  We model Earth's lower atmosphere as both incompressible and compressible
neutral fluid in motion on a rotating sphere and the viscosity of the fluid
depends on the density of the air. To justify the model, we carried out a
numerical computation of the thermodynamic and hydrodynamic properties of the
viscous atmospheric motion in two dimensions at some altitude using
Naiver$-$Stokes dynamics. To interpret the dynamics of a fluid, we considered
the governed partial differential equation (PDE) without any approximation then
we applied some sort of wind velocity, air mass density, and temperature
perturbation to the atmosphere at the boundary position $x = 0$ using the
finite difference method implemented in FORTRAN programing language. The
resulting dynamics of the atmosphere are governed by the competition among the
various dominant forces, namely, the viscous dissipation, the convective
motion, pressure gradient force, centripetal force, and Coriolis force. We
numerically obtain the resultant velocity field for the Naiver$-$Stokes
equation at a different time taken of simulation by changing the magnitude of
the Coriolis force. In addition to this, we consider the first law of the
thermodynamics applied to our model. The numerical results show that the
temperature, pressure, air mass density, and wind velocity variation with
longitude, latitude, and time in the atmospheric fluid due to viscous
dissipation, convective motion, work done by pressure gradient force, and the
bulk motion of the atmospheric fluid.
  Keywords: Naiver$-$ Stokes Equations; Finite difference method; Viscous
atmospheric motion; Viscous dissipation; convective motion.
