I'm looking to demonstrate the rotational behaviour of swirling storms with a numerical simulation. So far it includes pressure-gradient, Coriolis acceleration, frictional forces and the acceleration due to gravity.
In a non-inertial frame of reference (surface of the Earth), $(O', \vec{x'}, \vec{y'}, \vec{z'})$, $\vec{x'}$ points south, $\vec{y'}$ points eastward and $\vec{z'}$ upward. I've found that the acceleration of air masses is expressed as:
$\ddot{x'}=-\cfrac{1}{\rho}\cfrac{\partial p}{\partial x'} - \nu \dot{x'} + 2\omega\sin(\lambda)\dot{y'}$
$\ddot{y'}=- \cfrac{1}{\rho}\cfrac{\partial p}{\partial y'} - \nu \dot{y'} -2\omega(\sin(\lambda)\dot{x'} + \cos(\lambda)\dot{z'})$
$\ddot{z'}=- g - \cfrac{1}{\rho}\cfrac{\partial p}{\partial z'} - \nu \dot{z'} + 2\omega\cos(\lambda)\dot{y'}$
where the last terms are the deflections due to the Coriolis effect, the middle ones friction forces and the first ones pressure-gradient forces. $g$ is the acceleration due to gravity ($g=9.8 \ m.s^{-2}$).
I would work in 2 dimensions since my initial conditions (pressure maps) are given for a specific altitude. But then I would ignore some terms of the Coriolis and gravity, and many other effects I bet. I can go for a 3d description of the phenomenon, but it might be too difficult to deal with a collection of pressure maps at different altitudes.
Questions:
Is the set of equations physically correct to model accurately the spiral-ish rotation of hurricanes? If not, what are the other forces that would be worth adding?
I'm doing a numerical simulation, and I want to test it with pressure maps. The first thing I've done was removing the terms that act in the $\vec{z'}$ direction since my initial conditions (pressure maps) are given for a specific altitude. Is this approximation still correct to achieve my goal?
In addition to question 2, if the model should rather be described in a 3 dimensional coordinate system, it might be too difficult to deal with a collection of pressure maps at different altitudes, is it?
I'm definitely not an expert and my readings in the subject didn't help.