Radiant Heat Fluxes

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:

$$Q_{o}={S\cos ^{2}Z \over (\cos Z+2.7)e\times 10^{-5}+1.085\cos Z+0.10}$$

where the variables are as in the table below. The cosine of the zenith angle is computed using the formula:

$$\cos Z=\sin \phi \sin \delta +\cos \phi \cos \delta \cos H\!A.$$

The declination is

$$\delta =23.44^{\circ }\times \cos \left[(172-{\rm {day\,of\,year}})\times 2\pi /365\right]$$

and the hour angle is

$$H\!A=(12\,{\rm {hours-solar\,time}})\times \pi /12.$$

The correction for cloudiness is given by

$$SW\!\!\downarrow =Q_{o}(1-0.6c^{3}).$$

The cloud correction is optional since some sources of radiation contain it already.

Variable	Value	Description
$(a,b)$	(9.5, 7.66)	vapor pressure constants over ice
$(a,b)$	(7.5, 35.86)	vapor pressure constants over water
$c$		cloud cover fraction
$C_{E}$	$1.75\times 10^{-3}$	transfer coefficient for latent heat
$C_{H}$	$1.75\times 10^{-3}$	transfer coefficient for sensible heat
$c_{p}$	$1004~J~kg^{-1}~K^{-1}$	specific heat of dry air
$\delta$		declination
$e$		vapor pressure in pascals
$e_{s}$		saturation vapor pressure
$\epsilon$	0.622	ratio of molecular weight of water to dry air
$H\!A$		hour angle
$L$	$2.5\times 10^{6}~J~kg^{-1}$	latent heat of vaporization
$L$	$2.834\times 10^{6}~J~kg^{-1}$	latent heat of sublimation
$\phi$		latitude
$Q_{o}$		incoming radiation for cloudless skies
$q_{s}$		surface specific humidity
$q_{10{\rm {m}}}$		10 meter specific humidity
$\rho _{a}$		air density
$S$	$1353~W~m^{-2}$	solar constant
$\sigma$	$5.67\times 10^{-8}~W~m^{-2}~K^{-4}$	Stefan-Boltzmann constant
$T_{a}$		air temperature
$T_{d}$		dew point temperature
$T_{s\!f\!c}$		surface temperature of the water/ice/snow
$V_{wg}$		geostrophic wind speed
$Z$		solar zenith angle

Longwave Radiation

The clear sky formula for incoming longwave radiation is given by:

$$F\!\downarrow \,=\sigma T_{a}^{4}\left\{1-0.261\exp \left[-7.77\times 10^{-4}(273-T_{a})^{2}\right]\right\}$$

while the cloud correction is given by:

$$LW\!\downarrow \,=(1+0.275c)\,F\!\downarrow .$$

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:

$$H\!\downarrow \,=\rho _{a}c_{p}C_{H}V_{wg}(T_{a}-T_{s\!f\!c}).$$

Latent Heat

The latent heat depends on the vapor pressure and the saturation vapor pressure given by:

$${\begin{aligned}e&=611\times 10^{a(T_{d}-273.16)/(T_{d}-b)}\\e_{s}&=611\times 10^{a(T_{s\!f\!c}-273.16)/(T_{s\!f\!c}-b)}\end{aligned}}$$

The vapor pressures are used to compute specific humidities according to:

$${\begin{aligned}q_{10{\rm {m}}}&={\epsilon e \over p-(1-\epsilon )e}\\q_{s}&={\epsilon e_{s} \over p-(1-\epsilon )e_{s}}\end{aligned}}$$

The latent heat is also given by a standard aerodynamic formula:

$$LE\!\downarrow \,=\rho _{a}LC_{E}V_{wg}(q_{10{\rm {m}}}-q_{s}).$$

Note that these need to be computed independently for the ice-covered and ice-free portions of each gridbox since the empirical factors $a$ and $b$ and the factor $L$ differ depending on the surface type.