Radiant Heat Fluxes

Radiant Heat Fluxes

<wikitex>As was seen in Sea-Ice_Model#Thermodynamics, 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.</wikitex>

Shortwave Radiation

<wikitex> 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 - 0.6 c^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

</wikitex>

Longwave Radiation

<wikitex> 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 + 0.275 c)\, F\!\downarrow .

$$ Note that the CORE forcing files contain incoming longwave radiation so only the outgoing needs to be computed. </wikitex>

Sensible heat

<wikitex> 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}) .

$$ </wikitex>

Latent Heat

<wikitex> The latent heat depends on the vapor pressure and the saturation vapor pressure given by: $$ \eqalign{

  e &= 611 \times 10^{a(T_d - 273.16) / (T_d - b)} \cr 
  e_s &= 611 \times 10^{a(T_{s\!f\!c} - 273.16) / (T_{s\!f\!c} - b)}

} $$ The vapor pressures are used to compute specific humidities according to: $$ \eqalign{

  q_{10 \rm m} &= {\epsilon e \over p - (1 - \epsilon) e} \cr
  q_s &= {\epsilon e_s \over p - (1 - \epsilon) e_s}

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

  LE\!\downarrow\, = \rho_a L C_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. </wikitex>