ALBERT

Description: 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.

Description: For the first time, an analytical theory and a very high-resolution, frontal numerical model, both based on the unsteady, nonlinear, reduced-gravity shallow water equations on a β plane, have been used to investigate aspects of the migration of homogeneous surface, frontal warm-core eddies on a β plane. Under the assumption that, initially, such vortices are surface circular anticyclones of paraboloidal shape and having both radial and azimuthal velocities that are linearly dependent on the radial coordinate (i.e., circular pulsons of the first order), approximate analytical expressions are found that describe the nonstationary trajectories of their centers of mass for an initial stage as well as for a mature stage of their westward migration. In particular, near-inertial oscillations are evident in the initial migration stage, whose amplitude linearly increases with time, as a result of the unbalanced vortex initial state on a β plane. Such an initial amplification of the vortex oscillations is actually found in the first stage of the evolution of warm-core frontal eddies simulated numerically by means of a frontal numerical model initialized using the shape and velocity fields of circular pulsons of the first order. In the numerical simulations, this stage is followed by an adjusted, complex nonstationary state characterized by a noticeable asymmetry in the meridional component of the vortex’s horizontal pressure gradient, which develops to compensate for the variations of the Coriolis parameter with latitude. Accordingly, the location of the simulated vortex’s maximum depth is always found poleward of the location of the simulated vortex’s center of mass. Moreover, during the adjusted stage, near-inertial oscillations emerge that largely deviate from the exactly inertial ones characterizing analytical circular pulsons: a superinertial and a subinertial oscillation in fact appear, and their frequency difference is found to be an increasing function of latitude. A comparison between vortex westward drifts simulated numerically at different latitudes for different vortex radii and pulsation strengths and the corresponding drifts obtained using existing formulas shows that, initially, the simulated vortex drifts correspond to the fastest predicted ones in many realistic cases. As time elapses, however, the development of a β-adjusted vortex structure, together with the effects of numerical dissipation, tend to slow down the simulated vortex drift.

Description: New analytical nonstationary circular eddy solutions of the nonlinear, reduced-gravity shallow-water equations in a multilayer stratified rotating ocean are presented. The new solutions extend previous “pulson” analytical solutions, which describe circular oscillating lenslike warm core eddies in a reduced-gravity homogeneous ocean on the f plane, to arbitrary stable vertical stratifications within surface as well as intermediate vortices. As a result, cyclonic as well as anticyclonic horizontal swirl velocities can coexist on different vortex layers within parts of an inertial period, while vertical distributions of the vortex mean tangential velocity resembling observed vertical velocity distributions of surface as well as intermediate lenslike vortices are obtained. The dynamics of a two-layer pulson is discussed and it is shown that, for nonnegligible lower- and upper-layer thickness, its total water transport substantially differs from the total water transport of the corresponding homogeneous pulson. In linearly stratified warm core vortices it is found that the amplitude of the temporal oscillation of the azimuthal velocity is maximum in the bottom layer and decreases toward the surface, while the mean azimuthal velocity is minimum in the bottom layer and increases toward the surface. In linearly stratified intermediate vortices it is found that the mean azimuthal velocity is no longer a monotonic function of the water depth: it is maximum at the top and at the bottom of the vortex and minimum at the vortex central depth. The new solutions add realism to known analytical vortex solutions and elucidate aspects of the observed complexity of geophysical vortex motions.