The structure and evolution of elliptical barotropic modons

Abstract

Explicit stationary solutions to the equations of vorticity conservation on the $f$- and $\beta $-planes are considered, describing barotropic dipoles (modons) with elliptical frontiers. The far field – outside the elliptical separatrix demarcating the regions of closed and open streamlines – is given analytically, whereas the interior is solved numerically using a successive linearization algorithm. Both ellipses extended along the translation axis and those extended in the transverse direction are considered. In the latter case, it is shown that, among the possible solutions, there exist the so-called supersmooth modons marked by continuity of the vorticity derivatives at the separatrix. On the $\beta$-plane, the separatrix aspect ratio that allows a supersmooth solution varies depending on the modon translation speed and size, while on the $ f$-plane, there is only one such separatrix aspect ratio. In this context, the limiting transition from the $\beta $-plane to the $f$-plane is discussed. The dependence between the absolute (or relative) vorticity $q$ and the ‘co-moving’ streamfunction $\Psi $, which is nonlinear in the interior of non-circular modons, is analysed in detail for both $\beta $- and $f$-planes, the main concern being the relation between the separatrix form, on the one hand, and the shape of the $q$ against $\Psi $ scattergraph on the other. The stability of elliptical dipoles versus the separatrix aspect ratio is examined based on numerical simulations of the temporal evolution of the modons found. The supersmooth modons appear to be the most stable among all the elliptical dipoles.