Controlling the dual cascade of two-dimensional turbulence

Abstract

The Kraichnan-Leith-Batchelor (KLB) theory of statistically stationary forced homogeneous isotropic 2-D turbulence predicts the existence of two inertial ranges: an energy inertial range with an energy spectrum scaling of k −5/3 , and an enstrophy inertial range with an energy spectrum scaling of k −3 . However, unlike the analogous Kolmogorov theory for 3-D turbulence, the scaling of the enstrophy range in 2-D turbulence seems to be Reynolds number dependent: numerical simulations have shown that as Reynolds number tends to infinity the enstrophy range of the energy spectrum converges to the KLB prediction, i.e. E ∼ k −3 . The present paper uses a novel optimal control approach to find a forcing that does produce the KLB scaling of the energy spectrum in a moderate Reynolds number flow. We show that the time-space structure of the forcing can significantly alter the scaling of the energy spectrum over inertial ranges. A careful analysis of the optimal forcing suggests that it is unlikely to be realized in nature, or by a simple numerical model.

An adjoint-based approach for finding invariant solutions of Navier–Stokes equations

“…The key part of deriving the adjoint equation (3.3) is to find the adjoint operator L L L † 0 . Its derivation is standard and can be found in the literature on adjoint-based optimal control (see, e.g., Gunzburger (2002); our notation is closer to Farazmand et al (2011)). The difference is that, in optimal control, one seeks to minimize a cost functional with the constraint that the Navier-Stokes equations are satisfied.…”

Double Cascade Turbulence and Richardson Dispersion in a Horizontal Fluid Flow Induced by Faraday Waves

“…In contrast, enstrophy is passed to smaller wavelengths and in this range energy scales as E k / k À3 for k > k f (Z k / k À1 ). Kraichnans theory is strongly supported by recent numerical [13,14] and experimental results [10,[15][16][17], but has not been doubtlessly verified up to now [13,17].…”

Extreme Events: Mechanisms and Prediction

“…There are standard numerical methods for approximating the solutions of the constrained optimization problems of the form (19) that we do not review here but refer the interested reader to Refs. [138,139,140,141]. Letû 0 denote a solution of the problem (19) corresponding to an extreme event, i.e., f (S τ (û 0 )) > f e .…”