Appendix: Scaling modified Gaussians with fat tails

In Fig. 9 we showed the empirical probability distributions (Pr(ΔT > s), for the probability of a random (absolute) temperature difference ΔT exceeding a threshold s for time lags Δt increasing by factors of 2. Note that we loosely use the expression “distribution function” to mean Pr(ΔT > s). This is related to the more usual “cumulative distribution function” (CDF) by: CDF = Pr(ΔT < s) so that Pr(ΔT > s) = 1 − CDF. Two aspects of Fig. 9 are significant; the first is their near scaling with lag Δt: the shapes change little, this is the type of scaling expected for a monofractal “simple scaling” process, i.e. one with weak multifractality (as discussed in Lovejoy and Schertzer (2013), over these time scales, the parameter characterizing the intermittency near the mean, C 1 ≈ 0.02 so that this is a reasonable approximation).

This implies that there is a nondimensional distribution function P(s):

$$P(s) = \Pr \left( {\frac{{\Delta T(\Delta t)}}{{\upsigma_{{\Delta T}} }} > s} \right);\quad\upsigma_{{\Delta T}} = \left\langle {\Delta T(\Delta t)^{2} } \right\rangle^{1/2}$$

σ Δt is the standard deviation. Due to the temporal scaling, we have \(\upsigma_{{{\lambda \varDelta }t}} =\uplambda^{H}\upsigma_{{\Delta t}}\) where H is the fluctuation exponent and P(s) is independent of time lag Δt. From Fig. 9 it may be seen that as predicted by the RMS fluctuations (σ Δt , Fig. 7), H ≈ 0. This is a consequence of the slight decrease in the RMS Haar fluctuation (with exponent H Haar ≈−0.1; Fig. 8). Unlike the Haar fluctuation, the ensemble mean RMS differences cannot decrease but simply remains constant until the Haar fluctuations begin to increase again in the climate regime (compare Figs. 7, 8, beyond Δt ≈ 125 years).

The second point to note is that the lag invariant distribution function P(s) has roughly a Gaussian shape for small s, whereas for large enough s, it is nearly algebraic. This can be simply modelled as:

$$P_{qD} (s) = \begin{array}{*{20}l} {P_{G} (s);} \hfill & {s < s_{qD} } \hfill \\ {P_{G} (s_{qD} )\left( {\frac{s}{{s_{qD} }}} \right)^{ - qD} ;} \hfill & {s \ge s_{qD} } \hfill \\ \end{array}$$

where P G (s) is the cumulative distribution function for the absolute value of a unit Gaussian random variable. The simple way of determining s qD used here is to define s qD as the point at which the logarithmic derivative of P G is equal to −q D so that the plot of log P qD versus log s is continuous:

$$\left. {\frac{{d\log P_{G} (s)}}{d\log s}} \right|_{{s = s_{qD} }} = - q_{D}$$

this is an implicit equation for the transition point s qD .

In actual fact the only part of the model that is used for the statistical tests is the extreme large s “tail” which Fig. 9 empirically shows could be bracketed between:

$$P_{qD1} (s) < P(s) < P_{qD2} (s);\quad q_{D1} > q_{D2} ;\quad s > s_{qD1} > s_{qD2}$$