# Earth Cubed

## Hoskins and Simmons (1974) Coordinate System

sHoskins and Simmons (1974) Coordinate System

Hoskin and Simon used the following coordinate system:

$\mu = sin( \theta )$ where theta is the latitude.

$\sigma = pressure/P_*$ Where $P_*$ is the surface pressure and $\sigma$ is the verticle coordinate.

$\lambda$ is the longitude.

This coordinate system is also used in the following model which was recomended to me as a simple model:

As the complexity of general circulation models has been and still is growing considerably, it is not surprising that, for both education and research, models simpler than those comprehensive GCMs at the cutting edge of the development, are becoming more and more attractive. These medium complexity models do not simply enhance the climate model hierarchy. They support understanding atmospheric or climate phenomena by simplifying the system gradually to reveal
the key mechanisms. They also provide an ideal tool kit for students to be educated and to teach themselves, gaining practice in model building or modeling. Our aim is to provide such a model of intermediate complexity for the university environment: the PlanetSimulator. It
can be used for training the next GCM developers, to support scientists to understand climate processes, and to do fundamental research.

http://www.mi.uni-hamburg.de/216.0.html?&L=1

In order to use the coordinates given by Hoskins and Simmons in the Navier Stokes equations it is necissary to find the operations for Div Grad and Curl in this coordinate system. You can get a free PDF here:

http://www.scribd.com/doc/2590597/Lectures-on-Transformation-of-Coordinates

Which shows how to derive these operations for any coordinate system. These derivations express the apove operations in terms of scaling quantities. The scaling quantitys are the magnitude of the derivative of the change in postion, with respect to a change in one of the coordinates in the new coordinate system.

In our case for the coordinate $\mu$ the scalling quanity is:

$h_{\mu}=\left| \frac{ \partial \vec r}{\partial \mu} \right| = \left| \frac{ \partial (r \theta)}{\partial \mu} \right|$

because when you change latitude the distance moved is proportional to the radius multiplied by the change in latitudinal (measured in radians).

Now expressing the latitude in terms of $\mu$
$h_{\mu}=\left| \frac{ \partial (r \ sin^{-1}( \mu ))}{\partial \mu} \right| = \frac{r}{\sqrt{1-\mu^2}}$

Now for the $\lambda$ which is the longitude coordinate, $h_{\lambda}$ depends on how you define zero latitude. The distance changed for a given change in longitude is the product of the change in logitude (measured in radians) multiplied by the distance to the earths axis of rotation. If zero lattitude is the center of the earth then this distance is $r cos(\theta)$ if zero latitude is the north pole (typical in spherical coordinates) then this distance is $r sin(\theta)$.

We will see that taking zero latitude as the north pole will be simpler.

$h_{\lambda}=\left| \frac{ \partial \vec r}{\partial \lambda} \right| = \left| \frac{ \partial (r \lambda sin(\theta))}{\partial \lambda} \right|=r \ sin(\theta)$

But $\theta=sin^{-1}(\mu)$ Therefore:

$h_{\lambda}=r \mu$

The vertical coordinate $\sigma$ is the ratio of the hydrostatic pressure to the hydrostatic pressure at sea level.

In Hoskins & Simmons the hydrostatic pressure used in the coordinate ignores the lapse rate. In some models variations on this are used. For instance in a model used by Judith Curry this coordinate is somewhat adjusted based on the surface topology of the earth. (see my post: hybird Coordinate System)

In Curries model this is done to try and keep wind flow on the horizontal plane. Anyway, the equation for hydrostatic equilibrium is:

$\frac{dp}{dh} = - \frac {mpg}{RT}$

Now:

$\frac{\partial \sigma}{\partial h}=\frac{\partial \sigma}{\partial P}\frac{\partial P}{\partial h}=\frac{1}{P_*}\frac{\partial P}{h}=-\frac{1}{P_*} \frac {mpg}{RT} =-\sigma \frac{mg}{RT}$

Hmmmmmmm: From what I’ve done the only way I can see to make the constants $\frac{mg}{RT}$ disappear is to set $T=\frac{mg}{R}$

Here is what Hoskins and Simmons have to say about temperature scaling and such.

all non-dimensionalized using as length scale the radius of the planet, a, ; as time scale the reciprocal of its angular velocity $\Omega$ ; temperature scale $a^2\Omega^2/R$ (R as the gas constant) and pressure scale $P_o=1$ Bar.

In http://www.mi.uni-hamburg.de/fileadmin/files/forschung/theomet/planet_simulator/downloads/PS_ReferenceGuide.pdf $T_o$ was taken to be 250K. Some numbers of rellevence:

Gas constant of air: 287.058 J kg−1 K−1

and for the molar mass of air:

Standard temperature =273.16k

Source for the molar mass of air:

Standard pressure=1 Atm=1013.250*10^2 Pa.
The molar mass of air at standard temperature and pressure is 28.964*10^-3 kg/mole. Molar

http://www.blurtit.com/q139778.html

Standard Gravity: 9.80665 m/s2

August 27, 2009 - Posted by | GCM (General Circulation Models), Math

1. Just want to add a link to an old climate audit post by me for future reference.

It was just me trying to compile some information well I was learning but it has some links to some papers by Judith Curry.

Comment by s243a | August 27, 2009 | Reply

2. […] Operations in Hoskins and Simmons Coordinates In my post Hoskins and Simmons (1974) Coordinate System, I derived the following scaling quantities for the vector […]

Pingback by Vector Operations in Hoskins and Simmons Coordinates « Earth Cubed | August 29, 2009 | Reply

3. […] Just to recall from the post (Hoskins and Simmons (1974) Coordinate System): […]

Pingback by Coriolis Forces « Earth Cubed | August 29, 2009 | Reply