Abstract. The problem of constructing and classifying stationary equilibria of point vortices in the plane is studied. An ordinary differential equation that enables one to find positions of point vortices with circulations Γ1, Γ2, and Γ3 in stationary equilibrium is obtained. A necessary condition of an equilibrium existing is derived. The case of point vortex systems consisting of n + 2 point vortices with n vortices of circulation Γ 1 and two vortices of circulations Γ2 = aΓ1 and Γ3 = bΓ1, where a and b are integers, is considered in detail. The properties of polynomial solutions of the corresponding ordinary differential equation are investigated. A set of positivedimensional equilibrium configurations is found. A continuous free parameter is presented in the coefficients of corresponding polynomial solutions. These free parameters affect the positions of the roots and hence the vortex positions. Stationary equilibrium that could be derived from each other by rotation, extension, parallel translation is considered as equivalent. All found configurations seem to be new.

Abstract:In the present work, an analytical model of the vortex motion basic state of the dry atmosphere with nonzero air velocity divergence is constructed. It is shown that the air parcel moves along the open curve trajectory of spiral geometry. It is found that for the case of nonzero velocity divergence, the atmospheric basic state presents an unlimited sequence of vortex cells transiting from one to another. On the other hand, at zero divergence, the basic state presents a pair of connected vortices, and the trajectory is a closed curve. If in some cells the air parcel moves upward, then in the adjacent cells, it will move downward, and vice versa. Upon reaching the cell's middle height, the parcel reverses the direction of rotation. When the parcel moves upward, the motion is of anticyclonic type in the lower part of the vortex cell and of cyclonic type in the upper part. When the parcel moves downward, the motion is of anticyclonic type in the upper part of the vortex cell and of cyclonic type in the lower part.