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 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 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 Theoretical analysis of the zigzag instability of a vertical columnar vortex pair in a strongly stratified fluid

2000 ◽  
Vol 419 ◽  
pp. 29-63 ◽  
Author(s):  
PAUL BILLANT ◽  
JEAN-MARC CHOMAZ
Keyword(s):  
Froude Number ◽  
Evolution Equations ◽  
Vortex Pair ◽  
Horizontal Velocity ◽  
Phase Variable ◽  
Two Dimensional ◽  
Stratified Turbulence ◽  
Periodic Patterns ◽  
Phase Dynamics ◽  
Columnar Vortex

A general theoretical account is proposed for the zigzag instability of a vertical columnar vortex pair recently discovered in a strongly stratified experiment.The linear inviscid stability of the Lamb–Chaplygin vortex pair is analysed by a multiple-scale expansion analysis for small horizontal Froude number (Fh = U/LhN, where U is the magnitude of the horizontal velocity, Lh the horizontal lengthscale and N the Brunt–Väisälä frequency) and small vertical Froude number (Fv = U/LvN, where Lv is the vertical lengthscale) using the scaling of the equations of motion introduced by Riley, Metcalfe & Weissman (1981). In the limit Fv = 0, these equations reduce to two-dimensional Euler equations for the horizontal velocity with undetermined vertical dependence. Thus, at leading order, neutral modes of the flow are associated, among others, to translational and rotational invariances in each horizontal plane. To each broken invariance is related a phase variable that may vary freely along the vertical. Conservation of mass and potential vorticity impose at higher order the evolution equations governing the phase variables that we derive for Fh [Lt ] 1 and Fv [Lt ] 1 in the spirit of phase dynamics techniques established for periodic patterns. In agreement with the experimental observations, this asymptotic analysis shows the existence of an instability consisting of a vertically modulated rotation and a translation of the columnar vortex pair perpendicular to the travelling direction. The dispersion relation as well as the spatial eigenmode of the zigzag instability are determined. The analysis predicts that the most amplified vertical wavelength should scale as U/N and the maximum growth rate as U/Lh.Our main finding is thus that the typical thickness of the ensuing layers will be such that Fv = O(1) and not Fv [Lt ] 1 as assumed by Riley et al. (1981) and Lilly (1983). This implies that such strongly stratified flows are not described by two- dimensional horizontal equations. These results may help to understand the layering commonly observed in stratified turbulence and the fundamental differences with strictly two-dimensional turbulence.

Sensitivity of horizontal flows to forcing geometry

2001 ◽  
Vol 432 ◽  
pp. 419-441
Author(s):  
ISAO KANDA ◽  
P. F. LINDEN
Keyword(s):  
Reynolds Number ◽  
Analytical Solutions ◽  
Stokes Flow ◽  
Large Reynolds Number ◽  
Two Dimensional ◽  
Vortex Pattern ◽  
Inverse Energy Cascade ◽  
Horizontal Flows ◽  
Source Sink ◽  
Special Case

We investigate the horizontal flow produced by source–sink forcing in a stably stratified fluid. The forcing jets are kept laminar and are placed along the boundary of a square domain. We find that the resultant flow patterns are extremely sensitive to the forcing geometry. The single dominant vortex pattern, interpreted as the result of inverse energy cascade of two-dimensional turbulence in our previous work (Boubnov, Dalziel & Linden 1994), turns out to be a special case. We show that some of the steady patterns resemble the eigenmodes of the Helmholtz equation as the inviscid vorticity equation. Although there are significant discrepancies in the streamfunction vs. vorticity relations between the observed flows and the analytical solutions, we identify the differences as a result of viscous diffusion of vorticity from the source flows. We also study the transition from forced to decaying flow. The flow assumes the properties of Stokes flow at quite large Reynolds number, indicating transformation into patterns with small advective acceleration.

scholarly journals Haline Convection within a Fresh-Saline Water Interface in a Stratified Coastal Aquifer Induced by Tide

Water ◽  
2021 ◽  
Vol 13 (13) ◽  
pp. 1780
Author(s):  
Elad Ben-Zur ◽  
Haim Gvirtzman ◽  
Eyal Shalev
Keyword(s):  
Coastal Aquifer ◽  
Laboratory Experiments ◽  
Saline Water ◽  
Time Lag ◽  
Field Studies ◽  
Homogeneous System ◽  
Water Interface ◽  
Two Dimensional ◽  
Horizontal Movement ◽  
Horizontal Flows

Sea-tide effects on the fresh-saline water interface (FSI) in a stratified coastal aquifer are examined through laboratory experiments. The physical model, a two-dimensional rectangular flow tank, is filled with layered aquifers and aquitards. The aquifers serve as the main entrances/exits of water to/from the system through significant horizontal flows, creating unstable conditions of heavier saline water above lighter freshwater for short periods of time. Several processes create mixing; this instability results in haline convection, creating downward fingering, stable rising of horizontal saltwater front, and unstable upward fingerings of flushing freshwater. The time lag between the sea tide fluctuations and the emergence of adequate fresh- and saltwater is higher in a stratified system compared to a homogeneous system. Furthermore, longer tide cycles lead to the enlargement of the FSI’s toe horizontal movement range. The combination of tidal forcing with a layering aquifer structure leads to a wider FSI by creating a significant salt- and freshwater mixing inside each layer, vertical flows between the layers, and saltwater bodies at isolated areas. Haline convection within the FSI might be the reason for the wider fresh-saline interfaces that are found in field studies.

scholarly journals A Case Study of Two-Dimensional Stratified Turbulence

1999 ◽  
Vol 56 (7) ◽  
pp. 959-976 ◽  
Author(s):  
Ulf Högström ◽  
Ann-Sofi Smedman ◽  
Hans Bergström
Keyword(s):  
Two Dimensional ◽  
Stratified 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.

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.

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