EVAT 554OCEAN-ATMOSPHERE

DYNAMICS

OCEAN BOUNDARY CONDITIONS

LECTURE 17

(Reference: Peixoto & Oort, Chapter 8,10)

Consider an idealized ocean basin

Z=0

Z=

Z=-H

= E= W

= S

= N

= S

= NZ=0

Z=

Z=-H

= E= W

Consider an idealized ocean basin

An idealized global basin can be

envisioned as a system of coupled

such basins

Z=0

Z=

Z=-H

= E= W

= S

= N

Consider an idealized ocean basin

First, consider the boundary conditions on mass

(u)/=0 at W

No mass flux through the side boundaries,

(v)/=0 at NS

(w)/z=0 at z= ,-H

No mass flux through the vertical boundaries,

Z=0

Z=

Z=-H

= E= W

= S

= N

Consider an idealized ocean basin

Now, consider the boundary conditions on momentum

No normal flow at side boundaries,

w=0 at z=

‘Rigid lid’ approximation at upper boundary

v=0 at NS

u=0 at W

No normal flow at lower boundary

w=0 at z= -H

Z=0

Z=

Z=-H

= E= W

= S

= N

Consider an idealized ocean basin

Now, consider the boundary conditions on momentum

No lateral stress at side boundaries,

Atmospheric windstress forcing of upper boundary

No vertical stress at lower boundary

u/w/=0 at NS

v/w/=0 at , W

u/z=0, v/z=0, w/z=0 at z= H

V/f) u/z=V/f) v/z=w/z=at z=

Z=0

Z=

Z=-H

= E= W

= S

= N

Consider an idealized ocean basin

Now, consider the boundary conditions on temperature

No heat flux through side boundaries,

Heat flux at top boundary

No heat flux at lower boundary

T/z =0 at z= H

T/ =0 at NS

T/ =0 at EW

pCQzT

V // at z= z

Qq

1

Z=0

Z=

Z=-H

= E= W

= S

= N

Consider an idealized ocean basin

Now, consider the boundary conditions on salinity

No salt flux through side boundaries,

Salt flux at top boundary

No salt flux at lower boundary

S/z =0 at z= H

S/ =0 at NS

S/ =0 at EW

SthRPEzSIV

)/(/ at z=

This last term represents brine formation due to the freezing

of ocean water

Z=0

Z=

Z=-H

= E= W

= S

= N

Consider an idealized ocean basin

Return to the boundary conditions on temperature

No heat flux through side boundaries,

Heat flux at top boundary

No heat flux at lower boundary

T/z =0 at z= H

T/ =0 at NS

T/ =0 at EW

pCQzT

V // at z= z

Qq

1

Z=0

Z=

Z=-H

= E= W

= S

= N

Recall the temperature equation

Integrate from the base of the thermocline

Heat flux at top boundary

dtdp

CpCpq

zTz

T

zTwTtT

VH 1)/()(

/)(/

V

pCQzT

V // at z=

Z=-h

zQq

1

Recall the temperature equation

Integrate from the base of the thermocline

Heat flux at top boundary

dtdp

CpCpq

zTz

T

zTwTtT

VH 1)/()(

/)(/

V

pCQzT

V // at z=

Z=-h

Z

T

to the surface

0/00 ||

CpQ

zTV

zQq

1

Recall the temperature equation

Integrate from the base of the thermocline

dtdp

CpCpq

zTz

T

zTwTtT

VH 1)/()(

/)(/

V

Z=-hT

Z

T

to the surface

(steady state)

00 ||/CpQ

zTV

= Qrad - Qrad - Qconv - Qlat Q -sS

-(TS-TA)-=S(,)+aAQ

0/00 ||

CpQ

zTV

Z

T

= Qrad - Qrad - Qconv - Qlat Q

-(TS-TA)-=S(,)+aAQ

-sS

=S(,)+aAQ -(TS-TA)-

cbTaTQSA

00 ||/CpQ

zTV

00 ||/CpQ

zTV

SA

TTzT0|/

Z=-hT

-sS

Z

T -

sS=S(,)+aA

Q -(TS-TA)-

cbTaTQSA

00 ||/CpQ

zTV

SA

TTzT0|/

Consider departures from steady state,

''/'0|

SATTzT

)''(/'0|

SATTzT

This is sometimes approximated by a ‘restoring’ boundary condition,

Why ‘restoring’?

Z=-hT

Reconsider temperature equationIntegrate over the mixed layer (depth = h)

dtdp

CpCpq

zTz

T

zTwTtT

VH 1)/()(

/)(/

V

Z

T

0|/')'(')(/' zTThThtThVSSS H

V

Z=-h

Consider departures from steady state,

''/'0|

SATTzT

)''()'(')(/'SAVSSS

TTThThtThH

V

)''()'(')(/'SA

VSSS

TTTTtThH

V

zQq

1

Reconsider temperature equationIntegrate over the mixed layer (depth = h)

dtdp

CpCpq

zTz

T

zTwTtT

VH 1)/()(

/)(/

V

Z

T

0|/'/' zTtThVS

Z=-h

TS’/=-TS’+TA’

Consider departures from steady state,

''/'0|

SATTzT

)''(/'SAVS

TTtTh

)''(/'SA

VS

TTtTh

= t/t0

V

ht /0

/

Mixed Layer Temperatures

Z

T

Z=-h

TS’/=-TS’+TA’= t/t0

V

ht /0

We will assume that TA’ represents random atmospheric surface temperature variations due to e.g. mid-latitude storm

systems.

For simplicity, this can be approximated as Gaussian white noise

)0,(ˆ' wTTA

This model exhibits a “Red Noise” spectrum:

[Hasselmann, K. (1976) Stochastic climate models. Part I: Theory.

Tellus 28:473-485.]

/

Mixed Layer Temperatures

Z

T

Z=-h

TS’/=-TS’+TA’= t/t0

V

ht /0

This model exhibits a “Red Noise” spectrum:

frequency

220

20

2

t4

ˆ)(

ft

TfS

Pow

er Spectral D

ensity

This simple model can be generalized upon /

Mixed Layer Temperatures

Z

T

Z=-h

Generalizes upon the simple red noise model by allowing for both a mixed layer and

a “deep ocean” with exchange of heat by advection and diffusion between the two

layers

“Upwelling-diffusion” EBMZ=-hT

1

222

2221

ˆ)(

222

00

222

002

Wv

h

tft

Wv

h

t

h

WtTfS

= t/t0

V

ht /0

This simple model can be generalized upon

TS’/=-TS’+TA’

/

Mixed Layer Temperatures

Z

T

Z=-h1

222

2221

ˆ)(

222

00

222

002

Wv

h

tft

Wv

h

t

h

WtTfS

“Upwelling-diffusion” EBMZ=-hT

Wigley and Raper (1990) Natural Variability of the Climate System and Detection of the Greenhouse Effect.

Nature 344:324-327.