Abstract
[1] Riparian vegetation is part of one of the most diverse and fragile ecotones. The key role played by river discharge on the dynamics of riparian vegetation has been widely studied and documented. However, although randomness is a fundamental characteristic of river hydrology, very few quantitative vegetation studies take into account the random nature of river discharge. Here we propose a stochastic model of riparian vegetation ecosystem dynamics forced by random variations in river discharge. The model is solved, and the analytical expressions of the probability density function of the overall vegetation biomass and its first moments are obtained. These theoretical results are used to investigate the effect of river hydrology on the distribution of vegetation along the riparian transect transverse to the river. In particular, the influence of the type of riparian species and the statistical characteristics of discharge time series are discussed and compared with field observations. 
Introduction
[2] Riparian vegetation is a type of plant community growing close to river banks [Naiman and Decamps, 1997; Hughes, 1997; Mitsch and Gosselink, 2000] . It is part of a complex ecotone, called the riparian zone, which is ''a transitional semiterrestrial area regularly influenced by fresh water, usually extending from the edges of water bodies to the edges of upland communities'' [Naiman et al., 2005, p. 2] . Unlike upland vegetation, riparian vegetation is greatly influenced by the fluvial hydrological regime, through the control exerted by the river on water table depth, flooding, and hyporheic fluxes [Mitsch and Gosselink, 2000; Jones and Mulholland, 2000] . The spatial patterns of riparian vegetation are an evident indication of this strong influence: sparse vegetation is mainly associated with high river discharge variability, e.g., the vegetation is removed by intense floods or wilt during dry periods [Nanson et al., 2002] , while uniform vegetation is instead typical of more regular river flow conditions.
Introduction
[3] In recent years, several field studies have investigated the sensitivity of the riparian vegetation to river-induced disturbances. For example, Osterkamp and Hupp [1984] and Hupp and Osterkamp [1985] investigated the relationships between riparian vegetation patterns and fluvial landforms, Mahoney and Rood [1998] identified the geomorphological features of locations in which seedlings are most likely to germinate and survive, and Bendix and Hupp [2000] observed that the species are sorted along gradients in water table depth and unit stream power. Johnson [2000] carried out an extensive demographic analysis of the effects of river discharge on tree recruitment and seedling mortality [see also Bradley and Smith, 1986; Auble et al., 1994; Friedman and Auble, 1999; Lite et al., 2005] .
Introduction
[4] These field works have clearly shown that flooding and the depth of the water table are the two fundamental mechanisms through which river hydrology controls the riparian vegetation evolution. Flood events cause the socalled inundation of a plot, during which a vegetated site is submerged by the stream. Vegetation may benefit from floods, which supply moisture, seeds and nutrients [Naiman and Decamps, 1997] , though floods are more often known for their negative impact on vegetation, due to physical damage [Yanosky, 1982] , uprooting and sediment removal [Osterkamp and Costa, 1987] , anoxia [Kozlowski, 1984; Stevens and Waring, 1985; Naumburg et al., 2005] , and burial [Hupp, 1988; Friedman and Auble, 1999] . An exposure period instead occurs when the free surface of the river is lower than a vegetated site. In these conditions, the river affects the growth rate of vegetation by controlling the groundwater flow [Scott et al., 1999] . This aspect is very important in semiarid area, e.g., the southwest of the United States and Mediterranean regions, which support phreatophytes that tap the groundwater. On the other hand, in humid areas the groundwater levels have little bearing on vegetation distribution. In the following, we will focus on the former areas, even though, as shown in section 4.1, humid areas can be considered as a particular case where only the influence of the inundation periods is retained.
Introduction
[5] Even though random variability is a key characteristic of the river hydrology which plays a crucial role in the evolution of riparian vegetation [Tockner et al., 2000; Stromberg, 2001; Steiger et al., 2005] , the impact of stochastic hydrologic fluctuations on the dynamics of riparian vegetation ecosystems remains for most part poorly understood [Scott et al., 1999; Lytle and Merritt, 2004] . Some conceptual qualitative models [McKenney et al., 1995; Richter and Richter, 2000; Gurnell et al., 2001] and regression analyses between vegetation growth and river discharge [Stromberg and Patten, 1991] , or inundation duration [Franz and Bazzaz, 1977; Auble et al., 1994] , account for some probabilistic aspects of hydrological variability (e.g., frequency, intensity, or duration of the inundation). However, these studies do not explicitly address riparian vegetation dynamics using an approach that is both process based and probabilistic. Moreover, the lack of analytical models limits the study of riparian vegetation to numerical model simulations [e.g., Pearlstine et al., 1985; Brookes et al., 2000; Lytle and Merritt, 2004] . If this allows a detailed description of the processes, it makes very difficult to catch general aspects nevertheless. Here we propose a new approach to study the distribution of phreatophyte riparian vegetation, using a stochastic process to model the random forcing exerted by river flow on the dynamics of the overall biomass of vegetation.
Introduction
[6] The purposes of the present work are (1) to formulate and solve a process-based stochastic model of riparian vegetation dynamics and (2) to investigate the role of the flow variability on the vegetation distribution along a riparian transect. To these aims, we use a minimalistic approach which accounts for the key hydrologic and ecosystem processes, while keeping the model mathematically tractable. A number of simplifications are adopted in the representation of hydrologic and vegetation dynamics. In particular, we maintain steady river morphology and neglected sediment erosion and deposition. Analytical solutions for the steady state probability density function of the vegetation biomass and its main statistics are obtained. Their dependence on the distance from the river, the transect topography, the statistical properties of the river discharge, and the type of vegetation are then elucidated. These analytical results are used to assess to what extent river hydrology is able to affect the distribution of riparian vegetation and explain observed patterns of vegetation distribution.
Stochastic Model
[7] The scope of our study is to describe the key processes of riparian vegetation dynamics focusing on the stochastic influence of the river. In order to obtain a representative but mathematically tractable model, some simplifying hypotheses are introduced. First, we refer to the overall vegetation biomass of phreatophyte riparian species, neglecting interspecific interactions [see also Lytle and Merritt, 2004; Perucca et al., 2006] . Second, we consider a steady river morphology and, then, geomorphological processes, such as sedimentation and erosion, are neglected [e.g., Auble et al., 1994] . It follows that the model is not able to describe the feedback mechanisms of the vegetation on the site topography. Finally, we neglect the time delay between the vertical movements of the free surface in the river and the water level in the adjacent unconfined aquifer beneath the riparian vegetation. This assumption is reasonable because the delay is generally shorter than the typical timescale of vegetation-groundwater interactions.
Stochastic Model
[8] The links between the river and the processes that affect age, species and community structure generally develop at the reach scale, namely between the bar form scale and the corridor scale [Richards et al., 2002] . Thus we assume, as a typical spatial scale of the problem, a transversal width of the riparian zone of the order of the mean half width of the river, b* (the star indicates dimensioned quantities), corresponding to the mean water level in the river (see Figure 1) . The results of the model are therefore also valid for a curved reach provided the curvature is much lower than 1/b*, a condition that is very commonly met in meandering rivers [e.g., Ikeda and Parker, 1989; Camporeale et al., 2005] . The temporal scales described in the model range from a day, to account for the effect of flooding, to several years, to allow for vegetation growth. As some simplifications affect the shortest scales, the outcomes of the model are interpreted in the long-term, by investigating the steady state conditions.
Stochastic Model
[9] Let us consider the generic riparian transect shown in Figure 1 and assume the minimum water level in the river (which can be different from zero) as the vertical datum. The free surface oscillates randomly above this level, according to the hydraulic and hydrologic characteristics of the river. The following dimensionless variables are introduced
Stochastic Model
where x* is the transversal coordinate, with origin in the intersection between the minimum water level and the river bed, w* is the river width referred to the x* coordinate, while h*, z*, and h* are the river water level, the phreatic surface position and the bed topography elevation, respectively, referring to the vertical datum (see Figure 1 ). The bar in relationships (1) indicates the time averaged values; therefore w* marks the mean transversal position of the bank (w* b* when the minimum river discharge is zero) and h* is the average water level in the river. The function h = h(x) gives the transect topography, while function z = z(x, h) describes the spatial shape of the phreatic surface and its dependence on the water level in the river. Both h(x) and z(x, h) are input data of the problem. As h is a time random variable, z(x, h) and the local depth of the phreatic surface, d(x, h) = h(x) À z(x, h), are also random variables.
Stochastic Model
[10] The randomness of the river water levels is described by the probability distribution function (pdf), p(h), and the autocorrelation function, r h (s*) = h t* ð Þh t* þ s* ð Þ/h 2 t* ð Þ, where s* is the time delay. This latter function is a key characteristic of the hydrology of a river [e.g., Bras, 1990; Maidment, 1993] and it can be summarized by the integral scale t*, defined as the area of the autocorrelation function r h (s*), i.e., t* = R 1 0 r h (s*) ds*. This scale can be interpreted as the ''memory'' of the river discharge time series.
Stochastic Model
[11] The riparian vegetation dynamics are described according to the following model
Stochastic Model
where v is the dimensionless vegetation biomass (v = 1 in optimum steady conditions), t* is time, V c = V c (d) is the dimensionless carrying capacity, i.e., the maximum sustainable biomass, which depends on the depth of the aquifer water table, d, and a 1 is a coefficient that describes the magnitude of the damage to the vegetation by the flood. Finally, the exponents (n, m, p) and the intrinsic rate of growth, a 2 [Kot, 2001] , are numerical constants that depend on the characteristics of the vegetation.
Stochastic Model
[12] Equation (2a) models the decay of the vegetation biomass caused by flooding and assumes that the eventual beneficial influences are overcome by the detrimental processes (i.e., anoxia, burial, uprooting, etc.), as is usual in riparian environments with constant channel morphology [Auble et al., 1994] . Accordingly, a decay timescale, T d *, can be defined as the time necessary for the vegetation to decline by 90% (i.e., from v = 0.95 to v = 0.05). The exponent, n, modulates vegetation response to streaminduced disturbances. For example, n = 1 implies an exponential decrease, and when n < 1 the models allows the resistance of vegetation to disturbance to increase with the age of vegetation [Friedman and Auble, 1999; Lytle and Merritt, 2004] . The coefficient a 1 is assumed to be an increasing function of h, with a 1 = 0 at h = h. In fact, higher water levels are associated with stronger vegetation stress, due both to mechanical disturbance and to anoxic conditions. Because the mechanical effect of the stream water on vegetation is proportional to the tangential stress on the bed [Friedman and Auble, 1999] , while anoxic conditions increase with the water level, both mechanisms are modeled linearly dependent on the submerged depth h À h,
Stochastic Model
where K is a positive empirical coefficient that depends on the vegetation type.
Stochastic Model
[13] Equation (2b) is a generalization of the commonly used Verhulst-logistic function [Hunt, 1982] and it simulates the growth of a phreatophyte species tapping the groundwater [Botkin et al., 1972; Liu and Ashton, 1995] . Suitable choices of the exponents m and p and the growth rate a 2 allow equation (2b) to fit a wide variety of mathematical formulations proposed to describe the vegetation growth. An example is the JABOWA forest growth equation derived from Botkin et al. [1972; see also Shugart and West, 1977; Pearlstine et al., 1985] . These authors modeled vegetation growth under optimum conditions (i.e., V c = const = 1), as
Stochastic Model
where # = D*/D* m is the dimensionless diameter, with D* being the vegetation diameter at breast height H* 0 (conventionally set to 137 cm) and D* m its maximum value. H* m is the maximum vegetation height, and G and q are two parameters. Generally, G is taken in a way that the vegetation grows to 2H* m /3 at one-half of its maximum age, T* m , while q is an allometric exponent that depends on the species and site conditions and varies between 1.5 and 3 [Baskerville, 1965; Perry et al., 1969; Young et al., 1980] . Values of H* m , D* m , and T* m are listed in Table 1 for some typical riparian species.
Stochastic Model
[14] The generalized logistic equation (2b) fits very well the temporal behavior described by the differential equation (4). For example, Figure 2 shows a comparison between four different Botkin growth curves and the corresponding curves fitted to equation (2b). The values of a 2 obtained through curve fitting (fitting errors <10%) are reported in Table 1 for a number of riparian species in the case q = 2.5 and m = p = 1. Because of the versatility of equation (2b), other laws of vegetation growth can also be fitted by (2b) [e.g., Liu and Ashton, 1995] .
Stochastic Model
[15] The carrying capacity, V c , depends on the depth of water table: phreatophyte species generally have a maximum growth (i.e., V c = 1) when the water table is equal to an optimum depth, d* opt , while the water uptake capacity is significantly reduced for groundwater levels that are either too low or too high (e.g., for Salix Alba d* opt = 0.5 m, see section 4.2). In fact, because deeper groundwater is out of reach of tap roots, xylem cavitation and stomal closure occur [Naumburg et al., 2005] . On the other hand, shallow water tables are associated with water logging conditions and reduced respiration and gas exchange in the root zone [Kozlowski, 1984; Ridolfi et al., 2006] . It follows that there , that allows vegetation growth, while outside this range, plants are not able to survive. Coherently with this picture, the (dimensionless) carrying capacity reaches its maximum value (V c = 1) at the optimum depth, d* opt , and is zero outside the previously mentioned interval. We have modeled this behavior with a parabolic function [see also Phipps, 1979; Pearlstine et al., 1985] , which in the present dimensionless framework, reads as
Stochastic Model
with d opt = d opt * /h* and a = a * h* 2 , where a* is a constant that we take equal to 0.055 [e.g., Phipps, 1979] . Consequently,
Stochastic Model
[16] Through the dependence V c = V c (d) (i.e., equation (5)), model (2) accounts for the decay of vegetation biomass, due to water stress driven by changes in the phreatic water table, in addition to the decay induced by floods (equation (2a)). Considering the optimum condition, V c = 1, the growth timescale T* g is here defined as the time necessary for vegetation to undergo an overall growth, from v = 0.05 to v = 0.95 [Perucca et al., 2006] .
Stochastic Model
[17] The randomness of the hydrological forcing acts on the vegetation dynamics (equation (2)) in two different ways. First, the sign of hÀh controls the switching between equations (2a) and (2b), i.e., phases of inundation are alternated with phases of exposure. The statistical characteristics of the switching depend on the probability distribution and autocorrelation function, p(h) and r h (s*), of h. Second, both quantities a 1 and V c depend on the random variable h. Thus the model (2) -(5) describes the stochastic evolution of riparian vegetation at any point of a transversal transect for given probabilistic properties of the river flow (i.e., p(h) and r h (s*)), vegetation characteristics (n, m, p, K, a 2 , d 1 , and d 2 ), section geometry (h(x)), and geometry of the phreatic surface (z(x, h)).
Analytical Solution
[18] In order to obtain the analytical expression of the steady state pdf of the vegetation biomass, the functions a 1 = a 1 (h À h) and V c = V c (d) are approximated in any plot with their x-dependent probability-averaged values, that is,
Analytical Solution
where
Analytical Solution
are the probabilities of inundation and exposure, respectively. Since the pdf of the river levels, p(h), the crosssection geometry of the bank, h(x), and the phreatic surface, z(x, h), are the input data of the problem, the quantities ha 1 i and hV c i can be evaluated. These depend on (1) the coordinate x along the transect, (2) the probability distribution of water levels, p(h), and (3) the type of vegetation, through the parameters K, d opt and a. Once p(h) is assigned, the exposure probability, P E , is univocally linked to x, i.e., P E can be interpreted as a surrogate variable of the lateral coordinate, x. Finally, P I is the dimensionless fractional time in which a site at the bottomland is inundated, and is also known as the inundation duration The fitting parameters are also reported. Here m = p = 1; q = 2.5.
Analytical Solution
or flow duration by field workers [e.g., Auble et al., 1994; Hupp and Osterkamp, 1985] .
Analytical Solution
[19] By introducing relationships (7) and (8) into model (2a) and (2b), then dividing by a 2 , and scaling the time according to t = a 2 t* (and then t = a 2 t*), the following analytically more tractable model is obtained
Analytical Solution
are two known quantities that depend on position of the site along the transect. At the end of this section, it will be numerically verified how the approximations contained in model (9) do not significatively alter the steady state solution with respect to the original model (2).
Analytical Solution
[20] Model (9) retains the influence of the random oscillations of the river water level on the vegetation dynamics in the switching between decay phases and growth phases, that alternatively occur in a random (but correlated) way. Therefore, because of the stochastic character of the forcing, the outcome of model (9), herein adopted to describe the riparian vegetation biomass dynamics, is not a deterministic value but a random quantity described by a probability distribution function. The stochastic character of the switching permits model (9) to be expressed by a single stochastic differential equation
Analytical Solution
where x(t) is a dichotomic Markov process [Van Kampen, 1992] that switches between values D I (inundation) and D E (exposure), and functions f(v) and g(v) are such that equation (11) reduces to equations (9a) and (9b) depending on the value of x(t). This yields
Analytical Solution
[21] Thus we can use the theory of stochastic differential equations driven by multiplicative dichotomic noise [Kitahara et al., 1980; Van Den Broeck, 1983 ] to investigate the dynamics of riparian vegetation. To this aim and without any loss of generality, it can be assumed that the dichotomic noise, x(t), has a vanishing average value. Consequently, the average durations of the inundation and exposure periods (T I and T E , respectively) satisfy equations [Van Den Broeck, 1983 ]
Analytical Solution
where P DE and P DI (with P DI + P DE = 1) indicate the probability of the dichotomic process of being in state D E or D I , respectively, and t DP is its integral scale. Without any loss of generality, in the following we set D I = 1.
Analytical Solution
[22] The dichotomic noise, x(t), results to be characterized by only two independent parameters, that we chose to be t DP and P DI . In order for the noise to capture the stochastic structure of the hydrological forcing correctly we assume
Analytical Solution
namely the memory of the noise is the same as in the river discharge time series and the probability that x(t) = D I (i.e., model (11) coincides with equation (9a)) is equal to the inundation probability (hence P DE = P E ).
Analytical Solution
[23] The solution of the Fokker-Plank equation corresponding to (11) is the pdf of the vegetation density, P(v, t). Its steady state function (a fundamental property of the Fokker-Plank equation is that all solutions tend to the stationary solution when t ! 1 [Van Kampen, 1992, p. 104]) p(v) = P(v, t ! 1) can be obtained according to Kitahara et al. [1980] and reads
Analytical Solution
where N is a normalization constant (i.e., R 1 0 p(v)dv = 1) and
Analytical Solution
[24] The solution of the integral in equation (16) differs according to the value of the coefficients (n, m, p). In the following, we focus on the case that corresponds to the usual choice m = n = p = 1, which refers to the original Verhlust logistic model for growth and the exponential form for decay. Other cases are reported in Appendix A. By substituting equations (12) and (13) in (16) we obtain the steady state pdf of the vegetation biomass
Analytical Solution
with v 2 [0, b]. It is worth recalling that the pdf of the vegetation biomass depends on the transversal position, x, through the inundation probability, P I , the coefficient a, and the expected value of the carrying capacity, b.
Analytical Solution
Expression (18) is valid provided
Analytical Solution
otherwise N diverges and p(v) tends to d(v), where d(Á) is the Dirac delta function. Since a, b and P I depend on the bed elevation h through equations (7) and (8a), it results that condition (19) can be used to find the lowest topographic limit of the vegetated zone, and the extension of the riparian zone. It is interesting to observe the such limits do not depend on the correlation time, t.
Analytical Solution
[25] Relationship (18) allows one to obtain the analytical expression of its first four moments (i.e., mean, standard deviation, skewness and kurtosis), that are useful to summarize the main statistical characteristics of the stochastic outcome of model (9). For example, again for the case of m = n = p = 1 and under condition (19), the expected value is
Analytical Solution
where
Analytical Solution
whereas G[Á] is the Gamma function [Abramowitz and Stegun, 1965] . The expression of the standard deviation, s v , is given in Appendix B, while the third-and fourth-order moments are not reported due to space limitations.
Analytical Solution
[26] To obtain the analytical steady state solution of the stochastic model the time-dependent terms, V c = V c (d(t)) and a 1 = a 1 (h(t) À h), have been approximated with the respective average values defined by relationships (7). The influence of such approximations was tested by comparing the analytical results with the numerical simulation of the original model (2), and by investigating a wide range of hydrological and ecological conditions. Figures 3 and 4 show an example of these comparisons. This example refers to the case with n = m = p = 1, considering Salix Nigra as the riparian species, but the same behavior has been observed for all the other choices of the parameters. In particular, Figure 3 shows four comparisons of probability density functions of the vegetation biomass for different values of the inundation probability, P I , while the scatterplot reported in Figure 4 compares the predicted and simulated expected values m v (C h = m h /s h indicates the coefficient of variation of the water level). Figures 3 and 4 show an excellent agreement between model (2) and the simplified analytical model (9).
Riparian Vegetation Distribution Along a Transect
[27] The analytical framework developed in the previous section can be used to investigate the transversal distribution of the riparian vegetation. Two examples are shown in this section. We first discuss a simplified case where topography and river hydrology do not need to be specified; in this way, some general features of the vegetation distribution are discussed (section 4.1). We then consider a more specific case that refers to a class of rivers characterized by a quasi-trapezoidal transversal section (section 4.2). In both cases we assume n = m = p = 1.
Riparian Vegetation Distribution Along a Transect
[28] In the following, to avoid arbitrariness in setting a numeric value for K, we relate the ratio k = K/a 2 to the ratio of vegetation timescales, T* g and T* d . Thus, under the above assumptions, considering hh À hi of order of magnitude of 1, and recalling the definition of T* d and T* g (see section 2), we obtain
Riparian Vegetation Distribution Along a Transect
namely the ratio between the decay and the growth coefficients is equal to half the ratio of the corresponding timescales. Generally, T* g is greater than T* d , thus k > 2. 4.1. Case With a = k and b = 1 [29] Let us assume that neither the decay rate (a 1 ) nor the carrying capacity (V c ) depend on the transversal position, x. From an ecological point of view, these conditions correspond to vegetation which (1) suffers from inundation, regardless of the magnitude of the floods and (2) tolerates large excursions of the phreatic surface. This implies that the carrying capacity is always close to the optimum value regardless of the groundwater level, as in the case of riparian species in humid areas which do not tap groundwater. It follows that a 1 = K = const, V c = 1, and therefore a = k and b = 1. Coherently, model (9a) and (9b) becomes
Riparian Vegetation Distribution Along a Transect
where the stochastic hydrological forcing only influences the switching between the evolution equations, through P I , P E , and t, without affecting the value of the parameters of (25).
Riparian Vegetation Distribution Along a Transect
[30] The steady state pdf of model (25) and the corresponding moments are the same as those deduced in the previous section, with a = k and b = 1. In this case, because p(v) retains its dependence on x only through P E (or P I ), the statistics of v can be directly plotted as a function of the exposure probability, P E , which is a hydrologically based proxy for the transversal coordinate, x. In this way, the effect of the flow variability on the vegetation distribution along the transect can be described in general form, without any knowledge of the details of p(h).
Riparian Vegetation Distribution Along a Transect
[31] Figure 5 shows the dependence of the mean, m v , and standard deviation, s v , of vegetation biomass on the exposure probability, P E . It is observed that the integral scale of the water levels, t, does not significantly influence m v . The latter instead depends to a great extent on the ratio of decay and growth rate, k, and the exposure probability, P E . Moreover, the soil remains unvegetated until P E exceeds a threshold value that depends only on (i.e., increases with) k. Thus the exposure probability needs to exceed a threshold value for the growth phases to significantly affect vegetation dynamics. The high value of the threshold is due to the fact that the typical timescales of the vegetation decay are much shorter than the timescales of growth, as suggested by values of k greater than 2. For the same reason, even though in this case V c is constant (V c = 1) for any value of x, m v = 1 only for P E = 1, namely very far from the river. In fact, the random occurrence of even rare inundation events is always able to temporarily destroy part of the vegetation.
Riparian Vegetation Distribution Along a Transect
[32] The uniformity along the transect of the averaged value of the carrying capacity, b, implies a monotonic increase in the vegetation biomass, with the distance from the river. On the other hand, s v is never a monotonic function of the distance, regardless of the choice of the parameters. In fact, both for low (i.e., very close to the bank) and high values of P E (i.e., far from the bank), p(v) tends to a Dirac delta function, with s v = 0 and mean m = 0 and m = 1, respectively.
Quasi-trapezoidal Riparian Transects
[33] Let us consider the quasi-trapezoidal river section shown in Figure 6 , which schematizes the transversal topography that can often be observed along alluvial rivers. To obtain an analytical expression of the probability distribution, p(h), of the hydrologic forcing the following hypotheses are made (but a numerical form of p(h) can be obtained for more complicated conditions). (1) Because the timescale of the flow variability is much longer than the hydrodynamic timescale, the flow can be considered quasisteady. It follows that the local depth-averaged velocity satisfies Chezy's law throughout the cross section [Henderson, 2) two constant values of Manning friction coefficient (n r and n f ) are assumed for the central zone of the river bed and the riparian zone, respectively [Pettit et al., 2001] ; (3) the phreatic surface is assumed to be horizontal, so that z = h. Although not required by our analytical approach, this third assumption is justified by the relatively small slopes of the water table in the transverse direction.
Quasi-trapezoidal Riparian Transects
[34] We assume that flow discharge, Q, follows a lognormal distribution, p(Q) [Bras, 1990; Maidment, 1993] . Without any loss of generality, we define
Quasi-trapezoidal Riparian Transects
where the subscript ''0'' refers to the minimum water levels (i.e., h* = 0), c is a coefficient greater than unity, and Q, s Q , and C Q are the mean, standard deviation, and coefficient of variation of the river discharges, respectively. As p(Q) is lognormal, the pdf of the (dimensioned) water level in the river is
Quasi-trapezoidal Riparian Transects
where
Quasi-trapezoidal Riparian Transects
[35] Under the previously mentioned hydraulic hypotheses, the following relationship between the discharge and the water table level can be obtained
Quasi-trapezoidal Riparian Transects
where I t is the slope of the riparian transect, I r is the longitudinal river slope, y* 0 is the stream depth at the minimum water conditions, and b* 0 is the half width of the central zone of the river bed where the minimum discharge flows (see Figure 6 ). Moreover, from the dimensionless notation (1) in the present case, we obtain h*/x* = h/ x = I t and hence h x.
Quasi-trapezoidal Riparian Transects
[36] By substituting equation (29) in (27) we obtain
Quasi-trapezoidal Riparian Transects
[37] A sensitivity analysis was performed to test the dependence of equation (30) (2) the pdf of dimensionless water levels, p(h), is very well represented by a standard Gamma distribution which, using dimensionless variables, reads
Quasi-trapezoidal Riparian Transects
where l = 1/C h 2 . The latter point was proven by means the computation of the norm
Quasi-trapezoidal Riparian Transects
which gives an indication of the distance between functions p(h) and }(h). We always obtained jp, }j < 0.01 for each parameter combination, and therefore, for the sake of simplicity, we used the Gamma distribution in place of the actual pdf (30). As a consequence, p(h) ' }(h) results to be Figure 6 . Sketch of the cross section of a quasi-trapezoidal transect.
Quasi-trapezoidal Riparian Transects
only described by C h , which, along with the integral scale, t, completes the set of parameters and functions needed to describe the hydrologic forcing.
Quasi-trapezoidal Riparian Transects
[38] The transversal distributions of m v and s v are shown in Figures 7a -7d along with the dependence on hydrologic (C h , t) and biological (k, a) parameters that affect the dynamics.
Quasi-trapezoidal Riparian Transects
[39] The average value of vegetation biomass varies along the x direction from zero close to the river, up to a maximum value and, then, it slowly decreases asymptotically approaching the average value of carrying capacity, b(x), at high x. This behavior differs from the simplified case shown in Figure 5 due to two different factors. On the one hand, the monotonic decrease in the carrying capacity along x direction forces vegetation biomass, v, to decrease with increasing distances from the river bank. On the other hand, inundations frequently destroy vegetation close to the river. It follows that a maximum of the average value of the vegetation biomass will occur at a particular location, x = x M . Conversely, the dependence of s v on x (see Figure 7) maintains the same features discussed in section 4.1, apart from the fact that some curves lose the symmetry shown in Figure 5 because of the nonlinear transformation between P E and x.
Quasi-trapezoidal Riparian Transects
[40] As far as the role of the hydrologic parameters is concerned, Figures 7a and 7b show the effect of the coefficient of variation, C h , and the autocorrelation time, t, of the water levels, respectively. C h affects both the location of the vegetated zone and the peak value; in particular, the riparian vegetation shifts outward and the overall mean amount of riparian biomass (i.e., R 1 À1 m v dx) decreases when the discharge variability increases, because of the increasing occurrence of inundation events. Instead, the maximum value of s v remains almost constant. Conversely, different correlations do not lead to significant changes in the mean value of p(v), but significantly affect its shape, as shown by the standard deviation. In particular, s v increases with the autocorrelation of the hydrologic forcing. In fact, an increase in t corresponds to an increase in the mean duration of exposure and inundation phases, so the system is more likely close to the extreme stationary states, i.e., at v = 0 and v = b.
Quasi-trapezoidal Riparian Transects
[41] The role played by the biological parameters is shown in Figures 7c and 7d through the dependence on k and a. The former parameter influences m v in a similar way as C h : when k is high, the tolerance to the water stress by inundation decreases so the riparian zone shifts outward with a consequent increase in the unvegetated zone close to the river. However, in this case, the standard deviation increases with k. Conversely, an increment in coefficient a, which determines the range of water table depths tolerated by phreatophyte (see equations (6), (5)), induces a decrease both in m v and s v , and just a weak shift in the vegetated zone. An increase in a implies in fact a reduction of the growth phases, although the total time of exposure is unchanged.
Quasi-trapezoidal Riparian Transects
[42] It is also useful to make some comparisons of the different vegetation species. For example, we considered the six bottomland hardwood species listed in Table 1 , with values of D m , H m and T* m taken from Pearlstine et al. [1985] . The coefficients a 2 have been calculated for each species using the JABOWA model, as described in section 2, whereas the decay coefficient K was conventionally assumed equal to 0.0028 days À1 irrespective of the species, for the sake of simplicity. In short, three dimensionless parameters change with the vegetation: k = K/a 2 , d opt = d* opt / h, and t = a 2 t*.
Quasi-trapezoidal Riparian Transects
[43] The behavior of the mean vegetation biomass along the transect is reported in Figure 8 , where the picture does not describe a community of coexistent individuals (the interspecies interactions are not accounted for in our model) but rather different scenarios that correspond to single species. Figure 8 clearly shows that both the extent of the unvegetated zone and the fraction of biomass, with respect to optimal steady conditions (v is dimensionless) change significantly with the species. In particular, the model reproduces the tendency of Salix and Populus to grow close to the river [Naiman and Decamps, 1997] : due to their endurance and resistance to flooding, these species can stand a broad range of hydrological regimes. This explains the recognized competitive advantage of these species in the colonization of bare soil subsequent to sedimentation processes [e.g., Hughes, 1997] .
Quasi-trapezoidal Riparian Transects
[44] Further, the width of the unvegetated zone can be investigated, using the limit condition imposed by equation (19), which can be solved with respect to x under the hydraulic and geometric conditions considered in this case. The boundary of the riparian zone, x = x inf , is found to increase monotonically with C h , as shown in Figure 9 , irrespective of the value of t.
Discussion and Conclusions
[45] Riparian corridors are complex ecotones regulated by processes acting longitudinally (along the stream), laterally (along the transect), vertically (toward the groundwater), and in time [Tockner et al., 2000] . In our model we have focused on lateral and vertical directions and time, by stochastically modeling the main ecohydrologic processes affecting the control exerted by flow variability on riparian ecosystems. In a similar way to other ecohydrological studies in vegetation ecosystems (e.g., [RodriguezIturbe, 2000; Porporato and Rodriguez-Iturbe, 2002; D'Odorico et al., 2004 D'Odorico et al., , 2005 Rodriguez-Iturbe and Porporato, 2005] ), our aim was to quantitatively investigate how stream and near-stream hydrology is able to impact ecological processes. In this section, we discuss the limits and reliability of our modeling approach, in the light of some field studies.
Discussion and Conclusions
[46] The level of completeness of the analytical model developed in the previous sections is summarized in Table 2 , where both the considered and neglected processes are listed. Table 2 refers to the fundamental hydrogeomorphological factors that affect the riparian habitat, according to a recent review by Steiger et al. [2005] . The quantities used in our modeling framework are also reported in the same table. It confirms that several important aspects have been focused on and modeled. In particular, we considered all the main processes involving the hydrological and hydraulic characteristics of the river and the properties of riparian vegetation. Some aspects linked to woody debris, the effects of external forcings, and fluvial geomorphology instead were not considered. The latter is probably the most important one. Although some field studies have shown that sediment size characteristics can be of secondary importance [Osterkamp and Hupp, 1984] , other geomorphological processes such as erosion and sedimentation play an important role in the interaction with riparian vegetation and the consequent change in the riparian topography. Thus the main limitation of the present modeling approach is that it assumes the transect topography to be time-independent. This assumption precludes the application of our results to environments that develop high lateral migration rates, as is the case of very tortuous meandering rivers. Conversely, slightly sinuous planimetry permit the above results to be adopted, provided the timescale of river migration is longer than the time scale of the vegetation dynamics.
Discussion and Conclusions
[47] One of the main results of our analytic formulation is the description of the river-induced lateral distribution of riparian vegetation. A rigorous quantitative comparison with real riparian communities is very difficult since the literature on riparian systems, though wide and varied, rarely provides all the hydrological and biologic data necessary for a rigorous comparison with our model-based results. Furthermore, field studies usually investigate the transient response of vegetation to a man-induced change in the flow regime (e.g., dam regulation), rather than the long-term steady state. Nevertheless, a qualitative comparison is still possible on the basis of the results of several studies on the dependence of riparian vegetation biomass on the distance from the channel Figure 8 . Behavior of the expected value of the biomass along the riparian transect for different species: Carya tomentosa (Ct), Liriodendron tulipifera (Lt), Nyssa aquatica (Na), Populus deltoides (Pd), Prunus serotina (Ps), and Salix nigra (Sn). (C h = 0.5, t* = 30 days, h* = 2 m, and a 1 = 0.0028.) Figure 9 . Position of the lowest vegetated site versus C h according to the condition (19).
Discussion and Conclusions
[e.g., Nanson and Beach, 1977; Hupp and Osterkamp, 1985; Bradley and Smith, 1986; Richter and Richter, 2000] . In agreement with our results, peaked lateral distributions have been very frequently observed along riparian transects. For example, Johnson et al. [1995] investigated the occurrence of different types of vegetation along the Snake River (Idaho) as a function of plot elevation. A modified diagram of their results is given in Figure 10a . The transversal distribution of trees, forb shrubs, and transitional species shows typical peaked behavior, where both the position of the peak and the lateral extension of the habitat depend on the type of vegetation. van Coller [1993] described a similar distribution and abundance of mature individuals as a function of the elevational gradient of the Sabie River riparian forest (see Figure 10b ). Lytle and Merritt [2004] , through a numerical probabilistic model, also found that the cottonwood density peaks at intermediate levels of flood frequency (corresponding here to P I ). All these features qualitatively validate the results of our model (see Figure 8) .
Discussion and Conclusions
[48] The existence of a strip where riparian (phreatophyte) species develop, and where the interior and exterior margins are marked by hydrological processes, is easily recognizable along riparian corridors. Figures 11a and 11b show two examples, where the presence of a peak in vegetation density is also visible. The not monotonic behavior of the riparian vegetation distribution is due to the negative action (1) of floods, at plots close to the river, and (2) to a too deep phreatic surface, far from the river. In contrast, upland vegetation, which is not phreatophyte, is only affected by flood disturbance, which decreases in magnitude with an increasing distances from the river bank, while its carrying capacity does not depend on the depth of the water table (i.e., b = const). It follows that a monotonic increasing lateral distribution occurs, in agreement with the case discussed in section 4.1 ( Figure 5 ). The outcome of this model is also confirmed by examining field data by Johnson et al. [1995] (see Figure 10a) . Very similar findings were also presented by Auble et al. [1994] ; they investigated the dependence of the distribution of some herbaceous species on the probability P I , and detected a monotonic behavior that closely resembles the pattern described by our model in Figure 5 .
Discussion and Conclusions
[49] Several studies have pointed out the necessity of matching quantitative information of a functional response of riparian vegetation with quantitative information on the river hydrology [e.g., Scott et al., 1999; Lytle and Merritt, 2004] . The present work is an attempt in this direction, and it provides some process-based analytical tools to investigate the coupled ecosystem-hydrologic dynamics in riparian ecosystems. We are aware that some simplifying hypotheses have been made, but this was necessary to maintain the model both mathematically tractable and parsimonious. However, we believe that several fundamental processes have been properly taken into account and the resulted obtained with our closed form solutions are qualitatively consistent with field observations. The proposed model shows that rather simple interactions can produce complex patterns and, although this novel approach is not yet suitable for use in management and restoration of fluvial landscapes, it can be considered a starting point for subsequent works that address the stochastic features of the riparian vegetation dynamics.
Discussion and Conclusions
Appendix A: Some Analytical Expressions of the pdf, p(v)
Discussion and Conclusions
[50] Analytical solutions for steady vegetation biomass pdf are available for several different combinations of the exponents m, n, and p. In particular, here we report the cases with p = m, for which 
Discussion and Conclusions
if m = 1 and n 6 ¼ 1.
Discussion and Conclusions
[51] In the previous expressions, 2 F 1 [a, b; c, d ] is the Hypergeometric function (Abramowitz and Stegun [1965] ). 

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