conservation of salt: conservation of heat: equation of state: conservation of mass or continuity:...

Conservation of Salt:

zS

zK

zyS

yK

yxS

xK

xzS

wyS

vxS

utS

Conservation of Heat:

zT

zzyT

yyxT

xxzT

wyT

vxT

utT

Equation of State: ],,[ pTS

0

zw

yv

xu

Conservation of Mass or Continuity:

Equations that allow a quantitative look at the OCEAN

Conservation of Momentum (Equations of Motion)

mF

a

zw

wyw

vxw

utw

zv

wyv

vxv

utv

zu

wyu

vxu

utu

dtdw

dtdv

dtdu

dtVd

a

Fam

Newton’s Second Law:

Conservation of momentum Vm

as they describe changes of momentum in time per unit mass

adtVd

Vmdtd

m

1

Circulación típica en un fiordo

x

z

mFa

Aceleraciones

dtdu

zu

wyu

vxu

utu

x

z

Gradiente de presión

Debido a la pendiente del nivel del mar (barotrópico)

Debido al gradiente de densidad (baroclínico)

dzx

gx

gxp

H

01

x

z

Fricción

Debida a gradientes verticales de velocidad (divergencia del flujo de momentum)

zu

Az z

x

z

Coriolis

Debido a la rotación de la Tierra; proporcional a la velocidad

fv

x

z

Balance de momentum

zp

g

zv

Azy

pfu

dtdv

zu

Azx

pfv

dtdu

z

z

1

1

1x

z

mF

Pressure gradient + friction + tides+ gravity+ Coriolis

Pressure gradient: Barotropic and Baroclinic

Coriolis: Only in the horizontal

Gravity: Only in the vertical

Friction: Surface, bottom, internal

Tides: Boundary condition

REMEMBER, these are FORCES PER UNIT MASS

Forces per unit mass that produce accelerations in the ocean:

mF

Pressure gradient + friction+ tides+ gravity+ Coriolis

Pressure gradient: Barotropic and Baroclinic

Coriolis: Only in the horizontal

Gravity: Only in the vertical

Friction: Surface, bottom, internal

Tides: Boundary condition

REMEMBER, these are FORCES PER UNIT MASS

x

z

y

dy

dz

dx

p dxxp

p

Net Force in ‘x’ = dzdydxxp

Net Force per unit mass in ‘x’ = dzdydxxp

Vol

1xp

1

Total pressure force/unit mass on every face of the fluid element is: pzp

yp

xp

1

,,1

Illustrate pressure gradient force in the ocean

z

z

1 2

Pressure Gradient?Pressure Gradient

Pressure Gradient Force

Pressure of water column at 1 (hydrostatic pressure) : zgP 1

Hydrostatic pressure at 2 : zzgP 2

x

Pressure gradient force caused by sea level tilt:

xz

gxzg

xPP

xp

1211

BAROTROPIC PRESSURE GRADIENTBAROTROPIC PRESSURE GRADIENT

Descarga de Agua Dulce

Precipitación pluvial y Ríos

Aporte aproximado por lluvia: 2000 mm por año

area superficial: 350 km por 10 km = 3.5x109 m2

200 m3/s

Dirección Meteorológica de Chile

Aporte aproximado por ríos: 1000 m3/s

Milliman et al. (1995)

dzx

gx

gxp

H

01

mF

Pressure gradient + friction + tides+ gravity+ Coriolis

Pressure gradient: Barotropic and Baroclinic

Coriolis: Only in the horizontal

Gravity: Only in the vertical

Friction: Surface, bottom, internal

Tides: Boundary condition

REMEMBER, these are FORCES PER UNIT MASS

Acceleration due to Earth’s Rotation

Remember cross product of two vectors: ),,( 321 aaaA

),,( 321 bbbB

and

321

321

ˆˆˆ

bbb

aaa

kji

BA

)(ˆ 2332 babai )(ˆ 3113 babaj )(ˆ 1221 babak

),,( 122131132332 babababababaC

Now, let us consider the velocity of a fixed particle on a rotating body at the positionV

The body, for example the earth, rotates at a rate

r

r

V

r

, V

To an observer from space (us):E

Ef rdtrd

dtrd

This gives an operator that relates a fixed frame in space (inertial) to a moving object on a rotating frame on Earth (non-inertial)

EEf

dtd

dtd

This operator is used to obtain the acceleration of a particle in a reference frame on the rotating earth with respect to a fixed frame in space

EEf r

dtrd

dtrd

EEf

dtd

dtd

EEE

EEf r

dtrd

dtrd

rdtd

dtrd

dtd

dtrd

dtd

r

V

0

EEEf rV

dtVd

dtVd

2

Acceleration of a particle on a rotating Earth with respect to an observer in space

Coriolis Centripetal

forcesotherprVdtVd

EEE

1

2

The equations of conservation of momentum, up to now look like this:

Coriolis Acceleration

90

Cv

C h

vhvSNWE CC ,,0,,

cos90sin

hC

sin90cos

vC

sin,cos,0

uuvw

wvu

kji

V cos2,sin2,sin2cos2sincos0

ˆˆˆ

22

h

f

242

sin2

Making:

f is the Coriolis parameter

ufufvwV cos2,,cos22

This can be simplified with two assumptions:

1) Weak vertical velocities in the ocean (w << v, u)

2) Vertical component is ~5 orders of magnitude < acceleration due to gravity

0,,2 fufvV

0,,2 fufvV

Eastward flow will be deflected to the south

Northward flow will be deflected to the east

f increases with latitude

f is negative in the southern hemisphere

sin2f

mF

Pressure gradient + friction + tides+ gravity+ Coriolis

Pressure gradient: Barotropic and Baroclinic

Coriolis: Only in the horizontal

Gravity: Only in the vertical

Friction: Surface, bottom, internal

Tides: Boundary condition

forcesother

xp

xp

xp

dtdw

Cfudtdv

Cfvdtdu

y

x

1

1

1

0

mF

Pressure gradient + friction + tides+ gravity+ Coriolis

Pressure gradient: Barotropic and Baroclinic

Coriolis: Only in the horizontal

Gravity: Only in the vertical

Friction: Surface, bottom, internal

Tides: Boundary condition

REMEMBER, these are FORCES PER UNIT MASS

Centripetal acceleration and gravity

fg

r

r

forcesotherpgrVdtVd

f 1

2

fg

r

g

),0,0( gg

g has a weak variation with latitude because of the magnitude of the centrifugal acceleration

cos2 rg is maximum at the poles and minimum at the equator (because of both r and lamda)

Variation in g with latitude is ~ 0.5%, so for practical purposes, g =9.80 m/s2

forcesotherpgVdtVd

1

2

friction

gxp

xp

xp

dtdw

fudtdv

fvdtdu

1

01

01

0

Friction (wind stress)z

W

u

Vertical Shears (vertical gradients)

Friction (bottom stress)z

u

bottom

Vertical Shears (vertical gradients)

Friction (internal stress)z

u1

Vertical Shears (vertical gradients)

u2

Flux of momentum from regions of fast flow to regions of slow flow

x

z

y

dy

dz

dx

Shear stress has units of kg m-1 s-1 m s-1 m-1 = kg m-1 s-2

Shear stress is proportional to the rate of shear normal to which the stress is exerted zu

zu at molecular scales

µ is the molecular dynamic viscosity = 10-3 kg m-1 s-1 for water is a property of the fluid

or force per unit area or pressure: kg m s-2 m-2 = kg m-1 s-2

xu

dxxu

xxu

y

u

dyyu

yyu

zu

dzzu

zzu

x

z

y

dy

dz

dx

xu

dxxu

xxu

y

u

dyyu

yyu

zu

dzzu

zzu

Net force per unit mass (by molecular stresses) on u

zu

zyu

yxu

xFx

1

zu

zyu

yxu

x

sm26-10viscositymolecularkinematic

uzu

yu

xu

Fx2

2

2

2

2

2

2

If viscosity is constant,

zu

zyu

yxu

xFx becomes:

VpgVdtVd

)(1

2 2

And up to now, the equations of motion look like:

These are the Navier-Stokes equations

Presuppose laminar flow!

Compare non-linear (advective) terms to molecular friction

22

2

2

~

~

LU

xu

LU

xu

u

Inertial to viscous: Re2

2

UL

LULU Reynolds Number

Flow is laminar when Re < 1000

Flow is transition to turbulence when 100 < Re < 105 to 106

Flow is turbulent when Re > 106, unless the fluid is stratified

Low Re

High Re

Consider an oceanic flow where U = 0.1 m/s; L = 10 km; kinematic viscosity = 10-6 m2/s

610

100001.0Re 910

Is friction negligible in the ocean?

Frictional stresses from turbulence are not negligible but molecular friction is negligible at scales > a few m.

'TTT

T 0'' TT

0'

0'

TT

T

TT

- Use these properties of turbulent flows in the Navier Stokes equations-The only terms that have products of fluctuations are the advection terms- All other terms remain the same, e.g., tutututu

0

'

zu

wyu

vxu

uzu

wyu

vxu

u

'

''

''

'

dtud

z

wu

y

vu

x

uu

''''''

zw

uyv

uxu

uzu

wyu

vxu

u

'

''

''

''

''

''

'

zw

yv

xu

u'''

'

0

'','','' wuvuuu are the Reynolds stressesReynolds stresses

arise from advective (non-linear or inertial) terms

zu

Awu

yu

Avu

xu

Auu

z

y

x

''

''

''

This relation (fluctuating part of turbulent flow to the mean turbulent flow) is called a

turbulence closureturbulence closure

The proportionality constants (Ax, Ay, Az) are the eddy (or turbulent) viscositieseddy (or turbulent) viscosities and are a property of the flow (vary in space and time)

zu

Azy

uA

yxu

Ax

F zyxx

Ax, Ay oscillate between 101011 and 101055 mm22/s/s

Az oscillates between 1010-5-5 and 1010-1-1 mm22/s/s

zu

Azy

uA

yxu

Ax

F zyxx

Az << Ax, Ay but frictional forces in vertical are typically stronger

eddy viscosities are up to 1011 times > molecular viscosities

zw

Azy

wA

yxw

Ax

gzp

dtdw

zv

Azy

vA

yxv

Axy

pfu

dtdv

zu

Azy

uA

yxu

Axx

pfv

dtdu

zyx

zyx

zyx

1

1

1

Equations of motion – conservation of momentum

Fam

zp

g

zv

Azy

vA

yxv

Axy

pfu

dtdv

zu

Azy

uA

yxu

Axx

pfv

dtdu

zyx

zyx

1

1

1

0

zw

yv

xu

zS

zK

zyS

yK

yxS

xK

xzS

wyS

vxS

utS

],,[ pTS

zT

zzyT

yyxT

xxzT

wyT

vxT

utT

T

S

p

w

v

u