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Planet Formation Topic: Turbulence Lecture by: C.P. Dullemond
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Planet Formation. Topic: Turbulence Lecture by: C.P. Dullemond. The idea behind the α-formula. Viscosity is length times velocity:. Maximum height of an eddy: . Maximum velicity of an eddy: . The idea behind the α-formula. Time scales:. Maximum height of an eddy: . - PowerPoint PPT Presentation

### Transcript of Planet Formation

The formation of stars and planets

Planet FormationTopic:

Turbulence

Lecture by: C.P. Dullemond1The idea behind the -formula

Viscosity is length times velocity:Maximum height of an eddy:

Maximum velicity of an eddy:

The idea behind the -formula

Time scales:Maximum height of an eddy:

Maximum velicity of an eddy:

Simulations show:

Reynolds NumberReynolds Number:

Typically Re>>1Turbulent eddies cannot have smaller LV than:

i.e.because such eddies are quickly viscously dissipated.Kolmogorov Theory ofTurbulence:

The Turbulent CascadeKolmogorov turbulenceEddies contain eddies, which contain eddies, which contain eddies, which contain eddies, which contain eddies, which contain eddies, which contain eddies, which contain eddies, which contain eddies, which containlog(k)log(l)energy inputcascadethermal energy dissipation

Kolmogorov turbulencelog(E(k))log(k)energy input(turbulent driving)energy dissipation(molecular viscosity)at the Kolmogorov scaleKolmogorov turbulent cascade(must be a powerlaw!)

Kolmogorov turbulenceFor scales

i.e.we canuse dimensional analysis to get the powerlaw slope. The questionis: What combination of k and gives E? Dimensions (using erg=gram cm2 s-2):

Driving turbulence with energy input [erg/gram.s]

Only possible combination with the right dimensions:

Kolmogorov turbulence

Only possible combination with the right dimensions:

Now a similar dimensional analysis for the typical velocity v of turbulent eddies at each scale l=2/k:Kolmogorov turbulence

Only possible combination with the right dimensions:

Eddy turn-over time scale as a function of l=2/k:So while the biggest eddies (driving scale) have turn-over timescales ~ tkepler, the smaller eddies have shorter turn-over timescales. Kolmogorov turbulenceContribution of subsubsub-eddies to the viscosity:

As you see: for ever smaller l (ever bigger k) the contribution to theviscosity becomes smaller.

The viscosity is dominated by the biggest eddies!

However, the small eddies may play a role later, for the motion ofdust/rocky particles.Kolmogorov turbulence

This gives the Kolmogorov dissipation scale:

At which scale does the turbulence dissipate (i.e. what is thevalue of k)? Answer: at the scale where Re(k)=1:For a real Kolmogorov turbulent cascade to exist, one must have:

Kolmogorov turbulenceBack to the energy input : Let us check if this is consistent withthe viscous heating coefficient Q+ we derived in the previous chapter.In the cascade region we have (see few slides back):

Let us now make the bold step to assume that this also holds forthe biggest eddies (i.e. that the Kolmogorov powerlaw extentsto the largest eddies):For Veddy and Leddy we have expressions from -turbulence theory:

Kolmogorov turbulence

We also know from viscous disk theory:

It follows that the two formulae can only be mutually consistent if:

(keep in mind, however, the approximations made!)Estimates and numbersEstimates for disks & turbulence @ 1 AU

Typical accretion rate:Surface density powerlaw unknown, but from previous chapter theoretical considerations (viscous heating) give a good estimate:

With a mean molecular weight of 2.3 this leads to

We have no idea what the value of is (this is one of the big unknowns in the entire disk & planet formation theory), but simulations suggest =0.01, so let us take this value. The pressure scale height then becomes:

Estimates for disks & turbulence @ 1 AU

We can now calculate at 1 AU:

Large eddy size:

Large eddy velocity:

Estimates for disks & turbulence @ 1 AU

For a steady-state disk the surface density follows:The midplane density then follows with the scale height H:

(using =2.3)Estimates for disks & turbulence @ 1 AUMass between 0.8 AU and 1.3 AU (very rough estimate):

Note that this is in the form of gas + 1...2 % dust. Just about enough to form Earth. Seem thus to be ok!Mass within 1 large turbulent eddy:

Estimates for disks & turbulence @ 1 AU

The molecular viscosity is then:

Molecular cross section of H2 = 2x10-15 cm2Mean free path for gas is:

The Reynolds number of the turbulence is thus:

Estimates for disks & turbulence @ 1 AUNow calculate Komogorov scale. Remember:

At the largest eddies we have

At the smallest (Kolmogorov) scale (l=l) we have Re=1. So:

How turbulence is (presumably)driven:

The Magnetorotational Instability(ref: Book by Phil Armitage)Magnetorotational InstabilityHighly simplified pictographic explanation:If a (weak) pull exists between two gas-parcels A and B on adjacent orbits, the effect is that A moves inward and B moves outward: a pull causes them to move apart!ABThe lower orbit of A causes an increase in its velocity, while B decelerates. This enhances their velocity difference! This is positive feedback: an instability.ABCauses turbulence in the diskKelvin-Helmholtz InstabilityNow lets do this a bit better. We follow a discussion from thebook of Armitage.Kelvin-Helmholtz instability (shear instability):

Photo credit: Beverly Shannon (1999)Kelvin-Helmholtz InstabilityHowever, in a rotating system the rotation can stabilize the Kelvin-Helmholtz instability. The Rayleigh criterion says:

instabilityA Keplerian disk has:

Keplerian disksare Rayleigh-StableMagnetorotational InstabilityLet us study the stability of a disk with a weak vertical magneticfield. We will use perturbation theory and we will assumeideal MHD. The equations for ideal MHD are:

Magnetorotational InstabilityNow lets transform this to cylindrical coordinates. This is not trivial. But lets do this for the equation of motion of a singlefluid element under influence of a force-per-mass f:

Since and one could write let us write out:

Magnetorotational Instability

Taking our equations at =0:The momentum equations for the fluid parcel thus become:

f are the forces couplingthe gas to the B-field.Note that we can write out the gravity term:

which is the well-knowninverse square force lawMagnetorotational InstabilityNow define a local (x,y) coordinate system:

Inserting this into the previous equations, and discarding quadraticterms, yields (after some calculation):

Magnetorotational InstabilityNow lets look at an (x,y) displacement varying with height z and time t:

Remember now that gas displacements carry along the B-field.Lets assume a weak vertical initial B-field. Then the displacementscreate x- and y- components of this B-field:

Magnetorotational Instability

These produce a magnetic tension force (from ):

AlfvenvelocityMagnetorotational Instability

The equation of motion then becomes:Combining them yields the following dispersion relation:

Magnetorotational Instability

Most unstablemodeStableUnstableMost unstable mode:

Stable for:Magnetorotational Instability

Stable for:Conclusion: If the field is too strong, the disk is stable. So MRI works only for weak magnetic fields!Another conclusion: MRI does not work for too small wavelengths.There is a minimum scale that can be driven. There is also a certain scale where the driving is the strongest.Lets assume magnetic equipartition:Then the instability occurs at:

Scale larger thandisk thickness:Equipartition disk=stableMagnetorotational InstabilityNote: This instability works only if the disk is sufficiently ionizedfor ideal MHD equations to be valid.

Only a tiny bit of ionization is required.

But even that can be problematic, since dust grains veryefficiently vacuum clean away free electrons.