American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Attractor bifurcation and final patterns of the n-dimensional and generalized Swift-Hohenberg equations
  • DCDS-B Home
  • This Issue
  • Next Article
    Qualitative behavior of a family of delay-differential models of the Glucose-Insulin system
March  2007, 7(2): 425-440. doi: 10.3934/dcdsb.2007.7.425

Exponential approximations for the primitive equations of the ocean

Roger Temam 1, and  D. Wirosoetisno 2,

1. 

The Institute for Scientific Computing and Applied Mathematics, Indiana University, 831 E. 3rd St., Rawles Hall, Bloomington, IN 47405
2. 

Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, United Kingdom

Received  September 2006 Revised  November 2006 Published  December 2006

We show that in the limit of small Rossby number $\varepsilon$, the primitive equations of the ocean (OPEs) can be approximated by "higher-order quasi-geostrophic equations'' up to an exponential accuracy in $\varepsilon$. This approximation assumes well-prepared initial data and is valid for a timescale of order one (independent of $\varepsilon$). Our construction uses Gevrey regularity of the OPEs and a classical method to bound errors in higher-order perturbation theory.
Keywords: exponential asymptotics, Gevrey regularity, primitive equations, Singular perturbation.
Mathematics Subject Classification: Primary: 35B25, 76U0.
Citation: Roger Temam, D. Wirosoetisno. Exponential approximations for the primitive equations of the ocean. Discrete and Continuous Dynamical Systems - B, 2007, 7 (2) : 425-440. doi: 10.3934/dcdsb.2007.7.425
[1]

Hua Chen, Wei-Xi Li, Chao-Jiang Xu. Propagation of Gevrey regularity for solutions of Landau equations. Kinetic and Related Models, 2008, 1 (3) : 355-368. doi: 10.3934/krm.2008.1.355
[2]

Mei-Qin Zhan. Gevrey class regularity for the solutions of the Phase-Lock equations of Superconductivity. Conference Publications, 2001, 2001 (Special) : 406-415. doi: 10.3934/proc.2001.2001.406
[3]

Bixiang Wang, Shouhong Wang. Gevrey class regularity for the solutions of the Ginzburg-Landau equations of superconductivity. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 507-522. doi: 10.3934/dcds.1998.4.507
[4]

Feng Cheng, Chao-Jiang Xu. On the Gevrey regularity of solutions to the 3D ideal MHD equations. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6485-6506. doi: 10.3934/dcds.2019281
[5]

T. Tachim Medjo. The exponential behavior of the stochastic primitive equations in two dimensional space with multiplicative noise. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 177-197. doi: 10.3934/dcdsb.2010.14.177
[6]

M. Petcu, Roger Temam, D. Wirosoetisno. Existence and regularity results for the primitive equations in two space dimensions. Communications on Pure and Applied Analysis, 2004, 3 (1) : 115-131. doi: 10.3934/cpaa.2004.3.115
[7]

Hongjun Gao, Šárka Nečasová, Tong Tang. On weak-strong uniqueness and singular limit for the compressible Primitive Equations. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4287-4305. doi: 10.3934/dcds.2020181
[8]

Huaiyu Jian, Xiaolin Liu, Hongjie Ju. The regularity for a class of singular differential equations. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1307-1319. doi: 10.3934/cpaa.2013.12.1307
[9]

A. Rodríguez-Bernal. Perturbation of the exponential type of linear nonautonomous parabolic equations and applications to nonlinear equations. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 1003-1032. doi: 10.3934/dcds.2009.25.1003
[10]

Heather Hannah, A. Alexandrou Himonas, Gerson Petronilho. Anisotropic Gevrey regularity for mKdV on the circle. Conference Publications, 2011, 2011 (Special) : 634-642. doi: 10.3934/proc.2011.2011.634
[11]

Pawan Kumar Mishra, Sarika Goyal, K. Sreenadh. Polyharmonic Kirchhoff type equations with singular exponential nonlinearities. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1689-1717. doi: 10.3934/cpaa.2016009
[12]

Masaki Hibino. Gevrey asymptotic theory for singular first order linear partial differential equations of nilpotent type — Part I —. Communications on Pure and Applied Analysis, 2003, 2 (2) : 211-231. doi: 10.3934/cpaa.2003.2.211
[13]

Wei Wang, Yan Lv. Limit behavior of nonlinear stochastic wave equations with singular perturbation. Discrete and Continuous Dynamical Systems - B, 2010, 13 (1) : 175-193. doi: 10.3934/dcdsb.2010.13.175
[14]

Yangyang Shi, Hongjun Gao. Homogenization for stochastic reaction-diffusion equations with singular perturbation term. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2401-2426. doi: 10.3934/dcdsb.2021137
[15]

Gary Lieberman. A new regularity estimate for solutions of singular parabolic equations. Conference Publications, 2005, 2005 (Special) : 605-610. doi: 10.3934/proc.2005.2005.605
[16]

Cedric Galusinski, Serguei Zelik. Uniform Gevrey regularity for the attractor of a damped wave equation. Conference Publications, 2003, 2003 (Special) : 305-312. doi: 10.3934/proc.2003.2003.305
[17]

Alin Pogan, Kevin Zumbrun. Stable manifolds for a class of singular evolution equations and exponential decay of kinetic shocks. Kinetic and Related Models, 2019, 12 (1) : 1-36. doi: 10.3934/krm.2019001
[18]

Cheng Wang. The primitive equations formulated in mean vorticity. Conference Publications, 2003, 2003 (Special) : 880-887. doi: 10.3934/proc.2003.2003.880
[19]

Brian D. Ewald, Roger Témam. Maximum principles for the primitive equations of the atmosphere. Discrete and Continuous Dynamical Systems, 2001, 7 (2) : 343-362. doi: 10.3934/dcds.2001.7.343
[20]

Eduard Marušić-Paloka, Igor Pažanin. Homogenization and singular perturbation in porous media. Communications on Pure and Applied Analysis, 2021, 20 (2) : 533-545. doi: 10.3934/cpaa.2020279

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (55)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy