The Asymptotic Finite-dimensional Character of a Spectrally-hyperviscous Model of 3D Turbulent Flow

We obtain attractor and inertial-manifold results for a class of 3D turbulent flow models on a periodic spatial domain in which hyperviscous terms are added spectrally to the standard incompressible Navier–Stokes equations (NSE). Let P _m be the projection onto the first m eigenspaces of A =−Δ, let μ and α be positive constants with α ≥3/2, and let Q _m =I − P _m , then we add to the NSE operators μ A _φ in a general family such that A _φ≥Q _m A ^α in the sense of quadratic forms. The models are motivated by characteristics of spectral eddy-viscosity (SEV) and spectral vanishing viscosity (SVV) models. A distinguished class of our models adds extra hyperviscosity terms only to high wavenumbers past a cutoff λ _m0 where m ₀ ≤ m, so that for large enough m ₀ the inertial-range wavenumbers see only standard NSE viscosity.

We first obtain estimates on the Hausdorff and fractal dimensions of the attractor ${\mathcal{A}}$ (respectively $\dim_{\rm H}{\mathcal{A}}$ and $\dim_{\rm F}{\mathcal{A}}$). For a constant K _α on the order of unity we show if μ ≥ ν that $\dim_{\rm H} {\mathcal{A}} \leq \dim_{\rm F} {\mathcal{A}} \leq K_{\alpha} \left[\lambda_{m}/\lambda_{1}\right]^{9\left(\alpha - 1\right)/(10\alpha)} \left[l_{0}/l_{\epsilon}\right]^{(6\alpha+9)/(5\alpha)}$ and if μ ≤ ν that $\dim_{\rm H} {\mathcal{A}} \leq \dim_{\rm F} {\mathcal{A}} \leq K_{\alpha} \left(\nu/\mu\right)^{9/(10\alpha)}\left[\lambda_{m}/\lambda_{1}\right]^{9\left(\alpha -1\right)/(10\alpha)} \left[l_{0}/l_{\epsilon}\right]^{(6\alpha + 9)/(5\alpha)}$ where ν is the standard viscosity coefficient, l ₀ = λ ^−1/2₁ represents characteristic macroscopic length, and $l_{\epsilon}$ is the Kolmogorov length scale, i.e. $l_{\epsilon} = (\nu^{3}/\epsilon)$ where $\epsilon$ is Kolmogorov’s mean rate of dissipation of energy in turbulent flow. All bracketed constants and K _α are dimensionless and scale-invariant. The estimate grows in m due to the term λ _m /λ₁ but at a rate lower than m ^3/5, and the estimate grows in μ as the relative size of ν to μ. The exponent on $l_{0}/l_{\epsilon}$ is significantly less than the Landau–Lifschitz predicted value of 3. If we impose the condition $\lambda_{m} \leq (1/l_{\epsilon})^{2}$, the estimates become $K_{\alpha} \left[l_{0}/l_{\epsilon}\right]^{3}$ for μ ≥ ν and $K_{\alpha}\left(\nu/\mu \right)^{\frac{9}{10\alpha}}\left[l_{0}/l_{\epsilon}\right]^{3}$ for μ ≤  ν. This result holds independently of α, with K _α and c _α independent of m. In an SVV example μ ≥  ν, and for μ ≤  ν aspects of SEV theory and observation suggest setting $\mu \thicksim c\nu$ for 1/c within α orders of magnitude of unity, giving the estimate $c_{\alpha}K_{\alpha}\left[l_{0}/l_{\epsilon}\right]^{3}$ where c _α is within an order of magnitude of unity. These choices give straight-up or nearly straight-up agreement with the Landau–Lifschitz predictions for the number of degrees of freedom in 3D turbulent flow with m so large that (e.g. in the distinguished-class case for m ₀ large enough) we would expect our solutions to be very good if not virtually indistinguishable approximants to standard NSE solutions. We would expect lower choices of λ _m (e.g. $\lambda_{m}\thicksim a(1/l_{\epsilon})$ with a >  1) to still give good NSE approximation with lower powers on l ₀/l _ε, showing the potential of the model to reduce the number of degrees of freedom needed in practical simulations. For the choice $\epsilon \thicksim \nu^{\alpha}$, motivated by the Chapman–Enskog expansion in the case m =  0, the condition becomes $\lambda_{m}\leq \nu (1/l_{\epsilon})^{2}$, giving agreement with Landau–Lifschitz for smaller values of λ _m then as above but still large enough to suggest good NSE approximation. Our final results establish the existence of a inertial manifold ${\mathcal{M}}$ for reasonably wide classes of the above models using the Foias/Sell/Temam theory. The first of these results obtains such an ${\mathcal{M}}$ of dimension N >  m for the general class of operators A _φ if α >  5/2.

The special class of A _φ such that P _m A _φ =  0 and Q _m A _φ ≥ Q _m A ^α has a unique spectral-gap property which we can use whenever α ≥  3/2 to show that we have an inertial manifold ${\mathcal{M}}$ of dimension m if m is large enough. As a corollary, for most of the cases of the operators A _φ in the distinguished-class case that we expect will be typically used in practice we also obtain an ${\mathcal{M}}$, now of dimension m ₀ for m ₀ large enough, though under conditions requiring generally larger m ₀ than the m in the special class. In both cases, for large enough m (respectively m ₀), we have an inertial manifold for a system in which the inertial range essentially behaves according to standard NSE physics, and in particular trajectories on ${\mathcal{M}}$ are controlled by essentially NSE dynamics.