Search for: All records

Abstract The generation of broadband wave energy frequency spectra from narrowband wave forcing in geophysical flows remains a conundrum. In contrast to the long-standing view that nonlinear wave–wave interactions drive the spreading of wave energy in frequency space, recent work suggests that Doppler-shifting by geostrophic flows may be the primary agent. We investigate this possibility by ray tracing a large number of inertia–gravity wave packets through three-dimensional, geostrophically turbulent flows generated either by a quasigeostrophic (QG) simulation or by synthetic random processes. We find that, in all cases investigated, a broadband quasi-stationary inertia–gravity wave frequency spectrum forms, irrespective of the initial frequencies and wave vectors of the packets. The frequency spectrum is well represented by a power law. A possible theoretical explanation relies on the analogy between the kinematic stretching of passive tracer gradients and the refraction of wave vectors. Consistent with this hypothesis, the spectrum of eigenvalues of the background flow velocity gradients predicts a frequency spectrum that is nearly identical to that found by integration of the ray tracing equations. 

We derive and investigate numerically a reduced model for wave–vortex interactions involving non-dispersive waves, which we study in a two-dimensional shallow water system with an eye towards applications in atmosphere–ocean fluid dynamics. The model consists of a coupled set of nonlinear partial differential equations for the Lagrangian-mean velocity and the wave-related pseudomomentum vector field defined in generalized Lagrangian-mean theory. It allows for two-way interactions between the waves and the balanced flow that is controlled by the distribution of Lagrangian-mean potential vorticity, and for strong solutions it features a desirable exact energy conservation law for the sum of wave energy and mean flow energy. Our model goes beyond standard ray tracing as we can derive weak solutions that contain discontinuities in the pseudomomentum field, using the theory of weakly hyperbolic systems. This allows caustics to form without predicting infinite wave amplitudes, as would be the case in the standard ray-tracing theory. Suitable wave forcing and dissipation terms are added to the model and a numerical scheme for the model is implemented as a coupled set of pseudo-spectral and finite-volume integrators. Idealized examples of interactions between wavepackets and simple vortex structures are presented to illustrate the model dynamics. The unforced and non-dissipative simulations suggest a heuristic rule of ‘greedy’ waves, i.e. in the long run the wave field always extracts energy from the mean flow.