MHD waves with mixed properties / Alfvén waves with pressure variations: a review

TL;DR

This review surveys MHD waves in non-uniform plasmas, highlighting how spatial inhomogeneity couples wave variables so that Alfvén waves can carry pressure perturbations and magneto-sonic waves acquire vorticity. It develops the general theory in twisted and straight-field cylindrical geometries, emphasizing coupling via the functions C_A and C_S and the pivotal role of P' in mediating cross-field interactions. The text details resonant absorption as a central mechanism for wave damping, showing how waves in the Alfvén and cusp continua undergo strong radial transformations with dramatically enhanced parallel and perpendicular vorticity, and how the weak-damping regime preserves P' as a near-constant quantity at resonance. Collectively, the work demonstrates that non-uniformity fundamentally alters wave behavior in solar-like plasmas, with significant implications for energy transport, wave damping, and coronal loop dynamics, and provides a framework for interpreting mixed-property waves in observational contexts.

Abstract

Non-uniformity plays an important role for MHD waves. For a uniform plasma of infinite extent the MHD waves can be subdivided in two classes with distinct properties. The first class contains the Alfvén waves. The Alfvén waves are incompressible and propagate parallel vorticity. They do not have a parallel component of displacement, they do not cause variations in pressure and are driven by magnetic tension only. The second class contains the magneto-sonic waves. They are compressible and have a parallel component of displacement. They do not propagate parallel vorticity and are driven by pressure and magnetic tension. In non-uniform plasmas the situation can be very different. The clear division between Alfvén waves and magneto-sonic waves is no longer present. In a given part of the equilibrium an MHD wave can strongly resemble a magneto-sonic wave with little or no resemblance to Alfvén waves; while in another part of the equilibrium the MHD wave is practically an Alfvén wave, which has the amazing property of being accompanied by variations in pressure.

Figures (5)

• Figure 1: Top: schematic representation of the Alfvén frequency $\omega_A$ profile in the radial direction in models with a discontinuous jump (left) and with a continuous variation (right) of physical properties. The non-uniformity of the medium makes possible the resonance of the kink mode with the Alfvén continuum. Bottom: sketch (left) and radial dependence of physical conditions (right) in the classical cylindrical model representing a straightened coronal flux tube of length 2L and mean radius R modelled as a density enhancement. The magnetic field is uniform and parallel to the $z$-axis and the whole configuration is invariant in the $\varphi$-direction. The density (continuous line) and the Alfvén speed (dashed line) vary in a non-uniform boundary layer (light-shaded region) of length $l$ from their constant internal values, $\rho_i$ and $v_{Ai}$ to their constant external values, $\rho_e$ and $v_{Ae}$ in a nonuniform layer of thickness $l$ defined in the interval [$R-l/2$, $R+l/2$]. Note hat $\rho$ and $v_A$ are normalized to their internal values. In this model $\rho_i/\rho_e=3$, $k_z=\pi/50$ and $l=0.5 R$.
• Figure 2: Absolute value of the compression as a function of the radial position in a flux tube with $l/R= 0.5$, $k_{\rm z} R = 0.1$, and $\rho_{\rm i}/\rho_{\rm e} = 3$. The normalisation $\hbox{max}\left\{\mid \nabla \cdot \xi \mid \right \} = 1$ has been used. The shaded zone denotes the non-uniform region. Credit: goossens20, reproduced with permission © ESO.
• Figure 3: Absolute value of the radial (left), azimuthal (centre), and parallel (right) vorticity components as functions of the radial position in the same flux tube as in Figure \ref{['fig:f2']}. Credit: goossens20, reproduced with permission © ESO.
• Figure 4: Absolute value of the radial (left), azimuthal (centre), and parallel (right) vorticity components in the non-uniform part of a flux tube with $l/R=0.5$ and $\rho_{\rm i}/\rho_{\rm e} = 2$. The top panels are in linear scale, and the bottom panels are in logarithmic scale. The different line styles denote $k_{\rm z}R = 0.1$ (solid black line), $k_{\rm z}R=0.3$ (dotted red line), and $k_{\rm z}R=0.5$ (dashed blue line). Credit: goossens20, reproduced with permission © ESO.
• Figure 5: Same as Figure \ref{['fig:f4']}, but with $k_{\rm z}R = 0.1$ and different values of the density contrast: $\rho_{\rm i}/\rho_{\rm e}=2$ (solid black line), $\rho_{\rm i}/\rho_{\rm e}=5$ (dotted red line), and $\rho_{\rm i}/\rho_{\rm e}=10$ (dashed blue line). Credit: goossens20, reproduced with permission © ESO.