1. LONG WAVE RADIATION AT SEA SURFACE
Long wave radiation is the radiation emitted by the ocean surface at wavelengths greater than those of the visible light (at about 800 nanometres (nm)) but shorter than those of microwaves (at about 800,000 nm). Infrared radiation is associated with heat energy and not with visible light.
We will consider the NET long wave (LWnet) radiation at sea surface which represents the difference between the upward infrared radiation emitted by the ocean surface (LWs) and downward infrared radiation from the atmosphere (LWa):
LWnet = LWs - LWa (1)

2. Net LW radiation at sea surface comes out as a result of many complex processes
LWnet=???

3. The ocean surface irradiance consists of the emitted LW radiation
from the sea surface and the reflected atmospheric LW irradiance:
LWs = LWs0 + LLWa (2)
where LWs0 is LW irradiance of ocean surface
L is surface long wave albedo
Thus, the net LW radiation at sea surface can be expressed as:
LWnet = LWs0 - (1-L)LWa (3)

4. Upward long wave irradiance from sea surface
This is the major component of the longwave exitance!!!! For this part the physics is represented by a simple blackbody radiation
LWs0 = Ts (4)
where
 is the Stefan-Boltzmann constant (5.6710-8Wm-2K-4)
Ts is the sea surface !!!skin!!! temperature in degrees Kelvin
 is the emissivity of the sea surface
Emissivity of the sea surface  in a general case depends on the sea state and the optical properties of the sea water. For the fresh water  = 0.92 and varies from 0.89 to 0.98 for different conditions.

5. Restaurant Whose tea will be hotter at the moment of drinking?
Two industrial workers
go for a lunch:
They take each (besides the meals) a cup of tea simultaneously.
The first worker puts sugar into the cup immediately and starts to eat.
The second worker first starts to eat and puts the sugar into the cup just
before the drinking.
They start to drink their tea simultaneously
Whose tea will be hotter at
the moment of drinking?

6. Downward atmospheric long wave irradiance
Sea surface skin temperature Ts is not equivalent to SST, measured
at ships by buckets or engine intakes, it is a temperature of a very
thin (several to several hundreds ) surface skin layer, namely skin
temperature. Longwave absorption and emission both take place in
just about the top 0.5 mm of water, being dependent on the wavelength.
Back to the skin temperature issue – later !!!
Downward atmospheric long wave irradiance
Downwelling longwave radiation (longwave irradiance) originates from the emission by atmospheric gases (mainly water vapor, carbon dioxide and ozone), aerosols and clouds. Long-wave albedo is also poorly known and depends on the sea state and cloud conditions. Downwelling LW is the smallest, but the most uncertain term in the net LW radiation (longwave exitance).

7. Determining surface net long wave radiation
Measurements
Modelling - RTMs
Parameterization

8. Measurements of LW radiation:
The pyrgeometer is of a similar construction to the pyranometer, but the single dome is made from silicon or similar material transparent to the longwave band, coated on the inside with an interference filter to block shortwave radiation. The longwave irradiance passing through the dome, which we wish to determine, is only one component of the thermal balance of the thermopile. The remaining components come from various parts of the instrument. To isolate the geophysical component, the manufacturers provide the correction equations.

9. More detailed description of pyrgeometer
The PIR, or pyrgeometer, is sensitive to wavelengths in the range from 3000 to nm, which covers the span of temperatures (or thermal radiation) expected from the earth and atmosphere. The pyrgeometer works on the same principle as the pyranometer in that radiant energy is converted to heat energy which, in turn, is measured by a thermopile. However, protecting the sensor from the environment (e. g., solar radiation) is difficult. To do this, the dome is made of silicon, which is nearly opaque to solar wavelengths. The dome is also coated with a grayish interference filter that does not transmit wavelengths shorter than 3000 nm, but sharply increases to 50% transmission at 4000 nm. From 4000 to nm its transmittance slowly falls to about 30-40%. The detector senses a net signal from a number of sources which includes emissions from targets in its field of view, emission from the case of the instrument, and emission from the dome. To resurrect the true environmental thermal infrared irradiance, temperatures of the detector, case, and dome are monitored with thermistors. Because the case is shielded from the sun, its temperature represents the air temperature and therefore is a proxy for the degree of thermal emission by the atmosphere. The dome, however, is not protected from solar heating. Therefore, the difference between the thermal emissions of the case and dome represents an erroneous signal that must be removed. (As mentioned before, shading the dome would make this error negligible.) An empirical calibration equation accounts for all of these effects and converts the three measured temperatures to a true environmental thermal infrared irradiance in watts per square meter.
Requirements for calibration facilities
Calibrating facilities for infrared instruments are more complicated and therefore less common than those for solar radiation, and require careful technique. Field intercomparisons if several instruments are available are desirable. The use of on-site atmospheric soundings in clear-sky conditions can provide an absolute reference for longwave irradiance. The calculation of the downwelling longwave flux under a cloud requires knowledge of both cloud base height and emissivity. Cloud base height may be measured with active systems such as a MicroPulse lidar (Spinhirne, 1993) or a cloud profiling radar, although an infrared radiometer (such as the pyrgeometer) would still be needed to estimate cloud emissivity.

10. Modelling of the long wave radiation (RTMs)
The radiative transfer equation (RTE) states that the energy radiated by a parcel of material in a particular frequency range and particular direction (denoted by an increment of solid angle around the direction) is the sum of energy transmitted through the parcel and the energy emitted from within the parcel, in that frequency interval and direction.
RTE evaluates the effect of changes in temperature, humidity, cloud, aerosols, and chemical composition. Calculation of LW radiation should include all the major absorption bands of CO2, H2O, and O3, as well the weaker bands of CO2, N2O and CH4.
Structure of a simple RTM
Atmospheric column consists of N plane parallel layers, n=1,2,…,N. Temperatures Tn, n=0,1,….,N are denoted at layer edges. The mean optical thickness of the nth layer at a particular frequency  is n.

11. At frequency  and beam angle  the upward IR irradiance U,n and the downward IR irradiance D,n at the edges of the layer n are:
(5.6)
where  = cos. In (5,6): the first terms are the transmitted IR irradiances given by Lambert’s law, and the second terms are the IR irradiances emitted in the layer:
(7,8)
where B,n is the Planck function at frequency  and temperature Tn.

12. Planck Function
The Planck function represents the radiance as a function of frequency from
an object or gas at a given temperature, T, in thermodynamic equilibrium
It can be written in terms of wavenumber (ν) or wavelength (λ) as
B(T) is the energy (Joules) emitted per second
per unit lambda wavelength per steradian
from one square meter of a perfect
blackbody at temperature T
T is the temperature of the blackbody
h is Planck's constant = 6.63 × J*s
C is the speed of light = 3.00 x 108 m/s
λ is the wavelength
k is Boltzmann's constant = 1.38 x J/K
Integral of the Planck function gives you energy:

13. Theoretical Planck radiance curves for a number of Earth’s atmospheric
temperatures as a function of wavenumber and wavelenght
Thermal infrared emission
spectrum observed by Nimbus 4

14. (9,10) 11,12 Boundary conditions:  is the albedo,
=1- is the emissivity of the ocean surface.
Equation (9): no downward IR flux at the top of the atmosphere.
Equation (10): the upflux at the ocean surface is given by the sum of the
emission from the ocean plus the reflection of the downward flux.
For the frequency range [a,b] the total upward Un and downward Dn IR fluxes result from integrating U,n and U,n overall frequencies in [a,b] and beam angles:
11,12

15. Problems: Short summary:
Numerical solution of (5)-(12) is difficult and expensive due to:
The spectral complexity of the atmospheric constituents
Vertical inhomogeneity of the chemical composition of the atmosphere
There is a lack of measurements of basic parameters in the atmospheric column
Short summary:
Most accurate are the Line-By-Line Radiative Transfer Models (LBLRTM) which compute transfer of each constituent for each emission and absorption spectral line at many levels throughout the profile. Their computational burden is therefore large, which makes them unsuitable for routine use in numerical models.
Over the years, many broadband RTM's have been developed, increasing in accuracy and efficiency with improved parameterizations and increased computer power. Such models are widely applied in climate modelling, and in flux retrieval from direct and remotely sensed atmospheric variables.
The computation of LW flux with the best high spectral resolution codes under clear conditions is at an advanced state. For cloudy sky conditions, however, RTM's are not well validated. The calculation of the downwelling longwave flux under a cloud requires knowledge of both cloud base height and emissivity.

16. Parameterization of LW radiation:
LWnet = LWs0 - (1-L)LWa
What do we measure?
SST Ta, q, C (Cn, Cl)
1. No problem to parameterize the LWs0 , if we have SST and the emmissivity of the sea surface:
LWs0 = Ts4
However, you have to remember that Ts4 is a skin temperature and it is not equal to the bulk SST (LATER!)

17. LWnet = Ts4  - (1-L)LWa
2. LW albedo is more poorly known that a SW albedo. Some very tentative estimates give values in the range of (e.g. Clark et al. 1974). Since
the value of [1-LW albedo] is close (at least of the same
order as) to the emissivity of sea surface []
the accuracy of L and  is approximately the same
A reasonable approach is to establish an effective emissivity and to re-write the equation for the net LW as follows:
LWnet = Ts4  - (1-L)LWa
LWnet = (Ts4  - LWa)
(13)
where  is an effective emissivity and should not be understood as an emissivity of the sea surface (a typical mistake). From (2), (4), (13):
 =(LWs0 - LWa)/ (Ts4  - LWa)
(14)
Important:
   = LWs0 / Ts4 

18. 3. Parameterization of downwelling atmospheric LW radiation
LWnet =  (Ts4 - LWa )
The simplest approach is to measure downwelling LW radiation and to compare it with different combinations of surface parameters:
Tair, q, Cn, Cl
However, this approach results in a very uncertain dependency due to very different optical properties of clear sky and cloudy atmosphere. Air temperature blackbody radiation shows significant differences for clear skies and cloudy skies.

19. ! separate analysis should be performed for
Guest (1998) –
2 months of direct LW measurements in Weddell Sea:
Cloudy conditions
Clear sky
Processes are quite different under clear skies and clouds
! separate analysis should be performed for
atmospheric LW under clear sky and clouds

20. 1. Downwelling long wave radiation under clear skies
Since information about the atmospheric gases and aerosols is generally unavailable in routine observational practice, major efforts of researchers were concentrated on studying relationships between the clear sky downwelling atmospheric LW on
Surface humidity
Surface air temperature
Theoretically, from a physical view point, surface humidity should have a closer link with surface humidity.
Bignami et al. (1995) using results from direct observations in seven cruises in Mediterranean Sea, found close relationship between surface water vapor pressure and the ratio between atmospheric downwelling LW and air temperature blackbody radiation:
LWa = Ta4(a+be)
where
a=0.684, b=0.0056, =0.75

21. ln(LWa / Ta4) =a+ln(Ta) (14b)
However, in practice much better relationships are obtained for surface air temperature. Reasons for this (same as for the dependency of SW on Tair):
more easily available (more observations)
measurements are more accurate
Swinbank (1963) from Indian Ocean and lake observations:
LWa = Ta4(a+bTa) (14a) or
ln(LWa / Ta4) =a+ln(Ta) (14b)
where
a=-15.75, =0.75
Guest (1998) tested many formulations of clear sky atmospheric LW.
 No evidence of a better approximation for humidity than for air temperature.

22. Guest (1998) results (for your files):

23. LWa = 0.151e0.307 (16a) LWa = 1.2Ta4 -0.16 (16b)
Malevsky et al. (1992) from a very comprehensive data set collected in different World Ocean regions (incl. tropics and mid-latitudes), found the following relationship between the downwelling atmospheric LW and humidity (water vapor pressure):
LWa = 0.151e (16a)
Similar dependency for air temperature implies in the range of
temperatures less than 20º:
LWa = 1.2Ta (16b)
Or, including the role of humidity:
LWa = Ta4( e) (16c)
For the whole range of temperatures from -10º to +30º:
LWa = 1.026Ta210-5 – (17)

24. Obtain the dependence Ta4 (e)
Clear skies
There has been found considerably
smaller scatter for (17) than for (16a)
and a smaller RMS error.
INTERESTING:
equation (17) looks physically less reasonable than those which include
Ta4. However, analysis of empirical
data shows that formula (17) works
well in most conditions.
overcast
Dependence of atmospheric emissivity on temperature (Ta4) and humidity (e)
LWa = 0.151e0.307
LWa = 1.026Ta210-5 –0.541
Obtain the dependence Ta4 (e)

25. ? LWnet =  [ Ts4 - ( LWa0  F(c) )]
Thus, now we assume the following form of parameterization for the net LW radiation:
Sea surface !skin! temperature blackbody radiation. Since we do not have normally “skin” estimate, we account for the bulk effect in 
Effective emissivity of sea surface (accounts for the LW albedo and skin effect)  !!!
LWnet =  [ Ts4 - ( LWa0  F(c) )] ?
Atmospheric downwelling LW under clear skies:
Parameterized as a function of either surface humidity or surface temperature
Relationships with temperature give better results!
Function of cloud cover –
Should account for the effect of clouds

26. We DO NOT know (measure) We come actually to another RTM!
2. Downwelling long wave radiation under clouds – the cloud modification of LWa.
What should be parameterized from a theoretical view point is
a cloud temperature blackbody radiation: Tcl4
Lind and Katsaros (1982):
LWc(2) = n(2)(2) Tcl(2)4 + [1-n(2)] LWc(3)
LWc(1) = n(1)(1) Tcl(1)4 + [1-n(1)] LWc(2)
LWc(tot) = (1-(0)) LWc(1) +
+ LWa(sky) +(0)T0(1)4
n is fractional cloud cover of the
subscribed cloud layer
Tcl is cloud base temperature of the
 is effective emittance of the
(0) is emittance of layer from the surface
to lowest cloud base
Tcl is equivalent radiative temperature of
the lower layer
We DO NOT know (measure)
We come actually to another RTM!

27. This effect has to be parameterized
LWa + LWcl
LWa
The only available parameter is the
total fractional cloud cover and
sometimes is the fractional cover of the low-level cloudiness.
Typical approach: to make
measurements under the known
cloud conditions and to compare
clear sky atmospheric LW with that
measured under the cloudy sky.
Paramete-
rization of
F(n)
Bignami et al. (1995):
F(n) = c2 (Mediterranean Sea)
Clark et al. (1974):
F(n) = c (Pacific Ocean)
Efimova (1962):
F(n) = c (land data)
Typical expression is ac
for the total cloud cover
This effect has to be parameterized

28. LWcl+a = LWa (1+ktnt2) (19)
Malevsky et al. (1992) from his collection of field measurements for the total cloud cover found:
LWcl+a = 0.928Ta210-5 – (18)
He assumed that
LWcl+a = LWa (1+ktnt2) (19)
Coefficient kt can be derived from (19) under nt=1:
kt = (LWcl+a + LWa) / (LWa) (20)
where: LWa = 1.026Ta210-5 –0.541
Not surprisingly, in this formulation kt becomes dependent on the air temperature, since both LWcl+a and LWa are the functions of air temperature.

29. LWa = [LWa0  F(c)]=(1.026Ta210-5-0.541)(1+ktnt2)
Computation of the kt from a simple RTM:
kt = (1/LWa) [cla Tcl4 +
+LWa(0)(4h/Ta)-1]
cl – emissivity of the cloud base
cl – temperature of the cloud base
LWa(0) – irradiance below the cloud layer
cl – surface temperature
 - temperature gradient in the undercloud layer
h – cloud layer height
Thus, for the total cloud cover only:
LWa = [LWa0  F(c)]=(1.026Ta )(1+ktnt2)
kt = (-0.098Ta2 ) / (1.026Ta )

30. Dependence of the atmospheric emissivity
under different cloud conditions
Complete overcast (upper+mid
layer clouds)
Complete overcast (total cloudiness)
Complete overcast (lower layer clouds)
Complete overcast (mid layer clouds)
Complete overcast (upper layer clouds)
Clear skies

31. The effect of cloudiness for three different layers (low cloudiness, mid-level cloudiness and upper layer cloudiness).
For upper layer: LWclu+a = 0.995Ta210-5 – (21)
For mid-level: LWclm+a = 0.932Ta210-5 – (22)
For lower layer: LWcll+a = 0.921Ta210-5 – (23)
For the cloud coefficients:
For upper layer: ku = (LWclu+a+LWa)/(LWa) (24)
For mid-level: km = (LWclm+a+LWa)/(LWa) (25)
For lower layer: kl = (LWcll+a+LWa)/(LWa) (26)

32. LWnet =  [Ts4 - (LWa0  F(c))]
However, normally we have observations only for the fractional cloud cover of total and low-layer cloudiness. Thus, this 3-layer formulation has been simplified for the consideration of the total and low-layer cloudiness:
LWa = (1.026Ta )(1+klnl2)(1+ku+m(nt2-nl2) (27)
ku+m is the coefficient accounting for the total effect of the mid and upper layer cloudiness, which can be derived from the coefficients for the total and low-level cloudiness:
kl = (LWcll+a+LWa)/(LWa) (28)
ku+m = (ktnt2 - klnl2) / [(1+ktnt2)(nt2–nl2)] (29)
Now we can finally derive the parameterization of the net long-wave radiation at ocean surface in a general form:
LWnet =  [Ts4 - (LWa0  F(c))]

33. The number of parameterizations approaches several tens
Summary:
History is very long
The number of parameterizations approaches several tens
Formulations are similar
Differences are large
Brunt (1932)
LW=Ts4( ez1/2)(1-0.8nd), =0.98, d=1
Berliand and Berliand (1952)
LW=Ts4( ez1/2)(1-0.8nd)+4Ta3(Ts-Ta), =0.98, d=1
Anderson (1952)
LW=[Ts4-Ta4( ez)](1-0.8 nd), =0.98, d=1
Efimova (1961)
LW=Ta4( ez)(1-cnd)+4Ta3(Ts-Ta), =0.96, d=1
Swinbank (1963)
LW=[Ts4-9.3610-6Ta6](1-0.8 nd), =0.98, d=1
Clark et al. (1974)
LW=Ts4( ez1/2)(1-cnd)+4Ta3(Ts-Ta), =0.98, d=2
Bunker (1976)
LW=0.022[Ta4( ez)(1-0.68nd)+4Ta3(Ts-Ta), =0.96, d=1
Hastenrath and Lamb (1978)
LW=Ts4( q1/2)(1-0.53nd)+4Ta3(Ts-Ta), =0.98, d=2
Malevsky et al. (1992b)
LW=(Ts4-(1.026Ta )(1+cnd)), =0.91, d=2
Bignami et al. (1995)
LW=Ts4-[Ta4 ( ez)( nd)), =0.98, d=2
Josey et al. (2001)
LW=Ts4-(1-L)[Ta +an2 +bn +c +0.84(D+4.01)]4 , =0.98, a, b, c, D = empirical coefficients, L = 0.045

34. Variations in short-wave radiation and long-wave radiation due to the parameterizations
(North Atlantic SW and LW radiation budget)

35. Summary of LW radiation parameterizations:
Under clear sky and small cloudiness the accuracy is normally better 15 W/m2
Higher uncertainties occur under the moderate and high cloud cover
Uncertainties in the tropics are typically higher than in mid and high latitudes
and are primarily associated with the atmospheric clear sky IR irradiance
“Hot issues” of all parameterizations are the “skin temperature” and
representation of the multi-layer cloudiness of different types by fractional
[total] cloud cover
Recommendations:
Do not hesitate to use “old” parameterizations
Try to avoid the use of parameterizations based on water
vapor pressure and humidity
Do not use Bignami et al. (1995) except for Mediterranean sea
Be careful with the choice of emissivity value. Always remember
it is the effective emissivity and not the emissivity of the surface

36. RB = SW(1-) - LWnet (30)
Radiation balance of the ocean
RB = SW(1-) - LWnet (30)
defcit
surplus
Incoming
radiation
Outgoing radiation
At the top of atmosphere
At the ocean surface: SW LW

37. SW LW RB

38. Variations of SW and LW radiation due to different parameterizations

39. /helios/u2/gulev/handout/
longwave1.f – collection of LW radiation F77 codes
RIZL – Malevsky et al. (1992) scheme
RLWISI – Efimova (1961) as modified by Isemer et al (1989)
RLW_CLA – Clark et al. (1974)
RLW_BIG – Bignami et al. (1995)
Try to compare Malevsky, Efimova, Bignami and Clark schemes:
For Ts = 12C:
Clear sky, dependence on temperature, humidity
Cloud cover octa=4, dependence on temperature
Tair = 15C, dependence on cloud cover (in octas)

40. /helios/u2/gulev/handout/
swm_test.f – program to compute instantaneous values of SW radiation, using Malevsky et al. (1992) and Dobson and Simth (1988) schemes.
Compilation: f77 –o swm_test swm_test.f radiation.f
Results: sw.res
swr_test.f – program to compute daily values of SW radiation, using Reed (1977) scheme.
Compilation: f77 –o swr_test swr_test.f radiation1.f
Results: swr.res
lw_test.f – program to compute values of LW radiation, using Malevsky et al. (1992), Clark et al. (1974), Bignami et al. (1995) and Emivova (1962) schemes.
Compilation: f77 –o lw_test lw_test.f longwave1.f
Results: lw.res

41. READING
Angström, A.K., 1925: On the variation of the atmospheric radiation. Gerlands. Beitr. Geophys., 4,
Bignami, F., S.Marullo, R.Santorelly, and M.E.Schiano, 1995: Long-wave radiation budget in the Mediterranean Sea. J. Geophys. Res., 100,
Brunt, D., 1932: Notes on radiation in the atmosphere. Quart. J. Roy. Met. Soc., 58,
Bunker, A., 1976: A computation of surface energy flux and annual cycle over the North Atlantic Ocean. Mon. Wea. Rew., 104,
Clark, N.E., L.Eber, R.M.Laurs, J.A.Renner, and J.F.T.Saur, 1974: Heat exchange between ocean and atmosphere in the eastern North Pacific for NOAA Tech Rep. NMFS SSFR-682, US Dept. of Commer., Washington DC, 108 pp.
Efimova, N.A., 1961: On methods of calculating monthly values of net long-wave radiation. Meter. Hydr. 10,
Fung, I. Y., D.E.Harrison, and A. A. Lacis, 1984: On the variability of the net long-wave radiation at the ocean surface. Rev. Geophys., 22(2),
Gulev, S.K., 1995: Long-term variability of sea-air heat transfer in the North Atlantic Ocean. Int.J.Climatol., 15,
Isemer, H.-J., J.Willebrand, and L.Hasse, 1989: Fine adjustment of large-scale air-sea energy flux parameterizations by direct estimates of ocean heat transport. J.Climate, 2,
Josey, S., E.C.Kent, and P.K.Taylor, 1999: New insights into the ocean heat budget closure problem from analysis of the SOC air-sea flux climatology. J. Climate, 12,
Lind, R.J., K.B.Katsaros, and M.Gube, 1984: Radiation budget components and their parameterization in JASIN. Quart. J. Roy. Meteor. Soc., 110,
Malevsky, S.P., G.V.Girdiuk, and B.Egorov, 1992b: Radiation balance of the ocean surface. Hydrometizdat, Leningrad, 148 pp.
Oberhuber, J.M., 1988: An Atlas based on the COADS data set: the budgets of heat, buoyancy and turbulent kinetic energy at the surface of the global ocean. MPI fuer Meteorologie report, No. 15, 19pp. [Available from Max-Plank-Institute fuer Meteorologie, Bundesstrasse 55, Hamburg, Germany].
Rosati, A. and K. Miyakoda, 1988: A general circulation model for the upper ocean circulation. J. Phys. Oceanogr., 18,