1. Water, salt, and heat budget
●Conservation laws
application: box models
●Surface fresh water flux:
evaporation, precipitation, and river runoff
●Surface heat flux components:
sensible, latent, long and shortwave
●Ocean meridional transport

2. Mass Conservation
Continuity equation

3. Mass Conservation Continuity equation
, total mass in a column, we have
Integrating in ocean depth, ,
where .
Vertical boundary conditions:
E-evaporation, P-precipitation, R-river runoff
(measured in m/s, 1mm/day=1.1574x10-8 m/s).
Melting of sea ice may also be a factor
(neglected here)

4. Gaussian formula:
Integrating a two dimensional vector field over an area S with boundary L, we have
Where is a unit vector perpendicular to the boundary L.
Define the mass in a water column of bottom area as S:
Integrating the continuity equation in S with boundary L:
Using Gaussian formula

5. Lateral boundary conditions:
If L is a closed basin (e.g., the coastal line of an ocean domain):
no slip condition:
on L.
free slip condition:
In both cases 
Then

6. Salt Conservation The molecular diffusivity of salt is
, vertical eddy diffusion coefficient.
, horizontal eddy diffusion coefficient.
The molecular diffusivity of salt is
Ratio between eddy and molecular diffusivity:
Integrating for the whole ocean column,

7. We have known that
and
However, both E and P transfer the fresh water with S=0
There is a net salt influx into the oceans from river runoff (R), which is totally about 3 x 1012 kg/year. About 10% of that is recycled sea salt (salt spray deposited on land).
The turbulent salt flux through the surface and at the bottom of the sea are small
(entrainment of salt crystals into atmosphere)
(subsidence at the bottom,
underwater volcano-hydrothermal vents)
Compared to the total salt amount in the ocean: 5 x 1019 kg, the rate of annual salt increase is only one part in 17 million/year. As we know, the accuracy of present salinometer is ± Given average salinity 35 psu, the instrument uncertainty is in the order of ±0.003/35=1500/17 million.
The amount is small and negligible for salt budget.
Overall,

8. There is a net vertical salt flux near the sea surface driven by the fresh water flux.
Consider a thin interfacial layer, the balance of fresh water flow is
Where m is the rate of volume of the sea water entrained into the thin layer from its bottom
The corresponding turbulent salt flux is or
Where So is usually chosen as 36‰.
Usually, we neglect the effect of E-P on mass balance (i.e., w(z=0)=0) and take into its effect on salinity as
Apparent salt flux

9. Annual Mean Precipitation (mm/day)-COADS

10. Annual mean evaporation (mm/day)-COADS

11. Annual Mean E-P (mm/day)-COADS

12. Mean Sea Surface Salinity

13. Box Model
Under steady-state conditions, we apply the conservations of mass and salt to a box of volume V filled with sea water.
Conservation of volume:
Where Vi is inflow, Vo outflow; P precipitation, E evaporation, and R river runoff.
Salt conservation:
influx
outflux

14. (Vi , Vo) » X. Large exchange with the outside.
Denote excess fresh water as
Since
With
, we have 
and
If Si≈So,
(Vi , Vo) » X. Large exchange with the outside.
If Si » So,
Vi « X. Vo slightly larger than X. Small exchange.

15. Examples Basin Mediterranean Sea Black Sea Totoal volume (km3)
X=P-E(m3/s)
-7x104
6.5 x 103 Si
36.3 35 So
37.8 17
Vi (m3/s, km3/yr)
1.75x106, 5.5 x 104
6x103, 0.02x104
Vo(m3/s)
1.68 x 106
13x103
Flushing time (yr) 70
3000

16. Circulation Patterns

17. An evaporation rate of 1. 2 m/yr is equivalent to removing about 0
An evaporation rate of 1.2 m/yr is equivalent to removing about 0.03% of the total ocean volume each year. An equivalent amount returns to the ocean each year, about 10% by way of rivers and the remainder by rainfall.
The yearly salt exchange is less than 10-7 of the total salt content of the ocean.

18. Heat budget Temperature (Potential Temperature) Equation . where
: specific heat capacity at constant pressure.
, vertical eddy diffusion coefficient.
, horizontal eddy diffusion coefficient.
, Molecular thermal diffusivity
. ( )
Define
and
, we have

19. , heat storage.
, heat convergence by currents and sub-scale transport.
, penetrating solar radiation.
, surface heat flux.
Qsa: solar radiation absorbed 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).
We also take Qs=Qsp+Qsa as total solar radiation.
Then the heat budget is:

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

21. 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

23. If the incoming radiation is normalized to 100%,
then 16% are absorbed in the atmosphere
24% are reflected by clouds
7% are radiated back to space from the atmosphere
4% are reflected from the earth's surface (mainly from the sea)
The rest into the ocean (49%)

24. 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 θ:
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)

25. 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.

26. Another example: Reed (1977)
n~ fractional cloud cover (0.3 ≤n≤1). Otherwise Qs=Qso.
φ~ solar elevation in degrees.
cn~ cloud attenuation factor (≈0.62).
α~ albedo.

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

28. Annual Mean Cloud Cover-COADS

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

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

32. 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).

34. 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

35. 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
Longwave radiation is much smaller than the shortwave solar radiation

36. 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 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.

37. λ increases with latitude (0.5, equator; 0.73, 50o).
Another formula:
ε=0.98,
λ 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, leading to a “super greenhouse” effect.
It should be noted that this is still a highly speculated process, which has not been substantiated with a significant amount of measurements.

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

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

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

41. Properties of long wave radiation
• Qb does not change much daily, seasonally, or with location. This is because
(1) Qb ~T4, for T=283K, ΔT=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 8-13μm 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.

42. Evaporative heat flux (Qe)
51% of the heat input into the ocean is used for evaporation. Evaporation starts when the air over the ocean is unsaturated with moisture. Warm air can retain much more moisture than cold air.
The rate of heat loss:
Fe is the rate of evaporation of water in kg/(m2 s).
Lt is latent heat of evaporation in kJ.
For pure water,
. t~ water temperature (oC).
t=10oC, Lt=2472 kJ/kg.
t=100oC, Lt=2274 kJ/kg.
In general, Fe is parameterized with bulk formulae:
Ke is diffusion coefficient for water vapor due to turbulent eddy transfer in the atmosphere. It is dependent on wind speed, size of ripples, and waves at sea surface, etc.
de/dz is the gradient of water vapor concentration in the air above the sea surface.

43. In practice:
V wind speed (m/s) at 10 m height above sea.
es is the saturated vapor pressure over the sea-water (unit: kilopascals)
The saturated vapor pressure over the sea water (es) is smaller than that over distilled water (ed). For S=35, es=0.98ed(ts).
ea is the actual vapor pressure in the air at a height of 10 m above sea level. If the atmospheric variable is relative humidity (RH), ea=RH x ed(ta).
Example:
Ta=15oC, ed = 1.71 kPa = 12.8 mm Hg,
RH=85%, then ea= 1.71 x 0.85 kPa= 1.45 kPa.
This empirical formula is an approximation of eddy diffusion formula because:
, and
(very crude parameterization).

44. In most region, es > ea,
Fe and Qe are positive,
 there is a heat loss from the sea due to evaporation.
In general, if ts-ta > 0.3oC, Qe >0.
In some region, ts-ta<0oC (surface air is warmer than SST) and RH is high enough to cause condensation of water vapor from the air into the sea, which results in a gain of heat in the sea. Fogs occur in these regions due to the cooling of the atmosphere over the sea.

45. Annual Mean Latent Heat Flux (W/m2)-COADS

46. Mean Latent Heat Flux (W/m2), January, COADS

47. Mean Latent Heat Flux (W/m2), July, COADS

48. Sensible heat flux (Qh):
On average, the ocean surface is about °C warmer than the air above it (exception: upwelling regions). Direct heat transfer (transfer of sensible heat) therefore occurs usually from water to air and constitutes a heat loss. Heat transfer in that direction is achieved much more easily than in the opposite direction for two reasons:
1. It takes much less energy to heat air than water. The energy needed to increase the temperature of a layer of water 1 cm thick by 1°C is sufficient to raise the temperature of a layer of air 31 m thick by the same amount.
2. Heat input into the atmosphere from below causes instability (through a reduction of density at the ground) which results in atmospheric convection and turbulent upward transport of heat. In contrast, heat input into the ocean from above increases stability (through a reduction of density at the surface) and prevents efficient heat penetration into the deep layers.

49. Empirical formula of Qh
Bulk formula:
Wyrtki (1965):
ρa = 1.2 kg/m3 (density of air).
Cd = 1.55 x 10-3 (drag coefficient at sea surface).
V surface (10 m) wind speed in m/s.
Cp=1008 J/(kg K).
Then .

50. More recent bulk formula (Smith 1988):
Surface latent heat flux
where
is specific humidity
Surface sensible heat flux .
Ke and Kh are mainly functions of stability and wind speed.
Ke≈1.20Kh

51. Annual Mean Sensible Heat Flux (W/m2)-COADS

52. Mean Sensible Heat Flux (W/m2), January, COADS

53. Mean Sensible Heat Flux (W/m2), July, COADS

54. Annual Mean Net Surface Heat Flux (W/m2)-COADS

55. Mean Net Surface Heat Flux in January (W/m2)-COADS

56. Mean Net Surface Heat Flux in July (W/m2)-COADS

57. Magnitudes of heat budget terms

58. Although most of the solar radiation is rapidly absorbed near the oceanic surface in a layer that is centimeters to meters thick, the processes that control heat loss occur in a even thinner layer, i.e., the surface skin of the ocean.
As a consequence, there is a continuous upward heat flux in the top few millimeters of the ocean and a negative temperature gradient of a few tenths of a degree per millimeter in the surface skin of the ocean, which creates the surface skin (a few centimeters, measured by satellites) and bulk temperature (1 or 2 meters below the surface, measured by buoys or ships).
Their difference is around 0.1oC (day) to 0.3oC (night).

59. Where does the heat go in the ocean?
1. Globally, conservation for steady state is : heat in = heat out (It’s trivial!)
i.e., in global average,
• If the relation doesn’t hold, there should be long-term change.
• To achieve a steady state, we should at least average over a year.
2. Locally, ocean gains heat in low latitudes but loses heat in high latitude. To maintain a steady state, heat has to be transported from low to high latitudes to make it up. i.e.,

60. The horizontal heat convergence is
average for a constant latitude globally or from west to east coast of an ocean,
we have the meridional heat transport
where
and
are zonal averages.
Integrating from the northern most extent (yn) where the transport vanishes,
We can determine
from the net surface heat flux.
(The small sub-grid transport is negligible.)

62. Direct Transport Estimate
Let
and
where the bar over a variable represents vertical average the prime its departure
Then the meridional heat transport becomes
Barotropic
component
Baroclinic
component
The baroclinic term can be estimated from the relative geostrophic flow computed from hydrographic data along the section below the mixed layer and Ekman transport in the mixed layer.
The barotropic term is harder to estimate. However, in some locations, such as the25oN section in North Atlantic, reasonable estimate can be made based on information such as the measurements of the northward transport by the Florida Current.

65. Heat Storage
For oceanic variations (e.g., seasonal cycle), the heat storage is important
It can be contributed by all the surface fluxes and transport terms.

66. Seasonal heat storage
Heat is gained in the surface layer in the summer and then is released to the atmosphere in winter, which causes the formation of the seasonal thermocline.

67. Seasonal thermocline: develops in the upper zone in summer.
high stability within the seasonal thermocline
become shallower and stronger as summer progresses
weakens in fall, as daily loss exceeds the heat gain
is driven deeper in fall, as it becomes less stable and as winds increase
disappears in late winter (the cycle restarts in summer again)
Example: Seasonal thermocline at Ocean Weather Station “P” (50oN, 145oW)
March is nearly isothermal in upper 100 meters.
March-August, SST increases, (absorption of solar radiation). Mixed layer  30 m.
August-March, net loss of heat, seasonal thermocline eroding due to mixing.

68. Diurnal heat storage
This heat storage generates the diurnal thermocline.

69. Diurnal thermocline
• develops during the day at depth ~10-20 meters.
• can mix down a few meters
• further mix and cool (weaken) during the night
• anomalies often persists for many days