Inertial waves, Tides Precession, Dynamos, …

Andreas Tilgner, University of Göttingen

The Navier-Stokes equation in the boundary frame

Unit of length:

Unit of time:

Inertial Waves

Dispersion relation:

Rotation axis along z, wave vector along k

Phase and group velocities:

Zonal velocity

Meridional streamlines

Modes in spherical shells

The spin-over mode

A solution of the inviscid equation of motion in spheres, ellipsoids, and shells

The spin-over mode

Radial velocity, no slip boundaries

Superpose waves

• with frequency

• with wavevector

• in a wavepacket

The wavepacket

• is localized around a surface for an appropriate choice of

• stays localized around that surface as time evolves

• does not broaden in the course of time

Inertial waves can be superposed to form shear layers

These shear layers replace Ekman layers at “critical latitudes”where they are tangent to the boundary

Ekman layer: Balance between Coriolis and viscousterms

Critical latitudes: Balance between Coriolis term andtime derivative

Inertial waves are excited at critical latitudes !

Source driven inertial waves

Critical latitudes have enhanced Ekman pumps: They act as strong sourcesor sinks for the interior.

⇒Compute source driven flows in an infinite fluid as a simple model(A. Tilgner, Phys. Fluids 12, 1101 (2000))

Manageable algebra for point-, ring- or disk-sources:

Flow field of ring source

Wave reflection conserves the angle with the rotation axis(not the angle with the reflecting surface !)

In spherical shells, there are• closed cycles• “caustics” or “attractors”

A. Tilgner, Phys. Rev. E (1999)

Zonal velocity

Meridional streamlines

Ray pattern in the spin-over mode

Excitation of inertial waves by precession

Flow with elliptical streamlines is not a solution if there is an inner core witha different ellipticity than the outer boundary

meridional zonal

components of velocity

A. Tilgner, Geophys. J. Int. (1999)

Garrett & Kunze, Annu. Rev. Fluid Mech. (2007)

Inertial modes and zonal winds

Excitation mechanisms with well defined frequencies:• libration• precession• tides

Can a non-axisymmetric excitation drive an axisymmetric flow ?

H. Greenspan (1969): Nonlinear interaction of inviscid inertial modes donot drive “significant” axisymmetric zonal flows in full spheres

What about spherical shells, in which inertial modes have internal shear layers ?

In precessing flow, axisymmetriccomponents are

• observed experimentally (Malkus, Vanyo)

• predicted analytically (Busse 1968)

• computed numerically

Full equation of motion in corotating frame:

Assume a small Rossby number

Develop in powers of the Rossby number:

Imagine an inertial mode maintained at constant amplitude by some forcing(for example tides)

:

bar denotes azimuthal average

Solutions: inertial modes in the form

Azimuthal wind patterns

Ekman number 1E-6, different frequencies

0.88 1.23 -0.23

-0.80 0.69

Normalize:

Kinetic energy in differential rotation:

Thickness of internal oblique shear layers:

Typical velocity inside shear layers:

Velocity inside shear layers much larger than outside:

Estimate for

For normalized eigenmodes:

must diverge at small Ekman numbers

Extrapolation to Jupiter

Use

Estimate the energy in an inertial mode from the amplitude of the equilibrium tideraised by Io on Jupiter

A typical axisymmetric azimuthal velocity of 15 m/s is obtained for Ek=1E-15

And in the Earth’s core ?

G. Hulot et al., Nature (2002)

No slip boundaries => less differential rotation

Strong retrograde polar vortex known from experiments

Simulate a possible experiment:

• Rotating spherical shell

• Outer shell tidally deformed

• “Moon” is stationary or rotating in prograde direction

Vorticity at poles extrapolates to Earth’s values, but the vortex is too thin

10−8

10−7

10−6

10−5

10−4

Ek

100

102

104

106

|ω|

stationary

prograde

0.0 20.0 40.0 60.0θ

−104

0

ω

Ek=1E-7

prograde “Moon”

Vorticity at the pole

Summary

• Inclined shear layers appear in inertial modes in all but the simplest geometries

• Shear layers are focused on attractors

• Excitations occurring in geo- and astrophysics with well defined frequencies:tides, precession, libration

• There is significant nonlinear self-interaction of inertial modes due to the internalshear layers

• It is plausible from order of magnitude estimates that tidal forcing is responsiblefor the observed zonal winds in the atmospheres of the giant planets

• The same mechanism is less efficient in the Earth’s core

Planetary dynamos

Possible driving mechanisms

• Convection (thermal or chemical)

• Tides ?

• Precession ?

Dynamo driven by convection

Precession

The Poincaré solution

Look for a solution linear in x,y,z inside an ellipsoid

Stretch the coordinates to transform the ellipsoid in a sphere, assume the flow isa solid body rotation in the sphere, and transform back. The stretched velocity

is given by the vector product of any vectorwith

This flow is solenoidal, does not penetrate the boundaries, and has constant vorticity

In the curl of the Navier-Stokes equation:

Stationary state =>

Only the second equation is not trivial

No slip boundaries select a unique solution. Expect

Laminar flow

• This flow is inefficient as a dynamo

• It must become unstable before it can generatemagnetic fields

Precession experiments

Instability of the boundary layer

Instability of the bulk

Triad Resonances

Resonant Collapse

• Instability grows

• Laminar large scale mode suddenly decays into small scale turbulence

• The small scales draw energy from the large scale which they dissipate

• Once enough energy is dissipated: The flow becomes laminar again,new cycle

Resonant Collapse

Analogy Precession / Tides

Rotationaxis

Tidal body

Gledzer & Ponomarev J. Fluid Mech. (1992)

Stability depends on• amplitude of tidal deformation• orbital period of moon / rotation period of planet• viscosity

Excitation of inertial modes resonances

All prograde moons are believed to hit a resonance (?)

An upper bound on the growth rate of elliptical instabilities is known. Onecan exclude instability for some planets.

Tidal parameters

Tidal instability: Likely on all giant planets

Precessional instability:• Plausible for Neptune/Triton• Marginal for Earth

Libration ?

The martian dynamo

Arkani-Hamed et al., JGR E06003 (2008)

Infer (hypothesize ?) from martian geological features, that• the giant impact craters have been created by a single aseroid that brokeapart as it entered the Roche limit

• the asteroid had a mass 0.15% of the mass of Mars• the impact and the cessation of the martian dynamo occurred simultaneously

Compute the orbital history of that impacting body

Compute the tidal interaction between the impacting body and Mars

Conclude that these tides may have sustained the martian dynamo forseveral hundreds of millions of years

• Triad resonances need a non-axisymmetric ground state

• Ellipsoidal boundaries break axisymmetry of flow in the case ofprecession or tides

• The Ekman layers do the same for the spin-over mode (because therotation axes of fluid and boundaries are not identical)

• Expect triad resonances in a sphere with no slip boundaries

• Is there a precession driven dynamo in a sphere or spherical shell?

The induction equation

non-dimensional form:

magnetic Reynods number:

pure diffusion

stretching of magnetic field lines

stretching of magnetic field lines

Rm = 100 … 1000 for planets

Antidynamo theoremsaxisymmetric and 2D magnetic fields cannot begenerated by a dynamo

a toroidal velocity field cannot generate a magnetic field

The kinematic dynamo problem

Velocity u prescibed, are there growing magnetic fields B ?

The full (dynamic) dynamo problem

Forcing (for example thermal convection) prescribed, are there growing magnetic fields B ?

Problems with non-normal operators:

Transition to turbulence in shear flows => transient growth

Transient growth sustained growth

• transition to turbulence : non-linear terms

• kinematic dynamo : time dependent eigenstates

• slow time dependence of eigenvectors : no effect• fast time dependence of eigenvectors : averaging, decay• intermediate time dependence: growth is possible, even though all eigenvectors decay

Periodic 2D dynamo with drift

Propagating wave:

• Eigenfunctions at different times differ by a translation

• The scalar product between eigenfunctions is independent of time

• Propagation is equivalent to a “rotation” in function space

Roberts flow:

Many orthogonal eigenvectors because of symmetries

Non-orthogonal eigenvectors only within one symmetry class

Uniformly drifting velocity pattern is equivalent to a stationary flow in a co-moving frame:

Solved with periodic boundary conditions in the box

Compute solutions of the form:

Slow drift : Negligible effect.Fast drift : Distortions of magnetic field lines which occurred during one half period

are reverted during the next half period.

Propagating waves in convective dynamos

Vary the drift frequency of the velocity pattern

Find two kinematic dynamos, one of which satisfies the full dynamo problem

Bayliss et al.PRE (2007)

Higher Re introduces small scales, reduces Rm,crit (Tilgner, NJP (2007))

Conclusion

• An alternative view of magnetic field production: mixing ofnon-normal eigenmodes.

• Examples of time dependent dynamos, for which nosnapshot is a dynamo.

• Clearest demonstration for uniform drift (wave propagation),qualitatively the same must happen for more complicatedtime dependencies.

• Mean field MHD: The term responsible for the drift effect is of thesame order as other terms neglected in FOSA.

• Adjustment of time dependence is part of the saturation process.

The dynamo problem (precession)

vary and

Spectral method in a sphere

Hydrodynamic stability

indicates instability

0.2 0.4 0.6 0.8 1.0 1.2E x 10

3

0.0

2.0

4.0

6.0

Ea

/ Eki

n x

103

Kinematic dynamos

0.0 1.0 2.0 3.0Pm

−0.20

−0.15

−0.10

−0.05

0.00

0.05

0.10

p

Growth rates for Ekman numbers (squares) and

(circles)

0.0 0.5 1.0 1.5E x 10

3

0.0

5.0

10.0

15.0

Pmc

What is the relevant poloidal flow component?

• Ekman pumps? They are present even in stable flows

critical Rm is approx. 800 based on poloidal velocity

• Instability? If the critical Rm based on is constant (approx. 190)

Instability suppressedby enforcing symmetry

Time series of a self-consistent dynamo

0.0 500.0 1000.0 1500.0t

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

Ea

x 10

3

0.0 500.0 1000.0 1500.0t

0.0

0.5

1.0

1.5

2.0

EB x

103

Field structure

radial magnetic field at

the outer boundary mid-depth

The spectrum of the magnetic field

0 5 10 15 20 l

0

2x10−5

4x10−5

6x10−5

8x10−5

ε l , ε

sl x

100

Orientation of the dipole moment

Conclusion

• The orientation of the fluid axis is well known

• There are internal shear layers

• Inertial and viscous instabilities have been observed

• Are there parameters for which the internal shear layersbecome unstable first?

• Resonant collapse

• Precession driven dynamos exist at magnetic Reynoldsnumbers characteristic of the Earth’s core

Outlook

• Are there parameters for which the internal shear layersbecome unstable first?

• Are there observable effects in the orbit of the Moon?

• Are there observable signatures in the secular magneticvariations?

• Can precession driven dynamos produce dipole dominateddynamos? (Presumably yes, Roberts & Wu 2008)

References

J. Fluid Mech. 379, 303-318 (1999)

J. Fluid Mech. 447, 111-128 (2001)

J. Fluid Mech. 492, 363-379 (2003)

Phys. Fluids 17, 034104 (2005)

Geophys. Astrophys. Fluid Dynamics 101, 1-9 (2007)

Treatise on Geophysics, volume 8

A. Tilgner, Phys. Fluids 12, 1101 (2000)

Physical Review Letters 99, 194501 (2007)

New Journal of Physics 9, 270 (2007)

Physical Review Letters 100, 128501 (2008)

References