Model help:TopoFlow-Channels-Diffusive Wave
TopoFlow-Channels-Diffusive Wave
The module is used to compute flow routing in a D8-based, spatial hydrologic model with diffusive wave method.
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. It uses the "diffusive wave" method to compute flow velocities for all of the channels in a D8-based river network. This wave method is similar to the kinematic wave method for modeling flow in open channels, but instead of a simple balance between friction and gravity, this method includes the pressure gradient that is induced by a water-depth gradient in the downstream direction. This means that instead of using bed slope in Manning's equation or the law of the wall, the water-surface slope is used. One consequence of this is that water is able to move across flat areas that have a bed slope of zero. Local and convective accelerations in the momentum equations are still neglected, just as is done in the kinematic wave method.
Model parameters
Uses ports
• Meteorology
• Snow (Snowmelt)
• Evap (Evaporation)
• Infil (Infiltration)
• Satzone (Subsurface flow in saturated zone)
• Ice (Icemelt)
• Diversions (sources, sinks, canals)
Provides ports
• Channels (surface water flow in a network of channels)
• Configure (tabbed dialog GUI to change settings)
• Run (only if used as the Driver)
Main equations
- Mass conservation equation
[math]\displaystyle{ \Delta V \left (i,t \right)=\Delta t \ast [ R \left (i,t \right) \Delta x \Delta y -Q \left (i,t \right) +\Sigma_{k} Q \left (k,t \right) ] }[/math] (1)
- Mean water depth in channel segment (if θ > 0 )
[math]\displaystyle{ d=\{ [w^2 + 4 \tan \left (\theta\right) V / L] ^{\frac{1}{2}} -w\} / [ 2 \tan \left (\theta\right)] }[/math] (2)
- mean water depth in channel (if θ = 0)
[math]\displaystyle{ d= V / [ w \ast L] }[/math] (3)
- discharge of water
[math]\displaystyle{ Q=v \ast A_{w} }[/math] (4)
- section-averaged velocity (Manning's formula)
[math]\displaystyle{ v=n^{-1} \ast R_{h}^{\frac{2}{3}} \ast S^{\frac{1}{2}} }[/math] (5)
- section-averaged velocity (Law of the Wall)
[math]\displaystyle{ v=\left (g \ast R_{h} \ast S\right)^{\frac{1}{2}} LN\left (a \ast d / z_{0}\right) /\kappa }[/math] (6)
- hydraulic radius
[math]\displaystyle{ R_{h}= A_{w} /P_{w} }[/math] (7)
- wetted cross-sectional area of a trapezoid
[math]\displaystyle{ A_{w}= d \ast \left (w + \left (d \ast \tan \left (\theta\right)\right)\right) }[/math] (8)
- Wetted perimeter of a trapezoid
[math]\displaystyle{ P_{w}= w + \left ( 2 \ast d / cos\left (\theta\right)\right) }[/math] (9)
- wetted volume of a trapezoidal channel
[math]\displaystyle{ V_{w}=d^2 \ast \left (L \ast \tan \left (\theta\right)\right) +d \ast \left (L \ast w\right) }[/math] (10)
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 |
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 |
V_{w} | wetted volume of a trapezoid channel | 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 | - |
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
Notes on Input Parameters
The input variables for the diffusive 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 (i.e. [NE,E,SE,S,SW,W,NW,N] → [1,2,4,8,16,32,64,128]) 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 such as RiverTools.
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.
Notes on the Equations
All variables and their units can be seen by expanding the Nomenclature section above.
Conservation of mass, in integral form, is represented by the 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.
The diffusive wave method is similar to the kinematic wave method for modeling flow in open channels, but instead of a simple balance between friction and gravity, this method includes the pressure gradient that is induced by a water-depth gradient in the downstream direction. This means that instead of using bed slope in Manning's equation or the law of the wall, the water-surface slope is used. One consequence of this is that water is able to move across flat areas that have a bed slope of zero. Local and convective accelerations in the momentum equations are still neglected, just as is done in the kinematic wave method. For more information, see the help page for the kinematic wave method.
Note on the current version
In the current version of TopoFlow (1.5 beta), water-surface slopes are set to zero if they ever become negative (implying upstream flow).
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: https://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
- Schlicting, H. (1960) Boundary Layer Theory, 4th ed., McGraw-Hill, New York, 647 pp.
Links
- Model help:TopoFlow-Channels-Kinematic Wave
- Model help:TopoFlow-Channels-Dynamic Wave
- Model help:TopoFlow-Diversions
- Model:TopoFlow-Channels-Diffusive Wave (Model metadata)
- Model:TopoFlow