How does increasing temperature affect the sub-annual distribution of monthly rainfall?

This paper investigates the relationship between temperature and sub-annual rainfall patterns using long-term monthly rainfall and temperature data from 1920 to 2018 in Australia. A parameter (τ) is used to measure the evenness of temporal rainfall distribution within each year, with τ = 0 indicating a uniform pattern. The study examines the relationship between τ and temperature for each year, considering whether it was warmer or cooler than average across five climate zones (CZs) in Australia, including tropical, arid, and three temperate climate classes. This study discovered a considerable association between annual maximum temperature and the distribution of monthly rainfall, with high temperatures resulting in greater variation (as represented by larger τ values) in the sub-annual distribution of monthly rainfall throughout all CZs, particularly in arid regions with τ values ranging from 0.27 to 0.52. In contrast, regions with temperate climates without dry seasons had a lower and narrower range of τ, from 0.15 to 0.26. This variability in rainfall distribution makes managing water resources more challenging in arid regions in Australia.


Introduction
Increases in anthropogenic greenhouse gas concentrations have resulted in higher temperatures, creating changes in climatic and hydrological cycles across the world [1]. Of key interest to water resources management are the changes that have been witnessed in extreme precipitation [2][3][4][5][6], which have resulted in conflicting changes in global flood regimes [7][8][9][10]. While the impact on temperature and precipitation extremes has been the focus on most assessment studies, there is broader impact to the hydro-climatological cycle. Even though yearly rainfall totals appear to exhibit less significant change, the duration of dry spells has increased globally [11], creating additional water stress with reduced storage available in catchments and reservoirs. Arguments have been made for rethinking principles of hydrologic design, from the currently followed basis of 'resistance' , to a more amenable basis of 'resilience' [12]. One aspect of change that has received lesser attention is that of the sub-yearly temporal distribution of rainfall, which forms the focus of the present study.
The 'temporal pattern' of precipitation can represent a design temporal pattern associated with a rainfall burst. Such a design temporal pattern has been noted to be intensifying with increasing temperature [13], in turn accentuating urban flooding, intensified further through the increase in the storm intensity [14,15]. However, while there is a clear change in the seasonality of streamflow [16], the evidence for this in sub-annual precipitation is less evident. This paper investigates whether the variability in sub-annual precipitation patterns exhibits any association with temperature. If the hydrological cycle is intensifying as a result of climate change [17], the question we ask here is whether this intensification occurs mostly at the scale of individual storms or manifests itself down to sub-annual time scales.
The paper is organised as follows. The next section (section 2) presents the methodological basis for our investigation, along with the data used. Following that, results are presented along with a discussion of the implications they have on the hypotheses being assessed. This is followed by the key conclusions from our study.

Methodology
Our intent here is to assess change in the sub-annual temporal pattern of precipitation, as a function of a suitable covariate associated with warming. As the sub-annual pattern differs as a function of latitude, we simplify our assessment by arranging the year into an Australian 'water-year' (1 July to 30 June of the following year), commencing in the month representing the highest monthly precipitation on average, sequencing on for the ensuing 11 months. We then proceed to characterise the pattern by re-ordering the precipitation series from the highest to the lowest in the 12 month sequence. This re-ordered series is then characterised by a simple parametric exponential decay curve, giving access to the decay rate parameter, that is then assessed as a function of a warming surrogate. More details on this as well as the investigations that are reported, as provided in the following sub-sections.

Assessing the sub-annual temporal pattern of precipitation
The monthly precipitation sequence in a water year is re-ordered from the peak monthly rainfall to the lowest monthly rainfall, and an exponential model fitted [2]: where, P (t) represents ordered precipitation from maximum to minimum for the year, τ is the least squares recession coefficient of the exponential decay curve for the year, and t represents the month order (t = 0, …,11), equalling zero (0) for the month having the highest yearly rainfall. The exponential decay described by a single parameter (i.e. τ ) is often used to model declines that are proportional to current values. This type of model can be found in a variety of scientific fields [2,18,19]. In the present study, we applied this model to describe the exponential decay of monthly rainfall values that were arranged in descending order for a given year. One can visualise the above equation as representing the decay in the monthly rainfall series for each year, with τ = 0 representing the case where the precipitation is seasonally invariant. Our intent in using such a simplistic representation of seasonality in the precipitation time series is to allow investigation of change in this behaviour, especially as a function of a covariate (temperature). It should be noted that our representation does not capture change in the timing of seasonality, but simply the rate of decay in the temporal pattern for each year from the peak to the lowest rainfall total in the year.

Assessing the temperature ∼τ sensitivity
Assessing the hypothesis that the sub-annual precipitation temporal pattern is impacted by warming requires a model that can measure variation in τ as a function of a suitable warming surrogate. Following from the many investigations relating extreme precipitation to temperature as characterised by the Clausius-Clapeyron relationship, a log-linear relationship is used to relate the parameter τ to the average temperature for each year (denoted T) and the temperature-τ sensitivity generally called scaling α [20]. The derivation of α is initiated from equation (2) as below where τ 0 is the initial value and τ is the changed value by T at the rate of α. Taking the common logarithm on both sides as Then implementing the quantile regression for log τ values (y-axis) at a specific percentile (p) and T (x-axis) [21], obtaining the slope β P 1 , and solving it for α as where the parameter α can be also expressed as percentage, representing the sensitivity of τ to the temperature T (i.e. % • C −1 ).
Here the quantile regression for acquiring the slope β P 1 for log τ values at the percentile (p) is implemented as below. Considering data pairs (x i , y i ) for i = 1, . . . , n, where x i represents the temperature and y i , the logarithmic τ value obtained from the equation (3). Then the data pairs can be expressed as where p is a real number between 0 to 1 which represents the percentile, and the ε p i an error term with zero expectation. The parameters β p 0 (intercept) and β p 1 (slope) are chosen to minimize D as

Data
Monthly data are used for this study, with gridded precipitation and temperature records at 0.05 • × 0.05 • obtained from the Australian water availability project (AWAP) [22], spanning from 1920 to 2018. The monthly maximum and minimum temperature data from AWAP have been used to obtain three annual mean temperature datasets to find out which temperature is more related to τ . Those are the annual mean maximum temperature (i.e. average of the 12 monthly maximum temperatures), the annual mean minimum temperature (i.e. average of the 12 monthly minimum temperatures), and the annual mean temperature (i.e. average of the 24 monthly maximum and minimum temperatures).
In addition, the Köppen-Geiger climate type from [23] was used in collating the results presented next. The original climate zones (CZs) were grouped into five CZs representing tropical climate (Rainforest, Monsoon, and Savannah), arid climate (Hot Desert, Cold Desert, Hot Steppe, and Cold Steppe), temperate climate with dry summer (Dry& hot Summer, Dry & Warm Summer), temperate climate with dry winter (Dry Winter & Hot Summer) and temperate climate without dry season (Hot Summer and Warm Summer). These are termed CZ1 to 5 in the results that follow.
For every water year (30 June-1 July), a parameter τ at each grid cell is paired with a concurrent mean annual temperature to investigate the temperature-τ relationship. Then the data pairs are separated into two groups by medians of the three annual temperature datasets (denoted as high-and low-temperature groups), and furtherly separated by the five primary CZs. Figures 1(a) and (b) show the spatial distribution of τ along with the variability it exhibits. Also shown are the five CZs in figure 1(c) later results are arranged in.

Spatial distribution of τ
It is interesting to note that τ approaches values closest to 0 (representing a uniform temporal pattern) as one proceeds to the southern latitudes. The northern part of the continent exhibits low mean and standard deviation of τ values, which suggests that the annual distribution of monthly rainfall in this area is relatively even and stable. This finding is consistent with previous research that has reported low coefficients of variation in this region [21,24]. It is also interesting to note that the dry interior of Australia exhibits comparatively high variability in τ , possibly a result of the erratic rainfall patterns the region experiences.

Linking annual temperature to τ
One-tailed paired t-tests are applied for each set of data groups to test if the mean of τ values with higher temperatures is statistically different from that of τ values with lower temperatures at a significance level (0.05), and the results are presented in figure 2. The null hypothesis is that the means of the two data groups are equal, which is presented as greyish areas on the maps in figure 2, and the alternative hypothesis is that the mean of τ values with higher temperatures is larger (i.e. less uniform temporal distribution of monthly rainfall in a water year) than that of τ values with lower temperatures, which is presented as reddish area in figure 2. Figure 2(a), the t-test results between the two groups of τ separated by the annual average maximum temperatures, shows a similar pattern as the CZ map in figure 1(c). The reddish area on the east coast overlapped is CZ5, which indicates the statistically significant differences within τ conditioned by the accompanying annual average maximum temperatures. Additionally, the west coast and south coast overlap with CZ3. To the north, the alternative hypothesis area does not strictly coincide with CZ1, but several reddish areas exist in this CZ and take a comparatively large percentage. For CZ2 and CZ4, no valid evidence could prove the relationship between the t-test result and CZs, which requires other methods to determine  the relationship between high-temperature τ and low-temperature τ . However, in figures 3(b) and (c), which are the t-test between the annual average temperature and τ , and the annual average minimum temperature and τ , the existed pattern relationship is not obvious, though, for the mean temperature map ( figure 2(b)), the pattern exists in CZ5.
Hence, to analyse the specific differences between low-temperature τ and high-temperature τ , the annual mean maximum temperature is used now hereafter. Additionally, the analytical results of the CZ1, CZ3 and CZ5 in figure 2(a) show the same pattern as figures 1(a) and (b), which indicates that the CZ can be linked to the change of τ , or it can be considered as a covariate in the temperature-τ relationship.

Differences in τ conditioned by high and low temperature
Given the long length of record associated with each pixel, an assessment of the difference in the values of τ depending on whether the year was warm or cold (with respect to the average) was conducted. To undertake this assessment, the data for each pixel was split into two groups as a function of the annual mean maximum temperature (hereafter simply refer to as temperature). After sorting the τ -temperature pairs in descending order by temperature, it is selected that the warmest temperature group is the top 33% and the coldest temperature group is the bottom 33% of the dataset. The two groups were then compared using a one-tailed t-test at a 5% significance level to test if the mean of τ with high temperatures is greater than that of τ with low temperatures.
Box plots of τ as a function of the upper and lower temperature tercile, arranged by CZ, are provided in figure 3. Due to the number of CZs, there are a total of ten groups and the data is plotted in pairs of CZs, and the high-temperature τ and low-temperature τ were quantitatively compared under the same condition.
In figure 3, it is observed that the τ values with high temperatures are generally greater than those with low temperatures across all CZs. The one-tailed t-tests, at the significance level (0.05), also show that the mean of τ with high temperatures is greater than that of τ with low temperatures for each CZ, even including CZ2 and CZ4 where the statistical differences in their means of τ were not clearly identified in figure 2(a).  These results show that high temperatures tend to accompany high τ . The higher τ could result in a more uneven distribution of the annual rainfall. As the maximum temperature is the indicator used for analysis, the sensitivity of τ and annual maximum temperature is an important analytical object for water management and flood control.
The result in figure 2(a) highlights the importance of understanding the sensitivity of large-area τ values in the interior and coastal regions of the southwest of the Australian continent to the maximum annual mean temperature. These results are particularly relevant to the Murray-Darling Basin, which occupies a significant portion of this area and is a vital source of water for human consumption, irrigation, and other uses in Australia [25]. Further discussion follows in the next section.

The sensitivity of τ to temperature
The longer boxes in figure 3 would indicate higher sensitivity of τ to the temperature. For this, quantile regression is applied, by which a sensitivity parameter α in equation (4) can be acquired to represent the sensitivity of τ to the temperature. Here, the α value means what percentage τ increases (+) or decreases (−) when the maximum annual mean temperature increases by 1 • C. That is, the larger magnitude of the α value, the more sensitive the τ value is to the temperature increases. The results of α by three different percentiles three percentiles of τ (10, 50, and 90%tile) and five CZs (CZ1 to CZ5) are presented in table 1.
The calculated α values ranged from 1.9% to 8.1%, with the highest mean (7.1%) in CZ2 (arid) and the lowest mean (2.2%) in CZ1 (tropical). The magnitude of the mean values of α follows the order of temperate climates, CZ3 (6.5%), CZ4 (5.3%), and CZ5 (4.6%). What is interesting here is that CZ2, which covers most of Australia, including the Murray-Darling basin, has the highest α (8.1%) at the 90th percentile of τ . This means that as the temperature increases, the larger τ increases at a higher rate (α), indicating that annual rainfall tends to be more unevenly distributed, due to which difficulties in water resource management may arise. It should be noted that this tendency is also shown in CZ3 and CZ5, where much of the Australian population is located.

Conclusion
This study investigated the temporal pattern of annual rainfall, represented by a parameter τ , with relation to the annual temperatures across Australia.
The key findings are as follows.
(a) The annual distributions of the rainfall based on monthly data have more association with the annual mean maximum temperature. (b) High temperatures tend to accompany high τ values regardless of CZs, representing more uneven distributions of the annual rainfall. (c) The temperature-τ sensitivity, denoted by α, is considerably higher in the arid zone, followed by the temperate zone. This means that over most of Australia, annual rainfall tends to be more unevenly distributed as temperatures rise, indicating that the polarization of the temporal distribution of water resources can make management more difficult.
The result of this study can be helpful in water management, flood control, erosive hazard and more essentially, agricultural activities. The results of this study suggest that changes in the distribution of annual rainfall may occur due to climate change and can be one of the grounds for preparing measures for water management due to climate change.
This study analysed the temporal distribution of annual rainfall against temperature and potential shifting trends of the distribution caused by temperature change. However, the rainfall can be affected by various factors such as latitude, cloud moisture contends, relative humidity, and temperature [11,[26][27][28]. Additionally, spatial factors may also influence the rainfall distribution, as extreme rainfall events spread across an area rather than a single sector [2]. The analysis combined with these factors not considered in this study can increase the accuracy of the prediction of rainfall distribution changes due to climate change characterized by temperature rise.
In the introduction, we posed the question of whether intensification of rainfall primarily occurs at the scale of individual storms or extends to sub-annual time scales under the influence of climate change. Our findings indicate that rising temperatures can lead to intensification in the sub-annual distribution of rainfall. While the results of our study cannot be generalized to all of the world, it is likely that similar patterns would be observed in other regions, as our research was based on an analysis of long-term data from various CZs within Australia. Therefore, further study in other regions or on a global scale would be beneficial in gaining a comprehensive understanding of the relationship between temperature and annual rainfall patterns under climate change.

Data availability statement
The data analysed during the current study are publicly available from www.bom.gov.au/climate/data-services/maps.shtml.
The data that support the findings of this study are openly available at the following URL/DOI: www.bom.gov.au/climate/data-services/maps.shtml.