Anisotropic characteristics of the Kraichnan direct cascade in two-dimensional hydrodynamic turbulence

Statistical characteristics of the Kraichnan direct cascade for two-dimensional hydrodynamic turbulence are numerically studied (with spatial resolution $8192\times 8192$) in the presence of pumping and viscous-like damping. It is shown that quasi-shocks of vorticity and their Fourier partnerships in the form of jets introduce an essential influence in turbulence leading to strong angular dependencies for correlation functions. The energy distribution as a function of modulus $k$ for each angle in the inertial interval has the Kraichnan behavior, ${\sim}\,k^{-4}$, and simultaneously a strong dependence on angles. However, angle average provides with a high accuracy the Kraichnan turbulence spectrum $E_k=C_{\text{K}}\eta^{2/3} k^{-3}$, where $\eta$ is enstrophy flux and the Kraichnan constant $C_{\text{K}}\simeq 1.3$, in correspondence with the previous simulations. Familiar situation takes place for third-order velocity structure function $S_3^L$ which, as for the isotropic turbulence, gives the same scaling with respect to separation length $R$ and $\eta$, $S_3^L=C_3\eta R^3$, but the mean over angles and time $\bar {C_3}$ differs from its isotropic value.