Delta Formulation for Geophysical Fluid Flows

A novel formulation of the governing equations of fluid dynamics, called the delta formulation, is introduced and applied to a problem of interest. This formulation decomposes all simulation fields into background and perturbation components, then key assumptions about the separation of length and time scales are made to close the new system. The result is a set of one-way coupled differential equations for both the background fields and perturbation fields, whereby the background fields are able to influence evolution of the perturbation fields, and several possible solution algorithms are introduced. This decomposition aids in simplifying case setup as well as visualization of growth and evolution of a perturbation introduced into a domain, all while retaining most of the physics associated with the interaction between the background and perturbation fields. One such application of the delta formulation is in the field of geophysical fluid flows, which are subject to complex stratified flow physics. In this presentation, a 2D+t simulation focusing on the interaction of an axisymmetric current and a vertical shear profile whose setup would otherwise be far more complicated without the delta formulation. The governing incompressible RANS equations and turbulence model are decomposed using the delta formulation, then the shear profile initialized to the background fields of the simulation and the axisymmetric current initialized to the perturbation fields. Simulation of the domain shows that internal gravity waves produced by the decaying axisymmetric current propagate through and are affected by the background shear profile, and the interaction between these waves in the perturbation fields and the shear layer in the background fields yields a notable increase in turbulence. Overall, the delta formulation shows promise in geophysical flow problems and use in other disciplines of computational mechanics may be possible.