Lec02 Review of Equations

7/29/2019 Lec02 Review of Equations

1/20

Review of conservation equations

State, Mass and Momentum


2/20

Equation of State (EOS-80)

Determines water density from T, S, and P

),,(/1),,(/1)0,,(),,( PTSPTSKPTSPTS 22/3)0,,(/1)0,,( DSCSBSATSTS

22/32/3 )()(),,( PNSMPJSISHGSFSEPTSK A through Nare polynomials

Tis temperature in oC

S salinity

Ppressure in bars

Ksecant bulk modulus measure of compresibility


3/20

A B C D

T0 999.842594 8.24493E-1 -5.72466E-3 4.8314E-4

T1 6.793952E-2 -4.0899E-3 1.0227E-4

T2 -9.095290E-3 7.6438E-5 -1.6546E-6

T3 1.001685E-4 -8.2467E-7

T4 -1.120083E-6 5.3875E-9

T5 6.536332E-9

E F G

T0 19652.21 54.6746 7.944E-2

T1 148.4206 -0.603459 1.6483E-2

T2 -2.327105 1.09987E-2 -5.3009E-4

T3 1.360477E-2 -6.1670E-5

T4 -5.155288E-5

H I J

T0 3.239908 2.2838E-3 1.91075E-4

T1 1.43713E-3 -1.0981E-5

T2 1.16092E-4 -1.6078E-6

T3 -5.77905E-7

M N

T0 8.50935E-5 -9.9348E-7

T1 -6.12293E-6 2.0816E-8

T2 5.2787E-8 9.1697E-10

Check values:

489.1069)1000,5,35(

343.1023)0,25,35(

1000


4/20

Greater influence of salinity on density

90 % of Ocean Water

Mean T & S for

World Ocean


5/20

Effects of Temperature and Salinity on Density

T

1

S

1

Thermal Expansion Saline Contraction

x 10-4 oC-1 x 10-4 S-1

Density changes by 0.2 kg/m3 for a T change of 1oC,

and by 0.8 kg/m3 for a S change of 1.


6/20

x

z

y

dy

dz

dx

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

x

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

, v

u

Mass per area per time

(kg/(m2 s))


7/20

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

dzdydxw

zv

yu

xdzdydx

t

0 wzvyuxt which is the same as:

0

z

w

y

v

x

u

zwyvxut

01

z

w

y

v

x

u

Dt

D

or


8/20

This is the Continuity Equation or Equation ofConservation 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

z

w

y

v

x

u

Dt

D

0

Dt

DTaking

Boussinesq approximation

0 zw

y

v

x

u

x

u]O[10

km100

m/s0.1 6-

Dt

D

810O1 Dt

DBut


9/20

Continuity Equation in Bulk Form: b0 VEVPR

z

x


10/20

Conservation of Salt:

ydiffusivitDt

DS

z

S

zK

zy

S

yK

yx

S

xK

xz

Sw

y

Sv

x

Su

t

S

Conservation of Heat:

z

T

zzy

T

yyx

T

xxz

Tw

y

Tv

x

Tu

t

T

Equation of State:

SpTS 11000],,[


11/20

Continuity Equation in Bulk Form: b0 VEVPR

Sb

S0

Salt Conservation Equation in Bulk Form: VbSb =V0S0

z

x


12/20

Conservation of Momentum (Equations of Motion)

Fam

m

Fa

z

ww

y

wv

x

wu

t

w

z

vw

y

vv

x

vu

t

vz

uw

y

uv

x

uu

t

u

dt

Vda


13/20

mF Pressure gradient + friction + tides + gravity+ CoriolisPressure gradient: Barotropic and Baroclinic

Friction: Surface, bottom, internal

Tides: Boundary condition

Gravity: Only in the vertical

Coriolis: Only in the horizontal

REMEMBER, these are FORCES PER UNIT MASS


14/20








15/20

z

h dzgPHydrostatic Pressure

Pressure gradient force per unit mass

x

P

1

dzgPPz

a Total Pressure

dzxgxgxP za

1

Note that even if the density is constant with depth, the horizontal pressure gradient

increases with depth if there is a horizontal density gradient

Barometric Barotropic Baroclinic


16/20








17/20

Friction

z

uA

zy

uA

yx

uA

xzyx

@ surface:

WWC

z

uA

xdas

z

@ bottom: VCrruVuCVuC

z

uA

bb

bb

z

;

@ interior: 22

;?

z

v

z

u

z

g

RiRifzuA

z


18/20








19/20

Gravity

[0, 0, g] = [0, 0, 9.81]

Coriolis

[-fv, fu, 0]

h

f

24

2sin2


20/20

Fam

gz

P

z

vA

zy

vA

yx

vA

xdz

y

g

ygfu

z

vw

y

vv

x

vu

t

v

z

uA

zy

uA

yx

uA

xdz

x

g

xgfv

z

uw

y

uv

x

uu

t

u

z

zyx

z

zyx

10

0

z

w

y

v

x

u

z

S

z

K

zy

S

y

K

yx

S

x

K

xz

Sw

y

Sv

x

Su

t

S

z

T

zzy

T

yyx

T

xxz

Tw

y

Tv

x

Tu

t

T

SSpTS t 1000;11000],,[

Documents

Lec02 Review of Equations