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Lecture 11
Grad’s Equilibrium Analysis

Overview

This lecture is about equilibrium with a gyrotropic stress tensor, not about a particular closure.

1.
Starting from the static limit of the MHD momentum equation, we project force balance parallel and perpendicular to \(\B \).
2.
The parallel projection gives the mirror-force constraint; the perpendicular projection shows how anisotropy modifies field-line tension.
3.
In the isotropic limit one recovers \(p=p(\psi )\), which is the doorway to the Grad–Shafranov equation derived in the next lecture; in the slender-tube limit one recovers paraxial mirror balance.

We now stop evolving the full MHD system and instead ask what remains when the net force vanishes. The central point is that static equilibrium requires only a stress tensor and a magnetic field; it does not require the specific CGL evolution laws. This is why it is useful to separate the present lecture from the earlier closure lecture. The equilibrium problem is broader: any gyrotropic distribution produces a tensor of the form \[ \tens {P}=p_\perp \tens {I} + (p_\parallel -p_\perp )\vect {b}\vect {b}, \qquad \vect {b}\equiv \frac {\B }{B}, \] and the question is then how this tensor can balance magnetic pressure and magnetic tension.

Historical Perspective

The isotropic equilibrium problem led to the great toroidal-equilibrium literature and, ultimately, to the Grad–Shafranov equation. The anisotropic extension developed along a somewhat different path, through guiding-center and mirror-machine physics. Grad emphasized that weakly collisional plasmas should be formulated directly in terms of a gyrotropic pressure tensor, while Newcomb clarified the connection between global equilibrium and the paraxial mirror limit Grad [1967a,?], Newcomb [1981]. This is one of the places where magnetic confinement, kinetic theory, and classical MHD visibly overlap.

Caution

A conceptual boundary. This lecture treats static balance. It is not the place to do the full mirror and firehose wave calculations. Those dynamical instability problems belong later, when anisotropic MHD waves and the drift-kinetic appendix are in hand.

11.1 Static balance with a gyrotropic tensor

Start from the static limit of the momentum equation. The momentum equation stated in Eq. (1.8) becomes, when \(\vect {u}=0\),

\[-\divergence \tens {P}+\J \times \B =0. \tag{11.1}\]
It is often cleaner to collect the magnetic force into the Maxwell stress tensor
\[\tens {T}_B = -\frac {B^2}{2\muo }\tens {I} + \frac {\B \B }{\muo },\]
so that static equilibrium may be written as
\[\boxed { \divergence \left (\tens {P}-\tens {T}_B\right )=0. } \tag{11.3}\]
This says that plasma stress and magnetic stress must balance exactly.

Write out the divergence of a gyrotropic tensor once, slowly. Let \[ f \equiv p_\parallel -p_\perp , \qquad \tens {P}=p_\perp \tens {I}+f\,\vect {b}\vect {b}. \] Then

\[\begin {aligned} \divergence \tens {P} &= \grad p_\perp + \divergence \left (f\,\vect {b}\vect {b}\right ) \\ &= \grad p_\perp + (\vect {b}\cdot \grad f)\vect {b} + f\,\divergence \left (\vect {b}\vect {b}\right ). \end {aligned}\]
Now use
\[\divergence \left (\vect {b}\vect {b}\right ) = (\vect {b}\cdot \grad )\vect {b} + \vect {b}\,\divergence \vect {b} \tag{11.5}\]
and define
\[\vect {\kappa } \equiv (\vect {b}\cdot \grad )\vect {b}, \qquad \pp {}{\ell } \equiv \vect {b}\cdot \grad . \tag{11.6}\]
Because \(\divergence \B =0\), \[ 0=\divergence (B\vect {b})=\vect {b}\cdot \grad B + B\,\divergence \vect {b}, \] so
\[\divergence \vect {b}=-\frac {1}{B}\pp {B}{\ell }. \tag{11.7}\]
Substituting Eqs. (11.5) and (11.7) gives
\[\begin {aligned} \divergence \tens {P} &= \grad p_\perp + \vect {b}\pp {f}{\ell } + f\,\vect {\kappa } - \vect {b}\,\frac {f}{B}\pp {B}{\ell } \\ &= \grad _\perp p_\perp + (p_\parallel -p_\perp )\vect {\kappa } + \vect {b} \left [ \pp {p_\parallel }{\ell } - \frac {p_\parallel -p_\perp }{B}\pp {B}{\ell } \right ]. \end {aligned} \tag{11.8}\]
The last line uses \[ \pp {f}{\ell }=\pp {p_\parallel }{\ell }-\pp {p_\perp }{\ell }, \qquad \grad p_\perp = \grad _\perp p_\perp + \vect {b}\pp {p_\perp }{\ell }. \]

Kinetic examples. Appendix C gives three explicit kinetic realizations of the functions \(p_\perp (\psi ,B)\) and \(p_\parallel (\psi ,B)\): a fixed-pitch sloshing-ion distribution, a logarithmic mirror distribution generated by pitch-angle scattering, and the isotropic Maxwellian limit. The appendix makes the magnetic-field dependence of these pressures completely explicit and provides a useful bridge between the equilibrium formulas derived here and the underlying kinetic physics.

Write the magnetic force in the same geometric language. Using Eq. (1.11) together with Eq. (1.12), the Lorentz force may be written as

\[\begin {aligned} \J \times \B &= \frac {1}{\muo }(\curl \B )\times \B \\ &= -\grad \left (\frac {B^2}{2\muo }\right ) + \frac {1}{\muo }(\B \cdot \grad )\B \\ &= -\grad _\perp \left (\frac {B^2}{2\muo }\right ) + \frac {B^2}{\muo }\vect {\kappa }. \end {aligned} \tag{11.9}\]
The first term is magnetic-pressure gradient; the second is field-line tension.

Now project the force balance. Substitute Eqs. (11.8) and (11.9) into Eq. (11.1):

\[\begin {aligned} 0 &= -\grad _\perp \left (p_\perp + \frac {B^2}{2\muo }\right ) \\ &\quad + \left (\frac {B^2}{\muo }+p_\perp -p_\parallel \right )\vect {\kappa } \\ &\quad - \vect {b} \left [ \pp {p_\parallel }{\ell } + \frac {p_\perp -p_\parallel }{B}\pp {B}{\ell } \right ]. \end {aligned} \tag{11.10}\]
Since \(\vect {b}\cdot \vect {\kappa }=0\) and \(\vect {b}\cdot \grad _\perp =0\), the parallel and perpendicular projections follow immediately.

Parallel balance. Taking \(\vect {b}\cdot \)Eq. (11.10) yields

\[\boxed { \pp {p_\parallel }{\ell } + \frac {p_\perp -p_\parallel }{B}\pp {B}{\ell } =0. } \tag{11.11}\]
This is the static mirror-force constraint.

Perpendicular balance. Projecting Eq. (11.10) perpendicular to \(\B \) gives

\[Q \equiv \frac {B^2}{\muo }+p_\perp -p_\parallel , \tag{11.12}\]
so that
\[\boxed { \grad _\perp \left (p_\perp +\frac {B^2}{2\muo }\right ) = Q\,\vect {\kappa }. } \tag{11.13}\]
The coefficient \(Q\) is the effective field-line tension. Pressure anisotropy changes the restoring force associated with curvature, which is why it later reappears in anisotropic wave theory.

Caution

Why this is broader than CGL. Equations (11.11) and (11.13) follow from static force balance plus gyrotropy alone. They do not require the CGL evolution laws. CGL is one possible time-dependent closure that is compatible with these static relations, but the equilibrium relations themselves are more general.

11.2 Flux surfaces, sign conditions, and the isotropic bridge

Assume flux surfaces exist. Suppose there is a flux function \(\psi \) such that

\[\B \cdot \grad \psi =0.\]
If \(p_\parallel \) and \(p_\perp \) depend on position through \((\psi ,B)\), then along a field line \[ \pp {p_\parallel }{\ell } = \left .\pp {p_\parallel }{B}\right |_{\psi }\pp {B}{\ell }, \] because \(\vect {b}\cdot \grad \psi =0\). Insert this into Eq. (11.11):
\[\left [ \left .\pp {p_\parallel }{B}\right |_{\psi } + \frac {p_\perp -p_\parallel }{B} \right ] \pp {B}{\ell }=0.\]
For a generic mirror field with \(\pp {B}{\ell }\neq 0\), the bracket must vanish. Therefore
\[\boxed { p_\perp = p_\parallel - B\left .\pp {p_\parallel }{B}\right |_{\psi }. } \tag{11.16}\]
An equivalent form, often convenient in mirror geometry, is
\[\boxed { p_\perp = -B^2\left .\pp {}{B}\left (\frac {p_\parallel }{B}\right )\right |_{\psi }. } \tag{11.17}\]
The old label eq:cgl_constraint is preserved for continuity in these notes, but the relation is really a gyrotropic equilibrium constraint, not a specifically CGL one.

The isotropic limit is the bridge to Grad–Shafranov theory. If \(p_\perp =p_\parallel =p\), then Eq. (11.11) reduces to

\[\pp {p}{\ell }=0.\]
Thus the scalar pressure is constant along field lines and, in axisymmetry, becomes a flux function:
\[p=p(\psi ). \tag{11.19}\]
This is the key simplification that makes the isotropic axisymmetric equilibrium problem collapse to a single elliptic equation for \(\psi \). The next lecture is devoted to that reduction.

Positive effective tension. Equation (11.13) makes it clear that a bent field line only provides a restoring force when the coefficient \(Q\) is positive. Thus one requires

\[\boxed { \frac {B^2}{\muo }+p_\perp -p_\parallel >0. } \tag{11.20}\]
This is the static sign condition behind firehose behavior.

Include steady parallel flow if needed. A steady field-aligned flow adds an inertial curvature term \(-\rho u_\parallel ^2\vect {\kappa }\), so the effective coefficient becomes

\[Q_{\rm flow} = \frac {B^2}{\muo }+p_\perp -p_\parallel -\rho u_\parallel ^2.\]
The corresponding condition is
\[\boxed { \frac {B^2}{\muo }+p_\perp -p_\parallel -\rho u_\parallel ^2>0. } \tag{11.22}\]
This is the form that later becomes relevant in the solar-wind context when the flow approaches the Alfvén speed.

Positive compressive response. A second useful sign condition comes from asking how the total perpendicular pressure responds to a local increase in \(B\). Starting from Eq. (11.16),

\[Q = \frac {B^2}{\muo }+p_\perp -p_\parallel = \frac {B^2}{\muo }-B\left .\pp {p_\parallel }{B}\right |_{\psi },\]
so that
\[\frac {Q}{B} = \frac {B}{\muo }-\left .\pp {p_\parallel }{B}\right |_{\psi }. \tag{11.24}\]
Differentiate Eq. (11.24) with respect to \(B\) at fixed \(\psi \):
\[B\left .\pp {}{B}\left (\frac {Q}{B}\right )\right |_{\psi } = \frac {B}{\muo } - B\left .\pp {^2 p_\parallel }{B^2}\right |_{\psi }. \tag{11.25}\]
Now differentiate \[ p_\perp + \frac {B^2}{2\muo } = p_\parallel - B\left .\pp {p_\parallel }{B}\right |_{\psi } + \frac {B^2}{2\muo } \] with respect to \(B\):
\[\begin {aligned} \left .\pp {}{B}\left (p_\perp +\frac {B^2}{2\muo }\right )\right |_{\psi } &= \left .\pp {p_\parallel }{B}\right |_{\psi } - \left .\pp {p_\parallel }{B}\right |_{\psi } - B\left .\pp {^2 p_\parallel }{B^2}\right |_{\psi } + \frac {B}{\muo } \\ &= \frac {B}{\muo } - B\left .\pp {^2 p_\parallel }{B^2}\right |_{\psi }. \end {aligned}\]
Comparing with Eq. (11.25) gives
\[\boxed { \left .\pp {}{B}\left (p_\perp +\frac {B^2}{2\muo }\right )\right |_{\psi }>0. } \tag{11.27}\]
Again, this is being used here as a static sign condition, not yet as a wave-growth calculation.

11.3 Simple limits and the paraxial mirror bridge

The isotropic perpendicular balance. When \(p_\perp =p_\parallel =p\), Eq. (11.13) reduces to

\[\grad _\perp \left (p+\frac {B^2}{2\muo }\right ) = \frac {B^2}{\muo }\vect {\kappa }. \tag{11.28}\]
This is the form from which familiar one-dimensional equilibria follow.

Slab equilibrium. Take \[ \B =B(x)\,\ez , \qquad \vect {\kappa }=0. \] Then Eq. (11.28) becomes

\[\frac {d}{dx}\left (p+\frac {B^2}{2\muo }\right )=0. \tag{11.29}\]
Thus total pressure is constant across the slab.

The Z-pinch. For a cylindrical pinch with \[ \B =B_\theta (r)\,\etheta , \qquad \vect {\kappa }=-\frac {\er }{r}, \] Eq. (11.28) becomes

\[\boxed { \frac {d}{dr}\left (p+\frac {B_\theta ^2}{2\muo }\right ) + \frac {B_\theta ^2}{\muo r}=0. } \tag{11.30}\]
The second term is the inward hoop stress.

The screw pinch. If both \(B_\theta (r)\) and \(B_z(r)\) are present, only the azimuthal field contributes curvature. The equilibrium equation becomes

\[\boxed { \frac {d}{dr}\left (p+\frac {B_z^2+B_\theta ^2}{2\muo }\right ) + \frac {B_\theta ^2}{\muo r}=0. } \tag{11.31}\]
This cylindrical prototype is the natural warm-up for toroidal equilibrium.

Why the paraxial mirror limit is different. A mirror field is not really one-dimensional because the flux tube expands and contracts along the axis. But when the tube is slender, the geometry simplifies. Let \(a\) be the tube radius and \(Z\) the axial scale length, with

\[\epsilon \equiv \frac {a}{Z} \ll 1. \tag{11.32}\]
Write the axisymmetric field as \[ \B =B_r(r,z)\,\er + B_z(r,z)\,\ez . \] From \(\divergence \B =0\) and the estimates \[ \pp {}{r}\sim \frac {1}{a}, \qquad \pp {}{z}\sim \frac {1}{Z}, \] one finds
\[\frac {B_r}{B_z}=\mathcal O(\epsilon ). \tag{11.33}\]
If a field line is represented by \(r=a(z)\), its curvature in the meridional plane is
\[\kappa = \frac {|a''(z)|}{\left [1+(a'(z))^2\right ]^{3/2}} \simeq |a''(z)|. \tag{11.34}\]
Since \(a'\sim a/Z=\epsilon \) and \(a''\sim a/Z^2=\epsilon ^2/a\),
\[\kappa \sim \frac {\epsilon ^2}{a}. \tag{11.35}\]
So curvature is down by two powers of the slenderness parameter.

Leading-order paraxial balance. Because the curvature term is small, Eq. (11.13) reduces at leading order to

\[\boxed { \grad _\perp \left (p_\perp +\frac {B^2}{2\muo }\right )=0. } \tag{11.36}\]
Thus, on each axial slice, the sum of perpendicular plasma pressure and magnetic pressure is approximately constant across the tube.

Use the exterior field to reconstruct the tube. If the exterior vacuum field is \(B_{\rm ext}(z)\), then Eq. (11.36) implies

\[p_\perp (\psi ,z)+\frac {B^2(\psi ,z)}{2\muo } = \frac {B_{\rm ext}^2(z)}{2\muo }. \tag{11.37}\]
Hence
\[B(\psi ,z) = \sqrt {B_{\rm ext}^2(z)-2\muo p_\perp (\psi ,z)}. \tag{11.38}\]
For an axisymmetric flux function,
\[r^2(\psi ,z) = 2\int _0^{\psi }\frac {d\psi '}{B(\psi ',z)}. \tag{11.39}\]
This is the natural mirror-machine analogue of solving for flux surfaces in toroidal geometry.

11.4 Anisotropic pressure equilibrium

Now allow gyrotropic anisotropy,

\[\tens {P} = p_\perp \tens {I} + (p_\parallel - p_\perp ) {\vect {b}} {\vect {b}}, \qquad {\vect {b}} = \frac {\B }{B}.\]

Force balance becomes

\[\boxed { \J \times \B = \divergence \tens {P}. } \tag{11.41}\]

Parallel constraint

Projecting along \(\vect {b}\) gives

\[\boxed { \frac {\partial p_\parallel }{\partial \ell } + \frac {p_\perp - p_\parallel }{B} \frac {\partial B}{\partial \ell } = 0, } \tag{11.42}\]
where \(\partial /\partial \ell = {\vect {b}} \cdot \grad \).

Assuming \(p_{\perp ,\parallel } = p_{\perp ,\parallel }(\psi ,B)\),

\[\boxed { \pp {p_\parallel }{B} + \frac {p_\perp - p_\parallel }{B} = 0,\qquad \mbox { or } \qquad p_\perp = p_\parallel - B \frac {\partial p_\parallel }{\partial B} } \tag{11.43}\]

Perpendicular balance and cancellation algebra

Define

\[\sigma = \frac {1}{\muo } + \frac {p_\perp - p_\parallel }{B^2}.\]

Consider the following slight variation on \(\vect {J}\times \vect {B}\):

\[\B \times (\curl \sigma \B ) = B^2 \grad _\perp \sigma + \sigma \B \times (\curl \B ),\]
and expanding explicitly,
\[\begin {aligned} B^2 \grad _\perp \sigma &= \grad _\perp p_\perp - \grad _\perp p_\parallel - \frac {p_\perp - p_\parallel }{B^2} \grad _\perp B^2 , \\ \sigma \B \times (\curl \B ) &= \left ( 1 + \frac {\muo (p_\perp - p_\parallel )}{B^2} \right ) \left (\grad _\perp \frac {B^2}{2 \muo } - B^2 \sigma \vect {\kappa }\right ). \end {aligned}\]

Adding terms, the \(\grad _\perp B\) contributions cancel using (11.43), leaving

\[\begin {aligned} B^2 \grad _\perp \sigma + \sigma \B \times (\curl \B ) &= \grad _\perp p_\perp - \grad _\perp p_\parallel - \frac {p_\perp - p_\parallel }{B^2}\grad _\perp B^2 \\ &\quad + \left ( 1 + \frac {\muo (p_\perp - p_\parallel )}{B^2} \right ) \left (\grad _\perp \frac {B^2}{2 \muo } - B^2 \sigma \vect {\kappa }\right ) \\ &= - \nabla _\perp p_\parallel + \pp {p_\parallel }{B}\grad _\perp B \\ &= \pp {p_\parallel }{\psi }\grad _\perp \psi . \end {aligned}\]

Carrying through the axisymmetric representation yields the anisotropic Grad equation:

This proof now implies

\[\begin {aligned} (\nabla \times \vect {B})\times \vect {B} &= \frac {1}{\sigma }\left .\pp {p_\parallel }{\psi }\right |_B\,\nabla _\perp \psi + B^2 \nabla _\perp \ln \sigma \end {aligned}\]

From Eq. (11.50) we also note that

\[(\curl \vect {B})\times \vect {B} = -\frac {\Delta ^\star \psi }{R^2}\nabla _\perp \psi\]

Carrying this through with the axisymmetric representation yields and taking the dot product with \(\nabla \psi \)

\[\boxed { \Delta ^\star \psi = - \frac {R^2}{\sigma }\,\pp {p_\parallel }{\psi } \;-\; \, \nabla _\perp \psi \cdot \nabla _\perp \ln \sigma . } \tag{11.50}\]
For practical purposes it may be more useful to consider the dimensionless parameter
\[\begin {aligned} \bar {\sigma } & = \mu _0 \sigma \\ & = 1 + \frac {\beta _\perp - \beta _\parallel }{2} \\ & = 1 + \mu _0 \left (\frac {p_\perp - p_\parallel }{B^2}\right ) \\ & = 1 + \mu _0 \left (\frac {p_\parallel - B dp_\parallel /dB - p_\parallel }{B^2}\right ) \\ & = 1 - \frac {\mu _0}{B} \dd {p_\parallel }{B}. \end {aligned}\]
\[\boxed { \Delta ^\star \psi = - \mu _0 \frac {R^2}{\bar {\sigma }}\,\pp {p_\parallel }{\psi } \;-\; \, B^2 R^2 \frac { \partial \ln \bar {\sigma }}{\partial \psi } - \nabla _\perp \psi \cdot \nabla B \left . \frac { \partial \ln \bar {\sigma }}{\partial B}\right |_\psi . }\]

Equivalent Form.

\[\vect {J}\times \vect {B} = \nabla _\perp p_\perp + (p_\parallel - p_\perp ) \vect {\kappa }\]
that leads directly to all of the following equivalent formes for \(J_\phi \):
\[\begin {aligned} \mu _0 J_\phi & = -\frac {\Delta ^\star \psi }{ R} \\ J_\phi & = \frac {\vect {B}}{B^2} \times \left ( \nabla _\perp p_\perp + (p_\parallel - p_\perp ) \vect {\kappa } \right ) \\ \mu _0 J_\phi & = \frac {\vect {B}\times \nabla _\perp p_\perp }{B^2} + \frac { \beta _\parallel - \beta _\perp }{2} \vect {B}\times \vect {\kappa } \\ \mu _0 J_\phi & = - \mu _0 \frac {R}{\bar {\sigma }}\,\pp {p_\parallel }{\psi } \;-\; \, B^2 R \frac { \partial \ln \bar {\sigma }}{\partial \psi } - \frac {\nabla _\perp \psi \cdot \nabla B }{R} \left . \frac { \partial \ln \bar {\sigma }}{\partial B}\right |_\psi . \end {aligned}\]

Checks.

Bridge to the next lecture. The paraxial mirror problem is the cleanest anisotropic example because it stays close to one-dimensional intuition. The next lecture goes in the opposite direction: it specializes back to isotropic pressure, keeps full axisymmetric geometry, and derives the Grad–Shafranov equation for truly two-dimensional toroidal equilibria.

Experimental perspective. Equilibrium is where kinetic ideas first become impossible to hide from the experimentalist. In mirror devices, the dependence of pressure on \(B\) is built directly into confinement geometry. In toroidal devices, the same stress-balance logic survives but is reorganized by flux surfaces, current closure, and shaping coils. The point of doing the algebra carefully is that the diagnostic questions are always the same: what is balancing what, where does the tension sit, and which stress term is being inferred from the measurements?

Takeaways
  • Static anisotropic equilibrium begins with the gyrotropic tensor, not with a specific closure law.
  • Parallel balance gives Eq. (11.11); perpendicular balance gives Eq. (11.13).
  • If \(p_{\perp ,\parallel }=p_{\perp ,\parallel }(\psi ,B)\), then Eq. (11.16) follows. In the isotropic limit this collapses to \(p=p(\psi )\).
  • Slab, pinch, mirror, and toroidal equilibria are all different geometric faces of the same underlying stress-balance problem.

Bibliography

    James Clerk Maxwell. A Treatise on Electricity and Magnetism. Clarendon Press, Oxford, 1873. 2 vols.

    Archimedes. The Works of Archimedes. Cambridge University Press, Cambridge, 1897. Contains On Floating Bodies, Books I–II.

    Daniel Bernoulli. Hydrodynamica, sive de viribus et motibus fluidorum commentarii. Johann Reinhold Dulsecker, Argentorati, 1738.

    Leonhard Euler. Principes généraux du mouvement des fluides. Mémoires de l’Académie Royale des Sciences et des Belles-Lettres de Berlin, 11:274–315, 1757.

    Claude-Louis Navier. Mémoire sur les lois du mouvement des fluides. Mémoires de l’Académie des Sciences de l’Institut de France, 6, 1823. Read in 1822; the volume is commonly cited as 1823 and was issued in 1827.

    George Gabriel Stokes. On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids. Transactions of the Cambridge Philosophical Society, 8: 287–319, 1845.

    Olivier Darrigol. Worlds of Flow: A History of Hydrodynamics from the Bernoullis to Prandtl. Oxford University Press, Oxford, 2005.

    A. Tamburrino. From navier to stokes: Commemorating the bicentenary of navier’s equation on the lay of fluid motion. Fluids, 9(1):15, 2024. doi:10.3390/fluids9010015.

    Augustin-Louis Cauchy. Recherches sur l’équilibre et le mouvement intérieur des corps solides ou fluides, élastiques ou non élastiques. Bulletin de la Société Philomathique de Paris, pages 9–13, 1823.

    Harold Grad and H. Rubin. Hydromagnetic equilibria and force-free fields. Technical report, New York University, Institute of Mathematical Sciences, October 1958. Also circulated through the 2nd International Conference on the Peaceful Uses of Atomic Energy, Geneva.

    V. D. Shafranov. On magnetohydrodynamical equilibrium configurations. Soviet Physics JETP, 6:545–554, 1958.

    Julius Hartmann and Freimut Lazarus. Hg-dynamics II: Experimental investigations on the flow of mercury in a homogeneous magnetic field". Matematisk-fysiske Meddelelser, 15(7):1–45, 1937.

    Hannes Alfvén. Existence of electromagnetic-hydrodynamic waves. Nature, 150(3805):405–406, 1942.

    J. Hartmann. Hg Dynamics I: Theory of the laminar flow of an electrically conducting liquid in a homogeneous magnetic field". Mathematisk-fysiske Meddelelser, 15(6), 1937.

    G. K. Batchelor. An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge, 1967.

    S. Chandrasekhar. Hydrodynamic and Hydromagnetic Stability. Clarendon Press, Oxford, 1961.

    P. A. Davidson. An Introduction to Magnetohydrodynamics. Cambridge University Press, Cambridge, 2001.

    S. Molokov, R. Moreau, and H. K. Moffatt, editors. Magnetohydrodynamics: Historical Evolution and Trends. Springer, Dordrecht, 2007.

Problems

Problem 11.1. Derive the anisotropic equilibrium equations carefully

Part A. Starting from \(\tens {P}=p_\perp \tens {I}+(p_\parallel -p_\perp )\vect {b}\vect {b}\), reproduce Eq. (11.8). Do not skip the use of \(\divergence (B\vect {b})=0\).

Part B. Combine Eq. (11.8) with Eq. (11.9) and derive Eqs. (11.11) and (11.13).

Part C. Assume \(p_{\perp ,\parallel }=p_{\perp ,\parallel }(\psi ,B)\) and derive Eq. (11.16) step by step. Then show that the isotropic limit implies \(p=p(\psi )\).

Part D. Starting from Eq. (11.28), derive the slab, Z-pinch, and screw-pinch balances, Eqs. (11.29), (11.30), and (11.31).

Part E. In the paraxial ordering, show explicitly that the curvature term in Eq. (11.13) is smaller than the radial pressure-gradient term by \(\mathcal O(\epsilon ^2)\).