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
Literature
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
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
Literature
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
Literature
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
Literature
Midlatitude atmospheric jet
Midlatidude upper-tropospheric jet (left) and relatedsynoptic systems (right).
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
Literature
Oceanic currents : Gulfstream
Gulfstream (left) and related vortices (right). Velocityfollows isopleths of height anomaly in the firstapproximation.
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
Literature
Leeuwin current and associated vortices
Velocity (arrows) and temperature anomaly (colors) of theLeeuwin curent near Australian coast.
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
Literature
Coastal flows : Weddell sea
Instability of a coastal current in the Weddell sea.
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
Literature
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)
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
Literature
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)
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
Literature
Material surfaces
g f/2z
x
z2
z1w1= dz1/dt
w2= dz2/dt
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
Literature
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 ).
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
Literature
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)
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
Literature
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.
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
Literature
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.
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
Literature
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
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
Literature
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
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
Literature
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.
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
Literature
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
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
Literature
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)
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
Literature
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.
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
Literature
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)
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
Literature
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.
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
Literature
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.
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
Literature
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
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
Literature
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 ;
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
Literature
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
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
Literature
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
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
Literature
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
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
Literature
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
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
Literature
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
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
Literature
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.
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
Literature
−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).
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
Literature
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
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
Literature
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.
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
Literature
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.
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
Literature
Idealized coastal current configuration
y = 0
y = −L
ρ1
ρ2
f2
y
H1(y) U1(y)
H2(y) U2(y)
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
Literature
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.
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
Literature
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)
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
Literature
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.
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
Literature
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.
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
Literature
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.
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
Literature
The most unstable mode : Kelvin-Frontalresonance
y
x−1
0
Height and velocity fields of the most unstable modek = 3.5.
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
Literature
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.
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
Literature
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)
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
Literature
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).
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
Literature
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)
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
Literature
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.
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
Literature
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.
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
Literature
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.
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
Literature
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.
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
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.
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
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.
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
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.
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
Literature
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.
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
Literature
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.
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
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.
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
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)
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
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.
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
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
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
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.
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
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
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
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.
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
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.
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
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.
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
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).
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
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).
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
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.
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
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.
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
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,
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
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 →
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
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,
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
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.
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
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
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
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).
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
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.
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
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.
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
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.
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
Literature
Moist baroclinic instability in Nature
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
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).
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
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).