Session BG: Geophysical Fluid Dynamics I

Internal gravity wave generation by isolated topography in the laboratory: Limitations of linear theory

The internal wave field associated with steady flow past stationary isolated obstacles is studied in the laboratory in a recirculating stratified shear flume tank. The complete steady span-wise averaged wave field is determined using the synthetic schlieren technique. Typically, the obstacles are two-dimensional segments of cylinders, which are chosen in an attempt to satisfy the requirements for linear behaviour in the wave field. Such obstacles are predicted by linear theory to induce a steady (in the obstacle frame) lee wave field, with perturbations localized above, and slightly downstream of the obstacle, thus implying a relatively small dominant value of the streamwise wavenumber component. This is unsurprising, as the linear spectral response to an isolated obstacle is strongly peaked at small streamwise wavenumbers. Although there are many points of similarity between the observed and predicted wave fields, there are several non-trivial differences, in particular associated with the dominant beam angle for the lee waves, and also a clear decay in amplitude with distance from the obstacle. Possible reasons for these discrepancies will be discussed.   

Nonlinear internal wave generation by an oscillating cylinder

In a density stratified fluid a perturbation at a frequency below the Brunt-Vaisala frequency $N$ will generate internal waves. Internal waves play an important role in the atmosphere and oceans, but their generation mechanisms (especially nonlinear effects) are not well understood. We use particle image velocimetry to study internal waves generated by an oscillating cylinder in a stratified fluid with a linear density gradient ($N$=constant). When the cylinder is oscillated at $\omega$, both fundamental waves (frequency $\omega$) and harmonics at $n\omega$ are observed for integers $n$ such that $n\omega \le N$. The harmonics are generated by either nonlinear wave-wave interaction or cylinder-fluid interaction. For weak forcing, the intensity of the fundamental and second harmonic scales linearly and quadratically respectively with the forcing amplitude.   

Observations of inertial waves in spherical Couette flow

Dynamo action in the cores of planets and stars gives rise to their magnetic fields. We explore spherical Couette flows in liquid sodium as a laboratory model of the Earth's core. Our apparatus is comprised of two independently driven rotating spheres, radii 20 and 60~cm, with sodium filling the gap between them. We apply an axial magnetic field from external magnets and image the resulting induced magnetic field. For small Lundquist number $S \equiv B_0 l / \sqrt{\eta^2 \rho \mu_0}$ (where $l$ is a characteristic length scale, $\eta$ is the magnetic diffusivity, and $\rho$ is the density), the magnetic field is a passive probe of the internal flows. Data from a collection of Hall probes arrayed along a meridian and along the equator are used to characterize the induced magnetic field outside the outer sphere. Images of the field show patterns consistent with Coriolis-restored inertial modes. Moreover the frequencies and symmetries of the observed field are in good agreement with theoretical calculations of inertial modes in a sphere. Our results have implications for the geophysical behavior of Earth's core as well as other rotating fluid systems.   

Vertical propagation of submerged wakes in stratified fluids.

The vertical propagation of energy and momentum from vortex wakes immersed in a stratified ambient can be considered important either as a determinant of the vertical flux parameters required for effective mesoscale modeling in planetary-scale geophysical computations, or as an information transmission problem, where patterns from otherwise inaccessible objects can potentially be observed far from their origin, such as at a free surface. While internal waves can propagate this energy or information over relatively small timescales, questions have been raised as to how and whether the residual vortex motions themselves can propagate vertically through the atmosphere or ocean. A series of laboratory experiments was conducted in carefully-controlled conditions so that the long-time wake disturbances behind a towed sphere, and their vertical propagation through a uniform density gradient were measured. The propagation rates are compared with various models for the horizontal and vertical growth rates, and are parameterized with respect to the internal Froude number. The relative significance of the vertical diffusion can then be evaluated for field conditions.   

Stability of a tilted vortex in a stratified fluid

Intense cyclones on a slope and oceanic vortices shedded behind a coastal tip are two examples of tilted vortices in a linearly stratified fluid. The structure and the stability of such a basic flow are here analysed experimentally and theoretically when the maximum angular velocity of the vortex is larger than the buyoncy frequency. We first show by an asymptotic analysis in the limit of small tilt angles and large Reynolds numbers that tilting induces strong axial flow and density variations. These fields are found to exhibit a critical point singularity at the radius where the angular velocity of the vortex is equal to the buoyancy frequency. The axial velocity and density profiles obtained by smoothing this singularity in a viscous critical layer are compared to PIV measurements and shadowgraph visualisations and a good agreement is demonstrated. The strong axial shear generated in the critical layer is also shown to be unstable by a Kelvin-Helmholtz like instability. Shadowgraph visualizations and growth rate measurements are compared to theoretical results obtained from a linear stability analysis of the theoretical profiles.   

Three-dimensional instabilities and transient growth of trailing vortices

An aircraft wake is made of counter-rotating vortices and is known to be affected by a long (Crow) and a short (elliptic) wavelength instabilities. Numerical investigations on the three-dimensionnal instabilities and transient growth of such dipole are performed.By means of a three-dimensionnal linear stability analysis, we retrieve the instability bands corresponding to the Crow and elliptic modes but we also observe less unstable oscillatory modes with very broad peaks. The transient growth of this dipole, investigated by computing the optimal perturbations with a direct-adjoint\footnote{P. Corbett and A. Bottaro,Phys. Fluids {\bf 12}, 120 (2000)} technique, demonstrates the crucial role of the region of maximal strain. Further investigations on the dynamics of trailing vortices in stratified fluids will be performed. Indeed, as such dipoles propagate downwards, they evolve under the influence of the stratification of the atmosphere.   

Waves on a vortex in a stratified fluid

We investigate the waves of a vertical Rankine vortex in a strongly stratified fluid. These waves are very different from their well-known counterpart in homogeneous fluid (Kelvin waves). They exist only for non-axisymmetric azimuthal wavenumbers, their frequency is always positive and strikingly, they are weakly unstable because of radiation towards infinity. Asymptotic results that hint the instability mechanism will be presented.   

Low frequency acoustic waves from explosive sources in the atmosphere

In this study, a perturbative formulation of non linear euler equations is used to compute the pressure variation for low frequency acoustic waves from explosive sources in real atmospheres. Based on a Dispersion-Relation-Preserving (DRP) finite difference scheme, the discretization provides good properties for both sound generation and long range sound propagation over a variety of spatial atmospheric scales. It also assures that there is no wave mode coupling in the numerical simulation The background flow is obtained by matching the comprehensive empirical global model of horizontal winds HWM-93 (and MSISE-90 for the temperature profile) with meteorological reanalysis of the lower atmosphere. Benchmark calculations representing cases where there is downward and upward refraction (including shadow zones), ducted propagation, and generation of acoustic waves from low speed shear layers are considered for validation. For all cases, results show a very good agreement with analytical solutions, when available, and with other standard approaches, such as the ray tracing and the normal mode technique. Comparison of calculations and experimental data from the high explosive ``Misty Picture'' test that provided the scaled equivalent airblast of an 8 kt nuclear device (on May 14, 1987), is also considered. It is found that instability waves develop less than one hour after the wavefront generated by the detonation passes.