An important characteristic of geophysically turbulent flows is the
transfer of energy between scales. It is expected that balanced flows
pass energy from smaller to larger scales as part of the well-known
upscale cascade while submesoscale and smaller scale flows can transfer
energy eventually to smaller, dissipative scales. Much effort has been
put into quantifying these transfers, but a complicating factor in
realistic settings is that the underlying flows are often strongly
spatially heterogeneous and anisotropic. Furthermore, the flows may be
embedded in irregularly shaped domains that can be multiply connected.
As a result, straightforward approaches like computing Fourier spatial
spectra of nonlinear terms suffer from a number of conceptual issues. In
this paper, we endeavor to compute cross-scale energy transfers in
general settings, allowing for arbitrary flow structure, anisotropy and
inhomogeneity. We employ a Green's function approach to the kinetic
energy equation to relate kinetic energy at a point to its Lagrangian
history. A spatial filtering of the resulting equation naturally
decomposes kinetic energy into length scale dependent contributions and
describes how the transfer of energy between those scales takes place.
The method is applied to a numerical simulation of vortex merger,
resulting in the demonstration of the expected upscale energy cascade.
Somewhat novel results are that the energy transfers are dominated by
pressure work, rather than kinetic energy exchange, and dissipation is a
noticeable influence on the larger scale energy budgets.