# Show that the atmosphere is neutrally stable

1. Nov 29, 2011

### little_L_

1. The problem statement, all variables and given/known data
Assume that the atmosphere is dry, and that its temperature profile may be approximated by a linear function in height using a constant lapse rate:

T = T_0 - $\gamma$Z

where T_0 is the ground temperature. Also assume that the pressure can be approximated the following equations for a constant lapse rate atmosphere:

p = p_0((T_0 - $\gamma$Z)/T_0)^(g/$\gamma$R_d)

By using the definition of potential temperature, show that the atmosphere is neutrally stable (i.e d θ/d Z = 0 ) when the value of the constant lapse rate is equal to the dry adiabatic lapse rate, $\gamma$ = g/c_p

2. Relevant equations

equation of potential temperature:
θ = T(100/P)^k

3. The attempt at a solution
I believe i need to come up with an expression for the potential temperature using the definition given of T ( as a function of z and a constant lapse rate and the definition of p in a constant lapse rate) but I don't see how to do that to commute for dθ/dz = 0