Adiabatic lapse rate is defined as $$\Gamma_a \equiv -\left(\frac{dT}{dz}\right)_{\rm parcel}=-\frac{R_{\rm a} T}{c_p \,p}\left(\frac{dp}{dz}\right)_{\rm parcel} = \frac{g}{c_p}$$ $R_a$ is gas consntant divided by molar mass. If I have a parcel of dry air, I have dry adiabatic lapse rate.

Now, for standard atmosphere, I do not have $p(z)$ but I have $T(p)$ (in form of point data). Neither have I $c_p(p)$ and neither $g(p)$ and neither. So my idea to get the adiabatic lapse rate in general is: $$-\left(\frac{dT}{dz}\right)_{\rm parcel}=-\left(\frac{dT}{dp}\right)_{\rm parcel}\left(\frac{dp}{dz}\right)_{\rm parcel}=\left(\frac{\Delta T}{\Delta p}\right)_{\rm parcel} \rho g=\left(\frac{\Delta T}{\Delta p}\right)_{\rm parcel} \frac{p}{R_a T} g$$ Is this a correct approach?

Now to get dry adiabatic lapse rate I have absolutely no idea how to approach that because of I do not know $c_p(p)_{\rm dry-air}$ and neither e.g. $g(p)$. The only thing I know is the molar mass of dry air. Are those data about $T(p)$ for the atmosphere of any use for dry adiabatic lapse rate? What relations am I missing?

• While the same calculations appear in astronomy, the notation and context is different enough I might not know what I'm talking about. Still, feel like your first proposed equality is missing the rest of the chain rule: $dT/dz = (\partial T/\partial p)\vert_\rho (dp/dz) + (\partial T/\partial\rho)\vert_p (d\rho/dz)$. I suppose this all comes down to what $\Delta$ means, and whether you take $p$ to be a reparameterization of altitude (in which case what you wrote makes more sense) or one of necessarily two abstract thermodynamic abscissas to vary.
– user10851
Commented Feb 9, 2016 at 6:01

How much does $$g$$ vary between ground level and $$10 \, \mathrm{km}$$ altitude. The heat capacity of dry air is also pretty constant over the temperature range of the troposphere. At ground level, the globally averaged temperature is about $$25\sideset{^{\circ}}{}{\mathrm{C}} ,$$ and at about $$10 \, \mathrm{km} ,$$ the temperature of the air is on the order of about $$-60\sideset{^{\circ}}{}{\mathrm{C}} .$$ Look up the heat capacity of air at these temperatures and see how much or little it varies. Then you'll know what to do.
• The above problem relates to Jupiter. I have pressure levels and temperature at those levels. I am confused about the whole concept of dry air. If I have something called "dry air", I would assume, it has some given and, most importantly, constant composition, thus $c_p$ of dry air should be constant (independent of altitude or pressure). Then the lapse rate only changes as gravity changes. So to get adiabatic lapse rate for dry air on jupiter, I should know $c_p$ of dry air on jupiter which I do not know. The only thing I know related to the dry air on jupiter is its molar mass. Commented Feb 9, 2016 at 12:29