El Niño/Southern Oscillation
(ENSO)
Main Issues (contributed by E.
Tziperman):
• What is the mechanism of the El Nino cycle?
• Why is the mean period quite robustly 4 years?
• Is ENSO self-sustained or is it damped and
requires external forcing by weather noise for
example in order to be excited?
• Why are ENSO events irregular: is it due to
chaos? noise?
• Why do ENSO events tend to peak toward the
end of the calendar year (phase locking to the
seasonal cycle)?
ENSO Theory
Lower layer at
rest
pH = constant = patmos + ρ1 g ( h + zs ) + ρ 2 g ( H − h )
→ h ( ρ1 − ρ 2 ) + zs ρ1 = constant
Hydrostatic: At fixed depth within layer 1:
p = patmos + ρ1 g ( z + zs )
ρ 2 − ρ1
→ ∇p = g ∇z s = g
∇h ≡ g * ∇h
ρ1
ρ1
1
τs
dV
ˆ
+ β yk × V + g * ∇h =
− ε mV
dt
ρ1h
Mass continuity:
ww ·
§
³ h ¨© ’<V wz ¸¹dz 0
dh
o
h’<V # 0
dt
zs
Linear shallow water system:
wu
wh
E yv g *
wt
wx
wv
wh
E yu g *
wt
wy
§ wu wv ·
wh
H¨ ¸
wt
© wx wy ¹
Wx
H mu,
UH
Wy
H m v,
UH
H h h
“relaxation term”
Steady, undamped flow at equator:
∂h τ x
g* =
∂x ρ H
Easterly wind gives decreasing thermocline depth toward
east. Shallower thermocline is generally associated with
colder surface temperatures
For simplicity, take
εm = εh ≡ ε
Curl of momentum equations:
Small variability and damping:
βv
τ
s
ˆ
k i∇ ×
ρH
∂u
∂v
−
∂x
∂y
Sverdrup balance
Steady, undamped flow:
∂h τ x
g* =
+ β yv
∂x ρ H
τx
1 ∂τ x
=
−y
ρH
ρ H ∂y
Suppose
τx
= τ 0e
ρH
−y
2
2 L2
∂h
⎛ y
⎞
→ g * = τ 0 ⎜1 +
2 ⎟e
L ⎠
∂x
⎝
2
2
y
−
2 L2
ENSO Theories
• Delayed Oscillator (See E. Tziperman
notes)
• Ocean equatorial wave guide (stable or
unstable) stochastically forced by
atmopshere
• Unstable coupled modes
Rudiments of a local ENSO theory
Consider only (undamped) Kelvin
mode in ocean:
∂uo
∂h τ x
+ g* =
,
∂t
∂x ρ H
∂uo
∂h
+H
=0
∂t
∂x
Steady, undamped atmosphere with WISHE, forced
by ocean temperature anomalies:
∂s *
(Ts − T ) ∂x = − β yv,
∂s *
(Ts − T ) ∂y = β yu,
∂u ∂v
+ + α u = −γ s0 .
∂x ∂y
ε p CkU s0 − sb
εp
Ck | V |
α≡
, γ ≡
1 − ε p h | V | sb − sm
1 − ε p H sb − sm
(
)
Eliminate s* and v in favor of u:
∂s0
∂u
∂u
+ 2α u + α y
= −2γ s0 − γ y
∂x
∂y
∂y
Note that on equator:
∂u
+ 2α u = −2γ s0
∂x
Surface stress and ocean temperature:
τ x ≅ ρ a CD | V | u ,
s0 ≅ µ h
Ocean:
∂uo
∂h ρ a CD | V |
+ g* =
u,
∂t
∂x
ρH
∂uo
∂h
+H
=0
∂t
∂x
Atmosphere:
∂u
+ 2α u = −2γµ h
∂x
Eliminate variables in favor of h:
2
2
⎛
ρ a CD | V | ∂h
∂
∂
∂
h
h⎞
⎛
⎞
=0
⎜ + 2α ⎟ ⎜ 2 − g * H 2 ⎟ − 2γ H µ
∂x ⎠
ρ H ∂x
⎝ ∂x
⎠ ⎝ ∂t
Scalings:
c ≡ g*H
α ' ≡ aα
ρ a CD | V |
χ ≡ 2γ H µ
ρ Ha 2 c 2
x → ax
t→a t
c
∂h
⎛ ∂
⎞⎛ ∂ h ∂ h ⎞
=0
⎜ + α '⎟ ⎜ 2 − 2 ⎟ − χ
∂x ⎠
∂x
⎝ ∂x
⎠ ⎝ ∂t
2
h = h0 e
2
ikx −iωt
χ
ω =k −
1− iα '
2
2
k
No WISHE (α’=0):
ω =k −χ
2
2
Modes are either neutral and propagating (but
slowed down by interaction with atmosphere), or
stationary and amplifying/decaying.
With WISHE effect (unstable, eastward-propagating modes
only when α’ < 0):
These are only approximate solutions:
• Coupling produces mismatch of y
structure...no pure Kelvin modes in ocean
• Advection of temperature by ocean
currents ignored
• Mean slope of thermocline ignored
• No damping terms
• No influence of surface flux variations on
ocean temperature
MIT OpenCourseWare
http://ocw.mit.edu
12.811 Tropical Meteorology
Spring 2011
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.