<p>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.</p>

<p>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:</p>

<p>$\ddot{x'}=-\cfrac{1}{\rho}\cfrac{\partial p}{\partial x'} - \nu \dot{x'} + 2\omega\sin(\lambda)\dot{y'}$</p>

<p>$\ddot{y'}=- \cfrac{1}{\rho}\cfrac{\partial p}{\partial y'} - \nu \dot{y'} -2\omega(\sin(\lambda)\dot{x'} + \cos(\lambda)\dot{z'})$</p>

<p>$\ddot{z'}=- g - \cfrac{1}{\rho}\cfrac{\partial p}{\partial z'} - \nu \dot{z'} + 2\omega\cos(\lambda)\dot{y'}$</p>

<p>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 ($g=9.8 \ m.s^{-2}$).</p>

<p>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.</p>

<p>Questions:</p>

<ol>
<li><p>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?</p></li>
<li><p>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?</p></li>
<li><p>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?</p></li>
</ol>

<p>I'm definitely not an expert and my readings in the subject didn't help.</p>

CC BY-SA 3.0
2015-12-30T14:09:29.380Z
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?
Gordon Stanger
CC BY-SA 3.0
2015-12-30T15:37:48.873Z
No the centrifugal force isn't included.
tinyyy
CC BY-SA 3.0
2015-12-30T17:41:34.133Z
Take a look at http://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.
casey
CC BY-SA 3.0
2015-12-30T19:14:20.520Z
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).
milancurcic
CC BY-SA 3.0
2015-12-30T19:16:53.897Z
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 $d^2y/dt^2$.
milancurcic
CC BY-SA 3.0
2016-01-08T21:03:32.563Z
@milancurlic How that's defined in detail so that centrifugal acceleration is embedded in $d^2x/dt^2$ and $d^2y/dt^2$? As you pointed above, I'm absolutely not aware of the expression of friction forces, and I went for something linear to wind velocity with a constant equal to ~$10^{-6}s^{-1}$, which is jet stream drag.
tinyyy
CC BY-SA 3.0
2016-01-28T20:20:11.497Z
I don't think that it is that easy. You have 2 equations for at least 3 unknowns ($\dot{x}, \dot{y}, p$) - you'll need the mass continuity eqn in that case. If $\rho=\rho(T,p)$ instead of $\rho=\rho(p)$ I think you'll also need an energy eqn.
Lukas
<atmosphere-modelling><numerical-modelling><severe-weather>
Modeling swirling storms
tinyyy
# Modeling swirling storms

(Asked by: username_0 on 2015-12-30)

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.

**Tags:** atmosphere-modelling, numerical-modelling, severe-weather

> **Comments:**
> **username_1**: 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?  
> **username_0**: No the centrifugal force isn't included.  
> **username_2**: Take a look at http://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.  
> **username_3**: 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).  
> **username_3**: 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 $d^2y/dt^2$.  
> **username_0**: @milancurlic How that's defined in detail so that centrifugal acceleration is embedded in $d^2x/dt^2$ and $d^2y/dt^2$? As you pointed above, I'm absolutely not aware of the expression of friction forces, and I went for something linear to wind velocity with a constant equal to ~$10^{-6}s^{-1}$, which is jet stream drag.  
> **username_4**: I don't think that it is that easy. You have 2 equations for at least 3 unknowns ($\dot{x}, \dot{y}, p$) - you'll need the mass continuity eqn in that case. If $\rho=\rho(T,p)$ instead of $\rho=\rho(p)$ I think you'll also need an energy eqn.  

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## 0 answers

