Understanding large-scale instabilities of …gershwin.ens.fr/.../beamer-lectures_cambridge_2012.pdflayered models V. Zeitlin Unstable large-scale ﬂows Modeling large-scale processes

Instabilities inlayered models

V. Zeitlin

Unstablelarge-scale flows

Modelinglarge-scaleprocessesPrimitive equations on thetangent plane

Vertical averaging of theprimitive equations

Stability : generalLinear vs nonlinear(in)stability

Instabilities asphase-locking andresonance of the linearmodes

ExamplesBarotropic instability of a jet

Baroclinic instability of a jet

Coastal currents : passivelower layer

Coastal currents : activelower layer

Inertial vs baroclinicinstability

Moist baroclinic instability

Moist-convective RSWmodel

Moist vs dry baroclinicinstability

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Understanding large-scale instabilities ofatmospheric and oceanic flows and theirsaturation with layered rotating shallow

water models

V. Zeitlin

Laboratoire de Météorologie Dynamique, Paris

Fluid Dynamics of Sustainability and the Environment,Cambridge, September 2012


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PlanUnstable large-scale flowsModeling large-scale processes

Primitive equations on the tangent planeVertical averaging of the primitive equations

Stability : generalLinear vs nonlinear (in)stabilityInstabilities as phase-locking and resonance of thelinear modes

ExamplesBarotropic instability of a jetBaroclinic instability of a jetCoastal currents : passive lower layerCoastal currents : active lower layerInertial vs baroclinic instabilityMoist baroclinic instability

Moist-convective RSW modelMoist vs dry baroclinic instability

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Midlatitude atmospheric jet

Midlatidude upper-tropospheric jet (left) and relatedsynoptic systems (right).


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Oceanic currents : Gulfstream

Gulfstream (left) and related vortices (right). Velocityfollows isopleths of height anomaly in the firstapproximation.


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Leeuwin current and associated vortices

Velocity (arrows) and temperature anomaly (colors) of theLeeuwin curent near Australian coast.


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Coastal flows : Weddell sea

Instability of a coastal current in the Weddell sea.


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Primitive equations : ocean

Hydrostaticsgρ+ ∂zP = 0, (1)

P = P0 + Ps(z) + π(x , y , z; t),ρ = ρ0 + ρs(z) + σ(x , y , z; t), ρ0 � ρs � σ

Incompressibility

~∇ · ~v = 0, ~v = ~vh + zw . (2)

Euler :∂~vh

∂t+ ~v · ~∇~vh + f z ∧ ~vh = −~∇hφ. (3)

φ = πρ0

- geopotential.Continuity :

∂tρ+ ~v · ~∇ρ = 0. (4)


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Primitive equations : atmosphere,pseudo-height vertical coordinate

∂~vh

∂t+ ~v · ~∇~vh + f z ∧ ~vh = −~∇hφ, (5)

−gθ

θ0+∂φ

∂z= 0, (6)

∂θ

∂t+ ~v · ~∇θ = 0; ~∇ · ~v = 0. (7)

Identical to oceanic ones with σ → −θ, potentialtemperature.Vertical coordinate : pseudo-height, P - pressure.

z = z0

(1−

(PPs

) Rcp

)(8)


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Material surfaces

g f/2z

x

z2

z1w1= dz1/dt

w2= dz2/dt


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Vertical averaging and RSW modelsI Take horizontal momentum equation in conservative

form :

(ρu)t + (ρu2)x + (ρvu)y + (ρwu)z − fρv = −px , (9)

and integrate between a pair of material surfacesz1,2 :

w |zi=

dzi

dt= ∂tzi + u∂xzi + v∂yzi , i = 1,2. (10)

I Use Leibnitz formula and get :

∂t

∫ z2

z1

dzρu + ∂x

∫ z2

z1

dzρu2 + ∂y

∫ z2

z1

dzρuv −

f∫ z2

z1

dzρv = −∂x

∫ z2

z1

dzp − ∂xz1 p|z1+ ∂xz2 p|z2

.(11)

(analogously for v ).


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I Use continuity equation and get

∂t

∫ z2

z1

dzρ+ ∂x

∫ z2

z1

dzρu + ∂y

∫ z2

z1

dzρv = 0. (12)

I Introduce the mass- (entropy)- averages :

〈F 〉 =1µ

∫ z2

z1

dzρF , µ =

∫ z2

z1

dzρ. (13)

and obtain averaged equations :

∂t (µ〈u〉) + ∂x

(µ〈u2〉

)+ ∂y (µ〈uv〉)− fµ〈v〉

= −∂x

∫ z2

z1

dzp − ∂xz1 p|z1+ ∂xz2 p|z2

, (14)

∂t (µ〈v〉) + ∂x (µ〈uv〉) + ∂y

(µ〈v2〉

)+ fµ〈u〉

= −∂y

∫ z2

z1

dzp − ∂yz1 p|z1+ ∂yz2 p|z2

, (15)

∂tµ+ ∂x (µ〈u〉) + ∂y (µ〈v〉) = 0. (16)


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I Use hydrostatics and get, introducing mean constantdensity ρ :

p(x , y , z, t) ≈ −gρ(z − z1) + p|z1. (17)

I Use the mean-field (= columnar motion)approximation :

〈uv〉 ≈ 〈u〉〈v〉, 〈u2〉 ≈ 〈u〉〈u〉, 〈v2〉 ≈ 〈v〉〈v〉. (18)

and get master equation for the layer :

ρ(z2 − z1)(∂tvh + v · ∇vh + f z ∧ vh) =

− ∇h

(−gρ

(z2 − z1)2

2+ (z2 − z1) p|z1

)− ∇hz1 p|z1

+∇hz2 p|z2. (19)

I Pile up layers, with lowermost boundary fixed bytopography, and uppermost free or fixed.


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1-layer RSW, z1 = 0, z2 = h

∂tv + v · ∇v + f z ∧ v + g∇h = 0 , (20)

∂th +∇ · (vh) = 0 , (21)

In the presence of nontrivial topography b(x , y) :h→ h − b in the second equation.

g f/2z

h

v

x

y

Columnar motion.


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2-layer RSW, rigid lid : z1 = 0, z2 = h,z3 = H = const

∂tvi + vi · ∇vi + f z ∧ vi +1ρi∇πi = 0 , i = 1,2; (22)

∂th +∇ · (v1h) = 0 , (23)

∂t (H − h) +∇ · (v2(H − h)) = 0 , (24)

π1 = (ρ1 − ρ2)gh + π2 . (25)

g f/2

z

x

h

H

p2

p1

v2

v1 rho1

rho2


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2-layer rotating shallow water model with afree surface : z1 = 0, z2 = h1, z3 = h1 + h2

∂tv2 + v2 · ∇v2 + f z ∧ v2 = −∇(h1 + h2) (26)

∂tv1 + v1 · ∇v1 + f z ∧ v1 = −∇(rh1 + h2), (27)

∂th1,2 +∇ · (v1,2h1,2)

= 0 , (28)

where r = ρ1ρ2≤ 1 - density ratio, and h1,2 - thicknesses of

the layers.

g f/2

z

x

h1

v2

v1 rho1

rho2h2


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Useful notions

Balanced vs unbalanced motionsGeostrophic balance : balance between the Coriolis forceand the pressure force. In shallow-water model :

f z = −g∇h (29)

Valid at small Rossby numbers : Ro = U/fL, where U, L -characteristic velocity and horizontal scale. Balancedmotions at small Ro : vortices. Unbalanced motions :inertia-gravity waves.

Relative, absolute and potential vorticityRelative vorticity in layered models : ζ = z · ∇ ∧ v.Absolute vorticity : ζ + f . Potential vorticity (PV) : ζ+f

z2−z1for

the fluid layer between z2 and z1.


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Dynamical systems

U =M [U ] , (30)

U - dynamical variable(s),M - operator defined by thestructure of the model. Solutions : trajectories in thespace of U :

U(t0) −→ U(t) (31)

In hydrodynamics U = (v, ρ,p, ...).U0 : exact solution, i.e. state of restM [U0] = 0, or other.Linearization : U = U0 + u, ||u|| << 1 ⇒ linearequations :

u = L [U0] ◦ u, (32)

L - linear operator


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Linear and nonlinear (Lyapunov) stability

Linear stabilityLinearized system→ Fourier transform : u(t)→ u(ω)eiωt

→ eigenproblem for ω → spectrum of ω. Dispersionrelation : ω = ω(k), k - wavenumber.In general, complex eigenvalues : ω = ωr + iωiLinear stability (instability) : ωi ≥ 0(ωi < 0)↔ exponentialdecay (growth) of small perturbations of solution. Neutralstability : real ω.

Nonlinear stability (Lyapunov)

∀ε ∃δ : ||u||t=0 < δ ⇒ ||u||∀t>0 < ε. (33)


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Energy estimates (Hamiltonian systems)

I Linear (formal) stability : second variation of energyH : δ2

UH(U0) is sign-definite.I Nonlinear stability :

∀δU : 0 < const ≤ H(U0+δU)−H(U0)−δUH(U0)·δU .(34)

Remark :In the presence of other integrals of motionCα, α = 1,2, ...

H −→ HC = H +∑α

λαCα,

λα - Lagrange multipliers.


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Sufficient conditions of stability in multi-layerRSW (Ripa 1990)For a plane-parallel flows flow Ui(y) in geostrophicequilibrium in 2 layers of thickness Hi :

fUi(y) + gH ′i (y) = 0, i = 1,2. (35)

with potential vorticity : Qi(y) =f (y)−U′i (y)

Hi (y) :

∀y ∃α = const : (36)

1.(Ui(y)− α) Q′i (y) < 0, i = 1,2. (37)

2.

g >

2∑i=1

(Ui(y)− α)2

Hi(y)(38)


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Linear stability analysis : workflowI Choose a solution of the full system and linearize

about itI Make Fourier transform in timeI Solve the linear eigenproblem for the

eigenfrequencies and find correspondingeigenmodes (Find analytical solution, if you are lucky.Otherwise, use standard numerical routines. Mostpopular : shooting, pseudospectral collocation)

I Identify the physical nature of the eigensolutions(Necessary, helps to discard spurious modes oftenproduced by numerics)

Typical output : dispersion diagrams, giving the real andimaginary parts of the eigenvalues as a function ofmodes’ wavenumbers, and stability diagrams givingdependence of the growth rates on the parameters of theproblem.


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Instabilities as resonances of linear modes

Dispersion relation : D(ω, k) = 0 - implicit. For layer-wiseconstant mean velocity : polynomial with constantcoefficients. Real roots : propagating modes. Complexroots : unstable modes.

Physics : instability = phase-locking and resonance of thelinear waves propagating in the background flow.


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Main types of propagating waves in layeredmodels on the f -plane :

I Poincaré (P) or inertia-gravityI Baroclinic and barotropic Rossby (R)I Kelvin (K)

Physical origin : "elasticity" of the isopycnal (isentropic)surfaces (P), "elasticity" of the iso-PV surfaces (R),presence of boundary (K).

Reminder :I Barotropic motion : inter-layer columnarI Baroclinic motion : inter-layer shear


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Origins of the jet instabilities

"Standard" instabilitiesI Kelvin-Helmholtz : P - P (or K - P) resonanceI barotropic R - R resonance (standard : geostrophic)I baroclinic R - R resonance (standard : geostrophic)

"Non-standard" inertial instabilityTrapped waves with negative eigen square frequency.

I essentially ageostrophic : needs Ro = O(1) ;I symmetric with resp. to along-jet translations ;I baroclinic : needs vertical variations ;


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Instability of a barotropic jet on the f -planeProfiles of jet velocity, pressure and potential vorticity.

−1.5 −1 −0.5 0 0.5 1 1.5

−1

−0.5

0

x/Rd

v/V

−1.5 −1 −0.5 0 0.5 1 1.5

−1

−0.5

0

0.5

1

x/Rd

η/∆η

−1.5 −1 −0.5 0 0.5 1 1.5

0.95

1

1.05

x/Rd

PV/f


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Dispersion diagram : along-jet phase velocityand growth rate of the unstable modes

0 0.5 1 1.5 2 2.5−1

−0.8

−0.6

−0.4

−0.2

0

c p

0 0.5 1 1.5 2 2.50

0.05

0.1

0.15

σ


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The most unstable mode : pressure andvelocity distribution

x/Rd

y/Rd

−1.5 −1 −0.5 0 0.5 1 1.5

−1.5

−1

−0.5

0

0.5

1

1.5

−6

−4

−2

0

2

4

6

x 10−5


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Nonlinear evolution : anomaly of h andvelocity

x/Rd

y/Rd

40T

i

−1 0 1

−1

0

1

−1

−0.5

0

0.5

1

x 10−3

x/Rd

y/Rd

140T

i

−1 0 1

−1

0

1

−2

0

2

x 10−3

x/Rd

y/Rd

240T

i

−1 0 1

−1

0

1

−5

0

5x 10

−3

x/Rd

y/Rd

340T

i

−1 0 1

−1

0

1

−5

0

5

x 10−3

x/Rd

y/Rd

440T

i

−1 0 1

−1

0

1

−5

0

5x 10

−3

x/Rd

y/Rd

540T

i

−1 0 1

−1

0

1

−4

−2

0

2

4x 10

−3


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Nonlinear evolution : relative vorticity

x/Rd

y/R d

40T

i

−1 0 1

−1

0

1

−0.5

0

0.5

x/Rd

y/R d

140T

i

−1 0 1

−1

0

1

−0.5

0

0.5

x/Rd

y/R d

240T

i

−1 0 1

−1

0

1

−0.5

0

0.5

x/Rd

y/R d

340T

i

−1 0 1

−1

0

1

−0.5

0

0.5

x/Rd

y/R d

440T

i

−1 0 1

−1

0

1

−0.5

0

0.5

x/Rd

y/R d

540T

i

−1 0 1

−1

0

1

−0.5

0

0.5


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Baroclinic Bickley jet

Geostrophically balanced upper-layer jet on the f -plane.non-dimensional profiles of velocity and thicknessperturbations :

u1 = 0, η1 =1

α− 1tanh(y),

u2 = sech2(y), η2 =−1α− 1

tanh(y).

No deviation of the free surface : η1 + η2 = 0.Parameters : Ro = 0.1, Bu = 10 - typical for atmosphericjets.


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−3 −2 −1 0 1 2 30

0.5

1

u i/U

y/Rd

−3 −2 −1 0 1 2 3−10

0

10

η i/H0

y/Rd

−3 −2 −1 0 1 2 3−1

0

1

∆PV

i/fH0−

1

y/Rd

Bickley jet : zonal velocity ui , thickness deviation ηi andPV anomaly. Lower (upper) layer : solid black (dashed

gray).


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Linear stability diagram

0 0.5 1 1.5 2 2.50

0.2

0.4

k

c p

0 0.5 1 1.5 2 2.50

0.1

0.2

k

σ

Phase velocity (top) and growth rate (bottom) of theunstable modes


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The most unstable mode

0.5 1 1.5 2 2.5

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

ψ2

x/Rd

y/Rd

0.5 1 1.5 2 2.5

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

ψ1

x/Rd

y/Rd

Most unstable mode of the upper-layer Bickley jet.Upper(top) and lower (bottom) layer- geostrophic

streamfunctions and velocity (arrows) fields.


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Nonlinear saturation

250Ti

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

300Ti

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Relative vorticity in the lower (colors) and upper(contours) layers.


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Idealized coastal current configuration

y = 0

y = −L

ρ1

ρ2

f2

y

H1(y) U1(y)

H2(y) U2(y)


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RSW equations with coast (no bathymetry)

Equations of motion :

ut + uux + vuy − fv + ghx = 0,vt + uvx + vvy + fu + ghy = 0,

ht + (hu)x + (hv)y = 0. (39)

Boundary conditions :

u = 0|y=−L , (40)

H(y) + h(x , y , t) = 0, DtY0 = v |y=Y0. (41)

where Y0(x , t) is the position of the free streamline, Dt isLagrangian derivative.


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Balanced flows :u = U(y), v = 0, and h = H(y), exact stationary solution :

U(y) = −gf

Hy (y) (42)

−1 00

0.25

0.5

0.75

H1(y)

y−1 0

−0.5

0

0.5

U1(y)

y

Examples of the basic state heights (left) and velocities(right) for constant PV flows withU0 = −sinh(−1)/cosh(−1) (solid), U0 = 1/2 (dotted) anda zero PV flow (dash-dotted)


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Wave Number

U0

0 1 2 3 4 5 6 7 8 9 10

0.5

0.55

0.6

0.65

0.7

0.75

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Stability diagram in the (U0fL , k) plane for the constant PV

current. Values of the growth rates in the right column.


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Dispersion diagram : stable flow

0 1 2 3 4 5 6 7 8 9 10

0

1

c

k

K

F

Pn

Pn

Dispersion diagram for U0 = −sinh(−1)/cosh(−1) andQ0 = 1.


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Dispersion diagram : unstable flow

0

0.5

1

c K

F

Pn

Pn

2 4 6 8 100

0.01

0.02

0.03

0.04

0.05

0.06

0.07

k

σ

Dispersion diagram for U0 = 0.5 and Q0 = 1. Crossingsof the dispersion curves in the upper panel correspond toinstability zones in the lower panel.


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The most unstable mode : Kelvin-Frontalresonance

y

x−1

0

Height and velocity fields of the most unstable modek = 3.5.


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Saturation of the instabilityy

x

t= 0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

y

x

t= 33

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

Height and velocity fields of the perturbation at t = 0 (left)and t = 30 (right). Kelvin front is clearly seen at thebottom of the right panel.


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Kelvin wave breaking

1 2 3 4 5 6 7−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1t= 22.5026

u

x

Evolution of the tangent velocity at y = −L (at the wall) fort between 0 and 22.5 at the interval 2.5(from lower toupper curves)


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Reorganization of the mean flow

−1 −0.5 0 0.50

0.05

0.1

0.15

0.2

0.25

0.3

Hzonal

y

−1 −0.5 0 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

y

uzonal

Evolution of the mean zonal height (left) and mean zonalvelocity (right) : Initial state t = 0 (dashed line), primaryunstable mode saturated at t = 40 (dash-dotted line), latestage t = 300 (thick line).


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Equations of motion

Djuj − fvj = − 1ρj∂xπj ,

Djvj + fuj = − 1ρj∂yπj ,

Djhj +∇ · (hjvj) = 0,(43)

j = 1,2 : upper/lower layer, (x , y), hj(x , y , t) - depths ofthe layers, πj , ρj - pressures, densities of the layers,

∇πj = ρjg∇(sj−1h1 + h2), s = ρ1/ρ2. (44)


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Boundary conditions

I Upper layer : same as in 1.5-layer case,I Lower layer : for harmonic perturbations with

wavenumber k , a decay condition :

∂y (sh1 + h2) = −k(sh1 + h2)|y=0 .

= rigid lid beyond the outcropping↔ filtering of freeinertia-gravity waves and related (weak) radiativeinstabilities.

Key parameters :U0, the non-dimensional velocity of the upper layer at thefront location y = 0, equivalent to Rossby number, aspectratio r = H1(−1)/H2(−1), and stratification s = ρ1/ρ2.


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Configurations considered :

Stationary solutionsBalanced flow with depths Hj(y) and velocities Uj(y) :

∂yHj = (−1)j−1 fg′

(U2 − sj−1U1), (45)

I Bottom layer : initially at rest (U2 = 0),I Upper layer : with constant PV.

Two classes of flows : barotropically stable/unstable, i.e.stable/unstable in the 1.5 - layer limit.


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Barotropically stable case

Dispersion diagrams for s = .5. (a) r = 10, (b) r = 2, (c)r = 0.5. Horizontal scale of the bottom panel shrinked toshow short-wave KH instabilities.


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Barotropically unstable case

Dispersion diagrams for s = 0.5 and for Rd = 1. (a)r = 10, (b) r = 5, (c) r = 2 . The horizontal scale of thepanels shrinked to show short-wave KH instabilities.


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Literature

maxP2 / maxP

1 = 0.017044 maxP

2 / maxP

1 = 0.047946

maxP2 / maxP

1 = 0.015975 maxP

2 / maxP

1 = 0.35601

Typical unstable modes(left to right, top to bottom) : KF1,RF, RP, PF.


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Literature

Scenario of development of the baroclinic RFinstability as follows from DNS

1. Upper layer : frontal wave evolves into a series ofmonopolar vortices at certain spacing due to vortexlines clipping and reconnection following formation ofKelvin fronts

2. Lower layer : Rossby wave develops a series ofvortices of alternating signs

3. Lower-layer dipoles drive the vortex out of the shoreand are at the origin of the detachment.


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Literature

y

x

t= 160

0 1 2 3 4 5 6−1

0

1

2

y

x

t= 160

0 1 2 3 4 5 6−1

0

1

2

y

x

t= 200

0 1 2 3 4 5 6−1

0

1

2

y

x

t= 200

0 1 2 3 4 5 6−1

0

1

2

Levels of h1(x , y , t) in the upper layer (left) and isobars ofπ2(x , y , t) in the lower layer (right) at t = 150 and 200 forthe development of the unstable RF mode superposed onthe basic flow with a depth ratio r = 2.


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Kelvin front and dissipation duringdevelopment of RF instability

y

x

t= 155

3 3.5 4 4.5 5 5.5 6−1

−0.5

Before detachment : zoom of the wall region. Breaking ofthe Kelvin wave.


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Structure of the detached vortex

y

x

t= 225

2 3 4 5 6 7

0

1

2

3

4

Isobars of π1(x , y , t) in the upper layer (white lines) andπ2(x , y , t) in the lower layer (dark lines) at t = 250. Dark(light) background : anticyclonic (cyclonic) region.


V. Zeitlin














Literature

Barotropic Bickley jet in the 2-layer RSW

−10 −8 −6 −4 −2 0 2 4 6 8 100

0.5

1

1.5

H

−10 −8 −6 −4 −2 0 2 4 6 8 100

0.5

1

1.5

v

x/L

−10 −8 −6 −4 −2 0 2 4 6 8 10−10

0

1015

PV

x/L

Background jet profile as a function of nondimensional x ;solid : layer 1 ; dashed : layer 2.


V. Zeitlin














Literature

Linearization of the symmetric problem

Small perturbations of the jet :

hi = Hi(x)+η′i (x , t),ui = u′i (x , t), vi = V (x)+v ′i (x , t), i = 1,2,(46)

Solution is sought in the form(u′i , v

′i , η′i ) = (u0i(x), v0i(x), η0i(x)) e−iωt + c.c.→ a pair of

coupled Schrödinger equations for the across-frontvelocities of the layers :(−ω2 + f (f + ∂xV )

)( u01u02

)−g∂2

x

(H1u01 + H2u02rH1u01 + H2u02

)= 0.

(47)


V. Zeitlin














Literature

Eigenfrequencies

In terms of the barotropic and the baroclinic velocitycomponents : (

ubuB

)=

(u02 − u01

H1u01+H2u02H1+H2

). (48)

ω2 = f∫

(f + ∂xV )Hb|ub|2∫Hb|ub|2

+ g(1− r)

∫ |∂x (Hbub)|2∫Hb|ub|2

+ g(1− r)

∫Hbu∗b∂

2x (H1uB)∫

Hb|ub|2, (49)

where Hb = 11/H1+1/H2

.Relative vorticity of the jet ∂xV sufficiently negative⇒ω2 < 0⇒ inertial (symmetric) instability.


V. Zeitlin














Literature

A typical unstable mode atBu = 10, Ro = 5, d = 2, r = 0.5

−10 −5 0 5 10−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Re(

η 0)

x/L−10 −5 0 5 10

−0.2

−0.1

0

0.1

0.2

Re(

u 0)

x/L−10 −5 0 5 10

−0.5

−0.25

0

0.25

0.5

0.75

1

Re(

v 0)

x/L

−10 −5 0 5 10−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Im(η

0)

x/L−10 −5 0 5 10

−0.2

−0.1

0

0.1

0.2

Im(u

0)

x/L−10 −5 0 5 10

−0.5

−0.25

0

0.25

0.5

0.75

1

Im(v

0)x/L


V. Zeitlin














Literature

Stability diagram for small Ro

0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

k

Re(

w)/

k

0 1 2 3 4 5 60

0.025

0.05

0.075

0.1

Im(w

)

k

Left : phase speed Re(ω)/k as a function of k ; Right :Growth rate Im(ω) as a function of k . Quasi-geostrophicjet : H0 = 1,Bu = 10,Ro = 0.5,d = 2, r = 0.5.


V. Zeitlin














Literature

Cross-section of the most unstable mode

−5 −2.5 0 2.5 5

−0.1

−0.05

0

Re

(η

0)

x/L−5 −2.5 0 2.5 5

−0.6

−0.4

−0.2

0

0.2

Re

(u

0)

x/L−5 −2.5 0 2.5 5

−0.5

0

0.5

Re

(v

0)

x/L

−5 −2.5 0 2.5 5

−0.1

−0.05

0

Im

(η

0)

x/L−5 −2.5 0 2.5 5

−0.6

−0.4

−0.2

0

0.2

Im

(u

0)

x/L−5 −2.5 0 2.5 5

−0.5

0

0.5

Im

(v

0)

x/L

Solid (dashed) : layer 1 (2)⇒ barotropic instability


V. Zeitlin














Literature

2D structure of the most unstable mode

x/L

y/L

−5 −2.5 0 2.5 5

−2.5

0

2.5

x/L

y/L

−5 −2.5 0 2.5 5

−2.5

0

2.5

Left(Right) : upper(lower) layer⇒ barotropic instability.


V. Zeitlin














Literature

Stability diagram for large Ro

0 1 2 3 4 5 60

0.5

1

1.5

2

k

Re(

w)/

k

0 1 2 3 4 5 60

0.25

0.5

0.75

1

Im(w

)

k

Left : phase speed Re(ω)/k as a function of k ; Right :Growth rate Im(ω) as a function of k . Stronglyageostrophic jet :H0 = 1,Bu = 10,Ro = 5,d = 2, r = 0.5. Growth rate ofthe most unstable branch has non-zero limit at k → 0.


V. Zeitlin














Literature

Cross-section of the most unstable mode

−10 −5 0 5 10−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Re(

η 0)

x/L−10 −5 0 5 10

−0.1

0

0.1

0.2

0.3

Re(

u 0)

x/L−10 −5 0 5 10

−0.25

0

0.25

0.5

0.75

1

Re(

v 0)

x/L

−10 −5 0 5 10−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Im(η

0)

x/L−10 −5 0 5 10

−0.1

0

0.1

0.2

0.3

Im(u

0)

x/L−10 −5 0 5 10

−0.25

0

0.25

0.5

0.75

1

Im(v

0)

x/L

Striking resemblance to the 1D inertial instability mode.


V. Zeitlin














Literature

2D structure of the most unstable mode

x/L

y/L

−5 −2.5 0 2.5 5

−7.5

−5

−2.5

0

2.5

5

7.5

x/L y/L

−5 −2.5 0 2.5 5

−7.5

−5

−2.5

0

2.5

5

7.5

Essentially baroclinic, concentrated in the anticyclonicshear : asymmetric inertial instability(AII).


V. Zeitlin














Literature

Early stages of nonlinear evolution of AII :thickness field

x/L

y/L

−15 −10 −5 0 5 10 15

−15

−10

−5

0

5

10

15

x/L

y/L

−15 −10 −5 0 5 10 15

−15

−10

−5

0

5

10

15

x/L

y/L

−15 −10 −5 0 5 10 15

−15

−10

−5

0

5

10

15

x/L

y/L

−15 −10 −5 0 5 10 15

−15

−10

−5

0

5

10

15

x/L

y/L

−15 −10 −5 0 5 10 15

−15

−10

−5

0

5

10

15

x/L

y/L

−15 −10 −5 0 5 10 15

−15

−10

−5

0

5

10

15

Layer 1 (left), layer 2 (right).


V. Zeitlin














Literature

Late stages of nonlinear evolution of AII :thickness

x/L

y/L

−15 −10 −5 0 5 10 15

−15

−10

−5

0

5

10

15

x/L

y/L

−15 −10 −5 0 5 10 15

−15

−10

−5

0

5

10

15

x/L

y/L

−15 −10 −5 0 5 10 15

−15

−10

−5

0

5

10

15

x/L

y/L

−15 −10 −5 0 5 10 15

−15

−10

−5

0

5

10

15

x/L

y/L

−15 −10 −5 0 5 10 15

−15

−10

−5

0

5

10

15

x/L

y/L

−15 −10 −5 0 5 10 15

−15

−10

−5

0

5

10

15

Appearance of intence localized vortices.


V. Zeitlin














Literature

"Dry" primitive equations in pseudo-heightcoordinates

ddt

v + fk × v = −∇φddtθ = 0

∇ · v + ∂zw = 0

∂zφ = gθ

θ0

v = (u, v) and w - horizontal and vertical velocities,ddt = ∂t + v · ∇+ w∂z , f - Coriolis parameter, θ - potentialtemperature, φ - geopotential.


V. Zeitlin














Literature

Moisture and moist enthalpy

Condensation turned off : conservation of specifichumidity of the air parcel :

ddt

q = 0.

Condensation turned on : θ and q equations acquiresource and sink. Yet the moist enthalpy θ + L

cpq, where L -

latent heat release, cp - specific heat, is conserved forany air parcel on isobaric surfaces :

ddt

(θ +

Lcp

q)

= 0,


V. Zeitlin














Literature

Vertical averaging with convective fluxes

3 material surfaces :

w0 =dz0

dt, w1 =

dz1

dt+ W1, w2 =

dz2

dt+ W2.

W

W2

1

θ

θ1

2

0

2z

z

z

1

Mean-field + constant mean θ →


V. Zeitlin














Literature

Averaged momentum and mass conservationequations (master equations) :

{∂tv1 + (v1 · ∇)v1 + fk × v1 = −∇φ(z1) + g θ1

θ0∇z1,

∂tv2 + (v2 · ∇)v2 + fk × v2 = −∇φ(z2) + g θ2θ0∇z2 + v1−v2

h2W1,{

∂th1 +∇ · (h1v1) = −W1,∂th2 +∇ · (h2v2) = +W1 −W2,


V. Zeitlin














Literature

Linking convective fluxes to precipitationBulk humidity : Qi =

∫ zizi−1

qdz. Precipitation sink :

∂tQi +∇ · (Qiv i) = −Pi .

We assume "dry" stable background stratification :

θi+1 = θ(zi) +Lcp

q(zi) ≈ θi +Lcp

q(zi) > θi ,

with constant θ(zi) and q(zi). Integrating the moistenthalpy we get

Wi = βiPi , βi =L

cp(θi+1 − θi)≈ 1

q(zi)> 0

Last step : relaxation formula with relaxation time τ .

Pi =Qi −Qs

iτ

H(Qi −Qsi )

H(.) - Heaviside (step) function.


V. Zeitlin














Literature

Moist-convective 2-layer model with a dryupper layerVertical boundary conditions : upper surface isobaricz2 = const, geopotential at the bottom constant (ground)φ(z0) = const, Q2 = 0, Q1 = Q, α = θ2

θ1- stratification :

∂tv1 + (v1 · ∇)v1 + fk × v1 = −g∇(h1 + h2),

∂tv2 + (v2 · ∇)v2 + fk × v2 = −g∇(h1 + αh2) + v1−v2h2

βP,∂th1 +∇ · (h1v1) = −βP,∂th2 +∇ · (h2v2) = +βP,∂tQ +∇ · (Qv1) = −P, P = Q−Qs

τ H(Q −Qs)

θ

θ1

2h

h1

2W

P>0


V. Zeitlin














Literature

Baroclinic Bickley jet

−3 −2 −1 0 1 2 30

0.5

1

u i/U

y/Rd

−3 −2 −1 0 1 2 3−10

0

10

η i/H 0

y/Rd

−3 −2 −1 0 1 2 3−1

0

1

∆PV i/fH

0−1

y/Rd

Bickley jet : zonal velocity ui , thickness deviation ηi andPV anomaly. Lower (upper) layer : solid black (dashed

gray).


V. Zeitlin














Literature

Early stages : evolution of moisture

5Ti

(a)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−3

−2

−1

0

1

2

3x 10

−4

20Ti

(b)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−8

−6

−4

−2

0

2

4

6

8

x 10−4

200Ti

(c)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.11

−0.1

−0.09

−0.08

−0.07

−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

300Ti

(d)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

Evolution of the moisture anomaly Q −Q0 withsuperimposed lower-layer velocity. Black contour :condensation zones.


V. Zeitlin














Literature

Early stages : growth rates

0 100 200 300 400 500−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

t/Ti

σ/Ro

DryMoist

Red : moist, blue : dry simulations.⇒Transient increase in the growth rate due to condensation.


V. Zeitlin














Literature

Dry vs moist simulations : evolution of relativevorticity

Dry

200Ti

(a)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

250Ti

(c)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

300Ti

(e)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

350Ti

(g)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Moist

200Ti

(b)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

250Ti

(d)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

300Ti

(f)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

350Ti

(h)

0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Lower layer : colors, upper layer : contours.Condensation : solid black.


V. Zeitlin














Literature

Moist baroclinic instability in Nature


V. Zeitlin














Literature

Literature : general

Multi-layer RSW : derivation, dynamics, numericalmethods, experimentsNonlinear dynamics of rotating shallow water : methodsand advances, Zeitlin V., ed., Elsevier, 391pp, 2007.

Linear vs nonlinear stabilityHolm D.D., Marsden J.E., Ratiu T. and Weinstein, A.,Phys. Reports, v. 123, 1 - 116 (1985).

Stability in layered models

I Cairns R.A., J. Fluid. Mech., v. 92, 1 - 14 (1979).I Ripa P., J. Fluid Mech., v. 222, 119-137 (1990).


V. Zeitlin














Literature

Literature : RSW applications

Instabilities of barotropic and baroclinic jets, dry andmoistLambaerts J. ; Lapeyre G. and Zeitlin V., J. Atmos. Sci.,v.68, 1234-1252 (2011) ; v.69, 1405-1426 (2012).

Instabilities of coastal currentsI Reduced gravity : Gula J. and Zeitlin V., J. Fluid

Mech., v. 659, 69 - 93 (2010).I 2-layers : Gula J., Zeitlin V. and Bouchut F., J. Fluid

Mech., v. 665, 209 - 237 (2010).

Inertial instabilityBouchut F., Ribstein B. and Zeitlin V., Phys. Fluids v. 23,126601 (2011).

Documents

Understanding large-scale instabilities of …gershwin.ens.fr/.../beamer-lectures_cambridge_2012.pdflayered models V. Zeitlin Unstable large-scale ﬂows Modeling large-scale processes