A general method for conserving quantities related to potential vorticity in numerical models

Citation:
Salmon, R.  2005.  A general method for conserving quantities related to potential vorticity in numerical models. Nonlinearity. 18:R1-R16.

Date Published:

Sep

Keywords:

dynamics, enstrophy, mechanics, shallow-water equations

Abstract:

Nambu proposed a generalization of Hamiltonian dynamics in the form dF/dt = {F, H, Z}, which conserves H and Z because the Nambu bracket {F, H, Z} is completely antisymmetric. The equations of fluid dynamics fit Nambu's form with H the energy and Z a quantity related to potential vorticity. This formulation makes it easy, in principle, to construct numerical fluid-models that conserve analogues of H and Z; one need only discretize the Nambu bracket in such a way that the antisymmetry property is preserved. In practice, the bracket may contain apparent singularities that are cancelled by the functional derivatives of Z. Then the discretization must be carried out in such a way that the cancellation is maintained. Following this strategy, we derive numerical models of the shallow-water equations and the equations for incompressible flow in two and three dimensions. The models conserve the energy and an arbitrary moment of the potential vorticity. The conservation of potential enstrophy - the second moment of potential vorticity - is thought to be especially important because it prevents the spurious cascade of energy into high wavenumbers.

Notes:

n/a

Website

DOI:

10.1088/0951-7715/18/5/r01