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The Thickness equation (Hypsometric equation)

The Thickness equation (Hypsometric equation). Atms 4310 / 7310 Lab 4 Anthony R. Lupo. The Thickness equation (Hypsometric equation). This is where we’ll start. The ‘w’ or 3 rd equation of motion:. The Thickness equation (Hypsometric equation).

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The Thickness equation (Hypsometric equation)

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  1. The Thickness equation (Hypsometric equation) Atms 4310 / 7310 Lab 4 Anthony R. Lupo

  2. The Thickness equation (Hypsometric equation) • This is where we’ll start. The ‘w’ or 3rd equation of motion:

  3. The Thickness equation (Hypsometric equation) • Thus, the 3rd equation of motion reduces to hydrostatic balance. • Hydrostatic balance: Between PGF in the vertical and gravity.

  4. The Thickness equation (Hypsometric equation) • v PGF • gravity

  5. 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

  6. The Thickness equation (Hypsometric equation) •  Temperature is a variable (giving us an improper integral – darn it!), how to get around this?

  7. 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

  8. 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))

  9. 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.

  10. 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……. • Dz = [(-Rd Tv) / g] * ln (pu/pl)

  11. The Thickness equation (Hypsometric equation) • = [(-287.04 * 273.15)/9.81] * ln (700 / 1000) • = 2850 m, aha, the 285 line. This works well too!

  12. The Thickness equation (Hypsometric equation) • The End!

  13. The Thickness equation (Hypsometric equation) • Questions? • Comments? • Criticisms?

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