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Geophysical Fluid Dynamics / Dynamiques des fluides en geophysique
(Kevin Lamb and Richard Greatbatch, Organizers)

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LUCY CAMPBELL, Department of Mathematics, McGill University, Montreal, Quebec  H3A 2K6
Nonlinear critical layer development of forced wave packets in geophysical shear flows

In this talk, we present numerical simulations of the nonlinear evolution of a forced wave packet in the presence of a critical layer in a shear flow. Two different geophysical flows are considered: vertically propagating internal gravity wave packets in a stratified shear flow and Rossby wave packets propagating toward the equator in a zonal flow. It is believed that critical layer dynamics are pertinent to phenomena observed in the atmosphere and ocean, such as stratospheric sudden warmings in the northern hemisphere and the quasi-biennial oscillation. The theories associated with gravity wave critical layers may explain many aspects of the dynamics of waves forced by flow over topography, e.g., momentum transport, wave breaking and the generation of turbulence.

Most previous analyses of critical layer phenomena have dealt with spatially periodic, monochromatic waves. For a more realistic representation of wave activities in the atmosphere, we employ a forcing in the form of a spatially localized wave packet, rather than a monochromatic wave. We solve the nonlinear equations numerically using a pseudo-spectral Fourier approximation and a high order compact finite difference scheme. It is found that the spatial localization delays the onset of the nonlinear breakdown in the critical layer. The disturbance is absorbed at the critical layer and, unlike in the monochromatic case, this continues for large time because the convergence of momentum flux into the critical layer is balanced by an outward flux of momentum in the horizontal or zonal direction. In the gravity wave packet problem, we observe also that the prolonged absorption of the disturbance stabilizes the solution to the extent that it is always convectively stable; the local Richardson number remains positive well into the nonlinear regime.

MIKE FOREMAN, Institute of Ocean Sciences, Sidney, British Columbia  V8L 4B2
Summer current simulations for Juan de Fuca Strait and southwestern Vancouver Island

A three-dimensional finite element model is used to simulate summer currents for Juan de Fuca Strait and the southwestern continental margin of Vancouver Island. The calculations are forced and/or initialized with seasonal winds, climatological density fields, and elevation-specified boundary conditions that have been adjusted via inversion to more accurately represent the California Undercurrent. Tides are also included to correctly represent turbulent mixing, bottom friction, and the contribution of tidal rectification. The seasonal model currents are shown to compare favourably with multi-year, low-pass filtered current meter observations and to capture strong shears both vertically in Juan de Fuca Strait, and horizontally and vertically across the continental shelf and slope. The tidal currents are demonstrated to be more accurate than those computed by a three-dimensional model with homogeneous density.

RICHARD GREATBATCH, Department of Oceanography, Dalhousie University, Halifax, Nova Scotia  B3H 4J1
Who needs the Boussinesq approximation?

McDougall and Garrett have pointed out that there is an error of as much as 30% in the averaged tracer equations currently carried by Boussinesq ocean models. This is in addition to the error of about 5% usually associated with the Boussinesq approximation. New satellite mounted gravity measuring instruments will measure bottom pressure changes to within the accuracy of the Boussinesq approximation. In this talk, it is pointed out that by reinterpreting the model variables most of the 30% error is eliminated. Further, with very little extra cost in terms of cpu requirement, models can easily be made fully non-Boussinesq. Such models not only have the advantage that the averaged tracer equation is now accurately represented, but also conserve mass rather than volume and so accurately compute changes in bottom pressure and sea level. This is important for comparing model-computed sea level and bottom pressure with observations, and also for computing changes in sea level associated with climate change.

RICHARD KARSTEN, Massachusetts Institute of Technology, Cambridge, Massachusetts  02139, USA
A thermally and mechanically driven model of the circumpolar current

Inspired by laboratory experiments and eddy-resolving numerical simulations, we present a detailed theory of the structure of a circumpolar current. Using data from laboratory and numerical experiments, we derive empirical relationships for the depth of penetration, width, and strength of the current in terms of the surface buoyancy fluxes and wind stress.

In our laboratory and numerical experiments, a circumpolar current in response to a temperature gradient formed between cooled polar waters and warmed tropical waters. The current is enhanced by a wind stress which concentrates and pumps the warm, tropical surface water to depth while up-welling the deep, cold polar waters. Combined, the forcing acts to overturn the isotherms resulting in a strong horizontal temperature front. This front is baroclinically unstable, and growing waves feed off the available potential energy forming vigorous eddies. The resulting eddies play an essential role in transporting heat across the current by sweeping warm tropical water into the polar region where it is cooled, smoothing out the temperature front. When the heat flux across the front produced by the eddies balances the heat flux produced by the surface forcing, a statistically steady current is achieved.

In order to develop a simple theory of the circumpolar current, we examine a steady-state, two dimensional model in which the eddy-transport has been parameterized in terms of the mean buoyancy gradient and thermal wind. We are able to establish the general structure of the steady current from buoyancy budgets considerations. Using our experimental data to empirically determine the remaining constants of proortionality, we derive straight forward formulas that relate the depth of penetration, the width, and the speed of the current to the surface buoyancy flux and wind stress. Despite the simplicity of our assumptions, these formula give surprisingly reasonable predictions of the observed ACC when forced with observed surface fluxes.

BALU NADIGA, Earth and Environmental Sciences, Los Alamos National Laboratory, Los Alamos, New Mexico  87545, USA
Idealized studies of mesoscale-eddies in oceans

Mesoscale eddies (50-100 km scales) play an important role in determining the overall circulation of the world oceans and are intimately related to the transport of variables like heat, vorticity and momentum. Since a proper resolution of these eddy fields in long term simulations is computationally prohibitive, it is important to be able to parameterize their effects on mean flow. In this talk, we will consider two idealized setups to study such eddy fields. First, we will briefly present simulations of a bouyancy driven, baroclinically-unstable zonal flow in a channel and discuss the salient features of the associated meridional transport. Next, and in more detail, we will consider the case of a double-gyre wind forced barotropically-unstable flow in a rectangular midlatitude ocean basin. In this latter case-a much studied classical problem-we show that a new component of time-mean flow, which is eddy-driven, arises when dissipation is low enough. Finally, we present models for such a flow in which eddy effects are parameterized. One of these uses nonlinear dispersion to model the effects of eddies on mean flow and is based on the ideal mean flow fluid model of Holm, Marsden, and Raitu (PRL 80, 4273-4277, 1998).

MARTIN OBERLACK, RWTH Aachen, Templergraben 64, 52056  Aachen, Germany
Universal principles of turbulent wall-bounded flows

The logarithmic law of the wall is still considered one of the main building blocks in turbulence theory. It has been verified in innumerable experiments and has been implemented in almost all statistical turbulence models. Using group theoretical methods it has been shown that the law of the wall and a variety of new turbulence laws can be derived from first principles solely using Navier-Stokes equations.

W. SCHROEDER AND M. MEINKE, Aerodynamisches Institut, RWTH Aachen, 52062  Aachen, Germany
Large-eddy simulation of internal flows

The characteristic secondary flow pattern, so called Dean vortices, that develops in laminar flow through pipe bends is well known and equally well understood. The turbulent flow through pipe bends, however, is more complicated, and only a few numerical simulations were carried out to investigate this flow. An interesting flow phenomenon in a 90^°-bend is the so called swirl switching which, to the author's knowledge, has not yet been studied using numerical methods. >From an experimental investigation a tangential flow in alternating direction with a low frequency at the inner side of the curved pipe is reported. This paper investigates numerically such low frequency phenomena in pipe bends.

Large-eddy simulations are performed using an AUSM method with central differencing for the pressure term and a Runge-Kutta time stepping scheme. Results will be presented for two 90^°-bend radii at three Reynolds numbers. Time-averaged and statistical data will be compared with experimental findings.

DAVID STRAUB, Atmospheric and Oceanic Sciences, McGill University, Montreal, Quebec  H3A 2K6
Instability of chaotic 2 dimensional flow to 3D perturbations

We consider the growth of 3 dimensional perturbations to the hydrostatic equations linearized about a chaotic 2 dimensional base state. We first consider the case of a nonrotating, homogeneous layer of fluid. Based on an argument by Warn (personal communication), we show numerically that the 3D perturbations grow exponentially. We further argue that a secondary, nonhydrostatic instability occurs and acts to curb the growth. This leads to an estimate of the saturation level of the 3D perturbations. Next, the effect of rotation on the primary instability mechanism is considered. Preliminary results suggest that significant growth does not occur unless the maximum Rossby number is greater than unity. Since vorticity distributions in real turbulent flows show very non-Gaussian distributions (i.e., wide tails), this suggests that that instability occurs at r.m.s. Rossby numbers that are relatively small. Finally, we consider how stratification affects the initial growth of the 3D perturbations. The results have implications to the transfer of energy from ``balanced'' to unbalanced (or forward cascading) modes and, by extension, to the mechanisms whereby energy is dissipated from the general circulation of the oceans and atmosphere.