Analytical Solutions for Circular Stratified Eddies of the Reduced-Gravity Shallow-Water Equations

Abstract

A set of new analytical nonstationary solutions of the nonlinear, reduced-gravity shallow-water equations on an f plane in a vertically stably stratified active environment is presented. The solutions, which describe the dynamics of inertially pulsating surface as well as intermediate lens-like stratified vortices, represent an extension of previous analytical solutions to more realistic vortex shapes and structures of the vortex swirl velocity fields in the presence of an arbitrary stable vertical stratification within the active environment. To elucidate aspects of the novelty of the new set of solutions, examples are presented referring to a vertically stratified surface vortex and to a vertically stratified intermediate vortex, both characterized by azimuthal velocity fields that are nonlinear functions of the radius and by layer shapes that largely deviate from paraboloidal shapes. First, a solution describing a five-layer surface “warm core” eddy is analyzed and characteristics resembling characteristics observed in geophysical surface vortices are revealed: Within each layer, the obtained azimuthal velocity shows a realistic distribution, as its maximum is located far from the vortex rim, where it is negligible. Moreover, the obtained azimuthal velocity is largest in the surface layer and decreases monotonically toward the deeper layers. Integral properties of the new stratified solutions significantly differ from the corresponding properties of equivalent, known homogeneous solutions. Significant differences are found, for instance, between the shape of a five-layer vortex and that of its homogenous counterpart having the same mean azimuthal velocity structure. Second, a solution referring to a five-layer intermediate “meddy like” vortex is analyzed: while in each layer the obtained azimuthal velocity is maximum in the interior part of the vortex and decreases toward its center and its periphery, its magnitude is largest in the intermediate layer and it decreases toward the surface and toward the bottom layer, which are characteristics resembling characteristics of observed meddies. The quoted examples demonstrate that the new solutions add substantial realism to the analytical description of oceanic nonlinear geophysical vortices. In the context of the nonlinear reduced-gravity shallow-water equations on an f plane they seem to represent the most general analytical features achievable, which refer to vertically stratified circular vortices characterized by linear radial velocity fields.

Abyssal undular vortices in the Eastern Mediterranean basin

“…3b). Notably, anticyclonic mesoscale vortical features characterized by similar radial structures of their tangential velocities describing internal undular oceanic vortices have been recently discovered as robust solutions of the non-stationary, nonlinear shallow-water equations on an f-plane [23][24][25][26] and also demonstrated as being persistent features of both oceanic and laboratory flow fields 27,28 . Remarkably, they are all characterized by the presence of (quasi) inertial pulsations, as those observed here to be significantly enhanced during vortical activity (Fig.…”

Intrinsic dynamics and long‐term evolution of a convectively generated oceanic vortex in the Greenland Sea

“…During the last years a renewed attention has been devoted to the study of mesoscale as well as submesoscale oceanic surface and intermediate vortices [see, e.g., Lee and Niiler , 1998; Rubino et al , 1998, 2002; Wadhams et al , 2002; Gascard et al , 2002; Rubino and Brandt , 2003; Wadhams et al , 2004; Budéus et al , 2004; Dotsenko and Rubino , 2006; Zhai et al , 2007]. In fact, recent investigations have demonstrated the important role they exert in the formation and transformation of water masses [see, e.g., Gascard et al , 2002; Wadhams et al , 2002; Budéus et al , 2004; Ronski and Budéus , 2005].…”

“…In this paper the nonstationary structure, the intrinsic dynamics, as well as the long‐term evolution of a convectively generated vortex observed in the Greenland Sea are explored, under realistic conditions, using in situ data, a hierarchy of nested numerical models and a recently developed analytical theory. As a result, we recognize the simulated features to belong to intermediate, stratified, nonstationary pulson‐like vortices of high order [see Rubino et al , 1998; Dotsenko and Rubino , 2006]. By comparing the simulated water mass squared Brunt‐Väisälä frequency in the central Greenland Sea with and without assuming the presence of a convectively generated vortex in the area, the influence exerted by such rotating features on the convective activity in the central Greenland Sea is discussed.…”

On the Geostrophic Adjustment of an Isolated Lens: Dependence on Burger Number and Initial Geometry*

“…Analytical solutions under varying conditions have also been discussed in the literature (e.g., Csanady 1979;Flierl 1979;Ou 1986;McWilliams 1988, hereafter M88;Killworth 1992;Boss and Thompson 1995;Spall 1995;Kuo and Polvani 1997;Ungarish and Huppert 1998;Reznik et al 2001;Dotsenko and Rubino 2006), and a number of these are considered in more detail below. For the purposes of the present study, we will focus in particular on the semianalytical solutions of M88 for the adjustment of an axisymmetric lens in a continuously stratified rotating fluid, which is also discussed below.…”