Presence Of Background Rotation Research Articles

Generation of nonlinear internal waves by flow over topography: Rotational effects.

We use the forced Ostrovsky equation to investigate the generation of internal waves excited by a constant background current flowing over localized topography in the presence of background rotation. As is now well known in the absence of background rotation, the evolution scenarios fall into three cases, namely subcritical, transcritical, and supercritical. Here an analysis of the linearized response divides the waves into steady and unsteady waves. In all three cases, steady waves occur downstream but no steady waves can occur upstream, while unsteady waves can arise upstream only when there is a negative minimum of the group velocity. The regions occupied by the steady and unsteady waves are determined by their respective group velocities. When the background current is increased, the wave number of the steady waves decreases. In addition, the concavity (canyon or sill), the topographic width, and the relative strength of the rotation play an important role in the generation mechanism. Nonlinear effects modulate the wave amplitude and lead to the emergence of coherent wave packets. All these findings are confirmed by numerical simulations.

Path instabilities of light particles rising in a liquid with background rotation

The present paper reports on experimental observations performed on 2–6mm low-density polystyrene particles rising freely in a water-glycerol mixture for the liquid at rest as well as with slow background rotation (axisymmetric solid body rotation around the vertical axis). Particle trajectories were studied in the Reynolds-numbers range of 200<Re<800 at a fixed rotation rate of Ω=0.81/s. Both the trajectory measurements and the wake visualization indicate that there are differences in the wake structure of the particles as well as in the path instabilities compared to those in quiescent fluid. In the swirling flow, the two-threaded wake mode is dominant up to Reynolds numbers of about 500 with typical spiral-type trajectories until vortex shedding commences. In contrast, under quiescent flow conditions vortex shedding already occurs above Re=230, and the particles predominantly move in zig-zagging paths. Thus, the presence of background rotation is able to suppress vortex shedding and to delay the transition. For the co-rotating observer, the radius of the spiral-type path is comparable to that of the transversal motion under quiescent flow conditions. Therefore, the magnitude of lateral lift-forces is not greatly affected by the presence of the swirl.

A mathematically rigorous analysis of forced axisymmetric flows in the atmosphere

This note discusses a rigorous mathematical formulation for the evolution of the Eliassen balanced vortex. It is first shown that a stable balanced vortex of finite extent can be embedded in an ambient fluid at rest, and that such a vortex exists for prescribed angular momentum and potential temperature on fluid parcels. This uses a method developed by Shutts, Booth and Norbury. This is a different way of viewing the problem from the normal methods, which analyse the stability of a prescribed vortex. The stability of the vortex depends on the presence of background rotation and on the azimuthal velocity at the boundary of the vortex being less than that in the surrounding ambient fluid. It is then shown that the evolution of this vortex under axisymmetric forcing can be written as a conservation law for a potential pseudo‐density in the transformed coordinates introduced by Schubert and Hack. The stability of the vortex to non‐axisymmetric perturbations is also discussed.

Coupled Ostrovsky equations for internal waves in a shear flow

In the context of fluid flows, the coupled Ostrovsky equations arise when two distinct linear long wave modes have nearly coincident phase speeds in the presence of background rotation. In this paper, nonlinear waves in a stratified fluid in the presence of shear flow are investigated both analytically, using techniques from asymptotic perturbation theory, and through numerical simulations. The dispersion relation of the system, based on a three-layer model of a stratified shear flow, reveals various dynamical behaviours, including the existence of unsteady and steady envelope wave packets.

Coupled Ostrovsky Equations for Internal Waves, with a Background Shear Flow

We study the behaviour of weakly nonlinear oceanic internal waves in the presence of background rotation and shear flow, when two distinct linear long wave modes have nearly coincident phase speeds. The waves are described by a system of coupled Ostrovsky equations, derived from the full set of Euler equations for incompressible density stratified fluid with a free surface and rigid bottom boundary conditions. We report here some preliminary results obtained when there is a background shear flow.

On strongly interacting internal waves in a rotating ocean and coupled Ostrovsky equations

In the weakly nonlinear limit, oceanic internal solitary waves for a single linear long wave mode are described by the KdV equation, extended to the Ostrovsky equation in the presence of background rotation. In this paper we consider the scenario when two different linear long wave modes have nearly coincident phase speeds and show that the appropriate model is a system of two coupled Ostrovsky equations. These are systematically derived for a density-stratified ocean. Some preliminary numerical simulations are reported which show that, in the generic case, initial solitary-like waves are destroyed and replaced by two coupled nonlinear wave packets, being the counterpart of the same phenomenon in the single Ostrovsky equation.

Large-scale convective dynamos in a stratified rotating plane layer

We study the effect of stratification on large-scale dynamo action in convecting fluids in the presence of background rotation. The fluid is confined between two horizontal planes and both boundaries are impermeable, stress-free and perfectly conducting. An asymptotic analysis is performed in the limit of rapid rotation (τ ≫ 1 where τ is the Taylor number). We analyse asymptotic magnetic dynamo solutions in rapidly rotating systems generalising the results of Soward [A convection-driven dynamo I. The weak field case. Philos. Trans. R. Soc. Lond. A 1974, 275, 611–651] to include the effects of compressibility. We find that in general the presence of stratification delays the efficiency of large-scale dynamo action in this regime, leading to a reduction of the onset of dynamo action and in the nonlinear regime a diminution of the large-scale magnetic energy for flows with the same kinetic energy.

Three-dimensional stability of vortex arrays in a stratified and rotating fluid

This paper investigates numerically and through an asymptotic approach the three-dimensional stability of steady vertical vortex arrays in a stratified and rotating fluid. Three classical vortex arrays are studied: the Kármán vortex street, the symmetric double row and the single row of co-rotating vortices. The asymptotic analysis assumes well-separated vortices and long-wavelength bending perturbations following Billant (J. Fluid Mech., vol. 660, 2010, p. 354) and Robinson &amp; Saffman (J. Fluid Mech., vol. 125, 1982, p. 411). Very good agreement with the numerical stability analysis is found even for finite wavelength and relatively close vortices. For a horizontal Froude number Fh ≤ 1 and for a non-rotating fluid, it is found that the Kármán vortex street for a street spacing ratio (the distance h between the rows divided by the distance b between vortices in the same row) κ ≤ 0.41 and the symmetric double row for any spacing ratio are most unstable to a three-dimensional instability of zigzag type that vertically bends the vortices. The most amplified vertical wavenumber scales like 1/(bFh) and the growth rate scales with the strain Γ/(2πb2), where Γ is the vortex circulation. For the Kármán vortex street, the zigzag instability is symmetric with respect to the midplane between the two rows while it is antisymmetric for the symmetric double row. For the Kármán vortex street with well-separated vortex rows κ &gt; 0.41 and the single row, the dominant instability is two-dimensional and corresponds to a pairing of adjacent vortices of the same row. The main differences between stratified and homogeneous fluids are the opposite symmetry of the dominant three-dimensional instabilities and the scaling of their most amplified wavenumber. When Fh &gt; 1, three-dimensional instabilities are damped by a viscous critical layer. In the presence of background rotation in addition to the stratification, symmetric and antisymmetric modes no longer decouple and cyclonic vortices are less bent than anticyclonic vortices. However, the dominant instability remains qualitatively the same for the three vortex arrays, i.e. quasi-symmetric or quasi-antisymmetric and three-dimensional or two-dimensional. The growth rate continues to scale with the strain but the most unstable wavenumber of three-dimensional instabilities decreases with rotation and scales like Ro/(bFh) for small Rossby number Ro, in agreement with quasi-geostrophic scaling laws.

A laboratory study of mean flow generation in rotating fluids by Reynolds stress gradients

Laboratory experiments were conducted that demonstrate that a mean azimuthal flow can be produced by introducing Reynolds stress gradients to a rotating fluid with zero initial mean flow. This mechanism may play a role in the generation of mean currents in coastal regions. The experiments entail the establishment of turbulence in a thin annular‐shaped region centered within a cylindrical test cell through the use of a vertically oscillating grid. This region rests in a horizontal plane perpendicular to the vertical axis of the tank, and the entire system is placed on a turntable to simulate background rotation. Flow visualization techniques are used to depict qualitative features of the resulting flow field. Measurements of the mean and turbulent velocity fields are performed using a two‐component laser‐Doppler velocimeter. The results show how rectified currents (mean flows) can be generated via Reynolds stress gradients induced by periodic forcing of the grid. In the absence of background rotation, rectified flow is observed in the radial and vertical directions only. The presence of background rotation tends to organize these motions in that the flow tends to move parallel to the turbulent source, i.e., in the azimuthal direction, with the source (strong turbulence) located to the right, facing downstream. The influence of rotation on the Reynolds stresses and their gradients as well as on the ensuing mean flow is evaluated, and the observations are examined by considering individual contributions of the terms in the Reynolds‐averaged momentum equations.

Nonlinear Ekman effects in rotating barotropic flows

In the presence of background rotation, conventional two-dimensional models of geostrophic flow in a rotating system usually include Ekman friction – associated with the no-slip condition at the bottom – by adding a linear term in the vorticity evolution equation. This term is proportional to E1/2 (where E is the Ekman number), and arises from the linear Ekman theory, which yields an expression for the vertical velocity produced by the thin Ekman layer at the flat bottom. In this paper, a two- dimensional model with Ekman damping is proposed using again the linear Ekman theory, but now including nonlinear Ekman terms in the vorticity equation. These terms represent nonlinear advection of relative vorticity as well as stretching effects. It is shown that this modified two-dimensional model gives a better description of the spin-down of experimental barotropic vortices than conventional models. Therefore, it is proposed that these corrections should be included in studies of the evolution of quasi-two-dimensional flows, during times comparable to the Ekman period.

Horizontal wavenumber spectra of winds, temperature, and trace gases during the Pacific Exploratory Missions: 2. Gravity waves, quasi‐two‐dimensional turbulence, and vortical modes

We examine the horizontal wavenumber spectra of horizontal velocity and potential temperature collected by aircraft above the Pacific Ocean to determine whether gravity waves, quasi‐two‐dimensional (Q‐2‐D) turbulence, or vortical modes dominate atmospheric fluctuations at scale sizes of 1–100 km and altitudes of 2–12 km. We conclude from the study of Doppler‐shifting effects that Q‐2‐D turbulence and/or vortical modes are more prevalent than gravity waves over the ocean, except in the equatorial zone. The results are consistent with recent numerical simulations of Q‐2‐D turbulence, which show that the characteristic inverse cascade of energy is greatly facilitated by the presence of background rotation. Furthermore, a Stokes‐parameter analysis reveals the general paucity of coherent wavelike motions, although specific cases of gravity‐wave propagation are observed. Finally, a case study of a long flight segment displays a k−3 horizontal velocity variance spectrum at scales longer than about 100 km. A Stokes‐parameter analysis indicates that these large‐scale fluctuations were likely due to vortical modes rather than inertio‐gravity waves.

Three-dimensional destabilization of Stuart vortices: the influence of rotation and ellipticity

We investigate the influence of the ellipticity of a columnar vortex in a rotating environment on its linear stability to three-dimensional perturbations. As a model of the basic-state vorticity distribution, we employ the Stuart steady-state solution of the Euler equations. In the presence of background rotation, an anticyclonic vortex column is shown to be strongly destabilized to three-dimensional perturbations when background rotation is weak, while rapid rotation strongly stabilizes both anticyclonic and cyclonic columns, as might be expected on the basis of the Taylor–Proudman theorem. We demonstrate that there exist three distinct forms of three-dimensional instability to which strong anticyclonic vortices are subject. One form consists of a Coriolis force modified form of the ‘elliptical’ instability, which is dominant for vortex columns whose cross-sections are strongly elliptical. This mode was recently discussed by Potylitsin &amp; Peltier (1998) and Leblanc &amp; Cambon (1998). The second form of instability may be understood to constitute a three-dimensional inertial (centrifugal) mode, which becomes the dominant mechanism of instability as the ellipticity of the vortex column decreases. Also evident in the Stuart model of the vorticity distribution is a third ‘hyperbolic’ mode of instability that is focused on the stagnation point that exists between adjacent vortex cores. Although this short-wavelength cross-stream mode is much less important in the spectrum of the Stuart model than it is in the case of a true homogeneous mixing layer, it nevertheless does exist even though its presence has remained undetected in most previous analyses of the stability of the Stuart solution.

Development of a point plume in the presence of background rotation

This paper describes a laboratory study on the evolution of a point turbulent plume placed at the free surface of a homogeneous fluid layer in the presence of background rotation. It is shown that the plume initially evolves as if there is no rotation. However, the rotational effects become important after the plume descends a vertical distance hc1≈3.3(B/Ω3)1/4 for a normalized time Ωtc1≈2.4, whence the vertical descent rate of the plume is reduced while maintaining approximately the same lateral growth rate. Here Ω is the rate of background rotation and B is the specific buoyancy flux of the plume. The rotational effects inhibit the lateral growth of the plume at a time Ωtc2≈5.5, when the maximum plume width is bc≈1.4(B/Ω3)1/4. Thereafter, the vertical descent continues and the plume evolves into a cylindrical shape while developing a cyclonic circulation in and around it, except near the plume front. Upon reaching the bottom surface after traveling a fluid depth of H, the plume deflects, propagates horizontally, and becomes unstable breaking up into anticyclonic eddies. Studies carried out for the case of H&amp;lt;hc1 show that this instability is initiated at a horizontal length scale proportional to the Rossby deformation radius of the deflected flow, and hence it is of baroclinic type. These eddies appear to align vertically with the cyclonic eddies formed by the barotropic instability of the surface rim current, thus producing heton-like structures. The influence of the diameter d0 of the plume on the flow evolution is also studied, and it is shown that plumes with aspect ratio h/d0&amp;lt;12 (where h is the vertical extent) can be approximated as point plumes. Scaling arguments are advanced to explain the results. Some geophysical applications of the study are also discussed.

Narrow barotropic currents impinging on an isolated seamount

Laboratory and numerical experiments concerned with the interaction of homogeneous narrow currents (jets) and tall, isolated topographic features in the presence of background rotation were conducted. The laboratory jets (in an azimuthal direction) were produced in a rotating circular test cell using a circularly symmetric, free surface, source sink technique. By regulating the Rossby and Ekman numbers of the jets, the narrow currents so created can be either circularly symmetric (stable) or meandering (unstable). A theoretical analysis for a point sink at the test cell center is advanced to predict the strength and radial distribution of the azimuthal current in the absence of bottom topography. Laboratory observations are found to be in good agreement with the model. The interaction of stable currents with isolated topography was investigated. The system parameters are the Rossby and Ekman numbers, the normalized topographic width and height, the width of the current, and the position of the streamwise centerline. A numerical model was developed which employed a vertically averaged vorticity equation driven by Ekman layer forcing along the free surface and dissipated by Ekman pumping along the lower bounding surface. The model is supported by laboratory observations. Only stable currents are considered for the numerical calculations. Currents which pass either to the right or left of the obstacle center as well as those which impinge directly are considered. For all cases, anticyclonic motions are found atop the obstacle. In the numerical experiments the maximum values of cyclonic and anticyclonic vorticity above the topography are found to be monotonically increasing functions of the Rossby number, with other parameters fixed. For cases in which the current is to the right of the topographic feature, closed cyclonic vortices are formed in the left topography lee. The normalized area of these cyclones increases with increasing Rossby number, for the range of R0 investigated. For all cases the transport stream function lines in the right topography lee are deflected relatively more than the corresponding streamlines in cases for which the upstream flow is uniform and rectilinear (i.e., a broad current having no relative upstream vorticity).

The Motion of a Turbulent Thermal in the Presence of Background Rotation

The Motion of a Turbulent Thermal in the Presence of Background Rotation

Experiments on convection from a horizontal plate with and without background rotation

Laboratory experiments are described on convection driven by a finite-sized circular heating plate in the bottom of a horizontal fluid layer, both with and without background rotation. For the non-rotating case the various observations are arranged in a regime diagram, in which the different flow types are presented as a function of the governing parameters, being the Rayleigh number and the ratio δ of plate size and fluid depth. For the case δ = O(1) it is found that the finiteness of the heating disk has a significant effect on the convective flow structure, in the sense that an inflow is established which tends to stabilize the convection above the plate. The flow pattern changes drastically in the presence of background rotation, the effect of which is characterized by the Rossby number Ro. In the geostrophic limit (Ro « 1) one observes the flow to be organized in an array of axially-aligned vortices, as found in earlier studies. In the Ro = O(1) regime, however, the horizontal inflow along the bottom causes an intense vortex to arise above the heating disk. This vortex appears to be unstable, and it was seen to continuously change from a spiral mode to a toroidal mode and vice versa.

Thermals with background rotation and stratification

Scaling analysis and experiments are used to study the evolution of thermals in the presence of background rotation. When the ambient environment is homogeneous, the thermal rises and expands until it reaches a critical height where the Rossby number becomes ∼ 1. The thermal then stops expanding and rises in a column. Both the critical height and column radius scale with (F0f-2)1/4. F0 is the initial thermal buoyancy and f is the Coriolis frequency. The thermal vertical velocity is independent of f. When the background is stratified with buoyancy frequency N, the thermal rises to a neutral buoyancy level which scales with (F0N-2)1/4. For N/f [Lt ] 0.6 column formation occurs before the thermal reaches the neutral level. For N/f [Gt ] 0.6 the thermal reaches the neutral level before rotation is important. In both regimes, geostrophic adjustment eventually causes the formation of a baroclinic vortex consisting of an anticyclonic lens of thermal fluid at the neutral level and cyclonic circulation below. The lens has Nh/fl ∼ 1. The lens thickness 2h and the radius l obey relations of the form (F0N-2)1/4 (N/f)m. However, the exponents m are different in the two regimes. The relevance of these results to deep-ocean convection and hydrothermal venting is discussed.

Experimental studies of lift and drag forces upon cylindrical obstacles in homogeneous, rapidly rotating fluids

A series of laboratory measurements is described, in which simultaneous drag and lift forces have been measured directly on full and truncated cylinders moving uniformly through homogeneous rotating fluids. For the full cylinder experiments, it has been demonstrated that within the Reynolds ( Re ) and Rossby ( Ro ) number ranges 9.8 × 10 3 ≤ Re ≤5.6 × 10 4 and 0.1 ≤ Ro ≤0.9, the mean total lift coefficient and the mean lift coefficient because of wake asymmetry both increase with decreasing Rossby number Ro . For cases in which the fluid is non-rotating, the total lift coefficient is shown to be zero, and the values of the drag coefficient for the cylinder are shown to be in accord with existing drag for this geometry. When the value of the Reynolds number is sufficiently small, the presence of background rotation is shown to result in a two- to three-fold increase in the drag coefficient. Experiments with truncated cylinders show similar behaviour to the full cylinder cases with regard to the dependence of the drag coefficient upon Ro and Re . However, values of the lift coefficient are shown to be increased considerably over corresponding full cylinder cases throughout the Reynolds number range, particularly at low and intermediate values of the inverse Rossby number Ro −1 . The increase in lift is attributed to (1) the accentuation of wake asymmetry caused by vortex tube stretching as fluid passes below the truncated cylinder, and (2) the generation of an unbalanced Coriolis force on the volume of fluid displaced by the cylinder.

Motion of Oscillatory Currents Past Isolated Topography

Abstract With the aim of developing increasingly realistic physical models of the interaction of ocean currents with isolated seamounts, laboratory experiments concerning the flow of an oscillatory current past a cosine-squared body of revolution in the presence of background rotation and stratification are considered. The pertinent parameters governing the motion are the Rossby, temporal Rossby, Burger and Ekman numbers, the ratio of the magnitude of the oscillatory velocity component to the mean current and various geometrical parameters. With the exception that the present experiments distort the vertical coordinate, the studies are conducted in regions of parameter space of relevance to the real ocean; future communications will investigate relatively undistorted geometries. The experiments demonstrate that three fundamentally different flow regimes can be identified and that these are highly sensitive to the value of the Rossby number, Ro. These include, (i) at low Ro a regime in which the flow is fu...