5:15 PM - 6:30 PM
[MIS24-P01] Two-dimensional ideal MHD waves and the continuous Alfvén spectrum on a rotating sphere with a background toroidal magnetic field
Keywords:MHD waves, critical latitude, magnetic shear, continuous Alfvén spectrum, rotating fluid
Two-dimensional magnetohydrodynamic (MHD) linear waves on a rotating sphere are investigated for perfectly conducting fluids permeated by an azimuthal magnetic field whose magnitude may depend on the colatitude θ. If the toroidal background field is expressed as B0Φ = B0 sinθ, where B0 is constant and Φ is the azimuth, two types of waves exist: westward-propagating fast magnetic Rossby waves and eastward-propagating slow counterparts. Except for this profile of the imposed field, the wave equation of this system can possess regular singular points at which the zonal phase velocity of an eigenmode is equal to a local Alfvén wave velocity devided by sinθ. Since the positions of critical latitudes depend on the azimuthal phase speed of a chosen eigenmode, this means that continuous spectra appear due to Alfvén resonance in spite of the closed region such as a sphere surface. Similar situations include the linear problem in the inviscid parallel shear flow, which can yield critical layers where the phase velocity coincides with a local mean flow velocity (e.g. Case, 1960).
We solved the eigenvalue problem for B0Φ = B0 sinθcosθ numerically and confirmed the existence of the Alfvén continuum. Since the eigenfrequencies of continuous modes areally spread on the dispersion diagram, there is a possibility that the branches of discrete eigenmodes are buried under the continuous spectrum. We therefore scanned the buried branches by calculating the energy partition of each eigenmodes. Remarkably, discrete branches of slow magnetic Rossby waves were not found in the case of this main field configuration. Thus, the choice of the background field profile might have an influence of the appearance of these slow waves in the ideal MHD case.
Numerical study gives the structure of continuous eigenmodes and shows that these conform to the superposition of the two linearly independent solutions resulted from the Frobenius technique around a critical latitude and to their theoretical connection conditions. When the magnitude of the applied field is small, these eigenfunctions tend to be evanescent on the north side of the critical latitude in the north hemisphere for westward-propagating continuous modes and on the south side of the critical latitude in the north hemisphere for eastward-propagating continuous modes, although there is an exception. This feature is consistent with the WKBJ approximation in slowly varying magnetic shear (Acheson, 1972; Eltayeb & McKenzie, 1977), which suggests the critical level absorption of MHD waves with high wavenumber in rotating fluids. Although Acheson (1972) proposed the "valve" effect, which means that MHD waves can penetrate through the critical layer in limited circumstances, our problem is out of range of its condition, that is, horizontal angular velocity Ωθ vanishes.
We solved the eigenvalue problem for B0Φ = B0 sinθcosθ numerically and confirmed the existence of the Alfvén continuum. Since the eigenfrequencies of continuous modes areally spread on the dispersion diagram, there is a possibility that the branches of discrete eigenmodes are buried under the continuous spectrum. We therefore scanned the buried branches by calculating the energy partition of each eigenmodes. Remarkably, discrete branches of slow magnetic Rossby waves were not found in the case of this main field configuration. Thus, the choice of the background field profile might have an influence of the appearance of these slow waves in the ideal MHD case.
Numerical study gives the structure of continuous eigenmodes and shows that these conform to the superposition of the two linearly independent solutions resulted from the Frobenius technique around a critical latitude and to their theoretical connection conditions. When the magnitude of the applied field is small, these eigenfunctions tend to be evanescent on the north side of the critical latitude in the north hemisphere for westward-propagating continuous modes and on the south side of the critical latitude in the north hemisphere for eastward-propagating continuous modes, although there is an exception. This feature is consistent with the WKBJ approximation in slowly varying magnetic shear (Acheson, 1972; Eltayeb & McKenzie, 1977), which suggests the critical level absorption of MHD waves with high wavenumber in rotating fluids. Although Acheson (1972) proposed the "valve" effect, which means that MHD waves can penetrate through the critical layer in limited circumstances, our problem is out of range of its condition, that is, horizontal angular velocity Ωθ vanishes.