On Kolmogorov’s Third Order Structure Function Law, the Local Isotropy Hypothesis and the Pressure-Velocity Correlation

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).

Download Full-text

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

Journal of Fluid Mechanics ◽

10.1017/s0022112097007362 ◽

1997 ◽

Vol 353 ◽

pp. 67-81 ◽

Cited By ~ 55

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.

Download Full-text

Exact second-order structure-function relationships

Journal of Fluid Mechanics ◽

10.1017/s0022112002001696 ◽

2002 ◽

Vol 468 ◽

pp. 317-326 ◽

Cited By ~ 78

Author(s):

REGINALD J. HILL

Keyword(s):

Structure Function ◽

Scaling Laws ◽

Velocity Structure ◽

Structure Functions ◽

Energy Dissipation Rate ◽

Second Order ◽

Order Structure ◽

Third Order ◽

Navier Stokes Equation ◽

Local Isotropy

Equations that follow from the Navier–Stokes equation and incompressibility but with no other approximations are ‘exact’. Exact equations relating second- and third- order structure functions are studied, as is an exact incompressibility condition on the second-order velocity structure function. Opportunities for investigations using these equations are discussed. Precisely defined averaging operations are required to obtain exact averaged equations. Ensemble, temporal and spatial averages are all considered because they produce different statistical equations and because they apply to theoretical purposes, experiment and numerical simulation of turbulence. Particularly simple exact equations are obtained for the following cases: (i) the trace of the structure functions, (ii) DNS that has periodic boundary conditions, and (iii) an average over a sphere in r-space. Case (iii) introduces the average over orientations of r into the structure-function equations. The energy dissipation rate ε appears in the exact trace equation without averaging, whereas in previous formulations ε appears after averaging and use of local isotropy. The trace mitigates the effect of anisotropy in the equations, thereby revealing that the trace of the third-order structure function is expected to be superior for quantifying asymptotic scaling laws. The orientation average has the same property.

Download Full-text

Shallow water wave turbulence

Journal of Fluid Mechanics ◽

10.1017/jfm.2019.375 ◽

2019 ◽

Vol 874 ◽

pp. 1169-1196 ◽

Cited By ~ 1

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.

Download Full-text

Third-order structure function relations for quasi-geostrophic turbulence

Journal of Fluid Mechanics ◽

10.1017/s0022112006003697 ◽

2007 ◽

Vol 572 ◽

pp. 255-260 ◽

Cited By ~ 20

Author(s):

ERIK LINDBORG

Keyword(s):

Structure Function ◽

Three Dimensional ◽

Structure Functions ◽

Order Structure ◽

Third Order ◽

Longitudinal Structure ◽

The Third ◽

Inverse Cascade ◽

Geostrophic Turbulence ◽

Potential Enstrophy

We derive two third-order structure function relations for quasi-geostrophic turbulence, one for the forward cascade of potential enstrophy and one for the inverse cascade of energy. These relations are the counterparts of Kolmovorov's (1941) four-fifths law for the third-order longitudinal structure functions of three-dimensional turbulence.

Download Full-text

Impacts of Convergence on Structure Functions from Surface Drifters in the Gulf of Mexico

Journal of Physical Oceanography ◽

10.1175/jpo-d-18-0029.1 ◽

2019 ◽

Vol 49 (3) ◽

pp. 675-690 ◽

Cited By ~ 11

Author(s):

Jenna Pearson ◽

Baylor Fox-Kemper ◽

Roy Barkan ◽

Jun Choi ◽

Annalisa Bracco ◽

...

Keyword(s):

Gulf Of Mexico ◽

Structure Function ◽

Structure Functions ◽

Second Order ◽

Order Structure ◽

Third Order ◽

Second Order Statistics ◽

Surface Drifter ◽

Sign Change ◽

Surface Drifters

AbstractThere are limitations in approximating Eulerian statistics from surface drifters, due to biases from surface convergences. By contrasting second- and third-order Eulerian and surface drifter structure functions obtained from a model of the Gulf of Mexico, the consequences of the semi-Lagrangian nature of observations during the summer Grand Lagrangian Deployment (GLAD) and winter Lagrangian Submesoscale Experiment (LASER) are estimated. By varying launch pattern and location, the robustness and sensitivity of these statistics are evaluated. Over scales less than 10 km, second-order structure functions of surface drifters consistently have shallower slopes (~r2/3) than Eulerian statistics (~r), suggesting that surface drifter structure functions differ systematically and do not reproduce the scalings of the Eulerian fields. Medians of Eulerian and cluster release second-order statistics are also significantly different across all scales. Synthetic cluster release statistics depend on launch location and weakly on launch pattern. The observations suggest little seasonal difference in the second-order statistics, but the LASER third-order structure function shows a sign change around 1 km, while GLAD and the synthetic cluster releases show a third-order structure function sign change around 10 km. Further, synthetic surface drifter cluster releases (and therefore likely the GLAD observations) show robust biases in the negative third-order structure functions, which may lead to significant overestimation of the spectral energy flux and underestimation of the transition scale to a forward energy cascade. The Helmholtz decomposition, and curl and divergence statistics, of Eulerian and cluster releases differ, particularly on scales less than 10 km, in agreement with observations of drifters preferentially sampling convergences in coherent structures.

Download Full-text

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

Journal of Fluid Mechanics ◽

10.1017/s0022112099004851 ◽

1999 ◽

Vol 388 ◽

pp. 259-288 ◽

Cited By ~ 267

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.

Download Full-text

On Yaglom’s Law for the Interplanetary Proton Density and Temperature Fluctuations in Solar Wind Turbulence

Entropy ◽

10.3390/e22121419 ◽

2020 ◽

Vol 22 (12) ◽

pp. 1419

Author(s):

Giuseppe Consolini ◽

Tommaso Alberti ◽

Vincenzo Carbone

Keyword(s):

Solar Wind ◽

Structure Function ◽

Interplanetary Space ◽

Slow Solar Wind ◽

Order Structure ◽

Proton Density ◽

Linear Scaling ◽

Fast Solar Wind ◽

Third Order ◽

Passive Scalars

In the past decades, there has been an increasing literature on the presence of an inertial energy cascade in interplanetary space plasma, being interpreted as the signature of Magnetohydrodynamic turbulence (MHD) for both fields and passive scalars. Here, we investigate the passive scalar nature of the solar wind proton density and temperature by looking for scaling features in the mixed-scalar third-order structure functions using measurements on-board the Ulysses spacecraft during two different periods, i.e., an equatorial slow solar wind and a high-latitude fast solar wind, respectively. We find a linear scaling of the mixed third-order structure function as predicted by Yaglom’s law for passive scalars in the case of slow solar wind, while the results for fast solar wind suggest that the mixed fourth-order structure function displays a linear scaling. A simple empirical explanation of the observed difference is proposed and discussed.

Download Full-text