Esteban G. Tabak

A one-dimensional model for dispersive wave turbulence

SummaryA family of one-dimensional nonlinear dispersive wave equations is introduced as a model for assessing the validity of weak turbulence theory for random waves in an unambiguous and transparent fashion. These models have an explicitly solvable weak turbulence theory which is developed here, with Kolmogorov-type wave number spectra exhibiting interesting dependence on parameters in the equations. These predictions of weak turbulence theory are compared with numerical solutions with damping and driving that exhibit a statistical inertial scaling range over as much as two decades in wave number.It is established that the quasi-Gaussian random phase hypothesis of weak turbulence theory is an excellent approximation in the numerical statistical steady state. Nevertheless, the predictions of weak turbulence theory fail and yield a much flatter (|k|−1/3) spectrum compared with the steeper (|k|−3/4) spectrum observed in the numerical statistical steady state. The reasons for the failure of weak turbulence theory in this context are elucidated here. Finally, an inertial range closure and scaling theory is developed which successfully predicts the inertial range exponents observed in the numerical statistical steady states.

A two-dimensional model for quasigeostrophic flow: comparison with the two-dimensional Euler flow

Abstract A simple two-dimensional model for quasigeostrophic flow is contrasted with the two-dimensional incompressible Euler equations. The model arises under the assumptions of fast rotation, uniform stratification and uniform potential vorticity. It is found that the more loca feed-back of the quasigeostrophic model gives rise to strongly nonlinear front formation, as opposed to two-dimensional Euler, where the steepening process of mature fronts obeys a nonlocal, nearly linear mechanism.

Dispersive wave turbulence in one dimension

In this article, we study numerically a one-dimensional model of dispersive wave turbulence. The article begins with a description of the model which we introduced earlier, followed by a concise summary of our previous results about it. In those previous studies, in addition to the spectra of weak turbulence (WT) theory, we also observed another distinct spectrum (the “MMT spectrum”). Our new results, presented here, include: (i) A detailed description of coexistence of spectra at distinct spatial scales, and the transitions between them at different temporal scales; (ii) The existence of a stable MMT front in k-space which separates the WT cascades from the dissipation range, for various forms of strong damping including “selective dissipation”; (iii) The existence of turbulent cycles in the one-dimensional model with focusing nonlinearity, induced by the interaction of spatially localized coherent structures with the resonant quartets of dispersive wave radiation; (iv) The detailed composition of these turbulent cycles — including the self-similar formation of focusing events (distinct in the forced and freely decaying cases), and the transport by the WT direct and inverse cascades of excitations between spatial scales. This one-dimensional model admits a very precise and detailed realization of these turbulent cycles and their components. Our numerical experiments demonstrate that a complete theory of dispersive wave turbulence will require a full description of the turbulent field over all spatial scales (including those of the forcing and dissipation), and over extremely long times (as the nonlinear turnover time becomes very long in the weakly nonlinear limit). And, in the focusing case, a complete theory must also incorporate the interaction of localized coherent structures with resonant radiation.

Hamiltonian formalism and the Garrett-Munk spectrum of internal waves in the ocean.

Wave turbulence formalism for long internal waves in a stratified fluid is developed, based on a natural Hamiltonian description. A kinetic equation appropriate for the description of spectral energy transfer is derived, and its anisotropic self-similar stationary solution corresponding to a direct cascade of energy toward the short scales is found. This solution is very close to the high wave-number limit of the Garrett-Munk spectrum of long internal waves in the ocean. In fact, a small modification of the Garrett-Munk formalism includes a spectrum consistent with the one predicted by wave turbulence.

A PseudoSpectral Procedure for the Solution of Nonlinear Wave Equations with Examples from Free-Surface Flows

An algorithm for the solution of general isotropic nonlinear wave equations is presented. The algorithm is based on a symmetric factorization of the linear part of the wave operator, followed by its exact integration through an integrating factor in spectral space. The remaining nonlinear and forcing terms can be handled with any standard pseudospectral procedure. Solving the linear part of the wave operator exactly effectively eliminates the stiffness of the original problem, characterized by a wide range of temporal scales. The algorithm is tested and applied to several problems of three-dimensional long surface waves: solitary wave propagation, interaction, diffraction, and the generation of waves by flow over slowly varying bottom topography. Other potential applications include waves in rotating and stratified flows and wave interaction with more pronounced topographic features.

Internal hydraulic jumps and mixing in two-layer flows

Internal hydraulic jumps in two-layer flows are studied, with particular emphasis on their role in entrainment and mixing. For highly entraining internal jumps, a new closure is proposed for the jump conditions. The closure is based on two main assumptions: (i) most of the energy dissipated at the jump goes into turbulence, and (ii) the amount of turbulent energy that a stably stratified flow may contain without immediately mixing further is bounded by a measure of the stratification. As a consequence of this closure, surprising bounds emerge, for example on the amount of entrainment that may take place at the location of the jump. These bounds are probably almost achieved by highly entraining internal jumps, such as those likely to develop in dense oceanic over flows. The values obtained here are in good agreement with the existing observations of the spatial development of oceanic downslope currents, which play a crucial role in the formation of abyssal and intermediate waters in the global ocean.

Energy spectra of the Ocean's internal wave field: Theory and observations

The high-frequency limit of the Garrett and Munk spectrum of internal waves in the ocean and the observed deviations from it are shown to form a pattern consistent with the predictions of wave turbulence theory. In particular, the high-frequency limit of the Garrett and Munk spectrum constitutes an exact steady-state solution of the corresponding kinetic equation.

Drosophila insulin and target of rapamycin (TOR) pathways regulate GSK3 beta activity to control Myc stability and determine Myc expression in vivo

BackgroundGenetic studies in Drosophila melanogaster reveal an important role for Myc in controlling growth. Similar studies have also shown how components of the insulin and target of rapamycin (TOR) pathways are key regulators of growth. Despite a few suggestions that Myc transcriptional activity lies downstream of these pathways, a molecular mechanism linking these signaling pathways to Myc has not been clearly described. Using biochemical and genetic approaches we tried to identify novel mechanisms that control Myc activity upon activation of insulin and TOR signaling pathways.ResultsOur biochemical studies show that insulin induces Myc protein accumulation in Drosophila S2 cells, which correlates with a decrease in the activity of glycogen synthase kinase 3-beta (GSK3β ) a kinase that is responsible for Myc protein degradation. Induction of Myc by insulin is inhibited by the presence of the TOR inhibitor rapamycin, suggesting that insulin-induced Myc protein accumulation depends on the activation of TOR complex 1. Treatment with amino acids that directly activate the TOR pathway results in Myc protein accumulation, which also depends on the ability of S6K kinase to inhibit GSK3β activity. Myc upregulation by insulin and TOR pathways is a mechanism conserved in cells from the wing imaginal disc, where expression of Dp110 and Rheb also induces Myc protein accumulation, while inhibition of insulin and TOR pathways result in the opposite effect. Our functional analysis, aimed at quantifying the relative contribution of Myc to ommatidial growth downstream of insulin and TOR pathways, revealed that Myc activity is necessary to sustain the proliferation of cells from the ommatidia upon Dp110 expression, while its contribution downstream of TOR is significant to control the size of the ommatidia.ConclusionsOur study presents novel evidence that Myc activity acts downstream of insulin and TOR pathways to control growth in Drosophila. At the biochemical level we found that both these pathways converge at GSK3β to control Myc protein stability, while our genetic analysis shows that insulin and TOR pathways have different requirements for Myc activity during development of the eye, suggesting that Myc might be differentially induced by these pathways during growth or proliferation of cells that make up the ommatidia.

Interaction of Large-Scale Equatorial Waves and Dispersion of Kelvin Waves through Topographic Resonances

A new theoretical mechanism is developed in which large-scale equatorial Kelvin waves can modify their speed through dispersion and interaction with other large-scale equatorial waves, such as Yanai or Rossby modes, through topographic resonance. This resonance mechanism can prevent the breaking of a propagating nonlinear Kelvin wave, slow down its speed, and concentrate most of its energy in large-scale zonal wavenumbers while simultaneously generating large-scale Yanai or Rossby modes with specific zonal wavelengths. Simplified reduced dynamic equations for this resonant interaction are developed here via suitable asymptotic expansions of the equatorial shallow water equations with topography. Explicit exact solutions for the reduced equations and numerical experiments are utilized to display explicitly the features of large-scale dispersion and topographic resonance for equatorial Kelvin waves mentioned earlier. Two examples of this theory, corresponding to the barotropic and first baroclinic modes of the equatorial troposphere, are emphasized.

Evolving management of symptomatic chronic subdural hematoma: experience of a single institution and review of the literature.

Abstract Objective: Chronic subdural hematoma (cSDH) has an increasing incidence and results in high morbidity and mortality. We review here the 10-year experience of a single institution and the literature regarding the treatment and major associations of cSDH. Methods: We retrospectively reviewed all cSDHs surgically treated from 2000 to 2010 in the New York Harbor Health Care System to evaluate the duration from admission to treatment, type of treatment, length of stay (LOS) in critical care, LOS in the hospital, and recurrence. The literature was reviewed with regards to incidence, associations, and treatment of cSDH. Results: From 2000 to 2008, 44 patients were treated with burr holes (BHs). From 2008 to 2010, 29 patients were treated with twist-drill evacuation (subdural evacuating port system, SEPS). Four patients from each group were readmitted for reoperation (9% vs 14%; P  =  0·53). The average time to intervention for SEPS (11·2±15·3 hours) was faster than for BHs (40·3±69·1 hours) (P  =  0·02). The total hospital LOS was shorter for SEPS (9·3±6·8 days) versus BHs (13·4±10·2 days) (P  =  0·04); both were significantly longer than for a brain tumor patient undergoing craniotomy (7·0±0·5 days, n  =  94, P < 0·01). Conclusion: Despite decreasing LOSs as treatment for cSDH evolved from BHs to SEPS, the LOS for a cSDH is still longer than that of a patient undergoing craniotomy for brain tumor. We noted 11% recurrence in our series of patients, which included individuals who recurred as late as 3 years after initial diagnosis.