^{1,a)}

### Abstract

The purpose of this study is to examine the strongly rotating limit of a turbulent flowtheoretically and numerically. The goal is to verify the predictions of asymptotic theories. Given the limitations of experimental and dissipative numerical approaches to this problem, we use classical equilibrium statistical mechanics. We apply the statistical mechanics approach to the inviscid truncated model of strongly rotating turbulence (in the *small Rossby* number range) and derive the theoretical spectra of the decoupled model. We use numerical simulations to complement these derivations and examine the relaxation to equilibrium of the inviscid unforced truncated rotating turbulent system for different sets of initial conditions. We separate our discussion into two time domains: the discussion of the decoupled phase with time below a threshold time , for which a new set of invariants are identified, and the coupled phase with a time beyond , for which the quantities are no longer invariants. We obtain a numerical evaluation of which is coherent with the theoretical asymptotic expansions. We examine if the quantities play a constraining role on the coupled dynamics beyond . We find that the theoretical statistical predictions in the decoupled phase capture the horizontal dynamics of the flow. In the coupled phase, the invariants are found to still play a constraining role on the short-timescale horizontal dynamics of the flow. These results are discussed in the larger context of previous rotating turbulence studies.

This work was possible thanks to the financial support of the Natural Sciences and Engineering Research Council of Canada. The author is grateful to Dr. D. Straub, Dr. M. Mackey, and the anonymous referees for their comments.

I. INTRODUCTION

II. EQUATIONS AND ROTATING TURBULENCETHEORIES

A. Full equations

B. Two-timescale problem, resonance condition

C. Modal decomposition

III. INVARIANT QUANTITIES AND STATISTICAL EQUILIBRIUM OF THE SMALL Ro REGIME

A. Analysis of the full equations

B. Analysis of the decoupled reduced equations

IV. COMPARISON WITH NUMERICAL SOLUTIONS

A. Conservation and timescale

B. Horizontal dynamics

C. Vertical dynamics and anisotropy

V. DISCUSSION

### Key Topics

- Energy transfer
- 32.0
- Turbulent flows
- 26.0
- Integrated circuits
- 18.0
- Number theory
- 17.0
- Rotating flows
- 16.0

## Figures

Initial horizontal spectra of ICs, IC: I and IC: II.

Initial horizontal spectra of ICs, IC: I and IC: II.

Time series of the energy contributions and for , 0.2, and 0.01, initiated with IC: I (a) and IC: II (b). The time axis is the dimensional time.

Time series of the energy contributions and for , 0.2, and 0.01, initiated with IC: I (a) and IC: II (b). The time axis is the dimensional time.

Time series of the 2D enstrophy (a) and the energy of the vertical component of the 2D field (b) for , 0.2, and 0.01 for both IC: I and IC: II. The time axis is the dimensional time.

Time series of the 2D enstrophy (a) and the energy of the vertical component of the 2D field (b) for , 0.2, and 0.01 for both IC: I and IC: II. The time axis is the dimensional time.

Time series of the and for the simulation initialized with IC: I (a) and IC: II (b). The time axis is the nondimensional time .

Time series of the and for the simulation initialized with IC: I (a) and IC: II (b). The time axis is the nondimensional time .

Horizontal wavenumber spectra of (upper panel) and (lower panel) for and for IC: I (left column) and IC: II (right column). The theoretical spectra have been offset for clarity. The initial numerical spectra are denoted and multiple lines are for different times.

Horizontal wavenumber spectra of (upper panel) and (lower panel) for and for IC: I (left column) and IC: II (right column). The theoretical spectra have been offset for clarity. The initial numerical spectra are denoted and multiple lines are for different times.

Time series of [centroid of 2D energy spectra defined by Eq. (39)], with nondimensional time and for . Both simulations started with IC: I and IC: II are shown.

Time series of [centroid of 2D energy spectra defined by Eq. (39)], with nondimensional time and for . Both simulations started with IC: I and IC: II are shown.

Horizontal wavenumber spectra for and for IC: I (a) and IC: II (b). The theoretical spectra have been offset for clarity. The initial numerical spectra are denoted and the multiple lines are for different times.

Horizontal wavenumber spectra for and for IC: I (a) and IC: II (b). The theoretical spectra have been offset for clarity. The initial numerical spectra are denoted and the multiple lines are for different times.

Vertical spectra of 3D energy, , for simulations initiated with IC: I (a) and IC: II (b). The initial numerical spectra are denoted and the multiple lines are for different times. The theoretical spectra have been offset for clarity only for the simulation IC: II (a).

Vertical spectra of 3D energy, , for simulations initiated with IC: I (a) and IC: II (b). The initial numerical spectra are denoted and the multiple lines are for different times. The theoretical spectra have been offset for clarity only for the simulation IC: II (a).

Time averaged 3D energy spectrum in log-log scale at an initial time [(a) and (b)], an intermediate time range below such that [(c) and (d)], an intermediate time [(e) and (f)], and the end of the simulations with [(g) and (h)]. Both the and time ranges correspond to times larger than , i.e., beyond the decoupled phase. and the ICs are IC: I (left column) and IC: II (right column). The colors are normalized for each graph such that the maximum (minimum) value of the modal spectrum is represented by the brightest (darkest) color.

Time averaged 3D energy spectrum in log-log scale at an initial time [(a) and (b)], an intermediate time range below such that [(c) and (d)], an intermediate time [(e) and (f)], and the end of the simulations with [(g) and (h)]. Both the and time ranges correspond to times larger than , i.e., beyond the decoupled phase. and the ICs are IC: I (left column) and IC: II (right column). The colors are normalized for each graph such that the maximum (minimum) value of the modal spectrum is represented by the brightest (darkest) color.

## Tables

The timestep , the rotation rate , the final output time , the 2D Rossby number , and the Robert filter parameter for each of the selected simulations.

The timestep , the rotation rate , the final output time , the 2D Rossby number , and the Robert filter parameter for each of the selected simulations.

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