Section 3.4: Sea Surface Temperature Anomalies
39
which can even be tested when no simuItaneous information is available on
the weather forcing.
The correlation and the cross-spectrum between forcing and response can
also be calculated from (3.11), providing more stringent signatures than the
power spectra. In the univariate case, the covariance IYv (u) between atmospheric forcing and climate response obeys
(3.15)
which yields the cross correlation between X and Y. When Y leads, the
correlation is negligible, while when Y lags, the correlation has a maximum
at smalllags. The coherence
Coh~y(I) = Ir
XY
(1)1 ~ 1
Jr x (I)r Y (I)
(3.16)
and the phase lead of X(t) over Y(t), given by e XY (I)
-tan- 1 [QXY (I)/CXY (I)], can be obtained similarly from the crossspectrum
r XY (I) = C XY (I) - iQXY (I),
(3.17)
where C XY (I) = Rer XY (I) is the co-spectrum and QXY (I) = -I mr XY (I)
the quadrature spectrum. For f ~ t;l, the coherence between stochastic
forcing and climate response is unity and the phase given by -arctan(l j>..).
As illustrated below, the model (3.13) is consistent with the statistical
properties of the observed anomalies of mid-Iatitude sea surface temperature, soil moisture and sea ice extent, on the monthly to yearly time scales.
However, these are particular cases where the dynamics of the climate subsystem play little role. In other cases, the climate subsystem may exhibit
nonlinearities, resonances and more complex feedback mechanisms. Then,
the response spectra will differ from (3.12) and (3.14), reflecting primarily the
internal dynamics of the system. Stochastic climate models remain testable,
however, by focusing on energy levels and, if atmospheric observations are
available, cause-to-effect relationships (Müller and Frankignoul, 1981). Of
course, not all the climate changes can be explained by stochastic forcing,
and some climate variations are forced deterministically (e.g. by the changes
in the orbital parameters of the earth), or reflect the chaotic nature of the
climate subsystem.
3.4 Sea Surface Temperature Anomalies
Frankignoul and Hasselmann (1977) have shown that the stochastic climate
model successfully explains the main statistical properties of sea surface temperature (SST) anomalies in the midlatitudes, as they mainly reflect the response of the oceanic mixed layer to the day-to-day changes in the air-sea
39
which can even be tested when no simuItaneous information is available on
the weather forcing.
The correlation and the cross-spectrum between forcing and response can
also be calculated from (3.11), providing more stringent signatures than the
power spectra. In the univariate case, the covariance IYv (u) between atmospheric forcing and climate response obeys
(3.15)
which yields the cross correlation between X and Y. When Y leads, the
correlation is negligible, while when Y lags, the correlation has a maximum
at smalllags. The coherence
Coh~y(I) = Ir
XY
(1)1 ~ 1
Jr x (I)r Y (I)
(3.16)
and the phase lead of X(t) over Y(t), given by e XY (I)
-tan- 1 [QXY (I)/CXY (I)], can be obtained similarly from the crossspectrum
r XY (I) = C XY (I) - iQXY (I),
(3.17)
where C XY (I) = Rer XY (I) is the co-spectrum and QXY (I) = -I mr XY (I)
the quadrature spectrum. For f ~ t;l, the coherence between stochastic
forcing and climate response is unity and the phase given by -arctan(l j>..).
As illustrated below, the model (3.13) is consistent with the statistical
properties of the observed anomalies of mid-Iatitude sea surface temperature, soil moisture and sea ice extent, on the monthly to yearly time scales.
However, these are particular cases where the dynamics of the climate subsystem play little role. In other cases, the climate subsystem may exhibit
nonlinearities, resonances and more complex feedback mechanisms. Then,
the response spectra will differ from (3.12) and (3.14), reflecting primarily the
internal dynamics of the system. Stochastic climate models remain testable,
however, by focusing on energy levels and, if atmospheric observations are
available, cause-to-effect relationships (Müller and Frankignoul, 1981). Of
course, not all the climate changes can be explained by stochastic forcing,
and some climate variations are forced deterministically (e.g. by the changes
in the orbital parameters of the earth), or reflect the chaotic nature of the
climate subsystem.
3.4 Sea Surface Temperature Anomalies
Frankignoul and Hasselmann (1977) have shown that the stochastic climate
model successfully explains the main statistical properties of sea surface temperature (SST) anomalies in the midlatitudes, as they mainly reflect the response of the oceanic mixed layer to the day-to-day changes in the air-sea