Vortex Tubes, Local Anisotropy and Anomalous Scaling

Authors: Ayşe ERZAN

Abstract: It has recently been proposed [1] that in highly developed hydrodynamic turbulence the dissipation field has an anomalous scaling behaviour independent of the velocity fields. The anomalous scaling exponent is ``subcritical" so that the Kolmogorov scaling behaviour of the velocity fields is retained in the limit of Re\to \infty. On the other hand, the formation of vortex tubes leading to local isotropy-breaking has been shown to give rise to anomalous scaling behaviour of the moments of the derivatives of the velocity field, \partial_\alpha u_\beta, in terms of which one is able to define an "anisotropy factor" [2]. We argue that the formation of the locally two-dimensional structures, namely the vortex tubes, is also what gives rise to the anomalous behaviour of the dissipation field, since the dissipation field itself is defined in terms of the velocity gradients as \epsilon \equiv \langle \nu \vert \nabla u\vert ^2\rangle, so that the ``anisotropy factor" [2] and \epsilon can be related by operator algebra. We also examine the lowest order term in the expansion of \epsilon in terms of the nonlinearity in the Navier Stokes equations. It is the logarithmic singularity in this term that later sums, via ladder diagrams, up to a correction to the scaling behaviour. We show how vortex tubes and sheets may contribute.