Lower and Upper Bounds on Internal-Wave Frequencies in Stratified Rotating Fluids

According to classical theories, the frequencies of internal-gravity waves in stratified rotating fluids must lie between the Brunt-Väisälä frequency (a measure of the vertical density stratification) and the Coriolis frequency (equal to twice the rotation rate about the vertical axis). It is shown here that, in the case of the Earth\u27s rotation where the pole-to-pole axis of rotation is almost everywhere not parallel to the local vertical, the range of realizable frequencies is broader. New formulas are derived for the lower and upper bounds of the frequencies

Eulerian Derivation of the Coriolis Force

In textbooks of geophysical fluid dynamics, the Coriolis force and the centrifugal force in a rotating fluid system are derived by making use of the fluid parcel concept. In contrast to this intuitive derivation to the apparent forces, more rigorous derivation would be useful not only for the pedagogical purpose, but also for the applications to other kinds of rotating geophysical systems rather than the fluid. The purpose of this paper is to show a general procedure to derive the transformed equations in the rotating frame of reference based on the local Galilean transformation and rotational coordinate transformation of field quantities. The generality and usefulness of this Eulerian approach is demonstrated in the derivation of apparent forces in rotating fluids as well as the transformed electromagnetic field equation in the rotating system.Comment: Added references. Corrected typo

Modeling of internal tides in fjords

A previous model for the distribution of internal tides above irregular topography is generalized to include arbitrary stratification and a radiation condition at the open boundary. Thanks to a small amount of dissipation, this model remains valid in the presence of resonant internal tides, leading to intense wave-energy beams. An application to a Norwegian fjord correctly reproduces the observed energy pattern consisting of two beams both originating at the 60-meter deep entrance sill and extending in-fjord, one upward toward the surface, the other downward toward the bottom. After correction for the varying width of the fjord, the observed and modelled energy levels are in good agreement, especially in the upper levels where energy is the greatest. Furthermore, the substantial phase lag between these two energy beams revealed by the observations is correctly reproduced by the model. Finally, a third and very narrow energy spike is noted in the model at the level of a secondary bump inward of the sill. This beam is missed by the current meter data, because the current meters were placed only at a few selected depths. But an examination of the salinity profiles reveals a mixed layer at approximately the same depth. The explanation is that high-wave energy leads to wave breaking and vigorous mixing. The model\u27s greatest advantage is to provide the internal-tide energy distribution throughout the fjord. Discrepancies between observations and model are attributed to coarse vertical resolution in the vicinity of the sill and to unaccounted cross-fjord variations

Resonance of internal waves in fjords: A finite-difference model

After the time periodicity is removed from the problem, the spatial distribution of internal waves in a stratified fluid is governed by a hyperbolic equation. With boundary conditions specified all along the perimeter of the domain, information is transmitted in both directions (forward and backward) along every characteristic, and, unlike the typical temporal hyperbolic equation, the internal-wave equation is not amenable to a simple forward integration. The problem is tackled here with a finite-difference, relaxation technique by constructing a time-dependent, dissipative problem, the final steady state of which yields the solution of the original problem. Attempts at solving the problem for arbitrary topography then reveal multiple resonances, each resonance being caused by a ray path closing onto itself after multiple reflections. The finite-difference formulation is found to be a convenient vehicle to discuss resonances and to conclude that their existence renders the problem not only singular but also extremely sensitive to the details of the topography. The problem is easily overcome by the introduction of friction. The finite-difference representation of the problem is instrumental in serving as a guide for the investigation of the resonance problem. Indeed, it keeps the essence of the continuous problem and yet simplifies the analysis enormously. Although straightforward, robust and successful at providing a numerical solution to a first few examples, the relaxation component of the integration technique suffers from lack of efficiency. This is due to the particular nature of the hyperbolic problem, but it remains that numerical analysts could improve or replace the present scheme with a faster algorithm

Three-layer flows in the shallow water limit

We formulate and discuss the shallow water limit dynamics of the layered flow with three layers of immiscible fluids of different densities bounded above and below by horizontal walls. We obtain a resulting system of four equations, which may be nonlocal in the non‐Boussinesq case. We provide a systematic way to pass to the Boussinesq limit, and then study those equations, which are first‐order PDEs of mixed type, more carefully. We show that in a symmetric case the solutions remain on an invariant surface and using simple waves we illustrate that this is not the case for nonsymmetric cases. Reduced models consisting of systems of two equations are also proposed and compared to the full system

Nonlinear stability of two-layer shallow water flows with a free surface

The problem of two layers of immiscible fluid, bordered above by an unbounded layer of passive fluid and below by a flat bed, is formulated and discussed. The resulting equations are given by a first-order, four-dimensional system of PDEs of mixed-type. The relevant physical parameters in the problem are presented and used to write the equations in a non-dimensional form. The conservation laws for the problem, which are known to be only six, are explicitly written and discussed in both non-Boussinesq and Boussinesq cases. Both dynamics and nonlinear stability of the Cauchy problem are discussed, with focus on the case where the upper unbounded passive layer has zero density, also called the free surface case. We prove that the stability of a solution depends only on two ‘baroclinic’ parameters (the shear and the difference of layer thickness, the former being the most important one) and give a precise criterion for the system to be well-posed. It is also numerically shown that the system is nonlinearly unstable, as hyperbolic initial data evolves into the elliptic region before the formation of shocks. We also discuss the use of simple waves as a tool to bound solutions and preventing a hyperbolic initial data to become elliptic and use this idea to give a mathematical proof for the nonlinear instability

Heat transport in rotating convection without Ekman layers

Numerical simulation of rotating convection in plane layers with free slip boundaries show that the convective flows can be classified according to a quantity constructed from the Reynolds, Prandtl and Ekman numbers. Three different flow regimes appear: Laminar flow close to the onset of convection, turbulent flow in which the heat flow approaches the heat flow of non-rotating convection, and an intermediate regime in which the heat flow scales according to a power law independent of thermal diffusivity and kinematic viscosity.Comment: 4 pages, 4 figure

On the concentration of near-inertial waves in anticyclones

An overlooked conservation law for near-inertial waves propagating in a steady background flow provides a new perspective on the concentration of these waves in regions of anticyclonic vorticity. The conservation law implies that this concentration is a direct consequence of the decrease in spatial scales experienced by an initially homogeneous wave field. Scaling arguments and numerical simulations of a reduced-gravity model of mixed-layer near-inertial waves confirm this interpretation and elucidate the influence of the strength of the background flow relative to the dispersion

Geothermal Heating and its Influence on the Meridional Overturning Circulation

The effect of geothermal heating on the meridional overturning circulation is examined using an idealized, coarse-resolution ocean general circulation model. This heating is parameterized as a spatially uniform heat flux of 50 mW m-2 through the (flat) ocean floor, in contrast with previous studies that have considered an isolated hotspot or a series of plumes along the mid-Atlantic ridge. The equilibrated response is largely advective: a deep perturbation of the meridional overturning cell on the order of several Sv is produced, connecting with an upper-level circulation at high latitudes, allowing the additional heat to be released to the atmosphere. Risingmotion in the perturbation deep cell is concentrated near the equator. The upward penetration of this cell is limited by the thermocline, analogous to the role of the stratosphere in limiting the upward penetration of convective plumes in the atmosphere. The magnitude of the advective response is inversely proportional to the deep stratification; with a weaker background meridional overturning circulation and a less stratified abyss, the overturning maximum of the perturbation deep cell is increased. This advective response also cools the low-latitude thermocline. The qualitative behavior is similar in both a single hemisphere and double hemisphere configuration.The anomalous circulation driven by geothermal fluxes is more substantial than previously thought. We are able to understand the structure and strength of the response in the idealized geometry and further extend these ideas to explain the results of Adcroft et al. [2001], where the impact of geothermal heating was examined using a global configuration