Hydrology - Department of Mathematics, University of Utah

WATER RESOURCES RESEARCH, VOL. 34, NO. 12, PAGES 3303–3313, DECEMBER 1998

Stability and damping in a dynamical model of hillslope hydrology David Brandes and Christopher J. Duffy Department of Civil and Environmental Engineering, Pennsylvania State University, University Park

Joseph P. Cusumano Department of Engineering Science and Mechanics, Pennsylvania State University, University Park

Abstract. A stability analysis is performed on the two-state variable, integral-balance model of hillslope hydrology developed by Duffy [1996]. The motivation for the research is to develop a physically based, low-dimensional representation of inflow/outflow behavior within a hillslope-stream setting. Stability criteria are developed for the model equilibria and are evaluated using the results of numerical solutions of Richards’ equation for a convex-concave hillslope geometry. We show that for homogeneous hillslopes of three widely varying soil types the single moisture equilibrium is classified as a stable node for low precipitation rates and a stable spiral for wet conditions. The spiral equilibrium indicates that the hillslope system is lightly damped, and transient oscillations of the state variables are expected for high precipitation rates. The timescale of these oscillations is of the order of several days to weeks for the model hillslopes examined. Furthermore, we demonstrate that the model contains a Hopf bifurcation from a stable static equilibrium to a stable limit cycle. The amplitude, phase, and frequency of the limit cycle are determined analytically using second-order averaging. However, this behavior is shown to be nonphysical for the particular homogeneous soils and hillslope geometry investigated. Implications of lightly damped behavior in the hillslope system include moisture oscillations in the field under wet conditions and difficulty in numerical solution of Richards’ equation.

1.

Background

The effects of microscale nonlinearity of saturated/ unsaturated flow on macroscale, observable hydrologic behavior remain largely unknown. The highly nonlinear partial differential equation for variably saturated flow [Richards, 1931] indicates the possibility of complex dynamical behavior at the hillslope scale. However, because energy inputs and especially precipitation are strong and noisy, it is difficult to separate the response to such forcing from the dynamics that may be inherent in the soil/groundwater system. Within the framework of dynamical systems, hillslope soil moisture and associated hydrologic fluxes can be thought of as a noisy forced dynamical system. Consistent with this conceptualization, a two-state variable integral-balance dynamical model for hillslope hydrology has recently been introduced by Duffy [1996]. The purpose of this paper is to characterize the dynamical behavior of this model using both analytical methods and the results of numerical solutions of Richards’ equation. In particular, we explore conditions for stable and unstable moisture equilibria. The approach taken here differs from the “local” analysis of stability of flow in porous media, which is associated with a highly fingered spatial distribution of moisture during infiltration and redistribution because of gravitational, viscous, and capillary effects [Saffman and Taylor, 1958; Chuoke et al., 1959; Hill and Parlange, 1972; Glass et al., 1989]. Although flow fingering could have a significant effect on the volume averagCopyright 1998 by the American Geophysical Union. Paper number 98WR02532. 0043-1397/98/98WR-02532$09.00

ing inherent in our integrated state variables, here we analyze the temporal stability of the equilibria of the integrated hillslope; thus there is no explicit local spatial effect associated with stability or instability. Rather, stability refers to moisture equilibria that persist because of maintenance of a static balance of hillslope moisture fluxes. In addition, we adopt a dynamical systems approach to stability in that we are concerned with the qualitative dynamics of the model near the equilibria or fixed points in phase space [Guckenheimer and Holmes, 1986; Strogatz, 1994]. The global effects of stability/instability in the system are visualized through construction of phase portraits.

2. The Two-State Variable Integral-Balance Model At the point scale, flow in porous media can be described by combining the continuity equation with the unsaturated form of Darcy’s law [Richards, 1931]: C~ c !

c 5 ¹ z @K~ c !¹~ c 1 z!# t

(1)

where C( c ) 5 d u /d c is the specific moisture capacity, c is the suction or tension head, u is volumetric moisture content, K is hydraulic conductivity, and z is the elevation. Application of (1) at the hillslope scale requires detailed information on soil hydraulic properties and initial and boundary conditions, their variability in space and time, and a numerical solution method. The two-state variable integral-balance model provides a physically based alternative to simulation of Richards’ equa-

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BRANDES ET AL.: A DYNAMICAL MODEL OF HILLSLOPE HYDROLOGY

tion on the hillslope. The model is based on integration of the continuity equation over a hillslope control volume. This is made up of the saturated and unsaturated soil storage, with the water table serving as a moving boundary between these two storage volumes. The Reynolds transport theorem is used to relate local continuum equations for moisture to the system storages and fluxes, and the divergence theorem relates integrals of the spatial derivatives to integrals of the surface fluxes (see Duffy [1996] for details). A system of coupled ordinary differential equations with the following form results: dS 1 5 f 01 2 g 12 2 f 10 dt

(2)

dS 2 5 f 02 1 g 12 2 f 20 dt

where the state variables consist of the hillslope-integrated unsaturated moisture storage (S 1 ) (L), and the hillslopeintegrated saturated moisture storage (S 2 ) (L). The f 01 and f 10 terms (L/T) indicate fluxes between the external environment (designated with subscript 0) and S 1 ; likewise, f 02 and f 20 (L/T) indicate fluxes between the external environment and S 2 . The g 12 terms (L/T) indicate internal fluxes across the moving S 1 /S 2 boundary (i.e., the water table). The system is made dimensionless by scaling with a characteristic length scale which is the product of soil depth d and porosity n and the velocity scale K sat (the saturated hydraulic conductivity). This gives dimensionless variables and fluxes: S1 5

ˆ1 S , nd

S2 5

ˆ2 S , nd

f ij 5

ˆf ij , K sat

g 12 5

ˆg 12 , K sat

t5

ˆt K sat nd

where the circumflex refers to the unscaled variable. Note that scaling the state variables in this manner limits their values to the range [0 1]. The integral flux terms of (2) can be expressed as functions of the state variables S 1 and S 2 , and external forcing. Duffy [1996] developed these relations on the basis of numerical solutions [Lee, 1993] to Richards’ equation on model hillslopes with homogeneous soil properties. The following functions are used for the constitutive relations in (2): f 01 5 p@1 2 d 4~S 2 2 S 02!#

(3)

g 12 5 d 0~S 1 2 S 01! 1 d 1~S 1 1 d 2!~S 2 2 S 02! 2

(4)

f 20 5 d 3~S 2 2 S 02!

(5)

where p is the dimensionless precipitation rate (scaled by K sat), S 01 and S 02 are the residual storage volumes, and the d i s are parameters required to be positive. Note that the recharge term ( g 12 ) contains a nonlinear cross-term coupling the two state variables; however, the external flux terms ( f 01 and f 20 ) are linear. The f 01 term represents the net input to the hillslope. This “effective precipitation” is inversely proportional to the extent of surface saturation, which is itself a linear function of saturated storage. The model thus represents the saturationexcess runoff mechanism in that p 2 f 01 represents the surface runoff from the surface saturated area. The f 20 term represents outflow from the hillslope, also a linear function of saturated storage (S 2 ). At steady state, f 01 5 g 12 5 f 20 , the net flow through the hillslope. The justification for this form of the model is that it is consistent with numerical experiment results [Lee, 1993] and is of low order; however, other forms of the

model are certainly possible, for instance, quadratic or cubic external flux relations. In this simple case, external fluxes f 10 and f 02 are not included in the hillslope model. The model parameters S 01 and S 02 are the residual storage volumes for the gravity-drained hillslope ( p 5 0). S 01 increases with steep slopes and fine-grained soils, and S 02 increases with milder slopes. The parameters d 0 , d 1 , and d 2 describe the form of the hillslope-scale recharge relation coupling the state variables. This relation is a function of moisture retention characteristics of the hillslope soils and the hillslope geometry. In general, steeper slopes and greater unsaturated moisture retention result in larger values of these parameters. The parameters of the external flux terms each have a clear physical interpretation: d 3 is the subsurface flow rate constant, and d 4 relates the extent of the surface saturated area with saturated storage. In the following sections we explore the stability of the model equilibria under general conditions for a constant precipitation rate. It is important to realize throughout that an equilibrium in the integrated storage variables of the model corresponds to some equilibrium shallow water table/soil moisture spatial distribution. This configuration can be visualized through numerical simulations of Richards’ equation [Duffy, 1996; Brandes and Duffy, 1996].

3.

Local Linear Stability Analysis

The model (2) can be written in vector notation as dS/dt 5 f(S). Defining u 5 S 2 S*, as a small deviation from equilibrium, we write the variational equation for u as u ˙ < Df~S*!u

(6)

where Df(S*) is the Jacobian of f evaluated at the fixed point S*. The eigenvalues of the matrix Df(S*), hereafter designated A, govern the stability of the equilibrium S*. For the stability analysis it is convenient to shift the state variables to x 5 S 2 2 S 02 and y 5 S 1 2 S 01 and to define a new parameter d 5 d 2 1 S 01 . This eliminates the S 01 and S 02 parameters. Note that x must be positive; however, y may be negative. The system (2) then becomes ˙x 5 d 0 y 1 d 1~ y 1 d ! x 2 2 d 3 x ˙y 5 p~1 2 d 4 x! 2 d 0 y 2 d 1~ y 1 d ! x 2

(7)

with a single fixed point: x* 5

p d3 1 d4 p

y* 5

pd 3~d 3 1 d 4 p! 2 d d 1 p 2 d 0~d 3 1 d 4 p! 2 1 d 1 p 2

(8)

The Jacobian matrix A for (7) may be written as

F

a 2 d3 2pd 4 2 a

b 2b

G

with a 5 2d 1 ( y* 1 d ) x* and b 5 d 0 1 d 1 x* 2 . Because the eigenvalues of A are solely functions of its trace Tr(A) and determinant Det(A), the stability and type of fixed points can be determined by plotting the values of Tr(A) and Det(A), where in this case Tr(A) 5 a 2 b 2 d 3 and Det(A) 5 b (d 3 1 d 4 p). The possibilities are shown in Figure 1. In our case, because a , b , p, and the d i s are all positive quantities, it is obvious that Det(A) is always positive and the fixed point cannot be a saddle. Therefore the possible fixed points include stable and unstable nodes and spirals. Centers


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are not expected since the system is dissipative. An interesting question is under what conditions is Tr(A) positive, indicating an unstable fixed point? If the single fixed point is unstable, we expect another type of stable equilibrium, the limit cycle. For generic nonlinear systems, one can expect limit cycles to be created near a spiral-type equilibrium, as its stability changes smoothly from stable to unstable. In order to explore these possibilities the expressions for Det(A) and Tr(A) are developed by substitution of x* and y* into a and b. After some algebra they take the following forms Det(A) 5 ~d 0 1 d 1 x* 2!~d 3 1 d 4 p! 5 d 0~d 3 1 d 4 p! 1

d1 p2 d3 1 d4 p

(9)

2

Tr(A) 5 2d 0 2 d 3 2 d 1 x* 1 2d 1~ y* 1 d ! x* 5 2d 0 2 d 3 2 1

d1 p2 ~d 3 1 d 4 p! 2

which leads to the conclusion that the condition for instability (i.e., Tr(A) . 0) is simply

2d 1 p@d 0d ~d 3 1 d 4 p! 1 d 3 p# d 0~d 3 1 d 4 p! 2 1 d 1 p 2

Instability Condition on the y* 2 x* Relation

The instability condition on the relation y*( x*) of the integral-balance model can be developed by comparison of the expressions for Tr(A) and the slope d/dx* of y*( x*). First we express y* and Tr(A) as a function of the parameters and x*: y* 5

d 3 x* 2 d 1d x* 2 d 0 1 d 1 x* 2

Tr(A) 5 2d 0 2 d 3 2 d 1 x* 2 1

2 d d 0 d 1 x* 1 2d 1 d 3 x* 2 d 0 1 d 1 x* 2

(11) (12)

The expression for Tr(A) can be rewritten as Tr(A) 5 2d 0 2 d 1 x* 2 1

2d 0 d 3 1 2 d d 0 d 1 x* 1 d 1 d 3 x* 2 d 0 1 d 1 x* 2 (13)

Now we find the slope of y*( x*), resulting in d y* d 0 d 3 2 2 d d 0 d 1 x* 2 d 1 d 3 x* 2 5 dx* ~d 0 1 d 1 x* 2! 2

(14)

d y* ~d 1 d 1 x* 2! dx* 0

(15)

(16)

Therefore, for the fixed point to be unstable the relation between x* and y* (or S *2 and S *1 ) must have local slope , 21. It is noted that the d 0 and d 1 parameters of (15) also determine the slopes of this relation; therefore stability is completely controlled by the form of the steady state recharge relation between the state variables and not by the external flux terms of the model. 3.2.

Physical Implication of the Instability Condition

We now examine the physical implication of the instability condition (dS *1 /dS *2 ) , 21. For the assumptions of a rigid medium and incompressible fluid, mass conservation requires that the sum of the moisture state variables plus the empty (air-filled) pore space equals the total pore volume of the hillslope: S 1 1 S 2 1 S e 5 1,

(17)

where S e designates the empty pore space (scaled by nd). Taking the derivative of (17) with respect to S 2 and substituting the condition for instability, one is left with the expression (dS *e /dS *2 ) . 0. This inequality states that in the vicinity of an unstable fixed point the steady state empty pore volume must increase with increasing saturation of the hillslope. However, this scenario is physically impossible, as the empty pore space will always decrease as the water table rises. Therefore it is concluded that the condition for instability of the model equilibria, for the simple hillslope geometry and homogeneous soils investigated here, is nonphysical. The mass conservation constraint (17) does allow dS *1 /dS *2 to closely approach 21 without exceeding it, and in that case the system will be quite lightly damped, but not unstable. 3.3.

By substitution of this expression into Tr(A) we see that Tr(A) can be written as Tr~A! 5 2d 0 2 d 1 x* 2 2

d y* , 21 dx*

(10)

We see from (9) that Det(A) will increase as the precipitation rate increases, favoring spirals over nodes. The expression for Tr(A) contains five parameters determined by the hydraulic properties and form of the hillslope, and the control parameter, the precipitation rate p. By finding the zeros of this expression the stability boundaries can be determined as a function of p and the other parameters. However, the dimensionality of this parameter space makes visualization of the stability boundaries difficult. An alternative approach results in a useful instability condition on the relation between the steady state integrated state variables y*( x*), which incorporates all of the model parameters. This is detailed in the following section. 3.1.

Figure 1. Stability and type of fixed points for planar dynamical systems [see Strogatz, 1994].

Nodes and Spirals

The physically realizable equilibria for the model are either stable nodes or spirals. A spiral corresponds to a lightly damped equilibrium, and a node corresponds to a heavily damped equilibrium. In the time domain the spiral manifests itself as progressively smaller oscillations of the state variables until the static equilibrium point (fixed point) is reached. For our system this would correspond to a declining wave at the

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water table. On the other hand, for a node, there are no oscillations during relaxation; that is, the water table would rise or fall monotonically. Thus the degree of damping of the system has a significant effect on its time dynamics. For a lightly damped equilibrium as discussed in the previous section the system could make many oscillations before reaching the static fixed point. The transition from stable node to spiral occurs when 4Det(A) 5 Tr(A)2 (see Figure 1). Using (9) and (15), we can write this condition for the model as

S

x*~d 0 1 d 1 x* 2! 1 1 p5

d y* dx*

D

2

(18)

4

Given the curve y*( x*), for any value of x* one can use this expression to determine the critical precipitation rate for the transition from node to spiral. For precipitation rates less than this value the fixed point is a node, whereas for higher precipitation rates the fixed point is a spiral and oscillation occurs during relaxation. As p increases beyond this critical value, the system is progressively less damped. The system is most heavily damped when the slope d y*/dx* is positive. Representative values of the model parameters are needed to examine these stability and damping criteria for the integralbalance model. To develop these data, we rely on numerical solutions of the partial differential equations (PDEs) for a typical hillslope geometry and three soil types. The approach is detailed in the following section.

4.

Numerical Experiments

Numerical modeling experiments were conducted to solve the steady state Richards’ equation over a two-dimensional hillslope domain using the finite element code FEMWATER [Yeh, 1987]. Details of the numerical method are given by Lee [1993] and have been summarized by Duffy [1996]. Solutions of the boundary value problem were developed for a range of constant precipitation rates. For each simulation the integrated moisture state variables S *1 and S *2 are determined numerically by summing over the saturated and unsaturated hillslope volumes. The steady state position of the water table and associated surface saturated area are determined from the experiment. The geometry used here is that of a convex-concave hillslope of height H 25 m, length L 100 m, and soil depth d over an impermeable base of 2.5 m, as shown in Figure 2. These values are somewhat arbitrary but are typical of hillslopes in humidtemperate regions. The boundary conditions include no-flow (Neumann) boundaries along the sides (due to symmetry) and base of the hillslope, a variable infiltration/seepage boundary along the ground surface, and a single constant head (Dirichlet) node at the foot of the slope representing a first-order

Figure 2. Geometry and boundary conditions of the model hillslope used in the numerical experiments. stream. For the experiments summarized here, three soil types were used, a Guelph loam, a Plainfield sand, and a medium sand [Clapp and Hornberger, 1977] designated Cl&H sand. Although these soils were intended to provide a wide range of hydraulic characteristics, the purpose here is not to provide an exhaustive set of numerical results for calibration but rather to determine appropriate parameter values for analysis of the two-state variable model. Additional soil types and hillslope geometries have been investigated by Lee [1993], and those presented here reflect the general results. The numerical code requires input of analytical functions or tabular data describing soil moisture content, soil moisture capacity, and unsaturated hydraulic conductivity as a function of pore pressure. For the Guelph loam and Plainfield sand, soil hydraulic properties were described by the nonhysteretic parameterizations [Elrick et al., 1990]:

u ~c! 5

c 1c 2 1 c4 c 2 1 u c u c3

K~ c ! 5 K sate 2kc

(19) (20)

where u, c, and K are defined as previously, K sat is the saturated hydraulic conductivity, and c i and k are fitting parameters. For consistency the Cl&H sand property data were also fitted to these parameterizations. Table 1 summarizes the soil property parameter values used in the numerical experiments. The selected soils cover a range of almost 3 orders of magnitude in conductivity and a wide range in porosity. The water retention and relative conductivity functions for the soils are shown in Figure 3. Note in Figure 3 that the Guelph loam retains significantly more moisture than the sands and thus shows a gradual decrease in conductivity with pore tension,

Table 1. Summary of Hydraulic Property Data for the Soils Used in the Numerical Experiments Soil Type

K sat, cm s21

usat

k, m21

c1

c2

c3

c4

Guelph loam Plainfield sand Cl&H sand

3.67e-04 3.44e-03 1.76e-02

0.523 0.477 0.395

3.36 13.06 6.24

0.243 0.377 0.323

0.421 0.00154 0.927

2.0 4.0 0.85

0.28 0.1 0.086


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while the Plainfield sand exhibits the characteristics of a uniformly graded soil, with moisture content dropping sharply over a narrow range of pore tension. The Cl&H sand hydraulic behavior is intermediate to the Guelph and Plainfield soils. The numerical results for each soil type are shown in Figures 4, 5, and 6. Figures 4a, 5a, and 6a show the steady state relation S *1 (S *2 ) developed over a range of precipitation rates, i.e., the fixed points for different values of p. Figures 4b, 5b, and 6b show S 1 and S 2 against f 01 , the net steady flux through the hillslope. Figures 4c, 5c, and 6c show fractional saturated surface area ( fr SSA) against S 2 . Notice that the data show the similarity between the scaled S *1 -S *2 relations of all three soil types, the most significant difference being the lower unsaturated storage (S 1 ) of the Plainfield sand. In addition, the scaling of the flux quantities with saturated hydraulic conductivity results in the same range of dimensionless flux values (0 – 0.01) for all three soil types. The form of the steady state relation between the state variables S *1 and S *2 is an important result of these experi-

Figure 4. Numerical experiment results for the Guelph loam hillslope: (a) fixed points, (b) flux-storage relations, and (c) saturated surface area-storage relations. The f 01 term is the steady flux rate through the hillslope; fr SSA is the fractional surface saturated area. Fits to the model are shown as solid lines.

Figure 3. (a) Soil moisture characteristic curves and (b) hydraulic conductivity functions for the Guelph loam, Plainfield sand, and Cl&H sand (K rel 5 K( c )/K sat).

ments, as we have already seen that stability depends on the slopes of this relation. The inverse relation between unsaturated and saturated storage (except under very dry conditions) is a consequence of the state variables “competing” for the available hillslope pore space. For dry conditions, S *1 and S *2 are proportional because infiltration increases storage in both the initially dry unsaturated zone and the saturated zone near the stream channel. As precipitation rate increases, a moisture level is soon reached at which the relationship between the state variables becomes competitive, and their steady state relationship has negative slope. This is due to the water table (S 2 ) rising and beginning to capture the unsaturated soil (S 1 ) volume. For wetter conditions the competitive relationship remains until in the limit of saturation, S *2 approaches 1 and

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up the slope, the damping provided by the unsaturated soil decreases. The transition from node to spiral tends to occur at values of p near the inflection in the S *1 -S *2 relation, where the state variables become competitive. Therefore the hillslope system at high precipitation rates is not as highly damped as might be suspected and will show oscillations during relaxation at constant p. An important characteristic of the spiral fixed point oscillations is their period, particularly with respect to precipitation and solar forcing cycles. The dimensionless frequency v of the damped oscillation is given by the imaginary part of the complex eigenvalues, and the period is T 5 2 p / v . The periods of the oscillations for each soil type are shown in Table 3 for three precipitation rates. The very long periods for the Guelph loam

Figure 5. Numerical experiment results for the Plainfield sand hillslope: (a) fixed points, (b) flux-storage relations, and (c) saturated surface area-storage relations. The f 01 term is the steady flux rate through the hillslope; fr SSA is the fractional surface saturated area. Fits to the model are shown as solid lines. S *1 approaches 0. This general structure is seen for a variety of soil types and hillslope configurations [Lee, 1993]. Nonlinear least squares fitting of (5) and (6) with the data from Figures 4 – 6 was used to estimate the parameters for the two-state model. These fits are shown in Figures 4 – 6 as solid lines. The parameter values are summarized in Table 2. For each of the three soil types the fitted parameter values, used in conjunction with (18), result in a stable fixed point, a node for low precipitation rates and a spiral under wetter conditions. For dry conditions when the water table is near its lowest state the system is highly damped. This is due to the large adsorptive capacity of the hillslope for moisture. For example, when p 5 0 (interstorm periods) the equilibrium is always a node. As the hillslope becomes wetter and the capillary fringe moves higher

Figure 6. Numerical experiment results for the Cl&H sand hillslope: (a) fixed points, (b) flux-storage relations, and (c) saturated surface area-storage relations. The f 01 term is the steady flux rate through the hillslope; fr SSA is the fractional surface saturated area. Fits to the model are shown as solid lines.


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Table 2. Results of Nonlinear Fitting of the Numerical Solutions to the Two-State Variable Model Soil Type

d0

d1

d2

d3

d4

S 01

S 02

Guelph loam Plainfield sand Cl&H sand

0.06 0.035 0.05

0.15 0.13 0.095

0.4 0.16 0.5

0.013 0.012 0.012

0.9 0.85 0.85

0.51 0.21 0.49

0.164 0.169 0.168

and Plainfield sand are due to the low saturated hydraulic conductivities of these soils as determined in the laboratory. Flow timescales obtained using laboratory-derived matrix soil properties are much greater than for natural hillslope soils, which often include pathways of high conductivity. Inverse fitting of lumped parameter hillslope models to field baseflow data has indicated effective K sat values of the order of 1022– 1021 cm s21 for small catchments [Zecharius and Brutsaert, 1988; Troch et al., 1993]. Using these conductivity values and the dimensionless T values given in Table 3 results in oscillation periods of the order of several days to weeks, still rather long in comparison with typical storm durations. The slope of the S *1 -S *2 relation for each soil type (Figures 4a, 5a, and 6a) is not steep enough for unstable equilibria (i.e., (dS *1 /dS *2 ) . 2 1), as expected, since we have shown this condition to be nonphysical for the particular hillslope geometry and homogeneous soils investigated here. However, the relations for the Guelph loam and Cl&H sand approach the instability criteria, suggesting that an unstable fixed point may be possible for more complicated hillslope shapes with heterogeneous soil properties. One can easily construct a hypothetical S *1 -S *2 relation that will produce an unstable fixed point in the model for a certain range of p. Such a relation is shown in Figure 7 for the parameter values d 0 5 0.04, d 1 5 0.21, d 2 5 0.24, d 3 5 0.013, d 4 5 0.9, S 01 5 0.51, and S 02 5 0.165. These values are not far from those in Table 2, and the relation shown is similar to that of the Guelph loam, also shown in Figure 7.

5.

Phase Portraits

The phase portrait is a standard tool for analysis of dynamical systems and provides a useful way to visualize system behavior. In the phase space, in which each coordinate or dimension is a state variable of the system, qualitative aspects of the dynamics are revealed by the patterns of the timeimplicit trajectories evolving from various initial conditions.

These patterns tend to take a limited number of forms, each associated with distinct types of dynamical behavior. Phase portraits of the two-state variable model for a constant precipitation rate are constructed here using the VisualDSolve package [Schwalbe and Wagon, 1996]. On each phase portrait, in addition to solution trajectories, we also show the nullclines (curves upon which dS 1 /dt 5 0 and dS 2 /dt 5 0) and physical limits on the system states. The intersection of the two nullclines gives the fixed point. The physical limits on the system are given by two inequalities: (1) S 1 1 S 2 # 1 and (2) S 1 1 S 2 $ S 01 1 S 02 . The first limit is due to the constraint that total storage cannot exceed the hillslope pore volume, and the second is due to the constraint that total storage must exceed the residual storage value corresponding to p 5 0. Only system states that fall within these criteria are physically meaningful. Phase portraits and corresponding time domain behavior for one phase-space trajectory are illustrated in Figure 8 for each soil type discussed in the previous section. In each case the precipitation rate is p 5 0.004. For the Guelph loam (Figure 8a) the equilibrium is a stable spiral in phase space, and the time trajectories show a lightly damped oscillation about the fixed point (0.4, 0.44). For the Plainfield sand (Figure 8b) the equilibrium is also classified as a stable spiral, but only a single small oscillation occurs in the time domain. Because of the greater volume of dry soil (empty pore space) in the Plainfield hillslope, there is more damping in the system. For the Cl&H sand the phase portrait is quite similar to that of the Guelph loam, as might be expected from the similarity in the scaled steady state integral relations. Note, however, that the actual time represented by each trajectory is much less in this case because of the greater hydraulic conductivity of the Cl&H sand (see Table 3).

Table 3. Period of Transient Oscillations for Model Hillslopes of Three Soil Types Soil Type Guelph loam Plainfield sand Cl&H sand

p

T

ˆ , day T

0.002 0.004 0.006 0.002 0.004 0.006 0.002 0.004 0.006

269.9 195.7 170.7 432.3 301.6 254.2 324.0 230.4 199.3

1114 808 705 173 121 102 21 15 13

Dimensionless and dimensional values are given for three precipitation rates.

Figure 7. Hypothetical S *1 -S *2 relation for a hillslope with an unstable fixed point, showing the region where the slope exceeds 21. Also plotted is the relation for the Guelph loam hillslope.

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Figure 8. Phase portraits and corresponding time-domain dynamics for hillslopes of three soil types: (a) Guelph loam, (b) Plainfield sand, and (c) Cl&H sand. In all cases, p 5 0.004. In the phase portraits the thick lines are the system trajectories, the thin lines are the nullclines, and the diagonal dashed lines indicate physical limits on the system.

A phase portrait and time-domain trajectory for the hypothetical S *1 -S *2 relation of Figure 7 is shown in Figure 9. In this case the numerically obtained phase portrait indicates a limit cycle surrounding the unstable fixed point, and the time trajectory shows large amplitude oscillations. The significance of the limit cycle is that at steady state the state variables oscillate periodically with constant rainfall forcing, i.e., a “self-excited” oscillation. The limit cycle is due to the strong nonlinear interaction between the state variables in the recharge term (3) and unlike spirals and nodes, cannot occur in a linear system. Although limit cycle behavior is not possible for the homoge-

neous soils and convex-concave hillslope investigated above, the easily obtained lightly damped solutions indicate the possibility that such dynamics may occur under more complicated hillslope conditions. Therefore the limit cycle behavior is further analyzed in the following section.

6. 6.1.

The Stable Limit Cycle Solution Hopf Bifurcation

The transition from a static spiral fixed point to a periodic equilibrium or limit cycle as a parameter is smoothly varied is

Figure 9. Phase portrait and time-domain dynamics for the unstable fixed point case showing the stable limit cycle surrounding the unstable fixed point ( p 5 0.0045).


known as a Hopf bifurcation. Taking the precipitation rate p to be the bifurcation parameter (the other parameters are fixed by a particular hillslope geometry and soil type), the Hopf bifurcation theorem [Guckenheimer and Holmes, 1986] requires that (1) the matrix A has a pair of complex conjugate eigenvalues l ( p) 5 g ( p) 6 i v ( p) with g ( p) , 0, (2) at some value p*, g ( p*) 5 0, and (3) g ( p*)/p Þ 0. Using the parameters of the unstable fixed point case and a p value of 0.0035, we find complex conjugate eigenvalues of 20.00148 6 0.02834i. For p* 5 0.00438 and p* 5 0.00594 we find g ( p*) 5 0. Furthermore g(0.00438)/p 5 0.96 and g(0.00594)/p 5 20.78. So, there are, in fact, two Hopf bifurcations as p increases, a supercritical bifurcation to the limit cycle at p* 5 0.00438 and a subcritical bifurcation back to a static equilibrium at p* 5 0.00594. 6.2.

Finding the Limit Cycle by the Method of Averaging

The local linear stability analysis presented above does not give us any information about the phase portrait except at the fixed point. Although we can numerically generate phase portraits showing a limit cycle surrounding the unstable equilibrium (see Figure 9), to understand the way in which system parameters can be varied to give limit cycles requires additional analytical techniques. The classical perturbation method of averaging [Guckenheimer and Holmes, 1986; Murdock, 1991] is used to find the amplitude and phase of the limit cycle as a function of the model parameters. Averaging is based on the premise that small nonlinear terms introduce slow amplitude and phase drift from the linear solution C cos ( v t 1 f ) (i.e., simple harmonic motion), where C is a constant determined by initial conditions. Thus the contribution of the nonlinear terms can be represented by their average over one cycle time T 5 2 p / v . The result is a pair of “slow-time” differential equations, one each for the amplitude and phase drift of the motion on the limit cycle. Equilibria of the slow-time equations correspond to limit cycles, and the stability of these equilibria are readily assessed from the slow-time equations. Averaging is accomplished here by applying a general result for nonlinear oscillators to our specific problem. The method involves transformation of the original system (7) to normal or canonical form using the complex eigenvectors of A, then additional transformations to put the system into the proper form for averaging, followed by the averaging itself. These calculations are tedious and for brevity, full details are not included here; however, a description of the required transformations and application of the second-order averaging technique are discussed in Appendix A. Application of averaging yields equations of the following form (in polar coordinates) for the amplitude and phase drift r3 ˙r 5 g r 1 f 1~a, b, c, d, e, f, v ! v ˙ 5 f 2~a, b, c, d, e, f, v ! f

r2 v

(21) (22)

where r is the radius of the limit cycle, f is the phase drift, and the terms f 1 and f 2 are rather complicated rational functions shown in Appendix B. Equation (21) will have a positive real equilibrium r* (i.e., the system will have a limit cycle of radius r*) only if g and f 1 have opposite sign. Previously, we showed that the unstable fixed point occurs when Tr(A) is positive (see Figure 1). Since

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Figure 10. Limit cycle for the unstable fixed point case by the second-order averaging method with comparison to the numerically generated limit cycle of Figure 9.

g 5 Tr(A)/2 it follows that for stable limit cycles to exist, g must be positive and f 1 must be negative. We can determine that the limit cycle is stable by noting that for r , r*, the right-hand side of (21) is positive, and for r . r*, it is negative. Therefore small-amplitude oscillations (r , r*) are amplified, and largeamplitude oscillations (r . r*) are damped out. Though complex, the f 1 term can be used to investigate how the model parameters determine the amplitude of the oscillation. The limit cycle radius can be assessed for a given set of parameters by numerically evaluating (21) or (B1). Equation (22) indicates that the rate of phase drift on the limit cycle will be constant and dependent on r*. Because the phase f (r*) is equal to a constant ( f 2 r* 2 / v ) of time, the phase drift will cause the frequency of the motion on the limit cycle to be slightly different than v. For a precipitation rate of p 5 0.0045 (as used in Figure 9) the dimensionless period of the oscillation for the unstable fixed point case is 205.8. Using the saturated conductivity and porosity data for the Cl&H hillslope, this translates to a period of 13 days. In order to relate (21) and (22) back to the integrated moisture state space it is necessary to invert the series of transformations described above to recover the original x and y variables. This will result in equations for x and y on the limit cycle in terms of sin [ v t 1 f (t)] and cos [ v t 1 f (t)]. For the unstable fixed point case with p 5 0.0045 the result is shown in Figure 10, together with the numerically generated limit cycle (shown previously in Figure 9). It can be seen that the limit cycle found analytically using second-order averaging closely matches the limit cycle obtained numerically.

7.

Conclusions and Implications

In this paper we have explored the dynamical behavior of a new integral-balance model of hillslope hydrology and soil moisture [Duffy, 1996]. The basis of the model is the integration of the local continuity equation over the hillslope, and the constitutive relations are based on integration of numerical solutions to Richards’ equation. The form of the steady state relation coupling the state variables determines the stability of the model’s single equilibrium. This equilibrium is either a stable node at low precipitation rates or a spiral for wetter conditions. The physical reasoning for these types of equilibria is that under dry conditions, there is a large adsorptive capacity of the hillslope for additional moisture, and consequently, the

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system is highly damped. Under wet conditions the capillary fringe extends upslope and can quickly be converted to the saturated state. Therefore the internal damping provided by the unsaturated soil is much reduced. The fact that spiral equilibria are readily obtained for all three soil types indicates that the integrated hillslope system is not as strongly damped as may be assumed given the diffusive nature of Richards’ equation. Equation (18) can be used to predict the onset of lightly damped behavior in the integrated system. These types of equilibrium behavior indicate that the integrated hillslope system is not greatly influenced by the localscale nonlinearities in Richards’ equation. However, the model admits a stronger nonlinearity, a Hopf bifurcation to a limit cycle, when the slope of the S *1 -S *2 relation is ,21. For the particular hillslope configuration and homogeneous soil investigated here this behavior is shown to be nonphysical. However, we cannot preclude the bifurcation occurring for more complicated hillslope shapes or heterogeneous and hysteretic soils. Further numerical work is continuing to assess the effect of stratified soils and macropore soil characteristics on the constitutive relations of the model. Preliminary results indicate that higher-order functions will be required to accurately fit the numerical solutions for such hillslopes [Brandes, 1998]. Regardless of whether an unstable fixed point and limit cycle solution are physical, it is clear that the system can become lightly damped, and under these conditions, transient oscillations of the state variables are to be expected. For hillslopes of differing length, geometry, and soil properties, oscillations could occur at a wide range of periods. Considering the strong forcing typical of a natural watershed, an interesting question is whether such moisture oscillations could be detected independent of the response to external forcing. There are two likely explanations as to why they have not been reported: (1) relaxation in the field occurs during interstorm periods, when p 5 0 and the analysis shows the equilibrium is a node, and (2) the timescale of the oscillations (a few days to weeks) is quite long in comparison to the typical duration of rain events; for shorter duration events, only small portions of such trajectories could be observed. The presence of lightly damped solutions also has implications to modeling with Richards’ equation. In additional numerical modeling efforts [Brandes, 1998], difficulty was encountered obtaining convergence for intermediate moisture conditions. This often occurred for values of precipitation corresponding to near the peak in the S *1 -S *2 relation despite attempts with progressively finer grid sizes and time steps. The significance is that these convergence problems may not only be due to numerical instability, as is often claimed, but also to lightly damped physical solutions as predicted by (18). As a first step, this paper has focused on the autonomous hillslope system, that is, with constant precipitation forcing. Additional analysis of the dynamically forced model is underway to assess its behavior. We note that interactions between forcing and characteristic frequencies of lightly damped nonlinear systems are capable of producing a rich variety of dynamic responses [see Guckenheimer and Holmes, 1986; Strogatz, 1994].

Appendix A: Application of Second-Order Averaging to the Two-State Variable Model This appendix outlines the calculation needed to implement second-order averaging for the system. Because of the un-

wieldy form of the coefficients in the problem, extensive use was made of the Maple symbolic algebra system. We begin by expanding (7) in a Taylor series about the equilibrium ( x*, y*), with u5

H uu J 1

2

a small deviation from the equilibrium. Keeping terms up to third order in u 1 and u 2 results in a system with the form

FGF u ˙1

u ˙2

5

a

b

c

2b

GF G F u1

1

u2

du 21 1 eu 1u 2 1 fu 21u 2 2 du 21 2 eu 1u 2 2 fu 21u 2

G

(A1)

where a, b, c, d, e, and f are the following functions of the model parameters: a 5 2d 3 1 2d 1~ y* 1 d ! x* b 5 d 0 1 d 1 x* 2 c 5 2d 4 p 2 2d 1~ y* 1 d ! x* d 5 d 1~ y* 1 d ! e 5 2d 1 x* f 5 d1 and ( x*, y*) is the fixed point defined in (8). It is well known [Hirsch and Smale, 1974] that the eigenvectors of the linear part A can be used to define a variable transformation which will put A into its canonical form:

FG F n˙ 1 ˙n 2

5

g

v

2v

g

GF G n1

n2

1 NL~ n 1, n 2!

(A2)

where g is the real part and v is the imaginary part of the complex eigenvalues, and NL(n1, n2) refers to nonlinear terms in n1 and n2. In terms of the parameters defined above, g and v are given as

g5 v5

a2b 2

(A3)

Î2a 2 2 2ab 2 b 2 2 4bc

(A4)

2

A necessary condition for the Hopf bifurcation is that g vanishes, so that the eigenvalues are purely imaginary at the bifurcation point. To study the system near the bifurcation using perturbation methods, we let g 3 «2g, n1 3 «n1, and n2 3 «n2, to emphasize that n1, n2, and g are all assumed to be small. We then write the Taylor expansion of (21) in «, yielding

FG F ˙n 1

n˙ 2

5

1 «2

0

v

2v

0

F

GF G F n1

n2

1«

a 20n 21 1 a 11n 1n 2 1 a 02n 22 b 20n 21 1 b 11n 1n 2 1 b 02n 22

gn 1 1 a 30n 31 1 a 21n 21n 2 1 a 12n 1n 22 1 a 03n 32 gn 2 1 b 30n 31 1 b 21n 21n 2 1 b 12n 1n 22 1 b 03n 32

G

G

(A5)

The coefficients of the nonlinear terms are subscripted according to the powers on the corresponding variables (i.e., a 12 is the coefficient of n1 to the first power by n2 to the second power). Note that these coefficients are actually rational functions (after several variable transformations) of the model parameters. With the equation in this form, quadratic nonlinear


terms are order « small, and linear terms in g and cubic nonlinear terms are order «2 small. The system (A5) is clearly in the form of a perturbed harmonic oscillator. Weakly nonlinear equations of this form can be studied using a variety of perturbation techniques. We apply the method of averaging. Using the Van der Pol transformation

F G F w1

w2

5

cos ~ v t!

2sin ~ v t!

sin ~ v t!

cos ~ v t!

GF G n1

n2

system (A5) can be put directly into normal form for averaging: w ˙ 5 «G 1@w~t!, t# 1 « 2G 2@w~t!, t#

(A6)

Because of the presence of quadratic nonlinearities in (A5), averaging to first order yields no information, and thus secondorder averaging is required. This yields the slow-time equations shown in Appendix B.

Appendix B: Averaged Equations for the Limit Cycle This appendix presents the averaged (“slow-flow”) equations for the slow variables r and f. The equations for the limit cycle amplitude and phase drift in polar coordinates are the following: ˙r 5 gr

3

4

~a 1 b!~a 1 b 1 2c!@d~a 1 b! 1 2ce#@e~2c 2 a 2 b! 1 2d~a 1 b!# 16c3 v2 @d~a 1 b! 1 ce#@2d~a 1 b 1 c! 2 e~a 1 b!# 2c3

1 1

1

~a 1 2b! f ~2d 2 e!dv2 1 c c3

r3 v

z

(B1)

˙5 f

3

9~a 1 b!2 ~a 1 b 1 2c! f 5~a 1 b!2 ~a 1 b 1 2c!2 ~ad 1 bd 1 2ce!2 1 8c2 32c4 v2 1

F

z v2 1

10d2 v4 ~a 1 b!2 d2 ~10a 2 5b 1 7c! 1 c4 2c3

~a 1 b!de~23a 2 17b 1 14c! e2 ~4a 2 7b 1 c! 1 1 2c2 c

r2 z 24 v

4

G

15d2 ~a1b!2 10d2 ~a1b! 8d2 112de29e2 3~3a13b12c! f 1 1 1 4c4 c3 2c2 2c2

(B2)

where the parameters have been defined in Appendix A. Nontrivial equilibria of (B1) correspond to limit cycles. Acknowledgments. Funding for this research was provided to D.B. through a Graduate Research Assistantship from the Environmental

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Science Group, Los Alamos National Laboratory. The research was partially supported by grants from NASA (NAGW-4401) and NSF (EAR-9418674) to C.J.D. and J.P.C. C. Paniconi provided many helpful comments on the paper.

References Brandes, D., A low-dimensional dynamical model of hillslope soil moisture, with application to a semiarid field site, Ph.D. dissertation, 199 pp., Pa. State Univ., University Park, 1998. Brandes, D., and C. J. Duffy, Visualization of soil moisture dynamics on stratified hillslopes, Eos Trans. AGU, 77(17), Spring Meet. Suppl., S124, 1996. Chuoke, R. L., P. Van Meurs, and C. Van der Poel, The instability of slow, immiscible, viscous liquid-liquid displacements in permeable media, Trans. Am. Inst. Min. Metall. Pet. Eng., 216, 188 –194, 1959. Clapp, R. B., and G. M. Hornberger, Empirical equations for some soil hydraulic properties, Water Resour. Res., 14, 601– 604, 1978. Duffy, C. J., A two-state integral-balance model for soil moisture and groundwater dynamics in complex terrain, Water Resour. Res., 32, 2421–2434, 1996. Elrick, D. E., W. D. Reynolds, H. R. Geering, and K. A. Tan, Estimating steady infiltration rate times for infiltrometers and permeameters, Water Resour. Res., 26, 759 –769, 1990. Glass, R. J., J.-Y. Parlange, and T. S. Steenhuis, Wetting front instability, 1, Theoretical discussion and dimensional analysis, Water Resour. Res., 25, 1187–1194, 1989. Guckenheimer, J., and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 459 pp., Springer-Verlag, New York, 1986. Hill, D. E., and J.-Y. Parlange, Wetting front instability in layered soils, Soil Sci. Soc. Am. Proc., 36, 697–702, 1972. Hirsch, M. W., and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra, 358 pp., Academic, San Diego, Calif., 1974. Lee, D. H., On nonlinear dynamics of storage-flux relationships on a hillslope, Ph.D. dissertation, 120 pp., Pa. State Univ., University Park, 1993. Murdock, J. A., Perturbations, Theory and Methods, 509 pp., John Wiley, New York, 1991. Richards, L. A., Capillary conduction of liquids through porous mediums, Physics, 1, 318 –333, 1931. Saffman, P. G., and G. I. Taylor, The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid, Proc. R. Soc. London, Ser. A, 245, 312–331, 1958. Schwalbe, D., and S. Wagon, VisualDSolve New Frontiers in the Visualization of Differential Equations, Macalester College, St. Paul, Minn., 1996. Strogatz, S. H., Nonlinear Dynamics and Chaos, With Applications to Physics, Biology, Chemistry, and Engineering, 498 pp., AddisonWesley, Reading, Mass., 1994. Troch, P. A., R. P. De Troch, and W. Brutsaert, Effective water table depth to describe initial conditions prior to storm rainfall in humid regions, Water Resour. Res., 29, 427– 434, 1993. Yeh, G. T., FEMWATER: A finite element model of water flow through saturated-unsaturated porous media—first revision, Rep. ORNL-5567/R1, Oak Ridge Natl. Lab., Oak Ridge, Tenn., 1987. Zecharias, Y. B., and W. Brutsaert, Recession characteristics of groundwater outflow and base flow from mountainous watersheds, Water Resour. Res., 24, 1651–1658, 1988. D. Brandes and C. J. Duffy, Department of Civil and Environmental Engineering, Pennsylvania State University, University Park, PA 16802-1408. ([email protected]) J. P. Cusumano, Department of Engineering Science and Mechanics, Pennsylvania State University, University Park, PA 16802-1408.

(Received November 24, 1997; revised July 20, 1998; accepted July 30, 1998.)

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