Section 3.2: Spectral Chara.cteristics of Atmospheric Varia.bles
31
3.2 Spectral Characteristics of Atmospheric
Variables
Since the weather changes irregularly and is largely unpredictable beyond a
few days, the frequency spectra of extratropical atmospheric variables are
approximately white on time scales between about 10 days and a few years,
except for the seasonal peaks, and the variance is dominated by the white
noise extension ofthe daily weather fluctuations. This is shown in Figure 3.1
in a "log - log" presentation which emphasizes spectral slopes, and in Figure
3.2 (top) in a "frequency x spectrum - log frequency" plot which conserves
variance, therefore indicating more dearly the dominant time scales. The
level of the frequency spectra varies with both position and season, and high
resolution analysis may reveal weak spectral peaks superimposed on the white
background (e.g., Chave et al. , 1991). As one approaches the equator, the
frequency spectra become increasingly red and the level of high frequency
variability decreases.
Wavenumber-frequency analysis indicates that the time scale of the atmospheric fluctuations is larger at large scales, so the dominant period at the
planetary scale is of the order of 20-30 days while it is only a few days at large
wavenumbers (Figure 3.3). A more complete representation of the spectra
of atmospheric variables at a given level is thus given by the frequencywavenumber spectrum r(k, 1), which is related to the space-time covariance
function ')'( r, u) by
r(k, f) = (2'/1')-3 J~ ')'(r, u)e-i(k.r-Ju)drdu
(3.1)
and to the frequency spectrum by
(3.2)
Relation (3.2) indicates that the frequency spectra of the atmospheric fields
which are dominated by the largest scales (e.g. geopotential height, sea
level pressure) start falling off at larger periods than those which are dominated by small scales (e.g. wind stress curl). The turbulent air-sea fluxes
of momentum and energy in the extratropical ocean, which are quadratic or
higher-order functions of atmospheric variables, fall between these extremes
(see Figure 3.1) and are dominated at non-seasonal frequencies by short time
scale weather changes. At low frequencies, the exponential in (3.1) can then
be set to 1 and the spectrum is white, r(k, I) = r(k, 0). From the reality condition r(k, I) = r*(k, I) = r( -k, - 1), where the asterisk indicates
complex conjugate, it then follows that the spectra are symmetric, i.e. there
is no preferred propagation direction. At higher frequencies, however, the
extratropical fluctuations are dominated by eastward propagation, reflecting
the propagation of the storm systems (Figure 3.4).
31
3.2 Spectral Characteristics of Atmospheric
Variables
Since the weather changes irregularly and is largely unpredictable beyond a
few days, the frequency spectra of extratropical atmospheric variables are
approximately white on time scales between about 10 days and a few years,
except for the seasonal peaks, and the variance is dominated by the white
noise extension ofthe daily weather fluctuations. This is shown in Figure 3.1
in a "log - log" presentation which emphasizes spectral slopes, and in Figure
3.2 (top) in a "frequency x spectrum - log frequency" plot which conserves
variance, therefore indicating more dearly the dominant time scales. The
level of the frequency spectra varies with both position and season, and high
resolution analysis may reveal weak spectral peaks superimposed on the white
background (e.g., Chave et al. , 1991). As one approaches the equator, the
frequency spectra become increasingly red and the level of high frequency
variability decreases.
Wavenumber-frequency analysis indicates that the time scale of the atmospheric fluctuations is larger at large scales, so the dominant period at the
planetary scale is of the order of 20-30 days while it is only a few days at large
wavenumbers (Figure 3.3). A more complete representation of the spectra
of atmospheric variables at a given level is thus given by the frequencywavenumber spectrum r(k, 1), which is related to the space-time covariance
function ')'( r, u) by
r(k, f) = (2'/1')-3 J~ ')'(r, u)e-i(k.r-Ju)drdu
(3.1)
and to the frequency spectrum by
(3.2)
Relation (3.2) indicates that the frequency spectra of the atmospheric fields
which are dominated by the largest scales (e.g. geopotential height, sea
level pressure) start falling off at larger periods than those which are dominated by small scales (e.g. wind stress curl). The turbulent air-sea fluxes
of momentum and energy in the extratropical ocean, which are quadratic or
higher-order functions of atmospheric variables, fall between these extremes
(see Figure 3.1) and are dominated at non-seasonal frequencies by short time
scale weather changes. At low frequencies, the exponential in (3.1) can then
be set to 1 and the spectrum is white, r(k, I) = r(k, 0). From the reality condition r(k, I) = r*(k, I) = r( -k, - 1), where the asterisk indicates
complex conjugate, it then follows that the spectra are symmetric, i.e. there
is no preferred propagation direction. At higher frequencies, however, the
extratropical fluctuations are dominated by eastward propagation, reflecting
the propagation of the storm systems (Figure 3.4).