Radiant Heat Fluxes - Revision history

Robertson at 13:13, 18 May 2016

2016-05-18T13:13:28Z

Kate: /* Shortwave Radiation */ Rest of it is in and formatted

2009-09-23T21:29:56Z

Shortwave Radiation: Rest of it is in and formatted

← Older revision		Revision as of 21:29, 23 September 2009
Line 5:		Line 5:
	<wikitex>		<wikitex>
	The Zillman equation for radiation under cloudless skies is:		The Zillman equation for radiation under cloudless skies is:
	~~\begin{equation}~~		$$
	Q_o = {S \cos^2 Z \over (\cos Z + 2.7) e \times 10^{-5} + 1.085		Q_o = {S \cos^2 Z \over (\cos Z + 2.7) e \times 10^{-5} + 1.085
	\cos Z + 0.10}		\cos Z + 0.10}
	~~\end{equation}~~		$$
	where the variables are as in ~~Table \ref{radvars}~~. The cosine of the		where the variables are as in the table below. The cosine of the
	zenith angle is computed using the formula:		zenith angle is computed using the formula:
	~~\begin{equation}~~		$$
	\cos Z = \sin \phi \sin \delta + \cos \phi \cos \delta \cos H\!A .		\cos Z = \sin \phi \sin \delta + \cos \phi \cos \delta \cos H\!A .
	~~\end{equation}~~		$$
	The declination is		The declination is
	~~\begin{equation}~~		$$
	\delta = 23.44^{\circ} \times \cos \left[ (172 - {\rm day \, of \, year})		\delta = 23.44^{\circ} \times \cos \left[ (172 - {\rm day \, of \, year})
	\times 2 \pi / 365 \right]		\times 2 \pi / 365 \right]
	~~\end{equation}~~		$$
	and the hour angle is		and the hour angle is
	~~\begin{equation}~~		$$
	H\!A = (12 \, {\rm hours - solar \, time}) \times \pi / 12 .		H\!A = (12 \, {\rm hours - solar \, time}) \times \pi / 12 .
	~~\end{equation}~~		$$
	The correction for cloudiness is given by		The correction for cloudiness is given by
	~~\begin{equation}~~		$$
	SW\!\!\downarrow = Q_o ( 1 - 0.6 c^3) .		SW\!\!\downarrow = Q_o ( 1 - 0.6 c^3) .
	~~\end{equation}~~		$$
	~~An estimate of the cloud fraction~~ $c$ ~~will be provided by Jennifer~~		The cloud correction is optional since some sources of radiation contain it already.
	~~Francis (\cite{Francis00})~~.

	{\| border="1" cellspacing="0" cellpadding="5" align="center"		{\| border="1" cellspacing="0" cellpadding="5" align="center"
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	\|-		\|-
	\|}		\|}
			</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>		</wikitex>

Kate: rough - just saving before heading home

2009-09-23T01:06:05Z

rough - just saving before heading home

New page

==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 [[Bibliography#ParkinsonCL_1979a |Parkinson and Washington]] in computing these terms.</wikitex>

===Shortwave Radiation===
<wikitex>
The Zillman equation for radiation under cloudless skies is:
\begin{equation}
Q_o = {S \cos^2 Z \over (\cos Z + 2.7) e \times 10^{-5} + 1.085
\cos Z + 0.10}
\end{equation}
where the variables are as in Table \ref{radvars}. The cosine of the
zenith angle is computed using the formula:
\begin{equation}
\cos Z = \sin \phi \sin \delta + \cos \phi \cos \delta \cos H\!A .
\end{equation}
The declination is
\begin{equation}
\delta = 23.44^{\circ} \times \cos \left[ (172 - {\rm day \, of \, year})
\times 2 \pi / 365 \right]
\end{equation}
and the hour angle is
\begin{equation}
H\!A = (12 \, {\rm hours - solar \, time}) \times \pi / 12 .
\end{equation}
The correction for cloudiness is given by
\begin{equation}
SW\!\!\downarrow = Q_o ( 1 - 0.6 c^3) .
\end{equation}
An estimate of the cloud fraction $c$ will be provided by Jennifer
Francis (\cite{Francis00}).

{| border="1" cellspacing="0" cellpadding="5" align="center"
! 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>