Session GB: Turbulence: Fundamentals I

The bottleneck effect and the Kolmogorov constant in three-dimensional turbulence

A large database generated from direct numerical simulations (DNS) of isotropic turbulence, including recent simulations at up to $4096^3$ resolution and Taylor microscale Reynolds numbers of up to about $1000$, is used to explore the bottleneck effect in three-dimensional energy spectrum and in second-order structure functions, and to determine the Kolmogorov constant, $C_K$. The difficulties in estimating $C_K$ at any finite Reynolds number are examined. Our data from well-resolved simulations show that the bottleneck effect decreases with the Reynolds number and that its behavior is independent of the nature of the forcing scheme and is insensitive to small-scale resolution. This trend is seen in both spectral and physical spaces, though the effect is less noticeable in the latter. An alternative to the usual procedure for determining $C_K$ is suggested. The proposal does not depend on a particular choice of fitting ranges or power-law behavior in the inertial range. Within the resolution of the numerical data, $C_K$ thus determined is constant in the Reynolds number. A simple model including non-local energy transfer is proposed to reproduce the observed scaling. Further implications of the findings are discussed.   

Split energy cascade in turbulent flows

Hydrodynamic turbulence exhibits a remarkable dependence on the space dimension. This property manifests for instance in the direction of the kinetic-energy cascade, which is direct in 3D and inverse in 2D. A passive scalar transported by a turbulent flow shows an analogous behavior. In isotropic flows, the variance of the scalar field cascades either downwards or upwards depending on the dimension, the degree of compressibility, and the scaling exponent of the carrier flow. We undertake a geometrical approach to investigate the dependence of turbulence and turbulent transport on the space dimensionality. We first consider a system that is fully under analytical control, i.e. a passive scalar transported by a Gaussian short-correlated flow on a cylindrical surface, where the radius can be inflated or collapsed at will. For any finite radius, the variance cascade splits into a direct branch and an inverse one. This behavior is intimately connected to the existence of a non-degenerate invariant measure for the fluid-particle separations. Direct numerical simulations of the Navier-Stokes equations show that also the kinetic-energy cascade splits when the aspect ratio of the flow is less than one.   

Controlling the Dual Cascade of Two-dimensional Turbulence

It has been established that monoscale forcing cannot produce the dual cascades of energy and enstrophy with the scaling laws predicted by Kraichnan--Leith--Batchelor (KLB) theory. However, we have been able to find forcings which do produce the KLB scalings: $E(k)\propto k^{-5/3}$ for the inverse energy cascade and $E(k)\propto k^{-3}$ for the forward enstrophy cascade. We find these forcings using a novel adjoint-equation-based optimal control technique. First, the control problem is formulated and a method for controlling the energy spectrum of solutions of the incompressible Navier--Stokes equations is proposed. The control process is validated by several test cases. Then, this control method is applied to a pseudo-spectral numerical computation of the 2-D incompressible Navier-Stokes equations with doubly periodic boundary conditions in order to find the forcing that reproduces the scaling laws of KLB theory. Finally, we demonstrate that the flows we obtain are indeed dynamically active by measuring directly the energy and enstrophy fluxes. We also compare our forcing and the resulting turbulence with results obtained using a linear forcing recently proposed by Lundgren (2003). The results presented here show that the choice of forcing can fundamentally alter the dynamics and spectral properties of the turbulence, and that the theoretically attractive choice of band-width limited forcing is actually inconsistent with KLB theory.   

Forward/Inverse Energy Cascade in 2D and QG Turbulence

We perform numerical simulations to study two-dimensional and quasi-geostrophic turbulence. In all runs, small scale forcing injects energy at wave number, $ k_f $, and the inverse energy cascade is halted at large scale by linear drag. A new decomposition of spectral energy flux into aggregated transfer function cascading up and down (hereafter ATFu and ATFd) is introduced instead of Kraichnan's classical approach. Both functions are positive, monotonically increasing in spectral space and have the same power-law dependency in the energy inertial range. Amazingly, the ATFd has discontinuity at $ k_f $ and the jump equals to energy injection rate $ \dot{E}_{in} $ while the ATFu is always continuous. This implies that the energy injected is transferred first to small scale and then cascade inversely but not directly to large scale. The QG turbulence resembles the 2D turbulence if $ \beta $ effect is too weak to create zonal flow. If zonal jets are spotted, the exponent of the power-law dependency and the magnitude of ATFs are smaller compared to that in 2D turbulence indicates that $ \beta $ inhibits the inverse energy cascade. The properties of ATFs are strongly dependent on $ \dot{E}_{in} $ and the drag loss but not $ k_f $.   

Inverse Cascades and Zonal Flows on a Beta Plane

We examine the role of forward and inverse cascades in 2D turbulence in creating zonal jets on a $\beta$-plane. The magnitude of the characteristic velocity and the characteristic width of a zonal jet are set by the balance of the two cascades. The widths of the jets are strongly dependent on the value of the local Rossby deformation radius $L_R$. Kinetic energy is dominated by potential energy at length scales greater than $2 \pi L_R$. We find that little energy inverse cascades to scales greater than $2 \pi L_R$, and there is a break in the slope of the kinetic energy spectrum at that scale. Forcing at small scales produces large-scale zonal flows that resemble the widths, but not the magnitudes, of jet streams of Jupiter and Saturn. The magnitudes of the large-scale velocities of the computed zonal flows are much smaller than on Jupiter or Saturn. The transfer of energy from small scales to large scales involves many more wave number triads on an $f$-plane than on a $\beta$- plane, so the equilibrium energy of the large scale zonal flows is determined by only a few triads.   

What is turbulence and which way does it cascade?

Turbulence is defined as an eddy-like state of fluid motion where the inertial vortex forces of the eddies are larger than any of the other forces that tend to damp the eddies out. Inertial vortex forces vxw are zero for irrotational flows, so irrotational flows are not turbulent by definition even though they may be random and induced by turbulence. Because the vorticity w is always produced at small scales, turbulence always cascades from small scales to large. Turbulence growth is limited by vertical buoyancy forces at the Ozmidov scale of fossilization and by horizontal Coriolis forces at the Hopfinger scale of fossilization. Fossil turbulence is defined as a perturbation in any hydrophysical field produced by turbulence that persists after the fluid is no longer turbulent at the scale of the perturbation. Most turbulent mixing in the ocean and atmosphere occurs in fossil turbulence patches where most of the turbulent kinetic energy of the patch has radiated near vertically as fossil turbulence waves. Vertical heat, mass, momentum and information transport in the ocean is dominated by an intermittent generic process termed beamed zombie turbulence maser action mixing chimneys (see http: //maeresearch.ucsd.edu /\~{}cgibson /Documents2007 /GibsonBB08Nov26\_Alist.htm).   

Scale-locality of the energy cascade in turbulence using Fourier Analysis

We investigate the scale-locality of non-linear interactions which drive the energy cascade in a turbulent flow. The main picture that emerges from our work is that the primary participants in the cascade process are triplets of ``eddies'' comprised of adjacent \emph{logarithmic bands} of Fourier modes. We disprove in particular an alternate picture of ``local transfer by nonlocal triads'' by showing that such triads, due to their restricted number, make a vanishingly small contribution to the energy flux in the inertial range. We rigorously prove that it is only the aggregate effect of a geometrically increasing number of local wavenumber triads which can sustain the energy cascade to small scales. Our analysis shows that the SGS definition of the flux is the proper measure of the cascading energy and that the sharp spectral filter has a firm theoretical basis for use in LES modeling. It also demonstrates the danger in the widespread notion that the elementary interactions in turbulence are those involving triads of single Fourier modes. We support our results with numerical data from a $512^3$ pseudo-spectral simulation of isotropic turbulence with phase-shift dealiasing.   

Quantifying the locality of nonlinear interactions in MHD turbulence

The locality functions introduced by Kraichnan give the fraction of the energy flux across a given cutoff wavenumber $k_c$ that is due to nonlinear interactions with wavenumbers $k$ smaller than the cutoff (the infrared locality function) or greater than the cutoff (the ultraviolet locality function). Previous analysis of DNS data for hydrodynamic turbulence confirmed the theoretical scaling exponent of $n=4/3$ in the wavenumber ratio and in the limit of the infinite inertial range. We have extended the analysis to DNS data for MHD turbulence. Out of four nonlinear terms contributing to the energy transfer, two dominant ones, $b \cdot \nabla b$ and $b \cdot \nabla u$, lead to the locality functions that exhibit behavior that can be characterized by scaling exponents in the infrared. The extend of the inertial range is insufficient to determine the exponents uniquely but the data are indicative of values between 1/2 and 2/3, i.e., much less than for hydrodynamic turbulence. Therefore, the nonlinear energy transfer is significantly more nonlocal in MHD turbulence, with potential implications for theory and modeling.   

Energy flux in non-equilibrium energy spectra in steady turbulence

The energy spectrum $E(k)$ and energy flux function $\Pi(k)$ in non-equilibrium state are obtained using the spectral energy equation based on the Kovasznay, Leith diffusion and Heisenberg hypothesis. The derived models are assessed using the DNS data for forced homogeneous isotropic turbulence. Three different forcing schemes are used and compared. In all these forcing schemes, the base spectrum obeys the Kolmogorov law $E(k) \propto k^{-5/3}$, and $\Pi(k)=$ const, but the temporal development of the the deviatoric spectrum and flux is divided into the three phases. In the period in which $d\varepsilon/dt \equiv \dot{\varepsilon} > 0$, $E(k) \propto k^{-7/3}$ and $\Pi (k) \propto k^{-2/3}$ in the inertial subrange (Phase 1), while $E(k) \propto -k^{-7/3}$ and $\Pi(k) \propto -k^{-2/3}$ when $\dot{\varepsilon} < 0$ (Phase 2), where $\varepsilon$ is the dissipation rate. In the transient period between Phase 1 and Phase 2, $\dot{\varepsilon} \approx 0$ and $\ddot{\varepsilon}$ is large, and $E(k) \propto k^{-9/3}$ and $\Pi(k) \propto k^{- 4/3}$ (Phase T). On average, the deviatric spectrum induces the forward scatter of the energy into the small scale in Phase 1, and the backward scatter of the small scale energy into the large scale in Phase 2. These results are overall consistent with the prediction obtained using the closure models, but the eddy-viscosity Heisenberg model does not yields $E(k) \propto k^ {-9/3}$ in Phase T. Due to the effect of the intermittency, the energy spectrum and flux exhibit a slight deviation in the exponents.