EULERIAN AND LAGRANGIAN SCALING PROPERTIES OF RANDOMLY ADVECTED VORTEX TUBES

We investigate the Eulerian and Lagrangian spectral scaling properties of vortex tubes, and the consistency of these properties with Tenneke s' (1975) statistical advection analysis and universal equilibrium arg uments. We consider three different vortex tubes with power-law wavenu mber spectra: a Burgers vortex tube, an inviscid Lundgren single spira l vortex sheet, and a vortex tube solution of the Euler equation. Whil e the Burgers vortex is a steady solution of the Navier-Stokes equatio n, the other two are unsteady solutions of, respectively, the Navier-S tokes and the Euler equations. In our numerical experiments we study t he vortex tubes by subjecting each of them to external 'large-scale' s inusoidal advection of characteristic frequency f and length scale rho . Not only do we find that the Eulerian frequency spectrum Phi(E)(omeg a) can be derived from the wavenumber spectrum E(k) using the simple T ennekes advection relation omega similar to k for all finite advection frequencies f when the vortex is steady, but also when the vortex is unsteady, and in the Lundgren case even when f = 0 owing to the self-a dvection of the Lundgren vortex by its own differential rotation. An a nalytical calculation using the method of stationary phases for f = 0 shows that for large enough Reynolds numbers the combination of strain with differential rotation implies that Phi(L)(omega) similar to omeg a(-2)+Const for large values of omega. We verify numerically that Phi( L)(omega) does not change when f not equal 0. With the Burgers vortex tube we are in a position to investigate the spectral broadening of th e Eulerian frequency spectrum with respect to the Lagrangian frequency spectrum. A spectral broadening does exist but is different from the spectral broadening predicted by Tennekes (1975).