Porquerolles 2009

Porquerolles Island

Role of Tides in Planetary Systems

Cyprien Morize

• C. Morize1,

• M. Le Bars1,

• P. Le Gal1

• A. Tilgner2.

1IRPHE, Institut de Recherche sur les Phénomènes Hors Equilibre, Aix- Marseille University , France

2Institut of Geophysics, University of Göttingen, Germany

Context

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• Tidal forces deform planets

• Tidal forces = colossal values

Intense volcanism on Io

• Tides influence orbital trajectories and spin velocities of planets

• Elliptic instability

• Zonal flows

Rotating flows

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∂∂t

r u + (

r u ⋅

r ∇ )

r u = −

1ρ

r ∇ p − 2

r Ω ×

r u + ν∇2r u

Navier-Stokes equations in a rotating frame:

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∂∂t

r u + (

r u ⋅

r ∇ )

r u = −

1ρ

r ∇ p − 2

r Ω ×

r u + ν∇2r u

Rotating flows

Navier-Stokes equations in a rotating frame:

∂∂t

r u + (

r u ⋅

r ∇ )

r u = −

1ρ

r ∇ p − 2

r Ω ×

r u + ν∇2r u

Consequences :∂/∂z = 0 (2D flows)

Geostrophic equilibrium :

Taylor-Proudmann and Inertial waves

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∂∂t

r u + (

r u ⋅

r ∇ )

r u = −

1ρ

r ∇ p − 2

r Ω ×

r u + ν∇2r u

θσ cos2||

2)( // Ω=Ω=kkk r

r

Dispersion relation: Energy spread along a cone of angle

θθ π −= 2s

For σ << 2Ω, Taylor columns

cos(σt)

θs

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Consequences :∂/∂z = 0 (2D flows)

Geostrophic equilibrium :

Taylor-Proudmann and Inertial waves

∂∂t

r u + (

r u ⋅

r ∇ )

r u = −

1ρ

r ∇ p − 2

r Ω ×

r u + ν∇2r u

θσ cos2||

2)( // Ω=Ω=kkk r

r θθ π −= 2s

cos(σt)

θs

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)(0

tmieuu ωθ −=Azimuthal periodicity m

Temporal periodicity ω

Dispersion relation: Energy spread along a cone of angle

For σ << 2Ω, Taylor columns

Consequences :∂/∂z = 0 (2D flows)

Geostrophic equilibrium :

Taylor-Proudmann and Inertial waves

∂∂t

r u + (

r u ⋅

r ∇ )

r u = −

1ρ

r ∇ p − 2

r Ω ×

r u + ν∇2r u

θσ cos2||

2)( // Ω=Ω=kkk r

r θθ π −= 2s

cos(σt)

θs

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ωm

ode

)(0

tmieuu ωθ −=Azimuthal periodicity m

Temporal periodicity ω

Dispersion relation: Energy spread along a cone of angle

For σ << 2Ω, Taylor columns

Consequences :∂/∂z = 0 (2D flows)

Geostrophic equilibrium :

Taylor-Proudmann and Inertial waves

Experimental Apparatus

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Non-dimensional numbers:

• Eccentricity (Earth: ε = 10-7 ; Io: ε = 0.005)

• Ekman number

(Earth: Ek = 10-14 ; Io: Ek = 10-13)

• The ratio sorb ΩΩ /

2/ REk sΩ=ν

• Deformable sphere in silicone (planet) Ωs

• Rollers press the rotating sphere (moon) Ωorb

Radius = 10 cmΩspin ~ 3 HzΩorb ~ 2 Hz

Eccentricity : 0.01 0.07 Ekman ~ 10-5

ε

Elliptical instability

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Present in planetary systems suggested by Kerswell (1998)

3D destabilization of 2D rotating flows whose streamlines are elliptically distorted

Parametric resonance of 3 waves :

• Tidal wave, of temporal period T ~ Ωorb-1 and azimuthal period m=2

• 2 inertial waves of the rotating fluid

where frequencies is 0321 =++ ωωω

+ =

Rotation Stretch Elliptic streamlines

Spin-Over mode: fixed deformation (Ωorb=0)

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rpms 47;03.0 =Ω=ε rpms 67;04.0 =Ω=ε rpms 140;05.0 =Ω=ε

Spin-Over mode

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rpms 47;03.0 =Ω=ε rpms 67;04.0 =Ω=ε rpms 140;05.0 =Ω=ε

Spin-Over mode

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rpms 47;03.0 =Ω=ε rpms 67;04.0 =Ω=ε rpms 140;05.0 =Ω=ε

Ek62.22 −= εσ

Spin-Over mode

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rpms 47;03.0 =Ω=ε rpms 67;04.0 =Ω=ε rpms 140;05.0 =Ω=ε

Ek62.22 −= εσ

Spin-Over mode

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rpms 47;03.0 =Ω=ε rpms 67;04.0 =Ω=ε rpms 140;05.0 =Ω=ε

EARTH (α~1)

IO (α~270)

Tidal forces may excite intense axisymmetric flows

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Generation of zonal winds by tides

Tides (azimuthal period m=2) can force an inertial mode of azimuthal period m=2 (provided that the considered mode is symmetrical about equator) and of frequency twice the tidal frequency.

Adjusting the orbital frequency on the frequency of an eigenmodes

2mode

orbω

=Ω

The term of m=2 mode with its complex conjugate produce an intense axisymmetric flows m=0

vv rrr ).( ∇

Work motivated by the work of Andreas Tilgner

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δ ~ E 1/2

(ω 2 cos(θ))1/2

cos(θc) = ω/2

Divergence of δ at critical latitudes for

However, the non linear self-interaction of a non viscous mode does no produce a geostrophic mode (m=0) (Greenspan, 1969)

Precession: ω=1 θc=30°

Tides: ω=2 θc=90°(provided that Ωorb=0)

Generation of zonal winds by tides

Inertial waves are excited at critical latitudes

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Ωorb = 0.17 Ωorb = 0.38

Presence of a strong inner shear layer (Suess, 1971)

HOWEVER significant deviations to this theory: appearance of additional shear

layers at the axis of rotation

Ωorb = 0.54

Visualizations with kalliroscope flakes illuminating by a vertical laser sheet

Axisymmetric flows by excitation of modes A, B and C

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Ωorb = 0.17 Ωorb = 0.38 Ωorb = 0.54

Visualizations with kalliroscope flakes illuminating by a vertical laser sheet

Axisymmetric flows by excitation of modes A, B and C

Measure of the gradient of the luminosity intensity

Kinetic energy stored in the azimuthal mode m=2 by scanning Ωorbit.

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Kinetic energy stored in axisymmetric flows

(Tilgner, 2007)

Experimental observations (dotted lines) in good agreement with the peak of energy predicted numerically by Andreas Tilgner

Tilted forcing in comparison to rotation axis

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Perspective

Modes m=1 and m=2 are simultaneously present, leading to a multiplication of the number of inner shear layers.

ω = 2 (1- Ωorbit) = 0.92 ω = 0.886

RNL 2009 Cyprien Morize

Contour iso-uϕ

Excitation du mode C

Ωorb = 0.54

(Tilgner, 2007)

Observations expérimentales en bon accord avec les résultats numériques d’A. Tilgner

(Lacaze et al., 2004)

Modèle du Spin-Over

RNL 2009 Cyprien Morize

Equations d’Euler pour un solide ellipsoïde auquel on ajoute les effets visqueux fluides associés aux couches limites

Ek62.22 −= εσ

(Lacaze et al., 2004)

Le taux de croissance de l’instabilité est donné par :

Modèle avantageusement comparé aux expériences

Spin-Over = le fluide tourne autour d’un axe perpendiculaire à l’axe de rotation (comparable à l’instabilité de la rotation d’un ellipsoïde solide)

Déformation fixe dans le référentiel du labo.

Viscosity complicates the picture by creating Ekman layers in the vicinity of walls

The Poincaré Equation now reads:

where E stands for the Ekman numberz

ζθ

δ

Inside Ekman layer, dominant terms

ζ = z cos (θ) u = u0 e iωt e z/δ

i (ω 2 cos(θ)) u = E u

δ ~ E 1/2

(ω 2 cos(θ))1/2

Divergence of δ at critical angle: cos(θc) = ω/2

Critical latitude in the Sphere

Formation of Internal shear layers

ω=1, radial velocity contour plotsphere Tilgner 2007

ω=1, radial velocity contour plotspherical shell Tilgner 2007

Suplementary inner shear layersin the spherical shell

ω = 2 (1- Ωorbit) = 0.92 ω = 0.886

RNL 2009 Cyprien Morize

Contour iso-uϕ

Visualisation du mode C

Ωorb = 0.54

(Tilgner, 2007)

Observations expérimentales en parfait accord avec les résultats numériques d’A. Tilgner