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Using complementary numerical approaches at high resolution, we study the late-time behaviour of an inviscid incompressible two-dimensional flow on the surface of a sphere. Starting from a random initial vorticity field comprised of a small set of intermediate-wavenumber spherical harmonics, we find that, contrary to the predictions of equilibrium statistical mechanics, the flow does not evolve into a large-scale steady state. Instead, significant unsteadiness persists, characterised by a population of persistent small-scale vortices interacting with a large-scale oscillating quadrupolar vorticity field. Moreover, the vorticity develops a stepped, staircase distribution, consisting of nearly homogeneous regions separated by sharp gradients. The persistence of unsteadiness is explained by a simple point-vortex model characterising the interactions between the four main vortices which emerge.
Direct statistical simulation (DSS) solves the equations of motion for the statistics of turbulent flows in place of the traditional route of accumulating statistics by direct numerical simulation (DNS). That low-order statistics usually evolve slowly compared with instantaneous dynamics is one important advantage of DSS. Depending on the symmetry of the problem and the choice of averaging operation, however, DSS is usually more expensive computationally than DNS because even low-order statistics typically have higher dimension than the underlying fields. Here we show that it is in some cases possible to go much further by using a form of unsupervised learning, proper orthogonal decomposition, to address the ‘curse of dimensionality’. We apply proper orthogonal decomposition directly to DSS in the form of expansions in equal-time cumulants to second order. We explore two averaging operations (zonal and ensemble) and test the approach on two idealized barotropic models of fluid on a rotating sphere (a jet that relaxes deterministically towards an unstable profile and a stochastically driven flow that spontaneously organizes into jets). We show that the method offers the possibility of parameter continuation, in the reduced basis, for the lower-order statistics of the flow. Order-of-magnitude savings in computational cost are sometimes obtained in the reduced basis, potentially enabling access to parameter regimes beyond the reach of DNS.
We examine the effectiveness of the generalised quasilinear (GQL) approximation introduced by Marston et al. (Phys. Rev. Lett., vol. 116 (21), 2016, 214501). This approximation splits the variables into large and small scales in directions where there is a translational symmetry and removes nonlinear interactions involving only small scales. We utilise as a paradigm problem three-dimensional, turbulent, rotating Couette flow. We compare the results obtained from direct numerical solution of the equations with those from quasilinear (QL) and GQL calculations. In this three-dimensional setting, there is a choice of cutoff wavenumber for the GQL approximation both in the streamwise and in the spanwise directions. We demonstrate that the GQL approximation significantly improves the accuracy of mean flows, spectra and two-point correlation functions over models that are quasilinear in any of the translationally invariant directions, even if only a few streamwise and spanwise modes are included. We argue that this provides significant support for a programme of direct statistical simulation utilising the GQL approximation.
