Water, salt, and heat budget ●Conservation laws application: box models ●Surface fresh water flux: evaporation, precipitation, and river runoff ●Surface.

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

Mass Conservation

Continuity equation

0

z

w

y

v

x

u

t

Mass Conservation

0

z

w

y

v

x

u

t

Continuity equation

Integrating in ocean depth,

0

D

dzm , total mass in a column, we have

00

0

DzzD

HH wwdzVt

m

.

yxH ,

, ),( vuVH

RPEwz

0

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)

where

Vertical boundary conditions:

0 Dz

w

)(0

ERPdzVt

m

D

HH

Integrating the continuity equation in S with boundary L:

s

mdsM

s D

HH ERPdsdzVt

M)(

0

0

ˆD L

ERPdzdlnVt

M

n̂Where is a unit vector perpendicular to the boundary L.

Gaussian formula:

Integrating a two dimensional vector field over an area S with boundary L, we have

Define the mass in a water column of bottom area as S:

Using Gaussian formula

S L

dlnRdsR ˆ

R

0V

free slip condition: 0ˆ nV on L.

0

0ˆD L

dzdlnV

)( ERPt

M

Lateral boundary conditions:

If L is a closed basin (e.g., the coastal line of an ocean domain):

no slip condition:

In both cases

Then

Salt Conservation

z

S

zSA

z

Sw

y

Sv

x

Su

t

SHH

sm234 10~10 , vertical eddy diffusion coefficient.

smA 231 10~10 , horizontal eddy diffusion coefficient.

smS29105.1 The molecular diffusivity of salt is

Ratio between eddy and molecular diffusivity: 610~S

Integrating for the whole ocean column,

000

z

DzD

HHH

D z

SSwdzSASVSdz

t

RPEwz

0We have known that 0

Dzwand

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).

0

zz

SSF

The turbulent salt flux through the surface and at the bottom of the sea are small

Dzz

SBF

(entrainment of salt crystals into atmosphere)

The amount is small and negligible for salt budget.

(subsidence at the bottom, underwater volcano-hydrothermal vents)

00

z

Dzz

SSw Overall,

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 ±0.003. Given average salinity 35 psu, the instrument uncertainty is in the order of ±0.003/35=1500/17 million.

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

SmEP 1Where m is the rate of volume of the sea water entrained into the thin layer from its bottom

S

EPm

1

The corresponding turbulent salt flux is

S

SEPmS

z

S

1

oSPE

S

SPE

z

S)(

1

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

oz

SPEz

S)(

0

Apparent salt flux

Annual Mean Precipitation (mm/day)-COADS

Annual mean evaporation (mm/day)-COADS

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

Mean Sea Surface Salinity

Box ModelUnder 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.

EVPRV oi

Salt conservation: oooiii SVSV influx outflux

Denote excess fresh water as EPRVVX io

oi

XVV io

Since

With , we have oiii SXVSV

oi

oi SS

XSV

oi

io SS

XSV

1

oi

i

oi

o

SS

S

SS

S

i

oi S

SXV

ooii SVSV

and

If Si≈So,

(Vi , Vo) » X. Large exchange with the outside.

If Si » So,

Vi « X. Vo slightly larger than X. Small exchange.

Basin Mediterranean Sea Black Sea

Totoal volume (km3) 3.8 x 1060.6 x 106

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

Examples

Circulation Patterns

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.

Heat budgetTemperature (Potential Temperature) Equation

TCAz

TC

zQ

z

TCw

y

TCv

x

TCu

t

TCpHH

ppppp

where

yxH ,

.

CkgJC op 4000 : specific heat capacity at constant pressure.

sm234 10~10 , vertical eddy diffusion coefficient.

smA 231 10~10 , horizontal eddy diffusion coefficient.

smQ27105.1 . ( 410~

Q

), Molecular thermal diffusivity

Define TCh p , we have

hAz

h

zQhV

t

hHH

3

0

D

hdzHand

HAz

h

z

hQdzhdzV

t

HHH

DzzD D

HH

0

0 0

t

HQT

, heat storage.

0

D

HHv HAhdzVQ

, heat convergence by currents and sub-scale transport.

0

D

sp QdzQ , penetrating solar radiation.

hebsaz

sf QQQQz

hQ

0

, 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.

0~20

1 2mWz

hQ

DzD

, geothermal heat flux (neglected).

vehbST QQQQQQ

We also take Qs=Qsp+Qsa as total solar radiation.

Then the heat budget is:

Solar radiation: BasicsPlanck’s law: Black body irradiance (absorptance ) 1a

152 12),( TkhcehcTF

h~ Planck’s constant. k~ Boltzmann’s constant. c~ light speed in vacuum. T~ temperature (Kelvin), λ~wavelength.

4)( TTF 4281067.5 KWm

The wavelength of maximum irradiance (Wien’s law):

Tm , mK 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 S

S

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)

If the incoming radiation is normalized to 100%, then 16% are absorbed in the atmosphere

24% are reflected by clouds7% are radiated back to space from the atmosphere4% are reflected from the earth's surface (mainly from the sea)The rest into the ocean (49%)

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)

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)

10019.01 ncQQ nsos

n~ fractional cloud cover (0.3 ≤n≤1). Otherwise Qs=Qso.

φ~ solar elevation in degrees.cn~ cloud attenuation factor (≈0.62).

α~ albedo.

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

Annual Mean Cloud Cover-COADS

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

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

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).

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

F

F

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.

Another formula: aSSSb TTTneTQ 325.04 4105.039.0

ε=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, 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.

)(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, ΔT=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

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: tee LFQ

Fe is the rate of evaporation of water in kg/(m2 s).

Lt is latent heat of evaporation in kJ.

For pure water, kgkJtLt )2.22494( . t~ water temperature (oC).

t=10oC, Lt=2472 kJ/kg.t=100oC, Lt=2274 kJ/kg.

In general, Fe is parameterized with bulk formulae:dz

deKF ee

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.

In practice: )()(4.1 2daymkgeeVF ase

V wind speed (m/s) at 10 m height above sea.

23102.224944.1 mWteeVLFQ sastee

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.

m

ee

z

ee

dt

de asas

10

, and VK e 14 (very crude parameterization).

This empirical formula is an approximation of eddy diffusion formula because:

• 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.

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

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

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

Sensible heat flux (Qh):On average, the ocean surface is about 0.3-0.8°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.

Empirical formula of Qh

Bulk formula:

dz

dTKCQ hph

Wyrtki (1965): VCK dah

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

288.1 mWttVQ ash .

More recent bulk formula (Smith 1988):

asete qqVKLQ wh

ere p

e

ep

eq 622.0622.0

. asphh ttVCKQ Surface sensible heat flux

is specific humidity

Surface latent heat flux

Ke and Kh are mainly functions of stability and wind speed.

Ke≈1.20Kh

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

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

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

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

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

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

Magnitudes of heat budget terms

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).

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.

Where does the heat go in the ocean?

ehbSv QQQQQ

ehbS QQQQ • 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.

i.e., in global average,

1. Globally, conservation for steady state is : heat in = heat out (It’s trivial!)

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.,

0

D

HHv HAhdzVQ

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

y

TACdzvTC

yQ p

D

pv

0

where and vT T are zonal averages.

0

D

y

y

hebSp

n

dyQQQQdzvTC

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.)

Direct Transport EstimateTTT Let and vvv

where the bar over a variable represents vertical average the prime its departure

Then the meridional heat transport becomes

Barotropiccomponent

Barocliniccomponent

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.

dzTvCTvDCdzvTCD D

ppp

0 0

Heat Storage

For oceanic variations (e.g., seasonal cycle), the heat storage is important

vehbST QQQQQQ It can be contributed by all the surface fluxes and transport terms.

Seasonal heat storageHeat 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.

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.

Diurnal heat storageThis heat storage generates the diurnal thermocline.

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