Geophysical flows with anisotropic turbulence and dispersive waves: flows with stable stratification

Abstract

The quasi-normal scale elimination (QNSE) is an analytical spectral theory of turbulence based upon a successive ensemble averaging of the velocity and temperature modes over the smallest scales of motion and calculating corresponding eddy viscosity and eddy diffusivity. By extending the process of successive ensemble averaging to the turbulence macroscale one eliminates all fluctuating scales and arrives at models analogous to the conventional Reynolds stress closures. The scale dependency embedded in the QNSE method reflects contributions from different processes on different scales. Two of the most important processes in stably stratified turbulence, internal wave propagation and flow anisotropization, are explicitly accounted for in the QNSE formalism. For relatively weak stratification, the theory becomes amenable to analytical processing revealing just how increasing stratification modifies the flow field via growing anisotropy and gravity wave radiation. The QNSE theory yields the dispersion relation for internal waves in the presence of turbulence and provides a theoretical reasoning for the Gargett et al. (J Phys Oceanogr 11:1258-1271 scaling of the vertical shear spectrum. In addition, it shows that the internal wave breaking and flow anisotropization void the notion of the critical Richardson number at which turbulence is fully suppressed. The isopycnal and diapycnal viscosities and diffusivities can be expressed in the form of the Richardson diffusion laws thus providing a theoretical framework for the Okubo dispersion diagrams. Transitions in the spectral slopes can be associated with the turbulence-and wave-dominated ranges and have direct implications for the transport processes. We show that only quasi-isotropic, turbulence-dominated scales contribute to the diapycnal diffusivity. On larger, buoyancy dominated scales, the diapycnal diffusivity becomes scale independent. This result underscores the well-known fact that waves can only transfer momentum but not a scalar and sheds a new light upon the Ellison-Britter-Osborn mixing model. It also provides a general framework for separation of the effects of turbulence and waves even if they act on the same spatial and temporal scales. The QNSE theory-based turbulence models have been tested in various applications and demonstrated reliable performance. It is suggested that these models present a viable alternative to conventional Reynolds stress closures.

An analytical theory of the buoyancy–Kolmogorov subrange transition in turbulent flows with stable stratification

“…where the mixing efficiency Γ is a constant estimated in the QNSE theory at about 0.4 [2]. This value is somewhat larger than the widely accepted in the oceanographic literature estimate of 0.2 [61,[65][66][67].…”

“…A mathematically sharp delineation between these two entities can be stipulated by the fact that only waves have a dispersion relation. In flows on a β-plane, for example, turbulence and waves coexist on virtually all scales, which is evidenced by broadening of Rossby waves' spectral peaks with increasing wavenumbers [2]. One has yet to investigate whether or not turbulence causes degeneration of the Rossby wave dispersion relation.…”

“…The numerical value of the coefficient, C B = 0.214, is in agreement with DNS [108] (C B 0.1), large eddy simulations [109] (C B 0.2) and laboratory experiment [110] (C B 0.2). Equation (8.1) can be used to derive an expression for the spectrum of the vertical shear, 2k 2 3 E 1 (k 3 ), in the universal or canonical form used in physical oceanography [111,112], [12]. Note that for k 3 /k O > 1 the PATCHEX data were strongly affected by the molecular viscosity.…”

“…These flows are also turbulent, and the interaction between turbulence and waves has significant impact upon the transport of momentum and scalars on different scales [1,2]. Understanding and quantification of this interaction is one of the important unresolved problems of geophysical fluid dynamics and fluid dynamics in general [3,4].…”

Turbulence in the nocturnal boundary layer with light and variable winds

“…Mahrt (2011) inferred that the characteristics of the fluctuations with weak winds and strong stratification vary gradually with scale, suggesting that a sharp separation between turbulence and non-turbulent motions is not possible. This distinction is avoided in Galperin and Sukoriansky (2010), who model the fluctuations as a mix of turbulence and waves that merge in an intermediate buoyancy subrange (Sukorianski and Galperin, 2011). Such spectral approaches in the wave-number domain are difficult to evaluate in the stable atmospheric surface layer because of lack of fine-scale spatial information.…”