Introduction to barotropic and baroclinic instability

Introduction to barotropic and baroclinic instability

1. Introduction of
barotropic and
baroclinic instability
By :-Navinya chimurkar
Shukla poddar and
Girish Patidar

2. Concept of Instability
Therealatmospherenearlyalwayshas
horizontalandverticalwindshears.
Naturalforcesoperatinginatmospheretend
tocreatethesewindshears.
Itis,however,observedthatequilibrium
configurationofatmosphereinmotionbreaks
downwhentheseshearsexceedcertain
criticalvalues.
Thisisthebasicconceptof‘Dynamic
Instability’intheatmosphere.

3. Thisprocessgoesonandtheatmosphere
manifestscorrespondingchangesinwinds
andweatherconditionsondifferentscalesof
spaceandtime.
Awaveintheatmospherecanbecomeunstable,
providedithasanenergysource,uponwhichitcan
‘FEED’.
Wavesofmeteorologicalconcernhavetheenergy
sourceas‘Shear’inbackgroundwind.
Concept of Instability

4. Instability,ontheotherhand,isfavoredby
CoriolisParameter(f),largeverticalwindshear
andlowhydrostaticstability.
Ifafluidisleftafterforcingtomoveinthree
dimensions,itcaneithermoveinvertical,orin
horizontaldirection(withoutgainingheight),
ormoveonaninclinedplane,aftertheforce
hasbeenremoved duetoconditions
prevailingintheatmosphere.
Concept of Instability

5. •Convective instability :-Convective clouds grow as
parcels tap into the background CAPE
•Kelvin-Helmholtz instability :-Wave like clouds grow
(and “break”) as parcels tap into the background
vertical shear.
•Barotropic instability :-Disturbances grow by
extracting kinetic energy from the background flow
suction-vortices in tornadoes
meso-vortices in hurricanes
short-waves in jet stream.
•Baroclinic instability :-Disturbances grow by
extracting potential energy from the background
flow
Synoptic-scale waves
Different Types of Instability

6. Whentheunstablewindshearsare
essentiallyinthehorizontal
Barotropic Instability
Baroclinic Instability
Whentheunstablewindshearsare
essentiallyintheverticaland
associatedverticalmotionsaredry
adiabatic

7. Barotropic atmosphere
Theatmosphere,inwhich,thedensityisa
functionofpressurealone.
Surfacesofconstantofpressurearealso
surfacesofconstantdensity.
Thereisnohorizontalgradientoftemperature
andhencenoverticalwindshear.
Thus,thegeostrophicwindisindependentof
height.p

8. Baroclinic atmosphere
Theatmosphere,inwhich,thedensityisa
functionofpressureandtemperature.
Surfacesofconstantofpressurecutacross
surfacesofconstantdensity,forming
‘Solenoids’.
Horizontaltemperaturegradientexists,andso
doestheverticalwindshear.
RT
p
T,p

9. Baroclinic
instability

10. Baroclinic instability
BCIarisesfromverticalwindshearinarotating
atmosphere.
PotentialEnergy(PE)ofbasicstateflowis
convertedtoPEandKEofperturbation.
InBaroclinicatmosphere,isobaricsurfaces
intersectlinesofequaldensity(i.e.,Isosteric)at
someangle.
Presenceofverticalwindshearmeans
presenceofhorizontalthermalgradient.Thus,
thermalwindrelationshipisalsoimportant.

11. Baroclinic instability
ThereiszonalavailablePotentialEnergy(A
Z)in
theatmosphere.
ThisA
Zcangetaccumulatedinatmosphere
onlyuptocertainextent.Beyondthislimit,the
equilibriumisreadytobreakdownatthe
slightestprovocation.
Inaperturbationsuper-imposedonthisbasic
state,coldpolarairflowsequator-wardsand
relativelywarmersub-tropicalairflowspole-
wards,sothatcoldandwarmairispresent
sidebysidealongalatitudinalcircle.

12. Baroclinic instability
Here,A
ZisconvertedintoA
E(available
potentialenergyofeddy,orperturbation).
Generallyasituationprevails,inwhich,warm
airrisesandcoldairsinksinx-pplane.
So,a
eisconvertedintoKE.hence:-
A
z→a
e→k
e
ThestorehouseofA
Ziscontinuouslyfedby
differentialheatingofdifferentlatitudinal
zones(forwhich,solarheatingistheultimate
cause).
KEiscontinuouslydrainedoutofthesystemby
friction.

13. Baroclinic instability
IncaseofBTI,momentum istransferred(as
onlyhorizontalshearisinvolved).
IncaseofBCI,heatistransferred(asvertical
shearisinvolved,whichisrelatedtohorizontal
temperaturegradient).
BTIisstudiedonthebasisofconservationof
AbsoluteVorticity,whereas,theBCIisstudied
onthebasisofconservationofPotential
Vorticity,whichisgivenas. 
H
f
q




14. Baroclinic instability
Intherealatmosphere,therearemany
factorsthatmayinfluencethedevelopment
ofsynopticsystems,suchas:-
Instabilitiesduetolateralshear(inthejet
stream).
Non-linearinteractionoffiniteamplitude
perturbations.
ReleaseofLatentHeatinprecipitating
systems.
BaroclinicInstability.

15. Baroclinic instability
BCIarisesfromacombinationofverticalshear
androtationoftheearth.
Verticalshearofhorizontalwindimplies
meridionalgradientoftemperature.Thusthere
isazonalavailablepotentialenergyinthe
atmosphere.
BCIisrelatedtomeanmeridionalgradientof
potentialvorticityandmean meridional
gradientoftemperature.

18. Necessary condition for BCI
NecessaryConditionforBCItoexist
Whereintegralhasbeentakenforentire
domainfrom‘-L’to‘+L’.
Inthiscondition,‘c
i’shouldbenon-zerofor
unstablewaves.Thus,theentirequantityof
LHSmustvanishforBCItoexist.0
0
0
2
2
0
2
2
0














 



 

L
L
Z
L
L
*
*
i dy
cU
ψρ
Z
U
εdydZ
cU
ψρ
y
q
c
*

19. Necessary condition for BCI
Mostimportantterms,whichdecide
thepresenceofBCI,are:-
MeridionalGradientofPotential
Vorticity.
VerticaldependencyofbasicZonal
Flow‘U’.y
q

 *
Z
U


Othertermsgenerallyarenon-zero,while
integratedinthedomain.

20. Necessary condition for BCI
Inotherwords,forunstablewaves,‘c
i’must
benon-zero;thus,theintegralofthequantity
insquarebracketsmustvanish.
Sincethequantityincludingdisturbance
magnitudeisnon-negative,BCIispossible,
onlywhenthemeridionalgradientofthe
PotentialVorticityintheentiredomainmust
satisfycertainconstraints.

21. Barotropic
instability

22. Barotropic instability
BTIarisesmainlyfromexcessivehorizontalwind
shearsofflow(e.g.,incaseofJetStream).
ExchangesofKineticEnergy(KE)takeplace
betweenbasicflowandwaveperturbations
throughhorizontalnon-divergentmotions.
So,thereisexchangeofKEandwave
perturbationsgrowaftertakingKEfrombasic
zonalflow.
Intheprocess,zonaljet(i.e.,windmaxima)
getsdilutedanditsmomentum issharedbyits
adjacentlayers.

23. Barotropic instability
Whilethezonalmomentumofthejetflowasa
wholeisconserved,itsKEdecreases.
ThislossofKEbythejetisgainedbythewave
perturbation.

24. Necessary condition For BTI
EssentialConditionforBTItoexist:-
where‘A’isthewaveamplitudeandintegral
hasbeentakenforentiredomainfrom‘-d’to
‘+d’.
Inthiscondition,and arepositive.So,
inordertohaveBTIintheflow,either‘c
i’or
‘ ’mustbezero.










2
2
y
U
 
2
cU 2
A 
0
2
2
2
2
















dy
cU
y
U
A
c
d
d
i


25. WavenotGrowing/Decaying
Inthiscondition,‘c
i’iszero,thus,wavewill
neithergrow,nordecay.
Thus,wavewillmoveuniformlyalongx-
direction.
Necessary condition For BTI

26. WaveGrowing
Inthiscondition,‘c
i’is+ve.
Thus,forBTItoexist,itisnecessarythat
meridionalgradientofAbsoluteVorticitymustbe
zeroatsomelatitudewithinthedomain.0
2
2
2
2








































y
U
f
yy
U
y
f
y
U
 y
U
ff
y
U
y
U
x
v













  
0



















yy
f
y
U
f
y

Necessary condition For BTI

27. BTI in terms of KE equation & role
of tilt
 
dy
y
U
vu
dt
d
dy
vu
dt
d
dyKE
dt
d
d
d
''
d
d
d
d
''












  






2
22
RateofchangeinKEcanberepresentedas:-y
A
l
A
vu
''




22
22
WhereMomentum Fluxcanfurtherbewritten
as:-

28. BTI in terms of KE equation & role of
tilt dy
y
U
l
A
dyKE
dt
d
d
d
d
d
 















2
2

EssentialConditionforBTIintermsofKEis:-



y
l

TiltofTrough/Ridge
PositiveTilt(NE-SW)
(ForwardTilt):-
NegativeTilt(NW-SE)
(BackwardTilt):-00 



y
l
 00 



y
l


29. BTI in terms of KE equation & role
of tilt
Since‘A’and‘µ’are+vequantities,LHSwillbe
+veor–ve,dependingonsignsof‘Tilt’and
‘MeridionalShear’.
ThismeansthatintegralofKEincreasesor
decreaseswithtime,underspecificconditionsof
tiltedtroughs.
Increase/DecreaseinKEwillleadtoincrease/
decreaseofBTIinthedomain.
NecessaryconditionforBTIis:-0


y
U
l

30. BTI IN TERMS OF KE
EQUATION & ROLE OF TILT

31. BTI IN TERMS OF KE
EQUATION & ROLE OF TILT

32. BTIissatisfiedatsomelevelsinthevicinityof
ITCZ.Thus,BTIappearstobeaplausible
mechanismforthedevelopmentofweak
disturbancesalongITCZ.
Barotropicallyunstabledisturbanceswill
continuetogrow,onlyiftheshearofthe
meanzonalflowremainsunstable,sothat
wavescanextractenergyfromthemean
flow.
BTIisnotauniquelytropicalphenomenon.This
isalsopossibleinthevicinityofmid-latitudejet
stream.
Barotropic instability : additional
points

33. Barotropic instability : additional
points
However,inmid-latitudes,BaroclinicInstability
(BCI)isgenerallythemoreimportant
mechanism.
MeanmonsoonzonalcurrentovertheIndian
regionischaracterizedbyalow-levelW’lyjet,
locatedat850hPaandanupperlevelstrong
E’lyjet,locatedclosetothe150hPalevel.Itis
wellestablishedthatthesejetsaretheseatsof
thesynopticscaleactivedisturbancesduring
thesummermonsoon overIndia(e.g.,
MonsoonLows/Depressions,E’lyWaves,etc).

34. BTI & BCI : Summary
BTI BCI
1.Meridional shear in
zonal flow is relevant
(i.e., Horizontal
Shear).
Vertical shear of zonal
or meridional wind is
applicable (i.e., du/dz
or dv/dz).
2.No thermal wind
concept.
Shear is represented by
thermal wind (∂V
g/∂z).
3.Waves increase /
derive their KE from
KE of basic flow as a
result of vorticity
transport.
Waves grow as a result
of conversion of PE of
basic flow due to
baroclinicity in to KE.

35. BTI & BCI : Summary
BTI BCI
4.Dependsuponthe
concentration of
windandstrengthof
β.
Depends upon
concentration of
temperature with
respecttoheightand
strengthofwindwith
height.
5.Wavelength more
than300km&less
than 2500 km.
Maximum growth
ratefor2000km.
Wavelengthmorethan
2000km&lessthan
8000km.Maximum
growthratefor2500-
3000km.

36. BTI & BCI : Summary
BTI BCI
6.NecessaryCondition:
AbsoluteVorticity(ζ+f)
mustbeconstant,i.e.,
∂(ζ+f)/∂y=0
Necessary Condition :
PotentialVorticity[(ζ+f)/
∆H]mustbeconstant,i.e.,
[∂(ζ+f)/∆H]/∂y=0
7.Momentum transport
andhorizontalplane.
Heattransportandvertical
plane.
8.Nowaveformationofits
own.
Waveformationispresent.
9.Zoneofinterestnear
ITCZorEquator,shear
linewithperturbationin
trade.
NearPForSTR,whensharply
inclinedperturbationsin
mid-latW’lyandshort
waves.

37. BTI & BCI : Summary
BTI BCI
10.Horizontal(meridional)
shear exceeding
critical value
associatedwithE’ly
waveorLLJintrades
givingrisetounstable
perturbations(growth
oftropicalcyclones).
Under sub-critical
conditions,sinusoidal
wavesarelikelyto
continue without
breakingdown.
11.Low level jet,
Australianjetor
Africanjetgiveriseto
suchinstability.
PFJ/STJreinforcethe
instability.

38. BTI & BCI : Summary
BTI BCI
12.Temperature
gradientisverysmall;
sonosuchcriterion
exists,althoughE’ly
wave troughsare
examinedinrelation
totheirtemperature
characteristicsand
genesisoftropical
cyclones.
Thermalwaveshould
lagbehindpressure
wavefordeveloping
wave(andprecede
fordecayingsystems).
Coldtroughtravel
slowerthansurface
zonalflow.

39. •Bluestein, H. B, 1993: Synoptic-Dynamic Meteorology in
Midlatitudes. Volume II: Observations and Theory of
WeatherSystems. Oxford University Press, New York, 594
pp.
•Bretherton, F. P., 1966: Critical layer instability in
baroclinicflows, Quart. J. Roy. Meteor. Soc., 92, 325-334.
•Charney, J. G., 1947: the dynamics of long waves in a
baroclinicwesterly current. J. Meteor., 6, 56-60.
•Eady, E. T., 1949: Long waves and cyclone waves. Tellus,
1, 33-52.
•Lackmann, G., 2011: Mid-latitude Synoptic Meteorology
–Dynamics, Analysis and Forecasting, AMS, 343 pp.
•Orlanski, I., 1968; Instability of frontal waves. J. Atmos.
Sci., 25, 178-200.
References

40. THANKS