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1.
Abstract

It is shown that the linear equatorial dynamics of a shallow ocean is characterized by two boundary layers of width γ? L and γL (γ is the Ekman number of the flow, assumed small, and L is a horizontal dimension of the basin). In the γ? layer stress in the bottom Ekman layer is comparable to that in the surface Ekman layer. In the γ layer vertical friction is important throughout the depth of the ocean. Should the Rossby number ? be so large as to invalidate a linear theory (? > γ5/3), then inertial effects become important at a distance ?2/5 L from the equator. The role played in the circulation of the basin by the non-linear equatorial current first studied by Charney (1960) is shown to be similar to that of the γ layer of the linear theory. Though lateral friction is unimportant in a linear model of the flow, shear layers at the equator are found to be a necessary feature of non-linear flow.  相似文献   

2.
Abstract

The flow properties of an homogeneous fluid which is bounded by two concentric spheres and two meridional planes which intersect along a diameter of the spheres are investigated. The spheres rotate about this diameter with slightly different angular velocities. As in the axisymmetric case studied by Proudman (1956) and Stewartson (1966) the viscous terms in the equations of motion are important only in boundary layers on the spheres and on the cylinder C which circumscribes the inner sphere and which has generators parallel to the axis of rotation, provided the Ekman number E is small. In the inviscid region the velocities are independent of the coordinate measuring distance along the axis of rotation and are much weaker, by a factor 0(E ½), than the velocities in the Ekman layer on the driving surface (outer sphere). (It is assumed that the reference frame is fixed in the slower rotating inner sphere.) If the separation of the spheres is small compared to their radii then the asymmetric circulation inside C is characterized by an intense jet along the western wall. Loss of fluid from this jet sustains the eastward and northward flow in the inviscid interior where motion is driven by the suction of the Ekman layer on the outer sphere. (Geophysical conventions have been adopted.) Outside C an intense current is present on the eastern, not western, wall while motion in the inviscid region is westward, and away from the axis of rotation. Though there is no transport across C in the inviscid region, the meridional transport of the Ekman layer on the outer sphere is continuous across C and increases, through suction, as the equator is approached until it drains into an eastward flowing equatorial current of width 0(E 1/7). The eastern boundary current outside C and shear layers on C carry this fluid to the intersection of C and the western wall where it feeds the western boundary current inside C.

The relation between this study and the experiments of Baker and Robinson (1970) is discussed.  相似文献   

3.
Abstract

In a laboratory model ocean, fluid in a rotating tank of varying depth is subjected to “wind-stress”, For a certain range of the parameters, Ekman number E and Rossby number R, a homogeneous fluid displays steady, westward intensified flow. For the same range of E and R, a two-layer fluid can have baroclinic instabilities. The parameter range for the various kinds of instabilities is mapped in a regime diagram. The northward transport in the western boundary current is measured as it varies with Rossby number for both homogeneous and two-layer fluid.  相似文献   

4.
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.  相似文献   

5.
Abstract

A two gyre circulation and inertial western boundary currents have been observed in a sloping bottom laboratory model of a barotropic ocean circulation. Water of viscosity v is contained in a rotating (angular velocity ω), square basin of side L (30 cm) with a flat top and a bottom slope (tan θ) such that the depth (H) varies from 12 to 15 cm. The flow is driven by a distributed source and sink at the upper surface, a plate drilled with 342 holes. The hole distribution and size is arranged so that the average imposed vertical velocity, w = w 0 sin (2πy′/30), models the Ekman divergence from a two gyre zonal wind stress. Fluid flow is observed with the thymol blue technique over the ranges of Rossby numbers (w 0/2ωL tan θ) from 1.44 × 10?3 to 1.41 × 10?2 and Ekman numbers (v/2ωH 2) from 2.13 × 10?5 to 2.10 × 10?3. At the largest Rossby numbers the flow pattern changes markedly, but the non-uniformity of the imposed vertical velocity also penetrates deep into the fluid in this regime.  相似文献   

6.
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.  相似文献   

7.
Inertial waves as oscillatory motions in rotating fluids generate internal shear layers at critical latitudes. We investigated the nonlinear interaction of inertial waves for global flows (3D flows) in dependence on the Ekman number. When the value of the Ekman number decreases, the influence of the Ekman layers to the flow pattern increases. Critical latitudes, the attractor flow pattern and certainly internal shear layers are observable mainly at greater values of the Ekman number. Although, with decreasing the Ekman number smaller flow structures become visible, nonlinear interactions in shear layers drive an axisymmetric flow whose amplitude diverges at the limit of the vanishing Ekman number. We show that this conclusion is valid not only for zonal wind driven by inertial modes but also for similarly driven global flows.  相似文献   

8.
Abstract

The low Rossby number flow in a rotating cylinder with an inclined bottom, of small slope, is examined when part of the lid of the container is rotating at a slightly different rate. The resulting flow is calculated numerically by solving the governing equations for the two-dimensional geostrophic motion which approximates the flow in most of the fluid including the inertially-modified E ¼ -layers. The presence of ageostrophic regions, on the container walls and beneath the velocity discontinuity on the lid, is accounted for in the governing equations and their boundary conditions. This study supplements previous work on this configuration, in which the zero Rossby number flow was calculated and experimental results were presented, by enabling a direct comparison to be made between the results of the low Rossby number theory and the experiments. The numerical results for a range of Rossby and Ekman numbers compare well with those from the experiments despite a severe limitation on the size of the Rossby number arising from the analysis in the ageostrophic part of the detached shear layer.  相似文献   

9.
《Continental Shelf Research》1999,19(15-16):1851-1867
To investigate the instabilities of steady and oscillating Ekman layers, an 8 m×2 m horizontal plate was moved at controlled speed in homogeneous water at rest in solid body rotation in the “Coriolis” 13 m diameter rotating tank. For a steady Ekman layer two distinct wave types were found, in agreement with previous experimental or numerical studies. Type I was stationary, was oriented positively with respect to the flow direction and had a wavelength of about 10 times the Ekman layer thickness. Type II was oriented negatively with respect to the flow direction and had a wavelength which was more than 20 times the Ekman layer thickness and a phase-speed between 0.3 and 0.5 the forcing interior velocity. The growth rates of both type I and type II waves for various Reynolds numbers Re (computed with the Ekman layer thickness) were estimated and their Re-variations qualitatively agree with previous numerical results. For an oscillating Ekman layer, experimental results depended strongly on Rot, the temporal Rossby number: only when Rot<1 was it possible to observe either type I or type II instabilities. Moreover, for all Rot and average to high Re, there was a noticeable upward turbulent transport occurring during each cycle between the flow maximum and the flow reversal. Such an upward turbulent transport is consistent with observations in the English Channel where maximum upward benthic movements and maximum turbidity were recorded at the flow reversal, hence Ekman layer instabilities and transition to turbulence are likely to occur in shallow tidal seas where they may be relevant for sediment resuspension and transport as well as for some biological processes.  相似文献   

10.
Theory of wave boundary layers (WBLs) developed by Reznik (J Mar Res 71: 253–288, 2013, J Fluid Mech 747: 605–634, 2014, J Fluid Mech 833: 512–537, 2017) is extended to a rotating stratified fluid. In this case, the WBLs arise in the field of near-inertial oscillations (NIOs) driven by a tangential wind stress of finite duration. Near-surface Ekman layer is specified in the most general form; tangential stresses are zero at the lower boundary of Ekman layer and viscosity is neglected below the boundary. After the wind ceases, the Ekman pumping at the boundary becomes a linear superposition of inertial oscillations with coefficients dependent on the horizontal coordinates. The solution under the Ekman layer is obtained in the form of expansions in the vertical wave modes. We separate from the solution a part representing NIO and demonstrate development of a WBL near the Ekman layer boundary. With increasing time t, the WBL width decays inversely proportional to \( \sqrt{t} \) and gradients of fields in the WBL grow proportionally to \( \sqrt{t} \); the most part of NIO is concentrated in the WBL. Structure of the WBL depends strongly on its horizontal scale L determined by scale of the wind stress. The shorter the NIO is, the thinner and sharper the WBL is; the short-wave NIO with L smaller than the baroclinic Rossby scale LR does not penetrate deep into the ocean. On the contrary, for L?≥?LR, the WBL has a smoother vertical structure; a significant long-wave NIO signal is able to reach the oceanic bottom. An asymptotic theory of the WBL in rotating stratified fluid is suggested.  相似文献   

11.
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.  相似文献   

12.
Abstract

Models of a differentially rotating compressible convection zone are calculated, considering the inertial forces in the poloidal components of the equations of motion. Two driving mechanisms have been considered: latitude dependent heat transport and anisotropic viscosity. In the former case a meridional circulation is induced initially which in turn generates differential rotation, whereas in the latter case differential rotation is directly driven by the anisotropic viscosity, and the meridional circulation is a secondary effect.

In the case of anisotropic viscosity the choice of boundary conditions has a big influence on the results: depending on whether or not the conditions of vanishing pressure perturbation are imposed at the bottom of the convection zone, one obtains differential rotation with a fast (≥ 10 ms?1) or a slow (～ 1 ms?1) circulation. In the latter case the rotation law is mainly a function of radius and the rotation rate increases inwards if the viscosity is larger in radial direction than in the horizontal directions.

The models with latitude dependent heat transport exhibit a strong dependence on the Prandtl number. For values of the Prandtl number less than 0.2 the pole-equator temperature difference and the surface velocity of the meridional circulation are compatible with observations. For sufficiently small values of the Prandtl number the convection zone becomes globally unstable like a layer of fluid for which the critical Rayleigh number is exceeded.  相似文献   

13.
Abstract

The stability of a shear flow on a sloping bottom in a homogeneous, rotating system was investigated by means of a laboratory experiment.

The basic flow was driven near a vertical wall of a circular container by a ring-shaped plate that contacted with a free surface of the working fluid and rotated relative to the fluid container. The velocity profile was asymmetric in the radial direction and had only one inflection point. The velocity profile was well expressed by a linear theory for the vertical shear layer.

The effect of the circular geometry was checked by comparing experimental results obtained in two fluid systems in which only the sign of the curvature was opposite and it was confirmed that circular geometry was not essential for the shear flow on the sloping bottom in this experiment.

It was found that the sloping bottom stabilizes the basic flow only when the drift direction of the topographic Rossby wave is opposite to that of the basic flow. The viscous dissipation in both the Ekman layer and the interior region was also important in determining the critical Rossby number.

The eddy fields caused by the instability can be classified into two types: One is the stationary eddy field in which a row of eddies moves along the basic flow without changing form. The other is the flow pattern in which eddies have finite life times and their configuration is not well organized. When the sloping bottom does not stabilize the basic flow, the former flow pattern is realized, otherwise the latter flow pattern appears.

The wave numbers of the eddies in the regular flow pattern were observed as a function of the Rossby number. The relation did not fit to linear preferred modes predicted by an eigenvalue problem.  相似文献   

14.
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.  相似文献   

15.
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.  相似文献   

16.
Abstract

Chandrasekhar (1961) has summarized the stability results of Bénard convection in a rotating fluid for the cases where the boundary surfaces are both rigid and free, and for both exchange of stabilities and overstability. His analysis provides very accurate results for a limited range of Taylor number J. Bisshopp and Niiler (1965) presented an asymptotic analysis of the rigid boundary problem for exchange of stabilities which is valid for very large Taylor number. The present paper makes use of modern rotating fluid theory to develop an approximate scheme for evaluating the Rayleigh number and other parameters and variables. Known asymptotic results for the free boundary problem at large J are used and an expansion in powers of E1/6 (the Ekman number, E = 2J ?½) yields a sequence of equations and appropriate boundary conditions for the rigid boundary problem. After the algorithm for the calculation is developed, results are given for the problem to second order in the expansion parameter for the case of exchange of stabilities and to first order in the expansion parameters for the overstable case. Ekman boundary layers are important in the development as one might expect. However, an additional, diffusive boundary layer of thickness E? is necessary to provide the details of the temperature field. This boundary layer is the thermal response in the vertical direction to the horizontal spacing of the cells which is also order E?. The horizontal spacing of the cells is essentially a series of contiguous, Stewartson (1957) layers of thickness E?.  相似文献   

17.
Summary The time-dependent primitive equations for a shallow homogeneous ocean with a free surface are solved for a bounded basin on the sphere, driven by a steady zonal wind stress and subject to lateral viscous dissipation. These are the vertically integrated equations for a free-surface model, and are integrated to 60 days from an initial state of rest by an explicit centered-difference method with zero-slip lateral boundary conditions. In a series of comparative numerical solutions it is shown that at least a 2-deg resolution is needed to resolve the western boundary currents adequately and to avoid undue distortion of the transient (Rossby waves. The -plane formulation is shown to be an adequate approximation for the mean circulation in the lower and middle latitudes, but noticeably intensifies the transports poleward of about 50 deg and both slows and distorts the transients in the central basin. The influence of the (southern) zonal boundary on the transport solutions is confined to the southernmost gyre, except in the region of the western boundary currents where its influence spreads to the northern edge of the basin by 30 days. The total boundary current transport is shown to be approximately proportional to the zonal width of the basin and independent of the basin's (uniform) depth, while the elevation of the free water surface is inversely proportional to the basin depth, in accordance with linear theory. The introduction of bottom friction has a marked damping effect on the transient Rossby waves, and also reduces the maximum boundary-current transport. The solutions throughout are approximately geostrophic and are only slightly nonlinear.The root-mean-square (rms) transport variability during the period 30 to 60 days is concentrated in the southwest portion of the basin through the reflection of the transient Rossby waves from the western shore and has a maximum corresponding to an rms current variability of about 3 cm sec–1. The transport variabilities are about 10 percent of the mean zonal transport and more than 100 percent of the mean meridional transport over a considerable region of the western basin (outside the western boundary current regime). Some 99 percent of the total kinetic energy is associated with the zonal mean and standing zonal waves, which are also responsible for the bulk of the meridional transport of zonal angular momentum. Although the transient Rossby waves systematically produce a momentum flux convergence at the latitude of the maximum eastward current, much in the manner of their atmospheric counterparts, this is only a relatively small contribution to the zonal oceanic momentum balance; the bulk of the mean zonal stress is here balanced by a nearly stationary net pressure torque exerted against the meridional boundaries by the wind-raised water. In an ocean without such boundaries the role of the transient circulations may be somewhat more important.  相似文献   

18.
A seasonal ice edge zone is a unique frontal system with an air-ice-sea interface. This paper is a report on the numerical results from a quasi-three dimensional, time dependent, non-linear numerical model of circulation at a continental shelf-seasonal ice edge zone. The purpose of the experiments is to model the hydrography and circulation, including upwelling, baroclinic geostrophic flow, and inertial oscillations, at the ice edge with emphasis on examining the driving forces of wind and melting ice. It is suggested that the non-linear acceleration terms and vertical density diffusion terms are negligible and that the horizontal density diffusion terms are of secondary importance within the time and space scales of the experiments. The vertical eddy viscosity terms are important in a spin-up time scale and for Ekman transport and a bottom Ekman layer. The effects of the horizontal eddy viscosity terms are observable (a long-ice jet is diffused away from the ice edge) by the end (72 h) of the model runs.Model results are compared with available oceanographic and meteorological data for verification. The observed and modeled features of melt water induced water column stability, frontal structure, and ice edge upwelling are briefly discussed relative to observed ice edge primary production. Because the model is relatively general in nature, it is readily applicable to other seasonal or marginal ice edge zones in either hemisphere.  相似文献   

19.
Abstract

The flow of a rotating homogeneous, incompressible fluid past a long ridge is investigated. An analysis is presented for flows in which E ? 1, Ro ～ E½, H/D ～ E0, h/D ～ E½ and cosα ～ E0 where E is the Ekman number, Ro the Rossby number, H/D the fluid depth to ridge width ratio, h/D the ridge height to ridge width ratio and α the angle between the free stream flow and a line perpendicular to the ridge axis. The analysis includes effects of the nonlinear inertial terms. Particular examples of a ridge of triangular cross section and a sinusoidal topography are investigated in some detail. Experiments are presented for a triangular ridge which are in good agreement with the theory.  相似文献   

20.
Abstract

A laser Doppler velocimeter (LDV) has been successfully mounted on a high quality rotating turntable. The capability of this LDV is demonstrated by some detailed measurements of the relative flow during the spin-up of a homogeneous fluid in a cylinder. Local measurements in water of the zonal flow component of magnitude 0.1 cm/sec have been made with an error of about 0.003 cm/sec. The spatial resolution was about 0.1 cm and the temporal resolution about 0.5 Hz. Effects on the flow due to absorption of the low power laser beam (5 milliwatts) and to the low concentration (3 parts/million) of 0.5 micron diameter scattering particles were negligible. The results are compared with analytical theory and the agreement is good. For a Rossby number of 0.1, the weak inertial modes excited by the Ekman layer formation can be clearly seen and identified. The LDV offers great promise for checking numerical and analytical solutions against experiments. This is particularly true for contained flows where conventional probes often significantly disturb the flow.  相似文献   

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