1. Introduction

Hydrological cycling between the atmosphere, vegetation, and soil plays an important role in land surface processes. Understanding its variability and relationship to atmospheric processes on various time scales will help better understand the impact of soil moisture on weather and climate and better manage water resources. One of the important aspects of the surface hydrological cycle is how the land surface components (soil moisture, evapotranspiration, and runoff) respond to the precipitation forcing (Vinnikov and Yeserkepova 1991; Vinnikov et al. 1996; Scott et al. 1997; Koster and Suarez 1996, 2001; Wu et al. 2002; Dickinson et al. 2003).

Suppose that precipitation is the only water source of the land surface. The rainfall arriving at the top of vegetation either is intercepted by foliage, or falls through canopy to ground. The intercepted part reduces the supply of water to the soil. In addition, the vegetation foliage reduces soil evaporation by reduction of the net radiation through shadowing of leaves (Scott et al. 1995). On the other hand, water is extracted from the root zone to foliage for transpiration, and the moisture in soil surface layer and root zone also interact via infiltration. If the intensity of precipitation is large enough, some part of the rainfall can directly go into the root zone through macropores. In turn, evapotranspiration is always the main mechanism to feed back water into the atmosphere (Entekhabi et al. 1992), and it may affect the precipitation statistics (Koster and Suarez 1996). In general, some parts of precipitation are stored in the vegetation–soil reservoir, and other parts must be balanced by runoff and evapotranspiration (Milly 1993). These are the major mechanisms of hydrological interactions in the atmosphere–vegetation–soil system.

Because of the complexity of nonlinear and multiple interactive processes included in the hydrological system of atmosphere–vegetation–soil, few theoretical investigations have addressed the above mechanisms in the literature, and most of them considered the one-layer bucket model (Manabe 1969), except for some brief notes on general characters of time scales in multiple layers of soil (Dickinson et al. 2003). Many important questions are still open and need further study. The following questions are crucial: What are the essential differences between a model without vertical structure (one-layer model) and a model with vertical structure (a multiple-layer model)? How do the vertical structures and the mutual interactions between layers influence the temporal structure of the evolutionary process? How can we improve the estimation of soil moisture response time scales? With regard to these questions, studies using global models (Delworth and Manabe 1988, 1989; Koster and Suarez 2001) and observational data (Vinnikov and Yeserkepova 1991; Vinnikov et al. 1996; Entin et al. 2000; Wu et al. 2002) have shown that soil moisture memory is significantly longer than most of the atmospheric processes associated with weather systems. Further, soil moisture memory significantly depends on the characteristics of atmospheric processes such as the statistics of precipitation. In particular, Delworth and Manabe (1988) demonstrated that short-time-scale precipitation fluctuations could be filtered to provide long-time-scale anomalies of soil moisture. They also indicated that the time scale of soil moisture anomalies in the absence of precipitation forcing is proportional to the ratio of field capacity of soil layer to the potential evaporation, and this time scale is identical to the decay time scale of soil moisture when evaporation is a linear function of soil moisture. This theoretical conclusion was confirmed by using an empirical dataset in the Soviet Union (Vinnikov and Yeserkepova 1991).

In this paper, we introduce a three-layer model as an extension of the simple bucket model. It is simple enough to allow rigorous mathematical analysis and realistic enough to explain the important results from previous global modeling studies and the analysis of observational data. Section 2 analyzes the time-scale variations of precipitation and soil moisture in the one-layer model and improves the original theory of Delworth and Manabe (1988). Section 3 extends the one-layer model to a three-layer model. Section 4 develops theoretical methods suitable for multilayer models and describes their application to the analyses of temporal-vertical structures in the three-layer model. Section 5 provides the design of numerical simulations. Simulation results and their comparisons with theoretical analyses and previous studies in sections 2 and 4 are shown in section 6. Conclusions are given in section 7.

2. Time-scale analysis of one-layer model

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

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

To quantify the temporal variations of land surface components, 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 temporal variation of water stored in vegetation and soil for the given atmosphere–land condition in the absence of rainfall. Delworth and Manabe (1988) took W* and E* as the field capacity and the potential evaporation (or the potential evapotranspiration when the soil surface is covered by vegetation) respectively, and further assumed that evaporation is proportional to W/W* if the soil is not too wet. They also demonstrated that T* corresponds to the e-folding time for the damping of W in the linear system. The response time scale determined by this method is only a first-order approximation and should be further improved in the nonlinear system. In the following, T* denotes an inherent time scale since it does not depend on atmospheric forcing, and the e-folding denotes time as response time scale.

Under precipitation forcing, there is another time scale related to W variations in response to the external input, that is, precipitation P(t). Consider a highly idealized precipitation time series

where P_ω is the amplitude and ω is the frequency. It is assumed that E is a monotonic function of W. Substituting (2) into (1), it can be shown that in the case of 2π/ω < T*, the temporal variation of W cannot effectively adjust to such high-frequency input, and it maintains its inherent variation with time scale T* superimposed upon small fluctuations with time scale 2π/ω. However, in the case of 2π/ω ≥ T*, after the initial signal is sufficiently decayed, both W and E + R are mainly represented as forced oscillations with low frequency.

In general, P(t) can be described by its spectral distribution

The above analyses indicate that when the spectra of P(t) are in the high-frequency domain, that is, ω > 2π/T*, E (or E + R) also has almost the same spectral distribution as P_ω, and the high-frequency components in W are negligible. For the low-frequency components of P(t), that is, ω ≤ 2π/T*, W, E, and R all respond to P in their evolutionary processes.

From the above analyses, (1) can be taken as a “low pass” filter. The response of W to low-frequency inputs is more effective, and W always possesses low-frequency components even though the spectra of P are concentrated in the high-frequency domain. Therefore, the high-frequency input P leads to a statistical equilibrium state with small amplitude but fast fluctuations of E (or E + R) and relatively large amplitude but low-frequency harmonics of W. This is the major mechanism for the reddening of the response spectra relative to the high-frequency input.

If E + R is a linear function of W, the above analyses can be summarized by an analytical and compact mathematical formula. This has been provided in Delworth and Manabe (1988) and Dickinson et al. (2003) based on the concept and method developed by Hasselman (1976) in the design of statistical climate model. However, no compact analytical formula has been reported in the past for the general nonlinear case unless E is locally linearized.

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

With slight modification of the formula of Serafini and Sud (1987), the evaporation E from the available soil moisture W within a single column of vegetation and soil can be given as

where y ≡ W/W*, W* is the maximum available soil moisture (the soil moisture difference between the field capacity and wilting point of the soil), E* is the potential evapotranspiration, and the free parameter σ is usually 1–3 depending on vegetation types. The function h(y) can be linearized as h(y) ≈ h(y) + h₀(y)y′ with y′ ≡ W ′/W* and h₀(y) = dh(y)/dy, and the mean value, E, can be written as

where y ≡ W/W*.

Considering E = E + E′, we have

where α = σe^−σy/(1 − e^−σ). Denoting W as W** and the corresponding actual response time scale as T** for the convenience in the comparison, we have

While T* is a constant, T** varies with y and T** ≈ T* only when y is intermediate. Let W ′ and P′ be represented in the spectral form with amplitudes A_ω and A_p respectively along with the frequency ω and neglecting R′, then (4) yields

This is the same equation as obtained by Delworth and Manabe (1988) and Dickinson et al. (2003) except with T** replacing T*. Since α and T** depend on y, the separating frequency ω = (T**)⁻¹ is a slowly varying quantity in the nonlinear case.

3. A three-layer model

Water stored in different layers of the vegetation–soil system has different temporal variations for the precipitation forcing, and the one-layer model in (1) cannot quantitatively distinguish the response time scales. To adequately represent this complexity, a model should include 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 we develop a modified three-layer water balance model as a direct extension of (1) and (4) rather than as a simplification of current comprehensive and complicated land surface models (e.g., Zeng et al. 2002). Water in the top two layers directly interacts with the atmosphere. The bottom layer includes most of vegetation roots, and its water content is thus the main source for plant transpiration. Based on water mass conservation, the governing equations are

where W_i, P_i, and E_i, respectively, denote the water content, precipitation partition, and evapotranspiration (evaporation and transpiration) in different layers; R_c is the outflow from canopy; R_s includes two parts: surface runoff (1 − ν)R_s and infiltration from s layer to r layer νR_s. Here ν (0 ≤ ν ≤ 1) is a fraction depending on soil texture; R_r is the root zone drainage; and Q_sr represents the conductive (diffusive) transfer of water between the s layer and r layer.

Precipitation P is partitioned into three parts P_i (I = c, s, r), in which P_c = min(P*_c, P) is the part intercepted by canopy, P_s = min(P*_s, P − P_c) is the part held by surface soil layer, and P_r = P − P_c − P_s is the part that directly penetrates into root zone of soil in the case of heavy rain due to the macropores in the soil. Here, P*_c and P*_s are the maximum precipitation that c layer and s layer can hold. They depend on the vegetation characteristics and soil properties. Note that the P_r term is also a kind of infiltration but different from νR_s since they are from different sources and have different spectral distributions. The introduction of the P_r term implies that the direct influence of atmosphere to deep soil is taken into account in the model. Further discussion of P_r will be given in section 6.

Applying (5) to the three layers yields

where E*_i is the potential evapotranspiration, W_i* is upper bound of water content (field capacity) of ith layer, y_i ≡ W_i/W*_i, and σ_i could be different in different layers. In (10) and (13), E_c includes evaporation from wet leaves and transpiration from dry foliage, E_s is the evaporation from surface layer of soil, and E_r is the soil water extracted by roots and transferred to the canopy. Basically, E_r directly becomes transpiration in most cases, but some part of E_r can be also stored in the leaf in some cases.

The interaction term Q_sr between two soil layers can be parameterized as

where D*_s and D*_r are the thickness of s layer and r layer, respectively, D_sr ≡ D*_sD*_r/(D*_s + D*_r), and λ is the damping time scale. In the absence of the sources and sinks (i.e., P, E, and R are all absent), W_s/D*_s−W_r/D*_r decays to e⁻¹ of its initial value during λ time due to the conductive transfer Q_sr. It is also assumed that W*_s/D*_s = W*_r/D*_r, which implies that there is no conductive interaction when volumetric soil water contents in both layers are the same.

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

where W ≡ W_c + W_s + W_r, E ≡ E_c + E_s, and R ≡ (1 − ν)R_s + R_r. Equation (15) is equivalent to the one layer-model governed by (1).

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

In this three-layer model there are several characteristic time scales: an external forcing (P) time scale T*_f , an internal time scale T*_q determined by the conductive transport Q_sr, and three inherent time scales T*_i (i = c, s, and r), which is similar to T* in section 2. Furthermore, we can even define three other time scales T*_fi due to the partitioning of P into P_i (i = c, s, and r). It is obvious that T*_fc is the same as T*_f , and T*_fs and T*_fr depend on the spectral distribution of P_s and P_r, respectively. These multiple time scales imply that the temporal variation of the three-layer model is more complicated (than the one-layer model) and can be irregular.

Similar to the one-layer model, the first-order approximation of the inherent time scales T*_i for ith layer can be taken as

For the same atmospheric conditions, E*_i is of the same order of magnitude, but this does not mean E_c ≈ E_s ≈ E_r. On the other hand, W*_c is two orders of magnitude smaller 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:

From (16) and (17), we can derive the following relationship:

which means the one-layer model would shift some part of low-frequency variations from root zone into surface layer. Roughly, the time scale T*_q = λ from (14) is about 10–15 days, T*_s 10–30 days, and T*_r 2–6 months.

Equations (11) and (12) indicate that the temporal variation of W_s and W_r is affected by E_s and E_r as well as other processes. Therefore the response time scale of W_s and W_r can be different from T*_s and T*_r, respectively.

Also note that 1) the existence of the c layer does not directly affect the hydrological dynamics in the s layer and the r layer except reducing the precipitation available to the soil from P to P − P_c + R_c and (P_c − R_c)/P is very small; 2) the r layer directly affects the c layer by the water supply through E_r; and 3) the s layer and r layer are mutually interactive through conductive transfer and infiltration of water. Therefore we can modify the inherent time scales for s layer and r layer by the use of only (11) and (12) with a described P_s + R_c.

In the following analytical investigation of time scales, for generality, runoff R_s and R_r are parameterized as

where R*_i is the maximum value (upper bound) of R_i, y_i ≡ W_i/W*_i and g_i (0 ≤ g_i ≤ 1) is a continuous and monotonical function of y_i so that runoff is still allowed within unsaturated soil. Equations (11) and (12) then are locally linearized with respect to long time means (W_i, P_i), i = s, r. Thus we have the following nondimensional equations:

where

All the coefficients in the right-hand side are dimensionless, M_oi is the moisture index,

and

where

Each dimensionless parameter has a definite physical meaning. For instance, ϕ_rs is the ratio of potential evaporation 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 we transform them into two equations, each of which consists of one variable only:

where

In these equations −α₁ and −α₂ are the two eigenvalues of the matrix

and (α_1s, α_1r) and (α_2s, α_2r) are the two corresponding normalized eigenvectors, respectively. In general all α_j, α_js, and α_jr are real numbers, and α_j > 0, (j = 1, 2). From (20), (21), (24), and (25), the coefficients α_j (j = 1, 2) can be solved

The e-folding dimensionless time scale for z′_i is α_i⁻¹ (i = 1, 2), and the two response time scales in (29) are

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, various time scales and parameters can be computed, as given in Tables 2 and 3. Because α_2r is much larger than α_2s in (30), T₂** effectively provides a new estimate of the time scale of the r layer, and based on numerical simulations, it is better than T*_r from (16) (see section 6). On the other hand, the difference between T₁** and T*_s is small (see Table 2); both could represent the time scale of the s layer. It needs to be emphasized that while T*_s and T*_r in (16) are independent of precipitation input, our new time scale T₂** does depend upon the mean soil water [based on (26)–(28) and (33)] and hence precipitation.

Now we investigate the response of soil moisture anomaly in (29)–(31) to the input of precipitation anomaly in (22). Assume

where A_πs and A_πr are real numbers. Then (31) can be rewritten as

Equations (29) and (36) yield

where A_fj and A_zj are taken as real numbers:

Evidently, (38) is the direct extension of (9). Assuming π′_s/π′_r = P_s/P_r, and taking ω as the time scale of 2T₁**, we have ψ₁ = 0.46 and ψ₂ = 1.44 from (39). Taking ω as the time scale of 1.5T₂**, we have ψ₁ = 0.04 and ψ₂ = 0.59. In either case, T₂** > T₁** (Table 2), and hence ψ₂ > ψ₁ in agreement with the data analysis of Wu et al. (2002).

5. The design of numerical experiments

For illustration we have carried out some numerical experiments by using the three-layer model. Daily precipitation is the only model-required external input. For simplicity, the probability density function (PDF) of daily precipitation intensity in rainy days is represented by an exponential function

where p is the daily precipitation, and p̃ is the prescribed mean daily precipitation for the rainy days. It is also assumed that rainfall occurs 50% of the days over a typical midlatitude grassland region, so that the climatological mean p = p̃/2. For instance, (40) is found to reasonably fit the observed summer precipitation over Illinois (not shown). Under these assumptions, the random precipitation time series is generated. For other regions (e.g., monsoon regions), both the occurrence of precipitation (OP) in (40) and p̃ need to be adjusted. Furthermore, similar to Delworth and Manabe (1988), seasonal variation of precipitation is not considered here by taking a constant p.

Mean daily precipitation p and other required parameters in the numerical experiments are listed in Table 1. The time step is taken as 0.1 day, but we focus on the analysis of daily averaged evaporations and water contents for each layer. The model sensitivity to initial conditions is also studied by employing “dry” and “wet” initial conditions respectively. The initial water contents of all the three layers are set as slightly larger than zero in the dry condition, and as their saturated values in the wet condition. After the model has run for a long enough time (approximately after 12 months), the time series of water contents in each layer converge in these two simulations. Without loss of generality, in the following simulations the initial condition is set as wet, and the daily model output after the first 12 months is analyzed.

In our numerical simulations, only the runoff from oversaturation is considered. In other words, g_i = 1 for y_i ≥ 1 and g_i = 0 for y_i < 1 in (19). This is similar to the assumption used in Noilhan and Planton (1989). Sensitivity of our results to the runoff formulation will also be briefly addressed later.

Two statistical methods (i.e., spectral analysis and autocorrelation analysis) are adopted to analyze the temporal variations of the time series. The first one is used to analyze the spectra of precipitation, water content, and evaporation for different layers. The second is used to study the sensitivity of the time scale of soil moisture in root zone to some model parameters.

Delworth and Manabe (1988) used the first-order Markov process to demonstrate soil moisture memory when they modeled soil moisture variability in the Geophysical Fluid Dynamics Laboratory’s (GFDL) atmospheric general circulation model (AGCM). Even though the Markovian framework has some limitations (Koster and Suarez 2001), it captures the basic feature of soil moisture as a red-noise process responding to the short-term atmosphere forcing and provides a single parameter as the measure of soil moisture memory (Wu and Dickinson 2004).

By using the least squares method, the autocorrelation function r(t) of the simulated time series of soil moisture can be approximated as (Delworth and Manabe 1988; Vinnikov and Yeserkepova 1991; Vinnikov et al. 1996)

where t is the time lag, and T is the e-folding time scale. By using linearized equations, the temporal variation of soil moisture can be represented as a first-order Markov 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 moisture memory, it is empirically obtained from the soil moisture time series (rather than from land surface parameters). In contrast, inherent time scales [e.g., (16)] and response time scale [e.g., (34)] discussed in this paper are derived based on climate conditions and land parameters at a given region and hence have clear physical meanings. In the next section, T₂** for (34) and T*_r for (16) will be compared to see which is closer to T_r of the root zone soil water.

6. Analyses of the numerical results

7. Conclusions

Through theoretical analyses and numerical simulations of a three-layer soil–vegetation interaction model, this paper has investigated how land surface hydrology responds to precipitation forcing. The major results are as follows: (a) Different water reservoirs of the vegetation–soil system have different time scales, including very short time scales of canopy water, an intermediate time scale of soil moisture in the surface soil layer, and long time scale of root zone soil moisture. This is consistent with the theoretical conclusion of Dickinson et al. (2003), but cannot be represented properly by the one-layer bucket model. (b) Precipitation forcing is mainly concentrated on short time scales with some small components at low frequencies. However, it can cause the long time-scale disturbances in the soil moisture of root zone, because soil moisture responds more effectively to the low-frequency parts of precipitation spectra, and hence acts as a low-pass filter. (c) The inherent time scale determined by the ratio of the field capacity to the potential evapotranspiration is only a first-order approximation for the response time scale (as defined from the time series analysis). Our new time scale T₂** in (34) considers the vertical interactions between layers as well as the impact of mean soil water (and hence precipitation input), and is indeed closer to the actual response time scale of the root zone soil water (T_r) than the inherent time scale (T*_r). While T_r can be empirically computed from the soil water time series, it does not provide a clear physical interpretation. In contrast, T₂** is derived directly from our theoretical analysis, and it does provide a physical interpretation of the soil water response time. (d) Comparison of this three-layer model with the one-layer bucket model indicates that the time scale of evapotranspiration is quite different although the time scale of soil moisture itself is very close to each other. This suggests the need to consider the vertical structure in water reservoirs in land surface modeling. (e) The three time scales (i.e., T_r, T₂**, and T*_r) all increase with soil depth. The first two also increase with the characteristic interaction time scale (λ) between two soil layers (particularly when λ is small), while T*_r is independent of λ.

While the median T_r is not very sensitive to most model parameters or formulations, it does strongly depend on the mean precipitation rate p (particularly when p is increased). Therefore it needs to be emphasized that the quantitative results from numerical simulations in section 6 are valid only over midlatitude grassland and they may be different over different regions (e.g., under wetter conditions). However, the theoretical framework and the overall conclusion that T₂** better represents T_r than T*_r are expected to be valid over different regions.

In this study, we focus on the land surface response to precipitation forcing. When a land model is coupled to an AGCM, the time scales of soil moisture can be different from those in our uncoupled model due to the mutual interactions between land and atmosphere via the atmospheric boundary layer. Dickinson (2000) demonstrated the difference of the time scales for the variability of coupled and uncoupled systems, and concluded that coupling to the atmosphere lengthens the time scale of land. Also missing in our model, just as in many 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), horizontal interactions would introduce a new time scale. Further study is needed to determine this time scales as well as the impact of horizontal interactions on other time scales discussed in this paper.