A mathematical model for viscous flow dynamics of tropical cyclones

Abstract

A mathematical model for tropical cyclones’ winds, taking into account various crucial considerations makes the analysis of Cecil and Majdalani (2022) more realistic. Drawing inspiration from their work which obtains the axial velocity from the stream function, we incorporate the notion of viscous flow within cyclone dynamics, a modification that brings present model more closely aligned with the real-world conditions. Our key considerations include the absence of axial velocity at the ground, zero radial velocity at the cyclone’s centre, and outside the eye-wall. In order to derive pressure, we integrate axial pressure gradient with respect to axial coordinate; and as a consequence we get an arbitrary function of radial coordinate which we eliminate by using Vatistas (1991) velocity at the ground to meet the cyclostrophic balance. Azimuthal velocity and pressure are derived for viscous flows. The formulations hold good for arbitrary Reynolds number. The analysis demonstrates a positive relationship between Reynolds number and azimuthal velocity within cyclone’s eye. This trend persists within the inner eye-wall, characterized by a gradually diminishing velocity. An inflexion point is identified midway the eye and the eye-wall, where maximum azimuthal velocities are observed. The central focus of our study revolves around the influence of eye size on various velocity components and pressure. Our findings reveal that a larger eye size correlates with the development of more intense tropical storms. However, this increase in storm intensity reaches a peak and subsequently experiences a rapid decline within the rain band region compared to smaller eye cyclones. Regardless of the eye’s size, our analysis consistently demonstrates that atmospheric pressure increases as one moves away from the eye.