## Tuesday, October 8, 2013

### Why is Global Averaged Temperature a Non-Thermodynamic Construction?

draft Updated October 27, 2013

"Science is the belief in the ignorance of the experts."
-Richard P. Feynman

## Some Common Misconceptions in the Meaning and Uses of Temperature and Heat

First, temperature itself is not a measurement of heat.

Second, volume averaged temperature is not a measure of volume averaged heat.

Third, the atmosphere and ocean are not systems in thermodynamic states of equilibrium.

Fourth, the construction of an average temperature from a collection of disparate non-equilibrium thermodynamic systems gives a quantity that does not obey the laws of thermodynamics.

### The global average temperature as a non-thermodynamic construction.

A constructed quantity like global average temperature, is termed an "exterior quantity."   That is, "global average temperature," is not part of the physical theory of thermodynamics.  Global average temperature is exterior to the physics of thermodynamics and has no meaning expressible as a thermodynamic variable.

Another feature of the theory of thermodynamics is that time is not a variable in the theory. It simply does not appear.

So, a different, non-thermodynamic theory is needed to give global average temperature meaning. One approach is to model the real atmosphere as a collection of coupled thermodynamic systems each one of which has a well defined thermodynamic state. Seems reasonable. But such made-up systems need not behave like the real atmosphere. That is, model predictions will diverge from new data over time and not accurately predict future behavior. More on this later.

### So what is heat anyway?

In order to measure changes in the heat energy in the atmosphere, one also needs a physical quantity called the local specific heat. Heat is a form of energy and changes in the quantity of heat energy in a region of the Earth's atmosphere are measured using the product C x T of the "specific heat density product" C of atmospheric air at a particular position (longitude, latitude, altitude) with the local changes in temperature T and other thermodynamic variables. This product must be evaluated for every sufficiently small volume V of air in the region of interest in order to calculate the changes in the total thermal energy (heat) in a region of the atmosphere. However, this is a simplification of the thermodynamics. For more details read on, skip over the details if you like.

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Strictly speaking, the definition and use of specific heat needs some further technical details. Read on for more technical thermodynamics details, or skip the next four paragraphs, if you want.

Specific heats Cp, Cv are usually given in units of Joules per Kevin per kilogram of gas at constant pressure Cp, or Joules per Kelvin per cubic meter at constant volume Cv, or Cn Joules per Kelvin per particle at constant number density per species.

Mathematically, a thermodynamic state space is a topological manifold having a (local) differential structure. The coordinates of the manifold are the thermodynamic variables that appear in the equation of state of the system. The laws of thermodynamics are stated in terms of differentials associated with the local differential structure.

How does thermodynamics account for changes in which the system gains or loses heat?

In the theory of thermodynamics, some thermodynamic processes can be defined by curves in the state space of the system. The process starts at the initial state, at one end of the curve, and proceeds to the final state, the point at the other end of the curve. If the system remains in equilibrium states throughout the process, the process is represented by a continuous curve. For example, if a process takes place at constant pressure, then Cp is unchanged in the process, which makes easier the calculation of the amount of thermal energy lost or gained by the system, over the process.

Further, in thermodynamics the amount of heat in a system is not a state function. That is, only the difference in heat output or input "Delta Q" for a given thermodynamic process start to finish is available.   There is no heat function Q(P,V,T,N,...) of the coordinates of the manifold. The amount of heat gained or lost, Delta Q, depends on the shape of the curve or path in state space connecting the initial and final states of the process. One way to describe this, is to recognized that the differential heat change dQ is not an exact differential.

So what is needed to calculate the amount of heat gained or lost in a process?

In order to obtain dQ one uses the equation dQ = T dS where T is the absolute temperature and S is the entropy of the system at each point in the state space of the system. Unlike Q, both S and T are state functions.

Further, one needs the equation of state of the system. The equation of state defines a sub-manifold of equilibria embedded within the general abstract state space of the system. Thermodynamic processes of the system can be represented by curves on this sub-manifold. In order to compute the integral of dQ along such a curve one integrates TdS along the curve. One needs to know the entropy as a function of all state variables, S= S(P,V,T, N1, N2, N3, ...) restricted to the equation of state sub-manifold.

Of course, in the real world not all processes can be described by such continuous curves in state space.  We will not discuss such such non-quasi-static processes here.

In thermodynamics we have that S=0 when T=0, that is, at absolute zero temperature the entropy is zero.  This provides a "baseline reference" for the function S and the variable T.

Further, in a non-equilibrium system like the Earth's atmosphere and oceans, one needs to calculate the T dS integral for each sufficiently small volume (LTE volume) of the system and then sum the result over all such volumes. That is, one needs a state space integration and a physical volume integration to obtain net increases or decreases in heat content of the atmosphere and oceans.

The job of getting an accurate measure of warming or cooling from thermodynamics is a rather complex task.  Usually simplifying assumptions must be made, in order to make progress.

To simplify our discussion we define a variable C, a volumetric specific heat summed over gas species. For example, if rho is the mass density of a gas species at a given pressure p, then our C would equal rho x Cp giving our C in units of Joules per Kelvin per cubic meter for a gas sample at pressure p. One needs to know the density of each species and its specific heat at constant pressure and sum over all species in the atmosphere to get our C for that atmospheric composition...