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Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Wave resonances in coastal dynamics
V. Zeitlin
Laboratoire de Météorologie Dynamique, ENS/P6
"Coastal modelling", Toulon 2011
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
PlanRotating Shallow Water model
Derivation : vertical averaging of primitive equationsGeneral properties
Linear coastal waves in RSWKelvin waves and inertia-gravity wavesShelf waves
Wave interactions at the coastResonant excitation of waveguide modesLooking for IGW - KW resonancesAsymptotic expansions and removal of resonancesComments
Wave-resonances and instabilities of coastal currentsPassive lower layer
Linear stabilityNonlinear saturationInitial-value problem
Active lower layerLinear stabilityNonlinear saturationRole of the vertical shear (KH) instability
SummaryConclusions
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Introductory remarks
The RSW adRotating Shallow Water (RSW) model(s) - standardconceptual model in GFD. Introduced and used byclassics (Jeffreys, 1920s ; Obukhov, 1940s, Gill, 1970s),and massively used by GFD practitioners.Has all essential GFD ingredients : (differential) rotation,stratification, topography. Describes waves and vortices.Conserves potential vorticity (PV). Allows for barocliniceffects via superposition of layers.
The RSW essenceHydrostatics plus vertical averaging of "primitive"equations.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Primitive equations
Starting point : hydrostatic primitive equations
∂tvh + v · ∇vh + f z ∧ vh +∇Φ = 0,
∂zΦ +ρ
ρ0g = 0,
∂tρ+ v · ∇ρ = 0,∇ · v = 0, (1)
ρ - density (ocean), or minus potential temperature(atmosphere), Φ - geopotential, "h" denotes horizontalpart, v = (vh,w) = (u, v ,w), ∇ = (∇h, ∂z) = (∂x , ∂y , ∂z),z - height (ocean) or pseudo-height (atmosphere).Coriolis parameter f : f0 = 2Ω sinφ = const , on the f -plane, f = f0 + βy , on the β - plane.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
g f/2z
x
z2
z1w1= dz1/dt
w2= dz2/dt
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Vertical averaging and RSW modelsI Take horizontal momentum equation in conservative
form :
(ρu)t + (ρu2)x + (ρvu)y + (ρwu)z − fρv = −px , (2)
and integrate between a pair of material surfacesz1,2 :
w |zi=
dzi
dt= ∂tzi + u∂xzi + v∂yzi , i = 1,2. (3)
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
. (4)
(analogously for v ).
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
I Use continuity equation and get
∂t
∫ z2
z1
dzρ+ ∂x
∫ z2
z1
dzρu + ∂y
∫ z2
z1
dzρv = 0. (5)
I Introduce the mass- (entropy)- averages :
〈F 〉 =1µ
∫ z2
z1
dzρF , µ =
∫ z2
z1
dzρ. (6)
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
, (7)
∂t (µ〈v〉) + ∂x (µ〈uv〉) + ∂y
(µ〈v2〉
)+ fµ〈u〉
= −∂y
∫ z2
z1
dzp − ∂yz1 p|z1+ ∂yz2 p|z2
, (8)
∂tµ+ ∂x (µ〈u〉) + ∂y (µ〈v〉) = 0. (9)
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
I Use hydrostatics and get, introducing mean constantdensity ρ :
p(x , y , z, t) ≈ −gρ(z − z1) + p|z1. (10)
I Use the mean-field approximation :
〈uv〉 ≈ 〈u〉〈v〉, 〈u2〉 ≈ 〈u〉〈u〉, 〈v2〉 ≈ 〈v〉〈v〉. (11)
and get RSW equations for a 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. (12)
I Use as many layers as you wish, with lowermostboundary fixed by topography.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Examples1-layer RSW
∂tv + v · ∇v + f z ∧ v + g∇h = 0 , (13)
∂th +∇ · (vh) = 0 , (14)
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.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
2-layer RSW, rigid lid
∂tvi + vi · ∇vi + f z ∧ vi +1ρi∇πi = 0 , i = 1,2; (15)
∂th +∇ · (v1h) = 0 , (16)
∂t (H − h) +∇ · (v2(H − h)) = 0 , (17)
π1 = (ρ1 − ρ2)gh + π2 . (18)
g f/2
z
x
h
H
p2
p1
v2
v1 rho1
rho2
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Conservation laws - 1 layer
Equations in conservative form :
(hu)t + (hu2 +12
gh2)x + (huv)y − fhv = 0,
(hv)t + (huv)x + (hv2 +12
gh2)y + fhu = 0,
ht + (hu)x + (hv)y = 0. (19)
Remark :Coriolis force : stiff source.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Conserved quantities :
I Mass ∫dxdy h = const, (20)
I Energy ∫dxdy h
v2
2+ g
h2
2= const. (21)
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Potential vorticity (PV)
PV :
q =ζ + f
h, (22)
where ζ = vx − uy - relative vorticity, and ζ + f - absolutevorticity.Lagrangian conservation :
dqdt≡ (∂t + v · ∇) q = 0, (23)
follows by combining the equation for vorticity :
(∂t + v · ∇) (ζ + f ) + (ζ + f )∇ · v = 0, (24)
and the continuity equation.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Setup
H
g f/2z
x
shelf
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Non-dimensional RSW equations linearized in ahalf-plane (infinitely abrupt shelf) :
ut − v + ηx = 0,vt + u + ηy = 0,ηt + ux + vy = 0 (25)
Rectilinear meridional boundary : b. c. : u|x=0 = 0.Inhomogeneity in x ⇒ Fourier-transform in y , t :
(u, v , η) = (u0, v0, h0)ei(ly−ωt) ⇒
−iωu0 − v0 + h′0 = 0,−iωv0 + u0 + il h0 = 0,−iωh0 + il v0 + u′0 = 0, ⇒ (26)
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Reduction to a single equation (ω 6= 1)
h′′0 + (ω2 − 1− l2)h0 = 0, (27)
u0 = il h0 − ωh′0ω2 − 1
, ⇒ c.l. : l h0 − ωh′0∣∣x=0 = 0. (28)
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
SolutionsI Free inertia-gravity waves
ω2 − 1− l2 ≡ k2 > 0, (29)
h0 ∝ e±ikx , ω2 = 1 + k2 + l2. (30)
I Trapped Kelvin waves
ω2 − 1− l2 ≡ −κ2 < 0, (31)
h0 ∝ e−κx . (32)
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Reflexion of inertia-gravity waves
Incident wave plus reflected wave :
(u, v , η) = (ui , vi , ηi) + (ur , vr , ηr )
(ui , vi , ηi) = Ai
(kω + ilω2 − 1
,lω − ikω2 − 1
,1)
ei(kx+ly−ωt) + c.c.,
(ur , vr , ηr ) = Ar
(−kω + ilω2 − 1
,lω + ikω2 − 1
,1)
ei(−kx+ly−ωt) + c.c..
B.C. :
ui + ur |x=0 = 0, ⇒ Ar = Aikω + ilkω − il
, ω2 = 1+k2+l2. (33)
⇒ Snellius law.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Propagation of Kelvin waves
c.l. : l h0 − ωh′0∣∣x=0 = 0 ⇒ κ = − l
ω,
⇒ ω2 − 1− l2 +l2
ω2 = 0, ⇒ ω2 = l2 (ω 6= 1). (34)
κ > 0⇒ ω = −l , η ∝ e−x . (35)
Kelvin wave packet :
(u, v , η) = (0,K (y+t),−K (y+t))e−x , K−arbitrary function.(36)
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Dispersion relation of RSW with coast ( f -plane)
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Propagation of the Kelvin wave packet andformation of Kelvin fronts
t=0.0
y
−15 −10 −5 0 5 10 150
1
2
3
4
t=1.5
−15 −10 −5 0 5 10 150
1
2
3
4
t=3.0
−15 −10 −5 0 5 10 150
1
2
3
4
t=9.0
x
y
−15 −10 −5 0 5 10 150
1
2
3
4
t=21.0
x−15 −10 −5 0 5 10 150
1
2
3
4
t=30.0
x−15 −10 −5 0 5 10 150
1
2
3
4
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Abrupt shelf
Linearized equations in the presence of topography :
ut − v + ηx = 0,vt + u + ηy = 0,
ηt + (Hu)x + (Hv)y = 0 (37)
H - fluid depth. Abrupt shelf : typical scaleL << Rd ↔ L
Rd= ε.
Non-dimensional H :
H = H(xε
), H|x=0 = 0, H|x=∞ = 1
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Reduction to a single equationWave solution :
(u, v , η) = (u0, v0, h0)ei(ly−ωt) + c.c.
−iωu0 − v0 + h′0 = 0,−iωv0 + u0 + il h0 = 0,
−iωh0 + ilHv0 + (Hu0)′ = 0, ⇒ (38)
(Hh′0
)′+ (ω2 − 1− l2H − l
ωH ′)h0 = 0. (39)
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Asymptotic analysis
I Open ocean domain :
h′′0 + (ω2 − 1− l2)h0 = 0. (40)
Solution - trapped wave : h(h)0 = Ae−κx , κ > 0
κ2 = l2 + 1− ω2. (41)
Suppose : κ = κ0 + εκ1 + ..., ω = ω0 + εω1 + ....I Coastal domain :
1ε2
(H(ξ)h(c)
0 (ξ)′)′
+
(ω2 − 1− l2H(ξ)− 1
ε
lω
H ′(ξ)
)h(c)
0 = 0.
(42)h(c)
0 (ξ) = η(0)(ξ) + εη(1)(ξ) + ..., ξ =xε
(43)
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Hierarchy of equations for η(0)(n), n = 0,1, ... :
(H(ξ)η(0)(ξ)′
)′= 0,(
H(ξ)η(1)(ξ)′)′ − l
ω0H ′(ξ))η(0)(ξ) = 0,
.................................... (44)
Order zero
H(ξ)η(0)(ξ)′ = C = const. (45)
C 6= 0,⇒ singularity at x = 0, ⇒ η(0) = const.Matching with the domain (h) à x = εξ :
h(h)0 = A
(1− κ0εξ +
12κ2
0(εξ)2 − ε2κ1ξ + ....
), ⇒ (46)
η(0) = A.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Order 1(H(ξ)η(1)(ξ)′
)′ − lω0
H ′(ξ))A = C1 = const. (47)
Solution regulière pour u0, v0 C1 = 0⇒
η(1) =lω0
Aξ + const. (48)
Matching of η(0) + εη(1) with h(h)0 at x = εξ
⇒ lω0
= −κ0, const = 0.As κ2 = l2 + 1− ω2, ω2 6= 1 ⇒ κ0 = 1.⇒ Kelvin wave. Next corrections - correction to thedispersion relation.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
"Gentle" shell. Ball model
Non-dimensional profile of H :
H(x) = (1− e−ax ). (49)
Wave equation with ω, l for the perturbation of the freesurface h0(x) :
(Hh0
)′+ (ω2 − 1− l2H − l
ωH ′)h0 = 0. (50)
New variable s = e−ax - hypergeometric equation :
s[s(1− s)h0(s)′
]′+
[ω2 − f − l2 + (l2 − f
lω
s)
]h0(s) = 0
(51)with bars meaning renormalisation by a.B.C. : s = 1(x = 0) - regularity ; s = 0(x =∞) - h0 = 0(trapped waves).
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Dispersion relation for coastal waves in theBall model (n - number of zeros of thestructure function in x
- 1 0 - 8 - 6 - 4 - 2 0 2 4 6 8 1 0
1
2
3
4
5
6
SIGM
A
n = 0
n = 0
n = 1
n = 1
n = 1
f = 1
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
General properties of the coastal waves
I Unique Kelvin wave,I Discrete spectrum of trapped sub-inertial waves withω < f (shelf waves) with unique sens of propagation(left, looking at the coast)
I Discrete spectrum of trapped supra-inertial waveswith ω > f (edge waves) with double sens ofpropagation
I Continuous spectrum of inertia-gravity (Poincaré)incident/reflected waves
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Excitation of trapped waves by free waves : theideology
I Interactions between free waves and trapped waves(waveguide modes) may be resonant.
I If so the incoming free waves may resonantly excitewaveguide modes.
I The resonant growth should be nonlinearly ordissipatively saturated in one way or another leadingto coherent structures formation.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Non-dimensional RSW equations with idealizedcoast
ut − v + hx = −ε(uux + vuy )
vt + u + hy = −ε(uvx + vvy )
ht + ux + vy = −ε ((hu)x + (hv)y ) . (52)
Boundary condition : x ≥ 0, u|x=0 = 0.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
IGW - KW resonance
k2, l2
k1,l1
l
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Conditions of IGW - KW resonance
A pair of IGW with frequencies σ1,2 and along-coastwavenumbers l1,2 is in resonance with a KW withwavenumber l if
σ1 − σ2 = −l , l1 − l2 = l , l 6= 0. (53)
We choose l < 0 :
|l | =√
1 + k21 + l21 −
√1 + k2
2 + l22 , l2 = l1 + |l | , (54)
and √1 + k2
1 + l21 − |l | =√
1 + k22 + (l1 + |l |)2. (55)
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Algorithm for finding resonances :
1. Take any l and l1, then l2 = l1 + |l |,2. Take arbitrary k1 satisfying k2
1 ≥ 2|l |(√
1 + l22 + l2
),
3. Define k2 from
k22 = k2
1 − 2|l |(√
1 + k21 + l21 + l1
).
Therefore, a KW with wavenumber l may be resonantlyexcited by a continuum of incident IGW withwavenumbers l1 and
|k1| >√
2|l |(√
1 + (l1 + |l |)2 + l1 + |l |)
interacting with
another incident wave with k2, l2 :
k22 = k2
1 − 2|l |(√
1 + k21 + l21 + l1
), l2 = l1 + |l |. (56)
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Condition of absence of resonances in RSW equationswith coast : ∫ ∞
0dx e−x (Rh − Rv ) = 0, (57)
where Rh,v - r.h.s. of h- and v - equations.⇒Evolution equation for the amplitude of the Kelvinwave
KT + KKη = Seilη + S∗e−ilη, (58)
where η = y + t ,
S =
∫ ∞0
dx e−x [(H1U∗2 + U1H∗2)x − U1V ∗2x− V1x U2
+ il(H1V ∗2 + V1H∗2 − V1V ∗2 )] (59)
and (Ui ,Vi ,Hi), i = 1,2 are amplitudes of two IGW.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Final form of the evolution equationFrom the polarisation relations get :
S = iA1A2s, Im(s) = 0 (60)
and hence
KT + KKη = −2sA1A2 sin lη, (61)
where Ai - amplitudes of the two IGW,
s =4l
(k21 + 1)(k2
2 + 1)[1 + (k1 + k2)2][1 + (k1 − k2)2]·[
(σ1l2 + σ2l1 − l1l2)(1 + k21 + k2
2 )
+σ2l1k1(1 + k21 − k2
2 ) + σ1l2k2(1 + k22 − k2
1 )
+2k1k2(l1l2 − (1 + k21 )(1 + k2
2 ))]
(62)
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Isopleths of the interaction coefficients(l , k1, l1) for l = −1 at the interval 10
-10 -8 -6 -4 -2 0 2 4 6 8 10-10
-8
-6
-4
-2
0
2
4
6
8
10
1l
1k
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Integrability of the KW evolution equation
Forced simple-wave equation (after renormalizations) :
Kτ + KKχ = − sinχ. (63)
Lagrangian (characteristics) approach :
K = U = X ; ˙(...) = ∂τ + U∂χ(...)⇒ (64)
X + sinX = 0 (65)
Pendulum equation : integrable. Shock formation↔Lagrangian clustering (known in statistical physics :mean-field limit of the kinetics of particles with repulsivelong-range interaction on the circle)⇒ Implications fortransport and mixing.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Lagrangian trajectories
Two Lagrangian trajectories with different initialconditions. Intersection = clustering = shock.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Ramifications :I Including small dissipation⇒ harmonically forced
Burgers equation. Cole-Hopf change of variables→Mathieu equation for Laplace transform in time→integrable.
I Including small dispersion (long waves near a steep,but not vertical border)⇒ harmonically forced KdVequation.
SummaryResonant excitation of Kelvin waves by pairs of inertia -gravity waves near the coast is possible for a continuumof IGW - should be ubiquitous. The mechanism generatesKW "from nothing". Subsequent slow evolution of KWleads to nontrivial transport and mixing properties.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Motivation
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Approach
I Multi-layer shallow-water models ;I Exhaustive linear stability analysis by the collocation
method ;I High-resolution numerical simulations of nonlinear
evolution with new-generation finite-volume code.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Workflow
I Choose a model : 1.5 or 2 -layer (or more !) ;I Choose bathymetry ;I Choose balanced profiles of velocity/interface ;I Analyse linear stability : unstable modes, growth
rates ;I Initialise nonlinear simulations with the unstable
modes, study saturation ;I Look how instabilities manifest themselves in
initial-value problem.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Typical configuration
y = 0
y = −L
ρ1
ρ2
f2
y
H1(y) U1(y)
H2(y) U2(y)
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
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. (66)
Boundary conditions :
H(y) + h(x , y , t) = 0, DtY0 = v at y = Y0 , (67)
where Y0(x , t) is the position of the free streamline, Dt isLagrangian derivative.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Balanced flows :u = U(y), v = 0, and h = H(y),
U(y) = −gf
Hy (y) (68)
exact stationary solution.
−1 00
0.25
0.5
0.75
H1(y)
y−1 0
−0.5
0
0.5
U1(y)
y
FIG.: Examples of the basic state heights (left) and velocities(right) for constant PV flows with U0 = −sinh(−1)/cosh(−1)(thick line), U0 = 1/2 (dotted) and a zero PV flow (dash-dotted)
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Non-dimensional linearized system :
ut + Uux + vUy − v = − hx ,vt + Uvx + u = − hy ,
ht + Uhx = −(Hux + (Hv)y ).(69)
Linearized boundary conditions :
I
Y0 = − hHy
∣∣∣∣y=0
, (70)
I continuity equation evaluated at y = 0.
The only constraint is regularity of solutions at y = 0.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
PV of the mean flow
Q(y) =1− Uy
H(y), (71)
Geostrophic equilibrium⇒
Hyy (y)−Q(y)H(y) + 1 = 0, with
H(0) = 0Hy (0) = −U0,
(72)
U(0) = U0 is the mean-flow velocity at the front.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
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
FIG.: Stability diagram in the ( U0fL , k) plane for the constant PV
current. Values of the growth rates in the right column.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Dispersion diagram : stable flow
0 1 2 3 4 5 6 7 8 9 10
0
1
c
k
K
F
Pn
Pn
FIG.: Dispersion diagram for U0 = −sinh(−1)/cosh(−1) andQ0 = 1.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
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
FIG.: Dispersion diagram for U0 = 0.5 and Q0 = 1. Crossingsof the dispersion curves in the upper panel correspond toinstability zones in the lower panel.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
The most unstable mode : Kelvin-Frontalresonance
y
x−1
0
FIG.: Height and velocity fields of the most unstable modek = 3.5.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Saturation of the primary 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
FIG.: Height and velocity fields of the perturbation at t = 0 (left)and t = 30 (right). Kelvin front is clearly seen at the bottom ofthe right panel.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
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
FIG.: Evolution of the tangent velocity at y = −L (at the wall)for t = 0,2.5,5,7.5,10,12.5,15,17.5,20,22.5 (from lower toupper curves)
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Secondary instabilityy
x0.2 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
y
x0.2 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
FIG.: Height and velocity fields of the secondary perturbationat t = 335, t = 500 (right).
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
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
FIG.: 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).
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Stability diagram of the reorganized flow
FIG.: Dispersion diagram of the eigenmodes corresponding tothe basic state profile of the flow at t = 335, at the beginning ofthe secondary instability stage (see Fig. 9).
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Most unstable mode of the reorganized flow
y
x−1
0
FIG.: Height and velocity fields of the most unstable mode offigure 10 for k = k0. Only one wavelenght is plotted. Note thesimilarity with the mode observed in the simulation, Fig. 8
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Instability in Cauchy problem
c
y
−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−1
−0.5
0
−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02
0.025
0.03
c
y
−0.1 0 0.1 0.2 0.3−1
−0.5
0
−0.05
−0.04
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
0.04
0.05
FIG.: y − c diagram at t = 45 of the development of initiallylocalised perturbation (dotted) for linearly stable (upper) andunstable (lower) current
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Equations of motion
Djuj − fvj = − 1ρj∂xπj ,
Djvj + fuj = − 1ρj∂yπj ,
Djhj +∇ · (hjvj) = 0,(73)
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. (74)
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Stationary solutionsBalanced flow with depths Hj(y) and velocities Uj(y) :
∂yHj = (−1)j−1 fg′
(U2 − sj−1U1), (75)
Linearization/nondimensionalization :
∂tuj + Uj∂xuj + vj∂yUj − vj = −∂x (sj−1h1 + h2),∂tvj + Uj∂xvj + uj = −∂y (sj−1h1 + h2),
∂thj + Uj∂xhj + Hj∂xuj = −∂y (Hjvj).(76)
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Boundary conditions
I Upper layer : same as in 1.5-layer case,I Lower layer : for harmonic perturbations
(uj(x , y), vj(x , y),hj(x , y)) = (uj(y), vj(y), hj(y)) ei(kx−ωt),(77)
decay condition :
∂y (sh1 + h2) = −k(sh1 + h2)aty = 0
.
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.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Configurations considered :
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.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Barotropically stable case
FIG.: Dispersion diagrams for s = .5. (a) r = 10, (b) r = 2, (c)r = 0.5. Horizontal scale of the bottom panel shrinked to showshort-wave KH instabilities.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Barotropically unstable case
FIG.: Dispersion diagrams for s = 0.5 and for Rd = 1. (a)r = 10, (b) r = 5, (c) r = 2 . The horizontal scale of the panelsshrinked to show short-wave KH instabilities.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
maxP2 / maxP
1 = 0.017044 maxP
2 / maxP
1 = 0.047946
maxP2 / maxP
1 = 0.015975 maxP
2 / maxP
1 = 0.35601
FIG.: Typical unstable modes(left to right, top to bottom) : KF1,RF, RP, PF
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Synthetic stability diagram
FIG.: Growth rates (left) and wavenumbers of most unstablemodes (right) at s = 0.5
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
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.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
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
FIG.: 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 for thedevelopment of the unstable RF mode superposed on thebasic flow with a depth ratio r = 2.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Energetics
0 50 100 150 200 250 300
−8
−7
−6
−5
−4
−3
−2
−1
0
t
log
(Kp
er)
FIG.: Logarithm of the kinetic energy Kper of the perturbationfor the unstable mode in the upper layer (thick) and in the lowerlayer (dashed).
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
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
FIG.: Before detachment : zoom of the wall region.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Structure of the detached vortex 1
y
x
t= 225
2 3 4 5 6 7
0
1
2
3
4
FIG.: 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 forsimulation of figure 22. Dark (light) background : anticyclonic(cyclonic) region.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Structure of the detached vortex 2
3 4 5 6 7 8 90
0.5
1
1.5
2
2.5
h
x0 1 2 3 4 5 6
0
0.5
1
1.5
2
2.5
h
y
FIG.: The x (left) and y (right) cross-sections of the detachedvortex at t = 300
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Evolution of the total energy
0 50 100 150 200 250 3000.9996
0.9997
0.9998
0.9999
1
t
E/E0
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
y
x
t= 8
0 1 2 3 4−1
0
y
x
t= 8
0 1 2 3 4−1
0
y
x
t= 60
0 1 2 3 4−1
0
1
2
y
x
t= 60
0 1 2 3 4−1
0
1
2
FIG.: Levels of h1(x , y , t) in the upper layer (left) and isobars ofπ2(x , y , t) in the lower layer (right) at t = 20 and 60 for thedevelopment of the unstable RF mode superposed on thebasic flow with a depth ratio r = 0.5.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Energetics
0 20 40 60 80 100 120−10
−9
−8
−7
−6
−5
−4
−3
−2
−1
0
t
log
(Kp
er)
0 20 40 60 80 100 120
0.998
0.9985
0.999
0.9995
1
t
E/E0
FIG.: Left -logarithm of the kinetic energy of the perturbation forthe simulation of figure for mode k = k0 in the upper layer(solid) and in the lower layer (dashed), and for the sum ofmodes with k > 10 k0 (dashed-dotted ). Right -time-dependence of the total energy (thick line) and thedissipation rate (dashed line) for the evolution of the instability
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Loss of hyperbolicity
y
0.5 1 1.5 2 2.5 3 3.5 4−1
0
−6
−4
−2
0
2
4x 10
−3
x
y
0 1 2 3 4−1
0
−5
0
5
x 10−3
FIG.: Contours of π1(x , y , t) (upper panel) and π2(x , y , t) (lowerpanel) with mean zonal flow filtered out at t = 20 for thesimulation of figure ?? with a depth ratio r = 0.5. The whitelines indicate the boundaries of non-hyperbolic domains.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
I Physical nature of all instabilities as resonancesbetween various eigenmodes established,
I Nonlinear evolution of leading instabilities simulatedwith new high-resolution finite-volume code,
I An essential role of Kelvin fronts (breaking Kelvinwaves) in reorganization of the flow and coherentstructure formation highlighted,
I A mechanism of vortex detachment from theunstable baroclinic coastal current is identified,
I (Non-) Influence of short-scale shear instabilitiesunderstood.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Resonant interactions of waves⇒I Over the rest state : generation→ amplification→
breaking of coastal trapped waves⇒ Implications formixing in coastal zones
I Over the coastal current : destabilization of thecurrent, formation of secondary vortices⇒Implications for transport and mixing in coastalzones.
Waves - coast
V. Zeitlin
Rotating ShallowWater modelDerivation : verticalaveraging of primitiveequations
General properties
Linear coastalwaves in RSWKelvin waves andinertia-gravity waves
Shelf waves
Wave interactionsat the coastResonant excitation ofwaveguide modes
Looking for IGW - KWresonances
Asymptotic expansions andremoval of resonances
Comments
Wave-resonancesand instabilities ofcoastal currentsPassive lower layer
Linear stability
Nonlinear saturation
Initial-value problem
Active lower layer
Linear stability
Nonlinear saturation
Role of the vertical shear(KH) instability
Summary
Conclusions
Literature
Resonant excitation of trapped waves :
I Idealized shelf : G. Reznik and V. Zeitlin, Phys Lett.A., v. 373, 1019 -1021 (2009).
I Arbitrary shelf : G. Reznik and V. Zeitlin, J. FluidMech., v. 673, 349 - 394 (2011).
Instabilities of coastal currentsI Reduced gravity : J. Gula and V. Zeitlin, J. Fluid
Mech., v. 659, 69 - 93 (2010).I 2-layers : J. Gula, V. Zeitlin and F. Bouchut, J. Fluid
Mech., v. 665, 209 - 237 (2010).