The Thickness equation (Hypsometric equation)

Atms 4310 / 7310 Lab 4Anthony R. Lupo

The Thickness equation (Hypsometric equation)

This is where we’ll start. The ‘w’ or 3rd equation of motion:

77736 10101010101010

)cos(21

otherFricViscgzp

dtdw

The Thickness equation (Hypsometric equation)

Thus, the 3rd equation of motion reduces to hydrostatic balance.

Hydrostatic balance: Between PGF in the vertical and gravity.

The Thickness equation (Hypsometric equation)

v PGF

gravity

zp

1

The Thickness equation (Hypsometric equation)

Well we can take hydrostatic balance and invoke the Chain rule:

Then substitute ideal gas law for dry air, and invoke “the snake” from z1 to z2, thus bringing us to

tz

gtp

1

zu

zl

pu

pl tz

tp

pT

gR

The Thickness equation (Hypsometric equation)

Temperature is a variable (giving us an improper integral – darn it!), how to get around this?

plpu

gTR

zlzuz

or

tz

tp

pgTR zu

zl

pu

pl

ln

1

The Thickness equation (Hypsometric equation)

Trick Treat Temperature (and that’s soo sweet) as constant by averaging in the zu – zl layer

*** (Important!) The thickness between pl and pu increases with increasing mean temperature since the density decreases with increasing temperature at a given pressure. (If hydrostatic balance, then mass between two pressure levels is the same.)

To account for moisture, you can use Tv in place of T, and then use Rd = 287.04 J/K kg

The Thickness equation (Hypsometric equation)

Simple applications

1) Mean virtual Temperature increases in a layer 0.5 K for a 10 m increase in the 1000- 500 hPa thickness. I can use this to account for advection in forecasting!

Let’s see:

Tv = (Dz g) / (Rd ln (pu/pl))

The Thickness equation (Hypsometric equation)

= (10 m * 10) / (287 * 0.001) = 0.5 K!

2) We could use to calculate the “critical” thickness for the “mean” freezing of water in the 1000 – 700 hPa layer.

By appointing Tv = 273.15 K, we can calculate that 2850 meters is this “critical thickness, so this thickness value could represent a “rain – snow” line. Since we need the temperature to be below freezing in the cloud and snowfall layer.

The Thickness equation (Hypsometric equation)

This is where the “540” line comes from, grind out the calculation and it gives you “5436” m!!!

So for 850 – 1000 hPa thickness…….

z = [(-Rd Tv) / g] * ln (pu/pl)

The Thickness equation (Hypsometric equation)

= [(-287.04 * 273.15)/9.81] * ln (700 / 1000)

= 2850 m, aha, the 285 line. This works well too!

The Thickness equation (Hypsometric equation)

The End!

The Thickness equation (Hypsometric equation)

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