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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: .

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planet formation

Planet Formation



Lecture by: C.P. Dullemond

the idea behind the formula
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 formula1
The idea behind the α-formula

Time scales:

Maximum height of an eddy:

Maximum velicity of an eddy:

Simulations show:

reynolds number
Reynolds Number

Reynolds Number:

Typically Re>>1

Turbulent eddies cannot have smaller LV than:


because such eddies are quickly viscously dissipated.


Kolmogorov Theory of


The Turbulent Cascade

kolmogorov turbulence
Kolmogorov turbulence

Eddies containeddies, which contain eddies, which contain eddies, which containeddies, which containeddies, which contain eddies, which contain eddies, which containeddies, which containeddies, which contain

energy input

thermal energy dissipation




kolmogorov turbulence1
Kolmogorov turbulence

Kolmogorov turbulent cascade

(must be a powerlaw!)

energy input

(turbulent driving)


energy dissipation

(molecular viscosity)

at the „Kolmogorov scale“


kolmogorov turbulence2
Kolmogorov turbulence

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

For scales


we can

use dimensional analysis to get the powerlaw slope. The question

is: What combination of k and ε gives E? Dimensions (using erg=gram cm2 s-2):

Only possible combination with the right dimensions:

kolmogorov turbulence3
Kolmogorov turbulence

Now a similar dimensional analysis for the typical velocity v

of turbulent eddies at each scale l=2π/k:

Only possible combination with the right dimensions:

kolmogorov turbulence4
Kolmogorov turbulence

Eddy turn-over time scale as a function of l=2π/k:

Only possible combination with the right dimensions:

So while the biggest eddies (driving scale) have turn-over time

scales ~ tkepler, the smaller eddies have shorter turn-over time


kolmogorov turbulence5
Kolmogorov turbulence

Contribution of subsubsub-eddies to the viscosity:

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

viscosity becomes smaller.

The viscosity is dominated by the biggest eddies!

However, the small eddies may play a role later, for the motion of

dust/rocky particles.

kolmogorov turbulence6
Kolmogorov turbulence

At which scale does the turbulence dissipate (i.e. what is the

value of kη)? Answer: at the scale where Re(k)=1:

This gives the Kolmogorov dissipation scale:

For a real Kolmogorov turbulent cascade to exist, one must have:

kolmogorov turbulence7
Kolmogorov turbulence

Back to the energy input ε: Let us check if this is consistent with

the 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 for

the biggest eddies (i.e. that the Kolmogorov powerlaw extents

to the largest eddies):

For Veddy and Leddy we have expressions from α-turbulence theory:

kolmogorov turbulence8
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 for disks turbulence @ 1 au
Estimates 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

The pressure scale height then becomes:

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

estimates for disks turbulence @ 1 au1
Estimates for disks & turbulence @ 1 AU

We can now calculate at 1 AU:

Large eddy size:

Large eddy velocity:

estimates for disks turbulence @ 1 au2
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 μ=2.3)

estimates for disks turbulence @ 1 au3
Estimates for disks & turbulence @ 1 AU

Mass 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 au4
Estimates for disks & turbulence @ 1 AU

Molecular cross section of H2 = 2x10-15 cm2

Mean free path for gas is:

The molecular viscosity is then:

The Reynolds number of the turbulence is thus:

estimates for disks turbulence @ 1 au5
Estimates for disks & turbulence @ 1 AU

Now calculate Komogorov scale. Remember:

At the largest eddies we have

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


How turbulence is (presumably)


The Magnetorotational Instability

(ref: Book by Phil Armitage)

magnetorotational instability

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!



The lower orbit of A causes an increase in its velocity, while B decelerates. This enhances their velocity difference! This is positive feedback: an instability.



Magnetorotational Instability

Highly simplified pictographic explanation:

Causes turbulence in the disk

kelvin helmholtz instability
Kelvin-Helmholtz Instability

Now let‘s do this a bit better. We follow a discussion from the

book of Armitage.

Kelvin-Helmholtz instability (shear instability):

Photo credit: Beverly Shannon (1999)

kelvin helmholtz instability1
Kelvin-Helmholtz Instability

However, in a rotating system the rotation can stabilize the

Kelvin-Helmholtz instability. The Rayleigh criterion says:


A Keplerian disk has:

Keplerian disks

are Rayleigh-


magnetorotational instability1
Magnetorotational Instability

Let us study the stability of a disk with a weak vertical magnetic

field. We will use perturbation theory and we will assume

ideal MHD. The equations for ideal MHD are:

magnetorotational instability2
Magnetorotational Instability

Now let‘s transform this to cylindrical coordinates. This is not

trivial. But let‘s do this for the equation of motion of a single

fluid element under influence of a force-per-mass f:

Since and one could write let us write out:

magnetorotational instability3
Magnetorotational Instability

Taking our equations at ϕ=0:

The momentum equations for the fluid parcel thus become:

f are the forces coupling

the gas to the B-field.

Note that we can write out the gravity term:

which is the well-known

inverse square force law

magnetorotational instability4
Magnetorotational Instability

Now define a local (x,y) coordinate system:

Inserting this into the previous equations, and discarding quadratic

terms, yields (after some calculation):

magnetorotational instability5
Magnetorotational Instability

Now let‘s look at an (x,y) displacement varying with height z

and time t:

Remember now that gas displacements carry along the B-field.

Let‘s assume a weak vertical initial B-field. Then the displacements

create x- and y- components of this B-field:

magnetorotational instability6
Magnetorotational Instability

These produce a magnetic tension force (from ):



magnetorotational instability7
Magnetorotational Instability

The equation of motion then becomes:

Combining them yields the following dispersion relation:

magnetorotational instability8
Magnetorotational Instability



Most unstable


Most unstable mode:

Stable for:

magnetorotational instability9
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.

Let‘s assume magnetic equipartition:

Then the instability occurs at:

Scale larger than

disk thickness:

Equipartition disk=stable

magnetorotational instability10
Magnetorotational Instability

Note: This instability works only if the disk is sufficiently ionized

for ideal MHD equations to be valid.

Only a tiny bit of ionization is required.

But even that can be problematic, since dust grains very

efficiently „vacuum clean“ away free electrons.

This leads to so called „dead zones“ in disks.

The debate for what causes turbulence in disks remains wide open