rossby wave two-layer model with rigid lid η=0, p s ≠0 the pressures for the upper and lower...

Rossby Wave

gztyxpp s ρ−= ),,(1

Two-layer model with rigid lid η=0, ps≠0

The pressures for the upper and lower layers are

( )( )( ) ρρρρ

ρρρΔ−Δ+−=

+Δ+−+=ghzgp

zhgghpp

s

s2

spp ′=′1

The perturbations are

hgphgpp s ′−′=′⎟⎟⎠

⎞⎜⎜⎝

⎛Δ−′=′ ∗ρ

ρρρ 12

( )211

1pp

gHhh ′−′=−=′ ∗ρ

Potential vorticity conservation in upper and lower layers:

Linearize with respect to a rest state using

io

i pkf

v ′∇×=′rr

ρ1

021

2 =⎟⎟⎠

⎞⎜⎜⎝

⎛−+

+hHH

fdtd ζ

01 =⎟⎠

⎞⎜⎝

⎛ +hf

dtd ζ

io

i pf

′∇= 21ρ

ζ i=1,2

We have

021

1*

21

12 =⎟

⎠⎞

⎜⎝⎛

∂′∂

−∂

′∂−

∂′∂

+′∇∂∂

tp

tp

Hgf

xp

pt

oβ

021

1*

22

22 =⎟

⎠⎞

⎜⎝⎛

∂′∂

−∂

′∂+

∂′∂

+′∇∂∂

tp

tp

Hgf

xp

pt

oβ

Given plane wave solutions

( )tlykxiAep ω−+=′1( )tlykxiBep ω−+=′2

We have

( )A

Hg

fHg

fklk

Bo

o

1

21

222

∗

∗+++=

ω

ωβω

( )A

Hg

fklk

Hg

f

Bo

o

2

222

2

2

∗

∗

+++=

ωβω

ω

The dispersion relation is

( ) ( ) ( )

( ) ( )02 22

21

212

22

21

212

22222

=+⎥⎦

⎤⎢⎣

⎡ ++++

⎥⎦

⎤⎢⎣

⎡ ++++

∗

∗

ββω

ω

kHHg

HHflkk

HHg

HHflklk

o

o

The two solutions are

221 lk

k

+−=

βω A=B, barotropic or external mode

( )21

212

222

HHg

HHflk

k

o∗

+++

−=β

ω AH

HB

2

1−=

Out of phase between the upper and lower layers, baroclinic or internal mode

Let β=10-13 cm-1s-1, fo=10-4 s-1, H1+H2=4×105 cm, g*=0.002g=2 cm s-2

scmk

c /2511 −≈=

ωscm

kc /622 −≈=

ωThe phase speeds

Baroclinic InstabilityConsider a two-layer system with rigid lid and a mean slope interface H1. The vertical shear of the total velocity is

hkf

gvv

o

∇×=−rrr *

21

Assume H1=H1(x) only and the lower layer is at rest, we have

*1

g

Vf

dx

dHo= Where V is mean meridional current in the upper layer

The linearized potential vorticity equations are

01

1

1

11 =

′−⎟⎟⎠

⎞⎜⎜⎝

⎛ ′−′⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

dxdH

Huf

Hhf

yV

tooζ

01

1

2

12 =

−′

+⎟⎟⎠

⎞⎜⎜⎝

⎛−

′+′⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂+

∂∂

dxdH

HDuf

HDhf

yV

tooζ

For simplicity, H1 outside derivatives can be replaced by its mean Hm

Using the symbols we have used before, we have

( ) 0**

12

2112 =

′+⎟⎟⎠

⎞⎜⎜⎝

⎛′−′−′∇⎟⎟

⎠

⎞⎜⎜⎝

⎛∂

∂+

∂

∂

dy

pd

Hg

Vfpp

Hg

fp

yV

t m

o

m

o

( ) 0)(*)(*

22

2122 =

′

−−⎟⎟⎠

⎞⎜⎜⎝

⎛′−′

−+′∇⎟⎟

⎠

⎞⎜⎜⎝

⎛∂

∂+

∂

∂

dy

pd

HDg

Vfpp

HDg

fp

yV

t m

o

m

o

( )cVtyikeL

xnAp −⎟

⎠⎞

⎜⎝⎛=′π

sin11

Consider wave solutions as

( )cVtyikeL

xnAp −⎟

⎠⎞

⎜⎝⎛=′π

sin22

( ) ( ) 011

1 121

1221

12 =+⎥

⎦

⎤⎢⎣

⎡−+−− AAAAlc

λλ

( ) 011

221

1222

22 =−⎥

⎦

⎤⎢⎣

⎡−+−− AAAAlc

λλ

22

22

kL

nl +=

π

221

*

o

m

f

Hg=λ 2

22

)(*

o

m

f

HDg −=λ

where

The dispersion relation is

( ) ( ) 11

11

11 2

22

1 =⎥⎦⎤

⎢⎣⎡ −+⎥⎦⎤

⎢⎣⎡

−−+

cl

cl λλ

It can be shown that, if 4

22

21

4l≥

λλ, the equation has complex solution

( )22222

2

2

2

1

4 LknLL

+≥⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛π

λλThe instability relation can also be written as

2

2

21

2 πλλ

≥L

Take n=1 and k=0, we get minimum criteria as

Take Hm~ 800 m, we have λ1~ 50 km, λ2 ~ 100 km, the unstable eddies are in the scale of 100-200 km.

Meso-scale

eddies

Energy Diagram of Rossby Wave

Reflection of Rossby Wave

Equatorial Dynamics

1 and 1/2 layer model

Y North

X East

Equatorial Under Current

Equatorial Under Current

Core close to the equator, ~1m/s

Below mixed layer

Thickness ~ 100 m

Half-width 1-2 degrees (Rossby radius at the equator)

Forced by zonal pressure gradient established by equatorial easterlies

Equatorial Waves

Tropical and subtropical connections

Vertical structure of the ocean:Large meridional density gradient in the upper ocean, implying significant vertical shear of the currents with strong upper ocean circulation

Water mass formation by subduction occurs mainly in the subtropics.

Water from the bottom of the mixed layer is pumped downward through a convergence in the Ekman transport

Water “sinks” slowly along surfaces of constant density.

Subduction

Sketch of water mass formation by subduction

First diagram: Convergence in the Ekman layer (surface mixed layer) forces water downward, where it moves along surfaces of constant density. The 27.04 σt surface, given by the TS-combination 8°C and 34.7 salinity, is identified. Second diagram: A TS- diagram along the surface through stations A ->D is identical to a TS-diagram taken vertically along depths A´ - D´.

The Ventilated ThermoclineLuyten, Pedlosky and Stommel, 1983

Interaction between the Subtropical and Equatorial Ocean Circulation: The Subtropical Cell

The permanent thermocline and Central Water

• The depth range from below the seasonal thermocline to about 1000 m is known as the permanent or oceanic thermocline.

• It is the transition zone from the warm waters of the surface layer to the cold waters of great oceanic depth

• The temperature at the upper limit of the permanent thermocline depends on latitude, reaching from well above 20°C in the tropics to just above 15°C in temperate regions; at the lower limit temperatures are rather uniform around 4 - 6°C depending on the particular ocean.

• The water of the permanent thermocline is named as the Central Water, which is formed by subduction in the subtropics.