Radiant Heat Fluxes
As was seen in Sea-Ice_Model, the model thermodynamics requires fluxes of latent and sensible heat and longwave and shortwave radiation. We follow the lead of Parkinson and Washington in computing these terms.
Shortwave Radiation
The Zillman equation for radiation under cloudless skies is:
![{\displaystyle Q_{o}={S\cos ^{2}Z \over (\cos Z+2.7)e\times 10^{{-5}}+1.085\cos Z+0.10}}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/970803fa0ad6159492d08172cea0a7379539a6bb)
where the variables are as in the table below. The cosine of the
zenith angle is computed using the formula:
![{\displaystyle \cos Z=\sin \phi \sin \delta +\cos \phi \cos \delta \cos H\!A.}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/74b3746c07854460b1dec0af4812d998bcc610ce)
The declination is
![{\displaystyle \delta =23.44^{{\circ }}\times \cos \left[(172-{{\rm {day\,of\,year}}})\times 2\pi /365\right]}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/1723d5822496c66b57b2014f085ac530f3c6dd51)
and the hour angle is
![{\displaystyle H\!A=(12\,{{\rm {hours-solar\,time}}})\times \pi /12.}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/2a059182740ba4968cc38534f68cd966eea47ad9)
The correction for cloudiness is given by
![{\displaystyle SW\!\!\downarrow =Q_{o}(1-0.6c^{3}).}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/b035f76a65b75e3c7814ef748e744eaa45b93aa0)
The cloud correction is optional since some sources of radiation contain it already.
Variable
|
Value
|
Description
|
|
(9.5, 7.66)
|
vapor pressure constants over ice
|
|
(7.5, 35.86)
|
vapor pressure constants over water
|
|
|
cloud cover fraction
|
|
|
transfer coefficient for latent heat
|
|
|
transfer coefficient for sensible heat
|
|
|
specific heat of dry air
|
|
|
declination
|
|
|
vapor pressure in pascals
|
|
|
saturation vapor pressure
|
|
0.622
|
ratio of molecular weight of water to dry air
|
|
|
hour angle
|
|
|
latent heat of vaporization
|
|
|
latent heat of sublimation
|
|
|
latitude
|
|
|
incoming radiation for cloudless skies
|
|
|
surface specific humidity
|
|
|
10 meter specific humidity
|
|
|
air density
|
|
|
solar constant
|
|
|
Stefan-Boltzmann constant
|
|
|
air temperature
|
|
|
dew point temperature
|
|
|
surface temperature of the water/ice/snow
|
|
|
geostrophic wind speed
|
|
|
solar zenith angle
|
Longwave Radiation
The clear sky formula for incoming longwave radiation is given by:
![{\displaystyle F\!\downarrow \,=\sigma T_{a}^{4}\left\{1-0.261\exp \left[-7.77\times 10^{{-4}}(273-T_{a})^{2}\right]\right\}}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/3825ca447a43e521614b88362ed51142bbb3413e)
while the cloud correction is given by:
![{\displaystyle LW\!\downarrow \,=(1+0.275c)\,F\!\downarrow .}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/62b12c25181a0c616b3a90631dbad63e0458928d)
Note that the CORE forcing files contain incoming longwave radiation so only the outgoing needs to be computed.
Sensible heat
The sensible heat is given by the standard aerodynamic formula:
![{\displaystyle H\!\downarrow \,=\rho _{a}c_{p}C_{H}V_{{wg}}(T_{a}-T_{{s\!f\!c}}).}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/db86bb80e183874598fc2d71275d011805eb39b1)
Latent Heat
The latent heat depends on the vapor pressure and the saturation vapor
pressure given by:
![{\displaystyle {\begin{aligned}e&=611\times 10^{{a(T_{d}-273.16)/(T_{d}-b)}}\\e_{s}&=611\times 10^{{a(T_{{s\!f\!c}}-273.16)/(T_{{s\!f\!c}}-b)}}\end{aligned}}}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/46441e25a4b587fcb3c51b0f279b5826de033fe1)
The vapor pressures are used to compute specific humidities according
to:
![{\displaystyle {\begin{aligned}q_{{10{\rm {m}}}}&={\epsilon e \over p-(1-\epsilon )e}\\q_{s}&={\epsilon e_{s} \over p-(1-\epsilon )e_{s}}\end{aligned}}}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/931d06e07c307234ac4374b20ea30e6a90418a7e)
The latent heat is also given by a standard aerodynamic formula:
![{\displaystyle LE\!\downarrow \,=\rho _{a}LC_{E}V_{{wg}}(q_{{10{\rm {m}}}}-q_{s}).}](https://www.myroms.org/myroms.org/v1/media/math/render/svg/2a47a8a9c007d82e7cd4a4a3c655dcaaf020a951)
Note that these need to be computed independently for the ice-covered
and ice-free portions of each gridbox since the empirical factors
and
and the factor
differ depending on the surface type.