Two-dimensional isotropic inertia–gravity wave turbulence

2019 ◽  
Vol 872 ◽  
pp. 752-783 ◽  
Author(s):  
Jin-Han Xie ◽  
Oliver Bühler
Keyword(s):  
Total Energy ◽  
Isotropic Turbulence ◽  
Vertical Plane ◽  
Boussinesq Equations ◽  
Structure Functions ◽  
Wave Turbulence ◽  
Spectral Energy ◽  
Order Structure ◽  
Two Dimensional ◽  
Third Order

We present an idealized study of rotating stratified wave turbulence in a two-dimensional vertical slice model of the Boussinesq equations, focusing on the peculiar case of equal Coriolis and buoyancy frequencies. In this case the fully nonlinear fluid dynamics can be shown to be isotropic in the vertical plane, which allows the classical methods of isotropic turbulence to be applied. Contrary to ordinary two-dimensional turbulence, here a robust downscale flux of total energy is observed in numerical simulations that span the full parameter regime between Ozmidov and forcing scales. Notably, this robust downscale flux of the total energy does not hold separately for its various kinetic and potential components, which can exhibit both upscale and downscale fluxes, depending on the parameter regime. Using a suitable extension of the classical Kármán–Howarth–Monin equation, exact expressions that link third-order structure functions and the spectral energy flux are derived and tested against numerical results. These expressions make obvious that even though the total energy is robustly transferred downscale, the third-order structure functions are sign indefinite, which illustrates that the sign and the form of measured third-order structure functions are both crucially important in determining the direction of the spectral energy transfer.

Exact third-order structure functions for two-dimensional turbulence

2018 ◽  
Vol 851 ◽  
pp. 672-686 ◽  
Author(s):  
Jin-Han Xie ◽  
Oliver Bühler
Keyword(s):  
Structure Functions ◽  
Spectral Energy ◽  
Order Structure ◽  
Two Dimensional ◽  
Third Order ◽  
Transfer Rates ◽  
The Third ◽  
Random Forcing ◽  
Asymptotic Expressions ◽  
Special Case

We derive and investigate exact expressions for third-order structure functions in stationary isotropic two-dimensional turbulence, assuming a statistical balance between random forcing and dissipation both at small and large scales. Our results extend previously derived asymptotic expressions in the enstrophy and energy inertial ranges by providing uniformly valid expressions that apply across the entire non-dissipative range, which, importantly, includes the forcing scales. In the special case of white noise in time forcing this leads to explicit predictions for the third-order structure functions, which are successfully tested against previously published high-resolution numerical simulations. We also consider spectral energy transfer rates and suggest and test a simple robust diagnostic formula that is useful when forcing is applied at more than one scale.

scholarly journals Third-order structure functions for isotropic turbulence with bidirectional energy transfer

2019 ◽  
Vol 877 ◽  
Author(s):  
Jin-Han Xie ◽  
Oliver Bühler
Keyword(s):  
Energy Transfer ◽  
Isotropic Turbulence ◽  
Full Range ◽  
Structure Functions ◽  
Spectral Energy ◽  
Order Structure ◽  
Stratified Turbulence ◽  
Mhd Turbulence ◽  
Third Order ◽  
Transfer Rates

We derive and test a new heuristic theory for third-order structure functions that resolves the forcing scale in the scenario of simultaneous spectral energy transfer to both small and large scales, which can occur naturally, for example, in rotating stratified turbulence or magnetohydrodynamical (MHD) turbulence. The theory has three parameters – namely the upscale/downscale energy transfer rates and the forcing scale – and it includes the classic inertial-range theories as local limits. When applied to measured data, our global-in-scale theory can deduce the energy transfer rates using the full range of data, therefore it has broader applications compared with the local theories, especially in situations where the data is imperfect. In addition, because of the resolution of forcing scales, the new theory can detect the scales of energy input, which was impossible before. We test our new theory with a two-dimensional simulation of MHD turbulence.

scholarly journals Shallow water wave turbulence

2019 ◽  
Vol 874 ◽  
pp. 1169-1196 ◽  
Author(s):  
Pierre Augier ◽  
Ashwin Vishnu Mohanan ◽  
Erik Lindborg
Keyword(s):  
Reynolds Number ◽  
Structure Function ◽  
Shallow Water ◽  
Water Wave ◽  
Structure Functions ◽  
Wave Turbulence ◽  
Order Structure ◽  
Third Order ◽  
Shallow Water Wave ◽  
The Mean

The dynamics of irrotational shallow water wave turbulence forced at large scales and dissipated at small scales is investigated. First, we derive the shallow water analogue of the ‘four-fifths law’ of Kolmogorov turbulence for a third-order structure function involving velocity and displacement increments. Using this relation and assuming that the flow is dominated by shocks, we develop a simple model predicting that the shock amplitude scales as $(\unicode[STIX]{x1D716}d)^{1/3}$, where $\unicode[STIX]{x1D716}$ is the mean dissipation rate and $d$ the mean distance between the shocks, and that the $p$th-order displacement and velocity structure functions scale as $(\unicode[STIX]{x1D716}d)^{p/3}r/d$, where $r$ is the separation. Then we carry out a series of forced simulations with resolutions up to $7680^{2}$, varying the Froude number, $F_{f}=(\unicode[STIX]{x1D716}L_{f})^{1/3}/c$, where $L_{f}$ is the forcing length scale and $c$ is the wave speed. In all simulations a stationary state is reached in which there is a constant spectral energy flux and equipartition between kinetic and potential energy in the constant flux range. The third-order structure function relation is satisfied with a high degree of accuracy. Mean energy is found to scale approximately as $E\sim \sqrt{\unicode[STIX]{x1D716}L_{f}c}$, and is also dependent on resolution, indicating that shallow water wave turbulence does not fit into the paradigm of a Richardson–Kolmogorov cascade. In all simulations shocks develop, displayed as long thin bands of negative divergence in flow visualisations. The mean distance between the shocks is found to scale as $d\sim F_{f}^{1/2}L_{f}$. Structure functions of second and higher order are found to scale in good agreement with the model. We conclude that in the weak limit, $F_{f}\rightarrow 0$, shocks will become denser and weaker and finally disappear for a finite Reynolds number. On the other hand, for a given $F_{f}$, no matter how small, shocks will prevail if the Reynolds number is sufficiently large.

Can the atmospheric kinetic energy spectrum be explained by two-dimensional turbulence?

1999 ◽  
Vol 388 ◽  
pp. 259-288 ◽  
Author(s):  
ERIK LINDBORG
Keyword(s):  
Energy Spectrum ◽  
Structure Function ◽  
Structure Functions ◽  
Separation Distance ◽  
Inertial Range ◽  
Order Structure ◽  
Two Dimensional ◽  
Third Order ◽  
Kinetic Energy Spectrum ◽  
The Third

The statistical features of turbulence can be studied either through spectral quantities, such as the kinetic energy spectrum, or through structure functions, which are statistical moments of the difference between velocities at two points separated by a variable distance. In this paper structure function relations for two-dimensional turbulence are derived and compared with calculations based on wind data from 5754 airplane flights, reported in the MOZAIC data set. For the third-order structure function two relations are derived, showing that this function is generally positive in the two-dimensional case, contrary to the three-dimensional case. In the energy inertial range the third-order structure function grows linearly with separation distance and in the enstrophy inertial range it grows cubically with separation distance. A Fourier analysis shows that the linear growth is a reflection of a constant negative spectral energy flux, and the cubic growth is a reflection of a constant positive spectral enstrophy flux. Various relations between second-order structure functions and spectral quantities are also derived. The measured second-order structure functions can be divided into two different types of terms, one of the form r2/3, giving a k−5/3-range and another, including a logarithmic dependence, giving a k−3-range in the energy spectrum. The structure functions agree better with the two-dimensional isotropic relation for larger separations than for smaller separations. The flatness factor is found to grow very fast for separations of the order of some kilometres. The third-order structure function is accurately measured in the interval [30, 300] km and is found to be positive. The average enstrophy flux is measured as Πω≈1.8×10−13 s−3 and the constant in the k−3-law is measured as [Kscr ]≈0.19. It is argued that the k−3-range can be explained by two-dimensional turbulence and can be interpreted as an enstrophy inertial range, while the k−5/3-range can probably not be explained by two-dimensional turbulence and should not be interpreted as a two-dimensional energy inertial range.

scholarly journals Characterizing sand ripples at equilibrium phases

2013 ◽  
Vol 61 (4) ◽  
pp. 293-298 ◽  
Author(s):  
Jie Qin ◽  
Deyu Zhong ◽  
Guangqian Wang
Keyword(s):  
Time Series ◽  
Random Fields ◽  
Morphological Characteristics ◽  
Structure Functions ◽  
Second Order ◽  
Order Structure ◽  
Two Dimensional ◽  
Sand Ripples ◽  
Digital Elevation ◽  
Equilibrium Phases

Abstract Morphological characteristics of ripples are analyzed considering bed surfaces as two dimensional random fields of bed elevations. Two equilibrium phases are analyzed with respect to successive development of ripples based on digital elevation models. The key findings relate to the shape of the two dimensional second-order structure functions and multiscaling behavior revealed by higher-order structure functions. Our results suggest that (1) the two dimensional second-order structure functions can be used to differentiate the two equilibrium phases of ripples; and (2) in contrast to the elevational time series of ripples that exhibit significant multiscaling behavior, the DEMs of ripples at both equilibrium phases do not exhibit multiscaling behavior.

A note on Kolmogorov's third-order structure-function law, the local isotropy hypothesis and the pressure–velocity correlation

1996 ◽  
Vol 326 ◽  
pp. 343-356 ◽  
Author(s):  
Erik Lindborg
Keyword(s):  
Structure Function ◽  
Strain Tensor ◽  
Inertial Range ◽  
Order Structure ◽  
Two Dimensional ◽  
Mean Flow ◽  
Velocity Correlation ◽  
Third Order ◽  
The Third ◽  
Point Pressure

We show that Kolmogorov's (1941b) inertial-range law for the third-order structure function can be derived from a dynamical equation including pressure terms and mean flow gradient terms. A new inertial-range law, relating the two-point pressure–velocity correlation to the single-point pressure–strain tensor, is also derived. This law shows that the two-point pressure–velocity correlation, just like the third-order structure function, grows linearly with the separation distance in the inertial range. The physical meaning of both this law and Kolmogorov's law is illustrated by a Fourier analysis. An inertial-range law is also derived for the third-order velocity–enstrophy structure function of two-dimensional turbulence. It is suggested that the second-order vorticity structure function of two-dimensional turbulence is constant and scales with$\epsilon ^{2/3}_\omega$in the enstrophy inertial range, εωbeing the enstrophy dissipation. Owing to the constancy of this law, it does not imply a Fourier-space inertial-range law, and therefore it is not equivalent to thek−1law for the enstrophy spectrum, suggested by Kraichnan (1967) and Batchelor (1969).

scholarly journals Kappa Distributions and Isotropic Turbulence

Entropy ◽  
2019 ◽  
Vol 21 (11) ◽  
pp. 1093 ◽  
Author(s):  
Elias Gravanis ◽  
Evangelos Akylas ◽  
Constantinos Panagiotou ◽  
George Livadiotis
Keyword(s):  
Minimal Model ◽  
Model Comparison ◽  
Isotropic Turbulence ◽  
Order Structure ◽  
Kappa Distribution ◽  
Symmetric Part ◽  
Kappa Index ◽  
Third Order ◽  
Symmetry Properties ◽  
Temperature Parameter

In this work, the two-point probability density function (PDF) for the velocity field of isotropic turbulence is modeled using the kappa distribution and the concept of superstatistics. The PDF consists of a symmetric and an anti-symmetric part, whose symmetry properties follow from the reflection symmetry of isotropic turbulence, and the associated non-trivial conditions are established. The symmetric part is modeled by the kappa distribution. The anti-symmetric part, constructed in the context of superstatistics, is a novel function whose simplest form (called “the minimal model”) is solely dictated by the symmetry conditions. We obtain that the ensemble of eddies of size up to a given length r has a temperature parameter given by the second order structure function and a kappa-index related to the second and the third order structure functions. The latter relationship depends on the inverse temperature parameter (gamma) distribution of the superstatistics and it is not specific to the minimal model. Comparison with data from direct numerical simulations (DNS) of turbulence shows that our model is applicable within the dissipation subrange of scales. Also, the derived PDF of the velocity gradient shows excellent agreement with the DNS in six orders of magnitude. Future developments, in the context of superstatistics, are also discussed.

Structure functions of turbulence in the atmospheric boundary layer over the ocean

1970 ◽  
Vol 44 (1) ◽  
pp. 145-159 ◽  
Author(s):  
C. W. Van Atta ◽  
W. Y. Chen
Keyword(s):  
Boundary Layer ◽  
Atmospheric Boundary Layer ◽  
Fourth Order ◽  
Structure Functions ◽  
Inertial Subrange ◽  
Order Structure ◽  
Third Order ◽  
Wide Range ◽  
Theoretical Predictions ◽  
Order Quantities

Structure functions of turbulent velocity fluctuations up to fourth order have been measured at several heights in the atmospheric boundary layer over the open ocean, and the results are compared with theoretical predictions for separations in the inertial subrange. The behaviour of second- and third-order quantities shows substantial agreement with the predictions of Kolmogorov's original theory over a wide range of separations, but the results of a recent modification of the theory, attempting to account for intermittency in the local dissipation rate, are also consistent with the data over somewhat shorter separation intervals. The behaviour of the measured fourth-order structure function disagrees with that predicted from Kolmogorov's original work, but good agreement is found with the results of the modified theory.

Applicability of Kolmogorov's and Monin's equations of turbulence

1997 ◽  
Vol 353 ◽  
pp. 67-81 ◽  
Author(s):  
REGINALD J. HILL
Keyword(s):  
Structure Function ◽  
Structure Functions ◽  
Inertial Range ◽  
Homogeneous Turbulence ◽  
Order Structure ◽  
Third Order ◽  
Local Isotropy ◽  
The Third ◽  
Moderate Reynolds Number ◽  
Wide Range

The equation relating second- and third-order velocity structure functions was presented by Kolmogorov; Monin attempted to derive that equation on the basis of local isotropy. Recently, concerns have been raised to the effect that Kolmogorov's equation and an ancillary incompressibility condition governing the third-order structure function were proven only on the restrictive basis of isotropy and that the statistic involving pressure that appears in the derivation of Kolmogorov's equation might not vanish on the basis of local isotropy. These concerns are resolved. In so doing, results are obtained for the second- and third-order statistics on the basis of local homogeneity without use of local isotropy. These results are applicable to future studies of the approach toward local isotropy. Accuracy of Kolmogorov's equation is shown to be more sensitive to anisotropy of the third-order structure function than to anisotropy of the second-order structure function. Kolmogorov's 4/5 law for the inertial range of the third-order structure function is obtained without use of the incompressibility conditions on the second- and third-order structure functions. A generalization of Kolmogorov's 4/5 law, which applies to the inertial range of locally homogeneous turbulence at very large Reynolds numbers, is shown to also apply to the energy-containing range for the more restrictive case of stationary, homogeneous turbulence. The variety of derivations of Kolmogorov's and Monin's equations leads to a wide range of applicability to experimental conditions, including, in some cases, turbulence of moderate Reynolds number.

Exact relationship for third-order structure functions in helical flows

2000 ◽  
Vol 61 (5) ◽  
pp. 5321-5325 ◽  
Author(s):  
T. Gomez ◽  
H. Politano ◽  
A. Pouquet
Keyword(s):  
Structure Functions ◽  
Order Structure ◽  
Third Order ◽  
Exact Relationship
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