Reduced-gravity Shallow-water Equations Research Articles

Frictionally Decaying Frontal Warm-Core Eddies

Purpose. The dynamics of nonstationary, nonlinear, axisymmetric, warm-core geophysical surface frontal vortices affected by Rayleigh friction is investigated semi-analytically using the nonlinear, nonstationary reduced-gravity shallow-water equations. The scope is to enlarge the number of known (semi)analytical solutions of nonstationary, nonlinear problems referring to geophysical problems and even to pave the way to their extension to broader geometries and/or velocity fields. Methods and Results. The used method to obtain the solutions is based on the decomposition of the original equations in a part expressing their prescribed spatial structure, so that they can be transformed into ordinary differential equations depending on time only. Based on that analytical procedure, the solutions are then found numerically. In this frame, it is found that vortices characterized by linear distributions of their radial velocity and arbitrary structures of their section and azimuthal velocity can be described exactly by a set of nonstationary, nonlinear coupled ordinary differential equations. The first-order problem (i. e., that describing vortices characterized by a linear azimuthal velocity field and a quadratic section) consists of a system of 4 differential equations, and each further order introduces in the system three additional ordinary differential equations and two algebraic equations. In order to illustrate the behavior of the nonstationary decaying vortices and to put them in the context of observed dynamics in the World Ocean, the system’s solution for the first-order and for the second-order problem is then obtained numerically using a Runge-Kutta method. The solutions demonstrate that inertial oscillations and an exponential attenuation dominate the vortex dynamics: expansions and shallowings, contractions and deepenings alternate during an exact inertial period while the vortex decays. The dependence of the vortex dissipation rate on its initial radius is found to be non-monotonic: it is higher for small and large radii. The possibility of solving (semi)analytically complex systems of differential equations representing observed physical phenomena is rare and very valuable. Conclusions. Our analysis adds realism to previous theoretical investigations on mesoscale vortices, represents an ideal tool for testing the accuracy of numerical models in simulating nonlinear, nonstationary frictional frontal phenomena in a rotating ocean, and paves the way to further extensions of (semi-) analytical solutions of hydrodynamical geophysical problems to more arbitrary forms and more complex density stratifications.

Nonstationary Westward Translation of Nonlinear Frontal Warm-Core Eddies

Abstract 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.

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.

Nonlinear adjustment of a localized layer of buoyant, uniform potential vorticity fluid against a vertical wall

The nonlinear evolution of a localized layer of buoyant, uniform potential vorticity fluid with depth H, width w 0 and length L released adjacent to a wall in a rotating system is studied using reduced-gravity shallow-water theory and numerical modeling. In the interior, far from the two ends of the layer, the initial adjustment gives, after ignoring inertia–gravity waves, a geostrophic flow of width w ∞ and layer velocities parallel to the wall directed in the downstream direction (defined by Kelvin wave propagation). This steady geostrophic flow serves as the initial condition for a semigeostrophic solution using the method of characteristics. At the downstream end, the theory shows that the fluid intrudes along the wall as rarefaction terminating at a nose of vanishing width and depth. However, in a real fluid the presence of the lower layer leads to a blunt gravity current head. The theory is amended by introducing a gravity current head condition that has a blunt bore joined to the rarefaction by a uniform gravity current. The upstream termination of the initial layer produces a Kelvin rarefaction that propagates downstream, decreasing the layer depth along the wall, and initiating upstream flow adjacent to the wall. The theoretical solution compares favorably to numerical solutions of the reduced-gravity shallow-water equations. The agreement between theory and numerical solutions occurs regardless of whether the numerical runs are initiated with an adjusted geostrophic solution or with the release of a stagnant layer. The latter case excites inertia–gravity waves that, despite their large amplitude and breaking, do not significantly affect the evolution of the geostrophic flow. At times beyond the validity of the semigeostrophic theory, the numerical solutions evolve into a stationary array of vortices. The vortex formation can be interpreted as the finite-amplitude manifestation of a linear instability of the new flow established by the passage of the Kelvin wave. The Kelvin wave ultimately reduces the flux into the downstream gravity current and the vortices retain buoyant in the neighborhood of the initial layer.

The Stratified Pulson

Abstract 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.

Near-Inertial Oscillations of Geophysical Surface Frontal Currents

Intrinsic oscillations of stable geophysical surface frontal currents of the unsteady, nonlinear, reduced-gravity shallow-water equations on an f plane are investigated analytically and numerically. For frictional (Rayleigh) currents characterized by linear horizontal velocity components and parabolic cross sections, the primitive equations are reduced to a set of coupled nonlinear ordinary differential equations. In the inviscid case, two periodic analytical solutions of the nonlinear problem describing 1) the inertially reversing horizontal displacement of a surface frontal current having a fixed parabolic cross section and 2) the cross-front pulsation of a coastal current emerging from a motionless surface frontal layer are presented. In a linear and in a weakly nonlinear context, analytical expressions for field oscillations and their frequency shift relative to the inertial frequency are presented. For the fully nonlinear problem, solutions referring to a surface frontal coastal current are obtained analytically and numerically. These solutions show that the currents oscillate always superinertially, the frequency and the amplitude of their oscillations depending on the magnitude of the initial disturbance and on the squared current Rossby number. In a linear framework, it is shown that disturbances superimposed on the surface frontal current are standing waves within the bounded region, the frequencies of which are inertial/superinertial for the first mode/higher modes. In the same frame, a zeroth mode, which could be interpreted as the superposition of an inertial wave on a background vorticity field, would formally yield subinertial frequencies. For surface frontal currents affected by Rayleigh friction, it is shown that the magnitude of the mean current decays according to a power law and that the oscillations decay faster, because this decay follows an exponential law. Implications of the intrinsic oscillations and of their rapid dissipation for the near-inertial motion in an active ambient ocean are discussed.

Oceanic Teleconnections: Remote Response to Decadal Wind Forcing

Abstract The transhemispheric and interbasin response to time-dependent wind forcing localized in the Northern Hemisphere of a single basin is examined using the reduced-gravity shallow-water equations in domains of simple geometry. On decadal timescales, the pressure on the eastern boundary fluctuates synchronously in both hemispheres and thus communicates a signal to latitudes distant from the forcing. The signal then penetrates into the interior through westward radiation of Rossby waves. Associated with the eastern boundary pressure fluctuation is a time-dependent mass flux across the equator that, in a single basin, is balanced by a storage of mass in the unforced hemisphere. Two oceanic basins connected by a reentrant channel at the high-latitude edge of the Southern Hemisphere are then considered. Again the forcing is confined to the Northern Hemisphere of one basin only. In this geometry the time-dependent mass flux across the equator of the forced basin is not entirely balanced within the same basin, but induces a mass flux into the unforced basin, while the mass heaving within the periodic channel is negligible. This process is illustrated by considering winds oscillating at a period on the same order as the Rossby wave transit time in high latitudes. The interhemispheric and interbasin teleconnection is achieved by a combination of long Rossby waves and large-scale, low-frequency gravity waves forced by the Rossby signal. These disturbances share no characteristics of Kelvin waves; that is, they are not boundary trapped.

Propagation of a finite-amplitude potential vorticity front along the wall of a stratified fluid

A similarity solution to the long-wave shallow-water equations is obtained for a density current (reduced gravity = g′, Coriolis parameter = f) propagating alongshore (y = 0). The potential vorticity q = f/H1 is uniform in −∞ < x [les ] xnose(t), 0 < y [les ] L(x, t), and the nose of this advancing potential vorticity front displaces fluid of greater q = f/H0, which is located at L < y < ∞. If L0 = L(−∞, t), the nose point with L(xnose(t), t) = 0 moves with velocity Unose = √g′H0 φ, where φ is a function of H1/H0, f2L20/g′H0. The assumptions made in the similarity theory are verified by an initial value solution of the complete reduced-gravity shallow-water equations. The latter also reveal the new effect of a Kelvin shock wave colliding with a potential vorticity front, as is confirmed by a laboratory experiment. Also confirmed is the expansion wave structure of the intrusion, but the observed values of Unose are only in qualitative agreement; the difference is attributed to the presence of small-scale (non-hydrostatic) turbulence in the laboratory experiment but not in the numerical solutions.

Decadal Oceanic Response to Stochastic Wind Forcing

The low-frequency linear eigenmodes of the reduced-gravity shallow-water equations with weak friction are calculated numerically and using an analytic approximation. For basins with a large variation of the Coriolis parameter, large-scale eigenmodes emerge: the eigenfrequencies are integer multiples of the frequency for the gravest mode, which, in turn, has a period given by the transit time of the slowest long Rossby wave. The e-folding decay times are comparable to the period and independent of friction. These eigenmodes are excited by stochastic wind forcing and this leads to a weak peak in the spectral response near the frequency of the least-damped eigenmode. This decadal-frequency peak is most evident on the eastern and western boundaries and in the equatorial region of the basin.

Evolution of freely propagating, two‐dimensional surface gravity current fronts

This work addresses the two‐dimensional propagation and shape evolution of surface gravity current fronts with a surface density outcrop frontal line. The problem is formulated using the reduced gravity shallow water equations, and the gravity currents are assumed to advance into a fluid at rest. We formulate a nonlinear analytical model for the gravity current plume front morphology by applying the shock tube theory of compressible fluids, which casts the problem in the form of an initial value calculation to be solved numerically. The simulations are initiated by assuming three different plan forms for the initial plume front and their subsequent evolutions followed in time. The paper is concerned exclusively with gravity current fronts having initially a uniform frontal propagation speed locally normal to the plume front, and a number of interesting results emerge. We find that an initially concave region of the front can lead to a nonlinear focusing that results in an energetic bulge in the frontal plan view. These bulges form sharp angles, or kinks, where they are joined to the front at their edges on either side. As they evolve, these angles increase toward 180° (a straight line), and the front becomes smoother in time. The orientation of the bulge and kink features predicted by the model is in agreement with visual and radar imagery observations. The kinks are always oriented toward the lighter plume material. When a plume has two or more such concave regions, the resulting energetic bulges can interact at a later time. The issue of determining plume speeds by tracking these features on sequential images of gravity currents is also dealt with.

On the Westward Translation of Isolated Eddies

Abstract This paper examines the motion of an isolated lenslike eddy using the reduced-gravity shallow-water equations (SWE) on a β plane. In contrast to previous studies the author uses a correct first-order β-plane approximation of the SWE that posseses the same integrals of motion as the original system. The approximation includes terms proportional to (1 − y tanθ0/r0)−1, where y is the northward coordinate, θ0 the reference latitude, and r0 the earth’s mean radius. The center of mass of the eddy is found to oscillate meridionally and zonally with the inertial frequency f0 and drift westward at the same speed as in the case of the conventional β-plane approximation. The new terms arising from the non-Cartesian metric cancel each other. Therefore, the ad hoc procedure of replacing f0 by f0 + βy in the f-plane shallow-water equations does give the correct answer for the bulk motion of the eddy.