Dynamics of vorticity moments in shell models of turbulence: a comparison with the Navier-Stokes equations

Shell models allow much greater scale separations than those presently achievable with direct numerical simulations of the Navier–Stokes equations. Consequently, they are an invaluable tool for testing new concepts and ideas in the theory of fully developed turbulence. They also successfully display energy cascades and intermittency in homogeneous and isotropic turbulent flows. Moreover, they are also of great interest to mathematical analysts because, while retaining some of the key features of the Euler and the Navier–Stokes equations, they are much more tractable. A comparison of the mathematical properties of shell models and of the three-dimensional Navier–Stokes equations is therefore essential in understanding the correspondence between the two systems. Here, we focus on the temporal evolution of the moments, or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>m</mml:mi> </mml:mrow> </mml:msup> </mml:math> -norms, of the vorticity. Specifically, differential inequalities for the moments of the vorticity in shell models are derived. The contribution of the nonlinear term turns out to be much weaker than its equivalent for the three-dimensional Navier–Stokes equations. Consequently, pointwise-in-time estimates are shown to exist for the vorticity moments for shell models of any order. This result is also recovered via a high–low frequency slaving argument that highlights the scaling relations between vorticity moments of different orders. Finally, it is shown that the estimates for shell models formally correspond to those for the Navier–Stokes equations ‘on a point’.