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

Question

According to my education as a sailplane pilot our troposphere is in good approximation subject to adiabatic processes. Using adiabatic equations of (nearly ideal) gases, the temperature gradient with pressure can be derived easily:

$dp/dT = \frac{p}{RT}\cdot c_p$

On the other hand, I read recently, that in meteorology a so called potential temperature is defined with regard to standard pressure and adiabatic change:

$\theta = T \left( \frac{p}{p_0} \right)^{-\frac{R}{c_p}} $

According to my book (Principles of Planetary Climate, R. T. Pierrehumbert) this potential temperature rises with height:

This is, however, in contradiction to adiabatic changes, because in that case

$d\theta = \frac{d \theta }{dT} \cdot dT + \frac{d \theta}{dp} \cdot dp = \ldots = 0$

What does that mean with regard to the adiabatic assumption? Is it not so good as often stated? Or is there a mistake in my interpretation?

Answer 1

What does that mean with regard to the adiabatic assumption?

In the thermosphere, the environmental lapse rate is usually (almost always?) lower than is the dry adiabatic lapse rate. The environmental lapse rate averages to about 6.5 °C per kilometer while the adiabatic lapse rate is about 9.8 °C per kilometer. Once rising air reaches 100% humidity, the water transfers from the gas phase to the liquid phase, resulting in the release of latent heat. This results in an increase of potential temperature.

The stratosphere is the home of the ozone layer. The capture of incoming high frequency sunlight not only increases the potential temperature with increasing height, it also increases the actual temperature with increasing height.

Answer 2

The graphs means that that the troposphere and stratosphere are not well-mixed by turbulence. They are thermally stratified and there is not much turbulence that would mi the layers.

In the boundary layer, where turbulence mixes the air, you will find the potential temperature to be constant in the so-called mixed layer during the day in the convective boundary layer. This is because the layer with different potential temperature were mixed together by turbulence into a large mixed layer with almost uniform mean (in some Reynolds-averaging sense) potential temperature.

That also means that the mixing by turbulence is almost adiabatic.

Answer 3

Meanwhile I found my problem: I confused active transport of a particular parcel from z to z+Δz with two parcels 1, 2, spatially being separated by Δz at the same time. In the former case, assuming adiabatic state change, θ will not change while T changes by $$\Delta \theta =-g/c_p \cdot Δz$$

However, for the second case there is no limiting restriction regarding θ1 and θ2, since 2 could be energetically more "charged" as compared to 1. Of course I know, that a positive gradient of θ(z) yields a statically stable layer structure, while negative layered atmospheric structures are intrinsically unstable, but this is something else.