This study examines quasigeostrophic Rossby eigenmodes of homogeneous as well as stratified oceans. Analytical studies of Rossby basin modes are usually done by using simple basin geometries. Simple means that solutions are available by applying a separation ansatz in looking for basin modes. Here the focus is on half-trapezoidal geometry, in particular on a stratified ocean with a half-trapezoidal (zonal) cross section. In this case, separation is not possible. However, approximate solutions can be found. Different approximations are discussed: one related to linearization of the boundary conditions, one related to a sight change in boundary geometry, and one that uses an asymptotic expansion. It is shown that the widely used “linearized boundary conditions” give wrong results for low-frequency modes. The reason is that weak asymmetries of the basin are neglected and wave energy is therefore distributed too homogeneously in the basin. Although the basin's geometry used deviates only slightly from a square and the eigenvalue spectrum corresponds well with that of a square basin, eigenmodes can still differ greatly.
A B S T R A C T For quasi-geostrophic models, the beta plane approximation is well established and can be derived from spherical geographic coordinates.It has been argued that such a connection does not exist for a higher-order approximation, the so-called delta plane.Here it will be demonstrated that a quasi-geostrophic potential vorticity equation on the delta plane can formally be derived using rotated geographic instead of geographic coordinates.The rigorous derivation of such a model from the shallow-water equations leads to a correction of previous more intuitive-based formulations of the delta plane model.Some applications of the corrected delta plane model are given.It is shown that the delta plane model describes well the low-frequency basin modes of a polar plane shallow-water model.Moreover, it is found that the westward phase speed of the delta plane model shows a dependency on latitude comparable to a model on the sphere.The ratio of delta to beta plane zonal phase speed decreases monotonically with increasing latitude, in qualitative agreement with the phase speed ratio obtained by comparing a spherical to a beta plane model.Finally, it is demonstrated analytically that Rossby wave energy rays are curved on the delta plane, in contrast to the beta plane.Ray curvature is important for a realistic description of energy dispersion at high latitudes.The results suggest that the quasi-geostrophic delta plane model is a suitable tool for conceptual studies on Rossby wave dynamics at high latitudes.
Direct numerical simulations (DNS) of the flow in various rotating annular confinements have been conducted to investigate the effects of wall inclination on secondary fluid motions due to an unstable boundary layer. The inner wall resembles a truncated cone (frustum) whose apex half-angle is varied from to (straight cylinder). The large inner radius , the mean rotation rate and the kinematic viscosity were kept constant resulting in the constant Ekman number . Flows were excited by time-harmonic modulation of the inner wall’s rotation rate (so-called longitudinal libration) by prescribing the amplitude and the forcing frequency . By steepening the inner wall and hence reducing the effect of the local Coriolis force in the boundary layer three different flow regimes can be realized: a rotation-dominated, a libration-dominated and an intermediate regime. In the present study we focus on the libration-dominated regime. For small libration amplitudes (here ), a laminar Ekman–Stokes boundary layer (ESBL) is realized at the librating wall. With the aid of laminar boundary layer theory and DNS we show that the ESBL exhibits an oscillatory mass flux along the librating wall (Ekman property) and an oscillatory azimuthal velocity, which resembles a radially damped wave (Stokes property). For large libration amplitudes (here ), the DNS results exhibit an intermittently unstable ESBL, which turns centrifugally unstable during the prograde (faster) part of a libration period. This instability is due to the Stokes property and gives rise to Gortler vortices, which are found to be tilted with respect to the azimuth when the librating wall is at a finite angle relative to the axis of rotation. We show that this tilt is related to the Ekman property of the ESBL. This suggests that linear and nonlinear dynamics are equally important for this intermittent instability. Our DNS results indicate further that the Gortler vortices propagate into the fluid bulk where they generate an azimuthal mean flow. This mean flow is notably different from the mean flow driven in the case of the stable ESBL. A diagnostic analysis of the Reynolds-averaged Navier–Stokes (RANS) equations in the unstable flow regime hints at a competition between the radial and axial turbulent transport terms which act as generating and destructing agents for the azimuthal mean flow, respectively. We show that the balance of both terms depends on the wall inclination, that is, on the wall-tangential component of the Coriolis force.
Abstract Inertia–gravity waves (IGWs) play an essential role in the terrestrial atmospheric dynamics as they can lead to energy and momentum flux when propagating upward. An open question is to what extent IGWs contribute to the total energy and to the flattening of the energy spectrum observed at the mesoscale. In this work, we present an experimental investigation of the energy distribution between the large-scale balanced flow and the small-scale imbalanced flow. Weakly nonlinear IGWs emitted from baroclinic jets are observed in the differentially heated rotating annulus experiment. Similar to the atmospheric spectra, the experimental kinetic energy spectra reveal the typical subdivision into two distinct regimes with slopes k−3 for the large scales and k−5/3 for the small scales. By separating the spectra into the vortex and wave components, it emerges that at the large-scale end of the mesoscale the gravity waves observed in the experiment cause a flattening of the spectra and provide most of the energy. At smaller scales, our data analysis suggests a transition toward a turbulent regime with a forward energy cascade up to where dissipation by diffusive processes occurs.
The occurrence and source mechanism of inertia-gravity waves (IGWs) are studied in the differentially heated rotating annulus via laboratory experiments (BTU) and numerical simulations (GUF). Two differentially heated rotating annulus experiments are used for this purpose at the BTU laboratories. The first is a modified version of the classical baroclinic experiment in which a juxtaposition of convective and motionless stratified layers can be created by introducing a vertical salt stratification. The thermal convective motions are suppressed in a central region at mid depth of the rotating tank, therefore baroclinic waves can only build up in thin layers located at the top and bottom, where the salt stratification is weakest. This new experimental setup, coined instabil-ity, allows to study the exchange of momentum and energy between the layers, especially by the propagation of IGWs. Moreover, in contrast to the classical tank without salt stratification we have layers with N/f > 1. A ratio larger than unity implies that the IGW propagation in the experiment is expected to be qualitatively similar to the atmospheric case. Interestingly, we found local IGW packets along the jets in the surface and bottom layers where the local Rossby number is larger than 1, suggesting spontaneous imbalance as generating mechanism [1], and not boundary layer instability [2]. Theoretical considerations and numerical simulations have led to the identification of an annulus configuration, much wider and shallower, with a much larger temperature difference between the inner and outer cylinder walls, which is more atmosphere-like since it shows an N / f >1 even without the vertical salt stratification. Flow regime stability has been tested for this new differentially heated rotating annulus and compared with findings from the small tank. In view of the different geometries of the two experimental systems, their correspondence was excellent with respect to the large-scale. Moreover, direct numerical simulations were performed (GUF) for this atmosphere-like configuration of the experiment and possible regions of IGW activity were characterised by a Hilbert-transform algorithm. The simulations show a clear baroclinic wave structure exhibiting a realistic jet-front system superimposed by small-scale structures which are associated with IGWs occurring in wave packets [3]. The comparison of observations from a corresponding big tank experiment with numerical simulation shows that for both cases (as we already observed in the barostrat experiment), small scale wave packets are clearly correlated with an increased local Rossby number.
A water-filled differentially heated rotating annulus with initially prepared stable vertical salinity profiles is studied in the laboratory. Based on two-dimensional horizontal particle image velocimetry (PIV) data, and infrared camera visualizations, we describe the appearance and the characteristics of the baroclinic instability in this original configuration. First, we show that when the salinity profile is linear and confined between two non stratified layers at top and bottom, only two separate shallow fluid layers can be destabilized. These unstable layers appear nearby the top and the bottom of the tank with a stratified motionless zone between them. This laboratory arrangement is thus particularly interesting to model geophysical or astrophysical situations where stratified regions are often juxtaposed to convective ones. Then, for more general but stable initial density profiles, statistical measures are introduced to quantify the extent of the baroclinic instability at given depths and to analyze the connections between this depth-dependence and the vertical salinity profiles. We find that, although the presence of stable stratification generally hinders full-depth overturning, double-diffusive convection can yield development of multicellular sideways convection in shallow layers and subsequently to a multilayered baroclinic instability. Therefore we conclude that by decreasing the characteristic vertical scale of the flow, stratification may even enhance the formation of cyclonic and anticyclonic eddies (and thus, mixing) in a local sense.
