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

Lecture 1: GFDmodels andwave-vortex

paradigm

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

paradigm

Atmosphere

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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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

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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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

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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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

GFD : space view

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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

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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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

salt, ice ...)

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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

salt, ice ...)

paradigm

Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

salt, ice ...)

paradigm

Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

salt, ice ...)

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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

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

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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

Spherical coordinates

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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

Euler and continuity equations

v2λ + v2

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,

[1r2∂(r2vr )

1r cosφ

(∂(cosφvφ)

∂φ+∂vλ∂λ

∂t+ vr∂r +

vφr∂φ +

vλr cosφ

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

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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

Tangent plane approximation

∂~v∂t

+ ~v · ~∇~v + f z ∧ ~v = −~∇Pρ

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

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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

Mean oceanic stratification

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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

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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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

Mean atmospheric stratification

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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

Primitive equations : dry atmosphere,pseudo-height vertical coordinate

∂~vh

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

−gθ

θ0+∂φ

∂z= 0, (8)

∂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

), (10)

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

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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

Material surfaces

g f/2z

z1w1= dz1/dt

w2= dz2/dt

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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

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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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

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

Columnar motion.

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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

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)

v1 rho1

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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

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.

v1 rho1

rho2h2

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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

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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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

Characteristic parametres

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

√gH0

f0(22)

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

time-scales,

I Burger number : Bu =R2

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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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

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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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

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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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

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)

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

(32)J (A,B) ≡ ∂xA∂yB − ∂yA∂xB. (33)

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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

Order ε1

u(1) = −d (0)

dtv (0)−yu(0), v (1) =

dtu(0)−yv (0),⇒ (34)

∂xu(1) + ∂yv (1) = −d (0)

dt~∇2η − v (0),⇒ (35)

(η − ~∇2η

)− ∂xη = 0↔ d (0)

(η − ~∇2η − y

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

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

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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

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)

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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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

Non-dimensional PE under slow-motionscaling

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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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

QG equations

Asymptotic expansion in ε + elimination of w and σ →

(−∇2

hπ − y + ∂z

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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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

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.

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.

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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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

Dispersion relation and spectral gapHarmonic waves :

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

Dispersion relation

(ω2 −

(N2 k2

x + k2y

))= 0. (50)

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

dispersion relation :

ω = ±

k2x + k2

+ f 2, (51)

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

Spectral gap : ω ≥ f for IGW.

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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

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,

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.

paradigm

Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

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

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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

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

∂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± =

1±√

. (55)

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Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

Ubiquity of IGW : relaxation of pressureanomaly

t=1.650

−25 −20 −15 −10 −5 0 5

t=12.000

−25 −20 −15 −10 −5 0 5

−30 −25 −20 −15 −10 −5 0 5 10−5

20t=1.650

−30 −25 −20 −15 −10 −5 0 5 10−5

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.

paradigm

Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

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)

paradigm

Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

Dispersion relation of barotropic Rossbywaves on the β- plane

Negative values ω < 0 are not shown.

paradigm

Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

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.

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.

paradigm

Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

Baroclinic Rossby waves : continuousstratification

Formal linearisation

hπ − ∂z

ρ′s(z)∂zπ

)]+ ∂xπ = 0, ∂2

tzπ∣∣∣z=0,1

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)

ρ′s(z)S′(z)

∂xp(x , y ; t)S(z) = 0⇒

paradigm

Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

Equations in z and in x , y , t :

ρ′s(z)S′(z)

]′= κ2 (62)

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

κ - separation constant

paradigm

Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

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.

paradigm

Atmosphere

QG dynamics in RSW

QG in 2-layer RSW

QG models

Résumé

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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