The Madden–Julian oscillation (MJO) is a major source of
intraseasonal variability in the troposphere. Recently, studies have
indicated that also the solar 27-day variability could cause variability in
the troposphere. Furthermore, it has been indicated that both sources could
be linked, and particularly that the occurrence of strong MJO events could be
modulated by the solar 27-day cycle.
In this paper, we analyze whether the temporal evolution of the MJO phases
could also be linked to the solar 27-day cycle. We basically count the
occurrences of particular MJO phases as a function of time lag after the
solar 27-day extrema in about 38 years of MJO data. Furthermore, we develop a
quantification approach to measure the strength of such a possible
relationship and use this to compare the behavior for different atmospheric
conditions and different datasets, among others. The significance of the
results is estimated based on different variants of the Monte Carlo approach,
which are also compared.
We find indications for a synchronization between the MJO phase evolution and
the solar 27-day cycle, which are most notable under certain conditions: MJO
events with a strength greater than 0.5, during the easterly phase of the
quasi-biennial oscillation, and during boreal winter. The MJO appears to
cycle through its eight phases within two solar 27-day cycles. The phase relation
between the MJO and the solar variation appears to be such that the MJO
predominantly transitions from phase 8 to 1 or from phase 4 and 5 during the
solar 27-day minimum. These results strongly depend on the MJO index used
such that the synchronization is most clearly seen when using univariate
indices like the OLR-based MJO index (OMI) in the analysis but can hardly be
seen with multivariate indices like the real-time multivariate MJO index
(RMM). One possible explanation could be that the synchronization pattern is
encoded particularly in the underlying outgoing longwave radiation (OLR)
data. A weaker dependence of the results on the underlying solar proxy is
also observed but not further investigated.
Although we think that these initial indications are already worth noting, we do not claim to unambiguously prove this relationship in the
present study, neither in a statistical nor in a causal sense. Instead, we
challenge these initial findings ourselves in detail by varying underlying
datasets and methods and critically discuss resulting open questions to lay a
solid foundation for further research.
Introduction
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. .
Datasets and filtering
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.
Examples of the analyzed time series for the period January 2012 to
June 2013. The Lyman alpha solar proxy (a) shows pronounced 27-day
variability, particularly in the middle of this period. The MJO phase
evolution (b; here represented by the OMI index) shows the expected
periodic transitions from phase 1 to phase 8. However, it is obvious that the
solar 27-day signal as well as the MJO phase evolution are definitely not
perfectly periodic, but have to be considered as quasi-periodic processes
(see Sect. ). Strong variability also appears in the
MJO strength (c, OMI index). The figure also indicates the basic
steps of the analysis routine: identified solar minima are marked with red
stars in the top panel. The resulting 0- to 28-day time lag epochs are
indicated by alternating green and blue shaded areas in all three panels. As
an example of the following counting step, it is illustrated how often MJO
phase 3 occurred 10 days after solar minimum. For this, all 10-day time lags
are marked (all vertical red lines in panel b). During 4 of these
days, the MJO was in phase 3 (longer solid and dashed red lines). However,
only days during which the MJO strength exceeded a particular threshold (here
1, horizontal dashed line in panel c) are considered so that the
analysis results in two occurrences of phase 3 for time lag 10 (longer solid
vertical red lines).
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.
Essential nature of the potential relation between MJO phase evolution and solar 27-day cycle
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.
Number of occurrences of each MJO phase (one line per phase) as a
function of time lag after solar 27-day minima. The figure shows a particular
example: only solar minima during boreal winter and during a QBO easterly
phase were considered. After this filtering 26 epochs remain in the
analysis. Furthermore, the MJO strength on individual days has to be greater
than 1. The state of the MJO is characterized by the OMI index; the Lyman
alpha solar proxy was used to determine the solar minima.
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.
Position of the maximum occurrence numbers (measured in time lag
after solar minimum) for each MJO phase. Datasets and filtering conditions
are similar to those in
Fig. .
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.
Quantitative examination of the potential relation between MJO and solar cycle for different conditions
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.
Brief description of the quantification approach
Based on Fig. , it is
intuitive to define such a quantity as the similarity of the pattern
constituted by the eight data points to a sawtooth function. Numerically, this
similarity can be estimated by fitting a sawtooth function to the data
points. The goodness of fit χ2, which basically sums up the quadratic
deviations between data points and the fitted sawtooth function, could then
be a natural choice for such a measure; the smaller the value of χ2,
the better the similarity to a sawtooth function and the stronger the
relationship between the solar 27-day cycle and MJO evolution.
However, the common definition of χ2 has to be modified in two aspects
to be a suitable measure in the present context. This is described in detail
in Appendix and mentioned here
only briefly: first, the calculation of the individual deviations has to
account for the fact that the time lags are periodic with a periodicity of
27 days. This is considered in the quantity χper2 defined in
the Appendix . Second, the common
weighting of each data point with its reverse variance 1/σi2
works in the direction that a higher uncertainty (greater standard deviation
σi) leads to a smaller χ2. This is useful for the numerical
fitting routine, but works in the wrong direction for the present
application, the measurement of deviations. For this application, a higher
uncertainty should result in a greater value of the deviation, which reflects
a weaker certainty of the relationship found. Both aspects are considered in
the quantity X, which we defined as the measure of the deviation in the
Appendix . This quantity X,
simply called “deviation” in the following, is the measure used for the
strength of the relationship in this study; a lower deviation indicates a
stronger relationship between the solar 27-day cycle and the MJO phase evolution.
Altogether, our analysis routine comprises the following
steps.
Performing the analysis steps described in Sects.and. This consists of the
following:
identification of the solar extrema dates,
filtering of the input data according to the experimental
setup,
counting of the occurrences of the individual MJO phases as a function of time lag after the solar
extrema,
identification of the time lags with maximum occurrence number for each MJO phase.
Estimation of the uncertainty of the derived time lags using a bootstrap method. This is described in more detail in
Appendix .
Fitting the sawtooth function to the derived time lags of maximum occurrence for the eight MJO phases using the previously calculated bootstrap uncertainties as weights. As mentioned before, the fit is performed under consideration
of the 27-day periodicity of the time lags, hence by minimizing χper2 instead of χ2. For the same reason,
we have fixed the amplitude of the sawtooth function in the fit to a value of 27 days. Assuming, based on the previous
results, that the mean periodicity of the MJO is with 54 days twice the mean periodicity of the solar 27-day cycle, we
have also fixed the period to four MJO phases (only half of the eight MJO phases are experienced during one solar 27-day cycle).
The only free parameter of the fit is the phase ϕSt of the sawtooth function. As the fitting routine might not
directly find the global minimum of χper2 and, hence, the result might depend on the first guess of ϕSt,
the fitting procedure is repeated with the first guesses of ϕSt systematically varied between 1 and 8. The result with the minimal χper2 is then considered further on.
Calculation of the measure of deviationXbetween data and fit using the bootstrap uncertainties as weights.
Although the measure of the “deviation” is the direct quantitative result of the analysis, we are conceptually interested
in the opposite, the “similarity” of the pattern in the data and the sawtooth function. In the following, we will use both
terms equally in the sense that a small deviation means high similarity, which in turn means a stronger relation between
the solar 27-day cycle and the MJO phase evolution. Furthermore, we will not put emphasis on the physical units of X, which
depend on the weighting factors. Since only the variations in the results are of interest and not the absolute values, we will simply assume that X is given in arbitrary units.
Estimation of the significance of the quantified relationship. For this, the probability p that the value of
the deviation X could be the product of only random features in the data is calculated. This is achieved with a
Monte Carlo (MC) approach, which means that the input data are repeatedly modified with random numbers and the
complete analysis procedure is applied to a large number (1000) of such randomly modified input data representations.
The probability p is then simply calculated as the percentage of the random experiments, which resulted in a lower
deviation measure. This probability value was then used to quantify the significance of the respective result; the lower
the probability p that a low deviation can be reproduced with random numbers, the higher the significance of the
result. We have implemented different possibilities for the creation of the random data. These are outlined together
with the discussion of the respective results in Sect. . We use as the standard method in the
following the most conservative implementation, i.e., the one that indicates significance of the
results most rarely. This method is based on randomly shifting the original solar extrema dates by up to ±6 days and is also explained in more detail in Sect. .
An example of the fitting process, which corresponds to the case previously
discussed in Sect. , is shown
in Fig. . More
examples are included in the Supplement. After having performed this routine,
a measure of the deviation between the sawtooth pattern in the empirical data
and an analytical sawtooth function is known together with the fitted phase.
This deviation characterizes the strength of a possible systematic
relationship between the solar 27-day cycle and the MJO phase evolution. In
the following, we will apply this approach to a variety of different
experimental setups, which can then be compared among each other.
Time lags of the maximum occurrence for each MJO phase as in
Fig. , but including
results from the quantification approach: an estimation of the time lag
uncertainties, a fitted sawtooth function, the fit deviation X, and a
significance estimation.
Influence of the numerical setup
Before we discuss the results in detail, we note that the analysis is, like
most others, subject to well justified but strictly speaking arbitrary
choices. Wherever possible, we have repeated the analysis with different
realizations of these choices and have convinced ourselves that our main
conclusions do not depend on these choices.
One choice is the definition of the epoch period. We have defined an epoch to
start with a solar extremum and then last for 28 days. This is a natural
choice, since – if any relation can be substantiated – we expect the sun to
be the driver of the MJO phase evolution so that it makes sense to study the
atmospheric response in the period after the solar extremum. However, a
possible mechanism does not guaranty a direct response of the atmosphere in
the following 28 days. Instead the response could also manifest itself during
the solar cycles afterwards. Hence, no unambiguous starting point of an epoch
can be fixed and it would also be possible to, for example, center the solar
extremum in the epoch period, so that it covers the time lags from -14 to
14 days, like it is done in many studies. Interestingly, we found that some
of our conclusions appear even clearer using this alternative choice of the
epoch windows. Currently, we cannot decide whether this is a feature of the
studied relationship, or if it is a random effect. Therefore, we decided to
include the more conservative option of the 0- to 28-day epoch into the paper
but show the alternative results in the Supplement.
Another choice is the use of squared weights in the definition of the
deviation X as mentioned in Sect. . Hence, we have also repeated
the calculations with constant weights (we have chosen wi=1/8, so that
the sum over all 8 weights is unity), so that all data points are weighted
with the same factor. Although these results are not interesting from an
atmospheric point of view, we have also included them in the Supplement, to
convince the reader that the conclusions are not influenced by the
definition. However, for the interpretation of these alternative
calculations, one has to note that the significance analysis cannot lead to
very realistic results in the case of these arbitrary constant weights; since
the values of these weights directly influence the value of X, the choice
of weights directly influences the probability to gain a higher or lower X
based on a random dataset. And whereas the original calculation of the
weights considers the real spread of the data, leading to weights that
actually characterize the dataset, the constant weights are completely
unconnected with the dataset. What can still be seen from the results with
constant weights is that the qualitative comparison of results with different
experimental setups is similar and, hence, the conclusions are not dominated
by the kind of weighting.
As an example, we have also included the results of both alternative
calculations (the centered epoch definition as well as the constant
weighting) in the presentation of the first experiment
(Fig. , which is discussed in
Sect. ). Afterwards, all results will be
based on the 0- to 28-day epoch and the squared bootstrap uncertainties as
weights.
Calculated deviations X indicating the strength of the statistical
connection between the solar 27-day cycle and the MJO phase evolution; a
lower X indicates a stronger connection. The same data are shown in linear
scaling (a) for an overall visual impression and in logarithmic
scaling (b) for inspection of the smaller variations. The focus of this
experiment is the dependence of the deviation on the MJO strength threshold.
The green line shows these results for the standard numerical setup. Three
levels of significance in the sense of the MC experiment are shown: 10 %,
5 %, or 1 % chance of getting lower deviations with randomly modified
solar extremum dates. The other two lines give an impression of the reaction
of the results when the numerical setup is changed to, first, a different
epoch definition (blue line) or, second, the use of constant weights (red
line; see Sect. for details).
Influence of atmospheric conditions
In the following experiments, one parameter of the analysis will be varied,
while the others are kept constant with specific values. We used indications
from pretests and previous studies, to choose standard values for the
non-varied parameters, which lead to the clearest results and, hence, allow
the best conclusions concerning the particular influence of the varied
parameter. For the overall conclusions, the values of all filters have
finally, of course, to be considered at the same time. An example of more
relaxed filtering conditions is shown afterwards in
Sect. . An overview of the varied
parameters including the standard values in this and the following section
(Sect. ) is given in Table .
Parameters varied to quantify their influence on the possible
relation between the solar 27-day cycle and the MJO phase evolution.
ParameterPossible valuesStandard valueMinimum MJO strength0…2.5fully resolved in mostexperiments, otherwise 1QBO phaseeasterly, westerly, no filteringeasterlySeasonboreal winter (DJF), boreal winter and spring (DJFMAM),boreal winterboreal summer (JJA), no filteringSolar epoch trigger27-day maxima or minimaminimaMJO indexOMI, RMM, FMO, VPMOMISolar proxyLyman alpha, F10.7, NRLSSI2 205 nm, NRLSSI1 205 nmLyman alpha
Note that it would also be interesting to study the influence of the 11-year
solar cycle on the possible relationship between the solar 27-day cycle and the
MJO phase evolution in addition to the variation of the parameters listed in
Table . In principle, this is implemented in our
analysis as one further filter; however, it turns out that the number of
remaining samples becomes too small when this additional filter is applied.
For the standard filtering conditions, the number of remaining solar 27-day
cycles is already reduced to 19
(Sect. ) and is halved by
applying a solar 11-year maximum or minimum filter so that no significant and
reliable conclusions can be drawn anymore. Therefore, this aspect has to
remain open until longer datasets are available and one should keep in mind
that the solar 11-year minimum conditions are slightly underrepresented in
our analysis due to our 27-day extrema selection approach
(Sect. ).
MJO strength threshold
One major parameter for all MJO studies is the minimum MJO strength, which
has to be reached for an MJO event to be considered. Very often, a value of
1 is used, sometimes also a value of 2. We have examined in more detail the
influence of this threshold on our results first. For this we have varied the
MJO strength threshold between 0 and 2.5 in 0.1 steps. The other filter
criteria correspond to the standard of the example in
Sect. . The results for the standard
numerical setup are shown in Fig. (green line).
The results show comparatively low deviations, i.e., stronger indications for
a connection between the solar 27-day cycle and the MJO phase evolution,
between thresholds of 0.8 and 2.1. In this center range, most of the results
are significant at least at the 10 % level, many at the 5 % level, and some at
the 1 % level. This means that the chance to derive lower deviations with
randomly modified solar extremum dates is below this conservative estimate
(see Sect. ). Stronger deviations are evident
at both edges, which is the expected behavior. First, stronger deviations for
high MJO thresholds are directly caused by the low number of samples which
remain (the sample size starts with 26 for MJO strength threshold 0,
decreases to 19 for threshold 1, which corresponds to the example in Sect. , and decreases further down
to 1 in the present case with other restrictive filters for the threshold
2.5). Second, the stronger deviations for low MJO thresholds are caused by
the consideration of periods during which the MJO pattern (and with that the
value of the MJO phase) can hardly be identified and the analysis
incorporates mostly atmospheric variability not connected to the MJO.
We treat the deviation X between data and fit as the main outcome of our
analysis. Nevertheless, we also get values for the phase ϕSt, which
is the free fit parameter. It represents the phase of the MJO at the time of
the solar extremum. Using the solar minimum as the trigger, we get a value
for ϕSt of about 0.3. This means that the MJO predominantly
transitions from phase 8 to phase 1 or from phase 4 to phase 5 during solar
minimum (compare the fitted sawtooth function in
Fig. ,
particularly where it approaches time lags of 0 days). Consistently, we find
values for ϕSt of about 2.2 when we use the solar maximum as a
trigger, hence causing a shift by two MJO phases, which is a half of the sawtooth
period, as expected. This means that the MJO predominantly transitions from
phase 2 to phase 3 or from phase 6 to phase 7 during solar maximum. These
numbers for the fitted phase are quite stable among the different MJO
thresholds in this experiment, but also among the following experiments,
whenever a strong relationship between the solar 27-day cycle and the MJO
evolution is found. This stability of the fitted phase among the experiments
is remarkable, as it also supports some kind of synchronization between the
solar cycle and the MJO evolution in contrast to a hypothetical situation, in
which the fitted phase strongly jumps depending on the particular
experimental setup.
As mentioned before, Fig. also shows the
same results derived with slightly changed numerical setups, as described in
Sect. . First, for the alternative epoch
definition (blue line), it is seen that the significant range of low
deviations is somewhat shifted to lower MJO strength thresholds and shows a
bit less variability. Second, the deviations calculated with constant weights
are generally lower, which is, however, due to the fact that the arbitrarily
selected weights directly influence the value of the deviation X, so that a
comparison of absolute values of X is not reasonable. Only the variability
within the red curve can be compared to the variability in the other curves
and this looks quite similar. Overall, there are differences between the
numerical setups, but the observation that relatively low deviations are
found in the center of the MJO threshold range and higher deviations at the
edges is valid for all curves. In this sense, the following conclusions
will also be independent of the numerical setup, so that only the results derived
with the first setup will be shown in detail.
In the following we will present the results for the other numerical
experiments in a similar way. However, we will mostly show the results only
on the linear scale. That is because reading precise numbers of the
deviations is not really important on this arbitrary deviation scale.
Instead, the figures serve more as a visual comparison of the different
experiments, which is in our opinion easier with the linear scale in most
cases.
Phase of the QBO
showed that the MJO strength is
influenced by the QBO in a way that the MJO is stronger during the QBO
easterly phase. suggests that the solar influence (in this
case of the 11-year cycle) on the MJO activity might be masked by the QBO
influence if both work in opposite directions, so that the MJO activity is
strongest during solar minimum and under QBO easterly conditions.
finds that the influence of the solar 27-day variations on MJO strength is
also strongest for the QBO easterly phase.
We have also checked the influence of the QBO in the context of the MJO phase
evolution. For this, we excluded all epochs from the analysis, which do not
match the wanted QBO phase and repeated the analysis, again resolved for
different MJO strength thresholds. This has been done for boreal winter and a
solar minimum epoch trigger. The results (Fig. )
confirm a strong influence on the relationship between MJO phase evolution
and solar 27-day cycle, which is consistent with the previous studies. For
the QBO easterly phase, we find relatively low deviations X and
significance levels between 1 % and 10 % in the center range of MJO
thresholds. The deviations for QBO westerly periods are mostly more than 1 order of magnitude higher and the significance of all data points is worse
than 10 %. This means that there is no significant relationship between the
solar 27-day cycle and the MJO evolution based on the sawtooth-fitting
approach for QBO westerly phases in contrast to QBO easterly phases. If no
QBO filtering is applied, the deviations are, as expected, mostly between
those of the QBO easterly and westerly filtering. Almost no data points are
significant for this case.
Hence, we conclude that a possible relationship between the solar 27-day
cycle and the MJO phase evolution is only detectable for QBO easterly
conditions.
Note that alternatively defining the QBO phase by wind data at 30 hPa
instead of 50 hPa does not qualitatively affect this conclusion (see Fig. S15 in the Supplement).
As in Fig. , but with results
resolved for the QBO phase. The green line corresponds directly to the green
line in Fig. .
Seasons
It has been found before that the MJO strength modulation by both the QBO
and solar influences is mostly detectable during boreal winter, i.e., the
during the months December, January, and February ,
sometimes extended by the months March, April, and May .
We have checked the seasonality in the context of the MJO phase evolution by
restricting the considered epochs to the respective months. Indeed, our
results (Fig. ) also show that a strong relation
between the solar cycle and the MJO phase evolution is indicated
predominantly for boreal winter. A similarly strong relation is also seen for
boreal winter extended by the spring months, which we analyzed for the sake
of comparability to . During boreal summer, some of the deviations
are an order of magnitude higher and only rarely significant.
Data for boreal autumn have not been computed for reasons of computation
time. The unfiltered (i.e., year-round) data lead to deviations, which are
mostly located between the extremes and only rarely significant.
We have to note that the findings differ in this case somewhat among the
alternative numerical setups (see Sect. ).
In particular, that boreal winter extended by spring behaves similarly to winter
only is not true for the numerical setup, in which the epoch covers -14 to
14 days around the solar extremum (see Fig. S2 in the Supplement). In this
case only the boreal winter data show a clear, significant relationship, but
the data extended by spring do not. Although our results appear to be largely
consistent with , this detail is not consistent, as
is also based on the centered epochs.
As in Fig. , but with seasonally
resolved results. The green line corresponds directly to the green line in
Fig. . Boreal winter comprises the months
December, January, and February; boreal spring the months March, April, and
May; and boreal summer the month June, July, and August.
We conclude here that a possible relationship between the solar 27-day cycle
and the MJO phase evolution is detectable only during boreal winter, although
an extension into spring might be possible. The reasons for the seasonality
are speculative, but likely connected to the reasons for the seasonality
identified by and maybe also to that of the QBO influence
identified by , particularly the seasonality of the MJO itself
or the seasonality of the stratospheric residual circulation.
Solar minimum or maximum as epoch trigger
We have also checked whether it makes a difference to start the epochs with
the solar 27-day minimum or maximum. It turns out that, in the reasonable
range of MJO strength thresholds, the deviations X are on a similar order
of magnitude for both cases (Fig. ) and,
hence, that the choice of the trigger has no pronounced effect.
Nevertheless, looking more closely, one may note that the deviations are
mostly at least a bit lower and more data points are significant when the
solar minimum trigger is used. These differences should not be
overinterpreted, but we would like to at least mention them, because they
become more pronounced when the alternative experimental setup with centered
epochs is used (Fig. S3 in the Supplement). In this case, almost no data
points are significant using the solar maximum trigger, whereas a continuous
range over 13 data points is significant at the 5 % level for the solar
minimum trigger. Overall, the influence of the trigger must therefore remain
unclear in the present study. However, if a difference between both triggers
could be substantiated in the future, it could hint to possible mechanisms of a
synchronization between the solar 27-day cycle and the MJO; it could indicate
that the solar minimum is the actual trigger, which privileges certain MJO
phases and that the MJO phase evolution runs freely afterwards, so that the
results are more noisy when the analysis is started half of a cycle later
using the solar maximum trigger. Also the observation that the MJO is
predominantly before phase 1 during solar minimum would appear consistent in
this context (Sect. ).
As in Fig. , but also showing the
analysis results for the epochs being started with solar maximum. The green
line corresponds directly to the green line in
Fig. .
Relaxed atmospheric filter criteria
The previously presented experiments were designed such that one parameter
was varied, while the other parameters were set to the optimal values. This,
of course, limits the scope of the conclusions, since a clear relation
between the solar 27-day cycle and the MJO phase evolution is only indicated
when all conditions are met simultaneously, namely that the MJO strength
threshold is in a range around 1, the QBO is in easterly phase, and the
season is boreal winter. The previously shown negative results for other
filter setups demonstrate that no relationship that is significant at least
at the 10 % level is to be expected when one or more filter parameters are
relaxed. However, as stated before, we have applied a quite conservative
quantification approach in terms of the selected numerical approach
(Sect ) and the MC significance estimation
(Sect. and ), so
that it is still worth looking at an example with relaxed filter criteria to
get an impression.
A second reason for looking into this example is that it overcomes one major
drawback of the previous experiment design, namely that the number of samples
is relatively low. Because of these low numbers one may wonder whether the
relationship found is a particular feature of exactly this sample, even if this
risk is actually quantified by the MC analysis. In any case, it is worthwhile
to get an impression of results including more epochs.
As an example, the analysis has been repeated without the QBO and without
the season constraint. The derived time lags of maximum MJO phase occurrence
are shown in
Fig. ,
comparably with
Fig. . The MJO
strength threshold is set to 1, as in many other studies, but the results are
comparable for similar MJO thresholds. With these criteria, about 140 of
possible 243 epochs are considered instead of a few tens. As expected, the
deviation X is higher than in the optimally filtered case
(Fig. ) and is
not considered significant anymore, with the probability to derive a lower
deviation with random numbers being about 15 %. But still the data points do
not appear completely disordered. Instead they still remind the eye of the
sawtooth-like structure analyzed before.
As in Fig. , but
showing results of an experiment without filtering for QBO phase and season.
As described in the text, the results are not significant anymore on a level
better than 10 %. However, it is visually seen in this figure that there
are still indications for the sawtooth-like pattern instead of a totally
unstructured behavior.
On the one hand, this could indicate that the described relation may also be
there under different atmospheric conditions, but is superimposed by
different kinds of variability. On the other hand, it may mean that the
relation is so pronounced for specific atmospheric conditions that the
signature remains, even when other periods are included. In conclusion,
although we already analyzed 38 years of data, the period does not seem to be
long enough to significantly prove a statistical connection between the solar
27-day cycle and the MJO phase evolution for more general atmospheric
conditions than those described above, particularly not using our
conservative MC approach. However, this does not necessarily mean that the
relation is actually restricted to those conditions. Both a longer dataset
and refining the analysis approach to be less conservative, of course while
remaining scientifically strict, could help to answer this in the future.
Note that some more fit examples, which correspond to some cases of
particularly strong deviations in the previous experiments, are shown in the
Supplement. Also shown there, rudimentary indications of a sawtooth structure can
sometimes still be recognized although they are highly insignificant.
Influence of underlying datasets Influence of the MJO index
To describe the strength and phase evolution of the MJO, several independent
indices have been developed in the past (Sect. ). In
addition to the analysis based on the OMI index presented before, we have
repeated the analysis with other important indices, particularly the major
ones discussed in : RMM, VPM, and FMO.
We have recalculated, for example, the analysis described in
Sect. , that is for the standard conditions
boreal winter, QBO easterly phase, and solar minimum trigger. The results
(Fig. ) show clear differences
between two pairs of the indices; while the results for OMI and FMO show
relatively low deviations X, which are largely significant at the 5 %
level, the other two indices RMM and VPM show much larger deviations, which
are rarely significant. Hence, a relationship between the solar 27-day cycle
and the MJO phase evolution is only indicated by OMI and FMO. While this
result is somewhat surprising, the grouping of the indices appears plausible,
since the pairs also belong together conceptually, as they are either the
univariate indices OMI and FMO based only on OLR data or the multivariate
indices RMM and VPM, which also include circulation data.
As in Fig. , but also
showing the analysis results for alternative MJO indices. The green line
corresponds directly to the green line in
Fig. .
This can be interpreted in two ways. First, it could mean that a potential
connection of solar 27-day activity is not fully represented in the
circulation-based indices RMM and VPM. This appears plausible, because OMI
has a more precise representation of the convective center and, hence, might
better represent such subtle features we are looking for. This was exactly
the reason for using OMI as the primary index as also done in, for example,
. But second, it could mean that we did not
strictly identify a connection between the solar 27-day activity and the MJO
but only between the solar 27-day activity and OLR. The signature would then
appear more or less accidentally in the MJO indices and we would have found a
similar solar variability–OLR connection, as has been reported by
using different methods. In this respect, it is
appropriate to mention that and the
OMI and FMO description paper by
refer to the same OLR data basis, namely
. If this second interpretation were true, it would mean
that we have so far actually described the properties of the solar influence
on OLR and not directly on the MJO. Although this was not our original
objective, the results would still be interesting, as they underline the
possibility of solar 27-day influences on the tropospheric parameter OLR. As
in , the remaining open question of interest would
concern the mechanism of such a sun–OLR connection. An involvement of the
MJO in such a mechanism would still be likely, as the OLR is, of course,
influenced by the MJO.
Indeed, the fact that the properties of the relationship described so far are
largely consistent with other MJO-related studies suggests that the MJO is at
least involved in these interactions. Therefore, we propose to treat both
interpretations equally seriously for the time being. To be able to distinguish
both interpretations, future research should further examine the solar
influence on the data ingredients of the individual MJO indices and identify
processing steps in the computation of RMM and VPM, during which the solar
influence could get lost.
In conclusion, although the overall picture of the present results suggests
that the MJO is actually somehow involved, the present study must strictly
speaking remain inconclusive regarding the question of whether the MJO
is really influenced by the solar 27-day cycle or if only OLR is affected or
whether the OLR signal is maybe generated by a modulation of the MJO. What can be
stated, however, is that studies dealing with such subtle features of the MJO
should repeat the analyses with different MJO indices and not arbitrarily
select only one of them. This also concerns the aforementioned series of papers on
the solar influence on the MJO, which started with the RMM index
and switched to OMI while mentioning RMM results
before relying complete on OMI . In the
light of the current findings it would be of interest to know whether the results of
are reproducible also with RMM or not.
Influence of the solar proxy
We have also checked the influence of the solar proxy used in our study. In
addition to our standard proxy Lyman alpha, we have also included the F10.7
radio flux and for comparability with
the UV radiation data from NRL SSI models (Sect. ).
Generally, the solar proxy data are not as fundamental as the MJO index for
our analysis, since they are only used to generate a list of dates with solar
extrema, which define the epochs. The algorithm to find these local extrema
in the proxy time series
(Sect. ) depends on
thresholds, which are adjusted for the particular proxy. Variations in the
data among the proxies and the definition of these thresholds may cause the
resulting list of extrema to be a bit different depending on the proxy used.
Hence the present experiment basically checks the influence of a somewhat
different epoch sampling. If, hypothetically, an unambiguous list of the
solar 27-day extrema had been derived, this list would be used instead of the
proxy data and this kind of test would be obsolete.
As expected, the results
(Fig. ) show overall a similar
shape corresponding to the description in
Sect. , with higher deviations for low and high
MJO thresholds and lower deviations in the medium range. Nevertheless, there
is also considerable variability among the different curves, showing that the
analysis is still sensitive to the exact sampling of the epochs, although the
relatively long period of 38 years is analyzed. It appears that the range of
significance is somewhat different for Lyman alpha and the alternative
indices; where Lyman alpha shows
significant data points between MJO strength thresholds of roughly 1 to 2,
the significant range is located more between 0.3 and 1.3 for the other
indices. Also, this observation should, however, not be overinterpreted,
since it is not evident when using the alternative numerical setup with
centered epochs (Sect. , Fig. S5 in the
Supplement). In the Supplement, the significant range is more homogenous and
spans a broader range from roughly 0.3 to 1.6 for most solar proxies.
As in Fig. , but also showing the
analysis results for alternative solar proxies. The green line corresponds
directly to the green line in Fig. . Note
that this figure is shown in logarithmic scaling for an easier inspection of
the comparatively slight differences.
The fact that the variability introduced by a somewhat different epoch
sampling propagates into the final results suggests that it is currently
safer to repeat such subtle analyses of the solar influence in tropospheric
parameters with different proxies to check if the drawn conclusions are
robust.
Significance estimation with different Monte Carlo variants
We have estimated the significance of the individual results with a MC
approach as already outlined in Sect. ; for
each calculation result, the analysis is repeated 1000 times with randomly
modified input data. The significance is then indicated by the percentage of
runs, which resulted in equal or lower deviations X (stronger relationship
between the solar 27-day cycle and MJO phase evolution) compared to the original
calculation.
There is, however, a lot of freedom in the particular design of the random
modification of the input data and, to our knowledge, there is no unambiguous
argument for selecting a particular method. In contrast to this, the
particular implementation is usually only briefly described in many studies
and a comparison of the different results is difficult. In our case there is
not only freedom in how the random component is implemented, but also to
which of the three time series (MJO strength, MJO phase, list of solar
extrema) it is applied. Since we are analyzing here a very subtle feature,
the relationship of two quasi-periodic but still variable processes of the
sun–earth system, we decided to discuss different implementations here, so
that the spectrum of possible significance values becomes obvious.
The basic question for the investigation of a relationship between two
quasi-periodic processes is to what extent the random modification may
influence the internal temporal behavior of both processes. On the one hand,
it is exactly this internal structure that characterizes the inherent
nature of the processes (here, for example, the temporal evolution of the MJO) and
that should not be artificially modified. On the other hand, exactly this
temporal behavior has to be randomly disturbed in order to check whether the
relationship of both processes reacts to this disturbance. In other words, a
random modification has to be introduced, as this is the idea of the MC
technique, but is has to be kept so small that the nature of the analyzed
process remains comparable. This problem is also discussed in, for example,
or in the context of the bootstrap
method.
Since it is mostly not obvious which idea for the random modifications meets this compromise best, we have
tried different implementations, which are described in the following. The
results of all implementations are compiled in
Fig. for the standard
experiment conditions, which correspond to the example of
Sect. : QBO easterly phase,
boreal winter, MJO strength threshold 1, and solar minimum trigger. In
Fig. , the results are
ordered by a decreasing conservation of the internal structure of the
original time series. In addition to the standard MJO index OMI, we have also
calculated these experiments for RMM and FMO.
Results of the different MC implementations for the standard
experiment setup (QBO easterly, boreal winter, MJO strength threshold 1, and
solar minimum trigger), which corresponds to the example of
Sect. . The horizontal dashed
lines mark the significance levels of 1 %, 5 %, and 10 %,
respectively. The eight items are ordered according to a decreasing conservation
of the internal temporal structure by the random modifications. The results
are shown for the MJO indices OMI (green), FMO (blue), and RMM (red) for each
implementation method. The first and the last two methods, which are
separated by dashed lines, are included for completeness but do not really
characterize the significance of the experiment; see
Sect. for details.
To start with one extreme, it would be possible to replace one or both time
series with white noise, i.e., completely un-autocorrelated random data. It
is intuitively clear that the application of the described analysis procedure
to such a random time series would be very unlikely to result in a structured
pattern as seen in, for example, Fig. . Hence, low probabilities
of deriving lower deviations would be found and a high significance of the
original calculation would be indicated. But looking closer, this estimation
would not be very conclusive, since the characteristics of the original data,
which initially motivated the analysis, are not apparent anymore in the white
noise random time series. Nevertheless, we have conducted two related
experiments. First, we have replaced the MJO phase time series by a time
series in which the MJO phases are randomly distributed according to a
uniform distribution, without any autocorrelation. Indeed, the probability
to undercut the original deviation X with the random data is essentially
0 % (Fig. , on the very
right) for OMI and FMO. The probabilities for RMM are generally higher, since
the relationship was weak with this index anyway
(Sect. ). But it is, at about 1 %, still low
in this case. Hence, this experiment confirms the expectation that it is
unlikely to derive the sawtooth-pattern with a completely randomized MJO
phase distribution. In the second approach, we have left the MJO index values
untouched but have selected the dates for the solar extrema completely
randomly. For this, we have selected as many out of the about 14 000 possible
days as have been considered in the original analysis. Hence, the epochs are
randomly distributed over the complete analyzed period and are totally
independent of the actual temporal behavior of the solar proxy so that a
potential temporal relation between the solar proxy and the MJO index will be
broken. However, at least the temporal evolution of the MJO during the
individual epochs is conserved, since the MJO index is untouched. The
probability to find lower deviations with this random dataset is still below
1 % for OMI (Fig. ,
second from right), which has a somewhat stronger meaning than the first
experiment; it shows that the sampling of the MJO index with epochs has to be
largely systematic to reproduce the relationship found.
At the other end of the extreme, one could not touch the internal structure of
all time series at all. For example, the random component could be introduced
by randomly selecting subsets of the epochs originally considered, which is
similar to the bootstrap method. Hence, only different subsets of the same
data pairs (MJO phase and solar proxy) are evaluated and it is not very
surprising that this approach results in a comparatively high probability to
find similar low or lower deviations. Although we have included this result
for completeness (Fig. ,
on the very left), it is not very meaningful in this context, since this
experiment does not challenge the temporal relationship between both
processes at all. Instead, such an analysis evaluates the influence of the
particular sampling period on the result and could, for example, be used to compute
error bars for the deviations (which we have not extensively done due to a
limitation of computation time). Hence, this approach is not considered
further on.
As a good compromise between both extremes, we ended up with shifting the
originally considered solar extrema dates a bit (see also
Sect. ). Particularly, the extrema dates are
shifted by a few days, which are randomly selected from a uniform
distribution between -6 days and +6 days for each solar extremum
independently. Hence, this approach modifies the temporal relation between
both processes but is restrained to the effect that the evolution of the MJO
is not touched at all, while the mean periodicity of the solar 27-day cycle
is also conserved and only the deviations from this mean period are randomly
changed. Hence, also the inherent temporal mean structure of the solar proxy
is conserved when these random fluctuations are introduced. This approach
leads to the already-mentioned probability of about 8 % to undercut the
original deviation with the random data in the present example
(Fig. , second from left,
and Fig. ).
Considering the only slight changes of the solar extrema dates (less than
6 days compared to the large MJO period variability of a few tens of days)
the 8 % appear remarkably low; i.e., the significance was remarkably high.
Formulated the other way around, shifting the solar extrema dates randomly by
only a few days will already weaken the relationship found between solar
variability and the MJO phase evolution in 92 % of the cases. This low
probability to undercut the original deviation X with this conservative
approach indicates that coincidences of variations in the solar proxy and in
the MJO phase time series are not very tolerant against slight temporal
changes and, hence, that a synchronization of both variations might really
exist. Note that for some of the previously described experiments
(Sect. and )
significance values of better than 5 % and 1 % were also found using
this approach.
We are not aware of any unambiguous definition of the randomly generated data
but think that we have at least justified the latter approach, which has been
generally used as the standard method in this study.
However, we do not claim that this is the only possible approach. Aside from
the fact that the range for the random shifts of ±6 days is an arbitrary
definition, completely different approaches to generate the random data are
conceivable. We have implemented two further ideas (with two variants each),
which we will outline in the following. These approaches indicate an even
higher significance of our results. However, as we are carrying out this
subtle study as conservatively as possible, we have decided to use that
approach as the standard, which results in the lowest significance.
Both alternatives modify the MJO time series and leave the list of solar
extrema dates untouched. For the first approach the continuous MJO index time
series is completely shifted by a random number of days. The shifted period
can be each number of days between 0 and the length of the time series. The
ending period, which exceeds the original end date of the analysis after the
shift, is cut and pasted in place of the now missing starting period. To our
understanding, a comparable approach has also been used in . For our first variant of this approach, the shift is only applied
to the MJO phase evolution, whereas both phase and strength are modified
similarly in the second variant. Keep in mind that phase and strength have
different roles in the analysis; while the strength is only used as filter
criterion, the phase is basically the analyzed quantity. This approach almost
completely conserves the internal temporal structure of the MJO index except
at the two seams. The only thing disturbed is the direct temporal day-to-day
relation between solar variations and MJO variations. The disturbance is,
however, stronger than in our standard approach, since the resulting temporal
difference between originally coincident features of the solar proxy and the
MJO index can be many years instead of only ±6 days. The results show
(Fig. , third and forth
item) a very low probability to undercut the original deviation, which is
comparable to that of the totally random time series explained first. Hence,
this approach would indicate a high significance, if treated as the deciding
approach. The result of this approach further indicates that, in order to
explain the observed relation, it is not enough to have two processes, which
only act on related timescales in terms of the mean period. Instead, it
seems that a closer temporal linkage on the basis of individual solar and MJO
cycles could be necessary.
The second alternative is based on a random redistribution of individual MJO
events (i.e., continuous periods starting with phase 1 and lasting until
phase 1 is reached again), hence, the new MJO index time series are composed
by randomly redistributing MJO event pieces of the original time series. This
is also applied either to the MJO phase alone or to both phase and strength.
This approach also preserves the temporal structure of the MJO to a large
extent, since the mean periodicity as well as the temporal behavior of the
individual MJO events are not changed. But as in the previous case, the
temporal relation of the solar proxy and the MJO index is strongly disturbed,
since originally coincident cycles get randomly separated by possibly long
periods. The results
(Fig. , fifth and sixth
item) are comparable to those of the first alternative, which also indicates
that the temporal relation between both processes on the basis of individual
cycles seems to be important.
Note that also the result of the experiment with relaxed filter criteria (no
filtering for QBO and season; see
Sect. and
Fig. )
based on OMI, which was not considered significant with 14.7 % using the
standard approach, would be significant on at least a 5 % level if the latter
two alternative MC approaches would be treated as decisive.
Discussion and conclusions
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.
Code availability
The source code will be made available by the authors upon
request.
Data availability
The datasets used in this paper are publicly accessible.
The following MJO indices were obtained online from the NOAA Earth
System Research Laboratory: OMI
(https://www.esrl.noaa.gov/psd/mjo/mjoindex/omi.1x.txt, last access:
28 March 2019), VPM
(https://www.esrl.noaa.gov/psd/mjo/mjoindex/vpm.1x.txt, last access:
28 March 2019), and FMO
(https://www.esrl.noaa.gov/psd/mjo/mjoindex/fmo.1x.txt, last access:
28 March 2019). The MJO index RMM was obtained online from the
Australian Bureau of Meteorology
(http://www.bom.gov.au/climate/mjo/graphics/rmm.74toRealtime.txt, last
access: 28 March 2019). All solar proxy time series were obtained online
from the LASP Interactive Solar Irradiance Data Center
(http://lasp.colorado.edu/lisird/, last access: 28 March 2019). QBO
data were obtained online from the Institute for Meteorology at Freie
Universität Berlin
(http://www.geo.fu-berlin.de/met/ag/strat/produkte/qbo/qbo.dat, last
access: 28 March 2019).
Defining a measure for the strength of a relationship between the MJO phase evolution and the solar 27-day cycleCommon measure for the goodness of fit χ2
Commonly, analytical functions are fitted to measured data by minimizing the
quantity χ2=1ν∑iwi(yi-y(xi,a1,…,aM))2e.g.,. Here, the yi
are the data points, wi are weights (commonly defined as wi=1/σi2 with the σi being the standard deviations),
y(xi,a1,…,aM) is the analytical function fitted to the data points,
and a1,…,aM are the parameters, which are adjusted by the fit. The
number of degrees of freedom, ν, is the number of independent data points
minus the number of adjusted fit parameters. In the present case y=y(xi,ASt,PSt,ϕSt) is the sawtooth
function with the amplitude ASt≡27 days, the period
PSt≡4 MJO phases, and the phase ϕSt, which
is the only free parameter adjusted by the fit. The independent variable,
x, represents the eight MJO phases, and the dependent variable, y,
represents the time lags of maximum occurrence for each MJO phase.
After the fitting routine has determined the optimal parameter ϕSt,
i.e., the one that leads to a minimal value of χ2, this value of
χ2 summarizes the residual deviations between data and fit. Hence, it
could be used as the sought measure of the deviation between the data and a
sawtooth function. However, two pragmatic modifications have to be applied to
derive a suitable measure in the present case, which are described in the
following Sects.
and .
Accounting for periodicity in the fitting process
The calculation of the individual deviations has to account for the fact that
the time lags are periodic with a periodicity of 27 days. This means that,
for example, the deviation between the time lags 3 and 23.5 days is not the
comparatively large number of 20.5 days, but only 6.5 days, as exemplified in
Fig. . The largest
deviation that can occur is therefore 27days/2=13.5 days.
This has to be reflected by a modified quantity measuring the deviation
between data and fit, which we define as χper2=1ν∑iwiΔyi2. The Δyi are
initially defined to identically reproduce the original χ2, i.e,
Δyi=Δyi,orig=yi-y(xi,a1,…,aM). However, after
their initial calculation the values of the Δyi are restricted to
the range between ±13.5 days by subtracting multiples of 27 days from the
Δyi,orig, hence Δyi=Δyi,orig-ki⋅27 days, where ki counts the multiples of 27 days to be
subtracted.
Illustration of the calculation of deviations between the data
points and the fitted values. For reasons of clarity, only one data point has
been included (black cross). With the conventional approach, the deviation
Δyorig is simply the difference between the observed and
the fitted value (red cross), which is 20.5 days in this example. However,
the fitted relationship is not only periodic in the direction of the
abscissa, but also in the direction of the ordinate, since possible time lags
repeat themselves with a period of 27 days. The observed data can therefore
be virtually periodically continued and it becomes obvious that the relevant
deviation Δy is that between the fitted data point and the newly
introduced virtual data point (gray cross), which has a value of Δy=-6.5 days (=20.5–27 days) in this example.
Instead of the minimization of χ2 commonly used for curve fitting, we
use χper2 for the present study, so that the fitting routine finds
the optimal fit parameter ϕSt under consideration of the periodicity
of the fitted relationship.
Measuring the deviation including weights
For the calculation of χ2, the individual deviations are usually
weighted according to the uncertainty of the measurements yi. This is
adopted here also for the calculation of χper2. The weights
wi are calculated as usual as the reciprocal variances of the measured
data, i.e., wi=1/σi2 with the σi being the standard
deviations (the values of σi are estimated with the bootstrap method
described in Sect. ). This is, of
course, a useful definition for the originally intended application of
χ2 and χper2, being the quantities to be minimized
during the fitting process; the relative importance of data points with a
large uncertainty is reduced and the other way around.
However, such a quantity χper2 is not suitable for the intended
measure of similarity between the data points and a sawtooth function; good
similarity should be indicated by a small χper2 (small deviations
Δyi between data and fit). But using this kind of weighting, a
comparatively small χper2 is also produced by large uncertainties,
which is the opposite of the wanted behavior in this context.
A solution is to compute a different overall measure of the deviations
Δyi after the fitting (which remains based on the minimization of
χper2). A straightforward and pragmatic definition, which we
introduce here as deviation X, is similar to χper2 but uses
reciprocal weights: X=1ν∑iΔyi2wi=1ν∑iσi2Δyi2. With
this definition, large uncertainties lead to a higher value of X, which
indicates a weaker relation between the pattern of the data points and the
sawtooth function. And the other way around, smaller uncertainties work in
the same direction as small deviations between data and fit and lead to a
small value of X, which indicates a stronger relation between the pattern
of the data points and a sawtooth function. Hence, based on this value the
relation of the solar 27-day cycle and the MJO phase evolution can be
quantified and compared between different experimental setups (e.g.,
different filtering or underlying datasets).
We note that this definition also has disadvantages. First, it is a somewhat
arbitrary choice, particularly the power of 2, with which the standard
deviations σi contribute. It has been
chosen analogously to the definition of χ2 but could have also been
chosen differently. Second, this definition combines two factors which modify
the value of X, the deviations between data and fit and the
uncertainties. Hence, using this measure, it cannot
be distinguished whether differences of X between experimental setups are
dominated by the deviations or the uncertainties. The influence of this
choice on our conclusions is discussed in
Sect. and results derived with an
alternative choice are shown in the Supplement.
Estimating the uncertainty of the days with maximum MJO phase occurrence
Since the derived time lags of maximum MJO phase
occurrence are the result of a counting process that incorporates the
complete dataset, there is no possibility to directly determine the
corresponding uncertainties, i.e., the statistical distribution function and
its width. A well-established approach to estimate the uncertainties for such
cases is the bootstrap method e.g.,. Basically, random samples are drawn from the original sample
to generate additional virtual samples for which the complete analysis is
repeated a large number of times. This results in the distribution of
possible analysis results considering random effects in the original dataset.
From this distribution the uncertainty can be calculated as, for example, the
standard deviation.
In our case, we use the set of identified solar extrema dates as independent
members of the original sample. From these dates we draw 1000 random samples
with the same number of members (sampling with replacement) and repeat the
analysis for each random sample. This results in eight distribution functions of
the time lags of maximum MJO phase occurrence, one for each MJO phase.
Calculating the standard deviation of these distributions as uncertainty is
also somewhat more complicated than usual, again due to the periodicity of
the time lags (compare Sect. ); imagine
a distribution, which is centered at time lag 26 days and symmetric with wings
of a few days length on both sides. Because of the periodicity the right wing
will not be located around 29 days, but at time lags around 0 to 5 days,
whereas the left wing remains around 24 days. Hence, the distribution would
look like a bimodal distribution with two unconnected centers. The mean value
would be in the middle at about 13 days and the standard deviation would
represent a width, which spans the complete range from 0 to 27 days. To
overcome this problem, we shift each distribution function first, such that
the maximum is located in the middle at a time lag of about 13 days, and
calculate the standard deviation afterwards, which is, apart from that, not
affected by the shift.
The supplement related to this article is available online at: https://doi.org/10.5194/acp-19-4235-2019-supplement.
Author contributions
CGH outlined the project, designed the method, performed the
study and prepared the paper with substantial insight and interpretation
of results provided by CvS during all previously mentioned tasks.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
We would like to thank Indrani Roy and one anonymous reviewer for their
valuable comments, which helped to improve the paper significantly. This work was supported by the
University of Greifswald. We acknowledge support for the article processing
charge from the DFG (German Research Foundation, 393148499) and the Open
Access Publication Fund of the University of Greifswald.
Review statement
This paper was edited by Peter Haynes and reviewed
by Indrani Roy and one anonymous referee.
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