American Institute of Mathematical Sciences

Advanced
May  2009, 11(3): 717-740. doi: 10.3934/dcdsb.2009.11.717

The role of higher vorticity moments in a variational formulation of Barotropic flows on a rotating sphere

Chjan C. Lim 1, and  Junping Shi 2,

1. 

Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180, United States
2. 

Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187

Received  March 2008 Revised  December 2008 Published  March 2009

The effects of a higher vorticity moment on a variational problem for barotropic vorticity on a rotating sphere is examined rigorously in the framework of the Direct Method. This variational model differs from previous work on the Barotropic Vorticity Equation (BVE) in relaxing the angular momentum constraint, which then allows us to state and prove theorems that give necessary and sufficient conditions for the existence and stability of constrained energy extremals in the form of super and sub-rotating solid-body steady flows. Relaxation of angular momentum is a necessary step in the modeling of the important tilt instability where the rotational axis of the barotropic atmosphere tilts away from the fixed north-south axis of planetary spin. These conditions on a minimal set of parameters consisting of the planetary spin, relative enstrophy and the fourth vorticity moment, extend the results of previous work and clarify the role of the higher vorticity moments in models of geophysical flows.
Keywords: barotropic vorticity model, higher vorticity moment, variational method..
Mathematics Subject Classification: Primary: 76M30, Secondary: 76E20, 45K0.
Citation: Chjan C. Lim, Junping Shi. The role of higher vorticity moments in a variational formulation of Barotropic flows on a rotating sphere. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 717-740. doi: 10.3934/dcdsb.2009.11.717
[1]

Joachim Escher, Boris Kolev, Marcus Wunsch. The geometry of a vorticity model equation. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1407-1419. doi: 10.3934/cpaa.2012.11.1407
[2]

Delia Ionescu-Kruse. Variational derivation of the Camassa-Holm shallow water equation with non-zero vorticity. Discrete & Continuous Dynamical Systems, 2007, 19 (3) : 531-543. doi: 10.3934/dcds.2007.19.531
[3]

Cheng Wang. Convergence analysis of the numerical method for the primitive equations formulated in mean vorticity on a Cartesian grid. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1143-1172. doi: 10.3934/dcdsb.2004.4.1143
[4]

Walter A. Strauss. Vorticity jumps in steady water waves. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1101-1112. doi: 10.3934/dcdsb.2012.17.1101
[5]

Cheng Wang. The primitive equations formulated in mean vorticity. Conference Publications, 2003, 2003 (Special) : 880-887. doi: 10.3934/proc.2003.2003.880
[6]

Thomas Y. Hou, Zuoqiang Shi. Dynamic growth estimates of maximum vorticity for 3D incompressible Euler equations and the SQG model. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1449-1463. doi: 10.3934/dcds.2012.32.1449
[7]

Vikas S. Krishnamurthy. The vorticity equation on a rotating sphere and the shallow fluid approximation. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6261-6276. doi: 10.3934/dcds.2019273
[8]

H. Beirão da Veiga. Vorticity and regularity for flows under the Navier boundary condition. Communications on Pure & Applied Analysis, 2006, 5 (4) : 907-918. doi: 10.3934/cpaa.2006.5.907
[9]

Jifeng Chu, Joachim Escher. Steady periodic equatorial water waves with vorticity. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4713-4729. doi: 10.3934/dcds.2019191
[10]

Octavian G. Mustafa. On isolated vorticity regions beneath the water surface. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1523-1535. doi: 10.3934/cpaa.2012.11.1523
[11]

Vishal Vasan, Katie Oliveras. Pressure beneath a traveling wave with constant vorticity. Discrete & Continuous Dynamical Systems, 2014, 34 (8) : 3219-3239. doi: 10.3934/dcds.2014.34.3219
[12]

Sho Matsumoto, Jonathan Novak. A moment method for invariant ensembles. Electronic Research Announcements, 2018, 25: 60-71. doi: 10.3934/era.2018.25.007
[13]

Verónica Anaya, Mostafa Bendahmane, David Mora, Ricardo Ruiz Baier. On a vorticity-based formulation for reaction-diffusion-Brinkman systems. Networks & Heterogeneous Media, 2018, 13 (1) : 69-94. doi: 10.3934/nhm.2018004
[14]

Adrian Constantin. Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1397-1406. doi: 10.3934/cpaa.2012.11.1397
[15]

Mats Ehrnström. Deep-water waves with vorticity: symmetry and rotational behaviour. Discrete & Continuous Dynamical Systems, 2007, 19 (3) : 483-491. doi: 10.3934/dcds.2007.19.483
[16]

Calin Iulian Martin. Dispersion relations for periodic water waves with surface tension and discontinuous vorticity. Discrete & Continuous Dynamical Systems, 2014, 34 (8) : 3109-3123. doi: 10.3934/dcds.2014.34.3109
[17]

Philippe Bonneton, Nicolas Bruneau, Bruno Castelle, Fabien Marche. Large-scale vorticity generation due to dissipating waves in the surf zone. Discrete & Continuous Dynamical Systems - B, 2010, 13 (4) : 729-738. doi: 10.3934/dcdsb.2010.13.729
[18]

Hugo Beirão da Veiga. Navier-Stokes equations: Some questions related to the direction of the vorticity. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 203-213. doi: 10.3934/dcdss.2019014
[19]

J. F. Toland. Energy-minimising parallel flows with prescribed vorticity distribution. Discrete & Continuous Dynamical Systems, 2014, 34 (8) : 3193-3210. doi: 10.3934/dcds.2014.34.3193
[20]

Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (58)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy