Structure function of velocity in a geostrophic vortex under strong rotation

We consider analytically velocity pair structure function of turbulent pulsations on the background of a coherent geostrophic vortex in a fast rotating fluid. The statistics of the turbulent pulsations is determined by their dynamics that is the dynamics of inertial waves affected by the differential rotation in the vortex and weak viscous damping. Our consideration is restricted by the smallest scales, where the velocity field remains smooth. We establish the anisotropy of the structure function. The velocity gradient of the turbulent pulsations achieves its largest value for the nearly radial direction and its smallest value in the nearly streamwise direction. The inclination angle between the directions of extremum values of the gradient and the vortex-associated directions is determined by the interplay between the shear rate and the viscosity influence. The scaling laws resemble their behavior for a passive scalar in a steady shear flow.