Space-time self-organization of mesoscale rainfall and soil moisture
Paolo D'Odorico
a,*, Ignacio Rodr
õguez-Iturbe
a,ba
Water Resources and Environmental Engineering Program, Princeton University, Princeton, NJ 08544, USA b
Princeton Environmental Institute, Princeton University, Princeton, NJ 08544, USA
Received 3 January 1999; received in revised form 20 June 1999; accepted 3 July 1999
Abstract
The eects of mesoscale circulations induced by soil moisture heterogeneities are studied to assess the impact that the coupling of the land surface and the planetary boundary layer has on the rainfall and soil moisture dynamics at dierent scales. Our goal is to single out the most important physical mechanisms which aect the behaviour of the land-atmosphere system at the mesoscale leading to a dynamical evolution which shows the same features at dierent scales both in space and time. In particular we aim to understand if any hypothesis of self-organization in the system can ®nd its rationale in the mesoscale soil-atmosphere coupling. The main mechanisms of interaction have been simulated through a cellular automata model which incorporates in a soil water balance both the local and the large scale phenomena. Fractal features are shown to emerge in the simulated hydrological ®elds as an eect of the land-atmosphere coupling and this suggests the possible occurrence of self-organization. Some comparisons with real data tend to support such a hypothesis. Ó 2000 Elsevier Science Ltd. All rights reserved.
Keywords:Land-atmosphere interactions; Self-organization; Precipitation; Soil moisture
1. Introduction
This paper deals with an extended version of a model for the mesoscale land-atmosphere dynamics suggested by Rodrõguez-Iturbe et al. [1], who investigated if the emergence of scale invariance in the space-time structure of rainfall and soil moisture can result from local in-teractions between the land and the atmosphere. The cellular automata used by Rodrõguez-Iturbe et al. [1], concentrated on the most basic aspects of the feedback between soil and atmosphere treating them in an ex-tremely schematic manner without emphasizing a
real-istic description of the hydrological phenomena
involved. Our aim is here to study the emergence of scale invariance using a more realistic and physically moti-vated representation of the main physical phenomena involved in the dynamics.
Convective clouds and rain occurring in a spatial scale of several hundred kilometers play a crucial role in the dynamics of the earth's atmosphere aecting its en-ergy, momentum and moisture balances. Despite the eorts spent in the analysis of these phenomena, some
basic questions on the mesoscale dynamics still remain unresolved. Simulations by General Circulation Models (GCM's) cannot help the detailed understanding of mesoscale processes, because their resolution is larger than the scales at which convective clouds usually originate [2]. Mesoscale circulations are thus subgrid phenomena that are commonly parametrized within a GCM. This procedure could undoubtedly bene®t from a deeper understanding of the driving physical mecha-nisms.
The above problem is further complicated by the dependence of the rain and cloud space-time patterns on the landscape structure: sea-land breezes and other land-atmosphere interactions due to soil heterogeneities are among the main examples of how the earth surface may aect the climate of a region [3]. Here we study land-atmosphere interactions locally driven by landscape discontinuities. Such dynamics are believed to play a crucial role in the initiation and evolution of mesoscale convective processes [1,4]. Some recent analyses of data from Sahel [5,6] tend to support such hypothesis show-ing how the couplshow-ing between the land surface and the planetary boundary layer (PBL) occurs through the heat and moisture ¯uxes with soil-atmosphere interactions which have been recently detected [6] at scales as low as 20 km.
*
Corresponding author.
E-mail address:dodorico@princeton.edu (P. D'Odorico)
Heterogeneities in the landscape structure (e.g., in soil moisture, surface roughness, albedo, vegetal land cover, stomatal conductance, etc.) tend to enhance convective phenomena [2,3], aecting the Bowen ratio and inducing atmospheric circulations which may cause precipita-tions. Among the previous variables, soil moisture plays a predominant role because of its key in¯uence on the partitioning of energy into sensible and latent heat ¯uxes at the ground surface. Thus, soil moisture is commonly used as a representative variable to describe the land surface heterogeneity [2,5,7].
In wet ground regions the input of solar energy is partly used for evapotranspiration and returned to the atmosphere through latent heat ¯ux whereas in the dry areas it is dissipated through sensible heat ¯ux and long wave radiation. The dry soils reach higher temperatures yielding larger ¯uxes of sensible heat and thermal radi-ation to the atmosphere. Thus, as a result of the spatial variability of the soil moisture content the atmosphere develops local gradients in temperature and pressure that enhance mesoscale circulations from the wet (colder) areas to the dry (warmer) regions. These ground-induced circulations transfer humid air toward the dry areas and the convergence of these motions is conducive to a vertical transport of heat and moisture to the upper planetary boundary layer which may lead to local precipitation on the dry regions and the release in the atmosphere of (latent) heat of condensation [8].
The above dynamics is sustained by solar radiation over areas with strong spatial gradients in soil moisture. It occurs mainly in summer time when the radiation is stronger and its scale of evolution is of the order of one day. Thus it is during the morning when the input of the radiative energy, needed for the mesoscale circulations, takes place. This is followed by the formation of clouds, the possible occurrence of precipitations and the dissi-pation of the horizontal gradients of pressure and tem-perature throughout the early evening [2].
The role of land cover heterogeneities in the devel-opment of mesoscale circulations has also been exten-sively investigated [3,9,10]. In particular the numerical simulations of Segal et al. [8] point out the links between vegetation cover and thermally induced convective mo-tions at the mesoscale. Irrigation of vegetated areas in the semiarid environments of the American Southwest, the basin of Lake Chad, etc. has also been observed [8] to be responsible for important soil thermal contrasts leading to mesoscale circulations.
The spatial extent of the land heterogeneities needed for the development of mesoscale circulations has been a matter of extensive debate but the model studies of Chen and Avissar [4] and Lynn et al. [11], as well as the recent observational evidence reported by Taylor and Lebel [6] suggest that soil moisture gradients even at scales as small as 20 km may under certain conditions lead to the oc-currence of rainfall patterns of dierent characteristics.
External atmospheric forcings acting at larger scales activate the system. They are responsible for precipita-tion which ¯uctuates in spatial coverage and intensity and which in turn may initiate local circulations and the spontaneous internal activity of the system itself.
The dynamics includes a large number of feedback mechanisms which are part of the mass and energy balances between the soil and the atmosphere. When they occur, mesoscale induced precipitations locally in-crease the soil moisture, provide water for the evapo-transpiration process, aect the soil albedo in bare grounds [12] and provide a new scenario of spatially heterogeneous soil moisture gradients which may en-hance further mesoscale convections. The existence of such a strongly non linear and possibly self-sustaining dynamics may be an appealing physical mechanism for the explanation of numerous features of temporal per-sistence and spatial scale invariance observed in the space-time patterns of the hydrological ®elds [13]. Tay-lor and Lebel [6] argue that the in¯uence of vegetation on the spatial variability of moisture content and tem-perature of the PBL can be itself related to the self-sustained patterns in soil moisture and rainfall which may persist for several weeks. This itself induces spatial patterns in the vegetation growth and in the transpira-tion rates with a positive feedback to the above dy-namics.
Extensive data analyses in many dierent regions have detected the existence of long range correlations in the time series of rainfall and of other hydrologic vari-ables [13±15]. In a spatial context the size of the clouds and the areas of connected regions (e.g., clusters) having daily rainfall above a certain threshold are probabilis-tically distributed according to power laws in many dierent climatic regions [1,16]. Power laws valid through an extended range of scales have been shown to exist also in the relationship between perimeter and area for rain and clouds [17,18]. Although such scaling properties are commonly reproduced by algorithms in the generation of rainfall ®elds [19±21] or through par-ticular cumulus parametrizations in atmospheric models [22], a general physical understanding of the responsible dynamical processes which, acting in quite dierent sit-uations, lead to basically similar statistical signatures, has yet to be achieved.
phenomena the emergence of fractal features is shown to occur both in space and time through very simple models capturing the main driving mechanisms. Struc-tures of all sizes and durations were shown to occur in all cases displaying power law probability distributions without characteristic scales.
We attempt to ®nd a possible mechanism which, based on local dynamics and non-linear interactions, could provide an explanation for the temporal and spatial fractal signatures frequently found in rainfall and soil moisture. The mechanism focuses on mesoscale processes and is driven by soil moisture gradients re-sponsible for local dynamics and possible convective rain. The latter will in turn aect the spatial patterns of soil moisture and the global evolution of the system.
2. The model
A cellular automaton operating on a square lattice of
256256 pixels simulates the space-time evolution of
the water balance in a region subject to large scale at-mospheric external forcings. The cellular automata attempts a simplistic representation of local soil-atmo-sphere interactions, evapotranspiration, deep in®ltration and runo at every pixel. The water balance is carried out daily on a pixel scale of the order of 20 km20 km. We study the space-time evolution of soil moisture at each pixel,x x1;x2, through the water balance equation:
gZr
dS
dt RpRcÿEÿIÿD; 1
whereS is the relative soil moisture, g the porosity, Zr
the depth of the soil active in the moisture exchange with the atmosphere and in the storage of water volumes at the daily time scale;Rp the rain from the large scale
external forcings; Rc the rain resulting from the
meso-scale circulations driven by local spatial gradients in soil
moisture;Ethe water exchange from ground and
vege-tation to the atmosphere by evapotranspiration; I the
loss to deep in®ltration andDthe runo. The hydrologic representation of the terms in Eq. (1) is done in this paper with considerable more physical realism than the more schematic modeling of Rodrõguez-Iturbe et al. [1]. In what follows we give more details about the modeling of the hydrological processes, as well as of the soil properties.
2.1. The soil properties
The saturated hydraulic conductivity, KS x, is
modeled as a lognormally distributed random ®eld [28] with meanhKSi 1000 mm/day and coecient of vari-ation, CV0.1. The porosity at any point,g x, is then simulated as a random variable uniformly distributed around a mean value, hgi, (i.e., g hgi 0:1) which is
considered a function of the saturated hydraulic
con-ductivity through the relationship: hgi 0:375ÿ
0:0108 logKS [KS cm=s]. This is only an approximate relation obtained by ®tting the data given in Clapp and
Hornberger [29] and used with the sole purpose that g
andKS are not generated as independent random ®elds. The minimum water content that can be drained by gravity, fc x, is known as ®eld capacity and is here
expressed as a function of the porosity (e.g., from data in [30]), asfc x 1:25g x ÿ0:2625.
The dependence of the unsaturated hydraulic con-ductivity, K S, on the soil moisture is assumed to fol-low a power law [31]
K S KSSc; 2
where the exponentcis a function of the soil properties. The analysis of data from several soils shows thatc in-creases asKSdecreases and this dependence is expressed here ascaKÿb
S with the parametersa5:76 cm/s and
b0:148 estimated by ®tting the data reported in [29]. Again as in the case of the relationship betweenKS and
hgi all the approximate equations relating dierent
variables are used only to generate soil properties which are not independent from each other. They are not meant to be assigned any physical interpretation.
2.2. Evapotranspiration
The rate of evapotranspiration, E, is modeled as a
function of the soil moisture as shown in Fig. 1, in the hypothesis that, because of the high evaporative demand (summer time), the soil water content is the main lim-iting factor for evapotranspiration. This type of depen-dence is commonly assumed in hydrology [32], withEmax representing the maximum evapotranspiration which in general depends on both the climate and the vegetation and is here assumed to be constant throughout the
region. For relative soil moistures below a critical value
S, we considerEas a linear function ofSwhile, forS larger than theS, we assumeEas a constant and equal to Emax. S is in general a function of the soil and the vegetation but we assume ± as is frequently done ± that
Sis approximated by the ®eld capacity of the soil,f
c x,
at each speci®c pixel. It is important to remark that this is not conceptually correct since the soil moisture at which vegetation decreases its transpiration below the maximum possible value for a given climatic condition is dierent and lower than the ®eld capacity of the soil. Our assumption has no implications for the purposes of this paper and it is motivated only to avoid the gener-ation of another ®eld ofSvalues based on the type of assumed vegetation.
2.3. Deep in®ltration and surface runo
The rate of losses through deep leakage is a function of the soil water content and can be expressed as
I x;t KSSc: 3
When the relative soil moisture is above the ®eld ca-pacity,S, the values ofK Sallow drainage of the soil by gravity while for lower values ofSthe deep leakage is negligible and almost all losses of soil moisture are due to evapotranspiration.
Runo occurs in a pixel whenever its relative soil moisture calculated through Eq. (1) exceeds the unit value.
2.4. Rainfall
Rainfall in the model occurs according to two dif-ferent mechanisms operating at dierent scales. These two mechanisms attempt to represent the eect of (a) large scale synoptic events and (b) mesoscale precipita-tion induced by local land-atmosphere interacprecipita-tions.
Synoptic rainfall takes place in the region with probability,P, of occurrence in any given day. Its spatial structure at the daily time scale is modeled by a Poisson process in space [33] and consists of a set of rectangular
storm areas which may overlap. The number, N, of
storms at any day is a random variable sampled from a Poisson distribution with mean, hNi. The sides of each rectangular storm area are independent random vari-ables, l, exponentially distributed with mean hli. An uniform depth of rain,dp, is assigned to every storm and
is randomly generated from an exponential distribution. In those pixels where storms overlap the total depth of rain,Rp, is given by the sum of the depths of the storms.
The rain generated by mesoscale circulations is as-sumed to occur over those dry areas which have a soil moisture gradient with any wet neighbour exceeding a certain threshold, n. A rain depth, dc is then generated
from an exponential distribution with mean hdci. Thus
mesoscale convective rain is assumed to take place as a consequence of landscape discontinuities in the soil moisture which is considered as the most representative property of the ground characteristics aecting, through its spatial gradients, the mesoscale soil-atmosphere in-teraction. Simulations were also performed where, once the condition on the local gradients of soil moisture is satis®ed, rainfall is generated over the dry areas with a certain probability of occurrence. No substantial chan-ges were observed in the results.
Estimates of the values of the gradient in soil mois-ture needed in order to start and sustain mesoscale cir-culations are not readily available. Taylor et al., [5] during the HAPEX experiment in the semiarid region of southwest Niger, detected moisture gradients of 30± 50 mm/10 km in the top 150 cm of soil in days when self-sustained mesoscale circulations were observed. In terms of relative soil moisture this would mean that the
threshold n can assume values as low as 0.10 although
one needs to stress that the generation of mesoscale circulations and their sustainability depends on many interrelated factors.
3. Space-time patterns of hydrological variables
Although the modeling of the phenomena involved requires of a number of parameters, only two of them,P
and n, are needed to describe synthetically the relative role played by the external forcings and the local dy-namics in the water balance For a givenP, low values of
n mean relatively high amounts of locally driven meso-scale rain, while high n's allow almost exclusively rain-fall generated by synoptic events.
In what follows the dierent sets of values for the
parameters n, P and hNi in the local and synoptic
rainfall models have been chosen in a way to have in the region approximately the same average daily evapo-transpiration (2±2.5 mm/day) and average soil moisture (0:30) and a daily rainfall in the range 2.5±3.5 mm/ day. The following parameters were used for the soil and climate modeling: average soil depth,Zr40 cm,
aver-age hydraulic conductivity, hKSi 100 cm/day and
maximum rate of evapotranspiration,Emax4 mm/day.
The depth of the rainfall induced by mesoscale circula-tions is a random variable (exponentially distributed) with meanhdci 15 mm/storm, while the random depth
of the individual storms in the Poissonian model of synoptic rain is exponentially distributed with mean hdpi 10 mm/event. Every storm is a rectangle with
temporal) variability on the daily values of rainfall and evapotranspiration which is further enhanced by the heterogeneities of the soil properties.
The following sections show how in the presence of soil-atmosphere interactions the model displays fractal properties both in the temporal evolution of the hydrologic variables as well as in their spatial con®gu-ration. The emergence of these properties tends to sup-port the hypothesis of occurrence of self-organized dynamics due to the coupling between land surface and the atmosphere.
3.1. Analysis in time
We have studied the time series of average soil moisture and evapotranspiration in the whole region, as well as those of soil moisture at a point and the number of pixels evapotranspirating at the maximum rate. Power laws have been detected in the power spectra of these series (Figs. 2 and 3) irrespectively of the values of
Pandn. This implies that the time series of soil moisture and evapotranspiration generated by the model tend to show a lack of characteristic scales regardless of the occurrence or not of important interactions between the soil and the near-surface atmosphere. A dierent be-haviour has been found in the time series of average rainfall. In fact, with high values of n, the amount of
locally driven rain is negligible and the time series of the average rain show (Fig. 4(a)) a ¯at power spectrum (i.e., uncorrelated signal) as we would expect for the Poisson process here modelled for the synoptic rain. Decreasing
n, the crucial role of the local dynamics becomes
stronger inducing a dierent structure in the power spectra of rain which behaves as 1/fb noise (Fig. 4(b)).
Thus local land-atmosphere interaction could be re-sponsible for the scaling behaviour of rain which has been experimentally detected in real data [1,15].
Time series of the daily rain depths at a point were also studied through the analysis of the rescaled range [34] of the data sets yielded by the model. The rescaled range is de®ned as follows. LetfXigi1;2;...;T be the signal
(i.e., rain at a point) and let YkP
k
i1 XiÿX
i1;2;. . .;k be the cumulative deviations from the
mean. We de®ne as therangethe dierence between the
maximum and the minimum values of Yin the interval
0;T and we denote it by RT maxfYkgÿ
minfYkg 1<k<T. In order to compare values ofRT
related to dierent series and lags,T, we rescaleRT with
the standard deviation
ST
1
T
X
T
i1
XiÿX 2 v
u u
t :
Thus the rescaled range, R
T, is a dimensionless variable
which scales with Tas
Fig. 2. Power spectra of some time series generated by the model: average soil moisture, (A), and average evapotranspiration, (B). The parameter values used arehNi 40,P0:05 andn0:15.
RT RT ST
TH; 4
whereH is called the Hurst's exponent. For time series
with ®nite memory structure H0:5. A value of
H>0:5 is known as Hurst's phenomenon [35] and it implies that the integral in 0;1 of the correlation function ofXidoes not converge. Thus, the process has
an in®nite memory and is called persistent. We observe (Fig. 5(a)) that when soil-atmosphere interactions are present the local dynamics can induce persistence in the series of daily rain depth at a point, the Hurst's expo-nents being larger than 0.5. This favourably resembles the experimental ®ndings in long time series of rain in dierent parts of the world [14]. For high values ofnthe contribution due to the mesoscale dynamics becomes negligible and series of daily rainfall depth do not show any persistence (Fig. 5(b)). This could suggest the hy-pothesis that the persistence commonly observed in rainfall time series may ®nd a physical explanation in the interaction between land and atmosphere. The same analysis has been performed (Fig. 6) for the time series of relative soil moisture at a point which yield values of
H0:9 for all cases when locally driven rainfall be-comes dominant leading to the same conclusion made for the series of precipitation at a point. A similar analysis was undertaken for some real soil moisture data. Fig. 7 shows the range analysis for several time
series of daily relative soil moisture recorded at a depth of 5 cm in dierent points of the Central Plains in the
United States (data available on web site:
www.wcc.nrcs.usda.gov/smst/smst.html. Values of soil moisture have been determined at the hourly time scale through high frequency electrical measurements of the capacitive and conductive properties of the soil). The Hurst's exponent is found to fall in the range 0.94±0.98.
3.2. Spatial con®guration
The spatial structures of both the soil moisture ®eld and the rainfall ®eld were studied through the proba-bility distribution of the cluster sizes de®ned as the set of connected pixels of the domain where soil moisture (or rain) exceeds a certain threshold.
Fig. 8 shows the distribution of clusters in the soil moisture ®elds for dierent values of the parameters P
and n. Power laws PAPa aÿc are found with
ex-ponents in the range c0:6ÿ1:00. Unfortunately, the lack of spatial data of soil water content at this resolu-tion does not allow for a comparison of this result with real ®elds of soil moisture. What can be concluded here is that the occurrence of local land-atmosphere inter-action does not seem to be crucial to the statistical
Fig. 5. Rescaled range for the time series of daily rainfall depth (mm) at a point generated by the model. The parameter values used are
hNi 40,n0:15,P0:05, (A), andhNi 300,n0:9,P0:95, (B).
Fig. 4. Spectral analyses for the time series of average daily rainfall depth (mm), withhNi 300,P0:95,n0:90, (A), andhNi 40,
Fig. 7. Analysis of the rescaled range for the time series of daily values of relative soil moisture at a point recorded at a depth of 5 cm in dierent locations in the Central US Plains during the summers of 1995, 1996, 1997 and 1998 (June, 1st to September, 15th).
Fig. 8. Distribution of the sizes of the clusters in which soil moisture exceeds a certain threshold (SP0:3) for two dierent simulations: (A)
hNi 40,n0:15,P0:05 and (B)hNi 300,n0:90,P0:95. Areas, a, are in pixel units (20 km20 km).
Fig. 6. Rescaled range for the time series of daily values at a point of relative soil moisture generated by the model. The parameter values used arehNi 40, n0:15,P0:05, (A), andhNi 300,n0:9,
properties characterizing the spatial structure of soil moisture ®elds.
A similar analysis has been performed on the ®elds of daily rainfall occurrence where clusters have been de-®ned as connected areas receiving rain. Power laws are encountered in the distribution of cluster sizes but only for low values ofn(Fig. 9(a)), while for highn(i.e., with weak local dynamics) this scaling behaviour is gradually lost (Fig. 9(b)). Thus, land-atmosphere interactions may be important in explaining the scaling properties ob-served on the spatial con®guration of rainfall ®elds [1,16]. Fig. 10 shows a similar analysis carried out on
real data for a region of 1340 km 636 km in the
Central Plains of North America, where no major rele-vant orographic in¯uences are present. We observe that the sizes of the clusters of rain have well de®ned power-law distributions with exponents very similar to those found in the ®elds generated by the model for low values ofn.
The occurrence of local dynamics arising from land-atmosphere interaction may thus allow the emergence of the fractal properties in the space structure of rainfall ®elds which can be detected in real data.
4. Conclusions
We have simulated the mesoscale hydrologic dy-namics through a cellular automata model in which the evolution of the system is studied estimating local water balances in presence of interactions between the land surface and the atmosphere induced by heterogeneities in the surface soil moisture.
The simulations show how such interactions may lead to scale invariance in the space-time evolution of the system. When the local land-atmosphere interactions are important in the cellular automata, we observe 1=fbnoise
in the time series of the average daily rainfall depth and the occurrence of the Hurst's phenomenon in the time series of daily rainfall at a point. The analysis of images of daily precipitation shows that the distribution of the ar-eas experiencing rain are power laws where no charac-teristic sizes are found. All these fractal features in the space-time evolution of rainfall ®elds disappear whenever the model does not incorporate the local interactions.
The above results suggest that the soil-atmosphere coupling can explain the emergence of the fractal properties experimentally observed in the space-time
Fig. 9. Distribution of the sizes of the clusters of areas receiving rain in daily images generated with:hNi 40, n0:15,P0:05, (A), and
hNi 300, n0:90, P0:95, (B). Areas, a, are in pixel units (20 km20 km).
patterns of rainfall. The local and self-sustained inter-actions of the components of the system strongly aect its dynamics on a wide range of scales and the emer-gence of scale invariance both in time and in space is suggestive of self-organization.
Acknowledgements
This research was supported by grants from NASA (N A G WN A G W - 4 1 7 1 ,- 4 1 7 1 , N A G WN A G W - 4 7 6 6- 4 7 6 6) and NSF (E A R - 9 7 0 5 8 6 1E A R - 9 7 0 5 8 6 1). The research leading to this paper was carried out when the authors were at the Civil Engineering Department of Texas A&M University.
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