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An Alfvén wave is a magnetohydrodynamic wave in which magnetic tension restores a sideways disturbance of a conducting fluid. In the small-amplitude, uniform-background ideal magnetohydrodynamics model, its shear polarization of waves has displacement perpendicular to both the background magnetic field and the wavevector. It is an incompressible flow disturbance: the mass density and gas pressure have no first-order fluctuation. Its dispersion relation is ${\omega }^2=(\mathbf{k}\cdot {\mathbf{v}}_A{)}^2$, where ${\mathbf{v}}_A$ is the background Alfvén velocity.

The mechanism resembles a plucked taut string: magnetic flux freezing makes sideways fluid flow bend magnetic field lines, and their magnetic tension accelerates neighboring fluid. The fluid supplies inertia and the bent magnetic field supplies the restoring stress. This permits energy transport without a first-order compression, unlike an acoustic wave. Hannes Alfvén's 1942 paper, Existence of Electromagnetic-Hydrodynamic Waves, considered precisely a uniform axial magnetic field, infinite electrical conductivity and sideways velocity depending on axial position and time; it proposed a solar application. This was a theoretical prediction, followed by the distinct Lundquist liquid-metal Alfvén wave experiment.^[ref]

For oblique propagation, the shear Alfvén wave has velocity normal to the plane containing the wavevector and background magnetic field. The fast magnetosonic wave and slow magnetosonic wave instead have velocity in that plane and generally involve compression, with gas pressure and magnetic pressure contributing to their restoring forces. For exactly parallel propagation the plane is not unique: either transverse polarization of waves is allowed, and some modal speeds coincide. For $k_{\parallel }=0$ the shear branch has $\omega =0$, while the fast magnetosonic wave can still propagate perpendicular to the magnetic field. One should identify the polarization of waves as well as the phase velocity. The statement that the shear mode is incompressible concerns this linear mode; a finite linearly polarized disturbance can drive a quadratic compressive response, as in the magnetic-pressure obstruction to a linearly polarized Alfvén wave.

Flux expulsion is redistribution of magnetic flux out of the interiors of persistent circulating flows into their edges and surrounding regions. In two-dimensional incompressible flow, a magnetic flux function obeys ${\partial }_{t}A+\mathbf{u}\cdot \nabla A=\eta {\nabla }^{2}A$. At large but finite magnetic Reynolds number, circulation stretches gradients of $A$, enabling resistive smoothing and weak interior magnetic field with strong field near the circulation boundary. Finite magnetic diffusivity is essential to the eventual rearrangement; exact ideal magnetic flux freezing alone does not justify a flux-free steady cell. Magnetic back reaction can limit the process.

For an axisymmetric velocity $\mathbf{u}=(0,rf(r),g(r))$ independent of $z$, write $\mathbf{B}=B_z\hat{\mathbf{z}}+\nabla \times (A\hat{\mathbf{z}})$. Ideal induction becomesThe first equation expresses magnetic flux freezing of the transverse flux function, while the second is axial stretching $B_r{g}^{\prime }$. Starting from a uniform field $B_0\hat{\mathbf{x}}$ gives $A(r,\varphi ,t)=B_{0}r\sin (\varphi -f(r)t)$ and $B_z(r,\varphi ,t)=B_{0}t{g}^{\prime }(r)\cos (\varphi -f(r)t)$. In complex azimuthal-mode notation, $\hat{A}=B_{0}r{e}^{-ift}$ and ${\hat{B}}_z=i{B}_{0}t{g}^{\prime }{e}^{-ift}$, obtained by elementary integrating factors. The transverse field has $B_r=r^{-1}{A}_{\varphi }$ and $B_{\varphi }=-A_r$, preserving zero divergence.

Ideal magnetohydrodynamics describes a conducting fluid whose electromagnetic evolution neglects electrical resistivity. Its defining electric-field relation isso the electric field vanishes in the local nonrelativistic fluid rest frame. Together with Faraday's law this gives the ideal magnetohydrodynamic induction equationThe Lorentz force density couples the field back to fluid momentum. ^[ref]

For sufficiently smooth fields, the induction equation implies magnetic flux freezing: flux through a material surface moving with the fluid is conserved. Stretching the fluid can therefore amplify the magnetic field while preserving its connectivity. Magnetic tension supports Alfvén waves, providing a direct dynamical consequence of the coupling.

A large magnetic Reynolds number $\mathrm{Rm}=UL/\eta$ motivates neglecting diffusion on the chosen scales. Thin current sheets can still make nonideal terms important locally. The ideal induction condition alone does not set fluid viscosity; mechanical dissipation and magnetic diffusion are different approximations.

A horizontal flux tube displaced without bending preserves $B/\rho$ by magnetic flux freezing. If thermal exchange keeps its temperature equal to its new surroundings, lateral magnetic pressure balance gives $(c_T^2+v_A^2)({\rho }_i^{\prime }-{\rho }^{\prime })=\rho B(B/\rho {)}^{\prime }/{\mu }_0$. Here $c_T^2=p/\rho$, $v_A^2=B^2/({\mu }_0\rho )$, and primes denote the response to a small vertical displacement from an equilibrium. Its linear buoyancy acceleration is $-gB(B/\rho {)}^{\prime }\xi /[{\mu }_0(c_T^2+v_A^2)]$. Thus local interchange is unstable when $B(B/\rho {)}^{\prime }<0$, equivalently $(B^2/{\rho }^2{)}^{\prime }<0$. This fast-thermal-exchange criterion differs from the adiabatic localized interchange criterion for a magnetized atmosphere.

A magnetized long-wavelength plasma description with a common leading perpendicular flow, magnetic flux freezing and a gyrotropic pressure tensor. The kinetic magnetohydrodynamic momentum equation couples to a drift-kinetic equation for each species; parallel plasma pressure and perpendicular plasma pressure are calculated as velocity moments, rather than prescribed by an isotropic adiabatic closure. Relative species drifts enter at higher order in the ratio of gyroradius to macroscopic length.

For a static equilibrium and a fluid Lagrangian displacement $\xi$, conservation of mass, adiabatic entropy and magnetic flux freezing give $\delta p=-\xi \cdot \nabla p-\gamma p\nabla \cdot \xi$ together with the displayed relations. Linearizing the pressure/tension form of the momentum equation yields $\rho \ddot{\xi }=-\nabla \delta \Pi -g\delta \rho e_z+{\mu }_0^{-1}(\delta B\cdot \nabla B+B\cdot \nabla \delta B)$, where $\delta \Pi =\delta p+B\cdot \delta B/{\mu }_0$. Static force balance removes the zeroth-order terms; the density multiplying acceleration is the equilibrium density.

Magnetic flux freezing states that every material loop in an ideal conducting fluid encloses constant magnetic flux. A deformation perpendicular to a uniform field that preserves mass per unit length therefore preserves $B/\rho$.

Replacing the sound speed in a Jeans mass estimate by the Alfvén speed $B/\sqrt{{\mu }_0\rho }$ gives the displayed magnetic-support mass scale. A sphere whose diameter is the corresponding wavelength gives $C={\pi }^{5/2}/6$; the crossing-time cloud support mass instead gives $C=(4\pi /3)(3\pi /32{)}^{3/2}$. The coefficient is model dependent, and a real magnetic field supplies anisotropic stresses rather than isotropic gas pressure.

During isotropic magnetic flux freezing, $B\propto {\rho }^{2/3}$, so $M_B$ and the number of such masses in a fixed material cloud remain constant. This is another expression of a conserved mass-to-flux ratio. It contrasts with the decreasing thermal Jeans mass of isothermal fragmentation, and does not rule out motion or fragmentation along field lines.

Magnetic reconnection changes connectivity between magnetic field lines. It requires departure from smooth magnetic flux freezing, for example through finite resistivity or other nonideal terms concentrated in a current sheet. Ideal viscous magnetic relaxation dissipates mechanical energy while retaining frozen connectivity, so viscosity alone does not provide reconnection.

Magnetic relaxation converts magnetic energy into material motion through the Lorentz force density, and dissipates the resulting kinetic energy through viscosity. In a perfect conductor, magnetic flux freezing preserves field-line connectivity throughout every smooth stage. A regular limiting state has zero velocity and is a magnetostatic equilibrium, but its magnetic field can contain current sheets. Conservation of one global magnetic helicity is weaker than preservation of the complete frozen field-line structure.

For an axisymmetric poloidal magnetic field ${\mathbf{B}}_p=\nabla \Psi \times \nabla \varphi$, the ideal magnetohydrodynamic induction equation gives ${\partial }_t\Psi +{\mathbf{u}}_p\cdot \nabla \Psi =0$, after fixing the time-dependent additive constant in $\Psi$. A regular axial reference makes $2\pi \Psi$ the enclosed magnetic flux. This scalar statement of magnetic flux freezing holds even when rotation also produces a toroidal magnetic field.

Combine the material surface element evolution law with the product rule for $\mathbf{B}\cdot d\mathbf{S}$. The result isExpanding the curl gives the equivalent title formula. The ideal magnetohydrodynamic induction equation and Gauss's law for magnetism make the integrand zero and prove magnetic flux freezing, without requiring incompressible flow.

A material surface is transported with its fluid elements, including its boundary when open. Its oriented material surface element evolves by tangent transport. Magnetic flux freezing conserves the flux through such an open surface.

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

Magnetic flux freezing ties a transverse fluid displacement to a bend in the magnetic field lines. For the uniform background, the transverse displacement equation isFor a sinusoidal bend, ${\partial }_z^2{\mathbf{\xi }}_{\perp }=-k_{\parallel }^2{\mathbf{\xi }}_{\perp }$. Positive pressure-anisotropy-modified magnetic tension therefore gives a restoring force, and inertia produces oscillation. In an isotropic plasma, this is precisely the positive magnetic tension $B_0^2/(4\pi )$.

Particles moving predominantly along bent magnetic field lines contribute a transverse momentum flux. Their larger parallel pressure opposes the restoring curvature force supplied by the magnetic field and perpendicular pressure. Once $p_{\parallel 0}-p_{\perp 0}>B_0^2/(4\pi )$, the net force points along the original displacement: a larger bend produces a larger outward force, giving positive feedback. Equivalently, the bending contribution to the quadratic potential energy is $\frac{1}{2}{\mathcal{T}}_0|{\partial }_z{\mathbf{\xi }}_{\perp }{|}^2$ and has become negative.

In this static, homogeneous fluid model the unstable roots areWith $e^{-i\omega t}$ the root $+i\gamma$ grows. Its real angular frequency is zero, so the unstable pattern grows without propagation in the equilibrium rest frame; its partner decays. Advection by a background flow would change that frame-dependent statement. The growth increasing without bound as $|k_{\parallel }|$ increases is a limitation of this long-wavelength model: finite gyroradius and other kinetic corrections cannot be neglected at sufficiently short scales.

Write the physical magnetic field as ${\mathbf{B}}_{\mathrm{phys}}$ and use the constant mass density ${\rho }_0$ to introduce Alfvén velocity units, $\mathbf{B}={\mathbf{B}}_{\mathrm{phys}}/\sqrt{{\mu }_0{\rho }_0}$. If $P$ is the fluid's mechanical pressure, the magnetohydrodynamic total pressure per unit mass density is $p=P/{\rho }_0+B^2/2$. The ideal magnetohydrodynamic equations then reduce toThe first equation is momentum conservation: the magnetic pressure has been included in $p$, leaving magnetic tension explicitly. The second is the ideal magnetohydrodynamic induction equation, expressing magnetic flux freezing. The last two equations impose incompressibility and the solenoidal magnetic-field constraint.

The model assumes a smooth conducting-fluid continuum with constant mass density and constant magnetic permeability, nonrelativistic velocity, and scales on which a single-fluid description is appropriate. Negligible electrical resistivity removes magnetic diffusion; negligible viscosity removes viscous momentum diffusion. Equivalently, the magnetic Reynolds number and the Reynolds number are large on the scales being retained. The low-frequency approximation neglects the displacement current; Hall and other kinetic corrections to Ohm's law are absent. Incompressibility is an imposed approximation, commonly justified when the Mach number is small. There is no independently evolved thermodynamic pressure: $p$ enforces the incompressibility condition. We omit external body forces; a conservative one could instead be absorbed into $p$.

Linearized magnetic flux freezing gives the Eulerian magnetic field perturbationThe background is uniform, and incompressibility gives $\nabla \cdot \mathbf{\xi }=0$. For a horizontal displacement proportional to $e^{ikz}$, $\delta \mathbf{B}=ik{B}_0\mathbf{\xi }$ and $\delta B_z=0$. Thus the first-order magnetic pressure perturbation vanishes, and the magnetic-tension response to an incompressible displacement isBending a field line produces a restoring force proportional to displacement, like a spring obeying Hooke's law. Per unit mass, its squared spring frequency is $k^2{v}_A^2$, where $v_A=B_0/\sqrt{{\mu }_0\rho }$ is the Alfvén speed. The equilibrium can be independent of $z$ while the perturbation varies with $z$; neglect of background vertical structure does not set this perturbation's wavenumber to zero.

An Alfvén wave is a transverse disturbance restored by magnetic tension. In a uniform conducting fluid with background ${\mathbf{B}}_0=B_0{\mathbf{e}}_z$, a small transverse displacement ${\mathbf{\xi }}_{\perp }(z,t)$ bends the magnetic field lines. Magnetic flux freezing gives ${\mathbf{b}}_{\perp }=B_0{\partial }_z{\mathbf{\xi }}_{\perp }$, and the linear transverse Lorentz force density is $(B_0/{\mu }_0){\partial }_z{\mathbf{b}}_{\perp }$. Thus the Linear Alfvén wave equations reduce toFor this shear polarization, $\delta \rho =\delta p=0$ at linear order. The disturbance transports energy along the background magnetic field; a single traveling wave has equal perturbation magnetic energy and kinetic energy, with ${\mathbf{b}}_{\perp }=\mp \sqrt{{\mu }_0\rho }{\mathbf{u}}_{\perp }$ for propagation along $\pm {\mathbf{B}}_0$. These properties distinguish it from compressive magnetosonic waves.

The ideal magnetohydrodynamics approximation requires strong electrical coupling and wavelengths large compared with microscopic scales. Finite electrical resistivity and viscosity dissipate the wave's energy. In stellar winds and other astrophysical plasmas, Alfvén waves carry disturbances and can contribute to heating when dissipated. The Lundquist liquid-metal Alfvén wave experiment provides a laboratory counterpart: a conducting liquid in a strong applied magnetic field transmits a transverse magnetic disturbance, making the field's mechanical restoring action measurable.

For small isentropic perturbations of a fixed-composition gas, linearizing its equation of state gives $p_1=(\partial p/\partial \rho {)}_s{\rho }_1=c_s^2{\rho }_1$. The adiabatic sound speed is evaluated at the uniform background, so it is spatially constant at this order. If fast heat exchange instead enforces constant temperature, use the isothermal sound speed consistently. A completely arbitrary entropy perturbation would require an additional thermodynamic variable.

The uniform static infinite medium requires the Jeans swindle: its background gravitational field is subtracted or externally balanced, and only the perturbation sources the Poisson equation for Newtonian gravity. Linearizing the continuity equation, Euler equation and gravitational equation then givesTaking one more time derivative of the first equation and substituting the other two yieldsA Fourier mode proportional to $e^{i(\mathbf{k}\cdot \mathbf{x}-\omega t)}$ therefore obeys the Jeans instability relationFor $k>k_J$, the mode oscillates as a sound wave. For $0<k<k_J$, $\omega =\pm i\sqrt{4\pi G{\rho }_0-c_s^2{k}^2}$ gives a growing and a decaying mode; the static Jeans growth-rate spectrum has no preferred finite fastest-growing wavelength in the infinite-medium approximation. The threshold $k=k_J$ is marginal at linear order. The Jeans length is ${\lambda }_J=2\pi /k_J=c_s\sqrt{\pi /(G{\rho }_0)}$. Defining the Jeans mass as a sphere whose diameter is this wavelength givesOther radius conventions alter its numerical coefficient; the mass scale compares self-gravity with pressure communication.

For the pressureless finite cloud, the shell theorem gives an edge acceleration $\ddot{r}=-GM/r^2$, with fixed enclosed $M=4\pi r_0^3{\rho }_0/3$. Multiplication by $\dot{r}$ and integration from rest givesThe negative root describes collapse. Substituting $r=r_0{\sin }^2\theta$ gives the free-fall time of a uniform sphereThe surface and inner shells reach the origin together in pressureless homologous spherical collapse; pressure, rotation or shell crossing would change this ideal solution.

Equating this time to the specified radius-based sound-crossing time $r_0/c_s$ gives $r_0=c_s\sqrt{3\pi /(32G{\rho }_0)}$. The resulting crossing-time cloud support mass isThis is the same dimensional mass scale as $M_J$, with a different coefficient because the adopted length and time criteria differ. In particular, $t_{\mathrm{ff}}=r_0/c_s$ is not the exact marginal dispersion-relation condition.

With the requested isotropic support approximation, replace $c_s$ by the Alfvén speed $v_A=B/\sqrt{{\mu }_0{\rho }_0}$. In the crossing-time convention the Magnetic Jeans mass isUsing the wavelength-based Jeans mass convention instead replaces $C_{\mathrm{cross}}$ by ${\pi }^{5/2}/6$. The scaling, rather than a convention-independent coefficient, is the shared prediction.

For a fixed cloud mass undergoing isothermal homologous contraction, $\rho /{\rho }_0=(r_0/r{)}^3$ and $c_s$ is constant. Its number of thermal Jeans masses therefore grows asA cloud initially close to the support threshold has $N_{J,0}$ of order one, so smaller pieces can subsequently become independently unstable: this is isothermal fragmentation.

For approximately isotropic contraction with magnetic flux freezing, $B{r}^2$ stays constant, so $B\propto {\rho }^{2/3}$. Hence $M_B\propto B^3/{\rho }^2$ stays constant andThis ideal magnetic support therefore does not create progressively more unstable masses in the way thermal isothermal support does. The same conclusion follows from the conserved mass-to-flux ratio. A magnetically subcritical cloud cannot initiate unrestricted cross-field collapse under ideal flux freezing alone; flux loss or initially supercritical conditions are needed. A real magnetic field also gives anisotropic support: motion and fragmentation along its field lines are not excluded by this scalar-speed estimate.