Alfvén wave

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.

Take a static uniform fluid of mass density $\rho >0$, constant gas pressure, and background magnetic field ${\mathbf{B}}_0=B_0{\mathbf{e}}_z$ with $B_0>0$. Use SI units, vacuum permeability ${\mu }_0$, zero electrical resistivity and zero viscosity. Write $\mathbf{B}={\mathbf{B}}_0+{\mathbf{b}}_{\perp }(z,t)$ and $\mathbf{u}={\mathbf{u}}_{\perp }(z,t)$, with both disturbances perpendicular to ${\mathbf{e}}_z$. Retain only linear order in their amplitudes. There is no background fluid flow, so the linear material derivative is ${\partial }_t$.

The continuity equation has $\nabla \cdot {\mathbf{u}}_{\perp }=0$, hence ${\partial }_t\delta \rho =0$; choosing no initial mass density fluctuation gives $\delta \rho =0$. For this shear mode the gas pressure fluctuation is also zero. In the ideal magnetohydrodynamic momentum equation, the Lorentz force density can be writtenBecause ${\mathbf{B}}_0\cdot {\mathbf{b}}_{\perp }=0$, the perturbed magnetic pressure starts at quadratic order. The surviving transverse magnetic tension gives the first equation below. Expanding the ideal magnetohydrodynamic induction equation gives the second:Differentiate the first in time and substitute the second. Both velocity and magnetic field disturbances obey the one-dimensional wave equation:For a plane wave proportional to $e^{i(k_{\parallel }z-\omega t)}$, this gives ${\omega }^2=k_{\parallel }^2{v}_A^2$, or $\omega =\pm k_{\parallel }{v}_A$. The magnetic and velocity amplitudes satisfy ${\mathbf{b}}_{\perp }=-(B_0{k}_{\parallel }/\omega ){\mathbf{u}}_{\perp }$: they have opposite signs for propagation along $+{\mathbf{e}}_z$ and equal signs for propagation along $-{\mathbf{e}}_z$. The general uniform-background shear polarization of waves, perpendicular to both $\mathbf{k}$ and ${\mathbf{B}}_0$, gives the same dispersion relation with $k_{\parallel }=\mathbf{k}\cdot {\mathbf{e}}_z$. Its group velocity is $\pm v_A{\mathbf{e}}_z$, whereas its phase velocity normal to an oblique wavefront has magnitude $v_A|k_{\parallel }|/|\mathbf{k}|$.

Dot the two Linear Alfvén wave equations with ${\mathbf{u}}_{\perp }$ and ${\mathbf{b}}_{\perp }/{\mu }_0$, respectively, and add. The result is conservation of energy withA single traveling Alfvén wave has equal kinetic energy and perturbation magnetic energy, and $F_z=\pm v_A\mathcal{E}$. This equality need not hold instantaneously at each point of a superposed standing wave.

As an original ideal-model example, stipulate $B_0=1.00\mathrm{T}$, $\rho =10^4\mathrm{kg}{\mathrm{m}}^{-3}$ and a uniform path $L=0.20\mathrm{m}$. Using ${\mu }_0\simeq 4\pi \times 10^{-7}\mathrm{H}{\mathrm{m}}^{-1}$, the Alfvén speed is $8.92\mathrm{m}{\mathrm{s}}^{-1}$ and the transit time $L/v_A$ is $22.4\mathrm{ms}$. Doubling the magnetic field halves that time. A sinusoidal velocity with peak amplitude $0.010\mathrm{m}{\mathrm{s}}^{-1}$ has magnetic peak amplitude $1.12\mathrm{mT}$ and mean energy flux $⟨F_z⟩=\rho v_A{U}^2/2=4.46\mathrm{W}{\mathrm{m}}^{-2}$ in the positive direction. These are stipulated ideal parameters, not historical measurements or a prediction including vessel walls.

Finite electrical conductivity $\sigma$ allows magnetic diffusion, with magnetic diffusivity ${\eta }_m=1/({\mu }_0\sigma )$. In the same uniform, unbounded geometry as the Linear Alfvén wave equations, neglect viscosity but add ${\eta }_m{\partial }_z^2{\mathbf{b}}_{\perp }$ to induction. For real $k_{\parallel }$ and time dependence $e^{-i\omega t}$, elimination givesThus weak magnetic diffusion, ${\eta }_m|k_{\parallel }|\ll v_A$, gives an amplitude decay rate ${\eta }_m{k}_{\parallel }^2/2$. For ${\eta }_m|k_{\parallel }|>2v_A$, both temporal roots are purely decaying; they no longer describe an oscillating traveling mode. This temporal calculation at fixed real wavenumber is different from a driven vessel at fixed real frequency, where a complex spatial wavenumber describes penetration and attenuation. Viscosity also dissipates kinetic energy, and walls introduce transverse gradients and additional losses.

For an axial length $L$, comparing the magnetic diffusion time $L^2/{\eta }_m$ with the transit time $L/v_A$ gives the ratio $L{v}_A/{\eta }_m$. A large value favors propagation before diffusion, but a narrow vessel can introduce a shorter transverse diffusion scale. In the actual 1949 analysis, finite electrical conductivity and vessel dimensions restrict the useful frequency interval; above its upper limit the response becomes a skin wave.^[ref]

For fixed real positive angular frequency $\omega$, a transverse Alfvén wave proportional to $e^{i(kz-\omega t)}$ in a uniform fluid has $k^2={\omega }^2/(v_A^2-i{\eta }_m\omega )$ when magnetic diffusivity ${\eta }_m$ is retained and viscosity is neglected. Choose the branch with positive real and imaginary parts for propagation and attenuation along $+z$. For $ϵ\ll 1$, $k=(\omega /v_A)(1+iϵ/2+O({ϵ}^2))$. The amplitude attenuation length is $L_A=1/\mathrm{Im}k$, whereas the energy attenuation length is $L_E=1/(2\mathrm{Im}k)$. With wavelength $\lambda =2\pi /\mathrm{Re}k$, $L_A/\lambda \sim 1/(\pi ϵ)$ and $L_E/\lambda \sim 1/(2\pi ϵ)$. Fixed-real-wavenumber temporal Resistive damping of an Alfvén wave is a different boundary-value prescription.

Stig Lundquist, the author printed as S. Lundquist, reported an early laboratory test in 1949. The apparatus used mercury in a stainless-steel cylinder, a vibrating disk, and a floating mirror to indicate the motion at the liquid surface. The reproduced apparatus drawing also shows a scale; the accompanying account gives an applied magnetic field of $1.3\mathrm{T}$.^[ref]

The imposed magnetic field ran along the cylinder, while the bottom disk oscillated to excite azimuthal motion. Varying the drive frequency sought a standing wave resonance. A later authored experimental account describes this as an early hint: the resonance peak was not reached, and magnetic damping was substantial.^[ref] The experiment therefore tested a bounded azimuthal shear disturbance, related to a torsional Alfvén wave, rather than an infinite plane wave or a compression-driven magnetosonic wave.

The magnetic field provides the controlled physical parameter: the ideal Alfvén wave prediction makes a propagation speed proportional to its strength, while a resonance also depends on the vessel's boundary conditions. A quantitative test must compare the driven response with finite-electrical conductivity theory and account for mechanical coupling and wall losses. An observed surface motion alone cannot distinguish all these mechanisms. The reproduced apparatus account gives neither a numerical drive frequency nor a field-off comparison. Lundquist's paper itself calls the experiment preliminary and proposes further investigations.^[ref]

For an incompressible shear flow $U(z){\mathbf{e}}_x$ with uniform magnetic field $B{\mathbf{e}}_x$, define $S=\sigma +k_{x}U$ for time dependence $e^{i\sigma t}$ and $\zeta =i{k}_{x}v-i{k}_{y}u$. Eliminating the magnetic perturbations from the horizontal momentum equations gives the displayed product for $S\ne 0$. Away from the Alfvén wave resonance, it yields $\zeta =k_y{U}^{\prime }w/S$. At resonance one must retain the original system: for constant $U$, $k_y=0$, $S^2=k_x^2{B}^2/\rho$, any smooth transverse amplitude $v(z)$ with $b_y=k_{x}Bv/S$ and all other amplitudes zero is an Alfvén wave. Its nonzero vorticity disproves the unrestricted zero-vorticity reduction. Exponentially growing modes with real background coefficients cannot encounter these real-frequency resonances.

A kinetic Alfvén wave is a dispersive extension of the Alfvén wave in a plasma when perpendicular kinetic or two-fluid effects, including a pressure response, matter. Its particular dispersion relation depends on the ordering and plasma parameters. A similar product of parallel and perpendicular wavenumbers can occur in a whistler wave, but agreement of a scaling power does not identify the underlying physical equations.

Neighboring magnetic surfaces with different Alfvén speeds carry the same axial waveform at different rates. For a traveling profile $f(R,z+v(R)t)$, differentiation produces the displayed term, so transverse gradients grow even without axial wave steepening. Small magnetic diffusivity or viscosity can eventually act strongly on these gradients. The ideal process redistributes phase rather than directly dissipating wave energy.

A torsional Alfvén wave couples azimuthal velocity to a toroidal magnetic field around a fixed poloidal magnetic field. Axisymmetric induction gives ${\partial }_t{B}_{\varphi }=R{\mathbf{B}}_p\cdot \nabla \Omega$ and magnetic tension gives $\rho R^2{\mu }_0{\partial }_t\Omega ={\mathbf{B}}_p\cdot \nabla (R{B}_{\varphi })$. Their combination is the displayed wave equation. Its local dispersion relation is ${\omega }^2=(\mathbf{k}\cdot {\mathbf{B}}_p{)}^2/({\mu }_0\rho )$.

For uniform-density incompressible flow with $u_{\varphi }=f(R,z+vt)+g(R,z-vt)$, $B_{\varphi }/\sqrt{{\mu }_0\rho }=f-g$ and $B_z/\sqrt{{\mu }_0\rho }=v(R)$, the azimuthal magnetohydrodynamic momentum equation and ideal magnetohydrodynamic induction equation hold automatically. Let $\Pi =p/\rho +\Phi +[(f-g{)}^2+v^2]/2$. The remaining equations require ${\Pi }_z=0$ and ${\Pi }_R=4fg/R$, so ${\partial }_z(fg)=0$ is necessary and locally sufficient. A single traveling profile is unrestricted; two nonzero profiles on an unrestricted axial domain must be opposite exponentials or independent of their traveling arguments. Bounded profiles of both signs therefore reduce to steady solutions.

For a rotating conducting sphere, a geostrophic azimuthal velocity $U=rV$ couples to the cylinder-integrated magnetic stress $T=\int B_r{B}_{\varphi }dz$. Put $H=\int B_r^{2}dz$ and let $2z_b$ be the cylinder height. With an insulating boundary and no flow through it, cylindrical averaging removes the Coriolis force and gives $U_t=(2z_b{\mu }_0\rho r^2{)}^{-1}{\partial }_r(r^{2}T)$, $T_t=Hr{\partial }_r(U/r)$. The local squared wave speed is $H/(2z_b{\mu }_0\rho )$, the cylinder average of $B_r^2/({\mu }_0\rho )$.

For a cylinder-averaged torsional Alfvén wave with time-independent $H>0$, differentiating this quadratic form gives $dE/dt=[rUT{]}_0^a$. Regularity at the axis and vanishing magnetic torque at the outer endpoint therefore imply conservation. By Cauchy-Schwarz inequality, $T^2/H\le \int B_{\varphi }^{2}dz$; equality holds for the torsional-wave magnetic component proportional to $B_r$ along each cylinder. A stationary magnetic component orthogonal to $B_r$ is invisible to $T$.

The magnetic tension stress transports axial angular momentum along a poloidal magnetic field. In purely azimuthal ideal flow the poloidal energy flux is ${\mathbf{F}}_{E,p}=\Omega {\mathbf{F}}_{J,p}$. This proportionality is valid even at vanishing flux, where an ordinary ratio is undefined. The coupled fluxes explain how a torsional Alfvén wave redistributes rotation.

A finite-amplitude ideal-MHD Alfvén wave can rotate the transverse magnetic field at constant magnitude, with constant mass density, pressure and longitudinal components. Its transverse velocity obeys ${\mathbf{u}}_{\perp }=\mathbf{C}-s{\mathbf{B}}_{\perp }/\sqrt{{\mu }_0\rho }$ and its speed is constant along the wave. The constant magnetic pressure prevents compressive acceleration.

A finite sinusoidal transverse field $B_x=a{B}_0\cos [k(z-v_{a}t)]$, $B_y=0$, has varying magnetic pressure. In a purely transverse traveling Alfvén wave, longitudinal momentum requires ${\partial }_z[p+B_x^2/(2{\mu }_0)]=0$, but the adiabatic pressure equation with $u_z=0$ requires ${\partial }_{t}p=0$. A traveling compensating gas pressure cannot satisfy both conditions. The uncompensated force is quadratic in amplitude and drives a compressive response. A regular periodic one-dimensional traveling solution cannot evade this by varying mass density: its mass, transverse momentum and induction integrals force constant mass density at the zeros of the transverse field and hence throughout the nontrivial wave.

In homogeneous ideal magnetohydrodynamics, a constant longitudinal field $B_0>0$ and transverse field $a{B}_0(\cos \zeta ,\sin \zeta )$, $\zeta =k(z-v_{a}t)$, give an exact traveling wave with constant mass density and pressure, $u_z=0$, $v_a=B_0/\sqrt{{\mu }_0{\rho }_0}$ and the displayed transverse velocity. Induction and transverse momentum reduce to the same wave relation. Since $|{\mathbf{B}}_{\perp }{|}^2=a^2{B}_0^2$ is constant, there is no longitudinal magnetic pressure gradient. All nonlinear terms are compatible with the constant-density solution, even in a compressible fluid.

For propagation along $x$, the Alfvén right eigenvector has $\delta \rho =\delta p=\delta u_x=\delta B_x=0$, ${\mathbf{B}}_{\perp }\cdot \delta {\mathbf{u}}_{\perp }=0$ and $\delta {\mathbf{B}}_{\perp }=-s\sqrt{{\mu }_0\rho }\delta {\mathbf{u}}_{\perp }$. Its speed is $u_x+s{B}_x/\sqrt{{\mu }_0\rho }$, $s=\pm 1$. When the transverse background field vanishes, both transverse polarizations are allowed.

The Alfvén frequency is the frequency of an Alfvén wave with wavevector $\mathbf{k}$ in a uniform magnetic field $\mathbf{B}$ and mass density $\rho$. For a vertical field and vertical wavenumber $k$, its square is ${\omega }_a^2=k^2{B}_z^2/({\mu }_0\rho )$. Its competition with orbital shear determines the ideal magnetorotational dispersion relation.

An Alfvén wing is a stationary Alfvénic disturbance attached to a conducting obstacle moving through a magnetized plasma. In a super-Alfvénic flow its boundaries follow downstream characteristics.

For a stationary cold-plasma disturbance with obstacle speed $u_0>u_A$, the characteristic slope is $m=\sqrt{u_0^2/u_A^2-1}$. It approaches $u_0/u_A$ in the strongly super-Alfvénic limit.