Numerical Study on Vortices in the Middle Layer of Flow around a Large Mountain under Rotating Stratified Conditions

Generation of cyclonic vortices in the middle layer of flow around a large mountain like Tibet and Rocky was investigated by means of a 3-D nonhydrostatic meteorological prognostic model. Special attention was paid to the effects of the earth’s rotation and stratification on the vortices detached successively from the slope of a high and large horizontal scale mountain. It was found the successive formation and detachment of such ‘von Karman-like vortices’ occurred in the flow regime at high Rossby numbers Ro and low Froude numbers Fr. It was successfully divided by the criterion of baroclinic instability. This means that if the condition is unstable baroclinically, a lee vortex is destabilized into a three-dimensional one, while under baroclinically stable conditions the lee vortex with vertical axis retains its standing structure and remains long lasting in the middle layer.     

Influence of condensation and latent heat release upon barotropic and baroclinic instabilities of vortices in a rotating shallow water f-plane model

Analysis of the influence of condensation and related latent heat release upon developing barotropic and baroclinic instabilities of large-scale low Rossby-number shielded vortices on the f-plane is performed within the moist-convective rotating shallow water model, in its barotropic (one-layer) and baroclinic (two-layer) versions. Numerical simulations with a high-resolution well-balanced finite-volume code, using a relaxation parameterisation for condensation, are made. Evolution of the instability in four different environments, with humidity (i) behaving as passive scalar, (ii) subject to condensation beyond a saturation threshold, (iii) subject to condensation and evaporation, with three different parameterisations of the latter, are inter-compared. The simulations are initialised with unstable modes determined from the detailed linear stability analysis in the “dry” version of the model. In a configuration corresponding to low-level mid-latitude atmospheric vortices, it is shown that the known scenario of evolution of barotropically unstable vortices, consisting in formation of a pair of dipoles (dipolar breakdown) is substantially modified by condensation and related moist convection, especially in the presence of surface evaporation. No enhancement of the instability due to precipitation was detected in this case. Cyclone-anticyclone asymmetry with respect to sensitivity to the moist effects is evidenced. It is shown that inertia-gravity wave emission during the vortex evolution is enhanced by the moist effects. In the baroclinic configuration corresponding to idealised cut-off lows in the atmosphere, it is shown that the azimuthal structure of the leading unstable mode is sensitive to the details of stratification. Scenarios of evolution are completely different for different azimuthal structures, one leading to dipolar breaking, and another to tripole formation. The effects of moisture considerably enhance the perturbations in the lower layer, especially in the tripole formation scenario.     

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 exchange occurs if the initial ratio of the breadths is sufficiently extreme.

Finally the transport of passive fluid by a translating baroclinic pair is investigated. A pair of vortices in the top layer transports no lower layer fluid if the distance between the vortices is less than 1.72 deformation radii. By contrast, the size of the region trapped by a heton increases without bound as the spacing between the vortices increases.     

Submesoscale vortex structures at the entrance of the Gulf of Lions in the Northwestern Mediterranean Sea

The meanders of a baroclinic coastal current in the Northwestern Mediterranean Sea have already been reported in the literature. These meanders can be surrounded by vortices. Such vortices have been observed in the western part of the Gulf of Lions but the location and the mechanism of their formation are poorly documented. In this paper, we use the current measurements of a one-year experiment, which was conducted in the eastern part of the Gulf of Lions to detect and characterize the vortex activity. A vortex detection algorithm based on few velocity data was developed. Current measurements were available at the sea surface (HF radars) and in the water column from 50 to 140 m depth (four current meter moorings). SST images and hydrologic data were also used. Results focus on observations that are coherent 50 m and at the surface. Vortices are anticyclonic, of submesoscale size and present maximal velocities of 30–50 cm/s. The drift speed of the vortices is comparable to but less than the velocity of the Northern Current. These observations enable to estimate the minimum vortex occurrence in this area. The presence of vortex structures is strongly correlated with a specific sequence of wind patterns.     

Point vortex dynamics for coupled surface/interior QG and propagating heton clusters in models for ocean convection

Abstract

The dynamic behavior of baroclinic point vortices in two-layer quasi-geostrophic flow provides a compact model for studying the transport of heat in a variety of geophysical flows including recent heton models for open ocean convection as a response to spatially localized intense surface cooling. In such heton models, the exchange of heat with the region external to the compact cooling region reaches a statistical equilibrium through the propagation of tilted heton clusters. Such tilted heton clusters are aggregates of cyclonic vortices in the upper layer and anti-cyclonic vortices in the lower layer which collectively propagate almost as an elementary tilted heton pair even though the individual vortices undergo shifts in their relative locations. One main result in this paper is a mathematical theorem demonstrating the existence of large families of long-lived propagating heton clusters for the two-layer model in a fashion compatible to a remarkable degree with the earlier numerical simulations. Two-layer quasi-geostrophic flow is an idealization of coupled surface/interior quasi-geostrophic flow. The second family of results in this paper involves the systematic development of Hamiltonian point vortex dynamics for coupled surface/interior QG with an emphasis on propagating solutions that transport heat. These are novel vortex systems of mixed species where surface heat particles interact with quasi-geostrophic point vortices. The variety of elementary two-vortex exact solutions that transport heat include two surface heat particles of opposite strength, tilted pairs of a surface heat particle coupled to an interior vortex of opposite strength and two interior tilted vortices of opposite strength at different depths. The propagation speeds of the tilted elementary hetons in the coupled surface/interior QG model are compared and contrasted with those in the simpler two-layer heton models. Finally, mathematical theorems are presented for the existence of large families of propagating long-lived tilted heton clusters for point vortex solutions in coupled surface/interior QG flow.     

Influence of friction on the motion of quasigeostrophic vortices

Abstract

Who show through numerical experiments that friction may enhance westward motion of vortices on the beta-plane and establish a relation betwen the dissipation coefficients and westward aceleration. This suggests an explanation fot the dunamics of westward intensification.     

A steady nonlinear and singular vortex Rossby wave within a rapidly rotating vortex. Part II: Application to geophysical vortices

In a previous paper, Caillol [Geophys. Astrophys. Fluid Dyn., 2014, 108] investigated the steady nonlinear vortical structure of a singular vortex Rossby mode that has survived to a strong critical-layer-like interaction with a linearly stable, columnar, axisymmetric and dry vortex. We presented a general theory for this wave/mean flow interaction through the nonlinear critical layer theory and calculated the mean azimuthal and axial winds induced at the critical radius at the end of this interaction in the final stage. We here apply that theory to rapidly rotating geophysical vortices: tropical cyclones, cold-air mesocyclones and tornadoes. We find that the numerous assumptions invoked in that paper agree well with the reality of those intense vortices. We also find that in spite of a lack of moist-convection modelling, this dry vortex is fairly well accelerated at the critical radius by such a shear wave with a magnitude of order the square root of the damped-wave amplitude. The intensification level strongly depends on the aspect ratio, height of the system: rapid vortex and parent vortex, over core radius. The thinner the vortex is, the sharper the intensification is. This result is in sharp contrast to the numerous numerical simulations on VR wave/vortex interactions that yield a much smaller intensification of order the square of the wave amplitude. This weakly nonlinear approach nevertheless fails to model small vertical wavelength VR wave/vortex interactions for their related asymptotic expansions are divergent and for they yield strongly nonlinear VR waves coupled with evolving critical layers whose extent can no longer be considered as thin.     

Linear theory of the response of a two layer ocean to a moving hurricane‡

Abstract

This paper describes the linear response of an inviscid two‐layer model of a deep ocean on an f‐plane to a hurricane translating across the surface at constant speed. The forcing is a localized, radially‐symmetric pattern of positive wind stress curl and negative pressure anomaly. Only the steady state response is considered. The principal result is the identification of an internal wake in the lee of the storm, present when the translation speed of the storm exceeds the baroclinic long wave speed. The amplitude of the wake depends on the length of time over which the stress is experienced at a given point. The angle of the wedge filled by the wake is small, an effect due to the fact that the scale of a hurricane is typically larger than the baroclinic radius of deformation. After the wake disperses, a geostrophically balanced baroclinic ridge remains along the storm track.     

The effect of flow at Maud Rise on the sea-ice cover – numerical experiments

 The role of seamounts in the formation and evolution of sea ice is investigated in a series of numerical experiments with a coupled sea ice–ocean model. Bottom topography, stratification and forcing are configured for the Maud Rise region in the Weddell Sea. The specific flow regime that develops at the seamount as the combined response to steady and tidal forcing consists of free and trapped waves and a vortex cap, which is caused by mean flow and tidal flow rectification. The enhanced variability through tidal motion in particular modifies the mixed layer above the seamount enough to delay and reduce sea-ice formation throughout the winter. The induced sea-ice anomaly spreads and moves westward and affects an area of several 100 000 km². Process studies reveal the complex interaction between wind, steady and periodic ocean currents: all three are required in the process of generation of the sea ice and mixed layer anomalies (mainly through tidal flow), their detachment from the topography (caused by steady oceanic flow) and the westward translation of the sea-ice anomaly (driven by the time-mean wind).     

A Formal Theory for Vortex Rossby Waves and Vortex Evolution

A formal theory is presented for the balanced evolution of a small-amplitude, small-scale wave field in the presence of an axisymmetric vortex initially in gradient-wind balance and the accompanying changes induced in the vortex by the azimuthally averaged wave fluxes. The theory is a multi-parameter, asymptotic perturbation expansion for the conservative, rotating, f-plane, shallow-water equations. It extends previous work on Rossby-wave dynamics in vortices and more generally provides a new perspective on wave/mean-flow interaction in finite Rossby-number regimes. Some illustrative solutions are presented for a perturbed vortex undergoing axisymmetrization.     

The instability of barotropic circular vortices

Abstract

The linear, normal mode instability of barotropic circular vortices with zero circulation is examined in the f-plane quasigeostrophic equations. Equivalents of Rayleigh's and Fjortoft's criteria and the semicircle theorem for parallel shear flow are given, and the energy equation shows the instability to be barotropic. A new result is that the fastest growing perturbation is often an internal instability, having a finite vertical scale, but may also be an external instability, having no vertical structure. For parallel shear flow the fastest growing perturbation is always an external instability; this is Squire's theorem. Whether the fastest growing perturbation is internal or external depends upon the profile: for mean flow streamfunction profiles which monotonically decrease with radius, the instability is internal for less steep profiles with a broad velocity extremum and external for steep profiles with a narrow velocity extremum. Finite amplitude, numerical model calculations show that this linear instability analysis is not valid very far into the finite amplitude range, and that a barotropic vortex, whose fastest growing perturbation is internal, is vertically fragmented by the instability.     

Vortex evolution due to straining: a mechanism for dominance of strong,interior anticyclones

In this article we address two questions: Why do freely evolving vortices weaken on average, even when the viscosity is very small? Why, in the fluid's interior, away from vertical boundaries and under the influence of Earth's rotation and stable density stratification, do anticyclonic vortices become dominant over cyclonic ones when the Rossby number and deformation radius are finite? The context for answering these questions is a rotating, conservative, Shallow-water model with Asymmetric and Gradient-wind Balance approximations. The controlling mechanisms are vortex weakening under straining deformation (with a weakening that is substantially greater for strong cyclones than strong anticyclones) followed by a partially compensating vortex strengthening during a relaxation phase dominated by Vortex Rossby Waves (VRWs) and their eddy–mean interaction with the vortex. The outcome is a net, strain-induced vortex weakening that is greater for cyclones than anticyclones when the deformation radius is not large compared to the vortex radius and the Rossby number is not small. Furthermore, when the exterior strain flow is sustained, the vortex changes also are sustained: for small Rossby number (i.e., the quasigeostrophic limit, QG), vortices continue to weaken at a relatively modest rate, but for larger Rossby number, cyclones weaken strongly and anticyclones actually strengthen systematically when the deformation radius is comparable to the vortex radius. The sustained vortex changes are associated with strain-induced VRWs on the periphery of the mean vortex. It therefore seems likely that, in a complex flow with many vortices, anticyclonic dominance develops over a sequence of transient mutual straining events due to the greater robustness of anticyclones (and occasionally their net strengthening).     

Slow quasigeostrophic unstable modes of a lens vortex in a continuously stratified flow

This work addresses the linear dynamics underlying the formation of density interfaces at the periphery of energetic vortices, well outside the vortex core, both in the radial and axial directions. We compute numerically the unstable modes of an anticyclonic Gaussian vortex lens in a continuously stratified rotating fluid. The most unstable mode is a slow mode, associated with a critical layer instability located at the vortex periphery. Although the most unstable disturbance has a characteristic vertical scale which is comparable to the vortex height, interestingly, the critical levels of the successively fastest growing modes are closely spaced at intervals along the axial direction that are much smaller than the vortex height.     

The formation of new quasi-stationary vortex patterns from the interaction of two identical vortices in a rotating fluid

Within the framework of the quasi-geostrophic approximation, the interactions of two identical initially circular vortex patches are studied using the contour dynamics/surgery method. The cases of barotropic vortices and of vortices in the upper layer of a two-layer fluid are considered. Diagrams showing the end states of vortex interactions and, in particular, the new regime of vortex triplet formation are constructed for a wide range of external parameters. This paper shows that, in the nonlinear evolution of two such (like-signed) vortices, the filaments and vorticity fragments surrounding the merged vortex often collapse into satellite vortices. Therefore, the conditions for the formation and the quasi-steady motions of a new type of triplet-shaped vortex structure are obtained.     

On periodic vortices over a concave semicircular wall fixed in a rotating tank

Abstract

New periodic vortices were observed during a rotating tank experiment, to be described. The peculiarities and the formation mechanism of the present vortices are based on observations. The vortices reach the complete form in stages, viz. laminar boundary layer growth, sinuous motion and rolling oscillation. After being formed, the vortices are shed periodically and the diameter of the vortices grows as they advance.

The vortices appear when the flow over the concave semicircular wall, fixed in the rotating tank, is maintained by the shear stresses at the inner surface of the rotating tank. The vortices and the flow were visualized with thymol blue dye.     

利用GS流场重构方法研究磁尾等离子体片涡流

2000年9月30日Geotail卫星分别于17∶54∶36~18∶09∶00UT和18∶59∶00~19∶30∶00UT在磁尾晨侧等离子体片内(n≈0.4 cm^-3,T≈6 keV)观测到等离子体涡流事件.本文采用Grad-Shafranov (GS)流场重构技术再现了这些涡流的二维速度场、离子数密度和离子温度的分布图像.结果显示:从地心太阳磁层坐标系(GSM)赤道面上面看, 涡流的尺度约为5000 km×1400 km , 朝地球的运动速度约为15~25 km/s.所有5个涡流的旋转方向都为顺时针方向,旋转周期约为6~11 min.相邻涡流的相互作用导致它们之间的磁场强度增强.考察观测数据发现,涡流内不仅包含等离子体片热等离子体成分,也包含较大通量的类似源自磁鞘的冷等离子体成分(T<1 keV).这与观测到涡流等离子体的平均温度(T≈4 keV)较磁尾等离子体片等离子体的典型温度(T≈6 keV)明显偏低的事实是一致的.不仅如此,离子数密度和温度在结构内的分布也不均匀,数密度在涡流内部偏离中心的位置比较低而在每个涡流的边缘位置比较高,温度的分布大体上与密度相反.分析认为观测到的磁尾等离子体涡流事件可能由发生在低纬边界层的Kelvin-Helmholtz不稳定性引起,涡流结构内的冷等离子体可能来自磁层顶外部的磁鞘.     

The symmetric baroclinic instability of an equatorial current

Abstract

The stability of a baroclinic zonal current to symmetric perturbations on an equatorial β-plane is considered. The fluid is assumed to be Boussinesq, inviseid, adiabatic, hydrostatic, and stably stratified. The solutions exhibit the same stability properties as those on an f-plane: instability occurs whenever Ri < 1/(1 + d), where Ri is the Richardson number and d is a measure of the horizontal shear of the current; the most unstable motions tend to parallel the isotherms of potential temperature; and they have infinitely small scales of variation perpendicular to the isotherms. The variation of Coriolis parameter leads to one important difference in the structure of the eigenfunctions: the rapidly growing modes are concentrated in high latitudes, and the slowly growing ones in low latitudes.

The suggestion that the symmetric cloud bands observed at low latitudes in Jupiter's atmosphere are caused by symmetric instabilities is re-examined in the light of these results. These cloud bands would have to be associated with the slowly-growing, low-latitude modes. These modes consist of small scale motions parallel to the isotherms, with the magnitude of the motions having a large scale modulation as a function of latitude. The time scales of these modes and the latitude scales of their modulation agree qualitatively with the observations of Jupiter's cloud bands, so long as Ri is not very close to zero or to its critical value.     

Statistical equilibrium distributions of baroclinic vortices in a rotating two-layer model at low Froude numbers

The point-vortex equilibrium statistical model of two-layer baroclinic quasigeostrophic vortices in an unbounded f-plane is examined. A key conserved quantity, angular momentum, serves to confine the vortices to a compact domain, thereby justifying the statistical mechanics model, and also eliminating the need for boundary conditions in a practical method for its resolution. The Metropolis method provides a fast and efficient algorithm for solving the mean field non-linear elliptic PDEs of the equilibrium statistical theory. A verification of the method is done by comparison with the exact Gaussian solution at the no interaction limit of zero inverse temperature. The numerical results include a geophysically and computationally relevant power law for the radii at which the most probable vortex distribution is non-vanishing: For fixed total circulation, and fixed average angular momentum, the radii of both layers are proportional to the square root of the inverse temperature β. By changing the chemical potentials μ of the runs, one is able to model the most probable vorticity distributions for a wide range of total circulation and energy. The most probable vorticity distribution obtained at low positive temperatures are consistently close to a radially symmetric flat-top profiles. At high temperatures, the radially symmetric vorticity profiles are close to the Gaussian distribution.     

Vortex pairs and power station cooling water

Cooling water discharged from power stations in the U.K. is frequently released from an outlet in an estuary or the sea. The warm water forms a thermal plume which is slightly buoyant and which spreads horizontally over the water surface while mixing vertically downwards with the cooler ambient water. In this paper, the possibility of vortex pair production at the cooling water outlet is considered as a mechanism contributing to this spreading of the warm water.The motion of a vortex pair contained between two rigid plane boundaries is an idealization of the flow between the water surface and the sea bed. The resulting motion is calculated from potential theory and viscous effects are neglected. The problem of deciding what strength to assign the vortices is discussed and specific consideration of shear and buoyancy at the outlet is detailed. It is observed that bifurcation of the vortex pair is determined by the initial position of the vortices and is unlikely to occur in conditions relevant to U.K. power station discharges.It is calculated that, in the absence of turbulence, the motion of such vortex pairs would result in horizontal spreading of the warm water which is greater than that observed at site surveys. It is concluded that turbulence in the ambient receiving water is sufficient to destroy vortices produced by the discharge during the early stages of the plume development.     

Magnetic Rossby waves in a stably stratified layer near the surface of the Earth's outer core

Abstract

The stratification profile of the Earth's magnetofluid outer core is unknown, but there have been suggestions that its upper part may be stably stratified. Braginsky (1984) suggested that the magnetic analog of Rossby (planetary) waves in this stable layer (the ‘H’ layer) may be responsible for a portion of the short-period secular variation. In this study, we adopt a thin shell model to examine the dynamics of the H layer. The stable stratification justifies the thin-layer approximations, which greatly simplify the analysis. The governing equations are then the Laplace's tidal equations modified by the Lorentz force terms, and the magnetic induction equation. We linearize the Lorentz force in the Laplace's tidal equations and the advection term in the magnetic induction equation, assuming a zeroth order dipole field as representative of the magnetic field near the insulating core-mantle boundary. An analytical β-plane solution shows that a magnetic field can release the equatorial trapping that non-magnetic Rossby waves exhibit. A numerical solution to the full spherical equations confirms that a sufficiently strong magnetic field can break the equatorial waveguide. Both solutions are highly dissipative, which is a consequence of our necessary neglect of the induction term in comparison with the advection and diffusion terms in the magnetic induction equation in the thin-layer limit. However, were one to relax the thin-layer approximations and allow a radial dependence of the solutions, one would find magnetic Rossby waves less damped (through the inclusion of the induction term). For the magnetic field strength appropriate for the H layer, the real parts of the eigenfrequencies do not change appreciably from their non-magnetic values. We estimate a phase velocity of the lowest modes that is rather rapid compared with the core fluid speed typically presumed from the secular variation.