On the generation of baroclinic rossby waves in the presence of a semi-circular boundary

Abstract

An analytical model is constructed for the generation of baroclinic Rossby waves by a vorticity source in the presence of a semi-circular boundary. The vorticity source is used to represent the effect of the Agulhas retroflection to the south of Southern Africa. The displacement of the interface between the two layers of the model ocean consists of quantized waves near the coast and a train of Rossby waves drifting westward further offshore.     

Inertial effects in an oceanic circulation model

Abstract

The flow in a mechanically driven thin barotropic rotating fluid system is analysed. The linear theory of Baker and Robinson (1969) is modified and extended into the non-linear regime.

An internal parameter, the “local Rossby number”, is indicative of the onset of nonlinear effects. If this parameter is 0(1) then inertial effects are as important as Coriolis accelerations in the interior of the transport-turning western boundary layer and both of its Ekman layers. The inertial effects in the Ekman layers, ignored in previous explorations of non-linear wind driven oceanic circulation, are retained here and calculated using an approximation of the Oseen type. The circulation problem is reduced to a system of scalar equations in only two independent variables; the system is valid for non-small local Rossby number provided only that the approximate total vorticity is positive.

To complete the solution for small Rossby number a boundary condition for the inertially induced transport is needed. It is found by examining the dynamics controlling this additional transport from the western boundary layer as the transport recirculates through the rest of the ocean basin. The strong constraint of total recirculation within the western boundary layer (zero net inertial transport) is derived.

The calculated primary inertial effects are in agreement with the observations of the laboratory model of Baker and Robinson (1969).

The analysis indicates the extent to which three-dimensional non-linear circulation can be reduced to a two dimensional problem.     

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.     

The nonlinear spin-up of a stratified ocean

Abstract

The adjustment of a nonlinear, quasigeostrophic, stratified ocean to an impulsively applied wind stress is investigated under the assumption that barotropic advection of vortex tube length is the most important nonlinearity. The present study complements the steady state theories which have recently appeared, and extends earlier, dissipationless, linear models.

In terms of Sverdrup transport, the equation for baroclinic evolution is a forced advection-diffusion equation. Solutions of this equation subject to a “tilted disk” Ekman divergence are obtained analytically for the case of no diffusion and numerically otherwise. The similarity between the present equation and that of a forced barotropic fluid with bottom topography is shown.

Barotropic flow, which is assumed to mature instantly, can reverse the tendency for westward propagation, and thus produce regions of closed geostrophic contours. Inside these regions, dissipation, or equivalently the eddy field, plays a central role. We assume that eddy mixing effects a lateral, down-gradient diffusion of potential vorticity; hence, within the closed geostrophic contours, our model approaches a state of uniform potential vorticity. The solutions also extend the steady-state theories, which require weak diffusion, by demonstrating that homogenization occurs for moderately strong diffusion.

The evoiution of potential vorticity and the thermocline are examined, and it is shown that the adjustment time of the model is governed by dissipation, rather than baroclinic wave propagation as in linear theories. If dissipation is weak, spin-up of a nonlinear ocean may take several times that predicted by linear models, which agrees with analyses of eddy-resolving general circulation models. The inclusion of a western boundary current may accelerate this process, although dissipation will still play a central role.     

Baroclinic and topographic influences on the transport in western boundary currents

Abstract

One of the central unsolved theoretical problems of the large scale ocean circulation is concerned with explaining the very large transports measured in western boundary currents such as the Gulf Stream and the Kuroshio. The only theory up to now that can explain the size of these transports is that of non-linear recirculation in which the advective terms in the momentum equations became important near the western boundary. In this paper an alternative explanation is suggested. When bottom topography and baroclinic effects are included in a wind-driven ocean model it is shown that the western boundary current can have a transport larger than that predicted from the wind stress distribution even when the nonlinear advective terms are ignored. The explanation lies in the presence of pressure torques associated with bottom topography which can contribute to the vorticity balance in the same sense as the wind stress curl.

Three numerical experiments have been carried out to explore the nature of this process using a three dimensional numerical model. The first calculation is done for a baroclinic ocean of constant depth, the second for a homogeneous ocean with an idealized continental slope topography, and the third for a baroclinic ocean with the same continental slope topography. The nature of the vorticity balance and of the circulation around closed paths is examined in each case, and it is shown that bottom pressure torques lead to enhanced transport in the western boundary current only for the baroclinic case with variable depth.     

On the exact shape of the horizontal profile of a topographically rectified tidal flow

Abstract

Starting from the nonlinear shallow water equations of a homogeneous rotating fluid we derive the equation describing the evolution of vorticity by a fluctuating bottom topography of small amplitude, using a multiple scale expansion in a small parameter, which is the topographic length scale relative to the tidal wave length. The exact response functions of residual vorticity for a sinusoidal bottom topography are compared with those obtained by a primitive perturbation series and by harmonic truncation, showing the former to be invalid for small topographic length scales and the latter to be only a fair approximation for vorticity produced by planetary vortex stretching. In deriving the exact shape of the horizontal residual velocity profile at a step-like break in the bottom topography, it is shown that the Lagrangian profile only exists in a strip having the width of the amplitude of the tidal excursion at both sides of the break, and that it vanishes outside that interval. Moreover, in the limit of small amplitude topography at least, it vanishes altogether for the generation mechanism by means of planetary vortex stretching. The Eulerian profile is shown to extend over twice the interval of the Lagrangian profile both for production by vortex stretching and by differential bottom friction. These finite intervals over which the residual velocity profiles exist for a step-like topography are not reproduced by harmonic truncation of the basic equation. This method gives exponentially decaying profiles, indicating spurious horizontal diffusion of vorticity. In terms of orders of magnitude, the method of harmonic truncation is reliable for residual velocity produced by vortex stretching but it overestimates the residual velocity produced by differential bottom friction by a factor 2.     

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.     

An asymptotic model for resistive instability in the Earth's outer core

Abstract

Numerical work indicates that resistive instability may be the dominant mode of instability in the Earth's outer core for realistic core parameter regimes. In this paper, we assume that the Elsasser number is large in order to obtain an asymptotic analysis of resistive instability in an electrically conducting fluid confined to a rotating cylindrical shell of infinite extent in the axial direction. The dimensionless equations of motion are linearized about an ambient magnetic field which is purely azimuthal and depends only on the cylindrical radial variable. Applying the theory of ordinary differential equations with a large parameter, we obtain an asymptotic approximation to the solution. Relatively simple analytic expressions for the complex frequencies are obtained by applying the boundary conditions for insulating boundaries at the cylindrical sidewalls and then assuming that the ambient magnetic field vanishes at one or both of those sidewalls. The results appear to be consistent with previous numerical work.     

Finite-Amplitude thermal convection within a self-gravitating fluid sphere

Abstract

Finite-difference calculations have been carried out to determine the structure of finite-amplitude thermal convection within a self-gravitating fluid sphere with uniform heat release. For a fixed-surface boundary condition single-cell convection breaks up into double-cell convection at a Rayleigh number of 3 × 10⁴, at a Rayleigh number of 5 × 10⁵ four-cell convection is observed. With a free-surface boundary condition only single cell convection is obtained up to a Rayleigh number of 5 × 10⁶.     

On the steady mass transport induced by wave motions in a viscous rotating fluid

Abstract

The steady second-order motion induced by a first-order wave motion in a homogeneous, viscous and rotating fluid is examined. If the wave motion produces a steady Ekman layer suction by non-linear interactions, this suction must induce a steady component of interior, relative vorticity parallel to the axis of rotation in order to conserve mass. A boundary value problem for the determination of the induced, steady interior mass transport velocity is presented. The mass transport induced by a Kelvin wave is examined as an illustration and possible application of the theory.     

Studies on finite amplitude waves in bounded water bodies

The problem treated here is the dynamics of a bay where water is driven through its opening periodically in time. The basic equations are expressed in the two horizontal coordinates and time and they are obtained by an integration of the Navier-Stokes equations in the vertical coordinate. The equations are nonlinear because of the convective terms in the acceleration. The problem of harbor dynamics provides a natural parameter as the ratio of mass of water entering the bay through the waves to the total mass of water in the bay. This small parameter multiplies the nonlinear terms and thus the problem is ideally suited for a perturbation analysis. The nonlinear terms are responsible for the generation of secondary flows and are particularly important near resonant frequences. The analysis further indicates the existence of a time independent flow analogous to acoustic streaming, known from solutions of the Navier-Stokes equations. The question of vorticity is studied and is seen that: a constant dissipation coefficient precludes the generation of vorticity even for the nonlinear case: and that only a weak (second order) vorticity can exist in the case of a variable dissipation term expressed through the Chezy coefficient. The study suggests also a semi analytic-numerical scheme with savings of 0(10²) for irregular geometries through the separation of the various order harmonics as opposed to the usual integration in time.     

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.     

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.     

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.     

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.     

Residual vorticity induced by the action of tidal currents in combination with bottom topography in the Southern Bight of the North Sea

Abstract

Results are presented of calculations on the generation of residual vorticity by tidal currents over the bottom topography of the Southern Bight of the North Sea. A typical order of magnitude is 10^?6 to 10^?7 s ^?1. This is compared with current measurements on calm days, when similar magnitudes are found. At windspeeds less than about 5 m/s tidal generation of residual vorticity is important; at higher windspeeds wind effects begin to dominate. Our results are relevant in understanding the spatial variability of residual currents, because a non-zero vorticity implies the existence of horizontal gradients in the residual current field.     

Propagation of barotropic modons over topography

Abstract

This is a broad survey of the interaction of modons with topography in a one-layer, quasigeostrophic model. Numerical simulations of modons interacting with ridges, hills, random topography and other obstacles are presented. The behavior of the modon is compared to numerical simulations of a two-point-vortex model, which proves a useful guide to the basic trajectory deflection mechanism. Under sufficiently strong but quasigeostrophically valid topographic perturbations, the modon is shown to fission into two essentially independent, oppositely-signed vortices. In the breakup of a modon near a hill it is found that the positive vortex migrates to the top of the hill. The resulting correlation between the positive vorticity trapped over the hill and the topography is in sharp contrast with the theories of turbulent flow over topography and generation of flow over topography by large scale forcing, both of which describe the development of vorticity anticorrelated with topography. A heuristic explanation of this new behavior is provided in terms of the dynamics of β bT-plane vortices. Further, it is found that a modon travelling over rough topography homogenizes the field of potential vorticity in its vicinity. This behavior is explained in terms of the induced eddy activity near the modon.     

Superhelicity,helicity and potential vorticity

Abstract

‘‘Helicity'’ density H ≡ u · ω and other pseudo-scalar fields such as P ≡ ω · Vlnρ (which is related to Ertel potential vorticity) are useful quantities in theoretical fluid dynamics and magneto-fluid dynamics. Here u denotes the Eulerian flow velocity relative to the chosen frame of reference, ω ≡ V × u is the corresponding relative vorticity and ρ the mass density of the fluid. A general expression is readily obtained for ?H/?t (where t denotes time) in terms of P and the ‘‘superhelicity'’ density S ≡ ω · V × ω which, in fluids of low viscosity, has its highest values in boundary layers. One need for such a relationship became evident during an attempt to interpret the findings of laboratory experiments on thermal convection in rotating fluids in containers of various geometrical shapes and topological characteristics.

In electrodynamics an analogous expression can be found relating the time rate of change of ‘‘magnetic helicity'’ A · B to ‘‘magnetic superhelicity'’ B · ? × B (where B · ? × A is the magnetic field) and a scalar quantity analogous to P which involves non-Ohmic contributions to the relationship between the electric current density and the electric field.     

Nonlinear free thermal convection in a spherical shell:A variable viscosity model

Introduction The velocity field of surface plate motion can be split into a poloidal and a toroidal parts.At the Earth′s surface,the toroidal component is manifested by the existence of transform faults,and the poloidal component by the presence of convergence and divergence,i.e.spreading and subduc-tion zones.They have coupled each other and completely depicted the characteristics of plate tec-tonic motions.The mechanism of poloidal field has been studied fairly clearly which is related to …     

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.