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Lecture 5
Braginskii Closure and the Validity of MHD

Overview

This lecture answers a deceptively simple question: when is MHD actually trustworthy?

1.
MHD is obtained by taking moments of kinetic theory and then closing the resulting hierarchy.
2.
In a collisional, magnetized plasma that closure is anisotropic: transport along \(\B \) is not transport across \(\B \).
3.
The familiar one-fluid equations of Lectures 2–4 appear only after additional orderings suppress Hall, electron-inertia, and finite-Larmor-radius effects.

The phrase “validity of MHD” is slightly misleading, because MHD is not a yes-or-no proposition. It is a scale-separated description. The real question is which terms survive once the hierarchy of kinetic, two-fluid, and one-fluid equations has been ordered. This is why Braginskii theory is such an important checkpoint: it tells us what transport looks like when the plasma is collisional enough to be fluid-like but magnetized enough to be anisotropic.

Historical Perspective

Braginskii’s closure did not appear from nowhere. The general asymptotic strategy goes back to the Chapman–Enskog expansion and its systematic kinetic-theory treatment by Chapman and Cowling [Chapman and Cowling, 1952]. For fully ionized plasmas, important transport results already existed before the 1965 review: Spitzer and Härm obtained the classic unmagnetized conductivity and heat-flow coefficients [Spitzer and Härm, 1953], Braginskii’s own 1957/1958 paper extended the program to a two-temperature magnetized plasma [Braginskii, 1958], and on the finite-Larmor-radius side Kaufman and later Simon and Thompson isolated the “magnetic viscosity”/gyroviscous corrections associated with gyromotion [Kaufman, 1960, Thompson, 1961, Simon and Thompson, 1966]. What made Braginskii the standard reference was not that the subject ended there, but that he assembled these strands into the first broadly complete, practically usable closed fluid description of a collisional magnetized plasma—putting resistivity, thermal forces, heat fluxes, anisotropic viscosity, and gyroviscosity into one notation with coefficients that later MHD calculations could actually use. For weakly collisional plasmas, the double-adiabatic closure of Chew, Goldberger, and Low marks another important branch of the story [Chew et al., 1956, Kulsrud, 2005].

Caution

The important distinction is not “fluid” versus “kinetic” in the abstract. The important distinction is which length and time scales have been averaged away, and which closure survives that averaging. A plasma can be well described by MHD on the system scale and still require non-MHD physics in thin layers or fast transients.

5.1 From Kinetic Theory to Fluid Moments

Each species \(s\) is described by a distribution function \(f_s(\vect {x},\vect {v},t)\) satisfying

\[\pp {f_s}{t} + \vect {v}\cdot \grad f_s + \frac {q_s}{m_s} \left ( \E + \vect {v}\times \B \right )\cdot \pp {f_s}{\vect {v}} = \left ( \pp {f_s}{t} \right )_{\mathrm {coll}}. \tag{5.1}\]

Velocity moments define the fluid variables

\[\begin{aligned}n_s &= \int f_s\, d^3v, \\ \uvec _s &= \frac {1}{n_s} \int \vect {v}\, f_s\, d^3v, \\ \tens {P}_s &= m_s \int (\vect {v}-\uvec _s)(\vect {v}-\uvec _s)\, f_s\, d^3v.\end{aligned} \tag{5.4}\]

Taking moments of (5.1) produces the continuity and momentum equations for each species:

\[\begin{aligned}\pp {n_s}{t} + \divergence (n_s \uvec _s) &= 0, \\ m_s n_s \left ( \pp {\uvec _s}{t} + \uvec _s\cdot \grad \uvec _s \right ) &= q_s n_s \left ( \E +\uvec _s\times \B \right ) - \divergence \tens {P}_s + \vect {R}_s.\end{aligned} \tag{5.5}\]

Here \(\vect {R}_s\) is the collisional momentum exchange with the other species.

The crucial point is that the moment hierarchy does not close by itself: the evolution of \(\tens {P}_s\) depends on higher moments, the heat flux depends on still higher moments, and so on. MHD only emerges after a closure has been chosen.

5.2 From Two-Fluid Moments to One-Fluid MHD and Generalized Ohm’s Law

For a singly ionized plasma, define

\[n_e \approx n_i \equiv n, \qquad \J = en(\uvec _i-\uvec _e), \qquad \rho \equiv m_i n_i + m_e n_e \approx m_i n, \qquad \rho \uvec \equiv m_i n_i \uvec _i + m_e n_e \uvec _e . \tag{5.7}\]

The first step toward one-fluid MHD is to add the ion and electron momentum equations. The collisional forces cancel between species, and after using quasi-neutrality one obtains

\[\rho \left ( \pp {\uvec }{t} + \uvec \cdot \grad \uvec \right ) = \J \times \B - \grad (p_i+p_e) - \divergence (\tens {\Pi }_i+\tens {\Pi }_e), \tag{5.8}\]
up to small corrections from electron inertia and the relative drift \(\uvec _i-\uvec _e\). This is the real parent of the one-fluid MHD momentum equation: the Lorentz force appears only after the two species are combined into a bulk momentum balance.

The second step is to derive the induction-side closure. Using (5.7), one may write

\[\uvec _e = \uvec -\frac {\J }{en}, \qquad \uvec _i = \uvec +\mathcal {O}\!\left (\frac {m_e}{m_i}\right )\frac {\J }{en}. \tag{5.9}\]
Substituting \(\uvec _e=\uvec -\J /(en)\) into the electron momentum equation, and collecting the collisional, Hall, pressure-gradient, viscous, and inertia terms, yields the generalized Ohm law
\[\boxed { \E +\uvec \times \B = \eta \J + \frac {\J \times \B }{ne} - \frac {1}{ne}\grad p_e - \frac {1}{ne}\divergence \tens {\Pi }_e + \frac {m_e}{ne^2} \frac {d\J }{dt}. } \tag{5.10}\]

Equation (5.8) is the parent of the MHD momentum equation, while (5.10) is the parent of both the resistive Ohm law (1.9) and the ideal Ohm law (4.8). The single-fluid induction law of Lecture 1 is obtained by neglecting the Hall term, the electron-pressure term, the electron viscous term, and electron inertia relative to \(\uvec \times \B \). The ideal-MHD law of Lecture 4 further neglects \(\eta \J \).

That chain of reductions is worth keeping in mind: the momentum equation and the induction equation descend from different combinations of the two-fluid system, and (1.9) is already a reduced Ohm law, while (4.8) is a more restrictive reduction still.

5.3 Braginskii Closure

Braginskii theory assumes that each species remains close to a drifting Maxwellian and that collisions are frequent enough to justify a Chapman–Enskog-like expansion, while the magnetic field is strong enough to make transport anisotropic. In practice the ordering is

\[\omega \tau _s \ll 1, \qquad \omega _{cs}\tau _s \gg 1, \tag{5.11}\]
with gradients long compared to the mean free path:
\[\lambda _{\mathrm {mfp},s}\ll L. \tag{5.12}\]

Introduce the magnetic-field unit vector \[ \vect {b} \equiv \frac {\B }{B}. \] The pressure tensor is naturally decomposed as

\[\boxed { \tens {P}_s = p_{\perp s} \left ( \tens {I}-\vect {b}\vect {b} \right ) + p_{\parallel s}\vect {b}\vect {b} + \tens {\Pi }_s. } \tag{5.13}\]
The first two terms express anisotropic scalar pressures relative to the field; the remaining tensor \(\tens {\Pi }_s\) contains viscous corrections.

Viscosity Tensor Define the symmetric traceless strain tensor

\[W_{ij} = \pp {U_i}{x_j} + \pp {U_j}{x_i} - \frac {2}{3}\delta _{ij}\divergence \vect {U}.\]

Viscous stresses:

\[\begin{aligned}\Pi _{xx} &= -\frac {\eta _0}{2}(W_{xx}+W_{yy}) -\frac {\eta _1}{2}(W_{xx}-W_{yy}) -\eta _3 W_{xy}, \\ \Pi _{yy} &= -\frac {\eta _0}{2}(W_{xx}+W_{yy}) +\frac {\eta _1}{2}(W_{xx}-W_{yy}) +\eta _3 W_{xy}, \\ \Pi _{xy} &= -\eta _1 W_{xy} + \frac {\eta _3}{2}(W_{xx}-W_{yy}), \\ \Pi _{xz} &= -\eta _2 W_{xz} - \eta _4 W_{yz}, \\ \Pi _{yz} &= -\eta _2 W_{yz} + \eta _4 W_{xz}, \\ \Pi _{zz} &= -\eta _0 W_{zz}.\end{aligned}\]

Interpretation:

Compact coordinate-free form of the full viscous stress It is also useful to collect the full Braginskii stress into field-aligned tensor pieces rather than only Cartesian components. Define \[ \tens {I}_{\perp } \equiv \tens {I}-\vect {b}\vect {b}, \qquad W_{\parallel } \equiv \vect {b}\cdot \tens {W}\cdot \vect {b}, \qquad \tens {W}_{\perp } \equiv \tens {I}_{\perp }\cdot \tens {W}\cdot \tens {I}_{\perp }. \] Then one convenient decomposition is

\[\begin{aligned}\tens {W}^{(0)} &\equiv \frac {3}{2}\left (\vect {b}\vect {b}-\frac {\tens {I}}{3}\right )W_{\parallel }, \\ \tens {W}^{(1)} &\equiv \tens {W}_{\perp } -\frac {1}{2}\tens {I}_{\perp }\,\operatorname {tr}(\tens {W}_{\perp }), \\ \tens {W}^{(2)} &\equiv \tens {W}\cdot \vect {b}\vect {b} \, +\vect {b}\vect {b}\cdot \tens {W} -2\vect {b}\vect {b}\,W_{\parallel }.\end{aligned}\]

With these definitions, the full viscous stress may be organized as

\[\boxed { \tens {\Pi } = -\eta _0\,\tens {W}^{(0)} -\eta _1\,\tens {W}^{(1)} -\eta _2\,\tens {W}^{(2)} +\tens {\Pi }^{\mathrm {gv}}, } \tag{5.24}\]
where \(\tens {\Pi }^{\mathrm {gv}}\) contains the \(\eta _3\) and \(\eta _4\) finite-Larmor-radius pieces written below. For \(\vect {b}=\vect {e}_z\), \(\tens {W}^{(0)}\), \(\tens {W}^{(1)}\), and \(\tens {W}^{(2)}\) reduce exactly to the \(\eta _0\), \(\eta _1\), and \(\eta _2\) terms in the Cartesian formulas above.

Compact form of the gyroviscous stress It is often useful to write only the non-dissipative part in coordinate-free form. Define \[ \tens {I}_{\perp } \equiv \tens {I}-\vect {b}\vect {b}, \qquad \bigl (\vect {b}\times \tens {W}\bigr )_{ij}\equiv \epsilon _{ik\ell } b_k W_{\ell j}, \qquad \bigl (\tens {W}\times \vect {b}\bigr )_{ij}\equiv W_{ik}\epsilon _{jk\ell } b_\ell . \] Then the gyroviscous contribution may be written as

\[\boxed { \tens {\Pi }^{\mathrm {gv}} = \frac {\eta _3}{2} \left ( \vect {b}\times \tens {W}\cdot \tens {I}_{\perp } - \tens {I}_{\perp }\cdot \tens {W}\times \vect {b} \right ) + \eta _4 \left ( \vect {b}\times \tens {W}\cdot \vect {b}\vect {b} - \vect {b}\vect {b}\cdot \tens {W}\times \vect {b} \right ). } \tag{5.25}\]
For \(\vect {b}=\hat {z}\) this reduces to the Cartesian \(\eta _3\) and \(\eta _4\) terms written above. The interesting conceptual point is that these terms are not ordinary collisional viscosity: they are finite-Larmor-radius corrections that redistribute momentum through gyromotion, and in Braginskii theory they do not produce positive-definite viscous heating.

Likewise the heat flux takes the schematic form

\[\boxed { \vect {q}_s = -\kappa _{\parallel s}\, \vect {b}\vect {b}\cdot \grad T_s - \kappa _{\perp s} \left ( \tens {I}-\vect {b}\vect {b} \right )\cdot \grad T_s - \kappa _{\wedge s}\, \vect {b}\times \grad T_s. } \tag{5.26}\]
The last term is the cross-field or “Righi–Leduc” heat flux. Its exact coefficient is not the main point here; the main point is that magnetization turns scalar transport coefficients into tensorial ones.

The dominant transport scalings are

\[\begin{aligned}\eta _{\parallel s} &\sim p_s\tau _s, & \eta _{\perp s} &\sim \frac {p_s\tau _s}{1+(\omega _{cs}\tau _s)^2}, & \eta _{\mathrm {gv},s} &\sim \frac {p_s}{\omega _{cs}}, \\ \kappa _{\parallel s} &\sim \frac {n_s T_s\tau _s}{m_s}, & \kappa _{\perp s} &\sim \kappa _{\parallel s}(\omega _{cs}\tau _s)^{-2}.\end{aligned} \tag{5.27}\]

Thus parallel transport is usually much larger than perpendicular transport, while gyroviscous terms are nondissipative finite-Larmor-radius corrections.

Caution

Braginskii viscosity is not just Navier–Stokes viscosity with a magnetic field pasted on top. The decomposition in (5.13) means that a magnetized plasma has a preferred direction, so the closure retains the memory of the field geometry. That is the seed of later topics such as pressure anisotropy, mirror physics, firehose limits, and anisotropic damping.

Interactive Braginskii Formulary Calculator

Open a browser companion to the closure lecture. The calculator estimates gyro radii, skin depths, Coulomb mean free paths, Spitzer resistivity, Braginskii transport scales, Alfvén speed, magnetic Reynolds number, Lundquist number, and the regime checks that decide whether collisional single-fluid MHD is comfortable or already breaking down.

Open the formulary calculator

5.4 How One-Fluid MHD Emerges

The stress split used in Lecture 2,

\[\tens {P}=p\,\tens {I}-\tens {\Pi },\]
is the isotropized shadow of the more general tensor structure in (5.13). To recover the one-fluid MHD equations of Lectures 1–4, one needs several scale-separation assumptions in addition to quasi-neutrality.

Quasi-neutrality. The Debye length must be small compared with the system scale:

\[k\lambda _D \ll 1. \tag{5.29}\]

Fluidization. To justify a collisional closure of Braginskii type, the ion mean free path must be short and the dynamics slower than the collision rate:

\[\frac {\lambda _{\mathrm {mfp},i}}{L}\ll 1, \qquad \omega \tau _i\ll 1. \tag{5.30}\]

Magnetization and long wavelengths. To stay well above Larmor-radius physics,

\[\frac {\rho _i}{L}\ll 1, \qquad \frac {\omega }{\Omega _i}\ll 1. \tag{5.31}\]

Hall term. Using Ampère’s law (1.11) to estimate \(J\sim B/(\muo L)\), the Hall term in (5.10) satisfies

\[\frac {|(\J \times \B )/(ne)|}{|\uvec \times \B |} \sim \frac {J}{neU} \sim \frac {d_i}{L}\frac {V_A}{U}, \tag{5.32}\]
where \[ d_i = \sqrt {\frac {m_i}{\muo n e^2}} = \frac {c}{\omega _{pi}}, \qquad V_A = \frac {B}{\sqrt {\muo \rho }}. \] If the bulk speed is Alfvénic, \(U\sim V_A\), then the Hall correction is small when \(d_i/L\ll 1\).

Electron-pressure term. A similar estimate gives

\[\frac {|\grad p_e|/(ne)}{U B} \sim \frac {T_e}{e B L U} \sim \frac {\rho _s}{L}\frac {c_s}{U}, \tag{5.33}\]
where \(\rho _s=\sqrt {m_i T_e}/(eB)\) and \(c_s=\sqrt {T_e/m_i}\). For order-one Mach numbers and comparable ion and electron temperatures, this is again a small-gyroradius ordering.

Electron inertia. Estimating \(\partial _t\J \sim \omega J\) and taking \(\omega \sim U/L\) gives

\[\frac {m_e |\partial _t\J |/(ne^2)}{U B} \sim \frac {d_e^2}{L^2}, \qquad d_e = \sqrt {\frac {m_e}{\muo n e^2}} = \frac {c}{\omega _{pe}}. \tag{5.34}\]

Resistivity. The resistive term in (5.10) is small compared with \(\uvec \times \B \) when

\[\frac {\eta J}{U B} \sim \frac {\eta }{\muo U L} = \frac {1}{Rm} \ll 1, \tag{5.35}\]
where \(Rm\) is the magnetic Reynolds number from (4.3).

Collecting the most familiar orderings, one arrives at the heuristic MHD checklist

\[\boxed { k\lambda _D\ll 1, \qquad \frac {\lambda _{\mathrm {mfp},i}}{L}\ll 1, \qquad \frac {\rho _i}{L}\ll 1, \qquad \frac {d_i}{L}\ll 1, \qquad \frac {d_e^2}{L^2}\ll 1, } \tag{5.36}\]
with the additional condition \(Rm\gg 1\) if one wants the ideal limit and frozen-in flux (4.13).

5.5 What Survives When Collisions Are Weak?

One subtle but important point is that MHD-like behavior can survive even when the collisional Braginskii ordering fails. In a collisionless but magnetized plasma, the double-adiabatic or CGL closure gives

\[\begin{aligned}\frac {D}{Dt} \left ( \frac {p_\perp }{\rho B} \right ) &= 0, \\ \frac {D}{Dt} \left ( \frac {p_\parallel B^2}{\rho ^3} \right ) &= 0.\end{aligned} \tag{5.37}\]

These relations are already telling us that the right question is not simply whether MHD works, but which closure has replaced the isotropic pressure law (3.11). In other words, large-scale magnetic evolution can still look fluid-like while thermodynamics and stability become decisively anisotropic.

5.6 Experimental Perspective

This lecture is one of the places where laboratory intuition matters most. In liquid-metal experiments the conditions behind single-fluid resistive MHD are often satisfied so well that the model is almost the starting point. In magnetized plasma experiments the situation is subtler: the bulk flow may still obey the momentum and induction equations beautifully on the device scale, while parallel heat transport, pressure anisotropy, Hall corrections, or localized reconnection physics are already visible in special regions. The practical question is therefore not “MHD or nothing,” but rather “which reduced model is honest on the scale being measured?”

That perspective is especially useful for dynamos, self-organization experiments, and rotating plasma flows. Global evolution may be captured by MHD, but transport and topology change can still advertise the terms that were dropped on the way from (5.10) to (1.9) and then to (4.8).

Takeaways
  • MHD is not fundamental; it is a reduced moment system descended from (5.1).
  • Braginskii closure explains why transport in magnetized plasmas is strongly anisotropic, as in (5.13) and (5.26).
  • The road from generalized Ohm’s law (5.10) to resistive MHD (1.9) and ideal MHD (4.8) is paved with explicit ordering assumptions, not wishful thinking.

Bibliography

    Sydney Chapman and T. G. Cowling. The Mathematical Theory of Non-Uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases. Cambridge University Press, Cambridge, 2 edition, 1952.

    Jr. Spitzer, Lyman and Richard Härm. Transport phenomena in a completely ionized gas. Physical Review, 89(5):977–981, 1953. doi:10.1103/PhysRev.89.977.

    S. I. Braginskii. Transport phenomena in a completely ionized two-temperature plasma. Soviet Physics JETP, 6(2):358–369, 1958. English translation of Zh. Eksp. Teor. Fiz. 33, 459–472 (1957).

    Allan N. Kaufman. Plasma viscosity in a magnetic field. Physics of Fluids, 3(4):610–616, 1960. doi:10.1063/1.1706096.

    W. B. Thompson. The dynamics of high temperature plasmas. Reports on Progress in Physics, 24(1):363–424, 1961. doi:10.1088/0034-4885/24/1/308.

    A. Simon and W. B. Thompson. Hydromagnetic equations with viscosity for a collisionless plasma. Journal of Nuclear Energy, Part C: Plasma Physics, Accelerators, Thermonuclear Research, 8(4):373, 1966. doi:10.1088/0368-3281/8/4/302.

    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.

    Russell M. Kulsrud. Plasma Physics for Astrophysics. Princeton University Press, Princeton, NJ, 2005. ISBN 9780691120737.

Problems

Problem 5.1. Braginskii Viscosity — Parallel vs. Perpendicular Transport

Consider a strongly magnetized, collisional plasma with \(\omega _c \tau \gg 1\), described by Braginskii viscosity.

(a)
Using the scalings in (5.27), identify which transport coefficients correspond to parallel viscosity, perpendicular viscosity, and gyroviscosity.
(b)
Consider a shear flow parallel to the magnetic field, \[ \uvec = u_\parallel (x,z)\,\vect {b}, \qquad \B =B\,\vect {e}_z. \] Show that the dominant viscous stress is proportional to \(\eta _{\parallel }\) and explain physically why perpendicular viscosity does not contribute at leading order.
(c)
Now consider a shear flow perpendicular to the magnetic field, \[ \uvec = u_y(x)\,\vect {e}_y, \qquad \B =B\,\vect {e}_z. \] Estimate the magnitude of the dissipative viscous force and show that it is smaller than the field-aligned viscous force by a factor of order \((\omega _c\tau )^{-2}\).
(d)
Explain qualitatively why gyroviscosity is nondissipative even though it appears in the stress tensor.

Problem 5.2. From Generalized Ohm’s Law to Ideal MHD

Starting from (5.10):

(a)
Estimate the Hall, electron-pressure, electron-inertia, and resistive terms relative to \(\uvec \times \B \) using the orderings in (5.32), (5.33), (5.34), and (5.35).
(b)
State the conditions under which (5.10) reduces first to the resistive MHD Ohm law (1.9) and then to the ideal MHD law (4.8).
(c)
Explain why frozen-in flux (4.13) can fail even when the magnetic Reynolds number is large.