Lecture 6 CGL: Collisionless, Anisotropic MHD

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Lecture 6
CGL: Collisionless, Anisotropic MHD

Overview

This lecture introduces CGL as a closure for magnetized, weakly collisional plasma.
1.
The scalar pressure in Eq. (1.8) is replaced by a gyrotropic tensor with separate parallel and perpendicular pressures.
2.
The double-adiabatic laws are derived from adiabatic invariants, flux freezing, and particle conservation.
3.
The point of this lecture is the closure itself. The detailed geometry of the mirror force and the more general theory of anisotropic equilibrium are deferred to Lecture 11; the firehose and mirror wave instabilities are deferred still later.

The isotropic closure used in the introductory momentum equation, Eq. (1.8), and in the Newtonian stress law, Eq. (2.10), is appropriate when collisions rapidly drive the distribution toward a local Maxwellian. In many space and laboratory plasmas, however, the situation is more subtle: the particles remain strongly magnetized, so they remember the direction of \(\vect {B}\), while collisions are too weak to isotropize the pressure on the dynamical time. The leading-order pressure then becomes anisotropic.

Historical Perspective

Chew, Goldberger, and Low made the essential step in 1956: they did not abandon fluid theory in the collisionless limit, but instead derived a new closure directly from kinetic theory Chew et al. [1956]. That move made pressure anisotropy, adiabatic invariants, and magnetic geometry central. Later work by Northrop, Kulsrud, and many others clarified how the same ordering connects fluid MHD to guiding-center and drift-kinetic theory Northrop [1963], Kulsrud [1983].

6.1 Gyrotropic pressure tensor

What changes in the stress tensor? For a magnetized species \(s\), the leading pressure tensor may be written as

\tens {P}_s = p_{\perp s}\,\tens {I} + \left (p_{\parallel s}-p_{\perp s}\right )\vect {b}\vect {b} + \tens {\Pi }_s, \tag{6.1}

where \[ \vect {b}\equiv \frac {\vect {B}}{B} \] is the magnetic-field unit vector and \(\tens {\Pi }_s\) contains higher-order finite-Larmor-radius and collisional corrections. At leading CGL order one neglects \(\tens {\Pi }_s\), sums over species, and obtains

\boxed { \tens {P} = p_\perp \,\tens {I} + \left (p_\parallel -p_\perp \right )\vect {b}\vect {b}. } \tag{6.2}

In a field-aligned basis this tensor is diagonal, with entries \(p_\perp ,p_\perp ,p_\parallel \).

Moment definitions. In the local fluid frame,

n_s = \int f_s\,d^3v, \qquad p_{\perp s} = \int \frac {m_s v_\perp ^2}{2} f_s\, d^3v, \qquad p_{\parallel s} = \int m_s v_\parallel ^2 f_s\, d^3v. \tag{6.3}

The gyrophase has been averaged away, so the only distinguished direction is \(\vect {b}\). That is why the decomposition (6.2) is the natural collisionless analog of the isotropic stress split discussed earlier in Eq. (2.8).

6.2 Collisionless kinetic foundation

The collisionless Vlasov equation. The kinetic starting point is

\pp {f_s}{t} + \vect {v}\cdot \grad f_s + \frac {q_s}{m_s} \left ( \vect {E} + \vect {v}\times \vect {B} \right ) \cdot \pp {f_s}{\vect {v}} = 0. \tag{6.4}

Along phase-space characteristics,

\frac {D f_s}{Dt}=0. \tag{6.5}

For magnetized particles with \(\omega \ll \Omega _i\) and \(\rho _i/L\ll 1\), adiabatic invariants become available.

The adiabatic invariants. The first adiabatic invariant is the magnetic moment,

\mu \equiv \frac {m_s v_\perp ^2}{2B}, \tag{6.6}

and the second is the bounce or longitudinal action,

J \equiv \oint m_s v_\parallel \, d\ell . \tag{6.7}

The total kinetic energy may be written as

\epsilon \equiv \frac {1}{2}m_s v_\parallel ^2 + \mu B. \tag{6.8}

These are the basic ingredients behind the CGL equations.

6.3 The drift-kinetic equation

Expanded guiding-center form. To first order in the drift expansion, the guiding-center distribution \(f(\vect {R},v_\parallel ,\mu ,t)\) satisfies

\pp {f}{t} + \dot {\vect {R}}\cdot \grad f + \dot v_\parallel \,\pp {f}{v_\parallel } = 0, \tag{6.9}

with guiding-center velocity

\dot {\vect {R}} = v_\parallel \vect {b} + \vect {v}_E + \vect {v}_{\grad B} + \vect {v}_c +\cdots , \tag{6.10}

where

\vect {v}_E = \frac {\vect {E}\times \vect {B}}{B^2}, \qquad \vect {v}_{\grad B} = \frac {\mu }{qB}\,\vect {b}\times \grad B,

\vect {v}_c = \frac {m v_\parallel ^2}{qB}\,\vect {b}\times \vect {\kappa }, \qquad \vect {\kappa } \equiv (\vect {b}\cdot \grad )\vect {b}.

In the magnetostatic limit with \(E_\parallel =0\), the parallel acceleration reduces to the single-particle mirror force,

\dot v_\parallel \simeq -\frac {\mu }{m}\,\vect {b}\cdot \grad B. \tag{6.13}

The detailed geometry of this force is developed in Lecture 11; here it simply motivates why \(p_\parallel \) and \(p_\perp \) evolve differently.

Lowest-order reduced drift-kinetic equation. Under the ideal-MHD ordering, Ohm’s law reduces to \(\vect {E}=-\vect {u}\times \vect {B}\) as in Eq. (4.8), so the leading perpendicular motion is the frozen-in \(\vect {E}\times \vect {B}\) drift discussed in Eq. (4.13). A convenient reduced drift-kinetic equation is then

\pp {f}{t} + (\vect {v}_E+v_\parallel \vect {b})\cdot \grad f + \left ( -\vect {b}\cdot \frac {D\vect {v}_E}{Dt} -\frac {\mu }{m}\,\vect {b}\cdot \grad B +\frac {q}{m}E_\parallel \right ) \pp {f}{v_\parallel } = 0, \tag{6.14}

where

\frac {D}{Dt} \equiv \pp {}{t} + (\vect {v}_E+v_\parallel \vect {b})\cdot \grad . \tag{6.15}

This is the form we will return to later in the drift-kinetic appendix.

Equivalent velocity-space form. If one keeps \(v_\perp \) instead of \(\mu \), the same lowest-order equation becomes

\pp {f}{t} + (\vect {v}_E+v_\parallel \vect {b})\cdot \grad f - \frac {v_\perp }{2} \left [ \divergence \vect {v}_E - \bigl ((\vect {b}\cdot \grad )\vect {v}_E\bigr )\cdot \vect {b} + v_\parallel \divergence \vect {b} \right ] \pp {f}{v_\perp } - \left [ \vect {b}\cdot \frac {D\vect {v}_E}{Dt} - \frac {v_\perp ^2}{2}\,\divergence \vect {b} - \frac {q}{m}E_\parallel \right ] \pp {f}{v_\parallel } = 0. \tag{6.16}

In this form the betatron response appears explicitly in the \(v_\perp \)-derivative term. In the \((\mu ,v_\parallel )\) form, Eq. (6.14), the same physics is absorbed into the choice of variables because \(\mu \) is an adiabatic invariant at this order.

Equivalent energy-moment form. Because \((\mu ,\epsilon )\) does not distinguish \(v_\parallel >0\) from \(v_\parallel <0\), one supplements it with the sign \[ \sigma \equiv \sgn (v_\parallel ), \qquad v_\parallel = \sigma \sqrt {\frac {2(\epsilon -\mu B)}{m}}. \] Using \[ \left .\pp {}{v_\parallel }\right |_\mu = m v_\parallel \left .\pp {}{\epsilon }\right |_\mu , \] Eq. (6.14) becomes

\pp {f}{t} + (\vect {v}_E+v_\parallel \vect {b})\cdot \grad f + \left [ \mu \left (\pp {}{t}+\vect {v}_E\cdot \grad \right )B - m v_\parallel \vect {b}\cdot \frac {D\vect {v}_E}{Dt} + q E_\parallel v_\parallel \right ] \pp {f}{\epsilon } = 0. \tag{6.17}

This form is fully equivalent, but for later linear work the \((\mu ,v_\parallel )\) form is usually more transparent.

Linearization For the Lecture 17 we will look for waves and instabilities that depend upon the kinetic properties. S uppose \[ \vect {B}=B_0\ez +\delta B_\parallel \ez , \qquad \delta B_\parallel \propto e^{-i\omega t+i k_\parallel z}, \qquad E_{\parallel 1}=0. \] Then Eq. (6.14) linearizes to

-i(\omega -k_\parallel v_\parallel ) f_1 -\frac {\mu }{m}\, i k_\parallel \delta B_\parallel \, \pp {f_0}{v_\parallel } =0. \tag{6.18}

If one defines the magnetic moment per unit mass, \[ \mu _*\equiv \frac {v_\perp ^2}{2B}=\frac {\mu }{m}, \] then the same equation reads \[ -i(\omega -k_\parallel v_\parallel ) f_1 -\mu _*\, i k_\parallel \delta B_\parallel \, \pp {f_0}{v_\parallel } =0. \]

6.4 The CGL closure

From moments of the drift-kinetic equation. Chew, Goldberger, and Low showed that higher moments of the kinetic equation produce separate perpendicular and parallel heat fluxes. In the ordering

\nu _{ii} \ll \omega \ll \Omega _i, \tag{6.19}

the pressure equations may be written schematically as

\rho B\frac {D}{Dt}\!\left (\frac {p_\perp }{\rho B}\right ) = -\divergence (q_\perp \vect {b})-q_\perp \divergence \vect {b}, \tag{6.20}

\frac {\rho ^3}{B^2}\frac {D}{Dt}\!\left (\frac {p_\parallel B^2}{\rho ^3}\right ) = -\divergence (q_\parallel \vect {b})-2 q_\perp \divergence \vect {b}. \tag{6.21}

The heat fluxes are third moments of the distribution function. When they may be neglected, one obtains the double-adiabatic CGL laws.

The heat-flux-free limit. With \(q_\perp \to 0\) and \(q_\parallel \to 0\), Eqs. (6.20)–(6.21) reduce to

\frac {D}{Dt}\!\left (\frac {p_\perp }{nB}\right ) = 0, \tag{6.22}

\frac {D}{Dt}\!\left (\frac {p_\parallel B^2}{n^3}\right ) = 0. \tag{6.23}

These are the CGL closure relations. Pressure anisotropy has become an independent dynamical degree of freedom generated by compression, expansion, and field-strength change.

Caution

Important separation of ideas. The CGL equations are one particular closure for the evolution of \(p_\perp \) and \(p_\parallel \). The next two lectures use the more general gyrotropic tensor algebra without assuming that those pressures obey the double-adiabatic laws.

6.5 Heuristic derivation of the double-adiabatic laws

First adiabatic invariant and the perpendicular law. Conservation of \[ \mu = \frac {m v_\perp ^2}{2B} \] means that for a single particle \[ \frac {v_\perp ^2}{B}=\text {const.} \] If the distribution remains gyrotropic and its shape changes self-similarly, then the ensemble average obeys \[ \langle v_\perp ^2\rangle \propto B \qquad \Longrightarrow \qquad T_\perp \propto B. \] Since \(p_\perp =nT_\perp \), one obtains

\boxed {\frac {p_\perp }{nB}=\text {const. along a fluid element}.}

Writing the fluid-element derivative explicitly gives Eq. (6.22).

Second adiabatic invariant and the parallel law. For trapped or rapidly bouncing particles, the second invariant \[ J=\oint m v_\parallel \,d\ell \] implies, at the scaling level, \[ J\sim m v_\parallel L_\parallel = \text {const.} \qquad \Longrightarrow \qquad v_\parallel \propto \frac {1}{L_\parallel }. \] Therefore \[ T_\parallel \propto \langle v_\parallel ^2\rangle \propto \frac {1}{L_\parallel ^2}. \] To relate \(L_\parallel \) to \(n\) and \(B\), consider a thin flux tube moving with the fluid. Ideal flux freezing implies \[ \Phi = BA = \text {const.} \qquad \Longrightarrow \qquad A\propto \frac {1}{B}, \] while particle conservation implies \[ N=nAL_\parallel =\text {const.} \qquad \Longrightarrow \qquad L_\parallel \propto \frac {1}{nA}\propto \frac {B}{n}. \] Hence \[ T_\parallel \propto \frac {1}{L_\parallel ^2} \propto \left (\frac {n}{B}\right )^2, \] and therefore

\boxed {\frac {p_\parallel B^2}{n^3}=\text {const. along a fluid element}.}

Writing the convective derivative gives Eq. (6.23).

A differential derivation using earlier MHD equations. The heuristic argument above is memorable, but it is also useful to connect it directly to the fluid equations already derived earlier in the notes. In the ideal limit, Eq. (4.7) reduces to

\frac {D}{Dt}\left (\frac {\B }{\rho }\right ) = \left (\frac {\B }{\rho }\cdot \grad \right )\uvec . \tag{6.26}

Write \(\B = B\vect {b}\). Dot Eq. (6.26) with \(\vect {b}\):

\vect {b}\cdot \frac {D}{Dt}\left (\frac {B\vect {b}}{\rho }\right ) = \vect {b}\cdot \left (\frac {B\vect {b}}{\rho }\cdot \grad \right )\uvec .

Because \(\vect {b}\cdot \vect {b}=1\), differentiation gives

\frac {1}{\rho }\frac {DB}{Dt} - \frac {B}{\rho ^2}\frac {D\rho }{Dt} = \frac {B}{\rho }\,\vect {b}\vect {b}:\grad \uvec .

Multiplying through by \(\rho /B\) yields

\frac {1}{B}\frac {DB}{Dt} - \frac {1}{\rho }\frac {D\rho }{Dt} = \vect {b}\vect {b}:\grad \uvec .

Now use continuity, Eq. (1.7), in convective form,

\frac {1}{\rho }\frac {D\rho }{Dt} = -\divergence \uvec ,

so that

\frac {1}{B}\frac {DB}{Dt} = \vect {b}\vect {b}:\grad \uvec - \divergence \uvec . \tag{6.31}

This is the local field-strength evolution law hidden inside the ideal induction equation.

Starting from Eq. (6.22), \[ \frac {D}{Dt}\ln \left (\frac {p_\perp }{nB}\right )=0, \] one immediately gets

\frac {1}{p_\perp }\frac {Dp_\perp }{Dt} = \frac {1}{n}\frac {Dn}{Dt} + \frac {1}{B}\frac {DB}{Dt}. \tag{6.32}

For fixed ion mass, \(n\propto \rho \), so using continuity and Eq. (6.31) gives

\frac {1}{p_\perp }\frac {Dp_\perp }{Dt} = -\divergence \uvec + \left (\vect {b}\vect {b}:\grad \uvec -\divergence \uvec \right ) = \vect {b}\vect {b}:\grad \uvec - 2\divergence \uvec .

Therefore

\frac {Dp_\perp }{Dt} + p_\perp \left (2\divergence \uvec -\vect {b}\vect {b}:\grad \uvec \right ) = 0. \tag{6.34}

Likewise, starting from Eq. (6.23), \[ \frac {D}{Dt}\ln \left (\frac {p_\parallel B^2}{n^3}\right )=0, \] one finds

\frac {1}{p_\parallel }\frac {Dp_\parallel }{Dt} = 3\frac {1}{n}\frac {Dn}{Dt} - 2\frac {1}{B}\frac {DB}{Dt}. \tag{6.35}

Using continuity and Eq. (6.31) again,

\frac {1}{p_\parallel }\frac {Dp_\parallel }{Dt} = -3\divergence \uvec - 2\left (\vect {b}\vect {b}:\grad \uvec -\divergence \uvec \right ) = -\divergence \uvec - 2\vect {b}\vect {b}:\grad \uvec ,

so that

\frac {Dp_\parallel }{Dt} + p_\parallel \left (\divergence \uvec +2\vect {b}\vect {b}:\grad \uvec \right ) = 0. \tag{6.37}

Equations (6.34) and (6.37) are the differential forms most often used in actual calculations.

Caution

What the double-adiabatic model omits. CGL is not “collisionless plasma physics in general.” It is a specific asymptotic closure. Heat fluxes, pitch-angle scattering, Landau damping, and finite-Larmor-radius corrections are all omitted at leading order. The model is therefore excellent for showing how anisotropy is generated, but one should be cautious about using it as a quantitatively exact closure near kinetic instability thresholds.

Equivalent differential forms. Taking convective derivatives of Eqs. (6.22) and (6.23) yields

\frac {1}{p_\perp }\frac {dp_\perp }{dt} = \frac {1}{n}\frac {dn}{dt} + \frac {1}{B}\frac {dB}{dt},

\frac {1}{p_\parallel }\frac {dp_\parallel }{dt} = 3\frac {1}{n}\frac {dn}{dt} - 2\frac {1}{B}\frac {dB}{dt}.

These are often combined with the continuity equation, Eq. (1.7), and the ideal induction equation, Eqs. (1.13) and (4.7).

Scope of the heuristic argument. The derivation above assumes clear scale separation, negligible pitch-angle scattering, weak heat fluxes, and a distribution that remains close enough to bi-Maxwellian for \(p_\perp \) and \(p_\parallel \) to be meaningful state variables. Once heat fluxes, collisions, or FLR corrections matter, CGL is no longer the full story.

6.6 How CGL closes the fluid system

The closed set. The CGL model may be viewed as ideal MHD plus a gyrotropic pressure tensor and two extra pressure-evolution laws:

\pp {\rho }{t}+\divergence (\rho \vect {u}) = 0, \tag{6.40}

\rho \left (\pp {\vect {u}}{t}+\vect {u}\cdot \grad \vect {u}\right ) = -\divergence \tens {P} + \vect {J}\times \vect {B}, \tag{6.41}

\pp {\vect {B}}{t} = \curl (\vect {u}\times \vect {B}), \qquad \divergence \vect {B}=0, \tag{6.42}

\tens {P} = p_\perp \tens {I} + (p_\parallel -p_\perp )\vect {b}\vect {b}, \tag{6.43}

\frac {D}{Dt}\!\left (\frac {p_\perp }{nB}\right ) = 0, \qquad \frac {D}{Dt}\!\left (\frac {p_\parallel B^2}{n^3}\right ) = 0. \tag{6.44}

The next lecture expands \(-\divergence \tens {P}\) in detail. That algebra is general to gyrotropic theory and should not be confused with the special CGL closure itself.

6.7 Relation to Braginskii and to kinetic MHD

Two opposite collisional limits. Braginskii theory starts from a strongly magnetized but collisional plasma. There, anisotropy appears as a constitutive correction to an otherwise nearly Maxwellian fluid, and the viscous tensor is ordered as a transport term. CGL sits at the opposite end: collisions are weak enough that pressure anisotropy survives as a leading-order dynamical variable.

A bridge rather than a contradiction. These are not competing religions. They are different orderings of the same kinetic theory. Roughly speaking, \[ \text {Braginskii:} \qquad \Omega _i\tau _i\gg 1, \quad \nu _{ii}\ \text {still dynamically important}, \] whereas \[ \text {CGL:} \qquad \Omega _i\tau _i\gg 1, \quad \nu _{ii}\ll \omega . \] Kinetic MHD and Landau-fluid closures sit between and beyond these limits by retaining drift-kinetic information, heat fluxes, and resonant physics explicitly.

6.8 Physical perspective

What this lecture does and does not do. CGL is appropriate when

\(\rho _i\ll L\),
\(\omega \ll \Omega _i\),
the plasma remains magnetized, and
collisions and heat fluxes are weak enough that the double-adiabatic closure is a reasonable first approximation.

It does not capture FLR stabilization, heat-flux physics, Landau damping, or the correct kinetic mirror threshold. Those effects belong to later lectures and to the drift-kinetic appendix.

Takeaways

CGL is best thought of as ideal MHD plus a new pressure closure, not as a general theory of all anisotropic plasmas.

The two invariants \(p_\perp /(nB)\) and \(p_\parallel B^2/n^3\) encode perpendicular betatron heating and parallel cooling/heating by field-line shortening or stretching.

The geometry of the mirror force and the equilibrium consequences of anisotropy are broader than CGL and are treated separately in Lecture 11.

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.

Theodore G. Northrop. The Adiabatic Motion of Charged Particles. Interscience, New York, 1963.

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.

Problems

Problem 6.1. Estimating anisotropy generation with the CGL invariants

Consider a steady radial outflow with speed \(u_r(r)\approx \text {const}\) beyond a few solar radii. First take a purely radial magnetic field \(B_r\propto r^{-2}\). Assume the plasma is sufficiently collisionless that the CGL invariants, Eqs. (6.22) and (6.23), apply along streamlines.

Part A: radial field

(a): Using mass conservation for a steady radial flow, \[ \rho u_r r^2 = \text {const.}, \] show that \(\rho \propto r^{-2}\).
(b): Using \(B\propto r^{-2}\) together with Eqs. (6.22) and (6.23), show that \[ p_\perp (r)\propto r^{-4}, \qquad p_\parallel (r)\propto r^{-2}. \]
(c): Hence show that \[ \boxed {\frac {p_\perp }{p_\parallel }\propto r^{-2}.} \] Assuming isotropy at a reference radius \(r_0\), write \(p_\perp /p_\parallel \) at radius \(r\) in terms of \(r/r_0\).
(d): Using \(p_\perp =nT_\perp \) and \(p_\parallel =nT_\parallel \), deduce the scaling of \(T_\perp /T_\parallel \).

Part B: Parker-spiral asymptotics At sufficiently large radius a Parker spiral satisfies \(B\propto r^{-1}\) while the density still scales approximately as \(\rho \propto r^{-2}\).

(a): Show that in this limit \[ p_\perp (r)\propto r^{-3}, \qquad p_\parallel (r)\propto \text {const.} \] and therefore \[ \boxed {\frac {p_\perp }{p_\parallel }\propto r^{-3}.} \]
(b): Compare this result with the purely radial-field case. Which geometry drives stronger anisotropy at large radius?

Part C: conceptual reflection Give two reasons why the real solar wind need not follow the unrestricted CGL scalings all the way to arbitrarily large anisotropy. Do not quote threshold formulas; instead answer in terms of missing physics.