Statistical state dynamics analysis of buoyancy layer formation via the Phillips mechanism in two-dimensional stratified turbulence

2019 ◽  
Vol 864 ◽  
Author(s):  
Joseph G. Fitzgerald ◽  
Brian F. Farrell
Keyword(s):  
Equations Of Motion ◽  
Two Dimensional ◽  
Stratified Turbulence ◽  
Diffusion Property ◽  
Theoretical Understanding ◽  
Flux Gradient ◽  
Statistical State ◽  
The Operating Mechanism ◽  
Fundamental Equations ◽  
Negative Diffusion

Horizontal density layers are commonly observed in stratified turbulence. Recent work (e.g. Taylor & Zhou, J. Fluid Mech., vol. 823, 2017, R5) has reinvigorated interest in the Phillips instability (PI), by which density layers form via negative diffusion if the turbulent buoyancy flux weakens as stratification increases. Theoretical understanding of PI is incomplete, in part because it remains unclear whether and by what mechanism the flux-gradient relationship for a given example of turbulence has the required negative-diffusion property. Furthermore, the difficulty of analysing the flux-gradient relation in evolving turbulence obscures the operating mechanism when layering is observed. These considerations motivate the search for an example of PI that can be analysed clearly. Here PI is shown to occur in two-dimensional Boussinesq sheared stratified turbulence maintained by stochastic excitation. PI is analysed using the second-order S3T closure of statistical state dynamics, in which the dynamics is written directly for statistical variables of the turbulence. The predictions of S3T are verified using nonlinear simulations. This analysis provides theoretical underpinning of PI based on the fundamental equations of motion that complements previous analyses based on phenomenological models of turbulence.

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 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.

Two-dimensional leaps in near-critical flow over isolated orography

2005 ◽  
Vol 461 (2064) ◽  
pp. 3747-3763 ◽  
Author(s):  
E.R Johnson ◽  
G.G Vilenski
Keyword(s):  
Unsteady Flow ◽  
Equations Of Motion ◽  
Fold Increase ◽  
Flow Speed ◽  
Two Dimensional ◽  
Subcritical Flow ◽  
One Dimensional ◽  
Petviashvili Equation ◽  
Lee Wave ◽  
Supercritical Flows

This paper describes steady two-dimensional disturbances forced on the interface of a two-layer fluid by flow over an isolated obstacle. The oncoming flow speed is close to the linear longwave speed and the layer densities, layer depths and obstacle height are chosen so that the equations of motion reduce to the forced two-dimensional Korteweg–de Vries equation with cubic nonlinearity, i.e. the forced extended Kadomtsev–Petviashvili equation. The distinctive feature noted here is the appearance in the far lee-wave wake behind obstacles in subcritical flow of a ‘table-top’ wave extending almost one-dimensionally for many obstacles widths across the flow. Numerical integrations show that the most important parameter determining whether this wave appears is the departure from criticality, with the wave appearing in slightly subcritical flows but being destroyed by two-dimensional effects behind even quite long ridges in even moderately subcritical flow. The wave appears after the flow has passed through a transition from subcritical to supercritical over the obstacle and its leading and trailing edges resemble dissipationless leaps standing in supercritical flow. Two-dimensional steady supercritical flows are related to one-dimensional unsteady flows with time in the unsteady flow associated with a slow cross-stream variable in the two-dimensional flows. Thus the wide cross-stream extent of the table-top wave appears to derive from the combination of its occurrence in a supercritical region embedded in the subcritical flow and the propagation without change of form of table-top waves in one-dimensional unsteady flow. The table-top wave here is associated with a resonant steepening of the transition above the obstacle and a consequent twelve-fold increase in drag. Remarkably, the table-top wave is generated equally strongly and extends laterally equally as far behind an axisymmetric obstacle as behind a ridge and so leads to subcritical flows differing significantly from linear predictions.

EQUIVALENCE OF TWO-DIMENSIONAL GRAVITIES

1990 ◽  
Vol 05 (16) ◽  
pp. 1251-1258 ◽  
Author(s):  
NOUREDDINE MOHAMMEDI
Keyword(s):  
Gauge Theory ◽  
Explicit Solution ◽  
Light Cone ◽  
Equations Of Motion ◽  
Two Dimensional ◽  
Induced Gravity ◽  
Light Cone Gauge ◽  
Dimensional Gravity ◽  
The Relationship ◽  
Chern Simons

We find the relationship between the Jackiw-Teitelboim model of two-dimensional gravity and the SL (2, R) induced gravity. These are shown to be related to a two-dimensional gauge theory obtained by dimensionally reducing the Chern-Simons action of the 2+1 dimensional gravity. We present an explicit solution to the equations of motion of the auxiliary field of the Jackiw-Teitelboim model in the light-cone gauge. A renormalization of the cosmological constant is also given.

The Asymptotic Stability of Weakly Perturbed Two Dimensional Hamiltonian Systems

2001 ◽  
Author(s):  
Ludwig Arnold ◽  
Peter Imkeller ◽  
N. Sri Namachchivaya
Keyword(s):  
Lyapunov Exponent ◽  
Equations Of Motion ◽  
Perturbation Approach ◽  
Two Dimensional ◽  
White Noise Process ◽  
System A ◽  
Additive White Noise ◽  
Small Dissipation ◽  
Small Intensity ◽  
Single Well

Abstract The purpose of this work is to obtain an approximation for the top Lyapunov exponent, the exponential growth rate, of the response of a single-well Kramers Oscillator driven by either a multiplicative or an additive white noise process. To this end, we consider the equations of motion as dissipative and noisy perturbations of a two-dimensional Hamiltonian system. A perturbation approach is used to obtain explicit expressions for the exponent in the presence of small intensity noise and small dissipation. We show analytically that the top Lyapunov exponent is positive, and for small values of noise intensity ε and dissipation ε the exponent grows proportional to ε1/3.

The Use of Normal Modes in Problems of Forced Vibration and Impact

1957 ◽  
Vol 61 (560) ◽  
pp. 552-559
Author(s):  
R. P. N. Jones
Keyword(s):  
Forced Vibration ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Very High Frequency ◽  
Linear Elastic ◽  
Modes Of Vibration ◽  
Fundamental Equations ◽  
Very High ◽  
Simple Exposition

SummaryA simple exposition, using d'Alembert's principle and methods of virtual work, is given of the properties and applications of the normal modes of vibration of a linear elastic system. The use of the normal modes in problems of free and forced vibration and dynamic loading is discussed with the aid of simple examples, and it is shown that by these methods dynamical problems for any linear system may be solved without the use of the fundamental equations of motion, provided the natural frequencies and modes of the system are known. In most problems the solutions converge rapidly, so that only the first few modes of vibration need be considered, and in these cases the solution may be modified to give further improvement in convergence. Unsatisfactory convergence may be obtained, however, in problems where there is an exciting force of very high frequency, or an impact of short duration. An approximate allowance may be made for damping, provided this is small.

scholarly journals Construction And Test Of Thermostats And Twirlers For Molecular Rotations

2003 ◽  
Vol 58 (7-8) ◽  
pp. 377-391 ◽  
Author(s):  
Siegfried Hess
Keyword(s):  
Time Reversal ◽  
Rotational Diffusion ◽  
Equations Of Motion ◽  
Dynamical Variable ◽  
Two Dimensional ◽  
Polymer Molecules ◽  
Isotropic Harmonic Oscillator ◽  
Molecular Rotations ◽  
Basic Equations ◽  
Nonlinear Elastic

The equations of motion are coupled with a dynamical variable, referred to as twirler, which randomizes the angular momentum. The equations are time-reversal invariant, just as those for the standard Gaussian, Nosé-Hoover and configurational thermostats. The derivation of the basic equations is outlined. Test calculations are performed for the two-dimensional isotropic harmonic oscillator and for a nonlinear elastic dumbbell, used as a simple model to study properties of polymer molecules. Graphs of characteristic quantities and orbits, some of which are rather intriguing, are displayed. As applications, the rotational diffusion and the influence of a shear flow on the angular velocity and the deformation of the model polymer are analyzed.

scholarly journals THE CONSERVED CHARGES AND INTEGRABILITY OF THE CONFORMAL AFFINE TODA MODELS

1994 ◽  
Vol 09 (30) ◽  
pp. 2783-2801 ◽  
Author(s):  
H. ARATYN ◽  
L. A. FERREIRA ◽  
J. F. GOMES ◽  
A. H. ZIMERMAN
Keyword(s):  
Poisson Bracket ◽  
Equations Of Motion ◽  
Curvature Form ◽  
Two Dimensional ◽  
Conserved Charges ◽  
Zero Curvature ◽  
Algebraic Properties ◽  
Toda Model ◽  
Infinite Sets ◽  
Poisson Commute

We construct infinite sets of local conserved charges for the conformal affine Toda model. The technique involves the abelianization of the two-dimensional gauge potentials satisfying the zero-curvature form of the equations of motion. We find two infinite sets of chiral charges and apart from two lowest spin charges, all the remaining ones do not possess chiral densities. Charges of different chiralities Poisson commute among themselves. We discuss the algebraic properties of these charges and use the fundamental Poisson bracket relation to show that the charges conserved in time are in involution. Connections to other Toda models are established by taking particular limits.

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