Numerical solutions of the singular vortex problem

This study develops a finite-difference numerical formulation to describe the motion of a singular monopole in a quasigeostrophic beta-channel model with scale-selective frictional damping, using parameter values typical for the middle-latitude atmosphere and a wide range of viscosities. In this model, the theoretical singular vortex is replaced by the equivalent nonsingular vortex of a finite amplitude, consistent with the finite spatial resolution of the model. Numerical experiments demonstrate that at initial stages of the singular-vortex (SV) evolution, this model accurately reproduces the behavior expected from the theoretical considerations of the inviscid case. The numerical model also approximately conserves several invariants of motion derived from the continuous equations and accurately represents their modifications in the presence of friction. The evolution of a singular cyclone in the Northern Hemisphere starts with the development of the dipolar beta gyres in the regular component of the flow; these gyres induce initial northward displacement and subsequent westward bending of the cyclone trajectory. At larger times, the beta gyres gradually disintegrate, and the singular cyclone in the Northern Hemisphere continues to move northwestward by forming a dipolelike system with the localized secondary regular-field anticyclone northeast of the singular-cyclone center resulting eventually in a friction-assisted steady-state regime. The SV trajectories tend to become more zonally elongated for large vortices and small values of viscosity. These results lay a foundation for numerical consideration of systems of multiple singular vortices, which could provide further insights into our fundamental understanding of the processes underlying the multiscale atmospheric and oceanic variability.