Monthly sunspot number (SSN) and the global mean sea surface temperature (GMSST) for the period of 1854-1997 are analyzed by wavelet transform to study the nonstationary response of GMSST to SSN on the centennial, decadal-to-interdecadal, and interannual timescales. The dominant timescale of SSN is the well known 11-yr cycle, which length varies between 9 and 14 years. Beside this decadal timescale there are interdecadal and centennial (22-yr, 45-yr, 90-yr and longer) timescales on one side, and interannual (5-6yr and quasi-biennial) timescales on the other side. There is an overall inverse relationship between the intensity and the length of the 11yr cycle. The 11yr cycle is amplitude- and frequency- modulated by the interdecadal and centennial timescales. Similar multiple timescales are found in GMSST, which is dominated by interannual variability.
The low-frequency variability of GMSST is almost in-phase with the modulation envelopes of the 11yr cycle. There is an overall warming trend in GMSST since the beginning of this century, corresponding to a shortening trend in the length of the 11yr cycle. This coincidence suggests a long-term relationship between GMSST and solar activity on a centennial timescale. Superposed on the trend, there are two significantly warmer GMSST periods in the 1940s and 1990s, corresponding to the time periods when shorter 11yr cycles were observed. The relatively cooler GMSST period in the 1960s-1970s, which has not been explained by the theory of the greenhouse effect, coincides with the time period when relatively longer 11yr cycles were found. The warm-cool-warm GMSST and the shorter-longer-shorter length of the 11yr cycles show a possible response of GMSST to solar activity on the 45yr timescale, which is the first harmonic of the Gleissberg cycle.
On the interannual timescale, the 5-6yr component is stronger when the 11-yr cycles are stronger, suggesting that the 5-6yr component be the first harmonic of the 11yr cycle. The quasi-biennial variability (1.7-2.4yr) is usually stronger during the maximum and the transition period from the maximum to the minimum of the 11yr cycles. Interestingly, the timescale of the quasi-biennial variability in both GMSST and SSN is concurrently shortened from longer than 2yr in the late 1980s to less than 2yr in the mid-1990s. This is the period when Cycle 22 reaches its maximum and starts to decay and when the frequent El Nino events with less than 2yr timescales were observed in the first half of the 1990s.
There is no fixed phase relationship between GMSST and SSN on the 11yr timescale. It is understandable that when the intensity and the length of the 11yr cycle are modulated by longer timescales, the responsed phase of GMSST must vary. This may explain why most statistical methods designed for analyzing stationary data have failed to detect the solar-climate relationship from the nonstationary data such as the ones used here