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Convective mixing due to equatorial
inertial-parametric instability
Lien Hua Laboratoire de Physique des
Océans Ifremer Brest France
Coll M. dOrgeville, R. Schopp, C. Menesguen, S.
LeGentil
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1 Observations of deep extended layering at the
equator 2 Equatorial Inertial instability 3
Parametric subharmonic inertial instability 4
Evidence in basin simulation
3
Deep extended density layering as possible
imprints of equatorial inertial instability
dOrgeville, Hua, Schopp, Bunge, GRL, 2004
4
Why concentrate on the equatorial region for the
study of convective mixing in the ocean?

• 1/2 of the water moved by the thermohaline
  circulation is driven into the equatorial track
  where substantial mixing of its properties
  occurs.
• Significant reservoir of kinetic energy with its
  alternating zonal jets of about 20cm/s
  Energy that can potentially be used for mixing

• What are the characteristics of the equatorial
  fields?
• Where can we expect mixing?

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Atlantic, 10W
(Brunt-Vaisala)²
800
layering
DEPTH (m)
DEPTH (m)
salt
O2
?
1100
dOrgevilleal2004 Menesguenal2007
100m homogenized density
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(Brunt-Vaisala)² anomaly in equatorial Atlantic
at 10W Evidence of extended density layering
within 1-2 of the equator over the whole water
column 2 meridional extent 8 longitudinal
extent
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• Possible layering mechanisms
• double diffusion slow growth ( Edwards and
  Richards 2004)
• internal wave strain (Dengler and Quadfasel
  2002)
• meridional scales are small
• inertial instability fQ lt0 ( more easily
  verified inside westward jets)

Unlikely for extended layering
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Oceanic Equatorial Deep jets East-West flows
alternating with depth
Zonal velocity
depth
10 Sv transport!
latitude
Atlantic (instantaneous)
Gourioual2001
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Two vertical scales ( 700m and 100m) over the
whole depth
U
Atlantic, 10W
800
800
layering
DEPTH (m)
DEPTH (m)
600-700m
DEPTH (m)
?
1100
(dOrgevilleal2004)
100m homogenized density
2200
-1.5 0 1.5 latitude
-20cm/s
20cm/s
Interaction of the two scales layering is
embedded inside westward jets of the 600-700m
scale jets
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What is special about westward jets?
fßy PVf-uy
0
PV
f
U
S,T,spice
-1.5 0
1.5 latitude
Atlantic 10W
Homogenization of all tracers in weak ?yPV (ie
westward jets)
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Spatial correspondence of the two scales
600
For all Atlantic equatorial transects 90
extended layering occurs in blue regions spatial
coincidence of well-mixed PV and density layering
0
U
DEPTH (m)
dOrgeville et al. 2004
1600
-1 0 1
Atlantic 10W
latitude

• Density and tracers homogenized on small
  vertical scale, symmetrically
• about the equator, matching regions of weak
  ?yPV (i.e. westward jets).

• What mechanism could mix density and tracer in
  such a way?

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• What mechanism could mix density and tracer in
  such a way?

• Homogenization of PV to zero
• small vertical scales

Equatorial Inertial instability
Cellular structures instability (like
Rayleigh-Bénard or Taylor-Couette)
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Equatorial -plane
latitudinal gradient of Coriolis parameter
latitudinal distance from the equator
Angular momentum
earth radius
relative velocity
Horizontal Coriolis parameter
Height coordinate
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zonally symmetric
Meridional streamfunction of overturning motions
zonal velocity
zonal vorticity
density
Dissipative terms
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Diffusion/dissipation
Generalized Inertia operator
pure real growth rate
self-adjoint ?
Rayleigh-Bénard / Taylor-Couette like
? Cellular instability
Instability criterion
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Ertel Potential Vorticity
(Hoskins 1974)
Negative potential vorticity acts like unstable
density profile (cf convection)
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equatorial b -plane
Basic state angular momentum
Mean density (thermal wind balance)
Instability ? relative angle between iso-angular
momentum and iso-density surfaces
Maximum angular momentum displaced from the
equator triggers inertial instability
Instability favors smallest vertical scales (
diffusive scale selection, Dunkerton 1983)
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Steady shear
(strong supercriticality)
Initial conditions
angular momentum M
merid. streamf y
density
Depth (m)
latitude ()
latitude ()
latitude ()
Huaal 1997, Griffiths 2003
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Steady shear
Intermediate time
merid. streamf y
angular momentum M
density
Depth (m)
latitude ()
latitude ()
latitude ()
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Steady shear

• homogenization of M
• latitudinally-extended density layering

in a given hemisphere
Final state
angular momentum M
(Brunt-Vaisala)² anomaly
Depth (m)
latitude ()
latitude ()
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• homogenization of M
• latitudinally-extended density layering

in a given hemisphere
Staircase profile of density due to the growth
of cellular structures
Minitial
Madjusted
DEPTH
y
?
0
dOrgevilleHua2005
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latitudinal homogenization of PV to zeroon one
side of the equator
Steady shear
Minitial
Madjusted
PVadjusted
y
0
y
0
PVinitial
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Equatorial inertial instability of oscillating
shear flows
DOrgeville Hua, JFM 2005
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Inertial instability of oscillating shear flows
Motivation no clear observational evidence of
steady shear in deep equatorial track, instead
ubiquituous variability due to free waves
(especially Mixed Rossby Gravity waves 15-80
days)
Two zonally-symmetric problems with oscillating
shear forcing - stratified fluid with
barotropic oscillating shear - both time and
vertical dependence of shear Mixed Rossby
Gravity wave basic state
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Forcing by a geostrophic barotropic flow
Perturbations
inertial instability forced by steady shear
inertial-parametric instability
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Inertial-parametric instability subharmonic
resonance of free modes
Free modes are inertia-gravity waves with
vertical mode m meridional mode n
subharmonic resonance
Asymptotic results for
vertical scale selection
oscillating at 20 days
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n1
n0
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Steady versus oscillating shear
inviscid

• viscous
• n vertical diffusion

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Summary for
resonance of a high vertical mode
The higher the frequency of the forcing, the
lower the critical shear
Symmetric adjustment about the equator
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Zonally-symmetric Mixed Rossby Gravity wave
Simplest wave with oscillating shear in
(eg 15-20 days variability in all equatorial
oceans)
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Mixed Rossby Gravity wave
initial value simulation
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• Temporal evolution of initial value simulation of
  MRG wave
• adjustment will involve both small and large
  meridional scales motions we focus on phase of
  the adjustment response of maximal amplitude,
  involving low meridional modes that are
  influenced by Earth rotation.
• evidence of sustained growth of high vertical
  modes when compared to basic state MRG
  

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Clear growth of vertical modes 7 times higher
than basic state
Frequency spectrum and
Basic state MRGW filtered out
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Full initial-value problem
Truncated linearized system
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Summary for Zonally-symmetric Mixed Rossby
Gravity wave
Inertial parametric subharmonic instability
? clear growth of vertical modes 7 times higher
than basic state
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Basin simulation
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Formation of Oceanic Equatorial Deep jets
Multiple jets alternating with depth
Numerical Simulations in a basin
geometry Primitive Equations, constant
stratification, equatorial b- plane
Forcing Oscillating source inside Western
Boundary Layer
North
Equator
East
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Alternating zonal jets are generated by
destabilization of equatorial waves forced by an
oscillation in the western boundary.
(dOrgevilleal, Huaal 2007)
Zonal velocity at the equator for different
forcing periods
40 days
57 days
Depth
longitude
longitude
EARTH SIMULATOR (3D high-resolution 1/12-20
200-400 levels)
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Modelling of the equatorial velocity field
Atlantic, 10W
Numerical model
Primitive Equations model, in a 50 long basin
centered on the equator. 1/11, 400 levels.
600m
600m
DEPTH (m)
-1.5 0 1.5 latitude
-1.5 0 1.5 latitude
-20cm/s
-12cm/s
12cm/s
20cm/s

• Does the small vertical scale layering occur?
  Where?

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Layering in westward jets
Atlantic, 10W
Model
Weak ?yPV
600
DEPTH (m)
Strong ?yPV
1600
-1 0 1
-1 0 1

• With a velocity field able to excite
  parametric inertial instability,
• we can reproduce the layering observed in
  data.

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Summary Two distinct vertical scales -
Equatorial (stacked) deep jets (600-700m
Atlantic) - extended density layering at 100 m
vertical scale embedded in westward
jets Strong variability due to free waves and
oscillating shears in deep equatorial ocean can
trigger inertial-parametric instability -
latitudinal redistribution of angular momentum
? lateral mixing - homogenized density
layers 100 m vertical scale ? intermittent
vertical mixing
over the whole ocean depth
potential energy resupply of P0.1TW 10 of
the total needed to maintain the stratification
of the global ocean (WunshFerrari 2004)
(deposited in 1/50 of the total area of the
ocean). Privileged site for upwelling branch
of MOC?
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