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The rate of change of the internal energy:

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:. Specific heat c p at constant pressure is define as:. (1).

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The rate of change of the internal energy:

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  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 cp at constant pressure is define as: (1) On the other hand, from (2) we have

  3. Therefore, And we have Since

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

  5. Cp 4.0 X 103 J · kg-1 · °C-1 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.

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

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

  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. Total irradiance (Stefan-Boltzmann law): Stefan-Boltzmann constant: The wavelength of maximum irradiance (Wien’s law): Temperature at sun’s surface: T=5800K m=0.5m. , Solar radiation is in shortwave band: 50% visible, 0.35m ≤  ≤ 0.7m; 99%,  ≤ 4m

  9. Solar flux at the top of the atmosphere: FS=1365-1372 W/m2 Solar constant: (mean solar flux on 1 square meter of earth) Usually, we choose . Not all of the radiation received at the top of atmosphere is available to the ocean

  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.

  11. Factors influencing QS 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)

  12. Skylight is important at high latitudes Stockholm (59oN) direct sunlight skylight July 80% 20% December 13% 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

  13. 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

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

  15. Another example: Reed (1977) monthly mean shortwave radiation n~ fractional cloud cover (0.3 ≤n≤1). Otherwise Qs=Qso. ~ noon solar elevation in degrees. cn~ cloud attenuation factor (≈0.62). ~ albedo.

  16. 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.

  17. 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%.

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

  19. Annual Mean Solar Radiation at Sea Surface (W/m2)-COADS

  20. Annual Mean Cloud Cover-COADS

  21. Mean Surface Solar Radiation (W/m2), January, COADS

  22. Mean Surface Solar Radiation (W/m2), July, COADS

  23. Distribution daily inflow of solar radiation • The highest value (>300 W/m2) occur at 30oS and 30oN 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).

  24. 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

  25. Long-wave radiation (Qb) The difference between the energy radiated from the sea surface (T4, 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. Qb is a heat loss to ocean. The outgoing radiation is “longwave” Mean sea surface temperature is T= 12oC=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

  26. Empirical Formula of Qb tw=water temperature (oC). ea=relative humidity above the sea surface. C=cloud cover in oktas (1-8). Qbo=Qb(C=0) ranges from 70-120 W/m2. Qb (Qbo) decreases with tw and ea. ea increases exponentially with tw. Due to the faster increase of ea, inward atmospheric flux is larger than outgoing surface radiation). The net heat loss decreases with tw.

  27. Another formula: =0.98,  increases with latitude (0.5, equator; 0.73, 50o). e water vapor pressure (mb): Saturated water vapor pressure Nonlinearity in water vapor dependence: The water vapor content (humidity) increases exponentially with TS, which could result in a more rapid increase in the atmosphere’s radiation into the sea than the sea’s outward radiation (proportional to TS4. Thus Qb could decrease as TS increases. It should be noted that this is still a highly speculated process, which has not been substantiated with a significant amount of measurements.

  28. Annual Mean Longwave Radiation(W/m2)-COADS

  29. Longwave Radiation, January(W/m2)-COADS

  30. Longwave Radiation, July(W/m2)-COADS

  31. Properties of long wave radiation • Qb does not change much daily, seasonally, or with location. This is because (1) Qb ~T4, for T=283K, T=10K, , which is only 15% increase. (2) Inward radiation follows outgoing radiation. • 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. • Ice-albedo feedback Effect of ice and snow cover is relatively small for Qb but large for Qs due to large albedo (increase from normally 10-15% to 50-80%). Therefore, net gain (Qs-Qb) is reduced over ice. ice once formed tends to maintain.

  32. Accuracy of Radiative Fluxes Radiometers on ships, offshore platforms, and even small islands are used to make direct measurements of radiative fluxes.Wideband radiometers sensitive to radiation from 0.3 µm to 50 µm can measure incoming solar and infrared radiation with an accuracy of around 3% provided they are well calibrated and maintained.Satellite measurements may provide a better estimates of the radiative fluxes than the ship data. The satellite data accuracies are (Setwart 2008): VariableAverage Accuracy Net SW Monthly: ± 5% (± 15 W/m2) Daily: ± 10% Net LW Daily: ± 4-8% (± 15-27 W/m2)

  33. Radiative Fluxes at Surface Satellite Short wave Long wave In Situ

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