Tangential winds of a vortex system in a planetary surface layer
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Ó Indian Academy of Sciences (0123456789().,-volV)(0123456789( ).,-volV)
Tangential winds of a vortex system in a planetary surface layer SHEFALI UTTAM1,2 , DEEPAK SINGH1
and VARUN SHEEL1,*
1 Physical 2
Research Laboratory, Ahmedabad, India. Indian Institute of Technology, Gandhinagar, India. *Corresponding author. e-mail: [email protected] MS received 27 March 2019; revised 12 July 2019; accepted 18 July 2019
The planetary boundary layer (PBL) mediates interactions between the surface and free atmosphere. In Martian PBL, surface can force convective vortices leading to dust devils. We use the Navier–Stokes equations and the continuity equation to determine mean (with respect to time) tangential wind velocity in cylindrical co-ordinate system within the surface layer of a planetary atmosphere. We utilize Martian surface layer properties for theoretical derivation of our solution. However, our results remain valid for any planetary surface layer as long as all of our assumptions are valid. Our theoretical values of the tangential wind velocity lie well within the range of observed values. The derived equation represents the dependency of tangential velocity on both radial distances from the center of vortex, and the altitude. As we move further away from the vortex center, the effect of vortex becomes non-significant, and velocities start following the standard logarithmic profile. Due to dependency of tangential wind velocity on altitude, the tangential velocity increases as we move higher up in the vortex system. At 100 m altitude, for an order of magnitude increase in the radial distance, the mean tangential wind velocity drops by about a factor of 1.5 in magnitude. Keywords. Atmosphere dynamics; Mars atmosphere; planetary dynamics; terrestrial planets.
1. Introduction The circulation of wind in the atmosphere is driven by the pressure gradient developed between two locations, incoming energy from the Sun, and the rotation of the planet. The Navier–Stokes (NS) equations describe the motion of viscous fluid substances and are a formulation of the conservation of momentum of the system. NS equations or momentum equations in inertial frame of reference can be given as (Schlichting et al. 1955):
where ui is the component of fluid velocity, q is the density of the fluid, m is the kinematic viscosity of the fluid (= dynamic viscosity (l)/density of fluid (q)), p is the pressure in the surrounding, and g is the acceleration due to gravity of the planet. The Navier–Stokes equations in rotational frame of reference (tensorial notation) for incompressible fluid, are given as (Schlichting et al. 1955; Dong and Wu 2015): oui oui 1 op o2 ui ¼ þ m 2 g di3 2 X ijk gj uk ; þ vj q oxi ot oxj oxj ð2Þ
Dui 1 op o2 ui þ m 2 g di3 ; ¼ q oxi Dt oxj
ð1Þ
where X is the angular rotation of the planet. Though the Navier–Stokes equations represent
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J. Earth Syst. Sci. (2020)129:2
winds in any part of an atmosphere, modeling the wind profiles in the boundary layer is challenging (Petrosyan et al. 2011; Bryan et al.
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