1. The rate of change of the internal energy: e: internal energy q: heat flux  : viscous dissipation of kinetic energy The entropy  is related to e as: Since  is a function of state variables p and T:

2. Specific heat c p at constant pressure is define as: On the other hand, from we have (1) (2)

3. Therefore,And we have Since

4. The equation of heat conservation where F rad represents radiative flux is the irreversible internal energy fluxes driven by temperature gradient, i.e., diffusion of heat.

5. Specific heat of sea water at atmospheric pressure Cp in joules per gram per degree Celsius as a function of temperature in Celsius and salinity in practical salinity units, calculated from the empirical formula given by Millero et al., (1973) using algorithms in Fofonoff and Millard (1983). The lower line is the freezing point of salt water. C p  4.0 X 10 3 J · kg -1 · ° C -1

6. Heat budget Temperature (Potential Temperature) Equation where. : specific heat capacity, vertical eddy diffusion coefficient., horizontal eddy diffusion coefficient.. (), Molecular thermal diffusivity Define, we have as heat content and F S radiative heat flux

7. , heat storage., heat convergence by currents and sub-scale transport. turbulent surface heat flux. Q S : solar radiation at the sea surface. Q b : net heat loss due to long wave radiation. Q e : latent heat flux. Q h : sensible heat flux., geothermal heat flux (neglected). Then the heat budget is: Radiative fluxes at surface

8. Solar radiation: Basics Planck’s law: irradiance for absorptance h~ Planck’s constant. k~ Boltzmann’s constant. c~ light speed in vacuum. T~ temperature (Kelvin), ~wavelength. The wavelength of maximum irradiance (Wien’s law):, Total irradiance (Stefan-Boltzmann law): Stefan-Boltzmann constant: Solar radiation is in shortwave band: 50% visible, 0.35  m ≤ ≤ 0.7  m; 99%, ≤ 4  m Temperature at sun’s surface: T=5800K  m =0.5  m.

9. Solar flux at the top of the atmosphere: F S =1365-1372 W/m 2 Usually, we choose. Not all of the radiation received at the top of atmosphere is available to the ocean Solar constant: (mean solar flux on 1 square meter of earth)

10. Changes in solar constant (total solar irradiance) and global mean temperature of Earth’s surface over the past 400 years. Except for a period of enhanced volcanic activity in the early 19th century, surface temperature is well correlated with solar variability. From Stewart.

12. Factors influencing Q S 1). Length of the day (depending on season, latitude) 2). Atmospheric absorption. Absorption coefficient (gas molecules, dust, water vapor, etc). Elevation of the sun  : angle of the sun above the horizon. 3). Cloud absorption and scattering. 4). Reflection at the sea surface. direct sunlight (from one direction) reflection depends on elevation of the sun and the state of the sea (calm or waves). skylight (scattered sunlight from all directions) reflected about 8%. (A few percent of the radiation entering the sea may also be scattered back to the atmosphere)

13. Skylight is important at high latitudes Stockholm (59 o N)direct sunlightskylight July80%20% December13%87% However, total flux is less in December than in July. The 87% of skylight in December represents a smaller energy flow than the 20% in July

14. Effect of the elevation of the sun Absorption of the solar radiation in the atmosphere without cloud is due to the combined effect of gas molecules, dust in the atmosphere, water vapor etc. When the sun is overhead, the path in the atmosphere is shortest At lower elevation, the solar bean strikes the surface obliquely and is distributed over a larger area

15. Empirical Formula (Parameterization) (shortwave flux averaged over 24 hours): Example: 1). Clear sky radiation Q SO : clear sky radiation. A n : noon altitude of the sun in degree. t n : length of the day from sunrise to sunset in hours. is the solar flux arriving at the sea surface. C=8, 3). Reflection at the sea surface 4). Shortwave radiation into the sea 5). Original algorithm overestimates. Multiply by 0.7. Q so is clear sky solar radiation at sea surface. F is an empirical function of the fractional cloud cover. 2). Cloud reduction C=4,

16. Another example: Reed (1977) monthly mean shortwave radiation n~ fractional cloud cover (0.3 ≤n≤1). Otherwise Q s =Q so.  ~ noon solar elevation in degrees. c n ~ cloud attenuation factor (≈0.62).  ~ albedo.

17. Sea Surface Albedo the ratio of the radiation that is reflected from the surface to the incoming radiation, expressed in percent (also called reflectance) the solar zenith angle (or elevation, direct sunlight, table 5.1) type of sea state (calm or wavy, wind speed) aerosol/cloud optical depth (affect the partition of the direct and diffuse sunlight, ~8%) suspended matter in water (chlorophyll concentration) Reference: Jin, Z., T. Charlock, W. Smith Jr., and K. Rutledge, 2004: A parameterization of ocean surface albedo. Geophys. Res. Let., 31, L22301, doi:10.1029/2004GL021180.

18. Typical values of albedo The reflectance for water (most of the ocean) is about 10 - 12 % but can be higher or lower depending on the suspended matter. The albedo of sea ice can be much higher but depends strongly on ice type and whether it has snow cover. The reflectance for new ice can be as low as 5 - 20% while the albedo for first year ice can be as high as 60%. The albedo for multiyear ice without snow cover is about 70%. The reflectance for snow is between 60% and 90%. Land surface reflectance depends on vegetation and ranges between 0.5% and 30 %.

19. Source: http://www-cave.larc.nasa.gov/cave/sfc_albedo.html

21. Annual Mean Solar Radiation at Sea Surface (W/m 2 )-COADS

22. Annual Mean Cloud Cover-COADS

23. Mean Surface Solar Radiation (W/m 2 ), January, COADS

24. Mean Surface Solar Radiation (W/m 2 ), July, COADS

26. Distribution daily inflow of solar radiation The highest value (>300 W/m 2 ) occur at 30 o S and 30 o N in respective summer hemispheres. There is no shortwave input at high latitudes during the polar winter. The amount of energy input is greater in the southern hemisphere than in the northern hemisphere. (In its elliptic orbit, earth is closer to the sun in southern summer).

27. Absorption in the sea reduces the light level rapidly with depth. 73% reaches1 cm depth 44.5% reaches1 m depth 22.2% reaches10 m depth 0.53% reaches100 m depth 0.0062% reaches200 m depth

28. Long-wave radiation (Q b ) The difference between the energy radiated from the sea surface (  T 4, T ocean skin temperature) and that received from the sea by the atmosphere, mostly determined by water vapor in lower atmosphere. The outgoing radiation from the sea is always greater than the inward radiation from the atmosphere. Q b is a heat loss to ocean. The outgoing radiation is “longwave” Mean sea surface temperature is T= 12 o C=285K,  m =10.2  m. Most of the longwave radiation is in the range 3  m ≤ ≤ 80  m Longwave radiation is much smaller than the shortwave solar radiation

29. t w =water temperature ( o C). e a =relative humidity above the sea surface. C=cloud cover in oktas (1-8). Q bo =Q b (C=0) ranges from 70-120 W/m 2. Q b (Q bo ) decreases with t w and e a. Empirical Formula of Q b e a increases exponentially with t w. Due to the faster increase of e a, inward atmospheric flux is larger than outgoing surface radiation). The net heat loss decreases with t w.

30. Another formula:  =0.98, increases with latitude (0.5, equator; 0.73, 50 o ). e water vapor pressure (mb): Nonlinearity in water vapor dependence: The water vapor content (humidity) increases exponentially with T S, which could result in a more rapid increase in the atmosphere’s radiation into the sea than the sea’s outward radiation (proportional to T S 4. Thus Q b could decrease as T S increases. It should be noted that this is still a highly speculated process, which has not been substantiated with a significant amount of measurements. Saturated water vapor pressure

31. Annual Mean Longwave Radiation (W/m 2 )-COADS

32. Longwave Radiation, January (W/m 2 )-COADS

33. Longwave Radiation, July (W/m 2 )-COADS

34. Q b does not change much daily, seasonally, or with location. This is because (1) Q b ~T 4, for T=283K,  T=10K, Effect of cloud is significant. The big difference between clear and cloudy skies is because the atmosphere is transparent to radiation range from 8-13  m while clouds are not., which is only 15% increase. (2) Inward radiation follows outgoing radiation. Ice-albedo feedback Effect of ice and snow cover is relatively small for Q b but large for Q s due to large albedo (increase from normally 10-15% to 50-80%). Therefore, net gain (Q s -Q b ) is reduced over ice.  ice once formed tends to maintain. Properties of long wave radiation