A simplified model of precipitation enhancement over a heterogeneous surface

Soil moisture heterogeneities influence the onset of convection and subsequent evolution of precipitating systems through the triggering of meso-scale :::::::: mesoscale : circulations. However, local evaporation also plays a role in determining precipitation amounts. Here we aim at disentangling the effect of advection and evaporation on precipitation over the course of a diurnal cycle by formulating a simple conceptual model. The derivation of the model is inspired from the results of simulations performed with a high-resolution (250 m) Large-Eddy Simulation model over a surface with varying degrees of heterogeneity. 5 A key element of the conceptual model is the representation of precipitation as weighted sum of advection and evaporation, each weighted by its own efficiency. The model is then used to isolate the main parameters that control the variations of precipitation over a spatially drier patch. It is found that these changes surprisingly do not depend on soil moisture itself but instead purely on parameters that describe the atmospheric initial state. The likelihood for enhanced precipitation over drier soils is discussed based on these parameters. Additional experiments are used to test the validity of the model. 10


Minor Comments
1. Pg. 11, lines 8 -11: The interpretation on how convective updrafts are less affected by dry air entrainment in the ID cases than in the DA experiments is not sufficient. Besides the impact from moisture difference between the updrafts and the environment, their MSE differences also depend on the vertical profiles of convective mass flux which is associated with entrainment rate.
We computed the entrainment rate by considering the ratio between the vertical derivative of updraft MSE and the difference in MSE between updraft and the environment. When integrating the vertical profile of entrainment, larger values are obtained in the DA case with respect to the ID case, given the wider extension of the vertical column of the average updraft. Thus, this confirms our hypothesis that, in the DA case, the updraft are more influenced by the entrainment of dry air. This eventually suppress convection and the production of precipitation.

Introduction
Will a wetter soil lead to more or less precipitation? This apparently simple question inspired many studies over the course of the last 50 years. Over a homogeneous surface precipitation is expected to increase with surface evaporation, and thus with soil moisture in a soil moisture-limited regime (Manabe, 1969;Budyko, 1974), regardless of the atmospheric state (Cioni and Hohenegger, 2017) as long as convection can be triggered on both dry or wet surfaces (Findell and Eltahir, 2003). However, 15 the real world is far from being homogeneous. The presence of heterogeneity in surface soil moisture induces thermally-driven meso-scale ::::::::: mesoscale circulations (Segal and Arritt, 1992) which transport moist air from spatially wetter patches to spatially drier patches, acting against the initial perturbation of soil moisture, and which can then affect the distribution of precipitation.
Many idealized studies have investigated the effect of such circulations on convection and ensuing precipitation. Avissar and Liu (1996) found that the land-surface wetness heterogeneity (i.e. spatial gradients of soil moisture) controls the transition 20 from a randomly scattered state of convection to a more organized one where clouds form ahead of the front associated with the meso-scale ::::::::: mesoscale circulation. The presence of such circulations also tend to enhance the precipitation amount. Further analyses have shown that this basic response can be modified by many environmental factors. Yan and Anthes (1988) found that accumulated precipitation is maximized over spatially dry patches when the patch length is comparable to the local Rossby radius of deformation (⇠ 100 km in mid-latitudes), a result that was later confirmed by Chen and Avissar (1994) and Lynn et al. (1998). Robinson et al. (2008) proposed an alternative explanation by which the effect of surface hot spots is maximized for wavelength of roughly 50 km, that is when the aspect ratio of the applied heating matches the ratio of vertical and horizontal wavenumbers demanded by the dispersion relation for buoyancy (gravity) waves. Froidevaux et al. (2014) explored the interaction between horizontal soil moisture variations, wind and precipitation. They 5 found that, only when winds are too weak to control the propagation of thunderstorms, more precipitation is observed over drier surfaces. Finally, the response of precipitation also depends upon the background atmospheric profile. Chen and Avissar (1994) found that the presence of a moist atmospheric profile over a spatially drier surface reduces the precipitation advantage as the surface heat fluxes, which drive the surface heating and thus the circulation, are reduced. Hence, from such studies, an increase of precipitation over spatially drier patches is maximized when the gradient of surface wetness is high, the soil 10 moisture heterogeneity length-scale is around 50-100 km and no background wind is present.
These same mechanisms can be observed in some areas of the world, the so-called hot spots of land-atmosphere interactions (Koster et al., 2004). Several observational studies (e.g. Taylor et al. (2012)) :::::::::::::::::::: (e.g. Taylor et al., 2012) showed that in the Sahel region thunderstorms occur preferably over regions drier than their surroundings. In other areas of the world the synoptic forcing is usually so strong that a robust relationship of causality between soil moisture and precipitation cannot be found (Tuttle 15 and Salvucci, 2017). Instead of speaking of heterogeneous or homogeneous conditions, Guillod et al. (2015) have indicated that over most areas of the world, except the Sahel, a negative spatial coupling coexists together with a positive temporal coupling.
That is, areas drier than their surrounding (spatial component) but wetter than the climatological value (temporal component) may receive more precipitation than other ones.
Although the aforementioned studies have qualitatively shown how precipitation is influenced by soil moisture, soil moisture 20 gradients and by the atmospheric environment, here we aim at developing a simplified conceptual model to formally isolate the control of soil moisture on precipitation. In particular we aim at developing a mathematical expression for the derivative of precipitation with respect to soil moisture in the case of a heterogeneous surface to understand the response of precipitation to soil moisture changes. In this case, precipitation is not only affected by the advection of moisture due to the meso-scale :::::::: mesoscale : circulation but also by local evaporation (Wei et al., 2016). These two factors depend differently on soil moisture.

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The meso-scale :::::::: mesoscale : circulation triggered by the surface wetness heterogeneity strengthens with decreasing soil moisture of the dry patch, as this gives a larger spatial gradient of surface heat fluxes and thus of surface pressure. Instead local evaporation is limited with reduced local soil moisture. The superposition of local evaporation and remote moisture advection eventually contribute to the observed precipitation, with the atmosphere being the medium that weights these two different contributions.  Neelin and Zeng (2000)). Inspired by their work, we develop a theoretical model which is based on similar assumptions but simplifies the formulation of moisture advection and evaporation. In particular the fact that we consider the specific case of advection by a thermally-induced meso-scale :::::::: mesoscale : circulation, and not by the large-scale flow, will allow us to greatly 35 2 simplify the idealized framework.
Section 2 describes the model and experimental set-up that allows us to simulate the evolution of convective clouds and precipitation over a heterogeneous land-surface during a diurnal period. After a brief analysis of the features of the convective diurnal cycle in section 3.1 we estimate the various terms of the moisture balance and in particular the efficiencies of the conversion of evaporation and advection into precipitation in section 3.2. These results are used in section 4 to derive a simple conceptual 5 model of how precipitation responds to soil moisture changes over a heterogeneous surface. We will show that, at least to a first order, the change of precipitation with soil moisture does not depend on the soil moisture content itself but only on the atmospheric state. The results are concluded in section 5.

Methods
The modeling framework used in this work is, in terms of physical parametrizations and dynamical core, identical to the one 10 described in Cioni and Hohenegger (2017), to which the reader is referred for details. We use the ICOsahedral Non-hydrostatic -Large Eddy Model (ICON-LEM) as atmospheric model coupled to the land-surface model, TERRA-ML, to simulate the diurnal cycle of convection over idealized land surfaces from 6 Local Standard Time (LST) to 24 LST.
The horizontal periodic domain spans 1600⇥400 points with a resolution (in terms of the triangle edges, see Zängl et al. (2015)) of 250 m, which results in a size of approximately 400⇥100 km 2 . In the vertical dimension 150 levels are distributed from the 15 surface up to the model top located at 21 km: the spacing reaches 20 m in the lower levels and 400 m close to the model top.
In contrast to Cioni and Hohenegger (2017), heterogeneous surface conditions are used as bottom boundary condition. The heterogeneity is prescribed by dividing the domain's x-direction into two patches having the same surface area of 200⇥100 km 2 . Figure 1 displays a sketch of the domain setup, together with a visual representation of convective features that will be discussed later.

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The domain is rectangular in order to limit computational expenses and is elongated in the x-axis given that the front associated with the simulated meso-scale ::::::::: mesoscale circulation is expected to propagate with a direction parallel to the x-axis. The chosen patch size of 200 km is larger than the optimal value of the heterogeneity wavelength (⇠ 100 km) identified by Chen and Avissar (1994); Yan and Anthes (1988); Lynn et al. (1998). Therefore we do not expect to maximize, in terms of the strength of the meso-scale ::::::::: mesoscale circulation, the response of the atmosphere to the surface heterogeneity. This could eventually reduce 25 the dynamic contribution of advection on precipitation. The larger domain has nevertheless the advantage that the opposite fronts collide later in the day so that the daily precipitation amounts are less affected by what happens after the fronts have collided. The sensitivity of the diurnal evolution of precipitation to different y-axis size was tested, and found to not affect the results.
The surface heterogeneity is introduced by setting two different initial values of volumetric soil moisture [m 3 m 3 ] for 30 the two patches, wet and dry , respectively. The value is set to the entire soil column to ease the interpretation of the results.
The other parameters that characterize the land surface, including e.g. soil temperature, are horizontally homogeneous over the entire domain. Initially the soil temperature is prescribed using a linear profile which includes a climatological layer with a temperature of 281 K at 14.58 m below the surface and a surface layer which has the same temperature as the overlying lowermost level of the atmosphere (see Cioni and Hohenegger (2017)) :::::::::::::::::::::::::::: (see Cioni and Hohenegger, 2017).
The atmospheric initial state is spatially homogeneous except for random perturbations added to the vertical velocity and the virtual potential temperature in the lowermost three levels to break the perfectly symmetric initial state. The atmosphere is initialized using the dry-soil advantage profile of Findell and Eltahir (2003), albeit with winds set to zero to simplify the 5 analysis (see Fig. 2). This sounding, indicated throughout the manuscript as DA, was observed on 23 July 1999 in Lincoln (Illinois, USA) and was chosen as a typical example by Findell and Eltahir (2003) for cases when a strong heating of a homogeneous surface favors the triggering of convection.
To study the response of precipitation to variations in soil moisture, we perform a set of experiments by setting at the initial time wet to the saturation value and varying dry , with values ranging from the saturation to 20% of the saturation value. The 10 latter value is below the wilting point for the chosen soil type (loam). More details about the soil type can be found in Cioni and Hohenegger (2017) and Doms et al. (2011). The upper part of Tab. 1 summarizes the simulations performed with this basic configuration.
In order to test the validity of the theory proposed in section 4 based on this set of basic experiments we perform further sensitivity experiments. First, we decrease the initial value of wet to 70 % of the saturation value. Second, we change the 15 initial atmospheric profile. We tested the wet soil advantage sounding of Findell and Eltahir (2003) where, in contrast to the dry soil advantage sounding, convection triggering requires a strong moistening of the boundary layer. We also tested the sounding of Schlemmer et al. (2012), indicated as ID, which represents an idealization of the typical atmospheric state prone to convection in Europe (see Fig. 2 b). This sounding thus greatly differs from the conditions as observed in Lincoln. It has a lower surface temperature, a lower integrated water vapour ::::: vapor content but a larger initial instability. As the use of the wet soil advantage sounding of Findell and Eltahir (2003) yields very similar results as in DA, which is not the case when using ID, we only report here on the ID simulations.

General features of convection
Here, we describe the general features of the extreme case, DA_20_100, which reproduces the features expected from this 5 kind of simulations. The differential heating of the two patches, caused by the heterogeneity in soil moisture, manifests itself in a gradient of both sensible and latent heat fluxes. At 12 LST the difference in sensible heat flux between the two patches reaches almost 280 W m 2 . This results in a difference in near-surface virtual potential temperature of about 4 K at the same time (see the colored contours in Fig. 3). As a consequence, a pressure gradient of about 1 hPa develops close to the surface, which supports a thermally-driven circulation (Segal and Arritt, 1992). The circulation is constituted by a front of moist air 10 moving inland over the dry patch at lower levels (from the surface up to 1 km) and a return flow between 1 and 3 km, as shown by the wind vectors in Fig. 3. As a result of the circulation, and as found in past studies, convection preferentially develops over the dry patch and in particular at the edge of the front associated with the meso-scale ::::::::: mesoscale circulation.
In order to track the front associated with the meso-scale ::::::::: mesoscale circulation we use an algorithm designed to follow one of the front moving over the dry patch. The algorithm is based on the y-averaged zonal wind speed at 150 m of height. It is   Table 1. Overview of the performed simulations. The first column indicates the experiment name, whereas the second column indicates the sounding used for initialization: DA for dry soil advantage, after Findell and Eltahir (2003), and ID for idealized, after Schlemmer et al. (2012). Third and fourth columns indicate the value of soil moisture over the dry and wet patches, respectively, in percentage of the saturation value. Naming ::: The :::::: naming convention for the experiments follows SOUNDING_ dry_ wet. The vertical lines that characterize the ID cases are used to omit the repetition of the same experiments description, i.e. ID_30_100, ID_40_100, etc. . speed is performed in a box which is suitably chosen in order to maintain the focus of the tracking algorithm on the front.
More specifically, at the first two time instants the maximum is searched over the entire dry patch while from the third time step onward the maximum search is performed in a box centered on a first guess obtained from a simple linear extrapolation of the previous time instants. The size of the box is the only parameter that needs to be tuned when tracking the front in different simulations. Otherwise, the algorithm is robust. As an example, in the DA_20_100 case shown in Fig. 4, the box comprises 5   LST of y-averaged quantities for the DA_20_100 case. Zonal temperature anomaly (color contours), zonal wind (vectors, values between -0.5 and 1 m s 1 are masked) and cloud water mixing ratio (grey contour, only 10 5 g kg 1 isoline).
On the x-axis numbers indicate the distance from the center of the domain in km.
algorithm are displayed. The front starts to slowly propagate in the late morning with a velocity smaller than 2 m s 1 but is later accelerated by cold pools, in agreement with Rieck et al. (2015). The cold pools are formed after the first strong precipitation event between 12 and 13 LST. The speed of the front reaches values of up to 7 m s 1 before the front collides with the opposing front coming from the outer boundary due to the periodic domain. When the soil moisture of the dry patch exceeds 70% of the saturation value no circulation forms because the gradient in surface temperature is too weak to cause a pressure difference 5 between the patches. In this case the convection transitions to a randomly scattered state (Avissar and Liu, 1996) and we define the speed of the front to be 0 m s 1 .

Local and remote sources of precipitation
The diurnal cycle of precipitation can be inspected and compared to the one of evaporation and advection, using the methodology introduced in appendix A. This is needed to later formally express precipitation as : a function of soil moisture (see section 10 4). Figure 5 shows the various components of the moisture balance computed every 5 minutes from the model output and averaged over the dry patch as well as over the entire domain. It can be verified that the advection term averaged over the entire domain is zero, as expected. Instead, when considering the residual averaged over the dry patch, A dry , it is always positive, indicating a net transport of moisture from the wet to the dry patch.
The advection of moisture over the dry patch increases in the late morning as a result of the propagation of the front (see Fig.   15 4) and reaches a maximum at around 13 LST :::: LST. This behavior is similar to the one observed by Yan and Anthes (1988, their Fig. 9). The first deep convection event in DA_20_100 between 12 and 13 LST produces a strong cold pool which causes a strong surface divergence, explaining the minimum at about 14 LST in Fig. 5. Given that the maximum of precipitation asso- ciated with this event is located in the vicinity of the boundary between the wet and the dry patch, this induces a net negative effect on A dry .
In order to study the variation of the moisture budget terms as function of dry we conduct the same moisture balance analysis for every simulation and integrate the values over the entire diurnal cycle (18 hours). Results are reported in Tab. 2. As expected the advection term decreases with increasing local soil moisture whereas local evaporation increases. Overall the 5 accumulated precipitation averaged over the dry patch decreases when the soil moisture increases, as shown also in Fig. 6. The sharp decrease of precipitation with increasing value ::::: values : of soil moisture seems to suggest that advection and evaporation are characterized by different weights when producing precipitation. In fact, if the contribution of these processes would be the same, we would expect to observe a flattening of the precipitation values (blue asterisks in Fig. 6) instead than a decrease.
In other words, advection appears to be more efficient than evaporation in producing precipitation, as the increase of E dry with 10 soil moisture is followed by a sharp decrease of P dry .
These qualitative observations can be formalized by defining precipitation efficiencies :: the ::::::::::: precipitation ::::::::: efficiency. This ap- proach was first proposed by Budyko (1974) and later adopted by many studies including the one of Schär et al. (1999). The overall assumption underlying the pioneering work of Budyko (1974) is that moisture coming from inside (local evaporation) or outside (remote advection) of some closed domain is well mixed. Under this assumption one can express the precipitation over a certain area as: where ⌘ is the precipitation efficiency. All the terms are considered as areal averages and integrated over a certain time period.
The rightmost column of Tab. 2 shows the efficiency ⌘ computed according to Eq. 1. It can be seen that, in this case, convection is not so efficient in converting local and remote sources of moisture into precipitation as the values range from 16% to 9%. More importantly, the efficiency values vary by up to 7% depending on the initial dry . In fact, in the case DA_20_100, evaporation over the dry patch is negligible, i.e. E dry ' 0, so that Eq. 1 applied to the dry patch reads P dry ' ⌘A dry . Thus, the efficiency obtained in this case is representative of the advection process and can be interpreted as an advection efficiency ⌘ A .
On the other hand, in DA_100_100 the advection is negligible so that in this case we obtain an evaporation efficiency ⌘ E .
Taking all these findings together, we rewrite Eq. 1 as: where now ⌘ A 6 = ⌘ E . The values estimated from Tab. 2 are ⌘ A = 0.16 and ⌘ E = 0.11. Fig. 6 confirms that, regardless of the particular choice of a single efficiency ⌘, the decrease of precipitation over wetter soils cannot be captured . :::::: (orange :::: line :: in ::: Fig. ::: 6). In contrast, using the two efficiencies ⌘ A and ⌘ E gives a much better match with the simulated value of P dry (see blue line in Fig. 6). Also, by using two efficiencies, the latter become independent of soil moisture.

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The efficiencies can be alternatively estimated through a fit of Eq. 2 using all the values of A dry and E dry in Tab. 2. In this case we obtain the values ⌘ A = 0.15 and ⌘ E = 0.10 which, as expected, do not differ much from the ones computed using the two extreme cases.
The fact that one efficiency is not enough to describe the variations of precipitation, in contrast to previous studies, may be linked to the fact that we consider a small domain and a short time scale. The assumption of a well-mixed atmosphere likely 15 holds better on a continental (e.g. Europe) and seasonal scale, as in Schär et al. (1999). Using two efficiencies nevertheless requires data from at least two simulations with different values of advection, evaporation and precipitation.
Initializing the atmosphere with a different sounding will likely lead to different efficiencies. This is illustrated with the ID_ cases (see Tab. 3), where the idealized sounding of Schlemmer et al. (2012) is used to initialize the atmosphere (see section 2). For a given soil moisture, advection reaches smaller value ::::: values : that in the DA case. This is mainly an effect of larger precipitation amounts that fall on the wet patch which in turn prevents an efficient advection of moisture from the wet to the dry patch.
The efficiencies computed for this case range from 47% to 31%, indicating that the atmosphere is more efficient at converting  Table 3. As in Tab. 2 but for the ID sounding.

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advection and evaporation into precipitation than in DA. The higher efficiencies obtained with the ID sounding are due to a combination of different effects. One of those is the different convection triggering. With the ID sounding convection is triggered almost 1 hour before than with the DA sounding (not shown). This allows the atmosphere to fully exploit the instability caused by the morning heating which manifests itself in a stronger enhancement of precipitation at the front, as shown in Fig.   7. This is also corroborated by the fact that CAPE at 15 LST is larger than the one at the initial time over both patches in 10 DA_20_100 whereas it is depleted over the dry patch in the ID_20_100 case (not shown).
Despite the differences ::::::: between DA ::: and : ID, the ID case confirms that advection and evaporation exhibit distinct efficiencies and that precipitation decreases with increased local soil moisture. Here the decrease of precipitation is smaller than the one 20 obtained in the DA_ cases. Although this could be related to a weaker sensitivity of the ID atmospheric state to modifications in the land-surface heterogeneity, we note that the amount of precipitation strongly depends on the collision of the fronts. As shown in Fig. 7 the collision of the fronts in the center of the dry patch has different effects on precipitation depending on the atmospheric state. In the ID_20_100 case strong precipitation events with local maxima of 10 mm h 1 are produced in the center of the patch after the fronts' collision and several secondary events develop due to the fronts propagating away from the collision. Instead, in the DA_20_100 case, no strong precipitation event is produced when the fronts collide. 4 Theoretical :::::::::: Conceptual : model In section 3.2 we showed that precipitation can be expressed as a linear combination of advection and evaporation weighted by different efficiencies which are assumed independent of soil moisture (Eq. 2). Knowing this we can now try to answer 5 one of the first question that was posed in the introduction: What are the minimum parameters that control the variation of precipitation with soil moisture? In order to do so we first have to derive some functional forms of evaporation and advection in terms of soil moisture.

Surface evaporation
The simplest parametrization of evaporation (we will neglect the transpiration part given that our study does not include plants) 10 is the so-called bucket model introduced by Budyko (1961) and extended by Manabe (1969). Evaporation is defined as a potential term controlled by a limiting factor (also called stress factor). Here we use such a formulation to approximate first the  (3) where Q net [mm h 1 ] is the net incoming radiation at the surface (long-wave+short-wave), wp [m 3 m 3 ] the soil moisture at the permanent wilting point and crit [m 3 m 3 ] is the critical soil moisture at which evaporation does not increase any more with increasing soil moisture. As explained by Seneviratne et al. (2010) this does not usually correspond to the field capacity.

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A is a proportionality constant which needs to be introduced and specified given that, even in the extreme case of a saturated soil, non-zero sensible heat fluxes and ground heat flux prevent the entire conversion of Q net into LH. The constant A clearly depends on the particular soil model employed as well as on the different parameters that characterize the soil type considered (e.g. albedo, heat capacity) and partially also on the atmosphere.
In order to link Eq. 3 to the accumulated evaporation E dry needed in Eq 2 we average Eq. 3 over the dry patch and integrate 10 it over the accumulation period ⌧ . By doing so we assume a constant value for soil moisture and replace it with the value at the initialization time. Such assumption is motivated by the fact that changes in soil moisture over one diurnal cycle are not expected to be able to significantly feed back on evaporation and precipitation on such a short timescale. The assumption is also well justify as the daily average value of soil moisture remains similar to its initial value (not shown). This gives: where now E dry does not depend on time nor space. hQ net i denotes the net surface incoming radiation averaged over the dry patch and over the period ⌧ , whereas dry corresponds to the initial value of soil moisture.
Equation 4 can now be used to fit the values of E dry computed from the simulations (Tab. 2) to obtain an unambiguous value for the parameters A, wp and crit (see Fig. 8). These are estimated to be A = 0.663, wp = 0.213 m 3 m 3 and crit = 0.350 m 3 m 3 . Note that the latter estimate is not far from the field capacity of this soil type, i.e. 0.340 m 3 m 3 , while the estimated 20 wilting point is almost double the expected one, i.e. 0.110 m 3 m 3 . This is related to the fact that the employed bare soil evaporation scheme tends to shut down evaporation too early as noted by Schulz et al. (2016) and Cioni and Hohenegger (2017). Thus, both wp,crit depend not only on the employed soil type but also on the soil model. This is due to different cloud regimes which alter the surface radiation balance (Cioni and Hohenegger, 2017, Sec. 4b).
We note that our formulation of evaporation differs from the one used in the model of Lintner et al. (2013) where potential evaporation was used in place of Q net , which is the main difference between the original framework of Budyko (1961) and the one of Manabe (1969).

Advection
Our goal is to find a formulation of A dry as a function of soil moisture. This can be achieved starting from the definition of Eq.
A2 and assuming that the advection of every tracer is mainly due to the propagation of the front associated with the meso-scale 5 :::::::: mesoscale : circulation, hence H = H front . In this case where H front is the height of the front associated with the meso-scale :::::::: mesoscale : circulation or, equally, the Planetary Boundary Layer (PBL) height, v front its speed, ⇢ a is the air density and ⇢ w the water density. Eq. 5 has been already approximated given that the front propagates mainly in the x direction (see Sec. 3.1), so that there is no y component of rq tot .

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The propagation speed of the front u front and ⇢ a can be seen as constants in the vertical within the height H front , while q tot remains a function of x, z and time t. The time integration can be replaced by considering the average over time multiplied by the time scale :::::::: timescale : ⌧ to obtain 14 In Eq. 6 we approximated the derivative of q tot as the difference between the two patches q tot divided by the penetration length of the front, L front (Crosman and Horel, 2010).
To simplify the problem we assume q tot ' q v , which is viewed as the difference in specific humidity ahead of the front and behind it. As in studies which have viewed sea breezes as gravity current (Robinson et al., 2013), we assume that this difference is not directly affected by the circulation, which gives ::::: yields an upper bound estimate :::: given :::: that ::: the :::::::::: propagation :: of ::: the ::::: front 5 ::: over ::: the :::: dry :::: patch :::: will ::: act :: to :::::: reduce ::: the ::::::: gradient :: in :::::: specific :::::::: humidity :: in ::: the :::: PBL. The changes in q v due to surface evaporation accumulated up to a certain time ⌧ can then be written as where H moist is the vertical extent of the moistening process due to the accumulated surface evaporation E and q v (0) is the specific humidity at the initial time. By assuming that the moistening is confined to the PBL, so that H moist = H front we can 10 substitute Eq. 7 into Eq. 6 to obtain Our analysis thus indicates that the advection only depends on four terms: ⌧ , which is a constant, the difference in E between the two patches, which can be estimated from Eq. 4 and which depends on the soil moisture, as well as L front and u front . In all simulations the front has a constant inland propagation of L front ' 100 km, which corresponds to half of the patch size. More 15 importantly, the front speed does not vary much with different surface heterogeneity gradients, against our initial expectations that motivated this study (see Introduction). For example, between the DA_20_100 and the DA_60_100 cases only a 3% relative decrease in the front speed is observed (not shown).
This counter intuitive behaviour ::::::::::::: counter-intuitive :::::::: behavior is related to the fact that cold pools lead to a noticeable acceleration of the front, as seen in section 3.1. Although the front is initially triggered by the surface heterogeneity, and different surface 20 heterogeneities may lead to different initial propagation velocities, the much faster cold pools end up determining the front velocity, thus masking the effect of the surface heterogeneity. This stands in agreement with what found by Rieck et al. (2015), and in particular with the thermodynamic contribution of cold pools to the propagation speed of the front (their Eq. 1).
Moreover, cold pools are distributed along the front and continuously fed by precipitation events, similarly to what happens in squall-lines. Given this spatial organization, their strength and propagation do not depend on the surface state, as in the case 25 for isolated convection (Gentine et al., 2016). Instead they solely depend on the state of the mid-to upper-troposphere (Peters and Hohenegger, 2017) which is also not significantly modified by surface fluxes over the course of one diurnal cycle.
We can thus finally express advection simply as where B = ⌧ hu front i /L front is a proportionality constant that does not depend on soil moisture. Using the parameters A, wp , crit 30 obtained from the fit of Eq. 4 (see Fig. 8) we can compute the difference E( dry ). Together with the values of A dry obtained in the simulations the values of E( dry ) can be used to fit Eq. 9 and compute a value for the parameter B: in the DA cases B = 1.47. This is smaller than the value that would be obtained by estimating instead hu front i and L front directly, as this latter approximation does not take into account moisture losses due to advection. Figure 9 shows the values of A dry and the fit performed using Eq. 9 for the basic set of experiments and for further cases, the latter used to test the finding that B does not depend on but solely on the atmospheric state. Overall the fit matches the 5 variation of A dry with dry remarkably well given the various assumptions. Both the simulated decrease of A dry with higher DA_*_100 DA_*_70 ID_*_100 Figure 9. Fit of the advection in cases DA_ * _100, DA_ * _70 and ID_ * _100. Symbols indicate the values obtained from the simulations while lines represent the fit performed using Eq. 9. The obtained values of B are reported in the insets, together with the absolute error and the 2 value (see Fig. 8 for the definition). Note that for the DA_ * _100 and DA_ * _70 cases the fits yielded similar results: for this reason the obtained value for B is reported only once.
values of soil moisture and the flattening of advection by soil moisture lower than the wilting point are reproduced, although both effects seem to be overestimated by Eq. 9.
In the simulations where the initial value of wet is reduced to just 70% of saturation the estimated value of B is almost the same as the one of the default configuration, confirming that B does not depend on soil moisture. Instead, in the ID_ cases Equations 2, 4 and 9 can be combined in order to compute P dry . We are, however, interested in its variation with soil moisture, @Pdry @ dry , which can be computed as: Note that the derivation of Eq. 10 retains only one term of the difference given that E wet does not depend on dry . Using Eq. 4 5 it is straightforward to compute the derivative of E dry as which is a step-wise function constituted by constant values.
Equations 10 and 11 indicate that for dry < wp and dry > crit there is no change in precipitation with soil moisture independently of the value of the efficiencies. In contrast for wp  dry  crit then @Edry @ dry 6 = 0 but still the derivative of precipitation 10 with respect to soil moisture does not depend upon the soil moisture content itself. These findings contrast with the ones of Lintner et al. (2013), who found a minimum of the derivative for intermediate values of soil moisture. This is a consequence of the formulation of E dry as a linear function of dry and the fact that A dry also turned out to be a linear function of dry as u front is constant. This remains true as long as the convection is strongly organized by the front associated with the meso-scale :::::::: mesoscale : circulation and produces strong cold pools that end up determining the propagation velocity. It should be noted that, 15 although the derivative @Pdry @ dry does not depend on soil moisture, the absolute value of precipitation P dry does indeed depend on soil moisture, as we will shown :::: show later.
Coming back to Eqs. 10 and 11, we can now determine under which conditions P dry will increase or decrease.
The atmospheric conditions, through the terms ⌘ A , ⌘ E and B, determine whether increasing or decreasing the soil moisture of 20 the dry patch is needed to increase the precipitation amount. Inserting the values of the efficiencies and of B obtained from the DA_ simulations in Eq. 12 confirms that @Pdry @ dry < 0, which agrees with the simulated increase of precipitation with decreasing values of soil moisture. These results are generalized with the help of Fig. 10 for three different values of B.
In Fig. 10 positive values indicate an increase of precipitation over the dry patch with soil moisture, and vice-versa. Not surprisingly (see Eq. 12) using a value of B = 1 gives a symmetric picture where an increase of precipitation with soil moisture is obtained for those cases when ⌘ E > ⌘ A . This relationship is modified by the value of B. Figure 10 overall shows that, as long as ⌘ A > ⌘ E it is very unlikely to get a positive derivative. Only with values of B small enough, which would mean weaker and slower cold pools, the derivative may change sign even with ⌘ A > ⌘ E . This situation almost happens in the ID simulation, where the theory predicts a derivative close to zero. This agrees with the weaker sensitivity of precipitation to soil moisture observed in that case. Alternatively, to get a positive derivative, evaporation should become 5 much more efficient than advection, i.e. ⌘ E ⌘ A . This, however, did not happen in the performed simulations.
These findings already answer the main question posed in the introduction and can be further generalized to the case when both wet and dry are changed at the same time. This allows one to investigate the dependency of precipitation on the soil moisture values of the two patches when the ⌘ A , ⌘ E and B parameters are fixed. First of all, @Pdry @ wet can be computed with the same method as before: given that the evaporation over the dry patch does not depend on the soil moisture of the wet patch. Second, the two derivatives @Pdry @ wet , @Pdry @ dry can be combined to obtain the total precipitation change on ::: over : the dry patch.
Assuming that the soil type of both patches is the same, only the case wp < dry,wet < crit is of interest. The other cases either revert to the previously discussed case (Eq. 12) or reduce to the trivial solution where only wet is affecting P dry . For wp < dry,wet < crit we obtain @Edry @ dry = @Ewet @ wet . Thus, changes in precipitation in our idealized model can be formulated as The behaviour of Eq. 15 as function of dry , wet and B is investigated with the help of Fig. 11. In the default configuration Figure 11. Pdry as function of dry and wet for different values of the parameter B. The x and y axes represent the variation of dry and wet, respectively. Note that the maximum variation is crit wp, as Pdry is computed for the regime wp < dry,wet < crit. The red dashed lines indicate no variation of the soil moisture of either one of the patches. The efficiencies are set to (⌘A, ⌘E) = (0.16, 0.11) in (a) and to (⌘A, ⌘E) = (0.47, 0.35) in (b) and (c) to match the simulation results. The black arrow indicates the direction of maximum growth, i.e. when an increase of precipitation is expected. 5 described in section 3.1 the soil moisture of the wet patch was kept constant, i.e. wet = 0, while the soil moisture of the dry patch was increased, i.e. dry > 0. Figure 11 (a) shows that, in the aforementioned case, P dry is negative, as in our simulations. In this case decreasing dry and increasing wet is the most efficient way to increase precipitation. Figure 11 (b) presents the case characteristic of the simulations performed with the ID sounding. The flattening of the contour lines shows that there is little sensitivity on dry , as previously discussed. Mainly increasing wet would allow precipitation 10 to increase. In the extreme case where B is further reduced (Fig. 11 c) the picture partly reverses. Both soil moisture of the wet and of the dry patch should be increased to sustain an increase of precipitation, as evaporation becomes now relevant and advection has a negligible contribution. Figure 11 thus indicates that, in any case, the soil moisture of the wet patch should be increased to get more precipitation on the dry patch. The response to changes in soil moisture of the dry patch is more subtle, and the combination of the two responses can lead to positive or negative coupling depending on the atmosphere state. This may explain why in reality both signs of the coupling are observed with different atmospheric states.

Conclusions
Motivated by the ambiguous relationship between soil moisture, soil moisture heterogeneity and precipitation we designed 5 idealized simulations of a convective diurnal cycle that make use of a coupled configuration of an atmospheric Large Eddy Simulation (LES) model and a land-surface model. The heterogeneity in the land surface was prescribed by dividing the domain into two patches with different initial values of soil moisture. Inspired by the results of the simulations, we specifically wanted to derive a simple conceptual model that retains the minimum parameters that control precipitation over a spatially drier patch. Moreover, we wanted to use this model to understand which is the most efficient way to increase precipitation by 10 acting on soil moisture given the opposite control of soil moisture on advection and evaporation.
Since the main potential sources contributing to precipitation are constituted by remote moisture advection by the meso-scale :::::::: mesoscale : circulation triggered by the soil moisture heterogeneity and local evaporation, we first aim at disentangling the effects of these two on precipitation. Results from the simulations show, as expected, that the moisture advection over the dry patch decreases with increasing local soil moisture, while evaporation increases. The interplay between these two effects 15 produces a decrease of precipitation with increasing values of local soil moisture for the considered case.
More importantly the simulation results indicated that such a decrease can only be correctly reproduced by assuming that advection and evaporation processes contribute differently to precipitation. Hence we model precipitation as the sum of advection and evaporation each weighted by its own efficiency (see Eq. 2). By using two efficiencies they become independent of soil moisture and only dependent on the initial atmospheric state.

20
As a second step we conceptualize the variations of evaporation and advection with soil moisture. Evaporation can be approximated using the bucket model owing to Budyko (1961) (see Eq. 3). The advection is estimated as the product of the breeze front velocity and the gradient in near-surface specific humidity (see Eq. 5). A priori we would have expected a squared dependency of advection on soil moisture since both the velocity of the front and the gradient in specific humidity should be related to soil moisture. However, it turns out that the velocity of the front is independent of soil moisture as the development of convection 25 at the breeze front and the generation of strong cold pools lead to a strong acceleration of the front that fully masks the effect of the initial surface heterogeneity.
Putting all the results together indicates that the derivative of precipitation with respect to the soil moisture of the dry patch does not depend on the actual soil moisture value. This is due to the fact that the functional forms of advection and evaporation ends up to be linear functions in soil moisture. The idealized model is valid as long as the evaporation keeps its linearity as 30 function of soil moisture and the propagation speed of the front does not depend on the surface heterogeneity gradient, meaning strong enough cold pools.
The parameters that control the variations of precipitation with local soil moisture are the aforementioned efficiencies and a scale parameter that defines the magnitude of the advection. All these parameters depend solely on the atmospheric state.
According to the values of these parameters, as estimated from the simulations, the most efficient way to increase precipitation over the dry patch is to decrease the soil moisture of the dry patch. Thus, one can say that, in order to have more precipitation over spatially drier areas, more precipitation should first fall on spatially wetter ones. In other words, the most efficient way to obtain more precipitation over dry areas is to let them dry out for a long time so that a stronger gradient can build up and thus 5 produce more explosive convective events due to a stronger meso-scale ::::::::: mesoscale circulation.
However, if either the efficiency of evaporation becomes much larger than the one of advection or the scale parameter that defines the importance of advection decreases under a certain threshold then the response of precipitation can be reversed.
Although we did not find any evidence of this behaviour for the two atmospheric profiles tested in this work it would be interesting as a next step to derive the three parameters predicted by the conceptual model from more realistic simulations to infer where D indicates the total derivative, P [m] is the accumulated precipitation, E [m] the accumulated evaporation, q tot [kg kg 1 ] represents the sum of all tracers (water vapour q v , clouds q c , rain q r , snow q s , ice q i , graupel q g and hail q h ) mixing ratios, ⇢ w [kg m 3 ] the density of water, ⇢ a [kg m 3 ] the air density and v the velocity of air as a vector. H indicates the top of the simulation domain and ⌧ the lenght of the accumulation period (18 hours in our experiments). The total derivative can be divided into its advective term: and the local derivative: In both equations A2 and A3 the variables A, E, P are solely functions of (x, y) whereas q tot depends also on time t and on the vertical coordinate z. We can eliminate the dependency on (x, y) by applying an average operator over a certain area, indicated 25 with the subscript area : 1 ⇢ w ⌧ Z 0 H Z 0 @q tot @t area ⇢ a dz dt = A area + E area P area (A4) In the main text we use as area either the full domain, denoted with the suffix dom , or the dry patch only, denoted by the suffix dry . By indicating the weighted vertical integral of q tot as q tot ⌘ R H 0 q tot ⇢ a dz we can further simplify the previous equation to: 1 ⇢ w ⌧ Z 0 @q tot @t area dt = A area + E area P area (A5) Although other studies only considered the advection of water vapour, i.e. of q v , in order to close the balance it is necessary to consider all species. In fact, although q i , q g , q h , q s are order of magnitudes smaller than q v , q c , q r , their variations over time are 5 not, so that neglecting these terms in Eq. A5 would lead to an unbalance.
From the 5-min simulation output we use Eq. A5 and estimate the advection as the residual R area = 1 ⇢w R ⌧ 0 @q tot @t area dt + P area E area ⌘ A area . We verify that, when averaged over the entire domain, R dom = 0.
Competing interests. The authors declare that they have no competing interests. Acknowledgements.