Waves and related processes in Geophysical Fluid Dynamics · Geophysical Fluid Dynamics V. Zeitlin...

Lecture 1: GFDmodels andwave-vortex

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Large-scaleatmospheric andoceanic waves

Fluid dynamics onthe rotating sphereand on the tangentplane

Primitive equationson the tangentplaneOcean

Atmosphere

Getting rid ofvertical structure

Getting rid of fastmotionsSlow motions. Geostrophicequilibrium.

QG dynamics in RSW

QG in 2-layer RSW

QG in Primitive Equations

Waves vs vorticesPrimitive equations

Shallow-water models

QG models

Résumé

Waves and related processes inGeophysical Fluid Dynamics

V. Zeitlin

Laboratoire de Météorologie Dynamique, Sorbonne University and ÉcoleNormale Supérieure, Paris

Waves in Flows, Prague, August 2018


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QG dynamics in RSW

QG in 2-layer RSW




QG models

Résumé

PlanLarge-scale atmospheric and oceanic wavesFluid dynamics on the rotating sphere and on the tangentplanePrimitive equations on the tangent plane

OceanAtmosphere

Getting rid of vertical structureGetting rid of fast motions

Slow motions. Geostrophic equilibrium.QG dynamics in RSWQG in 2-layer RSWQG in Primitive Equations

Waves vs vorticesPrimitive equationsShallow-water modelsQG models

Résumé


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QG dynamics in RSW

QG in 2-layer RSW




QG models

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Atmospheric data : streamlines of the flowand velocity (colour) at the 200 mb level (left),and vorticity (colour)at the 500 mb level(right) of the atmosphere in the Northernhemisphere


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Internal waves in the atmosphere (left) and inthe ocean (right), as seen from satellite.

Coast line and an island give an idea of spatial scale.


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QG in 2-layer RSW




QG models

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GFD : space view


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GFD : what’s that ?

Hydrodynamics in all its complexity plus :

I Rotating frameI Thermal/stratification effectsI Spherical geometry (large- and meso-scales)I Fluid in the complex domains (coasts,

topography/bathymetry)I Multi-component, multi- phase fluids (water vapour,

salt, ice ...)

But !These additional effects often allow to simplify theanalysis


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Motion in the rotating frame

Euler equations in the rotating frame in the presenceof gravity :

∂~v∂t

+ ~v · ~∇~v + 2~Ω ∧ ~v = −~∇Pρ

+ ~g∗ (1)

Effective gravity :

~g∗ = ~g + m~Ω ∧(~Ω ∧~r

)(2)


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Spherical coordinates

r

r

1

r

r

1

r

r

1

r

r

1

r

r

1


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Euler and continuity equations

dvr

dt−

v2λ + v2

φ

r− 2Ω cosφvλ + g∗ = −1

ρ∂r P,

dvλdt

+vr vλ − vφvλ tanφ

r+ 2Ω (− sinφvφ + cosφvr )

= − 1ρr cosφ

∂λP,

dvφdt

+vr vφ + v2

λ tanφr

+ 2Ω sinφvλ = − 1ρr∂φP,

dρdt

+ ρ

[1r2∂(r2vr )

∂r+

1r cosφ

(∂(cosφvφ)

∂φ+∂vλ∂λ

)],

ddt

=∂

∂t+ vr∂r +

vφr∂φ +

vλr cosφ

∂φ

Traditional approx. : green + red→ out, r → R = constNon-traditional approx : green→ out.


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Tangent plane approximation

R

g

Ω

y

x

z

Θ

∂~v∂t

+ ~v · ~∇~v + f z ∧ ~v = −~∇Pρ

+ ~g

f - plane : f = const ; β - plane : f = f + βy ; f - Coriolisparameter : f = 2Ω sinφ


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Mean oceanic stratification


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Primitive equations : ocean

Hydrostaticsgρ+ ∂zP = 0, (3)

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

Incompressibility

~∇ · ~v = 0, ~v = ~vh + zw . (4)

Euler :∂~vh

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

φ = πρ0

- geopotential.Continuity :

∂tρ+ ~v · ~∇ρ = 0. (6)


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Mean atmospheric stratification


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Primitive equations : dry atmosphere,pseudo-height vertical coordinate

∂~vh

∂t+ ~v · ~∇~vh + f z ∧ ~vh = −~∇hφ, (7)

−gθ

θ0+∂φ

∂z= 0, (8)

∂θ

∂t+ ~v · ~∇θ = 0; ~∇ · ~v = 0. (9)

Identical to oceanic ones with σ → −θ, potentialtemperature, directly related to entropy. Verticalcoordinate : pseudo-height, P - pressure.

z = z0

(1−

(PPs

) Rcp

), (10)

R = cp − cv , Mayer relation for ideal gas.


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Material surfaces

g f/2z

x

z2

z1w1= dz1/dt

w2= dz2/dt


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Vertical averaging and RSW models

I Take horizontal momentum equation in conservativeform : and integrate between a pair of materialsurfaces z1,2,

I Use Leibniz formula and boundary conditions onmaterial surfaces to eliminate vertical velocity

I Introduce the vertical (mass-) averages : and getaveraged mass and horizontal momentum equations

I Use hydrostatics supposing mean constant meandensity

I Use the mean-field (= columnar motion)approximation and get shallow water momentumequation for the fluid layer.

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


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

∂tv + v · ∇v + f z ∧ v + g∇h = 0 , (11)

∂th +∇ · (vh) = 0 . (12)

⇒ 2d barotropic gas dynamics + Coriolis force.In the presence of nontrivial topography b(x , y) :h→ h − b in the second equation.

g f/2z

h

v

x

y

Columnar motion.


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

∂tvi + vi · ∇vi + f z ∧ vi +1ρi∇πi = 0 , i = 1,2; (13)

∂th +∇ · (v1h) = 0 , (14)

∂t (H − h) +∇ · (v2(H − h)) = 0 , (15)

π1 = (ρ1 − ρ2)gh + π2 . (16)

g f/2

z

x

h

H

p2

p1

v2

v1 rho1

rho2


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

∂tv2 + v2 · ∇v2 + f z ∧ v2 = −g∇(h1 + h2) (17)

∂tv1 + v1 · ∇v1 + f z ∧ v1 = −g∇(rh1 + h2), (18)

∂th1,2 +∇ ·(v1,2h1,2

)= 0 , (19)

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

the layers.

g f/2

z

x

h1

v2

v1 rho1

rho2h2


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Equations of horizontal motion

∂~vh

∂t+ ~v · ~∇~vh + f z ∧ ~vh = −~∇hΦ. (20)

f = f0(1 + βy), Φ = Φ0 + φ = g(H0 + h) (21)

h - geopotential (perturbation) height.

Scaling for eddy motions

I Velocity ~vh = (u, v), u, v ∼ U, w ∼W << UI Unique horizontal spatial scale L,I Vertical scale H << L,I Time-scale : turn-over time T ∼ L/U.


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Characteristic parametres

Intrinsic scale of the system : deformation (Rossby)radius :

Rd =

√gH0

f0(22)

I Rossby number : Ro = Uf0L - ratio of fast and slow

time-scales,

I Burger number : Bu =R2

dL2 ,

I Characteristic non-linearity : λ = ∆H/H0, where ∆His the typical value of h,

I Dimensionless gradient of f : β ∼ βL


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Geostrophic balance

Non-dimensional equations of horizontal motion

Ro (∂tvh + v · ∇vh) + (1 + β)z ∧ vh = −λBuRo∇hh , (23)

Geostrophic equilibriumBalance between the Coriolis force and the pressureforce→ geostrophic wind :

z ∧ vg = −∇h (24)

Conditions of realisation :I Ro → 0,I λ Bu ∼ Ro,I β → 0.


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QG regime in RSW

Non-dimensional RSW equations

Ro (∂tv + v · ∇v) + (1 + βy)z ∧ v = −λBuRo∇η , (25)

λ∂tη +∇ · (v(1 + λη)) = 0 . (26)

QG regime

λ ∼ Ro,⇒ Bu ∼ 1,⇒ L ∼ Rd , β ∼ Ro 1. (27)


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Asymptotic expansions

ε (∂tv + v · ∇v) + (1 + εy)z ∧ v = −∇η , (28)

ε∂tη +∇ · (v(1 + εη)) = 0 , ε ≡ Ro 1. (29)

v = v(0) + εv(1) + ε2v(2) + ... (30)

Order ε0

u(0) = −∂yη, v (0) = ∂xη ⇒ ∂xu(0) + ∂yv (0) = 0,(31)

d (0)

dt· · · = ∂t ...+ u(0)∂x ...+ v (0)∂y ... ≡ ∂t · · ·+ J (η, ...).

(32)J (A,B) ≡ ∂xA∂yB − ∂yA∂xB. (33)


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Order ε1

u(1) = −d (0)

dtv (0)−yu(0), v (1) =

d (0)

dtu(0)−yv (0),⇒ (34)

∂xu(1) + ∂yv (1) = −d (0)

dt~∇2η − v (0),⇒ (35)

d (0)

dt

(η − ~∇2η

)− ∂xη = 0↔ d (0)

dt

(η − ~∇2η − y

)= 0.

(36)Detailed writing with β = ˜beta/Ro restored, forconvenience.

∂tη − ~∇2∂tη − J (η, ~∇η)− β∂xη = 0. (37)


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Similar procedure and scaling→ equationsfor the pressures in the layers

d (0)idt

[∇2πi − (−1)iD−1

i η + y]

= 0 , i = 1,2. (38)

where

d (0)idt

(...) := ∂t (...) + J (πi , ...) , i = 1,2 , (39)

where Di = HiH , non-dimensional heights of the layers, and

π2 − π1 +N2

(π2 + π1) = η. (40)

N = 2ρ2−ρ1ρ2+ρ1

. Standard limit : weak stratification→ρ2 → ρ1 ⇒ η = π2 − π1


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Non-dimensional PE under slow-motionscaling

εddt

vh + (1 + βy)z ∧ vh = −~∇hπ. (41)

ddtσ + ρ′sw = 0, ∂zπ + σ = 0. (42)

~∇h · vh + λ∂zw = 0; (43)

where ddt = ∂t + vh · ∇h + λw∂z , λ - typical deviation of

the isopycnals σ = const Boundary conditions - rigidlid/flat bottom :

w |z=0,1 = 0. (44)

QG regime : ε ∼ λ ∼ β 1.


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QG equations

Asymptotic expansion in ε + elimination of w and σ →

d (0)

dt

(−∇2

hπ − y + ∂z

(1

ρ′s(z)∂zπ

))= 0, (45)

c.l. : w |z=0,1 = 0⇒ d (0)

dt∂zπ

∣∣∣∣∣z=0,1

= 0. (46)


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Linearising Primitive Equations on the f -planeabout the state of rest

ut − fv + φx = 0,vt + fu + φy = 0,φz + g

ρ0σ = 0,

σt + wρ′s = 0,ux + vy + wz = 0.

(47)

u, v , w are three components of velocity perturbation, φ -geopotential perturbation, σ - perturbation of the profile ofbackground density ρs. Successive elimination of σ andw :

ut − fv + φx = 0,vt + fu + φy = 0,ux + vy − N−2φzzt = 0.

(48)

If Brunt - Väisälä frequency N2 = −gρ′s

ρ0is constant, this is

a system of linear equations with constant coefficientswhich can be treated by the method of Fourier.


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Dispersion relation and spectral gapHarmonic waves :

(u, v , φ) = (u0, v0, φ0)ei(ωt−k·x) + c.c. (49)

Dispersion relation

ω

(ω2 −

(N2 k2

x + k2y

k2z

+ f 2

))= 0. (50)

Three roots : two different kinds of solutions :I Propagative inertia-gravity waves (IGW) with

dispersion relation :

ω = ±

√N2

k2x + k2

y

k2z

+ f 2, (51)

I Stationary solutions with ω = 0 ↔ linearisedconservation of Potential Vorticity, vortices.

Spectral gap : ω ≥ f for IGW.


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Linearising 1-layer RSW over the state of restSmall perturbations u, v , η about the state of rest withv = 0, h = H0 = const in the f -plane→

ut − fv + gηx = 0,vt + fu + gηy = 0,ηt + H0(ux + vy ) = 0,

(52)

Dispersion relation :

ω(ω2 − gH0k2 − f 2

)= 0. (53)

Negativevalues ω < 0 are not shown. Solution ω = 0 is displayed

in order to illustrate the spectral gap.


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Artist’s view of shallow vortex dynamics :vortices, waves, and topography (to appear inthe next lecture)in shallow water


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Linearising 2-layer RSW over the state of restηi , i = 1,2 - perturbations of free surface and interface :

∂t~vi + f z ∧ ~vi + g~∇(r i−1η1 + η2

)= 0,

∂tηi + Hi ~∇ · ~vi = 0.(54)

Simplest case : H1 = H2 = H2 . Barotropic-baroclinic

decomposition ~v± =√

r~v1 ± ~v2, η± = 2(√

rη1 ± η2).

Dispersion relation (zero roots - out) :

ω2± = c2

±k2 + f 2, c± =

√gH

1±√

r2

. (55)


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Ubiquity of IGW : relaxation of pressureanomaly

t=1.650

−25 −20 −15 −10 −5 0 5

0

2

4

6

8

10

12

14

16

18

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

t=12.000

−25 −20 −15 −10 −5 0 5

0

2

4

6

8

10

12

14

16

18

0.99

1

1.01

1.02

1.03

1.04

1.05

−30 −25 −20 −15 −10 −5 0 5 10−5

0

5

10

15

20t=1.650

−30 −25 −20 −15 −10 −5 0 5 10−5

0

5

10

15

20t=12.000

Relaxation of localised pressure anomaly in RSW. Leftpanel : initial stage of adjustment ; Right panel : advancedstage of adjustment. Upper row : pressure (thickness)field. Lower row : corresponding velocity field. Time ismeasured in f−1 and length in Rd . Initial perturbationconsists of a bump in thickness, with no velocity.


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1-layer (barotropic) QG modelf -planeQG equation

∂t ~∇2η − ∂tη + J (η, ~∇2η) = 0.

Linearisation : ∂t ~∇2η = 0 - no waves

β- plane

∂t ~∇2η − ∂tη + J (η, ~∇2η) + ∂xη = 0.

Linearisation :

∂tη − ∂t ~∇2η − ∂xη = 0. (56)

Waves η ∝ ei(kx+ly−ωt) with dispersion relation

ω = − kk2 + l2 + 1

. (57)


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Dispersion relation of barotropic Rossbywaves on the β- plane

Negative values ω < 0 are not shown.


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2-layer QG model with a rigid lid

Linearisation in the limit of weak stratification r → 1∂t[∇2π1 + F1(π2 − π1)

]+ ∂xπ1 = 0,

∂t[∇2π2 − F2(π2 − π1)

]+ ∂xπ2 = 0.

(58)

Looking for wave solutions πi = Aiei(k·x−ωt) + c.c. we getthe dispersion relation :

ω = − kx

2k2(k2 + F1 + F2)

[(2k2 + F1 + F2)± (F1 + F2)

].

(59)Two solutions correspond to :

I a faster barotropic mode : ωbt = − kxk2 ,

I a slower baroclinic mode : ωbc = − kx(k2+F1+F2)

.

As in the one-layer case, these waves are Rossby wavesarising due to the β - effect.


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Baroclinic Rossby waves : continuousstratification

Formal linearisation

∂t

[∇2

hπ − ∂z

(1

ρ′s(z)∂zπ

)]+ ∂xπ = 0, ∂2

tzπ∣∣∣z=0,1

= 0.

(60)

Separation of variables

π(x , y , z; t) = p(x , y ; t)S(z)⇒ (61)

∂t∇2hp(x , y ; t)S(z)− ∂tp(x , y ; t)

[1

ρ′s(z)S′(z)

]′+

∂xp(x , y ; t)S(z) = 0⇒


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Equations in z and in x , y , t :

I

1S(z)

[1

ρ′s(z)S′(z)

]′= κ2 (62)

I

∂t∇2hp(x , y ; t)−κ2∂tp(x , y ; t) +∂xp(x , y ; t) = 0, (63)

κ - separation constant


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Vertical modesSturm - Liouville problem :[

1ρ′s(z)

S′(z)

]′− κ2S(z) = 0, S′(z)

∣∣z=0,1 = 0 (64)

Eigenfunctions Sn(z) and eigenvalues κn,n = 0,1,2, ....

Rossby waves : p(x , y ; t) ∝ ei(k·x−ωt) :

ω = − kx

k2 + κ2n. (65)

The larger is the vertical wavenumber n (stronger verticalshear)→ the slower is the propagation. f - plane : nowaves.


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What have we seen :I Hierarchy of the GFD models : from PE to QG,

passing through RSWI Inertia-gravity wave-vortex dichotomy in the f - plane

approximation, and their time-scale separation(spectral gap) : waves - fast, vortices - slow

I Rossby waves appearing in the vortex-motion sectorin the β-plane

What we have not seen :Waves in the presence of

I boundariesI non-trivial topographyI mean flowI at the equator, where there is no f0

All this is coming up !

Waves and related processes in Geophysical Fluid Dynamics · Geophysical Fluid Dynamics V. Zeitlin...

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