While progress to understand the mechanisms of the MJO has improved over the years, global climate models still do not realistically reproduce all characteristics of the MJO (Zhang, 2005; Lin et al., 2006). Therefore, the question of how global warming will further impact the MJO has not been explored. The objective of this note is to show projections of possible changes in the activity of the MJO under one particular climate warming scenario. We refer to activity as the number of MJO events occurring in a period of time. The approach used here is probabilistic and employs previously published stochastic models of the MJO. Section 2 provides an overview of the stochastic models. Section 3 discusses the activity of the MJO in the current climate and presents projections of changes in the activity of the MJO. Section 4 presents the conclusions.

2. Stochastic models of the MJO

Two stochastic models have been developed to simulate the variability of the MJO. An overview of the methodology and main model features are summarized here, while additional details can be found in Jones (2009) and Jones and Carvalho (2011); some figures are reproduced here for completeness.

The stochastic models were built from observations of MJO occurrences. To identify MJO events, daily averages of zonal wind components at 850 hPa (U850) and 200 hPa (U200) from the National Centers for Environmental Prediction/National Center for Atmospheric Research (NCEP/NCAR) reanalysis (Kalnay et al., 1996) were used (1 January 1948–31 December 2008). Daily averages of outgoing long-wave radiation (OLR; Liebmann and Smith, 1996) were used to characterize the convective signal associated with the MJO (1 January 1979–31 December 2008)

To isolate the MJO signal, the seasonal cycle was removed and the time series were detrended and filtered in frequency domain to retain intraseasonal variations (20–200 days) (Matthews, 2000; Jones, 2009). MJO events were identified with combined empirical orthogonal function (EOF) analysis of equatorially averaged (15°S–15°N) U200 and U850. The phase diagram of the first two normalized principal components (PC1, PC2) approximately follows the convention of Wheeler and Hendon (2004) and was used to identify MJO events according to the following criteria: (i) the phase angle between PC1 and PC2 systematically rotated anti-clockwise indicating eastward propagation at least to phase 5 (maritime continent), (ii) the amplitude (PC1^{2} + PC2^{2})^{0.5} was always larger than 0.35, (iii) the mean amplitude during the event was larger than 0.9, and (iv) the entire duration of the event lasted between 30 and 90 days. A total of 239 MJO events were identified during 1948–2008. The overall conclusions in this study are not sensitive to the criteria defined above. Figure 1 shows the canonical life cycle of the MJO based on composites of OLR anomalies from events during 1979–2008.

2.1. A homogenous stochastic model of the MJO

Jones (2009) introduced a homogeneous stochastic model capable of simulating the temporal and spatial variability of the MJO. In this study, only the temporal component is relevant and additional details about other aspects of the model can be seen in Jones (2009). The temporal variability of the MJO uses the observed time series of phases, which consists of nine possible states S = 0,1,2,…8. S = 0 corresponds to days when the MJO is quiescent and S = [1,8] when it is active and in one of the eight phases of the PC1–PC2 diagram. According to observations of MJO events, phases increase from low to high values indicating eastward propagation. An event may propagate up to phase 8 or end prematurely before that (phases 4–7). Moreover, the MJO may persist in the same phase for several days (typically 5–8 days) and even retrograde to a lower phase for a few days (usually 1–2 days) before it continues eastward propagation. A primary MJO is defined when the oscillation initiates from a quiescent state, whereas a successive event occurs when one MJO initiates right after a previous event, i.e. the phase diagram shows a continuous evolution from phase 8 to phase 1 (Matthews, 2008).

Jones (2009) used a homogenous nine-state first-order Markov Chain approach (Wilks, 2006) and the time series of phases to model the temporal variability of the MJO. Eighty-one conditional probabilities are involved and correspond to all possible transitions: P_{ji},j = [0,8] and i = [0,8], where the first subscript indicates the state at time t and the second subscript the state at time t + 1 (Wilks, 2006). The conditional probabilities determine transitions from situations of non-MJO to MJO (and vice versa) and transitions through the eight phases that characterize its life cycle. The conditional probabilities of initiation of MJOs are given by P_{01},P_{02},P_{03} and P_{04} indicating that the MJO in this model can start in phases 1, 2, 3 and 4 (i.e. ‘primary’ event). The conditional probability of a successive event is P_{81} indicating transition from phase 8 to phase 1. In addition, P_{ji} is always higher for eastward propagation than westward propagation. The westward propagation simply indicates that the MJO can retrograde to lower phases usually for a few days. P_{ji} values equal to zero indicate that the MJO cannot ‘jump’ across non-adjacent phases. The conditional probabilities of termination of MJOs are P_{40},P_{50},P_{60},P_{70} and P_{80}. Figure 2 schematically shows all possible phase transitions.

The estimation of P_{ji} in the homogeneous model is done using the time series of phases (61 years at daily values). For instance, P_{01} is estimated as: P_{01} = (number of 1s following 0s)/(total number of 0s), and likewise for other conditional probabilities (Wilks, 2006). Jones (2009) discussed in detail the estimation of P_{ji} and how to minimize systematic errors in the MJO simulations. The optimal estimation for the homogeneous stochastic model was the mean P_{ji} values estimated in 30-year windows (Table II in Jones, 2009). The model is called homogeneous because, once estimated, the transition probabilities are constants.

The simulation of temporal variability of the MJO is accomplished with the following procedure. The model uses a daily time step and is initialized at phase S = 0 (i.e. quiescent MJO). An algorithm for uniform random numbers (r) continuously generates r ∈ [0,1], which is compared to the conditional probabilities until an MJO event initiates, i.e. a transition occurs from S = 0 to S = [1, 2, 3 or 4] (Figure 2). Once an event starts, a duration T_{k} ∈ [30,90] days is randomly generated from a gamma probability distribution function (p.d.f.) fitted to observed durations of MJOs. This step specifies that the given event will last T_{k} days. Random numbers r are generated and compared against P_{ji} values. The system then follows transitions through the eight MJO phases until the duration of T_{k} is reached. If a transition attempts to terminate the event before the duration T_{k} is reached, new r values are picked until it continues the transitions through the eight phases. If the duration T_{k} is reached and the system is in phase S = 4, 5, 6 or 7, the system returns to S = 0, since these are the only possibilities (Figure 2). If the system is in phase S = 8 when T_{k} is reached, the system may change to S = 0 or S = 1 depending on the value of r. This situation may initiate a successive MJO event.

Events generated by the homogeneous stochastic model occur irregularly in time and can appear as isolated events or successive MJOs in agreement with observations. The oscillations in the model can last between 30 and 90 days, have different zonal propagation characteristics and the eastward propagation is consistent with observations of the MJO.

2.2. A non-homogenous stochastic model of the MJO

Jones and Carvalho (2011) extended the homogeneous version to a non-homogeneous stochastic model in which the probabilities of MJO initiation vary in time. An overview of the main formulation is provided here and additional details are discussed therein.

First, the homogenous stochastic model provides the probabilities of primary and successive MJO initiations (Table II in Jones 2009). A constant parameter is defined as P_{HMG} = P_{01} + P_{02} + P_{03} + P_{04}, i.e. the sum of conditional probabilities of primary events in the homogeneous model. P_{HMG} = 0.015649 is the average value estimated in 30-year samples. The conditional probability of a successive MJO was estimated as P_{81} = 0.084706. (Note that the stochastic models need probabilities estimated in double precision but are shown rounded to six decimal digits.)

Next, to represent the non-stationarity of the MJO, the conditional probabilities of primary and successive events were estimated in shorter intervals throughout the time series of observed phases. A variable was then defined as K_{NHM}(t) = K_{01}(t) + K_{02}(t) + K_{03}(t) + K_{04}(t) which represents the sum of conditional probabilities of primary events estimated in 3-year moving windows. Likewise, the conditional probability of a successive MJO estimated in 3-year moving windows is K_{81}(t). The 3-year window size was deemed optimal and reproduces some important multi-year variations in the activity of the MJO during 1948–2008.

The non-stationarity of the MJO is represented by the ratios K_{NHM}(t)/P_{HMG} and K_{81}(t)/P_{81}, which show how the probabilities of primary and successive MJO initiations vary in time relative to a homogeneous stochastic process. The positive trend in K_{NHM}(t)/P_{HMG} (Figure 3(a)) indicates an increase in MJO activity after the 1970s. Moreover, multi-annual changes in K_{NHM}(t)/P_{HMG} suggest low-frequency variations in the MJO; those changes are more noticeable after removing the trend. Jones and Carvalho (2011) showed that the long-term trend and multi-annual changes in K_{NHM}(t)/P_{HMG} are significantly correlated with long-term warming and low-frequency changes in sea surface temperature (SST) anomalies in the Indian and western Pacific warm pool. Likewise, K_{81}(t)/P_{81} (Figure 3(b)) shows a positive trend to high values indicating an increase in successive MJO occurrences; if K_{81}(t)/P_{81} = 0 in a 3-year interval, it means that there was no successive MJO in that period, although isolated MJOs could have occurred. K_{81}(t)/P_{81} suggests that a change in the MJO occurred after the early 1970s such that the probability of successive MJO initiation is considerably higher than in the early part of the record.

Jones and Carvalho (2011) showed that the non-stationarity of the MJO during 1948–2008 can be empirically represented by a regression that expresses K_{NHM}(t)/P_{HMG} as a function of low-frequency SST changes in the tropical Indian and western Pacific warm pool:

(1)

where a_{j}, b_{j} are regression coefficients, M_{SST} and V_{SST} are detrended 3-year running means and variances of SST anomalies averaged over the warm pool (15°S–15°N; 50°E–150°W); τ_{j} (0, 156, 312, 468 and 624 days) and γ_{j} (0, 300, 600, 900 and 1200 days) are time lags; Tr_{30} is a non-dimensional 30-year running mean of SST anomalies to account for the long-term trend in the warm pool region. Tr_{30} is used (as opposed to a simple linear trend) to make the non-homogeneous stochastic model stable if large negative SST anomalies are used in the regression equation. The empirical constants (1.25 and 0.48) keep the bias in the model to a minimum.

It is opportune to mention that the relationship between K_{81}(t)/P_{81} and SST changes in the warm pool is not easily represented. During some decades, an increase in the number of successive MJOs leads warming in the warm pool by 1–2 years, whereas the relationship is not systematically seen in other decades. To simplify the problem, Jones and Carvalho (2011) used the fact that K_{81}(t)/P_{81} and K_{NHM}(t)/P_{HMG} are correlated (0.51) and defined K_{81}(t)/P_{81} as:

(2)

where P_{81} is the conditional probability in the homogeneous model (i.e. average value of 30-year samples) and K_{81}(t) is estimated in 3-year moving windows.

In the non-homogeneous stochastic model, only the probabilities of MJO initiation vary in time and the remaining ones are kept constant during the simulations. This simplifies the formulation and allows investigating the hypothesis that multi-annual to long-term changes in SST anomalies in the warm pool drive variations in the activity of the MJO. Another important feature is that uncertainties in SST anomalies in the warm pool and errors in the regression model are taken into account in the ensemble simulations.

An overview of the non-homogenous stochastic simulations for the observational record is as follows. The first step is to construct time series of K_{NHM}(t)/P_{HMG} and K_{81}(t)/P_{81}. Mean monthly SST anomalies averaged over the warm pool from four different datasets are used in Eqs (1) and (2). Uncertainties in SST anomalies are represented by the standard deviation of the datasets. Upper and lower bounds for K_{NHM}(t)/P_{HMG} and K_{81}(t)/P_{81} are constructed using Eqs (1) and (2) with SST anomalies expressed as SSTA ± δ, where SSTA is the mean anomaly and δ is the standard deviation. Errors associated with regression Eqs (1) and (2) are added to the upper and lower bounds of K_{NHM}(t)/P_{HMG} and K_{81}(t)/P_{81}. (Jones and Carvalho, 2011, provide full details on the error analysis.)

The non-homogeneous stochastic model is initialized at S = 0 (i.e. quiescent MJO) and observed mean SST anomaly in the warm pool on 1 January 1948. The probability ratios of primary and successive MJOs for that day are determined by Eqs (1) and (2). However, random Gaussian errors ε with zero mean and bounded by the upper and lower bounds of the K_{NHM}(t)/P_{HMG} and K81(t)/P81 curves are added to the value specified by Eqs (1) and (2) for that day: K_{NHM}(t)/P_{HMG} + ε, K_{81}(t)/P_{81} + ε. This process ensures that K_{NHM}(t)/P_{HMG} and K81(t)/P81 are randomly drawn between possible values that take into consideration uncertainties associated with SST anomalies in the warm pool and errors due to the regression models (Figure 6 below gives an example).

Next, since K_{NHM}(t)/P_{HMG} and K_{81}(t)/P_{81} are specified for that day, and P_{HMG} and P81 are constants, K_{NHM}(t) and K81(t) are also known which determine K_{01}(t), K_{02}(t), K_{03}(t), K_{04}(t) and K_{81}(t) for that day. For instance, if K_{NHM}(t)/P_{HMG} is 1.2 on that day, it means that K_{NHM}(t) = K_{01}(t) + K_{02}(t) + K_{03}(t) + K_{04}(t) has to increase by 20% and K_{00}(t) decrease by 20% (K_{00}(t) + K_{01}(t) + K_{02}(t) + K_{03}(t) + K_{04}(t) = 1) since these are the only possible transitions when the system is at state 0 at time t (Figure 2 also Table II in Jones, 2009). The increase (decrease) in each individual term is done maintaining the same relative ratios of the probabilities in the homogeneous stochastic model (likewise for the conditional probability of successive events). Since all conditional probabilities are then specified for that day, the model proceeds with the same rules described in the homogenous stochastic model simulation (section 2.1). Lastly, the simulation continues with a daily time step until the end of the observed record of K_{NHM}(t)/P_{HMG} and K_{81}(t)/P_{81}. The end result is a time series of phases in which MJO occurrences behave as in the observations.

3. The activity of the MJO in the present climate and stochastic projections

Observed multiannual changes and long-term trends in the activity of the MJO were investigated by computing the number of events in 5-year running windows (Figure 4). Variations in the MJO (solid curve with circles) show low activity (∼18 events in 5 years) until about 1972, when a regime shift appears to have occurred leading to a systematic long-term increase. In addition, multi-annual periods of high and low MJO activity are noticed throughout the record.

The observed activity of the MJO was compared against simulations from the homogeneous and non-homogeneous stochastic models in the following way (Figure 4). An ensemble run of 1000 members was performed with the homogeneous stochastic model and the number of MJO events in 5-year moving windows was computed. This ensemble run provides frequency distributions of activity of the MJO in each year. Since the transition probabilities in the homogeneous stochastic model are constant in time, the MJO in this model behaves as a stationary process. For this reason, trends and low-frequency changes cancel out in the computation of the ensemble mean, which is indicated by the horizontal dashed line (18 events in 5 years). The other two horizontal dashed lines show the 5th and 95th percentiles of the frequency distribution and indicate the spread of possible values in the homogeneous model.

Similarly, an ensemble run of 1000 members was performed with the non-homogeneous stochastic model and the solid smooth curve shows the ensemble mean (Figure 4). The ensemble mean from non-homogeneous stochastic simulations follows the observed trends in the MJO. The shaded region indicates the spread in the frequency distribution of number of events in 5 years. As discussed in detail by Jones and Carvalho (2011), the ensemble simulations with the non-homogeneous stochastic model are significantly correlated (0.68) with the observed long-term changes in the MJO (correlations greater/less than ±0.66 are significant at 5% level).

Probabilistic projections of changes in the activity of the MJO were investigated in the context of non-homogeneous stochastic model simulations. The focus was on the A1B global warming scenario (IPCC, 2007), which assumes balanced usage of fossil intensive and non-fossil energy sources in a future world of rapid economic growth and a global population that peaks in mid-century and declines thereafter. Projections of SST from coupled climate models for the period 2010–2099 were used to drive the non-homogeneous stochastic model of MJO activity. The following models were used: ECHAM5/MPI-OM (two runs), BCCR-BCM2.0 (Bjerknes Centre for Climate Research, Norway), CCSM3 (National Center for Atmospheric Research, USA), GFDL-CM2.1 (US Dept. of Commerce/NOAA/Geophysical Fluid Dynamics Laboratory, USA) and MRI-CGCM2.3.2 (Meteorological Research Institute, Japan). Monthly SST anomalies averaged over the Indian Ocean and western Pacific warm pool were used in the regression equations of conditional probabilities of MJO initiation (section 2).

Figure 5 shows each projection of SST anomalies and the ensemble mean (bold solid line). The ensemble mean SST anomaly in the warm pool increases from about 0.6°C in 2010 to 2.8°C in 2099. Although all models show systematic warming trends during 2010–2099, there is also a large spread among the model runs, particularly in the magnitudes of decadal variations. This uncertainty in changes in SST anomalies in the warm pool was represented as the standard deviation from the ensemble mean and used in the stochastic simulations of the MJO.

To derive projections of probabilities of primary and successive MJO initiations, the ensemble mean SST anomaly was used in Eqs (1) and (2) and upper/lower bounds calculated to account for uncertainties in projections of SST anomalies in the warm pool and errors associated with the regression model. The mean K_{NHM}(t)/P_{HMG} (Figure 6) starts from about 2.1 in the present climate and systematically increases to 3.7 by 2099. K_{NHM}(t)/P_{HMG} also shows periods of high and low activity and reflects the spread of decadal variations in SST anomalies projected by the coupled model runs. Projections of K_{81}(t)/P_{81} show long-term increases as well, since they were determined from Eq. (2) (not shown).

An ensemble of 1000 members was constructed such that each member runs for 90 years at daily resolution. Each simulation resulted in a time series of phases and represents situations of active MJO (S = [1,8]) and quiescent periods (S = 0). Next, we computed the number of MJO events per calendar year (2010–2099) (each year has a frequency distribution of 1000 data points). We note that the non-homogenous stochastic model simulates primary and successive MJOs. For simplicity, the results are presented here in terms of total (primary plus successive) events. Moreover, a statistical test (Rodionov, 2004) was performed to detect change points in the ensemble mean number of events per year.

Figure 7 displays the results as decadal shifts in the projected changes in the mean number of MJO events. The simulations suggest progressive increases from the current mean of 4.6 events year^{−1} to about 5.7 events year^{−1} by 2094–2099. It is interesting to note that, although the 95th percentiles of the distributions remain virtually unchanged in the projections, the 5th percentiles increase from 3 events (2010–207) to 4 events by 2038–2049.

An interesting feature of the stochastic model approach is the possibility of computing probabilistic projections of changes in the activity of the MJO. Jones and Carvalho (2011) derived cumulative probability curves of number of MJO events per year during 1880–2008. For the period of available observations, the probability of 5 or more events in a year was 0.19 in 1948–1972, 0.33 in 1973–1989 and 0.51 in 1990–2008. The same approach was applied to the projections of changes in the MJO (Figure 8). Note that the probability curves in any two adjacent periods are statistically different from each other at the 5% confidence level, since they were obtained after the identification of changepoints in the time series of number of events per year. For comparison with present climate, the probability of 5 or more events in a year increases as: 0.75 (2010–2027), 0.78 (2028–2037), 0.82 (2038–2049), 0.88 (2050–2093) and 0.92 (2094–2099).

4. Conclusions

The MJO is the most important mode of tropical intraseasonal variability. Observational studies based on reanalysis data indicate that the MJO shows regime changes on low-frequency time-scales (i.e. variations longer than ∼2 years) (Jones and Carvalho, 2006, 2011). In addition, the MJO exhibits a long-term trend in activity (Jones, 2009) including a trend toward greater event frequency after the mid-1970s (Jones and Carvalho, 2006; Pohl and Matthews, 2007). These changes in the MJO coincide with the long-term warming in the Indian Ocean and western Pacific warm pool. A possible linkage between increased MJO activity and warming in the tropical oceans is in agreement with the study of Slingo et al. (1999), although dynamical mechanisms need yet to be demonstrated. Slingo et al. (1999) performed general circulation model experiments forced with observed SSTs and partially reproduced the positive trends in tropical intraseasonal amplitudes since the mid-1970s.

This is the first study to derive quantitative projections of changes in MJO activity in the A1B global warming scenario. A non-homogeneous stochastic model was developed based on the empirical evidence that the probability of MJO initiation is associated with multi-annual changes and positive trend in SST anomalies in the Indian Ocean and western Pacific. Projections of SST anomalies in the warm pool from IPCC coupled models were used to drive the probabilities of MJO initiation in the non-homogeneous stochastic model. The results suggest decadal shifts in the number of MJO occurrences per year and the probability of very active years (5 or more events) significantly increases from the values in the present climate.

One significant challenge in the development of a comprehensive theory of the MJO has been to determine what dynamical mechanisms control the initiation of the oscillation. Kemball-Cook and Weare (2001), for instance, analyzed radiosonde data in the Indian Ocean, Maritime Continent and western Pacific Ocean to investigate three possible mechanisms of MJO initiation: (i) extratropical triggering, (ii) tropical circumnavigating triggering and (iii) recharge–discharge. They used an instability index defined as moist static energy (h) at 1000 hPa minus that at 500 hPa. (h = c_{p}T + Lq + gz; c_{p} = specific heat at constant pressure, T = temperature, L = latent heat of condensation, g = gravity acceleration, z = geopotential height). They showed that the instability index was well correlated with convective available potential energy (CAPE) and found support for the recharge–discharge mechanism. In this case, the initiation of organized convection during the MJO is initiated by a combination of low-level moist static energy build-up and concurrent drying of the mid-troposphere associated with subsidence from a previous cycle of MJO convection.

Thus, a possible hypothesis for the observed positive trend in MJO activity is that warming of the tropical Indian Ocean and western Pacific increases the background CAPE necessary for triggering MJO events. The instability index described above was computed with NCEP/NCAR reanalysis (not shown) and it clearly shows significant (5% level) positive trends in intraseasonal variance averaged over the warm pool during 1948–2008. However, Kemball-Cook and Weare (2001) warned that near-surface NCEP/NCAR reanalysis may be more influenced by model physics than station data over that region and may not be able to accurately represent the timing of MJO events. Nevertheless, the trend in the instability index is consistent with trends in CAPE and Convective Inhibition (CIN) discussed by Riemann-Campe et al. (2009) who analyzed ERA-40 reanalyses of the European Centre for Medium-Range Weather Forecasts for the period 1958–2001.

However it is important to mention other possible impacts that global warming might have on the MJO. These might include changes in longitudinal gradients of SST in the tropical Indian and Pacific Oceans, modifications in the mean state (Inness et al., 2003) or changes in extratropical stochastic forcing on the MJO (Ray and Zhang, 2010).

Lastly, it is relevant to note that this study investigated one aspect of a very complicated problem (i.e. number of MJO events). Additional studies need to be developed to examine likely modifications in amplitude, structure and differences in mechanisms associated with primary and successive MJO events. New reanalysis products and improved global climate models might bring new insights on the variability of the MJO in the present climate and how it might change in the future.

Acknowledgements

NCEP/NCAR reanalysis and OLR data were provided by the NOAA/OAR/ESRL PSD, Boulder, Colorado, USA, from their web site at http://www.cdc.noaa.gov. SST projections were obtained from the World Climate Research Programme's (WCRP's) Coupled Model Intercomparison Project phase 3 (CMIP3) multi-model dataset. We acknowledge the modelling groups, the Program for Climate Model Diagnosis and Intercomparison (PCMDI) and the WCRP's Working Group on Coupled Modelling (WGCM) for their roles in making available the WCRP CMIP3 multi-model dataset. Support of that dataset is provided by the Office of Science, US Department of Energy. This research was funded by NOAA Office of Global Programs (NOAA/NA05OAR4311129, NA07OAR4310211).