Plasmas and Magnetohydrodynamics

Magnetohydrodynamics (MHD) does not merely append a magnetic force to Euler flow. It replaces electron and ion distributions by one conducting continuum, so every MHD equation inherits a hierarchy of spatial, temporal, electromagnetic, and thermodynamic approximations. This page uses SI units for the plasma scales and three-vector MHD equations; $T_s$ is measured in energy units, consistent with $k_{\mathrm B}=1$.

Required background. Classical and Variational Mechanics supplies momentum and energy balance; States, Processes, and the Laws of Thermodynamics supplies pressure, entropy, and equations of state; Classical Fields and Their Actions supplies Maxwell equations, Lorentz force, and field energy; Continuum Kinematics and Balance Laws supplies material derivatives and continuum balance laws. Helpful background. Distribution Functions and the Boltzmann Equation supplies species moments and closure logic; Calculus and Linear Algebra for Relativity supplies vector identities and tensors; Differential Equations for Relativity supplies linear modes and scale separation.

Plasma scales and the one-fluid question

An ionized gas behaves collectively when charge separation is screened on a scale small compared with the system. For species $s$, useful scales are

$$\lambda_D^{-2} =\sum_s\frac{n_sq_s^2}{\epsilon_0T_s}, \qquad \omega_{ps}=\sqrt{\frac{n_sq_s^2}{\epsilon_0m_s}},$$ $$\Omega_{cs}=\frac{\lvert q_s\rvert B}{m_s}, \qquad \rho_s=\frac{v_{{\rm th},s}}{\Omega_{cs}}, \qquad v_{{\rm th},s}=\sqrt{\frac{T_s}{m_s}}, \qquad d_s=\frac{c}{\omega_{ps}}.$$

The Debye length $\lambda_D$ controls electrostatic shielding; $\omega_{ps}^{-1}$ is the collective charge-separation time; $\Omega_{cs}^{-1}$ and $\rho_s$ control gyromotion; and $d_s$ is the inertial or skin scale. The displayed screening formula assumes a small electrostatic perturbation of thermal species that can respond on the time scale considered; immobile species do not contribute to that response. The number of electrons in a sphere of radius $\lambda_D$ is

$$N_D=\frac{4\pi}{3}n_e\lambda_D^3.$$

A weakly coupled continuum plasma normally has $N_D\gg1$ and $\lambda_D\ll L$. Hutchinson derives shielding and the plasma parameter in Hutchinson 2006, §§1.2.3–1.4, printed pp. 10–14.

Standard single-fluid MHD additionally requires a scale ordering such as

$$L\gg d_i,\rho_i,\lambda_D, \qquad \omega\ll\Omega_{ci},\omega_{pe}, \qquad U,c_s,v_A\ll c.$$

The exact ordering depends on the problem. It supports:

• quasineutrality in the bulk, $n_e\simeq Zn_i$, while allowing small charge separation to sustain $\mathbf E$;
• omission of displacement current in low-frequency Ampère’s law;
• locking of species into one bulk velocity at leading order;
• a pressure and heat-flux closure, supplied by collisions or by an explicitly justified Chew–Goldberger–Low (CGL) double-adiabatic model, gyrofluid model, or kinetic model; strong magnetization alone instead motivates gyrotropy and can make anisotropy important;
• ideal MHD only when nonideal terms in the generalized Ohm law are small.

Failure is diagnostic. Debye-scale structure needs Poisson and sheath physics. Ion-inertial scales retain Hall drift. Gyroradius-scale or anisotropic dynamics need gyrofluid, gyrokinetic, or kinetic closure. A large collisional mean free path does not automatically destroy useful large-scale magnetic dynamics, but it does invalidate an unexamined scalar-pressure and dissipation law. Murphy summarizes the low-frequency, long-wavelength MHD approximation in Murphy 2014, printed slides 3–7.

One useful check comes directly from Ampère–Maxwell law. For an inductive mode of wave number $k$ and frequency $\omega$, Faraday’s law gives $E\sim(\omega/k)B$. Comparing displacement current with the curl of the magnetic field gives

$$\frac{\mu_0\epsilon_0|\partial_t\mathbf E|} {|\nabla\times\mathbf B|} \sim\frac{\omega^2}{k^2c^2}.$$

Thus a low-frequency mode with a large wavelength still requires $|\omega/k|\ll c$. In particular, the Newtonian expression $v_A=B_0/\sqrt{\mu_0\rho_0}$ must not be used to predict superluminal propagation; that limit signals the need for relativistic MHD.

From two fluids to total momentum

For each species $s$, the first two kinetic moments are

$$\partial_tn_s+\nabla\cdot(n_s\mathbf u_s)=0,$$ $$m_sn_s(\partial_t+\mathbf u_s\cdot\nabla)\mathbf u_s =q_sn_s(\mathbf E+\mathbf u_s\times\mathbf B) -\nabla\cdot\mathbf P_s+\mathbf R_s.$$

Here $\mathbf P_s$ is the species pressure tensor and $\mathbf R_s$ is collisional momentum transfer. Internal forces cancel when all species are summed:

$$\sum_s\mathbf R_s=0.$$

Define

$$\rho=\sum_s m_sn_s, \qquad \rho\mathbf u=\sum_s m_sn_s\mathbf u_s,$$ $$\rho_q=\sum_s q_sn_s, \qquad \mathbf J=\sum_s q_sn_s\mathbf u_s.$$

The total pressure tensor must include both random stress and species drift relative to the center-of-mass flow:

$$\mathbf P =\sum_s\left[ \mathbf P_s+m_sn_s(\mathbf u_s-\mathbf u) \otimes(\mathbf u_s-\mathbf u) \right].$$

Adding the species momentum equations now gives the exact conservative one-fluid balance

$$\boxed{ \partial_t(\rho\mathbf u) +\nabla\cdot(\rho\mathbf u\otimes\mathbf u+\mathbf P) =\rho_q\mathbf E+\mathbf J\times\mathbf B. }$$

In a quasineutral bulk with negligible drift stress and isotropic total pressure $\mathbf P=p\mathbf I$,

$$\rho\frac{D\mathbf u}{Dt} =-\nabla p+\mathbf J\times\mathbf B.$$

The electric force $\rho_q\mathbf E$ is small at leading quasineutral order, not identically forbidden. Hutchinson performs this species sum in Hutchinson 2006, §4.6, Eqs. (4.73)–(4.86), printed pp. 74–76.

Generalized Ohm law and its controlled reductions

For singly charged ions and electrons with $n_i\simeq n_e=n$, $m_e\ll m_i$, and $\mathbf u\simeq\mathbf u_i$,

$$\mathbf J=en(\mathbf u_i-\mathbf u_e).$$

Solve the electron momentum equation for the electric field and rewrite $\mathbf u_e=\mathbf u-\mathbf J/(en)$. A common generalized Ohm hierarchy is

$$\boxed{ \mathbf E+\mathbf u\times\mathbf B =\eta_\Omega\mathbf J +\frac{\mathbf J\times\mathbf B}{en} -\frac{\nabla\cdot\mathbf P_e}{en} -\frac{m_e}{e}\frac{D_e\mathbf u_e}{Dt}. }$$

$D_e/Dt=\partial_t+\mathbf u_e\cdot\nabla$ is the electron material derivative.

The terms are, respectively, resistive drag, Hall drift, electron-pressure force, and electron inertia. Anisotropic conductivity and additional kinetic terms may also matter.

• Ideal MHD drops the entire right side: $\mathbf E+\mathbf u\times\mathbf B=0$.
• Resistive MHD retains $\eta_\Omega\mathbf J$, where $\eta_\Omega=1/\sigma$ is electrical resistivity.
• Hall MHD retains $\mathbf J\times\mathbf B/(en)$.
• Electron-pressure or inertia scales require corresponding two-fluid or kinetic physics.

Hutchinson derives the generalized relation and the successive approximations in §4.6, Eqs. (4.87)–(4.100), printed pp. 76–77. The reduction is therefore a scale argument, not a definition of plasma.

The complete ideal-MHD conservation set

Use low-frequency Ampère and Faraday laws,

$$\mathbf J=\frac1{\mu_0}\nabla\times\mathbf B, \qquad \partial_t\mathbf B=-\nabla\times\mathbf E, \qquad \nabla\cdot\mathbf B=0.$$

For a gamma-law gas, no external force, and ideal Ohm law, a conservative system is

$$\partial_t\rho+\nabla\cdot(\rho\mathbf u)=0,$$ $$\partial_t(\rho\mathbf u) +\nabla\cdot\left[ \rho\mathbf u\otimes\mathbf u +\left(p+\frac{B^2}{2\mu_0}\right)\mathbf I -\frac{\mathbf B\otimes\mathbf B}{\mu_0} \right]=0,$$ $$\partial_t\mathbf B =\nabla\times(\mathbf u\times\mathbf B), \qquad \nabla\cdot\mathbf B=0,$$ $$\boxed{ \partial_t\mathcal E +\nabla\cdot\left[ \left(\mathcal E+p+\frac{B^2}{2\mu_0}\right)\mathbf u -\frac{(\mathbf u\cdot\mathbf B)\mathbf B}{\mu_0} \right]=0, }$$

where

$$\mathcal E =\frac12\rho u^2+\frac{p}{\gamma-1}+\frac{B^2}{2\mu_0}.$$

The magnetic part of the flux follows from the ideal Poynting vector

$$\frac{\mathbf E\times\mathbf B}{\mu_0} =\frac{B^2\mathbf u-(\mathbf u\cdot\mathbf B)\mathbf B}{\mu_0}.$$

It cancels the Lorentz-force work exchanged between field and matter. The equivalent adiabatic closure

$$\frac{D}{Dt}\left(\frac{p}{\rho^\gamma}\right)=0$$

is equivalent to the energy closure for smooth solutions of this ideal, nondissipative system. Across an ideal-MHD shock, the conservative total-energy equation and an admissibility condition are primary; the material adiabatic equation is not a weak jump law. Murphy presents the component ideal equations in Murphy 2014, printed slides 16–32; the combined energy flux above is the independent conservation check.

Resistive diffusion must include Joule heating

For constant scalar conductivity,

$$\mathbf E+\mathbf u\times\mathbf B =\frac{\mathbf J}{\sigma}, \qquad \eta_m=\frac{1}{\mu_0\sigma},$$

and Faraday’s law first gives the constraint-independent curl–curl form

$$\partial_t\mathbf B =\nabla\times(\mathbf u\times\mathbf B) -\eta_m\nabla\times(\nabla\times\mathbf B).$$

On the Maxwell constraint surface $\nabla\cdot\mathbf B=0$, the vector identity $\nabla\times\nabla\times\mathbf B=\nabla(\nabla\cdot\mathbf B)-\nabla^2\mathbf B$ reduces this to

$$\partial_t\mathbf B =\nabla\times(\mathbf u\times\mathbf B) +\eta_m\nabla^2\mathbf B.$$

The corresponding internal-energy equation contains positive Joule heating

$$\boxed{ Q_{\rm Joule}=\mathbf J\cdot(\mathbf E+\mathbf u\times\mathbf B) =\frac{J^2}{\sigma}\geq0. }$$

It is inconsistent to retain resistive magnetic diffusion while simultaneously asserting the source-free adiabatic law. Matter plus electromagnetic field still conserves total energy when Poynting flux and Joule transfer are included.

The magnetic Reynolds number

$$R_m=\frac{UL}{\eta_m}=\mu_0\sigma UL$$

compares advection with resistive diffusion. Large $R_m$ is necessary for ideal behavior on the chosen scale, but thin current sheets can make local gradients large enough for nonideal terms to matter.

Divergence, flux freezing, and magnetic stress

Taking the divergence of either the ideal induction equation or the unreduced resistive curl–curl equation gives

$$\partial_t(\nabla\cdot\mathbf B)=0.$$

Thus divergence-free initial data remain divergence free in the Maxwell continuum equations. Applying a divergence operator to the already reduced Laplacian form away from the constraint surface would instead produce a formal diffusion equation for the divergence error, but that off-constraint extension is not the equation derived from Faraday’s law. Numerical discretizations need their own constraint control; the continuum identity does not enforce itself after an inconsistent discretization.

For a smooth material surface $S(t)$ whose boundary moves with $\mathbf u$, the magnetic flux obeys

$$\frac{d}{dt}\int_{S(t)}\mathbf B\cdot d\mathbf S =-\oint_{\partial S(t)} (\mathbf E+\mathbf u\times\mathbf B)\cdot d\boldsymbol\ell.$$

Ideal Ohm law makes the right side zero:

$$\boxed{ \frac{d\Phi_B}{dt}=0. }$$

This is flux freezing. It concerns flux through material loops; “field lines move with the fluid” is a useful representation only where the field and flow remain regular. Nonideal terms, reconnection layers, or singular weak limits can break the field-line mapping. Hutchinson gives the moving-loop derivation in §4.11, Eqs. (4.154)–(4.157), printed pp. 87–88.

Finally, using $\nabla\cdot\mathbf B=0$,

$$\boxed{ \mathbf J\times\mathbf B =-\nabla\left(\frac{B^2}{2\mu_0}\right) +\frac{(\mathbf B\cdot\nabla)\mathbf B}{\mu_0}. }$$

The two terms are magnetic-pressure gradient and field-line tension. Since $B^2/\mu_0$ is an energy density or pressure, both have force-density units after one spatial derivative.

Alfvén and magnetosonic characteristics

Work in the rest frame of a uniform equilibrium, $\mathbf u_0=0$, and define

$$c_s^2=\left(\frac{\partial p}{\partial\rho}\right)_s, \qquad v_A^2=\frac{B_0^2}{\mu_0\rho_0}.$$

Let $\theta$ be the angle between the wavevector and $\mathbf B_0$. Linearized ideal MHD has a shear-Alfvén branch

$$\omega^2=k^2v_A^2\cos^2\theta$$

and a magnetosonic polynomial for $c^2=\omega^2/k^2$,

$$c^4-(c_s^2+v_A^2)c^2 +c_s^2v_A^2\cos^2\theta=0.$$

Therefore

$$\boxed{ c_{f,{\rm slow}}^2 =\frac12\left[ c_s^2+v_A^2 \pm\sqrt{(c_s^2+v_A^2)^2 -4c_s^2v_A^2\cos^2\theta} \right]. }$$

The upper sign gives $c_f^2$, and the lower sign gives $c_{\rm slow}^2$.

Two exact checks are

$$c_f^2+c_{\rm slow}^2=c_s^2+v_A^2, \qquad c_f^2c_{\rm slow}^2=c_s^2v_A^2\cos^2\theta.$$

For parallel propagation, the fast and slow speeds are the larger and smaller of $c_s$ and $v_A$; for perpendicular propagation, $c_f=\sqrt{c_s^2+v_A^2}$ while the slow and shear-Alfvén phase speeds vanish. Together with the entropy/contact family, the constrained system has seven physical branches: two fast, two Alfvén, two slow, and one entropy/contact branch. A divergence mode belongs only to an unconstrained or divergence-cleaning extension. In a uniform moving background, replace $\omega$ by the comoving frequency $\omega-\mathbf k\cdot\mathbf u_0$. There is no single “MHD wave speed.” Murphy derives the determinant and branches in Murphy 2016, Eqs. (39)–(70), printed slides 17–27.

Worked example: a shear-Alfvén wave

Take $\mathbf B_0=B_0\hat{\mathbf z}$ and uniform $\rho_0$, with transverse perturbations depending only on $z$. Linearized momentum and induction are

$$\rho_0\partial_t\delta\mathbf u_\perp =\frac{B_0}{\mu_0}\partial_z\delta\mathbf B_\perp, \qquad \partial_t\delta\mathbf B_\perp =B_0\partial_z\delta\mathbf u_\perp.$$

Eliminating either perturbation gives

$$\partial_t^2\delta\mathbf u_\perp =v_A^2\partial_z^2\delta\mathbf u_\perp, \qquad \omega^2=k^2v_A^2.$$

For one traveling wave,

$$|\delta B_\perp| =\sqrt{\mu_0\rho_0}\,|\delta u_\perp|.$$

Hence

$$\frac{\delta B_\perp^2}{2\mu_0} =\frac12\rho_0\delta u_\perp^2.$$

Magnetic and kinetic perturbation energies are equal, and magnetic tension supplies the restoring force. This normalization is a sharper check than remembering the phase speed alone.

Limits of nonrelativistic MHD

Hall MHD, two-fluid models, gyrokinetics, or full kinetic theory are needed near ion or electron scales, for pressure anisotropy, collisionless damping, or reconnection microphysics. Weak ionization introduces neutral drag and chemistry. When a bulk or characteristic speed approaches $c$, or gravity is strong, nonrelativistic MHD is invalid even if all its internal scale ratios look favorable.

The later Relativistic Magnetohydrodynamics and Plasmas chapter replaces the present mass, momentum, and electromagnetic split by covariant stress–energy and Maxwell balance, then treats ideal, resistive, force-free, and kinetic regimes. Fluid Waves, Shocks, and Instabilities now uses the characteristic viewpoint to explain nonlinear steepening and discontinuities.

Exercises

Choosing a model from scale ratios

A system has $L=10^6\,\mathrm m$, $d_i=10^2\,\mathrm m$, $\rho_i=10\,\mathrm m$, $\lambda_D=10^{-3}\,\mathrm m$, $U/c=10^{-3}$, and $\omega/\Omega_{ci}=10^{-2}$, but its collisional mean free path is $10^7\,\mathrm m$. Which MHD assumptions pass, and which closure remains doubtful?

Solution

The ratios $d_i/L=10^{-4}$, $\rho_i/L=10^{-5}$, and $\lambda_D/L=10^{-9}$ support the spatial MHD ordering; $\omega/\Omega_{ci}=10^{-2}$ is slow compared with ion gyration, and $U/c=10^{-3}$ makes the bulk motion nonrelativistic. These data do not specify $c_s/c$, $v_A/c$, or $\omega/\omega_{pe}$, so they do not establish the full electromagnetic and nonrelativistic ordering. Moreover, $\ell_{\rm mfp}/L=10$ does not support a collisionally isotropized scalar pressure or classical resistive transport. Large-scale ideal magnetic dynamics may still be useful, but the characteristic speeds, pressure tensor, and nonideal Ohm-law terms need independent checks.

Joule heat and magnetic diffusion

Show that a finite scalar conductivity cannot be combined with a source-free adiabatic law unless $\mathbf J=0$.

Solution

Resistive Ohm law gives

$$\mathbf J\cdot(\mathbf E+\mathbf u\times\mathbf B)=\frac{J^2}{\sigma}.$$

This is positive energy transferred from electromagnetic and bulk motion into internal energy. A source-free adiabatic law has no such term, so both statements can hold only where $J^2/\sigma=0$, in particular in the ideal limit $\sigma\to\infty$ or where $\mathbf J=0$.

References

• Hutchinson, Ian H. Introduction to Plasma Physics I, Chapter 1. MIT 22.611J course notes (2006 course copy), §§1.2.3–1.6.2, printed pp. 10–16. MIT OpenCourseWare PDF.
• Hutchinson, Ian H. Introduction to Plasma Physics I, Chapter 4: “Fluid Description of Plasma.” MIT 22.611J course notes (2006 course copy), §§4.1–4.6.3 and 4.10–4.13, especially Eqs. (4.73)–(4.100) and (4.151)–(4.157), printed pp. 64–90. MIT OpenCourseWare PDF.
• Murphy, Nicholas A. “Ideal Magnetohydrodynamics.” Harvard Astronomy 253 lecture notes (2014), printed slides 3–7 and 16–32. Official author PDF.
• Murphy, Nicholas A. “Magnetohydrodynamic Waves.” Harvard Astronomy 253 lecture notes (2016), Eqs. (39)–(70), printed slides 17–34. Official author PDF.