Show that the atmosphere is neutrally stable

[little_L_]

Homework Statement

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 - [itex]\gamma[/itex]Zwhere 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 - [itex]\gamma[/itex]Z)/T_0)^(g/[itex]\gamma[/itex]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, [itex]\gamma[/itex] = g/c_p

Homework Equations

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

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

 

[nate808]

To show that the atmosphere is neutrally stable, we need to find the condition where the potential temperature does not change with height (i.e. dθ/dZ = 0).

First, let's substitute the given equations for T and p into the definition of potential temperature:
θ = T(100/p)^k

θ = (T_0 - \gammaZ)(100/p)^k

Next, we can use the given equation for p to rewrite this as:
θ = (T_0 - \gammaZ)(100/p_0((T_0 - \gammaZ)/T_0)^(g/\gammaR_d))^k

θ = (T_0 - \gammaZ)(100/p_0)((T_0 - \gammaZ)/T_0)^(gk/\gammaR_d)

Now, we can simplify this expression by combining the terms inside the parentheses:
θ = (T_0 - \gammaZ)(100/p_0)((T_0 - \gammaZ)^{1+(gk/\gammaR_d)}/T_0^{1+(gk/\gammaR_d)})

Next, we can use the definition of potential temperature again to rewrite this as:
θ = T_0(100/p_0)^{1+(gk/\gammaR_d)}((T_0 - \gammaZ)^{1+(gk/\gammaR_d)}/T_0^{1+(gk/\gammaR_d)}) - \gammaZ(100/p_0)^{1+(gk/\gammaR_d)}((T_0 - \gammaZ)^{1+(gk/\gammaR_d)}/T_0^{1+(gk/\gammaR_d)})

θ = T_0(100/p_0)^{1+(gk/\gammaR_d)}(1 - \gammaZ/T_0)^{1+(gk/\gammaR_d)} - \gammaZ(100/p_0)^{1+(gk/\gammaR_d)}(1 - \gammaZ/T_0)^{1+(gk/\gammaR_d)}

Now, we can take the derivative of this expression with respect to Z to find dθ/dZ:
dθ/dZ = -T_0(100/p_0)^{1+(gk/\gammaR_d)}(1 - \gammaZ/T_0)^{(gk/\