32nd SIAM Southeastern-Atlantic Section Conference (SIAM-SEAS 2008)
March 14-15 2008, University of Central Florida, Orlando.

Special Session: Turbulence
organized by Eleftherios Gkioulekas


Turbulence is often referred to as the last open problem of classical mechanics. It is a problem of both fundamental importance and with a wide range of applications, in earth and atmospheric science, plasma physics, aerospace engineering, and many other areas. Despite nearly a century of efforts by the best minds, many questions remain open, and considerable progress has only been achieved for idealized situations, such as homogeneous and isotropic turbulence. The goal of the session is to bring together specialists from various subfields of turbulence research, so that ideas can be exchanged accross different subfields.


Below is a list of speakers that have agreed to attend, as of today. We can accomodate another 5 speakers. If you would like to attend please e-mail me the title and abstract of your talk.


Speaker: Brian Arbic
Title: Global energy dissipation rate of oceanic low-frequency flows by quadratic bottom drag: Results from observations and 1/32 degree models
Abstract: The energy budget of the deep ocean is a subject of great current interest, because of the potential implications for the strength of the overturning circulation. An important energy source is the 1 TW wind power input into geostrophic (low-frequency) flows. How these flows dissipate is largely unknown. Here we discuss one potential dissipation mechanism, namely bottom boundary layer drag. First, we show that eddy kinetic energy in idealized geostrophic turbulence models has realistic length scales, vertical structure, and amplitudes only if the nondimensional bottom friction strength is of order one. This is true for both linear and quadratic bottom drag (Arbic and Scott, in press). Next, we estimate the global energy dissipation rate of oceanic low-frequency flows by quadratic bottom drag (Sen et al., in review). We use data from 290 moored near-bottom current meters. We also utilize satellite altimetry data, 1) to estimate the bias in the global integral due to the poor sampling of the ocean by current meters, and 2) to estimate global maps of bottom velocity, given relationships computed at the mooring locations between surface and bottom flows. Finally, the dissipation rate is estimated from the output of a global ocean model run at 1/32 degree resolution (Arbic et al., in review). The model is validated by comparison to the current meter dataset. Our estimates of the dissipation rate range from 0.14 to 0.83 TW, with both the low and high ends likely being unrealistic. While uncertainty remains, we can say that bottom boundary layers very likely dissipate a substantial fraction of the 1 TW wind-power input into geostrophic flows.
Speaker: Bhimsen Shivamoggi
Title: Vortex Stretching in a Compressible Fluid and Applications to Turbulence:
Abstract: Stretching of axisymmetric vortices in a compressible fluid is considered. The flow associated with the vortex is perpendicular to the plane of the uniform straining flows. Compressibility effects are considered to be weak to facilitate an analytic solution. Applications to compressible turbulence are briefly considered.
Speaker: Francesco Fedele
Title: Rogue waves in oceanic turbulence and solitons in axisymmetric turbulent flow: an extreme view
Abstract: In the first part of the talk, a stochastic model of wave groups is presented to explain the formation of abnormal waves ( rogue waves) in oceanic turbulence. The latter state is defined as the chaotic state of a sea of weakly nonlinear coupled dispersive waves in evolution according to the Zakharov equation. Finally, the stochastic wave group theory is extended to characterize the nonlinear dynamics of disturbances in axisymmetric Poiseuille pipe turbulence. For large Reynolds numbers Re, it is shown that for disturbance's amplitudes of O(Re^{-3/2}), the latter state is characterized by the chaotic behavior of a sea of weakly nonlinear coupled cnoidal solitons obeying a non-Hamiltonian system of perturbed Korteweg-de Vries Equations.
Speaker: Mogens Melander
Title: Analysis of a Symmetry leading to an Inertial Range Similarity Theory for Isotropic Turbulence.
Abstract: PDF file
Speaker: John Bowman
Title: Casimir Cascades in Two Dimensional Turbulence
Abstract: The Kraichnan-Leith-Batchelor theory of two-dimensional turbulence is based on the fact that the nonlinear terms of the two-dimensional Navier-Stokes equation conserve both energy and enstrophy. In an infinite domain and in the limit of infinite Reynolds number, the net energy and enstrophy transfers out of a low-wavenumber forcing region must consequently be independent of wavenumber. The resulting dual cascade of energy to larger scales and enstrophy to smaller scales is readily observed in numerical simulations of two-dimensional turbulence in a finite domain.

While it is well known that the nonlinearity also conserves the global integral of any arbitrary C1 function of the scalar vorticity field, the direction of transfer of these quantities in wavenumber space remains unclear. Numerical investigations of this problem are hampered by the fact that pseudospectral simulations, which necessarily truncate the wavenumber domain, do not conserve these higher-order Casimir invariants. In this work we develop estimates for the degree of nonconservation of the Casimir invariants and demonstrate that with sufficiently well-resolved simulations, their cascade directions can in fact be numerically determined.

Speaker: Eleftherios Gkioulekas
Title: Locality and stability of the cascades of two-dimensional turbulence
Abstract: In my talk, I will discuss the notion of locality as it pertains to the cascades of two-dimensional turbulence. The mathematical framework underlying our analysis is the infinite system of balance equations that govern the generalized unfused structure functions, first introduced by L'vov and Procaccia. As a point of departure we use a revised version of the system of hypotheses that was proposed by Frisch for three-dimensional turbulence. We show that both the enstrophy cascade and the inverse energy cascade are local in the sense of non-perturbative statistical locality. We also investigate the stability conditions for both cascades. We have shown that statistical stability with respect to forcing applies unconditionally for the inverse energy cascade. For the enstrophy cascade, statistical stability requires large-scale dissipation and a vanishing downscale energy dissipation. I will conclude with a careful discussion of the subtle notion of locality.
Speaker: Minping Wan
Title: Physical mechanism of the inverse energy cascade of 2D turbulence: a numerical investigation
Abstract: We report an investigation of inverse energy cascade in steady-state 2D turbulence by direct numerical simulation of the 2D N-S equation, with small-scale forcing and large-scale damping. We employed several types of damping and dissipation mechanisms in simulations up to 20482 resolution. For all these simulations we obtained a wavenumber range for which the mean spectral energy flux is a negative constant and the energy spectrum scales as k^(-5/3), consistent with the predictions of Kraichnan (1967). To gain further insight, we investigated the energy cascade in physical space, employing a local energy flux defined by smooth filtering. We found that the inverse energy cascade is scale-local, but that the strongly local contribution vanishes identically. The mean flux across a length scale l was shown to be due mainly to interactions with modes 2 ~ 8 times smaller. A major part of our investigation was devoted to identifying the physical mechanism of the 2D inverse energy cascade. We made a quantitative study employing a precise topological criterion of merger events. Our statistical analysis showed that vortex mergers play a negligible direct role in producing mean inverse energy flux. Instead, we obtained considerable evidence in favor of a “vortex-thinning” mechanism, according to which the large-scale strains do negative work against turbulent stress as they stretch out the isolines of small-scale vorticity. In particular, we studied a Multi-Scale Gradient (MSG) expansion for the turbulent stress, whose contributions to inverse cascade can all be explained by “thinning.” The MSG expansion up to second-order in space gradients was found to predict very well the magnitude, spatial structure and scale distribution of the local energy flux. The majority of mean flux was found to be due to the relative rotation of strain matrices at different length-scales, a first-order effect of “thinning.” The remainder arose from two second-order effects, differential strain-rotation and vorticity-gradient stretching. Our findings give strong support to vortex-thinning as the fundamental mechanism of 2D inverse energy cascade.