Lagrangian evolution of velocity increments in rotating turbulence: The effects of rotation on non-Gaussian statistics
Introduction
Rotating turbulence plays an important role in many different areas, including geophysical, astrophysical and engineering applications. In rotating turbulence, the effects of rotation enter through the Coriolis force [1], [2]. The relative importance of the Coriolis force in homogeneous turbulence is qualitatively measured by the Rossby number, defined as the ratio of the nonlinear advection term to the Coriolis force. There has been a continuous effort to understand the effects of the Coriolis force that have led to the peculiar features of rotating turbulence. It is known that in rotating turbulence the energy transfer in Fourier space is weakened by the phase-scrambling effects generated by the inertial waves [3], [4]. As a consequence, a steeper energy spectrum is observed in simulations and experiments [5], [6], [7], [8], [9], which is also predicted by phenomenological and analytical models [10], [11]. On the other hand, it is argued that the nonlinear interaction between resonant waves is largely responsible for the generation of coherent columnar vortex structures, the tendency towards two-dimensionalization, inverse energy cascade, and a number of other phenomena [12], [13], [6], [14], [8], [15], although recently it has been shown that linear mechanisms may also make important contributions [16], [17].
For non-rotating turbulence, the small-scale structures of turbulence have received considerable attention. Measured by velocity increments and velocity gradients, the statistics at small scales have been shown to be highly non-Gaussian (see, e.g., [18]). The non-Gaussian statistics are generated by frequent, intense fluctuations in small-scale quantities, which presents great obstacles to the efforts to develop universal models. Similar approaches have been adopted in the study of the small-scale structure of rotating turbulence recently. The statistics of velocity gradients have been studied in [19]. It is observed that, generally, the statistics tend to become more Gaussian in rotating turbulence. A phenomenon that has received considerable attention is the observation that the vertical vorticity component displays positive skewness, which takes the maximum at some intermediate value of the Rossby number [13], [15], [20]. Visually, the observation is related to the prevalence of cyclonic vortices in rotating turbulent fields. It is observed that the maximum in the skewness coincides with the maximum in the three-dimensional to two-dimensional energy transfer [15]. The phenomenon has been attributed to the instability of anti-cyclonic vortices in [13]. On the other hand, [21] shows that the initial growth of the skewness is proportional to the product of the rotation rate of the frame of reference and the mean vortex stretching. Since the mean vortex stretching is positive in an isotropic turbulence, this will lead to an algebraic growth in the skewness when rotation is imposed. Using data generated by direct numerical simulation (DNS), [20] studies the problem in great detail and concludes that the stationary value of the skewness is affected by a number of other parameters. The properties of velocity increments have also been documented in the experimental and/or DNS studies reported in [22], [19], [9], [23], [24], [25]. The scaling law of the velocity increments is measured, showing reduced anomalous scaling. The skewness of the longitudinal velocity increments is also observed to be weakened by rotation. Several phenomenological models have been proposed to explain the observations regarding the scaling law in rotating turbulence [22], [25].
Thus, it appears that there is not yet a consensus as to the mechanisms of some observations regarding the non-Gaussian statistics in velocity increments and velocity gradients. In particular, an understanding based on the dynamics of the governing Navier–Stokes (NS) equations is desirable. In this paper, we intend to provide a partial yet unified explanation for a number of observations via a simple dynamical model. To provide the background for the model, we note that it is closely related to recent research on the so-called restricted Euler approximation and several models for the small-scale dynamics of turbulence. In the restricted Euler (RE) approximation, the equation for the velocity gradient is truncated, and only the nonlinear term and the isotropic part of the pressure Hessian are retained [26], [27]. The velocity gradient predicted from the RE approximation develops a finite time singularity. However, the tensorial structure of the gradient reproduces a number of important features observed in turbulence, such as the preferential alignment between the vorticity vector and the intermediate eigendirection of the strain rate tensor [26], [27], [28], [29]. Thus, the RE approximation has been used as a base model to understand the small-scale turbulence. A number of models for the pressure Hessian have been proposed to regularize the approximation. A useful idea is to follow the Lagrangian evolution of material elements, which has been pursued in [30], [31], [32], [33] (see also [34] for a recent model). The ideas are adopted to study the evolution of velocity increments in [35]. A simple dynamical model for the velocity increments is derived by following the Lagrangian evolution of a linear element [35]. The model is generalized to turbulence in two and four spatial dimensions, and to include the increments of passive scalars in [36]. These models reproduce quite a few important observations regarding the non-Gaussian statistics of the increments, and thus have helped clarify the origins of the observations from a dynamical point of view. It is also predicted that the increments of a passive scalar [36] are more intermittent in four spatial dimensions (compared with three spatial dimensions). In this paper, we applied the ideas to study the evolution of the non-Gaussian statistics of velocity increments in rotating turbulence. In order to incorporate the Coriolis force, a local coordinate system attached to an evolving material line is introduced. We show that, with the help of the coordinate system, a system of equations for the velocity increments over a fixed distance on the material line can be derived. The analysis of a restricted-Euler-type approximation of the system shows that several features of rotating turbulence can be reproduced, which thus provides explanations for some of the observations from a dynamical perspective.
The paper is organized as follows. In Section 2, the system of the equations is derived, and an a priori analysis is conducted. The numerical solution of the system is presented in Section 3. Conclusions are summarized in Section 4.
Section snippets
Derivation of the equations and a priori tests
Following the idea of [35], [36], we keep track of a line element and consider the velocity increments over a fixed distance along the direction of the element , where is the length of the line element. In [35], [36], where non-rotating turbulence is considered, a system of two equations for the longitudinal and transverse velocity increments over is derived. In rotating turbulence, however, we first need to define a local Cartesian coordinate system in order to better
Numerical results and discussions
In this section, we study the evolution of the velocity increments predicted by the model, starting from initial Gaussian random condition. , and are initialized as Gaussian random numbers with zero mean and unit variance, so that the velocity scale is . Initially the material line elements point to different directions with equal probabilities, so that is uniformly distributed in . At any time , statistics are accumulated from the evolving ensemble. Eqs. (9), (14), (15),
Conclusions
To summarize, we have derived a system of equations describing the evolution of velocity increments over a fixed distance on an evolving material element in rotating turbulence. The system extends the ideas presented in previous papers [35], [36] to include the effects of rotation. To do so, a Lagrangian local coordinate frame is defined. As a result, the nonlinear interaction between the increments as well as the Coriolis force is closed in the system, which thus allows detailed investigations
Acknowledgements
The author wishes to thank L. Chevillard, G.L. Eyink, C. Meneveau, and K. Ohkitani for helpful discussions.
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