Model help:TopoFlow-Snowmelt-Energy Balance
TopoFlow-Snowmelt-Energy Balance
This module is the snowmelt process component (Energy Balance method) for a D8-based, spatial hydrologic model
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.
Model parameters
Uses ports
• Meteorology (for air temperature, T_air, etc.)
Provides ports
• Snow (snowmelt)
• Configure (tabbed dialog GUI to change settings)
• Run (only if used as the Driver)
Main equations
- Meltrate
[math]\displaystyle{ M= \left ( 1000 \ast Q_{m}\right) / \left ( \rho_{water} \ast L_{f}\right) }[/math] (1)
- Max possible meltrate
[math]\displaystyle{ M_{max}= \left ( 1000 \ast h_{snow} / dt\right) \ast \left ( \rho_{water} / \rho_{snow}\right) }[/math] (2)
- Change in snow depth
[math]\displaystyle{ dh_{snow}= M \ast \left ( \rho_{water} / \rho_{snow}\right) \ast dt }[/math] (3)
- Energy flux used to melt snow
[math]\displaystyle{ Q_{m}= Q_{SW} + Q_{LW} + Q_{h} + Q_{e} - Q_{cc} }[/math] (4)
- Sensible heat flux
[math]\displaystyle{ Q_{h}= \rho_{air} \ast c_{air} \ast D_{h} \ast \left (T_{air} - T_{surf} \right) }[/math] (5)
- Latent heat flux
[math]\displaystyle{ Q_{e}= \rho_{air} \ast L_{v} \ast D_{e} \ast \left ( 0.662 / p_{0} \right ) \ast \left ( e_{air} - e_{surf} \right ) }[/math] (6)
- Bulk exchange coefficient
[math]\displaystyle{ D_{n}= \kappa^2 \ast u_{z} / LN [ \left ( z - h_{snow}\right) / z0_{air}]^2 }[/math] (7)
- Bulk exchange coefficient for heat
[math]\displaystyle{ D_{h}= \left\{\begin{matrix} D_{n} / [1 + \left (10 \ast Ri \right) ] & stable: T_{air} \gt T_{surf} \\ D_{n} \ast [ 1 - \left ( 10 \ast Ri \right)] & unstable: T_{air} \lt T_{surf} \end{matrix}\right. }[/math] (8)
- Bulk exchange coefficient for vapor
[math]\displaystyle{ D_{e}= D_{h} }[/math] (9)
- Richardson's number
[math]\displaystyle{ Ri= g \ast z \ast \left (T_{air}- T_{surf} \right) / [ u_{z}^2 \left ( T_{air} + 273.15 \right)] }[/math] (10)
- Initial cold content
[math]\displaystyle{ E_{cc} [ 0 ] = h0_{snow} \ast \rho_{snow} \ast c_{snow} \ast \left (T_{0} - T_{snow} \right) }[/math] (11)
- Vapor pressure of air
[math]\displaystyle{ e_{air}= e_{sat} \left ( T_{air} \right) \ast RH }[/math] (12)
- Vapor pressure at surface
[math]\displaystyle{ e_{surf}= e_{sat} \left ( T_{surf} \right) }[/math] (13)
- Saturation vapor pressure (mbar), Brutsaert (1975)
[math]\displaystyle{ e_{sat}=6.11 \ast exp [ \left (17.3 \ast T\right) /\left (T + 237.3 \right) ] }[/math] (14)
Symbol | Description | Unit |
---|---|---|
Q_{SW} | net shortwave radiation | W / m^3 |
Q_{LW} | net longwave radiation | W / m^2 |
T_{air} | air temperature | deg C |
T_{surf} | surface (snow) temperature | deg C |
RH | relative humidity in [0,1] | - |
P_{0} | atmospheric pressure | mbar |
u_{z} | wind velocity at height z | m / s |
z | reference height for wind | m |
z0_{air} | surface roughness height | m |
h0_{snow} | initial snow depth | m |
ρ_{snow} | density of the snow | kg / m^3 |
c_{snow} | wetted perimeter of a trapezoid | m |
ρ_{air} | density of the air | kg / m^3 |
c_{air} | specific heat capacity of air | kg deg C |
L_{f} | latent heat of fusion, water (334000) | J / kg |
L_{v} | latent heat of vaporization, water (2500000) | J / kg |
e_{air} | air vapor pressure at height z | mbar |
e_{surf} | vapor pressure at the surface | mbar |
g | gravitational constant, equals to 9.81 | m / s^{2} |
κ | von Karman's constant = 0.41 | - |
M | meltrate | mm / sec |
Q_{m} | energy flux used to melt snow | W / m^{2} |
M_{max} | max possible meltrate | mm / sec |
ρ_{water} | density of water | kg / m^{3} |
dh_{snow} | change in snow depth | m |
Q_{h} | sensible heat flux | W / m^{2} |
Q_{e} | latent heat flux | W / m^{2} |
Q_{cc} | cold content flux | W / m^{2} |
D_{h} | bulk exchange coefficient for heat | m / s |
D_{n} | bulk exchange coefficient (neutrally stable conditions) | m / s |
Ri | Richardson's number | |
E_{cc} | initial cold content | J / m^{2} |
T_{0} | initial temperature | degree C |
e_{sat} | saturation vapor pressure | mbar |
T | temperature | deg C |
T_{snow} | snow temperature | deg C |
D_{e} | bulk exchange coefficient for vapor | m / s |
Notes
Notes on Input Parameters
For each variable, you may choose from the droplist of data types. For the "Scalar" data type, enter a numeric value with the units indicated in the dialog. For the other data types, enter a filename. Values in files must also have the indicated units.
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.
Note: If net total radiation has been measured, it can be entered as Q_{SW} and then Q_{LW} can be set to zero in the Meteorology component that this component uses.
Notes on the Equations
All variables and their units can be seen by expanding the Nomenclature section above.
The cold content of the snow pack, E_{cc}, represents an energy deficit that must be overcome before snow begins to melt. First, Q_{net} is computed as the sum of all energy fluxes (the Q's). Wherever (Q_{net} < 0 and h_{snow} > 0) the snow cools and the cold content increases. Similarly, wherever (Q_{net} > 0 and h_{snow} > 0) the snow warms and the cold content decreases. In both cases the cold content changes according to: E_{cc} = [E_{cc} - (Q_{net} * dt)] and we have M=0 as long as (E_{cc} > 0). However, if warming continues long enough to consume the cold content (so that Ecc drops to zero), then the snow begins to melt (M > 0). In this case the meltrate is given by M = Q_{net} / (ρ_{water} * L_{f}).
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
Brutsaert, W. (1975) On a derivable formula for long-wave radiation from clear skies, Water Resources Research, 11, 742-744.
Dingman, S.L (2002) Physical Hydrology, 2nd ed., Prentice Hall, New Jersey. (see Chapter 7, pp. 285-299)
Schlicting, H. (1960) Boundary Layer Theory, 4th ed., McGraw-Hill, New York, 647 pp.
Zhang, Z., D.L. Kane and L.D. Hinzman (2000) Development and application of a spatially-distributed Arctic hydrological and thermal process model (ARHYTHM), Hydrological Processes, 14, 1017-1044.
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