Let T be temperature, t time and h(t) the net heat energy input to a body. Then

C(dT/dt) = h(t)
where C is the heat capacity of the body

Let u(t)=h(t)/C so

dT/dt = u(t)

The temperature at time t will then be the initial temperature T(0) plus the integral of u(s) from 0 to t. Equivalently T(t) will be the sum of the integrals of u(s) of any set of intervals terminating at time t.

If u(t) is a random variable with mean 0 then T(t) will be a random variable as well but one that appears to have trends and cycles.

Below is a simulation of such a variable.

Each time the REFRESH button is clicked a new sample of 2000 random numbers is generated and their cumulative sums calculated and displayed. In most cases the plot appears to show trends. However these apparent trends are not real. There are no long term trends in the model. The expected value of the temperature in the future is not an extrapolation of a recent trend. Instead for the variable described above the expected value of the temperature at time t given the information that is available at time s is the temperature at time s; i.e.,

E{T(t):s} = T(s)

The display below illustrates this point. The graph with the light blue background represents a history of recorded temperatures. The light green area represents the the expected future value of the temperature given the record. The expected future value given the temperature record is equal to the last recorded temperature.

In the above displays the expected values of the temperature changes were zero. Suppose it was suspected that the expected value of the temperature changes was nonzero and perhaps changing over time. The way to test this suspicion is not with the temperatures because these would display spurious trends but instead with the temperature changes.

Consider the record of global average temperatures.

The display seems to have an undeniable trend. If such a trend is there it would also be show up in terms of the temperature changes. These annual changes are shown below.

The results of a regression of the annual changes on time show that there is no statistically significant trend over time. The standard test is the t-ratio, the ratio of the regression coefficient to its standard deviation. For a regression coefficient to be statistically significant at the 95 percent level of confidence its value must be greater than about 2. As shown above, the t-ratio for the trend coefficient for the annual changes is 0.001.

For the temperatures to have a real trend there doesn't have to be been a trend in the annual changes. If the average change is nonzero there would be a trend in the cumulative sum of those annual changes. The question about the average change can be examined by looking at the distribution of annual changes.

A tabulation of the frequency distribution of these annual changes is

This distribution can be considered a finite sample from a normal distribution. Thus the annual temperature changes can considered a random variable. If the variation in the net heat input is a random variable then the temperature at any point in time is the cumulative sum of a random variable. There is a nonzero average but it is not statistically significantly different from zero.

The model above left out an important physical phenomena.

The Role of Thermal Radiation

A body at absolute temperature T radiates an amount of energy proportional to the fourth power of T, T⁴. The equation for the dynamics of temperature is

C(dT/dt) = H(t) − Aσ T⁴.

where A is the area of the body, H(t) is net heat inflow net of the thermal radiation, A is the surface area of the body, C is the heat capacity of the body and σ is the Stefan-Boltzmann constant.

Now H(t) can have a nonzero expected value, say H. This expected value H will generate an equilibrium temperature T according to the equation

H = AσT⁴
and hence
T = (H/(Aσ))^1/4

A simulation of the time path of temperature when there is thermal radiation is shown below.

The stochastic character of this display is essentially the same as for the case in which the effected value of the change is zero. This is because the temperature moves to the level where the expected value of the changes in the deviations from the equilibrium temperature is zero.

The relevant variable is thus ΔT=T−T. The dynamic equation for this variable is

dΔT/dt = (H−H) − Aσ(T⁴−T⁴)
which can be adequately approximated by
dΔT/dt = (H−H) − Aσ4T³ΔT

This latter equation can be represented as

dΔT/dt = ΔH − hΔT

where h=Aσ4T³.

This equation has a slightly different solution than the sum of random deviations. The deviation of temperature is the exponentially weighted sum of past deviation of heat inflow from its equilibrium level; i.e.,

ΔT(t) = exp(−ht)ΔT(0) + ∫₀^texp(−h(t-s))ΔH(s)ds

In this case the expected value of ΔT at time t given the value of ΔT at time s is

E{ΔT(t):s} = exp(−h(t-s))ΔT(s)

The Greenhouse Effect

With a greenhouse effect some proportion b of the outgoing thermal radiation is absorbed and re-radiated. A proportion c is radiated back to the surface. The parameter c might be one half, but because radiation could be sideways as well as up or down its value will not be specified as one half. The upward radiation proportion will be taken as equal to the downward proportion c.

The radiation from the atmosphere is proportional to T_a⁴, where T_a is the temperature of the atmosphere. There are now two dynamic equations

C(dT/dt) = H(t) − AσT⁴ + bcAσT_a⁴.
and
C_a(dT_a/dt) = bAσT⁴ − 2cAσT_a⁴.

Equilibrium surface and atmospheric temperatures are determined by setting H equal to its expected value H, then setting dT/dt and dT_a equal to zero. Let those values be denoted as T and T_a.

Subtracting the equations defining the equilibrium temperatures from the dynamic equations and approximating ΔT⁴ as 3T³ΔT result in

C(dΔT/dt) = ΔH(t) − Aσ4T³ΔT + bcAσ4T_a³ΔT_a.
and
 
C_a(dΔT_a/dt) = bAσ4T³ΔT − 2cAσ4T_a³ΔT_a.

These equations indicate that if it is suspected that ΔH is dependent upon time then the annual changes in temperatures, surface and atmospheric, should be regressed on time and ΔT and ΔT_a.

The deviation of the average global surface temperature, called the temperature anomalies, ΔT is readily available for 1959 to 2004. The average global atmospheric temperature since 1959 has be approximately constant. The levels of CO₂ are also readily available. The annual changes in temperature are easily computed from the temperature anomalies. The regression of dΔT/dt on ΔT and the concentrations of CO₂, ρ_CO2 yields the following result:

dΔT/dt = -1.0725 + 0.0004919ΔT + 0.00347ρ_CO2
   [-0.661]       [0.018]        [0.0988]
R² = 0.00216

The values in brackets are the t-ratios for the regression coefficients. None is significantly different from zero at the 95 percent level of confidence.

The Form of the Dependence

The greenhouse effect works through the absorption and re-radiation of thermal radiation. The thermal radiation is proportional to T⁴. Therefore if g(t) is the concentration of greenhouse gases the relevant variable would be g(t)T⁴.

Incredibly, despite the billion of dollars which have been spent on global warming research, there is not available a time series on the concentration of water vapor, the overwhelmingly predominant greenhouse gas. Such a time series would be difficult to construct became of its spatial and temporal variation but no more so than global average temperature. In lieu of data on the concentration of overall greenhouse gases the concentration of CO₂ may be used in the regression analysis under the presumption that the autonomous variation in the concentration of H₂O is zero. The results of this regression are:

dΔT/dt = -2.58332 + 0.08672ΔT + 0.00807ρ_CO2 − 0.000247ρ_CO2ΔT
   [-1.578]       [0.3621]      [0.2138]     [0.36424]     
R² = 0.005346

The inclusion of the ρ_CO2ΔT definitely improves the statistical performance of the regression but it is still minimal and the coefficients are not significantly different from zero statistically at the 95 percent level of confidence. And, two out of three are of the wrong sign.

There could be an upward trend in ΔT merely if the distribution of the annual temperature changes is positive. The average annual change in global temperature over the period from 1959 to 2004 was 0.12 44 C°, but the standard deviation of those annual changes was 1.569 C°. The standard deviation of an average temperature change based upon a sample of 45 is 1.569/√45=0.2392 so the t-ratio for the average change over the 45 year period is 0.53 and not statistically different from zero at the 95 percent level of confidence. In other words, the apparent trend in global temperature is well within the variation expected in a finite sample.

(To be continued.)