Hayashi spectra of the northern hemisphere mid-latitude atmospheric variability in the NCEP–NCAR and ECMWF reanalyses

Abstract

We compare 45 years of the reanalyses of National Center for Environmental Prediction–National Center for Atmospheric Research and European Center for Mid-Range Weather Forecast in terms of their representation of the mid-latitude winter atmospheric variability for the overlapping time frame 1957–2002. We adopt the classical approach of computing the Hayashi spectra of the 500 hPa geopotential height fields and we introduce an ad hoc integral measure of the variability observed in the Northern Hemisphere on different spectral subdomains. Discrepancies are found especially in the pre-satellite years of the records in the high frequency-high wavenumber propagating waves. This implies that in the pre-satellite period the two datasets have a different representation of the baroclinic available energy conversion processes. Minor differences are also found in the description of low frequency–low wavenumber standing waves. We observe a positive impact of the satellite data on the representation of wave activity over the oceanic sectors in the period starting from 1979, in particular on the description of high frequency variability. Since in the pre-satellite period the assimilated data are more scarce, predominately over the oceans, and of lower quality than found later on, they provide a weaker constraint to the model dynamics. Therefore, the resulting discrepancies in the reanalysis products may be mainly attributed to differences in the models’ behaviour.

Appendix

1.1 Space–time spectral analysis

The space–time spectral analysis introduced by Hayashi (1971) provides information about the direction or speed at which the eddies move. This information may be obtained firstly by Fourier-analysis of the spatial field, and then by computing the time-power spectrum of each spatial Fourier component. The difficulty here lies in the fact that straightforward space–time decomposition will not distinguish between standing and travelling waves: a standing wave will give two spectral peaks corresponding to travelling waves moving eastward and westward at the same speed and with the same phase. The problem can only be circumvented by making assumptions regarding the nature of the wave. For instance, we may assume complete coherence between the eastward and westward components of standing waves and attribute the incoherent part of the spectrum to real travelling waves (Pratt 1976; Fraedrich and Bottger 1978; Hayashi 1979).

In this formulation, for each winter considered, the power spectrum H _E/W (k,ω) at a zonal wavenumber k and temporal frequency ω for the eastward and westward propagating waves is:

$$H_E (k,\omega) = \frac{1}{4}\left\{{P_\omega (C_k) + P_\omega (S_k)} \right\} + \frac{1}{2}Q_\omega (C_k, S_k)$$

(2)

$$H_W (k,\omega) = \frac{1}{4}\left\{{P_\omega (C_k) + P_\omega (S_k)} \right\} - \frac{1}{2}Q_\omega (C_k, S_k)$$

(3)

P _ω and Q _ω are, respectively, the power and the quadrature spectra of the longitude (λ) and the time (t) dependent 500 hPa geopotential height Z(λ, t) expressed in terms of the zonal Fourier harmonics:

$$Z(\lambda, t) = Z_0 ( t) + \sum\limits_{k = 1}^\infty {\left\{{C_k (t)\cos (k\lambda) + S_k (t)\sin (k\lambda)} \right\}}. $$

(4)

The total variance spectrum H _T (k,ω) is given from the sum of the eastward and westward propagating components:

$$H_T (k,\omega) = \frac{1}{2}\left({P_\omega (C_k) + P_\omega (S_k)} \right)$$

(5)

while the propagating variance H _P (k,ω) is given by the difference between the components (A1a) and (A1b):

$$H_P (k,\omega) = |Q(k,\omega)|.$$

(6)

So, the standing variance spectrum H _S (k,ω) can be obtained by the difference:

$$H_S (k,\omega) = H_T (k,\omega) - |Q(k,\omega)|.$$

(7)

We emphasize that, for the sake of simplicity of the notation, we have neglected the indication of the winter under investigation denoted in the text by the superscript n. We emphasize that, customarily, Hayashi spectra are generally represented by plotting the quantities k·ω/2π·H _T (k,ω), k·ω/2π·H _S (k,ω), k·ω/2π·H _E (k,ω), and k·ω/2π·H _W (k,ω), as reported in Fig. 1a–d, in order for equal geometrical areas in the log–log plot to represent equal variance.