scholarly journals Statistical state dynamics of vertically sheared horizontal flows in two-dimensional stratified turbulence

2018 ◽  
Vol 854 ◽  
pp. 544-590 ◽  
Author(s):  
Joseph G. Fitzgerald ◽  
Brian F. Farrell
Keyword(s):  
Large Scale ◽  
Second Order ◽  
Two Dimensional ◽  
Stratified Turbulence ◽  
Stochastic Excitation ◽  
Mean Flow ◽  
Fully Nonlinear ◽  
Statistical State ◽  
Horizontal Flows ◽  
Nonlinear Simulations

Simulations of strongly stratified turbulence often exhibit coherent large-scale structures called vertically sheared horizontal flows (VSHFs). VSHFs emerge in both two-dimensional (2D) and three-dimensional (3D) stratified turbulence with similar vertical structure. The mechanism responsible for VSHF formation is not fully understood. In this work, the formation and equilibration of VSHFs in a 2D Boussinesq model of stratified turbulence is studied using statistical state dynamics (SSD). In SSD, equations of motion are expressed directly in the statistical variables of the turbulent state. Restriction to 2D turbulence facilitates application of an analytically and computationally attractive implementation of SSD referred to as S3T, in which the SSD is expressed by coupling the equation for the horizontal mean structure with the equation for the ensemble mean perturbation covariance. This second-order SSD produces accurate statistics, through second order, when compared with fully nonlinear simulations. In particular, S3T captures the spontaneous emergence of the VSHF and associated density layers seen in simulations of turbulence maintained by homogeneous large-scale stochastic excitation. An advantage of the S3T system is that the VSHF formation mechanism, which is wave–mean flow interaction between the emergent VSHF and the stochastically excited large-scale gravity waves, is analytically understood in the S3T system. Comparison with fully nonlinear simulations verifies that S3T solutions accurately predict the scale selection, dependence on stochastic excitation strength, and nonlinear equilibrium structure of the VSHF. These results constitute a theory for VSHF formation applicable to interpreting simulations and observations of geophysical examples of turbulent jets such as the ocean’s equatorial deep jets.

scholarly journals Vertically Sheared Horizontal Flow-Forming Instability in Stratified Turbulence: Analytical Linear Stability Analysis of Statistical State Dynamics Equilibria

2018 ◽  
Vol 75 (12) ◽  
pp. 4201-4227 ◽  
Author(s):  
Joseph G. Fitzgerald ◽  
Brian F. Farrell
Keyword(s):  
Stability Analysis ◽  
Large Scale ◽  
Fundamental Problem ◽  
Direct Analysis ◽  
Two Dimensional ◽  
Stratified Turbulence ◽  
Dynamical Equations ◽  
Flow Forming ◽  
Quasi Biennial Oscillation ◽  
Statistical State

Abstract Vertically banded zonal jets are frequently observed in weakly or nonrotating stratified turbulence, with the quasi-biennial oscillation in the equatorial stratosphere and the ocean’s equatorial deep jets being two examples. Explaining the formation of jets in stratified turbulence is a fundamental problem in geophysical fluid dynamics. Statistical state dynamics (SSD) provides powerful methods for analyzing turbulent systems exhibiting emergent organization, such as banded jets. In SSD, dynamical equations are written directly for the evolution of the turbulence statistics, enabling direct analysis of the statistical interactions between the incoherent component of turbulence and the coherent large-scale structure component that underlie jet formation. A second-order closure of SSD, known as S3T, has previously been applied to show that meridionally banded jets emerge in barotropic β-plane turbulence via a statistical instability referred to as the zonostrophic instability. Two-dimensional Boussinesq turbulence provides a simple model of nonrotating stratified turbulence analogous to the β-plane model of planetary turbulence. Jets known as vertically sheared horizontal flows (VSHFs) often emerge in simulations of Boussinesq turbulence, but their dynamics is not yet clearly understood. In this work S3T analysis of the zonostrophic instability is extended to study VSHF emergence in two-dimensional Boussinesq turbulence using an analytical formulation of S3T amenable to perturbation stability analysis. VSHFs are shown to form via an instability that is analogous in stratified turbulence to the zonostrophic instability in β-plane turbulence. This instability is shown to be strikingly similar to the zonostrophic instability, suggesting that jet emergence in both geostrophic and nonrotating stratified turbulence may be understood as instances of the same generic phenomenon.

scholarly journals WKB theory for rapid distortion of inhomogeneous turbulence

1999 ◽  
Vol 390 ◽  
pp. 325-348 ◽  
Author(s):  
S. NAZARENKO ◽  
N. K.-R. KEVLAHAN ◽  
B. DUBRULLE
Keyword(s):  
Large Scale ◽  
Isotropic Turbulence ◽  
Two Dimensional ◽  
Turbulent Spot ◽  
Mean Flow ◽  
Mean Velocity ◽  
Inhomogeneous Turbulence ◽  
Large Scale Vortex ◽  
The Mean ◽  
Mean Strain

A WKB method is used to extend RDT (rapid distortion theory) to initially inhomogeneous turbulence and unsteady mean flows. The WKB equations describe turbulence wavepackets which are transported by the mean velocity and have wavenumbers which evolve due to the mean strain. The turbulence also modifies the mean flow and generates large-scale vorticity via the averaged Reynolds stress tensor. The theory is applied to Taylor's four-roller flow in order to explain the experimentally observed reduction in the mean strain. The strain reduction occurs due to the formation of a large-scale vortex quadrupole structure from the turbulent spot confined by the four rollers. Both turbulence inhomogeneity and three-dimensionality are shown to be important for this effect. If the initially isotropic turbulence is either homogeneous in space or two-dimensional, it has no effect on the large-scale strain. Furthermore, the turbulent kinetic energy is conserved in the two-dimensional case, which has important consequences for the theory of two-dimensional turbulence. The analytical and numerical results presented here are in good qualitative agreement with experiment.

scholarly journals Phenomenology of two-dimensional stably stratified turbulence under large-scale forcing

2017 ◽  
Vol 18 (3) ◽  
pp. 219-239 ◽  
Author(s):  
Abhishek Kumar ◽  
Mahendra K. Verma ◽  
Jai Sukhatme
Keyword(s):  
Large Scale ◽  
Two Dimensional ◽  
Stratified Turbulence

scholarly journals Two-dimensional moist stratified turbulence and the emergence of vertically sheared horizontal flows

2012 ◽  
Vol 24 (3) ◽  
pp. 036602 ◽  
Author(s):  
Jai Sukhatme ◽  
Andrew J. Majda ◽  
Leslie M. Smith
Keyword(s):  
Two Dimensional ◽  
Stratified Turbulence ◽  
Horizontal Flows

scholarly journals Statistical State Dynamics of Jet–Wave Coexistence in Barotropic Beta-Plane Turbulence

2016 ◽  
Vol 73 (5) ◽  
pp. 2229-2253 ◽  
Author(s):  
Navid C. Constantinou ◽  
Brian F. Farrell ◽  
Petros J. Ioannou
Keyword(s):  
Large Scale ◽  
Cooperative Interaction ◽  
Small Scale ◽  
Mean Flow ◽  
Theoretical Understanding ◽  
Laminar Jet ◽  
Planetary Scale ◽  
Statistical State ◽  
Beta Plane ◽  
Scale Turbulence

Abstract Jets coexist with planetary-scale waves in the turbulence of planetary atmospheres. The coherent component of these structures arises from cooperative interaction between the coherent structures and the incoherent small-scale turbulence in which they are embedded. It follows that theoretical understanding of the dynamics of jets and planetary-scale waves requires adopting the perspective of statistical state dynamics (SSD), which comprises the dynamics of the interaction between coherent and incoherent components in the turbulent state. In this work, the stochastic structural stability theory (S3T) implementation of SSD for barotropic beta-plane turbulence is used to develop a theory for the jet–wave coexistence regime by separating the coherent motions consisting of the zonal jets together with a selection of large-scale waves from the smaller-scale motions that constitute the incoherent component. It is found that mean flow–turbulence interaction gives rise to jets that coexist with large-scale coherent waves in a synergistic manner. Large-scale waves that would exist only as damped modes in the laminar jet are found to be transformed into exponentially growing waves by interaction with the incoherent small-scale turbulence, which results in a change in the mode structure, allowing the mode to tap the energy of the mean jet. This mechanism of destabilization differs fundamentally and serves to augment the more familiar S3T instabilities in which jets and waves arise from homogeneous turbulence with the energy source exclusively from the incoherent eddy field and provides further insight into the cooperative dynamics of the jet–wave coexistence regime in planetary turbulence.

scholarly journals Transport and instability in driven two-dimensional magnetohydrodynamic flows

2016 ◽  
Vol 799 ◽  
pp. 541-578 ◽  
Author(s):  
Sam Durston ◽  
Andrew D. Gilbert
Keyword(s):  
Magnetic Field ◽  
Time Scale ◽  
Large Scale ◽  
Body Force ◽  
Passive Scalar ◽  
Fast Time ◽  
Two Dimensional ◽  
Mean Flow ◽  
The Mean ◽  
Background Shear

This paper concerns the generation of large-scale flows in forced two-dimensional systems. A Kolmogorov flow with a sinusoidal profile in one direction (driven by a body force) is known to become unstable to a large-scale flow in the perpendicular direction at a critical Reynolds number. This can occur in the presence of a ${\it\beta}$-effect and has important implications for flows observed in geophysical and astrophysical systems. It has recently been termed ‘zonostrophic instability’ and studied in a variety of settings, both numerically and analytically. The goal of the present paper is to determine the effect of magnetic field on such instabilities using the quasi-linear approximation, in which the full fluid system is decoupled into a mean flow and waves of one scale. The waves are driven externally by a given random body force and move on a fast time scale, while their stress on the mean flow causes this to evolve on a slow time scale. Spatial scale separation between waves and mean flow is also assumed, to allow analytical progress. The paper first discusses purely hydrodynamic transport of vorticity including zonostrophic instability, the effect of uniform background shear and calculation of equilibrium profiles in which the effective viscosity varies spatially, through the mean flow. After brief consideration of passive scalar transport or equivalently kinematic magnetic field evolution, the paper then proceeds to study the full magnetohydrodynamic system and to determine effective diffusivities and other transport coefficients using a mixture of analytical and numerical methods. This leads to results on the effect of magnetic field, background shear and ${\it\beta}$-effect on zonostrophic instability and magnetically driven instabilities.

Large-scale vortex structures in turbulent wakes behind bluff bodies. Part 1. Vortex formation processes

1987 ◽  
Vol 174 ◽  
pp. 233-270 ◽  
Author(s):  
A. E. Perry ◽  
T. R. Steiner
Keyword(s):  
Large Scale ◽  
Vector Fields ◽  
Vortex Formation ◽  
Three Dimensional ◽  
Point Theory ◽  
Shear Stresses ◽  
Bluff Bodies ◽  
Two Dimensional ◽  
Mean Flow ◽  
Turbulent Wakes

An investigation of turbulent wakes was conducted and phase-averaged velocity vector fields are presented, as well as phase-averaged and global Reynolds normal and shear stresses. The topology of the phase-averaged velocity fields is discussed in terms of critical point theory. Here in Part 1, the vortex formation process in the cavity region of several nominally two-dimensional bluff bodies is investigated and described using phase-averaged streamlines where the measurements were made in a nominal plane of symmetry. It was found that the flows encountered were always three-dimensional and that the mean-flow patterns in the cavity region were quite different from those expected using classical two-dimensional assumptions.

On the development of noise-producing large-scale wavelike eddies in a plane turbulent jet

1975 ◽  
Vol 70 (2) ◽  
pp. 353-368 ◽  
Author(s):  
Lee-Or Merkine ◽  
J. T. C. Liu
Keyword(s):  
Boundary Layer ◽  
Boundary Layer Thickness ◽  
Large Scale ◽  
Frequency Parameter ◽  
Turbulent Jet ◽  
Two Dimensional ◽  
Fine Scale ◽  
Mean Flow ◽  
Potential Core

In this paper we study the development of large-scale wavelike eddies in a two-dimensional turbulent jet, extending earlier work on the mixing region (Liu 1974). The basic mean flow develops from one of mixing-region type with an initially specified boundary-layer thickness into a fully developed jet. This study brings out the role of the varicose and sinuous modes as they develop in a growing mean flow. In general, it is found that, for a given frequency parameter, the varicose mode has a shorter streamwise lifetime than the sinuous mode. For lower frequencies, the latter persists past the end of the potential core only to become subject to dissipation by the enhanced fine-scale turbulent activity in that region.

scholarly journals On the Role of Clouds and Moisture in Tropical Waves: A Two-Dimensional Model Study

2006 ◽  
Vol 63 (8) ◽  
pp. 2140-2155 ◽  
Author(s):  
Danče Zurovac-Jevtić ◽  
Sandrine Bony ◽  
Kerry Emanuel
Keyword(s):  
Water Vapor ◽  
Large Scale ◽  
Active Role ◽  
Small Scale ◽  
Tropical Atmosphere ◽  
Two Dimensional ◽  
Mean Flow ◽  
Radiative Effects ◽  
Model Study ◽  
Cloud Radiative Effects

Abstract Observations show that convective perturbations of the tropical atmosphere are associated with substantial variations of clouds and water vapor. Recent studies suggest that these variations may play an active role in the large-scale organization of the tropical atmosphere. The present study investigates that possibility by using a two-dimensional, nonrotating model that includes a set of physical parameterizations carefully evaluated against tropical data. In the absence of cloud–radiation interactions, the model spontaneously generates fast upwind (eastward) moving planetary-scale oscillations through the wind-induced surface heat exchange mechanism. In the presence of cloud–radiative effects, the model generates slower upwind (eastward) propagating modes in addition to small-scale disturbances advected downwind (westward) by the mean flow. Enhanced cloud–radiative effects further slow down upwind propagating waves and make them more prominent in the spectrum. On the other hand, the model suggests that interactions between moisture and convection favor the prominence of moist Kelvin-like waves in tropical variability at the expense of small-scale advective disturbances. These numerical results, consistent with theoretical predictions, suggest that the interaction of water vapor and cloud variations with convection and radiation plays an active role in the large-scale organization of the tropical atmosphere.

scholarly journals Evolution and Stability of Two-Dimensional Anelastic Internal Gravity Wave Packets

2018 ◽  
Vol 75 (10) ◽  
pp. 3703-3724 ◽  
Author(s):  
Alain D. Gervais ◽  
Gordon E. Swaters ◽  
Ton S. van den Bremer ◽  
Bruce R. Sutherland
Keyword(s):  
Wave Packet ◽  
Gravity Wave ◽  
Linear Theory ◽  
Nonlinear Evolution ◽  
Wave Packets ◽  
Internal Gravity Wave ◽  
Two Dimensional ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
Nonlinear Simulations

The weakly nonlinear evolution, stability, and overturning of horizontally and vertically localized internal gravity wave packets is examined for a nonrotating, anelastic atmosphere that is stationary in the absence of waves. The weakly nonlinear evolution is examined through the derivation of their wave-induced mean flow, which is used to formulate a nonlinear Schrödinger equation. The induced flow is manifest as a long, hydrostatic, bow wake-like disturbance, whose flow direction transitions from positive on the leading flank of the wave packet to negative on the trailing flank of the wave packet. As such, two-dimensional wave packets are always modulationally unstable. This instability results in enhanced amplitude growth confined to either the leading or trailing flank. Hence, when combined with anelastic growth predicted by linear theory, we anticipate two-dimensional waves will overturn either somewhat below or just above the heights predicted by linear theory. Numerical solutions of the Schrödinger equation are compared with the results of fully nonlinear simulations to establish the validity of the weakly nonlinear theory. Actual wave overturning heights are determined quantitatively from a range of fully nonlinear simulations.

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