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Andrey Sheyko investigated rotating magnetohydrodynamics in a spherical shell using a spectral method. His thesis varied boundary conditions and control parameters in geodynamo models, supplied numerical benchmark solutions, and derived decay rates for magnetic fields in spherical geometries.^[ref]

Astrophysical fluid dynamics models gases and plasmas in stars, accretion discs, stellar winds and other astronomical flows. Its distinctive ingredients can include self-gravity, rotation, compressibility, radiative cooling and magnetic fields. ^[ref]

A useful scale comparison is the Mach number $\mathrm{Ma}=U/c_s$, with flow speed $U$ and sound speed $c_s$. Large compressions can produce shock waves; self-gravity can instead amplify density disturbances through Jeans instability. Conducting plasma adds magnetohydrodynamics. A continuum approximation must be justified: a very tenuous medium can require a kinetic description instead of a simple collisional fluid.

Hannes Alfvén predicted a wave coupling the motion of an electrically conducting fluid to a background magnetic field. In an Alfvén wave, bent magnetic-field lines provide a restoring effect while the material's inertia resists the displacement.

Linearizing ideal magnetohydrodynamics for a transverse disturbance along a uniform field gives the propagation speed $v_A=B_0/\sqrt{{\mu }_0\rho }$, with background field strength $B_0$, vacuum permeability ${\mu }_0$ and mass density $\rho$. A stronger field increases the restoring effect; greater material inertia slows propagation. The idea connects plasma dynamics in the laboratory with the behavior of astrophysical magnetic structures.^[ref]

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.

Awarded in equal shares to Hannes Olof Gösta Alfvén for work in magnetohydrodynamics and its applications in plasma physics, and Louis Eugène Félix Néel for discoveries concerning antiferromagnetism and ferrimagnetism and their applications in solid-state physics.^[ref]

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

Pascale Garaud is an applied mathematician and astrophysicist whose work connects fluid dynamics, turbulence and the evolution of stars. Her institutional biography identifies research on stellar flows, magnetohydrodynamics and double-diffusive convection, using both analytical and computational methods.^[ref]

With collaborators, she developed a model for mixing by fingering convection and tested it against three-dimensional direct numerical simulations. A stellar layer can resist ordinary overturning while still mixing because heat and composition diffuse at different rates. Predicting the resulting transport matters when a one-dimensional stellar calculation cannot resolve the individual turbulent motions.^[ref]

Her lecture notes explain how competing thermal and compositional density effects lead to double-diffusive convection. The connection is scientifically useful: a process familiar from salt water becomes relevant to material redistribution inside stars, with very different diffusivity ratios.^[ref]

Portrait: Simons Foundation speaker profile, identified portrait of Pascale Garaud.

Use the source's Gaussian units and neglect the displacement current, as in nonrelativistic magnetohydrodynamics. Write $\Delta p=p_{\perp }-p_{\parallel }$ for the pressure anisotropy. The pressure tensor is $\mathsf{P}=p_{\perp }\mathsf{I}-\Delta p\mathbf{bb}$, so its force density isAmpère's law and the solenoidal magnetic-field constraint give the Lorentz force decompositionAdding these two force densities and using $\mathbf{BB}=B^2\mathbf{bb}$ yieldsThe first term is the gradient of the modified magnetohydrodynamic total pressure; the second is the anisotropic Maxwell stress tensor contribution together with the directional pressure force. Their effective coefficient is the pressure-anisotropy-modified magnetic tension, $\mathcal{T}=B^2/(4\pi )+\Delta p$. At fixed magnetic field, positive pressure anisotropy increases resistance to bending, while negative pressure anisotropy reduces it. When $\mathcal{T}$ becomes negative, bending experiences an amplifying force rather than a restoring force. Gradients of $\Delta p$ can also drive motion, so the curvature interpretation refers specifically to the bending contribution.

A plasma contains mobile charged species whose collective electromagnetic response matters. On scales large compared with the Debye length, positive and negative charge densities nearly cancel: this is quasineutrality, not the absence of electric currents or electric fields. For electrons of temperature $T_e$ and number density $n_e$, the characteristic plasma screening and oscillation scales areA collective plasma description requires many particles within a screening volume; long-wavelength, sufficiently slow behavior can often be described by magnetohydrodynamics with an appropriate fluid closure.

An individual particle feels the Lorentz force $q(\mathbf{E}+\mathbf{v}\times \mathbf{B})$. In a uniform magnetic field, its transverse motion gyrates while its parallel motion remains free, giving a helical path when both components are present. The nonrelativistic cyclotron frequency and gyroradius areWhen the system scale is much larger than $r_L$ and the relevant time scale longer than ${\Omega }_c^{-1}$, the field strongly organizes transport across it, provided collisions do not interrupt the gyromotion. A perpendicular electric field also gives the common guiding-centre drift ${\mathbf{v}}_E=\mathbf{E}\times \mathbf{B}/B^2$.

For a conducting fluid, the macroscopic Lorentz force density has the useful decompositionThese terms are magnetic pressure and magnetic tension. Magnetic tension resists bending, while magnetic-pressure gradients resist nonuniform compression. The plasma beta $\beta =2{\mu }_{0}p/B^2$ compares thermal and magnetic stresses.

With uniform magnetic diffusivity ${\eta }_m$, the resistive induction equation isIf the magnetic Reynolds number $R_m=UL/{\eta }_m\gg 1$, ideal magnetohydrodynamics applies approximately and magnetic flux freezing links the field geometry to fluid motion. Finite diffusivity permits field slippage and magnetic reconnection. Field tension supports Alfvén waves with $v_A=B/\sqrt{{\mu }_0\rho }$; coupled magnetic and gas compression also supports magnetosonic waves. These effects allow a field to transmit forces and angular momentum between distant parts of an astrophysical plasma.

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.

Plasma turbulence is irregular, interacting multiscale motion of plasma particles and electromagnetic fields. At large scales a magnetohydrodynamics model can describe coupled velocity and magnetic fluctuations; near ion and electron scales their separate kinetic responses become important. A broadband spectrum alone is insufficient to establish how energy moves between scales.

A useful measurement combines spacecraft magnetometer data with plasma velocity and density measurements. Mixed third-order statistics of velocity and magnetic-field increments can test an energy-cascade relation. Ulysses observations found the predicted approximately linear scale dependence over an inertial range, providing evidence of transfer between scales rather than merely a noisy signal.^[ref]

For an idealized constant-density model, write the Elsässer fields $z^{\pm }=u\pm B/\sqrt{{\mu }_0\rho }$. Their nonlinear coupling shows why magnetic and velocity fluctuations should be measured together. Interpreting a spacecraft time lag as a spatial separation additionally requires a flow-advection approximation. At small scales, explaining plasma heating requires identifying how fluctuating field energy reaches particle distributions, beyond observing a spectral slope.