Multilayer primitive equations model with velocity shear and stratification

The purpose of this paper is to present a multilayer primitive equations model for ocean dynamics in which the velocity and buoyancy fields within each layer are not only allowed to vary arbitrarily with horizontal position and time, but also with depth--linearly at most. The model is a generalization of Ripa's inhomogeneous one-layer model to an arbitrary number of layers. Unlike models with homogeneous layers, the present model is able to represent thermodynamics processes. Unlike models with slab layers, i.e. those in which the layer velocity and buoyancy fields are depth-independent, the present model can represent explicitly the thermal-wind balance within each layer which dominates at low frequency. In the absence of external forcing and dissipation, energy, volume, mass, and buoyancy variance constrain the dynamics; conservation of total zonal momentum requires in addition the usual zonal symmetry of the topography and horizontal domain. The model further possesses a singular Hamiltonian structure. Unlike the single-layer counterpart, however, no steady solution has been possible to prove formally (or Arnold) stable using the above invariants. It is shown here that a model with only two layers provides an excellent representation of the exact gravest baroclinic mode phase speed. This suggests that configurations with only a small number of layers will be needed to tackle a large variety of problems with enough realism

Deep ocean influence on upper ocean baroclinic instability saturation

In this paper we extend earlier results regarding the effects of the lower layer of the ocean (below the thermocline) on the baroclinic instability within the upper layer (above the thermocline). We confront quasigeostrophic baroclinic instability properties of a 2.5-layer model with those of a 3-layer model with a very thick deep layer, which has been shown to predict spectral instability for basic state parameters for which the 2.5-layer model predicts nonlinear stability. We compute and compare maximum normal-mode perturbation growth rates, as well as rigorous upper bounds on the nonlinear growth of perturbations to unstable basic states, paying particular attention to the region of basic state parameters where the stability properties of the 2.5- and 3-layer model differ substantially. We found that normal-mode perturbation growth rates in the 3-layer model tend to maximize in this region. We also found that the size of state space available for eddy-amplitude growth tends to minimize in this same region. Moreover, we found that for a large spread of parameter values in this region the latter size reduces to only a small fraction of the total enstrophy of the system, thereby allowing us to make assessments of the significance of the instabilities.Comment: To appear \emph{in} O. U. Velasco-Fuentes et al. (eds.), \textit{Nonlinear Processes in Geophysical Fluid Dynamics}, Kluwer Academi

Coherent Lagrangian vortices: the black holes of turbulence

We introduce a simple variational principle for coherent material vortices in two-dimensional turbulence. Vortex boundaries are sought as closed stationary curves of the averaged Lagrangian strain. Solutions to this problem turn out to be mathematically equivalent to photon spheres around black holes in cosmology. The fluidic photon spheres satisfy explicit differential equations whose outermost limit cycles are optimal Lagrangian vortex boundaries. As an application, we uncover super-coherent material eddies in the South Atlantic, which yield specific Lagrangian transport estimates for Agulhas ring

Travel time stability in weakly range-dependent sound channels

Travel time stability is investigated in environments consisting of a range-independent background sound-speed profile on which a highly structured range-dependent perturbation is superimposed. The stability of both unconstrained and constrained (eigenray) travel times are considered. Both general theoretical arguments and analytical estimates of time spreads suggest that travel time stability is largely controlled by a property $\omega ^{\prime}$ of the background sound speed profile. Here, $2\pi/\omega (I)$ is the range of a ray double loop and $I$ is the ray action variable. Numerical results for both volume scattering by internal waves in deep ocean environments and rough surface scattering in upward refracting environments are shown to confirm the expectation that travel time stability is largely controlled by $\omega ^{\prime}$.Comment: Submitted to J. Acoust. Soc. Am., 30 June 200