Shallow-water modons on the f-plane

Solutions for steadily translating localized vortical structures, or modons, are sought in the framework of a 11/2-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.