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- Firehose Instability: Linear Theory
- - Mirror Instability: Linear Theory
+ - Mirror Instability: Linear Theory
+
- Origin of Pressure Anisotropy
-
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+
+ Comparison With Slow Mode
+
+- Driving Mechanism
+
+- Mirror Mode: Driven primarily by pressure anisotropy (\(T_\perp/T_\parallel > 1\)) in high-beta plasmas.
+- MHD Slow Mode: Driven by pressure gradients and magnetic field line curvature in low- to moderate-beta plasmas.
+
+- Propagation Characteristics
+
+- Mirror Mode: Primarily propagates parallel to the background magnetic field, but can have a small perpendicular component. It is a non-propagating mode in the fluid limit (zero frequency), but it can acquire a finite frequency due to kinetic effects.
+- MHD Slow Mode: Propagates obliquely to the magnetic field, with both parallel and perpendicular components. It is a propagating mode with a finite frequency.
+
+- Plasma Conditions
+
+- Mirror Mode: Typically found in high-beta (\(\gtrsim 1\)) plasmas, such as the Earth’s magnetosheath and the solar wind.
+- MHD Slow Mode: More common in low- to moderate-beta (\(\ll 1\)) plasmas, such as the solar corona and the Earth’s magnetosphere, where the magnetic pressure dominates.
+
+
+
Origin of Pressure Anisotropy
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"href": "contents/kmhd.html#sec-mirror",
"title": "14 Kinetic MHD",
"section": "14.2 Mirror Instability: Linear Theory",
- "text": "14.2 Mirror Instability: Linear Theory\nLet us go back to Equation 14.13 and Equation 14.14 and get apart from Alfvénic what other perturbations there are and when they are stable. We have already looked at the Alfvénic perturbation \\(\\delta\\hat{b}=\\delta \\mathbf{B}_\\perp/B\\). Now consider \\[\n\\frac{\\delta B}{B} = \\frac{\\delta B_\\parallel}{B}\n\\tag{14.17}\\]\nFrom Equation 14.13, we have the perpendicular compression increases B: \\[\n\\omega\\frac{\\delta B}{B} = \\mathbf{k}_\\perp\\cdot\\delta\\mathbf{u}_\\perp\n\\]\nTake \\(\\mathbf{k}_\\perp\\cdot\\) Equation 14.14: \\[\n\\omega\\rho\\mathbf{k}_\\perp\\cdot\\delta\\mathbf{u}_\\perp = \\rho\\omega^2 \\frac{\\delta B}{B} = k_\\perp^2\\Big( \\delta p_\\perp + \\frac{B\\delta B}{\\mu_0} \\Big) + k_\\parallel^2\\Big( p_\\perp - p_\\parallel + \\frac{B^2}{\\mu_0} \\Big) \\frac{\\delta B}{B}\n\\tag{14.18}\\]\nNote the \\(p_\\perp\\) term here: we need kinetic theory to calculate this! Fortunately we have Equation 14.12 ready for calculating \\[\n\\delta p_\\perp = \\int \\mathrm{d}\\mathbf{w}\\frac{m_sw_\\perp^2}{2}\\delta f_s(w_\\perp, w_\\parallel)\n\\]\n\\(\\delta f_s(w_\\perp, w_\\parallel)\\) can be obtained by calculating \\(F_s(\\mu,\\epsilon)\\) and transforming back to \\(w_\\perp,w_\\parallel\\).\nHere is a cute subtlety: our macroscopic equilibrium, around which we are expanding the distribution is \\[\nF_{0s}(\\mu,\\epsilon) = f_{0s}(w_\\perp,w_\\parallel) = f_{0s}\\Big( \\sqrt{\\frac{2B_0\\mu}{m_s}},\\sqrt{\\frac{2(\\epsilon-\\mu B_0)}{m_s}} \\Big)\n\\] which contains \\(B_0\\) the unperturbed magnetic field. \\(\\mu\\) in \\(F_0\\) contains \\(B_0+\\delta B\\), and this has to be taken into account when transforming to \\(w_\\perp,w_\\parallel\\). Now when we perturb everything: \\[\n\\begin{aligned}\nF_s(\\mu,\\epsilon) &= F_{0s}(\\mu,\\epsilon) + \\delta F_s \\\\\n&= f_{0s}(w_\\perp,w_\\parallel) + \\delta f_s \\\\\n&= f_{0s}\\Big( \\sqrt{\\frac{2\\mu(B_0+\\delta B)}{m_s}},\\sqrt{\\frac{2[\\epsilon-\\mu(B_0+\\delta B)]}{m_s}} \\Big) + \\delta f_s \\\\\n&= f_{0s}\\Big( \\sqrt{\\frac{2\\mu B_0}{m_s}}\\sqrt{1+\\frac{\\delta B}{B_0}},\\sqrt{\\frac{2(\\epsilon-\\mu B_0)}{m_s}}\\sqrt{1-\\frac{m_s\\mu\\delta B}{(\\epsilon-\\mu B_0)}} \\Big) + \\delta f_s \\\\\n&\\approx f_{0s}\\Big( \\sqrt{\\frac{2\\mu B_0}{m_s}},\\sqrt{\\frac{2(\\epsilon-\\mu B_0)}{m_s}} \\Big) + \\frac{2\\mu}{m_s}\\delta B\\Big(\\frac{\\partial f_{0s}}{\\partial w_\\perp^2}-\\frac{\\partial f_{0s}}{\\partial w_\\parallel^2} \\Big) + \\delta f_s\n\\end{aligned}\n\\]\nThus \\[\n\\delta f_s = \\delta F_s - w_\\perp^2\\frac{\\delta B}{B}\\Big(\\frac{\\partial f_{0s}}{\\partial w_\\perp^2}-\\frac{\\partial f_{0s}}{\\partial w_\\parallel^2} \\Big)\n\\]\nIf \\(f_{0s}\\) is a bi-Maxwellian, \\[\nf_{0s} = \\frac{n_s}{\\pi^{3/2}v_{\\text{th}\\perp s}^2 v_{\\text{th}\\parallel s}}\\exp\\Big(-\\frac{w_\\perp^2}{v_{\\text{th}\\perp s}^2}-\\frac{w_\\parallel^2}{v_{\\text{th}\\parallel s}^2}\\Big)\n\\] then this can be further written as \\[\n\\delta f_s = \\delta F_s + w_\\perp^2\\frac{\\delta B}{B}\\Big(\\frac{1}{v_{\\text{th}\\perp s}^2} - \\frac{1}{v_{\\text{th}\\parallel s}^2} \\Big) f_{0s} = \\delta F_s + w_\\perp^2\\frac{\\delta B}{B}\\frac{m_sn_s}{2}\\Big(\\frac{1}{p_{\\perp s}} - \\frac{1}{p_{\\parallel s}} \\Big) f_{0s}\n\\]\nWe can eliminate the partial derivatives via integration by parts: \\[\n\\begin{aligned}\n\\int \\mathrm{d}\\mathbf{w}\\frac{\\partial f_{0s}}{\\partial w_\\parallel^2} &= \\int_0^{2\\pi}\\mathrm{d}\\theta \\int dw_\\perp\\int \\frac{1}{2w_\\parallel}\\frac{\\partial f_{0s}}{\\partial w_\\parallel}dw_\\parallel \\\\\n&= \\int_0^{2\\pi}\\mathrm{d}\\theta \\int dw_\\perp\\int\\frac{1}{2w_\\parallel}\\mathrm{d} f_{0s} \\\\\n&= \\int_0^{2\\pi}\\mathrm{d}\\theta \\int dw_\\perp\\Big[\\cancel{\\frac{f_{0s}}{2w_\\parallel}\\bigg\\rvert_{-\\infty}^{\\infty}} - \\int f_{0s}\\mathrm{d}\\frac{1}{2w_\\parallel}\\Big] \\\\\n&= \\int_0^{2\\pi}\\mathrm{d}\\theta \\int dw_\\perp\\int\\frac{1}{2w_\\parallel^2}f_{0s}dw_\\parallel \\\\\n&= \\int \\mathrm{d}\\mathbf{w} \\frac{1}{2w_\\parallel^2}f_{0s} \\\\\n\\int \\mathrm{d}\\mathbf{w}w_\\perp w_\\perp^4\\frac{\\partial f_{0s}}{\\partial w_\\perp^2} &= 2\\pi\\int dw_\\parallel \\Big[ \\cancel{\\frac{1}{2}w_\\perp^4f_{0s}\\bigg\\rvert_{-\\infty}^{+\\infty}} - \\frac{1}{2}\\int f_{0s}dw_\\perp^4 \\Big] \\\\\n&= -2\\pi\\int dw_\\parallel 2w_\\perp^2 f_{0s}w_\\perp dw_\\perp \\\\\n&= -2 \\int \\mathrm{d}\\mathbf{w} w_\\perp^2 f_{0s}\n\\end{aligned}\n\\]\nThis then gives us \\[\n\\begin{aligned}\n\\delta p_{\\perp s} &= \\int \\mathrm{d}\\mathbf{w}\\frac{m_s w_\\perp^2}{2}\\delta f_s \\\\\n&= \\int \\mathrm{d}\\mathbf{w}\\frac{m_s w_\\perp^2}{2}\\delta F_s - \\int \\mathrm{d}\\mathbf{w}\\frac{m_s w_\\perp^4}{2}\\Big(\\frac{\\partial f_{0s}}{\\partial w_\\perp^2}-\\frac{\\partial f_{0s}}{\\partial w_\\parallel^2} \\Big)\\frac{\\delta B}{B} \\\\\n&= \\int \\mathrm{d}\\mathbf{w}\\frac{m_s w_\\perp^2}{2}\\delta F_s - \\int_0^{2\\pi}\\mathrm{d}\\theta \\int dw_\\perp w_\\perp \\int \\mathrm{d} w_\\parallel \\frac{m_s w_\\perp^4}{2}\\Big(\\frac{\\partial f_{0s}}{\\partial w_\\perp^2}-\\frac{\\partial f_{0s}}{\\partial w_\\parallel^2} \\Big)\\frac{\\delta B}{B} \\\\\n&= \\int \\mathrm{d}\\mathbf{w}\\frac{m_s w_\\perp^2}{2}\\delta F_s + 2\\int \\mathrm{d}\\mathbf{w}\\frac{m_s w_\\perp^2}{2}\\delta f_{0s}\\frac{\\delta B}{B} + \\int \\mathrm{d}\\mathbf{w}\\frac{2(\\frac{1}{2}m_sw_\\perp^2)^2}{m_sw_\\parallel^2}f_{0s}\\frac{\\delta B}{B} \\\\\n&= \\int \\mathrm{d}\\mathbf{w}\\frac{m_s w_\\perp^2}{2}\\delta F_s + \\frac{\\delta B}{B}\\Big( 2p_{\\perp s} - \\frac{2 p_{\\perp s}^2}{p_{\\parallel s}} \\alpha_s )\n\\end{aligned}\n\\tag{14.19}\\] where \\(\\alpha_s\\) is some coefficients of order 1 if \\(f_{0s}\\) is not bi-Maxwellian.\n\\(\\delta F_s\\) can be obtained by ignoring collisions and linearizing and Fourier-transforming Equation 14.12 (\\(\\mathbf{u}_s=0\\)): \\[\n\\begin{aligned}\n-i(\\omega - k_\\parallel w_\\parallel)\\delta F_s = - \\Big[ m_sw_\\parallel\\Big( \\frac{q_s}{m_s}E_\\parallel - i(\\omega-k_\\parallel w_\\parallel) \\delta u_{\\parallel s}\\Big) -i\\omega\\mu\\delta B \\Big]\\frac{\\partial F_{0s}}{\\partial \\epsilon} \\\\\n\\delta F_s = -i\\frac{w_\\parallel q_s E_\\parallel}{\\omega-k_\\parallel w_\\parallel}\\frac{\\partial F_{0s}}{\\partial \\epsilon} - \\delta u_{\\parallel s}m_sw_\\parallel\\frac{\\partial F_{0s}}{\\partial\\epsilon} - \\frac{\\omega}{\\omega-k_\\parallel w_\\parallel}\\mu\\delta B\\frac{\\partial F_{0s}}{\\partial\\epsilon}\n\\end{aligned}\n\\]\nThe first term can be ignored if \\(\\beta\\gg 1\\) (??? See the complete calculation in another note!); otherwise \\(E_\\parallel\\) can be got by imposing \\(\\sum_s q_s n_s = 0\\). The second term can be shown to be equivalent to \\(\\delta u_{\\parallel s}\\partial f_{0s}/\\partial w_\\parallel\\): \\[\n\\begin{aligned}\n\\frac{\\partial f_{0s}}{\\partial w_\\parallel} &= \\frac{\\partial f_{0s}}{\\partial \\epsilon}\\frac{\\partial \\epsilon}{\\partial w_\\parallel} + \\frac{\\partial f_{0s}}{\\partial\\mu}\\cancel{\\frac{\\partial \\mu}{\\partial w_\\parallel}} \\\\\n&= \\frac{\\partial F_{0s}}{\\partial\\epsilon}m_s w_\\parallel\n\\end{aligned}\n\\] so this will not contribute to \\(\\delta p_\\perp\\) because it integrates to 0.\nThe third term can be written as \\[\n\\frac{\\omega}{\\omega-k_\\parallel w_\\parallel}\\mu\\delta B\\frac{\\partial F_{0s}}{\\partial\\epsilon} = \\frac{\\omega}{\\omega-k_\\parallel w_\\parallel}\\frac{m_sw_\\perp^2}{2}\\frac{\\delta B}{B}\\frac{1}{w_\\parallel}\\frac{\\partial f_{0s}}{\\partial w_\\parallel} = \\frac{\\omega}{\\omega-k_\\parallel w_\\parallel} m_sw_\\perp^2\\frac{\\delta B}{B}\\frac{\\partial f_{0s}}{\\partial w_\\parallel^2}\n\\]\nThus, the “relevant” part of \\(\\delta F_s\\) is \\[\n\\delta F_s = -\\frac{\\omega}{\\omega-k_\\parallel w_\\parallel} m_sw_\\perp^2\\frac{\\delta B}{B}\\frac{\\partial f_{0s}}{\\partial w_\\parallel^2}\n\\] and its contribution to \\(\\delta p_{\\perp s}\\) is \\[\n\\int \\mathrm{d}\\mathbf{w}\\frac{m_sw_\\perp^2}{2}\\delta F_s = \\frac{\\delta B}{B}\\frac{\\omega}{|k_\\parallel|}\\int\\frac{dw_\\parallel}{w_\\parallel - \\frac{\\omega}{|k_\\parallel|}} \\Big[ \\frac{\\partial}{\\partial w_\\parallel^2}\\int \\mathrm{d}\\mathbf{w}_\\perp \\frac{m_s^2w_\\perp^4}{2} f_{0s} \\Big]\n\\]\nHere we have \\(|k_\\parallel|\\) because if \\(k_\\parallel <0\\), we can change the variable \\(w_\\parallel \\rightarrow -w_\\parallel\\). This involves the Landau integral, which can be evaluated with the residual theorem Equation 3.3 when integrate in the complex plane mostly along the real axis and the large semicircle in the upper half plane except for a small semicircle just below the pole (ADD FIGURE!): \\[\n\\frac{1}{w_\\parallel - \\frac{\\omega}{|k_\\parallel|}} = P\\frac{1}{w_\\parallel - \\frac{\\omega}{|k_\\parallel|}} + i\\pi\\delta\\Big(w_\\parallel - \\frac{\\omega}{|k_\\parallel|} \\Big)\n\\] so \\[\n\\begin{aligned}\n\\int \\mathrm{d}\\mathbf{w}\\frac{m_sw_\\perp^2}{2}\\delta F_s &= \\frac{\\delta B}{B}\\Big[ \\cancel{\\frac{\\omega}{|k_\\parallel|}P\\int\\frac{dw_\\parallel}{w_\\parallel - \\frac{\\omega}{|k_\\parallel|}} \\big[ ... \\big]} + i\\pi\\frac{\\omega}{|k_\\parallel|}\\big[ ... \\big]_{w_\\parallel}=\\omega/|k_\\parallel| \\Big]\n\\end{aligned}\n\\]\nThe first term is small when we assume \\(\\omega\\ll k_\\parallel v_{\\text{th}s\\parallel}\\); the second term must be kept because it is the lowest-order imaginary part which will lead to instability.\nFor a bi-Maxwellian, \\[\n\\Big[ \\frac{\\partial}{\\partial w_\\parallel^2}\\int \\mathrm{d}\\mathbf{w}_\\perp\\frac{m_s^2w_\\perp^4}{2}f_{0s} \\Big]_{w_\\parallel=\\omega/|k_\\parallel|} = -\\frac{2p_{\\perp s}^2}{p_{\\parallel s}}\\frac{e^{-\\frac{\\omega^2}{k_\\parallel^2 v_{\\text{th}\\parallel s}^2}}}{\\sqrt{\\pi}v_{\\text{th}\\parallel s}}\n\\]\nThe exponential term is nearly 1. If it is not a bi-Maxwellian, then we need to multiply by a coefficient \\(\\alpha_s \\sim 1\\).\nEquation 14.19 becomes \\[\n\\delta p_{\\perp s} = \\frac{\\delta B}{B}\\Big[ 2p_{\\perp s} - \\frac{2p_{\\perp s}^2}{p_{\\parallel s}}\\Big( \\alpha_s + i\\sqrt{\\pi}\\frac{\\omega}{|k_\\parallel|v_{\\text{th}\\parallel s}}\\sigma_s \\Big) \\Big]\n\\]\nThis goes into Equation 14.18: \\[\n\\rho \\omega^2 = k_\\perp^2\\frac{B^2}{\\mu_0}\\Big[ \\sum_s(1 - \\frac{p_{\\perp s}}{p_{\\parallel s}}\\alpha_s)\\beta_{\\perp s} - i\\sum_s \\sigma_s\\frac{p_{\\perp s}}{p_{\\parallel s}}\\beta_{\\perp s}\\sqrt{\\pi}\\frac{\\omega}{|k_\\parallel| v_{\\text{th} \\parallel s}} + 1 \\Big] + k_\\parallel^2 \\frac{B^2}{\\mu_0}\\Big[ \\sum_s\\frac{\\beta_{\\perp s}}{2}\\big( 1 - \\frac{p_{\\parallel s}}{p_{\\perp s}} \\big) + 1 \\Big]\n\\]\nThe left-hand side can be neglected because \\(\\omega\\ll k_\\parallel v_{\\text{th}\\parallel s}\\). The electron thermal velocity \\(v_{\\text{th}\\parallel e}\\) in the denominator can be neglected because \\(v_{\\text{th}\\parallel e} \\gg v_{\\text{th}\\parallel i}\\). The growth rate \\(\\gamma\\) is the imaginary part of \\(\\omega\\). Reorganize the last equation: \\[\n\\sigma_i\\frac{p_{\\perp i}}{p_{\\parallel i}}\\beta_{\\perp i}\\sqrt{\\pi}\\frac{\\gamma}{|k_\\parallel|v_{\\text{th}\\parallel i}} = \\sum_s \\Big(\\frac{p_\\perp s}{p_\\parallel s}\\alpha_s - 1 \\Big)\\beta_{\\perp s} - 1 -\\frac{k_\\parallel^2}{k_\\perp^2}\\Big[ \\sum_s\\frac{\\beta_{\\perp s}}{2}\\big( 1-\\frac{p_{\\parallel s}}{p_{\\perp s}}\\big) + 1 \\Big]\n\\tag{14.20}\\] where \\(\\Lambda\\equiv \\frac{k_\\parallel^2}{k_\\perp^2}\\sum_s \\Big(\\frac{p_\\perp s}{p_\\parallel s}\\alpha_s - 1 \\Big)\\beta_{\\perp s} - 1\\) triggers instability if this is positive: \\[\n\\sum_s \\Big(\\frac{p_\\perp s}{p_\\parallel s}\\alpha_s - 1 \\Big)\\beta_{\\perp s} > 1\n\\]\nExamining where this comes from, we see that this amounts to \\(\\delta p_\\perp\\) modifying the magnetic pressure force and turning it from positive to negative: \\[\n\\delta p_\\perp + \\frac{B\\delta B}{\\mu_0} = \\frac{B\\delta B}{\\mu_0}\\Big[ \\underbrace{1}_{\\text{B pressure}} - \\underbrace{\\sum_s\\Big( \\frac{p_{\\perp s}}{p_{\\parallel s}}\\alpha_s - 1 \\Big)\\beta_{\\perp s}}_{\\substack{\\text{non-resonant} \\\\ \\text{particle pressure}}} + \\underbrace{...}_{\\substack{\\text{resonant particle} \\\\ \\text{pressure}}} \\Big]\n\\]\nThus, fundamentally, pressure anisotropy makes it easier to compress or rarefy magnetic field — and things become unstable when the sign of the pressure flips and it becomes energetically profitable to create compressions and rarefications. (ADD FIGURE!) The dispersion relation Equation 14.20 is basically a statement of pressure balance between the magnetic pressure, the non-resonant particle pressure \\(\\delta p_\\perp\\) and the resonant particle pressure \\(\\propto\\gamma\\), which came from the betatron acceleration \\(\\mu \\mathrm{d}B/\\mathrm{d}t\\) in Equation 14.12. (See also Eq. 21 in Southwood and Kivelson (1993))\nThe betatron acceleration term refers to what happens in the stable case. When magnetic pressure opposes formation of \\(\\delta B\\) perturbations (say, troughs), to compensate it, we must have \\(\\gamma<0\\) and energy goes from \\(\\delta B\\) to resonant particles, which are accelerated by the mirror force. The corresponding decaying of \\(\\delta B\\) is the well-known Barnes damping (landau damping of “mirror field”, Barnes 1966, also known as transit-time damping from Stix’s book.)\nMore discussion on the physics is presented by Southwood and Kivelson (1993). They pointed out that although the instability is resonant, the role of resonant particle is unusual. The instability results from pressure imbalance between the bulk of the plasma and the magnetic field. For this to occur, the bulk (nonresonant) pressure response must be in antiphase with the magnetic pressure as occurs at low frequencies when the magnetic moment and the particle energy are conserved. The resonant particles produce a pressure perturbation, however, in phase with the field pressure change. A corollary is that unlike the nonresonant particles,the resonant particles experience energy changes as the instability develops. The linear growth rate of most resonant instabilities is proportional to the number of resonant particles. However, in the case of the mirror instability, the growth is inversely proportional to the number (and pressure contribution) of resonant particles.The reason for the anomalous result is that the fewer particles there are at small parallel velocity the higher the growth rate needs to be to balance the pressure imbalance generated by the nonresonant distribution.\nTo finish the job, note that, from Equation 14.20 (ADD FIGURE!) for a given \\(k_\\perp\\) \\[\n\\frac{\\partial\\gamma}{\\partial k_\\parallel}\\bigg\\rvert_{k_\\perp} \\propto \\Lambda - \\frac{k_\\parallel^2}{k_\\perp^2}\\Big[\\sum_s\\frac{\\beta_{\\perp s}}{2}\\Big(1-\\frac{p_\\parallel s}{p_\\perp s} \\Big) + 1 \\Big]\n\\]\nThe maximum growth rate is reached when the right-hand side goes to 0, which is equivalent to \\(\\frac{2}{3}\\Lambda\\), so the maximum growth rate \\[\n\\gamma_{\\text{max}} = \\frac{|k_\\parallel|v_{\\text{th}\\parallel i}}{\\sqrt{\\pi}}\\frac{2}{3}\\Lambda \\frac{p_{\\parallel i}}{p_{\\perp i}}\\frac{1}{\\sigma_i\\beta_{\\perp i}}\n\\]\nWe have assumed \\(\\gamma\\ll k_\\parallel v_{\\text{th}\\parallel s}\\), which is indeed true if \\[\n\\Lambda\\frac{1}{\\beta_{\\perp i}} = \\big(\\sum_s A_s \\beta_{\\perp s} - 1 \\big)\\frac{1}{\\beta_{\\perp i}}\\ll 1\n\\] so our approximations are consistent.\nIf we are close to marginal instablity, \\[\n\\frac{k_\\parallel}{k_\\perp}\\sim\\sqrt{\\Lambda}\\ll 1\n\\] so mirror modes are highly oblique near the threshold.\nAnother important point is that again we encounter the UV catastrophe since \\(\\gamma\\propto k_\\parallel\\). The mirror mode is a fast, microscale instability whose peak growth rate is outside KMHD regime. Including finite larmor radius gives (Hellinger 2007 PoP 14, 082105?) \\[\n\\gamma_{\\text{peak}}\\sim\\Big(A-\\frac{1}{\\beta}\\Big)^2\\beta\\Omega_i,\\quad k_{\\text{peak}}r_i\\sim\\Big( A-\\frac{1}{\\beta}\\Big)\\beta\n\\]\nThus, any high-\\(\\beta\\) macroscopic solution of KMHD with \\(p_\\perp>p_\\parallel\\) will blow up, just like the case for \\(p_\\parallel > p_\\perp\\), and again what happens next depends on how mirror instability saturates. Note that \\(A_e\\) is ignored since \\(A_e\\ll A_i\\) (?). The mirror instability condition is \\[\n\\begin{aligned}\n\\frac{p_{\\perp i}}{p_{\\parallel i}} - 1 > \\frac{1}{\\beta_{\\perp i}} = \\frac{1}{\\beta_{\\parallel i}}\\frac{p_{\\parallel i}}{p_{\\perp i}} \\\\\n\\frac{p_{\\perp i}}{p_{\\parallel i}}\\Big( \\frac{p_{\\perp i}}{p_{\\parallel i}}-1\\Big) > \\frac{1}{\\beta_{\\parallel i}}\n\\end{aligned}\n\\]\nFigure 14.2 shows observation from Wind spacecraft. The solar wind indeed seems to stay within these boundaries. (ADD REFS!)\n\n\n\n\n\n\nFigure 14.2: Collective solar wind observation data (\\(\\sim 1e6\\)) from Wind spacecraft. The lines represent the instability thresholds for mirror and firehose instability, respectively.",
+ "text": "14.2 Mirror Instability: Linear Theory\nLet us go back to Equation 14.13 and Equation 14.14 and get apart from Alfvénic what other perturbations there are and when they are stable. We have already looked at the Alfvénic perturbation \\(\\delta\\hat{b}=\\delta \\mathbf{B}_\\perp/B\\). Now consider \\[\n\\frac{\\delta B}{B} = \\frac{\\delta B_\\parallel}{B}\n\\tag{14.17}\\]\nFrom Equation 14.13, we have the perpendicular compression increases B: \\[\n\\omega\\frac{\\delta B}{B} = \\mathbf{k}_\\perp\\cdot\\delta\\mathbf{u}_\\perp\n\\]\nTake \\(\\mathbf{k}_\\perp\\cdot\\) Equation 14.14: \\[\n\\omega\\rho\\mathbf{k}_\\perp\\cdot\\delta\\mathbf{u}_\\perp = \\rho\\omega^2 \\frac{\\delta B}{B} = k_\\perp^2\\Big( \\delta p_\\perp + \\frac{B\\delta B}{\\mu_0} \\Big) + k_\\parallel^2\\Big( p_\\perp - p_\\parallel + \\frac{B^2}{\\mu_0} \\Big) \\frac{\\delta B}{B}\n\\tag{14.18}\\]\nNote the \\(p_\\perp\\) term here: we need kinetic theory to calculate this! Fortunately we have Equation 14.12 ready for calculating \\[\n\\delta p_\\perp = \\int \\mathrm{d}\\mathbf{w}\\frac{m_sw_\\perp^2}{2}\\delta f_s(w_\\perp, w_\\parallel)\n\\]\n\\(\\delta f_s(w_\\perp, w_\\parallel)\\) can be obtained by calculating \\(F_s(\\mu,\\epsilon)\\) and transforming back to \\(w_\\perp,w_\\parallel\\).\nHere is a cute subtlety: our macroscopic equilibrium, around which we are expanding the distribution is \\[\nF_{0s}(\\mu,\\epsilon) = f_{0s}(w_\\perp,w_\\parallel) = f_{0s}\\Big( \\sqrt{\\frac{2B_0\\mu}{m_s}},\\sqrt{\\frac{2(\\epsilon-\\mu B_0)}{m_s}} \\Big)\n\\] which contains \\(B_0\\) the unperturbed magnetic field. \\(\\mu\\) in \\(F_0\\) contains \\(B_0+\\delta B\\), and this has to be taken into account when transforming to \\(w_\\perp,w_\\parallel\\). Now when we perturb everything: \\[\n\\begin{aligned}\nF_s(\\mu,\\epsilon) &= F_{0s}(\\mu,\\epsilon) + \\delta F_s \\\\\n&= f_{0s}(w_\\perp,w_\\parallel) + \\delta f_s \\\\\n&= f_{0s}\\Big( \\sqrt{\\frac{2\\mu(B_0+\\delta B)}{m_s}},\\sqrt{\\frac{2[\\epsilon-\\mu(B_0+\\delta B)]}{m_s}} \\Big) + \\delta f_s \\\\\n&= f_{0s}\\Big( \\sqrt{\\frac{2\\mu B_0}{m_s}}\\sqrt{1+\\frac{\\delta B}{B_0}},\\sqrt{\\frac{2(\\epsilon-\\mu B_0)}{m_s}}\\sqrt{1-\\frac{m_s\\mu\\delta B}{(\\epsilon-\\mu B_0)}} \\Big) + \\delta f_s \\\\\n&\\approx f_{0s}\\Big( \\sqrt{\\frac{2\\mu B_0}{m_s}},\\sqrt{\\frac{2(\\epsilon-\\mu B_0)}{m_s}} \\Big) + \\frac{2\\mu}{m_s}\\delta B\\Big(\\frac{\\partial f_{0s}}{\\partial w_\\perp^2}-\\frac{\\partial f_{0s}}{\\partial w_\\parallel^2} \\Big) + \\delta f_s\n\\end{aligned}\n\\]\nThus \\[\n\\delta f_s = \\delta F_s - w_\\perp^2\\frac{\\delta B}{B}\\Big(\\frac{\\partial f_{0s}}{\\partial w_\\perp^2}-\\frac{\\partial f_{0s}}{\\partial w_\\parallel^2} \\Big)\n\\]\nIf \\(f_{0s}\\) is a bi-Maxwellian, \\[\nf_{0s} = \\frac{n_s}{\\pi^{3/2}v_{\\text{th}\\perp s}^2 v_{\\text{th}\\parallel s}}\\exp\\Big(-\\frac{w_\\perp^2}{v_{\\text{th}\\perp s}^2}-\\frac{w_\\parallel^2}{v_{\\text{th}\\parallel s}^2}\\Big)\n\\] then this can be further written as \\[\n\\delta f_s = \\delta F_s + w_\\perp^2\\frac{\\delta B}{B}\\Big(\\frac{1}{v_{\\text{th}\\perp s}^2} - \\frac{1}{v_{\\text{th}\\parallel s}^2} \\Big) f_{0s} = \\delta F_s + w_\\perp^2\\frac{\\delta B}{B}\\frac{m_sn_s}{2}\\Big(\\frac{1}{p_{\\perp s}} - \\frac{1}{p_{\\parallel s}} \\Big) f_{0s}\n\\]\nWe can eliminate the partial derivatives via integration by parts: \\[\n\\begin{aligned}\n\\int \\mathrm{d}\\mathbf{w}\\frac{\\partial f_{0s}}{\\partial w_\\parallel^2} &= \\int_0^{2\\pi}\\mathrm{d}\\theta \\int dw_\\perp\\int \\frac{1}{2w_\\parallel}\\frac{\\partial f_{0s}}{\\partial w_\\parallel}dw_\\parallel \\\\\n&= \\int_0^{2\\pi}\\mathrm{d}\\theta \\int dw_\\perp\\int\\frac{1}{2w_\\parallel}\\mathrm{d} f_{0s} \\\\\n&= \\int_0^{2\\pi}\\mathrm{d}\\theta \\int dw_\\perp\\Big[\\cancel{\\frac{f_{0s}}{2w_\\parallel}\\bigg\\rvert_{-\\infty}^{\\infty}} - \\int f_{0s}\\mathrm{d}\\frac{1}{2w_\\parallel}\\Big] \\\\\n&= \\int_0^{2\\pi}\\mathrm{d}\\theta \\int dw_\\perp\\int\\frac{1}{2w_\\parallel^2}f_{0s}dw_\\parallel \\\\\n&= \\int \\mathrm{d}\\mathbf{w} \\frac{1}{2w_\\parallel^2}f_{0s} \\\\\n\\int \\mathrm{d}\\mathbf{w}w_\\perp w_\\perp^4\\frac{\\partial f_{0s}}{\\partial w_\\perp^2} &= 2\\pi\\int dw_\\parallel \\Big[ \\cancel{\\frac{1}{2}w_\\perp^4f_{0s}\\bigg\\rvert_{-\\infty}^{+\\infty}} - \\frac{1}{2}\\int f_{0s}dw_\\perp^4 \\Big] \\\\\n&= -2\\pi\\int dw_\\parallel 2w_\\perp^2 f_{0s}w_\\perp dw_\\perp \\\\\n&= -2 \\int \\mathrm{d}\\mathbf{w} w_\\perp^2 f_{0s}\n\\end{aligned}\n\\]\nThis then gives us \\[\n\\begin{aligned}\n\\delta p_{\\perp s} &= \\int \\mathrm{d}\\mathbf{w}\\frac{m_s w_\\perp^2}{2}\\delta f_s \\\\\n&= \\int \\mathrm{d}\\mathbf{w}\\frac{m_s w_\\perp^2}{2}\\delta F_s - \\int \\mathrm{d}\\mathbf{w}\\frac{m_s w_\\perp^4}{2}\\Big(\\frac{\\partial f_{0s}}{\\partial w_\\perp^2}-\\frac{\\partial f_{0s}}{\\partial w_\\parallel^2} \\Big)\\frac{\\delta B}{B} \\\\\n&= \\int \\mathrm{d}\\mathbf{w}\\frac{m_s w_\\perp^2}{2}\\delta F_s - \\int_0^{2\\pi}\\mathrm{d}\\theta \\int dw_\\perp w_\\perp \\int \\mathrm{d} w_\\parallel \\frac{m_s w_\\perp^4}{2}\\Big(\\frac{\\partial f_{0s}}{\\partial w_\\perp^2}-\\frac{\\partial f_{0s}}{\\partial w_\\parallel^2} \\Big)\\frac{\\delta B}{B} \\\\\n&= \\int \\mathrm{d}\\mathbf{w}\\frac{m_s w_\\perp^2}{2}\\delta F_s + 2\\int \\mathrm{d}\\mathbf{w}\\frac{m_s w_\\perp^2}{2}\\delta f_{0s}\\frac{\\delta B}{B} + \\int \\mathrm{d}\\mathbf{w}\\frac{2(\\frac{1}{2}m_sw_\\perp^2)^2}{m_sw_\\parallel^2}f_{0s}\\frac{\\delta B}{B} \\\\\n&= \\int \\mathrm{d}\\mathbf{w}\\frac{m_s w_\\perp^2}{2}\\delta F_s + \\frac{\\delta B}{B}\\Big( 2p_{\\perp s} - \\frac{2 p_{\\perp s}^2}{p_{\\parallel s}} \\alpha_s )\n\\end{aligned}\n\\tag{14.19}\\] where \\(\\alpha_s\\) is some coefficients of order 1 if \\(f_{0s}\\) is not bi-Maxwellian.\n\\(\\delta F_s\\) can be obtained by ignoring collisions and linearizing and Fourier-transforming Equation 14.12 (\\(\\mathbf{u}_s=0\\)): \\[\n\\begin{aligned}\n-i(\\omega - k_\\parallel w_\\parallel)\\delta F_s = - \\Big[ m_sw_\\parallel\\Big( \\frac{q_s}{m_s}E_\\parallel - i(\\omega-k_\\parallel w_\\parallel) \\delta u_{\\parallel s}\\Big) -i\\omega\\mu\\delta B \\Big]\\frac{\\partial F_{0s}}{\\partial \\epsilon} \\\\\n\\delta F_s = -i\\frac{w_\\parallel q_s E_\\parallel}{\\omega-k_\\parallel w_\\parallel}\\frac{\\partial F_{0s}}{\\partial \\epsilon} - \\delta u_{\\parallel s}m_sw_\\parallel\\frac{\\partial F_{0s}}{\\partial\\epsilon} - \\frac{\\omega}{\\omega-k_\\parallel w_\\parallel}\\mu\\delta B\\frac{\\partial F_{0s}}{\\partial\\epsilon}\n\\end{aligned}\n\\]\nThe first term can be ignored if \\(\\beta\\gg 1\\) (??? See the complete calculation in another note!); otherwise \\(E_\\parallel\\) can be got by imposing \\(\\sum_s q_s n_s = 0\\). The second term can be shown to be equivalent to \\(\\delta u_{\\parallel s}\\partial f_{0s}/\\partial w_\\parallel\\): \\[\n\\begin{aligned}\n\\frac{\\partial f_{0s}}{\\partial w_\\parallel} &= \\frac{\\partial f_{0s}}{\\partial \\epsilon}\\frac{\\partial \\epsilon}{\\partial w_\\parallel} + \\frac{\\partial f_{0s}}{\\partial\\mu}\\cancel{\\frac{\\partial \\mu}{\\partial w_\\parallel}} \\\\\n&= \\frac{\\partial F_{0s}}{\\partial\\epsilon}m_s w_\\parallel\n\\end{aligned}\n\\] so this will not contribute to \\(\\delta p_\\perp\\) because it integrates to 0.\nThe third term can be written as \\[\n\\frac{\\omega}{\\omega-k_\\parallel w_\\parallel}\\mu\\delta B\\frac{\\partial F_{0s}}{\\partial\\epsilon} = \\frac{\\omega}{\\omega-k_\\parallel w_\\parallel}\\frac{m_sw_\\perp^2}{2}\\frac{\\delta B}{B}\\frac{1}{w_\\parallel}\\frac{\\partial f_{0s}}{\\partial w_\\parallel} = \\frac{\\omega}{\\omega-k_\\parallel w_\\parallel} m_sw_\\perp^2\\frac{\\delta B}{B}\\frac{\\partial f_{0s}}{\\partial w_\\parallel^2}\n\\]\nThus, the “relevant” part of \\(\\delta F_s\\) is \\[\n\\delta F_s = -\\frac{\\omega}{\\omega-k_\\parallel w_\\parallel} m_sw_\\perp^2\\frac{\\delta B}{B}\\frac{\\partial f_{0s}}{\\partial w_\\parallel^2}\n\\] and its contribution to \\(\\delta p_{\\perp s}\\) is \\[\n\\int \\mathrm{d}\\mathbf{w}\\frac{m_sw_\\perp^2}{2}\\delta F_s = \\frac{\\delta B}{B}\\frac{\\omega}{|k_\\parallel|}\\int\\frac{dw_\\parallel}{w_\\parallel - \\frac{\\omega}{|k_\\parallel|}} \\Big[ \\frac{\\partial}{\\partial w_\\parallel^2}\\int \\mathrm{d}\\mathbf{w}_\\perp \\frac{m_s^2w_\\perp^4}{2} f_{0s} \\Big]\n\\]\nHere we have \\(|k_\\parallel|\\) because if \\(k_\\parallel <0\\), we can change the variable \\(w_\\parallel \\rightarrow -w_\\parallel\\). This involves the Landau integral, which can be evaluated with the residual theorem Equation 3.3 when integrate in the complex plane mostly along the real axis and the large semicircle in the upper half plane except for a small semicircle just below the pole (ADD FIGURE!): \\[\n\\frac{1}{w_\\parallel - \\frac{\\omega}{|k_\\parallel|}} = P\\frac{1}{w_\\parallel - \\frac{\\omega}{|k_\\parallel|}} + i\\pi\\delta\\Big(w_\\parallel - \\frac{\\omega}{|k_\\parallel|} \\Big)\n\\] so \\[\n\\begin{aligned}\n\\int \\mathrm{d}\\mathbf{w}\\frac{m_sw_\\perp^2}{2}\\delta F_s &= \\frac{\\delta B}{B}\\Big[ \\cancel{\\frac{\\omega}{|k_\\parallel|}P\\int\\frac{dw_\\parallel}{w_\\parallel - \\frac{\\omega}{|k_\\parallel|}} \\big[ ... \\big]} + i\\pi\\frac{\\omega}{|k_\\parallel|}\\big[ ... \\big]_{w_\\parallel}=\\omega/|k_\\parallel| \\Big]\n\\end{aligned}\n\\]\nThe first term is small when we assume \\(\\omega\\ll k_\\parallel v_{\\text{th}s\\parallel}\\); the second term must be kept because it is the lowest-order imaginary part which will lead to instability.\nFor a bi-Maxwellian, \\[\n\\Big[ \\frac{\\partial}{\\partial w_\\parallel^2}\\int \\mathrm{d}\\mathbf{w}_\\perp\\frac{m_s^2w_\\perp^4}{2}f_{0s} \\Big]_{w_\\parallel=\\omega/|k_\\parallel|} = -\\frac{2p_{\\perp s}^2}{p_{\\parallel s}}\\frac{e^{-\\frac{\\omega^2}{k_\\parallel^2 v_{\\text{th}\\parallel s}^2}}}{\\sqrt{\\pi}v_{\\text{th}\\parallel s}}\n\\]\nThe exponential term is nearly 1. If it is not a bi-Maxwellian, then we need to multiply by a coefficient \\(\\alpha_s \\sim 1\\).\nEquation 14.19 becomes \\[\n\\delta p_{\\perp s} = \\frac{\\delta B}{B}\\Big[ 2p_{\\perp s} - \\frac{2p_{\\perp s}^2}{p_{\\parallel s}}\\Big( \\alpha_s + i\\sqrt{\\pi}\\frac{\\omega}{|k_\\parallel|v_{\\text{th}\\parallel s}}\\sigma_s \\Big) \\Big]\n\\]\nThis goes into Equation 14.18: \\[\n\\rho \\omega^2 = k_\\perp^2\\frac{B^2}{\\mu_0}\\Big[ \\sum_s(1 - \\frac{p_{\\perp s}}{p_{\\parallel s}}\\alpha_s)\\beta_{\\perp s} - i\\sum_s \\sigma_s\\frac{p_{\\perp s}}{p_{\\parallel s}}\\beta_{\\perp s}\\sqrt{\\pi}\\frac{\\omega}{|k_\\parallel| v_{\\text{th} \\parallel s}} + 1 \\Big] + k_\\parallel^2 \\frac{B^2}{\\mu_0}\\Big[ \\sum_s\\frac{\\beta_{\\perp s}}{2}\\big( 1 - \\frac{p_{\\parallel s}}{p_{\\perp s}} \\big) + 1 \\Big]\n\\]\nThe left-hand side can be neglected because \\(\\omega\\ll k_\\parallel v_{\\text{th}\\parallel s}\\). The electron thermal velocity \\(v_{\\text{th}\\parallel e}\\) in the denominator can be neglected because \\(v_{\\text{th}\\parallel e} \\gg v_{\\text{th}\\parallel i}\\). The growth rate \\(\\gamma\\) is the imaginary part of \\(\\omega\\). Reorganize the last equation: \\[\n\\sigma_i\\frac{p_{\\perp i}}{p_{\\parallel i}}\\beta_{\\perp i}\\sqrt{\\pi}\\frac{\\gamma}{|k_\\parallel|v_{\\text{th}\\parallel i}} = \\sum_s \\Big(\\frac{p_\\perp s}{p_\\parallel s}\\alpha_s - 1 \\Big)\\beta_{\\perp s} - 1 -\\frac{k_\\parallel^2}{k_\\perp^2}\\Big[ \\sum_s\\frac{\\beta_{\\perp s}}{2}\\big( 1-\\frac{p_{\\parallel s}}{p_{\\perp s}}\\big) + 1 \\Big]\n\\tag{14.20}\\] where \\(\\Lambda\\equiv \\frac{k_\\parallel^2}{k_\\perp^2}\\sum_s \\Big(\\frac{p_\\perp s}{p_\\parallel s}\\alpha_s - 1 \\Big)\\beta_{\\perp s} - 1\\) triggers instability if this is positive: \\[\n\\sum_s \\Big(\\frac{p_\\perp s}{p_\\parallel s}\\alpha_s - 1 \\Big)\\beta_{\\perp s} > 1\n\\]\nExamining where this comes from, we see that this amounts to \\(\\delta p_\\perp\\) modifying the magnetic pressure force and turning it from positive to negative: \\[\n\\delta p_\\perp + \\frac{B\\delta B}{\\mu_0} = \\frac{B\\delta B}{\\mu_0}\\Big[ \\underbrace{1}_{\\text{B pressure}} - \\underbrace{\\sum_s\\Big( \\frac{p_{\\perp s}}{p_{\\parallel s}}\\alpha_s - 1 \\Big)\\beta_{\\perp s}}_{\\substack{\\text{non-resonant} \\\\ \\text{particle pressure}}} + \\underbrace{...}_{\\substack{\\text{resonant particle} \\\\ \\text{pressure}}} \\Big]\n\\]\nThus, fundamentally, pressure anisotropy makes it easier to compress or rarefy magnetic field — and things become unstable when the sign of the pressure flips and it becomes energetically profitable to create compressions and rarefications. (ADD FIGURE!) The dispersion relation Equation 14.20 is basically a statement of pressure balance between the magnetic pressure, the non-resonant particle pressure \\(\\delta p_\\perp\\) and the resonant particle pressure \\(\\propto\\gamma\\), which came from the betatron acceleration \\(\\mu \\mathrm{d}B/\\mathrm{d}t\\) in Equation 14.12. (See also Eq. 21 in Southwood and Kivelson (1993))\nThe betatron acceleration term refers to what happens in the stable case. When magnetic pressure opposes formation of \\(\\delta B\\) perturbations (say, troughs), to compensate it, we must have \\(\\gamma<0\\) and energy goes from \\(\\delta B\\) to resonant particles, which are accelerated by the mirror force. The corresponding decaying of \\(\\delta B\\) is the well-known Barnes damping (landau damping of “mirror field”, Barnes 1966, also known as transit-time damping from Stix’s book.)\nMore discussion on the physics is presented by Southwood and Kivelson (1993). They pointed out that although the instability is resonant, the role of resonant particle is unusual. The instability results from pressure imbalance between the bulk of the plasma and the magnetic field. For this to occur, the bulk (nonresonant) pressure response must be in antiphase with the magnetic pressure as occurs at low frequencies when the magnetic moment and the particle energy are conserved. The resonant particles produce a pressure perturbation, however, in phase with the field pressure change. A corollary is that unlike the nonresonant particles,the resonant particles experience energy changes as the instability develops. The linear growth rate of most resonant instabilities is proportional to the number of resonant particles. However, in the case of the mirror instability, the growth is inversely proportional to the number (and pressure contribution) of resonant particles.The reason for the anomalous result is that the fewer particles there are at small parallel velocity the higher the growth rate needs to be to balance the pressure imbalance generated by the nonresonant distribution.\nTo finish the job, note that, from Equation 14.20 (ADD FIGURE!) for a given \\(k_\\perp\\) \\[\n\\frac{\\partial\\gamma}{\\partial k_\\parallel}\\bigg\\rvert_{k_\\perp} \\propto \\Lambda - \\frac{k_\\parallel^2}{k_\\perp^2}\\Big[\\sum_s\\frac{\\beta_{\\perp s}}{2}\\Big(1-\\frac{p_\\parallel s}{p_\\perp s} \\Big) + 1 \\Big]\n\\]\nThe maximum growth rate is reached when the right-hand side goes to 0, which is equivalent to \\(\\frac{2}{3}\\Lambda\\), so the maximum growth rate \\[\n\\gamma_{\\text{max}} = \\frac{|k_\\parallel|v_{\\text{th}\\parallel i}}{\\sqrt{\\pi}}\\frac{2}{3}\\Lambda \\frac{p_{\\parallel i}}{p_{\\perp i}}\\frac{1}{\\sigma_i\\beta_{\\perp i}}\n\\]\nWe have assumed \\(\\gamma\\ll k_\\parallel v_{\\text{th}\\parallel s}\\), which is indeed true if \\[\n\\Lambda\\frac{1}{\\beta_{\\perp i}} = \\big(\\sum_s A_s \\beta_{\\perp s} - 1 \\big)\\frac{1}{\\beta_{\\perp i}}\\ll 1\n\\] so our approximations are consistent.\nIf we are close to marginal instablity, \\[\n\\frac{k_\\parallel}{k_\\perp}\\sim\\sqrt{\\Lambda}\\ll 1\n\\] so mirror modes are highly oblique near the threshold.\nAnother important point is that again we encounter the UV catastrophe since \\(\\gamma\\propto k_\\parallel\\). The mirror mode is a fast, microscale instability whose peak growth rate is outside KMHD regime. Including finite larmor radius gives (Hellinger 2007 PoP 14, 082105?) \\[\n\\gamma_{\\text{peak}}\\sim\\Big(A-\\frac{1}{\\beta}\\Big)^2\\beta\\Omega_i,\\quad k_{\\text{peak}}r_i\\sim\\Big( A-\\frac{1}{\\beta}\\Big)\\beta\n\\]\nThus, any high-\\(\\beta\\) macroscopic solution of KMHD with \\(p_\\perp>p_\\parallel\\) will blow up, just like the case for \\(p_\\parallel > p_\\perp\\), and again what happens next depends on how mirror instability saturates. Note that \\(A_e\\) is ignored since \\(A_e\\ll A_i\\) (?). The mirror instability condition is \\[\n\\begin{aligned}\n\\frac{p_{\\perp i}}{p_{\\parallel i}} - 1 > \\frac{1}{\\beta_{\\perp i}} = \\frac{1}{\\beta_{\\parallel i}}\\frac{p_{\\parallel i}}{p_{\\perp i}} \\\\\n\\frac{p_{\\perp i}}{p_{\\parallel i}}\\Big( \\frac{p_{\\perp i}}{p_{\\parallel i}}-1\\Big) > \\frac{1}{\\beta_{\\parallel i}}\n\\end{aligned}\n\\]\nFigure 14.2 shows observation from Wind spacecraft. The solar wind indeed seems to stay within these boundaries. (ADD REFS!)\n\n\n\n\n\n\nFigure 14.2: Collective solar wind observation data (\\(\\sim 1e6\\)) from Wind spacecraft. The lines represent the instability thresholds for mirror and firehose instability, respectively.\n\n\n\n\n14.2.1 Comparison With Slow Mode\n\nDriving Mechanism\n\nMirror Mode: Driven primarily by pressure anisotropy (\\(T_\\perp/T_\\parallel > 1\\)) in high-beta plasmas.\nMHD Slow Mode: Driven by pressure gradients and magnetic field line curvature in low- to moderate-beta plasmas.\n\nPropagation Characteristics\n\nMirror Mode: Primarily propagates parallel to the background magnetic field, but can have a small perpendicular component. It is a non-propagating mode in the fluid limit (zero frequency), but it can acquire a finite frequency due to kinetic effects.\nMHD Slow Mode: Propagates obliquely to the magnetic field, with both parallel and perpendicular components. It is a propagating mode with a finite frequency.\n\nPlasma Conditions\n\nMirror Mode: Typically found in high-beta (\\(\\gtrsim 1\\)) plasmas, such as the Earth’s magnetosheath and the solar wind.\nMHD Slow Mode: More common in low- to moderate-beta (\\(\\ll 1\\)) plasmas, such as the solar corona and the Earth’s magnetosphere, where the magnetic pressure dominates.",
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