Group seminar on 1. November, 14:15 CET

A new consideration of equilibrium fluctuations and its link to Hasselmann’s stochastic climate model

Prof. Dr. Jin-Song von Storch

This talk discusses a new consideration of internal variability, further specified as equilibrium fluctuations,   and its link to Hasselmann’s stochastic climate model. The full paper can be found in https://doi.org/10.16993/tellusa.25

We consider a dynamical system described by a multidimensional state vector X. A component x of X evolves according to dx/dt=f(X). Equilibrium fluctuations are fluctuations of an equilibrium solution X(t) obtained when the system is in its equilibrium state reached under a constant external forcing.  The frequencies  of these fluctuations range from the major frequencies of the underlying dynamics to the lowest possible frequency,  the   frequency zero.    For such a system,  the known feature of the differential operator d/dt as a high-pass filter makes the spectrum of  f(X)  to vanish not only at frequency zero,  but de facto over an entire frequency range centered at frequency zero (when considering both positive and negative frequencies).  Consequently,    there is a non-zero portion of the total equilibrium variance of x that cannot  be determined by considering its differential forcing f(X).   Instead,   this portion of variance  arises  from many impulse-like interactions of x with other components of X  received by x  along an equilibrium  solution over time.    The effect of many impulse-like interactions can only be realized  by integrating  the evolution equations in form of   dx/dt=f(X) forward in time.  This integral effect  is not contained in,  and can hence not be explained by,  a differential forcing  f(X) defined at individual time instances.  While in line with the integral effect first pointed out by Hasselmann,  the new consideration makes clear that the integral effect cannot be derived by addressing f(X) at individual time instances. It is this integral effect that  makes the classical deterministic thinking (which relies on the one-to-one relationship  between x and f(X)) powerless.