Rotating modons over isolated topography in a two-layer ocean

Abstract

The nonlinear quasigeostrophic equations for two layers with finite scale topography in the lower layer are reduced to steady form by focusing on solutions that rotate at a constant rate and also decay monotonically outside the topography. The solutions, here called rotating baroclinic modons, are a composition of one radially dependent azimuthal component and a basic axisymmetric component. Smoothness conditions (global continuity of the streamfunction and its first derivatives), when applied to the azimuthal mode, result in a complicated eigenvalue problem. The consequence of this is a single azimuthal mode with arbitrary amplitude. When applied to the axisymmetric solution component, the smoothness conditions allow for a vortex of arbitrary amplitude, a rider, in each layer. These riders determine the rotation rate, which may be cyclonic or anticyclonic, as well as other modon parameters. Two special cases of the theory are discussed and numerical examples of the general solutions are given. The solutions exhibit a rich variety of behavior with a countable infinity of exact solution multipoles. The paper concludes with a comparison of the properties of rotating and rectilinear baroclinic modons and some speculations on applications of the theory.Mathematics Subject Classification (1991). 76C99 (Incompressible inviscid fluids, vorticity flows).

Well-Posedness and Long-Time Dynamics of the Rotating Boussinesq and Quasigeostrophic Equations

“…For example, as by observed Joyce et al [45,46,47] in their utilization of moored instruments as well as Hooker et al [40,41,42] in the remote sensing of warm-core rings, the ring system has bimodal distribution of spin-down with short-lived mean of 54 days and a long-lived mean of 229 days. It is not clear, however, if such results qualitatively and quantitatively match those generated by dynamically consistent translating modons of Flierl et al [20,17,18,79] and dynamically consistent rotating modons of Mied et al [62,54,53,68]. In the splendid report of Lipphardt et al [68,18,55,19,61], it is argued that there is a need for dissipative eddy-permitting models for the examination of mesoscale eddies that spin-down under the action of viscosity.…”

“…For example, since the quasigeostrophic equation is quadratic in the streamfunction, there is no general solution for this system of nonlinear partial differential equations. Additionally, the translating modons of Flierl et al [20,17,18,79] and the rotating modons of Mied et al [62,54,53,68] represent closed form vortex solutions of the quasigeostrophic model without dissipation, and as such the models have not accounted for the spin-down of mesoscale eddies. These limitations lead suggest the need for alternative perspectives for studying solutions of the models.…”

Quasi-two-layer finite-volume scheme for modelling shallow water flows over an arbitrary bed in the presence of external force. I. Algorithm development and validation

“…It is expected that solutions of depth-averaged equations have similar properties as depth-averaged solutions of initial fluid equations. The obtained equations are also rather complicated for constructing general analytical solutions because of their nonlinearity and bed complexity [33,34,35]. However, they can be successfully integrated numerically [10,50,53].…”

Quasi-two-layer finite-volume scheme for modelling shallow water flows over an arbitrary bed in the presence of external force. II. Algorithm applications and numerical results

“…It is expected that the solutions of depth-averaged equations have similar properties to the depth-averaged solutions of the initial uid equations. The obtained equations are rather complicated for constructing general analytical solutions because of their nonlinearity and the bed complexity [34][35][36]. However, they can be successfully integrated numerically [10,51,54].…”