This is very impressive work , my friend. ]]>

Thanks for that – it looks like the sort of thing I have been looking for. The following sentence however is a good reminder of how much things have been changing.

“The Arctic ocean is ice-covered all year, while the North Atlantic and the Southern Oceans experience seasonal melting.”

That was then, this is now.

]]>On the missing molar mass, I have now clarified the equation in a comment.

R = gas constant for dry air; and is in units of J kg^{-1}K^{-1}

Pekka,

Well spotted. I updated the graphic for figure 8.

]]>Pekka,

Good idea.

Refer to The Coriolis Effect and Geostrophic Motion for the geostrophic wind equations:

u_{g} = – 1/fρ . ∂p/∂y ….[4a]

v_{g} = 1/fρ . ∂p/∂x ….[4b]

where u_{g} = W-E wind speed, v_{g} = N-S wind speed

These equations are derived from:

- 1. Newton’s law of motion (net force = change in momentum) with the net force being the result of pressure gradients, gravity and friction
- 2. These laws then adapted to a rotating frame of reference which results in two additional forces “appearing” in the equation: the coriolis force and the centrifugal force
- 3. The approximation of low values of acceleration and friction (geostrophic approximation)

So from the equations we see that the E-W wind is proportional to the change in N-S pressure, and the N-S wind is proportional to the E-W change in pressure.

It is more convenient to consider the height of a pressure surface rather than the pressure at a given height.

Using the hydrostatic balance equation it’s easy to show that:

(∂p/∂x)_{z} = gρ (∂z/∂x)_{p} ….[5a]

(∂p/∂y)_{z} = gρ (∂z/∂y)_{p} ….[5b]

where (∂p/∂x)_{z} is the change in pressure (at constant height) in the x direction and (∂z/∂x)_{p} is the change in the height of a pressure surface (geopotential height) in the x direction

So combining 4 & 5 we get:

u_{g} = –g/f . ∂z/∂y ….[6a]

v_{g} = g/f . ∂z/∂x ….[6b]

This version of the equation doesn’t include density, ρ, and given that density varies in a compressible fluid like air this is a very helpful version of the equation. It is one reason why pressure is often preferred over height as the vertical coordinate.

And back to the thermal wind equation – if we differentiate equation 4 with respect to height we get the equation shown in the article although the derivation takes a little time to write out.

]]>Here you have a great resource on this topic.

http://kiwi.atmos.colostate.edu/group/dave/at605.html

I think that some further arguments could be added to explain your formulas. Now you don’t really present any justification for them. You tell the background but skip over the final essential step. Going through the full derivation is perhaps too much but the following basic idea behind the formulas can be given:

A possible situation and the one actually observed is that rather stable geostrophic winds persist. To have such a wind something must balance the force created by the pressure gradient of your Figure 6. The only horizontal “force” available is the Coriolis force. Your formulas represent the requirement that the “pressure force” and the Coriolis force are equal in size and have opposite directions.

]]>The caption of your Figure 8 (Figure 7.21 of Marshall and Plumb) refers erroneously to potential temperature while the values and thick contours present actual temperatures rather than potential temperatures. That’s not your error but may be confusing. You can find the same image with the correct caption as a Powerpoint file from the publishers web page:

http://www.elsevierdirect.com/v2/companion.jsp?ISBN=9780125586917

Check the ppt of Chapter 07.

(Similarly I think that the missing molar mass in the formula of your previous post may be confusing although it’s not essential for understanding your points. As the equation stands the units don’t match. Thus anyone trying to reproduce the formula may have unnecessary difficulties.)

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