On the Clark-alpha model of turbulence: global regularity and long--time dynamics

In this paper we study a well-known three--dimensional turbulence model, the filtered Clark model, or Clark-alpha model. This is Large Eddy Simulation (LES) tensor-diffusivity model of turbulent flows with an additional spatial filter of width alpha ($\alpha$). We show the global well-posedness of this model with constant Navier-Stokes (eddy) viscosity. Moreover, we establish the existence of a finite dimensional global attractor for this dissipative evolution system, and we provide an anaytical estimate for its fractal and Hausdorff dimensions. Our estimate is proportional to $(L/l_d)^3$, where $L$ is the integral spatial scale and $l_d$ is the viscous dissipation length scale. This explicit bound is consistent with the physical estimate for the number of degrees of freedom based on heuristic arguments. Using semi-rigorous physical arguments we show that the inertial range of the energy spectrum for the Clark-$\aa$ model has the usual $k^{-5/3}$ Kolmogorov power law for wave numbers $k\aa \ll 1$ and $k^{-3}$ decay power law for $k\aa \gg 1.$ This is evidence that the Clark$-\alpha$ model parameterizes efficiently the large wave numbers within the inertial range, $k\aa \gg 1$, so that they contain much less translational kinetic energy than their counterparts in the Navier-Stokes equations.Comment: 11 pages, no figures, submitted to J of Turbulenc

Estimates for the two-dimensional Navier-Stokes equations in terms of the Reynolds number

The tradition in Navier-Stokes analysis of finding estimates in terms of the Grashof number \bG, whose character depends on the ratio of the forcing to the viscosity $\nu$, means that it is difficult to make comparisons with other results expressed in terms of Reynolds number \Rey, whose character depends on the fluid response to the forcing. The first task of this paper is to apply the approach of Doering and Foias \cite{DF} to the two-dimensional Navier-Stokes equations on a periodic domain $[0,L]^{2}$ by estimating quantities of physical relevance, particularly long-time averages \left, in terms of the Reynolds number \Rey = U\ell/\nu, where U^{2}= L^{-2}\left and $\ell$ is the forcing scale. In particular, the Constantin-Foias-Temam upper bound \cite{CFT} on the attractor dimension converts to a_{\ell}^{2}\Rey(1 + \ln\Rey)^{1/3}, while the estimate for the inverse Kraichnan length is (a_{\ell}^{2}\Rey)^{1/2}, where $a_{\ell}$ is the aspect ratio of the forcing. Other inverse length scales, based on time averages, and associated with higher derivatives, are estimated in a similar manner. The second task is to address the issue of intermittency : it is shown how the time axis is broken up into very short intervals on which various quantities have lower bounds, larger than long time-averages, which are themselves interspersed by longer, more quiescent, intervals of time.Comment: 21 pages, 1 figure, accepted for publication from J. Math. Phys. for the special issue on mathematical fluid mechanic

Spectral scaling of the Leray-α\alpha model for two-dimensional turbulence

We present data from high-resolution numerical simulations of the Navier-Stokes-$\alpha$ and the Leray-$\alpha$ models for two-dimensional turbulence. It was shown previously (Lunasin et al., J. Turbulence, 8, (2007), 751-778), that for wavenumbers $k$ such that $k\alpha\gg 1$, the energy spectrum of the smoothed velocity field for the two-dimensional Navier-Stokes-$\alpha$ (NS-$\alpha$) model scales as $k^{-7}$. This result is in agreement with the scaling deduced by dimensional analysis of the flux of the conserved enstrophy using its characteristic time scale. We therefore hypothesize that the spectral scaling of any $\alpha$-model in the sub-$\alpha$ spatial scales must depend only on the characteristic time scale and dynamics of the dominant cascading quantity in that regime of scales. The data presented here, from simulations of the two-dimensional Leray-$\alpha$ model, confirm our hypothesis. We show that for $k\alpha\gg 1$, the energy spectrum for the two-dimensional Leray-$\alpha$ scales as $k^{-5}$, as expected by the characteristic time scale for the flux of the conserved enstrophy of the Leray-$\alpha$ model. These results lead to our conclusion that the dominant directly cascading quantity of the model equations must determine the scaling of the energy spectrum.Comment: 11 pages, 4 figure

Global solvability and blow up for the convective Cahn-Hilliard equations with concave potentials

We study initial boundary value problems for the convective Cahn-Hilliard equation \Dt u +\px^4u +u\px u+\px^2(|u|^pu)=0. It is well-known that without the convective term, the solutions of this equation may blow up in finite time for any $p>0$. In contrast to that, we show that the presence of the convective term u\px u in the Cahn-Hilliard equation prevents blow up at least for $0<p<\frac49$. We also show that the blowing up solutions still exist if $p$ is large enough ($p\ge2$). The related equations like Kolmogorov-Sivashinsky-Spiegel equation, sixth order convective Cahn-Hilliard equation, are also considered

Orthogonal, solenoidal, three-dimensional vector fields for no-slip boundary conditions

Viscous fluid dynamical calculations require no-slip boundary conditions. Numerical calculations of turbulence, as well as theoretical turbulence closure techniques, often depend upon a spectral decomposition of the flow fields. However, such calculations have been limited to two-dimensional situations. Here we present a method that yields orthogonal decompositions of incompressible, three-dimensional flow fields and apply it to periodic cylindrical and spherical no-slip boundaries.Comment: 16 pages, 2 three-part figure

Unstable recurrent patterns in Kuramoto-Sivashinsky dynamics

We undertake a systematic exploration of recurrent patterns in a 1-dimensional Kuramoto-Sivashinsky system. For a small, but already rather turbulent system, the long-time dynamics takes place on a low-dimensional invariant manifold. A set of equilibria offers a coarse geometrical partition of this manifold. A variational method enables us to determine numerically a large number of unstable spatiotemporally periodic solutions. The attracting set appears surprisingly thin - its backbone are several Smale horseshoe repellers, well approximated by intrinsic local 1-dimensional return maps, each with an approximate symbolic dynamics. The dynamics appears decomposable into chaotic dynamics within such local repellers, interspersed by rapid jumps between them.Comment: 11 pages, 11 figure

Efficient dynamical downscaling of general circulation models using continuous data assimilation

Continuous data assimilation (CDA) is successfully implemented for the first time for efficient dynamical downscaling of a global atmospheric reanalysis. A comparison of the performance of CDA with the standard grid and spectral nudging techniques for representing long- and short-scale features in the downscaled fields using the Weather Research and Forecast (WRF) model is further presented and analyzed. The WRF model is configured at 25km horizontal resolution and is driven by 250km initial and boundary conditions from NCEP/NCAR reanalysis fields. Downscaling experiments are performed over a one-month period in January, 2016. The similarity metric is used to evaluate the performance of the downscaling methods for large and small scales. Similarity results are compared for the outputs of the WRF model with different downscaling techniques, NCEP reanalysis, and Final Analysis. Both spectral nudging and CDA describe better the small-scale features compared to grid nudging. The choice of the wave number is critical in spectral nudging; increasing the number of retained frequencies generally produced better small-scale features, but only up to a certain threshold after which its solution gradually became closer to grid nudging. CDA maintains the balance of the large- and small-scale features similar to that of the best simulation achieved by the best spectral nudging configuration, without the need of a spectral decomposition. The different downscaled atmospheric variables, including rainfall distribution, with CDA is most consistent with the observations. The Brier skill score values further indicate that the added value of CDA is distributed over the entire model domain. The overall results clearly suggest that CDA provides an efficient new approach for dynamical downscaling by maintaining better balance between the global model and the downscaled fields

Semigroup evolution in Wigner Weisskopf pole approximation with Markovian spectral coupling

We establish the relation between the Wigner-Weisskopf theory for the description of an unstable system and the theory of coupling to an environment. According to the Wigner-Weisskopf general approach, even within the pole approximation (neglecting the background contribution) the evolution of a total system subspace is not an exact semigroup for the multi-channel decay, unless the projectors into eigesntates of the reduced evolution generator $W(z)$ are orthogonal. In this case these projectors must be evaluated at different pole locations $z_\alpha\neq z_\beta$. Since the orthogonality relation does not generally hold at different values of $z$, for example, when there is symmetry breaking, the semigroup evolution is a poor approximation for the multi-channel decay, even for a very weak coupling. Nevertheless, there exists a possibility not only to ensure the orthogonality of the $W(z)$ projectors regardless the number of the poles, but also to simultaneously suppress the effect of the background contribution. This possibility arises when the theory is generalized to take into account interactions with an environment. In this case $W(z)$, and hence its eigenvectors as well, are {\it independent} of $z$, which corresponds to a structure of the coupling to the continuum spectrum associated with the Markovian limit.Comment: 9 pages, 3 figure

Numerical solutions of the three-dimensional magnetohydrodynamic alpha-model

We present direct numerical simulations and alpha-model simulations of four familiar three-dimensional magnetohydrodynamic (MHD) turbulence effects: selective decay, dynamic alignment, inverse cascade of magnetic helicity, and the helical dynamo effect. The MHD alpha-model is shown to capture the long-wavelength spectra in all these problems, allowing for a significant reduction of computer time and memory at the same kinetic and magnetic Reynolds numbers. In the helical dynamo, not only does the alpha-model correctly reproduce the growth rate of magnetic energy during the kinematic regime, but it also captures the nonlinear saturation level and the late generation of a large scale magnetic field by the helical turbulence.Comment: 12 pages, 19 figure

Long-time Behavior of a Two-layer Model of Baroclinic Quasi-geostrophic Turbulence

We study a viscous two-layer quasi-geostrophic beta-plane model that is forced by imposition of a spatially uniform vertical shear in the eastward (zonal) component of the layer flows, or equivalently a spatially uniform north-south temperature gradient. We prove that the model is linearly unstable, but that non-linear solutions are bounded in time by a bound which is independent of the initial data and is determined only by the physical parameters of the model. We further prove, using arguments first presented in the study of the Kuramoto-Sivashinsky equation, the existence of an absorbing ball in appropriate function spaces, and in fact the existence of a compact finite-dimensional attractor, and provide upper bounds for the fractal and Hausdorff dimensions of the attractor. Finally, we show the existence of an inertial manifold for the dynamical system generated by the model's solution operator. Our results provide rigorous justification for observations made by Panetta based on long-time numerical integrations of the model equations