Degeneration of internal Kelvin waves in a continuous two-layer stratification

We explore the evolution of the gravest internal Kelvin wave in a two-layer rotating cylindrical basin, using direct numerical simulations (DNS) with a hyper-viscosity/diffusion approach to illustrate different dynamic and energetic regimes. The initial condition is derived from Csanady’s (J. Geophys. Res., vol. 72, 1967, pp. 4151–4162) conceptual model, which is adapted by allowing molecular diffusion to smooth the discontinuous idealized solution over a transition scale, ${\it\delta}_{i}$, taken to be small compared to both layer thicknesses $h_{\ell },\ell =1,2$. The different regimes are obtained by varying the initial wave amplitude, ${\it\eta}_{0}$, for the same stratification and rotation. Increasing ${\it\eta}_{0}$ increases both the tendency for wave steepening and the shear in the vicinity of the density interface. We present results across several regimes: from the damped, linear–laminar regime (DLR), for which ${\it\eta}_{0}\sim {\it\delta}_{i}$ and the Kelvin wave retains its linear character, to the nonlinear–turbulent transition regime (TR), for which the amplitude ${\it\eta}_{0}$ approaches the thickness of the (thinner) upper layer $h_{1}$, and nonlinearity and dispersion become significant, leading to hydrodynamic instabilities at the interface. In the TR, localized turbulent patches are produced by Kelvin wave breaking, i.e. shear and convective instabilities that occur at the front and tail of energetic waves within an internal Rossby radius of deformation from the boundary. The mixing and dissipation associated with the patches are characterized in terms of dimensionless turbulence intensity parameters that quantify the locally elevated dissipation rates of kinetic energy and buoyancy variance.

Trapped internal gravity waves in a geostrophic boundary current

The effect of a geostrophic boundary current on internal gravity waves is studied with a reduced-gravity model. We found that the boundary current not only modifies the coastal Kelvin wave, but also forms wave guides for short internal gravity waves. The combined effects of current shear, the boundary, and the slope of the interface create the trapping mechanism. These trapped internal gravity waves appear as groups of discrete zonal modes. They have wavelengths comparable to or shorter than the internal Rossby radius of deformation. Their phase speeds are close to that of the internal Kelvin wave. However, they can propagate both in, or opposite to, the direction of the Kelvin wave. The results of the present work suggest the possibility of finding an energetic internal gravity wave phenomenon with near-inertial frequency in a broad geostrophic boundary current.

Ageostrophic effects in rotating stratified flow

We consider the low Rossby number (R) flow of a stratified fluid in a long rotating channel, for which the bottom elevation varies in the downstream direction. The quasi-geostrophic response is shown to be singular at the side walls of the channel, and thus an ageostrophic analysis is necessary even for vanishingly small R. Part of the ageostrophic steady-state response is a modified quasi-geostrophic perturbation trapped near the bottom. A second component which is present even as R approaches zero is an internal Kelvin wave whose vertical wavelength adjusts so that the wave remains stationary with respect to the channel bottom and which propagates energy and momentum to infinite heights in an unbounded channel. The case of a bounded layer of fluid is also considered, and the resonance conditions are given. We also calculate the flow field when the bottom elevation varies in the cross-stream direction. We conclude that stagnation or flow reversal can be caused either by the modified quasi-geostrophic component or by the Kelvin wave and estimate the critical condition by an extrapolation of the perturbation velocity computed from linear theory.