Competition between nonlinear radial and transverse cascades in the dynamics of two-dimensional anticyclones in plane shear flows

**Authors:** Luka Burdiladze, George Chagelishvil **Affiliations:**Georgian National Astrophysical Observatory, Abastumani 0301, Georgia **Tbilisi State University, Faculty of Exact and Natural Sciences, Department of Physics, Tbilisi, Georgia We investigate the nonlinear dynamics of circular and elliptical anticyclonic vortices in two-dimensional (2D) hydrodynamic plane shear flows. The considered shear flow is spectrally stable - the extraction of shear flow energy by perturbations has a transient character. This imperfect linear growth mechanism is replenished by a variety of nonlinear cascades - in addition to the usual radial (direct and inverse) cascade, we identify a transverse cascade. We derive the spectral energy equation for perturbations from 2D incompressible, viscous fluid equations, explicitly accounting for the shear flow. Each term in the equation is analyzed, and the equations are numerically solved for anticyclonic perturbations (initially imposed in the flow) using the SNOOPY code. We construct and analyze the energy, nonlinear terms, Reynolds stresses, and other linear processes in the spectral plane at any given moment based on the numerical data. Our analysis reveals that nonlinearity induces a transverse cascade of spectral energy. For anticyclonic perturbations, the total energy monotonically increases due to positive nonlinear feedback during simulations. This suggests that the nonlinear transverse cascade plays a decisive role in the self-sustenance of the vortices in the flow. We further conduct a comparative analysis of the effectiveness of the radial (mainly inverse) and transverse cascades, introducing a parameter that quantifies the ratio of the nonlinear radial cascade to the transverse one. The analysis shows that as this parameter decreases, the degree of self-sustenance of the anticyclonic vortex in the flow increases. This study lays the groundwork for proposed studies of the dynamics of perturbations in more complex flows found in nature, such as astrophysical Keplerian and sub-Keplerian disks, where additional dynamical factors like Coriolis forces and radial pressure gradients play significant roles.