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

Addendum to "Coherent Lagrangian vortices: The black holes of turbulence"

In Haller and Beron-Vera (2013) we developed a variational principle for the detection of coherent Lagrangian vortex boundaries. The solutions of this variational principle turn out to be closed null-geodesics of the Lorentzian metric associated with a generalized Green-Lagrange strain tensor family. This metric interpretation implies a mathematical analogy between coherent Lagrangian vortex boundaries and photon spheres in general relativity. Here we give an improved discussion on this analogy.Comment: Revised 27 June 201

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 rings.Comment: To appear in JFM Rapid