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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.

Alfvénic turbulence is the turbulent interaction of fluctuations polarized primarily as Alfvén waves. In Elsässer variables ${\mathbf{z}}^{\pm }={\mathbf{u}}_{\perp }\pm \delta {\mathbf{B}}_{\perp }/\sqrt{4\pi {\rho }_0}$, each wave population is nonlinearly distorted by the oppositely propagating population. Equal populations give a balanced cascade; unequal populations give an imbalanced cascade. The absence of one population removes the leading nonlinear interaction in ideal reduced magnetohydrodynamics.

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.

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.

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.

For an incompressible flow with constant mass density, the Elsässer variables are ${\mathbf{z}}^{\pm }=\mathbf{u}\pm {\mathbf{v}}_a$, where ${\mathbf{v}}_a$ is the Alfvén velocity. Adding and subtracting the Euler momentum equation and ideal magnetohydrodynamic induction equation yields ${\partial }_t{\mathbf{z}}^{\pm }+({\mathbf{z}}^{\mp }\cdot \nabla ){\mathbf{z}}^{\pm }=-\nabla \psi$, with $\psi =p/\rho +\Phi +|{\mathbf{v}}_a{|}^2/2$ and $\nabla \cdot {\mathbf{z}}^{\pm }=0$. Thus each field is transported by the other, a useful way to expose interactions of oppositely directed Alfvén waves.

Ermanno Pietropaolo is a physicist whose research at the University of L'Aquila links observations of the Sun to measurements of the solar wind and the environment around Earth. His institutional research profile describes studies of dynamics in the photosphere and chromosphere, Alfvén wave fluctuations at different distances from the Sun, and instrumentation for ground-based observations relevant to space weather.^[ref]

He coauthored the 2007 study that tested an energy cascade using solar wind measurements from the Ulysses spacecraft.^[ref] The scientific connection is between a physical flow and a measurement procedure: instruments record velocity and magnetic field, and statistical analysis asks whether changes at different separations agree with magnetohydrodynamic turbulence. His profile's emphasis on apparatus and observing conditions is relevant because a change in a measured fluctuation must be distinguished from a change introduced by the observing system itself.

Parallel particle pressure can overcome magnetic tension. In the long-wavelength gyrotropic plasma limit, the Alfvén wave has ${\omega }^2=k_{\parallel }^2[B^2/(4\pi )+p_{\perp }-p_{\parallel }]/\rho$, so the condition in the title makes it unstable. The detailed kinetic threshold at shorter scales requires additional physics.

Goldreich–Sridhar turbulence combines critical balance, balanced counterpropagating Alfvén waves, a local perpendicular cascade, and constant energy flux per unit mass $\mathcal{E}$. With ${\tau }_{\mathrm{nl}}\sim {\ell }_{\perp }/\delta u_{\ell }$ and $\mathcal{E}\sim \delta u_{\ell }^2/{\tau }_{\mathrm{nl}}$, one obtains $\delta u_{\ell }\sim (\mathcal{E}{\ell }_{\perp }{)}^{1/3}$. Defining the one-dimensional perpendicular turbulent energy spectrum by $\delta u_{\ell }^2\sim k_{\perp }E(k_{\perp })$ gives $E(k_{\perp })\sim {\mathcal{E}}^{2/3}{k}_{\perp }^{-5/3}$. Critical balance also gives $k_{\parallel }\sim {\mathcal{E}}^{1/3}{k}_{\perp }^{2/3}/v_A$. This scaling argument neglects scale-dependent alignment and intermittency corrections, which are additional physical effects rather than consequences of the stated assumptions.

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]

For sound speed $v_s$ much larger than Alfvén speed $v_a$, and angle $\theta$ between the wavevector and magnetic field, the fast magnetosonic wave has $v_f^2=v_s^2+v_a^2{\sin }^2\theta +O(v_a^4/v_s^2)$. The slow magnetosonic wave has $v_{\mathrm{slow}}^2=v_a^2{\cos }^2\theta [1-(v_a^2/v_s^2){\sin }^2\theta +O(v_a^4/v_s^4)]$. The fast mode is predominantly longitudinal sound. The slow mode becomes almost incompressible, polarized in the plane of the wavevector and magnetic field; the Alfvén wave is polarized perpendicular to that plane. The latter two are predominantly magnetic-tension waves with equal leading phase speeds.

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.

For a uniform guide field, write the transverse perturbation in velocity units as $\mathbf{b}$. Linear momentum gives $\omega \mathbf{u}=-k_{\parallel }{v}_A\mathbf{b}$. Linear induction gives $\omega \mathbf{b}=-k_{\parallel }{v}_A\mathbf{u}+i{k}_{\parallel }{v}_A{d}_i\mathbf{k}\times \mathbf{b}$. The helicity decomposition of a transverse Fourier mode then yields the displayed dispersion relation. Its two positive frequency magnitudes are $|k_{\parallel }|v_A[\sqrt{1+(k{d}_i{)}^2/4}\pm k{d}_i/2]$, approaching the Alfvén wave frequency for $k{d}_i\ll 1$. For $k{d}_i\gg 1$, the fast branch is a whistler wave and the slow branch tends to $|k_{\parallel }|v_A/(k{d}_i)$.

For one-dimensional flow obeying an isothermal equation of state with $u_z=B_z=0$, the primitive variables $(\rho ,u_x,u_y,B_y)$ form a four-variable hyperbolic system and $B_x$ is constant. The speeds relative to $u_x$ satisfy $c^4-(c_s^2+v_a^2)c^2+c_s^2{v}_{ax}^2=0$. These are the fast and slow magnetosonic waves. The independent out-of-plane Alfvén wave polarization and a freely advected entropy mode are absent from this reduced isothermal system.

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.

The linearized ideal magnetohydrodynamic equations retain terms first order in perturbations about a chosen ideal-MHD background. Around a uniform state they describe Alfvén waves and magnetosonic waves.

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]

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.

Magnetohydrodynamics couples an electrically conducting fluid to a magnetic field. The electric current density $\mathbf{J}$ produces a Lorentz force density $\mathbf{J}\times \mathbf{B}$, and fluid motion changes the magnetic field through induction.

For constant magnetic diffusivity $\eta$, a common nonrelativistic model includeswith $\nabla \cdot \mathbf{B}=0$. Ideal magnetohydrodynamics neglects the diffusion term; resistive magnetohydrodynamics retains it. The ratio of induction $UB/L$ to diffusion $\eta B/L^2$ is the magnetic Reynolds number $\mathrm{Rm}=UL/\eta$.

Magnetic tension supplies a restoring force for an Alfvén wave, with speed $v_A=B/\sqrt{{\mu }_0\rho }$ in a homogeneous medium. Magnetic pressure and ordinary pressure can both oppose compression. These effects explain why conducting-fluid waves and equilibria differ from those of an unmagnetized gas. ^[ref]

Magnetohydrodynamic turbulence is irregular nonlinear motion of an electrically conducting fluid coupled to a magnetic field. Alfvén waves and magnetosonic waves give propagation times in addition to the nonlinear mixing time. A strong guide field distinguishes directions parallel and perpendicular to it, so the resulting cascades need not be isotropic.