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contents/fluid.html

@@ -818,7 +818,7 @@ <h3 data-number="5.4.3" class="anchored" data-anchor-id="fiveten-moment"><span c
 \]</span></p>
 <p>Again, the equations here must be closed by some approximation for the heat-flux tensor. Another option is to include evolution equations for even higher order moments, e.g., the ten independent components of the heat-flux tensor.</p>
 <p>Theoretically, the multifluid-Maxwell equations approach the Hall magnetohydrodynamics (MHD) under asymptotic limits of vanishing electron mass (<span class="math inline">\(m_e \rightarrow 0\)</span>) and infinite speed of light (<span class="math inline">\(c\rightarrow\infty\)</span>). All waves and effects within the two-fluid picture are retained, for example, the light wave, electron and ion inertial effects like the ion cyclotron wave and whistler wave. Particularly, through properly devised heat-flux closures, the ten-moment model could partially capture nonlocal kinetic effects like Landau damping, in a manner similar to the gyrokinetic models.</p>
-<p>Although multifluid-Maxwell models provide a more complete description of the plasma than reduced, asymptotic models like MHD, they are less frequently used. The reason for this is the fast kinetic scales involved. Retaining the electron inertia adds plasma-frequency and cyclotron time-scale, while non-neutrality adds Debye length spatial-scales. Further, inclusion of the displacement currents means that EM waves must be resolved when using an explicit scheme. Fortunately, the restrictions due to kinetic scales are introduced <em>only through the non-hyperbolic source terms</em> of the momentum equation, the Ampère’s law, and the pressure equation. Therefore we may eliminate these restrictions by <em>updating the source term separately either exactly</em> or <em>using an implicit algorithm</em> (Note: BATSRUS applies the point-implicit scheme.). This allows larger time steps and leads to significant speedup, especially with realistic electron/ion mass ratios. The speed of light constraint still exists, however, can be greatly relaxed, using reduced values for the speed of light and/or sub-cycling Maxwell equations. Of course, an implicit Maxwell solver, or a reduced set of electromagnetic equations like the Darwin approximation (???), can also relax the time-step restrictions.</p>
+<p>Although multifluid-Maxwell models provide a more complete description of the plasma than reduced, asymptotic models like MHD, they are less frequently used. The reason for this is the fast kinetic scales involved. Retaining the electron inertia adds plasma-frequency and cyclotron time-scale, while non-neutrality adds Debye length spatial-scales. Further, inclusion of the displacement currents means that EM waves must be resolved when using an explicit scheme. Fortunately, the restrictions due to kinetic scales are introduced <em>only through the non-hyperbolic source terms</em> of the momentum equation, the Ampère’s law, and the pressure equation. Therefore we may eliminate these restrictions by <em>updating the source term separately either exactly</em> or <em>using an implicit algorithm</em> (Note: BATSRUS applies the point-implicit scheme.). This allows larger time steps and leads to significant speedup, especially with realistic electron/ion mass ratios. The speed of light constraint still exists, however, can be greatly relaxed, using reduced values for the speed of light and/or sub-cycling Maxwell equations. Of course, an implicit Maxwell solver, or a reduced set of electromagnetic equations like the Darwin approximation<a href="#fn1" class="footnote-ref" id="fnref1" role="doc-noteref"><sup>1</sup></a>, can also relax the time-step restrictions.</p>
 </section>
 <section id="characteristic-wave-speeds" class="level3" data-number="5.4.4">
 <h3 data-number="5.4.4" class="anchored" data-anchor-id="characteristic-wave-speeds"><span class="header-section-number">5.4.4</span> Characteristic wave speeds</h3>
@@ -1239,7 +1239,7 @@ <h4 data-number="5.6.2.2" class="anchored" data-anchor-id="mhd-approximations-fo
 \]</span> On comparing the magnitude of the displacement current term in Ampère’s law to the left-hand side it is seen that <span class="math display">\[
 \frac{\mu_0\epsilon_0|\dot{\mathbf{E}}|}{|\nabla\times\mathbf{B}|} \sim \frac{c^{-2} E/t_{\mathrm{char}}}{B/l_{\mathrm{char}}} \sim \left( \frac{V_{\mathrm{ph}}}{c} \right)^2
 \]</span></p>
-<p>Thus, if <span class="math inline">\(V_{\mathrm{ph}} \ll c\)</span> the displacement current term can be dropped from Amperè’s law resulting in the so-called “pre-Maxwell” form <span id="eq-ampere-pre-maxwell"><span class="math display">\[
+<p>Thus, if <span class="math inline">\(V_{\mathrm{ph}} \ll c\)</span> the displacement current term can be dropped from Amperè’s law resulting in the so-called “pre-Maxwell” form (i.e.&nbsp;Darwin approximation) <span id="eq-ampere-pre-maxwell"><span class="math display">\[
 \nabla\times\mathbf{B} = \mu_0 \mathbf{J}
 \tag{5.47}\]</span></span></p>
 <p>The divergence of <a href="#eq-ampere-pre-maxwell">Equation&nbsp;<span>5.47</span></a> gives <span class="math inline">\(\nabla\cdot\mathbf{J}=0\)</span> so it is unnecessary to specify it separately.</p>
@@ -1421,6 +1421,12 @@ <h2 data-number="5.9" class="anchored" data-anchor-id="sec-plasma_approximation"
 </div>
 </div>
 </section>
+<section id="footnotes" class="footnotes footnotes-end-of-document" role="doc-endnotes">
+<hr>
+<ol>
+<li id="fn1"><p>the Darwin approximation ignores light waves by neglecting <span class="math inline">\(\partial\mathbf{E}\partial t\)</span> in Amperè’s law.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
+</ol>
+</section>
 
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contents/hybrid.html

@@ -411,6 +411,7 @@ <h2 data-number="24.1" class="anchored" data-anchor-id="classical-hybrid-model">
 \]</span> where <span class="math inline">\(n_e \approx n_i\)</span> and <span class="math inline">\(T_e=T_i\)</span>. Note however in a plasma electron pressure is usually higher than ion temperature, so this is a very crude assumption. Another commonly used assumption is an adiabatic process <span class="math display">\[
 P_e = n_e^\gamma k_B T_e = n_0(n/n_0)^\gamma k_B T_{e0}
 \]</span> where <span class="math inline">\(\gamma=5/3\)</span> is the adiabatic index for a monatomic ideal gas.</p>
+<p>A more complete review is given by <span class="citation" data-cites="winske2023hybrid">(<a href="#ref-winske2023hybrid" role="doc-biblioref">Winske et al. 2023</a>)</span> for the hybrid-kinetic model assuming massless electrons. The essential problem in all hybrid algorithms is how to calculate the electric field at the next time step.</p>
 <section id="pros-and-cons" class="level3" data-number="24.1.1">
 <h3 data-number="24.1.1" class="anchored" data-anchor-id="pros-and-cons"><span class="header-section-number">24.1.1</span> Pros and Cons</h3>
 <p>Strengths:</p>
@@ -743,6 +744,9 @@ <h3 data-number="24.8.4" class="anchored" data-anchor-id="collisionless-shock"><
 <div id="ref-rambo1995finite" class="csl-entry" role="listitem">
 Rambo, PW. 1995. <span>“Finite-Grid Instability in Quasineutral Hybrid Simulations.”</span> <em>Journal of Computational Physics</em> 118 (1): 152–58. <a href="https://doi.org/10.1006/jcph.1995.1086">https://doi.org/10.1006/jcph.1995.1086</a>.
 </div>
+<div id="ref-winske2023hybrid" class="csl-entry" role="listitem">
+Winske, Dan, Homa Karimabadi, Ari Yitzchak Le, Nojan Nick Omidi, Vadim Roytershteyn, and Adam John Stanier. 2023. <span>“Hybrid-Kinetic Approach: Massless Electrons.”</span> In <em>Space and Astrophysical Plasma Simulation: Methods, Algorithms, and Applications</em>, 63–91. Springer. <a href="https://doi.org/10.1007/978-3-031-11870-8_3">https://doi.org/10.1007/978-3-031-11870-8_3</a>.
+</div>
 </div>
 </section>
 </section>