Geophysical flows with anisotropic turbulence and dispersive waves: flows with stable stratification (original) (raw)
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Anisotropic turbulence and internal waves in stably stratified flows (QNSE theory
Physica Scripta, 2008
The quasi-normal scale elimination (QNSE) theory is an analytical spectral theory of turbulence based upon successive elimination of small scales of motion and calculating ensuing corrections to the viscosity and diffusivity. The main results of the theory are analytical expressions for eddy viscosity, eddy diffusivity, and kinetic energy and temperature spectra. Partial scale elimination yields a subgrid-scale representation for large-eddy simulations, whereas the elimination of all fluctuating scales is analogous to the Reynolds averaging. The scale-dependent analysis enables one to put processes on different scales at the spotlight and elucidate their contributions to eddy viscosities and eddy diffusivities. In addition, the method traces the modification of the flow with increasing stratification and recovers growing anisotropy and the effect of the internal waves. The QNSE-based Reynolds-averaged models present a viable alternative to conventional Reynolds stress models. A QNSE model of this kind was tested in the numerical weather prediction system HIRLAM instead of the existing reference Reynolds stress model. The performance of the QNSE model was superior in all simulations where stable stratification was noticeable.
Ocean Science, 2009
This paper extends a turbulence closure-like model for stably stratified flows into a new dynamic domain in which turbulence is generated by internal gravity waves rather than mean shear. The model turbulent kinetic energy (TKE, K) balance, its first equation, incorporates a term for the energy transfer from internal waves to turbulence. This energy source is in addition to the traditional shear production. The second variable of the new two-equation model is the turbulent enstrophy (Ω). Compared to the traditional shear-only case, the Ω-equation is modified to account for the effect of the waves on the turbulence time and space scales. This modification is based on the assumption of a non-zero constant flux Richardson number in the limit of vanishing mean shear when turbulence is produced exclusively by internal waves. This paper is part 1 of a continuing theoretical development. It accounts for mean shear- and internal wave-driven mixing only in the two limits of mean shear and no waves and waves but no mean shear, respectively. The new model reproduces the wave-turbulence transition analyzed by D'Asaro and Lien (2000b). At small energy density E of the internal wave field, the turbulent dissipation rate (ɛ) scales like ɛ~E2. This is what is observed in the deep sea. With increasing E, after the wave-turbulence transition has been passed, the scaling changes to ɛ~E1. This is observed, for example, in the highly energetic tidal flow near a sill in Knight Inlet. The new model further exhibits a turbulent length scale proportional to the Ozmidov scale, as observed in the ocean, and predicts the ratio between the turbulent Thorpe and Ozmidov length scales well within the range observed in the ocean.
Small-Scale Structure of Strongly Stratified Turbulence
Journal of Physical Oceanography, 2005
The small-scale structure of turbulence subjected to strong stratification is analyzed with rapid distortion theory to evaluate the performance of formulas for predicting dissipation of turbulent kinetic energy and dissipation of scalar variance. The approach is restricted to weak turbulence in strong stratification, like that in the thermocline or the abyssal ocean. Flows with and without mean shear are considered. For unsheared turbulence, the small scales are axisymmetric about the vertical axis, as others have previously assumed. The calculations here complement and extend previous work because they can be used to compute errors in dissipation estimates, develop simpler formulas, and examine the effects of shear and other parameters. For example, effects of the initial conditions can be significant. For sheared turbulence, the small-scale velocity and buoyancy fields are neither isotropic nor axisymmetric about the vertical axis. Although dissipation formulas based on isotropy w...
A quasi-normal scale elimination model of turbulence and its application to stably stratified flows
Nonlinear Processes in Geophysics, 2006
Models of planetary, atmospheric and oceanic circulation involve eddy viscosity and eddy diffusivity, K M and K H , that account for unresolved turbulent mixing and diffusion. The most sophisticated turbulent closure models used today for geophysical applications belong in the family of the Reynolds stress models. These models are formulated for the physical space variables; they consider a hierarchy of turbulent correlations and employ a rational way of its truncation. In the process, unknown correlations are related to the known ones via "closure assumptions" that are based upon physical plausibility, preservation of tensorial properties, and the principle of the invariant modeling according to which the constants in the closure relationships are universal. Although a great deal of progress has been achieved with Reynolds stress closure models over the years, there are still situations in which these models fail. The most difficult flows for the Reynolds stress modeling are those with anisotropy and waves because these processes are scaledependent and cannot be included in the closure assumptions that pertain to ensemble-averaged quantities. Here, we develop an alternative approach of deriving expressions for K M and K H using the spectral space representation and employing a self-consistent, quasi-normal scale elimination (QNSE) algorithm. More specifically, the QNSE procedure is based upon the quasi-Gaussian mapping of the velocity and temperature fields using the Langevin equations. Turbulence and waves are treated as one entity and the effect of the internal waves is easily identifiable. This model implies partial averaging and, thus, is scale-dependent; it allows one to easily introduce into consideration such parameters as the grid resolution, the degree of the anisotropy, and spectral characteristics, among others. Applied to tur-
Sheared turbulence in a weakly stratified upper ocean
Deep Sea Research Part I: Oceanographic Research Papers, 2006
Microstructure, ADCP and CTD profiles taken in the North Atlantic along 531N under moderate and high winds showed that the median of log-normally distributed kinetic energy dissipation rate e within the upper mixing layer is 1.5 Â 10 À7 W/kg and the layer depth, on the average, is $40 m. Assuming that mixing efficiency g is a constant (g ¼ 0:2), the following scaling is proposed for the normalized eddy diffusivity:
Turbulent mixing driven by mean-flow shear and internal gravity waves in oceans and atmospheres
2012
This study starts with balances deduced by Baumert and Peters (2004, 2005) from results of stratified-shear experiments made in channels and wind tunnels by Itsweire (1984) and Rohr and Van Atta (1987), and of free-decay experiments in a resting stratified tank by Dickey and Mellor (1980). Using a modification of Canuto's (2002) ideas on turbulence and waves, these balances are merged with an (internal) gravity-wave energy balance presented for the open ocean by Gregg (1989), without mean-flow shear. The latter was augmented by a linear (viscous) friction term. Gregg's wave-energy source is interpreted on its long-wave spectral end as internal tides, topography, large-scale wind, and atmospheric low-pressure actions. In addition, internal eigen waves, generated by mean-flow shear, and the aging of the wave field from a virginal (linear) into a saturated state are taken into account. Wave packets and turbulence are treated as particles (vortices, packets) by ensemble kinetics so that the loss terms in all three balances have quadratic form. Following a proposal by Peters (2008), the mixing efficiency of purely wave-generated turbulence is treated as a universal constant, as well as the turbulent Prandtl number under neutral conditions. It is shown that: (i) in the wind tunnel, eigen waves are switched off, (ii) due to remotely generated long waves or other non-local energy sources, coexistence equilibria of turbulence and waves are stable even at Richardson numbers as high as 10310^3103; (iii) the three-equation system is compatible with geophysically shielded settings like certain stratified laboratory flows. The agreement with a huge body of observations surprises. Gregg's (1989) wave-model component and the a.m. universal constants taken apart, the equations contain only one additional dimensionless parameter for the eigen-wave closure, estimated as Yapprox1.35.Y\approx 1.35.Yapprox1.35.
Quasi-geostrophic turbulence and generalized scale invariance, a theoretical reply
Atmospheric Chemistry and Physics, 2012
Lindborg et al. (2010) claim that the apparent spectrum power law E(k) ≈ k −3 on scales ≥ 600 km obtained with the help of commercial jetliner trajectory deviations (GASP and Mozaic databases) could not be brought into question (Lovejoy et al., 2009a), because this spectrum corresponds to "a well known theory of quasi-geostrophic turbulence developed by Charney (1971)". Lindborg et al. (2010) also claim that "limitations [of this theory] have been relaxed in many of the modern models of atmospheric turbulence". We show that both claims are irrelevant and that generalized scale invariance (GSI) is indispensable to go beyond the quasi-geostrophic limitations, to go in fact from scale analysis to scaling analysis in order to derive better analytical models. In this direction, we derive vorticity equations in a space of (fractal) dimension D = 2 + H z (0 ≤ H z ≤ 1), which corresponds to a first step in the derivation of a dynamical alternative to the quasi-geostrophic approximation and turbulence. The corresponding precise definition of fractional dimensional turbulence already demonstrates that the classical 2-D and 3-D turbulence are not the main options to understand atmospheric dynamics. Although (2+H z)-D turbulence (with 0 < H z < 1) has more common features with 3-D turbulence than with 2-D turbulence, it has nevertheless very distinctive features: its scaling anisotropy is in agreement with the layered pancake structure, which is typical of rotating and stratified turbulence but not of the classical 3-D turbulence.
Physical Review Fluids, 2020
Progress in the rapidly expanding exploration of planetary atmospheric and oceanic environments demands an adequate qualitative and quantitative representation of various processes in anisotropic turbulence. The existing analytical spectral theories are developed for homogeneous isotropic flows. They quickly become very complicated when expanded to anisotropic flows with waves. It is possible, however, to extend one such theory, the quasinormal scale elimination (QNSE), to stably stratified and rotating flows. Here the results of the theory are compared with a large variety of oceanic and atmospheric flows. These comparisons make it possible to clarify the physics of some processes governing the atmospheric and oceanic dynamics, quantify their spectra, and investigate their latitudinal and longitudinal variabilities. Some of the main results of this analysis are that vertical and horizontal spectra of atmospheric and oceanic turbulence can be derived within QNSE analytically; there exists a quantitative affinity between atmospheric and oceanic spectra; on large scales, spectral amplitudes are determined by the extra strains that cause flow anisotropization, rather than the energy or enstrophy fluxes; and planetary circulations appear to be amenable to classification as flows with compactified (compressed) dimensionality.