Resistive and Causal Dissipative MHD

Finite conductivity allows a comoving electric field, converts electromagnetic energy into heat, and breaks ideal flux freezing. It also introduces a short charge-relaxation timescale that can be much smaller than the fluid time step. A sound formulation must keep Maxwell propagation causal, specify an Ohm law and its frame, and distinguish the hyperbolic field equations from the parabolic magnetic-diffusion approximation. This page gives that continuum structure and a documented, unexecuted benchmark protocol.

Required background. Hilbert Stress–Energy from Metric Variation supplies total energy–momentum conservation; Exterior Algebra and Differential Forms supplies Maxwell’s homogeneous equation and magnetic constraint.

Helpful background. The Einstein Equation supplies spacetime coupling; Relativistic Matter and Compact Objects locates dissipative magnetofluids; and Computational Relativity supplies hyperbolic evolution and stiff-source methods.

Covariant Ohm law and energy conversion

Use Heaviside–Lorentz units and the Maxwell-source sign fixed in Covariant Electromagnetic Fluids: $\nabla_aF^{ba}=J^b$. Relative to the fluid velocity $u^a$, define the comoving electric field $e^a=F^{ab}u_b$ and decompose

$$J^a=q u^a+j^a, \qquad u_a j^a=0.$$

The simplest isotropic algebraic Ohm law is

$$j^a=\sigma e^a,$$

with nonnegative conductivity $\sigma$. Contracting the Lorentz force with $u_a$ shows that the comoving electromagnetic work converted into matter energy is

$$j_a e^a=\sigma e_a e^a\geq0.$$

For a perfect material fluid, the energy projection makes the heating explicit:

$$D\varepsilon+(\varepsilon+p)\theta=j_ae^a, \qquad D=u^a\nabla_a,\qquad\theta=\nabla_au^a.$$

With conserved baryon number and fixed composition, the first law gives $nT Ds=j_ae^a$. Thus nonnegative conductivity increases entropy. The closure assumes one scalar conductivity and omits Hall response, pressure anisotropy, battery terms, composition diffusion, and thermoelectric couplings. Any of those may be comparable in a collisionless or strongly magnetized plasma.

In the limit $\sigma\to\infty$, bounded current requires $e^a\to0$, recovering ideal RMHD. The limit is singular: it removes a fast relaxation mode rather than continuously setting every finite-conductivity correction to zero at fixed numerical time step. The covariant Ohm law, its laboratory-frame expression, and ideal limit appear in Palenzuela et al. 2009, § 2, Eqs. (21)–(24).

Telegraph propagation and diffusion

Consider a fixed fluid at rest in flat spacetime, constant $\sigma$, zero charge density, and linear electromagnetic fields. Maxwell plus $\boldsymbol J=\sigma\boldsymbol E$ gives

$$\partial_t\boldsymbol E=\boldsymbol\nabla\times\boldsymbol B-\sigma\boldsymbol E, \qquad \partial_t\boldsymbol B=-\boldsymbol\nabla\times\boldsymbol E.$$

Eliminating $\boldsymbol E$ yields the telegraph equation

$$\partial_t^2\boldsymbol B+\sigma\partial_t\boldsymbol B-\nabla^2\boldsymbol B=0.$$

The principal part propagates at the light speed, so the full system is causal. For a Fourier mode with wavenumber $k$, the two decay exponents are

$$r_\pm=\frac{-\sigma\pm\sqrt{\sigma^2-4k^2}}{2}.$$

For $k/\sigma\ll1$ and times long compared with $\sigma^{-1}$, the fast root decays away and the slow root approaches $-k^2/\sigma$. Both conditions matter: late times alone do not turn a short-wavelength underdamped wave into diffusion. Dropping the second time derivative gives magnetic diffusion,

$$\partial_t\boldsymbol B\simeq\frac1\sigma\nabla^2\boldsymbol B.$$

The diffusion equation has infinite mathematical signal speed and is only a long-wavelength, late-time approximation to the causal Maxwell–Ohm system. The same displacement-current comparison is made in Palenzuela et al. 2009, § 2, Eqs. (25)–(28). Notice that the algebraic relation $\boldsymbol J=\sigma\boldsymbol E$ itself has not made the full Maxwell system acausal.

Relaxing the current itself

An algebraic Ohm law assumes the conduction current adjusts instantaneously in the fluid frame. A causal material closure can instead promote $j^a$ to a dynamical variable, schematically

$$\tau h^a{}_b u^c\nabla_c j^b+j^a =\sigma e^a+\text{thermodynamic and magnetic couplings},$$

where $h^a{}_b=\delta^a{}_b+u^au_b$ projects into the material rest space and $\tau\gt0$ is a current-relaxation time. Such a law resolves material response that the algebraic closure omits and requires coefficients constrained by stability, entropy production, and microscopic physics. Merely inserting a relaxation term does not prove causality; the complete principal symbol must be checked.

The same caution applies to viscosity and heat flow. Conventional first-order gradient laws often produce parabolic equations, while transient theories enlarge the state space so dissipative fluxes relax over finite times. A kinetic example with current relaxation and explicit electric and magnetic couplings is Rocha et al. 2024, preprint § VII.B, Eq. (231); its one-species assumptions and Landau-frame coefficients must be translated before use in another plasma. Resistive RMHD should state whether “causal” refers only to retaining Maxwell displacement current or also to a causal material current closure.

Benchmark and rerun protocol

The continuum benchmark is the linear telegraph system above on $0\leq x<2\pi$ with periodic boundaries. Write the real fields as

$$B_y(x,t)=B_0 b(t)\cos(kx), \qquad E_z(x,t)=\frac{B_0}{k}\dot b(t)\sin(kx),$$

and choose $b(0)=1$ and $\dot b(0)=0$, so $B_y(x,0)=B_0\cos(kx)$ and $E_z(x,0)=0$. Maxwell’s equations then reduce exactly to

$$\ddot b+\sigma\dot b+k^2b=0.$$

The reference amplitude stays real in every damping regime. For $\sigma<2k$, define $\Omega=(k^2-\sigma^2/4)^{1/2}$; then

$$b(t)=e^{-\sigma t/2} \left[\cos(\Omega t)+\frac{\sigma}{2\Omega}\sin(\Omega t)\right].$$

At the repeated root $\sigma=2k$, the corresponding limit is

$$b(t)=e^{-kt}(1+kt).$$

For $\sigma>2k$, the two roots are real and distinct, and a numerically stable reference is

$$b(t)=\frac{-r_-e^{r_+t}+r_+e^{r_-t}}{r_+-r_-}.$$

For this prescribed stationary conductor, the electromagnetic benchmark is exactly linear at any amplitude. For a coupled-fluid comparison, choose background enthalpy density $w_0=1$ and $B_0=10^{-3}$ in the same units, so $B_0^2/w_0=10^{-6}$ and fluid feedback enters beyond the retained linear order. Use $k=1$, and test $\sigma=0.5$, $2$, and $20$ to cover underdamped, critical, and stiff overdamped regimes. Obtain the electric-field reference by differentiating $b(t)$ as specified above. A separate uniform-field check, $E_x(t)=E_0e^{-\sigma t}$, isolates the source update.

A possible implementation uses binary64 arithmetic, a finite-volume Maxwell update, and either an implicit–explicit Runge–Kutta method or an exactly integrated conductivity source. Record the executable, exact software versions, input data, and numerical method when performing the experiment. The following is a proposed convergence test; no finite-volume result is asserted here.

Run $N=64,128,256$ cells with Courant factors $0.4$ and $0.2$. Report $L_1$ and $L_\infty$ errors in $B_y$ and $E_z$, the observed refinement order, the discrete divergence constraints, and total electromagnetic energy lost versus integrated Joule heating. For $\sigma=20$, compare the slow decay rate with $k^2/\sigma$ only after several relaxation times; before then the diffusion approximation is not valid. A useful tolerance is convergence toward the scheme’s designed smooth order until roundoff or time integration dominates, not an arbitrary fixed decimal count.

Failure modes include instability when an explicit source step has $\sigma\Delta t\gg1$, excessive damping from a poorly balanced implicit solve, wrong decay rates from omitting displacement current, charge drift from inconsistent current and Gauss-law updates, and apparent ideal convergence produced by a conductivity-dependent floor. In full RMHD, negative pressure and failed primitive recovery can arise when electric and magnetic energies dominate material energy. Each safeguard must be varied to quantify its systematic effect.

Exercises

Derive the two telegraph exponents for a Fourier magnetic mode and identify the large-$\sigma$ slow timescale.

Solution

Insert $B_y\propto e^{rt+ikx}$ into the telegraph equation. The amplitude obeys

$$r^2+\sigma r+k^2=0,$$

so $r=r_\pm$. For $\sigma\gg k$, expand the square root to obtain $r_+\simeq-k^2/\sigma$ and $r_-\simeq-\sigma+k^2/\sigma$. The fast mode relaxes on $\sigma^{-1}$; the surviving magnetic mode diffuses on $\sigma/k^2$.

References

• Palenzuela, Carlos, Luis Lehner, Oscar Reula, and Luciano Rezzolla. “Beyond Ideal MHD: Towards a More Realistic Modelling of Relativistic Astrophysical Plasmas.” Monthly Notices of the Royal Astronomical Society 394, 1727–1740 (2009). DOI. Open preprint.
• Rocha, Gabriel S., David Wagner, Gabriel S. Denicol, Jorge Noronha, and Dirk H. Rischke. “Theories of Relativistic Dissipative Fluid Dynamics.” Entropy 26, 189 (2024). DOI. Open preprint.