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In vacuum-permeability magnetohydrodynamics, the magnetic energy in a volume is $E_B=\int B^2/(2{\mu }_0)dV$. Its volume density equals the magnetic pressure.

Consider uniform fields $B_{-}\hat{\mathbf{y}}$ below and $B_{+}\hat{\mathbf{x}}$ above an interface, with local constant densities ${\rho }_{-},{\rho }_{+}$. For ideal magnetohydrodynamics and local incompressible potential flow, the interface displacement $h{e}^{st+ikx+imy}$ has displacement potentials ${\chi }_{-}=h{e}^{Kz}/K$ and ${\chi }_{+}=-h{e}^{-Kz}/K$, $K=\sqrt{k^2+m^2}$. Magnetic flux freezing gives ${\mathbf{b}}_i=({\mathbf{B}}_i\cdot \nabla )\nabla {\chi }_i$. The linear magnetohydrodynamic total pressure is $-[{\rho }_i{s}^2+({\mathbf{k}}_h\cdot {\mathbf{B}}_i{)}^2/{\mu }_0]{\chi }_i$. Matching Lagrangian total pressure gives the displayed dispersion relation. For ${\rho }_{+}>{\rho }_{-}$ the most unstable orientation has its wavevector along the weaker field.

// Target: continuum-mechanics.bigb

In the rapidly rotating magnetohydrodynamics of a conducting core, the magnetostrophic balance neglects inertia and, initially, viscosity:Take a spherical or suitable axisymmetric impenetrable container, with gravity/buoyancy having no azimuthal component and no applied mechanical torque. Small Rossby number is the slow-motion assumption. It does not imply that an arbitrary magnetic field is compatible with the balance.

Use cylindrical coordinates $(s,\varphi ,z)$ and integrate the azimuthal momentum equation over the fluid portion $C_s$ of a cylinder. The pressure term integrates to zero because it is a periodic $\varphi$ derivative. The Coriolis contribution is proportional to the radial flux ${\int }_{C_s}{u}_{s}sd\varphi dz$. By incompressibility and impermeability it is zero: closing the cylinder with the container's caps and applying the divergence theorem produces no net flux. There is no azimuthal buoyancy term. ConsequentlyThis is Taylor's condition. Equivalently define the torque per radial thickness by $\mathcal{T}(s)=s^2{\int }_{C_s}(J\times B{)}_{\varphi }d\varphi dz$; it must vanish. In a shell, cylinders split by the inner boundary require the corresponding connected-region conditions. These compatibility conditions concern rotating-fluid torque, distinct from force-free magnetic relaxation in a plasma. Their primary origin is Taylor's rotating-fluid analysis.

A magnetostrophic Taylor state allows construction of the flow's ageostrophic part. Taking the curl givesIntegrate along rotation-axis lines and impose impermeability, then use incompressibility and periodicity to reconstruct a particular solution $u_a$. In a sphere the boundary geometry supplies the independent endpoint conditions; a cylinder with parallel planar end normals has additional degeneracies. The homogeneous momentum equation admits any differential cylindrical rotation $u_g=s\omega (s)\hat{\varphi }$. Its Coriolis force is a radial pressure gradient, so momentum balance alone cannot determine $\omega$.

The additional equation is dynamic preservation of the constraint. LetThis is the first variation of the cylindrical Lorentz torque. The resistive induction equation supplies $\dot{B}=\nabla \times (u\times B)+\eta {\nabla }^{2}B$. Define $C_a=\nabla \times (u_a\times B)+\eta {\nabla }^{2}B$ and $L_B[\omega ]=\nabla \times (s\omega \hat{\varphi }\times B)$. Then the geostrophic flow preserving the Taylor constraint solvesFor a given field this is linear in $\omega$; it prescribes the cylindrical rotation needed to keep the field on the Taylor-state manifold. The magnetic boundary conditions must also be imposed on $\dot{B}$. Boundary terms cannot generally be discarded for insulating exteriors; their role in a correct construction is described in Hardy and collaborators' primary analysis.

For example, when magnetic surface torque vanishes, as for compatible $B\cdot n=0$ conditions, the Maxwell stress tensor givesCylindrical rotation induces component changes ${\dot{B}}_s=-\omega {\partial }_{\varphi }{B}_s$ and ${\dot{B}}_{\varphi }=s{B}_s{\omega }^{\prime }-\omega {\partial }_{\varphi }{B}_{\varphi }$. Their product variation integrates to $s{\omega }^{\prime }\int B_s^{2}d\varphi dz$, since the azimuthal derivative integrates to zero. Thus the geostrophic part of the preservation equation reduces toRegularity and compatible boundary conditions fix the nontrivial integration freedom; total angular momentum fixes the solid-body rotation null mode. If $B_s$ vanishes on an entire cylinder, the magnetic coupling coefficient degenerates and generic uniqueness must not be asserted. The simplified expression is conditional on the surface-torque assumption, rather than a general replacement for the boundary-compatible operator equation.

With small positive viscosity the momentum equation adds $\rho \nu {\nabla }^{2}u$. Its cylinder integral becomes the viscous regularization of the Taylor constraintMagnetic torque need no longer vanish exactly; it can be transmitted by viscous stress. Bulk viscosity is order $E=\nu /(2\Omega L^2)$ at fixed velocity. For ordinary no-slip rotating boundaries, thin Ekman layers have thickness of order $L\sqrt{E}$ and shear stress of order $\rho \Omega LU\sqrt{E}$, larger than the bulk order-$E$ correction. Boundary pumping and this friction help select the otherwise undetermined geostrophic flow. In the inviscid interior only impermeability is imposed; no-slip matching is supplied by the thin layer, so simply imposing all viscous boundary conditions on the zero-viscosity equations is incorrect.

For bounded velocities, the viscous torque tends to zero as $E\to 0$, recovering the leading Taylor condition. A prescribed order-one uncompensated magnetic torque would instead require a velocity growing like $E^{-1/2}$ under Ekman friction, or $E^{-1}$ under bulk friction alone. Such growth eventually invalidates the assumed small Rossby number: the limit is singular, not a regular guarantee that every imposed magnetic field has an inviscid solution. With stress-free boundaries the leading no-slip friction is absent, and bulk stresses, magnetic boundary conditions and angular-momentum normalization require their own treatment. Retaining inertia permits torque-driven torsional motion rather than an exact magnetostrophic constraint. Thus zero viscosity gives a magnetic compatibility condition and an induction-based selection of cylindrical rotation, while small viscosity replaces it by a torque balance whose bounded solutions approach compatible Taylor states.

Write $c_T^2=\mathcal{R}{T}_0$ for the squared isothermal sound speed, distinguishing the gas constant $\mathcal{R}$ from a Rayleigh number, and put $H=c_T^2/g$. A uniform magnetic field has no bulk Lorentz force, so hydrostatic equilibrium and the ideal gas equation giveAt the interface, it is magnetohydrodynamic total pressure, not gas pressure alone, that is continuous:Define ${\Delta }_B=B_1^2-B_2^2>0$. The magnetic pressure discontinuity in an isothermal atmosphere therefore producesIn particular, the upper gas is denser at the interface. Both halves have the same scale height $H$, despite their different pressure normalizations.

For the disturbance calculation, write ${\rho }_{-}=\rho$, ${\rho }_{+}=\rho +\delta \rho$, where $\delta \rho ={\Delta }_B/(2{\mu }_0{c}_T^2)$, and ${\Sigma }_{\rho }={\rho }_{-}+{\rho }_{+}$. Work to leading order in the prescribed short-vertical-scale approximation: the bulk motion is locally incompressible potential flow, while the isothermal equation determines the equilibrium density jump. Its scale separation and small compressive correction will be checked at the maximizing mode.

Let the interface displacement be $h{e}^{st+ikx+imy}$ and $K=\sqrt{k^2+m^2}>0$. Using displacement rather than velocity potentials, write ${\mathbf{\xi }}_i=\nabla {\chi }_i$ and ${\mathbf{u}}_i=s{\mathbf{\xi }}_i$. Incompressibility and irrotational flow give $\Delta {\chi }_i=0$. Decay away from the interface and continuity of its normal displacement fixwith the common horizontal and temporal exponential understood. Both satisfy ${\partial }_z{\chi }_i(0)=h$. Thus the disturbance has vertical extent of order $K^{-1}$.

In ideal magnetohydrodynamics, magnetic flux freezing gives ${\mathbf{b}}_i=({\mathbf{B}}_i\cdot \nabla ){\mathbf{\xi }}_i$ for these locally incompressible displacements. If ${\pi }_i=p_i^{\prime }+{\mathbf{B}}_i\cdot {\mathbf{b}}_i/{\mu }_0$ is the total-pressure perturbation, linear momentum isSince $({\mathbf{B}}_i\cdot \nabla {)}^2=-({\mathbf{k}}_h\cdot {\mathbf{B}}_i{)}^2$ on the mode, integration in each half-space yieldsAt the displaced interface, continuity of Lagrangian magnetohydrodynamic total pressure requires ${\pi }_1-{\rho }_{-}gh={\pi }_2-{\rho }_{+}gh$. Inserting the two potentials gives the orthogonal-field magnetic Rayleigh-Taylor dispersion relation:The first term is the destabilizing Rayleigh-Taylor instability of the heavier upper gas; the second is stabilizing magnetic tension. A negative right-hand side gives an oscillatory mode rather than growth.

For fixed $K$, the magnetic penalty isIt is minimized by $m=0$: the wavevector is along the weaker upper magnetic field and perpendicular to the stronger lower field. If $B_2\ne 0$, define $A=g\delta \rho /{\Sigma }_{\rho }$ and $C=B_2^2/({\mu }_0{\Sigma }_{\rho })$. The growth law for this orientation is $s^2=AK-C{K}^2$, a concave quadratic with maximum at $K_{\ast }=A/(2C)$. HenceFor this orientation the unstable interval is $0<K<2K_{\ast }$; the maximum lies halfway along it.

The localization condition for the fastest orthogonal-field interface mode follows directly from the vertical decay length:This is the condition for ${\ell }_{\ast }\ll H$ on both sides. It also controls the locally incompressible approximation, sinceAcoustic adjustment is then fast compared with growth, so bulk density perturbations are higher-order corrections. If this localization condition fails, the constant-background approximation cannot be used to identify the physical fastest mode.

Finally, the printed assumptions allow $B_2=0$. In that case $m=0$ eliminates magnetic tension on both sides and gives $s^2=g\delta \rho |k|/{\Sigma }_{\rho }$. There is no finite maximum growth rate or maximizing wavenumber in the ideal sharp-interface model: growth is unbounded as $|k|\to \infty$. The finite maximum and localization formula above therefore require $0<|B_2|<|B_1|$.

In the growth curves, $\theta$ is the angle of the horizontal wavevector from the $x$-axis. The illustrated ratio $B_1^2/B_2^2=101$ gives $K_{\ast }H=25$, and shows how even a small rotation toward the stronger magnetic field lowers the maximum growth.

A plasma is a gas of mobile charged particles whose collective electromagnetic response is important to its dynamics. It commonly contains ions and Electrons, and may also contain a neutral component: complete ionization is not required. A multi-species fluid description has current density $\mathbf{J}={\sum }_s{q}_s{n}_s{\mathbf{u}}_s$. The relative motion of the species can matter even when a single-fluid magnetohydrodynamics approximation describes the large-scale dynamics.