The hydrostatic equilibrium equation including temperature, T(z)

[ndgoodburn]

Homework Statement

In this problem, you'll model the lower atmosphere of Venus. The atmospheric pressure reaches 1 bar (100 kPa) in the middle of the dense cloud deck, where T ~ 350 K. At the surface, the pressure is 90 bars (9000 kPa). From the surface to the 1 bar level, the temperature T(z) decreases at a rate of dT/dz = -8 K $km^{-1}$, close to the adiabatic lapse rate of -8 K $km^{-1}$.

(b)Write down the equation for hydrostatic equilibrium, including explicitly the variation T(z).

Homework Equations

We can estimate the adiabatic lapse rate with the approximation:
$\frac{dT}{dz}|_{ad}=-\frac{g}{c_p}$

The equation for hydrostatic equilibrium:
$\Delta P = -\rho \Delta z g$

The Attempt at a Solution

Part (a) asks me to confirm the adiabatic lapse rate for Venus, which is simple by plugging in g and c_p.

Part (b) (asking for the hydrostatic equilibrium equation) is what is confusing me. I am specifically having trouble including T(z). I can find an expression for T(z) by integrating the adiabatic lapse rate approximation:

$∫\frac{dT}{dz} dz|_{ad}=∫-\frac{g}{c_p} dz$

$T(z)=T_0-\frac{g}{c_p}z$

I called the constant of integration T₀ because that makes sense conceptually.

Now, what I tried was just solving T(z) for g and plugging it into the hydrostatic equilibrium equation because I'm just trying to find a way to include it, but that just yields

$\Delta P=-\rho \Delta z \frac{c_p}{z}(T_0-T(z))$

which seems useless to me, because T(z) would quickly reduce out with a tiny bit of simplification.

Maybe someone can help by telling me if what I've done so far makes sense and if I should continue, or if I'm just missing something that would make it more clear. Thanks in advance!

 

[haruspex]

$\frac{dT}{dz}|_{ad}=-\frac{g}{c_p}$

The equation for hydrostatic equilibrium:
$\Delta P = -\rho \Delta z g$

Just guessing here, but if $\frac{dT}{dz}|_{ad}=-\frac{g}{c_p}$ then can you not write $\Delta T = -\frac{g\Delta z}{c_p} = \frac{g\Delta P}{c_p\rho}$?