A General Method for Conserving Energy and Potential Enstrophy in Shallow-Water Models

The shallow-water equations may be posed in the form dF/dt {F, H, Z}, where H is the energy, Z is the potential enstrophy, and the Nambu bracket {F, H, Z} is completely antisymmetric in its three arguments. This makes it very easy to construct numerical models that conserve analogs of the energy and potential enstrophy; one need only discretize the Nambu bracket in such a way that the antisymmetry property is maintained. Using this strategy, this paper derives explicit finite-difference approximations to the shallowwater equations that conserve mass, circulation, energy, and potential enstrophy on a regular square grid and on an unstructured triangular mesh. The latter includes the regular hexagonal grid as a special case.

[1]  R. Salmon,et al.  Geophysical Fluid Dynamics , 2019, Classical Mechanics in Geophysical Fluid Dynamics.

[2]  A. Arakawa,et al.  A Potential Enstrophy and Energy Conserving Scheme for the Shallow Water Equations , 1981 .

[3]  R. Sadourny.  The Dynamics of Finite-Difference Models of the Shallow-Water Equations , 1975 .

[4]  Theodore G. Shepherd,et al.  Symmetries, Conservation Laws, and Hamiltonian Structure in Geophysical Fluid Dynamics , 1990 .

[5]  Yoichiro Nambu.  Generalized Hamiltonian dynamics , 1973 .

[6]  P. Morrison,et al.  Hamiltonian description of the ideal fluid , 1998 .

[7]  David A. Randall,et al.  Geostrophic Adjustment and the Finite-Difference Shallow-Water Equations , 1994 .

[8]  R. Blender,et al.  A Nambu representation of incompressible hydrodynamics using helicity and enstrophy , 1993 .

[9]  A. Arakawa.  Computational design for long-term numerical integration of the equations of fluid motion: two-dimen , 1997 .

[10]  Frank Abramopoulos.  Generalized Energy and Potential Enstrophy Conserving Finite Difference Schemes for the Shallow Water Equations , 1988 .

[11]  R. Sadourny,et al.  Compressible Model Flows on the Sphere. , 1975 .

[12]  Lynne D. Talley,et al.  Generalizations of Arakawa's Jacobian , 1989 .

[13]  Rick Salmon.  Poisson-Bracket Approach to the Construction of Energy- and Potential-Enstrophy- Conserving Algorithms for the Shallow-Water Equations , 2004 .

[14]  R. Salmon,et al.  Lattice Boltzmann solutions of the three-dimensional planetary geostrophic equations , 1999 .

[15]  Akio Arakawa,et al.  Computational Design of the Basic Dynamical Processes of the UCLA General Circulation Model , 1977 .

[16]  R. Salmon,et al.  A general method for conserving quantities related to potential vorticity in numerical models , 2005 .

[17]  G. L. Browning,et al.  Scaling and computation of smooth atmospheric motions , 1986 .

[18]  D. Randall,et al.  A Potential Enstrophy and Energy Conserving Numerical Scheme for Solution of the Shallow-Water Equations on a Geodesic Grid , 2002 .

[19]  Dale R. Durran,et al.  The Third-Order Adams-Bashforth Method: An Attractive Alternative to Leapfrog Time Differencing , 1991 .

[20]  Todd D. Ringler,et al.  Climate modeling with spherical geodesic grids , 2002, Comput. Sci. Eng..

[21]  Robert I McLachlan.  Spatial discretization of partial differential equations with integrals , 1998 .

[22]  Christopher S. Bretherton,et al.  A new approach for 3D cloud‐resolving simulations of large‐scale atmospheric circulation , 2005 .