Hydrostatic Lapse (Part 1 of 3)
In this series of articles called Criminal Clouds, we will be exploring how gravity effects the thermodynamics of gases. While not noticeable at a small scale, these properties become evident when dealing with large bodies of gas, such as the atmosphere. For instance, any time you find yourself packing a jacket for a journey up a mountain, your demonstrating that you understand temperature generally falls with increasing elevation.
It is this lapse in temperature, that will be the focus of this first article, where I will be making the case, for the existence of a particular type of lapse, that I will refer to as Hydrostatic Lapse, which is the temperature lapse that forms in a body of gas, that is at rest.
The reason gases should be expected to form a temperature differential, is that individual gas molecules are kinetic and move about in complete chaos. Like other objects, these molecules will slow when moving upwards and accelerate when moving downwards. As the temperature of a gas at a particular position, is proportional to the average velocity of its molecules at that location, gases are cooler in their upper portions.
Although this rationale would be viewed as common sense by some, others will no doubt question this logic and deny the existence of hydrostatic lapse. Luckily for those readers, I will go above and beyond your typical internet crackpot, in making the case for its existence.
Adiabatic Lapse
Let’s begin with a different type of lapse, which occurs to mobile parcels of gas, surrounded by an atmosphere. This type of lapse is well-accepted and driven by an adiabatic process, hence the name Adiabatic Lapse.
To understand adiabatic lapse, consider a hypothetical scenario, where a parcel of air is contained within a balloon, that does not apply pressure to the parcel, allowing it to expand and contract to match the pressure of the atmosphere, yet inhibiting thermal conduction. While where at it with ‘Idyllic’ balloons, let’s assume that the air is an ideal gas and that the atmosphere is hydrostatic.
If this parcel of air is lifted, the parcel will expand as it enters higher altitudes with lower pressures. This expansion requires the air parcel to use its internal heat energy to preform work, as it pushes out on the surrounding atmosphere; this results in the air parcel cooling as it is lifted. The lapse in temperature of the parcel can be quantified by the below formula:
With the Dry Adiabatic Lapse Rate calculated with the below formula:
If this is a well-accepted phenomenon, what if it were shown the Dry Adiabatic Lapse Rate formula, requires the hydrostatic atmosphere surrounding it to have an equal lapse rate? Would you conclude hydrostatic lapse is a legitimate phenomenon?
The Proof of Equivalence
This equivalence can be shown by direct proof. Let’s start by going back to our hypothetical scenario and beginning with an adiabatic process formula, which can also determine the temperature of the parcel, at its final altitude,
Because the pressure of the parcel matches that of the atmosphere, the pressure at its final altitude can be determined using the below formula:
The formula for the final altitude’s pressure can now be inserted into the parcel temperature formula and simplified:
As the index outside the brackets of this simplified formula is rather abstract, the index can be further modified by substituting the specific heat ratio with the below formula:
Which can be inserted into the index and simplified:
The numerator of the index can also be substituted with the result of the below formula:
Which can now be inserted into the index:
Because the formula for the temperature of the parcel contains an additional index within the outer bracket, these two indices can be consolidated by multiplication:
This consolidated index can be further simplified by substituting the result of below formula, which assumes the mass of the parcel is equal to 1 unit of mass:
This results in the index simplifying to:
This consolidated and simplified index can now replace the indices of the parcel temperature formula:
As the above formula is equal to the Adiabatic Lapse formula, we can now begin to solve for the Hydrostatic Atmosphere’s Lapse Rate:
As the index on the right side is equal to the lapse rate of the parcel, the right side can be further simplified:
The indices can be inversed and switched so that each side only contains one lapse rate type:
Because both sides are equal and identical in composition, with the exception of the lapse rate type, we can conclude that these different lapse rates are in fact equal.
What this proof has shown, is that the formula used to predict how much cooling a moving parcel of air will experience with elevation, will only be accurate if the surrounding hydrostatic air falls in temperature at the same rate.
Because the legitimacy of the Dry Adiabatic Lapse Rate is dependent on there being an equal Hydrostatic Lapse Rate, those who accept one must accept the other.
If you’re still sceptical, please drop a note where your concerns arise. Otherwise, let’s move to part two.
