Model help:TopoFlow-Meteorology
TopoFlow-Meteorology
The module is the meteorology process component for a D8-based, spatial hydrologic model
Model introduction
This component reads a variety of variables for the atmosphere and for the land surface from input files or as simple scalars. It then computes many additional variables, such as vapor pressure, e_{air}, and net shortwave (solar) radiation, Q_{SW}, using built-in shortwave and longwave radiation calculators that are based on celestial mechanics and widely-used empirical relationships. These additional variables are needed by the Snowmelt → Energy Balance and Evaporation → Energy Balance components. Direct, diffuse and back-scattered radiation fluxes are all modeled. Properties of the atmosphere (e.g. precipitation rate, P, air temperature, T_{air}, relative humidity, RH, and dust attenuation, γ), are used as well as surface/topographic properties (e.g. slope angle, aspect angle and surface albedo). The approach used here closely follows the one outlined in Appendix E of Dingman (2002). However, instantaneous vs. day-integrated radiation fluxes are used and the optical air mass is modeled using the widely used method of Kasten and Young (1989).
Model parameters
Uses ports
• Snow (Snowmelt)
Provides ports
• Meteorology
• Configure (tabbed dialog GUI to change settings)
• Run (only if used as the Driver)
Main equations
Equations Used to Compute Longwave Radiation
- Net longwave radiation
<math> Q_{LW} = \left (LW_{in} - LW_{out}\right ) </math> (1)
- Incoming longwave radiation (using Stefan-Bolzman law)
<math> LW_{in} = \varepsilon_{air} \cdot \sigma \cdot \left (T_{air} + 273.15 \right )^4 </math> (2)
- Outgoing longwave radiation (using Stefan-Bolzman law)
<math> LW_{out} = \varepsilon_{surf} \cdot \sigma \cdot \left (T_{surf} + 273.15 \right )^4 + \left ( 1 - \varepsilon_{surf} \right ) \, LW_{in} </math> (3)
- Saturation vapor pressure, Brutsaert (1975) method
<math> e_{sat} = 0.611 \cdot \exp [\left ( 17.3 \cdot T \right ) / \left ( T + 237.3 \right )] </math> (4)
- Saturation vapor pressure, Satterlund (1979) method
<math> e_{sat} = \frac{10^\left[11.4 - (2353 / (T + 273.15)) \right]}{1000} </math> (4)
- Vapor pressure of air
<math> e_{air} = e_{sat} \cdot RH </math> (5)
- Emissivity of air
<math> \varepsilon_{air} = \left[\left (1 - F \right ) \, 1.72 \, \left[e_{air}/ \left (T_{air} + 273.15 \right ) \right]^ \left ({\frac{1}{7}} \right ) \, \left (1 + 0.22 \cdot C^2 \right ) \right] + F </math> (6)
Equations Used to Compute Shortwave Radiation
- Day angle or fractional year
<math> \Gamma = 2 \, \pi \, \left ( J - 1 \right ) / 365 </math> (1)
- Earth orbit eccentricity correlation
<math> \begin{align} E_{0} &= \left ( r_{0} / r \right )^2 \\ &= 1.000110 + 0.034221 \cdot \cos \left ( \Gamma \right ) + 0.001280 \cdot \sin \left ( \Gamma \right ) + \\ & \qquad \qquad \qquad 0.000719 \cdot \cos \left ( 2 \, \Gamma \right ) + 0.000077 \cdot \sin \left ( 2 \, \Gamma \right ) \end{align} </math>
(2)
- Declination of the sun at a given latitude
<math> \begin{align} \delta &= \left( 180 / \pi \right) \, [ 0.006918 - 0.399912 \cdot \cos \left( \Gamma \right) + 0.070257 \cdot \sin \left( \Gamma \right) - \\ & \qquad \qquad \qquad \qquad 0.006758 \cdot \cos \left( 2 \, \Gamma \right) + 0.000907 \cdot \sin \left( 2 \, \Gamma \right) - \\ & \qquad \qquad \qquad \qquad 0.002697 \cdot \cos \left( 3 \, \Gamma \right) + 0.00148 \cdot \sin \left( 3 \, \Gamma \right) ] \end{align} </math>
(3)
- Zenith angle
<math> \theta = \arccos [\sin \left ( \Lambda \right ) \, \sin \left ( \delta \right ) + \cos \left ( \Lambda \right ) + \cos \left ( \Lambda \right ) \, \cos \left (\delta \right ) \, \cos \left ( \omega \, t_{h} \right )] </math> (4)
- Sunrise offset, horizontal plane (hours before true solar noon)
<math> T_{hr} = - {\frac{ \arccos \left[- \tan \left( \Lambda \right) \, \tan \left ( \delta \right ) )\right] }{\omega}} </math> (5)
- Sunset offset, horizontal plane (hours after true solar noon)
<math> T_{hs} = + {\frac{ \arccos \left[- \tan \left ( \Lambda \right ) \, \tan \left ( \delta \right ) \right]}{\omega}} </math> (6)
- Instantaneous extraterrestrial radiation flux on a horizontal plane
<math> k_{ET}^' = I_{sc} \, E_{0} \, [\cos \left ( \delta \right ) \, \cos \left ( \Lambda \right ) \, \cos \left ( \omega \, t \right ) + \sin \left ( \delta \right ) \, \sin \left ( \Lambda \right )] </math> (7)
- Dew point temperature
<math> T_{dew} = {\frac{ \ln \left ( e\right ) + 0.4926}{0.0708 - 0.00421 \, \ln \left ( e \right )}} </math> (8)
- Precipitable water content
<math> W_{p} = 1.12 \, \exp \left ( 0.0614 \, T_{dew} \right ) </math> (9)
- Optical air mass (from Kasten and Young, 1989)
<math> M_{opt} = 1 / [\cos \left ( \theta \right ) + 0.50572 \, \left ( 96.07995 - \theta \right ) ^ \left (-1.6364 \right )] </math> (10)
- Total atmospheric transmissivity
<math> \tau = \exp \left[ -0.124 - 0.0207 \, W_{p} - \left (0.0682 + 0.0248 \, W_{p} \right ) \, M_{opt} \right] </math> (11)
- Instantaneous direct radiation flux
<math> k_{dir}^ ' = \tau \, k_{ET}^ ' </math> (12)
- Scattering attenuation
<math> \gamma_{s} = 1 - \exp \left [ -0.0363 - 0.0084 \, W_{p} - \left (0.0572 + 0.0173 \, W_{p} \right ) \, M_{opt} \right ] + \gamma_{dust} </math> (13)
- Instantaneous diffuse radiation flux
<math> k_{dif}^ ' = 0.5 \, \gamma_{s} \, k_{ET} ^ ' </math> (13)
- Instantaneous global radiation flux
<math> k_{global}^' = k_{dir}^' + k_{dif}^' </math> (14)
- Backscattered radiation flux
<math> k_{bs}^' = 0.5 \, a \, \gamma_{s} \, k_{global}^' </math> (15)
- Longitude offset for sloped topography
<math> \Delta \Omega = \arctan \left[{\frac{\sin \left ( \beta \right ) \, \sin \left ( \alpha \right )}{\cos \left ( \beta \right ) \, \cos \left ( \Lambda \right ) - \sin \left ( \beta \right ) \, \sin \left ( \Lambda \right ) \, \cos \left ( \alpha \right )}} \right] </math> (16)
- Equivalent latitude for sloped topography
<math> \Lambda_{eq} = \arcsin \left[ \sin \left (\beta \right ) \, \cos \left ( \alpha \right ) \, \cos \left ( \Lambda \right ) + \cos \left ( \beta \right ) \, \sin \left ( \Lambda \right ) \right] </math> (17)
- Sunrise offset for sloped topography
<math> T_{sr} = - {\frac{ \arccos \left[ - \tan \left ( \Lambda_{eq} \right ) \, \tan \left ( \delta \right ) \right] - \Delta \Omega}{\omega}} </math> (18)
- Sunset offset for sloped topography
<math> T_{ss} = + {\frac{ \arccos \left[ - \tan \left ( \Lambda_{eq} \right ) \, \tan \left ( \delta \right ) \right] - \Delta \Omega}{\omega}} </math> (19)
- Instantaneous extra-terrestrial radiation flux for sloped topography
<math> k_{ET} = I_{sc} E_{\theta} \left[ \cos \left ( \delta \right ) \cos \left ( \Lambda_{eg} \right ) \cos \left ( \omega \, t_{h} + \Delta \Omega \right ) + \sin \left ( \delta \right ) \sin \left ( \Lambda_{eg} \right ) \right] </math> (20)
- Instantaneous clear sky radiation for sloped topography
<math> k_{cs} = \tau \, k_{ET} + k_{dif}^' + k_{bs} ^' </math> (21)
- Net shortwave radiation for sloped topography
<math> Q_{SW} = k_{cs} </math> (21)
- Earth rotation rate
<math> \Omega = 2 \, \pi \, {DPY}_{sidereal} </math> (22)
Symbol | Description | Unit |
---|---|---|
ρ_{H2O} | density of water | kg / m^3 |
Cp_{air} | heat capacity of air | J / kg / K |
ρ_{air} | density of air | kg / m^3 |
precip. rate | precipitation rate | mm / hr |
T_{air} | air temperature | deg C |
T_{surf} | surface temperature | deg C |
p_{0} | atmospheric pressure | mbar |
z | reference height for u_{z} | m |
u_{z} | wind velocity at reference height | m / s |
z_{0} | surface roughness length scale for wind | m |
albedo | surface albedo | unitless, in [0,1] |
em_{air} = ε_{air} | emissivity of air | unitless, in [0,1] |
dust atten. | dust attenuation factor | unitless, typically in [0,0.3] |
cloud factor | cloud factor | unitless, in [0,1], 0 for no clouds |
canopy factor | forest canopy factor | unitless, in [0,1], 0 for no canopy |
slope | topographic slope = rise/run | unitless or m/m, in [0, infinity] |
slope grid file | as flat binary, row-major file with 4-byte floats | radians |
aspect angle | aspect angle | radians measured CW from due North ##### CHECK |
aspect grid file | as flat binary, row-major file with 4-byte floats | radians measured CW from due North ##### CHECK |
time zone offset | offset from Greenwich Mean Time (GMT); <0 for east of prime meridian, >0 otherwise | hours |
start month | start month for solar radiation calculations | month number |
start day | start day for solar radiation calculations | day of the month number |
start hour | start hour for solar radiation calculations | decimal hours, 24-hour clock |
h | height above mean sea level | m |
R | mean earth radius | m |
M_{opt} | optical air mass | unitless, greater than 1 |
em_{surf} = ε_{surf} | emissivity of the surface (e.g. snow) | unitless, in [0,1] |
σ | Stefan-Boltzman constant, equal to 5.67e-8 | Watts / (m^{2} K^{4}) |
C | the fraction of sky covered by clouds | unitless, in [0,1] |
F | the fraction of sky covered by forest canopy | unitless, in[0,1] |
I_{sc} | solar constant, equal to 1367 | Watts / m^{2} |
J | Julian day | decimal days (J =1 at midnight on January 1) |
Γ | day angle or fractional year | radians |
E_{0} | Earth orbit eccentricity correction | unitless |
Λ | latitude for a given point on the earth's surface | radians |
Λ_{degree} | latitude for a given point on the earth's surface | degrees |
δ | declination of the sun | degrees |
ω | Earth angular velocity, equals to 0.2618 (15 degrees per hour) | radians/hour |
θ | zenith angle | radians |
t_{h} | number of hours before (-) or after (+) true solar noon (TSN) | hours |
T | temperature, measured | degrees Celsius |
RH | relative humidity, measured | unitless, in [0,1] |
γ_{dust} | dust attenuation (default is 0.08; typically < 0.2) | unitless |
α | aspect angle, measured clockwise from north | radians |
β | slope angle, equal to atan(slope) | radians |
DPY | Earth days per year, equals to 365.2425 | days |
DPY_{sidereal} | Earth days per year (sidereal), equals to 366.2425 | days |
angle | Earth tilt angle, equals to 23.4397 | degrees |
e (in shortwave radiation equations) | Earth orbit eccentricity, equal to 0.016713 | unitless |
Output
Symbol | Description | Unit |
---|---|---|
e_{air} | vapor pressure of air | mbar |
e_{surf} | vapor pressure at the surface | mbar |
Qn_{SW} | net shortwave radiation | W / m^2 |
Q_{LW} | net longwave radiation, equals to LW_{in} - LW_{out} | W / m^2 |
em_{air} | air emissivity | unitless, in [0,1] ##### CHECK |
LW_{in} | incoming longwave radiation | W/m^2 |
LW_{out} | outgoing longwave radiation | W/m^2 |
e_{sat} = e^{*} | saturation vapor pressure | mbar |
T_{hr} | sunrise offset | hours before true solar noon (TSN) |
T_{hs} | sunset offset | hours after true solar noon (TSN) |
k^{'} _{ET} | instantaneous extraterrestrial radiation flux on a horizontal plane | Watts / m^{2} |
e_{sat} | saturated vapor pressure | kPa |
e | vapor pressure | kpa |
T_{dew} | dew point | degree Celsius |
W_{p} | precipitation water content | cm |
M_{opt} | optical air mass (from Kasten and Young, 1989) | meters #### CHECK |
τ | total atmospheric transmissivity | unitless, in [0,1] |
k_{direct} ' | instantaneous direct radiation flux | Watts / m^{2} |
γ_{s} | scattering attenuation | unitless ### CHECK |
k_{diffuse}' | instantaneous diffuse radiation flux | Watts/m^{2} |
k_{global}' | instantaneous global radiation flux | Watts/m^{2} |
k_{bs}' | backscattered radiation flux | Watts/m^{2} |
ΔΩ | longitude offset for sloped topography | radians |
Λ_{eq} | equivalent latitude for sloped topography | radians |
noon_offset | noon offset for sloped topography, equal to - ΔΩ / ω | hours |
T_{sr} | sunrise offset for sloped topography | hours |
T_{ss} | sunset offset for sloped topography | hours |
day_length | day length for sloped topography, equal to T_{ss} - T_{sr} | hours |
k_{ET}' | instantaneous extraterrestrial radiation flux for sloped topography | Watts/m^{2} |
k_{cs} | instantaneous clear sky radiation | Watts/m^{2} |
Ω | earth rotation rate | radians/year |
Notes
Notes on Input Parameters
For each input 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 use 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.
For DEMs with pixel geometry and bounding box given in terms of Geographic coordinates, the latitude and longitude of each pixel is used in the calculations. For DEMs with a "fixed-length" pixel geometry (e.g. UTM coordinates), which tend to span smaller areas, the dialog prompts for a single lat/lon pair to be used in the calculations.
In the calculation, the timestep between frames in the new grid sequence (RTS file) should typically be about one hour and should match the timestep that will be used to model the Snowmelt or Evaporation processes. The number of frames in the RTS file will depend on the start and stop times, as well as the timestep. The start time, stop time and timestep should match those used to create the new shortwave radiation file with extension "*.Qn-SW".
Notes on the Equations
All variables and their units can be seen by expanding the Nomenclature section above.
time zone: Boundaries of time zones can be very irregular and a time zone map should be consulted if you are unsure. The time zone is not simply a function of the longitude. You can select an adjacent time zone to include the effect of Daylight Savings Time. Time zones with non-integer offsets from GMT are not yet supported.
slope: Topographic slopes (not slope angles) are specified as dimensionless numbers [m/m]. A RiverTools grid (RTG file) with extension "_slope.rtg", "_mf-slope.rtg" or "_dinf-slope.rtg" can be used.
aspect: Aspect is specified as an angle measured in radians counter-clockwise from due east (the standard convention). A RiverTools grid (RTG file) with extension "_mf-angle.rtg" or "_dinf-angle.rtg" can be used for the (continuous-angle) aspect grid.
Q_{SW} is set to zero between the times of local sunset and local sunrise, so frames in the RTS file that correspond to nighttime hours will contain only zeros.
Notes on the Equations to Compute Longwave Radiation
The value 237.3 in Brutsaert's equation for em_{air} is not a misprint.
Notes on the Equations to Compute Shortwave Radiation
The declination of the sun reaches its lowest value of -23.5 degrees on the Winter Solstice (Dec. 21/22) and reaches its highest value of 23.5 degrees on the Summer Solstice(June 21/22). It is zero for both the Vernal Equinox (Mar. 20/21) and the Autumnal Equinox (Sept. 22/23). The value of 23.4397 degrees is the fixed tilt angle of the Earth's axis from from the plane of the ecliptic.
The equation for declination of the sun is a Fourier series expansion that is supposed to have a maximum error of 0.0006 radians (less than 3 arcminutes). This should be double-checked. The equation shown here can also be found at: http://en.wikipedia.org/wiki/Declination, and is derived in the paper called "Fourier series representation of the position of the sun", by J.W. Spencer (1971), of CSIRO.
If (abs(lat_{deg}) gt 66.5) we are above Arctic circle or below Antarctic circle. This can also happen if lat_{deg} is an "equivalent latitude" for sloped topography. In this case, the absolute value of the argument to ACOS can exceed one and there is either no sunrise or no sunset at the given location. If the argument to ACOS is less than -1, it is set to -1. If it is greater than 1 it is set to 1.
True solar noon (TSN) for a given location is the time when the sun reaches its highest point above the horizon. At this time, the zenith angle attains its minimum value for the day and the sun is said to be "on the meridian." In order to compute the clock time at which true solar noon occurs, we start with the number 12, then add the following three corrections:
- (1) ΔT_{c} is the time difference between true solar noon and local clock noon, without accounting for any arbitrary time zone adjustments. It is computed from the so-called Equation of Time and is related to the figure-8-shaped "analemma". See: http://en.wikipedia.org/wiki/Equation_of_time and Whitman, A.M. (2003) A Simple Expression for the Equation of Time (online document: http://www58.homepage.villanova.edu/alan.whitman/eqoftime.pdf.
- (2) ΔT_{tz} is computed as the (signed) difference between the local longitude and the longitude at the center of the local time zone, divided by 15. The Earth turns through 15 degrees of longitude in one hour. The longitude at the center of a time zone is given by its GMT offset (and integer between -12 and 12) times 15.
- (3) ΔT_{dst} is a correction for Daylight Savings Time, generally an integer.
Sunrise and sunset occur when zenith angle, θ, is equal to π/2, so cos(θ) = 0. Using the equation for the zenith angle we can then solve for time offsets (e.g. sunrise and sunset offsets) as t_{h}, in hours.
Many of these equations are from Dingman, Appendix E. In his notation, a lower-case "k" indicates an instantaneous radiation flux. In expressions for "k", the argument "t_{h}" is the number of hours before (t_{h} < 0) or after (t_{h} > 0) true solar noon. Upper-case "K" then indicates the integral of "k" when t_{h} is allowed to vary from the sunrise offset to the sunset offset. Primes on "k" or "K" indicate values for a horizontal plane which must be corrected (as explained by Dingman) for sloped topography. For example, the sunrise and sunset times for a given location depend on the slope and aspect of the local topography.
The equation for k_{ET}' is similar to Dingman, App. E, E-25, but his was integrated over one day and this one is instantaneous. Need to double check that integrating the equation here over one day gives his equation E-25. Note that ω has units of radians/hour and ΔΩ (longitude offset) has units of radians. See the TopoFlow source code file: solar_funcs.py, which is imported by met_base.py.
TopoFlow provides two methods for computing the saturated vapor pressure, e_{sat} = e^{*}, as a function of temperature, T. One is from Brutsaert (1975) (and used in Dingman) and the other is from Satterlund (1979). Liston (1995, EnBal) uses the Satterlund method. When plotted, the two curves look almost identical for T between -40 and 20, but start to separate from one another for T > 20..
By definition, the relative humidity is the ratio of the actual vapor pressure to the saturated vapor pressure. That is, RH = e_{a} / e_{a} ^{*}. However, relative humidity is generally measured and used together with an equation for saturated vapor pressure to compute the vapor pressure as a function of temperature.
Optical air mass, M_{opt} is a dimensionless number that gives the relative path length (greater than 1) that radiation must travel through the atmosphere as the result of not entering at a right angle. The equation for optical air mass used here is from Kasten and Young (1989) and depends on latitude, declination and time of day. Instead of this equation, Dingman plots a family of curves in his Figure E-4 (p. 605) for *average daily* optical air mass. It would be interesting to integrate the equation given for M_{opt} over one day and compare the two by creating a contour plot.
Typical clear-sky values for γ_{dust} are between 0 and 0.2. See Dingman, p. 604-605.
The following variables are considered as input parameters (to be provided) that are used to compute many others: latitude (lat_deg), longitude (lon_deg or Λ), temperature (T), relative humidity (RH), slope angle (α), aspect angle (β), albedo, dust attenuation, cloudiness, canopy cover, solar constant (I_{sc}}), and various attributes of Earth such as angular velocity, tilt angle, eccentricity of orbit, days per year and perihelion.
A table of typical surface albedos is given by Dingman, Table D-2 on page 584.
The Earth orbit eccentricy *correction*, E_{0}, depends on the day angle, Γ, while the Earth orbit eccentricity is computed as: e = (b-a)/a, where a and b are the semi-major and semi-minor axes of the elliptical orbit.
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
Dingman, S.L (2002) Physical Hydrology, 2nd ed., Prentice Hall, New Jersey. (see Appendix E)
Brutsaert, W. (1975) On a derivable formula for long-wave radiation from clear skies, Water Resources Research, 11, 742-744.
Kasten and Young (1989) Revised optical air mass tables and approximation formula. Applied Optics, 28 (22): 4735~4738. (for the optical air mass equation)
Liston, G. *******
Marks and Dozier (1992) Climate and Energy exchange at the Snow Surface in the Alpine Region of the Sierra Nevada 1. Meteorological Measurements and Monitoring. Water Resources Research, 28(11), 3029~3042.
Whitman, A.M. (2003) A simple expression for the equation of time, online document, http://www58.homepage.villanova.edu/alan.whitman/eqoftime.pdf. Also see: http://en.wikipedia.org/wiki/Equation_of_time
Links
Related Help Pages
- Model help:TopoFlow-Snowmelt-Energy_Balance
- Model help:TopoFlow-Evaporation-Energy_Balance
- Model help:TopoFlow-Snowmelt-Degree-Day
- Model help:TopoFlow-Evaporation-Priestley_Taylor
- Model help:Gc2d
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