Spatio-temporal relationships between rainfall and convective clouds during Indian Monsoon through a discrete lens
aa r X i v : . [ phy s i c s . a o - ph ] A ug RESEARCH ARTICLE
Spatio-temporal relationships between rainfalland convective clouds during Indian Monsoonthrough a discrete lens
Arjun Sharma | Adway Mitra | Vishal Vasan | Rama Govindarajan Sibley School of Mechanical andAerospace Engineering, CornellUniversity, Ithaca, NY, 14853, USA Centre of Excellence in ArtificialIntelligence, Indian Institute ofTechnology Kharagpur, India International Centre for TheoreticalSciences (ICTS-TIFR) Shivakote,Hesaraghatta Hobli, Bengaluru560089, India
Correspondence
Arjun Sharma, Sibley School ofMechanical and AerospaceEngineering, Cornell University,Ithaca, NY, 14853, USAEmail: [email protected]
Funding information
Airbus Group Corporate FoundationChair in Mathematics of ComplexSystems
The Indian monsoon, a multi-variable process causing heavy rains duringJune-September every year, is very heterogeneous in space and time. Westudy the relationship between rainfall and Outgoing Longwave Radiation(OLR) -– a proxy for convective cloud cover -– for monsoon between 2004-2010. To identify, classify and visualize spatial patterns of rainfall and OLRwe use a discrete and spatio-temporally coherent representation of the data,created using a statistical model based on Markov Random Field. Our ap-proach clusters the days with similar spatial distributions of rainfall and OLRinto a small number of spatial patterns. We find that eight daily spatial pat-terns each in rainfall and OLR, and seven joint patterns of rainfall and OLR,describe over 90% of all days. Through these patterns, we find that OLRgenerally has a strong negative correlation with precipitation, but with signif-icant spatial variations. In particular, peninsular India (except for the westcoast) is under significant convective cloud cover over a majority of days butremains rainless. We also find that much of the monsoon rainfall co-occurswith low OLR, but some amount of rainfall in Eastern and North-western In-dia in June occurs on OLR days, presumably from shallow clouds. To studyday-to-day variations of both quantities, we identify spatial patterns in thetemporal gradients computed from the observations. We find that changes inconvective cloud activity across India most commonly occur due to the estab-lishment of a north-south OLR gradient which persists for 1-2 days and shiftsthe convective cloud cover from light to deep or vice versa. Such changesare also accompanied by changes in the spatial distribution of precipitation.The present work thus provides a highly reduced description of the complexspatial patterns and their day-to-day variations, and could form a useful toolfor future simplified descriptions of this process.
Keywords — Observational data analysis, Statistical methods, Mesoscale,Seasonal, Clouds, Radiation, Rainfall, Atmosphere | INTRODUCTION
The South Asian monsoon season is active from June to September, bringing heavy rains to India, Pakistan, Sri Lanka,Bangladesh, Burma, Nepal and other neighbouring countries. According to present wisdom, during summer, precipitation overa zonal spread including the Indian landmass is caused by the northward shift of a giant mass of clouds called the Inter-TropicalConvergence Zone (ITCZ) Schneider et al. (2014), Gadgil (2003), Sikka & Gadgil (1980). Outside of the Indian Summer mon-soon season and the Indian Ocean region, it is wrapped around the Equator. In this paper, we focus on the dynamics of themonsoon over the landmass of India and the surrounding seas (the Bay of Bengal and the Arabian Sea). On one hand, devastat-ing floods result due to excess rainfall as it forces the sudden release of water from a large number of dams and reservoirs, andon the other hand only 10% less rain than average leads to severe droughts Gadgil & Gadgil (2006). The Indian Summer mon-soon rainfall (ISMR) has a profound impact on agriculture, and hence the economy of India Gadgil & Gadgil (2006). Hence,understanding the characteristics of the process is of utmost importance. Several aspects of this process, such as strong couplingbetween ocean and atmosphere processes Wang et al. (2005), Pottapinjara et al. (2016) and absence of any long term trends inISMR over several decades Goswami & Xavier (2005), make it a challenging problem.It is known that a typical season of the Indian monsoon has intra-seasonal oscillations, which are manifested as “activespells" of high rainfall activity across the landmass, and “break spells" where most of the landmass remains dry Rajeevan et al.(2006). Such intra-seasonal oscillations causes day-to-day changes in the spatial distribution of cloud cover and rainfall. Spa-tial patterns of daily rainfall across the landmass have been studied using Empirical Orthogonal Functions and their vari-ants Krishnamurthy & Shukla (2007), Mishra et al. (2012), Suhas et al. (2013). Krishnamurthy & Shukla (2007) used multi-channel singular spectrum analysis of the daily rainfall anomalies and identified a seasonal component and two major sub-seasonal components corresponding to active and break spells. Rajeevan et al. (2010) empirically defined active and breakspells and spatial distribution of rainfall around them. Suhas et al. (2013) identified eight phases of the monsoon, each charac-terized by a spatial pattern of rainfall, appearing in near-cyclic order in each monsoon season.The northward propagation of ITCZ from the equator to the tropical region as the cause of monsoon is well established.The seminal paper by Sikka & Gadgil (1980) theorized that there are waves of northward propagation of cloud bands across theIndian landmass. They demonstrated it using Hovmöller diagrams along specific longitudes. Chattopadhyay et al. (2009) alsoidentified northward propagation of the stratiform component of rainfall (not the convective component), through Hovmöllerdiagrams around active and break spells. Several studies have considered the role of clouds, especially synoptic-scale bandsduring monsoon. Though it is difficult to measure or quantify clouds, Outgoing Longwave Radiation (OLR) is a useful proxyfor convection, which plays a major role in the Indian monsoon. It is readily available from many satellites. The propagationpatterns of OLR anomalies were studied by Krishnamurthy & Shukla (2008), who once again tied their analysis around activeand break spells to visually demonstrate northward propagation. They also performed spectral analysis of the OLR time-seriesto identify two pairs of north-south oscillatory patterns of OLR anomalies. They too used Hovmöller diagrams to identify asmall eastward component and a considerable northward component of the propagation of OLR anomalies over and around theIndian landmass. Rajeevan et al. (2013) studied vertical cloud structures during the active and break spells.We avoid the usual emphasis on “active” and “break” spells, and instead look to identify typical daily spatial patterns ofrainfall and OLR over the Indian region during monsoon. Recently, Mitra et al. (2018) identified 10 spatial patterns of rain-fall distribution over the Indian landmass during the monsoon season (June-September) on a low-resolution ( ◦ ) precipitationdataset Rajeevan et al. (2006). Each pattern is representative of a cluster of days of “similar” rainfall distribution, such thateach day’s spatial distribution of rainfall may be considered to be a noisy version of the pattern of its cluster. Their approach isbased on Markov Random Field (MRF) that creates a discrete representation of any spatio–temporal dataset while maintainingspatio-temporal coherence among the discrete state values. Each discrete state represents a probability distribution over thespace of observations. The model then partitions the dataset into spatial and temporal clusters that maintain spatio–temporal coherence. Mitra et al. (2019) identified temporal relationships between the patterns corresponding to each cluster. The numberof clusters is estimated implicitly by the model and studying the corresponding patterns allows new physical insights to beinferred. In this paper, we use the same model as a tool to obtain clusters of patterns on high-resolution satellite-based precip-itation (TRMM) and Outgoing Longwave Radiation data over and around the Indian landmass to obtain the spatial patterns ofconvective cloud and rainfall over the region. We also extend the Mitra et al. (2018) model to detect joint patterns of rainfalland OLR, to explain the relationship between convective cloud cover and rainfall. To the best of our knowledge, this is thefirst attempt at identifying joint spatial patterns of OLR and rainfall. Our approach identifies eight prominent spatial patterns ofprecipitation, eight prominent spatial patterns of OLR and seven prominent joint spatial patterns of both OLR and precipitation.We then study their inter-relationship. While low OLR is frequently associated, as would be expected, with high rainfall, andvice versa in most places, we find that this does not hold in the south-eastern peninsula, where low OLR (heavy cloud cover)does not cause significant rainfall.Our approach to identify spatial OLR patterns indicates that patterns associated with increased convective cloud activityover the entire spatial region studied are positively correlated with increased precipitation over the landmass. To investigate thetemporal variation of convective clouds and rainfall we computed one-day anomalies for OLR and precipitation and once againemployed our framework to determine spatial patterns of the one-day anomaly. Considering transition frequencies of OLRspatial patterns conditioned on different one-day anomaly patterns, we identified three families of convective cloud cover: high,low and transition. The one-day OLR anomaly analysis also indicates the most common means of shifting convective cloudactivity across India is by establishing a north-south one-day OLR anomaly gradient. The direction of the gradient dictatesthe direction of the shift in convective activity, either high to low or low to high. Due to the association of higher convectivecloud activity with higher precipitation, the direction of the one-day OLR anomaly gradient indicates a period of higher/lowerprecipitation. | DATA AND METHODOLOGY
In this section, we briefly discuss the model, based on MRF, proposed by Mitra et al. (2018), which is used to generate discretespatial patterns of rainfall and OLR. We present only the necessary features for this study and for further details we refer thereader to Mitra et al. (2018).The model replaces the observation X ( s , t ) at any spatial location s and day t by an integer Z ( s , t ) . For example, in thejoint patterns of rainfall and OLR, X is a × vector containing daily rainfall and daily OLR data, and Z is for low OLRand rainy, for low OLR and dry, for high OLR and dry and for high OLR and rainy. Note that we use the terms “wet" and“rainy" interchangeably. The model designs Z for spatial and temporal coherence, i.e. neighboring locations are likely to havethe same value of Z . Thus for each day during the study period, we have a spatial pattern of Z over the geographical domain.The model collects days with “similar" Z patterns and puts them into one cluster. A cluster is represented by a discrete spatialpattern (denoted by a vector θ d ) which is the mode of the spatial distribution of Z across its constituent days, and a continuousspatial pattern (denoted by a vector θ ) which is the mean of the spatial distribution of X across its constituent days. Each clusteris identified by an integer U , and hence each day belongs to a given value of U as elaborated below. | Methodology
Let there be S spatial locations (e.g. a rectangular grid system) and T time-points (here these are days). X ( s , t ) denotes theobservation of any continuous field (e.g. rainfall or OLR) at location s ∈ [ , S ] and time t ∈ [ , T ] .For every observation X ( s , t ) (which is a vector for the joint patterns but a scalar for the individual patterns), as mentioned above, the model provides a discrete scalar variable Z ( s , t ) . Each value of Z is the index to a probability distribution overthe range of X . At most locations, the distributions of rainfall and OLR over time are bimodal, with distinctive peaks. Therainfall distribution at each location is modelled with two Gamma distributions and the OLR with two Gaussian distributionscorresponding to lower and higher ranges of the respective variable. In this study, the parameters defining these distributionsare made location-specific, but the model allows them to be time-specific as well. In the individual representation for OLR orrainfall at each ( s , t ) value the variable Z ( s , t ) can take one of the two possible values (high or low), but in the joint OLR-rainfallrepresentation, one of the four possible values for Z ( s , t ) , as explained above, are possible.The model has a discrete variable U ( t ) for each day, indicating the cluster membership of the day. Each U corresponds toa spatial pattern of Z , represented by an S × discrete vector, θ d . These vectors are called spatial patterns , and the Z -vector ofeach day t is equal (except for a small number of locations) to any one of the spatial patterns.We now estimate the parameters of the distributions of the model, the spatial patterns ( θ d vectors) and also find appropriatevalues of all the Z and U variables. In doing so, the model ensures that Z -variables are spatio-temporally coherent, i.e., Z ( s , t ) is likely to be equal to Z ( s ′ , t ′ ) if s ′ ∈ Ω ( s ) or t ′ ∈ { t − , t , t + 1 } , where Ω ( s ) is the set of neighboring locations of s .Also, the number of spatial patterns is not fixed beforehand but the model finds an appropriate number from the data. Theyare chosen in such a way that prominent spatial patterns , which appear repeatedly and in multiple years are identified. Theseare achieved by considering Z and U as random variables with prior distributions on them, and also their joint distributionsspecified by pairwise edge potential functions as per the convention of Markov Random Fields. The optimal values of all therandom variables and parameters are obtained by a procedure known as Gibbs Sampling, which is based on Markov ChainMonte Carlo Methods. | Datasets
We study the monsoon months (June- September) from 2004 to 2010. The geographical region analysed is between ◦ N- ◦ N and ◦ E- ◦ E with a resolution of . ◦ in either direction. However, to collect daily patterns into clusters, only aregion roughly coinciding with the Indian landmass is considered. For precipitation, we use data from the Tropical RainfallMeasurement Mission (TRMM). Our data-set consists of 5772 spatial locations and 854 days. These 5772 spatial locations arerestricted to the roughly the Indian and Bangladesh landmass, i.e., the coloured regions in figure 1 or any of the other discretepatterns. The rainfall activity over the Bay of Bengal is much more intense than on the landmass. Thus we have ignored thesurrounding seas to allow the clustering process to bring out salient features within the landmass as these features are otherwisesubdued in the clustering process. Although the surrounding seas have been ignored during the clustering process, we includethese locations in the continuous patterns in figures 2, 4, 7, 8 and 13. From these figures the spatial features, i.e., regions of high/low rainfall or OLR appearing over landmass can be observed to smoothly extend over the surrounding oceans. On repeatingthe experiments with daily precipitation data compiled by India Meteorological Department (IMD, Pai et al. (2014)) we findsimilar results (not shown) as those based on TRMM.One of the best-known proxy variables for convective clouds is OLR, and such clouds play an important role in the Indianmonsoon. Several works such as Sikka & Gadgil (1980), Krishnamurthy & Shukla (2008), Utsav et al. (2017), Chakravarty et al.(2018) have used this proxy in the context of the Indian monsoon. Moreover, it is readily available from many earth monitoringsatellites. We use the OLR data measured on-board the Kalpana satellite Mahakur et al. (2013) with a resolution of . ◦ ineither direction. The OLR data-set consists of the same 5772 spatial locations as the precipitation data but only 821 days (dataon some of the days during 2004-2010 monsoon months is not available). We consider these 821 days for both the rainfall andOLR clustering. | SPATIAL PATTERNS OF OLR AND RAINFALL
By applying the Markov Random Field model described in section 2 on rainfall and OLR datasets, we obtain three types ofdiscrete spatial patterns: precipitation (PPT) patterns, OLR patterns and joint PPT-OLR patterns. The first two are obtained byusing rainfall and OLR respectively as the observed variables for the model, and the discrete latent variable Z takes on one oftwo values, for “wet” and “dry” in rainfall or “deep clouds” and “shallow/no clouds” in OLR. But in the third case, the observedvariable is a 2-dimensional vector with both rainfall and OLR. Similarly, the discrete latent variable has 4 states rather than 2. | Rainfall and OLR Patterns (obtained separately)
We obtain eight “prominent" spatial rainfall patterns by applying the model to rainfall data. By “prominent", we imply patternswhich appear at least over 20 days across the seasons on which the model is trained. These patterns, denoted by R1 to R8, areshown in figure 1, where the locations marked in blue are “wet", i.e., satisfying θ d ( s ) = 1 , while those marked in yellow are“dry", satisfying θ d ( s ) = 2 . These patterns are similar to the ones obtained by Mitra et al. (2018). The small differences aredue to the difference in the data source. As mentioned in Section 2, we use satellite-based TRMM data for precipitation, whileMitra et al. (2018) had used ground-sensed data published by Indian Meteorological Department. TRMM data is not only ofhigher spatial resolution but also includes Bangladesh here unlike the other dataset which is restricted to the political bordersof India.FIGURE 1 Discrete rainfall patterns arranged in increasing order of mean daily rainfall. Orange square markers indicate theCentre of Mass (Def. 2 in the supplementary material) specific to the pattern and the green diamonds are the Centre of Mass(Def. 2 in the supplementary material) across all days. The green diamond is hence a fixed reference point against which wecompare the migration of the centre of mass for each pattern of rain.Among the eight patterns in figure 1, the first two only have a few wet locations in the north-east, while in the remaining sixthere are a considerable number of wet locations across the landmass. All of these six patterns have several wet locations along FIGURE 2 Continuous Rainfall patterns corresponding to the discrete rainfall patterns of figure 1. Black circles and redsquares are the Centre of Mass (Def. 1 in the supplementary material) of each pattern’s rainfall over the Indian landmass andthe full area respectively. The purple diamond and green crosses are the Centre of Mass (Def. 1 in the supplementary material)of rainfall over the Indian landmass and the full area respectively, across all the days.Rainfall pattern number R1 R2 R3 R4 R5 R6 R7 R8Number of days 160 279 90 42 96 35 37 42Mean landmass rainfall (mm/day) 4.39 6.42 7.47 8.31 8.67 8.83 9.07 10.04TABLE 1 Number of attributed days over seven monsoon seasons and the daily mean rainfall corresponding to the discreterainfall patterns of figure 1. These patterns account in total for 781 of the 854 monsoon days in the seven seasons, which isover % . The rest of the days are classified into non-prominent patterns.the coast in the south-west. Patterns R3 and R4 have most of their rainy locations in the north-eastern part of the landmass,including the Gangetic plain. Pattern R5, where parts of the west coast, east-central India (around Odisha) and north-east arewet at the same time, was not observed by Mitra et al. (2018). We attribute this to the fact that we employ a different data setfrom that used earlier. Pattern R6 to R8 involve rain mostly on the west coast and central India with small parts of north-east andnorthern India being wet. Table 1 shows the number of days in the period 2004-2010 (June-September) which were assignedto these eight patterns, as well as the mean aggregate rainfall (daily, across the landmass) associated with these patterns. Morethan 91 % of all the days in these seven seasons are classified into one of the patterns in figure 1. The remaining 9 % of daysdisplay rainfall patterns that are not prominent, i.e. they appear in only a minority of the seasons. It is clear that the maximumnumber of days are in the first two “dry" patterns, while the “Gangetic Plain patterns" R3 and R4 are slightly more frequent thanthe “Central India patterns" R6, R7 and R8. These discrete rainfall patterns correlate well with the corresponding continuousrainfall patterns shown in figure 2. These continuous patterns show the mean rainfall quantity at each location, over land orsea, of the days corresponding to a given cluster (or spatial pattern). It can be seen that though our clusters were made by only considering landmass rain, each cluster (or pattern) corresponds to a characteristic pattern over the Bay of Bengal. Broadly,higher landmass rain corresponds to higher activity over the Bay. Remarkably the patterns extend seamlessly into the Bay andthe Arabian Sea, indicating that the patterns we have detected are parts of larger weather systems characteristic to each of ourpatterns of rainfall over land. The composite figures of the discrete rainfall patterns, their corresponding continuous rainfallpatterns, and the mean spatial distributions of OLR are shown in figure 1 in the supplementary material where it may be visuallyobserved that regions of low OLR often correspond to regions of high rainfall in each pattern.Next, we investigate the spatial patterns obtained on OLR data. Once again we obtain eight prominent patterns, O1 to O8,shown in discrete and continuous forms in figures 3 and 4 respectively. The discrete and continuous patterns correlate wellwith each other. Since OLR is a proxy for deep convection, the regions with low OLR (shown in blue) are taken to be underdense convective cloud cover, while the regions with high OLR (shown in yellow) have a clear sky or shallow clouds. Thepatterns are arranged in increasing order of mean rainfall on the days assigned to each OLR cluster. Table 2 shows the numberof days during 2004-2010 assigned to each OLR pattern, and the mean aggregate rainfall associated with them. Again the OLRpatterns of a vast majority of days in the seven monsoon seasons is represented by these eight prominent patterns, leaving outa negligible number of days which display rare patterns. The first two patterns are mostly devoid of convective clouds andcorrespond to relatively low rainfall. The other six OLR patterns depict strong convective cloud cover over a significant part ofthe landmass. Note that low OLR in the northernmost part of the landmass (especially the province of Jammu and Kashmir)may not necessarily correspond to convective cloud cover as this region contains snow-capped mountains. Pattern O4 representsdays when primarily the southern part of the landmass is covered by convective clouds. Patterns O3 to O8 show significantpresence of convective cloud cover in the south-east, where rainfall during the monsoon season is low. We will return to thispoint. Patterns O6, O7 and O8 appear similar in their discrete representations, but are progressively more intense in convectivecloud cover, as observed in figure 4. The composite figures of the discrete OLR patterns, their corresponding continuous OLRpatterns, and the mean spatial distributions of rainfall are shown in figure 2 in the supplementary material where once again, itmay be visually observed that regions of low OLR often correspond to regions of high rainfall for each pattern.OLR pattern number O1 O2 O3 O4 O5 O6 O7 O8Number of days 159 107 67 86 102 159 103 34Mean landmass rainfall (mm/day) 4.40 5.70 6.62 7.17 7.28 8.44 9.10 10.44TABLE 2 Number of attributed days and daily mean rainfall for the discrete OLR patterns of figure 3. These OLR patternstogether account for 817 days (95% of the monsoon days).Next, we examine the relationship between the aforementioned rainfall and OLR patterns. Since both sets of patterns havebeen identified from the same period, each of the 821 days that is assigned to an O-pattern is also represented by one R-pattern.33 days from R-patterns are left out as OLR data is not available on these days. This allows us to study the correspondencebetween these two sets of patterns. Histograms of the distribution of OLR patterns across the days attributed to a given rainfallpattern are available in figure 5. The high-OLR pattern 1 in figure 3 corresponds entirely to the driest rainfall pattern R1 infigure 1 and vice versa. The rainfall pattern R2, which consists of a small number of wet locations in the north-east, coincidesmostly with the OLR patterns O2, O3 or O4, i.e., the ones with patches of low OLR in the north-east. For all the higher rainfallpatterns (R3 to R8 in figure 1) the corresponding OLR patterns are seen to be O5 to O8 of figure 3. From the continuousversions of these rain-providing (to the landmass) OLR patterns in figure 4 we can observe an intense convective cloud band(a trough in OLR) in the south-eastern portion of the Bay of Bengal that partially extends to the landmass. OLR patterns O3and O4 also share this feature. The convective cloud band in O3 is thinner and more diagonal covering only central India in thelandmass part. Pattern O4 has a convective cloud band sitting on the southern part of landmass (also extending to the bay). It FIGURE 3 Discrete cloud (OLR) patterns arranged in increasing order of mean daily rainfall. Orange square markers are theCentre of Mass (Def. 2 in the supplementary material) of high cloud cover locations specific to the patterns and the greendiamonds are the Centre of Mass (Def. 2 in the supplementary material) of high cloud cover locations across all days.FIGURE 4 Continuous OLR patterns corresponding to the discrete OLR patterns of figure 3. Red squares are Centre of Mass(Def. 1 in the supplementary material) of the full area-weighted with 1/OLR of the pattern. The green diamonds are the Centreof Mass (Def. 1 in the supplementary material) of the full area across all the days sampled. mainly coincides with the dry rainfall pattern R2 (it does provide rain in Kerala on a very small number of days through rainfallpattern R6). Large areas of high OLR in the north-west make this a non-raining cloud pattern. Hence, in most parts of India,there is a clear anti-correlation between OLR and rainfall, but this is not true in southern India where low OLR is often notaccompanied by rainfall. This will be a particular feature of the joint patterns discussed in Section 3.2.FIGURE 5 The number of days in which each OLR pattern (figure 3) occurs for a given rainfall pattern (figure 1). Note thevariable vertical axis.Rainfall patterns R3 and R4 are qualitatively similar but the corresponding OLR patterns are different. OLR patternsO5 and O6 are responsible for the scanty rainfall pattern R3 and the OLR pattern O7 is responsible for the more intense butrarer rainfall pattern R4 (table 1). Rainfall pattern R5 occurs mostly on the days of OLR pattern O6. OLR pattern O6 is alsoresponsible for rainfall pattern R6 and is a major part for rainfall pattern R7. The most intense rainfall pattern R8 is broughtabout in almost equal proportion by the OLR patterns O7 and O8. OLR pattern O7 is thus partly responsible for the intenseversions of particular rainfall patterns, i.e., rainfall over the north-east via pattern R4 and rainfall over the west coast and centralIndia via pattern R8. Unlike other rain-providing OLR patterns, O7 has two intense peaks in convective cloud cover (or troughsin OLR): one in the south-east Bay and other in central India. OLR pattern O8, which is responsible for rainfall pattern R8 onabout 50% of the days, has three intense peaks of convective cloud cover: in the south-east Bay, over central India and in theNorth East, but also forms a continuous band from the south-east Bay up to the landmass. In general, the low OLR patternscover a broader part of the landmass than high rainfall patterns, which implies that on an average, it rains over a subset of cloudyregions, as is reasonable to expect. | Joint Patterns
Next, we consider the patterns obtained by considering OLR and rainfall together as observations, as discussed in Section 2.Seven prominent patterns, denoted by J1 to J7, where J stands for “joint", are obtained. The latent variable Z can now take fourvalues (high OLR-high rain, low OLR-high rain, low-OLR-low rain, high OLR-low rain), instead of 2 as in the previous cases. FIGURE 6 Discrete joint patterns of OLR and rainfall arranged in increasing order of mean daily rainfall. Yellow circle andorange square markers are the centre of Mass (Def. 2 in the supplementary material) of high rainfall and high cloud cover (lowOLR) locations respectively specific to the patterns. The green diamond and purple cross are the Centres of Mass (Def. 2 inthe supplementary material) of high rainfall and high cloud cover (low OLR) locations respectively over all days.Figure 6 shows these patterns, where the four values of Z are suitably color-coded. Table 3 shows the number of days in whicheach joint pattern is displayed.Joint OLR-rainfall pattern number J1 J2 J3 J4 J5 J6 J7Number of days 156 263 124 123 77 44 30Mean landmass rainfall (mm/day) 4.39 6.38 7.52 8.46 8.71 9.71 10.41TABLE 3 Number of attributed days and the corresponding daily mean rainfall for the discrete joint patterns of figure 6. of the monsoon days are classified into these patterns.For all the patterns, the number of high OLR-high rainfall locations is insignificant. In patterns J1 and J2, most locationshave high OLR and low rainfall. For the patterns with a significant number of rainy locations, i.e., joint patterns J3 to J7 in figure6, we observe that the west coast is always rainy and cloudy (the wet rainfall patterns of Mitra et al. (2018) also had severalwet locations on the west coast). Also, southern India is always cloudy in patterns J3 to J7, but includes no rainfall, except forthe west coast. Patterns J3 and J5 correspond to convective rainfall in the north-eastern side while the western side remaininguncovered by convective clouds. Similarly, patterns J4 and J6 correspond to a high OLR in north-east, cloudy north-west, andrainy west coast and central India. Pattern J7 corresponds to convective cloud cover all over India with rain over the west coastand the central and north eastern regions.Figures 7 and 8 shown the continuous rainfall and OLR patterns corresponding to the discrete joint patterns of figure 6,where a good correlation between the continuous and discrete information can be observed. The continuous OLR patternsreveal further relation between the specific OLR and rainfall distributions. Joint patterns J3 and J5 which correspond to more FIGURE 7 Continuous Rainfall patterns corresponding to the discrete joint patterns of figure 6. Black circle and red squaresare Centres of Mass (Def. 1 in the supplementary material) of the Indian landmass and the full area corresponding to thepatterns. The purple diamonds and green crosses are the Centres of Mass (Def. 1 in the supplementary material) of the Indianlandmass and the full area across all the days sampled.rain in the north-east and less in central India, consist of convective cloud bands that are aligned roughly diagonally (fromnorth-west to south-east). The joint patterns J4, J6 and J7 which bring significant rainfall over central India, corresponding to amore horizontal convective cloud band over the same region. The last of these also consists of more intense cloud band over theBay of Bengal that stretches towards the south-east. There is a separate disjoint convective peak over Kerala along the south-western coast in patterns J3 and J5 in figure 8. Perhaps this is the topographical effect of the Western Ghats mountain rangethat stretches along the western coast, causing an increased convective cloud intensity and rainfall in this region even for thediagonally aligned cloud-bands. The composite figures of the discrete joint rainfall and OLR patterns and their correspondingcontinuous rainfall and OLR patterns are shown in figure 3 in the supplementary material.In the literature, it is common to study daily anomalies of rainfall or OLR instead of their absolute values. For consistencywith the literature, we also carried out the same analysis using the daily anomaly values of both quantities and obtained verysimilar patterns as shown above (not shown). | Spatio-temporal OLR-rainfall relationship
At each spatial location, we now identify the probability of occurrence of each of the four states, namely LOW (low OLR withhigh chance of rainfall), LOD (low OLR but no rainfall), HOD (high OLR and dry), and HOW (high OLR and rainy). We countthe number of days with each of the four states as computed by the joint analysis. This number divided by the total number ofdays for all the four states is used to define a 2 × Z -variables whose binary values are assigned by the inference algorithm,or from the joint patterns computed above. The difference between these two approaches is that the second represents themodes, i.e. most frequent behaviors, while the first approach can account for infrequent behaviors also. FIGURE 8 Continuous OLR patterns corresponding to the discrete joint patterns of figure 6. Red squares are Centres ofMass (Def. 2 in the supplementary material) of the full area-weighted with 1/OLR corresponding to the patterns. The greendiamonds are Centres of Mass (Def. 2 in the supplementary material) of the full area across all the days sampled.First, we study the approach of estimating the OLR-rainfall relation directly from Z and X . In the period under analysis(7 monsoon seasons spanning 854 days over 5772 locations), there are 23.6% LOW, 26.7% LOD, 47.1% HOD and only 2.6%HOW events. An event refers to a spatio-temporal point (s,t) which is assigned a states Z(s,t) by the proposed model (withrespect to both OLR and rainfall) before we calculate the mode across the days constituting a cluster (this mode defines thediscrete pattern of figure 6). The spatial and temporal distribution of these events are shown in Figure 9.According to the mode-based analysis, each element of the matrix at each location is shown on a map in figure 10a.This provides further credence to observations already made while revealing some new features. Broadly, the landmass maybe divided into four regions in terms of this behaviour: Kerala and the Western Ghats; the rest of South India apart fromthis region; the north east, the Ganga basin, central India; and the northwest. All the regions have a significant but varyingprobability of being in a high OLR and dry state (“break spells"). In the Kerala and Western Ghats, it is cloudy and rainy withhigh probability, with a small number of high OLR and dry days in parts. Over the North-East, the Ganga basin and centralIndia, it is almost always either LOW or HOD, with the probability of each varying with location. It is to be remembered fromour earlier discussion, however, that the individual days on which a given state occurs are quite different in central India andthe North East. The predominant state in the northwest is HOD. Surprisingly, the most likely state in the rest of South India isLOD, with a smaller probability of HOD. Thus, convective clouds most often cover the rest of South India but do not cause rain.The 2.6% HOW events get suppressed upon taking the mode, and hence do not show up in Figure10. Although rainfall fromnon-convective clouds does occur in Indian monsoon, especially in the Western Ghats Utsav et al. (2017), this is not a modalfeature as it does not appear anywhere in any pattern.The above analysis makes it clear that rainfall is almost always associated with reduced values of OLR. We know that OLRis a proxy for deep convection which is likely to cause rainfall, but it is possible for local rainfall can occur from shallow non-convective clouds too, which do not cause low OLR. There is no cut-off on OLR values to differentiate between convective and FIGURE 9 (a) Percentage of days at each location corresponding to the four OLR-rainfall states: LOW (low OLR and wet),LOD (low OLR and dry), HOW (high OLR and wet) and HOD (high OLR and dry), (b) Total number of monthly instances ofeach OLR-rainfall state.non-convective rainfall. How may OLR be connected to significant rainfall? The answer may once again vary spatially sincerainfall happens due to different mechanisms in different parts of the landmass. At each location, we note raw OLR values forthe binary “wet" days there (as identified by the model). We calculate the mean and standard deviation of these values, and plotthem in figure 10b. It can be observed that generally, rainfall occurs at lower OLR values (180-185 W / m ) along the easterncoast, adjacent to the Bay of Bengal, and parts of the Gujarat coast along the Arabian Sea, while for Central India this valueis in the range 190-195 W / m . For the long south-western coast and the hilly north-eastern region, this range is even higher,around 200 W / m , while in North-western India it is even higher, 210 W / m or more. However, the variability of these valuesis quite high along the eastern coast and Gujarat coast, very high in the north-west, but low in the North-eastern region, CentralIndia and the South-western coast. The low mean values of OLR along the eastern coast is consistent with the fact that muchof the rainfall there occurs due to deep convective events that happen over the Bay of Bengal, resulting in frequent lows anddepressions. The same is not true about much of the western coast, except its northern parts in Gujarat (Kathiawar peninsula). | Comparison with known spatial patterns
Several papers have identified various spatial patterns related to rainfall and other climatic variables related to the Indianmonsoon. These include the study of active and break spells by Rajeevan et al. (2010), and that of Monsoon Intra-seasonalOscillation (MISO) by Suhas et al. (2013). In the case of Rajeevan et al. (2010), a list of active and break spells over Indiais provided, as is the mean spatial distribution of rainfall across the landmass on days leading up to or following such spells.We now ask whether our assignment of each day to a pattern is consistent with the classification of active and break days. Weobtain frequency distributions over the patterns for active and break days as identified by Rajeevan et al. (2010), and show their FIGURE 10 How does OLR relate to rainfall?: a) Conditional matrices from the joint clustering. Yellow corresponds to highlikelihood while deep blue indicates a low probability, b) Mean and standard deviation of OLR values on rainy days.histograms in figures 11 for PPT and OLR. We find that most of the “active" days are assigned to PPT patterns 7 and 8 whichcorrespond to heavy rainfall over central India and the west coast, which is expected during active spells. Similarly, the daysidentified as “break" are predominantly assigned to PPT patterns 1 or 2 which are mostly dry. Similar features are visible in theOLR patterns too. Thus our results not only agree with previously known ones but present spatial patterns in a more detailedand quantitative fashion. For example, we have shown that every day’s OLR and rainfall are one among a small number ofspatial patterns.FIGURE 11 Frequency distribution of PPT (a, b) and OLR (c, d) patterns identified by our method during the active (a, c)and break spells (b, d) identified by Rajeevan et al. (2010). They supply the list up to 2007. Note that only a minority of daysare classified as either active or break. The work of Suhas et al. (2013) uses Extended Empirical Orthogonal Functions (EEOF) to identify eight phases of mon-soon, and each day is assigned to one of those phases, except for those days when the signal is weak. We collect the days in eachof their MISO (Monsoon Intra-Seasonal Oscillations) phases, compute the mean spatial rainfall maps of each phase, and showthem in figure 12. While they differ in detail from our rainfall patterns, we see broad similarity in that the eight MISO phasestoo broadly fall into the three “families" (similar to those found by Mitra et al. (2019)) - i) where the rainfall is restricted to thewestern coast and the north-eastern regions, ii) rainfall covers central India, iii) rainfall covers eastern India and the Gangeticplains. By counting the pattern assignments from our method in a particular phase of MISO, we find that each pattern roughlyoverlaps with 3 MISO phases, generally from the same “family". For example, the 1st pattern overlaps with MISO phases1,2,3 and in all cases, the rainfall is concentrated in the North-east and foothills of the Himalayas and the western coast. MRFpattern 5 overlaps with MISO phases 6 and 7, where the rainfall occurs mostly in Central India. To put this conclusion fromvisual comparison on firmer footing, we calculated the correlation coefficient of the spatial rainfall distribution on each day ina particular MISO phase with each of eight PPT patterns ( θ ) identified by us, and this corroborated the visual comparison.FIGURE 12 Mean rainfall maps of the eight MISO phases identified by Suhas et al. (2013). Each plot is obtained byaveraging the rainfall across all days in a given MISO phase.There are other studies which discuss spatial patterns of rainfall over India, such as Chattopadhyay et al. (2008) who carriedout a clustering based on Self-Organizing Maps to identify the modes of intra-seasonal variations of monsoon, and identifiedspatial patterns of rainfall anomalies. Some of these spatial patterns have similarity with some of the patterns identified by ourmodel, especially the modes (1,1), (2,1), (3,1) and (3,2) where the rainfall is primarily over Central India or Gangetic plains.However, since their reported results are for an earlier period (1980-1996), we could not make a direct comparison. | TEMPORAL VARIATION IN OLR SPATIAL PATTERNS
Next, we proceed to investigate the temporal dynamics of convective cloud bands and its consequences on rainfall over thelandmass. We first compute daily fluctuations or one-day anomalies of the OLR and precipitation data as follows: dX ( s , t ) = X ( s , t ) − X ( s , t − ) , where X denotes any measurement - either precipitation or OLR. A positive value indicates an increaseof the variable relative to the previous day, while a negative value indicates a decrease. We then ran our MRF model on thisone-day anomaly data, where the binary latent variables dZ ( s , t ) indicate either positive or negative gradients. The data-edgepotential functions are now defined using the sigmoid function as follows: ψ ( dZ ( s , t ) , dX ( s , t )) = 11 + exp ((− ) dZ ( s , t ) dX ( s , t )) , dZ ∈ [ , ] . (1)We find eight prominent patterns in one-day OLR anomaly, each of which appears on at least 35 days during the periodunder consideration. The continuous spatial patterns of OLR, the mean one-day anomalies for a particular spatial pattern, θ areshown in figure 13. Red indicates that OLR is higher, i.e. less convective cloud cover on the following day than on the currentday, and blue indicates vice versa. Their discrete counterparts θ d produced by this approach are shown in the supplementarymaterial in figure 4. Frequencies of the patterns and their relationships with daily aggregate rainfall are shown in table 4. Pattern1 appears on relatively dry days. Patterns 2 and 3 appear mostly in June and September when the convective cloud cover isrelatively low and sporadic. Patterns 4 and 5 are complementary, indicating increasing OLR over South India and decreasingover North India, and vice versa. Both patterns 4 and 5 appear in all Monsoon months evenly. Similarly, the patterns 6 and 7 arecomplementary, one showing increase of OLR all over India except the north-eastern corner, and the other showing the reverse.Pattern 8 shows a decrease of OLR over Western India, with an increase over the rest of the landmass.FIGURE 13 Continuous spatial patterns of one-day OLR anomaly, during the period 2004-10. Red indicates an increase inOLR (less convective cloud on the following day), blue indicates a decrease in OLR (more convective cloud on the followingday). One-day OLR anomaly pattern number OG1 OG2 OG3 OG4 OG5 OG6 OG7 OG8Number of days per season 5.6 15.1 12.7 18.7 18.1 16.1 15.1 8.4Mean landmass rainfall (mm/day) 4.4 4.7 5.2 6.8 7.0 8.0 9.0 9.14TABLE 4 Number of attributed days and daily mean rainfall for the discrete rainfall temporal gradient patterns of figure 13.These one-day OLR anomaly patterns together account for 770 of the 854 monsoon days.O1 O2 O3 O4 O5 O6 O7 O8O1 2 (45) 1 (8) 3 (1) 3 (8) 3 (3) 3 (1) 3 (2) X (0)O2 2 (17) 4 (14) 2 (1) 5 (9) 5 (5) 4 (1) 3 (4) X (0)O3 5 (3) 2 (2) 4 (10) 5 (1) 5 (7) 4 (13) 5 (1) 4 (1)O4 2 (3) 4 (6) 4 (5) 5 (15) 5 (5) 4 (3) 4 (1) X (0)O5 5 (1) 6 (7) 4 (3) 6 (4) 5 (15) 4 (6) 7 (6) 4 (1)O6 6 (2) 6 (4) 6 (11) 6 (4) 6 (11) 6 (35) 7 (16) 6 (3)O7 X (0) 8 (1) 8 (4) 8 (1) 7 (5) 8 (19) 8 (18) 8 (7)O8 X (0) X (0) X (0) X (0) X (0) 8 (2) 8 (2) 8 (1)TABLE 5 The most common one-day anomaly pattern of OLR (OG), through which the transition occurs from one OLRspatial pattern (O) to another in successive days. Each row indicates the current day’s spatial pattern, each column indicatesthe following day’s spatial pattern, and the entry contains the most frequent one-day anomaly pattern. The frequency of such atransition is shown in brackets. The rows and columns labeled as O1- O8 refer to the OLR patterns of figure 3 and 4, but thedata entries refer to the OLR gradient patterns of figure 13.Next we study how the patterns in figure 4 vary in time. We construct two matrices shown in table 5. The (i,j) element ofthe matrix in table 5 indicates the most frequently occurring one-day OLR anomaly pattern (depicted in figure 13) associatedwith the transition from OLR spatial pattern O i on a given day to OLR spatial pattern O j on the following day. For example,row 1 of this matrix indicates the transition from OLR pattern O to all other OLR patterns on the following day, and thediagonal indicates the maintaining of a given pattern on two consecutive days. X indicate transitions that were never observed.We observe that the transition from pattern O to O most frequently occurs through one-day OLR anomaly pattern OG . Wecan summarize this symbolically as ( O , OG ) → O . The number of times the transition event ( O i , OG k ) → O j occurred, isindicated within brackets along with the corresponding matrix entry, i.e. the one-day OLR anomaly pattern. Higher numbersindicate more frequent events.We now summarise our interpretation of the results in table 5. The one-day OLR anomaly pattern OG most often occurs when transitioning from O into another pattern. This isreasonable, since O represents the least convective cloud activity and any other state of OLR over India would involve anincrease in convective cloud activity at all spatial locations, as represented by OG . The one-day OLR anomaly patterns OG and OG (representing a negative OLR gradient over most of the Indian land-mass) mostly occur with OLR patterns O , O , O (last three rows of table 5). This suggests that the highest cloud coverpatterns ( O , O , O ) behave differently from the other patterns. Indeed, for the other OLR patterns, the most commonone-day anomaly pattern is either OG or OG , the patterns that depict a north-south one-day anomaly gradient. Patterns O , O and O form a family by virtue of frequent transitions (supplementary material, section 2) between them-selves. Likewise patterns O , O and O also form a family. Note that ( O , O , O ) generally coincide with more convec-tive cloud activity whereas ( O , O , O ) with lower convective cloud activity. From Table 2, we infer that ( O , O , O ) is a ‘high rainfall’ family whereas ( O , O , O ) is a ‘low rainfall’ family. Looselywe can think of the days labeled with the former as representing an active spell, while those labeled with the latter as eithera break spell, or the onset or withdrawal in June and September respectively. Pattern O is particularly interesting since it behaves like a transition between the families of higher and lower convectivecloud activity. It has almost equal likelihood between the two families. Days labeled pattern O also occurred betweenactive and break spells as determined by the rainfall record, using thresholds such as those used by Rajeevan et al. (2010). Pattern O is also a transition pattern, most frequently occurring after a spell of low OLR (high convective cloud activity).However, despite a lower level of rainfall (see table 2), it is not necessarily indicative of a transition from the high convectivefamily ( O , O , O ) to the low convective family ( O , O , O ) as O has a small bias to transition into O . By and large, OG , OG , OG represent transition into the high convective family ( O , O , O ) and OG , OG , OG represent transition into the low convective family ( O , O , O ) . This split is best observed in pattern O which representsthe bridge between higher and lower convective cloud activity. Evidently the one-day OLR anomaly patterns may present transitions between themselves. The most common transitionfor all patterns was the self-transition and the mean duration for each of them is between 1-2 days (supplementary material,section 2).The upshot of the above findings is that the most common means of changing the convective cloud activity across India isby establishing roughly a north-south OLR gradient. Furthermore, recall that the OLR spatial patterns depicted in figure 4 areordered according to increasing daily rainfall over the Indian landmass (see table 2). Thus the above analysis may be interpretedto draw the following claims: establishing an increased convective activity in north India/monsoon zone leads, more often thannot, to a period of time associated with higher rainfall overall. On the other hand, suppressing convective cloud activity inthe monsoon zone/north India leads, more often than not, to a period of time associated with reduced overall rainfall. Such arelationship between OLR spatial pattern and rainfall, at this stage, is purely a statistical inference of our method. However, aspointed out in Chakravarty et al. (2018), significant deep convection is present during active phases and not present during breakphases. That work was limited to the Western Ghats region but does suggest that convection has some direct relationship withprecipitation. Our work is an attempt to extrapolate this inference over a wider region using the OLR satellite data. Indeed OLRpatterns showing more convective cloud activity are associated with greater rainfall over the Indian landmass. In the presentwork we adopt this relationship to give a succinct interpretation of our findings listed above, through a probabilistic model forthe one-day anomaly. The precise nature of this relationship is the object of future studies.A similar study of dynamics can also be done with precipitation. However, the one-day anomaly patterns are less clear inthis case. The results are shown in the supplementary material, section 2. | CONCLUSION
We have identified dominant patterns in daily OLR and rainfall in and around the Indian subcontinent by using the MarkovRandom Field model developed by Mitra et al. (2018). We find a consistent picture emerging from three independently obtainedsets of patterns: rainfall alone, OLR alone and rainfall and OLR together. Several of the patterns from one set are directly andstrongly correlated to a particular pattern from another set, such as the least rainfall, highest OLR and joint lows in rainfalland convective cloud cover, which occur almost always on the same days. The monsoon days had been classified (see e.g. Mitra et al. (2018)) into three families: one where the entire landmass is relatively dry, another where it rains on Kerala, theGanga basin and the north-east, and the third when it rains in Kerala and central India. From the continuous patterns emergingfrom the joint patterns, we find that OLR is high over most of the landmass during the first kind of day, convective cloud bandsrelated to the second kind are oriented diagonally (from north-west to south-east direction) whereas convective cloud bandsrelated to the third kind are aligned in the zonal direction.We also study the conditional relationship between OLR and rainfall, and its spatial variation across the landmass. We findall locations on the landmass to have a significant probability of having high OLR and low rainfall even during monsoon (pre-onset, post-withdrawal and break spells), and much of southern India, except for the west coast, remains cloudy (low OLR) butdry through a significant part of the monsoon season. This is much less common in the west coast, central India and North-east,where low OLR and high rainfall often co-occur. It turns out that OLR is lowest for the rainy days in eastern India and Bay ofBengal coast and northern parts of the western coasts, suggesting that most of the rainfall occurs there from deep convectiveprocesses. But most of the rainfall in north-western India occurs under relatively high OLR.The OLR spatial patterns that we identified are categorised into three types or families associated with low convective cloudactivity, high convective cloud activity and transition pattern between high and low. The MRF-based model also determinedthat the OLR spatial patterns are correlated with mean rainfall over the Indian landmass: patterns associated with higher cloudconvective activity were correlated with higher precipitation. Moreover, by considering patterns in the one-day anomaly wedetermined how the spatial patterns in convective cloud activity changed in time. Specifically, we identified a north-south OLRanomaly gradient with higher convective cloud activity in the monsoon zone, corresponded to the start of a period of higherconvective cloud activity overall. Contrariwise, an anomaly with higher convective cloud activity in southern India led to aperiod of lower convective cloud activity overall.The main contribution of this paper is to utilize an MRF-based statistical model to identify a small number of staticdaily patterns and their temporal variation in OLR and rainfall during monsoon over India. This approach will allow us to i)visualize spatio-temporal characteristics of monsoon variables in a more comprehensive way through discrete representations,ii) potentially identify previously unknown patterns, iii) explore a framework by which the relations between multiple spatio-temporal variables can be studied, iv) develop a low-dimensional model for monsoon climate which can be used to parameterizeclimate models. This framework may be expanded to study other aspects of Indian monsoon such as the influences of winds,land and sea surface temperature etc, moisture transport, and also other large spatio-temporal climatic processes such as IndianOcean Dipole, North Atlantic Dipole, El Nino-Southern Oscillation etc. | ACKNOWLEDGMENTS
AS and AM were at ICTS-TIFR, Bangalore, India during the early stages of this work. AM acknowledges the support of theAirbus Group Corporate Foundation Chair in Mathematics of Complex Systems established in ICTS-TIFR and TIFR-CAM. Weacknowledge support of the Department of Atomic Energy, Government of India, under project no. 12-R&D-TFR-5.10-1100.We also thank Drs. E. Suhas, N. Joseph and Prof. B. N. Goswami for providing valuable suggestions and data. R EFERENCES
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Spatio-temporal relationships between rainfalland convective clouds during Indian Monsoonthrough a discrete lens: SupplementaryInformation
Arjun Sharma | Adway Mitra | Vishal Vasan | Rama Govindarajan | COMPOSITE ANALYSIS OF THE SPATIAL PATTERNS
In section 3 of the main paper, we identify several spatial patterns of rainfall and OLR. For a more comprehensive understandingbetween OLR and rainfall, we visualize these patterns through composite figures. We compile the mean rainfall and OLR overthe days associated with each such pattern. The composites for the rainfall (R), OLR(O) and joint (J) patterns shown in figures1, 3 and 6 of the main paper are shown here in figures 1, 2 and 3. | TEMPORAL VARIATION IN PRECIPITATION SPATIAL PATTERNS
In section 4 of the main paper we discussed the day-to-day variation of OLR using the continuous patterns of temporal gradientsor one-day anomalies. The discrete patterns corresponding to those continuous patterns in figures 13 of the main paper areshown in figure 4.The transition matrices for one-day OLR anomaly patterns along with the mean duration (residence time) and frequencyof each patterns are shown in figure 5. These figures are the basis of the interpretation in Section 4 of the main paper.We have also repeated the analysis of the one-day rainfall anomaly and show the continuous one-day rainfall anomalypatterns in Figure 7 and the discrete counterparts in figure 6 with relevant statistics in table 1.PPT gradient pattern number RG1 RG2 RG3 RG4 RG5 RG6 RG7 RG8Number of days 145 39 198 107 158 105 49 46Mean landmass rainfall (mm/day) 4.40 4.8 6.5 7.1 7.9 8.8 9.6 9.9TABLE 1 Number of attributed days and daily mean rainfall for the discrete PPT temporal gradient patterns of figure 6.These rainfall gradient patterns together account for 847 of the 854 days. | PROPAGATION BASED ON COM ANALYSIS
To deduce the movement of the cloud bands, and their relationship to the evolution of the rainfall distribution, we calculate theCentre of Mass (c.o.m.) of the rainfall and the inverse OLR (1/OLR) for each day (where rainfall and 1/OLR values at a givenlocation form the weights). We consider two definitions of the location ( Y com ( t ) = [ l at com ( t ) , l on com ( t )] ) of centre of mass (CoM) on each day: mean of the geospatial coordinates (latitude and longitude) within the area, weighted by volume of rainfall or 1/OLR, l at com ( t ) = Í s X ( s , t ) l at ( s ) Í s X ( s , t ) , l on com ( t ) = Í s X ( s , t ) l on ( s ) Í s X ( s , t ) , (1) mean of the geospatial coordinates (latitude and longitude) within the area, weighted by the binary state of rainfall or1/OLR as calculated by the model, l at com ( t ) = Í s Z ( s , t ) l at ( s ) Í s Z ( s , t ) , l on com ( t ) = Í s Z ( s , t ) l on ( s ) Í s Z ( s , t ) . (2)The velocity of CoM, with components in the meridional and zonal direction, on each day is defined as V com ( t ) = Y com ( t ) − Y com ( t − ) . The statistics related to the angle of velocity vector are shown in figures 8, 9 and 10. Movement of both convectiveclouds and rainfall is primarily restricted to a dominant direction, and the two differ by about 30-40 degrees. FIGURE 1 Composites associated with the rainfall patterns. Left, middle and right column represent the discrete rainfallpattern, continuous rainfall and continuous OLR pattern.
FIGURE 2 Composites associated with the OLR patterns. Left, middle and right column represent the discrete OLR pattern,continuous rainfall and continuous OLR pattern.
FIGURE 3 Composites associated with the joint patterns. Left, middle and right column represent the discrete joint pattern,continuous rainfall and continuous OLR pattern.
FIGURE 4 Discrete spatial patterns corresponding to the temporal gradient data of OLR, during the period 2004-10. Yellowindicates an increase in OLR (less cloud on the following day), blue indicates a decrease in OLR (more cloud on the followingday).
Mean DurationFrequency
FIGURE 5 Statistics related to the one -day OLR anomaly patterns: a) Transition matrix and b) Mean duration andResidence time.
FIGURE 6 Discrete spatial patterns corresponding to the temporal gradient data of precipitation, during the period 2004-10.Yellow indicates a decrease in rainfall, blue indicates an increase in rainfall.FIGURE 7 Continuous spatial patterns corresponding to the temporal gradient data of precipitation, during the period2004-10. Red indicates a decrease in rainfall, blue indicates an increase in rainfall.