waves and related processes in geophysical fluid dynamics · geophysical fluid dynamics v. zeitlin...
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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
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
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
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
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
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
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.
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
GFD : space view
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
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
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
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
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
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
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
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
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
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
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
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)
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
Spherical coordinates
r
r
1
r
r
1
r
r
1
r
r
1
r
r
1
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
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.
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
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
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
Mean oceanic stratification
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
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)
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
Mean atmospheric stratification
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
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.
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
Material surfaces
g f/2z
x
z2
z1w1= dz1/dt
w2= dz2/dt
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
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.
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
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.
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
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
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
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
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
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.
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
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
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
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.
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
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)
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
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)
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
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)
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
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
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
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.
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
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)
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
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.
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
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.
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
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.
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
Artist’s view of shallow vortex dynamics :vortices, waves, and topography (to appear inthe next lecture)in shallow water
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
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)
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
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.
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
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)
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
Dispersion relation of barotropic Rossbywaves on the β- plane
Negative values ω < 0 are not shown.
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
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.
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
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⇒
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
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
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
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.
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
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 !