Doesn't Increase of Potential Temperature with Height contradict Adiabatic Nature of Processes within Troposphere?

You may be forgetting that pressure also decreases with height (exponentially). Also, because $P=rho R T$, $frac{dP}{dT}=rho R$ (that is, $c_p$ does not appear). But I digress in answering your question.

Let's break down why potential temperature increases with height. Let's start with the equation: $$theta=Tleft(frac{P_0}{P}right)^{frac{R_d}{c_p}}tag{1}$$ Now, if we want to ask why $theta$ increases with height, let's rephrase $(1)$ as a function of height: $$theta(z)=T(z)left[frac{P_0}{P(z)}right]^{frac{R_d}{c_p}}tag{2}$$ Now if you follow my derivation for finding $$frac{partial theta}{partial z}=frac{theta}{T}left(frac{partial T}{partial z}+frac{g}{c_p}right)$$

You may note that the $frac{partialtheta}{partial z}=0$ iff. $frac{partial T}{partial z}=-frac{g}{c_p}=-9.8 textrm{ K km}^{-1}$ (dry adiabatic lapse rate). If $frac{partial T}{partial z}>-frac{g}{c_p}$ then $frac{partial theta}{partial z}>0$.

This does not mean that the parcel is not adiabatic, though. For a parcel to be truly adiabatic, $frac{partial theta}{partial t}+ufrac{partial theta}{partial x}+vfrac{partial theta}{partial y}+wfrac{partial theta}{partial z}=0$ Therefore, the only time a certain location can be considered adiabatic and have $frac{partial theta}{partial z}=0$ is when there is no heating ($frac{partial theta}{partial t}=0$), and no horizontal advection of $theta$ ($ufrac{partial theta}{partial x}+vfrac{partial theta}{partial y}=0$). Note, I did not exclude vertical advection of $theta$, because that is implied in the condition that $frac{partial theta}{partial z}=0$.

I'll also take pause here and list a few diabatic processes, and why there may be some justification to ignore them for your mental image.

• Radiation: with the exception of greenhouse gases, the atmosphere is mostly transparent.
• Surface heating/ turbulent transport: more important near the surface (note that $theta$ is a minimum at the surface).
• Clouds/microphysical processes: discrete and are never everywhere all the time and moisture is a completely separate variable that varies all the time. There is a way around that (to some extent- the equivalent potential temperature, for latent heat exchange).
• Chemical processes: See below about how I remember $theta$ increases with height.

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If you want a good reminder of why $theta$ increases with height, it might be easier to remember that $sgn left( frac{partial theta}{partial z}right)$ is a good indicator of static stability. And thunderstorms need to stop at some point, even if that is the tropopause (where $frac{partial theta}{partial z}>0$ due to ozone heating).