applet-magic.com Thayer Watkins Silicon Valley & Tornado Alley USA

The Effect of Autocorrelated Heat Flows on the Spectrum of Temperature

The point is made elsewhere that the temperature of the Earth is the
cumulative sum of the net energy inflows. The net flows have a stochastic element. As a result the
temperature statistics will appears to be following trends and/or cycles even if there are no such
long term patterns. The following simulation illustrates this phenomenon

However the net heat flows are not strictly white noise. They are autocorrelated. To illustrate the possible effects
of autocorrelations consider the special case in which a moving average of a white noise is taken of white noise
and this is used as the net heat inflow to generate a temperature series. This simulation is shown below for
the case of the moving average of length four.

Generally the effect of the moving averaging is to make the trends to appear more persistent, more like some
underlying development is been played out.

Analysis

The Case of a Simple Moving Average

Suppose u(t) is white noise and the net heat inflow is h(t) = (1/H)∫_{t-H}^{t}u(s)ds.
Then

T(t) = (1/C)∫_{0}^{t}h(s)ds

where T is the temperature and C is the heat capacity.

The spectrum of a series is the absolute value of its Fourier transform, which generally is a complex-valued function
of frequency.

There are some general properties of the Fourier transform that show relationships. Consider a simple
moving average for a series.

h(t) = (1/H)∫_{t-½H}^{t+½H}u(s)ds
which can be represented as
h(t) = (1/H)[∫_{0}^{t+½H}u(s)ds − ∫_{0}^{t-½H}u(s)ds]

The Fourier transform of h(t) is

F_{h}(ω) = (1/H)[(1/iω)(e^{-ωH/2}F_{u} − e^{+ωH/2}F_{u}]
which reduces to
F_{h}(ω) = (2/H)(1/ω)sin(½ωH)F_{u}

The |sin(½ωH)| portion of the above product looks like the following.

The division by ω² has such a powerful effect that the fluctuations in |sin(½ωH)| are
obscured. However if the spectrum is plotted at different scales over different ranges the peaks are revealed.

The General Case

Let the net heat flow h(t) be a weighted sum of a stochastic variable u(t); i.e.,

h(t) = ∫_{0}^{t}g(s)u(t-s)ds

It then follows from the Convolution Theorem for Fourier transforms that

F_{h}(ω) = F_{g}(ω)*F_{u}(ω)

Then for u(t) being white noise of strength k and
T(t) = ∫_{0}^{t}h(s)ds

F_{T}(ω) = (1/iω)F_{h}(ω) = (k/iω)F_{g}(ω)
and hence
|F_{T}(ω)| = (k/ω)|F_{g}(ω)|

So the spectrum of T is the spectrum of the function g multiplied by the factor (k/ω).