The kinematics of quasi-2-D freely decaying turbulence in a stratified fluid

Abstract

We have studied experimentally the dynamics of freely decaying turbulence in a stratified fluid. At low Froude number this flow has a quasi-2-D behavior1 that modifies the mixing (Fr=0.004, Re=1000; based on the integral scales). We characterize the vorticity evolution in the range scales where the enstrophy cascade is taking place. The quasi-2-D period of this turbulence is studied by photographic analysis of the velocity field V, which is interpolated on a regular grid (128×128). We are able to decompose it into its internal wave and turbulent component, and to compute its derivative fields up to the vector η=curlHΩV (related to the enstrophy cascade; where ΩV is the vertical vorticity). An enstrophy transfer model is developed by generalizing the model of Weiss to slightly divergent flows: the tensor F=∇HVH−DivH VHI2 (where H denotes the horizontal component, and I2 is the identity tensor of order 2) enters the equation of evolution for η as dtη=ηF+ΩV curlH (DivHVH). (1) An elongation of η corresponds to an increase of the local vorticity gradient, which is the mechanism of the enstrophy cascade to small scales. We neglect the second term of (1) since it occurs at the measurement noise scale. Thus eigenvalues λi and eigenvectors of F characterize Eq. (1) and we present maps of these functions. The analysis of these eigenvalues of F gives us the ‘‘coherent’’ zones of the flow where λ is complex, i.e., where the local effect of rotation counteracts the local strain induced by the whole vorticity field. In the remaining zones the λi are generally of opposite sign. Under this condition, the histogram of the angle α between the elongating (V+) and the contracting (V−) eigenvectors shows broad peaks centered on ±π/2, Where the separating curve of ΩV=0 on an Ω map corresponds to α=±π/2. In these regions the strain provokes a deformation of the vorticity field, which corresponds to the enstrophy cascade phenomenon. Statistics on the angles β±=∠(η,V±) reveals a mean alignment of η with V+ (elongation) and a mean right angle with V−. Thus, the relations between F and η are clearly shown in our experimental case. The main effect of vertical compression or expansion at every point of the field, due for instance to a weak internal wave, is to modify the eigenvalues of F, thus inhibiting or favoring the local enstrophy cascade. In some cases, it is able to impose its sign onto the two eigenvalues: this is observed at the border of the coherent zone.