x

z

y

dy

dz

dxFlux of mass in (kg/s) = dzdyu Flux of mass out (kg/s) = dzdyu

dzdydxux

Net Flux of mass in ‘x’ = dzdydxux

Net Flux of mass in ‘y’ = dzdydxvy

Net Flux of mass in ‘z’ = dzdydxwz

dxux

u

, u

, w

, vu

Mass per volume per time(kg/(m2 s)

The change of mass per unit time going through the volume element is:

And the change of mass per unit time per unit volume is:

dzdydxtt

M

dzdydxwz

vy

ux

dzdydxt

0

w

zv

yu

xt

which is the same as:

0

zw

yv

xu

zw

yv

xu

t

01

zw

yv

xu

DtD

or

This is the Continuity Equation or Equation of Conservation of Mass

How valid is the Boussinesq approximation in the OCEAN?

How would you determine that?

1 sigma-t throughout one day = 1 / (24*3600.) = 1.1510-5

01

zw

yv

xu

DtD

0DtDTaking

Boussinesq approximation

0

zw

yv

xu

xu ]O[10

km 100m/s 0.1 6-

DtD

810O1 DtD

But

0

u

zw

yv

xu

zyx,, Nabla or Del operator

Special case of variations in the horizontal only:

yxh ,

econvergenc 0 uh

x

zu u

ww ww

xdivergence 0 uh

z

u u

ww ww

Example:

What is the vertical velocity in an upwelling region 30 m deep, where the flow accelerates southward by 0.10 m/s in 10 km, and westward by 0.20 m/s in 10 km?

xu

yv

54 102

102.0

54 101

101.0

5103

zw

30

0

5103 dzw

smzw / 10930103|103 45300

5

daymw / 8.77)(86400)109( 4

Continuity Equation in Bulk Form: b0 V E V P R

Continuity assuming: steady stateone dimensional motionflow integrated over a closed surface

x

z

y

dy

dz

dx

Flux of salt into dydz = dzdyxSKdzdyuS

Net Flux of Salt in ‘x’ =

dzdydxxSK

xdzdydxuS

x

, u

, w

, v

uS

Advective flux of salt

Conservation of Salt

xSK

Diffusive flux of salt

dxuSx

uS

dxxSK

xxSK

Flux of salt out of dydz = dzdydxuSx

dzdyuS

dzdydxxSK

xdzdy

xSK

Net Flux of Salt in ‘y’ =

dzdydxySK

ydzdydxvS

y

Net Flux of Salt in ‘z’ = dzdydxzSK

zdzdydxwS

z z

Net Salt Flux per unit volume =tS

zSK

zySK

yxSK

xwS

zvS

yuS

xtS

z

zw

yv

xuS

zSw

ySv

xSu

0

zSK

zySK

yxSK

xzSw

ySv

xSu

tS

z

Salt Conservation

Conservation of Salt:

zS

zKzy

SKyx

SKxz

SwySv

xSu

tS

ydiffusivitDtDS

At steady state and assuming constant diffusivities:

2

2

2

2

2

2

zS

zKySK

xSK

zSw

ySv

xSu

Advection-Diffusion Equation

Further assuming motion in one direction and integrated over the volume considered,the statement of CONSERVATION OF SALT may be given as:

Vin Sin = Vout Sout

Continuity Equation in Bulk Form: b0 V E V P R

Sb

S0

Salt Conservation Equation in Bulk Form: VbSb = V0S0

Example of Conservation Principles

What is the volume inflow and outflow at the Chesapeake Bay entrance if the mean river discharge is 2,200 m3/s, and the outflowing salinity is 24 and the inflowing salinity is 30?

Conservation of Mass: Vout = Vin + R

Conservation of Salt: Vout Sout = Vin Sin

ininoutin SVSRV

ininoutoutin SVRSSV

outoutinin RSSSV

outin

outin SS

SRV

sVin /m 88006

242200 3

sVout /m 11000 3

Residence Time

Time it takes to flush entire volume of system (this is one definition).

May be determined from volume of basin and volume of water that enters the basin per unit time. From the above example:

days 53/m 8800

m 1043

310

sRT

Conservation of Heat:

zT

zzyT

yxT

xzTw

yTv

xTu

tT

Earth doesn’t gain or lose heat, but the ocean exchanges it with the atmosphere

pa

iBls

pa CQQQQ

CQ

zT

z

Q heat flux through air-water interface (Watts/m2)

rhoa density of air (1.2 kg/m3)

Cp specific heat of moist air [~1000 joules/(kg oC)]

Qs sensible heat or direct thermal transfer

Ql latent heat flux or evaporation

QB long wave radiation from the ocean

Qi short-wave radiation or incident solar radiation

Heating and Cooling Processes

100

30 reflected from clouds, by water and land, and backscattered by air(shortwave radiation)

19

51

51 absorbedin ocean

To Space

21 long-waveradiation

6 to space

23 Evaporation 7 Sensible

Absorbed byAtmosphere

15

64 emitted byclouds, water vapor and CO2

(long-wave radiation)

QiQl Qs

QB

1370 W/m2

)7.01)(1(

)6.01(

)(

)(

42

10

10

csioi

scB

sl

ass

QQ

TQ

WqqQ

WTTQ

zC

zzyC

yxC

xzCw

yCv

xCu

tC

Advection-Diffusion for a Non-conservative Tracer

C could be chlorophyll_a

dissolved oxygen

nutrient species

pollutant

planktonic species

sediment concentration

zC

zzyC

yxC

xzCw

yCv

xCu

tC

At oceanic station S1 the [DIN] is 200 µg/l and at station S2, 5 km to the E, it is 300 µg/l. If biochemical processes add 100 µg/l to the entire area each day and mixing/diffusion can be neglected, how much would you expect [DIN] to change in time if the region is swept by a 0.5 m/s eastward current?

slg

xCu

tC

86400/100

slg

mlgsm

tC

864/ 1

5000/100/ 5.0

slg

tC

/ 009.0