Surface zone

The storage in the surface zone satisfies the mass balance equation

\[ \frac{ds_{sf}}{dt} = q_{sf}^{-} - q_{sf}^{+} - r_{sf \rightarrow rz} \]

where surface storage is increased by lateral downslope flow from upslope HRUs. Water flows to the root zone at a constant rate \(r_{sf \rightarrow rz}\), unless limited by the available storage at the surface or the ability of the root zone to receive water (for example if the saturation storage deficit is 0 and the root zone storage is full). In cases where lateral flow in the saturated zone produce saturation storage deficits water may be returned from the root zone to the surface giving negative values of \(r_{sf \rightarrow rz}\).

The Muskingham-Cunge method offers a parsimonious method for relating inflow, outflow and stroage that can capture a range of surface dynamics. This section outlines the key equations and shows how they may be used to capture a range of conceptual models. Muskingham-Cunge routing can be derived in a number of ways. Reggiani et al. give the following equation for the surface storage \(s_{sf}\) in a reach or hillslope of length \(\Delta x\) in terms of a representative velocity \(v_{sf}\), celerity \(c_{sf}\) and dispersion \(D_{sf}\) as \[ s_{sf} = \kappa \left(\eta q_{sf}^{-} + \left(1-\eta\right) q_{sf}^{+} \right) \] where \[ \kappa = \frac{\Delta x}{v_{sf}} \] and \[ \eta = \frac{1}{2}\left( 1 - \frac{2D_{sf} v_{sf}}{c_{sf}^{2} \Delta x} \right) \] The dispersion \(D_{sf}\) varies depending upon the formulation required. A diffusion wave approximation takes \[ D_{sf} = \frac{q_{sf}}{2BS_f} \approx \frac{q_{sf}}{2BS_0} \] For a Kinematic wave formulation \(D_{sf}=0\) while a linear tank with time constant \(\frac{\Delta x}{v_{sf}}\) is given by \(\eta=0\).

In the numerical solution we consider the representaive values to be computed with reference to the representative flow \(q_{sf} = \frac{q_{sf}^{+}+q_{sf}^{-}}{2}\)

Root zone

The root zone gains water from precipitation and the surface zone. It loses water through evaporation and to the unsaturated zone. All vertical fluxes are considered to be spatially uniform. The root zone storage satisfies

\[\begin{equation} 0 \leq s_{rz} \leq s_{rzmax} \end{equation}\]

with the governing ODE \[\begin{equation} \frac{ds_{rz}}{dt} = p - \frac{e_p}{s_{rzmax}} s_{rz} + r_{sf\rightarrow rz} - r_{rz \rightarrow uz} \end{equation}\]

Fluxes from the surface and to the unsaturated zone are controlled by the level of root zone storage along with the state of the unsaturated and saturated zones.

For \(s_{rz} \leq s_{rzmax}\) then \(r_{rz \rightarrow uz} \leq 0\). Negative values of \(r_{rz \rightarrow uz}\) may occur only when water is returned from the unsaturated zone due to saturation caused by lateral inflow to the saturated zone.

When \(s_{rz} = s_{rzmax}\) then \[\begin{equation} p - e_p + r_{sf\rightarrow rz} - r_{rz \rightarrow uz} \leq 0 \end{equation}\] In this case \(r_{rz \rightarrow uz}\) may be positive if \[\begin{equation} p - e_p + r_{sf\rightarrow rz} > 0 \end{equation}\] so long as the unsaturated zone can receive the water. If \(r_{rz \rightarrow uz}\) is ‘throttled’ by the rate at which the unsaturated zone can receive water, then \(r_{sf\rightarrow rz}\) is adjusted (potentially becoming negative) to ensure the equality is met.

Unsaturated Zone

The unsaturated zone acts as a non-linear tank subject to the constraint \[\begin{equation} 0 \leq s_{uz} \leq s_{sz} \end{equation}\]

The governing ODE is written as \[\begin{equation} \frac{ds_{uz}}{dt} = r_{rz \rightarrow uz} - \frac{A s_{uz}}{T_d s_{sz}} \end{equation}\]

If water is able to pass freely to the saturated zone, then it flows at the rate \(\frac{A s_{uz}}{T_ds_{sz}}\). In the situation where \(s_{sz}=s_{uz}\) the subsurface below the root zone can be considered saturated, as in there is no further available storage for water, but separated into two parts: an upper part with vertical flow and a lower part with lateral flux.

It is possible that the downward flux is constrained by the ability of the saturated zone to receive water. If this is the case \(r_{uz \rightarrow sz}\) occurs at the maximum possible rate and \(r_{rz \rightarrow uz}\) is limited to ensure that \(s_{uz} \leq s_{sz}\).

Saturated Zone

<Beven et al.> propose a Kinematic subsurface approximation. As outlined for the surface zone a kinematic model can be represented using a Muskingum-Cunge formulation. Storage in the saturated zone is given in terms of the storage deficit \(s_{sz}\); evolving as

\[ \frac{d}{dt}s_{sz} = q_{sz}^{+} - r_{uz \rightarrow sz} - q_{sz}^{-} \]

The flow across a given cross section is a function of the depth to the saturated layer \(h_{sz}\) with the functional form \(\mathcal{G}\left(h_{sz}, \Theta_{sz}\right)\) taking its maximum value \(q_{sz}^{max}\) when \(h_{sz}=0\). In Appendix A we show that the Muskingham-Cunge storage equation can be expressed using a representative flow \(q_{sz}\) and corresponding depth \(h_{sf}\) as

\[ s_{sf} = \frac{\Delta x}{v_{sf}}\left( q_{sz}^{max} - q_{sz} \right) = w h_{sz} \Delta x = A h_{sz} \]

Table 2 gives \(\mathcal{G}\left(h_{sz},\Theta_{sz}\right)\) for various the transmissivity profiles present as options within the dynatop package, the corresponding value to use for the type value in the saturated zone specification and the additional parameters required. The additional parameters are defined in Table 3.

Table 2: Transmissivity profiles available with the dynatop package .

Name	\(G\left(\bar{s}_{sz},\Theta_{sz}\right)\)	\(\Theta_{sz}\)	type value	Notes
Exponential	\(T_{0}w\sin\left(\beta\right)\exp\left(-\frac{\cos\beta}{Am}s_{sz}\right)\)	\(T_0,\beta,m\)	exp	Originally given in Beven & Freer 2001
Bounded Exponential	\(T_{0}w\sin\left(\beta\right)\left(\exp\left(-\frac{\cos\beta}{Am}s_{sz}\right) - \exp\left(-\frac{\cos\beta}{m}D\right)\right)\)	\(T_0,\beta,m,D\)	bexp	Originally given in Beven & Freer 2001
Constant Celerity	\(\frac{c_{sz}w}{A}\left(AD-s_{sz}\right)\)	\(c_{sz},D\)	cnst
Double Exponential	\(T_{0}w\sin\left(\beta\right)\left( \omega\exp\left(-\frac{\cos\beta}{Am}s_{sz}\right) + \left(1-\omega\right)\exp\left(\frac{\cos\beta}{Am_2}s_{sz}\right)\right)\)	\(T_0,\beta,m,m_2,\omega\)	dexp

Table 3: Additional parameters used in the transmissivity profiles.

Symbol	Description	unit
\(T_0\)	Transmissivity at saturation	m\(^2\)/s
\(m\),\(m_2\)	Exponential transmissivity constants	m\(^{-1}\)
\(\beta\)	Angle of hill slope	rad
\(c_{sz}\)	Saturated zone celerity	m/s
\(D\)	Storage deficit at which zero lateral flow occurs	m
\(\omega\)	weighting parameter	-

Rebalance Inflows

Limit the inflow to the saturated zone to the maximum lateral saturated zone flow giving

\[ q_{sz}^{-} = \min\left( q_{sz}^{max}, \left. q_{sz}^{-} \right\rvert_{t+\Delta t} \right) \] \[ q_{sf}^{-} = \left. q_{sf}^{-} \right\rvert_{t+\Delta t} + \left. q_{sz}^{-} \right\rvert_{t+\Delta t} - q_{sz}^{-} \]

Downward Pass

Upward Pass

Given the results from the downward pass the upward pass starts with mass balance across the stores giving

1. \(v_{uz \rightarrow sz} = \left. s_{sz}\right.\rvert_{t} +\Delta t \left(\left. \tilde{q}_{sz}^{-}\right.\rvert_{t+\Delta t} - \left. \tilde{q}_{sz}^{+} \right\rvert_{t+\Delta t}\right) - \left. s_{sz}\right.\rvert_{t+\Delta t}\)

2. \(\left. s_{uz}\right.\rvert_{t+\Delta t} = \min\left(\left. s_{sz}\right.\rvert_{t+\Delta t},\left. s_{uz}\right.\rvert_{t} + \hat{v}_{rz \rightarrow uz} - v_{uz \rightarrow sz}\right)\)

3. \(v_{rz \rightarrow uz} = \left. s_{uz}\right.\rvert_{t+\Delta t} -\left. s_{uz}\right.\rvert_{t} + v_{uz \rightarrow sz}\)

4. \(v_{sf \rightarrow rz} = \min\left(\hat{v}_{sf \rightarrow rz}, s_{rzmax} - \left. \bar{s}_{rz} \right.\rvert_{0} - \Delta t \left( p - e_p \right) + {v}_{rz \rightarrow uz} \right)\)

5. Solve for \(\left. s_{rz}\right.\rvert_{t+\Delta t}\) using the equation already given.

6. Compute the volume loss to evapotranspiration as \(\left. v_{e_{p}}\right.\rvert_{t+\Delta t} = \left. s_{rz}\right.\rvert_{t} + v_{sf \rightarrow rz} - v_{rz \rightarrow uz} + \Delta t p - \left. s_{rz}\right.\rvert_{t+\Delta t}\)

This leaves the solution of the surface zone

Approximating Tank Models

Taking \(\eta=0\) gives \[ \frac{ds}{dt} = q^{-} + r - \frac{sv}{\Delta x} \] which is the equation of a tank model with possibly varying time constant \(T = \frac{\Delta x}{v}\)

Approximating Diffusion Wave Routing

Diffuse routing with lateral inflow can be expressed as the parabolic equation (see for example doi:10.1016/j.advwatres.2005.08.008)

\[ \frac{dq}{dt} + c\frac{dq}{dx} - D \frac{d^2 q}{dx^2} = cl - D \frac{dl}{dx} \] where \(l\) is lateral inflow per unit length. Simplify this by considering that the lateral inflow \(r\) uniformly distributed so that \(l=\frac{r}{\Delta x}\) and \(\frac{dl}{dx}=0\) to give

\[ \frac{dq}{dt} + c\frac{dq}{dx} - D \frac{d^2 q}{dx^2} = cl \]

Let us relate this to the Muskingham Solution using the relationship \[ q = \eta q^{-} + \left(1-\eta\right) q^{+} \]

Taking Taylor series expansions for \(q^{-}\) and \(q^{+}\) based on \(q\) gives

\[ q^{+} \approx q + \eta \Delta x \frac{dq}{dx} + \frac{1}{2}\eta^2\Delta x^2 \frac{d^2 q}{dx^2} \] and \[ q^{-} \approx q - \left(1-\eta\right) \Delta x \frac{dq}{dx} + \frac{1}{2}\left(1-\eta\right)^2 \Delta x^2 \frac{d^2 q}{dx^2} \]

Subtracting the expression for \(q^{-}\) from that for \(q^{+}\) gives

\[ q^{-} - q^{+} \approx -\Delta x \frac{dq}{dx} + \frac{1}{2}\left(1-2\eta\right) \Delta x^2 \frac{d^2 q}{dx^2} \]

From the mass balance condition \[ \frac{ds}{dt} \approx \Delta x l -\Delta x \frac{dq}{dx} + \frac{1}{2}\left(1-2\eta\right)\Delta x^2 \frac{d^2 q}{dx^2} \]

Using the expression for storage and relationship between flow and area \[ \frac{dq}{dt} = \frac{dq}{da} \frac{da}{dt} = \frac{c}{\Delta x} \frac{ds}{dt} \]

Then combining this with the approximation produces \[ \frac{dq}{dt} + c\frac{dq}{dx} - \frac{c}{2}\left(1-2\eta\right)\Delta x \frac{d^2 q}{dx^2} \approx c l \]

Comparison to the original diffuse routing equation shows that \[ \eta \approx \frac{1}{2} - \frac{D}{c \Delta x} \]

The value of \(\eta\) is limited to between 0 and \(\frac{1}{2}\). Firstly \(\eta = 1/2\) results from \(D=0\); that is the Kinematic wave equation. Taking \(\eta=0\) produces the maximum dispersion of \(D = \frac{c\Delta x}{2}\), which is given by the linear tank.

Compound Channels

Compound channels are considered as channels with two regimes corresponding to different reltionships between the \(s_{sf}\), \(q_{sf}\) and \(\eta\) depending upon whether \(s_{sf} \leq s_{c}\), for some storage threshold \(s_{1}\). This allows a basic representation of surface storage such as runoff attenuation features, or out of bank channel flow.

In solving this inflow is partitioned to ensure, where possible, \(s_sf > s_{c}\)

Available Formulations

The different formulation available in the model are given in Table 4 with their parameters outlined in Table 5

Table 4: Outline formulations for the storage-flow relationship in the surface zone.

Type	Description	Parameters	\(\mathcal{F}\left(s_{sf},q^{-}_{sf},r_{sf \rightarrow rz},\Theta_{sf}\right)\)
cnst	Linear Tank coupled below a constant parameter diffusive wave solutions	\(t_{raf}, s_{raf},c_{sf}, d_{sf}\)	\[\begin{equation}\begin{array}{l} \eta &=& \frac{1}{2} - \frac{d_{sf}}{c_{sf}\Delta x} \\ q^{+}_{sf} &=& \frac{\min\left(s_{sf},s_{raf}\right)}{t_raf} + \max\left(0, \frac{1}{1-\eta}\left( \frac{c_{sf}}{\Delta x}\max\left(0,s_{sf}-s_{raf}\right) - \eta \hat{q}_{sf}^{-} \right) \right) \end{array}\end{equation}\]
kin	Linear tank with Kinematic Wave for remaining flow	\(t_{raf}, s_{raf}, n, w_{sf}, g_{sf}\)	\[\begin{equation} \begin{array}{l} q_{sf} &=& \frac{g_{sf}^{1/2}}{n w_{sf}^{2/3}} \left(\frac{\max\left(0,s_{sf}-s_{raf}\right)}{\Delta x}\right)^{5/3} \\ q^{+}_{sf} &=& \frac{\min\left(s_{sf},s_{raf}\right)}{t_raf} + \max\left(0,2\left( q_{sf} - \frac{1}{2} \hat{q}_{sf}^{-} \right) \right) \end{array}\end{equation}\]
comp	Two constant parameter diffusive wave solutions	\(v_{sf,1}, d_{sf,1}, s_{1}, v_{sf,2}, d_{sf,2}\)	\[\begin{equation}\begin{array}{l} \eta_{1} &=& \frac{1}{2} - \frac{d_{sf,1}}{v_{sf,1}\Delta x} \\ \eta_{2} &=& \frac{1}{2} - \frac{d_{sf,2}}{v_{sf,2}\Delta x} \\ q^{+}_{sf} &=& \max\left(0, \frac{1}{1-\eta_{1}}\left( \frac{v_{sf,1}}{\Delta x}\max\left(s_{sf},s_{1}\right) - \eta_{1} \hat{q}_{sf,1}^{-} \right) \right) + \max\left(0, \frac{1}{1-\eta_{2}}\left( \frac{v_{sf,2}}{\Delta x}\max\left(0,s_{sf}-s_{1}\right) - \eta_{2} \hat{q}_{sf,2}^{-} \right) \right) \end{array}\end{equation}\]

Table 5: Parameters used in the surface zone storage-flow relationships.

Symbol	Description	unit
\(t_{raf}\)	Time constant of the runoff attenuation feature	s
\(c_{sf}\)	Free flowing surface water velocity	ms\(^{-1}\)
\(s_{raf}\)	storage of the runoff attenuation feature	m\(^3\)
\(d_{sf}\)	Free flowing surface water diffusion rate	m\(^2\)s\(^{-1}\)
\(w_{sf}\)	Width of surface channel	m
\(g_{sf}\)	Gradient of surface channel	-
\(n\)	Manning \(n\) coefficent for roughness	sm\(^{-1/3}\)