Model help:TopoFlow-Channels-Dynamic Wave
TopoFlow-Channels-Dynamic Wave
The module is used to compute flow routing in a D8-based, spatial hydrologic model with dynamic wave process.
Model introduction
This process component is part of a spatially-distributed hydrologic model called TopoFlow, but it can now be used as a stand-alone model. The dynamic wave method is the most complete and complex method for modeling flow in open channels. This method retains all of the terms in the full, 1D momentum equation, including the gravity, friction and pressure gradient terms (as used by the diffusive wave method) as well as local and convective acceleration (or momentum flux) terms. This full equation is known as the St. Venant equation. In the current version of TopoFlow it is assumed that the flow directions are static and given by a D8 flow grid. In this case, integral vs. differential forms of the conservation equations for mass and momentum can be used.
Model parameters
Uses ports
This will be something that the CSDMS facility will add
Provides ports
This will be something that the CSDMS facility will add
Main equations
- Mass conservation equation
[math]\displaystyle{ \Delta V \left (i,t \right)=\Delta t \left (R \left (i,t \right) \Delta x \Delta y -Q \left (i,t \right) +\Sigma_{k} Q \left (k,t \right) \right) }[/math] (1)
- Mean water depth in channel segment (if θ > 0 )
[math]\displaystyle{ d=\left (\left (w^2 + 4 \tan \left (\theta\right) V / L\right)^{\frac{1}{2}} -w\right) / \left ( 2 \tan \left (\theta\right)\right) }[/math] (2)
- Mean water depth in channel segment (if θ = 0)
[math]\displaystyle{ d= V / \left (w L \right) }[/math] (3)
- Momentum conservation equation
[math]\displaystyle{ \Delta v \left (i,t \right)=\Delta t \left ( T_{1} + T_{2} + T_{3} + T_{4} + T_{5} \right) / \left (d \left (i,t \right) A_{w}\right) }[/math] (4)
- Efflux term
[math]\displaystyle{ T_{1}=v \left (i,t \right) Q \left (i,t \right) \left ( C - 1 \right) }[/math] (5)
- Influx term
[math]\displaystyle{ T_{2}=\Sigma_{k} \left ( v \left (k,t \right) - v \left (i,t \right) C \right) Q \left (k,t \right) }[/math] (6)
- "New mass" momentum term
[math]\displaystyle{ T_{3}=- v \left (i,t \right) C R \left (i,t \right) \Delta x \Delta y }[/math] (7)
- Gravity term
[math]\displaystyle{ T_{4}=A_{w} \left (g d \left (i,t \right) S \left (i,t \right)\right) }[/math] (8)
- Friction term
[math]\displaystyle{ T_{5}=- A_{w} \left (f\left (i,t \right) v \left (i,t \right)^2 \right) }[/math] (9)
- Discharge of water
[math]\displaystyle{ Q=v A_{w} }[/math] (10)
- Friction factor (for law of the wall)
[math]\displaystyle{ f \left (i,t \right)= \left (/kappa / LN \left (a d \left (i,t \right) / z_{0}\right)\right)^2 }[/math] (11)
- Friction factor for Manning's equation
[math]\displaystyle{ f \left (i,t \right)=g n^2 / R_{h}\left (i,t \right)^{\frac{1}{3}} }[/math] (12)
- Area ratio appearing
[math]\displaystyle{ C=A_{w} / A_{t} }[/math] (13)
- Top surface area of a channel segment
[math]\displaystyle{ A_{t}=w_{t} L }[/math] (14)
- hydraulic radius
[math]\displaystyle{ R_{h}= A_{w} /P_{w} }[/math] (15)
- wetted cross-sectional area of a trapezoid
[math]\displaystyle{ A_{w}= d \left (w + \left (d \tan \left (\theta\right)\right)\right) }[/math] (16)
- Wetted perimeter of a trapezoid
[math]\displaystyle{ P_{w}= w + \left ( 2 d / cos\left (\theta\right)\right) }[/math] (17)
- wetted volume of a trapezoidal channel
[math]\displaystyle{ V_{w}=d^2 \left (L \tan \left (\theta\right)\right) +d \left (L w\right) }[/math] (18)
Symbol | Description | Unit |
---|---|---|
ΔV | change in water volume | m^3 |
Δt | time step | sec |
R | effective rain rate / excess rainrate, represents the sum of all vertical contributions to a grid cell's mass balance | m / s |
Δx | distance change in x direction | m |
Δy | distance change in y direction | m |
w | width of channel | m |
θ | bank angle for trapezoid | deg |
V | water volume | m^3 |
L | channel length | m |
T_{1 } | efflux term | m^4 / s^2 |
T_{2 } | influx term | m^4 / s^2 |
T_{3 } | "new mass" monmentum term | m |
T_{4 } | gravity term | m^4 / s^2 |
T_{5 } | friction term | m |
A_{w} | wetted cross-sectional area of a trapezoid | m^2 |
n | Manning's n | s / m^{1/3} |
P_{w} | wetted perimeter of a trapezoid | m |
S | bed slope | m / m |
g | gravity acceleration | m / s^2 |
z_{0} | roughness length | m |
κ | Von Karman's constant, equals to 0.41 | - |
a | constant | - |
C | constant | - |
A_{t} | top surface area of a channel segment | m^2 |
W_{t} | top width of a wetted trapezoidal cross section | m |
V_{w} | wetted volume of a trapezoidal channel | m |
Output
Symbol | Description | Unit |
---|---|---|
Q | discharge of water | m^3 / s |
v | flow velocity | m / s |
d | mean channel flow depth | m |
f | friction factor | - |
R_{h} | hydraulic radius | m |
S_free | free-surface slope | m / m |
Notes
- Note on input parameters
The input variables for the dynamic wave method should usually be specified as grids, except in special cases.
Single grids and grid sequences are assumed to be stored as RTG and RTS files, respectively. Time series are assumed to be stored as text files, with one value per line. For a time series or grid sequence, the time between values must coincide with the timestep provided.
Flow directions are determined by a grid of D8 flow codes. All grids are assumed to be stored as RTG (RiverTools Grid) files and flow codes are assumed to follow the Jenson (1984) convention (see above) that is also used for RiverTools D8 flow grids. Flow grids and slope grids can be created by RiverTools or a similar program and the other grids can be created using tools in the TopoFlow Create menu.
Bed slope, S, can be computed from a DEM by using the Create → Profile-smoothed DEM dialog or by using hydrologic GIS software. The current version assumes that all channels have trapezoidal cross-sections (see Notes below) but allows bottom-width and bank angle to vary spatially as grids. TopoFlow has pre-processing tools in the Create menu for creating grids of bed width, bank angle and bed roughness. The Create → Channel Geometry Grids → With Area Grid tool allows you to parameterize these variables as power-law functions of contributing area. The Create → Channel Geometry Grids → With HS Order Grid tool allows you to assign values based on Horton-Strahler order.
Each pixel is classified as either a hillslope pixel (overland flow) or a channel pixel (channelized flow) and appropriate parameters must be used for each. For overland flow, w >> d, Rh → d, and bank angle drops out. Overland flow can then be modeled with a large value of Manning's n, such as 0.3. For channelized flow, the variation of n with bed grain size can be modeled with Strickler's equation as explained in the Notes below.
If a sinuosity greater than 1 is specified, then bed slopes are reduced by dividing them by this value. As with the other variables, it is most appropriate to specify a grid in this case.
It is physically unrealistic to specify a spatially uniform initial flow depth by entering a scalar value greater than zero for init_depth. This will result in a very large peak in the hydrograph and may cause TopoFlow to crash. The Create → RTG File for Initial Depth tool can be used to create a grid of initial flow depths that varies spatially and is in steady-state equilibrium with a specified baseflow recharge rate.
- Note on the equations
The dynamic wave method is the most complete and complex method for modeling flow in open channels. This method retains all of the terms in the full, 1D momentum equation, including the gravity, friction and pressure gradient terms (as used by the diffusive wave method) as well as local and convective acceleration (or momentum flux) terms. This full equation is known as the St. Venant equation. In the current version of TopoFlow it is assumed that the flow directions are static and given by a D8 flow grid. In this case, integral vs. differential forms of the conservation equations for mass and momentum can be used.
Conservation of mass, in integral form, is represented by the first three equations above. The quantity, R, that appears in the first equation is known as the effective rainrate or excess rainrate and represents the sum of all vertical contributions to a grid cell's mass balance. R is computed as R = (P + M + G) - (I + E), where P = precipitation, M = snowmelt, G = seepage from subsurface, I = infiltration and E = evapotranspiration. (Note that R is technically not the same as the runoff, since runoff includes horizontal fluxes.) The summation sign in the first equation adds up all horizontal inflows to a grid cell from its neighbor grid cells. Mean channel flow depth, d, is then computed from channel geometry and the water volume that is computed for the corresponding grid cell. Note that channel length depends on distance between grid cell centers and sinuosity, while cross-sections are trapezoidal. When the bank angle, θ is greater than zero, the flow depth required to accomodate the water volume is computed by solving the last equation (a quadratic) for d to get the second equation.
Conservation of momentum, in integral form, is represented by the remaining equations above. The term T3 results from using the equation for mass conservation to simplify the one for momentum conservation. It represents the change in a grid cell's momentum balance due to the addition or subtraction of mass within the cell from vertical fluxes. Recall that momentum represents a product of mass and velocity, so changes in momentum can result from changes in mass or changes in velocity (chain rule).
Notice that boundary-layer theory (which leads to the law of the wall) allows us to compute the loss of momentum due to friction directly as a separate process and isn't really restricted to the special case of steady, uniform flow. By contrast, Manning's equation is empirical and was found by studying steady, uniform flows. It does not allow us to cleanly separate the effects of friction and gravity; it assumes that they are in balance. However, if we use the fact that Manning's equation is really just a power-law approximation to the law of the wall, we can back-calculate a Manning's-equation version of the friction term (and friction factor) which has been freed from the assumption of steady, uniform flow.
In the current version of TopoFlow (1.5 beta), water-surface slopes are set to zero if they ever become negative (implying upstream flow).
- Note on the current version
It has also been assumed here that the water is incompressible; this allows us to take the density of water, ρ, to be a constant.
Examples
An example run with input parameters, BLD files, as well as a figure / movie of the output
Follow the next steps to include images / movies of simulations:
- Upload file: http://csdms.colorado.edu/wiki/Special:Upload
- Create link to the file on your page: [[Image:<file name>]].
See also: Help:Images or Help:Movies
Developer(s)
References
Peckham, S.D. (2008) Geomorphometry and spatial hydrologic modeling, In: Geomorphometry, chapter 25, Elsevier, New York, in press.
Peckham, S.D. and J.D. Smith (2008) Manning's equation and the best power-law approximation to the logarithmic law of the wall, in preparation.
Schlicting, H. (1960) Boundary Layer Theory, 4th ed., McGraw-Hill, New York, 647 pp.