What Determines the Spectrum of a Climate Variable at Zero Frequency?

Abstract

In order to understand the spectrum Γx(?) of a climate variable xt, the relation between Γx(?) and its forcing has to be considered. If the evolution of xt over (discretized) time is determined by ft, that is, ?x/?t ≡ (xt ? xt?1)/?t = ft, the only existing relation is the one between Γx(?) and the spectrum Γf(?) of ft. The gain function G(?) of the difference operator ?/?t, which acts as a high-pass filter, controls the relation between Γx(?) and Γf(?). For Γx(?), which is bounded at zero frequency, G(?) completely suppresses the variations of ft at zero frequency, so that Γx(0) cannot be related to Γf(0). In practice, the efficiency of the difference operator as a high-pass filter can make the detection of the low-frequency spectral relation between xt and ft difficult.