Quasinormal scale elimination theory of the anisotropic energy spectra of atmospheric and oceanic turbulence


Circulations of planetary atmospheres and oceans feature huge Reynolds numbers and are turbulent on almost all scales of motion. In addition, the action of density stratification, planetary rotation, streamline curvature, geometric confinement and other factors render these circulations anisotropic. A systematic description of turbulence behavior in response to the impact exerted by extra strains is one of the important outstanding problems of fluid dynamics. Attempts to address this problem ran into difficulty from the outset due to the proliferation of dimensional variables that hinder the application of dimensional analysis. Moreover, direct numerical simulations are limited by the value of the resolvable Reynolds number, even in relatively simple cases of channel flows. Linear and nonlinear theories of anisotropic turbulence have been proposed over the years, yet a clear understanding of the spectral behavior of such flows remains elusive. Further progress relies upon the development of basic self-consistent theories that make verifiable predictions suitable for testing against a large variety of data.

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 were initially developed for homogeneous isotropic flows. They quickly become very complicated when expanded to anisotropic flows with waves. Nonetheless, it is possible to extend one such theory: i.e. the quasinormal scale elimination (QNSE), to stably stratified and rotating flows.

On this account, Professor Boris Galperin from the University of South Florida, in collaboration with Professor Semion Sukoriansky at the Ben-Gurion University of the Negev in Israel, employed QNSE to analyze on a scale-by-scale basis the interactions between different physical processes, under the assumption of an infinite Reynolds number. Their work is currently published in the research journal, Physical Review Fluids.

In their approach, theoretical predictions of various spectra were compared with measurements in atmospheric and oceanic flows. Particular attention was given to one-dimensional (1D) spectra that quantify the turbulence anisotropy. The researchers were motivated by the realization that in existing literature, there is a lack of clarity with regard to spectral slopes and amplitudes of atmospheric and oceanic turbulence, their dependence on latitude, and generally their governing physics.

The authors reported that vertical and horizontal spectra of atmospheric and oceanic turbulence could be derived within QNSE analytically and, furthermore, there exists a quantitative affinity between atmospheric and oceanic spectra. They established that on large scales, spectral amplitudes are likely determined by the extra strains that cause flow anisotropization, rather than the energy or enstrophy fluxes. Overall, planetary circulations were reported to appear to be amenable to classification as flows with compactified (compressed) dimensionality.

By comparing the results of the QNSE theory with a large number of oceanic and atmospheric flows, Galperin and Sukoriansky were not only able to clarify processes governing the atmospheric and oceanic dynamics, but also to quantify their spectral characteristics, including latitudinal, longitudinal and seasonal variabilities. When generalized for spherical coordinates, the QNSE results were in good agreement with numerical simulations, replicated the well-known but poorly understood Nastrom-Gage spectra in the spherical geometry, and provided guidelines for deriving the subgrid-scale parametrizations suitable for implementation in numerical models. In a statement to Advances in Engineering, Professor Boris Galperin highlighted that despite large differences in visual appearances of the atmospheric and oceanic circulations, their spectra are remarkably congruent. This important property of the spectra is accurately predicted by the QNSE theory.

Quasinormal scale elimination theory of the anisotropic energy spectra of atmospheric and oceanic turbulence - Advances in Engineering
The disparity in the visual appearance of atmospheric and oceanic circulations vs. the congruence of their kinetic energy spectra.
Quasinormal scale elimination theory of the anisotropic energy spectra of atmospheric and oceanic turbulence - Advances in Engineering
While the snapshots of high-resolution computer simulations of the atmospheric (left top corner; source: https://earth.nullschool.net/) and oceanic (right top corner; source: http://www.nasa.gov/topics/earth/features/perpetual-ocean.html) circulations in the area of the North Atlantic ocean that encompasses the Gulf Stream exhibit diverse patterns, the spectra of their kinetic energies (corresponding bottom panels) are remarkably congruent as they are described by the same QNSE expressions given in Galperin and Sukoriansky (2020). On the left bottom panel, the black dots show the data and the black lines correspond to the observed Nastrom-Gage spectra while red lines are given by the QNSE theory. On the right bottom panel, the solid red and blue lines show the observational longitudinal and transverse spectra south of the Gulf Stream while the red and blue dotted lines show the spectra predicted by the QNSE theory.

About the author

Dr Boris Galperin
College of Marine Science
University of South Florida
St Petersburg, FL 33701, USA
+1 727 553 1249

Dr Boris Galperin is an Associate Professor at the College of Marine Science, University of South Florida. He received his PhD in Civil Engineering from the Technion, the Israel Institute of Technology, where he studied turbulence in Marine Boundary Layers. After graduating, he spent 6 years at Princeton University, first with the Program of Atmospheric and Ocean Sciences and then with the Program of Applied and Computational Mathematics. He has been with USF since 1989.

Dr Galperin has worked at the intersection of the branches of Physical Sciences encompassing Fluid Turbulence and Atmospheric, Oceanic and Planetary Circulations striving to improve our understanding of the dynamics of planetary atmospheres and oceans from the perspective of fundamental turbulence.

Whilst at Princeton, he improved the Mellor-Yamada turbulence closure model widely used in the modeling of geophysical and planetary flows. These flows feature strong anisotropy and turbulence-wave interaction. To enhance the fidelity of turbulence description, a spectral theory was required. Dr Galperin became involved in the development of a theory known today as Quasi-Normal Scale Elimination (QNSE). The theory deals with anisotropic turbulence with waves on a fundamental physical level.
This research led to the discovery of the regime of zonostrophic turbulence that establishes the qualitative and quantitative framework for the process of the formation of zonal jets in rotating flows with a variable Coriolis parameter. After exposing this regime in computer simulations, Boris and his colleagues detected it in planetary environments, particularly, gaseous giant planets. Furthermore, they elaborated an analogy between the observed zonal jets on giant planets and in the Earth’s oceans.

One of the hallmark results of the application of the QNSE theory to stably stratified flows was the evidence of the absence of the critical Richardson number at which turbulence is fully extinguished.

More recently, Dr Galperin used the QNSE theory to explain and quantify the kinetic energy spectra observed in the terrestrial atmospheric and oceanic circulations. He played the key role in explaining the Nastrom & Gage spectrum of the atmospheric turbulence, quantifying kinetic energy spectra of oceanic flows including those obtained with satellite altimetry, and exposing the important analogy between the atmospheric and oceanic spectra. Dr Galperin wrote and coauthored about 100 scientific papers and book chapters and coedited two books.


Boris Galperin, Semion Sukoriansky. Quasinormal scale elimination theory of the anisotropic energy spectra of atmospheric and oceanic turbulence. Physical Review Fluids; volume 5, 063803.

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