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[MIS21-P01] Two-dimensional ideal MHD waves over a rotating sphere with continuous spectra of slow magnetic Rossby and Alfvén waves
Keywords:MHD waves, stably stratified layer at the Earth's outermost core, magnetic Rossby waves, continuous spectra, critical layers
Two-dimensional ideal magnetohydrodynamic (MHD) linear waves on a rotating sphere are focused on as a model of a stably stratified layer at the top of the Earth's core. This thin stable layer can accommodate MHD waves which may induce various geomagnetic and geodetic variations (e.g. Gillet et al., 2021[1]; Triana et al., 2021[2]). Under the Malkus background field B0Φ = B0 sinθ, where B0 is a constant, Φ is the longitude and θ denotes the colatitude, linear waves in a thin layer are categorized into two types of branches, which correspond to westward-propagating fast magnetic Rossby waves and eastward-propagating slow counterparts, and on which both modes gradually become Alfvén waves as this field strengthens (Zaqarashvili et al., 2007[3]; Márquez-Artavia et al., 2017[4]). Slow magnetic Rossby waves are especially put weight on as a cause of decadal and subdecadal geomagnetic fluctuations (e.g. Chulliat et al., 2015[5]; Chi-Durán et al., 2021[6]). On the other hand, non-Malkus fields such as plausible toroidal fields within the Earth's core make our linear problem complicated due to the advent of regular singular points in its equation. At these singular latitudes for a given wave, the zonal phase velocity of the wave is equal to the local Alfvén speed at the latitude. In the present presentation, we demonstrate that a non-Malkus field B0Φ = B0 sinθcosθ yields a continuous spectrum and an infinite number of singular eigenmodes instead of slow magnetic Rossby discrete modes, which are worthy of attention in studying Earth's magnetism, and Alfvén ones.
Slowly varying wavetrains in inhomogeneous fields are also investigated to compare with our numerical calculation seeking eigenmodes. This implies that a wave packet of the continuous modes propagates toward the critical latitude corresponding to the wave with its pseudomomentum conserved and is ultimately absorbed there. In addition, we found that the behavior that westward- (eastward-) propagating packets approach the latitudes from the equatorial (polar) side is consistent with the tendency which the profiles of their eigenfunctions have. Further thorough discussions of the continuous spectra possibly improve the theory of wave-mean field interaction and one's understanding of the dynamics in the layer.
The geophysically traditional approximation in which one ignores the inertia term is useful to discuss slow MHD waves including slow magnetic Rossby waves (e.g. Buffett and Matsui, 2019[7]). This approximation can also conveniently rule out the continuous modes, which lead to difficulties with our problem. However, this can have a severe influence on the existence of the critical latitudes, hence eigenmodes significantly different from the original. We therefore urge that one should not carelessly drop the inertial effect in the system in which main fields possess spatial dependence.
[ Reference ]
[1] Gillet, N., Gerick, F., Angappan, R., Jault, D. (2021) Surv. Geophys., doi: 10.1007/s10712-021-09664-2
[2] Triana, S.A., Dumberry, M., Cébron, D., Vidal, J., Trinh, A., Gerick, F., Rekier, J. (2021) Surv. Geophys., doi: 10.1007/s10712-021-09668-y
[3] Zaqarashvili, T.V., Oliver, R., Ballester, J.L., Shergelashvili, B.M. (2007) Astron. Astrophys., 470, 815–820, doi: 10.1051/0004-6361:20077382
[4] Márquez-Artavia, X., Jones, C.A., Tobias, S.M. (2017) Geophys. Astrophys. Fluid Dyn., 111, 282–322, doi: 10.1080/03091929.2017.1301937
[5] Chulliat, A., Alken, P., Maus, S. (2015) Geophys. Res. Lett., 42, 3321-3329, doi: 10.1002/2015GL064067
[6] Chi-Durán, R., Avery, M.S., Buffett, B.A. (2021) Geophys. Res. Lett., 48, e2021GL094692, doi: 10.1029/2021GL094692
[7] Buffett, B., Matsui, H. (2019) Geophys. J. Int., 218, 1210–1225, doi: 10.1093/gji/ggz233
Slowly varying wavetrains in inhomogeneous fields are also investigated to compare with our numerical calculation seeking eigenmodes. This implies that a wave packet of the continuous modes propagates toward the critical latitude corresponding to the wave with its pseudomomentum conserved and is ultimately absorbed there. In addition, we found that the behavior that westward- (eastward-) propagating packets approach the latitudes from the equatorial (polar) side is consistent with the tendency which the profiles of their eigenfunctions have. Further thorough discussions of the continuous spectra possibly improve the theory of wave-mean field interaction and one's understanding of the dynamics in the layer.
The geophysically traditional approximation in which one ignores the inertia term is useful to discuss slow MHD waves including slow magnetic Rossby waves (e.g. Buffett and Matsui, 2019[7]). This approximation can also conveniently rule out the continuous modes, which lead to difficulties with our problem. However, this can have a severe influence on the existence of the critical latitudes, hence eigenmodes significantly different from the original. We therefore urge that one should not carelessly drop the inertial effect in the system in which main fields possess spatial dependence.
[ Reference ]
[1] Gillet, N., Gerick, F., Angappan, R., Jault, D. (2021) Surv. Geophys., doi: 10.1007/s10712-021-09664-2
[2] Triana, S.A., Dumberry, M., Cébron, D., Vidal, J., Trinh, A., Gerick, F., Rekier, J. (2021) Surv. Geophys., doi: 10.1007/s10712-021-09668-y
[3] Zaqarashvili, T.V., Oliver, R., Ballester, J.L., Shergelashvili, B.M. (2007) Astron. Astrophys., 470, 815–820, doi: 10.1051/0004-6361:20077382
[4] Márquez-Artavia, X., Jones, C.A., Tobias, S.M. (2017) Geophys. Astrophys. Fluid Dyn., 111, 282–322, doi: 10.1080/03091929.2017.1301937
[5] Chulliat, A., Alken, P., Maus, S. (2015) Geophys. Res. Lett., 42, 3321-3329, doi: 10.1002/2015GL064067
[6] Chi-Durán, R., Avery, M.S., Buffett, B.A. (2021) Geophys. Res. Lett., 48, e2021GL094692, doi: 10.1029/2021GL094692
[7] Buffett, B., Matsui, H. (2019) Geophys. J. Int., 218, 1210–1225, doi: 10.1093/gji/ggz233