On the influence of gravity in the dynamics of geophysical flows

Daniele Del Santo; Francesco Fanelli; Gabriele Sbaiz; Aneta Wróblewska-Kamińska; Daniele Del Santo; Francesco Fanelli; Gabriele Sbaiz; Aneta Wróblewska-Kamińska

doi:10.3934/mine.2023008

Mathematics in Engineering

2023, Volume 5, Issue 1: 1-33. doi: 10.3934/mine.2023008

Previous Article Next Article

Research article Special Issues

On the influence of gravity in the dynamics of geophysical flows

1.
Dipartimento di Matematica e Geoscienze, Università degli Studi di Trieste, Via Valerio 12/1, 34127 Trieste, Italy
2.
Institut Camille Jordan, Université de Lyon, Université Claude Bernard Lyon 1, 43 Boulevard du 11 novembre 1918, F-69622 Villeurbanne, France
3.
Institute of Mathematics, Polish Academy of Sciences, ul.Śniadeckich 8, 00-656 Warszawa, Poland

Received: 20 September 2021 Revised: 29 December 2021 Accepted: 29 December 2021 Published: 24 January 2022

In the present paper, we study a multiscale limit for the barotropic Navier-Stokes system with Coriolis and gravitational forces, for vanishing values of the Mach, Rossby and Froude numbers ($ {\rm{Ma}} $, $ {\rm{Ro}} $ and $ {\rm{Fr}} $, respectively). The focus here is on the effects of gravity: albeit remaining in a low stratification regime $ {\rm{Ma}}/{\rm{Fr}}\, \rightarrow\, 0 $, we consider scaling for the Froude number which go beyond the "critical" value $ {\rm{Fr\, = \, \sqrt{\rm{Ma}}}} $. The rigorous derivation of suitable limiting systems for the various choices of the scaling is shown by means of a compensated compactness argument. Exploiting the precise structure of the gravitational force is the key to get the convergence.
- barotropic Navier-Stokes-Coriolis system,
- gravity,
- stratification effects,
- low Mach,
- Froude and Rossby numbers,
- multiscale limit
Citation: Daniele Del Santo, Francesco Fanelli, Gabriele Sbaiz, Aneta Wróblewska-Kamińska. On the influence of gravity in the dynamics of geophysical flows[J]. Mathematics in Engineering, 2023, 5(1): 1-33. doi: 10.3934/mine.2023008

Related Papers:

Abstract

In the present paper, we study a multiscale limit for the barotropic Navier-Stokes system with Coriolis and gravitational forces, for vanishing values of the Mach, Rossby and Froude numbers ($ {\rm{Ma}} $, $ {\rm{Ro}} $ and $ {\rm{Fr}} $, respectively). The focus here is on the effects of gravity: albeit remaining in a low stratification regime $ {\rm{Ma}}/{\rm{Fr}}\, \rightarrow\, 0 $, we consider scaling for the Froude number which go beyond the "critical" value $ {\rm{Fr\, = \, \sqrt{\rm{Ma}}}} $. The rigorous derivation of suitable limiting systems for the various choices of the scaling is shown by means of a compensated compactness argument. Exploiting the precise structure of the gravitational force is the key to get the convergence.

References

[1]	H. Bahouri, J.-Y. Chemin, R. Danchin, Fourier analysis and nonlinear partial differential equations, Heidelberg: Springer, 2011. http://dx.doi.org/10.1007/978-3-642-16830-7
[2]	E. Bocchi, F. Fanelli, C. Prange, Anisotropy and stratification effects in the dynamics of fast rotating compressible fluids, arXiv: 2011.02411.
[3]	D. Bresch, M. Gisclon, C.-K. Lin, An example of low Mach (Froude) number effects for compressible flows with nonconstant density (height) limit, ESAIM: M2AN, 39 (2005), 477–486. http://dx.doi.org/10.1051/m2an:2005026 doi: 10.1051/m2an:2005026
[4]	J.-Y. Chemin, B. Desjardins, I. Gallagher, E. Grenier, Mathematical geophysics: An introduction to rotating fluids and the Navier-Stokes equations, Oxford: Oxford University Press, 2006.
[5]	D. Cobb, F. Fanelli, On the fast rotation asymptotics of a non-homogeneous incompressible MHD system, Nonlinearity, 34 (2021), 2483–2526. http://dx.doi.org/10.1088/1361-6544/abb929 doi: 10.1088/1361-6544/abb929
[6]	B. Cushman-Roisin, Introduction to geophysical fluid dynamics, Englewood Cliffs: Prentice-Hall, 1994.
[7]	D. Del Santo, F. Fanelli, G. Sbaiz, A. Wróblewska-Kamińska, A multiscale problem for viscous heat-conducting fluids in fast rotation, J. Nonlinear Sci., 31 (2021), 21. http://dx.doi.org/10.1007/s00332-021-09677-6 doi: 10.1007/s00332-021-09677-6
[8]	D. G. Ebin, Viscous fluids in a domain with frictionless boundary, In: Global analysis-analysis on manifolds, Leipzig: Teubner, 1983, 93–110.
[9]	F. Fanelli, Highly rotating viscous compressible fluids in presence of capillarity effects, Math. Ann., 366 (2016), 981–1033. http://dx.doi.org/10.1007/s00208-015-1358-x doi: 10.1007/s00208-015-1358-x
[10]	F. Fanelli, A singular limit problem for rotating capillary fluids with variable rotation axis, J. Math. Fluid Mech., 18 (2016), 625–658. http://dx.doi.org/10.1007/s00021-016-0256-7 doi: 10.1007/s00021-016-0256-7
[11]	F. Fanelli, Incompressible and fast rotation limit for barotropic Navier-Stokes equations at large Mach numbers, Physica D, 428 (2021), 133049. http://dx.doi.org/10.1016/j.physd.2021.133049 doi: 10.1016/j.physd.2021.133049
[12]	F. Fanelli, I. Gallagher, Asymptotics of fast rotating density-dependent fluids in two space dimensions, Rev. Mat. Iberoam., 35 (2019), 1763–1807. http://dx.doi.org/10.4171/rmi/1101 doi: 10.4171/rmi/1101
[13]	F. Fanelli, E. Zatorska, Low Mach number limit for degenerate Navier-Stokes equations in presence of strong stratification, submitted for publication.
[14]	E. Feireisl, Dynamics of viscous compressible fluids, Oxford: Oxford University Press, 2004. http://dx.doi.org/10.1093/acprof:oso/9780198528388.001.0001
[15]	E. Feireisl, I. Gallagher, D. Gérard-Varet, A. Novotný, Multi-scale analysis of compressible viscous and rotating fluids, Commun. Math. Phys., 314 (2012), 641–670. http://dx.doi.org/10.1007/s00220-012-1533-9 doi: 10.1007/s00220-012-1533-9
[16]	E. Feireisl, I. Gallagher, A. Novotný, A singular limit for compressible rotating fluids, SIAM J. Math. Anal., 44 (2012), 192–205. http://dx.doi.org/10.1137/100808010 doi: 10.1137/100808010
[17]	E. Feireisl, R. Klein, A. Novotný, E. Zatorska, On singular limits arising in the scale analysis of stratified fluid flows, Math. Mod. Meth. Appl. Sci., 26 (2016), 419–443. http://dx.doi.org/10.1142/S021820251650007X doi: 10.1142/S021820251650007X
[18]	E. Feireisl, Y. Lu, A. Novotný, Rotating compressible fluids under strong stratification, Nonlinear Anal. Real, 19 (2014), 11–18. http://dx.doi.org/10.1016/j.nonrwa.2014.02.004 doi: 10.1016/j.nonrwa.2014.02.004
[19]	E. Feireisl, A. Novotný, Singular limits in thermodynamics of viscous fluids, Basel: Birkhäuser, 2009. http://dx.doi.org/10.1007/978-3-7643-8843-0
[20]	E. Feireisl, A. Novotný, Scale interactions in compressible rotating fluids, Annali di Matematica, 193 (2014), 1703–1725. http://dx.doi.org/10.1007/s10231-013-0353-7 doi: 10.1007/s10231-013-0353-7
[21]	E. Feireisl, A. Novotný, Multiple scales and singular limits for compressible rotating fluids with general initial data, Commun. Part. Diff. Eq., 39 (2014), 1104–1127. http://dx.doi.org/10.1080/03605302.2013.856917 doi: 10.1080/03605302.2013.856917
[22]	I. Gallagher, L. Saint-Raymond, Weak convergence results for inhomogeneous rotating fluid equations, J. Anal. Math., 99 (2006), 1–34. http://dx.doi.org/10.1007/BF02789441 doi: 10.1007/BF02789441
[23]	P. Kundu, I. Cohen, D. Dowling, Fluid mechanics, 5 Eds., Oxford: Academic Press, 2012.
[24]	Y.-S. Kwon, D. Maltese, A. Novotný, Multiscale analysis in the compressible rotating and heat conducting fluids, J. Math. Fluid Mech., 20 (2018), 421–444. http://dx.doi.org/10.1007/s00021-017-0327-4 doi: 10.1007/s00021-017-0327-4
[25]	Y.-S. Kwon, A. Novotný, Derivation of geostrophic equations as a rigorous limit of compressible rotating and heat conducting fluids with the general initial data, Discrete Contin. Dyn. Syst., 40 (2020), 395–421. http://dx.doi.org/10.3934/dcds.2020015 doi: 10.3934/dcds.2020015
[26]	P.-L. Lions, Mathematical topics in fluid mechanics. Vol. 2. Compressible models, New York: Oxford University Press, 1998.
[27]	N. Masmoudi, Rigorous derivation of the anelastic approximation, J. Math. Pures Appl., 88 (2007), 230–240. http://dx.doi.org/10.1016/j.matpur.2007.06.001 doi: 10.1016/j.matpur.2007.06.001
[28]	J. Pedlosky, Geophysical fluid dynamics, New York: Springer, 1987. http://dx.doi.org/10.1007/978-1-4612-4650-3
[29]	G. Sbaiz, Fast rotation limit for the $2$-D non-homogeneous incompressible Euler equations, submitted for publication.
[30]	S. Scrobogna, Highly rotating fluids with vertical stratification for periodic data and vanishing vertical viscosity, Rev. Mat. Iberoam., 34 (2018), 1–58. http://dx.doi.org/10.4171/RMI/980 doi: 10.4171/RMI/980

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)