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
, regardless of the atmospheric
state as long as convection can be triggered on both
dry or wet surfaces . However, the real world
is far from being homogeneous. The presence of heterogeneity in surface soil
moisture induces thermally driven mesoscale circulations
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. found that the
land-surface wetness heterogeneity (i.e., spatial gradients of soil moisture)
controls the transition from a randomly scattered state of convection to a
more organized one where clouds form ahead of the front associated with the
mesoscale circulation. The presence of such circulations also tends to enhance
the precipitation amount. Further analyses have shown that this basic
response can be modified by many environmental factors.
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 and .
proposed an alternative explanation by which the
effect of surface hot spots is maximized for wavelengths 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.
explored the interaction between horizontal
soil moisture variations, wind and precipitation. They 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.
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 in
precipitation over spatially drier patches is maximized when the gradient of
surface wetness is high, the soil 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
. Several observational studies
e.g., 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 . Instead of speaking of
heterogeneous or homogeneous conditions, 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 surroundings (spatial component) but wetter than
the climatological value (temporal component) may receive more precipitation
than other areas.
Although the aforementioned studies have qualitatively shown how
precipitation is influenced by soil moisture, soil moisture 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 mesoscale circulation but also by local
evaporation . These two factors depend differently on soil moisture.
The 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 weighs these two different contributions.
already derived an equation for the derivative
of precipitation with respect to soil moisture based on a model of
intermediate-level complexity of the tropical atmosphere the
Quasi-equilibrium Tropical Circulation Model 1,. 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 mesoscale circulation, and not by the
large-scale flow, will allow us to greatly simplify the idealized framework.
Section describes the model and experimental setup 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
Sect. we estimate the various terms of the moisture balance
and in particular the efficiencies of the conversion of evaporation and
advection into precipitation in Sect. . These results
are used in Sect. to derive a simple conceptual 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 Sect. .
Methods
The modeling framework used in this work is, in terms of physical
parametrizations and dynamical core, identical to the one described in
, to which the reader is referred for details. We use
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 06:00 to 24:00 LST (Local Solar Time).
The horizontal periodic domain spans 1600 × 400 points with a resolution
(in terms of the triangle edges; see ) of 250 m, which
results in a size of approximately 400 km × 100 km. In the vertical
dimension 150 levels are distributed from the 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 , heterogeneous
surface conditions are used as bottom boundary conditions. The heterogeneity
is prescribed by dividing the domain's x direction into two patches having
the same surface area of 200 × 100 km2. Figure
displays a sketch of the domain setup, together with a visual representation
of convective features that will be discussed later.
Idealized sketch of the employed experimental framework. The initial
condition for soil moisture and the expected initial development of
convection are also sketched in order to ease the interpretation of the
results.
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
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
, and . Therefore, we do not
expect to maximize, in terms of the strength of the mesoscale circulation,
the response of the atmosphere to the surface heterogeneity. This could
eventually reduce 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 ϕ (m3 m-3) for the two
patches, ϕwet and ϕdry, 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 for example 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.
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 , albeit with winds set to zero to
simplify the analysis (see Fig. ). 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
for cases when a strong heating of a
homogeneous surface favors the triggering of convection.
Skew-T diagrams of the two soundings used to initialize the
atmosphere in the simulations. (a) Shows the dry soil advantage
sounding of , DA, while
(b) shows the idealized sounding of ,
ID. The upper inset in both panels show the value of pressure at the
LCL (lifting condensation level), temperature at the LCL, precipitable water
and CAPE (convective available potential energy).
To study the response of precipitation to variations in soil moisture,
we perform a set of experiments
by setting ϕwet to the saturation value at the initial time and varying ϕdry,
with values ranging from the saturation to 20 % of the saturation value. The
latter value is below the wilting point for the chosen soil type (loam). More
details about the soil type can be found in and
. The upper part of Table summarizes the
simulations performed with this basic configuration.
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
, and ID for idealized, after
. Third and fourth columns indicate the value of
soil moisture over the dry and wet patches, respectively, in percentage of
the saturation value. The naming convention for the experiments follows
SOUNDING_ϕdry_ϕwet. 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.
Experiment
Sounding
ϕdry
ϕwet
Basic configuration
DA_20_100
DA
20
100
DA_30_100
30
100
DA_40_100
40
100
DA_50_100
50
100
DA_60_100
60
100
DA_65_100
65
100
DA_70_100
70
100
DA_80_100
80
100
DA_100_100
100
100
Unsaturated wet patch
DA_20_70
DA
20
70
DA_30_70
30
70
DA_40_70
40
70
DA_50_70
50
70
DA_60_70
60
70
DA_70_70
70
70
Idealized sounding
ID_20_100
ID
20
100
|
|
|
|
ID_100_100
ID
100
100
In order to test the validity of the theory proposed in
Sect. ,
based on this set of basic experiments, we perform
further sensitivity experiments. First, we decrease the initial value
of ϕwet to 70 % of the saturation value. Second, we change the
initial atmospheric profile. We tested the wet soil advantage sounding of
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 , indicated
as ID, which represents an idealization of the typical atmospheric
state prone to convection in Europe (see Fig. b). This
sounding thus greatly differs from the conditions as observed in Lincoln. It
has a lower surface temperature and a lower integrated water vapor content but a
larger initial instability. As the use of the wet soil advantage sounding of
yields very similar results as in DA,
which is not the case when using ID, we only report here on the
ID simulations.
Results
General features of convection
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.
x–z diagram at 12:00 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.
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.
Tracking of the front associated with the mesoscale circulation for
the case DA_20_100. (a) Hovmöller diagram (distance
from domain center vs. time) of the y-averaged zonal wind at a height of
150 m above the surface. Dots indicate the position of the front tracked
every 15 min (see text for details). (b) Front inland propagation
(black line) with respect to the center of the domain (km) and front speed
(red line) derived using finite differences
(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 . 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.
Different terms of the moisture balance (Eq. )
computed for the entire domain (subscript dom, solid lines) and for
the dry patch (subscript dry, dashed lines) in the DA_20_100 case.
A indicates advection, E evaporation and P precipitation. Units
are millimeters per hour. Note that all variables in this figure are instantaneous.
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.
Values of advection (green line and crosses), evaporation (purple
line and plus symbols) and precipitation (blue asterisks) from
Table as a function of soil moisture. The orange line
represents an estimate of precipitation obtained as a sum of advection and
evaporation weighted by the same efficiency, i.e.,
0.16 (Adry + Edry), while the blue line represents
a similar estimate obtained by using two different efficiencies,
i.e., 0.16 Adry + 0.11 Edry.
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.
Values of soil moisture (m3 m-3), advection (mm),
evaporation (mm) and precipitation (mm) over the dry patch accumulated over
the diurnal cycle. The rightmost column shows the precipitation efficiency
(dimensionless) computed as PdryAdry+Edry.
Case
ϕdry
Adry
Edry
Pdry
η
DA_20_100
0.0908
7.796
0.0008
1.255
0.161
DA_30_100
0.1362
7.467
0.0113
1.077
0.144
DA_40_100
0.1816
7.223
0.1140
1.005
0.137
DA_50_100
0.2270
6.673
0.6373
0.912
0.125
DA_60_100
0.2724
4.665
2.0271
0.888
0.133
DA_65_100
0.2951
3.222
3.0393
0.805
0.129
DA_70_100
0.3178
1.734
4.0920
0.708
0.122
DA_80_100
0.3632
-0.412
5.2770
0.533
0.094
DA_100_100
0.4540
0.031
5.0800
0.560
0.110
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.
As in Table but for the ID sounding.
Case
Adry
Edry
Pdry
η
ID_20_100
3.814
0.008
1.789
0.468
ID_30_100
3.861
0.028
1.912
0.492
ID_40_100
3.920
0.143
1.671
0.411
ID_50_100
3.557
0.659
1.740
0.413
ID_60_100
2.631
2.054
1.759
0.376
ID_70_100
0.865
4.080
1.542
0.312
ID_80_100
-0.068
4.884
1.652
0.334
ID_100_100
0.022
4.776
1.662
0.346
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.
Hovmöller diagram of precipitation rate (mm h-1) in cases
(a) DA_20_100 and
(b) ID_20_100.
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.
Conceptual model
In Sect. 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. ). Knowing this we can now try to answer the
first question that was posed in the Introduction: what are the minimum
parameters that control the variation in 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) 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.
Fit of Edry with values obtained from the simulations of
the default configuration (DA_20_100 to DA_100_100).
Crosses indicate values obtained from simulations while the line indicates the
fit performed using Eq. (). The upper left inset
shows the values obtained by the fit together with absolute errors and the
residual sum of squares χ2, i.e., the sum of the squared difference
between the values predicted by the fit and the ones obtained in the
simulations.
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 .
Advection
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.
Computing the derivative of precipitation
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.
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. (). The obtained values of B are
reported in the insets, together with the absolute error and the
χ2 value (see Fig. 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.
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.
Contour plot of ∂Pdry∂ϕdry
(mm m3 m-3) as a function of ηA and ηE for different
values of the parameter B. The black points in (a) and
(c) are placed using the efficiencies obtained in the ID_
and DA_ cases, respectively. The dashed red line distinguishes the
areas where ηA > ηE and vice versa. Note the symmetric
color scale and the thicker zero contour line.
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.
ΔPdry as a function of Δϕdry
and Δϕwet for different values of the
parameter B. The x and y axes represent the variation in
ϕdry and ϕwet, respectively. Note that the
maximum variation is ϕcrit - ϕwp, as
ΔPdry is computed for the regime
ϕwp < ϕdry,wet < ϕcrit.
The red dashed lines indicate no variation in the soil moisture of either one
of the patches. The efficiencies are set to (ηA,
ηE) = (0.16, 0.11) in (a) and to (ηA,
ηE) = (0.47, 0.35) in (b) and (c) to match the
simulation results. The black arrows indicate the direction of maximum
growth, i.e., when an increase in precipitation is
expected.
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.
Conclusions
Motivated by the ambiguous relationship between soil moisture, soil moisture
heterogeneity and precipitation we designed idealized simulations of a
convective diurnal cycle that make use of a coupled configuration of an
atmospheric LES (large eddy simulation) 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 acting on soil moisture given the opposite control of soil moisture on
advection and evaporation.
Since the main potential sources contributing to precipitation are
consisted of remote moisture advection by the 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 produces a decrease in
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 weighed by its
own efficiency (see Eq. ). By using two
efficiencies, they become independent of soil moisture and only dependent on
the initial atmospheric state.
As a second step we conceptualize the variations in evaporation and advection
with soil moisture. Evaporation can be approximated using the bucket
model owing to (see Eq. ). The
advection is estimated as the product of the breeze front velocity and the
gradient in near-surface specific humidity (see Eq. ). 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 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 end up being linear functions
of soil moisture.
The idealized model is valid as long as the evaporation
keeps its linearity as a 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 in 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 produce more explosive convective
events due to a stronger 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
behavior 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 the frequency of
occurrence of the various precipitation regimes.