The thermodynamic equation for seawater where is the irreversible internal energy fluxes driven by temperature gradient, i.e., diffusion of heat. € − ρ c p κ Q ∇ T € α T =− 1 ρ ∂ ρ ∂ p ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ T € dT dt − α T T ρ c p dp dt = Q ρ c p + κ Q ∇ 2 T € Q =− ∂ F S ∂ z F S radiative heat flux € dT dt = Q ρ c p + κ Q ∇ 2 T If we ignore the amount of work done by pressure, the temperature equation becomes s m Q 2 7 10 5 . 1 − × ≈ κ Molecular thermal diffusivity s m S 2 9 10 5 . 1 − × ≈ κ Molecular diffusivity of salt
Transcript
The thermodynamic equation for seawater
where
is the irreversible internal energy fluxes driven by temperature gradient, i.e., diffusion of heat.€ −ρcpκQ∇T €
€ dTdt=Qρcp+κQ∇2TIf we ignore the amount of work done by pressure, the temperature
equation becomes
smQ27105.1 −×≈κ Molecular thermal diffusivity
smS29105.1 −×≈κ Molecular diffusivity of salt
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.
For a column of sea water, let QT be its rate of heat change
Qv heat convergence by currents and sub-scale transport.
QS: solar radiation at the sea surface.Qb: net heat loss due to long wave radiation.Qe: latent heat flux.Qh: sensible heat flux.
QD is geothermal heat flux from the bottom (negligible).
vehbST QQQQQQ −−−−=Then the heat budget is:
Solar radiation: BasicsPlanck’s law: irradiance for absorptance 1=λa
( )152 12),(−− −= TkhcehcTF λλλ
h~ Planck’s constant. k~ Boltzmann’s constant. c~ light speed in vacuum. T~ temperature (Kelvin), λ~wavelength.
4)( TTF σ=4281067.5 −−−×= KWmσ
The wavelength of maximum irradiance (Wien’s law):
Tmαλ = , mKμα 8.2897=
Total irradiance (Stefan-Boltzmann law):
Stefan-Boltzmann constant:
Solar radiation is in shortwave band:50% visible, 0.35μm ≤ λ ≤ 0.7μm; 99%, λ ≤ 4μm
Temperature at sun’s surface: T=5800K λm=0.5μm.
Solar flux at the top of the atmosphere:
FS=1365-1372 W/m2
22
2
0 34325.341~44
mWF
R
RFS SS −==
ππ
Usually, we choose 23424
1370mWSo ==
.
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)
Changes in 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.
Recent evidence based on variability of sunspots and faculae (bright spots) shows that the output varied by ± 0.2% over centuries, and that this variability is correlated with changes in global mean temperature of Earth's surface of ± 0.4oC.
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)
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
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
Empirical Formula (Parameterization)(shortwave flux averaged over 24 hours): FQQ sos =
Example:1). Clear sky radiation 24.0 mWtAQ nnSO =
QSO: clear sky radiation. An: noon altitude of the sun in degree.
tn: length of the day from sunrise to sunset in hours.
( )30012.01 CQQ SOS −=′
SQ′ is the solar flux arriving at the sea surface.
SOS QQ 92,0=′ C=8, SOS QQ 39.0=′
3). Reflection at the sea surface 2)01.0(15.0 SSr QQQ ′−′=
4). Shortwave radiation into the sea ( ) 2241085.0 mWQQQQQ SSrSS ′−′=−′= −
5). Original algorithm overestimates. Multiply by 0.7.
Qso is clear sky solar radiation at sea surface.F is an empirical function of the fractional cloud cover.
2). Cloud reduction
C=4,
Another example: Reed (1977) monthly mean shortwave 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).
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
Long-wave radiation (Qb)The difference between the energy radiated from the sea surface (σT4, 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, λm=10.2μm.
Most of the longwave radiation is in the range 3μm ≤ λ ≤ 80μm
171527~285
58004
⎟⎠
⎞⎜⎝
⎛=E
S
FF
Longwave radiation is much smaller than the shortwave solar radiation
2)1.01)(46.09.0143( mWCetQ awb −−−=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.
Empirical Formula of Qb
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.
=0.98, λ increases with latitude (0.5, equator; 0.73, 50o).
e water vapor pressure (mb):
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 TS
4. 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.
)(TeRHe d×=
Saturated water vapor pressure ⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−=
ad TTe
16.273185.19exp108.6)(
Annual Mean Longwave Radiation(W/m2)-COADS
Longwave Radiation, January(W/m2)-COADS
Longwave Radiation, July(W/m2)-COADS
• Qb does not change much daily, seasonally, or with location. This is because
(1) Qb ~T4, for T=283K, T=10K,
15.1~283
29344
⎟⎠
⎞⎜⎝
⎛=⎟⎠
⎞⎜⎝
⎛ +T
TT
• Effect of cloud is significant. The big difference between clear and cloudy skies is because the atmosphere is transparent to radiation range from 8-13μm while clouds are not.
, which is only 15% increase.
(2) Inward radiation follows outgoing radiation.
• Ice-albedo feedbackEffect 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.
Properties of long wave radiation
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):
Variable Average AccuracyNet SW Monthly: ± 5% (± 15 W/m2)
