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Hypsometric Equation with Lapse Rate


hm8

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So I'm having trouble with this homework problem...

 

Calculate the 1000- to 850-hPa thickness for an atmosphere with the constant lapse of Γ = 6.5 K km-1. The temperature and pressure at the surface (z0 = 0 m) are T0 = 273 K and ps = 1000 hPa.

 

I did a couple problems with an isothermal atmosphere fine...but I can't seem to figure out how to incorporate a lapse rate into the equation correctly.

 

Thickness = (Rd/g)* (T/P)*dP

So when I plug in the lapse rate (Γ) I get

Thickness = (Rd/g)* [(T0 - Γz)/P]*dP

But then I have that z variable in there, and I'm not sure how to change it to P so I can integrate

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So I'm having trouble with this homework problem...

 

Calculate the 1000- to 850-hPa thickness for an atmosphere with the constant lapse of Γ = 6.5 K km-1. The temperature and pressure at the surface (z0 = 0 m) are T0 = 273 K and ps = 1000 hPa.

 

I did a couple problems with an isothermal atmosphere fine...but I can't seem to figure out how to incorporate a lapse rate into the equation correctly.

 

Thickness = (Rd/g)* (T/P)*dP

So when I plug in the lapse rate (Γ) I get

Thickness = (Rd/g)* [(T0 - Γz)/P]*dP

But then I have that z variable in there, and I'm not sure how to change it to P so I can integrate

 

 

Slide 10 may help:

 

web.missouri.edu/~lupoa/atms4310lab5.ppt

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So I'm having trouble with this homework problem...

 

Calculate the 1000- to 850-hPa thickness for an atmosphere with the constant lapse of Γ = 6.5 K km-1. The temperature and pressure at the surface (z0 = 0 m) are T0 = 273 K and ps = 1000 hPa.

 

I did a couple problems with an isothermal atmosphere fine...but I can't seem to figure out how to incorporate a lapse rate into the equation correctly.

 

Thickness = (Rd/g)* (T/P)*dP

So when I plug in the lapse rate (Γ) I get

Thickness = (Rd/g)* [(T0 - Γz)/P]*dP

But then I have that z variable in there, and I'm not sure how to change it to P so I can integrate

This might not be the most realistic way of doing this, but you could plot your temperature profile on a skew-T and then use the method of equal areas to find a mean temperature of the layer. Then that way temperature isn't a variable anymore. You would make T - the lapse rate a constant that you can therefore pull out of the integral. Then you just have to integrate (1/P)dP.

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