6 results
Nonlinear Rossby adjustment in a channel: beyond Kelvin waves
- Albert J. Hermann, Peter B. Rhines, E. R. Johnson
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- Journal:
- Journal of Fluid Mechanics / Volume 205 / August 1989
- Published online by Cambridge University Press:
- 26 April 2006, pp. 469-502
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Nonlinear advective adjustment of a discontinuity in free-surface height under gravity and rotation is considered, using the method of contour dynamics. After linear wave-adjustment has set up an interior jet and boundary currents in a wide ([Gt ] one Rossby radius) channel, fluid surges down-channel on both walls, rather than only that wall supporting a down-channel Kelvin wave. A wedgelike intrusion of low potential vorticity fluid on this wall, and a noselike intrusion of such fluid on the opposite wall, serve to reverse the sign of relative vorticity in the pre-existing currents. For narrower channels, a coherent boundary-trapped structure of low potential vorticity fluid is ejected at one wall, and shoots ahead of its parent fluid. The initial tendency for the current to concentrate on the ‘right-hand’ wall (the one supporting a down-channel Kelvin wave in the northern hemisphere) is defeated as vorticity advection shifts the maximum to the left-hand side. Ultimately fluid washes downstream everywhere across even wide channels, leaving the linearly adjusted upstream condition as the final state. The time necessary for this to occur grows exponentially with channel width. The width of small-amplitude boundary currents in linear theory is equal to Rossby's deformation radius, yet here we find that the width of the variation in velocity and potential vorticity fields deviates from this scale across a large region of space and time. Comparisons of the contour dynamics solutions, valid for small amplitude, and integration of the shallow-water equations at large amplitude, show great similarity. Boundary friction strongly modifies these results, producing fields more closely resembling the linear wave-adjusted state. Observed features include those suggestive of coastally trapped gravity currents. Analytical results for the evolution of vorticity fronts near boundaries are given in support of the numerical experiments.
Buoyant inhibition of Ekman transport on a slope and its effect on stratified spin-up
- Parker MacCready, Peter B. Rhines
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- Journal:
- Journal of Fluid Mechanics / Volume 223 / February 1991
- Published online by Cambridge University Press:
- 26 April 2006, pp. 631-661
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The unsteady boundary layer of a rotating, stratified, viscous, and diffusive flow along an insulating slope is investigated using theory, numerical simulation, and laboratory experiment. Previous work in this field has focused either on steady flow, or flow over a conducting boundary, both of which yield Ekman-type solutions. After the onset of a circulation directed along constant-depth contours, Ekman-type flux up or down the slope is opposed by buoyancy forces. In the unsteady, insulating case, it is found that the cross-slope transport decreases in time as (t/τ)−½ where \[ \tau = \frac{1}{S^2f\cos\alpha}\left(\frac{1/\sigma + S}{1+S}\right), \] may be called the ‘shut-down’ time. Here S = (N sin α/f cos α)2, f is the Coriolis frequency, α is the slope angle, N is the buoyancy frequency, and σ is the Prandtl number. Subsequently the along-slope flow, $\hat{v}$, approximately obeys a simple diffusion equation \[ \frac{\partial\hat{v}}{\partial t} = \nu\left(\frac{1/\sigma + S}{1+S}\right)\frac{\partial^2\hat{v}}{\partial\hat{z}^2}, \] where t is time, ν is the kinematic viscosity, and $\hat{z}$ is the coordinate normal to the slope. By this process the boundary layer diffuses into the interior, unlike an Ekman layer, but at a rate that may be much slower than would occur with simple non-rotating momentum diffusion. The along-slope flow, $\hat{v}$, is nevertheless close to thermal wind balance, and the much-reduced cross-slope transport is balanced by stress on the boundary. For a slope of infinite extent the steady asymptotic state is the diffusively driven ‘boundary-mixing’ circulation of Thorpe (1987). By inhibiting the cross-slope transport, buoyancy virtually eliminated the boundary stress and hence the ‘ fast’ spin-up of classical theory in laboratory experiments with a bowl-shaped container of stratified, rotating fluid.
Homogenization of potential vorticity in planetary gyres
- Peter B. Rhines, William R. Young
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- Journal:
- Journal of Fluid Mechanics / Volume 122 / September 1982
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- 20 April 2006, pp. 347-367
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The mean circulation of planetary fluids tends to develop uniform potential vorticity q in regions where closed time-mean streamlines or closed isolines of mean potential vorticity exist. This state is established in statistically steady flows by geostrophic turbulence or by wave-induced potential-vorticity flux. At the outer edge of the closed contours the expelled gradients of q are concentrated. Beyond this transition lies motionless fluid, or a different flow regime in which the planetary gradient of q may be dominant. The homogenized regions occur where direct forcing by external stress or heating within the closed isoline is negligible, upon the potential-density surface under consideration. In the stably stratified ocean such regions are found at depths greater than those of direct wind-induced stress or penetrative cooling. In ‘channel’ models of the atmosphere we again find constant q when mesoscale eddies cause the dominant potential-vorticity flux. In the real atmosphere the results here can apply only where internal heating is negligible. The derivations given here build upon the Prandtl–Batchelor theorem, which applies to non-rotating, steady two-dimensional flow. Supporting evidence is found in numerical circulation models and oceanic observations.
Waves and turbulence on a beta-plane
- Peter B. Rhines
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- Journal:
- Journal of Fluid Mechanics / Volume 69 / Issue 3 / 10 June 1975
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- 29 March 2006, pp. 417-443
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Two-dimensional eddies in a homogeneous fluid at large Reynolds number, if closely packed, are known to evolve towards larger scales. In the presence of a restoring force, the geophysical beta-effect, this cascade produces a field of waves without loss of energy, and the turbulent migration of the dominant scale nearly ceases at a wavenumber kβ = (β/2U)½ independent of the initial conditions other than U, the r.m.s. particle speed, and β, the northward gradient of the Coriolis frequency.
The conversion of turbulence into waves yields, in addition, more narrowly peaked wavenumber spectra and less fine-structure in the spatial maps, while smoothly distributing the energy about physical space.
The theory is discussed, using known integral constraints and similarity solutions, model equations, weak-interaction wave theory (which provides the terminus for the cascade) and other linearized instability theory. Computer experiments with both finite-difference and spectral codes are reported. The central quantity is the cascade rate, defined as \[ T = 2\int_0^{\infty} kF(k)dk/U^3\langle k\rangle , \] where F is the nonlinear transfer spectrum and 〈k〉 the mean wavenumber of the energy spectrum. (In unforced inviscid flow T is simply U−1d〈k〉−1/dt, or the rate at which the dominant scale expands in time t.) T is shown to have a mean value of 3·0 × 10−2 for pure two-dimensional turbulence, but this decreases by a factor of five at the transition to wave motion. We infer from weak-interaction theory even smaller values for k [Lt ] kβ.
After passing through a state of propagating waves, the homogeneous cascade tends towards a flow of alternating zonal jets which, we suggest, are almost perfectly steady. When the energy is intermittent in space, however, model equations show that the cascade is halted simply by the spreading of energy about space, and then the end state of a zonal flow is probably not achieved.
The geophysical application is that the cascade of pure turbulence to large scales is defeated by wave propagation, helping to explain why the energy-containing eddies in the ocean and atmosphere, though significantly nonlinear, fail to reach the size of their respective domains, and are much smaller. For typical ocean flows, $k_{\beta}^{-1} = 70\,{\rm km} $, while for the atmosphere, $k_{\beta}^{-1} = 1000\,{\rm km}$. In addition the cascade generates, by itself, zonal flow (or more generally, flow along geostrophic contours).
Nonlinear stratified spin-up
- LEIF N. THOMAS, PETER B. RHINES
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- Journal:
- Journal of Fluid Mechanics / Volume 473 / 10 December 2002
- Published online by Cambridge University Press:
- 13 December 2002, pp. 211-244
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Both a weakly nonlinear analytic theory and direct numerical simulation are used to document processes involved during the spin-up of a rotating stratified fluid driven by wind-stress forcing for time periods less than a homogeneous spin-up time. The strength of the wind forcing, characterized by the Rossby number ε, is small enough (i.e. ε[Lt ]1) that a regular perturbation expansion in ε can be performed yet large enough (more specifically, ε∝E1/2, where E is the Ekman number) that higher-order effects of vertical diffusion and horizontal advection of momentum/density are comparable in magnitude. Cases of strong stratification, where the Burger number S is equal to one, with zero heat flux at the upper boundary are considered. The Ekman transport calculated to O(ε) decreases with increasing absolute vorticity. In contrast to nonlinear barotropic spin-up, vortex stretching in the interior is predominantly linear, as vertical advection negates stretching of interior relative vorticity, yet is driven by Ekman pumping modified by nonlinearity. As vertical vorticity is generated during the spin-up of the fluid, the vertical vorticity feeds back on the Ekman pumping/suction, enhancing pumping and vortex squashing while reducing suction and vortex stretching. This feedback mechanism causes anticyclonic vorticity to grow more rapidly than cyclonic vorticity. Strict application of the zero-heat-flux boundary condition leads to the growth of a diffusive thermal boundary layer E−1/4 times thicker than the Ekman layer embedded within it. In the Ekman layer, vertical diffusion of heat balances horizontal advection of temperature by extracting heat from the thermal boundary layer beneath. The flux of heat extracted from the top of the thermal boundary layer by this mechanism is proportional to the product of the Ekman transport and the horizontal gradient of the temperature at the surface. The cooling caused by this heat flux generates density inversions and intensifies lateral density gradients where the wind-stress curl is negative. These thermal gradients make the potential vorticity strongly negative, conditioning the fluid for ensuing symmetric instability which greatly modifies the spin-up process.
Laboratory studies of equatorially trapped waves using ferrofluid
- DANIEL R. OHLSEN, PETER B. RHINES
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- Journal:
- Journal of Fluid Mechanics / Volume 338 / 10 May 1997
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- 10 May 1997, pp. 35-58
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We introduce a new technique to model spherical geophysical fluid dynamics in the terrestrial laboratory. The local vertical projection of planetary vorticity, f, varies with latitude on a rotating spherical planet and allows an important class of waves in large-scale atmospheric and oceanic flows. These Rossby waves have been extensively studied in the laboratory for middle and polar latitudes. At the equator f changes sign where gravity is perpendicular to the planetary rotation. This geometry has made laboratory studies of geophysical fluid dynamics near the equator very limited. We use ferrofluid and static magnetic fields to generate nearly spherical geopotentials in a rotating laboratory experiment. This system is the laboratory analogue of those large-scale atmospheric and oceanic flows whose horizontal motions are governed by the Laplace tidal equations. As the rotation rate in such a system increases, waves are trapped to latitudes near the equator and the dynamics can be formulated on the equatorial β-plane. This transition from planetary modes to equatorially trapped modes as the rotation rate increases is observed in the experiments. The equatorial β-plane solutions of non-dispersive Kelvin waves propagating eastward and non-dispersive Rossby waves propagating westward at low frequency are observed in the limit of rotation fast compared to gravity wave speed.