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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:

  1. 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?

  2. 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?

  3. 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.

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    $\begingroup$ I'm not an expert either, but shouldn't there be M.r.ω^^2 terms for conservation of angular momentum? Or is this sufficiently covered by the 'Coriolis' term? $\endgroup$ – Gordon Stanger Dec 30 '15 at 14:09
  • $\begingroup$ No the centrifugal force isn't included. $\endgroup$ – tinyyy Dec 30 '15 at 15:37
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    $\begingroup$ Take a look at www2.mmm.ucar.edu/people/bryan/cm1/cm1_equations.pdf which details the equations used in CM1 which is an idealized model used for severe storms and hurricanes. Also note by convention you'll typically see positive x as as east and positive y as north. $\endgroup$ – casey Dec 30 '15 at 17:41
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    $\begingroup$ What you described in your question are viscous shallow water equations. They can be expressed in single- (2-d) or multi-layer (3-d) form. Please specify in your question whether you intend to study the 2-d or 3-d system. The system of equations is correct to the limits of the physical processes that it describes - gravity-inertial waves. In case of 3-d system, your linear friction $\nu u$ will be inapropriate, as $\nu$ is very different between $x, y$ and $z$, and is very different near the surface and aloft. Your solution will permit nearly-geostrophic vortices (not hurricanes). $\endgroup$ – milancurcic Dec 30 '15 at 19:14
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    $\begingroup$ Also note that centrifugal force is in there, you just don't see it as an explicit term in the Cartesian reference frame. It would pop out as a term in cylindrical reference frame, but in this form it is embedded in $d^2x/dt^2$ and $d^2y/dt^2$. $\endgroup$ – milancurcic Dec 30 '15 at 19:16

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