Vortex Dipole Formation by Baroclinic Instability of Boundary Currents

2007 ◽  
Vol 37 (6) ◽  
pp. 1661-1677 ◽  
Author(s):  
L. Chérubin ◽  
X. Carton ◽  
D. G. Dritschel
Keyword(s):  
Potential Vorticity ◽  
Baroclinic Instability ◽  
Point Vortex ◽  
Wave Dispersion ◽  
Finite Amplitude ◽  
Layer Potential ◽  
Weakly Nonlinear ◽  
In Situ Data ◽  
Vortex Dipole ◽  
Quasigeostrophic Model

Abstract In situ data of the Mediterranean Water undercurrents and eddies south of Portugal indicate that the undercurrents have a tubelike structure in potential vorticity and that dipole formation can occur when the lower undercurrent extends seaward below an offshore upper countercurrent. A two-layer quasigeostrophic model is used to determine the dynamical conditions under which dipole formation is possible. With piecewise-constant potential vorticity, the flow exhibits two linear modes of instability comparable to those found in the Phillips model with topography. Weakly nonlinear analysis and fully nonlinear simulations of the flow evolution agree on the regimes of either finite-amplitude perturbation saturation, corresponding to filamentation, or amplification, corresponding to vortex or dipole formation. This latter regime is more specifically studied: vortex dipole formation and ejection from the coast is obtained for long waves, with opposite-signed but similar amplitude layer potential vorticities. A simple point vortex model reproduces this phenomenon under the same conditions. It is then shown that dipole formation occurs for minimal wave dispersion, and hence for weak horizontal velocity shears. As observed at sea, dipoles are formed when the lower potential vorticity core extends seaward below a countercurrent.

On the instability of geostrophic vortices

1988 ◽  
Vol 197 ◽  
pp. 349-388 ◽  
Author(s):  
Glenn R. Flierl
Keyword(s):  
Potential Vorticity ◽  
Large Scale ◽  
Baroclinic Instability ◽  
Finite Amplitude ◽  
Outer Radius ◽  
The Other ◽  
Nonlinear Stabilization ◽  
Weakly Nonlinear ◽  
Outer Annulus ◽  
Baroclinic Currents

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.

Finite-Amplitude Baroclinic Instability of Time-Varying Abyssal Currents

2006 ◽  
Vol 36 (1) ◽  
pp. 122-139 ◽  
Author(s):  
Seung-Ji Ha ◽  
Gordon E. Swaters
Keyword(s):  
Baroclinic Instability ◽  
Soliton Solutions ◽  
Zonal Flow ◽  
Finite Amplitude ◽  
Marginal Stability ◽  
Time Variability ◽  
Time Varying ◽  
Weakly Nonlinear ◽  
Abyssal Current ◽  
The Stability

Abstract The weakly nonlinear baroclinic instability characteristics of time-varying grounded abyssal flow on sloping topography with dissipation are described. Specifically, the finite-amplitude evolution of marginally unstable or stable abyssal flow both at and removed from the point of marginal stability (i.e., the minimum shear required for instability) is determined. The equations governing the evolution of time-varying dissipative abyssal flow not at the point of marginal stability are identical to those previously obtained for the Phillips model for zonal flow on a β plane. The stability problem at the point of marginally stability is fully nonlinear at leading order. A wave packet model is introduced to examine the role of dissipation and time variability in the background abyssal current. This model is a generalization of one introduced for the baroclinic instability of zonal flow on a β plane. A spectral decomposition and truncation leads, in the absence of time variability in the background flow and dissipation, to the sine–Gordon solitary wave equation that has grounded abyssal soliton solutions. The modulation characteristics of the soliton are determined when the underlying abyssal current is marginally stable or unstable and possesses time variability and/or dissipation. The theory is illustrated with examples.

scholarly journals On the Dynamics of Flows Induced by Topographic Ridges

2015 ◽  
Vol 45 (3) ◽  
pp. 927-940 ◽  
Author(s):  
Changheng Chen ◽  
Igor Kamenkovich ◽  
Pavel Berloff
Keyword(s):  
Potential Vorticity ◽  
Mesoscale Eddies ◽  
Dynamical Analysis ◽  
Far Field ◽  
Layer Potential ◽  
Zonal Jets ◽  
Vorticity Source ◽  
Quasigeostrophic Model ◽  
Stable Current

AbstractThis study describes a nonlocal mechanism for the generation of oceanic alternating jets by topographic ridges. The dynamics of these jets is examined using a baroclinic quasigeostrophic model configured with an isolated meridional ridge. The zonal topographic slopes of the ridge lead to the formation of a system of currents, consisting of mesoscale eddies, meridional currents over the ridge, and multiple zonal jets in the far field. Dynamical analysis shows that transient eddies are vital in sustaining the deep meridional currents over the ridge, which in turn play a key role in the upper-layer potential vorticity (PV) balance. The zonal jets in the rest of the domain owe their existence to the eddy forcing over the ridge but are maintained by the local Reynolds and form stress eddy forcing. The analysis further shows that a broad stable current that either becomes locally nonzonal or encounters a topographic ridge tends to become unstable. This instability provides a vorticity source and generates multiple zonal jets in the far field through a nonlocal mechanism.

On the radiation damping of finite-amplitude progressive edge waves

1990 ◽  
Vol 431 (1883) ◽  
pp. 419-431 ◽  
Keyword(s):  
Wave Theory ◽  
Wave Dispersion ◽  
Wave Mode ◽  
Finite Amplitude ◽  
Edge Wave ◽  
Water Model ◽  
Edge Waves ◽  
Transfer Energy ◽  
Oblique Waves ◽  
Weakly Nonlinear

Using small-amplitude expansions, it is demonstrated that weakly nonlinear periodic edge waves, travelling along the shoreline of a beach, can be attenuated owing to radiation of oblique waves out to sea. A few beach profiles, for which edge-wave dispersion relations are known in closed form, are discussed, and necessary conditions are determined for such radiation to occur due to nonlinear self-interactions. In particular, it is shown that quadratic nonlinear interactions cause the second edge-wave mode on a uniformly sloping beach of slope α to radiate when 1/18π < α < ⅙π; a detailed derivation to find the amplitude of the radiated wave and the attendant decay rate of the edge wave is presented, using the full water-wave theory. Also, it is pointed out that a concomitant nonlinear mechanism can transfer energy from incoming oblique waves to subharmonic edge waves – a plausible mechanism for the generation of travelling edge waves in coastal waters – and the details of this process are discussed within the framework of a shallow-water model.

scholarly journals On the Poleward Motion of Midlatitude Cyclones in a Baroclinic Meandering Jet

2013 ◽  
Vol 70 (8) ◽  
pp. 2629-2649 ◽  
Author(s):  
Ludivine Oruba ◽  
Guillaume Lapeyre ◽  
Gwendal Rivière
Keyword(s):  
Potential Vorticity ◽  
Statistical Study ◽  
Rossby Waves ◽  
Lower Layer ◽  
Finite Amplitude ◽  
Synoptic Scale ◽  
Beta Plane ◽  
The Cross ◽  
Quasigeostrophic Model ◽  
Pv Gradient

Abstract The motion of surface depressions evolving in a background meandering baroclinic jet is investigated using a two-layer quasigeostrophic model on a beta plane. Synoptic-scale finite-amplitude cyclones are initialized in the lower and upper layer to the south of the jet in a configuration favorable to their baroclinic interaction. The lower-layer cyclone is shown to move across the jet axis from its warm-air to cold-air side. It is the presence of a poleward-oriented barotropic potential vorticity (PV) gradient that makes possible the cross-jet motion through the beta-drift mechanism generalized to a baroclinic atmospheric context. The potential vorticity gradient associated with the jet is responsible for the dispersion of Rossby waves by the cyclones and the development of an anticyclonic anomaly in the upper layer. This anticyclone forms a PV dipole with the upper-layer cyclone that nonlinearly advects the lower-layer cyclone across the jet. In addition, the background deformation is shown to modulate the cross-jet advection. Cyclones evolving in a deformation-dominated environment (south of troughs) are strongly stretched while those evolving in a rotation-dominated environment (south of ridges) remain quasi isotropic. It is shown that the more stretched cyclones trigger a more efficient dispersion of energy, create a stronger upper-layer anticyclone, and move perpendicularly to the jet faster than the less stretched ones. Both the intensity and location of the upper-layer anticyclone explain the distinct cross-jet speeds. A statistical study consisting in initializing cyclones at different locations south of the jet core confirms that the cross-jet motion is faster for the more meridionally elongated cyclones evolving in areas of strongest barotropic PV gradient.

scholarly journals Quasigeostrophic Turbulence with Explicit Surface Dynamics: Application to the Atmospheric Energy Spectrum

2009 ◽  
Vol 66 (2) ◽  
pp. 450-467 ◽  
Author(s):  
Ross Tulloch ◽  
K. Shafer Smith
Keyword(s):  
Surface Temperature ◽  
Potential Vorticity ◽  
Baroclinic Instability ◽  
Temperature Gradients ◽  
Vertical Shear ◽  
Model Driven ◽  
Function Solution ◽  
Mean Temperature ◽  
Small Scales ◽  
Quasigeostrophic Model

Abstract The horizontal wavenumber spectra of wind and temperature near the tropopause have a steep −3 slope at synoptic scales and a shallower −5/3 slope at mesoscales, with a transition between the two regimes at a wavelength of about 450 km. Here it is demonstrated that a quasigeostrophic model driven by baroclinic instability exhibits such a transition near its upper boundary (analogous to the tropopause) when surface temperature advection at that boundary is properly resolved and forced. To accurately represent surface advection at the upper and lower boundaries, the vertical structure of the model streamfunction is decomposed into four parts, representing the interior flow with the first two neutral modes, and each surface with its Green’s function solution, resulting in a system with four prognostic equations. Mean temperature gradients are applied at each surface, and a mean potential vorticity gradient consisting both of β and vertical shear is applied in the interior. The system exhibits three fundamental types of baroclinic instability: interactions between the upper and lower surfaces (Eady type), interactions between one surface and the interior (Charney type), and interactions between the barotropic and baroclinic interior modes (Phillips type). The turbulent steady states that result from each of these instabilities are distinct, and those of the former two types yield shallow kinetic energy spectra at small scales along those boundaries where mean temperature gradients are present. When both mean interior and surface gradients are present, the surface spectrum reflects a superposition of the interior-dominated −3 slope cascade at large scales, and the surface-dominated −5/3 slope cascade at small scales. The transition wavenumber depends linearly on the ratio of the interior potential vorticity gradient to the surface temperature gradient, and scales with the inverse of the deformation scale when β = 0.

Stability of a potential vorticity front: from quasi-geostrophy to shallow water

1996 ◽  
Vol 315 ◽  
pp. 65-84 ◽  
Author(s):  
E. Boss ◽  
N. Paldor ◽  
L. Thompson
Keyword(s):  
Shallow Water ◽  
Potential Vorticity ◽  
Lower Layer ◽  
Baroclinic Instability ◽  
Resonant Interaction ◽  
Short Wave ◽  
Intersection Point ◽  
Rossby Number ◽  
Layer Potential ◽  
Long Wave

The linear stability of a simple two-layer shear flow with an upper-layer potential vorticity front overlying a quiescent lower layer is investigated as a function of Rossby number and layer depths. This flow configuration is a generalization of previously studied flows whose results we reinterpret by considering the possible resonant interaction between waves. We find that instabilities previously referred to as ‘ageostrophic’ are a direct extension of quasi-geostrophic instabilities.Two types of instability are discussed: the classic long-wave quasi-geostrophic baroclinic instability arising from an interaction of two vortical waves, and an ageostrophic short-wave baroclinic instability arising from the interaction of a gravity wave and a vortical wave (vortical waves are defined as those that exist due to the presence of a gradient in potential vorticity, e.g. Rossby waves). Both instabilities are observed in oceanic fronts. The long-wave instability has length scale and growth rate similar to those found in the quasi-geostrophic limit, even when the Rossby number of the flow is O(1).We also demonstrate that in layered shallow-water models, as in continuously stratified quasi-geostrophic models, when a layer intersects the top or bottom boundaries, that layer can sustain vortical waves even though there is no apparent potential vorticity gradient. The potential vorticity gradient needed is provided at the top (or bottom) intersection point, which we interpret as a point that connects a finite layer with a layer of infinitesimal thickness, analogous to a temperature gradient on the boundary in a continuously stratified quasi-geostrophic model.

scholarly journals Mathematical Modeling of Vortex Interaction Using a Three-Layer Quasigeostrophic Model. Part 1: Point-Vortex Approach

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1228 ◽  
Author(s):  
Mikhail A. Sokolovskiy ◽  
Xavier J. Carton ◽  
Boris N. Filyushkin
Keyword(s):  
Mathematical Modeling ◽  
Model Theory ◽  
Numerical Experiments ◽  
Point Vortex ◽  
Point Vortices ◽  
The Self ◽  
Vortex Interaction ◽  
Vortex Dipole ◽  
Quasigeostrophic Model ◽  
Surface Vortex

The theory of point vortices is used to explain the interaction of a surface vortex with subsurface vortices in the framework of a three-layer quasigeostrophic model. Theory and numerical experiments are used to calculate the interaction between one surface and one subsurface vortex. Then, the configuration with one surface vortex and two subsurface vortices of equal and opposite vorticities (a subsurface vortex dipole) is considered. Numerical experiments show that the self-propelling dipole can either be captured by the surface vortex, move in its vicinity, or finally be completely ejected on an unbounded trajectory. Asymmetric dipoles make loop-like motions and remain in the vicinity of the surface vortex. This model can help interpret the motions of Lagrangian floats at various depths in the ocean.

Evolution of baroclinic wave packets in a flow with continuous shear and stratification

1981 ◽  
Vol 377 (1771) ◽  
pp. 379-404 ◽  
Keyword(s):  
Evolution Equations ◽  
Gordon Equation ◽  
Wave Packets ◽  
Baroclinic Instability ◽  
Wave Dispersion ◽  
Baroclinic Wave ◽  
Sine Gordon Equation ◽  
Weakly Nonlinear ◽  
Boundary Slope ◽  
Continuous Shear

Evolution equations are developed for weakly nonlinear baroclinic wave packets in a flow with continuous shear and stratification. When wave dispersion is forced by using sloping upper and lower boundaries it is shown that, for boundary slope equal to isothermal slope in the basic state, these evolution equations are transformable to the sine-Gordon equation. The relation between behaviour in this model and that in the discretelayer models of baroclinic instability is discussed, and finally solutions of the sine-Gordon equation appropriate to a periodic domain are presented and compared with experimental results.

Nonlinear evolution of the elliptical instability: an example of inertial wave breakdown

1999 ◽  
Vol 396 ◽  
pp. 73-108 ◽  
Author(s):  
D. M. MASON ◽  
R. R. KERSWELL
Keyword(s):  
Finite Amplitude ◽  
Complex Conjugate ◽  
Small Scale ◽  
Conjugate Pair ◽  
Weakly Nonlinear ◽  
Elliptical Instability ◽  
Elliptical Flow ◽  
Complex Conjugate Pair ◽  
Generic Instability ◽  
Secondary Instabilities

A direct numerical simulation is presented of an elliptical instability observed in the laboratory within an elliptically distorted, rapidly rotating, fluid-filled cylinder (Malkus 1989). Generically, the instability manifests itself as the pairwise resonance of two different inertial modes with the underlying elliptical flow. We study in detail the simplest ‘subharmonic’ form of the instability where the waves are a complex conjugate pair and which at weakly supercritical elliptical distortion should ultimately saturate at some finite amplitude (Waleffe 1989; Kerswell 1992). Such states have yet to be experimentally identified since the flow invariably breaks down to small-scale disorder. Evidence is presented here to support the argument that such weakly nonlinear states are never seen because they are either unstable to secondary instabilities at observable amplitudes or neighbouring competitor elliptical instabilities grow to ultimately disrupt them. The former scenario confirms earlier work (Kerswell 1999) which highlights the generic instability of inertial waves even at very small amplitudes. The latter represents a first numerical demonstration of two competing elliptical instabilities co-existing in a bounded system.

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