Here we describe the general features of the extreme case, DA_20_100, which reproduces the features expected from these kinds of simulations. The differential heating of the two patches, caused by the heterogeneity in soil moisture, manifests itself as a gradient of both sensible and latent heat fluxes. At 12:00 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. ). As a consequence, a pressure gradient of about 1 hPa develops close to the surface, which supports a thermally driven circulation . The circulation consists of a front of moist air 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. . 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 mesoscale circulation.

In order to track the front associated with the mesoscale circulation we use an algorithm designed to follow one of the fronts moving over the dry patch. The algorithm is based on the y-averaged zonal wind speed at 150 m of height. It is triggered when the wind speed in the middle of the domain reaches 1 m s-1 and automatically stops when the opposite fronts collide in the center of the dry patch. At every output time step (15 min) a search of the maximum value of zonal wind 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. , the box comprises five grid points, thus approximately 1.25 km.

Figure a shows the Hovmöller diagram of the zonal wind and the tracked position of the front every 15 min with shaded circles for the case DA_20_100. In Fig. b the position and speed of the front obtained with the aforementioned 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 . The cold pools are formed after the first strong precipitation event between 12:00 and 13:00 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 between the patches. In this case the convection transitions to a randomly scattered state and we define the speed of the front to be 0 m s-1.

The diurnal cycle of precipitation can be inspected and compared to the one of evaporation and advection, using the methodology introduced in Appendix . This is needed to later formally express precipitation as a function of soil moisture (see Sect. ). Figure  shows the various components of the moisture balance computed every 5 min 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, Adry, 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. ) and reaches a maximum at around 13:00 LST. This behavior is similar to the one observed by their Fig. 9. The first deep convection event in DA_20_100 between 12:00 and 13:00 LST produces a strong cold pool which causes a strong surface divergence, explaining the minimum at about 14:00 LST in Fig. . Given that the maximum of precipitation associated 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 Adry.

In order to study the variation in the moisture budget terms as a function of ϕdry we conduct the same moisture balance analysis for every simulation and integrate the values over the entire diurnal cycle (18 h). Results are reported in Table . As expected the advection term decreases with increasing local soil moisture, whereas local evaporation increases. Overall the accumulated precipitation averaged over the dry patch decreases when the soil moisture increases, as shown also in Fig. . The sharp decrease in precipitation with increasing 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. ) instead of a decrease. In other words, advection appears to be more efficient than evaporation in producing precipitation, as the increase in Edry with soil moisture is followed by a sharp decrease in Pdry.

These qualitative observations can be formalized by defining the precipitation efficiency. This approach was first proposed by and later adopted by many studies including the one of . The overall assumption underlying the pioneering work of 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 Parea=ηAarea+Earea, where η is the precipitation efficiency. All the terms are considered as areal averages and integrated over a certain time period. The rightmost column of Table  shows the efficiency η computed according to Eq. (). 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 ϕdry. In fact, in the case DA_20_100, evaporation over the dry patch is negligible, i.e., Edry ≃ 0, so that Eq. () applied to the dry patch reads Pdry ≃ ηAdry. Thus, the efficiency obtained in this case is representative of the advection process and can be interpreted as an advection efficiency ηA. On the other hand, in DA_100_100 the advection is negligible so that in this case we obtain an evaporation efficiency ηE. Taking all these findings together, we rewrite Eq. () as Parea=ηA⋅Aarea+ηE⋅Earea, where now ηA ≠ ηE. The values estimated from Table  are ηA = 0.16 and ηE = 0.11.

Figure  confirms that, regardless of the particular choice of a single efficiency η, the decrease in precipitation over wetter soils cannot be captured (orange line in Fig. ). In contrast, using the two efficiencies, ηA and ηE, gives a much better match with the simulated value of Pdry (see blue line in Fig. ). Also, by using two efficiencies, the latter become independent of soil moisture. The efficiencies can be alternatively estimated through a fit of Eq. () using all the values of Adry and Edry in Table . In this case we obtain the values ηA = 0.15 and ηE = 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 in precipitation, in contrast to previous studies, may be linked to the fact that we consider a small domain and a short timescale. The assumption of a well-mixed atmosphere likely holds better on a continental (e.g., Europe) and seasonal scales, as in . 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 Table ), where the idealized sounding of is used to initialize the atmosphere (see Sect. ). For a given soil moisture, advection reaches smaller 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 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 h before compared to the DA sounding (not shown). This allows the atmosphere to fully exploit the instability caused by the morning heating which manifests itself as a stronger enhancement of precipitation at the front, as shown in Fig. . This is also corroborated by the fact that convective available potential energy (CAPE) at 15:00 LST is larger than the one at the initial time over both patches in DA_20_100, whereas it is depleted over the dry patch in the ID_20_100 case (not shown).

Moreover, as indicated by Fig. , the dew-point depression in the ID sounding is smaller than in the DA sounding throughout most of the atmospheric column. This suggests that, in the ID case, convective updrafts are less affected by the entrainment of environmental dry air. We verify this by computing the average difference in MSE (moist static energy) between updrafts, defined as grid points with vertical velocities greater than 1 m s-1 and cloud water content greater than 10-4 kg kg-1, and the environment (not shown). Results show that this difference in the ID case is less than 50 % of the values observed in the DA case. Our goal, however, is not to determine how the efficiencies depend on the atmospheric state but rather how precipitation depends on the efficiencies.

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 in precipitation is smaller than the one 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.  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.

The simplest parametrization of evaporation (we will neglect the transpiration part given that our study does not include plants) is the so-called bucket model introduced by and extended by . Evaporation is defined as a potential term controlled by a limiting factor (also called stress factor). Here we use such a formulation to first approximate the surface latent heat flux LH (mm h-1) at a certain point in space and time as a function of soil moisture ϕ (m3 m-3): LH(ϕ)=AQnet×0forϕ<ϕwpϕ-ϕwpϕcrit-ϕwpforϕwp≤ϕ≤ϕcrit1forϕ>ϕcrit, where Qnet (mm h-1) is the net incoming radiation at the surface (longwave + shortwave), ϕwp (m3 m-3) the soil moisture at the permanent wilting point and ϕcrit (m3 m-3) is the critical soil moisture at which evaporation does not increase any more with increasing soil moisture. As explained by this does not usually correspond to the field capacity.

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 Qnet 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. () to the accumulated evaporation Edry needed in Eq. () we average Eq. () over the dry patch and integrate it over the accumulation period τ. By doing so we assume a constant value for soil moisture and replace it with the value at the initialization time. Such an 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 justified as the daily average value of soil moisture remains similar to its initial value (not shown). This gives Edryϕdry=τA〈Qnet〉×0forϕdry<ϕwpϕdry-ϕwpϕcrit-ϕwpforϕwp≤ϕdry≤ϕcrit1forϕdry>ϕcrit, where now Edry does not depend on time nor space. 〈Qnet〉 denotes the net surface incoming radiation averaged over the dry patch and over the period τ, whereas ϕdry corresponds to the initial value of soil moisture.

Equation () can now be used to fit the values of Edry computed from the simulations (Table ) to obtain an unambiguous value for the parameters A, ϕwp and ϕcrit (see Fig. ). These are estimated to be A = 0.663, ϕwp = 0.213 m3 m-3 and ϕcrit = 0.350 m3 m-3. Note that the latter estimate is not far from the field capacity of this soil type, i.e., 0.340 m3 m-3, while the estimated wilting point is almost double the expected one, i.e., 0.110 m3 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 and . Thus, both ϕwp,crit depend not only on the employed soil type but also on the soil model.

Figure  shows the fit of Eq. (), together with the values obtained in the simulations. It reveals an excellent agreement between theory and simulations. The small discrepancies mainly come from the fact that we assume a constant value of 〈Qnet〉 = 300 W m-2 = 0.43 mm h-1 across the simulations, although the simulated value depends on soil moisture and varies by about 7 %. This is due to different cloud regimes which alter the surface radiation balance Sect. 4b.

We note that our formulation of evaporation differs from the one used in the model of where potential evaporation was used in place of Qnet, which is the main difference between the original framework of and the one of .

Our goal is to find a formulation of Adry as a function of soil moisture. This can be achieved starting from the definition of Eq. () and assuming that the advection of every tracer is mainly due to the propagation of the front associated with the mesoscale circulation, hence H = Hfront. In this case Adry=-1ρw∫0τ∫0Hfrontvfront⋅∇qtot|dryρadzdt,≃-1ρw∫0τ∫0Hfrontufront∂qtot∂x|dryρa.dzdt, where Hfront is the height of the front associated with the mesoscale circulation or, equally, the PBL (planetary boundary layer) height, vfront its speed, ρa is the air density and ρw the water density. Equation () has been already approximated given that the front propagates mainly in the x direction (see Sect. ), so that there is no y component of ∇qtot.

The propagation speed of the front ufront and ρa can be seen as constants in the vertical within the height Hfront, while qtot remains a function of x, z and time t. The time integration can be replaced by considering the average over time multiplied by the timescale τ to obtain Adry=-τρaρw〈ufront〉dry〈∫0Hfront∂qtot∂x|drydz〉,=-τρaρw〈ufront〉dry〈∫0HfrontΔqtotLfrontdz〉,=-τρaρw〈ufront〉dryHfrontLfront〈Δqtot〉. In Eq. () we approximated the derivative of qtot as the difference between the two patches Δqtot divided by the penetration length of the front, Lfront .

To simplify the problem we assume Δqtot ≃ Δqv, 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 , we assume that this difference is not directly affected by the circulation, which yields an upper bound estimate given that the propagation of the front over the dry patch will act to reduce the gradient in specific humidity in the PBL. The changes in qv due to surface evaporation accumulated up to a certain time τ can then be written as qv(τ)=qv(0)+ρwEρaHmoist⇒Δqv,=-ρwρaHmoistEwet-Edry︸ΔE, where Hmoist is the vertical extent of the moistening process due to the accumulated surface evaporation E and qv(0) is the specific humidity at the initial time. By assuming that the moistening is confined to the PBL, so that Hmoist = Hfront we can substitute Eq. () into Eq. () to obtain Adry=τ〈ufront〉LfrontΔE. Our analysis thus indicates that the advection only depends on four terms: τ, which is a constant; the difference in E between the two patches, which can be estimated from Eq. () and which depends on the soil moisture; Lfront; and ufront. In all simulations the front has a constant inland propagation of Lfront ≃ 100 km, which corresponds to half of the patch size. More 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 counterintuitive behavior is related to the fact that cold pools lead to a noticeable acceleration of the front, as seen in Sect. . Although the front is initially triggered by the surface heterogeneity, and different surface 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 was found by , 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 for isolated convection . Instead they solely depend on the state of the mid- to upper-troposphere , which is also insignificantly modified by surface fluxes over the course of one diurnal cycle.

We can thus finally express advection simply as Adryϕdry=BΔEϕdry, where B = τ〈ufront〉/Lfront is a proportionality constant that does not depend on soil moisture. Using the parameters A, ϕwp and ϕcrit obtained from the fit of Eq. () (see Fig. ) we can compute the difference ΔE(ϕdry). Together with the values of Adry obtained in the simulations, the values of ΔE(ϕdry) can be used to fit Eq. () 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 〈ufront〉 and Lfront directly, as this latter approximation does not take into account moisture losses due to advection.

Figure  shows the values of Adry and the fit performed using Eq. () for the basic set of experiments and for further cases, the latter used to test the finding that B does not depend on ϕ but solely on the atmospheric state. Overall the fit matches the variation in Adry with ϕdry remarkably well given the various assumptions. Both the simulated decrease in Adry with higher 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. ().

In the simulations where the initial value of ϕwet 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 (Table ), which use a different atmospheric profile and hence support distinct cold pool strength, the value of B is reduced by about half.

Equations (), () and () can be combined in order to compute Pdry. We, however, are interested in its variation with soil moisture, ∂Pdry∂ϕdry, which can be computed as ∂Pdry∂ϕdry=ηA∂Adry∂ϕdry+ηE∂Edry∂ϕdry,=ηA∂∂ϕdryBEwet-Edry+ηE∂Edry∂ϕdry,=-ηAB∂Edry∂ϕdry+ηE∂Edry∂ϕdry. Note that the derivation of Eq. () retains only one term of the difference given that Ewet does not depend on ϕdry. Using Eq. () it is straightforward to compute the derivative of Edry as ∂Edry∂ϕdry=τA〈Qnet〉×0forϕdry<ϕwpϕcrit-ϕwp-1forϕwp≤ϕdry≤ϕcrit0forϕdry>ϕcrit, which is a step-wise function consisting of constant values.

Equations () and () indicate that for ϕdry < ϕwp and ϕdry > ϕcrit there is no change in precipitation with soil moisture independently of the value of the efficiencies. In contrast for ϕwp ≤ ϕdry ≤ ϕcrit, the ratio ∂Edry∂ϕdry ≠ 0 but still the derivative of precipitation with respect to soil moisture does not depend upon the soil moisture content itself. These findings contrast with the ones of , who found a minimum of the derivative for intermediate values of soil moisture. This is a consequence of the formulation of Edry as a linear function of ϕdry and the fact that Adry also turned out to be a linear function of ϕdry as ufront is constant. This remains true as long as the convection is strongly organized by the front associated with the mesoscale circulation and produces strong cold pools that end up determining the propagation velocity. It should be noted that, although the derivative ∂Pdry∂ϕdry does not depend on soil moisture, the value of precipitation Pdry does indeed depend on soil moisture, as we will show later.

Coming back to Eqs. () and (), we can now determine under which conditions Pdry will increase or decrease. ∂Pdry∂ϕdry0⇔-ηAB+ηE0⇔ηEηAB The atmospheric conditions, through the terms ηA, ηE and B, determine whether increasing or decreasing the soil moisture of 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. () confirms that ∂Pdry∂ϕdry < 0, which agrees with the simulated increase in precipitation with decreasing values of soil moisture. These results are generalized with the help of Fig.  for three different values of B.

In Fig.  positive values indicate an increase in precipitation over the dry patch with soil moisture, and vice versa. Not surprisingly (see Eq. ) using a value of B = 1 gives a symmetric picture where an increase in precipitation with soil moisture is obtained for those cases when ηE > ηA. This relationship is modified by the value of B.

Figure  overall shows that, as long as ηA > ηE, 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 ηA > ηE. 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 much more efficient than advection, i.e., ηE ≫ ηA. 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 ϕwet and ϕdry 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 ηA, ηE and B parameters are fixed. First of all, ∂Pdry∂ϕwet can be computed with the same method as before: ∂Pdry∂ϕwet=ηAB∂Ewet∂ϕwet given that the evaporation over the dry patch does not depend on the soil moisture of the wet patch. Second, the two derivatives ∂Pdry∂ϕwet and ∂Pdry∂ϕdry can be combined to obtain the total precipitation change over the dry patch. ΔPdry=∂Pdry∂ϕdryΔϕdry+∂Pdry∂ϕwetΔϕwet,=ηE-ηAB∂Edry∂ϕdryΔϕdry+ηAB∂Ewet∂ϕwetΔϕwet. Assuming that the soil type of both patches is the same, only the case ϕwp < ϕdry,wet < ϕcrit is of interest. The other cases either revert to the previously discussed case (Eq. ) or reduce to the trivial solution where only Δϕwet is affecting ΔPdry. For ϕwp < ϕdry,wet < ϕcrit we obtain ∂Edry∂ϕdry = ∂Ewet∂ϕwet. Thus, changes in precipitation in our idealized model can be formulated as ΔPdry=τA〈Qnet〉ϕcrit-ϕwpηE-ηABΔϕdry+ηABΔϕwet. The behavior of Eq. () as a function of Δϕdry, Δϕwet and B is investigated with the help of Fig. . In the default configuration described in Sect.  the soil moisture of the wet patch was kept constant, i.e., Δϕwet = 0, while the soil moisture of the dry patch was increased, i.e., Δϕdry > 0. Figure a shows that, in the aforementioned case, ΔPdry is negative, as in our simulations. In this case decreasing ϕdry and increasing ϕwet is the most efficient way to increase precipitation.

Figure 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 to ϕdry, as previously discussed. Mainly increasing ϕwet would allow precipitation to increase. In the extreme case where B is further reduced (Fig. c) the picture partly reverses. Both soil moisture of the wet and of the dry patch should be increased to sustain an increase in precipitation, as evaporation becomes now relevant and advection has a negligible contribution.

Figure  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.