Mathematics in Geosciences, September 2001 - June 2002

Talk Abstracts:

Reduced Descriptions of Coupled GFD Systems (Slow Manifolds in the Ocean and Atmosphere)

February 11-15, 2002

Material from Talks

William K. Dewar (Department of Oceanography, Florida State University)  bill@vincent.ocean.fsu.edu

A nonlinear, conceptual mid-latitude climate model

Recent models of mid-latitude climate have speculated on the role of the North Atlantic ocean in modulating the North Atlantic Oscillation (NAO) Here this role is examined by means of numerical experimentation with a quasi-geostrophic ocean model underneath a highly idealized atmosphere. It is argued the dominant mid-latitude oceanic influence is due to the so-called inertial recirculations.

David G. Dritschel (Mathematical Institute, University of St. Andrews)  dgd@mcs.st-and.ac.uk   

An explicit potential-vorticity conserving approach to modelling three-dimensional Boussinesq flows    Slides

The Boussinesq equations are used to describe the dynamical behaviour of a rotating, stratified fluid, a prime example being the oceans. These equations consist of momemtum and mass conservation, together with the condition of incompressibility. As normally written, they obscure the underlying material conservation of a quantity called "potential vorticity," given by the product of the absolute vorticity (including the background rotation of the Earth) and the gradient of the density, itself materially conserved. For a stably-stratified fluid, density decreases monotonically with height everywhere, a situation typical of most of the oceans. Then, material conservation of potential vorticity amounts to the conservative advection of potential vorticity on surfaces of constant density (isopycnals).

This mathematical result is well known but, in practise, little exploited. Using potential vorticity explicitly poses two major problems: (1) it forces one to solve a nonlinear diagnostic equation for one of the "primitive" variables (velocity, density, pressure or a combination thereof); and (2) numerical methods are traditionally not suited for conservative advection. In this talk, a new approach is presented which overcomes these two problems. Theoretically, the equations are reformulated in a mathematically convenient way, revealing the existence of an underlying Monge-Ampere equation, a nonlinear diagnostic equation for one of the primitive variables. The reformulation uses the ageostophic horizontal vorticity, a first-order estimate for the "imbalanced" (wave-part) of the flow. Numerically, explicit potential vorticity conservation is handled by "contour advection," which tracks potential vorticity contours in a grid-free way on density surfaces. These contours are converted to gridded values for the purpose of solving the Monge-Ampere equation, and the remaining part of the numerical algorithm uses conventional methods (e.g. pseudo-spectral).

An example of a strongly anticyclonic vortex is presented. We focus on the behaviour of a notoriously difficult field, the vertical velocity, which is typically 10,000 times weaker than the horizontal velocity. Our solutions are shown to be highly accurate, as judged indirectly by comparison with the vertical velocity diagnosed from an approximate balance relation (the "quasi-geostrophic omega equation"). The method also appears able to accurately quantify the radiation of internal-gravity waves.

Darryl D. Holm (Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, MS B284, Los Alamos, NM 87545)  dholm@lanl.gov   http://cnls.lanl.gov/~dholm/

Lagrangian averages, averaged Lagrangians and dimension reduction in modeling GFD turbulence    Slides

By using Averaged/Approximated Lagrangians one obtains a well known series of GFD model equations. Each of these preserves energetics and potential-vorticity/Kelvin-circulation dynamics at its own level of approximation. One is then faced with the additional task in these multiscale problems of reducing the number of degrees of freedom by finding an average description of the motion that incorporates the mean effects of the small scales on the large scales. This is the problem of subgrid-scale modeling.

Lagrangian averaging (LA) also preserves energetics and fluid transport properties in the process of averaging over fast times scales following Lagrangian fluid trajectories. This compatibility allows LA to be imposed at any level of this series of GFD model equations, in either order of approximation. The resulting nonlinear Lagrangian mean equations, however, are not closed.

We shall obtain closure by introducing a small amplitude expansion and Taylor's hypothesis of frozen-in turbulence into the Lagrangian at a given level of approximation, before averaging. This closure approximation also reduces the number of degrees of freedom by smoothing the solution.

C. David Levermore (Department of Mathematics, University of Maryland, College Park)  lvrmr@math.umd.edu

A Shallow Water Model with Eddy Viscosity for Basins with Varying Bottom Topography    Slides

The motion of an incompressible fluid confined to a shallow basin with a varying bottom topography is considered. We introduce appropriate scalings into a three dimensional anisotropic eddy viscosity model to derive derive a two dimensional shallow water model. The global regularity of the resulting model is proved. The anisotropic form of the stress tensor in our three dimensional eddy viscosity model plays a critical role in ensuring the resulting shallow water model dissipates energy.

Alex Mahalov (Department of Mathematics, Arizona State University)  mahalov@asu.edu

Fast Singular Oscillating Limits and Global Regularity for the 3D Primitive Equations of Geophysics    pdf    postscript

Fast singular oscillating limits of the three-dimensional "primitive'' equations for stably stratified rotating geophysical fluid flows are analyzed. We prove existence on infinite time intervals of regular solutions to the 3D "primitive'' Navier-Stokes equations for strong stratification (large stratification parameter). This uniform existence is proven for all domain aspect ratios, including the case of all three wave resonances in the limit resonant equations; smoothness assumptions for initial data are the same as for local existence theorems, that is initial data in H_s, s > 3/4. The global existence is proven using techniques of the Littlewood-Paley dyadic decomposition. Infinite time regularity for solutions of the 3D "primitive'' Navier-Stokes equations is obtained by bootstrapping from global regularity of the limit resonant equations and strong convergence theorems. Algebraic geometry of resonant Poincare curves is also used to obtain regularity results in generic cases for solutions of 3D Euler "primitive'' equations.

James C. McWilliams (IGPP, UCLA)  jcm@atmos.ucla.edu

The Theory and Theology of Nonlinear Balance    Slides

A didactic presentation will be given on the premises, mathematical structure, regimes of validity, historical experience, evolutionary singularities, and unbalanced instabilities for the reduced fluid-dynamical system, the Balance Equations. The Balance Equations are an asymptotically consistent (but nonunique) set of approximations for rotating, stably stratified flows, built around the quasi-static momentum balances of hydrostacy in the vertical and gradient-wind balance in the horizontal divergence.

It is widely agreed that the vast majority of the energy in the general circulations of the ocean and atmosphere is in balanced motions. In this context, the organizing focus of the talk will be on the mystery of large-scale energy dissipation in the ocean and atmosphere: planetary forcing energizes large-scale balanced motions, including balanced instabilities of the directly forced flows; balanced flows are asymptotically characterized by an inverse energy cascade towards larger scales, but there are few and relatively inefficient dissipation mechanisms available are large scales (e.g., bottom drag and radiative cooling); thus, somehow the route to dissipation at small scales remains to be elucidated.

Leslie M. Smith (Departments of Mathematics and Mechanical Engineering, University of Wisconsin, Madison, WI, USA)   lsmith@math.wisc.edu   http://www.math.wisc.edu/~lsmith

Population of Slow Manifolds in Strongly Stratified, Rotating Flows with Unbalanced Turbulent Forcing    Slides

Numerical simulations are used to study the population of slow manifolds in rotating, stably stratified flow in the Boussinesq approximation, with rotation and stratification both in the vertical direction. Energy is injected through a three-dimensional, isotropic, white-noise forcing localized at small scales. The parameter range studied corresponds to Froude numbers smaller than an O(1) critical value, below which energy is transferred to scales larger than the forcing scales. The values of the ratio N/f range from 1/2 to infinity.

For purely stratified flows, there exist two distint classes of non-wave modes: the Vertically Sheared Horizontal Flow (VSHF) modes with dependence only on the vertical wavenumber, and the Potential Vorticity (PV) modes, existing for all wavevectors and with zero vertical velocity. For strongly stratified flows with N/f >> 1, the only non-wave modes are the PV modes, while the VSHF modes have (small) wave frequency f or -f. Somewhat surprisingly, for all strongly stratified flows including the purely stratified case, our simulations show that the large scales generated by the turbulence are the VSHF modes. In this case, the PV modes play a secondary role, acting to inhibit the transfer of energy to large scales. On the other hand, for N/f between 1/2 and 2, our simulations show that the inertial-gravity waves are insignificant and that the dynamics are completely dominated by the PV modes. This is quasi-geostrophic turbulence characterized by the inviscid conservation of two quadratic invariants and a -5/3 inverse energy cascade. The region 1/2 < N/f < 2 is also exactly the region where resonant triad interactions cannot occur. These results suggest that 1/2 < N/f < 2 is the domain of validity of the quasi-geostrophic model (for moderate aspect ratios), and that resonant wave interactions play an important role in the population of the slow, VSHF motions in strongly stratified flow.

Edriss S. Titi (Department of Mathematics, Mechanical and Aerospace Engineering, University of California, Irvine, CA 92717-3875, USA)   etiti@math.uci.edu   http://www.math.uci.edu/~etiti/

Global Well-posedness and Long-Term Dynamics for Certain Geophysical Models

The basic problem faced in geophysical fluid dynamics is that a mathematical description based only on fundamental physical principles, which are called the "Primitive Equations,'' is often prohibitively expensive computationally, and hard to study analytically. In this talk I will present a formal derivation of more manageable shallow water approximate models for the three dimesional Euler equations in a basin with slowly spatially varying topography, the so called "Lake Equation" and "Great Lake Equation," which should represent the behavior of the physical system on time and length scales of interest. These approximate models will be shown to be globally well-possed. I will also show that the Charney-Stommel model of the gulf-stream, which is a two dimensional damped driven shallow water model for ocean circulation, has a global attractor. Whether this attractor is finite or infinite dimensional is still an open question. Other results concerning the global well-posedess of three dimensional viscous planetary geostrophic models will be presented.

Jacques Vanneste (Department of Mathematics and Statistics, University of Edinburgh)  vanneste@maths.ed.ac.uk

Dirac-bracket approach to Hamiltonian balanced models

Balanced models can be viewed as constrained systems, obtained from the primitive equations by projection on a (slow) manifold devoid of inertia-gravity waves. Salmon showed how balanced models naturally inherit the Hamiltonian structure of the primitive equations if the constraints are implemented in the variational principle associated with the primitive equations. This, however, requires the introduction of an extended state space, using Lagrangian variables either as dependent or independent variables. Here, we demonstrate how this can be avoided and we derive Hamiltonian balanced models using exclusively the standard Eulerian formulation of the primitive equations. This is achieved by applying Dirac's theory of constrained systems to the Poisson structure of the fluid equations in Eulerian form. We consider multilayer primitive equations and implement general constraints which prescribe the velocity field as a pseudo-differential function of the mass field. This leads to the Poisson structure of a general class of balanced models which include (multilayer versions of) Salmon's L1 model, the semi-geostrophic model, and higher-order balanced models. The well-posedness of models in this class depends on certain invertibility issues which will be discussed.

Beth A. Wingate (Los Alamos National Laboratory)  wingate@lanl.gov

The alpha-model of turbulence for GFD applications

In this talk I discuss results related to using $\alpha$-models for the primitive equations used in ocean modeling. Two thrusts are discussed, the first is baroclinic instability, the other the numerical simulations of rotating shallow water equations. We look at the wall-bounded, wind-forced double gyre problem and periodic decaying shallow water turbulence.

Djoko Wirosoetisno (Faculty of Applied Mathematics, University of Twente)  djoko@maths.ed.ac.uk

Nonlinear Averaging in GFD Systems and Higher-Order Balance Dynamics    Slides:    pdf    postscript

An averaging (or renormalization) procedure is used to obtain slow evolution equations for all degrees of freedom of a parent model which contains fast and slow dynamics. In a GFD context, the parent model is the primitive equations, the slow dynamics consists of vortical motion and the fast dynamics consists of inertia-gravity waves. This procedure can be carried out (formally) to any order in the timescale separation parameter giving higher-order slow equations.

We will show a close connection between these slow equations and the more familiar classical balance models, which are obtained using a singular perturbation expansion and have a reduced number of degrees of freedom. Issues of convergence of these asymptotic procedures will be considered.