Internal hydraulic control in rotating fluids—applications to oceans

Abstract

Laboratory experiments and analysis of shallow water equations in a rotating fluid show that channel flow is governed by the ratio of the width of the channel to the Rossby radius of deformation R= √[g&Delta;ρH/ρf ²]. Flows through narrow ocean openings exhibit blocking and clear evidence of hydraulic control. These imply that formulae can be derived for width, volume flux, and velocity scales of the currents. A new version of the constant potential vorticity problem is solved, and it is shown to predict volume flux within 22% of the zero potential vorticity results. Next a systematic method of predicting volume flux through ocean passages is described. Some examples are given from the Denmark Straits overflow and the flow of Antarctic Bottom Water into the western Atlantic Ocean. Two-layer flows and counter-flows with rotation in a narrow passage, the so-called lock exchange flow problem, duplicate flows at a number of important straits and openings to bays. A potential vorticity formulation is reviewed. The flows in the mouths of various bays such as Funka Bay in Hokkaido, Japan, Spencer Gulf in South Australia, and Chesapeake Bay in the United States has R < width of the mouth, and the two currents are separated by a front. The width of the front and the density difference can be predicted with good results.     

Laboratory experiments on fronts

Abstract

Supercritically unstable density fronts near a vertical wall in a rotating, two-layer fluid were created on a laboratory turntable by withdrawing the outer wall of an annulus with a narrow gap, and allowing buoyant fluid from within the annulus to collapse toward a state of quasi-geostrophic balance. The resulting “coastal” current has a nearly uniform potential vorticity and is bounded by a front on which ageostrophic, wave-like disturbances grow. If the current width is comparable to the Rossby radius of deformation, the dominant length scale of disturbances is proportional to the width of the current. On the other hand, if the upper layer is much wider than the Rossby radius, then the observed length scale is a constant multiple of the Rossby radius. If the vertical boundary is omitted in the experiments, so that we are left with a circular anticyclonic vortex, the observed length scales and large-amplitude behaviour of disturbances are identical to those for the boundary currents, indicating that the wall has no significant influence on the flow.

At very large amplitude the growing waves lead to the formation of cyclone-anticyclone vortex pairs. For very wide currents, both the mean flow and the disturbances are first confined to a region within a few Rossby radii of the front. However, both the mean flow and the turbulent eddy motions slowly propagate into the previously stationary upper layer until, eventually, the whole of the upper layer is turbulent.     

Topographic eddies in temporally varying oceanic flows

Abstract

In this paper we examine the behaviour of oceanic unsteady flow impinging on isolated topography by means of numerical simulation. The ocean model is quasigeostrophic and forced by an oscillatory mean flow. The fluid domain is of the channel type and open-boundary numerical conditions are used to represent downstream and upstream flow.

In certain cases, vortex shedding, either cyclonic or anticyclonic, is observed in the lee of obstacles. Such shedding can be explained as the consequence of both an enhanced process of vorticity dissipation over the topography which locally affects the balance of potential vorticity on the advective timescale, and a periodic dominance of advective effects which sweep the fluid particles trapped on the seamount. For refined resolution and smallest viscosity the model will predict flows in which the shed eddies are coherent structures with closed streamlines.

The model suggests a mechanism by which topographically generated eddies may be swept away from a seamount in the ocean.     

Rapid formation of taylor columns: Obstacles against sidewalls

Abstract

The problem of topographic forcing by an obstacle against the boundary of a rotating flow is considered in various parameter regimes. The timescale for the motion is the topographic vortex-stretching time, which is inversely proportional to the background rotation rate and the fractional height of the obstacle. For slow flows this time is short compared with the advection time and the governing equation of conservation of potential vorticity is linear. The final state satisfies the non-linear equation for the advection of potential vorticity, however, and so time dependence has given a specific solution to a non-linear problem. The presence of the sidewall causes a stagnant Taylor column to be set up far more rapidly than cases with no sidewall. It is shown that viscosity and mixing arrests the inviscid evolution at some stage, thus some fluid still crosses the obstacle in the steady state. These solutions suggest that experimental results on separation obtained by Griffiths and Linden (1983) can tentatively be ascribed to entrainment and expulsion of fluid through vertical shear layers at the edge of the topography.     

A depth-dependent study of the topographic rectification of tidal currents

Abstract

A depth-dependent model for the topographic rectification of tidal currents in a homogeneous rotating fluid is used to examine the dependence of the rectified mean flow on various tidal, topographic and frictional parameters. Friction is parameterized through a vertically-uniform, time-independent vertical eddy viscosity and a bottom stress law applied near the top of the constant stress layer. The model neglects the interaction of mean and tidal currents, assumes uniformity along isobaths, and is closed with the assumption of zero depth-averaged mean flow across isobaths.

In the limit of depth-independence, the model reduces to that considered by Huthnance (1973) and Loder (1980) which, for weak friction, favours anticyclonic mean circulation around shallow regions and Lagrangian flow which is significantly reduced from the Eulerian. With the inclusion of vertical structure, the magnitude of the anticyclonic flow is amplified suggesting that depth-independent models may underestimate the along-isobath flow. For strong friction the direction of the mean flow depends on the orientation of the tidal ellipse relative to the isobaths. The depthindependent model again underestimates the magnitude of the along-isobath flow, but this can be offset with an appropriate reduction of the bottom friction coefficient.

The cross-isobath mean flows are one to two orders of magnitude weaker than the along-isobath flows and generally have more vertical structure. There is also a significant Stokes drift in the cross-isobath direction. Although there is some tendency for the cross-isobath mean bottom current to be down the cross-isobath mean pressure gradient, it appears that it is not generally possible to infer this current from depth-independent models.     

Comment on rotating hydraulics

Abstract

Numerical solutions of the axisymmetric flows during the relatively early phase of spin-up from rest of a stratified fluid in a cylinder are presented. Detailed results are given for a cylinder of aspect ratio of O(l) and for a minute Ekman number, showing axisymmetric spin-up for three values of the stratification parameter. As the stratification increases, the meridional circulation is confined to a region closer to the Ekman layers. An axisymmetric shear wave propagates radially inward from the sidewall, but, unlike the strictly vertical front for a homogeneous fluid, the interface which separates rotating from nonrotating fluid is bow-shaped. For a stratified fluid, the axial vorticity distribution is nonuniform both in the vertical and in the radial directions. With increasing stratification, diffusive vorticity production near the sidewall is more pronounced. Axisymmetric flows in the early phase of spin-up of a stratified fluid are controlled by both the inviscid dynamic effect and the viscous diffusion effect. At a location close to the Ekman layers, the inviscid effect outweighs the viscous effect, in much the same way as in a homogeneous fluid. However, at a location close to mid-depth, the viscous diffusion effect, enhanced by substantial flow gradients in that region, is dominant. This points to the necessity of including the direct effect of viscous diffusion in the interior in formulating an analytical model of stratified spin-up problems.     

On reduced-gravity flow through sills

Abstract

The hydraulic flow of a reduced-gravity fluid of non-negative potential vorticity through a sill is considered. It is shown that for any flow with a reversal of current, another, physically realisable, flow exists which is unidirectional and/or resting, and carries more flux than the original flow. Thus only non-negative flows need be considered when examining maximal hydraulic fluxes. Then, for a simple sill (one which slopes downward on the left and upward on the right, looking downstream), it is shown that zero potential vorticity flow, possibly modified by having a region of motionless fluid at its right, carries the maximum flux possible for that sill shape. This makes the calculation of maximal fluxes for a given sill considerably simpler, and examples of parabolic and V-shaped sills are computed.     

Flows within the Ekman layer during spin-up of a thermally stratified fluid

Abstract

Finite-difference numerical solutions were obtained to present the flow and temperature field details within the transient Ekman layer during spin-up of a thermally stratified fluid in a cylinder. This complements the earlier studies on stratified spin-up which examined the flows in the interior core region. As the stratification increases, the following changes in the flow field are noticeable. The radial velocity in the Ekman layer decreases in magnitude. The azimuthal flows adjust smoothly from the interior region to the endwall boundary, and the Ekman layer in the azimuthal flow field fades. Vertical motions are inhibited, resulting in a weakened Ekman pumping. The axial vorticity field behaves similarly to the azimuthal flows. The temperature deviation from the equilibrium profile decreases, and the heat transfer flux from the endwall to the fluid decreases. The thickness of the thermal layer is larger than the velocity layer thickness. Illustrative comparisons of the relative sizes of the terms in the governing equations are conducted in order to assess the stratification effect in the adjustment process of the fluid.     

Instability of a geostrophic front and its energetics

Abstract

The instability of a current with a geostrophic surface density front is investigated by means of a reduced gravity model having a velocity profile with nearly uniform potential vorticity. It is shown that currents are unstable when the mean potential vorticity decreases toward the surface front at the critical point of the frontal trapped waves investigated by Paldor (1983). This instability is identical with that demonstrated by Killworth (1983) in the longwave limit.

The cross-stream component of mass flux and the rates of energy conversions among the five energy forms defined by Orlanski (1968) are also calculated. The main results are as follows, (a) The mass flux toward the surface front is positive near the front and negative around the critical point. The positive mass flux near the front does not vanish at the position of the undisturbed surface front, so that the mean position of the front moves outward and the region of the strong current spreads. (b) The potential energy of the mean flow integrated over the fluid is released through the work done by the force of the pressure gradient of the mean flow on the fluid, and is converted into the kinetic energy of the mean flow. (c) In the critical layer, the mean flow is rapidly accelerated with the growth of the unstable wave. This acceleration is caused by the rapid phase shift of the unstable wave in the critical layer.     

风驱动下f-平面准地转三维海洋环流的形成及总质量守恒的应用

将理想化的南中国海海盆在垂直方向上划分为Ekman层、惯性层和摩擦层. Ekman层中的运动由大气风应力驱动,其底部的扰动压力将作为其下惯性层中运动的上边界条件. 惯性层中的运动是由f 平面三维非线性方程在准地转近似下位势涡度守恒控制,由此得到控制惯性层中运动关于扰动压力的三维椭圆型方程. 在惯性层以下考虑到深层的海盆水平尺度很小,由此引进带有底部摩擦的线性控制方程,方程的边界条件为惯性层和摩擦层交界面上的扰动压力连续,沿海盆边界假定海水与相邻的固壁间无热量交换,由此设在海盆边界上扰动温度为零. 在此基础上分别利用惯性层和摩擦层中的椭圆型控制方程计算了相应层次上冬、夏季的扰动压力和准地转流. 结果表明冬季各层上以气旋式环流为主,且随深度的增加流速减小;夏季各层上以反气旋式环流为主,流速也随深度增加而减小. 这在一定程度上和观测事实相符.     

Frontal upwelling in a rotating two-layer fluid

Abstract

This paper describes the source-sink driven flow in a two-layer fluid confined in a rotating annulus. Light fluid is injected at the inner wall, while denser fluid is withdrawn at the outer wall. The interface between the immiscible fluids intersects the bottom and thus produces a front. The net transport from the source to the sink is carried by Ekman layers at the bottom and at the interface, and by Stewartson layers at the side walls. A detached Stewartson layer arises at the front, leading to a pronounced upwelling circulation.     

On the inadequacies of Long's model for steady two-dimensional stratified flows

Nonlinear analysis of two-dimensional steady flows with density stratification in the presence of gravity is considered. Inadequacies of Long's model for steady stratified flow over topography are explored. These include occurrence of closed streamline regions and waves propagating upstream. The usual requirements in Long's model of constant dynamic pressure and constant vertical density gradient in the upstream condition are believed to be the cause of these inadequacies. In this article, we consider a relaxation of these requirements, and also provide a systematic framework to accomplish this. As illustrations of this generalized formulation, exact solutions are given for the following two special flow configurations: the stratified flow over a barrier in an infinite channel; the stratified flow due to a line sink in an infinite channel. These solutions exhibit again closed-streamline regions as well as waves propagating upstream. The persistence of these inadequacies in the generalized Long's model appears to indicate that they are not quite consequences of the assumptions of constant dynamic pressure and constant vertical density gradient in Long's model, contrary to previous belief.

On the other hand, solutions admitted by the generalized Long's model show that departures from Long's model become small as the flow becomes more and more supercritical. They provide a nonlinear mechanism for the generation of columnar disturbances upstream of the obstacle and lead in subcritical flows to qualitatively different streamline topological patterns involving saddle points, which may describe the lee-wave-breaking process in subcritical flows and could serve as seats of turbulence in real flows. The occurrences of upstream disturbances in the presence of lee-wave-breaking activity described by the present solution are in accord with the experiments of Long (Long, R.R., “Some aspects of the flow of stratified fluids, Part 3. Continuous density gradients”, Tellus 7, 341--357 (1955)) and Davis (Davis, R.E., “The two-dimensional flow of a stratified fluid over an obstacle”, J. Fluid Mech. 36, 127–143 ()).     

A note on the use of zero potential vorticity models

Abstract

The acceptability of zero potential vorticity models as approximations for natural systems of small, but finite, potential vorticity is studied for bounded frontal flows of arbitrary profile. It is demonstrated that all (infinitely) long-wave solutions of the zero potential vorticity front are asymptotic limits for some (not necessarily long-wave) solutions of the small potential vorticity front. In contrast, for downstream-varying solutions there is no simple way of demonstrating this property. These findings suggest that the use of zero potential vorticity models should be carefully examined in other, non-frontal, problems as well. Finally we show that the longwave solutions of the zero potential vorticity flow are at most neutral (quasi-stable).     

The instability of precessing flow

Abstract

An explanation is put forward for the instability observed within a precessing, rotating spheroidal container. The constant vorticity solution for the flow suggested by Poincaré is found to be inertially unstable through the parametric coupling of two inertial waves by the underlying constant strain field. Such resonant couplings are due either to the elliptical or shearing strains present which elliptically distort the circular streamlines and shear their centres respectively. For the precessing Earth's outer core, the shearing of the streamlines and the ensuing shearing instability are the dominant features. The instability of some exact, linear solutions for finite precessional rates is established and used to corroborate the asymptotic analysis. A complementary unbounded analysis of a precessing, rotating fluid is also presented and used to deduce a likely upperbound on the growth rate of a small disturbance. Connection is made with past experimental studies.     

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.     

Stability and stable modes of coastal fronts

Abstract

The stability of a single layer, geostrophic front of zero potential vorticity bounded by a vertical coast (wall) is investigated by means of a Rayleigh integral. It is proved that the flow of the density-driven current is stable at all wavenumbers provided the mean velocity of basic flow exceeds fL (where f is the Coriolis parameter and L is the distance between the wall and the free streamline). The frequency of the stable long waves is either zero or super-inertial.     

A 3-D coastal tidal rectification process: observations, theory and experiments

In homogeneous rotating fluid, when there is an oscillating forcing in the interior fluid with a period long enough for an Ekman layer to develop, there is an interaction between the oscillatory Ekman layer and the vertical wall, since the latter imposes an alternating adjustment flow confined near the wall. As a result, this coastal rectification process leads to a Lagrangian transport along the coast. The Ekman number, the Rossby number and the temporal Rossby number of the forcing flow are the governing parameters of that mechanism which can be described by a simplified analytical model taking into account both the vertical time-dependent structure of the current and the presence of the wall. The model shows that the residual (rectified) current flowing with the coast to its right results from the strong nonlinear interaction between along- and cross-shore tidal currents leading to asymmetrical momentum exchanges between the Ekman bottom layer and the coastal boundary layer. The model provides simple scaling laws for the maximum intensity and width of the residual current. The latter is significantly larger than the friction (Stokes) lateral boundary layer of the forcing flow. A comprehensive set of experiments is performed in the 13 m diameter rotating tank by oscillating an 8 m×2 m horizontal plate and vertical wall in a homogeneous fluid at rest in solid-body rotation and measuring the two horizontal components of the current at several locations and depths above the central part of the plate. The predicted and experimentally measured maximum intensity and width of the residual current are in very good agreement, within the range of validity of the model, i.e. when the Ekman number is sufficiently small. However experiments also show that the residual current still occurs when the Ekman layer thickness is of the same order as the fluid depth, but it is then confined to a narrower band along the vertical wall. The flow structure found experimentally is also correctly described by a numerical model developed by Zhang et al. (1994). Current measurements in the Eastern part of the English Channel near the French coast reveal a significant coastal residual current flowing Northward and the coastal rectification process described here may account for part of it.     

The effects of vertical shear and stratification on stationary rossby waves

Abstract

The generation of stationary Rossby waves by sources of potential vorticity in a westerly flow is examined here in the context of a two-layer, quasi-geostrophic, β-plane model. The response in each layer consists of a combination of a barotropic Rossby wave disturbance that extends far downstream of the source, and a baroclinic disturbance which is evanescent or wave-like in character, depending on the shear and degree of stratification. Contributions from each of these modes in each layer are strongly dependent on the basic flows in each layer; the degree of stratification; and the depths of the two layers. The lower layer response is dominated by an evanescent baroclinic mode when the upper layer westerlies are much larger than those in the lower layer. In this case, weak stationary Rossby waves of large wavelengths are confined to the upper layer and the disturbance in the lower layer is confined to the source region.

Increasing the upper layer flow (with the lower layer flow fixed) increases the Rossby wavelength and decreases the amplitude. Decreasing the lower layer flow (with the upper layer flow fixed) decreases the wavelength and increases the amplitude. Stratification increases the contribution from the barotropic wave-like mode and causes the response to be confined to the lower layer.

The finite amplitude response to westerly flow over two sources of potential vorticity is also considered. In this case stationary Rossby waves induced by both sources interact to reinforce or diminish the downstream wave pattern depending on the separation distance of the sources relative to the Rossby wavelength. For fixed separation distance, enhancement of the downstreatm Rossby waves will only occur for a narrow range of flow variables and stratification.     

Wave patterns generated by disturbances travelling horizontally in rotating stratified fluids

The pattern and propagation of waves generated by steady or oscillatory disturbances travelling horizontally in a rotating, stratified fluid are studied following a technique developed by Lighthill. Both two‐ and three‐dimensional distrubances are investigated. The results show how rotation modifies internal wave patterns in a stratified fluid and how stratification modifies inertial wave patterns in a rotating fluid. The results are used to compute the effective diminution of Taylor column length due to the presence of density stratification. They also show that the appearance of wave crests upstream of a disturbance is possible only when the disturbance is unsteady and that observations of upstream blocking in a two‐dimensional stratified flow can be explained by the existence of a certain class of plane waves as modified by viscosity.     

An experimental study of Stewartson layer separation

Abstract

The separation of sidewall boundary layers in a rotating annulus of homogeneous fluid is studied experimentally. The flow is driven by a differentially rotating lid, and a very small uniform slope of the bottom causes a weak mountain vortex pair to form in the interior, away from the sidewalls. A necessary condition for aerodynamic separation of the sidewall boundary layers is derived and compared with the experimental results. The laboratory flow separates for parameters that are just slightly more inviscid than those required by the necessary condition for the existence of adverse pressure gradients at the wall. As the bottom friction is decreased further, the flow becomes unsteady and chaotic. The most interesting aspect of this problem is that chaotic interior behavior, associated with the separated boundary layer, is observed for parameter values for which the interior topographically forced flow is, by itself, essentially linear.