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 Incoming links: Pseudo-Alfvén wave

Solving the polarization quadratic gives all four signed roots:

ω_{h, s} = \frac{q}{2} (hk d_{i} + s 4 + k^{2} d_{i}^{2}), h, s = \pm 1.

A sign-independent description is the pair of positive frequency magnitudes

Ω_{fast, slow} = ∣ k_{∥} ∣ v_{A} [1 + \frac{k ^{2} d _{i}^{2}}{4} \pm \frac{k d _{i}}{2}] .

For wavelengths long compared with the ion skin depth,

k d_{i} ≪ 1

, both magnitudes approach

∣ k_{∥} ∣ v_{A}

. The Hall correction is then small and the usual Alfvén wave dispersion

ω^{2} = k_{∥}^{2} v_{A}^{2}

is recovered. In an incompressible system the two transverse polarizations include the shear and pseudo-Alfvén wave sectors; away from this limit Hall coupling selects circular helicity combinations. A small parallel wavenumber alone does not remove the relative Hall splitting: the expansion parameter is the full

k d_{i}

ϵ = v_{a}^{2} / c^{2} ≪ 1

. Expanding the square root in part (a) gives

(1 + ϵ)^{2} - 4 ϵ cos^{2} θ = 1 + ϵ (1 - 2 cos^{2} θ) + O (ϵ^{2}) .

Hence, to first absolute order in

ϵ

v_{f}^{2} = c^{2} + v_{a}^{2} sin^{2} θ + O (v_{a}^{4} / c^{2}), v_{slow}^{2} = v_{a}^{2} cos^{2} θ + O (v_{a}^{4} / c^{2}) .

The high-pressure limit of magnetosonic waves can also be expressed as nonnegative speeds. Retaining the next squared-speed term gives the first relative correction for the slow mode:

v_{f} = c [1 + \frac{ϵ}{2} sin^{2} θ + O (ϵ^{2})], v_{slow} = v_{a} ∣ cos θ ∣ [1 - \frac{ϵ}{2} sin^{2} θ + O (ϵ^{2})] .

The exact Alfvén speed along the propagation direction remains

v_{a} ∣ cos θ ∣

k

and appreciable mass density and gas pressure variations. Its magnetic pressure correction is weak. The Alfvén wave is exactly incompressible, polarized perpendicular to the

k

B_{0}

k

. Indeed, the leading sound speed part of the matrix in (a) annihilates a vector perpendicular to

k

, so only the smaller magnetic restoring force acts on that MHD wave polarization. Its compression is smaller by order

ϵ

, with gas pressure changes almost cancelling magnetic pressure changes. It is often called the pseudo-Alfvén wave MHD wave polarization in this high pressure limit. Thus two distinct almost-incompressible magnetic MHD wave polarizations have the same leading speed, while the third is predominantly sound. For exactly perpendicular propagation the Alfvén wave and slow magnetosonic wave frequencies vanish; for parallel propagation the two transverse magnetic MHD wave polarizations are degenerate.