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Lecture 17
Kinetic MHD: Collisionless Pressure Response

Overview

Kinetic MHD keeps the low-frequency MHD force law but replaces the pressure closure.

1.
The magnetic geometry, field-line bending, and frozen-in motion are still governed by the same low-frequency framework developed in the flux-freezing lecture, in Lecture 6, and in Lecture 16.
2.
The new physics enters through the collisionless pressure response, obtained from the drift-kinetic equation rather than from a fluid closure.
3.
The main lesson is beautifully sharp: the firehose mode is already visible in anisotropic fluid theory, while the mirror mode is the place where the kinetic closure changes the quantitative answer.
Historical Perspective

The CGL closure showed in 1956 that a magnetized collisionless plasma can still be described by a fluid system, provided one respects the magnetic-field direction and the adiabatic invariants Chew et al. (1956). The next step was to recognize that not every low-frequency problem is captured accurately by a local fluid closure. Kulsrud’s treatment made the low-frequency bridge from MHD to drift-kinetic theory especially clear Kulsrud (1983). For anisotropy-driven instabilities, the classic mirror calculations of Vedenov and Sagdeev and the later drift-mirror analysis of Hasegawa showed explicitly where the kinetic response changes the answer Vedenov and Sagdeev (1958a,b); Hasegawa (1969). The physical mechanism of the mirror mode was later clarified in a particularly transparent way by Southwood and Kivelson Southwood and Kivelson (1993). The modern language “kinetic MHD” is therefore well chosen: it is still MHD in its force balance, but kinetic in its closure.

Kinetic MHD sits in the gap between ideal fluid theory and full kinetic theory. It is still a long-wavelength, low-frequency theory, so the magnetic field and bulk motion are treated in an MHD spirit. But it refuses to guess the pressure response. Instead it computes that response from the kinetic equation. That is why this lecture is worth separating from the ordinary Alfvén-wave lecture: it is not merely another dispersion relation, but a different lesson about how plasma theory is organized.

17.1 What kinetic MHD keeps and what it changes

Ordering. We retain the low-frequency, magnetized ordering

\[\omega \ll \Omega _i, \qquad k_\perp \rho _i \ll 1, \tag{17.1}\]
so the frozen-in relation of Eq. (4.13) and the basic low-frequency force balance of Eq. (1.8) remain useful. What is not assumed is rapid collisional relaxation to a local Maxwellian.

Force law versus closure. In a uniform anisotropic equilibrium with \(\B _0=B_0\vect {e}_z\) and \(\vect {k}=k_\perp \vect {e}_x+k_\parallel \vect {e}_z\), the linearized momentum equation still has the same geometric pieces as in the previous lecture. For the compressive branch follows directly from perpendicular and parallel force balance: one may write schematically

\[\begin{aligned}-\omega ^2 \rho _0 \vect {\xi }_\perp &= - i\vect {k}_\perp \left ( \delta p_\perp + \frac {B_0\delta B}{\muo } \right ) - k_\parallel ^2 \left ( \frac {B_0^2}{\muo }+p_{\perp 0}-p_{\parallel 0} \right )\vect {\xi }_\perp , \\ -\omega ^2 \rho _0 \xi _\parallel &= - i k_\parallel \left [ \delta p_\parallel + \left (p_{\perp 0}-p_{\parallel 0}\right )\frac {\delta B}{B_0} \right ].\end{aligned} \tag{17.2}\]

These are still MHD-looking equations. The real question is: what are \(\delta p_\perp \) and \(\delta p_\parallel \)? In CGL they come from the double-adiabatic laws. In kinetic MHD they come from the drift-kinetic equation.

The lesson in one sentence. If a mode mainly tests magnetic tension, CGL and kinetic MHD often agree. If a mode mainly tests compressive pressure balance, the kinetic closure matters. That is why the firehose threshold is already visible in Eq. (16.46), while the mirror threshold needs to be re-derived.

17.2 Drift-kinetic pressure response of a bi-Maxwellian plasma

Geometry and variables. Take a uniform equilibrium field

\[\B _0 = B_0 \vect {e}_z, \qquad \delta B \equiv \delta B_\parallel , \qquad E_\parallel = - i k_\parallel \tilde {\Phi }, \qquad \propto e^{-i\omega t + i k_\parallel z}. \tag{17.4}\]
For compressive perturbations,
\[\frac {\delta B}{B_0} = - i \vect {k}_\perp \cdot \vect {\xi }_\perp . \tag{17.5}\]
Use guiding-center variables \((\mu ,v_\parallel )\), where
\[\mu \equiv \frac {m_s v_\perp ^2}{2B_0}. \tag{17.6}\]
The equilibrium distribution for each species is taken to be bi-Maxwellian:
\[f_{0s}(\mu ,v_\parallel ) = n_0 \frac {m_s}{2\pi T_{\perp s}} \left (\frac {m_s}{2\pi T_{\parallel s}}\right )^{1/2} \exp \left ( -\frac {m_s v_\parallel ^2}{2T_{\parallel s}} -\frac {\mu B_0}{T_{\perp s}} \right ). \tag{17.7}\]
It is convenient to define
\[A_s \equiv \frac {T_{\perp s}}{T_{\parallel s}}, \qquad v_{{\rm th}\parallel s} \equiv \sqrt {\frac {2T_{\parallel s}}{m_s}}, \qquad x_s \equiv \frac {v_\parallel }{v_{{\rm th}\parallel s}}, \qquad \zeta _s \equiv \frac {\omega }{k_\parallel v_{{\rm th}\parallel s}}. \tag{17.8}\]

Linearized drift-kinetic equation. The parallel part of the drift-kinetic equation is

\[-i(\omega -k_\parallel v_\parallel ) f_{1s} + \left ( \frac {q_s}{m_s}E_\parallel - \frac {\mu }{m_s} i k_\parallel \delta B \right ) \pp {f_{0s}}{v_\parallel } =0. \tag{17.9}\]
For the bi-Maxwellian equilibrium,
\[\pp {f_{0s}}{v_\parallel } = -\frac {m_s v_\parallel }{T_{\parallel s}} f_{0s}. \tag{17.10}\]
Insert \(E_\parallel =-ik_\parallel \tilde {\Phi }\) and Eq. (17.10) into Eq. (17.9):
\[\begin{aligned}f_{1s} &= \frac {k_\parallel v_\parallel }{\omega -k_\parallel v_\parallel } \left ( \frac {q_s\tilde {\Phi }}{T_{\parallel s}} + \frac {\mu \delta B}{T_{\parallel s}} \right )f_{0s} \nonumber \\ &= \frac {x_s}{\zeta _s-x_s} \left ( \frac {q_s\tilde {\Phi }}{T_{\parallel s}} + \frac {\mu B_0}{T_{\parallel s}}\frac {\delta B}{B_0} \right )f_{0s}.\end{aligned} \tag{17.11}\]

This formula already shows the central resonance: \(\omega -k_\parallel v_\parallel \).

Plasma-dispersion notation. Introduce

\[Z(\zeta ) = \frac {1}{\sqrt {\pi }} \int _{-\infty }^{\infty } \frac {e^{-x^2}}{x-\zeta }\,dx, \qquad R(\zeta )\equiv 1+\zeta Z(\zeta ), \tag{17.12}\]
and define \(R_s\equiv R(\zeta _s)\). The required Gaussian integrals are
\[\frac {1}{\sqrt {\pi }} \int _{-\infty }^{\infty } \frac {x e^{-x^2}}{\zeta -x}\,dx = - R(\zeta ), \tag{17.13}\]
and
\[\frac {1}{\sqrt {\pi }} \int _{-\infty }^{\infty } \frac {x^3 e^{-x^2}}{\zeta -x}\,dx = -\left (\frac {1}{2}+\zeta ^2 R(\zeta )\right ). \tag{17.14}\]

Density and pressure moments. At fixed \((\mu ,v_\parallel )\), the phase-space measure is

\[d^3v = \frac {2\pi B_0}{m_s}\,d\mu \,dv_\parallel . \tag{17.15}\]
Evaluating the moments gives
\[\begin{aligned}\frac {\delta n_s}{n_0} &= \left [1-A_s R_s\right ]\frac {\delta B}{B_0} - R_s\frac {q_s\tilde {\Phi }}{T_{\parallel s}}, \\[4pt] \delta p_{\perp s} &= 2p_{\perp s}\left [1-A_s R_s\right ]\frac {\delta B}{B_0} - p_{\perp s} R_s \frac {q_s\tilde {\Phi }}{T_{\parallel s}}, \\[4pt] \delta p_{\parallel s} &= p_{\parallel s} \left [1-A_s\left (1+2\zeta _s^2R_s\right )\right ]\frac {\delta B}{B_0} - p_{\parallel s}\left (1+2\zeta _s^2R_s\right )\frac {q_s\tilde {\Phi }}{T_{\parallel s}}.\end{aligned} \tag{17.16}\]

For singly charged ions and electrons, quasineutrality \(\delta n_i=\delta n_e\) gives

\[\boxed { e\tilde {\Phi } = \frac {A_eR_e-A_iR_i} {R_i/T_{\parallel i}+R_e/T_{\parallel e}} \frac {\delta B}{B_0}. } \tag{17.19}\]
Equations (17.16)–(17.19) are the core of kinetic MHD for this problem.

17.3 Slow mirror ordering

Low-frequency limit. The mirror and long-wavelength firehose instabilities live in the regime

\[|\zeta _s| = \left |\frac {\omega }{k_\parallel v_{{\rm th}\parallel s}}\right |\ll 1, \qquad \Longrightarrow \qquad R_s \to 1. \tag{17.20}\]
Then Eq. (17.19) becomes
\[e\tilde {\Phi } = \frac {A_e-A_i} {1/T_{\parallel i}+1/T_{\parallel e}} \frac {\delta B}{B_0}. \tag{17.21}\]

Equal anisotropy or vanishing \(E_\parallel \). A particularly clean limit is

\[A_i=A_e\equiv A, \qquad \tilde {\Phi }=0. \tag{17.22}\]
Then Eqs. (17.16)–(17.18) reduce to
\[\begin{aligned}\frac {\delta n}{n_0} &= (1-A)\frac {\delta B}{B_0}, \\ \delta p_{\parallel s} &= p_{\parallel s}(1-A)\frac {\delta B}{B_0}, \\ \delta p_{\perp s} &= 2p_{\perp s}(1-A)\frac {\delta B}{B_0}.\end{aligned} \tag{17.23}\]

These are the formulas used below to derive the simplest collisionless mirror threshold.

17.4 Firehose: loss of field-line tension

Transverse polarization. Take a purely shear perturbation

\[\vect {\xi } = \xi _y \vect {e}_y, \qquad \vect {k}\cdot \vect {\xi }=0, \qquad \frac {\delta B}{B_0}=0. \tag{17.26}\]
Then the kinetic pressure response drops out because the perturbation is incompressible and contains no \(\delta B_\parallel \). The perturbed magnetic field is
\[\B _1 = i k_\parallel B_0 \xi _y \vect {e}_y, \tag{17.27}\]
so the field-direction perturbation is
\[\vect {b}_1 = i k_\parallel \xi _y \vect {e}_y. \tag{17.28}\]

Force balance. The magnetic tension force is

\[\frac {1}{\muo }(i\vect {k}\times \B _1)\times \B _0 = - \frac {B_0^2}{\muo }k_\parallel ^2 \xi _y \vect {e}_y. \tag{17.29}\]
The anisotropic pressure tensor contributes
\[-\left (\divergence \tens {P}_1\right )_y = -\left (p_{\perp 0}-p_{\parallel 0}\right )k_\parallel ^2\xi _y. \tag{17.30}\]
Hence
\[-\omega ^2 \rho _0 \xi _y = - k_\parallel ^2 \left ( \frac {B_0^2}{\muo }+p_{\perp 0}-p_{\parallel 0} \right )\xi _y. \tag{17.31}\]
Therefore
\[\boxed { \omega ^2 = k_\parallel ^2 \frac {B_0^2/\muo +p_{\perp 0}-p_{\parallel 0}}{\rho _0}. } \tag{17.32}\]
This is exactly the same result as Eq. (16.45). The firehose threshold is therefore
\[\boxed { p_{\parallel 0}-p_{\perp 0} > \frac {B_0^2}{\muo }. } \tag{17.33}\]

The lesson. The firehose mode is already present in anisotropic fluid theory because its physics is the sign of the field-line tension. The closure hardly enters. In that sense the firehose instability is the easiest anisotropy-driven mode to understand.

17.5 Mirror: failure of compressive balance

Oblique compressive perturbation. Now take a compressive perturbation in the \(x\)–\(z\) plane. From Eq. (17.5),

\[\frac {\delta B}{B_0} = - i k_\perp \xi _x, \qquad \frac {\delta \B _\perp }{B_0} = i k_\parallel \vect {\xi }_\perp . \tag{17.34}\]
The perturbed curvature of an initially straight field line is
\[\vect {\kappa }_1 = (\vect {b}\cdot \grad )\vect {b}_1 = i k_\parallel \vect {b}_1 = - k_\parallel ^2 \vect {\xi }_\perp . \tag{17.35}\]
Take the divergence of the perpendicular force balance (17.2) and use Eq. (17.5). One obtains
\[\boxed { \omega ^2 \frac {\delta B}{B_0} = k_\perp ^2 \left ( \frac {\delta p_\perp }{\rho _0} + v_A^2 \frac {\delta B}{B_0} \right ) + k_\parallel ^2 \left ( v_A^2 + c_\perp ^2 - c_\parallel ^2 \right ) \frac {\delta B}{B_0}. } \tag{17.36}\]
The parallel force balance is
\[-\omega ^2 \rho _0 \xi _\parallel = - i k_\parallel \left [ \delta p_\parallel + \left (p_{\perp 0}-p_{\parallel 0}\right )\frac {\delta B}{B_0} \right ]. \tag{17.37}\]
In the slow limit this says that the parallel force is nearly balanced, so the mirror mode is oblique rather than exactly perpendicular.

Insert the kinetic closure. Use the slow-ordering response (17.25):

\[\delta p_\perp = 2p_{\perp 0}(1-A)\frac {\delta B}{B_0}, \qquad A \equiv \frac {T_\perp }{T_\parallel }. \tag{17.38}\]
Equation (17.36) becomes
\[\begin{aligned}\omega ^2 &= k_\perp ^2 \left [ v_A^2 + 2\frac {p_{\perp 0}}{\rho _0}(1-A) \right ] + k_\parallel ^2 \left ( v_A^2+c_\perp ^2-c_\parallel ^2 \right ) \nonumber \\ &= k_\perp ^2 v_A^2 \left [ 1+\beta _\perp ^\ast (1-A) \right ] + k_\parallel ^2 \left ( v_A^2+c_\perp ^2-c_\parallel ^2 \right ),\end{aligned} \tag{17.39}\]

where for compactness we defined

\[\beta _\perp ^\ast \equiv \frac {2p_{\perp 0}}{\rho _0 v_A^2} = \frac {2\muo p_{\perp 0}}{B_0^2}. \tag{17.40}\]
The mirror mode is most unstable for \(k_\perp \gg k_\parallel \), so the first term controls the threshold:
\[1+\beta _\perp ^\ast (1-A) < 0. \tag{17.41}\]
Thus the simplest kinetic-MHD mirror criterion is
\[\boxed { \frac {T_\perp }{T_\parallel } > 1+\frac {1}{\beta _\perp ^\ast }. } \tag{17.42}\]

It is also interesting that this "magnetosonic" branch also has a second type of firehose instability. It, like the mirror is oblique but becomes large when the parallel wavelength is short and \(k_\parallel \gg k_\perp \). The firehose condition governing stability is exactly the same as before but the character of the wave is no longer a shear but rather compressional.

Comparison with CGL. Compare Eq. (17.42) with the CGL result (16.54). The magnetic part of the force balance is the same in both cases. The difference is entirely in the compressive pressure response. That is why it is worth separating this lecture from the ordinary wave lecture: the mirror mode is the cleanest example of a low-frequency problem whose geometry is fluid-like but whose closure is kinetic.

A useful physical picture. The mirror mode bunches magnetic flux so that some regions have larger \(B\) and some smaller \(B\). When \(T_\perp >T_\parallel \), particles with large magnetic moment prefer the weaker-\(B\) regions. That pile-up raises the perpendicular pressure where the field is already weak, which further deepens the magnetic well. The mode is therefore a compressive anti-restoring response. Southwood and Kivelson’s discussion makes this picture especially transparent Southwood and Kivelson (1993).

PIC

Figure 17.1: Solar-wind proton anisotropy data with mirror and firehose thresholds overlaid. The point of the figure is not merely observational: nature actually populates the kinetic-MHD marginality boundaries. Bale et al. (2009)

Observational perspective. The expanding solar wind provides the cleanest natural laboratory for these anisotropy-driven instabilities. As the plasma expands, \(T_\perp /T_\parallel \) is driven away from unity, but the proton distribution is observed to remain bounded by the mirror and oblique-firehose thresholds rather than wandering freely in anisotropy space Hellinger et al. (2006); Bale et al. (2009). That is a beautiful confirmation of the kinetic-MHD point of view: the plasma evolves under low-frequency MHD-like dynamics, but kinetic microinstabilities regulate the pressure tensor.

Caution

Kinetic MHD is still a reduced theory. It assumes \(\omega \ll \Omega _i\) and \(k_\perp \rho _i\ll 1\). Once finite-Larmor-radius corrections, cyclotron resonances, or strongly non-Maxwellian particle populations become essential, one must go beyond kinetic MHD as well.

Takeaways

This lecture gives the cleanest lesson in anisotropic plasma theory.

1.
The force law remains MHD-like: field-line bending, magnetic compression, and anisotropic stress appear exactly where fluid intuition says they should.
2.
The closure is where the kinetic information lives. Equations (17.16)–(17.19) replace the double-adiabatic laws.
3.
The shear firehose mode is basically a tension problem and is already captured by anisotropic fluid theory, whereas the mirror mode and a second firehose is a compressive response problem and therefore tests the closure.

Bibliography

    G. F. Chew, M. L. Goldberger, and F. E. Low. The Boltzmann equation and the one-fluid hydromagnetic equations in the absence of particle collisions. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 236(1204):112–118, 1956. doi:10.1098/rspa.1956.0116.

    R. M. Kulsrud. Mhd description of plasma. In A. A. Galeev and R. N. Sudan, editors, Handbook of Plasma Physics, Volume 1: Basic Plasma Physics I, chapter 1.4, pages 115–145. North-Holland, Amsterdam, 1983. Series editors: M. N. Rosenbluth and R. Z. Sagdeev.

    A. A. Vedenov and R. Z. Sagdeev. Some properties of a plasma with an anisotropic ion velocity distribution in a magnetic field. In M. A. Leontovich, editor, Plasma Physics and the Problem of Controlled Thermonuclear Reactions, volume 3, pages 332–339. Pergamon Press, New York, 1958a. This classic mirror-instability reference is cited in the literature with variant publication years (1958, 1959, or 1961) depending on whether one cites the original Russian proceedings or the translated Pergamon edition.

    A. A. Vedenov and R. Z. Sagdeev. On some properties of a plasma with an anisotropic ion-velocity distribution in a magnetic field. Soviet Physics Doklady, 3:278, 1958b. Common shorthand citation used in later plasma-physics literature for the original mirror-instability paper.

    Akira Hasegawa. Drift mirror instability in the magnetosphere. Physics of Fluids, 12(12): 2642–2650, 1969. doi:10.1063/1.1692407.

    David J. Southwood and Margaret G. Kivelson. Mirror instability: 1. physical mechanism of linear instability. Journal of Geophysical Research: Space Physics, 98(A6):9181–9187, 1993. doi:10.1029/92ja02837.

    S. D. Bale, J. C. Kasper, G. G. Howes, E. Quataert, C. Salem, and D. Sundkvist. Magnetic fluctuation power near proton temperature anisotropy instability thresholds in the solar wind. Physical Review Letters, 103:211101, 2009. doi:10.1103/PhysRevLett.103.211101.

    P. Hellinger, P. Trávníček, J. C. Kasper, and A. J. Lazarus. Solar wind proton temperature anisotropy: Linear theory and WIND/SWE observations. Geophysical Research Letters, 33: L09101, 2006. doi:10.1029/2006GL025925.

Problems

Problem 17.1.
Starting from Eq. (17.9), derive Eq. (17.11). Keep the signs of \(E_\parallel \) and \(\delta B\) explicit throughout.
Problem 17.2.
Use the Gaussian integrals (17.13)–(17.14) to derive Eqs. (17.16)–(17.18).
Problem 17.3.
Show directly that the firehose dispersion (17.32) does not depend on the kinetic pressure response for the shear polarization \(\vect {\xi }=\xi _y\vect {e}_y\).
Problem 17.4.
Starting from Eq. (17.36), re-derive the mirror threshold (17.42). Identify clearly where the ordering \(k_\perp \gg k_\parallel \) enters.
Problem 17.5.
Compare the kinetic mirror threshold (17.42) with the CGL threshold (16.54). Which part of the calculation is the same, and which part changes?
Problem 17.6.
Repeat the derivation of the slow-ordering pressure response for the case of isotropic electrons and anisotropic ions. How do the coefficients of \(\delta p_\perp \) and \(\delta p_\parallel \) change?