QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY Q. J. R. Meteorol. Soc. 135: 2168–2178 (2009) Published online 1 October 2009 in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/qj.512 On time-evolution of a balanced circular vortex M. V. Kurgansky a,b * a Department of Geophysics, Faculty of Physics and Mathematics, University of Concepci´ on, Concepci´ on, Chile b A. M. Obukhov Institute of Atmospheric Physics, Russian Academy of Sciences, Moscow, Russia ABSTRACT: A simplified version of Eliassen’s cyclostrophically and hydrostatically balanced circular vortex is considered, when the relative distribution of the tangential velocity is the same across the vortex at all altitudes (the similarity assumption). This vortex model mimics the high-wind core of an intense tropical cyclone and might also be applicable to major tornadoes and to Martian giant dust devils. A self-consistent problem is analytically solved to specify a slow time evolution of the primary and the secondary circulation in the vortex under the influence of the diabatic heating and the surface friction. The results show that a self-similar balanced warm-core vortex can either be stationary or decay (algebraically) in time, the latter for a vortex flow with incomplete similarity of the velocity field when the characteristics of the primary and the secondary circulation are scaled with height differently. Possible applications to the axisymmetric spin-down dynamics of tropical cyclones are briefly discussed. Copyright c  2009 Royal Meteorological Society KEY WORDS Sawyer-Eliassen equation; tropical cyclones; axisymmetric spin-down of hurricanes Received 25 September 2008; Revised 26 March 2009; Accepted 11 August 2009 1. Introduction Arnt Eliassen (1951) was the first to determine a slow meridional circulation in a cyclostrophically and hydro- statically balanced vortex, caused by sources of heat and angular momentum. For vortices which are stable with respect to axisymmetric disturbances, the stream function of this secondary circulation was found to sat- isfy a generalized Poisson equation in the meridional plane. In the modern literature, especially related to the- ories of frontogenesis, this equation is often named the Sawyer-Eliassen (SE) equation (Sawyer, 1956; Eliassen, 1962). To this day, Eliassen’s (1951) model remains a useful starting point for understanding the axisymmetric behaviour of intense mesoscale vortices in the presence of local sources of heat and angular momentum; see e.g. Hendricks et al. (2004) and Montgomery et al. (2006) and references therein. Closer to the end of his fundamental paper, Eliassen (1951) stated that after determination of the secondary circulation one can reuse the governing equations to calculate the rate of change of the primary circulation. In a vortex which is not in a stationary state, the primary circulation, and hence the sources of heat and angular momentum and the secondary circulation, must continually change in time. About such a vortex flow, Eliassen (1951) wrote: ‘Very little can be said about the characteristics of such changes. If a stationary state exists, one would perhaps expect that the vortex would approach this stationary state asymptotically. If there are no stationary states, then the changes must go on forever’. ∗ Correspondence to: M. V. Kurgansky, Department of Geophysics, Faculty of Physics and Mathematics, University of Concepci ´ on, Casilla- 160C, Concepci´ on, Chile. E-mail: kurgansk@udec.cl In the subsequent decades, many studies based essentially on Eliassen’s ideas have been directed at understanding axisymmetric hurricane intensification and quasi-steady equilibrium, see e.g. an extensive but not of course complete reference list on the subject in Dunker- ton et al. (2008), but comparatively little work has been directed at a further theoretical analysis of non- stationary Eliassen’s balanced vortices from the above- outlined broader perspective. In a more restricted sense this issue was addressed by A. Eliassen himself (Eliassen, 1971; Eliassen and Lystad, 1977) with the focus on the frictionally driven secondary circulation and its mutual interaction with the primary circulation in a barotropic vortex flow leading in particular to the vortex spin-down. Montgomery et al. (2001) overviewed these ideas and underlined in summary that future work should consider the effects of stable stratification on the spin-down pro- cess. Following this pathway, a simplified model of cyclostrophically and hydrostatically balanced circular vortex is considered in this study, when the relative dis- tribution of the tangential velocity is the same across the vortex at all altitudes (the similarity assumption). The considered vortex model mimics the high-wind core of an intense tropical cyclone and might also be applicable to major tornadoes and Martian giant dust devils. In par- ticular, it captures such elements of reality as decrease of the cyclonic circulation with height and moderate expansion of the vortex core with height. The above similarity assumption, which makes the system of fluid dynamical equations manageable, eventually goes back to Schl¨ uter and Temesv´ ary (1958), who solved a mathemat- ically similar magneto-hydrostatic problem, and was used by Kurgansky (2005) in a study of steady dust-devil-like Copyright c  2009 Royal Meteorological Society