Vortex evolution

Abstract

Friedmann's equation and the potential vorticity equation are generalised for turbulent motion. The generalised equations incorporate some new phenomena connected with turbulent transport of mass. It is proved that, if ?×[S×Ω+S(?·S)]≠0 where Ω is the absolute vorticity of the velocity and S is the turbulent density flux, then the Helmholtz-Kelvin theorem concerning the conservation of the velocity circulation around a closed path is violated and the potential vorticity is not a Lagrangian adiabatic invariant. The effects of this turbulent transport of mass on the creation or dissipation of vorticity discussed here is not equivalent to effects of baroclinicity or viscosity. Some possible implications of the new circulation theorem in geophysical and astrophysical fluid dynamics are discussed.     

长时间强度维持台风内核区位涡的中尺度分布特征及诊断

本文分析长时间强度维持"菲特"台风不同发展阶段的位涡分布特征发现：台风内核区中尺度高值PV带及其变化与台风强度变化具有伴随关系,即高值PV区与内核区强对流不仅具有对应关系;而且其生命史与台风强盛维持期一致;此外在眼墙区附近位涡梯度最大.分析还指出：垂直剖面上的高值PV呈现由单极位涡态（台风发展加强期）向中空位涡态（台风强盛维持期）的转变,到台风快速衰减期,又形成PV量值较小的单极位涡态.位涡收支方程诊断表明：内核区域水平平流、垂直输送和凝结加热的初始增强和大值收支带不仅对台风内中尺度高值位涡分布及长时间强度维持具有重要影响,而且具有伴随关系.此外,位涡收支各项对位涡态的转变起着不同的作用,其中凝结加热在台风强盛期中空位涡塔的建立中作用明显,水平平流项则在眼墙区的位涡塔中上层有着正贡献,垂直输送在高值PV分布的再分配中起中介作用.     

Quasi-stationary striations in basin-scale oceanic circulation: vorticity balance from observations and eddy-resolving model

Recently, prominent jet-like features of the ocean circulation, called striations, with a meridional scale of O(300–500 ;km) and extending for thousands of kilometers in length, have been detected in satellite and in situ observations and in high-resolution numerical models. In this paper, we study quasi-stationary striations, which are best seen in the multi-year time-averaged velocity fields. Using 1993–2002 mean dynamic ocean topography, satellite altimeter observations, and output of the Ocean General Circulation Model for the Earth Simulator, the dynamics of the quasi-stationary striations in the eastern parts of the subtropical North and South Pacific are examined by assessing individual terms in the time-averaged equations of relative and potential vorticity. While non-linear effects are found to be essential in the dynamics of the striations, rejecting some linear hypotheses forwarded in the earlier studies, the relevance of the Rhines mechanism is not confirmed. Eddy flux does not act as a relative vorticity source for the striations. Using the potential vorticity (PV) diagnostics, we show that the time-mean PV is not conserved along the time-mean streamlines, and on the scale of the striations these changes in PV are largely induced by the eddy flux of layer thickness. The fact that eddy fluxes contribute to the striations’ time-mean PV budget suggests that the striations are not a kinematical artifact of time-averaging of westward-propagating eddies.     

First considerations on the structure and development of the Iberian thermal low-pressure system

During the summer a thermal low-pressure system is locked over the Iberian Peninsula. We present a first analysis of such a system using the potential vorticity approach. Our results show that its main characteristic is the existence of a negative potential vorticity (PV) dome and a funnel-like structure for potential temperature, both located at the centre of the low. The build-up and evolution of this PV dome can be understood in terms of the dot products of the absolute vorticity and the gradient of diabetic heating vectors and the curl of the friction forces and the gradient of potential temperature vectors. The inhibition of the Algerian Mediterranean cyclogenesis during the summer seems to bear some relation to the existence of this kind of low-pressure disturbance over the Iberian Peninsula.A part of this paper was presented as a poster at Session OA11 on “Cyclone Structure and Development”, EGS General Assembly in Wiesbaden, 3 - 7 May, 1993     

Piecewise uniform potential vorticity pancake shielded vortices

Shielded vortices consist of a core of potential vorticity (PV) of a given sign surrounded (or shielded) by a layer of opposite-signed PV. Such vortices have specific properties and have been the focus of numerous studies, first in two dimensional geometries (where PV is just the vertical component of the vorticity vector) and in geophysical applications (mostly in layered models). The present paper focuses on three-dimensional, spheroidal shielded vortices. In particular, we focus on vortical structures whose overall volume-integrated PV is zero. We restrict attention to vortices of piecewise uniform PV in the present research. We first revisit the problem within the quasi-geostrophic model, then we extend the results to the non-hydrostatic regime. We show that the stability of the structure depends on the ratio of PV between the inner core and the outer shield. In particular it depends on the polarity of the core and of the wavenumber of the azimuthal mode perturbed.     

The effects of asymmetric potential vorticity forcing on the instability of South Asia High and Indian summer monsoon onset

Based on the theory of potential vorticity(PV),the unstable development of the South Asia High(SAH)due to diabatic heating and its impacts on the Indian Summer Monsoon(ISM)onset are studied via a case diagnosis of 1998.The Indian Summer Monsoon onset in 1998 is related to the rapidly strengthening and northward moving of a tropical cyclone originally located in the south of Arabian Sea.It is demonstrated that the rapid enhancement of the cyclone is a consequence of a baroclinic development characterized by the phase-lock of high PV systems in the upper and lower troposphere.Both the intensification of the SAH and the development of the zonal asymmetric PV forcing are forced by the rapidly increasing latent heat released from the heavy rainfall in East Asia and South East Asia after the onsets of the Bay of Bengal(BOB)monsoon and the South China Sea(SCS)monsoon.High PV moves southwards along the intensified northerlies on the eastern side of the SAH and travels westwards on its south side,which can reach its northwest.Such a series of high PV eddies are transported to the west of the SAH continuously,which is the main source of PV anomalies in the upper troposphere over the Arabian Sea from late spring to early summer.A cyclonic curvature on the southwest of the SAH associated with increasing divergence,which forms a strong upper tropospheric pumping,is generated by the anomalous positive PV over the Arabian Sea on 355 K.The cyclone in the lower troposphere moves northwards from low latitudes of the Arabian Sea,and the upper-layer high PV extends downwards and southwards.Baroclinic development thus occurs and the tropical low-pressure system develops into an explosive vortex of the ISM,which leads to the onset of the ISM.In addition,evolution of subtropical anticyclone over the Arabian Peninsula is another important factor contributing to the onset of the ISM.Before the onset,the surface sensible heating on the Arabian Peninsula is very strong.Consequently the subtropical anticyclone which dominated the Arabian Sea in spring retreats westwards to the Arabian Peninsula and intensifies rapidly.The zonal asymmetric PV forcing develops gradually with high PV eddies moving southwards along northerlies on the eastern side of the anticyclone,and a high PV trough is formed in the middle troposphere over the Arabian Sea,which is favorable to the explosive barotropic development of the tropical cyclone into the vortex.Results from this study demonstrate that the ISM onset,which is different from the BOB and the SCS monsoon onset,is a special dynamical as well as thermodynamic process occurring under the condition of fully coupling of the upper,middle,and lower tropospheric circulations.     

On the three-dimensional oceanic density distribution in a millennium integration with an AO-GCM

 The three-dimensional time-mean density distribution in the ocean is determined not only by the time-mean fluxes of heat and freshwater at the sea surface, but also by time-mean vertical currents and time-mean density fluxes due to oceanic transients excited by fluctuating fluxes at the sea surface. The effects of these various processes on the global density fields are assessed using a balance equation of the variance of spatial density anomalies and a millennium integration with an atmosphere–ocean general circulation model. It is found that spatial density anomalies are generated by the time-mean heat fluxes at the sea surface and destroyed by the time-mean surface freshwater flux, by sinking of dense water and rising of less dense water, and finally by density fluxes associated with transients. The last two processes take place essentially in the oceanic interior. Since density fluxes of transient eddies act to reduce the existing density differences between the Atlantic/Southern Oceans and the other oceans, their presence could affect the global density balance, and from that the thermohaline circulation and the stability of this circulation. Received: 4 October 2001 / Accepted: 10 October 2002 Responsible Editor: Richard J. Greatbatch Acknowledgements I thank Ulrich Cubasch and his colleagues for providing me with the ECHAM3/LSG integration, Peter Müller and Richard Greatbatch for valuable suggestions.     

Solitary Rossby Waves in Zonally Varying Jet Flows

The dynamics of solitary Rossby waves (SRWs) embedded in a meridionally sheared, zonally varying background flow are examined using a non-divergent barotropic model centered on a midlatitude g -plane. The zonally varying background flow, which is produced by an external potential vorticity (PV) forcing, yields a modified Korteweg-de Vries (K-dV) equation that governs the spatial-temporal evolution of a disturbance field that contains both Rossby wave packets and SRWs. The modified K-dV equation differs from the classical equation in that the zonally varying background flow, which varies on the same scale as the disturbance field, directly affects the disturbance linear translation speed and linear growth characteristics. In the limit of a locally parallel background flow, equations governing the amplitude and propagation characteristics of SRWs are derived analytically. These equations show, for example, that a sufficiently large (small) translation speed and/or a sufficiently weak (strong) background zonal shear favor transmission (reflection) of the SRW through (from) the jet. Conservation equations are derived showing that time changes in the domain averaged amplitude ("mass") or squared amplitude ("momentum") are due to zonal variation in both the linear, long-wave phase speed and linear growth; dispersion and nonlinearity do not affect the "mass" or "momentum". Provided (1) the background PV forcing is sufficiently small, or (2) the background PV forcing is meridionally symmetric and the disturbance is a SRW, the dynamics of the disturbance field is Hamiltonian and mass and energy are thus conserved. Numerical solutions of the K-dV equation show that the zonally varying background flow yields three general classes of behavior: reflection, transmission, or trapping. Within each class there exists SRWs and Rossby wave packets. SRWs that become trapped within the zonally localized jet region may exhibit the following behaviors: (1) an oscillatory decay to a steady state at the jet center, (2) the creation of additional SRWs within the jet region, or (3) a steady-state wherein the solution has a smoothed step-like structure located downstream along the jet axis.     

Frontal circulation induced by up-front and coastal downwelling winds

Two-dimensional (cross-shelf and depth) circulation by downwelling wind in the presence of a prograding front (with isopycnals that slope in the same direction as the topographic slope) over a continental shelf is studied using high-resolution numerical experiments. The physical process of interest is the cross-shelf circulation produced by northeasterly monsoon winds acting on the Kuroshio front over the East China Sea outer shelf and shelfbreak where upwelling is often observed. However, a general problem is posed and solved by idealized numerical and analytical models. It is shown that upwelling is produced shoreward of the front. The upwelling is maintained by (1) a surface bulge of negative vorticity at the head of the front; (2) bottom offshore convergence beneath the front; and (3) in the case of a surface front that is thin relative to water depth, also by upwelling due to the vorticity sheet under the front. The near-coast downwelling produces intense mixing due to both upright and slant-wise convection in regions of positive potential vorticity. The analytical model shows that the size and on-shore propagating speed of the bulge are determined by the wind and its shape is governed by a nonlinear advection–dispersion equation which yields unchanging wave-form solutions. Successive bulges can detach from the front under a steady wind. Vertical circulation cells develop under the propagating bulges despite a stable stratification. These cells can have important consequences to vertical exchanges of tracers and water masses.     

Using a dynamical advection to reconstruct a part of the SSH evolution in the context of SWOT,application to the Mediterranean Sea

The main oceanographic objective of the future SWOT mission is to better characterize the ocean mesoscale and sub-mesoscale circulation, by observing a finer range of ocean topography dynamics down to 20 km wavelength. Despite the very high spatial resolution of the future satellite, it will not capture the time evolution of the shorter mesoscale signals, such as the formation and evolution of small eddies. SWOT will have an exact repeat cycle of 21 days, with near repeats around 5–10 days, depending on the latitude. Here, we investigate a technique to reconstruct the missing 2D SSH signal in the time between two satellite revisits. We use the dynamical interpolation (DI) technique developed by Ubelmann et al. (2015). Based on potential vorticity (hereafter PV) conservation using a one and a half layer quasi-geostrophic model, it features an active advection of the SSH field. This model has been tested in energetic open ocean regions such as the Gulf Stream and the Californian Current, and has given promising results. Here, we test this model in the Western Mediterranean Sea, a lower energy region with complex small scale physics, and compare the SSH reconstruction with the high-resolution Symphonie model. We investigate an extension of the simple dynamical model including a separated mean circulation. We find that the DI gives a 16–18% improvement in the reconstruction of the surface height and eddy kinetic energy fields, compared with a simple linear interpolation, and a 37% improvement in the Northern Current subregion. Reconstruction errors are higher during winter and autumn but statistically, the improvement from the DI is also better for these seasons.     

论被动源磁电勘探理论

本文从麦氏方程出发,运用电流磁场的等效理论阐明了在磁电勘探中,当矿体的二次极化电流具有垂直对称轴分布时,在地表面及以上空间中任一点的磁异常场（H）均等于零。因而,具有任何垂直对称状电化学矿产模式（石油、天然气、地下水及金属矿床等）的磁电勘探法异常场线积分均不存在。此外,由于对各种形状和任何极化倾角的矿体情况,地表面上均无电流密度垂直分量（i_n）存在,故地表面上异常场线积分均为零（∮_L H·dL=0）。因此,磁电勘探法不具备在地面上寻找此种矿产的物理前提。 本文还对倾斜极化球体和柱体的磁电勘探异常进行了计算,给出了主剖面上理论曲线的异常分布规律,指出了磁电异常的数量级。同时,还对井中磁电异常的分布规律给出了计算资料。为了对比,也对相应条件下的自然电场异常做了计算,指出了电场异常与磁场异常之间的不同特点及其对实际找矿的意义。     

Second order potential vorticity and its potential applications

A new invariant, the second order potential vorticity (SPV), is derived in this paper. SPV is the dot product of vorticity and the potential vorticity (PV) gradient, and is proven conservative for a compressible, adiabatic and frictionless atmosphere. Research shows that the new invariant may be used to indicate the evolution of PV, because SPV includes all the information that determines PV evolution: the wind field, and the PV gradient. Furthermore, SPV is capable of diagnosing heavy precipitation because of the strong signals it presents in areas of heavy rainfall. SPV also shows great potential as a comprehensive conserved quantity for indicating the dynamical tropopause and baroclinic instability.     

Models of interacting pairs of thin,quasi-geostrophic vortices: steady-state solutions and nonlinear stability

We study pairwise interactions of elliptical quasi-geostrophic (QG) vortices as the limiting case of vanishingly thin uniform potential vorticity ellipsoids. In this limit, the product of the vertical extent of the ellipsoid and the potential vorticity within it is held fixed to a finite non-zero constant. Such elliptical “lenses” inherit the property that, in isolation, they steadily rotate without changing shape. Here, we use this property to extend both standard moment models and Hamiltonian ellipsoidal models to approximate the dynamical interaction of such elliptical lenses. By neglecting non-elliptical deformations, the simplified models reduce the dynamics to just four degrees of freedom per vortex. For simplicity, we focus on pairwise interactions between identical elliptical vortices initially separated in both the horizontal and vertical directions. The dynamics of the simplified models are compared with the full QG dynamics of the system, and show good agreement as expected for sufficiently distant lenses. The results reveal the existence of families of steadily rotating equilibria in the initial horizontal and vertical separation parameter space. For sufficiently large vertical separations, equilibria with varying shape exist for all horizontal separations. Below a critical vertical separation (stretched by the constant ratio of buoyancy to Coriolis frequencies N / f), comparable to the mean radius of either vortex, a gap opens in horizontal separation where no equilibria are possible. Solutions near the edge of this gap are unstable. In the full QG system, equilibria at the edge of the gap exhibit corners (infinite curvature) along their boundaries. Comparisons of the model results with the full nonlinear QG evolution show that the early stages of the instability are captured by the Hamiltonian elliptical model but not by the moment model that inaccurately estimates shorter-range interactions.     

Gravity Anomalies of Arbitrary 3D Polyhedral Bodies with Horizontal and Vertical Mass Contrasts

During the last 15 years, more attention has been paid to derive analytic formulae for the gravitational potential and field of polyhedral mass bodies with complicated polynomial density contrasts, because such formulae can be more suitable to approximate the true mass density variations of the earth (e.g., sedimentary basins and bedrock topography) than methods that use finer volume discretization and constant density contrasts. In this study, we derive analytic formulae for gravity anomalies of arbitrary polyhedral bodies with complicated polynomial density contrasts in 3D space. The anomalous mass density is allowed to vary in both horizontal and vertical directions in a polynomial form of \(\lambda =ax^m+by^n+cz^t\), where m, n, t are nonnegative integers and a, b, c are coefficients of mass density. First, the singular volume integrals of the gravity anomalies are transformed to regular or weakly singular surface integrals over each polygon of the polyhedral body. Then, in terms of the derived singularity-free analytic formulae of these surface integrals, singularity-free analytic formulae for gravity anomalies of arbitrary polyhedral bodies with horizontal and vertical polynomial density contrasts are obtained. For an arbitrary polyhedron, we successfully derived analytic formulae of the gravity potential and the gravity field in the case of \(m\le 1\), \(n\le 1\), \(t\le 1\), and an analytic formula of the gravity potential in the case of \(m=n=t=2\). For a rectangular prism, we derive an analytic formula of the gravity potential for \(m\le 3\), \(n\le 3\) and \(t\le 3\) and closed forms of the gravity field are presented for \(m\le 1\), \(n\le 1\) and \(t\le 4\). Besides generalizing previously published closed-form solutions for cases of constant and linear mass density contrasts to higher polynomial order, to our best knowledge, this is the first time that closed-form solutions are presented for the gravitational potential of a general polyhedral body with quadratic density contrast in all spatial directions and for the vertical gravitational field of a prismatic body with quartic density contrast along the vertical direction. To verify our new analytic formulae, a prismatic model with depth-dependent polynomial density contrast and a polyhedral body in the form of a triangular prism with constant contrast are tested. Excellent agreements between results of published analytic formulae and our results are achieved. Our new analytic formulae are useful tools to compute gravity anomalies of complicated mass density contrasts in the earth, when the observation sites are close to the surface or within mass bodies.     

Wind-induced subduction at the South Atlantic subtropical front

The South Atlantic Subtropical Front, associated with the eastward-flowing South Atlantic Current, separates the colder, nutrient-rich waters of the subpolar gyre from the warmer, nutrient-poor waters of the subtropical gyre. Perturbations to the quasi-geostrophic, eastward flow generate meanders and filaments which induce cross-frontal exchange of water properties. Down-front winds transport denser waters from the South over warm waters from the North, inducing convective instability and subduction. Such processes occur over spatial scales of the order of 1 km and thus require high horizontal spatial resolution. In this modeling study, a high-resolution (4 km) regional grid is embedded in a basin-wide configuration (12 km) of the South Atlantic Ocean in order to test the importance of submesoscale processes in water mass subduction along the subtropical front. Stronger and more numerous eddies obtained in the high-resolution run yield more intense zonal jets along the frontal zone. Such stronger jets are more susceptible to instabilities, frontogenesis, and the generation of submesoscale meanders and filaments with \(\mathcal {O}(1)\) Rossby number. As a consequence, vertical velocities larger than 100 md ¹ are obtained in the high-resolution run, one order of magnitude larger than in the low-resolution run. Wind-driven subduction occurs along the frontal region, associated with negative Ertel potential vorticity in the surface layer. Such processes are not observed in the low-resolution run. A passive tracer experiment shows that waters with density characteristics similar to subtropical mode waters are preferentially subducted along the frontal region. The wind-driven buoyancy flux is shown to be much larger than thermal or haline fluxes during the wintertime, which highlights the importance of the frictional component in extracting PV from the surface ocean and inducing subduction, a process that has been overlooked in subtropical mode water formation in the region.     

Airside velocity measurements over the wind-sheared water surface using particle image velocimetry

An experimental investigation of the airflow structure in the near surface region over the wind-sheared air–water interface is reported. The two-dimensional velocity fields in a plane perpendicular to the water surface were measured using particle image velocimetry (PIV) technique over a wind speed range from 1.5 to 4.4 m s⁻¹. The results show a reduction in the mean velocity magnitudes and the tangential stresses when gravity waves appear on the surface. An enhanced vorticity layer was observed immediately above the water surface that extended to a height of approximately 2 cm. The vorticity was enhanced by an order of magnitude, and the energy dissipation rate was enhanced by a factor of 7 in this layer at all wind speeds. The vertical profiles of Reynolds stress, energy production, and dissipation indicate the contribution of surface waves in the enhanced transfer of momentum and energy between the two fluids. The results in this study show that the flow dynamics in a layer immediately adjacent to the water surface whose thickness is of the order of the significant wave height is significantly different from that at greater heights. Thus, it is concluded that the quantitative investigation of the flow in the immediate vicinity of the interface is vital for an improved understanding of the heat, mass, and momentum exchange between air and water. The present study demonstrates that PIV is an effective technique to accurately measure the velocity fields in this region.     

Statistical properties of an enstrophy conserving finite element discretisation for the stochastic quasi-geostrophic equation

ABSTRACT

A framework of variational principles for stochastic fluid dynamics was presented by Holm, and these stochastic equations were also derived by Cotter, Gottwald and Holm. We present a conforming finite element discretisation for the stochastic quasi-geostrophic equation that was derived from this framework. The discretisation preserves the first two moments of potential vorticity, i.e. the mean potential vorticity and the enstrophy. Following the work of Dubinkina and Frank, who investigated the statistical mechanics of discretisations of the deterministic quasi-geostrophic equation, we investigate the statistical mechanics of our discretisation of the stochastic quasi-geostrophic equation. We compare the statistical properties of our discretisation with the Gibbs distribution under assumption of these conserved quantities, finding that there is an agreement between the statistics under a wide range of set-ups.     

The role of boundary conditions in the simulation of rotating,stratified turbulence

Abstract

In this paper we use the CASL method to explore the role of boundary conditions in determining the long-time behaviour of rotating, stratified, quasi-geostrophic turbulence. We find that initially two-dimensional (sufficiently tall) columns of potential vorticity (PV) break down through three-dimensional instability to give a fully three-dimensional flow consisting of ellipsoidal structures. This is the case both for rigid-lid (isothermal) vertical boundary conditions and for vertically periodic boundaries. However, the rigid boundary case gives rise to semi-ellipsoids at both the top and bottom boundaries, and, for sufficient domain depths, preferred depths for the formation of ellipsoids in the interior. By contrast, in the vertically periodic case, the distribution of ellipsoids is homogeneous in depth.

The role of the horizontal boundaries is indirect, but still significant. In all cases doubly periodic horizontal boundary conditions are imposed. We consider a range of initial conditions where in each case equal numbers of two-dimensional columns of positive and negative vorticity are used, taking up a fixed, but relatively small fraction of the domain (approximately 5%). Thus when there is only a small number of vortices, they have larger radius. When the initial number of vortices is small enough (i.e., when the radius is not small compared with the horizontal domain width), at long time there is a two-dimensionalisation giving rise to a single column of positive PV and a single column of negative PV, as has been observed in some previous simulations. We find the same phenomenon for both vertically periodic and rigid lid boundary conditions, but it occurs over a broader range of initial conditions in the vertically periodic case. However, in all cases fully three-dimensional final states are regained when the number of vortices is increased while keeping the fraction of the domain occupied by vortices fixed, i.e., when the vortex radius/domain width ratio is sufficiently small.     

Observational evidence of deep convection over Indonesian sector in relation with major stratospheric warming events of 2003–04 and 2005–06

The major sudden stratospheric warming (SSW) events of 2003–04 and 2005–06 are considered to investigate changes in equatorial convection due to circulation changes associated with the SSW events. It is observed that the SSW events are accompanied by a considerable decrease in Outgoing Longwave Radiation (OLR), a proxy for tropical convection, over equatorial latitudes (15°N–15°S) in the Indonesian sector (90°E–150°E). However, unlike noted by earlier observations, the zonal mean OLR does not show any notable relationship with the SSW events. It can be explained from the latitude–longitude map of potential vorticity (PV) at 100 hPa, which shows a tongue of high PV emanating from high latitudes towards equator and converges in the longitude band of 90°E–150°E on the day of peak warming at 1 hPa in the case of 2003–04 and 10 hPa in the case of 2005–06. The latitude-height map of Eliassen–Palm (EP) vector and its divergence show convergence of EP flux in the upper troposphere at latitudes even lower than 20°N on these days. Further, vertical winds computed from the convergence of momentum flux are upward indicating convective activity at low-latitudes and downward at mid-latitudes.