METR 2413 - Homework 7                                       Name: _____________________________

Date assigned: 5 March, 2004
Date due: 12 March, 2004

1. Using the equation for potential temperature                                 with κ = R/cp

derive the following expression for the vertical gradient of potential temperature:   .

 


Take ln [                         ]   è  ln(θ) = ln(T) + κln(1000) – κln(p)   (now take d/dz)

(1/θ)dθ/dz = (1/T)dT/dz  + 0 – (κ/p)dp/dz  

(use hydrostatic approx: dp/dz = -ρg, ideal gas law: p = ρRT and κ = R/cp)

(1/θ)dθ/dz = (1/T)dT/dz – [(R/cp)/(ρRT)](-ρg)    (now simplify the final term)

(1/θ)dθ/dz = (1/T)dT/dz + (1/T)(g/cp)

(multiply by θ and note that g/cp is the dry adiabatic lapse rate: Γd)

dθ/dz = (θ/T)dT/dz + (θ/T)Γd

or:    (4 points)

 

What does this equation imply about the change of potential temperature with height for dry adiabatic conditions?

 

If dry adiabatic then dT/dz = - Γd  è  From the equation this gives dθ/dz = 0

Or, the potential temperature (θ) is constant with height for dry adiabatic conditions. (1 point)

 

What is the sign of the vertical gradient of potential temperature for unstable dry adiabatic motion?

For unstable dry adiabatic motion we must have dT/dz < -Γd

From , dθ/dz must therefore be negative, since the term will be less

than zero for this case.  (1 point)



2.      If the morning environmental temperature at 850 hPa is 10oC, estimate the afternoon maximum temperature at sea level assuming a mixed layer depth of 150 hPa.  Under what environmental conditions is this estimate most valid?  Would you expect the maximum temperature to be higher or lower if the mixed layer depth was shallower than 150 hPa?


The maximum temperature will be equal to θ(850mb), assuming that the boundary layer completely mixes out and becomes dry adiabatic.

 


Use                              Max Temperature = θ(850mb) = (283K)(1000mb/850mb)(287/1004)


Max Temperature ~ 296.5K = 23.5°C  (2 points)

This estimate will be most valid with clear skies and moderate/strong winds.  These conditions will provide the greatest amount of mixing, resulting in a dry adiabatic layer.  (1 point)

If the mixed layer depth was shallower, a lower maximum temperature would be expected.  (1 point)