scholarly journals The effect of asymmetric large-scale dissipation on energy and potential enstrophy injection in two-layer quasi-geostrophic turbulence

2012 ◽  
Vol 694 ◽  
pp. 493-523 ◽  
Author(s):  
Eleftherios Gkioulekas
Keyword(s):  
Large Scale ◽  
Bottom Layer ◽  
Two Dimensional ◽  
Interlayer Interaction ◽  
Open Questions ◽  
Small Scales ◽  
Scale Range ◽  
Geostrophic Turbulence ◽  
Enstrophy Cascade ◽  
Potential Enstrophy

AbstractIn the Nastrom–Gage spectrum of atmospheric turbulence, we observe a${k}^{\ensuremath{-} 3} $energy spectrum that transitions into a${k}^{\ensuremath{-} 5/ 3} $spectrum, with increasing wavenumber$k$. The transition occurs near a transition wavenumber${k}_{t} $, located near the Rossby deformation wavenumber${k}_{R} $. The Tung–Orlando theory interprets this spectrum as a double downscale cascade of potential enstrophy and energy, from large scales to small scales, in which the downscale potential enstrophy cascade coexists with the downscale energy cascade over the same length scale range. We show that, in a temperature-forced two-layer quasi-geostrophic model, the rates with which potential enstrophy and energy are injected place the transition wavenumber${k}_{t} $near${k}_{R} $. We also show that, if the potential energy dominates the kinetic energy in the forcing range, then the Ekman term suppresses the upscale cascading potential enstrophy more than it suppresses the upscale cascading energy, a behaviour contrary to what occurs in two-dimensional turbulence. As a result, the ratio$\eta / \varepsilon $of injected potential enstrophy over injected energy, in the downscale direction, decreases, thereby tending to decrease the transition wavenumber${k}_{t} $further. Using a random Gaussian forcing model, we reach the same conclusion, under the modelling assumption that the asymmetric Ekman term predominantly suppresses the bottom layer forcing, thereby disregarding a possible entanglement between the Ekman term and the nonlinear interlayer interaction. Based on these results, we argue that the Tung–Orlando theory can account for the approximate coincidence between${k}_{t} $and${k}_{R} $. We also identify certain open questions that require further investigation via numerical simulations.

Charney isotropy and equipartition in quasi-geostrophic turbulence

2010 ◽  
Vol 656 ◽  
pp. 448-457 ◽  
Author(s):  
ANDREAS VALLGREN ◽  
ERIK LINDBORG
Keyword(s):  
High Resolution ◽  
Kinetic Energy ◽  
Horizontal Velocity ◽  
Energy Cascade ◽  
Two Dimensional ◽  
Inverse Energy Cascade ◽  
Wide Range ◽  
Geostrophic Turbulence ◽  
Decaying Turbulence ◽  
Enstrophy Cascade

High-resolution simulations of forced quasi-geostrophic (QG) turbulence reveal that Charney isotropy develops under a wide range of conditions, and constitutes a preferred state also in β-plane and freely decaying turbulence. There is a clear analogy between two-dimensional and QG turbulence, with a direct enstrophy cascade that is governed by the prediction of Kraichnan (J. Fluid Mech., vol. 47, 1971, p. 525) and an inverse energy cascade following the classic k−5/3 scaling. Furthermore, we find that Charney's prediction of equipartition between the potential and kinetic energy in each of the two horizontal velocity components is approximately fulfilled in the inertial ranges.

Geostrophic Turbulence

1998 ◽  
Author(s):  
Rick Salmon
Keyword(s):  
Large Scale ◽  
Bottom Topography ◽  
Single Layer ◽  
Two Dimensional ◽  
Homogeneous Fluid ◽  
Coriolis Parameter ◽  
Strongly Nonlinear ◽  
Geostrophic Turbulence ◽  
Large Scale Flow ◽  
Quasigeostrophic Equations

Strongly nonlinear, rapidly rotating, stably stratified flow is called geostrophic turbulence. This subject, which blends ideas from chapters 2,4, and 5, is relevant to the large-scale flow in the Earth’s oceans and atmosphere. The quasigeostrophic equations form the basis of the study of geostrophic turbulence. We view the quasigeostrophic equations as a generalization of the vorticity equation for two-dimensional turbulence to include the important effects of stratification, bottom topography, and varying Coriolis parameter. Thus the theory of geostrophic turbulence represents an extension of the theory of two dimensional turbulence. However, its richer physics and greater applicability to real geophysical flows make geostrophic turbulence a much more interesting and important subject. This chapter offers a very brief introduction to the theory of geostrophic turbulence. We illustrate the principal ideas by separately considering the effects of bottom topography, varying Coriolis parameter, and density stratification on highly nonlinear, quasigeostrophic flow. We make no attempt at a comprehensive review. In every case, the theory of geostrophic turbulence relies almost solely on two now-familiar components: a conservation principle that energy and potential vorticity are (nearly) conserved and an irreversibility principle in the form of an appealing assumption that breaks the time-reversal symmetry of the exact (inviscid) dynamics. This irreversibility assumption takes a great many superficially dissimilar forms, fostering the misleading impression of a great many competing explanations for the same phenomena. However, broadminded analysis inevitably reveals that these competing explanations are virtually equivalent. We begin by considering the quasigeostrophic flow of a single layer of homogeneous fluid over a bumpy bottom. No case better illustrates how diverse forms of the irreversibility principle lead to the same conclusions.

scholarly journals The vortex instability pathway in stratified turbulence

2013 ◽  
Vol 716 ◽  
pp. 1-4 ◽  
Author(s):  
Michael L. Waite
Keyword(s):  
Large Scale ◽  
Three Dimensional ◽  
Small Scale ◽  
Nonlinear Interactions ◽  
Stratified Turbulence ◽  
Transfer Energy ◽  
Columnar Vortex ◽  
Open Questions ◽  
Small Scales ◽  
Flow Transitions

AbstractThe parameter regime of strong stable density stratification and weak rotation is an important one in geophysical fluid dynamics. These conditions exist at intermediate length scales in the atmosphere and ocean (mesoscale and sub-mesoscale, respectively), and turbulence here links large-scale quasi-geostrophic motions with small-scale dissipation. While major advances in the theory of stratified turbulence have been made over the last few decades, many open questions remain, particularly about the nature of the energy cascade. Recent numerical experiments and analysis by Augier, Chomaz & Billant (J. Fluid Mech., vol. 713, 2012, pp. 86–108) present a remarkably vivid illustration of the nonlinear interactions that drive such turbulence. They consider a columnar vortex dipole, which naturally three-dimensionalizes under the influence of strong stratification. Kelvin–Helmholtz instabilities subsequently transfer energy directly to small scales, where the flow transitions into three-dimensional turbulence. This direct link between large and small scales is quite distinct from the usual picture of a turbulent cascade, in which nonlinear interactions are local in scale. But how important is this mechanism in the atmosphere and ocean?

Coherent vortices and tracer cascades in two-dimensional turbulence

2007 ◽  
Vol 574 ◽  
pp. 429-448 ◽  
Author(s):  
ARMANDO BABIANO ◽  
ANTONELLO PROVENZALE
Keyword(s):  
Inertial Range ◽  
Two Dimensional ◽  
Passive Tracer ◽  
Simultaneous Presence ◽  
Inverse Cascade ◽  
Coherent Vortices ◽  
Small Scales ◽  
Elliptic Regions ◽  
Enstrophy Cascade

We study numerically the scale-to-scale transfers of enstrophy and passive-tracer variance in two-dimensional turbulence, and show that these transfers display significant differences in the inertial range of the enstrophy cascade. While passive-tracer variance always cascades towards small scales, enstrophy is characterized by the simultaneous presence of a direct cascade in hyperbolic regions and of an inverse cascade in elliptic regions. The inverse enstrophy cascade is particularly intense in clusters of small-scales elliptic patches and vorticity filaments in the turbulent background, and it is associated with gradient-decreasing processes. The inversion of the enstrophy cascade, already noticed by Ohkitani (Phys. Fluids A, vol. 3, 1991, p. 1598), appears to be the main difference between vorticity and passive-tracer dynamics in incompressible two-dimensional turbulence.

Late time evolution of unforced inviscid two-dimensional turbulence

2009 ◽  
Vol 640 ◽  
pp. 215-233 ◽  
Author(s):  
DAVID G. DRITSCHEL ◽  
RICHARD K. SCOTT ◽  
CHARLIE MACASKILL ◽  
GEORG A. GOTTWALD ◽  
CHUONG V. TRAN
Keyword(s):  
Energy Spectrum ◽  
High Resolution ◽  
Slow Growth ◽  
Inviscid Limit ◽  
Small Scale ◽  
Recent Theory ◽  
Late Time ◽  
Two Dimensional ◽  
Small Scales ◽  
Scale Range

We propose a new unified model for the small, intermediate and large-scale evolution of freely decaying two-dimensional turbulence in the inviscid limit. The new model's centerpiece is a recent theory of vortex self-similarity (Dritschel et al., Phys. Rev. Lett., vol. 101, 2008, no. 094501), applicable to the intermediate range of scales spanned by an expanding population of vortices. This range is predicted to have a steep k−5 energy spectrum. At small scales, this gives way to Batchelor's (Batchelor, Phys. Fluids, vol. 12, 1969, p. 233) k−3 energy spectrum, corresponding to the (forward) enstrophy (mean square vorticity) cascade or, physically, to thinning filamentary debris produced by vortex collisions. This small-scale range carries with it nearly all of the enstrophy but negligible energy. At large scales, the slow growth of the maximum vortex size (~t1/6 in radius) implies a correspondingly slow inverse energy cascade. We argue that this exceedingly slow growth allows the large scales to approach equipartition (Kraichnan, Phys. Fluids, vol. 10, 1967, p. 1417; Fox & Orszag, Phys. Fluids, vol. 12, 1973, p. 169), ultimately leading to a k1 energy spectrum there. Put together, our proposed model has an energy spectrum ℰ(k, t) ∝ t1/3k1 at large scales, together with ℰ(k, t) ∝ t−2/3k−5 over the vortex population, and finally ℰ(k, t) ∝ t−1k−3 over an exponentially widening small-scale range dominated by incoherent filamentary debris.Support for our model is provided in two parts. First, we address the evolution of large and ultra-large scales (much greater than any vortex) using a novel high-resolution vortex-in-cell simulation. This verifies equipartition, but more importantly allows us to better understand the approach to equipartition. Second, we address the intermediate and small scales by an ensemble of especially high-resolution direct numerical simulations.

Numerical simulation of relative dispersion in two-dimensional, homogeneous, decaying turbulence

1981 ◽  
Vol 109 ◽  
pp. 45-61 ◽  
Author(s):  
A. D. Kowalski ◽  
R. L. Peskin
Keyword(s):  
Single Particle ◽  
Large Scale ◽  
Local Strain ◽  
Relative Dispersion ◽  
Two Dimensional ◽  
Temporal Behaviour ◽  
Relative Separation ◽  
Non Local ◽  
Decaying Turbulence ◽  
Enstrophy Cascade

Lagrangian statistical results are presented from numerical simulations of an ensemble of fluid particles which were generated from a two-dimensional pseudospectral code. The single-particle results are in qualitative agreement with previous simulations on a lower-resolution grid. The two-particle, relative velocity correlations were found to fall off more rapidly than the single-particle correlations for short to intermediate times due to large-scale eddy advection in the single-particle case. The temporal behaviour of the mean square relative separation, 〈rl2〉, is analysed for short to intermediate times and is found to be consistent with scaling arguments based on Kraichnan's expression for the non-local strain acting in the high-wavenumber enstrophy cascade spectral range. For longer times, 〈rl2〉 exhibits tn behaviour. The power-law region is associated with the locally determined strain rates which characterize a backward energy-cascade spectrum.

The emergence of isolated coherent vortices in turbulent flow

1984 ◽  
Vol 146 ◽  
pp. 21-43 ◽  
Author(s):  
James C. Mcwilliams
Keyword(s):  
Turbulent Flow ◽  
Turbulent Flows ◽  
Large Scale ◽  
Initial Conditions ◽  
Turnaround Time ◽  
Large Reynolds Number ◽  
Two Dimensional ◽  
Dominant Component ◽  
Geostrophic Turbulence ◽  
Large Scale Flow

A study is made of some numerical calculations of two-dimensional and geostrophic turbulent flows. The primary result is that, under a broad range of circumstances, the flow structure has its vorticity concentrated in a small fraction of the spatial domain, and these concentrations typically have lifetimes long compared with the characteristic time for nonlinear interactions in turbulent flow (i.e. an eddy turnaround time). When such vorticity concentrations occur, they tend to assume an axisymmetric shape and persist under passive advection by the large-scale flow, except for relatively rare encounters with other centres of concentration. These structures can arise from random initial conditions without vorticity concentration, evolving in the midst of what has been traditionally characterized as the ‘cascade’ of isotropic, homogeneous, large-Reynolds-number turbulence: the systematic elongation of isolines of vorticity associated with the transfer of vorticity to smaller scales, eventually to dissipation scales, and the transfer of energy to larger scales. When the vorticity concentrations are a sufficiently dominant component of the total vorticity field, the cascade processes are suppressed. The demonstration of persistent vorticity concentrations on intermediate scales - smaller than the scale of the peak of the energy spectrum and larger than the dissipation scales - does not invalidate many of the traditional characterizations of two-dimensional and geostrophic turbulence, but I believe it shows them to be substantially incomplete with respect to a fundamental phenomenon in such flows.

The enstrophy cascade in forced two-dimensional turbulence

2011 ◽  
Vol 671 ◽  
pp. 168-183 ◽  
Author(s):  
ANDREAS VALLGREN ◽  
ERIK LINDBORG
Keyword(s):  
Large Scale ◽  
Stokes Equations ◽  
Cascade Range ◽  
Navier Stokes ◽  
Logarithmic Correction ◽  
Computational Domain ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Grid Points ◽  
Enstrophy Cascade

We carry out direct numerical simulations of two-dimensional turbulence with forcing at different wavenumbers and resolutions up to 327682 grid points. In the absence of large-scale drag, a state is reached where enstrophy is quasi-stationary while energy is growing. In the enstrophy cascade range the energy spectrum has the form E(k) = εω2/3k−3, without any logarithmic correction, where εω is the enstrophy dissipation and is of the order of unity. However, is varying between different simulations and is thus not a perfect constant. This variation can be understood as a consequence of large-scale dissipation intermittency, following the argument by Landau (Landau & Lifshitz, Fluid Mechanics, 1959, Pergamon). In the presence of a large-scale drag, we obtain a slightly steeper spectrum. When forcing is applied at a scale which is somewhat smaller than the computational domain, no vortices are formed, and the statistics remain close to Gaussian in the enstrophy cascade range. When forcing is applied at a smaller scale, long-lived coherent vortices form at larger scales than the forcing scale, and intermittency measures become very large at all scales, including the scales of the enstrophy cascade. We conclude that the enstrophy cascade with a k−3-spectrum is a robust feature of the two-dimensional Navier–Stokes equations. However, there is a complete lack of universality of higher-order statistics of vorticity increments in the enstrophy cascade range.

Turbulent diffusion in the geostrophic inverse cascade

2002 ◽  
Vol 469 ◽  
pp. 13-48 ◽  
Author(s):  
K. S. SMITH ◽  
G. BOCCALETTI ◽  
C. C. HENNING ◽  
I. MARINOV ◽  
C. Y. TAM ◽  
...  
Keyword(s):  
Drag Coefficient ◽  
Large Scale ◽  
Turbulent Transport ◽  
Passive Scalar ◽  
Turbulent Heat ◽  
Two Dimensional ◽  
Inverse Cascade ◽  
Random Forcing ◽  
Linear Drag ◽  
Geostrophic Turbulence

Motivated in part by the problem of large-scale lateral turbulent heat transport in the Earth's atmosphere and oceans, and in part by the problem of turbulent transport itself, we seek to better understand the transport of a passive tracer advected by various types of fully developed two-dimensional turbulence. The types of turbulence considered correspond to various relationships between the streamfunction and the advected field. Each type of turbulence considered possesses two quadratic invariants and each can develop an inverse cascade. These cascades can be modified or halted, for example, by friction, a background vorticity gradient or a mean temperature gradient. We focus on three physically realizable cases: classical two-dimensional turbulence, surface quasi-geostrophic turbulence, and shallow-water quasi-geostrophic turbulence at scales large compared to the radius of deformation. In each model we assume that tracer variance is maintained by a large-scale mean tracer gradient while turbulent energy is produced at small scales via random forcing, and dissipated by linear drag. We predict the spectral shapes, eddy scales and equilibrated energies resulting from the inverse cascades, and use the expected velocity and length scales to predict integrated tracer fluxes.When linear drag halts the cascade, the resulting diffusivities are decreasing functions of the drag coefficient, but with different dependences for each case. When β is significant, we find a clear distinction between the tracer mixing scale, which depends on β but is nearly independent of drag, and the energy-containing (or jet) scale, set by a combination of the drag coefficient and β. Our predictions are tested via high- resolution spectral simulations. We find in all cases that the passive scalar is diffused down-gradient with a diffusion coefficient that is well-predicted from estimates of mixing length and velocity scale obtained from turbulence phenomenology.

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