Shallow-water modons on the f-plane

Abstract

Solutions for steadily translating localized vortical structures, or modons, are sought in the framework of a 1½-layer rotating shallow-water (RSW) model on the f-plane. In this model, the fluid is assumed to rotate at a constant rate and to be composed of an active finite-depth layer and a passive infinitely deep layer. The focus is on the smooth intense modons, in which the potential vorticity field is continuous, and the pressure (hence, the active-layer thickness) and velocity are smooth, while inertial effects and deviations of the active-layer thickness from the static level are considerable. The problem is solved numerically employing a Newton--Kantorovich iterative procedure, Fourier–Chebyshev spectral expansion and collocations. The numerics are preceded by a theoretical modon design discussion that includes: derivation of fundamental modon invariants; distinction between the flow in the trapped-fluid region and the flow outside it; and the boundary conditions at the separatrix, the streamline demarcating the two regions. Also, some basic distinctions from the quasi-geostrophic modons are discussed, and an asymptotic analysis of the RSW modon far-field characteristics is carried out. This analysis reveals that an RSW modon must propagate more slowly than inertia–gravity waves. In smooth modons, the requirement that the active-layer thickness should be positive imposes an even stronger restriction on the allowed translational speed. To enable the use of Fourier–Chebyshev series, only the modons with circular separatrices are considered. The numerical iterative procedure is initialized by an analytical quasi-geostrophic dipolar modon solution; accordingly, the obtained RSW modons appear as cyclone–anticyclone pairs. Computations show that the allowed maximal translational speed monotonically decreases as a function of the modon size and, for reasonable sizes, is appreciably smaller than the gravity-wave limit. As distinct from quasi-geostrophic modons, the RSW modons with circular separatrices display nonlinearity of the potential vorticity (PV) vs. streamfunction relation, and the cyclone–anticyclone asymmetry: while the integral mass anomaly in the modon is zero, the cyclone is more intense and compact than the anticyclone.