EVAT 554 OCEAN-ATMOSPHERE DYNAMICS FILTERING OF EQUATIONS FOR OCEAN LECTURE 10 (Reference: Peixoto & Oort, Chapter 3,8)

EVAT 554OCEAN-ATMOSPHERE

DYNAMICS

FILTERING OF EQUATIONS FOR OCEAN

LECTURE 10

(Reference: Peixoto & Oort, Chapter 3,8)

Note that the major horizontal ocean

circulation systems mirror closely the semi-

permanent high and low pressure

systems

Scale AnalysisThe Ocean

)/()(/cosˆ1'v

/)(//

zuz

upa

wff

zuwutudtdu

VH

V

Zonal Momentum Balance:

)/v()v(/ˆ1

/vv)(/v/v

zz

pa

fu

zwtdtd

VH

VMeridional Momentum Balance:

Vertical Momentum Balance:

)/()w(/ˆ1

ˆ'

/)(//

zwz

zpguf

zwwwtwdtdw

VH

V

Continuity: 01 zw

dtd V

),,( TSpEquation of State:

dtdp

CpCplatq

Cpradq

zTz

T

zTwTtTdtdT

VH 1)/()(

/)(//

VHeat Equation:

(incompressible!)

How many equations?

How many variables?

)/()(/cosˆ1'v/ zu

zup

awffdtdu VH

Zonal Momentum Balance:

222 //ˆ/'v/ huluLpwffLuVH

Length scale: L106m, l105m

Depth scale: H103m, h 102m

Horizontal velocity scale: u,v 10-1 ms-1

Vertical velocity scale: w 10-4 ms-1

Horizontal pressure scale: p 100 mb = 104 Pa

Time Scale: L/u 107s H/w 107s

Radius of Earth: a=6.37x 106m

Coriolis parameter: f,f' 10-4 s-1

Density of Water: 1000 kg m-3

Horizontal Eddy Viscosity: H 105 m2s-1

Vertical Eddy Viscosity: V 10-1 m2s-1

10-8 ms-2

/cosˆ1v0 pa

f

10-5 ms-2 10-8 ms-2 10-5 ms-2 10-6 ms-2 10-6 ms-2

Meridional Momentum Balance:

10-8 ms-2 10-5 ms-2 10-5 ms-2 10-6 ms-2 10-6 ms-2

)/v()v(/ˆ1/v z

zp

afudtd VH

222 //ˆ/u/ huluLpfLuVH

/ˆ10 pa

fu

Length scale: L106m, l105m

Depth scale: H103m, h 102m

Horizontal velocity scale: u,v 10-1 ms-1

Vertical velocity scale: w 10-4 ms-1

Horizontal pressure scale: p 100 mb = 104 Pa

Time Scale: L/u 107s H/w 107s

Radius of Earth: a=6.37x 106m

Coriolis parameter: f,f' 10-4 s-1

Density of Water: 1000 kg m-3

Horizontal Eddy Viscosity: H 105 m2s-1

Vertical Eddy Viscosity: V 10-1 m2s-1

Horizontal Momentum Balance

10-8 ms-2 10-5 ms-2 10-5 ms-2 10-6 ms-2 10-6 ms-2

Geostrophic Balance

“Rossby Number”

Geostrophic Balance Holds when Ro << 1

(zonal) (meridional)

|u||/| 2

fLuRo

3

64

1

101010

10

Lfu||||

222 //ˆ/u/ huluLpfLuVH

/cosˆ1v pa

f

/ˆ1 pa

fu

Length scale: L106m, l105m

Depth scale: H103m, h 102m

Horizontal velocity scale: u,v 10-1 ms-1

Vertical velocity scale: w 10-4 ms-1

Horizontal pressure scale: p 100 mb = 104 Pa

Time Scale: L/u 107s H/w 107s

Radius of Earth: a=6.37x 106m

Coriolis parameter: f,f' 10-4 s-1

Density of Water: 1000 kg m-3

Horizontal Eddy Viscosity: H 105 m2s-1

Vertical Eddy Viscosity: V 10-1 m2s-1

Horizontal Momentum Balance

10-8 ms-2 10-5 ms-2 10-5 ms-2 10-6 ms-2 10-6 ms-2

Geostrophic Balance

Geostrophic Balance Holds when Ek << 1

(zonal) (meridional)

|u|

|/| 2

f

luEk H

H

1

104

5

101010

10

2|| lf

vH

222 //ˆ/u/ huluLpfLuVH

/cosˆ1v pa

f

/ˆ1 pa

fu

“Ekman Number”

Length scale: L106m, l105m

Depth scale: H103m, h 102m

Horizontal velocity scale: u,v 10-1 ms-1

Vertical velocity scale: w 10-4 ms-1

Horizontal pressure scale: p 100 mb = 104 Pa

Time Scale: L/u 107s H/w 107s

Radius of Earth: a=6.37x 106m

Coriolis parameter: f,f' 10-4 s-1

Density of Water: 1000 kg m-3

Horizontal Eddy Viscosity: H 105 m2s-1

Vertical Eddy Viscosity: V 10-1 m2s-1

Horizontal Momentum Balance

10-8 ms-2 10-5 ms-2 10-5 ms-2 10-6 ms-2 10-6 ms-2

Geostrophic Balance

Geostrophic Balance Holds when Ek << 1

(zonal) (meridional)

|u|

|/| 2

f

huEk V

V

1

44

1

101010

10

2|| hf

vV

222 //ˆ/u/ huluLpfLuVH

/cosˆ1v pa

f

/ˆ1 pa

fu

“Ekman Number”

Length scale: L106m, l105m

Depth scale: H103m, h 102m

Horizontal velocity scale: u,v 10-1 ms-1

Vertical velocity scale: w 10-4 ms-1

Horizontal pressure scale: p 100 mb = 104 Pa

Time Scale: L/u 107s H/w 107s

Radius of Earth: a=6.37x 106m

Coriolis parameter: f,f' 10-4 s-1

Density of Water: 1000 kg m-3

Horizontal Eddy Viscosity: H 105 m2s-1

Vertical Eddy Viscosity: V 10-1 m2s-1

Horizontal Momentum Balance

Geostrophic Balance

(zonal) (meridional)

/cosˆ1v pa

f

/ˆ1 pa

fu

Length scale: L106m, l105m

Depth scale: H103m, h 102m

Horizontal velocity scale: u,v 10-1 ms-1

Vertical velocity scale: w 10-4 ms-1

Horizontal pressure scale: p 100 mb = 104 Pa

Time Scale: L/u 107s H/w 107s

Radius of Earth: a=6.37x 106m

Coriolis parameter: f,f' 10-4 s-1

Density of Water: 1000 kg m-3

Horizontal Eddy Viscosity: H 105 m2s-1

Vertical Eddy Viscosity: V 10-1 m2s-1

110 1

HVEkEk110 3 Ro

Note that these approximations are only appropriate for “interior solutions” and will break down in boundary layers, where

horizontal or vertical shear are large!

Or near the equator!!

Horizontal Momentum Balance

Geostrophic Balance

(zonal) (meridional)

/cosˆ1v pa

f

/ˆ1 pa

fu

Horizontal Momentum Balance

Geostrophic Balance

(zonal) (meridional)

/cosˆ1v pa

f

/ˆ1 pa

fu

Dynamic Topography

Horizontal Momentum Balance

Geostrophic Balance

(zonal) (meridional)

/cosˆ1v pa

f

/ˆ1 pa

fu

Dynamic Topography

Horizontal Momentum Balance

Geostrophic Balance

(zonal) (meridional)

/cosˆ1v pa

f

/ˆ1 pa

fu

Dynamic Topography

the dynamic typography is not a simple consequence of

the overlying sea level pressure

requires an understanding of ocean

dynamics and its relation with atmospheric

windstress

22 //ˆ/u'/ HwLwHpgfLwuVHb

Vertical Momentum Balance:

Length scale: L106m, l105m

Depth scale: H103m, h 102m

Horizontal velocity scale: u,v 10-1 ms-1

Vertical velocity scale: w 10-4 ms-1

Horizontal pressure scale: p 100 mb = 104 Pa

Time Scale: L/u 107s H/w 107s

Radius of Earth: a=6.37x 106m

Coriolis parameter: f,f' 10-4 s-1

Density of Water: 1000 kg m-3

Horizontal Eddy Viscosity: H 105 m2s-1

Vertical Eddy Viscosity: V 10-1 m2s-1

10-11 ms-2 10-5 ms-2 10 ms-2 10 ms-2 10-11 ms-2 10-11 ms-2

)/()w(/ˆ1

ˆ'/ zw

zzpgufdtdw VH

zpg /ˆ1

ˆ0

fag

zu

cosv

fag

z

But now, depends on T,S,p

As with the atmosphere, we can combine geostrophic and hydrostatic balance to get

We can’t go proceed until we develop the equation of state for ocean water…

Thermal Wind Balance