[11] To demonstrate the potential utility of the time subordination approach, we restrict our attention to the particular phenomenon of subsurface lateral flow over an impermeable base. In particular, we assume that the slope of this base is sufficiently steep that the topographic gradient is the dominant driver of flow. This allows us to ignore certain nonlinearities in the flow (as discussed below), and base the theory on the kinematic wave equation proposed by *Beven* [1981], a simplified form of the modified Boussinesq equation. We will also make the assumption that recharge is uniform over the hillslope. In future work we hope to be able to relax some of these assumptions and broaden the framework.

#### 2.1. Kinematic Wave Equation

[13] The Boussinesq description of lateral subsurface flow is based on a form of Darcy's law vertically integrated over the saturated zone, under the assumption that flow is primarily oriented parallel to an impervious hillslope base. Continuity in the saturated zone requires that

where *h* = *h*(*x*, *y*, *t*) is the thickness of the aquifer at a point given by coordinates (*x*, *y*) defined in the plane of the sloping base, at time *t*; ϕ_{d} is the drainable porosity and *r* = *r*(*x*, *y*, *t*) is the recharge rate. The components of the flux = (*x*, *y*, *t*) in the *i* direction are given by

which is a function of the hydraulic conductivity *K*_{i}, water table gradient *dh*/*dx*_{i}, and angle of depression θ_{i} in the *i* direction. Note that equation (5) is a modified form of the traditional Boussinesq equation which is purely diffusional and does not include a convective term.

[14] The resulting equation is a nonlinear PDE due to the multiplication of the *h* and *dh*/*dx* terms in the expression for the flux. However, conditions have been described under which this nonlinearity can be ignored, most recently by *Harman and Sivapalan* [2009a]. The *dh*/*dx* term describes the component of the potential gradient driving flow given by the gradient of the aquifer thickness. When the ratio of the mean aquifer thickness and the product of hillslope length *L* and slope tanθ is small,

the potential gradient driving the flux is dominated by the topographic gradient. The expression for flux in the downslope direction in a straight, uniform hillslope (with *i* subscripts dropped, since flow is only downslope), then reduces to

and the PDE is then linear. We can rewrite this in terms of the velocity of an impulse moving through the hillslope *v* as

where the velocity is defined by

Considering only 1-D flow down the hillslope, the conservation equation for flow can then be written as

[15] *Beven* [1981] used this approximation to derive an expression for the response of a sufficiently steep rectangular hillslope (with uniform conductivity) to an impulse of recharge applied uniformly across the hillslope. The discharge per unit depth of recharge is given by

Because the system is linear, this result is the impulse response function (or unit hydrograph), and the output of the system for an arbitrary time series of (spatially uniform) recharge inputs *r*(*t*) can be obtained by convolving the unit hydrograph, equation (11), with the input:

where the ☆ represents the convolution operator. A buildup of storage in low-conductivity areas within heterogeneous hillslopes can create large gradients in the aquifer thickness, violating the assumption that these gradients have a negligible influence on flow. However, *Harman and Sivapalan* [2009b] demonstrated that these gradients do not have a significant effect on the flow when the condition in equation (6) is met. If we can assume that the behavior is still linear, it should be possible to find a new unit hydrograph for a heterogeneous conductivity field *IUH*_{K} that allows us to obtain the discharge result for arbitrary input in the same manner as for the uniform *K* field:

[16] We test this assumption using synthetic data generated from a numerical model that solves the full Boussinesq equation (equations (4) and (5)). The model is run first for an impulse of recharge to produce a “unit hydrograph,” and then for a hypothetical time-variable recharge event of 24 h duration. By convolving the unit hydrograph with the recharge event, it should be possible to obtain a similar discharge time series as the numerical model if the hillslope is behaving linearly, and a different response if the linearity assumption is not valid.

#### 2.2. Subordination of the Kinematic Wave Equation

[17] The kinematic wave equation (equation (7)) provides a basis for the development of an analytical framework for modeling flow through heterogeneous hillslopes. According to equation (7), the flux *q* [L/t] at the base of a hillslope for an instantaneous and evenly distributed recharge impulse of magnitude *R* for a hillslope of length *L* is a piston of magnitude *Rv*/*L* with duration *L*/*v* (Figure 2). The piston flow response arises from a collection of flow impulses at the hillslope base, where the impulses all move at the same velocity but have different starting locations. The arrival time of an individual impulse *t*_{i} with a travel distance of *L *− *x*_{i}, where *x*_{i} is the initial location and values of *x* increase in the direction of flow, is equal to (Figure 2). The study by *Harman* [2007] demonstrates that the motion of impulses within heterogeneous hillslopes is not subject to a constant or equivalent velocity. Rather, impulses are subject to a wide range of velocities as they migrate through a hillslope, and flow responses of heterogeneous hillslopes signify an assemblage of both fast and slow arrival times. As a consequence, the flow behavior of heterogeneous hillslopes clearly violates the equivalent velocity assumption of a piston. By randomizing the time the impulses spend in motion, and consequently arrival times *t*_{i} of impulses at the base of a hillslope via subordination, we broaden the applicability of equation (7) which was previously limited to only the least heterogeneous hillslopes.

[18] Subordination is a standard tool in the theory of Markov and Lévy processes [e.g., *Baeumer and Meerschaert*, 2007; *Bertoin*, 1996; *Bochner*, 1949; *Feller*, 1971; *Meerschaert and Scheffler*, 2004, 2008; *Sato*, 1999]. The process of subordination refers to the replacement of linear time with operational time [e.g., *Baeumer and Meerschaert*, 2007; *Meerschaert and Scheffler*, 2004, 2008]. In our work, operational time refers to the random amount of time that a flow impulse “operates” or participates in the motion process. Other earth science applications of subordination include transport of sediment particles in river systems [*Ganti et al.*, 2009] and transport of solutes through heterogeneous aquifers subject to differential advection [*Baeumer et al.*, 2001] or retention in immobile zones [e.g., *Baeumer and Meerschaert*, 2007; *Benson and Meerschaert*, 2009; *Schumer et al.*, 2003].

[19] We focus our application on hillslope flow responses with heavy tails (i.e., exhibit a linear decay of flux over time on a log-log plot). The heavy tails imply that the heterogeneity within a hillslope soil exerts a strong time influence on the motion of flow impulses; that is, the time between motions is heavy tailed. However, it is well known that the classical advection-diffusion equation fails to take into account the heavy tails in the waiting times or the velocities of transport [see, e.g., *Benson et al.*, 2000, 2001]. It has been specifically shown that the governing equation for the transport with a heavy-tailed distribution with a power law decay in the tails of waiting times and/or transport distances can be recast into a fractional advection-dispersion equation [e.g., *Meerschaert et al.*, 1999; *Meerschaert and Scheffler*, 2001; *Schumer et al.*, 2003]. Specifically, it has been shown that heavy tails in waiting times can be taken into account by time fractional derivatives and heavy tails in travel distances by space fractional derivatives [e.g., *Benson et al.*, 2001; *Ganti et al.*, 2010; *Schumer et al.*, 2003, 2009]. In the above case we focus our attention on heavy-tailed waiting times and in the scaling limit these motions (without recharge) satisfy the fractional in-time differential equation

where the fractional time derivative of order *γ* and scale factor *s* (with units of [t^{γ−1}]) describe an inverse stable distribution of waiting times between impulse motion. Note that the selection of a stable subordinator is motivated by mathematical limit theorems that describe the convergence of the sum of *iid* heavy-tailed random variables to a Lévy stable density [*Meerschaert and Scheffler*, 2001]. In order to incorporate recharge *r*(*x*, *t*) which modifies ∂*h*/∂*t*, equation (14) needs to be recast (differentiating (1 − *γ*) times both sides in *t*) to expose a first-order temporal derivative on the left hand side [*Baeumer et al.*, 2005]:

Note that equation (15) is a generalized form of equation (10). Equations (10) and (15) are equivalent for the special case of *γ* = 1 and *s* = 1 which describes piston flow through a homogenous hillslope. The function *H* is given by

where *h* is the piston flow solution to an impulse, and *p*(*u*, *t*) is an inverse stable density of index gamma:

where *g*_{γ} is a standard stable density (*e*^{−λt}*g*_{γ}(*t*) *dt* = exp(−*λ*^{γ})). Our approach essentially randomizes the response for a homogenous hillslope *Q*_{0}, equation (11), with a heavy-tailed subordinator to generate unit responses for heterogeneous hillslopes *Q*_{K} (equation (13)).

[20] We can find the discharge (flux) *F* at a point *x* by looking at the rate of change of total mass to the right of *x*:

To model the flux of impulses at the base of a hillslope the following steps are required. First, an instantaneous recharge flux *f*_{0} is distributed evenly for all *x* along a hillslope of length *L* according to

where *R* is total recharge applied to the entire hillslope, and Φ(·) is the Heaviside step function. Then *r*(*x*, *t*) = *δ*(*t*)*f*_{0}(*x*). As *h*(*x*, *t*) = *δ*(*x *− *vt*), we obtain

Hence equation (18) becomes

where an *x* = *L* substitution describes the flux of recharge impulses at the base of the hillslope and *v*/*s* is an effective velocity used in calibration to the numerical data. Dividing this by the size of the original impulse gives the dimensionless unit hydrograph (equation (13)):

Parameters *γ* and *s* will depend on the properties of the hillslope, though analytical expressions for this relationship have not been derived at this time. Solutions to the subordinated kinematic wave equation (equation (21)) for a wide range of *γ* are shown in Figure 3. Values of *γ* describe the tail index of a *γ* stable subordinator used to regulate the times that the piston impulses participate in the motion process. As values of *γ* decrease, more probability mass is shifted to the tail of the subordinator and the likelihood of extreme events increases. This translates into shorter amounts of time that the recharge impulses actively spend in motion, greater impulse arrival times at the hillslope base, and greater deviations from the homogeneous piston flow response (Figure 3). If heterogeneity imparts a temporal memory on the motion of recharge impulses in hillslopes, then increases in the heterogeneity should be linked to decreases in *γ*.

[21] The mass of the initial piston impulse is conserved by equation (21) for all parameter values. The subordinated solution at early time has a decay rate of −1 + *γ*. Once the piston impulse is depleted, the decay of the subordinated solutions is more rapid (i.e., steeper slope) and transitions to −1 − *γ*. Transition times between the two slopes depend on values of both *γ* and *s*. For example, the transition time for *γ* = 0.9 when *s* = 1 is close to the duration of the classical piston (*γ* = 1) (Figure 3). Lower values of *γ* temporally spread the mass of the piston, and slope transitions are not observed in Figure 3 when *γ* ≤ 0.3.

[22] Calibration of the analytical solutions to data is facilitated by changing values of *s* to apply a temporal scale to the velocity of the piston input. This temporal scale changes the velocity of the piston, thereby changing the magnitude and duration of the piston input (Figure 3). The quotient of *v* and *s* can be viewed as an overall or effective velocity, where *s* > 1 and *s* < 1 represent lower and greater velocities than originally assigned to the piston, respectively (Figure 3). If *s* = 1, the original piston velocity and effective velocity are equal. This method of varying *s* allows us to efficiently calibrate the subordinated solution to the numerical data without changing the piston parameters directly. Since the hydraulic conductivity fields are random and all other hillslope parameters are held equal, the use of *s* is equivalent to dividing the mean piston *K* value (and hence *v*) by *s*.