barotropic wave

7/30/2019 Barotropic wave

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Ch 9: Barotropic wave


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Outline

Scale analysis

Linearization

(shallow water)

Fundamental Eq

(Linearized)

Local scaleKelvin wave

Poincare wave

Planetary scale Rossby wave

Topography wave

9.1

9.2

9.3

9.4

9.5

What are we

focusing on?


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What are we focusing on?

Symbol Unit Meaning

T time time scale we are interested in

1/ time Earths rotation time scale

L length length scale we are interested in

U length/time the L scale systems average moving speed

C length/time =L/T, which means the wave speed we are tracking

H,H length water average height and its deviation

Ro U

L1, RoT

1

T

1

T~ 1

C

L

T~ L U

Important

relation


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Outline

Scale analysis

Linearization

(shallow water)

Fundamental Eq

(Linearized)


Poincare wave

Planetary scaleRossby wave

Topography wave

9.1

9.2

9.3

9.4

9.5

What are we

focusing on?


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From 7.3 in vector form

d

dtHv

Vv

V 0

d

V

dt f

k

v

V gMomentum

Continuity

3 unknowns(u, v, ), 3 equations(momentum x 2 + continuity)


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From 7.3 for each axis

t

v

V Hv

Vv

V 0

u

t u

u

x v

u

y w

u

z fv

1

p

x g

x

v

t uv

x v v

y w v

z fu 1

p

y g

y

Momentum

Continuity



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LT

( U ) (L) (gHU)

u

t u

u

x v

u

y w

u

z fv g

x

Scale Analysis 1

U

T ( UU

L ) (U) (gH

L )

multiplyL

U

CL

T~ L U


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Scale Analysis 1

u

t u

u

x v

u

y w

u

z fv g

x

u

t fv g

x

vt

fu gy

V

t f k

v

V g


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Scale Analysis 2

t

v

V Hv

V v

V 0

( HT

) (UHL

) (HUL

) (HUL

)

multiplyL

H

( L

T) ( U ) (

H

HU) ( U )

CL

T~ L U


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Scale Analysis 2

t

v

V Hv

V v

V 0

tH

v

V 0


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Reformulate, and we get

t

Hv

V 0

u

t fv g

x

vt

fu gy

Momentum

Continuity



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t

Hv

V 0

Momentum

Continuity


V

t f k

v

V g


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Outline

Scale analysis

Linearization

(shallow water)

Fundamental Eq

(Linearized)


Poincare wave


Topography wave

9.1

9.2

9.3

9.4

9.5

What are we

focusing on?


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We first using fourier analysis

u

v

u

v

e

ikxx ikyy it

kx ky

gikx i f

giky f ii iHkx iHky

uv

kx ky eikxx ikyy it

0


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2 f2 gH(kx2 ky

2) 0

0 or f2

gHk2

define R2 gH

f2

c

f

2

f

2

1 (kR)2


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Solution I : =0

1. Timeless (it doesnt change with time!)

2. It means theres no constraint on

wavenumber (any shape!)

Geostrophic solution

f

k

V g

v

Vg

f

k


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f

2

1 (kR)2

k gH when f 0

When f0, it degeneratesto a line (non-dispersive) Poincare

waves(dispersive)

Solution II :


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R Rossby radius of deformation

R gH

fc

f

R

L

1f

Lc

TEarth

Twave

R Less rotation effect

R More rotation effect

R means in what scale of length will Earths rotation comes into play


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Question

Is there any way to create a situation in which

the effect of rotation can be eliminated?


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Outline

Scale analysis

Linearization

(shallow water)

Fundamental Eq

(Linearized)


Poincare wave


Topographiy wave

9.1

9.2

9.3

9.4

9.5

What are we

focusing on?


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Answer: wall

Coriolis force

Coriolis force Force from wall

Rotation takes effect

Rotation canceled

(Kelvin wave)

The Kelvin wave is a traveling disturbance that

requires the support of a lateral boundary.


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2v

t2

gH2v

y2

c 22v

y2

where c gH

u 0 everywhere, using

u

t fv g

x

vt

fu gy

t H

v

V 0

Condition (in NH)


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Lets do some math

2v

t2 c2

2v

y2

v V1(x,y ct) V2 (x,y ct)

H

gV1(x,y ct)

H

gV2(x,y ct)

usingu

t fv g

x

V1

x

f

gHV1 ,

V2

x

f

gHV2


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V1 V1(y ct) exf / gH

,V2 V2(y ct) exf / gH

V2 is not a physically accepted solution (V exploded as x )

defineR gH

f

c

f

the solution becomes

u 0

v cF(y ct) ex /R

HF(y ct) ex /R


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Equation tells us

v cF(y ct) ex /R

HF(y ct) ex /R

1. If R is large no rotation effect pure gravity wave with c2 = (gH)2

2. If R is smallwave doesnt exists geostrophic condition

Again, R means in what scale of length will Earths rotation comes into play or

more precisely, reciprocal length


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Visualize

v gf

x


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Conclusion

The Kelvin wave is non-dispersive, the phasespeed is equal to the group speed of the waveenergy for all frequencies.

Thus its said to be trapped. In the longshoredirection, the wave travels without distortionat the speed of surface gravity waves.

Although the direction of propagation is unique,the sign ofvis arbitrary.


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f

2

1 (kR)2 if we let f and k be const ,

kR kc /f kR c Less rotation effec

More like

pure gravity

wave

More like

pure gravity

wave

More like

pure gravity

wave

More like

pure gravity

wave

Affected by

rotation

Affected by

rotation


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But wait, theres something wrong

A BIG assumption of previous solution:

but in fact f 2sin non constant

f constant!!


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Please remember this

In the previous ppt, we make f const and get

two solutions, correspond to

1.

V

t 0 (Geostrophic)

2.v

V

t 0 (Kelvin and Poincarewaves)

In the next section, we will using Quasi-Geostrophic condition:

V

t

fkv

VU

fL1 (Low Rossby number)

v

V

t

fkv

V (Quasi Geostrophic)


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Outline

Scale analysis

Linearization

(shallow water)

Fundamental Eq

(Linearized)

Local scale Kelvin wave

Poincare wave

Planetary scale

Rossby wave

Topography wave

9.1

9.2

9.3

9.4

9.5

What are we

focusing on?


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From f-plane to -plane

For simplicity, we use a Taylor 1st order

expansion to represent non-const f.

f f0 df

dyy0

y f0 0y

planetary number0L

f01


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t

Hv

V 0

Momentum

Continuity


V

t

(f0 0y)k

v

V g


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Using 1st order appoximation

Vt

(f0 0y)k vV g

v

Vgk

f0 (and iterate again)

gk

f0

t f0

kv

V0g

f0y g

v

Vgk

f0

g

f02

t

0g

f02 y

k


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v

Vgk

f0

g

f02

t

0g

f02 y

k

tH

v

V 0

t H

gk

f0

g

f02

t

0g

f02 y

k

t

R2

t

2 0R2

x

0

Evaluate into continuity equation:



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Using fourier analysis

eikxxikyyit

kx ky

i R2(i)(kx2 ky

2) 0R

2(ikx ) 0

0R

2kx

1R2 kx2 ky

2


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Result I : Verify

0R

2kx

1R2 kx2 ky

2 , k~ 1/L

Shortwave :L R, ~ 0L

Long wave:L R, ~0R

2

L 0R 0L0L


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Result II : cx, cy

0R

2kx

1R2

kx2

ky2

cx

kx

0R2

1R2 kx2 ky

2 westward only

cy

ky

0R2

ky 1R2 kx

2 ky2

no constraint


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Result III : cg

0R

2kx

1R2 kx2 ky

2

, k~ 1/L

kx 0

2

2

ky2

02

42

1

R2

max

0

R

2

figure 9-4,9-5


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Conclusion: Rossby wave

1. Quasi-Geostrophic

2. Exists in beta-plane

3. Has frequency maximum limit

4. Single freq wave can only move westward.

5. Energy can be transport in ANY direction!!


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Video ref

http://www.youtube.com/watch?v=iuZ2Zc5x1ZUNotice that wave propagate backward while it moves forward
http://www.youtube.com/watch?v=iuZ2Zc5x1ZUhttp://www.youtube.com/watch?v=iuZ2Zc5x1ZUhttp://www.youtube.com/watch?v=iuZ2Zc5x1ZUhttp://www.youtube.com/watch?v=iuZ2Zc5x1ZU


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Question

Weve already expand f to analysis the 1st order

(-plane) phenomenon Rossby wave.

Then, what if we expand the depth of water into

1st order? (dH/dy=-plane)


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Outline

Scale analysis

Linearization

(shallow water)

Fundamental Eq

(Linearized)

Local scale Kelvin wave

Poincare wave

Planetary scale

Rossby wave

Topography wave

9.1

9.2

9.3

9.4

9.5

What are we

focusing on?


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Topography waves

Figure 9-6

HH0 0y

0L

H01

h H0 0y also, QG condition


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Rewrite continuity equation

h

t

hu

x

hv

y 0, h H0 0y

t

u0y

x

H0 0y u

x

v0y

y

H0 0y v

y

0

t

v

V H0 0y v

V 0v 0H

T

H

T

tH0

v

V 0v 0


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tH0

v

V 0v 0

Momentum

Continuity


V

t

f kv

V g


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V

t f k

v

V g

v

V

gk

f (and iterate again)

gk

f

t f k

v

V g

v

V g

f2

t

gk

f


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v

V g

f2

t

gk

f

tH0

v

V 0v 0

t H gf2 tg

kf

0v 0

t

R2

t

20g

f

x

0

Evaluate into continuity equation:



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Wow, thats fimiliar

(Topography)

t

R2

t

20g

f

x

0

(Rossby)

tR2

t20R

2

x 0

(Topography)0g

f 0R

2(Rossby)


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Result : Compare with Rossby

0R

2kx

1R2 kx2 ky

2 cx

kx

0R2

1R2 kx2 ky

2

cy

ky

0R2

ky 1R2 kx

2 ky2

m ax 0R

2

0g

f

kx

1R2 kx2 ky

2 cx

kx

0g f

1R2 kx2 ky

2

cy

ky 0g f

ky 1R2 kx

2 ky2

m ax 0g

2fR

Topography waves Rossby waves


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Planetary waves and Topography waves

q f

h

f0 0y v x u y H0 0y

1

H0

f0 0y 0f0

H0

y v x u y f0H0


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Figure 9-7

V vU


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Review

Scale analysisLinearization

(shallow water)

Fundamental Eq

(Linearized)

Local scale

(0th order vorticity)

Kelvin wave

Poincare wave

Planetary scale

(1st order vorticity,

vorticity conservation)

Rossby wave

Topography wave

9.1

9.2

9.3

9.4

9.5

What are wefocusing on?

(nondispersive)

(dispersive)

(-plane)

(-plane)

V

t f k

v

V g

tH

v

V 0

Ro U

L 1,

CL

TU

q f

h

1

H0f0 0y

0f0

H0y ...

~

kx

1R2 kx2 ky

2

R gH

f c

f

R

L

1f

Lc

TEarthTwave

f

2

1 (kR)2


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Reference

Introduction to Geophysical Fluid Dynamics,

Benoit Cushman-Roisin and Jean-Marie

Beckers(Ch.9)

Geophysical Fluid Dynamics Laboratory

http://www.ocean.washington.edu/research/

gfd/
http://www.ocean.washington.edu/research/gfd/http://www.ocean.washington.edu/research/gfd/http://www.ocean.washington.edu/research/gfd/http://www.ocean.washington.edu/research/gfd/http://www.ocean.washington.edu/research/gfd/http://www.ocean.washington.edu/research/gfd/


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Thanks for your attention!

barotropic wave

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