The wavelet cross spectra of the yearly sunspot index and homogeneous-zone Indian monsoon rainfall time series are examined over the 120 a period 1871–1990 using Morlet continuous wavelets. The cross spectra of sunspot numbers with synthetic noise ensembles, including those matching the spectrum and probability distribution function of the rainfall time series, are used as reference to assess the significance of the cross spectrum between sunspot numbers and rainfall; their differences are studied using standard statistical tests. It is found particularly revealing to consider the two test periods 1878–1913 and 1933–1964, each comprising three complete solar cycles, which between them exhibit maximum contrast in solar activity since the beginning of accurate rainfall data collection in India . It is shown that the cumulative distribution of the 9–13 a band averaged cross power between sunspots and rainfall, derived from yearly values over either test period, differs from that between sunspots and spectrally matched noise in the same period at confidence levels of 99.5% or higher by the χ2 test. Further, the cross power of the sunspot index with rainfall, averaged over either test period, exceeds that with synthetic noise at z test confidence levels exceeding 99.99% over scale bands covering the 11.6 a sunspot cycle. The results further show variations across the different homogeneous rainfall zones of India, northeast India exhibiting a dramatically different phase relationship compared to the western regions of the country. The strong connections demonstrated here between Indian rainfall and solar activity are found to be organized approximately along quasi-longitudinal bands tilting eastward at their northern end. This suggests that higher solar activity causes intensification of the Walker and Hadley circulations; the mechanisms that may be responsible and the evidence from models are briefly discussed.
 That solar activity influences climate and long-term variability is now a well-accepted idea [Beer et al., 1990; Mehta and Lau, 1997; Haigh, 2001; Neff et al., 2001]. However, recent work [van Loon et al., 2004; Kodera, 2004; Bhattacharyya and Narasimha, 2005] has also provided evidence for association between solar processes and climate indices on shorter multidecadal scales; in particular, Bhattacharyya and Narasimha [2005, hereinafter referred to as BN05] present an analysis showing significant connections between solar activity and Indian monsoon rainfall by comparing rainfall in two test periods 1878–1913 and 1933–1964 (each comprising three complete solar cycles), between which the average sunspot number showed highest contrast since the beginning of accurate rainfall data collection in India (1871). (The choice of such relatively long test periods has the advantage that the sample size of about 30 in each period is now sufficiently large for drawing statistical conclusions with considerable confidence. This advantage gives weight to changes on multidecadal scales, and was judged to more than compensate for the consequent inability to discuss shorter timescale (say interdecadal) variations, although these may be important.) This analysis showed that rainfall was higher in the more solar active period. However, apart from a secondary role in identifying the test periods, no quantitative indicator of solar activity takes direct part in the BN05 analysis. This leaves open the possibility that some other parameter (such as greenhouse gases, for example) may be responsible for the differences observed in rainfall between the two test periods, especially as the period of higher solar activity was also one of higher CO2 concentrations.
 We examine this problem here by a wavelet cross spectral analysis of solar activity and homogeneous-zone Indian rainfall data. We prefer this procedure to an analysis solely of the often used all-India summer monsoon rainfall, as the geographical area it covers is not meteorologically homogeneous, and so physically coherent signals will tend to get diluted because of heterogeneity. Furthermore, spatial variations in solar-rainfall connections (of the kind we shall describe below) will obviously not be revealed by analysis of any all-India index. The statistical significance of a cross spectrum is usually estimated using the testing procedure outlined by Torrence and Compo . It will be shown here that this procedure suggests unduly high significance levels for connections between even white noise and sunspot index. This is due to the definition adopted for the cross spectrum, and to the use therein of a reference spectrum which utilizes lag-1 and lag-2 autocorrelations of the signals being analyzed. When one of these (solar activity in the present case) has a strong long-period oscillation the test is misleading, as is easily demonstrated by replacing the rainfall with white noise. A new Monte Carlo type procedure is proposed here to overcome this limitation of the Torrence-Compo procedure. The principle behind the new procedure is to compare the cross spectrum under question with those between solar activity and three specially generated synthetic noise signals, one white (WN) and the other two matching respectively the frequency spectrum (SN, for “spectrally matched”) and amplitude probability distribution (AN, for “amplitude matched”) of the rainfall time series being analyzed (the last two will be called “rainfall-matched”). The difference in solar-rainfall and solar-noise cross spectra can then be subjected to standard statistical tests to assess significance. It is incidentally found that spectrally matched noise SN offers the most stringent test, so results will often be quoted only for this case.
2. Data Analyzed and Methods Used
 We consider the annual rainfall of the six homogeneous Indian rainfall zones (homogeneous Indian monsoon (HIM) region, west central India (WCI), northwest India (NWI), central northeast India (CNEI), northeast India (NEI) and peninsular India (PENSI), and also the all India summer monsoon (AISM) rainfall, as analyzed and tabulated by Parthasarathy et al. [1993, 1995]. This data set covers the 120 a period from 1871 to 1990. Data beyond 1990 have not been considered because they have not yet been analyzed on the basis of the classification scheme introduced by Parthasarathy et al. [1993, 1995].
 We present results for only one solar activity index, namely the yearly sunspot number (SS) obtained from Rai Choudhuri  and Fligge et al. ; in our earlier work (BN05) it has been found that this is an excellent representative measure of solar activity for the kind of analysis we are undertaking.
 The spectrally matched synthetic noise signal (SN) is generated to match the estimated power spectrum of rainfall, using the spectral representation method of Grigoriu . The amplitude-matched noise (AN) is generated from the amplitude probability distribution function of rainfall using the stationary non-Gaussian translation process of Grigoriu . Figure 1 shows the normalized time series for sunspot number and HIM rainfall, and one (randomly selected) realization of each of the three noise time series. (We shall comment on the contents of Figure 1 in greater detail in section 3.) Figure 2 shows a comparison of the spectrum of SN and the probability distribution of AN generated to match the characteristics of HIM rainfall. It is seen that the agreement is reasonable. Incidentally, we see that the contribution of the frequency band around the solar period of 11.6 a is of the order 10%, much lower than around 2 a but rather higher than around 4 a.
 All cross spectra with rainfall-matched noise are computed over an ensemble of 1000 realizations of 120 samples each. This ensemble size was found to be large enough to yield robust estimates for the cross spectra, by comparing with results from a larger ensemble of 4000 realizations. This assures us that we have a close approximation to the true population statistics involving synthetic noise.
2.1. Wavelet Cross Power Spectrum
 Following the definitions of Torrence and Compo , the wavelet cross spectrum WnSS–RF(s) between two time series Rn(t) and Sn(t), having respective wavelet transforms WnRF(s) and WnSS(s), is taken as
where indicates a complex conjugate. The cross wavelet power is ∣WnSS–RF(s)∣.
2.2. Torrence-Compo Test
 This test first introduces a background or reference spectrum given by
where k = 0,1,…N–1 is the frequency index and α = (α1 + )/2, where α1 and α2 are the lag-1 and lag-2 autocorrelation coefficients of the process under consideration. If the two time series have background spectra given respectively by PkSS and PkRF, then the cross wavelet power distribution will be given by
where σSS and σRF are the standard deviations of SS and RF respectively, ν is the number of degrees of freedom with which χ2 is distributed, p denotes the level of confidence and Zν(p) denotes the value of the χ2 distribution with ν degrees of freedom at the confidence level p. For a complex wavelet ν = 2.
2.3. Comparison Between Rainfall and Noise Cross Spectra
where sj = s02jδj, j = 0, 1,., J and J = δ−1log2(Nδt/s0). We choose δj = 0.25y in order to obtain a reasonably fine resolution in scale. The constant Cδ = 0.776 is a scale-independent reconstruction factor for the Morlet function.
 Using the above formulation the average cross power over the 10–12 a, 9–13 a and 8–16 a bands is computed as a function of running time. These three bands, of progressively increasing width, are all centered around the well-known sunspot cycle with a period of about 11.6 a, and serve to allow for the variations noticed in the sunspot period. (According to Lassen and Friis-Christensen , the solar cycle length varies from 9 to 13 a, so the 9–13 a band is often preferred in the present analysis; it accounts for 11.4, 10.4, 12.1, 6.1, 6.2, 5.0 and 7.8% of rainfall spectral power in HIM, WCI, NWI, CNEI, NEI, PENSI and AISM rainfall respectively.) Over each of the two test periods of low and high solar activity identified, namely 1878–1913 and 1933–1964 respectively (BN05), the band-averaged cross power between the rainfall and sunspot index RbSS–RF(t) and those between each of the noise samples and sunspot index RbSS–SN(t) are compared. The reasons for confining attention to such test periods are (1) to obtain maximum possible contrast in any solar activity signal that may be present in rainfall and (2) to ensure that the statistical nonstationarity, which characterizes both rainfall and solar activity on longer timescales (BN05), does not affect the analysis.(We may recall here that choosing two five-cycle test periods did not alter the nature of the BN05 conclusions, but the confidence levels were lower.) This comparison demands that we test whether differences between band-averaged coefficients of different kinds are significant. Two are relevant for the present investigation: (1) the difference between rainfall and noise cross spectra with sunspots, in any given period, and (2) the difference in rainfall-sunspot cross spectra between the two test periods.
 We now introduce the following notation for the cross-spectral averages that form the chief objects of our analysis: (1) average over a band b, RbSS–SN(t), function of running time, subscript b denoting band; (2) average over ensemble, 〈RbSS–SN(t)〉, function of running time and band; and (3) average over test period k, [RbSS–SN(t)]k, function of band, k = 1, 2. For instance the cross spectrum between sunspots and noise, averaged over the 9–13 a band as well as over the ensemble for test period 1, will be denoted as [〈R9–13SS–SN(t)〉]1, which is just one number.
 If 1 and 2 are the means of the two band-averaged cross spectra, we form the z test statistic
where n1, n2 are the number of samples used to obtain the concerned averages. The z test is adequate if the averages cover at least one test period, as we have more than 30 samples in each period. We adopt the null hypothesis that there is no significant difference between the two means. The hypothesis is rejected at confidence level 100(1–α)% if Z > zα for a one sided test against the alternative hypothesis that the difference is greater than zero [see Crow et al., 1960].
3. Preliminary Test Results
 Using the one-tailed z test, the null hypothesis that the difference in the averages of the concerned rainfall and noise time series over the two test periods is zero is rejected at the confidence levels marked in Figure 1. It is seen that the mean annual HIM rainfall during 1933–1964 is higher than that during the period 1878–1913 at a confidence level of 96.8% (confirming the earlier results of BN05). The confidence levels for the noise time series WN, SN and AN are respectively 55%, 65% and 70%, with the mean being slightly higher in the second test period for WN and SN and lower for AN. Over a number of realizations of up to 4000, the confidence levels for the noise time series vary over the range 50–85%, the differences being either higher or lower between the two test periods at random. This indicates with high probability that the rainfall results are genuine.
 To investigate whether there is a quantitative association of solar activity with rainfall we present, in Figure 3, data on wavelet cross spectra in the form of color-coded contour maps as functions of time and Fourier period (henceforth referred to as period).
 Both rainfall and noise cospectra are drawn to the same scale to facilitate comparison. In both cases the maximum cross power is seen in the 9–13 a period band, particularly over the higher solar activity period; significantly, however, the maximum power in the case of SS-HIM is approximately twice that in the SS-WN case.
 The global wavelet cross power spectrum is shown in Figure 4 for the same two cases as shown in Figure 3. It is seen that both noise and rainfall curves peak in the 9–13 a period band. The 95% confidence lines according to the Torrence-Compo test are also shown in Figure 4. It is seen that the difference between the two global cross spectra (unlike that between local maxima) is small, both being statistically significant to greater than 95% confidence level according to the Torrence-Compo test. Furthermore, by the same test, the individual cross spectra are significant at confidence levels of 98% and 93% for rainfall and WN respectively. At first sight this result seems to contradict the conclusion from Figure 1. However the results indicate the limitations of a test procedure that produces cross spectra of high significance between white noise and sunspots.
 The first of these limitations is connected with the very definition of the cross spectrum (see equation (1)), which is dominated by the strong periodic component in the sunspot time series even when the other signal in the pair is noise. The second is due to the model underlying the red noise background spectrum Pk (equation (2)) adopted for a signal with relatively long periods like that of sunspot number. Such limitations have been commented upon also by Maraum and Kurths . Now for a periodic signal the autocorrelation function is also periodic. So the basic premise of modeling the reference background spectrum using lag-1 and lag-2 autocorrelations for a signal dominated by a strong longer-period component is questionable. One option for overcoming this problem is to model the background spectra using the actual solar activity index spectrum. However, we propose a different method here that is more direct and effective. In this method, the primary variable is the difference in the cross spectra between the sunspot number and rainfall on the one hand, and between sunspot number and synthetic noise on the other.
 The relevant differences in cross spectra are shown in Figure 5. They range in all cases from small negative to large positive values, the highest positive differences again being in the 9–13 a band in the test period 1933–1964 in all three cases. Table 1 lists the set of mean values μ1, μ2 for [RbSS–HIM]1,2, [〈RbSS–WN〉]1,2, [〈RbSS–SN〉]1,2 and [〈RbSS–AN〉]1,2 for all the three bands considered in the analysis. (It is incidentally seen here that the ratio μ2/μ1 is approximately the same in the 9–13 and 8–16 a bands, but smaller in the 10–12 a band. The last is clearly too narrow to fully capture the variability of sunspot cycles. We shall give greatest weight here to conclusions from the 9–13 a band, for reasons already discussed in section 2.3, but results for the 8–16 a band will not be very different.)
Table 1. Mean Wavelet Cross Power Over Two Test Periods for HIM Rainfall
Mean cross power in each band over three cycles of low solar activity, 1878–1913.
Mean cross power in each band over three cycles of high solar activity, 1933–1964.
9–13 a Band
10–12 a Band
4. Results From Direct Monte Carlo Tests
 Using rainfall-matched noise time series, ensembles of 1000 realizations of their wavelet cross spectra with sunspot number (SS) have been generated, and found to yield satisfactorily convergent population statistics for the cross spectra (as verified from a few runs with ensembles of size up to 4000). We use the most stringent case of spectrally matched noise (SN) as reference to assess the significance of the SS-rainfall (SS-RF) cross spectra, and illustrate our approach by presenting analyses of some of the rainfall data in two distinct ways. (We have carried out the same analysis using WN and AN as well, but the corresponding results are even stronger.)
4.1. Distributions of Band-Averaged Cross Power
 In the first method we compare the cumulative distributions (CDF) of the 9–13 a band-average SS-RF cross power R9–13SS–RF(t) and SS-SN cross power 〈R9–13SS–SN(t)〉 over each test period. That is, where SN is involved the ensemble average is taken at each year; thus we have one number for each year in the test period for both rainfall and noise cross spectra, and 36 and 32 samples respectively in test periods 1 and 2 to define the CDFs. We can now test the hypothesis that the SS-RF data are from the same population as defined by the SS-SN ensemble, by making a standard χ2 goodness-of-fit test. As Figure 6 shows the two distributions for HIM are vastly different, and the χ2 test rejects the hypothesis at confidence levels exceeding 99.5% in both test periods.
Figure 7 shows the cumulative distributions for the case of NEI rainfall. The rainfall curve here is now well to the right of the noise curve in test period 1, and well to the left in test period 2, demonstrating once again a dramatically different behavior from HIM. Figures 6 and 7 show that rainfall and noise cross spectra have widely different distributions, and furthermore that there are strong qualitative differences between them in different regions. It is these large differences that are responsible for the extremely high confidence levels that characterize the present results.
 The mean over each test period of the b band-averaged SS-RF cross-power spectrum, [RbSS–RF]1,2, is now compared with the similar mean of SS–SN CDF and tested for significance using the standard z test. For all the cases we find Z > zα for α = 0.0001. Over both the test periods, the SS–HIM cross power is greater than the mean SS–WN, SS–SN and SS–AN cross powers, at confidence levels exceeding 99.99% for all the three bands considered in this analysis (10–12 a, 9–13 a, 8–16 a). As an illustration, the 9–13 a band averaged cross power for SS–RF averaged over each test period, [R9–13SS–RF]1,2, is shown in Table 2 in comparison with that for SS–SN, [〈R9–13SS–SN〉]1,2, for all the homogeneous zone and AISM rainfall time series considered. It can be seen that for all cases except SS–NEI, the mean 9–13 a band averaged cross power for SS–RF over test period 2 is greater than that over test period 1, the difference being significant at 99.99% or higher. For HIM, WCI, and NWI, the mean value over the test period of the band averaged cross power SS–RF [R9–13SS–RF]1,2 is appreciably higher than for SS–SN [〈R9–13SS–SN〉]1,2, the differences being significant at 99.8% or greater in test period 1 and at 99.99% or greater in test period 2. In the case of NEI, SS–RF cross power is significantly higher than that for SS–SN over the first test period, and significantly lower over the second test period, the confidence levels being much higher than 99.99% in both cases. This once again confirms that there is an inherent phase difference between NEI on the one hand, and HIM, WCI, NWI, and to some extent even CNEI, on the other. The values of SS–RF in CNEI and PENSI are lower as compared to the SS–SN values in both test periods, although the mean cross power over the second test period is higher than that over the first test period for all the time series considered. Over both the test periods, the means for SS–AISM are greater than those for SS–SN over the second test period, as in the case of HIM, but lower than for the test signal in the first test period (unlike HIM).
Table 2. The 9–13 a Band Averaged Wavelet Cross Power Between Sunspot Number and (1) Rainfall and (2) Noise
% Confidence [R9–13SS–RF]2, −[R9–13SS–RF]1
Cases where values for noise cross spectra are higher than for rainfall cross spectra.
Cases with mean band averaged cross power over test period 2 (TP2) lesser than that over test period 1 (TP1).
 The mean, standard deviation and range of 〈R9–13SS–SN〉, in comparison with the respective values for R9–13SS–RF, are listed for each test period in Table 3. Table 3 shows how relatively small the standard deviations often are, and serves to support the high confidence levels of Table 2.
Table 3. The 9–13 a Band Averaged Wavelet Cross Power
Rainfall Time Series
Test Period 1 (36 Samples)
Test Period 2 (32 Samples)
4.2. Mean Band-Averaged Cross Spectra for the Test Periods
 We now turn to the second analysis. Its chief objective is to depict, through some simple index, the regional variations in solar connections with rainfall that we have already noticed. This can in principle be done in many ways, but a particularly simple one exploits the Monte Carlo results already available over an ensemble of synthetic rainfall-matched noise realizations. The method we adopt is the following. We compare the SS–SN and SS–RF cross spectra [〈R9–13SS–SN(t)〉]1,2 and [R9–13SS–RF(t)]1,2, each averaged over the 9–13 a band in scale, and over each test period in time. This gives one number for each realization (1000 for SN, one for RF) for each test period. The average over the test period is justified as the rainfall time series are approximately stationary over such periods (confirmed by the F test [Azad et al., 2007]). In Figure 8, we first present the CDF for SS–SN cross power over the 1000-member ensemble. Also marked on Figure 8, as illustration, are the corresponding numbers for SS–HIM, indicating the percentile values attained by the rainfall points for the relevant test periods on the SS–SN CDF curves. These percentile values locate the rainfall process in relation to the distribution of the cross spectrum between sunspots and noise. It turns out that they provide a sensitive index of the nature of the solar-rainfall connection.
 To see this, we present a similar plot for the case of SS–NEI also in Figure 8. It is seen that the HIM points are located respectively at the 66.5 and 79.4 percentiles of the SS–SN CDF curve over the two test periods, while in the case of NEI the location percentiles are 79.3 and 24.1, indicating once again a nearly opposite phase relationship over the two test periods. (As rainfall numbers are means over samples of 36 and 32 a in the two test periods respectively, the 95% confidence bands for them are ±6.4% and ±8.5% respectively for the two test periods for HIM, and ±1.5% and ±3.2% respectively for the two test periods for NEI. The corresponding 95% confidence bands on these percentile locations for the two test periods are respectively 60.1, 73.5 and 70.2, 85.6 for HIM and 77.8,80.7 and 21.5, 26.6 for NEI.)
Table 4 lists the percentiles of the SS–RF points in the SS–SN CDFs over each of the test periods for all the homogeneous rainfall and AISM time series under consideration. It is seen that there is significant variation among the different homogeneous rainfall zones. Particularly in the more solar active test period 2, HIM, WCI and NWI are all placed toward the high end of the reference curve (greater than 79th percentile). CNEI and PENSI are located lower (respectively at the 69th and 36th) percentiles. NEI shows a dramatically different behavior from the rest, being located at a high percentile in test period 1 and low percentile in test period 2, and hence opposite in phase to the rest of the rainfall time series. From a plot of these results in Figure 9, a crudely quasi-longitudinal organization, tilting slightly east toward the north, is indicated, supporting a similar conclusion suggested by the work of BN05 (see their Figure 5). AISM, as may be expected, exhibits a behavior between the western and northeastern extremes.
Table 4. Percentile Location of [R9–13SS–RF]1,2 on CDF of [R9–13SS–SN]1,2 From Direct Monte Carlo
Percentile location, on SS–SN CDF, of SS–RF cross spectrum averaged over 9–13 a period band and over test period 1 [R9–13SS–RF]1.
Percentile location, on SS–SN CDF, of SS–RF cross spectrum averaged over 9–13 a period band and over test period 2 [R9–13SS–RF]2.
 At this stage we should perhaps more generally summarize the special features of the present analysis responsible for the high confidence levels of our results in a field that for long has been so controversial. (1) We select two periods respectively of highest and lowest solar activity, to provide maximum contrast in any solar process effect on rainfall. (2) We look for effects in bands around the dominant solar cycle period of 11.6 a. (3) We compare rainfall cross spectra against those with reference rainfall-matched synthetic noise, making the present results more rigorous than simple procedures using as reference red or blue noise based on short-term autocorrelations. At the same time we make the comparisons more broad based using Monte Carlo simulations with such synthetic noise. Such simulations enable us to seek differences not merely in means but in strikingly different characteristics of the whole probability distribution. These new techniques help us to tackle effectively an enduring problem in solar/terrestrial climate relations: namely the relatively low confidence levels at which statistical conclusions can be drawn because of the high variability of surface meteorological parameters (pointed out also by Shindell et al.  and Labitzke et al. , contrasting surface with stratospheric variables).
5. Implications for Mechanisms
 The results reported here raise the important question whether they can shed any light on possible mechanisms for the observed solar/rainfall connections. The most important conclusions from the present work in this connection are (1) the regional differentiation in the observed effect, (2) its organization along bands with a marked longitudinal component and a tilt eastward at the northern end, and (3) the presence of strong evidence for a direct connection between recorded rainfall and solar activity established through an analysis of cross spectra, thus distinguishing solar from other causes operating at the same time, such as increasing greenhouse gas levels.
 Before considering possible mechanisms it is worth presenting a brief review of recent work in the light of the present findings.
 In the nature of observational/modelling evidence, van Loon et al.  investigate results of the NCEP/NCAR reanalysis data produced for the period beginning in 1958. They find that the major climatological tropical precipitation maxima are intensified during solar maxima. From a consideration of temperature and OLR data they conclude that an increase in solar forcing intensifies the Hadley and Walker circulations. This is particularly true of what has been called the western Walker cell by Meehl and Arblaster ; its greater strength is associated with stronger upward motion, negative OLR differences and enhanced precipitation.
Van Loon et al.  also find that an all-India monsoon rainfall index, defined by Meehl and Arblaster  as area-averaged rainfall over land points in what may be called an Indian monsoon box bounded by 5–40°N, 60–100°E, showed an enhancement of rainfall by 0.13 mm d−1 in a solar max minus solar min composite. Our work, based on direct rainfall data, shows that between the high minus low solar activity test periods, the difference is slightly less in AISMR (0.08 mm d−1 or 3.5%), but reveals a regional variation (not noticed in the work of van Loon et al. ) from a high of 0.22 mm d−1 (7.3%) in WCI to lows of 0.06 mm d−1 (5.0%) in NWI and 0.08 mm d−1 (1.4%) in NEI. However, van Loon et al. [2004, Figure 4a] show OLR data which reveal an appreciable reduction over an area that roughly corresponds to the HIM region (northwest and central India). Although the confidence levels and spatial resolution are not very high [van Loon et al., 2004, Figure 4b], this conclusion is qualitatively consistent with the present work. The other important conclusion, also illustrated in the same figures, is the anomalously stronger western Walker cell in the solar maxima years, which again is consistent with the strong element of longitudinal organization found here. The present work provides direct evidence on rainfall that is consistent with the above observations.
Kodera  has suggested that the solar influence on the Indian ocean monsoon arises through stratospheric dynamical processes. More recently, using the reanalysis data from 1958 to 2004, Kodera and Shibata  report a direct relationship between the stratospheric meridional circulation and the Indian Ocean monsoon circulation in the troposphere. They point out that solar UV heating around the tropopause may itself be a primary cause. This is transmitted to lower levels through vertical propagation of planetary waves and the meridional circulation. The resulting heating in the lower stratosphere enhances the meridional Hadley circulation in the tropics. They do not rule out contributions from indirect radiative influences in the stratosphere through variations in ozone concentration, which we shall return to below.
 In a recent study published after this paper was submitted, Kodera et al.  show that, during periods of high solar activity, there is a shift in the location of a descending branch of the anomalous Walker circulation associated with the El Nino/Southern Oscillation (ENSO). In particular ENSO and the Indian Ocean dipole (IOD) are separated during high solar activity periods, whereas they are connected during low solar activity periods. They further argue that the shift in the Walker circulation is not controlled from the Pacific, but through interactions with the basic monsoon flow in the Indian sector. These ideas are consistent with the findings of the present work suggesting a solar effect on the Walker circulation, and also with the connections between ENSO and solar activity found by cross-spectral analysis by Bhattacharyya .
 An increasing number of model simulation studies exploring the mechanisms that might be responsible for solar-terrestrial links are now being reported. Such models are of two types. In the first are coupled stratosphere-troposphere global climate models [Haigh, 1994; Shindell et al., 1999; Haigh et al., 2005]. Now the changes in total solar radiation due to variations in solar activity amount only to about 0.1%. However the changes in ultraviolet (UV) radiation can be as high as 20% [Lean et al., 1997], which is high enough to have a strong effect on stratospheric chemistry as UV is largely absorbed by ozone. According to Shindell et al.  such UV absorption affects upper atmospheric zonal wind, which in turn affects planetary wave propagation and hence the troposphere. It is found that the net effect is to induce a circulation cell with a descending branch at 40°N and an ascending branch at 60°N. There is also a slight poleward shift in the northern subtropical jet, in the simulations of Haigh , which is confirmed by observations. These are well to the north of the Indian monsoon region, and so are not directly relevant to the present findings. Furthermore, these studies examine zonal averages, which are useful for unraveling stratospheric mechanisms but not the longitude-wise differentiation we report here. However, the greater effect on regional surface temperatures than on the global mean surface temperature found by Shindell et al.  is similar to our conclusion on Indian rainfall; for example, high minus low solar activity change in WCI rainfall (in mm d−1) is about 2.7 times higher than in AISMR.
 The second class of studies is illustrated by the work of Meehl et al. , who use a global coupled model which is forced only by changes in total solar irradiance, with no wavelength dependence or other effects. The results of the simulation exhibit intensified climatological precipitation regimes in the tropics, and stronger regional Hadley and Walker circulations. These effects are attributed to such tropospheric mechanisms as greater heating in relatively cloud-free subtropical high-pressure regions; stronger meridional temperature gradients; and warmer oceans producing more evaporation and latent heat flux leading to larger sources of low-level moisture, which is transported by low-level trade winds into monsoon zones, resulting in greater precipitation there.
 Our work supports the conclusions of Meehl et al.  to the extent that they also suggest a regional variation due to solar activity in the precipitation pattern, especially over the tropics and subtropics, and show a strengthening of Walker as well as Hadley cells. Although they further conclude that the enhancement of meridional temperature gradients due to greater solar forcing over land regions contributes to stronger West African and south Asian monsoons, they are unable to say anything definitively on changes in Indian monsoon precipitation. As we have shown here, AISMR is too heterogeneous an index to show strong effects of solar activity (cf. also the conclusion of Mehta and Lau ). Furthermore, it is difficult to see how small changes in total solar irradiance are amplified in this model.
 As in much of the present work we use a band-average around the decadal solar oscillation, the question arises whether the effect of other decadal signals is filtered out in the present analysis or not. Recent work has offered several candidates for such decadal signals, e.g., decadal variability in the West Pacific [Clark and Serreze, 2000; Holland and Raphael, 2006] and the Indian Ocean [Hurrell et al., 2004; Kucharski et al., 2005]. The present technique does not rule out such mechanisms without a separate analysis, but it does assert, with considerable confidence, a connection with sunspot numbers as they directly figure in the cross spectrum.
 In summary, the regional differentiation and the pattern of organization of the effect of solar activity on Indian monsoon rainfall reported in the present work are consistent with the idea that the chief mechanism responsible is connected with the Walker and Hadley circulations, which in one form or other is found in all recent models. Thus changes reported by Meehl et al.  are generally consistent with this result, but their model relies entirely on tropospheric dynamics and so is without a credible amplification mechanism. The coupled models, on the other hand, describe such an amplification mechanism, and although they also suggest changes in circulation, they do not yet exhibit the kind of enhanced precipitation we report here. Clearly a more comprehensive coupled model at a finer resolution is required before any firm conclusions can be drawn.
 In contrast, the dynamical mechanisms inferred by Van Loon, Kodera and their coworkers based on reanalysis data are broadly consistent with our conclusions, which together present a target for model simulations.
 The present authors have previously presented (BN05) a time domain analysis of rainfall, based on a comparison of means in two test periods, each comprising three complete solar cycles but between them exhibiting a strong contrast in solar activity. This analysis showed significantly higher rainfall in the period of higher solar activity. The analysis was however open to the objection that other factors (e.g., greenhouse gas levels) that also have a high contrast between the same periods could be responsible for the observed effect.
 This objection is removed in the present work, on the basis of an analysis of the cross spectra between rainfall and sunspot numbers. It is shown by Monte Carlo simulations using rainfall-matched synthetic noise signals, that the observed cross spectra are consistent with a strong connection (i.e., at high confidence levels) between directly measured rainfall data (not proxies) and solar activity parameters. The present analysis therefore complements and reinforces the authors' earlier result.
 One of the major outcomes of the present work is a much more quantitative analysis of the regional differentiation within the Indian land mass with regard to solar effects on rainfall. For example, the differences between northeast and western India are striking. (Of course the northeast is orographically and meteorologically different from the rest of India; it is one of the wettest areas in the world, so a 1.4% difference in rainfall translates to 0.08 mm d−1.) It is found that the observed effects exhibit a strong component of longitudinal organization, in the form of bands that tilt eastward toward the north. This longitudinal organization will be missed in 2D model simulations or results on zonally averaged fields. An important consequence of this result is that it indicates that solar processes affect the intensity of the Walker circulation; the tilt in the bands mentioned above suggests that there must be an effect on the Hadley cell as well.
 These conclusions are compared here with outputs from model simulations. The work of Meehl et al. , without a spectral differentiation in solar irradiance, is unable to say anything definitive about all-India monsoon rainfall. However, their conclusion about how regional effects on temperature are stronger than global effects is echoed here with regard to precipitation as well, suggesting that a proper assessment of model physics requires a regional analysis at relatively high resolutions. As an analysis of AMIP results has shown [Gadgil and Sajani, 1998], capturing the temporal and spatial variations in Indian monsoon rainfall remains a major challenge to modelers [see, e.g., Gadgil and Sajani, 1998, Figures 8 and 9], even in the absence of any attempt to simulate the effects of varying solar activity.
 Results from this class of models are difficult to interpret when they do not contain a mechanism that can amplify the small changes (of order 0.1%) in solar irradiance to the much larger changes in rainfall (of order 1–7%) that they seem to generate. This gap is filled by coupled stratospheric-tropospheric models [Shindell et al., 1999; Haigh et al., 2005], which utilize the considerable increase in UV radiation (of up to 20%) in periods of high solar activity to demonstrate changes in the stratospheric chemistry of ozone, and proceed to derive dynamical effects in the troposphere through coupling with the stratosphere. Purely dynamical connections through lower stratospheric heating are also possible [Kodera, 2004].
 Both classes of model predict that higher solar activity tends to intensify the Walker and Hadley circulations (qualitatively consistent with the present results), but neither can yet depict the regionally differentiated effects we report here. Progress here will demand first of all better models for the Indian monsoons even without an explicit consideration of the effects of varying solar activity, and secondly development of coupled stratotropospheric models that have sufficient resolution to handle regional differentiation on the scale of the order of a degree in latitude and longitude.
 The authors would like to thank A. Rai Choudhuri of the Physics Department of Indian Institute of Science for his help on the solar data and P. G. Vaidya of the National Institute of Advanced Studies for useful discussions. The authors thank the reviewers for their insightful comments and are grateful to the Centre for Atmospheric and Oceanic Sciences of the Indian Institute of Science for their continued hospitality.