rossby wave two-layer model with rigid lid η=0, p s ≠0 the pressures for the upper and lower...
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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
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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
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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
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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
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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
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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
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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.
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Meso-scale
eddies
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Energy Diagram of Rossby Wave
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Reflection of Rossby Wave
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Equatorial Dynamics
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1 and 1/2 layer model
Y North
X East
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Equatorial Under Current
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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
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Equatorial Waves
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Tropical and subtropical connections
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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
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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
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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´.
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The Ventilated ThermoclineLuyten, Pedlosky and Stommel, 1983
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Interaction between the Subtropical and Equatorial Ocean Circulation: The Subtropical Cell
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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.