The HRU
dynatop Conceptual Model
One of the underlying principles of dynamic TOPMODEL is that the landscape can
be broken up into hydrologically similar regions, or Hydrological Response
Units (HRUs), where all the area within a HRU behaves in a hydrologically
similar fashion. Initially hydrological similarity was determined through
classifying the topographic wetness index ; though this rapidly developed to
more complex classifications <panola, most subsequent work> making use of, for
example, soil properties or land cover. Alongside this different schemes have
been used fro the spatial discretisation of the catchment prior to
classification into HRUs <ref Manc, Newcastle>; as discussed in <Bene etal
2021? this can be particularly important for ensuring the hyrological ordering
of the HRUs which is significant in infering the impact of potential catchment
interventions.
The methodology presented starts, as with
the orignal work, with a gridded catchment DEM, from this we derive a
distributed model the states of which can then be simplified using a
classification approach in the spirit of the original dynamic TOPMODEL. This
is not the first attempt to produce a distributed hydrological model based on
the underlying hydrological assumptions of TOPMODEL; particularly the
approximation of kinematic subsurface routing; see for example [.]
HRU conceptual model
This document outlines the conceptual structure and computational
implementation of the HRU; which is initially considered as a single grid cell
of the catchment DEM.
The grid cell HRU is considered to have fluxes occuring with it’s eight
adjacent cells.
It is formed of four zones representing the surface water, which
passes water between HRUs and drains to the root zone. The root zone characterises
the interactions between evapotranspiration and precipitation and when full spills to
the unsaturated zone. This drains to the saturated zone which also interacts
with other HRUs. The behaviour of the saturated zone is modelled using a kinematic wave
approximation. The zones and variables used below are shown in the schematic diagram.
In the following sections the governing equations of the HRU are
given in a finite volume form. Approximating equations for the solution of the
governing equations are then derived with associated implicit and semi-implicit numerical
schemes. These are valid for the wide range of surface zone and saturated zone
transmissivity profiles presented in the vignette. Appendices provide supporting information.
Notation
Table 1 outlines the notation used for describing an
infinitesimal slab across the hillslope HRU.
The following conventions are used:
All vertical fluxes \(r_{\star \rightarrow \star}\) are considered to
spatially uniform and to be positive when
travelling in the direction of the arrow, but can, in some cases be negative.
Gradients are positive when going downslope.
\(\left.y\right\rvert_t\) indicates the value of the
variable \(y\) at time \(t\)
Table 1: Outline of notation for describing a cross
section of the hillslope HRU
| Storage |
\(s_{sf}\) |
Surface excess storage |
m\(^3\) |
|
\(s_{rz}\) |
Root zone storage |
m\(^3\) |
|
\(s_{uz}\) |
Unsaturated zone storage |
m\(^3\) |
|
\(s_{sz}\) |
Saturated zone storage deficit |
m\(^3\) |
| Vertical fluxes |
\(r_{sf \rightarrow rz}\) |
Flow from the surface excess store to the root zone |
m\(^3\)/s |
|
\(r_{rz \rightarrow uz}\) |
Flow from the root zone to the unsaturated zone |
m\(^3\)/s |
|
\(r_{uz \rightarrow sz}\) |
Flow from unsaturated to saturated zone |
m\(^3\)/s |
| Lateral fluxes |
\(q_{sf}\) |
Lateral flow in the hillslope surface zone |
m\(^3\)/s |
|
\(q_{sz}\) |
Lateral inflow to the hillslope surface zone |
m\(^3\)/s |
| Vertical fluxes to the HRU |
\(p\) |
Precipitation rate |
m\(^3\)/s |
|
\(e_p\) |
Potential Evapotranspiration rate |
m\(^3\)/s |
| Other HRU properties |
\(A\) |
Plan area |
m\(^2\) |
|
\(w\) |
Width of a hill slope cross section |
m |
|
\(\Delta x\) |
Effective length of the hillslope HRU (x = A/w$) |
m |
|
\(\Theta_{sf}\) |
Further properties and parameters for the solution of the surface zone |
:-: |
|
\(\Theta_{rz}\) |
Further properties and parameters for the solution of the root zone |
:-: |
|
\(\Theta_{uz}\) |
Further properties and parameters for the solution of the saturated zone |
:-: |
|
\(\Theta_{sz}\) |
Further properties and parameters for the solution of the saturated zone |
:-: |
Finite Volume Formulation
In this section a finite volume formulation of the Dynamic TOPMODEL equations
is derived for the evolution of the states over the time step \(t\) to \(t +
\Delta t\). The lateral flux rates are considered at the start and end of the
time step. The vertical flux rates are considered as constant throughout the
time step, allowing for a semi-analytical solution of some storage zones.
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 \(q_{sf} = \frac{q_{sf}^{+}+q_{sf}^{-}}{2}\)
Compound Channels
Compound channels are considered as channels with two regimes corresponding to
different reltionships between the \(q_{sf}\), \(\kappa\) and \(\eta\) depending
upon whether \(s_{sf} \leq s_{c}\), for some storage threshold(s).
This allows a basic representation of surface storage such as runoff attenuation
features, or out of bank channel flow. Examples are given later.
\[
q = a_{1}v_{1} + a_{2}v_{2}
v = q / \left(a_{1} + a_{2}\right)
a_{2} = \frac{q - q_{1}}{v_{2}}
\]
\[
v_{2} = \frac{1}{n}R_{h}^{2/3}S_{0}^{0.5}
\]
\[
R_{h} \approx h
\]
\[
q_{2} = \frac{w}{n}h^{5/3}S_{0}^{0.5}
h^{5/3} = q_{2}\frac{n}{wS_{0}^{0.5}}
\]
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 .
| 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.
| \(T_0\) |
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 |
- |
Numerical Solution
No analytical solution yet exists for simultaneous integration of the system of
ODEs outlined above. In the following an semi-implicit scheme, fluxes
between stores are considered constant over the time step and an ordering of
the HRUs exists which allows a solution to be derived based on the premis that
the inflows can be determined before evolving the HRU states.
The basis of the solution is twofold:
- That a gravity driven system will maximise
the downward flow of water within each timestep of size \(\Delta t\). Hence on
the first downward pass through the stores the maximum downward volumes of the vertical
fluxes \(\hat{v}_{* \rightarrow *} = \Delta t \hat{r}_{* \rightarrow *}\) are determined. These then moderated by the
solution of the saturated zone, with an upward pass giving the final states.
- The upstream inflow to the saturated zone is capped at the receving HRU’s
\(q_{sz}^{max}\) allowing for the generation of “run on” surface flow.
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
Surface excess
The maximum downward flux \(\hat{v}_{sf \rightarrow rz}\) is acheived by
draining the surface store giving
\[
\hat{v}_{sf \rightarrow rz} = \left.s_{sf}\right\rvert_{t} + \Delta t
q_{sf}^{-}\
\]
Root Zone
The implicit formulation for the root zone gives
\[\begin{equation}
\left. s_{rz} \right.\rvert_{t + \Delta t} =
\left( 1 + \frac{e_{p}\Delta t}{s_{rzmax}} \right)^{-1}
\left(
\left. s_{rz} \right.\rvert_{0}
+ \Delta t p +
v_{sf \rightarrow rz} - v_{rz \rightarrow uz} \right)
\end{equation}\]
Since flow to the unsaturated zone occur only to keep
\(s_{rz} \leq s_{rzmax}\) then
\[\begin{equation}
\hat{v}_{rz \rightarrow uz} = \max\left(0,
\left. s_{rz} \right.\rvert_{0}
+ \Delta t \left( p -e_{p}\right)
+ \hat{v}_{sf \rightarrow rz}
- s_{rzmax}
\right)
\end{equation}\]
Unsaturated Zone
The implicit expression for the unsaturated zone is given in terms of
\(\left. s_{uz} \right.\rvert_{t+\Delta t}\) and
\(\left. s_{sz} \right.\rvert_{t+\Delta t}\) as
\[\begin{equation}
\left. s_{uz} \right.\rvert_{t+\Delta t} =
\frac{T_{d} \left. s_{sz} \right.\rvert_{t+\Delta t}}{
T_{d} \left. s_{sz} \right.\rvert_{t+\Delta t} + A \Delta t}
\left( \left. s_{uz} \right.\rvert_{t} +
v_{rz \rightarrow uz}\right)
\end{equation}\]
Since \(\left. s_{uz} \right.\rvert_{t+\Delta t} \leq \left. s_{sz}
\right.\rvert_{t+\Delta t}\) this places an additional condition of
\[\begin{equation}
\left. s_{uz} \right.\rvert_{t} + v_{rz \rightarrow uz} \leq
\left. s_{sz} \right.\rvert_{t+\Delta t} + \frac{A \Delta t}{T_d}
\end{equation}\]
By mass balance this gives the maximum downward flux volume as
\[
A\Delta t \min\left(
\frac{\left( \left. s_{uz} \right.\rvert_{t} + \hat{v}_{rz \rightarrow uz}\right)}
{T_{d}\left. s_{sz} \right.\rvert_{t+\Delta t} + A \Delta t}, \frac{1}{T_{d}} \right)
\]
Saturated Zone
The solution of the saturated zone is complicated by the formulation of the
unsaturated zone solution. The iterative numerical scheme, based on a
semi-implicit step can be formulated to provide a solution
Take an initial outflow estimate \(q^* = q_{sz}^{-}\)
For a number of iterations:
Compute \(q_{sz} = \frac{1}{2}\left(q^* + q_{sz}^{-}\right)\)
Solve for \(h_{sz}\) based on \(q_{sz}\).
Approximate the saturated storage deficit by \(Ah_{sf}\) to give
\[ v_{uz \rightarrow sz}^{*} = \Delta t \min\left(
\frac{\left( \left. s_{uz} \right.\rvert_{t} + \hat{v}_{rz \rightarrow
uz}\right)}
{T_{d}h_{sz} + \Delta t}, \frac{A}{T_{d}} \right)
\]
Take
\[q^{*} = \min \left( q_{sz}^{max}, \max\left(0,
\frac{ Ah_{sz} - \left.s_{sz}\right\rvert_{t} + v_{uz \rightarrow sz}^{*} +
\Delta t q_{sz}^{-} }{ \Delta t } \right) \right)
\]
Set \(\left.q_{sz}^{+}\right\rvert_{t+\Delta t} = q^{*}\) and
compute
\[\left.s_{sz}\right\rvert_{t+\Delta t} = \max\left( 0,\left.s_{sz}\right\rvert_{t} +
\Delta t \left(\left.q_{sz}^{+}\right\rvert_{t+\Delta t} - q_{sz}^{-}
\right) - v_{uz \rightarrow sz}^{*}\right)\]
Upward Pass
Given the results from the downward pass the upward pass starts with mass
balance across the stores giving
\(v_{uz \rightarrow sz} = \left. s_{sz}\right.\rvert_{t} +\Delta t
\left(\left. \tilde{q}_{sz}^{-}\right.\rvert_{t+\Delta t} - \left. \tilde{q}_{sz}^{+}
\right\rvert_{t+\Delta t}\right) - \left. s_{sz}\right.\rvert_{t+\Delta t}\)
\(\left. s_{uz}\right.\rvert_{t+\Delta t} =
\min\left(\left. s_{sz}\right.\rvert_{t+\Delta
t},\left. s_{uz}\right.\rvert_{t} + \hat{v}_{rz \rightarrow uz} - v_{uz
\rightarrow sz}\right)\)
\(v_{rz \rightarrow uz} = \left. s_{uz}\right.\rvert_{t+\Delta t}
-\left. s_{uz}\right.\rvert_{t} + v_{uz \rightarrow sz}\)
\(v_{sf \rightarrow rz} = \min\left(\hat{v}_{sf \rightarrow rz}, s_{rzmax} -
\left. \bar{s}_{rz} \right.\rvert_{0} - \Delta t \left( p - e_p \right) + {v}_{rz
\rightarrow uz} \right)\)
Solve for \(\left. s_{rz}\right.\rvert_{t+\Delta t}\) using the equation
already given.
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
Surface Zone
Surface flow can only occur if there is water in the saturated zone, therfor
if
\[
\left.s_{sf}\right\rvert_{t} + \Delta t q_{sf}^{-} - v_{sf \rightarrow rz} = 0
\]
then \(\left.s_{sf}\right\rvert_{t + \Delta t} =
\left.q_{sf}^{+}\right\rvert_{t + \Delta t} = 0\).
Otherwise proceeding in a similar fashion to the saturated zone
Take an initial outflow estimate \(q^* = q_{sf}^{-}\)
For a number of iterations:
Compute \(q_{sf} = \frac{1}{2}\left(q_{sf}^{-} + q^{*} \right)\)
Solve for \(\kappa_{sf}\) and \(\eta_{sf}\) based on \(q_{sf}\).
Take
\[q^{*} = \max\left(0, \frac{
\left.s_{sf}\right\rvert_{t} - v_{sf \rightarrow rz}^{*} +
\left( \Delta t - \kappa_{sf}\eta_{sf}\right)
q_{sf}^{-} }{
\Delta t + \kappa_{sf}\left(1-\eta_{sf}\right) } \right)
\]
Set \(\left.q_{sz}^{+}\right\rvert_{t+\Delta t} = q^{*}\) and
compute \(\left.s_{sf}\right\rvert_{t+\Delta t} =
\left.s_{sf}\right\rvert_{t} +
\Delta t \left(q_{sf}^{-} -
\left.q_{sf}^{+}\right\rvert_{t+\Delta t}\right) - v_{sf
\rightarrow rz}\)
Surface Zone Representations
Recall that The storage in the surface zone satisfies the mass balance equation
\[
\frac{ds_{sf}}{dt} = q^{-}_{sf} - r_{sf \rightarrow rz} - q^{+}_{sf}
\]
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).
The following subsections present various options for the surface zone
solution based on the Muskingham approximation method.
The Muskingham method offers a parsimonious for capturing 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 routing can be derived in a number of ways. The surface storage \(s_{sf}\) in a reach or hillslope of length \(\Delta
x\) can be expressed in terms of the representative flow \(q_{sf}\) and velocity \(v_{sf}\)
as
\[
s_{sf} = \frac{q_{sf} \Delta x}{v_{sf}}
\]
The representative flow is related to the inflow \(q_{sf}^{-}\) and outflow
\(q_{sf}^{+}\) using the parameter \(\eta\) through
\[
q_{sf} = \eta q_{sf}^{-} + \left(1-\eta\right) q_{sf}^{+}
\]
which gives
\[
s_{sf} = \frac{\Delta x}{v_{sf}} \left(\eta q_{sf}^{-} + \left(1-\eta\right) q_{sf}^{+} \right)
\]
Taking vertical inflow \(r\) the mass balance can be written
\[
\frac{ds}{dt} = q^{-} + r - q^{+}
\]
where substitution of
\[
q_{sf}^{+} = \max\left(0, \frac{1}{1-\eta}\left(\frac{sv}{\Delta x} - \eta q^{-} \right) \right)
\]
gives
\[
\frac{ds}{dt} = q^{-} + r - \max\left(0, \frac{1}{1-\eta}\left(\frac{sv}{\Delta x} - \eta q^{-} \right) \right)
\]
where the maximum ensures that outflow is positive.
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, \(s_sf > s_{c}\)
Appendix A - Monotonicity of \(\mathcal{H}\left(z\right)\)
By definition \(\left. s_{uz} \right.\rvert_{t}\), \(\Delta t\), A, \(T_d\), \(\hat{v}_{rz
\rightarrow uz}\) and \(z\) are all greater or equal to 0. To show that the
gradient of
\[
\mathcal{H}\left(z\right) =
z -
\left. s_{sz}\right.\rvert_{t} +
\Delta t \left. q_{sz}^{-} \right\rvert_{t+\Delta t} +
A\Delta t \min\left(
\frac{\left( \left. s_{uz} \right.\rvert_{t} + \hat{v}_{rz \rightarrow uz}\right)}
{T_{d}z + A \Delta t}, \frac{1}{T_{d}} \right) - \\
\Delta t \min\left(q_{sz}^{max},\max\left( 0, 2 \mathcal{G}\left(z,\Theta_{sz}\right) - \left. q_{sz}^{-}
\right\rvert_{t+\Delta t}
\right)\right)
\]
is strictly positive consider first the gradient of the last term which takes
the value
\[
\frac{d}{dz} \min\left(q_{sz}^{max},\max\left( 0, 2 \mathcal{G}\left(z,\Theta_{sz}\right) - \left. q_{sz}^{-}
\right\rvert_{t+\Delta t}
\right)\right) = \left\{ \begin{array}{cl}
0 & 2 \mathcal{G}\left(z,\Theta_{sz}\right) < \left. q_{sz}^{-} \right\rvert_{t+\Delta t}\\
0 & 2 \mathcal{G}\left(z,\Theta_{sz}\right) - \left. q_{sz}^{-} \right\rvert_{t+\Delta t}> q_{sz}^{max} \\
2 \frac{d}{dz} \mathcal{G}\left(z,\Theta_{sz}\right) & \mathrm{otherwise}
\end{array}
\right.
\]
By definition \(-\frac{d}{dz} \mathcal{G}\left(z,\Theta_{sz}\right) \geq 0\)
hence
\[
\frac{d}{dz} \mathcal{H}\left(z\right) \geq
1 +
A\Delta t \frac{d}{dz} \min\left(
\frac{\left( \left. s_{uz} \right.\rvert_{t} + \hat{v}_{rz \rightarrow uz}\right)}
{T_{d}z + A \Delta t}, \frac{1}{T_{d}} \right)
\]
Separating the range of \(z\) into two sections gives
\[
\frac{d}{dz} \mathcal{H}\left(z\right) \geq \left\{ \begin{array}{cl}
1 & \frac{\left( \left. s_{uz} \right.\rvert_{t} + \hat{v}_{rz \rightarrow uz}\right)}
{T_{d}z + A \Delta t} \geq \frac{1}{T_{d}} \\
1 - T_{d} A\Delta t
\frac{\left. s_{uz} \right.\rvert_{t} + \hat{v}_{rz \rightarrow uz}}
{\left(T_{d}z + A \Delta t\right)^2} & frac{\left( \left. s_{uz} \right.\rvert_{t} + \hat{v}_{rz \rightarrow uz}\right)}{T_{d}z + A \Delta t} \leq
\frac{1}{T_{d}}
\end{array}
\right.
\]
Further using the range of \(z\) valid for the second case
\[
\frac{d}{dz} \mathcal{H}\left(z\right) \geq \left\{ \begin{array}{cl}
1 & \frac{\left( \left. s_{uz} \right.\rvert_{t} + \hat{v}_{rz \rightarrow uz}\right)}
{T_{d}z + A \Delta t} \geq \frac{1}{T_{d}} \\
1 - \frac{A\Delta t}{T_{d}z + A \Delta t} & \frac{\left( \left. s_{uz} \right.\rvert_{t} + \hat{v}_{rz \rightarrow uz}\right)}{T_{d}z + A \Delta t} \leq
\frac{1}{T_{d}}
\end{array}
\right.
\]
From this it can be seen that \(\frac{d}{dz} \mathcal{H}\left(z\right) \geq 0\) with
equality possible when \(z=0\).
