Quasi-two-dimensional turbulence in shallow fluid layers: The role of bottom friction and fluid layer depth

The role of bottom friction and the fluid layer depth in numerical simulations and experiments of freely decaying quasi-two-dimensional turbulence in shallow fluid layers has been investigated. In particular, the power-law behavior of the compensated kinetic energy E-0(t)=E(t)e(2lambdat), with E(t) the total kinetic energy of the flow and lambda the bottom-drag coefficient, and the compensated enstrophy Omega(0)(t)=Omega(t)e(2lambdat), with Omega(t) the total enstrophy of the flow, have been studied. We also report on the scaling exponents of the ratio Omega(t)/E(t), which is considered as a measure of the characteristic length scale in the flow, for different values of lambda. The numerical simulations on square bounded domains with no-slip boundaries revealed bottom-friction independent power-law exponents for E-0(t), Omega(0)(t), and Omega(t)/E(t). By applying a discrete wavelet packet transform technique to the numerical data, we have been able to compute the power-law exponents of the average number density of vortices rho(t), the average vortex radius a(t), the mean vortex separation r(t), and the averaged normalized vorticity extremum omega(ext)(t)/rootE(t). These decay exponents proved to be independent of the bottom friction as well. In the experiments we have varied the fluid layer depth, and it was found that the decay exponents of E-0(t), Omega(0)(t), Omega(t)/E(t), and omega(ext)(t)/rootE(t) are virtually independent of the fluid layer depth. The experimental data for rho(t) and a(t) are less conclusive; power-law exponents obtained for small fluid layer depths agree with those from previously reported experiments, but significantly larger power-law exponents are found for experiments with larger fluid layer depths.