Improved bounds on linear instability of barotropic zonal flow within the shallow water equations

Here we develop mathematical results to describe the location of linear instability of a parallel mean flow within the framework of the shallow water equations; growth estimates of near neutral modes (for disturbances subcritical with respect to gravity wave speed) in the cases of non-rotating and rotating shallow water. The bottom topography is taken to be one-dimensional and the isobaths are parallel to the mean flow. In the case of a rotating fluid, the isobaths and the mean flow are assumed to be zonal. The flow is front-like: there is a monotonic increase of mean flow velocity. Our results show that for barotropic flows the location of instabilities will be a semi-ellipse region in the complex wave velocity plane, that is based on the wave-number, Froude number, and depth of the fluid layer. We also explore the instability region for the case of spatially unbounded mean velocity profiles for non-rotating shallow water.     

Radiating Vortices in Geophysical Fluid Dynamics

The dynamics of a single vortex on a beta-plane is discussed in this paper. A barotropic, an equivalent barotropic, one-and-a half and two-layer models are considered. The momentum and energy balances are used to describe the evolution of a vortex. A quasi-stationary balance of the Rossby, Zhukovsky-Kutta forces and the force induced by Rossby-wave radiation, describes the dynamics of the barotropic vortex. A net Coriolis force occurs if the fluid is stratified. The difference between the dynamics of cyclones and anticyclones results directly from the Coriolis force acting on a single vortex in a stratified fluid.All vortices radiate Rossby waves in the quasigeostrophic approximation but intense anticyclones propagate steadily in a one-and-a half layer model. A critical amplitude that bounds radiating and steadily propagating anticyclones is found. Steady propagation of anticyclones in general is impossible in a two-layer fluid due to the radiation of a barotropic Rossby-wave. Some solutions of solitary wave type which are known for a two-layer model, survive owing to wave interference.A single vortex can extract energy from a Rossby wave if synchronism conditions are satisfied. The wave interference again plays a crucial role in this case. The wave interference also determines the energy exchange of vortices located at larger distances. If the distance between the vortices is shorter than the length of the radiated waves, modon may be formed due to a small energy loss.The unbounded monotonic variation of the planetary vorticity is a characteristic feature of a beta-plane approximation. As a result, a single vortex propagates up to a 'rest latitude' where it disappears. The evolution of a single barotropic vortex over bottom topography provides another example of a background vorticity distribution with a local extremum above hills (valleys) or ridges (troughs). Physics of its movement differs from a beta-plane case, but if a vortex lies over broad topography, equations are similar and the evolution of a vortex manifests the same typical features. Particularly, a cyclonic vortex tends to drift to the top of a hill or a ridge. An anticyclonic vortex, on the contrary, slides to the bottom of a valley or a trough.An interaction of a barotropic vortex with a broad mean flow is tractable qualitatively on the basis of previous results. Numerical examples illustrating absorption of a small vortex by a larger one and a vortex movement across the flow, are direct analogies of the vortex evolution over a hill and a ridge, respectively. At the same time, strong influence of strain drastically changes the vortex structure.     

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.     

Influence of bottom frictional effects in sill regions upon lee wave generation and implications for internal mixing

A cross-sectional nonhydrostatic model using idealized sill topography is used to examine the influence of bottom friction upon unsteady lee wave generation and flow in the region of a sill. The implications of changes in shear and lee wave intensity in terms of local mixing are also considered. Motion is induced by a barotropic tidal flow which produces a hydraulic transition, associated with which are convective overturning cells, wave breaking, and unsteady lee waves that give rise to mixing on the lee side of the sill. Calculations show that, as bottom friction is increased, current profiles on the shallow sill crest develop a highly sheared bottom boundary layer. This enhanced current shear changes the downwelling of isotherms downstream of the sill with an associated increase in the hydraulic transition, wave breaking, and convective mixing in the upper part of the water column. Both short and longer time calculations with wide and narrow sills for a number of sill depths and buoyancy frequencies confirm that increasing bottom friction modifies the flow and unsteady lee wave distribution on the downstream side of a sill. Associated with this increase in bottom friction coefficient, there is increased mixing in the upper part of the water column with an associated decrease in the vertical temperature gradient. However, this increase in mixing and decrease in temperature gradient in the upper part of the water column is very different from the conventional change in near-bed temperature gradient produced by increased bottom mixing that occurs in shallow sea regions as the bottom drag coefficient is increased.     

Internal solitary wave-induced flow over a corrugated bed

A combined numerical and experimental study of the propagation of an internal solitary wave (ISW) over a corrugated bed is presented, in which the amplitude and the wavelength of the corrugated bed, together with the wave amplitude and wave speed of the ISW, have been varied parametrically. Both ISWs of elevation and depression have been considered. The wave-induced currents over the corrugated bed cause flow separation at the apex of the corrugations and a sequence of lee vortices forms as a result. These vortices develop fully after the main wave has passed over the topographic feature, resulting in deformation of the overlying pycnocline and, in some instances, significant vertical mixing. It is found that the intensity of the vortex formation is dependent on both the amplitude and wavelength of the bottom topography. In the case of an ISW of depression, the generation of vertically (upward)-propagating vortices is seen to result in entrainment of fluid from a bottom boundary jet (Carr and Davies, Phys Fluids 18:016601, 2006), while, in the elevation case, a second mechanism is present to induce significant turbulent mixing in the water column. It occurs when the bottom corrugations reach into, or are very near, the pycnocline at rest. Large waves of elevation that are stable on approach to the corrugations exhibit evidence of a spatio-temporally developing shear instability as they interact with the bottom corrugation. The shear instability takes the form of billows that have a vertical extent that can reach 50% of the wave amplitude.     

Barotropic waves along an eastern continental shelf

Abstract

An analysis is presented of the propagation of barotropic non-divergent oscillations along the western side of an ocean basin along which the persistent circulation in the basin is strongly intensified and laterally sheared. Because the Rossby number of a western boundary current is near unity, the properties of these waves are strongly affected by the steady circulation pattern. It is shown that for relatively long wavelengths, these waves can travel along the shelf in both directions; however, for a small range of short wavelengths they can only propagate northward and are unstable. Along the southeastern coast of North America, the unstable waves have wavelengths of order 150 km and periods of order 10 days. However, these waves can become stable oscillations in the deeper water northeast of Cape Hatteras. These oscillations are a possible explanation of the initiation of Gulf Stream meanders along the continental rise.     

Experiments on instability of columnar vortex pairs in rotating fluid

We present results from a new series of experiments on the geophysically important issue of the instability of anticyclonic columnar vortices in a rotating fluid in circumstances such that the Rossby number exceeds unity. The vortex pair consisting of a cyclonic and an anticyclonic vortex is induced by a rotating flap in a fluid which is itself initially in a state of solid-body rotation. The anticyclonic vortex is then subject to either centrifugal or elliptical instability, depending on whether its initial ellipticity is small or large, while the cyclone always remains stable. The experimental results demonstrate that the perturbations due to centrifugal instability have a typical form of toroidal vortices of alternating sign (rib vortices). The perturbations due to elliptical instability are of the form of sinuous deformation of the vortex filament in the plane of maximal stretching which corresponds to the plane of symmetry for the vortex pair. The initial perturbations in both cases are characterized by a definite wave number in the vertical direction. The characteristics of the unstable anticyclone are determined by the main nondimensional parameter of the flow - the Rossby number. The appearance of both centrifugal and elliptical instabilities are in accord with the predictions of theoretical criteria for these cases.     

Resonance induced by Rossby Lee waves

Abstract

The mutual interaction of fields induced by spatially separated potential vorticity sources in a quasi-geostrophic barotropic flow is investigated using the weakly nonlinear approach. It is found that a powerful nonlinear response can be triggered by Rossby lee waves. This resonance phenomenon which dominates all other nonlinear corrections depends on certain global resonance conditions and on the change in the phase of the Rossby lee wave across the distance separating the sources. The response is particularly strong for topographic forcing possessing δ-function characterisitics.     

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.     

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.     

Role of barotropic mechanism in the development of a monsoon depression: A MONEX study

The role of barotropic processes in the development of a monsoon depression, formed on 5 July 1979 during MONEX observational period, is studied by considering it as a quasi-geostrophic divergent barotropic instability problem of zonal flow of 3 July 1979 at 700 mb level. Numerical solutions are obtained by initial value approach. The preferred wave has a wavelength of 2750 km, an e-folding time of 4.3 days, a period of 6.5 days and an eastward phase speed of 4.9 ms^–1. Structure of preferred wave is found to be in good agreement with the observed horizontal structure of the depression at 700 mb. Poleward momentum transports are found to predominate over equatorward transports.Parts of this paper were presented at the National Symposium on Early Results of MONEX-1979. 9–12 March 1981, in New Delhi, India.     

Flow and transport in channels with submerged vegetation

This paper reviews recent work on flow and transport in channels with submerged vegetation, including discussions of turbulence structure, mean velocity profiles, and dispersion. For submerged canopies of sufficient density, the dominant characteristic of the flow is the generation of a shear-layer at the top of the canopy. The shear-layer generates coherent vortices by Kelvin-Helmholtz (KH) instability. These vortices control the vertical exchange of mass and momentum, influencing both the mean velocity profile, as well as the turbulent diffusivity. For flexible canopies, the passage of the KH vortices generates a progressive wave along the canopy interface, termed monami. The KH vortices formed at the top of the canopy penetrate a distance δ _e into the canopy. This penetration scale segregates the canopy into an upper layer of rapid transport and a lower layer of slow transport. Flushing of the upper canopy is enhanced by the energetic shear-scale vortices. In the lower layer turbulence is limited to length-scales set by the stem geometry, and the resulting transport is significantly slower than that of the upper layer.     

Convective and absolute instability of baroclinic eddies

Abstract

The long time response to an arbitrary unstable disturbance in a two layer model on an f-plane is sought. It has been found that depending on the ratio of the shear to the average speed of the mean flow two types of baroclinic instabilities exist: convective and absolute. When the system supports convective instabilities the long time response to an initial pulse excitation decreases with time at a fixed point in space. When such a system is excited by a wave maker the steady state frequency of response of the system corresponds to a spatially amplifying wave oscillating with the frequency of the wave maker. If the dispersion relation yields a saddle point of the frequency in the wave number complex plane with positive imaginary part of the frequency the system supports absolute instabilities. The response of the system at any point in space excited by an arbitrary signal grows exponentially with time at a rate determined by the properties of the system at the saddle point. This response is different from that of unstable normal modes. It is conjectured that absolute instability may be responsible for local cyclogenesis activity in certain geographical regions.     

近磁尾TC-1观测到伴随有高速流的ULF波及不稳定性分析

2004年8月3日近地TC-1卫星在磁尾X_GSM~－12R_E的等离子体片内,观测到了伴随着高速流的低于离子回旋频率的波,即超低频波(ULF,Ultra Low Frequency).该波垂直分量的振幅在高速流及其振荡减速期间大致相当;而平行分量振幅在高速流时明显大于其振荡减速时. 利用一个扰动双流模型对完全磁化离子横场漂移驱动的电磁不稳定性计算后,预测结果表明:(1)对于垂直分量来说,横场漂移速度与Alfvén速度的比值影响不稳定性增长率和激发波频率,随其比值增加,增长率变大,激发波频率从负值增加到正值.(2)对于平行分量来说,温度各向异性时等离子体热速度与Alfvén速度比值只影响不稳定性增长率和激发波频率,未改变不稳定性模类别;而温度各向同性时离子横场漂移速度与Alfvén速度比值既影响不稳定性模的种类及其分支,又影响激发波频率.进一步将卫星观测到的等离子体密度、温度、整体流速和磁场代入模型方程,进行数值计算与上述预测结果对比后发现:卫星观测中垂直分量的功率谱密度(PSD,Power Spectrum Density)增强时间和频段与理论模型中由β_//、β_⊥和v_⊥/V_A引起不稳定性激发的波一致;卫星观测中平行分量的功率谱密度增强时间与理论模型基本相符,但是前者的频率明显地低于后者.因此,除了需考虑平行磁场的离子整体流速对不稳定性激发波频率的可能影响,还需要统计上进一步核实伴随有高速流的ULF波与不稳定性的相关性.     

On resonant over-reflection of waves by jets

It is well known that internal or Rossby waves propagating across a jet can be amplified, a phenomenon usually referred to as over-reflection. In some cases, over-reflection can be infinitely strong – physically, this means that the reflected and transmitted waves can exist without an incident one, i.e. they are spontaneously emitted by the mean flow. In this article, it is shown that infinitely strong over-reflection (resonant over-reflection) occurs for gravity-wave scattering by ageostrophic jets in a rotating barotropic ocean and Rossby-wave scattering by a two-jet configuration on the quasigeostrophic beta-plane. It is further demonstrated that, generally, a resonantly over-reflected wave is always marginal to instability, i.e. either an increase or a decrease of its wavenumber transforms it into an unstable eigenmode localised near the jet.     

Shear flow driven magnetized planetary wave structures in the ionosphere

The generation and further linear and nonlinear dynamics of planetary ultra-low-frequency (ULF) waves are investigated in the rotating dissipative ionosphere in the presence of inhomogeneous zonal wind (shear flow). Planetary ULF magnetized Rossby type waves appear as a result of interaction of the medium with the spatially inhomogeneous geomagnetic field. An effective linear mechanism responsible for the intensification and mutual transformation of large scale magnetized Rossby type and small scale inertial waves is found. For shear flows, the operators of the linear problem are not self-conjugate, and therefore the eigenfunctions of the problem may not be orthogonal and can hardly be studied by the canonical modal approach. Hence, it becomes necessary to use the so-called nonmodal mathematical analysis. The nonmodal analysis shows that the transformation of wave disturbances in shear flows is due to the non-orthogonality of eigenfunctions of the problem in the conditions of linear dynamics. Using numerical modeling, the peculiar features of the interaction of waves with the background flow as well as the mutual transformation of wave disturbances are illustrated in the ionosphere. It has been shown that the shear flow driven wave perturbations effectively extract an energy of the shear flow increasing the own energy and amplitude. These perturbations undergo self-organization in the form of the nonlinear solitary vortex structures due to nonlinear twisting of the perturbation’s front. Depending on the features of the velocity profiles of the shear flows the nonlinear vortex structures can be either monopole vortices or vortex streets and vortex chains.     

Influence of open boundary conditions and sill height upon seiche motion in a gulf

A cross-sectional model of an idealised constant depth gulf with a sill at its entrance, connected to a deep ocean, is used to examine the barotropic and baroclinic response of the region to wind forcing. The role of the oceanic boundary condition is also considered. Calculations show that in the case of a tall sill, where the pycnocline intersects the sill, the baroclinic response of the gulf is similar to that of a lake, and internal waves cannot radiate energy out of the gulf. The barotropic response shows free surface oscillations, with nodes located close to the centre of the oceanic basin and entrance to the gulf, with associated barotropic resonant periods. As the sill height is reduced, baroclinic wave energy is radiated from the gulf into the ocean, and the form of the baroclinic response changes from a standing wave (tall sill) as in a lake to a progressive wave (no sill). The location of sea surface elevation nodes and resonant periods changes as the sill height is reduced. Calculations of the barotropic resonant periods with and without stratification could not determine if they were influenced by the presence of stratification, although published analytical theory suggests that they should be able to when energy is lost from the gulf by internal wave radiation. This inability to detect changes in barotropic resonant period due to stratification effects is due to the small change in resonant frequency produced by baroclinic effects, as shown by analytical results, and the broad peak nature of the computed resonant frequency. In the case of a closed offshore boundary (an offshore island), there is a stronger and narrower energy peak at the resonant frequency than when a barotropic radiation condition is applied. However, the influence of stratification upon the resonant frequency could not be accurately determined. Although the offshore boundary was well removed from the gulf to such an extent that any baroclinic waves reflected from it could not reach the gulf within the integration period, it did, however, slightly influence the gulf baroclinic response due to its influence on the barotropic response.     

Hemispheric simulation of the Asian summer monsoon

A three-level, -plane, filtered model is used to simulate the Northern Hemisphere summer monsoon. A time-averaged initial state, devoid of sub-planetary scale waves, is integrated through 30 days on a 5° latitude-longitude grid. Day 25 through day 30 integrations are then repeated on a 2.5° grid. The planetary-scale waves are forced by time-independent, spatially varying diabatic heating. Energy is extracted via internal and surface frictional processes. Orography is excluded to simplify synoptic-scale energy sources.During integration the model energy first increases, but stabilizes near day 10. Subsequent flow patterns closely resemble the hemisphere summer monsoon. Climatological features remain quasi-stationary. At 200 mb high pressure dominates the land area, large-scale troughs are found over the Atlantic and Pacific Oceans, the easterly jet forms south of Asia, and subtropical jets develop in the westerlies. At 800 mb subtropical highs dominate the oceans and the monsoon trough develops over the Asian land mass. The planetary scales at all levels develop a realistic cellular structure from the passage of transient synoptic-scale features, e.g., a baroclinic cyclone track develops near 55°N and westward propagating waves form in the easterlies.Barotropic redistribution of kinetic energy is examined over a low-latitude zonal strip using a Fourier wave-space. In contrast to higher latitudes where the zonal flow and both longer and shorter waves are fed by barotropic energy redistribution from the baroclinically unstable wavelengths, the low-latitude waves have a planetary-scale kinetic energy source. Wave numbers 1 and 2 maintain both the zonal flow and all shorter scales via barotropic transfers. Transient and standing wave processes are examined individually and in combination.Wave energy accumulates at wave numbers 7 and 8 at 200 mb and at wave number 11 in the lower troposphere. The 800-mb waves are thermally indirect and in the mean they give energy to the zonal flow. These characteristics agree with atmospheric observation. The energy source for these waves is the three wave barotropic transfer. The implications of examining barotropic processes in a Fourier wave-space, vice the more common approach of separating the flow into a mean plus a deviation are discussed.     

Taylor columns in horizontally sheared flow

Abstract

Solutions of the steady, inviscid, non-linear equations for the conservation of potential vorticity are presented for linearly sheared geostrophic flow over a right circular cylinder. The indeterminancy introduced by the presence of closed streamline regions is removed by requiring that the steady flow retains above topography a given fraction of that fluid initially present there, assuming the flow to have been started from rest. Those solutions which retain the largest fraction in uniform and negatively sheared streams satisfy the Ingersoll (1969) criterion (that, in the limit of vanishingly small viscosity, closed streamline regions are stagnant) and so are unaffected by Ekman pumping. These flows are set up on the advection time scale. In positively sheared flows the maximum retention solutions do not satisfy the Ingersoll criterion and thus would be slowly spun down on the far longer viscous spin-up time.

For arbitrary isolated topography, both the partial retention and Ingersoll problems are reduced to a one-dimensional non-linear integral equation and the solution of the Ingersoll problem obtained in the limit of strong positive shear. The stagnant region is symmetric about the zero velocity line and extends to infinity in the streamwise direction. Its cross-stream width is proportional to the rotation rate and fractional height occupied by the obstacle and inversely proportional to the strength of the shear, decreasing inversely as the square of distance upstream and downstream.     

The evolution of sub-mesoscale,coherent vortices on the β-plane

Abstract

An investigation is made of the evolution of small-scale, axisymmetric vortices in a stratified fluid with spatially variable Coriolis parameter. The criteria for smadness are a horizontal scale less than or equal to the first internal radius of deformation and a vertical scale less than or equal to that of the ambient stratification. These circumstances match those of Sub-mesoscale, Coherent Vortices frequently observed in the oceans. The dynamical model is the balance equations, which include various effects of finite Rossby number. The principal topics are the regime of nearly uniform propagation, the development of an equilibrium ratio of vertical and horizontal scales (i.e., Burger number selection), and the occurrence of various types of instability for vortices with extremes in amplitude or shape.