Mon, 08/16/2010 - 17:27 — apsmith

**Note:** I wrote the following at the request of climate blogger "Science of Doom" - go there to read the full article in context! Also, please make comments over there, not here, thanks.

[The following is in regard to models of the high temperature of the surface of Venus which ascribe that high temperature essentially to the high atmospheric pressure on that planet, without properly recognizing the role of infrared absorption by greenhouse gases. In particular, Leonard Weinstein, in previous discussions there, had essentially claimed that even if all solar radiation were absorbed by a thin high layer in Venus' atmosphere at a relatively cool temperature, the surface temperature would still be very high. The following is why I disagree.]

On the question of convective heat flow from heating above: I agree some such heat flow is possible, but it is difficult. Goody and Walker were wrong if they felt this could explain high Venusian surface temperatures.

The foundation for my certainty on this lies in the fundamental laws of thermodynamics, which I'll start by reviewing in the context of the general problem of heat flow in planetary atmospheres (and the “Very Tall Room Full of Gas”). Note that these laws are very general and based in the properties of energy and the statistics of large numbers of particles, and have been found applicable in systems ranging from the interior of stars to chemical solutions and semiconductor devices and the like. External forces like gravitational fields are a routine factor in thermodynamic problems, as are complex intermolecular forces that pose a much thornier challenge. The laws of thermodynamics are among the most fundamental laws in physics - perhaps even more fundamental than gravitation itself.

I'm going to discuss the laws out of order, since they are of various degrees of relevance to the discussion we've had. The third law (defining behavior at zero temperature) is not relevant at all and won't be discussed further.

**The First Law**

The first law of thermodynamics demands conservation of energy:

Energy can be neither created nor destroyed.

This means that in any isolated system the total energy embodied in the particles, their motion, their interactions, etc. must remain constant. Over time such an isolated system approaches a state of thermodynamic equilibrium where the measurable, statistically averaged properties cease changing.

In our previous discussion I interpreted Leonard's “Very Tall Room Full of Gas” example as such a completely isolated system, with no energy entering or leaving. Therefore it should, eventually at least, approach such a state of thermodynamic equilibrium. Scienceofdoom above interpreted it as being in a condition where the top of the room was held at a given specific temperature. That condition would allow energy to enter and leave over time, but eventually the statistical properties would also stop changing, and then energy flow through that top surface would also cease, total energy would be constant, and you would again arrive at an equilibrium system (but with a different total energy from the starting point).

That would also be the case in Leonard's original thought experiment concerning Venus if the temperature of the “totally opaque enclosure” was a uniform constant value. The underlying system would reach some point where its properties ceased changing, and then with no energy flow in or out, it would be effectively isolated from the rest of the universe, and in its own thermodynamic equilibrium. However, Leonard allows the temperature of his opaque enclosure to vary with latitude and time of day which means that strictly such a statistical constancy would not apply and the underlying atmosphere would not be completely in thermodynamic equilibrium. I'll look at that later in discussing the restrictions imposed by the second law.

In a system like a planetary atmosphere with energy flowing through it from a nearby star (or from internal heat) and escaping into the rest of the universe, you are obviously not isolated and would not reach thermodynamic equilibrium. Rather, if a condition where averaged properties cease changing is reached, this is referred to as a steady state. Under steady state conditions the first law must still be obeyed. Since internal statistical properties are unchanging, that means the system must not be gaining or losing any internal energy. So in steady state you have a balance between incoming and outgoing energy from the system, enforced by the first law of thermodynamics.

If such an atmospheric system is NOT in steady state, if there is, say, more energy coming in than leaving, then the total energy embodied in the particles of the system will increase. That higher average energy per particle can be measured as an increase in temperature - but that gets us to the definition of temperature.

**The Zeroth Law**

The zeroth law essentially defines temperature:

If two thermodynamic systems are each in thermal equilibrium with a third, then they are in thermal equilibrium with each other.

Here thermal equilibrium means that when the systems are brought into physical proximity so that they may exchange heat, no heat is actually exchanged. A typical example of the zeroth law is to make the “third system” a thermometer, something that you can use to read out a measurement of its internal energy level. Any number of systems can act as a thermometer: the volume of mercury liquid in an evacuated bulb, the resistance of a strip of platinum, or the pressure of a fixed volume of helium gas, for example.

If you divide a system “A” in thermodynamic equilibrium into two pieces, “A1″ and “A2″, and then bring those two into physical proximity again, no heat should flow between them, because no heat was flowing between them before separating them since neither one's statistical properties were changing. I.e. Any two subsystems of a system in thermodynamic equilibrium must be in thermal equilibrium with each other. That means that if you place a thermometer to measure the temperature of subsystem “A1″, and find a temperature “T” for thermal equilibrium of the thermometer with “A1″, then subsytem “A2″ will also be in thermal equilibrium with “T”, i.e. its temperature will also read out as the same value.

That is, the temperature of a system in thermodynamic equilibrium is the same as the temperature of every (macroscopic) subsystem - temperature is constant throughout. The zeroth law implies temperature must be a uniform property of such equilibrium systems.

This means that in both “Very Large Room” examples and for my version of Leonard's original thought experiment for Venus (with a uniform enclosing temperature), the thermodynamic equilibrium that the atmosphere must eventually reach must have a constant and uniform temperature throughout the system. Temperature in the room or in the pseudo-Venus' atmosphere would be independent of altitude - an isothermal, not adiabatic, temperature profile.

The zeroth law can actually be derived from the first and second laws - this is done for example in* Statistical Physics*, 3rd Edition Part 1, Landau and Lifshiftz (Vol. 5) (Pergamon Press, 1980) – Chapter II, *Thermodynamics Quantities*, Section 9 – “Temperature” - and again the conclusion is the same:

Thus, if a system is in a state of thermodynamic equilibrium, the [absolute temperature] is the same for every part of it, i.e. is constant throughout the system.

**The Second Law**

The first and zeroth laws tell us what happens in the cases where the atmosphere can be characterized as in thermodynamic equilibrium, i.e. actually or effectively isolated from the rest of the universe after sufficient period of time that quantities cease changing. Under those conditions it must have a uniform temperature. But what about Leonard's actual Venus thought experiment, where there are constant fluxes of energy in and out due to latitudinal and time-of-day variations in the temperature of the opaque enclosure? What can we say about the temperatures in the atmosphere below given heating from above under those conditions? Here the second law provides the primary constraint, and in particular the Clausius formulation:

Heat cannot of itself pass from a colder to a hotter body.

A planetary atmosphere is not driven by machines that move the air around, there are no giant fans pushing the air from one place to another. There is no incoming chemical or electrical form of non-thermal energy that can force things to happen. The driving force is the flux of incoming energy from the local star that brings heat when it is absorbed. All atmospheric processes are driven by the resulting temperature differences. Thanks to the first law of thermodynamics each incoming chunk of energy can be accounted for as it is successively absorbed, reflected, re-emitted and so forth until it finally leaves again as thermal radiation to the rest of the universe. In each of these steps the energy is spontaneously exchanged from a portion of the atmosphere at one temperature to another portion at another temperature.

What the second law tells us, particularly in the above Clausius form, is that the net spontaneous energy exchange describing the flow of each chunk of incoming energy to the atmosphere MUST ALWAYS BE IN THE DIRECTION OF DECREASING TEMPERATURE. Heat flows “downhill” from high to low temperature regions. The incoming energy starts from the star - very high temperature. If it's absorbed it's somewhere in the atmosphere or the planetary surface, and from that point it must go through successfully colder and colder portions of the system before it can escape to space (where the temperature is 2.7 K).

There can be no net flow of energy from colder to hotter regions. And that means, if the atmosphere below Leonard's “opaque enclosure” is at a higher temperature than any point on the enclosure, heat must be flowing out of the atmosphere, not inward. The enclosure, no matter the distribution of temperatures on its surface, cannot drive a temperature below it that is any higher than the highest temperature on the enclosure itself.

So even in the non-equilibrium case represented by Leonard's original thought experiment, while the atmosphere's temperature will not be everywhere the same, it will nowhere be any hotter than the highest temperature of the enclosure, after sufficient time has passed for such statistical properties to stop changing.

The thermodynamic laws are the fundamental governing laws regarding temperature, heat, and energy in the universe. It would be extraordinary if they were violated in such simple systems as these gases under gravitation that we have been discussing. Note in particular that any violation of the second law of thermodynamics allows for the creation of a “perpetual motion machine”, a device legions of amateurs have attempted to create with nothing but failure to show for it. Both the first and second laws seem to be very strictly enforced in our universe.

**Approach to Equilibrium**

The above results on temperatures apply under equilibrium or steady state conditions, i.e. after the “measurable, statistically averaged properties cease changing.” That may perhaps take a long time - how long should we expect?

The heat content of a gas is given by the product of the heat capacity and temperature. For the Venus case we're starting at 740 K near the surface and, under either of the “thought experiment” cases, dropping to something like 240 K in the end, about 500 degrees. Surface pressure on Venus is 93 Earth atmospheres, so in every square meter we have a mass of close to 1 million kg of atmosphere above it. [Quick calculation: 1 earth atmosphere = 101 kPa, or 10,300 kg of atmosphere per square meter, or 15 pounds per square inch. On Venus it's 1400 pounds/sq inch.] The atmosphere of Venus is almost entirely carbon dioxide, which has a heat capacity of close to 1 kJ/kgK (see this reference). That means the heat capacity of the column of Venus' atmosphere over 1 square meter is about one billion (10^{9}) J/K.

So a temperature change of 500 K amounts to 500 billion joules = 500 GJ for each square meter of the planetary surface. This is the energy we need to flow out of the system in order for it to move from present conditions to the isothermal conditions that would eventually apply under Leonard's thought experiment.

Now from scienceofdoom's previous post we expect at least an initial heat flow rate out of the atmosphere of 158 W/m² (that's the outgoing flow that balances incoming absorption on Venus - since we've lost incoming absorption to the opaque shell, this ought to be roughly the initial net flow rate). Dividing this into 500 GJ/m² gives a first-cut time estimate for the cooling: 3.2 billion seconds, or about 100 years. So the cool-down to isothermal would be hardly immediate, but still pretty short on the scale of planetary change.

Now we shouldn't expect that 158 W/m² to hold forever. There are four primary mechanisms for heat flow in a planetary atmosphere: conduction (the diffusion of heat through molecular movements), convection (movement of larger parcels of air), latent heat flow (movement of materials within air parcels that change phases - from liquid to gas and back, for example, for water) and thermal radiation. The heat flow rate for conduction is simply proportional to the gradient in temperature. The heat flow rate for radiation is similar except for the region of the atmospheric “window” (some heat leaves directly to space according to the Planck function for that spectral region at that temperature). Latent heat flow is not a factor in Venus' present atmosphere, though it would come into play if the lower atmosphere cooled below the point where CO2 liquefies at those pressures.

For convection, however, average heat flow rates are a much more complex function of the temperature gradient. Getting parcels of gas to displace one another requires some sort of cycle where some areas go up and some down, a breaking of the planet's symmetry. On Earth the large-scale convective flows are described by the Hadley cells in the tropics and other large-scale cells at higher latitudes, which circulate air from sea level to many kilometers in altitude. On a smaller scale, where the ground becomes particularly warm then temperature gradients exceeding the adiabatic lapse rate may occur, resulting in “thermals”, local convective cells up to possibly several hundred meters. If the temperature difference between high and low altitudes is too low, the convective instability vanishes and heat flow through convection becomes much weaker.

So as temperatures come closer to isothermal in an atmosphere like Venus', except for the atmospheric “window” for radiative heat flow, we would expect all the heat flow mechanisms to decrease, and convection in particular to almost cease after the temperature difference gets too small. So we might expect the cool-down to isothermal conditions to slow down and end up much longer than this 100-year estimate. How long?

Another of the thought experiment versions discussed in the previous thread involved removing radiation altogether; with both radiation and convection gone, that leaves only conduction as a mechanism for heat flow through the atmosphere. For an ideal gas the thermal conductivity increases as the 2/3 power of the density (it's proportional to density times mean free path) and the square root of temperature (mean particle speed). While CO2 is not really ideal at 93 atmospheres at 740 K, using this rule gives us a rough idea of what to expect - at 1 atmosphere and 273 K we have a value of 14.65 mW/(m.K) so at 93 atmospheres and 740 K it should be about 500 mW/(m . K). For a temperature gradient of 10 K/km that gives a heat flux of 0.005 W/m². 500 GJ would then take about 10^{14} seconds, or 3 million years.

So the approach to an isothermal equilibrium state for these atmospheres would take between a few hundred and a few million years, depending on the conditions you impose on the system. Still, the planets are billions of years old, so if heating from above was really the mechanism at work on Venus we should see the evidence of it in the form of cooler surface temperatures there by now, even if radiative heat flow were not a factor at all.

**The View From a Molecule**

Leonard in our previous discussion raised the point that an individual molecule sees the gravitational field, causing it to accelerate downwards. So molecular velocities lower down should be higher than velocities higher up, and that means higher temperatures.

Leonard's picture is true of the behavior of a molecule in between collisions with the other molecules. But if the gas is reasonably dense, the “mean free path” (the average distance between collisions) becomes quite short. At 1 atmosphere and room temperature the mean free path of a typical gas is about 100 nanometers. So there's very little distance to accelerate before a molecule would collide with another; to consider the full effect you need to look at the effect of collisions due to gas pressure along with the acceleration by gravity.

An individual molecule in a system in thermodynamic equilibrium at temperature T has available a collection of states in phase space (position, momentum and any internal variables) each with some energy E. In the case of our molecule in a gravitational field, that energy consists of the kinetic energy ½mv² (m = mass, v = velocity) plus the gravitational potential energy = gmz (where z = height above ground). The Boltzmann distribution applies in equilibrium, so that the probability of the molecule being in a state with energy E is proportional to:

e^{(-E/kT)} = e^{(-(½mv² + gmz)/kT)}.

So the Boltzmann distribution in this case specifies both the distribution of velocities (the standard Maxwell-Boltzmann distribution) and also an exponential decrease in gas density (and pressure) with height. It is very unlikely for a molecule to be at a high altitude, just as it is very unlikely for a molecule to have a high velocity. The high energy associated with rare high velocities comes from occasional random collisions building up that high speed. Similarly the high energy associated with high altitude comes from random collisions occassionally pushing a molecule to great heights. These statistically rare occurences are both equally captured by the Boltzmann distribution. Note also that since the temperature is uniform in equilibrium, the distribution of velocities at any given altitude is that same Maxwell-Boltzmann distribution at that temperature.

**Force Balance**

The decrease in pressure with height produces a pressure-gradient force that acts on “parcels of gas” in the same way that the gravitational force does, but in the opposite direction. At equilibrium or steady-state, when statistical properties of the gas cease changing, the two forces must balance.

That leads to the equation of hydrostatic balance equating the pressure gradient force to the gravitational force:

dp/dz = -mng

(here p is pressure and n is the number density of molecules - N/V for N molecules in volume V). In equilibrium n(z) is given by the Boltzmann distribution:

n(z) = c.e^{(-mgz/kT)};

for the ideal gas pressure is given by p = nkT, so the hydrostatic balance equation becomes:

dp/dz = k T dn/dz = k T c (-mg/kT) e^{(-g m z/kT)} = -mg c e^{(-mgz/kT)} = - m n(z) g

I.e. the Boltzmann distribution for this ideal gas system automatically ensures the system is in hydrostatic equilibrium.

Another approach to this sort of analysis is to look at the detailed flow of molecules near an imaginary boundary. This is done in textbook calculations of the thermal conductivity of an ideal gas, for example, where a gradient in temperature results in net flow of energy (necessarily from hotter to colder). In our system with gravitational force and pressure gradients both must be taken into account in such a calculation. Such calculations are somewhat complex and depend on assumptions about molecular size and neglecting other interactions that would make the gas non-ideal, but the net effect must always satisfy the same thermodynamic laws as every other such system: in thermodynamic equilibrium temperature is uniform and there is no net energy flow through any imagined boundary.

In conclusion, after sufficient time that statistical properties cease changing, all these examples of a system with a Venus-like atmosphere must reach essentially the same isothermal or near-isothermal state. The gravitational field and adiabatic lapse rate cannot explain the high surface temperature on Venus if incoming solar radiation does not reach (at least very close to) the surface.