Time Scales of Land Surface Hydrology - Dr. Sam Shen · 2015-11-16 · of the hydrological cycle between the atmosphere, vegetation, and soil. A three-layer model for land surface

Time Scales of Land Surface Hydrology

AIHUI WANG

Department of Atmospheric Sciences, The University of Arizona, Tucson, Arizona, and Institute of Atmospheric Physics, ChineseAcademy of Sciences, Beijing, China

XUBIN ZENG

Department of Atmospheric Sciences, The University of Arizona, Tucson, Arizona

SAMUEL S. P. SHEN

Department of Mathematics and Statistics, University of Alberta, Edmonton, Alberta, Canada

QING-CUN ZENG

Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, China

ROBERT E. DICKINSON

School of Earth and Atmospheric Sciences, Georgia Institute of Technology, Atlanta, Georgia

(Manuscript received 1 November 2005, in final form 21 December 2005)

ABSTRACT

This paper intends to investigate the time scales of land surface hydrology and enhance the understandingof the hydrological cycle between the atmosphere, vegetation, and soil. A three-layer model for land surfacehydrology is developed to study the temporal variation and vertical structure of water reservoirs in thevegetation–soil system in response to precipitation forcing. The model is an extension of the existingone-layer bucket model. A new time scale is derived, and it better represents the response time scale of soilmoisture in the root zone than the previously derived inherent time scale (i.e., the ratio of the field capacityto the potential evaporation). It is found that different water reservoirs of the vegetation–soil system havedifferent time scales. Precipitation forcing is mainly concentrated on short time scales with small low-frequency components, but it can cause long time-scale disturbances in the soil moisture of root zone. Thistime scale increases with soil depth, but it can be reduced significantly under wetter conditions. Althoughthe time scale of total water content in the vertical column in the three-layer model is similar to that of theone-layer bucket model, the time scale of evapotranspiration is very different. This suggests the need toconsider the vertical structure in land surface hydrology reservoirs and in climate study.

1. Introduction

Hydrological cycling between the atmosphere, veg-etation, and soil plays an important role in land surfaceprocesses. Understanding its variability and relation-ship to atmospheric processes on various time scaleswill help better understand the impact of soil moisture

on weather and climate and better manage water re-sources. One of the important aspects of the surfacehydrological cycle is how the land surface components(soil moisture, evapotranspiration, and runoff) respondto the precipitation forcing (Vinnikov and Yeserkepova1991; Vinnikov et al. 1996; Scott et al. 1997; Koster andSuarez 1996, 2001; Wu et al. 2002; Dickinson et al.2003).

Suppose that precipitation is the only water source ofthe land surface. The rainfall arriving at the top of veg-etation either is intercepted by foliage, or falls throughcanopy to ground. The intercepted part reduces the

Corresponding author address: Aihui Wang, Dept. of Atmo-spheric Sciences, 1118 E. 4th St., P.O. Box 210081, The Universityof Arizona, Tucson, AZ 85721.E-mail: [email protected]

868 J O U R N A L O F H Y D R O M E T E O R O L O G Y VOLUME 7

© 2006 American Meteorological Society

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supply of water to the soil. In addition, the vegetationfoliage reduces soil evaporation by reduction of the netradiation through shadowing of leaves (Scott et al.1995). On the other hand, water is extracted from theroot zone to foliage for transpiration, and the moisturein soil surface layer and root zone also interact via in-filtration. If the intensity of precipitation is largeenough, some part of the rainfall can directly go intothe root zone through macropores. In turn, evapotrans-piration is always the main mechanism to feed backwater into the atmosphere (Entekhabi et al. 1992), andit may affect the precipitation statistics (Koster andSuarez 1996). In general, some parts of precipitationare stored in the vegetation–soil reservoir, and otherparts must be balanced by runoff and evapotranspira-tion (Milly 1993). These are the major mechanisms ofhydrological interactions in the atmosphere–vegeta-tion–soil system.

Because of the complexity of nonlinear and multipleinteractive processes included in the hydrological sys-tem of atmosphere–vegetation–soil, few theoretical in-vestigations have addressed the above mechanisms inthe literature, and most of them considered the one-layer bucket model (Manabe 1969), except for somebrief notes on general characters of time scales in mul-tiple layers of soil (Dickinson et al. 2003). Many impor-tant questions are still open and need further study.The following questions are crucial: What are the es-sential differences between a model without verticalstructure (one-layer model) and a model with verticalstructure (a multiple-layer model)? How do the verticalstructures and the mutual interactions between layersinfluence the temporal structure of the evolutionaryprocess? How can we improve the estimation of soilmoisture response time scales? With regard to thesequestions, studies using global models (Delworth andManabe 1988, 1989; Koster and Suarez 2001) and ob-servational data (Vinnikov and Yeserkepova 1991; Vin-nikov et al. 1996; Entin et al. 2000; Wu et al. 2002) haveshown that soil moisture memory is significantly longerthan most of the atmospheric processes associated withweather systems. Further, soil moisture memory signifi-cantly depends on the characteristics of atmosphericprocesses such as the statistics of precipitation. In par-ticular, Delworth and Manabe (1988) demonstratedthat short-time-scale precipitation fluctuations could befiltered to provide long-time-scale anomalies of soilmoisture. They also indicated that the time scale of soilmoisture anomalies in the absence of precipitation forc-ing is proportional to the ratio of field capacity of soillayer to the potential evaporation, and this time scale isidentical to the decay time scale of soil moisture whenevaporation is a linear function of soil moisture. This

theoretical conclusion was confirmed by using an em-pirical dataset in the Soviet Union (Vinnikov and Yes-erkepova 1991).

In this paper, we introduce a three-layer model as anextension of the simple bucket model. It is simpleenough to allow rigorous mathematical analysis and re-alistic enough to explain the important results from pre-vious global modeling studies and the analysis of ob-servational data. Section 2 analyzes the time-scalevariations of precipitation and soil moisture in the one-layer model and improves the original theory of Del-worth and Manabe (1988). Section 3 extends the one-layer model to a three-layer model. Section 4 developstheoretical methods suitable for multilayer models anddescribes their application to the analyses of temporal-vertical structures in the three-layer model. Section 5provides the design of numerical simulations. Simula-tion results and their comparisons with theoreticalanalyses and previous studies in sections 2 and 4 areshown in section 6. Conclusions are given in section 7.

2. Time-scale analysis of one-layer model

To investigate the temporal variations of land surfacehydrology, we first examine the simplest model, theone-layer bucket water balance model (Manabe 1969).For a column of soil and vegetation, its water balanceequation can be written as

dW

dt� P � E � R, �1�

where W is the total water stored within the column, Pis the precipitation, E is the evapotranspiration, and Ris the surface runoff and drainage. The evapotranspira-tion term E is always one of the leading terms and isassumed to be a function of W; R only occurs whenwater in the vegetation–soil reservoir is supersaturated.Therefore, (1) indicates that the vegetation–soil waterreservoir is an open (due to P) and diffusive (due to Eand R) system, which typically includes multiple timescales. Here we would extend previous studies to give amore systematic analysis of these time scales.

To quantify the temporal variations of land surfacecomponents, we introduce characteristic scales W*, E*,and P* for W, E, and P, respectively. Then, T* � W*/E* represents the characteristic time scale in the tem-poral variation of water stored in vegetation and soil forthe given atmosphere–land condition in the absence ofrainfall. Delworth and Manabe (1988) took W* and E*as the field capacity and the potential evaporation (orthe potential evapotranspiration when the soil surfaceis covered by vegetation) respectively, and further as-sumed that evaporation is proportional to W/W* if the

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soil is not too wet. They also demonstrated that T*corresponds to the e-folding time for the damping of Win the linear system. The response time scale deter-mined by this method is only a first-order approxima-tion and should be further improved in the nonlinearsystem. In the following, T* denotes an inherent timescale since it does not depend on atmospheric forcing,and the e-folding denotes time as response time scale.

Under precipitation forcing, there is another timescale related to W variations in response to the externalinput, that is, precipitation P(t). Consider a highly ide-alized precipitation time series

P�t� � P�ei�t, �2�

where P� is the amplitude and � is the frequency. It isassumed that E is a monotonic function of W. Substi-tuting (2) into (1), it can be shown that in the case of2�/� � T*, the temporal variation of W cannot effec-tively adjust to such high-frequency input, and it main-tains its inherent variation with time scale T* superim-posed upon small fluctuations with time scale 2�/�.However, in the case of 2�/� � T*, after the initialsignal is sufficiently decayed, both W and E � R aremainly represented as forced oscillations with low fre-quency.

In general, P(t) can be described by its spectral dis-tribution

P�t� � ��

��

P�ei�t d�. �3�

The above analyses indicate that when the spectra ofP(t) are in the high-frequency domain, that is, � 2�/T*, E (or E � R) also has almost the same spectraldistribution as P�, and the high-frequency componentsin W are negligible. For the low-frequency componentsof P(t), that is, � � 2�/T*, W, E, and R all respond toP in their evolutionary processes.

From the above analyses, (1) can be taken as a “lowpass” filter. The response of W to low-frequency inputsis more effective, and W always possesses low-fre-quency components even though the spectra of P areconcentrated in the high-frequency domain. Therefore,the high-frequency input P leads to a statistical equi-librium state with small amplitude but fast fluctuationsof E (or E � R) and relatively large amplitude butlow-frequency harmonics of W. This is the majormechanism for the reddening of the response spectrarelative to the high-frequency input.

If E � R is a linear function of W, the above analysescan be summarized by an analytical and compact math-ematical formula. This has been provided in Delworthand Manabe (1988) and Dickinson et al. (2003) based

on the concept and method developed by Hasselman(1976) in the design of statistical climate model. How-ever, no compact analytical formula has been reportedin the past for the general nonlinear case unless E islocally linearized.

Here we extend the above analysis to the generalnonlinear case by using the local linearization method.It is convenient to divide variables into the perturba-tions (W, E, R, and P) and the long time means (W,E, R, and P), and rewrite (1) as

dW�

dt� P� � E� � R�. �4�

With slight modification of the formula of Serafiniand Sud (1987), the evaporation E from the availablesoil moisture W within a single column of vegetationand soil can be given as

E � E*�1 � e��W�W*��1 � e��1 � E*h�y�, �5�

where y � W/W*, W* is the maximum available soilmoisture (the soil moisture difference between the fieldcapacity and wilting point of the soil), E* is the poten-tial evapotranspiration, and the free parameter � is usu-ally 1–3 depending on vegetation types. The functionh(y) can be linearized as h(y) h(y) � h0(y)y withy � W/W* and h0(y) � dh(y)/dy, and the mean value,E, can be written as

E � E*h�y�, �6�

where y � W/W*.Considering E � E � E, we have

E� � E*�yW�

W� E**

W�

W, �7�

where � � �e��y/(1 � e��). Denoting W as W** andthe corresponding actual response time scale as T** forthe convenience in the comparison, we have

T** �W**E**

� ��1T*. �8�

While T* is a constant, T** varies with y and T** T*only when y is intermediate. Let W and P be repre-sented in the spectral form with amplitudes A� and Ap

respectively along with the frequency � and neglectingR, then (4) yields

Aw2 � ApT**2

1 � ��T**�2 . �9�

This is the same equation as obtained by Delworth andManabe (1988) and Dickinson et al. (2003) except withT** replacing T*. Since � and T** depend on y, the


separating frequency � � (T**)�1 is a slowly varyingquantity in the nonlinear case.

3. A three-layer model

Water stored in different layers of the vegetation–soilsystem has different temporal variations for the precipi-tation forcing, and the one-layer model in (1) cannotquantitatively distinguish the response time scales. Toadequately represent this complexity, a model shouldinclude at least three layers: the vegetation canopy, c,the surface soil layer, s, and the root zone of the soil, r(hereafter the subscript i � c, s, and r denote canopy,surface soil layer, and root zone, respectively). Here wedevelop a modified three-layer water balance model asa direct extension of (1) and (4) rather than as a sim-plification of current comprehensive and complicatedland surface models (e.g., Zeng et al. 2002). Water inthe top two layers directly interacts with the atmo-sphere. The bottom layer includes most of vegetationroots, and its water content is thus the main source forplant transpiration. Based on water mass conservation,the governing equations are

dWc

dt� Pc � Er � Ec � Rc, �10�

dWs

dt� Ps � Rc � Es � Rs � Qsr, and �11�

dWr

dt� Pr � Rs � Er � Rr � Qsr, �12�

where Wi, Pi, and Ei, respectively, denote the watercontent, precipitation partition, and evapotranspiration(evaporation and transpiration) in different layers; Rc isthe outflow from canopy; Rs includes two parts: surfacerunoff (1 � �)Rs and infiltration from s layer to r layer�Rs. Here � (0 � � � 1) is a fraction depending on soiltexture; Rr is the root zone drainage; and Qsr representsthe conductive (diffusive) transfer of water between thes layer and r layer.

Precipitation P is partitioned into three parts Pi (I �c, s, r), in which Pc � min(P*c , P) is the part interceptedby canopy, Ps � min(P*s , P � Pc) is the part held bysurface soil layer, and Pr � P � Pc � Ps is the part thatdirectly penetrates into root zone of soil in the case ofheavy rain due to the macropores in the soil. Here, P*cand P*s are the maximum precipitation that c layer ands layer can hold. They depend on the vegetation char-acteristics and soil properties. Note that the Pr term isalso a kind of infiltration but different from �Rs sincethey are from different sources and have different spec-tral distributions. The introduction of the Pr term im-

plies that the direct influence of atmosphere to deepsoil is taken into account in the model. Further discus-sion of Pr will be given in section 6.

Applying (5) to the three layers yields

Ei � E*i �1 � e��iWi�W*i ��1 � e��i��1 � E*i hi�yi�,

�i � c, s, r�, �13�

where E*i is the potential evapotranspiration, Wi* is up-per bound of water content (field capacity) of ith layer,yi � Wi/W*i , and �i could be different in different layers.In (10) and (13), Ec includes evaporation from wetleaves and transpiration from dry foliage, Es is theevaporation from surface layer of soil, and Er is the soilwater extracted by roots and transferred to the canopy.Basically, Er directly becomes transpiration in mostcases, but some part of Er can be also stored in the leafin some cases.

The interaction term Qsr between two soil layers canbe parameterized as

Qsr � ��1Dsr�Ws

D*s�

Wr

D*r�, �14�

where D*s and D*r are the thickness of s layer and rlayer, respectively, Dsr � D*s D*r /(D*s � D*r ), and � is thedamping time scale. In the absence of the sources andsinks (i.e., P, E, and R are all absent), Ws/D*s �Wr/D*rdecays to e�1 of its initial value during � time due to theconductive transfer Qsr. It is also assumed that W*s /D*s� W*r /D*r , which implies that there is no conductiveinteraction when volumetric soil water contents in bothlayers are the same.

To compare the three-layer model with the one-layermodel, we combine (10)–(12) to obtain

dW

dt� P � E � R, �15�

where W � Wc � Ws � Wr, E � Ec � Es, and R �(1 � �)Rs � Rr. Equation (15) is equivalent to the onelayer-model governed by (1).

4. Time-scale analysis of the three-layer model

In this three-layer model there are several character-istic time scales: an external forcing (P) time scale T*f ,an internal time scale T*q determined by the conductivetransport Qsr, and three inherent time scales T*i (i � c,s, and r), which is similar to T* in section 2. Further-more, we can even define three other time scales T*fidue to the partitioning of P into Pi (i � c, s, and r). It isobvious that T*fc is the same as T*f , and T*fs and T*frdepend on the spectral distribution of Ps and Pr, respec-tively. These multiple time scales imply that the tem-


poral variation of the three-layer model is more com-plicated (than the one-layer model) and can be irregu-lar.

Similar to the one-layer model, the first-order ap-proximation of the inherent time scales T*i for ith layercan be taken as

T*i � W*i �E*i �i � c, s, r�. �16�

For the same atmospheric conditions, E*i is of the sameorder of magnitude, but this does not mean Ec Es Er. On the other hand, W*c is two orders of magnitudesmaller than W*s and W*r , and W*s � W*r due to D*s �D*r . Hence T*c � T*s � T*r .

On the other hand, the inherent time scale T* for(15) is as follows:

T* � �W*c � W*s � W*r ��E*c � E*s �

�W*s � W*r ��E*c � E*s �. �17�

From (16) and (17), we can derive the following rela-tionship:

T*s T* T*r , �18�

which means the one-layer model would shift some partof low-frequency variations from root zone into surfacelayer. Roughly, the time scale T*q � � from (14) is about10–15 days, T*s 10–30 days, and T*r 2–6 months.

Equations (11) and (12) indicate that the temporalvariation of Ws and Wr is affected by Es and Er as wellas other processes. Therefore the response time scale ofWs and Wr can be different from T*s and T*r , respec-tively.

Also note that 1) the existence of the c layer does notdirectly affect the hydrological dynamics in the s layerand the r layer except reducing the precipitation avail-able to the soil from P to P � Pc � Rc and (Pc � Rc)/Pis very small; 2) the r layer directly affects the c layer bythe water supply through Er; and 3) the s layer and rlayer are mutually interactive through conductive trans-fer and infiltration of water. Therefore we can modifythe inherent time scales for s layer and r layer by theuse of only (11) and (12) with a described Ps � Rc.

In the following analytical investigation of timescales, for generality, runoff Rs and Rr are parameter-ized as

Ri � R*i gi�yi� �i � s, r�, �19�

where R*i is the maximum value (upper bound) of Ri,yi � Wi/W*i and gi (0 � gi � 1) is a continuous andmonotonical function of yi so that runoff is still allowedwithin unsaturated soil. Equations (11) and (12) thenare locally linearized with respect to long time means

(Wi, Pi), i � s, r. Thus we have the following nondi-mensional equations:

dy�sd�

� Moims��s � �sy�s � �sy�r, �20�

dy�rd�

� Moimr��r � �ry�s � �ry�r, �21�

where

y�s � W�s �W*s , y�r �� W�r �W*r , ��s � P�s�P, ��r � P�r �P,

�22�

� � t�T*rs, and T*rs � �T*s T*r ��T*s � T*r �. �23�

All the coefficients in the right-hand side are dimen-sionless, Moi is the moisture index,

Moi � P�E*s , �24�

and

ms � ��*s ��1, mr � �rs��*r , �25�

�s � ��s1 � r*s �s � �*�sds�1��*s, �s � ��*�swsrdr

�1��*s ,

�26�

�r � ��r1 � r*r �r � �*�rdr�1��*r,

�r � ��rsr*s �s � �*�rwsr�1ds

�1��*r , �27�

where

�*s � T*s �T*rs, �*r � T*r �T*rs, �*�s � T*s ��, �*�r � T*r ��,

�rs � E*s �E*r , wsr � W*r �W*s , r*s � R*s �E*s , r*r � R*r �E*r ,

�s1 � dhs�ys��dys, �r1 � dhr�yr��dyr, �s � dgs�ys��dys,�r � dgr�yr��dyr, ds

�1 � Dsr�D*s , and dr�1 � Dsr �D*r.

�28�

Each dimensionless parameter has a definite physicalmeaning. For instance, �rs is the ratio of potentialevaporation of the s layer to that of the r layer.

It is difficult to compute the e-folding time scale from(20) or (21) that contains two variables. Therefore wetransform them into two equations, each of which con-sists of one variable only:

dz�jd�

� ��jz�j � f �j � j � 1, 2�, �29�

where

z�j � �jsy�s � �jry�r and �30�

f �j � ��jsms��s � �jrmr��r�Moi � j � 1, 2�. �31�

In these equations ��1 and ��2 are the two eigenval-ues of the matrix


��s �r

�s ��r�,

and (�1s, �1r) and (�2s, �2r) are the two correspondingnormalized eigenvectors, respectively. In general all �j,�js, and �jr are real numbers, and �j 0, ( j � 1, 2).From (20), (21), (24), and (25), the coefficients �j ( j �1, 2) can be solved

�1 �12��s � �r� � ��s � �r�

2 � 4�s�r�1�2�, �32�

�2 �12��s � �r� � ��s � �r�

2 � 4�s�r�1�2�. �33�

The e-folding dimensionless time scale for zi is �i�1

(i � 1, 2), and the two response time scales in (29) are

T1** � �1�1T*rs and T2** � �2

�1T*rs,

rather than the first-order approximation T*s and T*r in(16). Note that the functional form of (34) is similar to(8).

Taking the parameters as listed in Table 1, varioustime scales and parameters can be computed, as givenin Tables 2 and 3 . Because �2r is much larger than �2s

in (30), T2** effectively provides a new estimate of thetime scale of the r layer, and based on numerical simu-lations, it is better than T*r from (16) (see section 6). Onthe other hand, the difference between T1** and T*s issmall (see Table 2); both could represent the time scaleof the s layer. It needs to be emphasized that while T*sand T*r in (16) are independent of precipitation input,our new time scale T2** does depend upon the mean soilwater [based on (26)–(28) and (33)] and hence precipi-tation.

Now we investigate the response of soil moistureanomaly in (29)–(31) to the input of precipitationanomaly in (22). Assume

��s � A�sei�t, ��r � A�re

i�t, �35�

where A�s and A�r are real numbers. Then (31) can berewritten as

f �j � Afjei�t, Afj � Moi��jsmsA�s � �jrmrA�r�.

�36�

Equations (29) and (36) yield

z�j � Azjei��j�, �37�

where Afj and Azj are taken as real numbers:

Azj2 � �j�2 Afj2

�1 � ��j�2�

and �38�

�j � arctan��

�j�. �39�

Evidently, (38) is the direct extension of (9). Assuming�s/�r � Ps/Pr, and taking � as the time scale of 2T1**,we have �1 � 0.46 and �2 � 1.44 from (39). Taking � asthe time scale of 1.5T2**, we have �1 � 0.04 and �2 �0.59. In either case, T2** T1** (Table 2), and hence�2 �1 in agreement with the data analysis of Wu et al.(2002).

5. The design of numerical experiments

For illustration we have carried out some numericalexperiments by using the three-layer model. Daily pre-cipitation is the only model-required external input.For simplicity, the probability density function (PDF)

TABLE 3. Dimensionless quantities in the s layer and r layerusing the parameters listed in Table 1.

Dimensionless parameters Values

�s 1.2�r 0.11�s 0.56�r 0.07�1 1.2�2 0.076�1s 0.9�1r �0.44�2s 0.06�2r 0.98

TABLE 1. Parameters used in the three-layer model, as inferredfrom observational data in the temporal semiarid grassland innorthern China (Jiang 1988; Zeng et al. 2004).

P (cmday�1) �

�(day) i

Pi* (cmday�1)

W*i(cm)

Ei* (cmday�1)

D*i(cm) �i

c 0.1 0.02 0.2 — 1.30.33 0.3 15 s 1.0 2 0.2 5 1.3

r — 16 0.1 40 1.3

TABLE 2. Derived time scales using the parameters listed inTable 1.

Time scales Values (days)

T*s 10T*r 160T*q 15T*rs 9T1** 8T2** 121


of daily precipitation intensity in rainy days is repre-sented by an exponential function

f�p� �1p̃

exp��p

p̃�, �40�

where p is the daily precipitation, and p̃ is the pre-scribed mean daily precipitation for the rainy days. It isalso assumed that rainfall occurs 50% of the days overa typical midlatitude grassland region, so that the cli-matological mean p � p̃/2. For instance, (40) is found toreasonably fit the observed summer precipitation overIllinois (not shown). Under these assumptions, the ran-dom precipitation time series is generated. For otherregions (e.g., monsoon regions), both the occurrence ofprecipitation (OP) in (40) and p̃ need to be adjusted.Furthermore, similar to Delworth and Manabe (1988),seasonal variation of precipitation is not consideredhere by taking a constant p.

Mean daily precipitation p and other required pa-rameters in the numerical experiments are listed inTable 1. The time step is taken as 0.1 day, but we focuson the analysis of daily averaged evaporations and wa-ter contents for each layer. The model sensitivity toinitial conditions is also studied by employing “dry” and“wet” initial conditions respectively. The initial watercontents of all the three layers are set as slightly largerthan zero in the dry condition, and as their saturatedvalues in the wet condition. After the model has run fora long enough time (approximately after 12 months),the time series of water contents in each layer convergein these two simulations. Without loss of generality, inthe following simulations the initial condition is set aswet, and the daily model output after the first 12months is analyzed.

In our numerical simulations, only the runoff fromoversaturation is considered. In other words, gi � 1 foryi � 1 and gi � 0 for yi � 1 in (19). This is similar to theassumption used in Noilhan and Planton (1989). Sensi-tivity of our results to the runoff formulation will alsobe briefly addressed later.

Two statistical methods (i.e., spectral analysis andautocorrelation analysis) are adopted to analyze thetemporal variations of the time series. The first one isused to analyze the spectra of precipitation, water con-tent, and evaporation for different layers. The second isused to study the sensitivity of the time scale of soilmoisture in root zone to some model parameters.

Delworth and Manabe (1988) used the first-orderMarkov process to demonstrate soil moisture memorywhen they modeled soil moisture variability in the Geo-physical Fluid Dynamics Laboratory’s (GFDL) atmo-spheric general circulation model (AGCM). Even

though the Markovian framework has some limitations(Koster and Suarez 2001), it captures the basic featureof soil moisture as a red-noise process responding to theshort-term atmosphere forcing and provides a singleparameter as the measure of soil moisture memory (Wuand Dickinson 2004).

By using the least squares method, the autocorrela-tion function r(t) of the simulated time series of soilmoisture can be approximated as (Delworth andManabe 1988; Vinnikov and Yeserkepova 1991; Vinni-kov et al. 1996)

r�t� � exp��t�T�, �41�

where t is the time lag, and T is the e-folding time scale.By using linearized equations, the temporal variation ofsoil moisture can be represented as a first-orderMarkov process. Then r(t) is exactly governed by (41),and T is indeed the time scale of the soil moisture.While T represents a reasonable measure of soil mois-ture memory, it is empirically obtained from the soilmoisture time series (rather than from land surface pa-rameters). In contrast, inherent time scales [e.g., (16)]and response time scale [e.g., (34)] discussed in thispaper are derived based on climate conditions and landparameters at a given region and hence have clearphysical meanings. In the next section, T2** for (34) andT*r for (16) will be compared to see which is closer to Tr

of the root zone soil water.

6. Analyses of the numerical results

a. Analysis of Wr and precipitation

Figure 1 shows the variations of the response timescales of Wr [as computed from (41)] to the precipita-tion forcing in 300 independent simulations. Thesesimulations differ only in the random number used todescribe f(p) in (40), while keeping the daily mean pre-cipitation constant. The time scale of Wr is sensitive tothe random sequence of precipitation (Fig. 1a), but inmost of the simulations it is concentrated within therange of 50 to 100 days (Fig. 1b). The median responsetime scale (denoted as Tr) is 74 days with the 10th and90th percentiles being 35 and 128 days, respectively.Using the soil observation data over the midlatitudegrassland and agricultural areas in Eurasia and Ameri-can continents, Entin et al. (2000) estimated that thetime scales for top 1 m vary from 1 to 2.5 months,compatible with the response time scales in Fig. 1. Themedian value of 74 days in Fig. 1 is just half the inherenttime scale T*r � W*r /E*r � 160 days. This is not surpris-ing because T*r does not explicitly consider the influ-ence of precipitation on the response time scale of soil


water. Our new time scale T2** � 121 days from (34) isalso larger than Tr but is within the 90th percentilepartly because the input of precipitation is consideredthrough the mean soil water, as mentioned earlier. Thisdemonstrates that T2** is a better representation of Tr

(i.e., the e-folding response time scale of the root zonesoil water) than T*r . This conclusion is also supportedby additional sensitivity tests in section 6c.

To see more clearly the relationship between thetemporal variation of precipitation and those of watercontent, we randomly take one from the 300 simula-tions. Time series of daily precipitation partitions arenormalized by their critical values (Table 1). Fordrizzles, all rainfall is intercepted by vegetation. Onlyfor heavy-precipitation events (occurring less than 8%of the time in Fig. 2a) can some rainfall directly reachthe root zone through large gaps in the soil. Under thisrainfall forcing, the temporal variations of water con-tents in three layers are quite different (Fig. 2b). Thevariation of water in the canopy is the fastest among the

three layers and closely follows the variation of precipi-tation in Fig. 2a, while water in the root zone variesslowest.

b. Spectra analysis of the multiple scales

To quantify the time scales in the vertical structure ofvegetation and soil, power spectra of precipitation, wa-ter content, and evapotranspiration are calculated. Thespectra are normalized by its total spectra accumulatedin all periods, and then divided into four segments: pe-riods �0.2, 0.2 � 1, 1 � 3, and �3 months. Results fromone simulation are given in Fig. 3. Over 70% spectra ofthe total precipitation and its partitions are concen-trated in the high-frequency domain (periods �0.2months), and only a very small part of spectra are dis-tributed in the low-frequency domain (periods 3months).

The water contents in different layers show the mul-tiple-time-scale phenomena (Fig. 3b). Most of the spec-tra (over 60%) of canopy water are concentrated in theshort periods. The vegetation canopy holds a small

FIG. 1. (a) The response time scale of Wr from (41) for differentprecipitation input. Each point denotes the time scale calculatedfrom one of the 300 simulations. Simulations differ only in thedaily precipitation distribution, while the mean precipitation re-mains the same as given in Table 1. (b) Normalized occurrenceprobability of the time scale of Wr in (a).

FIG. 2. Normalized time series of (a) precipitation and (b) watercontent in each layer. Precipitation is normalized by its charac-teristic value given in Table 1 and the normalized values are sepa-rated by increments of 1 from the c layer to r layer for clarity.Water content is normalized by its saturated value (Table 1). Forclarity, only the last 180 days (from 1620 to 1800 days) in a 5-yrmodel integration are shown.


amount of water in comparison with the soil, and thiswater is quickly removed as evaporation. The waterstored in the root zone Wr has the longest time scaleand most of its spectra are concentrated in the longperiods (over 3 months). The water stored in the s layeris of intermediate time scale in the three layers. Be-cause the total water is dominated by Wr, its time scaleis close to that of Wr. However, the time scale for totalwater is still smaller than that of Wr, but larger than thatof Ws, in agreement with (18).

Figure 3c shows the spectral distributions of evapo-ration and transpiration. Evaporation from leaves(Ec � Er) includes a large part of short-term compo-nents and the intercepted water is quickly evaporatedback to the atmosphere. Scott et al. (1997) indicatedthat time scales associated with evaporation of inter-cepted water are around one day. Therefore, the spec-

tral distribution of both Wc and (Ec � Er) closely followthat of precipitation in Fig. 3a. Compared with (Ec �Er) and Es, the variation of Er has the longest timescale. Even if precipitation is absent, Er itself couldprovide some water to canopy. In comparison with Er

and Es, the spectra of the total evapotranspiration(Ec � Es) are more evenly distributed due to the con-tribution of the low-frequency Er and high-frequencycanopy evaporation. As the evapotranspiration directlyinfluences the atmospheric processes, therefore, thesemultiple time scales indeed have different contributionsto climate variability on different time scales (Kosterand Suarez 1996).

Figure 4 compares the spectra of total water content(i.e., W � Wc � Ws � Wr) and evapotranspiration (i.e.,E � Ec � Es) from the three-layer model with those inthe one-layer model. The spectra distributions of Wfrom both models are nearly the same with most of thespectra concentrated in the low-frequency domain dueto the dominance of water content in the root zone ofsoil. However, the spectra of evapotranspiration fromthe two models are significantly different. Since theevapotranspiration is a function of water content only,the distribution of E exactly follows that of W in theone-layer model. However, due to the nonlinear inter-actions among vertical layers in the three-layer model,E and W have very different spectral distributions.Since an important hydrological feedback from vegeta-

FIG. 3. Normalized power spectra of (a) precipitation and itspartitions, (b) water content in each layer, and (c) evapotranspi-ration, using the same simulation as in Fig. 2.

FIG. 4. Normalized power spectra of total water and evapotrans-piration produced by (a) the three-layer model [i.e., (10)–(12)]and (b) the one-layer model (15).


tion and soil to the atmosphere is via evapotranspira-tion, Fig. 4 implies that the one-layer model might in-correctly yield climatic variances in the lower perioddomain, and a multilayer model is more appropriate forclimate modeling.

c. Sensitivity of time scales of soil moisture tomodel parameters

Previous sections have shown that the time scale ofsoil moisture in the root zone (i.e., Tr) has a relativelylong memory in comparison with those of precipitationand surface evaporation, and this time scale is betterrepresented by our new time scale T2** than by thefirst-order approximation of T*r . Additional simulationsare preformed here using the same 300 precipitationseries as in section 6a to address the sensitivity of soilmoisture time scales to various model parameters.

Figure 5a shows that T*r and the median Tr and T2**all increase with the soil depth of root zone D*r , consis-tent with the results from the analyses of observationaldata (Vinnikov et al. 1996; Entin et al. 2000; Wu et al.2002). For the same D*r , T2** is always closer to Tr thanT*r . Both (T*r � Tr) and (T2** � Tr) increase with D*rbut the increase of the former is faster. The muchsmaller interquartile range (i.e., the difference betweenthe 75th and 25th percentiles) of T2** than that of Tr in

Fig. 5a also suggests that T2** is not sensitive to therandom number used to generate the precipitation se-ries in (40).

Figure 5b shows that, while T*r is independent of �(i.e., the interaction time scale of the s layer and rlayer), both Tr and T2** increase with �. Since Qsr 0in (11) and (12) most of the time, increasing � wouldreduce Qsr (i.e., reduce the amount of higher-frequencysoil moisture in the s layer going into the r layer).Therefore, the increasing rate of Tr and T2** with � isrelatively fast when � is small. When � is too large, Qsr

is nearly zero, and the increase of Tr and T2** with � isrelatively slow. Similar to Fig. 5a, the interquartilerange of T2** is smaller than that of Tr.

Table 4 shows that, when the OP is increased from50% (control) to 60%, the median value and the inter-quartile range of Tr vary less than those when OP isdecreased from 50% to 40%. Overall, the median Tr isnot very sensitive to the variation of OP. When thedaily mean precipitation P is reduced by 0.05 cm day�1

(to 0.28 cm day�1), the median Tr varies little. In con-trast, when P is increased by 0.05 cm day�1 (i.e., whensoil is wetter) the median Tr is significantly reduced.This asymmetrical behavior agrees with the observeddata analysis in Wu et al. (2002) and Wu and Dickinson(2004), which indicated that the dryness signal extendsdeeper into the soil than the wetness signal and soilmoisture time scale can be several times longer for drierconditions than for sufficiently rainy conditions. Table4 also shows that the median Tr is nearly independentof the infiltration index �.

We have also analyzed the sensitivity of Tr to somemodel formulations. Substituting the formula of evapo-ration (13) by the formulation of Delworth and Manabe

FIG. 5. The variation of time scales of Wr using the same 300precipitation time series as Fig. 1 along with their interquartileranges (i.e., the difference between the 75th and 25th percentiles;vertical lines) with respect to (a) depth of r layer D*r and (b)interaction time scale (�) between the s layer and r layer.

TABLE 4. Sensitivity of the 25th, 50th (i.e., median), and 75thpercentiles of the response time scale (Tr) to various model pa-rameters and formulations. The results are based on the same 300independent simulations as in Fig. 1. The parameters for the con-trol simulations are OP � 50%, P � 0.33 cm day�1, � � 0.3, and� � 0.0. See the text for additional explanations.

Case

Tr (days)

25th 50th 75th

Control 58 74 97OP: 40% 47 62 81OP: 60% 61 77 106P � 0.28 cm day�1 60 75 99P � 0.38 cm day�1 40 54 69� � 0.1 58 74 94� � 0.5 54 73 94Delworth and Manabe’s (1989)

evaporation equation53 68 88

Add extra runoff ��wr

(� � 0.001)55 71 90


(1989), the median Tr varies by less than 10% (Table 4).As mentioned in section 5, runoff is only consideredunder supersaturated conditions in the above simula-tions. A continuous runoff for unsaturated soil can beconsidered by adding the term ��wr (� � 0.001) in theright–hand side of (12). Again, Table 4 shows that themedian Tr varies little partly because runoff is overallsmall in the control simulations.

7. Conclusions

Through theoretical analyses and numerical simula-tions of a three-layer soil–vegetation interaction model,this paper has investigated how land surface hydrologyresponds to precipitation forcing. The major results areas follows: (a) Different water reservoirs of the vegeta-tion–soil system have different time scales, includingvery short time scales of canopy water, an intermediatetime scale of soil moisture in the surface soil layer, andlong time scale of root zone soil moisture. This is con-sistent with the theoretical conclusion of Dickinson etal. (2003), but cannot be represented properly by theone-layer bucket model. (b) Precipitation forcing ismainly concentrated on short time scales with somesmall components at low frequencies. However, it cancause the long time-scale disturbances in the soil mois-ture of root zone, because soil moisture responds moreeffectively to the low-frequency parts of precipitationspectra, and hence acts as a low-pass filter. (c) Theinherent time scale determined by the ratio of the fieldcapacity to the potential evapotranspiration is only afirst-order approximation for the response time scale(as defined from the time series analysis). Our new timescale T2** in (34) considers the vertical interactions be-tween layers as well as the impact of mean soil water(and hence precipitation input), and is indeed closer tothe actual response time scale of the root zone soilwater (Tr) than the inherent time scale (T*r ). While Tr

can be empirically computed from the soil water timeseries, it does not provide a clear physical interpreta-tion. In contrast, T2** is derived directly from our theo-retical analysis, and it does provide a physical interpre-tation of the soil water response time. (d) Comparisonof this three-layer model with the one-layer bucketmodel indicates that the time scale of evapotranspira-tion is quite different although the time scale of soilmoisture itself is very close to each other. This suggeststhe need to consider the vertical structure in water res-ervoirs in land surface modeling. (e) The three timescales (i.e., Tr, T2**, and T*r ) all increase with soil depth.The first two also increase with the characteristic inter-action time scale (�) between two soil layers (particu-larly when � is small), while T*r is independent of �.

While the median Tr is not very sensitive to mostmodel parameters or formulations, it does strongly de-pend on the mean precipitation rate p (particularlywhen p is increased). Therefore it needs to be empha-sized that the quantitative results from numerical simu-lations in section 6 are valid only over midlatitudegrassland and they may be different over different re-gions (e.g., under wetter conditions). However, thetheoretical framework and the overall conclusion thatT2** better represents Tr than T*r are expected to bevalid over different regions.

In this study, we focus on the land surface response toprecipitation forcing. When a land model is coupled toan AGCM, the time scales of soil moisture can be dif-ferent from those in our uncoupled model due to themutual interactions between land and atmosphere viathe atmospheric boundary layer. Dickinson (2000)demonstrated the difference of the time scales for thevariability of coupled and uncoupled systems, and con-cluded that coupling to the atmosphere lengthens thetime scale of land. Also missing in our model, just as inmany other land surface models (e.g., Zeng et al. 2002),are the horizontal interactions within the soil layers.Similar to the vertical interaction term (14), horizontalinteractions would introduce a new time scale. Furtherstudy is needed to determine this time scales as well asthe impact of horizontal interactions on other timescales discussed in this paper.

Acknowledgments. This work was supported byNASA (NNG04G061G), NSF (ATM0301188), and theChina Natural Science Foundation (40233027). Thefirst author also acknowledges support from World Me-teorology Organization fellowship. Dr. Xiaodong Zengis thanked for helpful discussions, and three anony-mous reviewers are thanked for their constructive com-ments that helped improve the manuscript.

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