In Defense of the Adiabatic Lapse Rate

For some reason, the venerable Adiabatic Lapse Rate, ubiquitous throughout the solar system, has confused a small contingent of the WUWT gang, despite the fact that 97% of the science is settled.

Nevertheless, an explanation is in order for why a planetary atmosphere, or the compressed oxygen in a welder's tank, will try to arrange itself under gravity so that the temperature at the bottom is greater than the temperature at the top, with a uniform lapse rate as we examine higher elevations.

Earth's atmosphere always has things like planetary rotation, differential heating, butterfly wings, etc., upsetting its efforts to achieve an ideal lapse rate, but it is always trying to correct in that direction.

Back in 2012, Professor Robert Brown of Duke University ("rgbatduke" in comment attire) attempted to refute this universal truth in a WUWT post that offers some persuasive arguments.  Recommended reading, very persuasive.

His post has two scenarios:

  1. The usual adiabatically isolated column of air with the usual adiabatic lapse rate
  2. A more thought-provoking air column with a high-thermal-conductivity insulated silver wire running from bottom to top, with ends exposed to the air at top and bottom, potentially setting us 97 percenters in the untenable position of defending an analog to perpetual motion

We'll examine the usual situation first.

The Usual Scenario

Temperature & density of air column

Column A - Let's start with an adiabatically isolated column of air with no gravity.  Pressure, density, and temperature are uniform throughout.

Column B - When we turn on gravity, the air heads for the bottom, increasing the pressure there and decreasing the pressure at the top.  Compressing the air increases temperature at the bottom, decreasing pressure cools air at the top, uncontroversial.  The fall may result in an initial overshoot of the standard lapse rate.

Column C - Deviations from the standard lapse rate will cause convection, which will bring the column temperatures back to standard.  This is the stable condition to which planetary atmospheres aspire.

Column D - Professor Brown believes that the thermal conductivity of the air, though small, will eventually restore the even temperature along the column.  As we will see, this cannot happen, but for the sake of argument let us assume the temperature is uniform within our pressure and density-lapsed column, like Column d.

The professor states that nothing as simple as gravity could cause this column to spontaneously separate into stable regions of different temperatures.  Why, indeed, would a quiescent column of air bestir itself and shuffle its hotter molecules lower?

(We can ignore moving parcels of air or the convection arguments of 'work against a gravity field' or compression and expansion here.  We need look only at the actual mechanism of thermal conductivity in a gas.)

How It Works

Robert Brown, botanist
Robert Brown, not at Duke

In a solid, the molecules are fixed, connected to each other, transmitting heat as vibration through the lattice.  In a gas, the individual molecules are always in motion (look up Robert Brown 1773-1858, and Brownian motion).  Absent the motion, the gas would condense to liquid or solid.  Heat is conducted through a gas by the molecules bumping into each other, and to do that, they must travel.  The temperature we measure is the average energy of motion, the kinetic energy, of the molecules.  The faster they travel, the higher their kinetic energy, the higher their temperature.

At any level in our column, molecules are traveling randomly in every direction, up, down, sideways.  The molecules traveling upward do so against the pull of gravity.  They lose speed – they cool.  When they collide with another molecule above them, they do so at a lower temperature than when they started.  Conversely, gravity accelerates them when they descend.  Their temperature rises, and when they collide with a molecule below, they do so at a higher temperature than when they started.  This trading of energy continues until the adiabatic lapse rate is reached, where the lower molecules are as hot as the downward moving molecules, and conversely upwards.

The method of heat conduction in a gas moves heat downward, and cannot move heat upward against gravity.

Professor Brown's statement that "The gas is perfectly capable of conducting heat from the bottom of the container to the top all by itself!" is simply not the case.  The gas has no mechanism to conduct net heat upward against gravity.

If heat must move upward in a gas (for whatever reason), the only means available is through convection, which can rapidly move large amounts of heat great distances.

The Paradoxical Scenario

Professor Brown's Paradox
air column with silver wire

The other scenario required more thought to solve.  Professor Brown takes our adiabatically-lapsed column and runs an insulated silver wire from top to bottom, making silver/air contact at only the uninsulated top and bottom.

While air is an insulator (k = 0.024 watts/meter∙°C), silver is an excellent conductor of heat (k = ~430 watts/meter∙°C).  Thus, at first glance, the air is trying to warm the bottom and cool the top, while the silver is busily conducting that heat back up where it came from.  Perpetual something or other, and an unsustainable argument.

This is where we must to look at the numbers.  The dry adiabatic lapse rate on this planet is 1°C/100 meters.  That means a silver wire/rod/telephone pole has to move heat 100 meters (that's an NFL football field plus one end zone) with a temperature difference of only 1°C.  Unlikely.

A 0.01°C/meter temperature gradient – that's down in the temperature noise region (HadCRUT notwithstanding).  Silver is an excellent conductor, but not a perfect one.  It has a small but non-zero thermal resistivity that the gradient must be strong enough to overcome.  Upshot is that there is probably a minimum temperature gradient required for any material to move heat, and even for silver it's probably well above 0.01°C/meter.

Conductivities of 430 and 0.024 are four orders of magnitude apart.  Why bother insulating the wire?  My guess is that if we left it bare in the column we would still have the lapse rate, and the wire locally would be the temperature of the surrounding air.

I haven't found anything on the web about minimum gradients yet, so maybe they don't exist.  Possible.  Let's look around.

Real World Temperature Gradients

Rather than search at the conductivity extremes, let's look for a minimum required gradient in a material two orders of magnitude, i.e., midway, between silver and air:  Ice, with k = ~2 watts/meter∙°C .

Take a column of ice 3310 meters tall, scatter a variety of temperatures within a 13°C range along its length, and give it time to even out.  How long will it take to go isothermal?

Well, it's been more than a third of a million years at Vostok, and it still hasn't evened out.  We're talking about actual borehole thermometer readings, not the isotopically-derived proxy temperatures we usually see charted.

Below the bubble-sealing region (50-120 meters), there are at least 47 one-meter levels maintaining a gradient of 1°C/meter or greater, with one gradient above 2°C/meter.  That something as commonplace as ice could maintain gradients that steep for tens to hundreds of thousands of years is frankly amazing, but the evidence is there.

Silver's minimum gradient would be much lower, but there is little reason to doubt it exists.  If it is greater than a tiny hundredth of a degree per meter, the silver wire won't have any effect at all on the column of air, and the Adiabatic Lapse Rate will rule.

Vostok data:
https://cdiac.ess-dive.lbl.gov/ftp/trends/temp/vostok/vostok.1999.temp.dat

Vostok bubble sealing level:
http://www.globalchange.umich.edu/globalchange1/current/labs/Lab10_Vostok/Vostok.htm Figure 6

One meter gradients:
http://www.rockyhigh66.org/stuff/Vostok_gradients.txt