The solar electromagnetic radiation is the major energy source of the earth system. Although usually described with the solar constant (1361 W m-2), the total solar irradiance (TSI) is subject to variations on different timescales, with the most prominent one being the solar 11-year cycle. While the variation of the TSI is only on the order of 0.1 %, it differs among the spectral regions and is particularly strong in the UV e.g.,and references therein. Of interest for the present study is the solar 27-day cycle, which is a combined result of the differential rotation of the sun and irradiance inhomogeneities on the solar disc. The amplitude of the 27-day cycle is generally smaller than that of the 11-year cycle, but can be on the order of 50 % of the 11-year amplitude in the UV during strong events. The 27-day cycle is not perfectly periodic, but exhibits some variability, so that the 27 days have to be seen as a mean period of a quasi-periodic process. When using terms like the “solar cycle”, “solar maximum”, etc., we always refer to the 27-day variations in this paper if not stated otherwise.

The solar variations introduce atmospheric variability and many effects have been identified in the past, particularly in the middle atmosphere, where the strongly varying UV is important. Signatures of the 27-day cycle have been found in, for example, temperature , trace gases e.g.,, polar mesospheric clouds e.g.,, and very recently in radio wave reflection heights . The interactions between solar and atmospheric variability are still subject to ongoing research, which aims at both identifying more affected parameters and elucidating the underlying mechanisms. The attribution of 27-day signals in the atmosphere to a solar cause is thereby complicated by the fact that internal variability of the atmosphere can itself also produce signals with periods around 27 days, as pointed out by, for example, , which becomes even more important for lower altitudes.

Nevertheless, in addition to implications in the middle atmosphere, a discussion of possible of 27-day signatures in the troposphere came up recently, mostly in the context of convection and clouds , but also related to temperature . Even more than for the middle atmospheric effects, questions concerning the mechanisms behind tropospheric signatures arise. Two major classes of conceivable mechanisms are summarized by and mentioned here only briefly: on the one hand the “bottom-up” mechanisms detailed by, which assume that the only slight variations of the TSI produce strong enough heating changes directly in the troposphere to generate the observed modulations in the upper troposphere, and on the other hand the “top-down” mechanisms, which consider the stratospheric effects of the stronger UV variations as starting point. The modulations could result via a chain of effects in a change of upper tropospheric static stability and with that in a change of tropospheric deep convection, with implications for clouds and temperature. Another mechanism, particularly for a connection between clouds and the solar variability, has been proposed in a few variants (e.g., ; ) but has also been heavily criticized e.g., and is mentioned here only for completeness. It considers a connection of cloud condensation nuclei and incoming galactic cosmic rays, whose flux is affected by solar activity.

Independent of a possible solar influence, there is a known important source of tropospheric variability on the intraseasonal timescale, the Madden–Julian oscillation (MJO). It was first reported by , and more recent reviews of properties and implications are found in and . In brief, it is a planetary-scale pattern in the tropics consisting of a region with anomalous strong deep convection flanked by two regions of weak deep convection to the east and to the west. This pattern evolves over the Indian Ocean and travels eastward across the Maritime Continent until it decays in the Pacific. This temporal evolution is usually split into eight phases as originally suggested by Fig. 16. The MJO pattern reappears periodically; however, the period is strongly variable in a range between 30 and 100 days . The MJO is the dominant component of intraseasonal variability in the tropical troposphere with strong influences on rainfall and the genesis of tropical cyclones in the respective regions, for example. In addition, there are also increasing indications for an entanglement of the MJO in teleconnections and, hence, for an influence of the MJO in the extratropics e.g.,. Due to its intraseasonal timescale and the large spatial scales, one important motivation for MJO research is that it could help to push the limits of weather forecasting skills towards longer periods .

In addition to the tropospheric implications, indications for interdependencies with the middle atmosphere have also been brought up, particularly with ozone and temperature e.g.,. Of particular interest for the present study is the finding that the MJO depends on the state of quasi-biennial oscillation (QBO) e.g.,. The QBO represents a quasi-periodic reversal of the stratospheric equatorial zonal winds with a mean period of 28 months e.g.,. Briefly, it affects the MJO strength, particularly during boreal winter, such that the MJO is stronger during the QBO easterly phase.

In the context of solar-induced tropospheric variability there is a two-level interest in the MJO. First, it might be difficult to distinguish both possible causes for intraseasonal tropospheric variability, since the MJO acts on timescales (starting with 30 days) close to the solar 27-day variation. Hence, suspected 27-day signatures in the troposphere might in reality be connected to the MJO. Second, in the light of recent publications, which are outlined below, it appears at least conceivable that the MJO is itself influenced by the 27-day cycle. From this point of view, the MJO might be a pathway for the 27-day solar signal into the troposphere. Hence, the three topics solar 27-day variation, MJO, and tropospheric variability on intraseasonal timescales might be interconnected.

An example for the first level is the publication by , which reports on 27-day variations found in the cloud amount over the western Pacific region. These results are based on a frequency analysis of OLR data in place of direct cloud amount data. The authors are cautious with speculating on possible mechanisms but also briefly mention that the spectral analysis shows indications of MJO activity and that some kind of interdependency cannot be ruled out.

The second level, a possible modulation of the MJO itself by the 27-day solar cycle, has been proposed by a series of studies . The study by is actually focused on a tropospheric temperature response to solar 27-day variations. A modulation of the MJO is discussed as part of the mechanism, which brings the temperature signal into the troposphere. An initial investigation of this hypothesis shows a change of the occurrence of the particular MJO phases 1, 7, and 8 after solar 27-day extrema, which is considered to be consistent with the tropospheric temperature change. directly deals with a solar modulation of the MJO, but is focused on the solar 11-year cycle and the occurrence rate of strong MJO events instead of MJO phase occurrences. The study indicates that the MJO is influenced by solar 11-year variations during boreal winter. This influence is roughly as important as the previously mentioned QBO modulation and might work with a similar mechanism: the modification of upper tropospheric stability. This also means that both influences have to work in the same direction (e.g., QBO easterly phase and solar minimum) to get a detectable MJO change. also analyzes the occurrence of strong MJO events but returns to the solar 27-day variations. A statistical relationship between the solar 27-day variations and the occurrence of strong MJO events is indeed found during the boreal winter and spring months from December to May. Particularly, strong MJO events (amplitudes greater than 2) are decreased following solar maxima and vice versa. As before, this effect is stronger under QBO east conditions.

The analysis presented here contributes to the critical examination of a possible linkage between the solar 27-day cycle and the MJO based on the analysis of about 38 years of MJO data. It is complementary to the previous studies, as it deals with the temporal MJO phase evolution instead of MJO strength. Analyzing the temporal evolution focuses on a special aspect: the relation of the periods of both processes; first, the range of possible MJO periods starts close to the period of the solar 27-day cycle. And second, the mean periodicity of the MJO is with 50 to 60 days approximately twice that of the solar 27-day variability, which turns out to be of interest in the following. Overall, it is analyzed here if there are similarities and regularities in the temporal evolution of both processes and we will show that a kind of coincident behavior can indeed be found in a statistical sense, which is partly surprisingly clear. However, we would like to emphasize that we do not try to prove a causal relationship between the solar 27-day cycle and the MJO phase evolution at this early stage. Likewise, we do not try to establish a particular mechanism. Instead, we aim at describing the statistical features of a combined inspection of both quasi-period processes as a basis for future research.

In Sect.  we describe the analyzed datasets and the initial filtering of the data. In Sect.  the basic analysis idea is outlined first, before the essence of the statistical relationship found between the solar 27-day cycle and MJO phase evolution is demonstrated based on a particularly clear example. In Sect.  questions concerning the generalizability of this example are addressed. For this a numerical approach to measure the strength of the relationship is developed first, before the analysis is applied to different selections of the underlying data. A discussion of major open questions and the conclusions are found in Sect. .

Basically two pieces of information are needed to perform the present analysis: the time series of the solar activity and the MJO in the past.

The solar activity is represented by several proxy time series. We primarily use the Lyman alpha flux as an indicator for solar activity. In addition, we have also performed the same analysis with the F10.7 radio flux index e.g.,and references therein and UV radiation at 205.5 nm simulated by the NRLSSI2 model , as well as similar data from a previous model version.

Many different indices have been developed to compactly describe the strength and phase of the MJO at a given time. These indices are usually calculated from either circulation data or information on cloudiness. The latter is usually represented by outgoing longwave radiation (OLR) data. Some approaches also combine both aspects to form multivariate indices . One of the latter indices is the real-time multivariate MJO index (RMM), which became the standard after its publication by . A variant of RMM is the velocity potential MJO index (VPM) introduced by , in which the OLR information is replaced by a velocity potential. This leads to a better MJO representation during boreal summer, among other advantages. More recently, introduced the OLR-based MJO index (OMI), which is a univariate index solely based on OLR data. It overcomes drawbacks of RMM at the expense of the real-time capability. This disadvantage is, however, not of importance for retrospective analyses, so that OMI has become an important index at least for these cases. also introduce a second univariate OLR-based index, the filtered MJO OLR index (FMO), which is easier to calculate than OMI. point out that all these different indices lead to similar results concerning the statistical gross features of the MJO, but differences are to be expected when working on the basis of individual MJO events.

Since our analysis does not depend on real-time information, we use primarily the OMI index. An example of the OMI data as well as of the Lyman alpha solar proxy is shown in Fig. . Additionally, we have also applied our analysis to the RMM index, the VPM index, and the FMO index. All these indices provide two coefficients each, which are transformed into MJO phase and strength by basically applying a transformation from Cartesian coordinates, in which the index coefficients are given, to polar coordinates. The radius and phase angle of the polar coordinates then correspond to the MJO strength and phase, respectively. Note that there are different conventions among the different indices for the attribution of the index coefficients to the Cartesian coordinate system . The phase angle is then divided into eight ranges of 45∘ each, which represent the eight MJO phases mentioned before e.g.,.

The availability of the MJO indices is the limiting factor for the temporal extent of the analysis. The OMI index starts in 1979 and ends in August of 2017 at the time of the analysis and hence covers about 38 years. The other MJO indices cover roughly a similar period. All datasets are available with daily resolution so that the analysis is performed on a daily resolved grid.

As part of the analyses described in Sects.  and , the datasets are filtered with respect to geophysical properties: first, only days during which the MJO strength exceeds a particular threshold are considered. Second, as the marker for the start of a new solar 27-day cycle, the solar minimum is used mostly, but the solar maximum can also be selected. Third, from the detected solar cycles, the relevant ones can be selected according to the season and, fourth, they can be filtered according to the state of the QBO. For the latter, 50 hPa (and 30 hPa as alternative) zonal wind data from radiosondes in the tropics have been used . For the determination of the QBO phase, simply the sign of the wind data is used with positive values denoting the westerly phase and negative values denoting the easterly phase.

As the basic analysis step, we check whether individual MJO phases appear preferentially at a particular state of the solar 27-day cycle. The idea is to count the number of occurrences of the individual MJO phases as a function of time lag after solar extrema. We analyze 28 days after each solar extremum; these temporal windows are called the epochs. This analysis is related to the approach of , but we treat all eight MJO phases separately, while focused on a combination of a few of them. We will demonstrate in the following that a preference for particular MJO phases depending on the solar 27-day state appears indeed to be present under certain conditions. We chose an experimental setup in this section for demonstration purposes, with which this relationship appears comparatively clear, and we will discuss the ability to generalize these findings in Sect. .

The following explanation of the analysis approach is also illustrated in Fig. . The analysis starts with identifying the solar 27-day minima in the Lyman alpha solar proxy time series (solar 27-day maxima are calculated likewise for other experimental setups). For this, the anomaly of the Lyman alpha time series is calculated by subtracting the smoothed time series (35-day moving average), which removes the variations greater than 35 days. Also the shorter-term variations are removed from the anomaly by smoothing it with a 5-day moving average. In the resulting proxy anomaly time series, the local extrema are identified. Only extrema with anomaly values of at least 0.2×1011 photons cm-2 s-1 above or below 0 are considered. This is a relative conservative filtering of extrema candidates; to make sure that only clear cases are considered in the analysis, we risk that some actual extrema are missed by the algorithm. This approach leads therefore to a slight underrepresentation of solar 11-year minimum conditions, since 27-day minima are also less pronounced during these periods and are more likely to be rejected (quantitatively, no 27-day minima have been selected by the algorithm for Lyman alpha values below 3.57×1011 photons cm-2 s-1, and a reduced 27-day minima selection is visually seen roughly below 3.8×1011 photons cm-2 s-1). In total, the algorithm finds 243 solar minima in the 38-year period, which means that about 6500 days out of the about 14 000 days are covered with considered epochs. From this set only solar minima are selected, which occurred during boreal winter (December, January, February) and during the QBO easterly phase. This results in a set of only remaining 26 epochs. However, the filter criteria in this example are among the most restrictive ones, so that the number of 26 samples is roughly a lower boundary for the sample size of the following experiments.

For all remaining solar minima days, we count how often each of the eight MJO phases has occurred. A phase occurrence is only taken into account if the MJO strength exceeds the threshold of 1 in the current example, so that the sum of all phase occurrences is usually lower than the number of considered epochs (19 occurrences in this example). This is not only done for exactly those days with the solar minima, but it is repeated for all time lags between 1 and 28 full days after each solar minimum. This results in one curve for each of the eight MJO phases describing the number of occurrences as a function of time lag after solar minimum. These curves are shown for the current example in Fig. .

Considering that the MJO shows a great variability and that also the solar 27-day cycle is only a quasi-periodic process, one would expect that these curves are basically constant with strong noise contributions. This would mean that each MJO phase occurs without any preference similarly often at each time lag after the solar minimum, which in turn means that the MJO phases evolve independently of the solar 27-day cycle. And the first overall impression of the functions in Fig.  might apparently reflect this expected chaotic nature to a certain extent.

However, a closer look reveals some structure in the functions. First, each of the eight curves exhibits a maximum at a particular time lag. The maxima are partly quite pronounced (e.g., for MJO phase 1) and partly somewhat broader (e.g., for MJO phase 7), but a kind of maximum is recognizable for each of the MJO phases. This indicates that the MJO phases occur preferentially at a certain time lag after the solar minimum. Second, the positions of the maxima reveal a specific ordering. Starting with the maximum of MJO phase 1 at time lag 3 days, the maxima of the phases 2, 3, and 4 follow monotonically with increasing time lag. MJO phase 5 starts again with a low time lag of 8 days followed again monotonically by the phases 6, 7, and 8.

This structure is more clearly visualized in Fig. , where the time lags of the phase occurrence maxima are shown for each MJO phase. The two sequences of monotonically increasing time lags for the phases 1 to 4 and 5 to 8 are clearly visible and constitute a sawtooth-like pattern.

The appearance of this clear pattern is the major qualitative result of this study and the essential characterization of the possible relationship between the solar 27-day cycle and MJO phase evolution. We think that this result is quite remarkable considering that the MJO, although showing some kind of periodicity, is a highly variable phenomenon.

Based on the clearness of this pattern, it appears attractive to directly assume a causal synchronizing mechanism between the solar 27-day cycle and the MJO phase evolution, which would, however, be premature. Nevertheless, the mere appearance of this sawtooth pattern has at least two requirements. First, the mean period of the MJO should be twice as large as the mean period of the solar 27-day variation. Hence, it should be about 54 days, which is well in the already known range of periods. Taking into account, though, that the instantaneous MJO period varies strongly, the second requirement is needed, namely that there should be a predominant phase relation of the solar 27-day variations and the MJO phase evolution during the complete analyzed period, i.e., the MJO is predominantly around phase 1 or around phase 5 at solar minimum. Otherwise, the sawtooth shape would be arbitrarily shifted over the MJO phases for certain subperiods, so that the pattern would finally be averaged out when taking the complete analyzed period into account. These requirements are obviously to a large extent fulfilled in the present example; however, the question remains whether this fact really demands a causal mechanism or if it could also be a coincidence in the analyzed period. Furthermore, a possible causal mechanism would have to explain why the solar 27-day variation produces a variation with a doubled period, i.e., why there are two possible MJO phases at each solar state. We emphasize again that it is not our aim to prove such a causal connection in this study. Instead, we aim at carving out more statistical characteristics of this connection from the dataset itself in Sect.  as a first step. This helps to get a clearer picture of the conditions under which such a connection might exist.

In the light of the present findings, it is in order to briefly comment on some results in , which are also based on counting the occurrences of MJO phases as a function of time lag after solar 27-day maxima or minima. However, in contrast to the present study, the MJO phases are not treated individually, but only the cumulative occurrence of MJO phases 1, 7, and 8 is evaluated, which is motivated by the particular questions in this analysis. The author finds that the cumulative occurrence of these phases is enhanced in the days after solar minimum and reduced about 10 days after the solar minimum. With the present results in mind, it does not seem to be a very reasonable choice to combine the particular phases 1, 7, and 8 as their positions of maximum occurrence represent three (of possible four) different time lag ranges (Fig. ). Instead, if one wants to group the phases with respect to the solar cycle, it would be more plausible to overlay the two lines of the sawtooth pattern, which means that the following pairs of MJO phases belong together in their relation to the solar 27-day cycle: 1 and 5, 2 and 6, 3 and 7, 4 and 8. Additionally, it should not be expected that the phase package 1, 7, and 8 behaves contrarily to the “opposite” phase package, consisting of the MJO phases 3, 4, and 5 (which does not claim, but what the reader might intuitively think). Instead this package represents similar maximum occurrence time lags as the first package 1, 7, and 8 (Fig. ) and should behave similarly. With this in mind, the conclusions drawn based on these results in should be reconsidered, especially because the author has pointed out the initial character of these results himself.

It is our aim to challenge the hypothesis of a relationship between the solar 27-day cycle and the MJO phase evolution by diversifying the setups of the numerical experiments. That means that the same analysis is repeated for different choices of atmospheric conditions, underlying datasets, and also implementation details. To do so, a quantity is needed first that measures the strength of the relationship and, hence, makes the results for different setups comparable.

The MJO has been known to be a major source of tropospheric variability on the intraseasonal timescale for some decades. More recently, studies indicated that the solar 27-day cycle could introduce variability not only in the upper and middle atmosphere, but also in the troposphere. At first, this raises questions on how these sources can be unambiguously attributed to observed variability. But even more interestingly, there have been indications that both sources are actually linked. In particular, it has been suggested that the occurrence of strong MJO events is modulated by the solar 27-day cycle.

We have analyzed a complementary aspect, namely whether the temporal evolution of the MJO phases is potentially linked to the temporal evolution of the solar 27-day cycle. For this, we have analyzed about 38 years of MJO indices and solar proxies in combination. We have basically counted the occurrences of particular MJO phases as function of time lag after the solar 27-day extrema. To achieve comparability between different experiments, we have developed a quantification approach based on the standard least-squares fitting routine to measure the strength of such a possible relationship. We have used this to analyze the relationship under different atmospheric conditions (state of the QBO, seasons, MJO strengths), different solar cycle triggers, and different MJO indices and solar proxies. Furthermore we have applied different implementations of a MC significance analysis and compared the results.

We have indeed found indications for a synchronization between the MJO phase evolution and the solar 27-day cycle under certain conditions, which are summarized below. Overall, the relation is such that the MJO cycles through its eight phases within two solar cycles, i.e., the mean period of the MJO is twice that of the solar variation. Hence, it should be approximately 54 days, which fits well into the broad range of possible periods between 30 and 90 days known before. The phase relation between the MJO and the solar variation is such that the MJO is predominantly either between phase 8 and 1 or between phase 4 and 5 at the times of solar 27-day minimum. Consistently, the MJO transitions either from phase 2 to phase 3 or from phase 6 to phase 7 during solar maximum.

We have found that this relation is most pronounced during QBO easterly phases (defined by either 50 or 30 hPa winds) and during boreal winter, which is consistent with previous studies. The relationship can then be identified for a broad range of MJO strength thresholds between approximately 0.5 and 2.0. The upper limit is, however, probably only the artificial result of a very low number of samples and might increase with the availability of longer datasets. For these conditions combined, the relation is surprisingly clear, as shown in Fig. . For relaxed atmospheric filter criteria, the relation is still recognizable (e.g., Fig. ) but is not significant anymore according to a conservative estimation. It has to be kept in mind that our selection of the 27-day extrema leads to a slight overrepresentation of solar 11-year maximum conditions in our analysis. Furthermore, interconnections between the solar 11-year state, the QBO phase, and other middle atmospheric parameters have been described in the literature before and could be relevant for the interpretation of the presented results. Unfortunately, the analyzed time series is not long enough to differentiate between 11-year maximum and minimum conditions in the present analysis.

As we have been trying to carve out a very subtle potential feature of the sun–earth system, we have implemented not only one MC experiment as significance analysis, but several variants. The basic difference among these implementations is the extent to which the random modifications may alter the original internal temporal structure of the time series. As our standard method, we have selected the most conservative one, i.e., the one which needs only modest modifications of the original time series and which has comparatively low significance values. In particular, we leave the MJO index time series as it is and randomly shift the solar extrema dates by up to ±6 days each. It is, however, difficult to find the only unambiguously correct variant for this particular problem, so that we have also discussed other implementations. With some of these variants, the relation between the solar cycle and the MJO phase evolution would actually be considered significant under more diverse conditions, including the previously mentioned relaxed atmospheric filter criteria.

Although we think that the partially surprising clarity of the results justifies already reporting on this topic now, we would like to emphasize that we do not consider the relationship to be already proven: first, not in a statistical sense, since there are many open questions left and since our analysis still suffers from a low number of samples despite the 38 analyzed years, and second, even less is clear in a causal sense, on which we have not worked so far. Even if the statistical connection is confirmed in the future, it appears difficult to undoubtedly extract the exact mechanism, which would also have to explain why the mean period of the affected process is twice that of the forcing.

One major question, which has been brought up by the present study, is why the relationship appears so clearly when using univariate OLR-based MJO indices like OMI and is almost not present when using multivariate indices like RMM. As OMI is known to better represent the convective center, one explanation could be that RMM simply fails to reproduce this subtle feature. However, another possibility, which cannot be neglected, is that the relationship is not really a property of the real MJO but only of its representation in OMI. Since OMI is only based on OLR data, this could mean that we have analyzed a relationship between the solar 27-day cycle and OLR. Despite not being our original focus, this would also be of interest, as it would be an additional indication for the presumption that upper tropospheric parameters are influenced by solar variability. And the directly following question on the mechanism of such a potential sun–OLR relationship might refer back to the MJO. In any case, the triad solar 27-day cycle, OLR, and MJO should be subject to further studies in the future.

Another major question, which could not be clearly answered by the present study, concerns the origin and the consequence of the period relation of both processes, i.e., the factor 2, which apparently connects the mean periods of the solar 27-day cycle and the MJO (54 days). This factor appears remarkable and might also support the assumption that a synchronization between the solar 27-day cycle and the MJO phase evolution really exists. However, one could also argue the other way around that this factor could be a random feature of the sun–earth system, which accidentally produces the results of the present analysis. Indeed, if one assumes that the analysis is applied to two perfect harmonic oscillations, with a factor 2 between the periods, then one would expect exactly the same sawtooth-like pattern in the results. In this case, a statistical relationship that has no causal counterpart at all would be found. However, this implicitly assumes that the phase between the two oscillations is constant, or at least that a particular phase relationship dominates during the analyzed period. This can, unfortunately, not be excluded based on the present analysis, but it appears at least questionable if such a dominant phase relation is plausible for such a variable phenomenon as the MJO without any synchronization mechanism. Hence, it was one aim of the conducted MC experiments to also quantify the influence of random variability in the context of these two quasi-periodic processes. The results of different MC implementations consistently indicated that it is not sufficient to have a constant relation of the mean periods of the two processes. Instead, the results indicated that a connection on a nearly day-to-day basis is important to reproduce such a close relationship between the processes, as seen in the real data. Nevertheless, such MC experiments might indicate that the probability for a pure coincidence of two processes with doubled periods is low, but they cannot disprove this possibility, so that this question remains open.

An additional outcome of this study is that the particular importance of the influences of the applied datasets and methods was emphasized. Studies on this topic should be repeated with different MJO indices and the precise meaning of applied MC analyses should also be discussed. In this respect, further efforts in method development would also be valuable, which could lead to a standardization of approaches to make the results more comparable. This should also include frequency analyses, which we have not applied here, but which could also help to better understand the appearance of the factor 2 between the periods of the solar 27-day cycle and the MJO phase evolution.