Physical processes driving intensification of future precipitation in the mid- to high latitudes

Precipitation is changing as the climate warms, and downpours can become more intense due to the increased water holding capacity of the atmosphere. However, the exact nature of the precipitation response and its characteristics is still not well understood due to the complex nature of the physical processes underlying the formation of clouds and precipitation. In this study, present and future Norwegian climate is simulated at convection-permitting scales with a regional climate model. The future climate is a high emission scenario at the middle of the century. Hourly precipitation is separated into three categories (convective, stratiform, and orographically enhanced stratiform) using a physically-based algorithm. We investigate changes in the frequency, intensity and duration of precipitation events for each category, delivering a more nuanced insight into the precipitation response to a changing climate. Results show very strong seasonality, with significant intensification of autumn precipitation. An increase in convective precipitation frequency and intensity dominates the climate change signal regardless of season. While changes in winter and summer are well explained by thermodynamical theory, the precipitation response in autumn and spring deviates from the idealised thermodynamic response, partly owing to changes in cloud microphysics. These results show that changes in the precipitation distribution are affected in complex ways by the local climatology, terrain, seasonality and cloud processes. They illustrate the need for further and more detailed investigations about physical processes underlying projected precipitation changes and their seasonal and regional dependence.


Introduction
Thanks to rain gauge observations, particularly hourly observations, it is now clear that there has been a significant increase in the intensity of extreme precipitation since the preindustrial period (Min et al 2011). There is 'medium confidence' that this is a consequence of increasing temperature due to anthropogenic emissions of greenhouse gases (IPCC 2013). This confirms early model predictions (Manabe and Wetherald 1975). However, observed increases are larger than those predicted by climate models in Europe (Fischer and Knutti 2016) and the Northern Hemisphere (Min et al 2011).
This fact is of particular importance since precipitation has major knock-on effects on a range of economic sectors, and can be responsible for damage affecting human life and health, ecosystems, infrastructure and cultural heritage.
Precipitation can arise from a variety of mechanisms and can be classified into types, such as convective, stratiform, and orographic (Houze 1981, Houze 2012. These processes might respond differently to the increase of global temperatures. A host of different physical mechanisms have been invoked to explain observed changes in precipitation characteristics. Among these are: increases in atmospheric moisture according to the Clausius-Clapeyron relationship and subsequent increased latent heating in precipitating systems (Westra et al 2014), transitions from stratiform to convective rainfall (Berg et al 2013, Molnar et al 2015, diminution of precipitation area , Benestad 2018, changes to the eddy-driven jet leading to a reduction of the Rossby deformation radius affecting both position and intensity of mid-latitude disturbances (Dwyer and O'Gorman 2017), vertical shift of the melting level (Prein and Heymsfield 2020). These involve different temporal and spatial scales as well as physical processes, which can, and often are, studied separately.
Nevertheless, these processes are actually coupled. Insofar as a complete theoretical and physical approach of these changes is not possible, models are required to describe the full range of mechanisms. Recent advances in computational power have enabled regional climate models (RCMs) to simulate the climate at convection-permitting resolutions (Prein et al 2015). Convection-permitting RCMs remove many uncertainties related to precipitation as convective processes are no longer parameterised. Instead, they are represented by the primitive dynamic and thermodynamic equations (Brisson et al 2016). Although it is now known that models permit convection at grid spaces as large as 25 km (Vergara-Temprado et al 2020), an improved representation of fundamental precipitation characteristics by kilometer-scale models is acknowledged and has been broadly evaluated and confirmed (Ban et al 2014, Kendon et al 2017, Berthou et al 2018b, Chan et al 2018, Keller et al 2018, Finney et al 2019. More recently, the added value of these models has been demonstrated over Scandinavia by Lind et al (2020). However, the use of these models for studying climate change impacts on precipitation is relatively new and little is known about the physical processes driving these changes.
Early studies  showed that these convection-permitting models are able to reproduce the expected strong increase in precipitation intensity with global warming, and this response is strongly dependent on location and season. However, most existing convection-permitting modeling studies are limited to a description of the output. We believe that these models can also help us understand the physical processes responsible for the observed and simulated changes, since these processes are explicitly present in the model output.
This work aims to elucidate the changing characteristics of precipitation in order to better understand the dominant processes that drive the response in middle and high latitudes, with Norway as an example. The letter is structured as follows: section 2 describes the model data, and section 3 presents the methodological techniques. Section 4 briefly describes the physical theory underpinning the precipitation response to warming. Results are presented in section 5 and discussed in section 6, where we also examine physical processes that elucidate the results.

Model setup
We use the Advanced Weather Research and Forecasting (WRF) model version 3.9.1.1 (Skamarock et al 2005) with 3 km grid spacing. The model uses 1-way nesting, i.e. the boundary conditions are first given to an intermediate simulation at 15 km resolution (in which convection is parameterised), running over Europe, and then this simulation gives boundary conditions to the final simulation at 3 km resolution over Scandinavia. In the 3 km domain, the convective parameterisation is switched off and all precipitation is produced by the microphysical scheme, described in Thompson et al (2008). Other parameterisations include the RRTMG radiation scheme (Iacono et al 2008), the Yonsei University planetary boundary layer scheme (Hong et al 2006), the revised Monin-Obukhov surface layer scheme (Jiménez et al 2012), and the Noah multi-physics land surface model (Noah-MP; (Niu et al 2011)). Please see Mooney et al (2021) for further details about the model configuration and evaluation.

Surrogate warming experiment
Two simulations are conducted. In a control simulation, the European Centre for Medium Range Weather Forecasting Interim Reanalysis (ERA-Interim, (Dee et al 2011)) provides boundary conditions including sea surface temperatures (SST) during the period 1996-2005. The second simulation is a surrogate warming experiment (Schär et al 1996) under the RCP8.5 scenario at the middle of the century which corresponds to approximately 1.5 • C of warming globally. Boundary conditions are still given by the ERA-Interim dataset of 1996-2005, but they are perturbed by adding a monthly change that corresponds to the difference in the monthly mean of temperature, moisture, wind, geopotential height and SST in the CMIP5 (Coupled Model Intercomparison Project) simulations listed in appendix D between the periods 1976-2005 and 2035-2065. The experimental setup implies that only weak large-scale dynamical changes are present in the data, thus the precipitation changes will be due mainly to thermodynamical and microphysical processes. We also note that while the moisture is perturbed at the lateral boundaries, within the domain all aspects of the water cycle are allowed to occur freely. Domain-mean temperature and precipitation increases between the control and surrogate warming simulations are 2.1 • C (2.3 • C over land only) and 82 mm yr −1 (120 mm yr −1 over land only).

Analysis of the rain rate distribution
We analyse the full distribution of precipitation rates using the Analyzing Scales of Precipitation (ASoP, (Klingaman et al 2017)) technique. For a detailed description of the technique, the reader is referred to Berthou et al (2018b) (see their figure 2 for a summary). This method has become widely used in the convection-permitting modeling community (Berthou et al 2018b, Lind et al 2020, Vergara-Temprado et al 2020, because it enables a comprehensive view of precipitation changes across the whole distribution of precipitation rates, from drizzle to heavy downpours. A histogram of precipitation rates is calculated, providing a frequency of occurrence f i for each bin (1 ⩽ i ⩽ n). This frequency can be multiplied by the value of the bin centre b i to obtain the actual contribution of this bin to total precipitation f i b i . This actual contribution is then averaged over all the grid points of the sub-region of interest. The mean precipitation rate is then P = n i=0 f i b i and the fractional contribution from each bin to total precipitation is f i b i /P. The actual contribution changes show the changes in the amount of precipitation produced by each precipitation rate, whereas the fractional changes show the changes in the shape of the distribution only.

Uncertainty quantification
For uncertainty quantification, a bootstrapping technique by yearly blocks is used (Hesterberg 2015): 10 000 random data samples are produced by choosing with replacement 10 random years within the 10 years of both the control and the future simulations. Each sample provides a curve/histogram and future-present differences and, depending on the confidence level, appropriate percentiles of these histograms are taken as the limits of the uncertainty interval.

Classification of precipitation
Precipitation is separated into different categories in order to provide a better physical understanding of how precipitation changes in a warming climate. Three precipitation types are considered: convective, stratiform and orographically enhanced stratiform precipitation (hereafter orographic). Because there are no convective parameterisations in convectionpermitting models, precipitation is not classified in the model output as it is in state-of-the-art regional climate models. A separation algorithm is needed. We use a physically-based algorithm that is suited for mountainous areas. It is based on three dimensional wind speed. Convection is detected via identification of its overturning circulation and its horizontal potential vorticity dipoles. The description of the algorithm, as well as precise definitions of the three categories of precipitation, can be found in Poujol et al (2020). This reference also contains an evaluation and validation of the algorithm with various datasets and time scales ranging from the hourly to the climatological.

Region of interest
The domain covers most of Scandinavia, however, this analysis only focuses on Norway (figure 1(a)). The country presents strong climatic contrasts. Southern Norway is separated into east and west by the Scandes, a wide mountain ridge with peaks above 2000 m. The western region (Vestlandet) is strongly influenced by the North Atlantic storm track, with precipitation from land-falling storms and frontal systems often enhanced by uplift over the Scandes, leading to annual precipitation amounts of 1500-6000 mm, peaking in winter. In contrast the Eastern part of the mountains (Østlandet) is a more continental climate, with stratiform precipitation coming from the South and the South East peaking in fall, and predominantly convective precipitation in summer. Annual precipitation in the eastern part is 500-1000 mm. Moving north, the mountains are lower and the climate transitions from a mid-latitude climate towards a sub-Arctic climate in Hålogaland with precipitation amounts of 1000 to 2500 mm. In this study, Norway is separated into five sub-regions, that are defined by merging precipitation regions defined in Hanssen-Bauer et al (2009). As shown in figures 1(b) and (c), the simulated annual precipitation compares favourably with the seNorge2018 gridded observational product. Also plotted on figure 1(c) is the stations that underlie the seNorge2018 product, which is produced on a 1 km grid and covers the period from 1957 to present (Lussana et al 2019). WRF underestimates the contrasts of precipitation in Norway, with an underestimation of precipitation on the West coast and a slight overestimation in the East. However, gauge basedgridded precipitation datasets are suspect in mountainous regions (Lundquist et al 2019), partly due to poor station density (figure 1(c)). For a more thorough evaluation of the model please refer to Mooney et al (2021). The remainder of this paper only considers the precipitation increases shown in figure 1(d) for Western Norway (Vestlandet), Eastern Norway (Østlandet) and Northern Norway (Hålogaland). Results for Trøndelag and Sørlandet are available in the supplement (available online at stacks.iop.org/ERL/16/034051/mmedia).

Theoretical background
Basic physical laws and a simple representation of a cloud show that the precipitation rate is roughly the product of three terms (see appendix B for calculations): • The moisture content at the surface (q sat ). This increases with temperature at a rate of 7% per degree of warming in agreement with the Clausius-Clapeyron equation (Clausius 1850);  • The vertical velocity in the cloud (w) that depends mainly on water condensation in the cloud; • The precipitation efficiency (ε) which represents the efficiency of cloud microphysical processes to convert condensed water into rain drops or snow flakes large enough to precipitate. It varies from 0 (totally inefficient) to 1 (100% efficient).
Whereas increasing water vapor always corresponds to an intensification of precipitation under atmospheric warming, the other two factors can increase or decrease.

Results
Figure 2 depicts the changes in the fractional contributions of each precipitation rate to the total precipitation (defined in section 2.3). For all precipitation types, the relative contribution of heavy precipitation increases at the expense of the contribution of lower precipitation rates, leading to an overall intensification. The strongest intensification is found for convective precipitation, with the strongest(weakest) shift found in Eastern(Northern) Norway.
The distributions of the intensity and the duration of precipitation events, as well as their modeled changes, are presented in figure 3. A precipitation event is defined, on a given grid point, as a consecutive sequence of precipitation rates exceeding 0.1 mm hr −1 . Its peak intensity is the maximal hourly rain rate that is attained during the event. Note that because of this Eulerian approach, the number of grid points and the duration of events associated with a moving precipitating system will depend on its speed. Changes in the distribution are strongly dependent on precipitation type. Convective precipitation becomes more frequent across the entire distribution (figures 3(a)-(c)), and changes are larger in the tail, for the most intense and longest events. This might indicate an increased frequency of organized convection and orographically enhanced convection lasting for several hours. Conversely, the distribution of orographically enhanced stratiform precipitation exhibits a pronounced shift towards higher peak intensities in a future climate. The frequency of events with a peak intensity above 10 mm hr −1 more than doubles. Stratiform precipitation is less affected. However, one can see an extension of this distribution towards short and intense events (which might be an artifact due to the misclassification of shallow convection as stratiform precipitation as noted in Poujol et al (2020)).
The changes in the actual contributions of each precipitation rate to total precipitation are depicted in figure 4. Unlike fractional contributions, the changes in actual contributions provide information on the change in the amount of each precipitation type, which is the area under the curve. A comprehensive view of changes in the frequency, intensity and distribution of precipitation can be reached only if figures 3 and 4 are considered together. In the interest of brevity we summarise the key points arising from such a examination: • All precipitation types exhibit increased amounts and intensities in autumn, mostly coming from moderate to heavy rainfall (brown curves on figure 4). Figures 3(a)-(c) shows that the main contribution comes from an increase in both the frequency and the intensity of convective precipitation, even for long duration events, although intense orographic precipitation also has a significant contribution (figures 3(g)-(i)). • Convective precipitation exhibits the strongest response to global warming. Both frequency (figures 3(a)-(c)) and intensity (figures 2(d)-(f)) increase across all seasons. Further, all precipitation rates contribute to the increase, except summer in the East where low and moderate rates decrease (figures 4(a)-(c)). Intensification of convection is strongest in summer and autumn, and weakest in spring (figures 4(a)-(c)). • Orographic precipitation exhibits little change in overall frequency (figures 3(g)-(i)) but moderate and heavy precipitation become more frequent at the expense of light precipitation (figures 2(d)-(f) and 3(g)-(i)). Contribution from rates over 1 mm hr −1 is increased, except summer in the East (figures 4(g)-(i)). A pronounced intensification is found (figures 3(g)-(i)), mainly in winter and autumn (figures 4(g) and (h)). In Northern Norway the intensification is found all year round (figure 4(i)). • Stratiform precipitation exhibits weak changes that are generally insignificant(figures 3(d)-(f) and 4(d)-(f)), except in spring where all precipitation rates contribute to a statistically significant increase (figures 4(d) and (e)).

Discussion
In this section, we explore the consistency of our results season by season with other studies. We also provide some insight to the physical processes that could explain these results.

Winter
The winter changes correspond relatively well to the expected theoretical response of precipitation (section 4). The increase in orographic precipitation comes mainly from the contribution from the high rates. The constant relative humidity assumption is almost satisfied (figure 5). A significant increase in convective precipitation intensity and amount is found, and this is consistent with the observations reported by Rulfová and Kyselý (2013), who show that the convective activity is related to surface temperature. This was also found by Gentine et al (2013) in a theoretical onedimensional boundary layer model, where a high surface temperature clearly helped the onset of convection. Recently, Kendon et al (2020) also found an increased frequency of wintertime convection over the United Kingdom of Great Britain and Northern Ireland, attributable to an increased sea surface temperature and more convective systems advected onshore from the sea. This is consistent with figure 4 that shows significant increases in wintertime convective precipitation in coastal regions (North and West).

Summer in Eastern Norway
In Eastern Norway in summer, a decrease in the the frequency of low and moderate precipitation rates is found, and only the highest precipitation rates (over 10 mm hr −1 ) contribute to increasing amounts of convective and orographic precipitation. This is similar to summer results in other mid-latitude continental regions, such as the central United States , or Southern Europe (Berthou et al 2018a) where a strong drying is found in summer. Dai et al (2017) propose the following mechanism: global warming leads to increased surface atmospheric stability because of the decreasing moist adiabatic lapse rate, and thus more convective inhibition (CIN). Convective available potential energy (CAPE) can therefore accumulate over a stable surface layer. Only precipitation events with the strongest forcing can then be initiated. They have a higher intensity because they can retrieve more CAPE from the upper troposphere, and remove more moisture from the boundary layer. Thus, moisture takes more time to recover by surface evaporation, which implies decreasing precipitation frequency.
This theory is in good agreement with our simulations, which exhibit an increase in CAPE and CIN, however, these are not statistically significant (figure 5).

Autumn and summer
The strongest responses are found in autumn (as well as summer in Western and Northern Norway) with dramatic increases in precipitation amounts. Strong increases in midlatitude autumn precipitation amounts have also been found in other studies, e.g. de Luis et al (2010) in Spain, Berthou et al (2018a) in Southern and Western Europe, and Liu et al (2017) in summer and autumn over the Rocky mountains. Berthou et al (2018a) also found a shift in the seasonality of extreme precipitation from summertime to autumn in Western, Southern and Central Europe. By separating precipitation in this study, a more thorough comprehension of the mechanisms responsible for autumn changes can be reached.
Our results show a very strong increase in convective precipitation frequency and intensity, as well as an increase in orographic precipitation intensity (figure 3). Substantial changes (over a doubling) are simulated for long-lasting, intense convective events (figure 3), and these occur mainly in summer and autumn (not shown). These more frequent long-lasting convection events are probably associated with orographically-sustained convection (i.e. convection fed by the uplift of a moist air layer over a mountain ridge), that can persist for several hours at the same place in opposition to isolated convective events that are usually short-lived. This would be consistent with the results of Cannon et al (2012).
An increase of relative humidity is also evident (figure 5). This might partly explain why such a strong intensification of precipitation occurs in autumn.

Spring
Spring exhibits weak responses in precipitation intensity and frequency. However, a significant increase in the amount of stratiform precipitation is found. To the best of our knowledge, such a result has not been noticed previously in the literature. This change cannot be explained by dynamical changes such as a latitudinal shift of the storm tracks, because the surrogate warming framework conserves weather (geopotential changes are only used to avoid breaking of the thermal wind relation). A possible reason would be that more rain than snow occurs in warmer conditions. Because the fall speed of rain is faster, it has less time to evaporate, and so precipitation would be more likely to reach the ground. However, we note that this result could also be due to more frequent shallow convection that is misclassified as stratiform precipitation because convection is not deep enough (this is a known shortcoming of the classification algorithm (Poujol et al 2020)).

Microphysical explanations
Basic thermodynamic considerations fail to fully explain the seasonality of the precipitation response. Since large-scale dynamical changes are almost not taken into account in these simulations, this suggests that changes in the precipitation characteristics are likely due to microphysical processes. Such microphysical changes have already been noticed by Singh and O'Gorman (2014). To investigate these changes we make an estimation of the precipitation efficiency, which represents the efficiency of the conversion of water sources into precipitation by cloud processes. This estimation is presented in appendix C. When looking at the results, presented in figure 6, several points emerge: • Stratiform precipitation is by far the most efficient. It is formed by supersaturated water vapor deposition on the snow flakes, which then fall. The mechanism is supposed to be almost 100% efficient (Findeisen 1938, Cannon et al 2012. Stratiform precipitation shows the highest efficiency in winter, when temperatures are colder and when the condensation of vapor on snow flakes occurs more frequently. • Orographic precipitation is not efficient. An explanation is that when moist air is forced to ascend over mountains, water vapor condenses close to the ground. Clouds are therefore mostly liquid, and the precipitation is very inefficient (Findeisen 1938) because most of the water condensate stays in the water column and does not precipitate. • Precipitation efficiency for stratiform and orographic changes little in a future warmer climate. However, the efficiency of convective precipitation increases considerably. This increase is concentrated during the warm season when temperatures are above freezing in the lower troposphere (from late spring to early autumn in Northern and Eastern Norway where the warm season is short, but almost all year round in western Norway where winters are milder). As suggested by Prein and Heymsfield (2020), increases in the freezing level height can lead to more efficient riming in deep convective clouds, leading to more intense precipitation.
These changes in cloud microphysics and in precipitation efficiency shed some light on the mechanisms responsible for the strong response of convective precipitation in summer and autumn. Indeed, the intensity of convective precipitation increases well above the Clausius-Clapeyron rate (not shown), likely owing in part to these processes.

Conclusions
Precipitation is changing in a warming climate and has major impacts on human activities. This study investigated the physical processes driving the response of hourly precipitation to climate change through the use of convection-permitting climate simulations of the current and future climate (represented by the surrogate warming experiment). Results are consistent with previous work, and exhibit an increase in precipitation amounts in a future climate, mainly due to an increase in event intensity rather than frequency. This study brings new knowledge insofar as it delves into the separation of precipitation types. The precipitation response strongly depends on both seasonality and precipitation type, with convective precipitation being the most sensitive to warming. Whereas the response of precipitation intensity during winter and summer is well explained by thermodynamical theory, more complex changes occur during the transition seasons. These changes result from an interplay between various thermodynamic and microphysical mechanisms. In particular, a dramatic increase in autumn rainfall is found which likely results from the interaction between increased relative humidity and an increase of convective precipitation efficiency. While the evidence presented here suggests super Clausius-Clapeyron scaling , particularly in summer-autumn, the theory relies on a complex interplay of processes. Further work that focuses on these is needed before a robust conclusion can be determined.
Due to the prohibitive computational costs of running multiple simulations, this study relies on a single model which limits the reliability of our results. An ensemble of convection-permitting simulations would be preferable to assess model uncertainty, and support or contradict our results. One promising initiative is the World Climate Research Program Coordinated Regional Downscaling Experiment Flagship Pilot Studies (WCRP-CORDEX-FPS) (Coppola et al 2019) which benefits from the coordination and combined computational resources of multiple institutions. Perhaps one of the most important issues raised by this work is the striking difference between precipitation response in spring and in autumn. Changes are very weak in spring, whereas a large intensification was found in autumn, consistent with many other studies. This likely indicates that there is some fundamental thermodynamic or microphysical reason for this autumn precipitation increase, but this reason has not been elucidated yet. Future work could extend the physical analysis by using idealised models to investigate responses during the transition seasons.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.

Acknowledgment
B P warmly thanks Nicolas Rochetin for evaluating and giving feedback on an internship report from which this letter is largely inspired. Analysis was performed on resources provided by UNINETT Sigma2-the National Infrastructure for High-Performance Computing and Data Storage in Norway (NN9280K, NN9468K, NS9599K). The authors gratefully acknowledge support from the Research Council of Norway (Grant No. 268243) and its basic institute support of their strategic project on Climate Services.

Appendix B. Theoretical expectations
Let us consider an air particle moving upward from the surface, eventually producing precipitation through adiabatic lifting. According to the Clausius-Clapeyron relation (Clausius 1850), the water holding capacity of the parcel increases almost exponentially with temperature (approximately 7% per degree of warming): (All mathematical notations are defined in appendix A.) Moreover, precipitation P is the vertical integral of condensation C in the air column multiplied by a precipitation efficiency ε (equality 2.a): where d * represents a Lagrangian derivative along a moist adiabat. Since q sat depends only on temperature, and temperature can change only by vertical motion (in an adiabatic movement), it can be transformed into a vertical derivative (equality 2.c). Then, precipitation efficiency, air density and vertical velocity are assumed to vary slowly in the vertical compared to moisture, and are therefore taken out of the integral, following Muller et al (2011) (approximation 2.d). The result is that the surface precipitation intensity change is the sum of a thermodynamic term, a dynamical term and a microphysical term: Whereas the first term always corresponds to an intensification, the two other ones can be either positive or negative depending on situations.

Appendix C. Precipitation efficiency definition
There are many different ways of defining the precipitation efficiency. This quantity is supposed to represent the efficiency of the thermodynamical and microphysical processes that convert water supply into precipitation. To estimate it, different definitions have been suggested. In this study, we revisit one of the two definitions provided by Sui et al (2007). Their first definition corresponds to a microphysical precipitation efficiency, traducing the efficiency of the conversion of condensate into precipitation. This definition, the most common one, was used in section 4 of the main text presenting the theoretical response of precipitation to warming. We focus here on the second one, which is a large scale precipitation efficiency and represents the efficiency of conversion of water (vapor or condensate) into precipitation. This second definition is the following: where sgn is the Heaviside function (sgn(x) = 1 if x > 0 and 0 otherwise), and ⃗ Q is the vector containing four source terms of water content (liquid, solid and vapor) in an air column: where [·] denotes a vertical integration over the troposphere depth, q v is the water vapor mixing ratio, E is the evaporation, ⃗ u is the wind, and C is the cloud water mixing ratio. The four terms account respectively for (1) the local loss of water vapor in the air column, (2) the advection and convergence of water vapor into the air column, (3) the surface evaporation, and (4) the advection, convergence of cloud water and its local loss. Each of these terms can contribute to the precipitation if, and only if, it is positive. This is why they are multiplied by a Heaviside function.
It is possible to decompose the second term: We argue that in most of the cases, the advection of water vapor by the wind −[⃗ u · ⃗ ∇q v ] is roughly balanced with the local time derivative of precipitable water − ∂ [qv] ∂t , since precipitable water has generally very strong values compared to condensed water. Thus, one of these two terms is generally highly positive and the other one is highly negative (approximately the opposite), and that leads to a strong underestimation of the precipitation efficiency. In this study, we thus gather these two terms in order to consider the contribution of the residual only, which is the Lagrangian time derivative of the column precipitable water. The remaining term, the convergence of moisture into the air column, is kept separated.
Moreover, the cloud water and the water vapor are taken together into these two terms to make the expression simpler.
Because of the limited data of the simulation, the water vapour and cloud water contents that we use are vertically integrated, i.e. the wind is considered to vary slowly in the vertical compared to the atmospheric water content in the integral. This is compatible with the precedent definition only if the wind profile is uniform, which is what we supposed. We thus used the wind at 700 hPa to calculate the Lagrangian derivative and the convergence, assuming most of the water is concentrated in the lower troposphere. Results obtained using a higher level (500 hPa) were similar.
The definition of precipitation efficiency used in this study is thus: where: C4) and D/Dt = ∂/∂t + ⃗ u 700 hPa · ⃗ ∇ is the Lagrangian derivative computed with the mid-tropospheric horizontal wind ⃗ u 700 hPa . Because of these approximations, the precipitation efficiency can exceed the value of 1. The main errors might occur for disorganized convection, where the moisture of the cells is provided by low-level convergence, which is not well accounted for here because of the wind taken constant at its 700 hPa value. Errors might also come from the fact the the cloud data is only 6 hourly, which is clearly long compared to the time scale of convective precipitation and some orographic precipitation. To avoid large deviations in the mean because of the extreme values that ε can reach, the precipitation efficiencies exceeding 10 have been considered as missing data.