Singular Vectors for Tropical Cyclone–Like Vortices in a Nondivergent Barotropic Framework

Abstract

Abstract In this study, singular vectors (SVs) are calculated for tropical cyclone (TC)–like vortices on an f plane and β plane using a barotropic model, and the structure and time evolution of the SVs are investigated. In the f-plane study, SVs are calculated for TC-like vortices that do and do not satisfy a necessary condition of barotropic instability of normal modes, in which the vorticity gradient changes sign. It is found that, in the case where the initial vortices do not meet the condition, 1) the SVs are tilted against the shear of the background angular velocity as found earlier by Nolan and Farrell, indicating the growth of SVs through the Orr mechanism; 2) the leading singular value increases with the maximum tangential wind speed Vmax and decreases with the radius of the maximum wind (RMW); and 3) the locations of SVs move outward with increasing RMW, Vmax, and the optimization time. In the case where the initial vortex allows for barotropic instability, the SV is initially tilted against the background shear and exhibits transient growth for a limited period. At a certain time during the initial growth, the SV “locks in” to a normal mode structure and remains in that structure so that it may grow exponentially with time. In contrast to the SVs on an f plane, the azimuthal distribution of the SVs on a β plane becomes more asymmetric, and the extent of the asymmetry increases as the strength of the beta gyres increases. On the β plane, all first and second SVs calculated in this study have an azimuthal wavenumber-1 structure at the optimization time, regardless of whether the vorticity gradient of initial TC-like vortices changes sign and the TC-like vortices include the beta gyres at initial time. It is found that when the first and second SVs are used as ensemble initial perturbations, the linear combination of the initial first and second SVs can shift the vortex toward any direction at the optimization time. This is true even when SVs with a low horizontal resolution are used as initial perturbations, as in the European Centre for Medium-Range Weather Forecasts (ECMWF) and Japan Meteorological Agency (JMA) ensemble prediction system. Such wavenumber-1 perturbations could be useful for generating sufficient spread among the tropical cyclone tracks in ensemble forecasts.