# Computation of geostrophic wind

I am attempting to calculate the zonal and meridional components of geostrophic wind using the equations below:

$$u_g = -\frac{g_0}{f} \cdot \frac{1}{r} \cdot \frac{\partial \Phi }{\partial \phi}$$

$$v_g = \frac{g_0}{f} \cdot \frac{1}{r \cdot cos(\phi)} \cdot \frac{\partial \Phi}{\partial \lambda}$$

In the equations:

$$\phi, \lambda$$ are the latitude and longitude respectively, $$u, v$$ is the zonal and meridional components respectively, $$f$$ is the Corlilios parameter, $$r$$ is the mean radius of the Earth, and $$\Phi$$ is geopotential height.

I got these equations from the link below:

http://colaweb.gmu.edu/dev/clim301/lectures/wind/wind-geo

My question is primarily concerned with the calculation of geopotential height. Based on my understanding thus far, geopotential is calculated using this equation:

$$|\nabla \Phi| \approx 9.80665 + (1 - 0.00259 \cdot cos(\phi)(1 - 3.14 \times10 ^ {7} z)$$

Note: $$|\nabla \Phi|$$ is the local gravitational constant. It is sometimes I believe represented as $$g_A$$.

Which plugs into the equation for geopotential height below, where $$g_o$$ is the standard gravity at mean sea level:

$$\Phi = \frac{|\nabla \Phi|}{g_o}$$

I think this works fine for the zonal component, however, it results (again I think) in an output of 0 for the meridional component. This cannot be true, can it? Am I missing something crucial?

I do know one extra piece of information. Under the continuity equation:

$$\frac{\partial u_g}{\partial \phi} + \frac{\partial v_g}{\partial \lambda} = 0$$

Therefore:

$$\frac{\partial u_g}{\partial \phi} = -\frac{\partial v_g}{\partial \lambda}$$

In short, I am looking for an equation that considers latitudinal and longitudinal variation in acceleration due to gravity.

I am extremely new (complete newb) to the world of atmospheric science/dynamics, so, any assistance would be greatly appreciated.

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• Hi and welcome to physics SE. Please, notice that many people here might not be familiar with this branch of physics. You should tell what each letter is, and explain "non widely used" concepts, like "where $u$ and $v$ are the velocity components of wind, $Z$ is the geopotential height defined as the height that would have the same potential if gravity were kept constant, that is: $Z=g\cdot z/g_0$, and so on. – FGSUZ Apr 25 at 22:07
• @FGSUZ I think I added your suggested changes to the question. I believe the local gravitational constant is meant to change with changes in either longitude, latitude, or, altitude, which would allow I believe the calculation of the zonal, and meridional components of geostrophic wind. – Conor Casey Apr 25 at 22:22
• we assume latitudinal dependence of g is weak. height dependence of g is not considered. Can you show me a book in meteorology where they do take these dependencies into account ? – gansub Apr 28 at 0:31

## 1 Answer

If you consider latitudinal variation in acceleration due to gravity, force balance with centrifugal force and true gravity will give

$$mg_{\phi} = mg - m \omega^{2}r \cos \phi$$, $$r = R\cos \phi$$

$$g_{\phi} = g - R\omega^{2} \cos^{2} \phi$$

$$\Phi = \int_{0}^{z} g_{\phi} dz$$

$$r$$ is radius of Earth and $$R$$ is perpendicular distance from the axis of rotation to the point on surface under consideration

• I understand that there is a latitudinal variation in acceleration due to gravity, however, what I am looking for is an equation that includes longitudinal variation as well. – Conor Casey Apr 27 at 16:58
• are you talking about longitudinal variation in g ? that can occur due to topography I guess and could be explained by some empirical relation. nonetheless, if you have proper expression of g you can integrate over height surfaces to get geopotential – Arijeet Dutta Apr 27 at 18:36
• Yeah, I am talking about longitudinal variation in g, and I am looking for an equation that can encapsulate that. Could you expand your point on "if you have proper expression of g you can integrate over height surfaces to get geopotential" please? – Conor Casey Apr 27 at 18:44
• I am not aware of any such expression. What I meant if you get g as function of latitude and longitude you can use the last equation from my answer to get geopotenital and consequently find geostrophic wind – Arijeet Dutta Apr 27 at 18:52
• @ArijeetDutta Can you provide a source for your first equation ? – gansub Apr 28 at 6:21