ALBERT

Description: When a rotating fluid over sloping topography is heated from below and/or cooled from above, horizontal temperature gradients develop which drive convection cells aligned with isobaths. We refer to these cells as topographic Hadley cells. Laboratory experiments reveal that sinking occurs in small cyclonic vortices situated in relatively shallow regions. This is balanced by slower upwelling in adjacent deeper regions. The cross-isobath motions which connect the upwelling and downwelling are accelerated by Coriolis forces, resulting in strong jets which follow isobathic contours. For anticlockwise rotation, the surface jets keep the shallows to their left when looking in the direction of flow, which is opposite to both Kelvin and Rossby wave propagation. The width of the jets scales with the Rossby deformation radius and if this is much less than the width of the slope region then a number of parallel jets form. Motions on the deeper side of the jets where the flow is accelerating are adequately described by linear inviscid theory. However, the strong shears generated by this acceleration lead to baroclinic instability. The resulting cross-stream momentum fluxes broaden and flatten the velocity profile, allowing the flow on the shallow side of the jet to decelerate smoothly before sinking. Topographic Hadley cells are dynamically similar to terrestrial atmospheric Hadley cells and may also be relevant to the zonal jet motions observed on Jupiter and Saturn. It is also suggested that in coastal seas they may represent an important mode of heat (or salt) transfer where surface cooling (or evaporation) drives convection. © 1994, Cambridge University Press. All rights reserved.

Description: The entire free-surface elevation field of a rotating fluid in the laboratory can be imaged and analysed, by using it as a parabolic Newtonian telescope mirror. This 'optical altimetry' readily achieves a precision of better than 1 μm of surface elevation. The surface topography corresponds to the pressure field just beneath the surface. It is the streamfunction for the geostrophic hydrostatic circulation, which can be resolved to better than 0.1 mm s-1. Still and animated images thus produced, of the entire surface elevation field, are of value in themselves, and using a projected image (a speckle pattern), have the promise of providing quantitative slope and height field data recovered by PIV (particle imaging velocimetry) techniques. With homogeneous fluid, geostrophic flow is the same at all depths. Yet of equal interest are sheared stratified rotating flows where the surface pressure is associated with inertial waves, convection, and other motions, geostrophic or ageostrophic. Although the technique is designed for experiments in which Coriolis effects are strong, it is possible to use reflective imaging for flows at such high Rossby number that Coriolis effects are negligible, and hence this becomes a tool of more general interest in non-rotating fluid dynamics (for example, illuminating surface gravity waves). Examples are given, involving (i) the Taylor-Proudman effect with very slow flows over topography; (ii) quasi-geostrophic and inertial-wave flows over a mountain (f-plane); (iii) inertial waves generated by oscillatory forcing; (iv) Kelvin waves (v) free oscillatory Rossby waves on a polar β-plane; and (vi) stationary waves, blocking, jets and wakes with β-plane zonal flow past a mountain. Movies are available with the online version of the paper. © 2007 Cambridge University Press.

Description: Results from new experiments on baroclinic instability of a coastal jet demonstrate that this almost balanced flow spontaneously emits inertial waves when the Rossby radius of deformation is relatively small such that the characteristics of baroclinic meanders match the dispersion relation for the inertial waves. The energy of the waves is small compared to the energy of the flow. A single event of wave emission is identified in the experiment with larger radius of deformation and is interpreted in terms of vorticity dynamics. The flows are generated on a laboratory polar β plane where the Coriolis parameter varies quadratically with latitude. A new method for imaging the rotating flows, which the authors call “altimetric imaging velocimetry,” is employed. Optical color coding of slopes of the free-surface elevation field allows the authors to derive the fields of pressure, surface elevation, geostrophic velocity, or the “gradient wind” velocity with very high spatial resolution (typically several million vectors) limited largely by the pixel resolution of the available imaging sensors. The technique is particularly suited for the investigations of small-amplitude waves, which are often difficult to detect by other methods.

Description: This paper describes qualitative features of the generation of jetlike concentrated circulations, wakes, and blocks by simple mountainlike orography, both from idealized laboratory experiments and shallow-water numerical simulations on a sphere. The experiments are unstratified with barotropic lee Rossby waves, and jets induced by mountain orography. A persistent pattern of lee jet formation and lee cyclogenesis owes its origins to arrested topographic Rossby waves above the mountain and potential vorticity (PV) advection through them. The wake jet occurs on the equatorward, eastern flank of the topography. A strong upstream blocking of the westerly flow occurs in a Lighthill mode of long Rossby wave propagation, which depends on βa2/U, the ratio of Rossby wave speed based on the scale of the mountain, to zonal advection speed, U (β is the meridional potential vorticity gradient, f is the Coriolis frequency, and a is the diameter of the mountain). Mountains wider (north–south) than the east–west length scale of stationary Rossby waves will tend to block the oncoming westerly flow. These blocks are essentially β plumes, which are illustrated by their linear Green function. For large βa2/U, upwind blocking is strong; the mountain wake can be unstable, filling the fluid with transient Rossby waves as in the numerical simulations of Polvani et al. For small values, βa2/U ≪ 1 classic lee Rossby waves with large wavelength compared to the mountain diameter are the dominant process. The mountain height, δh, relative to the mean fluid depth, H, affects these transitions as well. Simple lee Rossby waves occur only for such small heights, δh/h ≪ aβ/f, that the f/h contours are not greatly distorted by the mountain. Nongeostrophic dynamics are seen in inertial waves generated by geostrophic shear, and ducted by it, and also in a texture of finescale, inadvertent convection. Weakly damped circulations induced in a shallow-water numerical model on a sphere by a lone mountain in an initially simple westerly wind are also described. Here, with βa2/U ∼1, potential vorticity stirring and transient Rossby waves dominate, and drive zonal flow acceleration. Low-latitude critical layers, when present, exert strong control on the high-latitude waves, and with no restorative damping of the mean zonal flow, they migrate poleward toward the source of waves. While these experiments with homogeneous fluid are very simplified, the baroclinic atmosphere and ocean have many tall or equivalent barotropic eddy structures owing to the barotropization process of geostrophic turbulence.

Description: This paper is part of a study of quasigeostrophic waves, which depend on the topography of the ocean floor and the curvature of the earth.In a homogeneous, β-plane ocean, motion of the fluid across contours of constantf/hreleases relative vorticity (fis the Coriolis parameter andhthe depth). This well-known effect provides a restoring tendency for either Rossby waves (withhconstant) or topographic waves over a slope. The long waves in general obey an elliptic partial differential equation in two space variables. Because the equation has been integrated in the vertical direction, the exact inviscid bottom boundary condition appears in variable coefficients.When the depth varies in only one direction the equation is separable at the lowest order in ω, the frequency uponf. With a simple slope, |[xdtri ]h/h| = constant, the transition from Rossby to topographic waves occurs at |[xdtri ]h| ∼h/Re, where Reis the radius of the earth. Isolated topographic features are considered in §2. It is found that a step of fractional height δ on an otherwise flat ocean floor reflects the majority of incident Rossby waves when δ 〉 2ω. In the ocean ω is usually small, due to continental barriers, so even slight depth variations are important. A narrowridgedoes not act as a great obstruction but calculations show, for example, that the Mid-Atlantic Ridge is broad enough to reflect all but the lowest mode Rossby waves in the North Atlantic.Besides isolating oceanic plains from one another, steps and ridges support trapped topographic waves of greatest frequency ∼ δ/2, analogous to the potential well solutions in quantum mechanics. These waves cannot carry energy alongabrupttopography, but they disperse more rapidly over broader slopes; the phase and group speeds may be hundreds of cm/sec. The continentalshelf waves found by Robinson are an example of the latter case. There are many such wave guides, where thef/hcontours are crowded, in the deep ocean.The theory suggests that measurement of Rossby waves will rarely be possible at the coast of a continent.

Description: The homogenization of a passive ‘tracer ’ in a flow with closed mean streamlines occurs in two stages: first, a rapid phase dominated by shear-augmented diffusion over a time ≈ P⅓ (L/U), where the Peclet number P=Lu/K (L, U And Kare lengthscale, velocity scale and diffusivity), in which initial values of the tracer are replaced by their (generalized) average about a streamline; second, a slow phase requiring the full diffusion time ≈ L2/K.The diffusion problem for the second phase, where tracer isopleths are held to streamlines by shear diffusion, involves a generalized diffusivity which is proportional to K, but exceeds it if the streamlines are not circular. Expressions are also given for flow fields that are oscillatory rather than steady. © 1983, Cambridge University Press. All rights reserved.

Description: We consider slow oscillations trapped about axisymmetric islands and seamounts. ω is in the range [lsim ] δ/2 (ω is the frequency divided by f, the Coriolis parameter, and δ the fractional change in depth). The periods, for example, are [gsim ] 2·4 days for an island with a sloping ‘skirt’, h [vprop ] r½, where h is the depth and (r, θ) are polar co-ordinates in the plane tangent to the mean sea surface. Energy leaks slowly away from the topography in Rossby waves. In the limiting case of a cylindrical island with vertical walls there are no such trapped motions, but incident Rossby waves are scattered anisotropically. If γ, the ratio of the island radius, a, to the Rossby wavelength, is small, the scattering cross-section ∼ γ3a. The free oscillations at seamounts and islands with skirts allow much stronger scattering (with cross-section ∼ a/γ, ∼ a wavelength), when one of their frequencies is near that of the incident wave.The theory suggests that measurements of Rossby waves will be possible at small islands, but that the many local oscillations in the same frequency range will add some confusion.