wave resonances in coastal dynamics - directory listing of...

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


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


General properties


Shelf waves




Comments


Linear stability



Active lower layer

Linear stability



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


General properties


Shelf waves




Comments


Linear stability



Active lower layer

Linear stability



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

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General properties


Shelf waves




Comments


Linear stability



Active lower layer

Linear stability



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


General properties


Shelf waves




Comments


Linear stability



Active lower layer

Linear stability



Summary

Conclusions

g f/2z

x

z2

z1w1= dz1/dt

w2= dz2/dt

Waves - coast

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General properties


Shelf waves




Comments


Linear stability



Active lower layer

Linear stability



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 ).

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Linear stability



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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)

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Linear stability



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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.

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Active lower layer

Linear stability



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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.

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General properties


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Comments


Linear stability



Active lower layer

Linear stability



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

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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.

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Conserved quantities :

I Mass ∫dxdy h = const, (20)

I Energy ∫dxdy h

v2

2+ g

h2

2= const. (21)

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Linear stability



Active lower layer

Linear stability



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


General properties


Shelf waves




Comments


Linear stability



Active lower layer

Linear stability



Summary

Conclusions

Setup

H

g f/2z

x

shelf

Waves - coast

V. Zeitlin


General properties


Shelf waves




Comments


Linear stability



Active lower layer

Linear stability



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

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General properties


Shelf waves




Comments


Linear stability



Active lower layer

Linear stability



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

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General properties


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Linear stability



Active lower layer

Linear stability



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

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Linear stability



Active lower layer

Linear stability



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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.

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Comments


Linear stability



Active lower layer

Linear stability



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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


General properties


Shelf waves




Comments


Linear stability



Active lower layer

Linear stability



Summary

Conclusions

Dispersion relation of RSW with coast ( f -plane)

Waves - coast

V. Zeitlin


General properties


Shelf waves




Comments


Linear stability



Active lower layer

Linear stability



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


General properties


Shelf waves




Comments


Linear stability



Active lower layer

Linear stability



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


General properties


Shelf waves




Comments


Linear stability



Active lower layer

Linear stability



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


General properties


Shelf waves




Comments


Linear stability



Active lower layer

Linear stability



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)

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Linear stability



Active lower layer

Linear stability



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.

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Linear stability



Active lower layer

Linear stability



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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.

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Linear stability



Active lower layer

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"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

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General properties


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Linear stability



Active lower layer

Linear stability



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

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Linear stability



Active lower layer

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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

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Linear stability



Active lower layer

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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.

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Linear stability



Active lower layer

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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.

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IGW - KW resonance

k2, l2

k1,l1

l

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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)

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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)

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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.

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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)

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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

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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.

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Lagrangian trajectories

Two Lagrangian trajectories with different initialconditions. Intersection = clustering = shock.

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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.

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Motivation

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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.

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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.

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Typical configuration

y = 0

y = −L

ρ1

ρ2

f2

y

H1(y) U1(y)

H2(y) U2(y)

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RSW equations with coast (no bathymetry)

Equations of motion :

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

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

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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)

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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.

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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.

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Wave Number

U0

0 1 2 3 4 5 6 7 8 9 10

0.5

0.55

0.6

0.65

0.7

0.75

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

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

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

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Dispersion diagram : stable flow

0 1 2 3 4 5 6 7 8 9 10

0

1

c

k

K

F

Pn

Pn

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

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Dispersion diagram : unstable flow

0

0.5

1

c K

F

Pn

Pn

2 4 6 8 100

0.01

0.02

0.03

0.04

0.05

0.06

0.07

k

σ

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.

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The most unstable mode : Kelvin-Frontalresonance

y

x−1

0

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

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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.

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Kelvin wave breaking

1 2 3 4 5 6 7−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1t= 22.5026

u

x

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)

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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).

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Reorganization of the mean flow

−1 −0.5 0 0.50

0.05

0.1

0.15

0.2

0.25

0.3

Hzonal

y

−1 −0.5 0 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

y

uzonal

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).

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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).

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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

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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

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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)

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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)

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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.

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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.

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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.

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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.

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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

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Synthetic stability diagram

FIG.: Growth rates (left) and wavenumbers of most unstablemodes (right) at s = 0.5

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Scenario of development of the baroclinic RFinstability as follows from DNS

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

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

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

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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.

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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).

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Kelvin front and dissipation duringdevelopment of RF instability

y

x

t= 155

3 3.5 4 4.5 5 5.5 6−1

−0.5

FIG.: Before detachment : zoom of the wall region.

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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.

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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

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Evolution of the total energy

0 50 100 150 200 250 3000.9996

0.9997

0.9998

0.9999

1

t

E/E0

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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.

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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

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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.

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General properties


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Linear stability



Active lower layer

Linear stability



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.

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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.

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Linear stability



Active lower layer

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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).

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