Geostrophic Vortices Research Articles

The vortices of geostrophic turbulence

A solution for decaying geostrophic turbulence at large Reynolds number is analysed by means of an automated vortex census. The census identifies the flow structures that approximately conform to the idealized shape of an isolated, coherent geostrophic vortex. It also determines vortex characteristics, such as amplitude, size, and shape. The distributions of these characteristics within the vortex population are examined, as are their time evolutions. These distributions are interpreted with reference to the dynamical processes of vortices, and they are compared to analogous distributions from a solution for two-dimensional turbulence.

Sensitivity to initial conditions in the merging of two-layer baroclinic vortices

Merging of geostrophic vortices in a two-layer stratified fluid is studied by means of direct numerical simulations. It is shown that the influence of density stratification on merging depends strongly on the initial shape of the vortices. When potential vortices are initially confined to the upper layer, with zero potential vorticity in the lower layer, the strength of the stratification is found to have no effect on merging. On the contrary, when the vortices initially have all their relative vorticity in the upper layer, with the lower layer at rest, the merging conditions are radically different. In the latter case, the most striking phenomenon is the pronounced scale selectivity that stratification has on merging. This behavior is provisionally interpreted as a consequence of the different far-field flow patterns for the different vortex profiles.

Two-layer geostrophic vortex dynamics. Part 1. Upper-layer V-states and merger

We generalize the methods of two-dimensional contour dynamics to study a twolayer rotating fluid that obeys the quasi-geostrophic equations. We consider here only the case of a constant-potential-vorticity lower layer. We derive equilibrium solutions for monopolar (rotating) and dipolar (translating) geostrophic vortices in the upper layer, and compare them with the Euler case. We show that the equivalent barotropic (infinite lower layer) case is a singular limit of the two-layer system. We also investigate the effect of a finite lower layer on the merger of two regions of equalsign potential vorticity in the upper layer. We discuss our results in the light of the recent laboratory experiments of Griffiths & Hopfinger (1986). The process of filamentation is found to be greatly suppressed for equivalent barotropic dynamics on scales larger than the radius of deformation. We show that the variation of the critical initial distance for merger as a function of the radius of deformation and the ratio of the layers at rest is closely related to the existence of vortex-pair equilibria and their geometrical properties.

On the instability of geostrophic vortices

The instabilities of barotropic and baroclinic, quasi-geostrophic, f-plane, circular vortices are found using a linearized contour dynamics model. We model the vortex using a circular region of horizontally uniform potential vorticity surrounded by an annulus of uniform, but different, potential vorticity. We concentrate mostly upon isolated vortices with no circulation in the basic state outside the outer radius b. In addition to linear analyses, we also consider weakly nonlinear waves. The amplitude equation has a cubic nonlinearity and, depending upon the sign of the coefficient of the cubic term, may give nonlinear stabilization or nonlinear enhancement of the growth. Barotropic isolated eddies are unstable when the outer annulus is narrow enough; on the other hand, if the scale of the whole vortex is sufficiently small compared to the radius of deformation of a baroclinic mode, the break up may be preferentially to a depth-varying disturbance corresponding to a twisting and tilting of the vortex. As the vortex becomes more baroclinic, we find that large-scale vortices show an elliptical mode baroclinic instability as well which is relatively insensitive to the scale of the outer annulus. When the baroclinic currents in the basic state dominate, the twisting mode disappears, and we see only the instabilities associated with either strong enough shear in the annular region or sufficiently large vortices compared with the deformation radius. The finite amplitude results show that the baroclinic instability mode for large enough vortices is nonlinearly stabilized while in most cases, the other two kinds of instability are nonlinearly destabilized.

The Stability and Genesis of Rossby Vortices

Abstract The stability and genesis of the vortices associated with long solitary divergent Rossby waves-the Rossby vortices–are studied numerically using the single-layer (SL) model with Jovian parameters. Vortex behavior depends on location and on balances among the translation, twisting, steepening, dispersion and advection processes. Advection is the main preserver of vortices. The solutions provide an explanation for the origin, uniqueness and longevity of the Great Red Spot (GRS). In midlatitudes, stable anticyclones exist in a variety of sizes and balances: from the large planetary-geostrophic (PG) and medium intermediate-geostrophic (IG) vortices that propagate westward, to the small quasi-geostrophic (QG) vortices that migrate equatorward. These vortices all merge during encounters. Geostrophic vortices in the f0-plane system adjust toward symmetry by rotating; those on the sphere adjust by rotating and propagating. Stable cyclones exist mainly at the QG scale or on the f0-plane. In low latitudes ...

The singular geostrophic vortices on β-plane as a model for synoptic vortices: Gryanik, V.M., 1986. Okeanologiia, 26(2):174–179. (In Russian, English abstract.)

The singular geostrophic vortices on β-plane as a model for synoptic vortices: Gryanik, V.M., 1986. Okeanologiia, 26(2):174–179. (In Russian, English abstract.)

Some interactions between small numbers of baroclinic, geostrophic vortices

Abstract Various interactions between small numbers (two and four) of baroclinic, geostrophic point vortices in a two-layer system are studied with attention to the qualitative changes in behavior which occur as size of the deformation radius is varied. A particularly interesting interaction, which illustrates the richness of baroclinic vortex dynamics, is a collision between two hetons. (A heton is a vortex pair in which the constituent vortices have opposite signs and are in opposite layers. The “breadth” of a heton is the distance between its constituent vortices. A translating heton transports heat.) When two hetons, which initially have different breadths, collide, the result is either an exchange of partners, or a “slip-through” collision in which the initial structures are preserved. It is shown here that the outcome is always an exchange, provided the deformation radius is sufficiently small. This strongly contrasts with a collision between pairs of classical, one-layer vortices in which no exchan...

The heton, an elementary interaction between discrete baroclinic geostrophic vortices, and its implications concerning eddy heat-flow

Among the interactions of two discrete baroclinic geostrophic vortices in a two-layer system there is one class of interaction that is non-trivial; when the two vortices are of opposite sign and in different layers, and close enough together, they transport heat. Because this particular interaction can transport heat, we propose to call it the heton. It is a tilted baroclinic pair. In the Northern Hemisphere it transports heat to the left of the direction toward which its top tilts. Two warm or two cold hetons repel one another when outside the radius of deformation. A warm and a cold heton attract one another. A simple two-heton engine that exhibits vortex splitting, loss of available potential energy, and meridional heat transport is presented.

Solitons in a Convective Motion of a Low-β Plasma

The present paper shows that in a plane normal to an applied magnetic field, envelope solitary vortices comprising convective cells of small scale propagate through a nonuniform bounded plasma of low β as envelope solitons of a nonlinear Schrodinger equation. It is shown further that the modulational instability is enhanced resonantly for certain wavenumbers and vortices of large scale is excited by the ponderomotive force. Discussions are based on the model equation of Hasegawa and Mima that is the geostrophic vortex equation, and a generation of zonal flows by the Rossby wave is noted.

Stationary Spectrum of Pseudo-Three-Dimensional Electrostatic Turbulence in Magnetized Plasmas

We present a stationary k -spectrum of pseudo-three-dimensional electrostatic plasma turbulence in a uniform, loss-less, magnetized plasma based on the model equation of Hasegawa and Mima; \(\partial/\partial t(\phi-\mbi{\nabla}^{2}\phi)+[\mbi{\nabla}\phi\times\hat{z}{\cdot}\mbi{\nabla}]\mbi{\nabla}^{2}\phi{=}0\). The k -spectrum for a potential φ k is given by =(1+ k 2 ) -1 (α+β k 2 ) -1 , where α and β are constants. The existence of the two constants, α and β, in the spectrum is the consequence of the two rugged constants of motion. The spectrum is obtained using three different methods, namely, the method of Gibbs distribution, of Hopf's equation and of Wiener Hermite expansion. Because of the similarity of the equation, the obtained spectrum should also apply to the hydrodynamic turbulence of geostrophic vortices.

Stability of rectilinear geostrophic vortices in stationary equilibrium

A hemispherical barotropic model of global-scale atmospheric motions is studied in a planar approximation using three K0 (Bessel) vortices of equal strength γ and length scale κ−1 to represent the subtropical semipermanent high pressure systems. In equilibrium these anticyclonic vortices are equally spaced on a latitude circle of radius a. One K0 vortex of strength γ0 is used to simulate the polar (cyclonic) vortex; γ0 is chosen so that the equilibrium configuration remains stationary with respect to the rotating earth. Over the range of the length parameter 0⩽κa⩽5, two distinct stable regimes are found: (1) a ’’soft’’-vortex regime, 0?κa<κcr⋅a=1.44763516... and (2) a ’’hard’’-vortex (climatological) regime, κ≳κcr. The nonlinear part of this stability investigation has been carried out mainly by numerical computation.

Interacting Motion of Rectilinear Geostrophic Vortices

The kinematic motion of (N+1) geostrophic (Bessel) vortices is studied. Initially, the vortices are positioned near a uniformly rotating equilibrium configuration which consists of N vortices of equal strength γ equally spaced on a circle of radius a, and one vortex of strength γ0 at the center of the circle. One length scale κ−1 is associated with each of the vortices, the limiting case κ = 0being the classical logarithmic vortex. The linearized stability of this equilibrium configuration is studied over the range of the three parameters: N≥2, 0≤ (κa) < ∞, and − ∞ <(γ0/κ) < ∞. Over the entire range of the parameters, there are double-zero eigenvalues, which imply that the motion of the center vortex is inherently nonlinear. Numerical computations show that nonlinear effects are important in a parametric transition region embracing both the stable and unstable sides of the neutral stability curves for the circle vortices, even for small initial perturbations. The stability of the center vortex is not well represented by the neutral stability curves. In addition, the nonlinear motion away from an unstable equilibrium toward a stable equilibrium configuration is studied numerically.

Interacting Motion of Rectilinear Geostrophic Vortices

The kinematic motion of (N+1) geostrophic (Bessel) vortices is studied. Initially, the vortices are positioned near a uniformly rotating equilibrium configuration which consists of N vortices of equal strength γ equally spaced on a circle of radius a, and one vortex of strength γ0 at the center of the circle. One length scale κ−1 is associated with each of the vortices, the limiting case κ = 0being the classical logarithmic vortex. The linearized stability of this equilibrium configuration is studied over the range of the three parameters: N≥2, 0≤ (κa) < ∞, and − ∞ <(γ0/κ) < ∞. Over the entire range of the parameters, there are double‐zero eigenvalues, which imply that the motion of the center vortex is inherently nonlinear. Numerical computations show that nonlinear effects are important in a parametric transition region embracing both the stable and unstable sides of the neutral stability curves for the circle vortices, even for small initial perturbations. The stability of the center vortex is not well...

Interaction of a uniform wind stress with a geostrophic vortex : Stern Melvin E., 1965. Deep-Sea Res., 12 (3): 355–367

Interaction of a uniform wind stress with a geostrophic vortex : Stern Melvin E., 1965. Deep-Sea Res., 12 (3): 355–367

Interaction of a uniform wind stress with a geostrophic vortex

We consider the weak (small Rossby numbers) non-linear interaction between a quasi-geostrophic vortex and the Ekman flow which is produced by a uniform wind stress acting at the top of the mixed layer. The differential advection of geostrophic vorticity by the undisturbed Ekman components tends to tilt the axis of the vortex away from the vertical but, because of the strong rotational constraint, the vortex maintians vertical ‘rigidity’ by developing vertical velocities which balance the advective tendency (and frictional rotation). In this approximate vorticity balance for a ‘frictional’ component of the total motion, the vertical integral implies a net vertical velocity (‘Ekman suction’) at the free surface. To maintain the proper boundary condition an equal and opposite velocity must be applied to a second (geostrophic) component of the motion. This coupling condition has been determined as a function of the surface wind stress, the gradient of geostrophic vorticity, and the depth of the homogeneous mixed layer. If the base of the mixed layer were rigid the vortex would move as a unit with the velocity of the mean Ekman drift. The suction velocity formula is then generalized to include the special case ( Charney 1955) of a global wind variation and variable Coriolis parameter ( Sverdrup 1947). The effect of the local wind in supplying kinetic energy to the deeper ocean is also discussed, and suggestions are made for including the baroclinicity of the friction layer into the geostrophic dynamics.