After a brief review of oceanic mid -latitude zonal jets observations and of their currently proposed formation mechanisms, the characteristics of oceanic equatorial zonal jets are presented. A new mechanism for zonal jets formation is proposed based on the destabilization of short Mixed Rossby Gravity waves, providing a rationale for the difference of scales observed in the Pacific and in the Atlantic deep equatorial jets. Results of numerical simulations performed on the Earth Simulator are presented to validate the analytical theory

HUA, Bach Lien

I-Revues

18 ème  Congrès  Français  de  Mécanique Grenoble,  27- 31  août  
2007
Equatorial  zonal  Jets  formation
Bach Lien Hua
Laboratoire  de  Physique  des  Océans,  
Ifremer,  BP 70  29280,  
Plouzané  France
email: lien@ifremer.fr  
Abtract   :
After  a brief  review  of  oceanic  mid  - latitude  zonal  jets  observations   and  of  their  
currently  proposed  formation  mechanisms,  the  characteristics  of  oceanic  
equatorial  zonal  jets  are  presented.   A  new  mechanism  for  zonal  jets  formation  is  
proposed   based  on  the  destabilization  of  short  Mixed  Rossby  Gravity  waves,  
providing  a rationale   for  the  difference  of  scales  observed  in  the  Pacific and  in  
the  Atlantic  deep  equatorial   jets.   Results  of  numerical  simulations  performed  on  
the  Earth  Simulator  are  presented   to validate  the  analytical  theory.
Key- words  : 
Zonal  jets,  Floquet,  Mixed  Rossby  Gravity  wave
1 Introduction
Flows  exhibiting  multiple  zonal  jets   are  ubiquitous  in   observations   of  the  
atmospheres  of  giant  planets  and  in  the  oceans,  as  well  as  in  laboratory  
experiments.  Such  systems  share  in  common  east - west  velocity  directions   that  
alternate   either   with  latitude  or  with  depth  and  present  a  marked  asymmetry,  
with  the  eastward  jets  being  narrower  and  faster  than  the  westward  jets.
Besides  involving  a  considerable  portion  of  the  total  kinetic  energy,  the  
existence  of  multiple  zonal  jets  has  profound  implications  for  the  transport  and  
mixing  of  tracers  by  setting  up  barriers  to  meridional  transpor t.  Such  barriers,  
which  can  be  intermittent  in  space  and  time,  partition  the  flow  into  well- mixed  
regions  of  distinct  properties.
Zonal  jets  (ZJ)  can  be  viewed  as  coherent  structures  that  spontaneously  
arise  when,  in  a  differentially  rotating  medium,   the  well- known  inverse  cascade  
of  two- dimensional  turbulence  is  halted  by  wave  dispersion  effects  or  by  
friction:  jets  represent  a spatial  organization  of  a turbulent  flow.  One  possible  ZJ 
formation  mechanism  occurs  through  the  destabilization  by  batropic  shear  
instability  of  short  meridional  velocity  fluctuations  on  a  -plane:  the  kineticβ  
energy  of  the  short  waves  is  transferred  through  a  negative  viscosity  effect  to  
zonal  jets  of  very  long  zonal  scale,  typical  of  the  inverse  energy  cascade  
[Sivashinsky  (1985)  , Gill (1974),   Legras  et  al. (1999),  Manfroi  & Young  (1999)] .
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18 ème  Congrès  Français  de  Mécanique Grenoble,  27- 31  août  
2007
While  oceanic  observations  (Maximenko  et  al.  (2006))  as  well  as   most  recent  
high- resolution  ocean  circulation  models  (Richards  et  al.  (2006))  confirm  that  
mid- latitude  zonal  jets  tend  to  be  barotropic,  i.e.  correspond  to  the  largest  
vertical   scale  of  motions,  equatorial  observations  show  that,   in  contrast,  
equatorial  zonal  jets  present  very  small  vertical  scales,  typically   a  vertical  
baroclinic  mode  15- 20  for   the  Atlantic  and  30- 40  for   the  Pacific  (Gouriou  et  al.  
(2001),  Firing  (1988)).   The  formation  mechanism  of  these  depth - alternating  
zonal  jets  - -  the  so- called  Equatorial  Deep  Jets  - -  have  puzzled  oceanographers  
for  decades.
2 Short Rossby and Mixed Rossby-Gravity waves destabilization
For  a  rotating  planet,  the  rotation  rate  about  the  local  vertical  axis   varies  
with  latitude:  this  is  the    β- effect,  where    is  the  northward  gradient   of  theβ  
Coriolis  parameter  due  to  the  planet’s  rotation  . This  effect  triggers   the  most  
important  class  of    wave   motions    for  large- scale  meteorological  and  oceanic  
processes,   namely  the   Rossby   waves.  The   β- effect  sets  the  elasticity  of  the  
wave   for  north - south   displacements  of  a   chain  of  fluid  parcels  about  an  
equilibrium   latitude  (meridional  motions  are  inherently  necessary  for  the  
Rossby  shear   wave  to  propagate  westward).
Lorenz  (1972)  and  Gill (1974)  have  studied  the  stability  of  barotropic  Rossby  
waves  and  shown  that  they  are   prone  to  destabilization  by  shear  instability.  For  
a  mid–latitude  –plane,  Gill  (1974)  demonstrated  that  a  short  Rossby  wave  canβ  
be  unstable  and  generate  low–frequency  zonal  flows,  the  mechanism  being  
dependent  on  the  stability  of  the  meridional  component  of  velocity  (cf.  Manfroi  
and  Young  (1999),   Lee  and  Smith  (2002)   for  an  in- depth  discussion).  Gill’s  
study  was  based  on  a  Floquet  analysis  of   a  small- scale  sinusoidal  flow  field  
dominated  by  meridional   velocities  that  are  characteristic  of   zonally- short  
Rossby  waves.  
At  low  latitudes,  for  an  equatorial  -plane,  the  free  wave  whoseβ  
characterisitics  are  the  closest  to  those  of  the  mid- latitude  barotropic  Rossby  
wave  is  the  so- called  Mixed  Rossby  Gravity  (MRG) wave  .  A  short  westward–
propagating  mixed  Rossby–gravity  wave,  like  a  mid–latitude  Rossby  wave  with  a  
similar  zonal  wavenumber,  has  predominantly  meridional  motions.  Moreover,  its  
group  velocity  is  eastward  like  that  of   a  short  mid–latitude  Rossby  wave.
Following  Gill  (1974),  a  Floquet  analysis  of  the  flow  field  of  a  zonally- short  
MRG wave  has  been  performed.  The  analysis  establishes  the  characterisitics  of  
the  production  of  zonally  coherent  flows  with  low zonal  wavenumber.
The  zonal  jet–like  structures  are  formed  through  a  barotropic  instability  
associated  with  the  zonal  derivative  of  the  meridional  velocity  field.  This  process  
enables  an  inverse  cascade  to  zonally  coherent  low  and  high  vertical  modes,  
which  is  conjectured  to  correspond  to  the  extra–tropical  low  vertical  mode  jets  
and  the  equatorial  deep- jets,  respectively.  
The  vertical  scale  is  selected  as  follows:  barotropically  unstable  meridional  
flow  favors  a  certain  meridional  scale,  which  in  turn  determines  the  vertical  
scale  through  the  equatorial  radius  of  deformation  relationship.  This  implies  
that  the  dominant  vertical  mode  number  scales  like  the  square  of  the  
dimensional  period  of  the  short  MRG wave  (Hua  et  al. (2007)).
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18 ème  Congrès  Français  de  Mécanique Grenoble,  27- 31  août  
2007
3 Numerical  simulations
Numerical  simulations  using  primitive  equations  for  a  continously  stratfied  
fluid  on  an  equatorial  -plane  have  been  performed  on  the  Earth  Simulatorβ  
(Japan)  for  several  types  of  configurations,  a  re- entrant  channel,  a  basin  and  an  
initially  zonally- localized  MRG wave  in  a very  long  channel,  to  sort  out  whenever  
possible  the  different  signals  that  are  excited  through  their  eastward /westward  
propagation.  
The  basin  simulations  concerned  a  bihemispheric  basin  of  idealized  
geometry,  centered  about  the  equator  and  of  comparable  width  to  either  the  
Atlantic  or  Pacific.  Typical   resolutions  vary  between  1/12°and  1/24°  in  the  
horizontal  and   between  200  and  400  levels  in  the  vertical.  
We have  analyzed  the  basin  response  to  an  oscillating  forcing  located  in  the  
western  boundary  layer.   The  symmetry  of  the  forcing   about  the  equator  is  such  
that  mixed  Rossby- gravity  (MRG) waves  are  excited  in  the  western  part  of  the  
equatorial  ocean.  The  forcing  frequency  for  short  MRG  waves  has  typical  
intraseasonal  periods  of  40- 80  days.
These  short  waves,  as  expected  from  theoretical  considerations,  will 
destabilize   in  the  vicinity  of  the  western  boundary.  This  leads  to  the  formation  
of  finite  amplitude,  nonlinear  jets  in  the  entire  equatorial  basin.  The  space  and  
time  patterns  of  the  jets  correspond  to  low- frequency  oscillating  equatorial  
basin- modes,  the  period  of  which  is  set  by  the  dominant  vertical  mode  of  the  
MRG wave  instability.  The  vertical  scale  of  the  jets  is  a  function  mainly  of  the  
forcing  period  and  is  independent  of  the  forcing  vertical  mode,  as  long  as  the  
excited  MRG waves  are  in  a sufficiently  short  regime  to  be  unstable.  
As a result,  unlike  a steady  western  boundary  current,  an  oscillatory  western  
boundary  current  leads  to  a  permanent  equatorial  zonal  circulation,.  But  most  
importantly,  MRG waves  destabilization  appears  to  be  a  plausible  formation  
mechanism  for  the  observed  Equatorial  Deep  Jets.  The  spatial  and  dynamical  
characteristics  of  the  zonal  circulation  achieved  with  a  60  days  forcing  period  
are  indeed  compatible  with  the  observations  in  the  Atlantic  Ocean  (d’Orgeville  et  
al.  (2007)), while  those  with  a 75  days  forcing  period  agree  with  the  Pacific  ocean  
observations.
FIG. 1  – LONGITUDE- DEPTH SECTION ALONG THE EQUATOR  OF THE ZONAL VELOCITY FIELD FOR A 40  DAYS 
(A), 60  DAYS (B) AND 75  DAYS FORCING FREQUENCY INSIDE THE WESTERN BOUNDARY LAYER.
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18 ème  Congrès  Français  de  Mécanique Grenoble,  27- 31  août  
2007
4 Conclusions
A Floquet  analysis  predicts  the   destabilization  of  a  westward- propagating  
Mixed  Rossby  Gravity   wave  producing  zonally  coherent  jets,   provided  the  wave  
is  short  enough.  The  main  result  concerning  the  vertical  scale  selection  of  high  
baroclinic  mode  zonal  jets  is  that  the  dominant  vertical  mode  scales  like  the  
square  of  the  dimensional  period  of  the  short  MRG wave.
Applying  these  theoretical  considerations  to  the  response  of  an  equatorial  
basin  to  an  oscillating  forcing  located  in  the  western  boundarylLayer,  we  show  
that  intraseasonal  variability  can  excite  short  MRG waves  whose  destabilization  
leads  to  the  formation  of  Equatorial  Deep  Jets  with  east - west  velocity  directions  
that  alternate  with  depth  at  vertical  scales  comparable  to  observations
References
D’Orgeville,  M., B. L. Hua,  and  H. Sasaki  2007:  Equatorial  deep  jets  triggered  by  a  
large  vertical  scale  variability  within  the  western  boundary  layer.  J. Mar.  Res. , 
(in  press).
Gill, A. E. 1974  The  stability  of  planetary  waves  on  an  in nite  -plane.  ﬁ β Geophys.  
Astrophys.  Fluid  Dynamics     6,  29–47.
Firing,  E. 1987:  Deep  zonal  currents  in  the  central  equatorial  Pacific.  J. Mar.  Res ., 
45 , (4), 791- 812.
Gouriou,  Y., B. Bourlès,  H. Mercier,  R. Chuchla,  1999:  Deep  jets  in  the  equatorial  
Atlantic  Ocean.  J. Geophys.  Res. , 104 , C9, 21217- 21226.
Hua,  B. L., M. D’Orgeville,  C. Menesguen,  and  H. Sasaki,  2007:  Destabilization  of  
mixed  Rossby- gravity  waves  and  equatorial  zonal  jets  formation.  J.  Fluid  
Mech.  (submitted).  
Lee,  S.  Y.  &  Smith,  L.  2002  Stability  of  Rossby  waves  in  the  -planeβ  
approximation.  Physica  D, 162.
Legras  B.,  Villone  B. ,   Frisch  U.  1999  Dispersive  stabilization  of  the  inverse  
cascade  for  the  Kolmogorov  flow,  Phys.  Rev.  Lett.  82  4440—4443.
 Manfroi  A.J., W.R. Young,   1999  Slow evolution  of  zonal  jets  on  the   plane,   β J. 
Atmos.  Sci. 56  784- - 800.
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Equatorial zonal jets formation

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