A mathematically rigorous framework for singular limits of the magnetohydrodynamic rotating shallow water equations with ill-prepared data is developed when the Rossby and Froude numbers tend to zero at different rates. The reduced systems are derived, respectively, for the stratification-dominant and the rotation-dominant cases by means of the developed three-scale fast averaging method.
Yue FANG
,
Jiawei WANG
,
Xin XU
. THREE-SCALE SINGULAR LIMITS OF THE MHD ROTATING SHALLOW WATER SYSTEM[J]. Acta mathematica scientia, Series B, 2025
, 45(2)
: 695
-714
.
DOI: 10.1007/s10473-025-0223-6
[1] Bourgeois A, Beale J. Validity of the quasigeostrophic model for large-scale flow in the atmosphere and ocean. SIAM Journal on Mathematical Analysis, 1994, 25: 1023-1068
[2] Cheng B, Ju Q, Schochet S. Three-scale singular limits of evolutionary PDEs. Archive for Rational Mechanics and Analysis, 2018, 229: 601-625
[3] Dutrifoy A. Fast averaging for long-and short-wave scaled equatorial shallow water equations with coriolis parameter deviating from linearity. Archive for Rational Mechanics and Analysis, 2015, 216: 261-312
[4] Dutrifoy A, Majda A. The dynamics of equatorial long waves: a singular limit with fast variable coefficients. Communications in Mathematical Sciences, 2006, 4: 375-397
[5] Dutrifoy A, Majda A. Fast wave averaging for the equatorial shallow water equations. Communications in Partial Differential Equations, 2007, 32: 1617-1642
[6] Dutrifoy A, Majda A, Schochet S. A simple justification of the singular limit for equatorial shallow-water dynamics. Communications on Pure and Applied Mathematics, 2009, 62: 322-333
[7] Embid P, Majda A. Low froude number limiting dynamics for stably stratified flow with small or finite rossby numbers. Geophysical & Astrophysical Fluid Dynamics, 1998, 87: 1-50
[8] Feireisl E, Lu Y, Novotnỳ A. Rotating compressible fluids under strong stratification. Nonlinear Analysis: Real World Applications, 2014, 19: 11-18
[9] Feireisl E, Novotnỳ A. Multiple scales and singular limits for compressible rotating fluids with general initial data. Communications in Partial Differential Equations, 2014, 39: 1104-1127
[10] Feireisl E, Novotnỳ A. Scale interactions in compressible rotating fluids. Annali di Matematica Pura ed Applicata, 2014, 193: 1703-1725
[11] Gill A E. Atmosphere-Ocean Dynamics.New York: Academic Press, 1982
[12] Gilman P. Magnetohydrodynamic "shallow water" equations for the solar tachocline. The Astrophysical Journal, 2000, 544: L79-L82
[13] Heng K, Spitkovsky A. Magnetohydrodynamic shallow water waves: linear analysis. The Astrophysical Journal, 2009, 703: 1819-1831
[14] Hunter S.Waves in shallow water magnetohydrodynamics[D]. Leeds: University of Leeds, 2015
[15] Ju Q, Mu P. Low froude and rossby number three-scale singular limits of the rotating stratified boussinesq equations. Zeitschrift für angewandte Mathematik und Physik, 2019, 70: 1-20
[16] Lahaye N, Zeitlin V. Coherent magnetic modon solutions in quasi-geostrophic shallow water magnetohydrodynamics. Journal of Fluid Mechanics, 2022, 941: A15
[17] Lillo R, Mininni P, Gómez D. Toward a dynamo model for the solar tachocline. Physica A: Statistical Mechanics and its Applications, 2005, 349: 667-674
[18] Majda A.Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. New York: Springer-Verlag: 1984
[19] Majda A.Introduction to PDEs and Waves for the Atmosphere and Ocean. Providence, RI: Amer Math Soc, 2003
[20] Majda A, Embid P. Averaging over fast gravity waves for geophysical flows with unbalanced initial data. Theoretical and Computational Fluid Dynamics, 1998, 11: 155-169
[21] Mak J, Griffiths S, Hughes D. Shear flow instabilities in shallow-water magnetohydrodynamics. Journal of Fluid Mechanics, 2016, 788: 767-796
[22] Mu P, Ju Q. Three-scale singular limits of the rotating stratified boussinesq equations. Applicable Analysis, 2021, 100: 2405-2417
[23] Nečasová Š, Tang T. On a singular limit for the compressible rotating euler system. Journal of Mathematical Fluid Mechanics, 2020, 22: 1-14
[24] Ngo V S, Scrobogna S.Dispersive effects of weakly compressible and fast rotating inviscid fluids. arXiv:1611.06112
[25] Schecter D, Boyd J, Gilman P. "Shallow-water" magnetohydrodynamic waves in the solar tachocline. The Astrophysical Journal, 2001, 551: L185-L188
[26] Schochet S. Singular limits in bounded domains for quasilinear symmetric hyperbolic systems having a vorticity equation. Journal of Differential Equations, 1987, 68: 400-428
[27] Schochet S. Fast singular limits of hyperbolic pdes. Journal of Differential Equations, 1994, 114: 476-512
[28] Spiegel E, Zahn J. The solar tachocline. Astronomy and Astrophysics, 1992, 265: 106-114
[29] Vreugdenhil C B.Numerical Methods for Shallow-Water Flow. Dordrecht: Springer Science & Business Media, 1994
[30] Wingate B, Embid P, Holmes-Cerfon M, Taylor M. Low rossby limiting dynamics for stably stratified flow with finite froude number. Journal of Fluid Mechanics, 2011, 676: 546-571
[31] Wu K C. Low froude number limit of the rotating shallow water and Euler equations. Proceedings of the American Mathematical Society, 2014, 142: 939-947
