonstationary.

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21.4.4.2 Simulation studies

+

Perpendicular or quasi-perpendicular collisionless shocks are relatively easy to treat in numerical simulations. Already from the first one-dimensional numerical experiments on collisionless shocks it became clear that such shocks have a very particular structure. This structure, which we have describe in simplified version in Figure 21.8 and which could to some extent also be inferred from the observations of Figure 21.9, becomes ever more pronounced the more refined the resolution becomes and the better the shorter scales can be resolved.

+

As already mentioned, collisionless shocks are in thermodynamic non-equilibrium and therefore can only evolve if a free energy source exists and if the processes are violent enough to build up and maintain a shock. Usually in a freely evolving system the free energy causes fluctuations which serve dissipating and redistributing the free energy towards thermodynamic and thermal equilibria. Thermal equilibria are characterised by equal temperatures among the different components, e.g. \(T_e = T_i\) which is clearly not given in the vicinity of a shock as seen from Figure 21.9. Thermodynamic equilibria are characterized by Gaussian distributions for all components of the plasma. To check this requires information about the phase space distribution of particles. Shocks contain many differing particle distributions, heated, top-flat, beam distributions, long energetic tails, and truncated as well as gyrating distributions which we will encounter later. Consequently, they are far from thermodynamic equilibrium.

+

For a shock to evolve the amount of free energy needed to dissipate is so large that fluctuations are unable to exercise their duty. This happens at large Mach numbers. The shock itself takes over the duty of providing dissipation. It does it in providing all kinds of scales such short that a multitude of dissipative processes can set on.

+

Scales

+
@@ -1157,7 +1184,7 @@

-Figure 21.9: The patchwork model of J. Schwartz Steven and David (1991) of a quasi-parallel supercritical shock reformation. Left: Magnetic pulsations (SLAMS) grow in the ion foreshock and are convected toward the shock where they accumulate, thereby causing formation of an irregular shock structure. Note also the slight turning of the magnetic field into a direction that is more perpendicular to the shock surface with the shock surface itself becoming very irregular. Right: The same model with the pulsations being generated in the relatively broad ULF-wave-unstable region in greater proximity to the ion-foreshock boundary. When the ULF waves evolve to large amplitude and form localised structures these are convected toward the shock, grow, steepen, overlap, accumulate and lead to the build up of the irregular quasi-parallel shock structure which overlaps into the downstream direction. +Figure 21.10: The patchwork model of J. Schwartz Steven and David (1991) of a quasi-parallel supercritical shock reformation. Left: Magnetic pulsations (SLAMS) grow in the ion foreshock and are convected toward the shock where they accumulate, thereby causing formation of an irregular shock structure. Note also the slight turning of the magnetic field into a direction that is more perpendicular to the shock surface with the shock surface itself becoming very irregular. Right: The same model with the pulsations being generated in the relatively broad ULF-wave-unstable region in greater proximity to the ion-foreshock boundary. When the ULF waves evolve to large amplitude and form localised structures these are convected toward the shock, grow, steepen, overlap, accumulate and lead to the build up of the irregular quasi-parallel shock structure which overlaps into the downstream direction.
@@ -1257,14 +1284,14 @@

warm beams (\(v\sim v_b\))
  • hot beams (\(v\gg v_b\))
    • -

    Figure 21.10 shows the beam configurations for these three cases and the location of the wave resonances respectively the position of the unstable frequencies.

    +

    Figure 21.11 shows the beam configurations for these three cases and the location of the wave resonances respectively the position of the unstable frequencies.

    -Figure 21.10: The three cases of ion beam-plasma interaction and the location of the unstable frequencies. Shown is the parallel (reduced) distribution function \(F_i(k_\parallel v_\parallel)\), where for simplicity the (constant) parallel wavenumber \(k_\parallel\) has been included into the argument. Right handed resonant modes (RH) are excited by a cool not too fast beam. When the beam is too fast the interaction becomes nonresonant. When the beam is hot, a resonant left hand mode (LH) is excited. In addition the effect of temperature anisotropy is shown when a plateau forms on the distribution function (after Gary 1993). +Figure 21.11: The three cases of ion beam-plasma interaction and the location of the unstable frequencies. Shown is the parallel (reduced) distribution function \(F_i(k_\parallel v_\parallel)\), where for simplicity the (constant) parallel wavenumber \(k_\parallel\) has been included into the argument. Right handed resonant modes (RH) are excited by a cool not too fast beam. When the beam is too fast the interaction becomes nonresonant. When the beam is hot, a resonant left hand mode (LH) is excited. In addition the effect of temperature anisotropy is shown when a plateau forms on the distribution function (after Gary 1993).
    @@ -1291,7 +1318,7 @@

    21.6.3 Whistlers

    -

    Gary (1993) has investigated the case of whistler excitation by an electron beam. He finds from numerical solution of the full dispersion relation including an electron beam in parallel motion that with increasing beam velocity \(v_b\) the real frequency of the unstable whistler decreases, i.e. the unstably excited whistler shifts to lower frequencies on the whistler branch while remaining in the whistler range \(\omega_{ci} < \omega < \omega_{ce}\). Both the background electrons and beam electrons contribute resonantly. The most important finding is that the whistler mode for sufficiently large \(\beta_i \sim 1\) (which means low magnetic field), \(n_b/n_e\) and \(T_b/T_e\) has the lowest beam velocity threshold when compared with the electrostatic electron beam instabilities as shown in Figure 21.11. This finding implies that in a relatively high-β plasma a moderately dense electron beam will first excite whistler waves. In the shock environment the conditions for excitation of whistlers should thus be favourable whenever an electron beam propagates across the plasma along the relatively weak magnetic field. The electrons in resonance satisfy \(v_\parallel = (\omega - \omega_{ce})/k_\parallel\) and, because \(\omega \ll \omega_{ce}\) the resonant electrons move in the direction opposite to the beam. Enhancing the beam temperature increases the number of resonant electrons thus feeding the instability.

    +

    Gary (1993) has investigated the case of whistler excitation by an electron beam. He finds from numerical solution of the full dispersion relation including an electron beam in parallel motion that with increasing beam velocity \(v_b\) the real frequency of the unstable whistler decreases, i.e. the unstably excited whistler shifts to lower frequencies on the whistler branch while remaining in the whistler range \(\omega_{ci} < \omega < \omega_{ce}\). Both the background electrons and beam electrons contribute resonantly. The most important finding is that the whistler mode for sufficiently large \(\beta_i \sim 1\) (which means low magnetic field), \(n_b/n_e\) and \(T_b/T_e\) has the lowest beam velocity threshold when compared with the electrostatic electron beam instabilities as shown in Figure 21.12. This finding implies that in a relatively high-β plasma a moderately dense electron beam will first excite whistler waves. In the shock environment the conditions for excitation of whistlers should thus be favourable whenever an electron beam propagates across the plasma along the relatively weak magnetic field. The electrons in resonance satisfy \(v_\parallel = (\omega - \omega_{ce})/k_\parallel\) and, because \(\omega \ll \omega_{ce}\) the resonant electrons move in the direction opposite to the beam. Enhancing the beam temperature increases the number of resonant electrons thus feeding the instability.

    On the other hand, increasing the beam speed shifts the particles out of resonance and decreases the instability. Hence for a given beam temperature the whistler instability has a maximum growth rate a few times the ion cyclotron frequency.

    @@ -1299,7 +1326,7 @@

    -Figure 21.11: The regions of instability of the electron beam excited whistler mode in density and beam velocity space for two different β compared to the ion acoustic and electron beam modes. Instability is above the curves. The whistler instability has the lowest threshold in this parameter range (after Gary 1993). +Figure 21.12: The regions of instability of the electron beam excited whistler mode in density and beam velocity space for two different β compared to the ion acoustic and electron beam modes. Instability is above the curves. The whistler instability has the lowest threshold in this parameter range (after Gary 1993).
    @@ -1321,18 +1348,18 @@

    <
    -Figure 21.12: The two cases of shock reflection. Left: Reflection from a potential well \(\Phi(x)\). Particles of energy higher than the potential energy \(e\Phi\) can pass while lower energy particles become reflected. Right: Reflection from the perpendicular shock region at a curved shock wave as the result of magnetic field compression. Particles move toward the shock like in a magnetic mirror bottle, experience the repelling mirror force and for large initial pitch angles are reflected back upstream. +Figure 21.13: The two cases of shock reflection. Left: Reflection from a potential well \(\Phi(x)\). Particles of energy higher than the potential energy \(e\Phi\) can pass while lower energy particles become reflected. Right: Reflection from the perpendicular shock region at a curved shock wave as the result of magnetic field compression. Particles move toward the shock like in a magnetic mirror bottle, experience the repelling mirror force and for large initial pitch angles are reflected back upstream.
    -

    Due to this simplistic picture the shock ramp should contain a steep increase in the electric potential \(\Delta\Phi\) which will reflect any ion which has less kinetic energy \(m_i V_N^2/2 < e\Delta\Phi\) (Figure 21.12).

    +

    Due to this simplistic picture the shock ramp should contain a steep increase in the electric potential \(\Delta\Phi\) which will reflect any ion which has less kinetic energy \(m_i V_N^2/2 < e\Delta\Phi\) (Figure 21.13).

    -

    Another simple possibility for particle reflection from a shock ramp in magnetised plasma is mirror reflection. An ion approaching the shock has components \(v_{i\parallel}\). Assume a curved shock like Earth’s bow shock. Close to its perpendicular part where the upstream magnetic field becomes tangential to the shock the particles approaching the shock with the stream and moving along the magnetic field with their parallel velocities experience a mirror magnetic field configuration that results from the converging magnetic field lines near the perpendicular point (Figure 21.12). Conservation of the magnetic moment \(\mu = T_{i\perp}/B\) implies that the particles become heated adiabatically in the increasing field; they also experience a reflecting mirror force \(-\mu \nabla_\parallel B\) which tries to keep ions away from entering the shock along the magnetic field. Particles will mirror at the perpendicular shock point and return upstream when their pitch angle becomes 90◦ at this location. (??? Leroy & Mangeney, 1984; Wu, 1984)

    +

    Another simple possibility for particle reflection from a shock ramp in magnetised plasma is mirror reflection. An ion approaching the shock has components \(v_{i\parallel}\). Assume a curved shock like Earth’s bow shock. Close to its perpendicular part where the upstream magnetic field becomes tangential to the shock the particles approaching the shock with the stream and moving along the magnetic field with their parallel velocities experience a mirror magnetic field configuration that results from the converging magnetic field lines near the perpendicular point (Figure 21.13). Conservation of the magnetic moment \(\mu = T_{i\perp}/B\) implies that the particles become heated adiabatically in the increasing field; they also experience a reflecting mirror force \(-\mu \nabla_\parallel B\) which tries to keep ions away from entering the shock along the magnetic field. Particles will mirror at the perpendicular shock point and return upstream when their pitch angle becomes 90◦ at this location. (??? Leroy & Mangeney, 1984; Wu, 1984)

    Specular reflection from shocks is the extreme case of shock particle reflection. Other mechanisms like turbulent reflection are, however, not well elaborated and must in any case be investigated with the help of numerical simulations.

    @@ -1344,7 +1371,7 @@

    21.8 Shock Particle Acceleration

    In the context of cosmic rays that have been observed in the interstellar space, medium energy particles refer to ~ few GeV ions and ~ few MeV electrons. Above these ranges relativistic shocks must be considered. Near the Earth’s bow shock the solar wind hydrogen kinetic energy is ~ 1 keV; ~ 10 keV is about the low threshold for energetic ions. Here we limit our discussions first to the non-relativistic case.

    -

    Figure 21.13 shows schematically the process of particle acceleration. Based on early estimations by Fermi (1949), a large number of shock crossings and reflections back and forth is required for the particles to reach energetic cosmic ray level. The scattering process is a stochastic process that is assumed to conserve energy; in particular they should not become involved into excitation of instabilities which consume part of their motional energy. The only actual dissipation that is allowed in this process is dissipation of bulk motional energy from where the few accelerated particles extract their energy gain. This dissipation is also attributed to direct particle loss by either convective transport or the limited size of the acceleration region. Thus this mechanism works until the gyro-radius of the accelerated particle becomes so large that it exceeds the size of the system.

    +

    Figure 21.14 shows schematically the process of particle acceleration. Based on early estimations by Fermi (1949), a large number of shock crossings and reflections back and forth is required for the particles to reach energetic cosmic ray level. The scattering process is a stochastic process that is assumed to conserve energy; in particular they should not become involved into excitation of instabilities which consume part of their motional energy. The only actual dissipation that is allowed in this process is dissipation of bulk motional energy from where the few accelerated particles extract their energy gain. This dissipation is also attributed to direct particle loss by either convective transport or the limited size of the acceleration region. Thus this mechanism works until the gyro-radius of the accelerated particle becomes so large that it exceeds the size of the system.

    The stochastic process implies that the basic equation that governs the process is a phase space diffusion equation in the form of a Fokker-Planck equation \[ \frac{\partial F(\mathbf{p}, \mathbf{x}, t)}{\partial t} + \mathbf{v}\cdot\nabla F(\mathbf{p},\mathbf{x},t) = \frac{\partial}{\partial \mathbf{p}}\cdot\mathbf{D}_{pp}\cdot\frac{\partial F(\mathbf{p}, \mathbf{x}, t)}{\partial \mathbf{p}},\quad \mathbf{D}_{pp}=\frac{1}{2}\left< \frac{\Delta \mathbf{p}\Delta \mathbf{p}}{\Delta t} \right> \] where \(\Delta \mathbf{p}\) is the variation of the particle momentum in the scattering process which happens in the time interval \(\Delta t\), and the angular brackets indicate ensemble averaging. \(\mathbf{D}_{pp}\) is the momentum space diffusion tensor. It is customary to define \(\mu = \cos\alpha\) as the cosine of the particle pitch angle \(\alpha\) and to understand among \(F(p,\mu)\) the gyro-phase averaged distribution function, which depends only on \(p = |\mathbf{p}|\) and \(\mu\).

    @@ -1354,7 +1381,7 @@

    -Figure 21.13: Schematic of the acceleration mechanism of a charged particle in reflection at a quasi-parallel (\(\theta_{Bn} < 45^\circ\)) supercritical shock. The upstream plasma flow (left, \(\mathbf{V}_1 \gg \mathbf{V}_2\)) contains the various upstream plasma modes: upstream waves, shocklets, whistlers, pulsations. The downstream (right) is turbulent. The energetic particle that is injected at the shock to upstream is reflected in an energy gaining collision with upstream waves, moves downstream where it is reflected in an energy loosing collision back upstream. It looses energy because it overtakes the slow waves, but the energy loss is small. Returning to upstream it is scattered a second time again gaining energy. Its initially high energy is successively increased until it escapes from the shock and ends up in free space as an energetic Cosmic Ray particle. The energy gain is on the expense of the upstream flow which is gradually retarded in this interaction. However, the number of energetic particles is small and the energy gain per collision is also small. So the retardation of the upstream flow is much less than the retardation it experiences in the interaction with the shock-reflected low energy particles and the excitation of upstream turbulence. +Figure 21.14: Schematic of the acceleration mechanism of a charged particle in reflection at a quasi-parallel (\(\theta_{Bn} < 45^\circ\)) supercritical shock. The upstream plasma flow (left, \(\mathbf{V}_1 \gg \mathbf{V}_2\)) contains the various upstream plasma modes: upstream waves, shocklets, whistlers, pulsations. The downstream (right) is turbulent. The energetic particle that is injected at the shock to upstream is reflected in an energy gaining collision with upstream waves, moves downstream where it is reflected in an energy loosing collision back upstream. It looses energy because it overtakes the slow waves, but the energy loss is small. Returning to upstream it is scattered a second time again gaining energy. Its initially high energy is successively increased until it escapes from the shock and ends up in free space as an energetic Cosmic Ray particle. The energy gain is on the expense of the upstream flow which is gradually retarded in this interaction. However, the number of energetic particles is small and the energy gain per collision is also small. So the retardation of the upstream flow is much less than the retardation it experiences in the interaction with the shock-reflected low energy particles and the excitation of upstream turbulence.
    @@ -1367,14 +1394,14 @@

    Figure 21.14. Theoretically ((Balogh and Treumann 2013)) any particle which returns from downstream to upstream is accelerated in the upstream flow, even in the absence of any upstream turbulence and scatterings. If the upstream medium is magnetised and is sufficiently extended to host the upstream gyration orbit, pick-up ion energization can happen via the convection electric field \(\mathbf{E}=-\mathbf{V}\times\mathbf{B}\) all along their upstream half-gyrocircles. Alternatively, the upstream turbulence can also cause ion energization.

    +

    In terms of particle acceleration the shock appears as a boundary between two independent regions of different bulk flow parameters which are filled with scattering centres for the particles as sketched in Figure 21.15. Theoretically ((Balogh and Treumann 2013)) any particle which returns from downstream to upstream is accelerated in the upstream flow, even in the absence of any upstream turbulence and scatterings. If the upstream medium is magnetised and is sufficiently extended to host the upstream gyration orbit, pick-up ion energization can happen via the convection electric field \(\mathbf{E}=-\mathbf{V}\times\mathbf{B}\) all along their upstream half-gyrocircles. Alternatively, the upstream turbulence can also cause ion energization.

    -Figure 21.14: Cartoon of the diffusive shock acceleration (left) and shock heating mechanisms [after an sketch by M. Scholer and Hoshino]. In diffusive shock acceleration the particle is scattered around the shock being much faster than the shock. The requirement is the presence of upstream waves and downstream turbulence or waves. In shock heating the particle is a member of the main particle distribution, is trapped for a while at the shock and thereby thermalised and accelerated until leaving the shock. +Figure 21.15: Cartoon of the diffusive shock acceleration (left) and shock heating mechanisms [after an sketch by M. Scholer and Hoshino]. In diffusive shock acceleration the particle is scattered around the shock being much faster than the shock. The requirement is the presence of upstream waves and downstream turbulence or waves. In shock heating the particle is a member of the main particle distribution, is trapped for a while at the shock and thereby thermalised and accelerated until leaving the shock.
    diff --git a/images/Quasi_perp_bow_shock_crossing_ISEE.png b/images/Quasi_perp_bow_shock_crossing_ISEE.png new file mode 100644 index 00000000..17b16d48 Binary files /dev/null and b/images/Quasi_perp_bow_shock_crossing_ISEE.png differ diff --git a/search.json b/search.json index 698f4f88..d094d0bc 100644 --- a/search.json +++ b/search.json @@ -1724,7 +1724,7 @@ "href": "contents/shock.html", "title": "21  Shock", "section": "", - "text": "21.1 MHD Theory\nThe conserved form of MHD equations can be written as: \\[\n\\begin{aligned}\n\\nabla\\cdot\\mathbf{B}=0 \\\\\n\\frac{\\partial \\mathbf{B}}{\\partial t} - \\nabla\\times (\\mathbf{U}\\times \\mathbf{B})=0 \\\\\n\\frac{\\partial\\rho}{\\partial t} + \\nabla\\cdot(\\rho\\,\\mathbf{U})=0 \\\\\n\\frac{\\partial (\\rho\\,\\mathbf{U})}{\\partial t} + \\nabla\\cdot\\mathbf{S}=0 \\\\\n\\frac{\\partial K}{\\partial t} + \\nabla\\cdot\\mathbf{w}=0\n\\end{aligned}\n\\tag{21.1}\\] where \\[\n\\mathbf{S} = \\rho\\,\\mathbf{V}\\,\\mathbf{V} + \\left(p+ \\frac{B^2}{2\\mu_0}\\right)\\mathbf{I}- \\frac{\\mathbf{B}\\mathbf{B}}{\\mu_0}\n\\tag{21.2}\\] is the total (i.e., including electromagnetic, as well as plasma, contributions) stress tensor, \\(\\mathbf{I}\\) the identity tensor, \\[\nK = \\frac{1}{2}\\rho V^2 + \\frac{p}{\\gamma-1} + \\frac{B^2}{2\\mu_0}\n\\tag{21.3}\\] the total energy density, and \\[\n\\mathbf{w} = \\left(\\frac{1}{2}\\rho V^2+ \\frac{\\gamma}{\\gamma -1}p\\right)\\mathbf{V} + \\frac{\\mathbf{B}\\times (\\mathbf{V}\\times \\mathbf{B})}{\\mu_0}\n\\tag{21.4}\\] the total energy flux density.\nLet us move into the rest frame of the shock. For a 1D shock, suppose that the shock front coincides with the \\(y\\)-\\(z\\) plane. Furthermore, let the regions of the plasma upstream and downstream of the shock, which are termed regions 1 and 2, respectively, be spatially uniform and time-static, i.e. \\(\\partial/\\partial t = \\partial/\\partial y = \\partial/\\partial z = 0\\). Moreover, \\(\\partial/\\partial x=0\\), except in the immediate vicinity of the shock. Finally, let the velocity and magnetic fields upstream and downstream of the shock all lie in the x-y plane. The situation under discussion is illustrated in the figure below.\nHere, \\(\\rho_1\\), \\(p_1\\), \\(\\mathbf{U}_1\\), and \\(\\mathbf{B}_1\\) are the downstream mass density, pressure, velocity, and magnetic field, respectively, whereas \\(\\rho_2\\), \\(p_2\\), \\(\\mathbf{U}_2\\), and \\(\\mathbf{B}_2\\) are the corresponding upstream quantities.1\nThe basic RH relations are listed in MHD shocks. In the immediate vicinity of the planar shock, Equation 21.1 reduce to \\[\n\\begin{aligned}\n\\frac{\\mathrm{d}B_{x}}{\\mathrm{d}x}=0 \\\\\n\\frac{\\mathrm{d}}{\\mathrm{d}x}(U_x\\,B_y-U_y\\,B_x)=0 \\\\\n\\frac{\\mathrm{d} (\\rho\\, U_x)}{\\mathrm{d}x}=0 \\\\\n\\frac{\\mathrm{d} S_{xx}}{\\mathrm{d}x}=0 \\\\\n\\frac{\\mathrm{d} S_{xy}}{\\mathrm{d}x}=0 \\\\\n\\frac{\\mathrm{d} w_x}{\\mathrm{d}x}=0\n\\end{aligned}\n\\]\nIntegration across the shock yields the desired jump conditions: \\[\n\\begin{aligned}\n\\lfloor B_x\\rceil=0 \\\\\n\\lfloor U_x\\,B_y-U_y\\,B_x\\rceil=0 \\\\\n\\lfloor \\rho\\,U_x\\rceil=0 \\\\\n\\lfloor \\rho\\,U_x^{\\,2}+p + B_y^{\\,2}/2\\mu_0\\rceil=0 \\\\\n\\lfloor \\rho\\,U_x\\,U_y - B_x\\,B_y/\\mu_0\\rceil=0 \\\\\n\\Big\\lfloor \\frac{1}{2}\\,\\rho\\,U^2\\,U_x + \\frac{\\gamma}{\\gamma-1}\\,p\\,U_x + \\frac{B_y\\,(U_x\\,B_y-U_y\\,B_x)}{\\mu_0}\\Big\\rceil=0\n\\end{aligned}\n\\tag{21.5}\\] where \\(\\lfloor A \\rceil = A_2 - A_1\\) is the difference across the shock. These relations are often called the Rankine-Hugoniot relations for MHD. There are 6 scalar equations and 12 (6 upstream and 6 downstream) scalar variables all together. Assuming that all of the upstream plasma parameters are known, there are 6 unknown parameters in the problem–namely, \\(B_{x\\,2}\\), \\(B_{y\\,2}\\), \\(U_{x\\,2}\\), \\(U_{y\\,2}\\), \\(\\rho_2\\), and \\(p_2\\). These 6 unknowns are fully determined by the six jump conditions. If we loose the planar assumption, then we typically write the \\(x\\)-component as the normal component (\\(U_n, B_n\\)) and the combined y- and z-components as the tangential component (\\(U_t,B_t\\)): \\[\n\\begin{aligned}\n\\lfloor B_n\\rceil=0 \\\\\n\\lfloor U_n\\,B_t-U_t\\,B_n\\rceil=0 \\\\\n\\lfloor \\rho\\,U_n\\rceil=0 \\\\\n\\lfloor \\rho\\,U_n^{\\,2}+p + B_t^{\\,2}/2\\mu_0\\rceil=0 \\\\\n\\lfloor \\rho\\,U_n\\,U_t - B_n\\,B_t/\\mu_0\\rceil=0 \\\\\n\\Big\\lfloor \\frac{1}{2}\\,\\rho\\,U^2\\,U_n + \\frac{\\gamma}{\\gamma-1}\\,p\\,U_n + \\frac{B_t\\,(U_n\\,B_t-U_t\\,B_n)}{\\mu_0}\\Big\\rceil=0\n\\end{aligned}\n\\tag{21.6}\\]\nLuckily this is still deterministic. However, as you can see later, the general case is very complicated.\nA clear exposition of the two types of strong discontinuities, namely the shock wave and the tangential discontinuity can be found in §84, Landau & Lifshitz. By definition\nMathematically, let the shock plane speed be \\(U_s\\), then \\[\n\\begin{aligned}\n\\lfloor\\rho (U_x - U_s)\\rceil&\\neq 0\\quad \\text{for shock} \\\\\n\\lfloor\\rho (U_x - U_s)\\rceil&=0\\quad \\text{for discontinuity}\n\\end{aligned}\n\\tag{21.7}\\]\nThus in shocks \\(\\lfloor U_n \\rceil \\neq 0\\), and in discontinuities \\(\\lfloor U_n \\rceil = 0\\). Take a reference frame fixed to the discontinuity with x-axis along the normal. Since mass, momentum and energy is conserved across the discontinuity, we must have from Equation 21.5 for inviscid flows (no magnetic field, y-direction represents the tangential direction), \\[\n\\begin{aligned}\n\\lfloor \\rho U_x\\rceil=0 \\\\\n\\lfloor \\rho U_x^2 + p \\rceil = 0, \\lfloor \\rho U_x U_y \\rceil=0 \\\\\n\\lfloor \\rho U_x\\left(\\frac{1}{2}U^2 + H\\right)\\rceil=0\n\\end{aligned}\n\\] where \\(H\\) is the enthalpy. In tangential discontinuities, no particle transport means \\(U_{1x} = U_{2x} = 0\\). Then the x-momentum jump implies \\(\\lfloor p \\rceil=0\\), where the y-momentum jump sets no restrictions on \\(U_y\\). There is also no restriction on \\(\\rho\\). Energy equation is also satisfied. Thus, in tangential discontinuities, the density and tangential velocity components can be discontinuities, whereas the pressure must be continuous and the normal velocity component must be zero.\nThe categories of the solution of Equation 21.5 are shown in Table 21.1. The \\(\\pm\\) signs denote the changes of the downstream compared with the upstream (\\(+\\) means increase, \\(-\\) means decrease).\nThe solutions can also be summarized in the context of Riemann problem or visually in Figure 21.2.", + "text": "21.1 MHD Theory\nThe conserved form of MHD equations can be written as: \\[\n\\begin{aligned}\n\\nabla\\cdot\\mathbf{B}=0 \\\\\n\\frac{\\partial \\mathbf{B}}{\\partial t} - \\nabla\\times (\\mathbf{U}\\times \\mathbf{B})=0 \\\\\n\\frac{\\partial\\rho}{\\partial t} + \\nabla\\cdot(\\rho\\,\\mathbf{U})=0 \\\\\n\\frac{\\partial (\\rho\\,\\mathbf{U})}{\\partial t} + \\nabla\\cdot\\mathbf{S}=0 \\\\\n\\frac{\\partial K}{\\partial t} + \\nabla\\cdot\\mathbf{w}=0\n\\end{aligned}\n\\tag{21.1}\\] where \\[\n\\mathbf{S} = \\rho\\,\\mathbf{U}\\,\\mathbf{U} + \\left(p+ \\frac{B^2}{2\\mu_0}\\right)\\mathbf{I}- \\frac{\\mathbf{B}\\mathbf{B}}{\\mu_0}\n\\tag{21.2}\\] is the total (i.e., including electromagnetic, as well as plasma, contributions) stress tensor, \\(\\mathbf{I}\\) the identity tensor, \\[\nK = \\frac{1}{2}\\rho U^2 + \\frac{p}{\\gamma-1} + \\frac{B^2}{2\\mu_0}\n\\tag{21.3}\\] the total energy density, and \\[\n\\mathbf{w} = \\left(\\frac{1}{2}\\rho U^2+ \\frac{\\gamma}{\\gamma -1}p\\right)\\mathbf{U} + \\frac{\\mathbf{B}\\times (\\mathbf{U}\\times \\mathbf{B})}{\\mu_0}\n\\tag{21.4}\\] the total energy flux density.\nLet us move into the rest frame of the shock. For a 1D shock, suppose that the shock front coincides with the \\(y\\)-\\(z\\) plane. Furthermore, let the regions of the plasma upstream and downstream of the shock, which are termed regions 1 and 2, respectively, be spatially uniform and time-static, i.e. \\(\\partial/\\partial t = \\partial/\\partial y = \\partial/\\partial z = 0\\). Moreover, \\(\\partial/\\partial x=0\\), except in the immediate vicinity of the shock. Finally, let the velocity and magnetic fields upstream and downstream of the shock all lie in the x-y plane. The situation under discussion is illustrated in the figure below.\nHere, \\(\\rho_1\\), \\(p_1\\), \\(\\mathbf{U}_1\\), and \\(\\mathbf{B}_1\\) are the downstream mass density, pressure, velocity, and magnetic field, respectively, whereas \\(\\rho_2\\), \\(p_2\\), \\(\\mathbf{U}_2\\), and \\(\\mathbf{B}_2\\) are the corresponding upstream quantities.1\nThe basic RH relations are listed in MHD shocks. In the immediate vicinity of the planar shock, Equation 21.1 reduce to \\[\n\\begin{aligned}\n\\frac{\\mathrm{d}B_{x}}{\\mathrm{d}x}=0 \\\\\n\\frac{\\mathrm{d}}{\\mathrm{d}x}(U_x\\,B_y-U_y\\,B_x)=0 \\\\\n\\frac{\\mathrm{d} (\\rho\\, U_x)}{\\mathrm{d}x}=0 \\\\\n\\frac{\\mathrm{d} S_{xx}}{\\mathrm{d}x}=0 \\\\\n\\frac{\\mathrm{d} S_{xy}}{\\mathrm{d}x}=0 \\\\\n\\frac{\\mathrm{d} w_x}{\\mathrm{d}x}=0\n\\end{aligned}\n\\]\nIntegration across the shock yields the desired jump conditions: \\[\n\\begin{aligned}\n\\lfloor B_x\\rceil=0 \\\\\n\\lfloor U_x\\,B_y-U_y\\,B_x\\rceil=0 \\\\\n\\lfloor \\rho\\,U_x\\rceil=0 \\\\\n\\lfloor \\rho\\,U_x^{\\,2}+p + B_y^{\\,2}/2\\mu_0\\rceil=0 \\\\\n\\lfloor \\rho\\,U_x\\,U_y - B_x\\,B_y/\\mu_0\\rceil=0 \\\\\n\\Big\\lfloor \\frac{1}{2}\\,\\rho\\,U^2\\,U_x + \\frac{\\gamma}{\\gamma-1}\\,p\\,U_x + \\frac{B_y\\,(U_x\\,B_y-U_y\\,B_x)}{\\mu_0}\\Big\\rceil=0\n\\end{aligned}\n\\tag{21.5}\\] where \\(\\lfloor A \\rceil = A_2 - A_1\\) is the difference across the shock. These relations are often called the Rankine-Hugoniot relations for MHD. There are 6 scalar equations and 12 (6 upstream and 6 downstream) scalar variables all together. Assuming that all of the upstream plasma parameters are known, there are 6 unknown parameters in the problem–namely, \\(B_{x\\,2}\\), \\(B_{y\\,2}\\), \\(U_{x\\,2}\\), \\(U_{y\\,2}\\), \\(\\rho_2\\), and \\(p_2\\). These 6 unknowns are fully determined by the six jump conditions. If we loose the planar assumption, then we typically write the \\(x\\)-component as the normal component (\\(U_n, B_n\\)) and the combined y- and z-components as the tangential component (\\(U_t,B_t\\)): \\[\n\\begin{aligned}\n\\lfloor B_n\\rceil=0 \\\\\n\\lfloor U_n\\,B_t-U_t\\,B_n\\rceil=0 \\\\\n\\lfloor \\rho\\,U_n\\rceil=0 \\\\\n\\lfloor \\rho\\,U_n^{\\,2}+p + B_t^{\\,2}/2\\mu_0\\rceil=0 \\\\\n\\lfloor \\rho\\,U_n\\,U_t - B_n\\,B_t/\\mu_0\\rceil=0 \\\\\n\\Big\\lfloor \\frac{1}{2}\\,\\rho\\,U^2\\,U_n + \\frac{\\gamma}{\\gamma-1}\\,p\\,U_n + \\frac{B_t\\,(U_n\\,B_t-U_t\\,B_n)}{\\mu_0}\\Big\\rceil=0\n\\end{aligned}\n\\tag{21.6}\\]\nLuckily this is still deterministic. However, as you can see later, the general case is very complicated.\nA clear exposition of the two types of strong discontinuities, namely the shock wave and the tangential discontinuity can be found in §84, Landau & Lifshitz. By definition\nMathematically, let the shock plane speed be \\(U_s\\), then \\[\n\\begin{aligned}\n\\lfloor\\rho (U_x - U_s)\\rceil&\\neq 0\\quad \\text{for shock} \\\\\n\\lfloor\\rho (U_x - U_s)\\rceil&=0\\quad \\text{for discontinuity}\n\\end{aligned}\n\\tag{21.7}\\]\nThus in shocks \\(\\lfloor U_n \\rceil \\neq 0\\), and in discontinuities \\(\\lfloor U_n \\rceil = 0\\). Take a reference frame fixed to the discontinuity with x-axis along the normal. Since mass, momentum and energy is conserved across the discontinuity, we must have from Equation 21.5 for inviscid flows (no magnetic field, y-direction represents the tangential direction), \\[\n\\begin{aligned}\n\\lfloor \\rho U_x\\rceil=0 \\\\\n\\lfloor \\rho U_x^2 + p \\rceil = 0, \\lfloor \\rho U_x U_y \\rceil=0 \\\\\n\\lfloor \\rho U_x\\left(\\frac{1}{2}U^2 + H\\right)\\rceil=0\n\\end{aligned}\n\\] where \\(H\\) is the enthalpy. In tangential discontinuities, no particle transport means \\(U_{1x} = U_{2x} = 0\\). Then the x-momentum jump implies \\(\\lfloor p \\rceil=0\\), where the y-momentum jump sets no restrictions on \\(U_y\\). There is also no restriction on \\(\\rho\\). Energy equation is also satisfied. Thus, in tangential discontinuities, the density and tangential velocity components can be discontinuities, whereas the pressure must be continuous and the normal velocity component must be zero.\nThe categories of the solution of Equation 21.5 are shown in Table 21.1. The \\(\\pm\\) signs denote the changes of the downstream compared with the upstream (\\(+\\) means increase, \\(-\\) means decrease).\nThe solutions can also be summarized in the context of Riemann problem or visually in Figure 21.2.", "crumbs": [ "21  Shock" ] @@ -1764,7 +1764,7 @@ "href": "contents/shock.html#supercritical-perpendicular-shock", "title": "21  Shock", "section": "21.4 Supercritical Perpendicular Shock", - "text": "21.4 Supercritical Perpendicular Shock\nQuasi-perpendicular shocks are the first and important family of collisionless magnetised shocks which reflect particles back upstream in order to satisfy the shock conditions. Discussion of the particle dynamics gives clear definition for distinguishing them from quasi-parallel shocks by defining a shock normal angle with respect to the upstream magnetic field. They exist for shock normal angles \\(< 45^\\circ\\). Reflected particles at quasi-perpendicular shocks cannot escape far upstream along the magnetic field. After having performed half a gyro-circle back upstream they return to the shock ramp and ultimately traverse it to become members of the downstream plasma population; they also form a foot in front of the shock ramp. We discuss the reflecting shock potential and the explicit shock structure. Most theoretical insight is provided by numerical simulations which confirm reflection, foot formation and reformation of the shock. The latter being caused by steeping of the foot disturbance until the foot itself becomes the shock transition, reflecting particles upstream. Reformation modulates the shock temporarily but on the long terms guarantees its stationarity. Ion and electron dynamics are explicitly discussed in view of the various instabilities involved as well as particle acceleration and shock heating. Finally, a sketchy model of a typical quasi-perpendicular shock transition is provided.\nIn order to help maintain a shock in the supercritical case the shock must forbid an increasing number of ions to pass across its ramp, which is done by reflecting some particles back upstream. This is not a direct dissipation process, rather it is an emergency act of the shock. It throws a fraction of the incoming ions back upstream and by this reduces both the inflow momentum and energy densities. Clearly, this reflection process slows the shock down by attributing a negative momentum to the shock itself. The shock slips back and thus in the shock frame also reduces the difference velocity to the inflow, i.e. it reduces the Mach number. In addition, however, the reflected ions form an unexpected obstacle for the inflow and in this way reduce the Mach number a second time.\n\n21.4.1 Particle Dynamics\nLet’s return to the orbit a particle interacting with a supercritical shock when it becomes reflected from the shock. In the simplest possible model one assumes the shock to be a plane surface, and the reflection being specular turning the component \\(v_n\\) of the instantaneous particle velocity \\(\\mathbf{v}\\) normal to the shock by \\(180^\\circ\\), i.e. simply reflecting it. Here we follow the explicit calculation for these idealised conditions as given by S. J. Schwartz, Thomsen, and Gosling (1983) who treated this problem in the most general way. One should, however, keep in mind that the assumption of ideal specular reflection is the extreme limit of what happens in reality, which is no more than a convenient assumption. In fact, reflection must by no means be specular because\n\nThe shock ramp is not a rigid wall; the particles penetrate into it at least over a distance of a fraction of their gyroradius.\nParticles interact with waves and even excite waves during this interaction and during their approach of the shock.\n\nFigure 5.1 shows the coordinate frame used at the planar (stationary) shock, with shock normal \\(\\hat{n}\\), magnetic \\(\\hat{b}\\) and velocity \\(\\hat{v}\\) unit vectors, respectively. Shown are the angles \\(\\theta_{Bn},\\,\\theta_{Vn},\\,\\theta_{BV}\\). The velocity vector \\(\\mathbf{V}_\\text{HT}\\) is the de Hoffmann-Teller velocity which lies in the shock plane and is defined in such a way that in the coordinate system moving along the shock plane with velocity \\(\\mathbf{V}_\\text{HT}\\) the plasma flow is along the magnetic field, \\(\\mathbf{U} − \\mathbf{V}_\\text{HT} = −v_\\parallel \\hat{b}\\). Because of the latter reason it is convenient to consider the motion of particles in the de Hoffmann-Teller frame. The guiding centers of the particles in this frame move all along the magnetic field. Hence, using \\(\\mathbf{U} = −U\\hat{v}\\), \\(\\hat{n}\\cdot\\hat{v} = \\cos\\theta_{Vn}\\), \\(\\hat{n}\\cdot(\\hat{b},\\,\\hat{x},\\,\\hat{y})=(\\cos\\theta_{Bn},\\sin\\theta_{Bn},0)\\), \\[\nv_\\parallel = U\\frac{\\cos\\theta_{Vn}}{\\cos\\theta_{Bn}},\\quad \\mathbf{V}_\\text{HT} = U\\left( -\\hat{v} + \\frac{\\cos\\theta_{Vn}}{\\cos\\theta_{Bn}}\\hat{b} \\right) = \\frac{\\hat{n}\\times\\mathbf{U}\\times\\mathbf{B}}{\\hat{n}\\cdot\\mathbf{B}},\\quad V_\\text{HT,n}\\equiv 0\n\\tag{21.24}\\]\n\n\n\n\n\n\nFigure 21.6: The shock coordinate system showing the shock normal \\(\\hat{n}\\), velocity and magnetic field directions \\(\\hat{v}\\), \\(\\hat{b}\\), the three angles \\(\\theta_{Bn},\\,\\theta_{Vn},\\,\\theta_{BV}\\) between \\(\\hat{b}\\) and \\(\\hat{n}\\), velocity \\(\\mathbf{U}\\) and \\(\\hat{n}\\), and velocity \\(\\mathbf{U}\\) and \\(\\hat{b}\\), respectively. The velocity \\(\\mathbf{V}_\\text{HT}\\) in the shock plane is the deHoffmann-Teller velocity (after S. J. Schwartz, Thomsen, and Gosling 1983).\n\n\n\nThe de Hoffmann-Teller velocity is the same to both sides of the shock ramp, because of the continuity of normal component \\(B_n\\) and tangential electric field \\(\\mathbf{E}_t\\). Thus, in the de Hoffmann-Teller frame there is no induction electric field \\(\\mathbf{E} = -\\hat{n}\\times\\mathbf{U}\\times\\mathbf{B}\\). The remaining problem is two-dimensional (because trivially \\(\\hat{n}\\), \\(\\hat{b}\\) and \\(-v_\\parallel\\hat{b}\\) are coplanar, which is nothing else but the coplanarity theorem holding under these undisturbed idealized conditions).\nIn the de Hoffmann-Teller (primed) frame the particle velocity is described by the motion along the magnetic field \\(\\hat{b}\\) plus the gyromotion of the particle in the plane perpendicular to \\(\\hat{b}\\): \\[\n\\mathbf{v}^\\prime(t) = v_\\parallel^\\prime\\hat{b} + v_\\perp\\left[ \\hat{x}\\cos(\\omega_{ci}t +\\phi_0) \\mp\\hat{y}\\sin(\\omega_{ci}t + \\phi_0) \\right]\n\\tag{21.25}\\]\nThe unit vectors \\(\\hat{x}\\), \\(\\hat{y}\\) are along the orthogonal coordinates in the gyration plane of the ion, the phase \\(\\phi_0\\) accounts for the initial gyro-phase of the ion, and \\(\\pm\\) accounts for the direction of the upstream magnetic field being parallel (+) or antiparallel (-) to \\(\\hat{b}\\).\nIn specular reflection (from a stationary shock) the upstream velocity component along \\(\\hat{n}\\) is reversed, and hence (for cold ions) the velocity becomes (???) \\[\n\\mathbf{v}^\\prime = -v_\\parallel^\\prime\\hat{b} + 2v_\\parallel \\cos\\theta_{Bn}\\hat{n}\n\\] which (with \\(\\phi_0 = 0\\)) yields for the components of the velocity \\[\n\\frac{v_\\parallel^\\prime}{U} = \\frac{\\cos\\theta_{Vn}}{\\cos\\theta_{Bn}}\\left( 2\\cos^2\\theta_{Bn} - 1 \\right) \\quad \\frac{v_\\perp^\\prime}{U} = 2\\sin\\theta_{Bn}\\cos\\theta_{Vn}\n\\]\nThese expressions can be transformed back into the observer’s frame by using \\(\\mathbf{V}_\\text{HT}\\). It is, however, of greater interest to see under which conditions a reflected particle turns around in its upstream motion towards the shock. This happens when the upstream component of the velocity \\(v_x = 0\\) of the reflected ion vanishes. For this we need to integrate Equation 21.25 which for \\(\\phi_0 = 0\\) yields \\[\n\\mathbf{x}^\\prime(t) = v_\\parallel^\\prime t \\hat{b} + \\frac{v_\\perp}{\\omega_{ci}}\\left[ \\hat{x}\\sin\\omega_{ci}t \\pm\\hat{y}(\\cos\\omega_{ci}t -1) \\right]\n\\]\nScalar multiplication with \\(\\hat{n}\\) yields the ion displacement normal to the shock in upstream direction. The resulting expression \\[\n\\mathbf{x}_n^\\prime(t^\\ast) = v_\\parallel^\\prime t \\cos\\theta_{Bn} + \\frac{v_\\perp}{\\omega_{ci}}\\sin\\theta_{Bn}\\sin\\omega_{ci}t^\\ast = 0\n\\tag{21.26}\\] vanishes at time \\(t^\\ast\\) when the ion re-encounters the shock with normal velocity \\[\nv_n(t^\\ast) = v_\\parallel^\\prime \\cos\\theta_{Bn} + v_\\perp\\sin\\theta_{Bn}\\cos\\omega_{ci}t^\\ast\n\\]\nThe maximum displacement away from the shock in normal direction is obtained when setting this velocity to zero, obtaining for the time \\(t_m\\) at maximum displacement (again including the initial phase here) \\[\n\\omega_{ci}t_m + \\phi_0 = \\cos^{-1}\\left( \\frac{1 - 2\\cos^2\\theta_{Bn}}{2\\sin^2\\theta_{Bn}} \\right)\n\\tag{21.27}\\]\nThis expression must be inserted in \\(\\mathbf{x}_n\\) yielding for the distance a reflected ion with gyroradius \\(r_{ci} = V/\\omega_{ci}\\) can achieve in upstream direction \\[\n\\Delta x_n = r_{ci}\\cos\\theta_{Vn}\\left[ (\\omega_{ci}t_m + \\phi_0)(2\\cos^2\\theta_{Bn} - 1) + 2\\sin^2\\theta_{Bn}\\sin(\\omega_{ci}t_m +\\phi_0) \\right]\n\\tag{21.28}\\]\nFor a perpendicular shock \\(\\theta_{Bn} = 90^\\circ\\) and \\(\\phi_0 = 0\\) this distance is \\(\\Delta x_n \\simeq 0.7 r_{ci} \\cos\\theta{Vn}\\), less than an ion gyro radius. The distance depends on the shock normal angle, decreasing for non-planar shocks. Note that the argument of \\(\\cos^{-1}\\) in Equation 21.27 changes sign for \\(\\theta_{Bn} \\le 45^\\circ\\). Equation 21.26 has solutions for positive upstream turning distances only for shock normal angles \\(\\theta_{Bn} > 45^\\circ\\), for an initial particle phase \\(\\phi_0 = 0\\).9 Reflected ions can return to the shock in one gyration time only when the magnetic field makes an angle with the shock normal that is larger than this value. For less inclined shock normal angles the reflected ions escape along the magnetic field upstream of the shock and do not return within one gyration. This sharp distinction between shock normal angles \\(\\theta_{Bn} < 45^\\circ\\) and \\(\\theta_{Bn} > 45^\\circ\\) thus provides the natural (kinematic specular) discrimination between quasi-perpendicular and quasi-parallel (planar) shocks we were looking for.\nThe theory of shock particle reflection holds, in this form, only for cold ions, which implies complete neglect of any velocity dispersion and proper gyration of the ions. The ions are considered of just moving all with one and the same oblique flow velocity \\(\\mathbf{U}\\). In a warm plasma each particle has a different speed, and it is only the group of bulk velocity ions which are described by the above theory. Fortunately, these are the particles which experience the reflecting shock potential strongest and are most vulnerable to specular reflection. When temperature effects will be included, the theory is more involved in a number of ways:\n\nThe de-Hoffmann-Teller velocity must be redefined to include the microscopic particle motion.\nThe assumption of ideal specular reflection becomes questionable, as the particles themselves become involved into the generation of the shock potential. However, observations in space suggest that, for high flow velocities and supercritical Mach numbers, the simple kinematic reflection is a sufficiently well justified mechanism.\n\nThe formal treatment for warm ions are shown in P153 of Balogh and Treumann (2013). The result is that including thermal effect may\n\nmodify the angle of transition, and\nwill substantially affect the distance up to that a specularly reflected particle at the quasi-perpendicular shock can penetrate the upstream flow, i.e. it affects the width of the quasi-perpendicular shock-foot, even in the case when the reflection process is genuinely specular.\n\n\n\n21.4.2 Foot Formation and Acceleration\nShock reflected ions in a quasi-perpendicular shock cannot escape far upstream (see Figure 21.7). Their penetration into the upstream plasma is severely restricted by Equation 21.28. Within this distance the ions perform a gyration orbit before returning to the shock.\n\n\n\n\n\n\nFigure 21.7: Top: Reflected ion orbits in the foot of a quasi-perpendicular shock in real space. The ion impacts under an instantaneous angle \\(\\theta_{Vn}\\), is reflected from the infinitely thin shock, performs a further partial gyration in the upstream field \\(\\mathbf{B}_1\\) where it is exposed to the upstream convection electric field \\(\\mathbf{E} = -\\mathbf{U}_1 \\times \\mathbf{B}_1\\) in which it is accelerated as is seen from the non-circular section of its orbit in the shock foot. It hits the shock ramp a second time now at energy high enough to overcome the shock potential, passing the ramp and arriving in the compressed downstream magnetic field behind the shock where it performs gyrations of reduced gyro-radius. Bottom: The ion distribution function mapped into velocity space \\(v_x\\), \\(v_y\\) for the indicated regions in real space, upstream in the foot, at the ramp, and downstream of the shock ramp. Upstream the distribution consists of the incoming dense plasma flow (population 1, dark circle at \\(v_y = 0\\)) and the reflected distribution 2 at large negative \\(v_y\\). At the ramp in addition to the incoming flow 1 and the accelerated distribution \\(2^\\prime\\) there is the newly reflected distribution 3. Behind the ramp in the downstream region the inflow is decelerated \\(1^\\prime\\) and slightly deflected toward non-zero \\(v_y\\), and the energised passing ions exhibit gyration motions in different instantaneous phases, two of them (\\(2^{\\prime\\prime}\\), 4) directed downstream, one of them (\\(2^{\\prime\\prime\\prime}\\)) directed upstream. (redrawn after Sckopke et al. 1983).\n\n\n\nSince the reflected ions are about at rest with respect to the inflowing plasma they are sensitive to the inductive convection electric field \\(\\mathbf{E} = -\\mathbf{U}_1 \\times \\mathbf{B}_1\\) behaving very similar to pick-up ions and becoming accelerated in the direction of this field to achieve a higher energy (S. J. Schwartz, Thomsen, and Gosling 1983). When returning to the shock their maximum (minimum) achievable energy is \\[\n\\epsilon_\\text{max} = \\frac{m_i}{2}\\left[ (v_\\parallel^\\prime + V_\\text{HT})^2 + (V_\\text{HT} \\pm v_\\perp)^2 \\right]\n\\tag{21.29}\\]\nThis energy is larger than their initial energy with that they have initially met the shock ramp and, under favourable conditions, they now might overcome the shock ramp potential and escape downstream. Otherwise, when becoming reflected again, they gain energy in a second round until having picked up sufficient energy for passing the shock ramp.\nIn addition to this energization of reflected ions which in the first place have not made it across the shock, the reflected ions when gyrating and being accelerated in the convection electric field constitute a current layer just in front of the shock ramp of current density \\(J_y\\sim e n_\\text{i,refl} v_\\text{y, refl}\\) which gives rise to a foot magnetic field of magnitude \\(B_\\text{z, foot}\\sim \\mu_0 j_y \\Delta x_n\\). It is clear that this foot ion current, which is essentially a drift current in which only the reflected newly energized ion component participates, constitutes a source of free energy as it violates the energetic minimum state of the inflowing plasma in its frame. Being the source of free energy it can serve as a source for excitation of waves via which it will contribute to filling the lack of dissipation. However, in a quasi-perpendicular shock there are other sources of free energy as well which are not restricted to the foot region.\nFigure 21.8 shows a sketch of some of the different free-energy sources and processes across the quasi-perpendicular shock. In addition to the shock-foot current and the presence of the fast cross-magnetic field ion beam there, the shock ramp is of finite thickness. It contains a charge separation electric field \\(E_x\\) which in the supercritical shock is strong enough to reflect the lower energy ions. In addition it accelerates electrons downstream thereby deforming the electron distribution function.\n\n\n\n\n\n\nFigure 21.8: Geometry of an ideally perpendicular supercritical shock showing the field structure and sources of free energy. The shock is a compressive structure. The profile of the shock thus stands for the compressed profile of the magnetic field strength \\(|B|\\), the density N, temperature T, and pressure NT of the various components of the plasma. The inflow of velocity \\(\\mathbf{V}_1\\) and outflow of velocity \\(\\mathbf{V}_2\\) is in the x direction, and the magnetic field is in the z direction. Charge separation over an ion gyroradius \\(r_{ci}\\) in the shock ramp magnetic field generates a charge separation electric field \\(E_x\\) along the shock normal which reflects the low-energy ions back upstream. These ions see the convection electric field \\(E_y\\) of the inflow, which is along the shock front, and become accelerated. The magnetic field of the current carried by the accelerated back-streaming ions causes the magnetic foot in front of the shock ramp. The shock electrons are accelerated antiparallel to \\(E_x\\) perpendicular to the magnetic field. The shock electrons also perform an electric field drift in y-direction in the crossed \\(E_x\\) and compressed \\(B_{z2}\\) fields which leads to an electron current \\(J_y\\) along the shock. These different currents are sources of free energy which drives various instabilities in different regions of the perpendicular shock. (Balogh and Treumann 2013)\n\n\n\nThe presence of this field, which has a substantial component perpendicular to the magnetic field, implies that the magnetized electrons with their gyro-radii being smaller than the shock-ramp width experience an electric drift \\(V_{ye} = -E_x/ B_{z2}\\) along the shock in the ramp which can be quite substantial giving rise to an electron drift current \\(J_{ye} = -en_\\text{e,ramp}V_{ye} = en_\\text{e,ramp}E_x/B_{z2}\\) in the y-direction. This current has again its own contribution to the magnetic field, which at maximum is roughly given by \\(B_z \\sim \\mu_0 J_{ye}\\Delta x_n\\). Here we use the width of the shock ramp. The electron current region might be narrower, of the order of the electron skin depth \\(d_e = c/\\omega_{pe}\\). However, as long as we do not know the number of magnetized electrons which are involved into this current nor the width of the electric field region (which must be less than an ion gyro-radius because of ambipolar effects) the above estimate is good enough.\nThe magnetic field of the electron drift current causes an overshoot in the magnetic field in the shock ramp on the downstream side and a depletion of the field on the upstream side contributing to the steepness of the ramp. When this current becomes strong it contributes to current-driven cross-field instabilities like the modified two-stream instability.\nFinally, the mutual interaction of the different particle populations present in the shock at its ramp and behind provide other sources of free energy. A wealth of instabilities and waves is thus expected to be generated inside the shock. To these micro-instabilities add the longer wavelength instabilities which are caused by the plasma and field gradients in this region. These are usually believed to be less important as the crossing time of the shock is shorter than their growth time. However, some of them propagate along the shock and have therefore substantial time to grow and modify the shock profile. In the following we will turn to the discussion of numerical investigations of some of these processes reviewing their current state and provide comparison with observations.\n\n\n21.4.3 Shock Potential Drop\nOne of the important shock parameters is the electric potential drop across the shock ramp – or if it exists also across the shock foot. This potential drop is not necessarily a constant but changes with location along the shock normal. We have already noted that it is due to the different dynamical responses of the inflowing ions and electrons over the scale of the foot and ramp regions. Its theoretical determination is difficult, however when going to the de Hoffmann-Teller frame the bulk motion of the particles is only along the magnetic field, and in the stationary electron equation of motion the \\(\\mathbf{V}_e \\times \\mathbf{B}\\)-term drops out and, to first approximation, the cross shock potential is given by the pressure gradient (when neglecting any contributions from wave fields). The expression is then simply \\[\n\\Delta \\Phi(x) = \\int_0^x \\frac{1}{eN_e(n)}\\left[ \\nabla\\cdot \\overleftarrow{P}_e(n) \\right]\\mathrm{d}\\mathbf{n}\n\\tag{21.30}\\]\nIntegration is over \\(n\\) along the shock normal \\(\\hat{n}\\). For a gyrotropic electron pressure, valid for length scales longer than an electron gyroradius, \\(\\overleftrightarrow{P}_e = P_{e\\perp}\\mathbf{I} + (P_{e\\parallel} - P_{e\\perp})\\mathbf{B}\\mathbf{B}/BB\\) one obtains (Goodrich and Scudder 1984), taking into account that \\(\\mathbf{E}\\cdot\\mathbf{B}\\) is invariant, \\[\n\\frac{\\mathrm{d}}{\\mathrm{d}n}\\phi(n) = -\\frac{E_\\parallel}{\\cos\\theta_{Bn}} = \\frac{1}{eN_e}\\left[ \\frac{\\mathrm{d}}{\\mathrm{d}n}P_{e\\parallel} - (P_{e\\parallel} - P_{e\\perp})\\frac{\\mathrm{d}}{\\mathrm{d}n}(\\ln B) \\right]\n\\] which, when used in the above expression, yields \\[\ne\\Delta \\Phi(x) = \\int_0^x \\mathrm{d}n\\left\\{ \\frac{\\mathrm{d}T_{e\\parallel}}{\\mathrm{d}n} + T_{e\\parallel}\\frac{\\mathrm{d}}{\\mathrm{d}n}\\ln\\left[ \\frac{N(n)}{N_1}\\frac{B_1}{B(n)} \\right] + T_{e\\perp}\\frac{\\mathrm{d}}{\\mathrm{d}n}\\ln\\left[ \\frac{B(n)}{B_1} \\right] \\right\\}\n\\]\nThis expression can approximately be written in terms of the gradient in the electron magnetic moment \\(\\mu_e = T_{e\\perp}/B\\) as follows: \\[\ne\\Delta\\Phi(x) \\simeq \\Delta(T_{e\\parallel} + T_{e\\perp}) - \\int_0^x \\mathrm{d}n \\frac{\\mathrm{d}\\mu_e(n)}{\\mathrm{d}n}B(n)\n\\] with \\(T_e\\) in energy units. When the electron magnetic moment is conserved, the last term disappears, yielding a simple relation for the potential drop \\(e\\Delta\\Phi(x) \\simeq \\Delta(T_{e\\parallel} + T_{e\\perp})\\) as the sum of the changes in electron temperature. The perpendicular temperature change can be expressed as \\(\\Delta T_{e\\perp} = T_{e\\perp,1}\\Delta B/B_1\\) which is in terms of the compression of the magnetic field. Non-adiabatic effects contribute via the dropped integral term, which breaks the adiabatic invariant.\nThe parallel change in temperature is more difficult to express. One could express it in terms of the temperature anisotropy \\(A_e = T_{e\\parallel} /T_{e\\perp}\\) as has been done by Kuncic et al [2002], and then vary \\(A_e\\). But this depends on the particular model. It is more important to note that this adiabatic estimate of the potential drop does not account for any dynamical process which generates waves and substructures in the shock. It thus gives only a hint on the order of magnitude of the potential drop across the foot-ramp region in quasi-perpendicular shocks.\n\n\n21.4.4 Shock Structure", + "text": "21.4 Supercritical Perpendicular Shock\nQuasi-perpendicular shocks are the first and important family of collisionless magnetised shocks which reflect particles back upstream in order to satisfy the shock conditions. Discussion of the particle dynamics gives clear definition for distinguishing them from quasi-parallel shocks by defining a shock normal angle with respect to the upstream magnetic field. They exist for shock normal angles \\(< 45^\\circ\\). Reflected particles at quasi-perpendicular shocks cannot escape far upstream along the magnetic field. After having performed half a gyro-circle back upstream they return to the shock ramp and ultimately traverse it to become members of the downstream plasma population; they also form a foot in front of the shock ramp. We discuss the reflecting shock potential and the explicit shock structure. Most theoretical insight is provided by numerical simulations which confirm reflection, foot formation and reformation of the shock. The latter being caused by steeping of the foot disturbance until the foot itself becomes the shock transition, reflecting particles upstream. Reformation modulates the shock temporarily but on the long terms guarantees its stationarity. Ion and electron dynamics are explicitly discussed in view of the various instabilities involved as well as particle acceleration and shock heating. Finally, a sketchy model of a typical quasi-perpendicular shock transition is provided.\nIn order to help maintain a shock in the supercritical case the shock must forbid an increasing number of ions to pass across its ramp, which is done by reflecting some particles back upstream. This is not a direct dissipation process, rather it is an emergency act of the shock. It throws a fraction of the incoming ions back upstream and by this reduces both the inflow momentum and energy densities. Clearly, this reflection process slows the shock down by attributing a negative momentum to the shock itself. The shock slips back and thus in the shock frame also reduces the difference velocity to the inflow, i.e. it reduces the Mach number. In addition, however, the reflected ions form an unexpected obstacle for the inflow and in this way reduce the Mach number a second time.\n\n21.4.1 Particle Dynamics\nLet’s return to the orbit a particle interacting with a supercritical shock when it becomes reflected from the shock. In the simplest possible model one assumes the shock to be a plane surface, and the reflection being specular turning the component \\(v_n\\) of the instantaneous particle velocity \\(\\mathbf{v}\\) normal to the shock by \\(180^\\circ\\), i.e. simply reflecting it. Here we follow the explicit calculation for these idealised conditions as given by S. J. Schwartz, Thomsen, and Gosling (1983) who treated this problem in the most general way. One should, however, keep in mind that the assumption of ideal specular reflection is the extreme limit of what happens in reality, which is no more than a convenient assumption. In fact, reflection must by no means be specular because\n\nThe shock ramp is not a rigid wall; the particles penetrate into it at least over a distance of a fraction of their gyroradius.\nParticles interact with waves and even excite waves during this interaction and during their approach of the shock.\n\nFigure 5.1 shows the coordinate frame used at the planar (stationary) shock, with shock normal \\(\\hat{n}\\), magnetic \\(\\hat{b}\\) and velocity \\(\\hat{v}\\) unit vectors, respectively. Shown are the angles \\(\\theta_{Bn},\\,\\theta_{Vn},\\,\\theta_{BV}\\). The velocity vector \\(\\mathbf{V}_\\text{HT}\\) is the de Hoffmann-Teller velocity which lies in the shock plane and is defined in such a way that in the coordinate system moving along the shock plane with velocity \\(\\mathbf{V}_\\text{HT}\\) the plasma flow is along the magnetic field, \\(\\mathbf{U} − \\mathbf{V}_\\text{HT} = −v_\\parallel \\hat{b}\\). Because of the latter reason it is convenient to consider the motion of particles in the de Hoffmann-Teller frame. The guiding centers of the particles in this frame move all along the magnetic field. Hence, using \\(\\mathbf{U} = −U\\hat{v}\\), \\(\\hat{n}\\cdot\\hat{v} = \\cos\\theta_{Vn}\\), \\(\\hat{n}\\cdot(\\hat{b},\\,\\hat{x},\\,\\hat{y})=(\\cos\\theta_{Bn},\\sin\\theta_{Bn},0)\\), \\[\nv_\\parallel = U\\frac{\\cos\\theta_{Vn}}{\\cos\\theta_{Bn}},\\quad \\mathbf{V}_\\text{HT} = U\\left( -\\hat{v} + \\frac{\\cos\\theta_{Vn}}{\\cos\\theta_{Bn}}\\hat{b} \\right) = \\frac{\\hat{n}\\times\\mathbf{U}\\times\\mathbf{B}}{\\hat{n}\\cdot\\mathbf{B}},\\quad V_\\text{HT,n}\\equiv 0\n\\tag{21.24}\\]\n\n\n\n\n\n\nFigure 21.6: The shock coordinate system showing the shock normal \\(\\hat{n}\\), velocity and magnetic field directions \\(\\hat{v}\\), \\(\\hat{b}\\), the three angles \\(\\theta_{Bn},\\,\\theta_{Vn},\\,\\theta_{BV}\\) between \\(\\hat{b}\\) and \\(\\hat{n}\\), velocity \\(\\mathbf{U}\\) and \\(\\hat{n}\\), and velocity \\(\\mathbf{U}\\) and \\(\\hat{b}\\), respectively. The velocity \\(\\mathbf{V}_\\text{HT}\\) in the shock plane is the deHoffmann-Teller velocity (after S. J. Schwartz, Thomsen, and Gosling 1983).\n\n\n\nThe de Hoffmann-Teller velocity is the same to both sides of the shock ramp, because of the continuity of normal component \\(B_n\\) and tangential electric field \\(\\mathbf{E}_t\\). Thus, in the de Hoffmann-Teller frame there is no induction electric field \\(\\mathbf{E} = -\\hat{n}\\times\\mathbf{U}\\times\\mathbf{B}\\). The remaining problem is two-dimensional (because trivially \\(\\hat{n}\\), \\(\\hat{b}\\) and \\(-v_\\parallel\\hat{b}\\) are coplanar, which is nothing else but the coplanarity theorem holding under these undisturbed idealized conditions).\nIn the de Hoffmann-Teller (primed) frame the particle velocity is described by the motion along the magnetic field \\(\\hat{b}\\) plus the gyromotion of the particle in the plane perpendicular to \\(\\hat{b}\\): \\[\n\\mathbf{v}^\\prime(t) = v_\\parallel^\\prime\\hat{b} + v_\\perp\\left[ \\hat{x}\\cos(\\omega_{ci}t +\\phi_0) \\mp\\hat{y}\\sin(\\omega_{ci}t + \\phi_0) \\right]\n\\tag{21.25}\\]\nThe unit vectors \\(\\hat{x}\\), \\(\\hat{y}\\) are along the orthogonal coordinates in the gyration plane of the ion, the phase \\(\\phi_0\\) accounts for the initial gyro-phase of the ion, and \\(\\pm\\) accounts for the direction of the upstream magnetic field being parallel (+) or antiparallel (-) to \\(\\hat{b}\\).\nIn specular reflection (from a stationary shock) the upstream velocity component along \\(\\hat{n}\\) is reversed, and hence (for cold ions) the velocity becomes (???) \\[\n\\mathbf{v}^\\prime = -v_\\parallel^\\prime\\hat{b} + 2v_\\parallel \\cos\\theta_{Bn}\\hat{n}\n\\] which (with \\(\\phi_0 = 0\\)) yields for the components of the velocity \\[\n\\frac{v_\\parallel^\\prime}{U} = \\frac{\\cos\\theta_{Vn}}{\\cos\\theta_{Bn}}\\left( 2\\cos^2\\theta_{Bn} - 1 \\right) \\quad \\frac{v_\\perp^\\prime}{U} = 2\\sin\\theta_{Bn}\\cos\\theta_{Vn}\n\\]\nThese expressions can be transformed back into the observer’s frame by using \\(\\mathbf{V}_\\text{HT}\\). It is, however, of greater interest to see under which conditions a reflected particle turns around in its upstream motion towards the shock. This happens when the upstream component of the velocity \\(v_x = 0\\) of the reflected ion vanishes. For this we need to integrate Equation 21.25 which for \\(\\phi_0 = 0\\) yields \\[\n\\mathbf{x}^\\prime(t) = v_\\parallel^\\prime t \\hat{b} + \\frac{v_\\perp}{\\omega_{ci}}\\left[ \\hat{x}\\sin\\omega_{ci}t \\pm\\hat{y}(\\cos\\omega_{ci}t -1) \\right]\n\\]\nScalar multiplication with \\(\\hat{n}\\) yields the ion displacement normal to the shock in upstream direction. The resulting expression \\[\n\\mathbf{x}_n^\\prime(t^\\ast) = v_\\parallel^\\prime t \\cos\\theta_{Bn} + \\frac{v_\\perp}{\\omega_{ci}}\\sin\\theta_{Bn}\\sin\\omega_{ci}t^\\ast = 0\n\\tag{21.26}\\] vanishes at time \\(t^\\ast\\) when the ion re-encounters the shock with normal velocity \\[\nv_n(t^\\ast) = v_\\parallel^\\prime \\cos\\theta_{Bn} + v_\\perp\\sin\\theta_{Bn}\\cos\\omega_{ci}t^\\ast\n\\]\nThe maximum displacement away from the shock in normal direction is obtained when setting this velocity to zero, obtaining for the time \\(t_m\\) at maximum displacement (again including the initial phase here) \\[\n\\omega_{ci}t_m + \\phi_0 = \\cos^{-1}\\left( \\frac{1 - 2\\cos^2\\theta_{Bn}}{2\\sin^2\\theta_{Bn}} \\right)\n\\tag{21.27}\\]\nThis expression must be inserted in \\(\\mathbf{x}_n\\) yielding for the distance a reflected ion with gyroradius \\(r_{ci} = V/\\omega_{ci}\\) can achieve in upstream direction \\[\n\\Delta x_n = r_{ci}\\cos\\theta_{Vn}\\left[ (\\omega_{ci}t_m + \\phi_0)(2\\cos^2\\theta_{Bn} - 1) + 2\\sin^2\\theta_{Bn}\\sin(\\omega_{ci}t_m +\\phi_0) \\right]\n\\tag{21.28}\\]\nFor a perpendicular shock \\(\\theta_{Bn} = 90^\\circ\\) and \\(\\phi_0 = 0\\) this distance is \\(\\Delta x_n \\simeq 0.7 r_{ci} \\cos\\theta{Vn}\\), less than an ion gyro radius. The distance depends on the shock normal angle, decreasing for non-planar shocks. Note that the argument of \\(\\cos^{-1}\\) in Equation 21.27 changes sign for \\(\\theta_{Bn} \\le 45^\\circ\\). Equation 21.26 has solutions for positive upstream turning distances only for shock normal angles \\(\\theta_{Bn} > 45^\\circ\\), for an initial particle phase \\(\\phi_0 = 0\\).9 Reflected ions can return to the shock in one gyration time only when the magnetic field makes an angle with the shock normal that is larger than this value. For less inclined shock normal angles the reflected ions escape along the magnetic field upstream of the shock and do not return within one gyration. This sharp distinction between shock normal angles \\(\\theta_{Bn} < 45^\\circ\\) and \\(\\theta_{Bn} > 45^\\circ\\) thus provides the natural (kinematic specular) discrimination between quasi-perpendicular and quasi-parallel (planar) shocks we were looking for.\nThe theory of shock particle reflection holds, in this form, only for cold ions, which implies complete neglect of any velocity dispersion and proper gyration of the ions. The ions are considered of just moving all with one and the same oblique flow velocity \\(\\mathbf{U}\\). In a warm plasma each particle has a different speed, and it is only the group of bulk velocity ions which are described by the above theory. Fortunately, these are the particles which experience the reflecting shock potential strongest and are most vulnerable to specular reflection. When temperature effects will be included, the theory is more involved in a number of ways:\n\nThe de-Hoffmann-Teller velocity must be redefined to include the microscopic particle motion.\nThe assumption of ideal specular reflection becomes questionable, as the particles themselves become involved into the generation of the shock potential. However, observations in space suggest that, for high flow velocities and supercritical Mach numbers, the simple kinematic reflection is a sufficiently well justified mechanism.\n\nThe formal treatment for warm ions are shown in P153 of Balogh and Treumann (2013). The result is that including thermal effect may\n\nmodify the angle of transition, and\nwill substantially affect the distance up to that a specularly reflected particle at the quasi-perpendicular shock can penetrate the upstream flow, i.e. it affects the width of the quasi-perpendicular shock-foot, even in the case when the reflection process is genuinely specular.\n\n\n\n21.4.2 Foot Formation and Acceleration\nShock reflected ions in a quasi-perpendicular shock cannot escape far upstream (see Figure 21.7). Their penetration into the upstream plasma is severely restricted by Equation 21.28. Within this distance the ions perform a gyration orbit before returning to the shock.\n\n\n\n\n\n\nFigure 21.7: Top: Reflected ion orbits in the foot of a quasi-perpendicular shock in real space. The ion impacts under an instantaneous angle \\(\\theta_{Vn}\\), is reflected from the infinitely thin shock, performs a further partial gyration in the upstream field \\(\\mathbf{B}_1\\) where it is exposed to the upstream convection electric field \\(\\mathbf{E} = -\\mathbf{U}_1 \\times \\mathbf{B}_1\\) in which it is accelerated as is seen from the non-circular section of its orbit in the shock foot. It hits the shock ramp a second time now at energy high enough to overcome the shock potential, passing the ramp and arriving in the compressed downstream magnetic field behind the shock where it performs gyrations of reduced gyro-radius. Bottom: The ion distribution function mapped into velocity space \\(v_x\\), \\(v_y\\) for the indicated regions in real space, upstream in the foot, at the ramp, and downstream of the shock ramp. Upstream the distribution consists of the incoming dense plasma flow (population 1, dark circle at \\(v_y = 0\\)) and the reflected distribution 2 at large negative \\(v_y\\). At the ramp in addition to the incoming flow 1 and the accelerated distribution \\(2^\\prime\\) there is the newly reflected distribution 3. Behind the ramp in the downstream region the inflow is decelerated \\(1^\\prime\\) and slightly deflected toward non-zero \\(v_y\\), and the energised passing ions exhibit gyration motions in different instantaneous phases, two of them (\\(2^{\\prime\\prime}\\), 4) directed downstream, one of them (\\(2^{\\prime\\prime\\prime}\\)) directed upstream. (redrawn after Sckopke et al. 1983).\n\n\n\nSince the reflected ions are about at rest with respect to the inflowing plasma they are sensitive to the inductive convection electric field \\(\\mathbf{E} = -\\mathbf{U}_1 \\times \\mathbf{B}_1\\) behaving very similar to pick-up ions and becoming accelerated in the direction of this field to achieve a higher energy (S. J. Schwartz, Thomsen, and Gosling 1983). When returning to the shock their maximum (minimum) achievable energy is \\[\n\\epsilon_\\text{max} = \\frac{m_i}{2}\\left[ (v_\\parallel^\\prime + V_\\text{HT})^2 + (V_\\text{HT} \\pm v_\\perp)^2 \\right]\n\\tag{21.29}\\]\nThis energy is larger than their initial energy with that they have initially met the shock ramp and, under favourable conditions, they now might overcome the shock ramp potential and escape downstream. Otherwise, when becoming reflected again, they gain energy in a second round until having picked up sufficient energy for passing the shock ramp.\nIn addition to this energization of reflected ions which in the first place have not made it across the shock, the reflected ions when gyrating and being accelerated in the convection electric field constitute a current layer just in front of the shock ramp of current density \\(J_y\\sim e n_\\text{i,refl} v_\\text{y, refl}\\) which gives rise to a foot magnetic field of magnitude \\(B_\\text{z, foot}\\sim \\mu_0 j_y \\Delta x_n\\). It is clear that this foot ion current, which is essentially a drift current in which only the reflected newly energized ion component participates, constitutes a source of free energy as it violates the energetic minimum state of the inflowing plasma in its frame. Being the source of free energy it can serve as a source for excitation of waves via which it will contribute to filling the lack of dissipation. However, in a quasi-perpendicular shock there are other sources of free energy as well which are not restricted to the foot region.\nFigure 21.8 shows a sketch of some of the different free-energy sources and processes across the quasi-perpendicular shock. In addition to the shock-foot current and the presence of the fast cross-magnetic field ion beam there, the shock ramp is of finite thickness. It contains a charge separation electric field \\(E_x\\) which in the supercritical shock is strong enough to reflect the lower energy ions. In addition it accelerates electrons downstream thereby deforming the electron distribution function.\n\n\n\n\n\n\nFigure 21.8: Geometry of an ideally perpendicular supercritical shock showing the field structure and sources of free energy. The shock is a compressive structure. The profile of the shock thus stands for the compressed profile of the magnetic field strength \\(|B|\\), the density N, temperature T, and pressure NT of the various components of the plasma. The inflow of velocity \\(\\mathbf{V}_1\\) and outflow of velocity \\(\\mathbf{V}_2\\) is in the x direction, and the magnetic field is in the z direction. Charge separation over an ion gyroradius \\(r_{ci}\\) in the shock ramp magnetic field generates a charge separation electric field \\(E_x\\) along the shock normal which reflects the low-energy ions back upstream. These ions see the convection electric field \\(E_y\\) of the inflow, which is along the shock front, and become accelerated. The magnetic field of the current carried by the accelerated back-streaming ions causes the magnetic foot in front of the shock ramp. The shock electrons are accelerated antiparallel to \\(E_x\\) perpendicular to the magnetic field. The shock electrons also perform an electric field drift in y-direction in the crossed \\(E_x\\) and compressed \\(B_{z2}\\) fields which leads to an electron current \\(J_y\\) along the shock. These different currents are sources of free energy which drives various instabilities in different regions of the perpendicular shock. (Balogh and Treumann 2013)\n\n\n\nThe presence of this field, which has a substantial component perpendicular to the magnetic field, implies that the magnetized electrons with their gyro-radii being smaller than the shock-ramp width experience an electric drift \\(V_{ye} = -E_x/ B_{z2}\\) along the shock in the ramp which can be quite substantial giving rise to an electron drift current \\(J_{ye} = -en_\\text{e,ramp}V_{ye} = en_\\text{e,ramp}E_x/B_{z2}\\) in the y-direction. This current has again its own contribution to the magnetic field, which at maximum is roughly given by \\(B_z \\sim \\mu_0 J_{ye}\\Delta x_n\\). Here we use the width of the shock ramp. The electron current region might be narrower, of the order of the electron skin depth \\(d_e = c/\\omega_{pe}\\). However, as long as we do not know the number of magnetized electrons which are involved into this current nor the width of the electric field region (which must be less than an ion gyro-radius because of ambipolar effects) the above estimate is good enough.\nThe magnetic field of the electron drift current causes an overshoot in the magnetic field in the shock ramp on the downstream side and a depletion of the field on the upstream side contributing to the steepness of the ramp. When this current becomes strong it contributes to current-driven cross-field instabilities like the modified two-stream instability.\nFinally, the mutual interaction of the different particle populations present in the shock at its ramp and behind provide other sources of free energy. A wealth of instabilities and waves is thus expected to be generated inside the shock. To these micro-instabilities add the longer wavelength instabilities which are caused by the plasma and field gradients in this region. These are usually believed to be less important as the crossing time of the shock is shorter than their growth time. However, some of them propagate along the shock and have therefore substantial time to grow and modify the shock profile. In the following we will turn to the discussion of numerical investigations of some of these processes reviewing their current state and provide comparison with observations.\n\n\n21.4.3 Shock Potential Drop\nOne of the important shock parameters is the electric potential drop across the shock ramp – or if it exists also across the shock foot. This potential drop is not necessarily a constant but changes with location along the shock normal. We have already noted that it is due to the different dynamical responses of the inflowing ions and electrons over the scale of the foot and ramp regions. Its theoretical determination is difficult, however when going to the de Hoffmann-Teller frame the bulk motion of the particles is only along the magnetic field, and in the stationary electron equation of motion the \\(\\mathbf{V}_e \\times \\mathbf{B}\\)-term drops out and, to first approximation, the cross shock potential is given by the pressure gradient (when neglecting any contributions from wave fields). The expression is then simply \\[\n\\Delta \\Phi(x) = \\int_0^x \\frac{1}{eN_e(n)}\\left[ \\nabla\\cdot \\overleftarrow{P}_e(n) \\right]\\mathrm{d}\\mathbf{n}\n\\tag{21.30}\\]\nIntegration is over \\(n\\) along the shock normal \\(\\hat{n}\\). For a gyrotropic electron pressure, valid for length scales longer than an electron gyroradius, \\(\\overleftrightarrow{P}_e = P_{e\\perp}\\mathbf{I} + (P_{e\\parallel} - P_{e\\perp})\\mathbf{B}\\mathbf{B}/BB\\) one obtains (Goodrich and Scudder 1984), taking into account that \\(\\mathbf{E}\\cdot\\mathbf{B}\\) is invariant, \\[\n\\frac{\\mathrm{d}}{\\mathrm{d}n}\\phi(n) = -\\frac{E_\\parallel}{\\cos\\theta_{Bn}} = \\frac{1}{eN_e}\\left[ \\frac{\\mathrm{d}}{\\mathrm{d}n}P_{e\\parallel} - (P_{e\\parallel} - P_{e\\perp})\\frac{\\mathrm{d}}{\\mathrm{d}n}(\\ln B) \\right]\n\\] which, when used in the above expression, yields \\[\ne\\Delta \\Phi(x) = \\int_0^x \\mathrm{d}n\\left\\{ \\frac{\\mathrm{d}T_{e\\parallel}}{\\mathrm{d}n} + T_{e\\parallel}\\frac{\\mathrm{d}}{\\mathrm{d}n}\\ln\\left[ \\frac{N(n)}{N_1}\\frac{B_1}{B(n)} \\right] + T_{e\\perp}\\frac{\\mathrm{d}}{\\mathrm{d}n}\\ln\\left[ \\frac{B(n)}{B_1} \\right] \\right\\}\n\\]\nThis expression can approximately be written in terms of the gradient in the electron magnetic moment \\(\\mu_e = T_{e\\perp}/B\\) as follows: \\[\ne\\Delta\\Phi(x) \\simeq \\Delta(T_{e\\parallel} + T_{e\\perp}) - \\int_0^x \\mathrm{d}n \\frac{\\mathrm{d}\\mu_e(n)}{\\mathrm{d}n}B(n)\n\\] with \\(T_e\\) in energy units. When the electron magnetic moment is conserved, the last term disappears, yielding a simple relation for the potential drop \\(e\\Delta\\Phi(x) \\simeq \\Delta(T_{e\\parallel} + T_{e\\perp})\\) as the sum of the changes in electron temperature. The perpendicular temperature change can be expressed as \\(\\Delta T_{e\\perp} = T_{e\\perp,1}\\Delta B/B_1\\) which is in terms of the compression of the magnetic field. Non-adiabatic effects contribute via the dropped integral term, which breaks the adiabatic invariant.\nThe parallel change in temperature is more difficult to express. One could express it in terms of the temperature anisotropy \\(A_e = T_{e\\parallel} /T_{e\\perp}\\) as has been done by Kuncic et al [2002], and then vary \\(A_e\\). But this depends on the particular model. It is more important to note that this adiabatic estimate of the potential drop does not account for any dynamical process which generates waves and substructures in the shock. It thus gives only a hint on the order of magnitude of the potential drop across the foot-ramp region in quasi-perpendicular shocks.\n\n\n21.4.4 Shock Structure\nFigure 21.9 shows observations from one of the first unambiguous satellite crossings of a quasi-perpendicular supercritical (magnetosonic Mach number \\(M_\\text{ms} \\sim 4.2\\)) shock at the Earth’s bow shock.\n\n\n\n\n\n\nFigure 21.9: Time profiles of plasma and magnetic field parameters across a real quasi-perpendicular shock that had been crossed by the ISEE 1 and 2 spacecraft on November 7, 1977 in near-Earth space (Sckopke et al. 1983). \\(N_E\\) is the electron density, \\(N_i\\) the reflected energetic ion density, both in \\(\\text{cm}^{-3}\\), \\(T_P\\), \\(T_E\\) are proton and electron in K. \\(V_P\\) is the proton (plasma) bulk velocity in \\(\\text{km}\\,\\text{s}^{-1}\\), \\(P_e\\) electron pressure in \\(10^{-9}\\,\\text{N}\\,\\text{m}^{-2}\\), B the magnitude of the magnetic field in nT, and \\(\\theta_{Bn}\\). The vertical lines mark the first appearance of reflected ion, the outer edge of the foot in the magnetic profile, and the ramp in the field magnitude, respectively. The abscissa is the Universal Time UT referring to the measurements. The upper block are observations from ISEE 1, the lower block observations from ISEE 2.\n\n\n\n\n21.4.4.1 Observational evidence\nThe crossing occurred on an inbound path of the two spacecraft ISEE 1 (upper block of the figure) and ISEE 2 (lower block of the figure) from upstream to downstream in short sequence only minutes apart. In spite of some differences occurring on the short time scale the two shock crossings are about identical, identifying the main shock transition as a spatial and not as a temporal structure. Temporal variations are nevertheless visible on the scale of a fraction of a minute.\nIn this case, \\(\\theta_{Bn}\\) is close to \\(90^\\circ\\) prior to shock crossing (in the average \\(\\theta_{Bn} \\sim 85^\\circ\\)), and fluctuates afterwards around \\(90^\\circ\\) identifying the shock as quasi-perpendicular. Accordingly, the shock develops a foot in front of the shock ramp as can be seen from the slightly enhanced magnetic field after 22:51 UT in ISEE 1 and similar in ISEE 2, and most interestingly also in the electron pressure. At the same time the bulk flow velocity starts decreasing already, as the result of interaction and retardation in the shock foot region. The foot is also visible in the electron density which increases throughout the foot region, indicating the presence of electrons which, as is suggested by the increase in pressure, must have been heated or accelerated.\nThe best indication of the presence of the foot is, however, the measurement of energetic ions (second panel from top). These ions are observed first some distance away from the shock but increase drastically in intensity when entering the foot. These are the shock-reflected ions which have been accelerated in the convection electric field in front of the shock ramp. Their occurrence before entrance into the foot is understood when realising that the shock is not perfectly perpendicular. Rather it is quasi-perpendicular such that part of the reflected ions having sufficiently large parallel upstream velocities can escape along the magnetic field a distance larger than the average upstream extension of the foot. For nearly perpendicular shocks, this percentage is small.\nThe shock ramp in Figure 21.9 is a steep wall in B and \\(P_E\\), respectively. The electron temperature \\(T_E\\) increases only moderately across the shock while the ion temperature \\(T_P\\) jumps up by more than one magnitude, exceeding \\(T_E\\) downstream behind the shock. This behaviour is due to the accelerated returning foot-ions which pass the shock. \\(P_E\\), \\(B\\), and \\(N_E\\) exhibit overshoots behind the shock ramp proper. Farther away from the shock they merge into the highly fluctuating state of lesser density, pressure, and magnetic field that can be described as some kind of turbulence. Clearly, this region is strongly affected by the presence of the shock which forms one of its boundaries, the other boundary being the obstacle (Earth) which is the main responsible for the formation of the shock.\nThe evidence provided by the described measurements suggests that the quasi-perpendicular shock is a quasi-stationary entity. This should, however, not been taken apodictic. Stationarity depends on the spatial scales as well as the time scales. A shock is a very inhomogeneous subject containing all kinds of spatial scales; being stationary on one scale does not imply that it is stationary on another scale. For a shock like the Earth’s bow shock considered over times of days, weeks or years the shock is of course a stationary subject. However on shorter time scales of the order of flow transition times this may not be the case. A subcritical shock may well be stationary on long and short time scales. However, for a supercritical shock the conditions for forming a stationary state are quite subtle. From a single spacecraft passage like that described above it cannot be concluded to what extent, i.e. on which time scale and on which spatial scale and under which external conditions (Mach number, angle, shock potential, plasma-β,…) the observed shock can be considered to be stationary (a discussion of the various scales has been given, e.g, by Galeev et al, 1988). Comparison between the two ISEE spacecraft already shows that the small-scale details as have been detected by both spacecraft are very different. This suggests that – in this case – on time scales less than a minute variations in the shock structure must be expected.\nGenerally spoken, one must be prepared to consider the shock locally (on the ion gyroscale) and temporarily (on the ion-cyclotron frequency scale) as a non-stationary phenomenon (this has been realised first by Morse et al, 1972) which depends on many competing processes and, most important though only secondarily related to non-stationarity, a shock as a whole is not in thermal equilibrium. It needs to be driven by some energy source external to the shock in order to be maintained. It will thus be very sensitive to small changes in the external parameters and will permanently try to escape the non-equilibrium state and to approach equilibrium. Since its non-equilibrium is maintained by the conditions in the flow, it is these conditions which determine the time scales over which a shock evolves, re-evolves and changes its state. Real supercritical shocks, whether quasi-perpendicular or quasi-parallel, are in a permanently evolving state and thus are intrinsically nonstationary.\n\n\n21.4.4.2 Simulation studies\nPerpendicular or quasi-perpendicular collisionless shocks are relatively easy to treat in numerical simulations. Already from the first one-dimensional numerical experiments on collisionless shocks it became clear that such shocks have a very particular structure. This structure, which we have describe in simplified version in Figure 21.8 and which could to some extent also be inferred from the observations of Figure 21.9, becomes ever more pronounced the more refined the resolution becomes and the better the shorter scales can be resolved.\nAs already mentioned, collisionless shocks are in thermodynamic non-equilibrium and therefore can only evolve if a free energy source exists and if the processes are violent enough to build up and maintain a shock. Usually in a freely evolving system the free energy causes fluctuations which serve dissipating and redistributing the free energy towards thermodynamic and thermal equilibria. Thermal equilibria are characterised by equal temperatures among the different components, e.g. \\(T_e = T_i\\) which is clearly not given in the vicinity of a shock as seen from Figure 21.9. Thermodynamic equilibria are characterized by Gaussian distributions for all components of the plasma. To check this requires information about the phase space distribution of particles. Shocks contain many differing particle distributions, heated, top-flat, beam distributions, long energetic tails, and truncated as well as gyrating distributions which we will encounter later. Consequently, they are far from thermodynamic equilibrium.\nFor a shock to evolve the amount of free energy needed to dissipate is so large that fluctuations are unable to exercise their duty. This happens at large Mach numbers. The shock itself takes over the duty of providing dissipation. It does it in providing all kinds of scales such short that a multitude of dissipative processes can set on.\nScales", "crumbs": [ "21  Shock" ] @@ -1774,7 +1774,7 @@ "href": "contents/shock.html#supercritical-parallel-shock", "title": "21  Shock", "section": "21.5 Supercritical Parallel Shock", - "text": "21.5 Supercritical Parallel Shock\nFrom MHD or double adiabatic theory, parallel shocks are more special in that the magnetic field strength remains unchanged so the equations effectively describe pure gasdynamic solutions. (Kuznetsov and Osin 2018) presents a simplified solution in a 1D parallel shock case with parallel and perpendicular thermal energy heat fluxes \\(S_\\parallel\\) and \\(S_\\perp\\) included. Note again the original CGL theory assumes 0 heat fluxes.\nHowever, as have been indicated in Section 21.1.7, this does not cover the real physics involved into parallel shocks which must be treated on the basis of kinetic theory and with the simulation tool at hand. These shocks possess an extended foreshock region with its own extremely interesting dynamics for both types of particles, electrons and ions, reaching from the foreshock boundaries to the deep interior of the foreshock. Based mostly on kinetic simulations, the foreshock is the region where dissipation of flow energy starts well before the flow arrives at the shock. This dissipation is caused by various instabilities excited by the interaction between the flow and the reflected particles that have escaped to upstream from the shock. Interaction between these waves and the reflected and accumulated particle component in the foreshock causes wave growth and steeping, formation of shocklets and pulsations and causes continuous reformation of the quasi-parallel shock that differs completely from quasi-perpendicular shock reformation. It is the main process of maintaining the quasi-parallel shock which by its nature principally turns out to be locally nonstationary and, in addition, on the small scale making the quasi-parallel shock close to becoming quasi-perpendicular for the electrons. This process can be defined as turbulent reformation, with transient phenonmena like hot flow anomalies, foreshock bubbles, and the generation of electromagnetic radiation. Foreshock physics is important for particle acceleration.\nThe turbulent nature implies that the quasi-parallel shock transition is less sharp than the quasi-perpendicular shock transition and thus less well defined; there exists an extended turbulent foreshock instead of a shock foot. This foreshock consists of an electron and an ion foreshock. The main population is a diffuse ion component. The turbulence in the foreshock is generated by the reflected and accelerated foreshock particle populations. An important point in quasi-parallel shock physics is the reformation of the shock which works completely differently from quasi-perpendicular shocks; here it is provided by upstream low-frequency electromagnetic waves excited by the diffuse ion component. Steeping of these waves during shockward propagation and addition of the large amplitude waves at the shock transition reforms the shock front. The old shock front is expelled downstream where it causes downstream turbulence. During the reformation process the shock becomes locally quasi-perpendicular for the ions supporting particle reflection.\n\n21.5.1 Turbulent Reformation\n\n\n\n\n\n\nFigure 21.9: The patchwork model of J. Schwartz Steven and David (1991) of a quasi-parallel supercritical shock reformation. Left: Magnetic pulsations (SLAMS) grow in the ion foreshock and are convected toward the shock where they accumulate, thereby causing formation of an irregular shock structure. Note also the slight turning of the magnetic field into a direction that is more perpendicular to the shock surface with the shock surface itself becoming very irregular. Right: The same model with the pulsations being generated in the relatively broad ULF-wave-unstable region in greater proximity to the ion-foreshock boundary. When the ULF waves evolve to large amplitude and form localised structures these are convected toward the shock, grow, steepen, overlap, accumulate and lead to the build up of the irregular quasi-parallel shock structure which overlaps into the downstream direction.\n\n\n\nLucek+ [2002] checked this expectation by determining the local shock-normal angle \\(\\theta_{Bn}\\) and comparing it with the prediction estimated from magnetic field measurements by the ACE spacecraft which was located farther out in the upstream flow. The interesting result is that during the checked time-interval of passage of the quasi-parallel shock the prediction for the shock normal was around \\(20-30^\\circ\\), as expected for quasi-parallel shocks. However, this value just set a lower bound on the actually measured shock normal angle. The measured \\(\\theta_{Bn}\\) was highly fluctuating around much larger values and, in addition, showed a tendency to be close to \\(90^\\circ\\). It strengthens the claim that quasi-parallel shocks are locally, on the small scale, very close to perpendicular shocks, a property that they borrow from the large magnetic waves by which they are surrounded. In fact, we may even claim that locally, on the small scale (\\(\\leq \\mathcal{O}(d_i)\\)), quasi-parallel shocks are quasi-perpendicular.\nThus, the quasi-parallel shock is the result of a build-up from upstream waves which continuously reorganise and reform the shock. The shock transition region turns out to consist of many embedded magnetic pulsations (SLAMS) of very large amplitudes. These pulsations have steep flanks and quite irregular shape, exhibit higher frequency oscillations probably propagating in the whistler mode while sitting on the feet or shoulders of the pulsations.\nAnother interesting property of magnetic pulsations in the shock transition region is the electric cross-SLAMS potential, which corresponds to a steep pressure gradient. The pulsations are subject to a fairly large number of high frequency, Debye scale structures in the electric field. These intense nonlinear electrostatic electron plasma waves indicates that the quasi-parallel shocks are sources of electron acceleration into beams which are capable to move upstream along the magnetic field over a certain distance and excite electron plasma waves at intensity high enough to enter into the nonlinear regime, forming solitons and electron holes (BGK modes). This is possible only if quasi-parallel shocks are quasi-perpendicular as well on the electron scale.\n\n\n21.5.2 Parallel Shock Particle Reflection\nThere are two possible mechanisms:\n\nA quasi-parallel shock is capable of generating a large cross-shock potential, or it is capable of stochastically – or nearly stochastically – scattering ions in the shock transition region in pitch angle and energy in such a way that part of the incoming ion distribution can escape upstream.\nOn a scale that affects the ion motion, a quasi-parallel shock close to the shock transition becomes sufficiently quasi-perpendicular that ions are reflected in the same way as if they encountered a quasi-perpendicular shock.", + "text": "21.5 Supercritical Parallel Shock\nFrom MHD or double adiabatic theory, parallel shocks are more special in that the magnetic field strength remains unchanged so the equations effectively describe pure gasdynamic solutions. (Kuznetsov and Osin 2018) presents a simplified solution in a 1D parallel shock case with parallel and perpendicular thermal energy heat fluxes \\(S_\\parallel\\) and \\(S_\\perp\\) included. Note again the original CGL theory assumes 0 heat fluxes.\nHowever, as have been indicated in Section 21.1.7, this does not cover the real physics involved into parallel shocks which must be treated on the basis of kinetic theory and with the simulation tool at hand. These shocks possess an extended foreshock region with its own extremely interesting dynamics for both types of particles, electrons and ions, reaching from the foreshock boundaries to the deep interior of the foreshock. Based mostly on kinetic simulations, the foreshock is the region where dissipation of flow energy starts well before the flow arrives at the shock. This dissipation is caused by various instabilities excited by the interaction between the flow and the reflected particles that have escaped to upstream from the shock. Interaction between these waves and the reflected and accumulated particle component in the foreshock causes wave growth and steeping, formation of shocklets and pulsations and causes continuous reformation of the quasi-parallel shock that differs completely from quasi-perpendicular shock reformation. It is the main process of maintaining the quasi-parallel shock which by its nature principally turns out to be locally nonstationary and, in addition, on the small scale making the quasi-parallel shock close to becoming quasi-perpendicular for the electrons. This process can be defined as turbulent reformation, with transient phenonmena like hot flow anomalies, foreshock bubbles, and the generation of electromagnetic radiation. Foreshock physics is important for particle acceleration.\nThe turbulent nature implies that the quasi-parallel shock transition is less sharp than the quasi-perpendicular shock transition and thus less well defined; there exists an extended turbulent foreshock instead of a shock foot. This foreshock consists of an electron and an ion foreshock. The main population is a diffuse ion component. The turbulence in the foreshock is generated by the reflected and accelerated foreshock particle populations. An important point in quasi-parallel shock physics is the reformation of the shock which works completely differently from quasi-perpendicular shocks; here it is provided by upstream low-frequency electromagnetic waves excited by the diffuse ion component. Steeping of these waves during shockward propagation and addition of the large amplitude waves at the shock transition reforms the shock front. The old shock front is expelled downstream where it causes downstream turbulence. During the reformation process the shock becomes locally quasi-perpendicular for the ions supporting particle reflection.\n\n21.5.1 Turbulent Reformation\n\n\n\n\n\n\nFigure 21.10: The patchwork model of J. Schwartz Steven and David (1991) of a quasi-parallel supercritical shock reformation. Left: Magnetic pulsations (SLAMS) grow in the ion foreshock and are convected toward the shock where they accumulate, thereby causing formation of an irregular shock structure. Note also the slight turning of the magnetic field into a direction that is more perpendicular to the shock surface with the shock surface itself becoming very irregular. Right: The same model with the pulsations being generated in the relatively broad ULF-wave-unstable region in greater proximity to the ion-foreshock boundary. When the ULF waves evolve to large amplitude and form localised structures these are convected toward the shock, grow, steepen, overlap, accumulate and lead to the build up of the irregular quasi-parallel shock structure which overlaps into the downstream direction.\n\n\n\nLucek+ [2002] checked this expectation by determining the local shock-normal angle \\(\\theta_{Bn}\\) and comparing it with the prediction estimated from magnetic field measurements by the ACE spacecraft which was located farther out in the upstream flow. The interesting result is that during the checked time-interval of passage of the quasi-parallel shock the prediction for the shock normal was around \\(20-30^\\circ\\), as expected for quasi-parallel shocks. However, this value just set a lower bound on the actually measured shock normal angle. The measured \\(\\theta_{Bn}\\) was highly fluctuating around much larger values and, in addition, showed a tendency to be close to \\(90^\\circ\\). It strengthens the claim that quasi-parallel shocks are locally, on the small scale, very close to perpendicular shocks, a property that they borrow from the large magnetic waves by which they are surrounded. In fact, we may even claim that locally, on the small scale (\\(\\leq \\mathcal{O}(d_i)\\)), quasi-parallel shocks are quasi-perpendicular.\nThus, the quasi-parallel shock is the result of a build-up from upstream waves which continuously reorganise and reform the shock. The shock transition region turns out to consist of many embedded magnetic pulsations (SLAMS) of very large amplitudes. These pulsations have steep flanks and quite irregular shape, exhibit higher frequency oscillations probably propagating in the whistler mode while sitting on the feet or shoulders of the pulsations.\nAnother interesting property of magnetic pulsations in the shock transition region is the electric cross-SLAMS potential, which corresponds to a steep pressure gradient. The pulsations are subject to a fairly large number of high frequency, Debye scale structures in the electric field. These intense nonlinear electrostatic electron plasma waves indicates that the quasi-parallel shocks are sources of electron acceleration into beams which are capable to move upstream along the magnetic field over a certain distance and excite electron plasma waves at intensity high enough to enter into the nonlinear regime, forming solitons and electron holes (BGK modes). This is possible only if quasi-parallel shocks are quasi-perpendicular as well on the electron scale.\n\n\n21.5.2 Parallel Shock Particle Reflection\nThere are two possible mechanisms:\n\nA quasi-parallel shock is capable of generating a large cross-shock potential, or it is capable of stochastically – or nearly stochastically – scattering ions in the shock transition region in pitch angle and energy in such a way that part of the incoming ion distribution can escape upstream.\nOn a scale that affects the ion motion, a quasi-parallel shock close to the shock transition becomes sufficiently quasi-perpendicular that ions are reflected in the same way as if they encountered a quasi-perpendicular shock.", "crumbs": [ "21  Shock" ] @@ -1784,7 +1784,7 @@ "href": "contents/shock.html#instabilities-and-waves", "title": "21  Shock", "section": "21.6 Instabilities and Waves", - "text": "21.6 Instabilities and Waves\nIn the context of collisionless shocks the instabilities of interest can be divided into two classes. The first class contains those waves which can grow themselves to become a shock. It is clear that these waves will be of low frequency and comparably large scale because otherwise they would not evolve into a large macroscopic shock. The primary candidates are magnetosonic, Alfvén and whistler modes. A number of waves can form secondarily after an initial seed shock ramp and grow in some way out of one of these wave modes: these are ion modes which have now been identified to be responsible for structuring, shaping and reforming the shock. In fact real oblique shocks — which are the main class of shocks in interplanetary space and probably in all space and astrophysical objects — cannot survive without the presence of these ion waves which can therefore be considered the wave modes that really produce shocks in a process of taking and giving between shock and waves.\nThe second class includes waves that accompany the shock and provide anomalous transport coefficients like anomalous collision frequencies, friction coefficients, heat conductivity and viscosity. These waves are also important for the shock as they contribute to entropy generation and dissipation. However, they are not primary in the sense that they are not shock-forming waves.\nAmong them there is another group that only carries away energy and information from the shock. These are high-frequency waves, mostly electrostatic in nature, produced by electrons or when electromagnetic they are in the free-space radiation modes. In the latter case they carry the information from remote objects as radiation in various modes, radio or x-ray to Earth, informing of the existence of a shock. In interplanetary space it is only radio waves which fall into this group as the radiation measure of the heliospheric shocks is too small to map them into x-rays.\nHere we restrict mostly to low frequency EM waves in warm plasma, \\(\\omega\\le \\omega_{ci}\\), while only mention the high frequency EM waves in the end. Such waves are excited by plasma streams or kinetic anisotropies in one or the other way. A simple summary is given in Table 21.2.\n\n\n\nTable 21.2: Types of instabilities and waves related to shocks\n\n\n\n\n\n\n\n\n\n\n\nMode\nWave Type\nHandedness\nOther Properties\n\n\n\n\nFirehose\nAlfvén\nleft\nParallel prefered anisotropy\n\n\nIon-Ion Beam\nFast\nright\nCool beam\n\n\nIon cyclotron\nAlfvén\nleft\nWarm beam\n\n\nKAW\nAlfvén\nright\nElectron, parallel electric field\n\n\nWhistler\nAlfvén\nright\nHall term, electron\n\n\n\n\n\n\n\n21.6.1 Ion Instabilities \\(\\omega \\leq \\omega_{ci}\\)\n\n21.6.1.1 Firehose mode\nThe simplest instability known which distorts the magnetic field by exciting Alfvén waves that propagate along the magnetic field is the firehose mode. The wave excited are ordinary Alfvén waves, however, and are not suited for shock formation.\nWhen the ion beam is fast and cold it does not go into resonance because its velocity is too high. In this case all ions participate in a nonresonant instability which in fact is a thermal firehose mode where the ion beam has sufficient energy to shake the field line. This mode propagates antiparallel to the ion beam, has small phase speed and negative helicity. This mode has large growth rate for large \\(n_b / n_e\\) and \\(v_b / v_A\\) simply because then there are many beam ions and the centrifugal force is large while the beam velocity lies outside any resonant wave speed. This instability becomes stronger when the ion beam is composed of heavier ions as the larger mass of these increases the centrifugal force effect.\n\n\n21.6.1.2 Kinetic Alfvén waves\nKAWs (Section 10.9.4) possess a finite \\(E_\\parallel\\) which can accelerate electrons; in the other way, electron moving along the magnetic field in the opposite direction become retarded and feed their energy into KAWs.\nNormally this is likely to be a minor effect, as the interaction of ions which are reflected from a solitary pulse and move back upstream ahead of the pulse will cause a stronger instability. The reflected ions will represent a beam that is moving against the initial plasma inflow which by itself is another ion beam neutralised by the comoving electrons. The free energy presented in the two counter-streaming beams leads to various instabilities as viewed by Gary (1993).\n\n\n21.6.1.3 Kinetic growth rate\nAt low frequencies it suffices for our purposes of understanding shock physics to deal with a three-component plasma consisting of two ion species and one neutralising electron component which we assume to follow a Maxwellian velocity distribution. Moreover, we assume that the drifting ion components are Maxwellians as well. In conformity with the above remarks on a resonant instability we assume that the dominant ion component has large density \\(n_i\\gg n_b\\), and the second component represents a weak fast beam of density \\(n_b\\) propagating on the ion-electron background with velocity \\(v_b \\gg v_i \\approx = 0\\). Following Gary (1993) it is convenient to distinguish the three regimes:\n\ncool beams (\\(0<v<v_b\\))\nwarm beams (\\(v\\sim v_b\\))\nhot beams (\\(v\\gg v_b\\))\n\nFigure 21.10 shows the beam configurations for these three cases and the location of the wave resonances respectively the position of the unstable frequencies.\n\n\n\n\n\n\nFigure 21.10: The three cases of ion beam-plasma interaction and the location of the unstable frequencies. Shown is the parallel (reduced) distribution function \\(F_i(k_\\parallel v_\\parallel)\\), where for simplicity the (constant) parallel wavenumber \\(k_\\parallel\\) has been included into the argument. Right handed resonant modes (RH) are excited by a cool not too fast beam. When the beam is too fast the interaction becomes nonresonant. When the beam is hot, a resonant left hand mode (LH) is excited. In addition the effect of temperature anisotropy is shown when a plateau forms on the distribution function (after Gary 1993).\n\n\n\n\n\n21.6.1.4 Cold Ion Beam: Right-Hand Instability\nAssume that the ion beam is thermally isotropic and cool, i.e. its velocity relative to the bulk plasma is faster than its thermal speed. In this case a right-handed resonant instability occurs. In the absence of a beam \\(v_b=0\\) the parallel mode is a right-circularly polarised magnetosonic wave propagating on the lowest frequency whistler dispersion branch with \\(\\omega\\approx k_\\parallel v_A\\). In presence of a drift this wave becomes unstable, and the fastest growing frequency is at frequency \\(\\omega \\simeq k_\\parallel v_b - \\omega_{ci}.\\) This mode propagates parallel to the beam, because \\(\\omega>0\\), \\(k_\\parallel > 0\\), and \\(v_b > 0\\). The numerical solution of this instability for densities \\(0.01 \\le n_b/n_i \\le 0.1\\) at the wave-number \\(k_\\parallel\\) of fastest growth rate identifies a growth rate of the order of the wave frequency \\(\\gamma\\sim\\omega\\) and \\[\n\\gamma_m \\simeq \\omega_{ci}\\left( \\frac{n_b}{n_e} \\right)^{1/3}\n\\] for the maximum growth rate \\(\\gamma_m\\), where \\(n_e = n_i + n_b\\) is the total density from quasi-neutrality. This instability drives waves propagating together with the beam in the direction of the ion beam on the plasma background which has been assumed at rest. If applied for instance, to shock reflected ions then for 2% reflected ions the maximum growth rate is \\(\\gamma_m\\sim 0.2\\omega_{ci}\\), and \\(v_b\\sim 1.2 \\omega_{ci}/k_\\parallel\\), \\(k_\\parallel\\sim 0.2 \\omega_{ci}/v_A\\) which gives \\(v_b \\sim 6 v_A\\). In the solar wind the Alfvén velocity is about \\(v_A ≈ 30\\,\\mathrm{km/s}\\). Hence the velocity difference between shock reflected ions and solar wind along the magnetic field should be roughly \\(\\sim 180\\,\\mathrm{km/s}\\).10 The thermal velocity of the ion beam must thus be substantially less than this value, corresponding to a thermal beam energy less than \\(T_b \\ll 100\\,\\mathrm{eV}\\) which in the solar wind, for instance, is satisfied near the tangential field line. The solar wind travels at 300–1200 km/s. Complete reflection should produce beam speeds twice these values.11\nThe cyclotron resonance condition associated with the generated fast magnetosonic mode is \\[\n\\omega = v_b k_\\parallel - \\omega_{ci}\n\\tag{21.31}\\] where \\(v_b\\) is the beam velocity and \\(\\omega_{ci}\\) the ion gyrofrequency. It can be approximated as \\(\\omega=v_A k_\\parallel\\).\n\n\n21.6.1.5 Warm Ion Beam: Left-Hand Instability\nWhen the temperature of the ion beam increases and the background ions remain to be cold, then beam ions appear on the negative velocity side of the bulk ion distribution and go into resonance there with the left-hand polarised ion-Alfvén wave. The maximum growth rate is a fraction of the growth rate of the right-hand low frequency whistler mode.12 Nevertheless it can excite the Alfvén-ion cyclotron wave which also propagates parallel to the beam. For this instability the beam velocity must exceed the Alfvén speed \\(v_b > v_A\\).\nAt oblique propagation both the right and left hand instabilities have smaller growth rates. But interestingly, it has been shown by Goldstein et al. (1985) that the fastest growing modes then appear for oblique \\(\\mathbf{k}\\) and harmonics of the ion cyclotron frequency \\(\\omega\\sim n\\omega_{ci}, n=1,2,...\\).13\n\n\n\n21.6.2 Electron Instabilities and Radiation \\(\\omega \\sim \\omega_{pe}\\)\nOther than ion beam excited instabilities electron-beam instabilities are not involved in direct shock formation (unless the electron beams are highly relativistic which in the entire heliosphere is not the case). The reason is that the frequencies of electron instabilities are high. However, just because of this reason they are crucial in anomalous transport being responsible for anomalous collision frequencies and high frequency field fluctuations. The reason is that the high frequency waves lead to energy loss of the electrons retarding them while for the heavier ions they represent a fluctuating background scattering them. In this way high frequency waves may contribute to the basic dissipation in shocks even though this dissipation for supercritical shocks will not be sufficient to maintain a collisionless shock or even to create a shock under collisionless conditions. This is also easy to understand intuitively, because the waves need time to be created and to reach a substantial amplitude. This time in a fast stream is longer than the time the stream needs to cross the shock. So waves will not accumulate there; rather the fast stream will have convected them downstream long before they have reached substantial amplitudes for becoming important in scattering.\nWhen we are going to discuss electromagnetic waves which can be excited by electrons we also must keep in mind that such waves can propagate only when there is an electromagnetic dispersion branch in the plasma under consideration. These electromagnetic branches in \\((\\omega,\\mathbf{k})\\)-space are located at frequencies below the electron cyclotron frequency \\(\\omega_{ce}\\). The corresponding branch is the whistler mode branch. Electrons will (under conditions prevailing at shocks) in general not be able to excite electromagnetic modes at higher frequencies than \\(\\omega_{ce}\\). We have seen before that ion beams have been able to excite whistlers at low frequencies but above the ion-cyclotron frequency. This was possible only because of the presence of the high frequency electron whistler branch as a channel for wave propagation. EM waves excited by electrons propagate on the whistler branch or its low frequency Alfvénic extension, both of which are right-handed. They also excite a variety of electrostatic emissions.\n\n\n21.6.3 Whistlers\nGary (1993) has investigated the case of whistler excitation by an electron beam. He finds from numerical solution of the full dispersion relation including an electron beam in parallel motion that with increasing beam velocity \\(v_b\\) the real frequency of the unstable whistler decreases, i.e. the unstably excited whistler shifts to lower frequencies on the whistler branch while remaining in the whistler range \\(\\omega_{ci} < \\omega < \\omega_{ce}\\). Both the background electrons and beam electrons contribute resonantly. The most important finding is that the whistler mode for sufficiently large \\(\\beta_i \\sim 1\\) (which means low magnetic field), \\(n_b/n_e\\) and \\(T_b/T_e\\) has the lowest beam velocity threshold when compared with the electrostatic electron beam instabilities as shown in Figure 21.11. This finding implies that in a relatively high-β plasma a moderately dense electron beam will first excite whistler waves. In the shock environment the conditions for excitation of whistlers should thus be favourable whenever an electron beam propagates across the plasma along the relatively weak magnetic field. The electrons in resonance satisfy \\(v_\\parallel = (\\omega - \\omega_{ce})/k_\\parallel\\) and, because \\(\\omega \\ll \\omega_{ce}\\) the resonant electrons move in the direction opposite to the beam. Enhancing the beam temperature increases the number of resonant electrons thus feeding the instability.\nOn the other hand, increasing the beam speed shifts the particles out of resonance and decreases the instability. Hence for a given beam temperature the whistler instability has a maximum growth rate a few times the ion cyclotron frequency.\n\n\n\n\n\n\nFigure 21.11: The regions of instability of the electron beam excited whistler mode in density and beam velocity space for two different β compared to the ion acoustic and electron beam modes. Instability is above the curves. The whistler instability has the lowest threshold in this parameter range (after Gary 1993).", + "text": "21.6 Instabilities and Waves\nIn the context of collisionless shocks the instabilities of interest can be divided into two classes. The first class contains those waves which can grow themselves to become a shock. It is clear that these waves will be of low frequency and comparably large scale because otherwise they would not evolve into a large macroscopic shock. The primary candidates are magnetosonic, Alfvén and whistler modes. A number of waves can form secondarily after an initial seed shock ramp and grow in some way out of one of these wave modes: these are ion modes which have now been identified to be responsible for structuring, shaping and reforming the shock. In fact real oblique shocks — which are the main class of shocks in interplanetary space and probably in all space and astrophysical objects — cannot survive without the presence of these ion waves which can therefore be considered the wave modes that really produce shocks in a process of taking and giving between shock and waves.\nThe second class includes waves that accompany the shock and provide anomalous transport coefficients like anomalous collision frequencies, friction coefficients, heat conductivity and viscosity. These waves are also important for the shock as they contribute to entropy generation and dissipation. However, they are not primary in the sense that they are not shock-forming waves.\nAmong them there is another group that only carries away energy and information from the shock. These are high-frequency waves, mostly electrostatic in nature, produced by electrons or when electromagnetic they are in the free-space radiation modes. In the latter case they carry the information from remote objects as radiation in various modes, radio or x-ray to Earth, informing of the existence of a shock. In interplanetary space it is only radio waves which fall into this group as the radiation measure of the heliospheric shocks is too small to map them into x-rays.\nHere we restrict mostly to low frequency EM waves in warm plasma, \\(\\omega\\le \\omega_{ci}\\), while only mention the high frequency EM waves in the end. Such waves are excited by plasma streams or kinetic anisotropies in one or the other way. A simple summary is given in Table 21.2.\n\n\n\nTable 21.2: Types of instabilities and waves related to shocks\n\n\n\n\n\n\n\n\n\n\n\nMode\nWave Type\nHandedness\nOther Properties\n\n\n\n\nFirehose\nAlfvén\nleft\nParallel prefered anisotropy\n\n\nIon-Ion Beam\nFast\nright\nCool beam\n\n\nIon cyclotron\nAlfvén\nleft\nWarm beam\n\n\nKAW\nAlfvén\nright\nElectron, parallel electric field\n\n\nWhistler\nAlfvén\nright\nHall term, electron\n\n\n\n\n\n\n\n21.6.1 Ion Instabilities \\(\\omega \\leq \\omega_{ci}\\)\n\n21.6.1.1 Firehose mode\nThe simplest instability known which distorts the magnetic field by exciting Alfvén waves that propagate along the magnetic field is the firehose mode. The wave excited are ordinary Alfvén waves, however, and are not suited for shock formation.\nWhen the ion beam is fast and cold it does not go into resonance because its velocity is too high. In this case all ions participate in a nonresonant instability which in fact is a thermal firehose mode where the ion beam has sufficient energy to shake the field line. This mode propagates antiparallel to the ion beam, has small phase speed and negative helicity. This mode has large growth rate for large \\(n_b / n_e\\) and \\(v_b / v_A\\) simply because then there are many beam ions and the centrifugal force is large while the beam velocity lies outside any resonant wave speed. This instability becomes stronger when the ion beam is composed of heavier ions as the larger mass of these increases the centrifugal force effect.\n\n\n21.6.1.2 Kinetic Alfvén waves\nKAWs (Section 10.9.4) possess a finite \\(E_\\parallel\\) which can accelerate electrons; in the other way, electron moving along the magnetic field in the opposite direction become retarded and feed their energy into KAWs.\nNormally this is likely to be a minor effect, as the interaction of ions which are reflected from a solitary pulse and move back upstream ahead of the pulse will cause a stronger instability. The reflected ions will represent a beam that is moving against the initial plasma inflow which by itself is another ion beam neutralised by the comoving electrons. The free energy presented in the two counter-streaming beams leads to various instabilities as viewed by Gary (1993).\n\n\n21.6.1.3 Kinetic growth rate\nAt low frequencies it suffices for our purposes of understanding shock physics to deal with a three-component plasma consisting of two ion species and one neutralising electron component which we assume to follow a Maxwellian velocity distribution. Moreover, we assume that the drifting ion components are Maxwellians as well. In conformity with the above remarks on a resonant instability we assume that the dominant ion component has large density \\(n_i\\gg n_b\\), and the second component represents a weak fast beam of density \\(n_b\\) propagating on the ion-electron background with velocity \\(v_b \\gg v_i \\approx = 0\\). Following Gary (1993) it is convenient to distinguish the three regimes:\n\ncool beams (\\(0<v<v_b\\))\nwarm beams (\\(v\\sim v_b\\))\nhot beams (\\(v\\gg v_b\\))\n\nFigure 21.11 shows the beam configurations for these three cases and the location of the wave resonances respectively the position of the unstable frequencies.\n\n\n\n\n\n\nFigure 21.11: The three cases of ion beam-plasma interaction and the location of the unstable frequencies. Shown is the parallel (reduced) distribution function \\(F_i(k_\\parallel v_\\parallel)\\), where for simplicity the (constant) parallel wavenumber \\(k_\\parallel\\) has been included into the argument. Right handed resonant modes (RH) are excited by a cool not too fast beam. When the beam is too fast the interaction becomes nonresonant. When the beam is hot, a resonant left hand mode (LH) is excited. In addition the effect of temperature anisotropy is shown when a plateau forms on the distribution function (after Gary 1993).\n\n\n\n\n\n21.6.1.4 Cold Ion Beam: Right-Hand Instability\nAssume that the ion beam is thermally isotropic and cool, i.e. its velocity relative to the bulk plasma is faster than its thermal speed. In this case a right-handed resonant instability occurs. In the absence of a beam \\(v_b=0\\) the parallel mode is a right-circularly polarised magnetosonic wave propagating on the lowest frequency whistler dispersion branch with \\(\\omega\\approx k_\\parallel v_A\\). In presence of a drift this wave becomes unstable, and the fastest growing frequency is at frequency \\(\\omega \\simeq k_\\parallel v_b - \\omega_{ci}.\\) This mode propagates parallel to the beam, because \\(\\omega>0\\), \\(k_\\parallel > 0\\), and \\(v_b > 0\\). The numerical solution of this instability for densities \\(0.01 \\le n_b/n_i \\le 0.1\\) at the wave-number \\(k_\\parallel\\) of fastest growth rate identifies a growth rate of the order of the wave frequency \\(\\gamma\\sim\\omega\\) and \\[\n\\gamma_m \\simeq \\omega_{ci}\\left( \\frac{n_b}{n_e} \\right)^{1/3}\n\\] for the maximum growth rate \\(\\gamma_m\\), where \\(n_e = n_i + n_b\\) is the total density from quasi-neutrality. This instability drives waves propagating together with the beam in the direction of the ion beam on the plasma background which has been assumed at rest. If applied for instance, to shock reflected ions then for 2% reflected ions the maximum growth rate is \\(\\gamma_m\\sim 0.2\\omega_{ci}\\), and \\(v_b\\sim 1.2 \\omega_{ci}/k_\\parallel\\), \\(k_\\parallel\\sim 0.2 \\omega_{ci}/v_A\\) which gives \\(v_b \\sim 6 v_A\\). In the solar wind the Alfvén velocity is about \\(v_A ≈ 30\\,\\mathrm{km/s}\\). Hence the velocity difference between shock reflected ions and solar wind along the magnetic field should be roughly \\(\\sim 180\\,\\mathrm{km/s}\\).10 The thermal velocity of the ion beam must thus be substantially less than this value, corresponding to a thermal beam energy less than \\(T_b \\ll 100\\,\\mathrm{eV}\\) which in the solar wind, for instance, is satisfied near the tangential field line. The solar wind travels at 300–1200 km/s. Complete reflection should produce beam speeds twice these values.11\nThe cyclotron resonance condition associated with the generated fast magnetosonic mode is \\[\n\\omega = v_b k_\\parallel - \\omega_{ci}\n\\tag{21.31}\\] where \\(v_b\\) is the beam velocity and \\(\\omega_{ci}\\) the ion gyrofrequency. It can be approximated as \\(\\omega=v_A k_\\parallel\\).\n\n\n21.6.1.5 Warm Ion Beam: Left-Hand Instability\nWhen the temperature of the ion beam increases and the background ions remain to be cold, then beam ions appear on the negative velocity side of the bulk ion distribution and go into resonance there with the left-hand polarised ion-Alfvén wave. The maximum growth rate is a fraction of the growth rate of the right-hand low frequency whistler mode.12 Nevertheless it can excite the Alfvén-ion cyclotron wave which also propagates parallel to the beam. For this instability the beam velocity must exceed the Alfvén speed \\(v_b > v_A\\).\nAt oblique propagation both the right and left hand instabilities have smaller growth rates. But interestingly, it has been shown by Goldstein et al. (1985) that the fastest growing modes then appear for oblique \\(\\mathbf{k}\\) and harmonics of the ion cyclotron frequency \\(\\omega\\sim n\\omega_{ci}, n=1,2,...\\).13\n\n\n\n21.6.2 Electron Instabilities and Radiation \\(\\omega \\sim \\omega_{pe}\\)\nOther than ion beam excited instabilities electron-beam instabilities are not involved in direct shock formation (unless the electron beams are highly relativistic which in the entire heliosphere is not the case). The reason is that the frequencies of electron instabilities are high. However, just because of this reason they are crucial in anomalous transport being responsible for anomalous collision frequencies and high frequency field fluctuations. The reason is that the high frequency waves lead to energy loss of the electrons retarding them while for the heavier ions they represent a fluctuating background scattering them. In this way high frequency waves may contribute to the basic dissipation in shocks even though this dissipation for supercritical shocks will not be sufficient to maintain a collisionless shock or even to create a shock under collisionless conditions. This is also easy to understand intuitively, because the waves need time to be created and to reach a substantial amplitude. This time in a fast stream is longer than the time the stream needs to cross the shock. So waves will not accumulate there; rather the fast stream will have convected them downstream long before they have reached substantial amplitudes for becoming important in scattering.\nWhen we are going to discuss electromagnetic waves which can be excited by electrons we also must keep in mind that such waves can propagate only when there is an electromagnetic dispersion branch in the plasma under consideration. These electromagnetic branches in \\((\\omega,\\mathbf{k})\\)-space are located at frequencies below the electron cyclotron frequency \\(\\omega_{ce}\\). The corresponding branch is the whistler mode branch. Electrons will (under conditions prevailing at shocks) in general not be able to excite electromagnetic modes at higher frequencies than \\(\\omega_{ce}\\). We have seen before that ion beams have been able to excite whistlers at low frequencies but above the ion-cyclotron frequency. This was possible only because of the presence of the high frequency electron whistler branch as a channel for wave propagation. EM waves excited by electrons propagate on the whistler branch or its low frequency Alfvénic extension, both of which are right-handed. They also excite a variety of electrostatic emissions.\n\n\n21.6.3 Whistlers\nGary (1993) has investigated the case of whistler excitation by an electron beam. He finds from numerical solution of the full dispersion relation including an electron beam in parallel motion that with increasing beam velocity \\(v_b\\) the real frequency of the unstable whistler decreases, i.e. the unstably excited whistler shifts to lower frequencies on the whistler branch while remaining in the whistler range \\(\\omega_{ci} < \\omega < \\omega_{ce}\\). Both the background electrons and beam electrons contribute resonantly. The most important finding is that the whistler mode for sufficiently large \\(\\beta_i \\sim 1\\) (which means low magnetic field), \\(n_b/n_e\\) and \\(T_b/T_e\\) has the lowest beam velocity threshold when compared with the electrostatic electron beam instabilities as shown in Figure 21.12. This finding implies that in a relatively high-β plasma a moderately dense electron beam will first excite whistler waves. In the shock environment the conditions for excitation of whistlers should thus be favourable whenever an electron beam propagates across the plasma along the relatively weak magnetic field. The electrons in resonance satisfy \\(v_\\parallel = (\\omega - \\omega_{ce})/k_\\parallel\\) and, because \\(\\omega \\ll \\omega_{ce}\\) the resonant electrons move in the direction opposite to the beam. Enhancing the beam temperature increases the number of resonant electrons thus feeding the instability.\nOn the other hand, increasing the beam speed shifts the particles out of resonance and decreases the instability. Hence for a given beam temperature the whistler instability has a maximum growth rate a few times the ion cyclotron frequency.\n\n\n\n\n\n\nFigure 21.12: The regions of instability of the electron beam excited whistler mode in density and beam velocity space for two different β compared to the ion acoustic and electron beam modes. Instability is above the curves. The whistler instability has the lowest threshold in this parameter range (after Gary 1993).", "crumbs": [ "21  Shock" ] @@ -1794,7 +1794,7 @@ "href": "contents/shock.html#shock-particle-reflection", "title": "21  Shock", "section": "21.7 Shock Particle Reflection", - "text": "21.7 Shock Particle Reflection\nThe process of particle reflection from a shock wave is one of the most important processes in the entire physics of collisionless shocks. However, the mechanism of particle reflection has not yet been fully illuminated.\nParticle reflection is required in supercritical shocks as it is the only process that can compensate for the incapability of dissipative processes inside the shock ramp to digest the fast inflow of momentum and energy into the shock. Shock particle reflection is not dissipative by itself even though in a fluid picture which deals with moments of the distribution function it can be interpreted as kind of an ion viscosity, i.e. it generates an anomalous viscosity coefficient which appears as a factor in front of the second derivative of the ion velocity in the ionic equation of motion. As such it also appears in the ion heat-transport equation. The kinematic ion viscosity can be expressed as \\[\n\\mu_\\mathrm{vis} = m_i n \\nu_i \\lambda_\\mathrm{mfp} \\simeq P_i / 2\\omega_{ci}\n\\] through the ion pressure \\(P_i\\) and the ion-cyclotron frequency \\(\\omega_{ci}\\) when replacing the mean free path through the ion gyro-radius. In this sense shock particle reflection constitutes by itself a very efficient non-dissipative dissipation mechanism. However, its direct dissipative action is to produce real dissipation as far as possible upstream of the shock in order to dissipate as much energy of motion as remains to be in excess after formation of a shock ramp, dissipation inside the ramp, and reflection of ion back upstream. The shock does this by inhibiting a substantial fraction of inflow ions to pass across the shock from upstream into the downstream region. It is sending these ions back into the upstream region where they cause a violently unstable upstream ion beam-plasma configuration which excites a large amplitude turbulent wave spectrum that scatters the uninformed plasma inflow, heats it and retards it down to the Mach number range that can be digested by the shock. In this way the collisionless shock generates a shock transition region that extends far upstream with the shock ramp degrading to the role of playing a subshock at the location where the ultimate decrease of the Mach number from upstream to downstream takes place.\nShock reflection has another important effect on the shock as the momentum transfer from the reflected particle component to the shock retards the shock in the region of reflection thereby decreasing the effective Mach number of the shock.\n\n21.7.1 Specular Reflection\nSpecular reflection of ions from a shock front is the simplest case to be imagined. It requires that the ions experience the shock ramp as an impenetrable wall. This can be the case when the shock itself contains a positive reflecting electric potential which builds up in front of the approaching ion. Generation of this electric potential is not clarified yet. In a very naive approach one assumes that in flowing magnetised plasma a potential wall is created as the consequence of charge separation between electrons and ions in penetrating the shock ramp. It occurs over a scale typically of the spatial difference between an ion and an electron gyro-radius, because in the ideal case the electrons, when running into the shock ramp, are held temporarily back in the steep magnetic field gradient over this distance while the ions feel the magnetic gradient only over a scale longer than their gyro-radius and thus penetrate deeper into the shock transition.\n\n\n\n\n\n\nFigure 21.12: The two cases of shock reflection. Left: Reflection from a potential well \\(\\Phi(x)\\). Particles of energy higher than the potential energy \\(e\\Phi\\) can pass while lower energy particles become reflected. Right: Reflection from the perpendicular shock region at a curved shock wave as the result of magnetic field compression. Particles move toward the shock like in a magnetic mirror bottle, experience the repelling mirror force and for large initial pitch angles are reflected back upstream.\n\n\n\n\nReflection from Shock Potential\n\nDue to this simplistic picture the shock ramp should contain a steep increase in the electric potential \\(\\Delta\\Phi\\) which will reflect any ion which has less kinetic energy \\(m_i V_N^2/2 < e\\Delta\\Phi\\) (Figure 21.12).\n\nMirror Reflection\n\nAnother simple possibility for particle reflection from a shock ramp in magnetised plasma is mirror reflection. An ion approaching the shock has components \\(v_{i\\parallel}\\). Assume a curved shock like Earth’s bow shock. Close to its perpendicular part where the upstream magnetic field becomes tangential to the shock the particles approaching the shock with the stream and moving along the magnetic field with their parallel velocities experience a mirror magnetic field configuration that results from the converging magnetic field lines near the perpendicular point (Figure 21.12). Conservation of the magnetic moment \\(\\mu = T_{i\\perp}/B\\) implies that the particles become heated adiabatically in the increasing field; they also experience a reflecting mirror force \\(-\\mu \\nabla_\\parallel B\\) which tries to keep ions away from entering the shock along the magnetic field. Particles will mirror at the perpendicular shock point and return upstream when their pitch angle becomes 90◦ at this location. (??? Leroy & Mangeney, 1984; Wu, 1984)\nSpecular reflection from shocks is the extreme case of shock particle reflection. Other mechanisms like turbulent reflection are, however, not well elaborated and must in any case be investigated with the help of numerical simulations.\n\n\n21.7.2 Consequences of Shock Reflection\nHow far the reflected ions return upstream depends on the direction of the magnetic field with respect to the shock, i.e. on the shock normal angle \\(\\theta_{Bn}\\). For perpendicular shocks the reflected ions only pass just one gyro-radius back upstream. Seeing the convection electric field \\(\\mathbf{E} = -\\mathbf{v}_\\mathrm{flow}\\times\\mathbf{B}\\) they become accelerated along the shock forming a current, the velocity of which in any case exceeds the inflow velocity (which is zero in the perpendicular direction) and for sufficiently cold ions also the ion acoustic velocity \\(c_{ia}\\) in which case the ion-beam plasma instability will be excited in the shock foot region where the ion current flows. This may generate anomalous collision in the shock foot region. Moreover, since the excited waves accelerate electrons along the magnetic field other secondary instabilities can arise as well.\nIn quasi-perpendicular and oblique shocks the ions can escape along the magnetic field. In this case an ion two-stream situation arises between the upstream beam and the plasma inflow with the consequence of excitation of a variety of electromagnetic and electrostatic instabilities. In addition, however, an ion-electron two-stream situation is caused between the upstream ions and the inflow electrons which because of the large upstream electron temperatures probably excites mainly ion-acoustic modes but can also lead to Buneman two-stream mode excitation. These modes contribute to turbulence in the upstream foreshock region creating a weakly dissipative state in the foreshock where the plasma inflow becomes informed about the presence of the shock. The electromagnetic low frequency instabilities on the other hand, which are excited in this region, will grow to large amplitude, form localised structures and after being convected by the main flow towards the shock ramp interact with the shock and modify the shock profile or even contribute to shock formation and shock regeneration.", + "text": "21.7 Shock Particle Reflection\nThe process of particle reflection from a shock wave is one of the most important processes in the entire physics of collisionless shocks. However, the mechanism of particle reflection has not yet been fully illuminated.\nParticle reflection is required in supercritical shocks as it is the only process that can compensate for the incapability of dissipative processes inside the shock ramp to digest the fast inflow of momentum and energy into the shock. Shock particle reflection is not dissipative by itself even though in a fluid picture which deals with moments of the distribution function it can be interpreted as kind of an ion viscosity, i.e. it generates an anomalous viscosity coefficient which appears as a factor in front of the second derivative of the ion velocity in the ionic equation of motion. As such it also appears in the ion heat-transport equation. The kinematic ion viscosity can be expressed as \\[\n\\mu_\\mathrm{vis} = m_i n \\nu_i \\lambda_\\mathrm{mfp} \\simeq P_i / 2\\omega_{ci}\n\\] through the ion pressure \\(P_i\\) and the ion-cyclotron frequency \\(\\omega_{ci}\\) when replacing the mean free path through the ion gyro-radius. In this sense shock particle reflection constitutes by itself a very efficient non-dissipative dissipation mechanism. However, its direct dissipative action is to produce real dissipation as far as possible upstream of the shock in order to dissipate as much energy of motion as remains to be in excess after formation of a shock ramp, dissipation inside the ramp, and reflection of ion back upstream. The shock does this by inhibiting a substantial fraction of inflow ions to pass across the shock from upstream into the downstream region. It is sending these ions back into the upstream region where they cause a violently unstable upstream ion beam-plasma configuration which excites a large amplitude turbulent wave spectrum that scatters the uninformed plasma inflow, heats it and retards it down to the Mach number range that can be digested by the shock. In this way the collisionless shock generates a shock transition region that extends far upstream with the shock ramp degrading to the role of playing a subshock at the location where the ultimate decrease of the Mach number from upstream to downstream takes place.\nShock reflection has another important effect on the shock as the momentum transfer from the reflected particle component to the shock retards the shock in the region of reflection thereby decreasing the effective Mach number of the shock.\n\n21.7.1 Specular Reflection\nSpecular reflection of ions from a shock front is the simplest case to be imagined. It requires that the ions experience the shock ramp as an impenetrable wall. This can be the case when the shock itself contains a positive reflecting electric potential which builds up in front of the approaching ion. Generation of this electric potential is not clarified yet. In a very naive approach one assumes that in flowing magnetised plasma a potential wall is created as the consequence of charge separation between electrons and ions in penetrating the shock ramp. It occurs over a scale typically of the spatial difference between an ion and an electron gyro-radius, because in the ideal case the electrons, when running into the shock ramp, are held temporarily back in the steep magnetic field gradient over this distance while the ions feel the magnetic gradient only over a scale longer than their gyro-radius and thus penetrate deeper into the shock transition.\n\n\n\n\n\n\nFigure 21.13: The two cases of shock reflection. Left: Reflection from a potential well \\(\\Phi(x)\\). Particles of energy higher than the potential energy \\(e\\Phi\\) can pass while lower energy particles become reflected. Right: Reflection from the perpendicular shock region at a curved shock wave as the result of magnetic field compression. Particles move toward the shock like in a magnetic mirror bottle, experience the repelling mirror force and for large initial pitch angles are reflected back upstream.\n\n\n\n\nReflection from Shock Potential\n\nDue to this simplistic picture the shock ramp should contain a steep increase in the electric potential \\(\\Delta\\Phi\\) which will reflect any ion which has less kinetic energy \\(m_i V_N^2/2 < e\\Delta\\Phi\\) (Figure 21.13).\n\nMirror Reflection\n\nAnother simple possibility for particle reflection from a shock ramp in magnetised plasma is mirror reflection. An ion approaching the shock has components \\(v_{i\\parallel}\\). Assume a curved shock like Earth’s bow shock. Close to its perpendicular part where the upstream magnetic field becomes tangential to the shock the particles approaching the shock with the stream and moving along the magnetic field with their parallel velocities experience a mirror magnetic field configuration that results from the converging magnetic field lines near the perpendicular point (Figure 21.13). Conservation of the magnetic moment \\(\\mu = T_{i\\perp}/B\\) implies that the particles become heated adiabatically in the increasing field; they also experience a reflecting mirror force \\(-\\mu \\nabla_\\parallel B\\) which tries to keep ions away from entering the shock along the magnetic field. Particles will mirror at the perpendicular shock point and return upstream when their pitch angle becomes 90◦ at this location. (??? Leroy & Mangeney, 1984; Wu, 1984)\nSpecular reflection from shocks is the extreme case of shock particle reflection. Other mechanisms like turbulent reflection are, however, not well elaborated and must in any case be investigated with the help of numerical simulations.\n\n\n21.7.2 Consequences of Shock Reflection\nHow far the reflected ions return upstream depends on the direction of the magnetic field with respect to the shock, i.e. on the shock normal angle \\(\\theta_{Bn}\\). For perpendicular shocks the reflected ions only pass just one gyro-radius back upstream. Seeing the convection electric field \\(\\mathbf{E} = -\\mathbf{v}_\\mathrm{flow}\\times\\mathbf{B}\\) they become accelerated along the shock forming a current, the velocity of which in any case exceeds the inflow velocity (which is zero in the perpendicular direction) and for sufficiently cold ions also the ion acoustic velocity \\(c_{ia}\\) in which case the ion-beam plasma instability will be excited in the shock foot region where the ion current flows. This may generate anomalous collision in the shock foot region. Moreover, since the excited waves accelerate electrons along the magnetic field other secondary instabilities can arise as well.\nIn quasi-perpendicular and oblique shocks the ions can escape along the magnetic field. In this case an ion two-stream situation arises between the upstream beam and the plasma inflow with the consequence of excitation of a variety of electromagnetic and electrostatic instabilities. In addition, however, an ion-electron two-stream situation is caused between the upstream ions and the inflow electrons which because of the large upstream electron temperatures probably excites mainly ion-acoustic modes but can also lead to Buneman two-stream mode excitation. These modes contribute to turbulence in the upstream foreshock region creating a weakly dissipative state in the foreshock where the plasma inflow becomes informed about the presence of the shock. The electromagnetic low frequency instabilities on the other hand, which are excited in this region, will grow to large amplitude, form localised structures and after being convected by the main flow towards the shock ramp interact with the shock and modify the shock profile or even contribute to shock formation and shock regeneration.", "crumbs": [ "21  Shock" ] @@ -1804,7 +1804,7 @@ "href": "contents/shock.html#shock-particle-acceleration", "title": "21  Shock", "section": "21.8 Shock Particle Acceleration", - "text": "21.8 Shock Particle Acceleration\nIn the context of cosmic rays that have been observed in the interstellar space, medium energy particles refer to ~ few GeV ions and ~ few MeV electrons. Above these ranges relativistic shocks must be considered. Near the Earth’s bow shock the solar wind hydrogen kinetic energy is ~ 1 keV; ~ 10 keV is about the low threshold for energetic ions. Here we limit our discussions first to the non-relativistic case.\nFigure 21.13 shows schematically the process of particle acceleration. Based on early estimations by Fermi (1949), a large number of shock crossings and reflections back and forth is required for the particles to reach energetic cosmic ray level. The scattering process is a stochastic process that is assumed to conserve energy; in particular they should not become involved into excitation of instabilities which consume part of their motional energy. The only actual dissipation that is allowed in this process is dissipation of bulk motional energy from where the few accelerated particles extract their energy gain. This dissipation is also attributed to direct particle loss by either convective transport or the limited size of the acceleration region. Thus this mechanism works until the gyro-radius of the accelerated particle becomes so large that it exceeds the size of the system.\nThe stochastic process implies that the basic equation that governs the process is a phase space diffusion equation in the form of a Fokker-Planck equation \\[\n\\frac{\\partial F(\\mathbf{p}, \\mathbf{x}, t)}{\\partial t} + \\mathbf{v}\\cdot\\nabla F(\\mathbf{p},\\mathbf{x},t) = \\frac{\\partial}{\\partial \\mathbf{p}}\\cdot\\mathbf{D}_{pp}\\cdot\\frac{\\partial F(\\mathbf{p}, \\mathbf{x}, t)}{\\partial \\mathbf{p}},\\quad \\mathbf{D}_{pp}=\\frac{1}{2}\\left< \\frac{\\Delta \\mathbf{p}\\Delta \\mathbf{p}}{\\Delta t} \\right>\n\\] where \\(\\Delta \\mathbf{p}\\) is the variation of the particle momentum in the scattering process which happens in the time interval \\(\\Delta t\\), and the angular brackets indicate ensemble averaging. \\(\\mathbf{D}_{pp}\\) is the momentum space diffusion tensor. It is customary to define \\(\\mu = \\cos\\alpha\\) as the cosine of the particle pitch angle \\(\\alpha\\) and to understand among \\(F(p,\\mu)\\) the gyro-phase averaged distribution function, which depends only on \\(p = |\\mathbf{p}|\\) and \\(\\mu\\).\n\n\n\n\n\n\nFigure 21.13: Schematic of the acceleration mechanism of a charged particle in reflection at a quasi-parallel (\\(\\theta_{Bn} < 45^\\circ\\)) supercritical shock. The upstream plasma flow (left, \\(\\mathbf{V}_1 \\gg \\mathbf{V}_2\\)) contains the various upstream plasma modes: upstream waves, shocklets, whistlers, pulsations. The downstream (right) is turbulent. The energetic particle that is injected at the shock to upstream is reflected in an energy gaining collision with upstream waves, moves downstream where it is reflected in an energy loosing collision back upstream. It looses energy because it overtakes the slow waves, but the energy loss is small. Returning to upstream it is scattered a second time again gaining energy. Its initially high energy is successively increased until it escapes from the shock and ends up in free space as an energetic Cosmic Ray particle. The energy gain is on the expense of the upstream flow which is gradually retarded in this interaction. However, the number of energetic particles is small and the energy gain per collision is also small. So the retardation of the upstream flow is much less than the retardation it experiences in the interaction with the shock-reflected low energy particles and the excitation of upstream turbulence.\n\n\n\nThe dependence on the gyro-radius imposes a severe limitation on the acceleration mechanism, i.e. the injection problem. In order to experience a first scattering, i.e. in order to being admitted to the acceleration process, the particle must initially already possess a gyro-radius much larger than the entire width of the shock transition region. Only when this condition is given, the shock will behave like an infinitesimally thin discontinuity separating two regions of vastly different velocities such that the particle when crossing back and forth over the shock can become aware of the bulk difference in speed and take an energetic advantage of it. This restriction rules out any particles in the core of the upstream inflow distribution from participation in the acceleration process: in order to enter the Fermi shock-acceleration mechanism a particle must be pre-accelerated or pre-heated until its gyro-radius becomes sufficiently large. This condition poses the injection problem, where an unresolved seed population of energetic particles are needed for further acceleration, that has not yet been resolved.\nForeshock transients (Section 21.9), especially HFAs and FBs, can accelerate particles and contribute to the primary shock acceleration. These can form secondary shocks which leads to several possible acceleration mechanisms; they can also cause local magnetic reconnection that accelerate particles. The interaction with foreshock transients provides a possible solution to Fermi’s injection problem and increase the acceleration efficiency of primary shocks.\n\nAs foreshock transients convect with the upstream flow, particles enclosed within their boundary and the primary shock can experience Fermi acceleration.\nSecondary shocks have also been observed to accelerate upstream particles on their own through the shock drift acceleration (SDA)14 and even to form a secondary foreshock.\nSecondary shocks can also capture and further energize primary shock-accelerated electrons through betatron acceleration.\nMagnetic reconnection inside foreshock transients contributes to the electron and ion acceleration/heating.\n\nAnother problem awakens attention is that how the shocks are modulated by the presence of energetic particles.\nIn terms of particle acceleration the shock appears as a boundary between two independent regions of different bulk flow parameters which are filled with scattering centres for the particles as sketched in Figure 21.14. Theoretically ((Balogh and Treumann 2013)) any particle which returns from downstream to upstream is accelerated in the upstream flow, even in the absence of any upstream turbulence and scatterings. If the upstream medium is magnetised and is sufficiently extended to host the upstream gyration orbit, pick-up ion energization can happen via the convection electric field \\(\\mathbf{E}=-\\mathbf{V}\\times\\mathbf{B}\\) all along their upstream half-gyrocircles. Alternatively, the upstream turbulence can also cause ion energization.\n\n\n\n\n\n\nFigure 21.14: Cartoon of the diffusive shock acceleration (left) and shock heating mechanisms [after an sketch by M. Scholer and Hoshino]. In diffusive shock acceleration the particle is scattered around the shock being much faster than the shock. The requirement is the presence of upstream waves and downstream turbulence or waves. In shock heating the particle is a member of the main particle distribution, is trapped for a while at the shock and thereby thermalised and accelerated until leaving the shock.", + "text": "21.8 Shock Particle Acceleration\nIn the context of cosmic rays that have been observed in the interstellar space, medium energy particles refer to ~ few GeV ions and ~ few MeV electrons. Above these ranges relativistic shocks must be considered. Near the Earth’s bow shock the solar wind hydrogen kinetic energy is ~ 1 keV; ~ 10 keV is about the low threshold for energetic ions. Here we limit our discussions first to the non-relativistic case.\nFigure 21.14 shows schematically the process of particle acceleration. Based on early estimations by Fermi (1949), a large number of shock crossings and reflections back and forth is required for the particles to reach energetic cosmic ray level. The scattering process is a stochastic process that is assumed to conserve energy; in particular they should not become involved into excitation of instabilities which consume part of their motional energy. The only actual dissipation that is allowed in this process is dissipation of bulk motional energy from where the few accelerated particles extract their energy gain. This dissipation is also attributed to direct particle loss by either convective transport or the limited size of the acceleration region. Thus this mechanism works until the gyro-radius of the accelerated particle becomes so large that it exceeds the size of the system.\nThe stochastic process implies that the basic equation that governs the process is a phase space diffusion equation in the form of a Fokker-Planck equation \\[\n\\frac{\\partial F(\\mathbf{p}, \\mathbf{x}, t)}{\\partial t} + \\mathbf{v}\\cdot\\nabla F(\\mathbf{p},\\mathbf{x},t) = \\frac{\\partial}{\\partial \\mathbf{p}}\\cdot\\mathbf{D}_{pp}\\cdot\\frac{\\partial F(\\mathbf{p}, \\mathbf{x}, t)}{\\partial \\mathbf{p}},\\quad \\mathbf{D}_{pp}=\\frac{1}{2}\\left< \\frac{\\Delta \\mathbf{p}\\Delta \\mathbf{p}}{\\Delta t} \\right>\n\\] where \\(\\Delta \\mathbf{p}\\) is the variation of the particle momentum in the scattering process which happens in the time interval \\(\\Delta t\\), and the angular brackets indicate ensemble averaging. \\(\\mathbf{D}_{pp}\\) is the momentum space diffusion tensor. It is customary to define \\(\\mu = \\cos\\alpha\\) as the cosine of the particle pitch angle \\(\\alpha\\) and to understand among \\(F(p,\\mu)\\) the gyro-phase averaged distribution function, which depends only on \\(p = |\\mathbf{p}|\\) and \\(\\mu\\).\n\n\n\n\n\n\nFigure 21.14: Schematic of the acceleration mechanism of a charged particle in reflection at a quasi-parallel (\\(\\theta_{Bn} < 45^\\circ\\)) supercritical shock. The upstream plasma flow (left, \\(\\mathbf{V}_1 \\gg \\mathbf{V}_2\\)) contains the various upstream plasma modes: upstream waves, shocklets, whistlers, pulsations. The downstream (right) is turbulent. The energetic particle that is injected at the shock to upstream is reflected in an energy gaining collision with upstream waves, moves downstream where it is reflected in an energy loosing collision back upstream. It looses energy because it overtakes the slow waves, but the energy loss is small. Returning to upstream it is scattered a second time again gaining energy. Its initially high energy is successively increased until it escapes from the shock and ends up in free space as an energetic Cosmic Ray particle. The energy gain is on the expense of the upstream flow which is gradually retarded in this interaction. However, the number of energetic particles is small and the energy gain per collision is also small. So the retardation of the upstream flow is much less than the retardation it experiences in the interaction with the shock-reflected low energy particles and the excitation of upstream turbulence.\n\n\n\nThe dependence on the gyro-radius imposes a severe limitation on the acceleration mechanism, i.e. the injection problem. In order to experience a first scattering, i.e. in order to being admitted to the acceleration process, the particle must initially already possess a gyro-radius much larger than the entire width of the shock transition region. Only when this condition is given, the shock will behave like an infinitesimally thin discontinuity separating two regions of vastly different velocities such that the particle when crossing back and forth over the shock can become aware of the bulk difference in speed and take an energetic advantage of it. This restriction rules out any particles in the core of the upstream inflow distribution from participation in the acceleration process: in order to enter the Fermi shock-acceleration mechanism a particle must be pre-accelerated or pre-heated until its gyro-radius becomes sufficiently large. This condition poses the injection problem, where an unresolved seed population of energetic particles are needed for further acceleration, that has not yet been resolved.\nForeshock transients (Section 21.9), especially HFAs and FBs, can accelerate particles and contribute to the primary shock acceleration. These can form secondary shocks which leads to several possible acceleration mechanisms; they can also cause local magnetic reconnection that accelerate particles. The interaction with foreshock transients provides a possible solution to Fermi’s injection problem and increase the acceleration efficiency of primary shocks.\n\nAs foreshock transients convect with the upstream flow, particles enclosed within their boundary and the primary shock can experience Fermi acceleration.\nSecondary shocks have also been observed to accelerate upstream particles on their own through the shock drift acceleration (SDA)14 and even to form a secondary foreshock.\nSecondary shocks can also capture and further energize primary shock-accelerated electrons through betatron acceleration.\nMagnetic reconnection inside foreshock transients contributes to the electron and ion acceleration/heating.\n\nAnother problem awakens attention is that how the shocks are modulated by the presence of energetic particles.\nIn terms of particle acceleration the shock appears as a boundary between two independent regions of different bulk flow parameters which are filled with scattering centres for the particles as sketched in Figure 21.15. Theoretically ((Balogh and Treumann 2013)) any particle which returns from downstream to upstream is accelerated in the upstream flow, even in the absence of any upstream turbulence and scatterings. If the upstream medium is magnetised and is sufficiently extended to host the upstream gyration orbit, pick-up ion energization can happen via the convection electric field \\(\\mathbf{E}=-\\mathbf{V}\\times\\mathbf{B}\\) all along their upstream half-gyrocircles. Alternatively, the upstream turbulence can also cause ion energization.\n\n\n\n\n\n\nFigure 21.15: Cartoon of the diffusive shock acceleration (left) and shock heating mechanisms [after an sketch by M. Scholer and Hoshino]. In diffusive shock acceleration the particle is scattered around the shock being much faster than the shock. The requirement is the presence of upstream waves and downstream turbulence or waves. In shock heating the particle is a member of the main particle distribution, is trapped for a while at the shock and thereby thermalised and accelerated until leaving the shock.", "crumbs": [ "21  Shock" ]