P.I.M.S. Summer School in Fluid Dynamics

Abstracts of Invited Talks

Geophysical Plumes
J. W. M. Bush, M.I.T.

We review the dynamics of buoyant fluid released from point and line sources. Both continuous (plume) and instantaneous (thermal) releases are considered. Particular attention is given to the combined influence of rotation and stratification on the dynamics, and on the case where the buoyancy is associated with suspended particulate matter. Simple scaling arguments are developed to describe this class of problems, and supporting experimental results presented. A number of geophysical and environmental applications are discussed, including hydrothermal venting on the seafloor, thermohaline convection forced by arctic leads in the polar seas, and the dumping of dredged material.

10:30am, Thursday August 8 in room 243 Central Academic Building

Large Eddy Simulations
J.-L. Guermond, L.I.M.S.I.

The main objective of this lectures is to review and report on key mathematical issues related to the theory of Large Eddy Simulation of turbulent flows. Key mathematical results on the Navier--Stokes equations will be reviewed and put in parallel with the common heuristic understanding of turbulence. It will be shown that most LES models have mathematical counterparts that have been introduced to resolve questions such as uniqueness, existence of a maximum principle, convergence to entropy solutions, and convergence in graph norms.

3pm, Monday July 29 and 3pm, Tuesday July 30 in room 243 Central Academic Building

Fluid Dynamics of the Solid Earth
U. Hansen, Institut fur Geophysik, Muenster

Convective transport plays a dominant role in many geological systems. In Magma chambers convection does not only govern the transport of heat and matter but also determines widely the evolution of structures like the appearance of prominent layered systems. On larger scale, convection turns heat from the Earth's interior into mechanical work, thus being the driving mechanism of plate tectonics. According to our present knowledge, convective currents within the Earth's molten metallic core generate and maintain the Earth's magnetic field through a dynamo process.Convection in these system is driven by thermally or compositionally induced density differences. In all relevant scenarios the control parameters are far above their critical values, i.e. convection takes place in the highly nonlinear regime and displays motion with large amplitude. Especially for the extreme parameters found in geological systems, laboratory experiments are difficult to perform, such that numerical experiments play an important role. In this lecture I will present results of numerical experiments, carried out to investigate some features of convective flows in geological systems. With respect to global circulation in the Earth's rocky mantle, a type of convection is considered which is characterized by a high Rayleigh number (Ra > 10**7) and an infinite Prandtl number, i.e. inertia doesn't play a role. Nevertheless the flow typically shows a chaotic temporal evolution. Unlike in th atmosphere and the oceans the viscosity in Earth's material is strongly dependent on the temperature - hot material is in general less viscous than cold material. This gives convection a characteristic style, which will be described and discussed. Thermo-chemical, or double-diffusive convection takes place if at least one driving and one restoring force act on the system. Im magma chambers the temperature distribution may be the driving force, while a stabilizing compositional stratification can act as a restorting force. Under such circumstances layers do typically evolve. finger-like structures are found in situations where the fast diffusing component (i.e. heat) acts as a stabilizing element against a slowly diffusing destabilizing compositional stratification. Experiments from both regimes will be presented in the lecture.

10:15am, Thursday August 1 in room 243 Central Academic Building

Talk 1: Overturning circulations in oceans and atmospheres
P. Rhines, School of Oceanography, University of Washington

There are many circumstances in which fluid exhibits a subtle mode of circulation, often transverse to the primary direction of flow, which arises to fulfill three-dimensional force balance, and acts to transfer momentum and other properties from boundar y to interior. Familiar examples are the Ekman layer-pumping circulation in a rotating fluid moving stressed by its boundaries, the Hadley/Ferrel meridional circulation of the atmospheric troposphere, the Brewer/Dobson meridional circulation of the atmospheric stratosphere, the meridional overturning of the global ocean, the buoyancy-driven overturning of an ocean estuary, and secondary flows in curved-channel flows and rivers. Characteristic of overturning circulations is that they are often weak, difficult to observe, yet essential to the dynamics. Overturning circulations often carry out essential tranport of Lagrangian tracers, trace chemicals as well as dynamical quantities. As such they are sometimes called 'pumps', for example carrying tropospheric moisture into the stratosphere or biologically active carbon deep into the ocean. Here we review the known mechanisms for these circulations, and some of the vorticity dynamics associated with them. There are many challenging ideas relating to constraints of potential vorticity, wave/mean-flow interaction, the omega equation, potential vorticity transport and symmetric instability. Recent work is described, on the overturning in the small-scale stratified spin-up problem (Thomas and Rhines, J.Fluid Mech. in press) and in the large-scale Antarctic Circumpolar Current (MacCready and Rhines, J.Phys.Oceanography 2002).

10:15am, Wednesday August 7 in room 243 Central Academic Building

Talk 2: Mountainous flows in rotating fluids: vorticity dynamics, form drag, induced circulation
P. Rhines, School of Oceanography, University of Washington
Flow over mountains, in the atmosphere or the sea, is a key problem in which pressure forces on the upstream and downstream faces of the topography express a horizontal force on the fluid...a form drag. This form drag works equally well at hilly density interfaces within the fluid, and is a prime agent for the vertical flux of ho rizontal momentum. Topography provides delta-function sheets of potential vorticity that can be stripped off into the fluid to produce vortex wakes and broad regions of new circulation (Rhines, 1979; Hallberg and Rhines 2000; Schneider, Held and Garner 2002). Here we describe some of the vortex dynamics involved in form drag; these are applied to laboratory experiments involving shedding of vorticity and induction of mean circulation in steady and oscillatory flows over mountains. The fluid is rotating and may be stratified. The effects are sensitive to the large-scale potential vorticity field, which can readily control the nature of induced circulation. The 'oscillatory Charney-deVore' flow shown in which Rossby-wave dynamics controls the wave-drag a nd induced circulation. With oscillating flow over the mountains, control by Coriolis effects is summarized in a Ro3.5 variation of turbulent kinetic energy, and Ro-1/2 dependence of spin-up time, where Ro is the Rossby number of the flow. The induced mean circulation switches sign as Ro increases, and shedding of vorticity changes from negative to positive. A large-scale topographic slope provides large-scale potential vorticity gradient and Rossby wave dynamics which control the induced mean circulation. For large (/Ro the m ountains are a 'perfect rectifier', driving a mean flow (pseudo-westward, directed with large potential vorticity to its right); ( is the fractional topographic height. Density stratification traps these effects vertically.

9:00am, Thursday August 8 in room 243 Central Academic Building

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Bruce R. Sutherland, May 2002.