 # Topic: Turbulence Lecture by: C.P. Dullemond

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Topic: Turbulence Lecture by: C.P. Dullemond
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: Maximum velicity of an eddy: Simulations show:

Reynolds Number Reynolds Number: Typically Re>>1
Turbulent eddies cannot have smaller LV than: i.e. because such eddies are quickly viscously dissipated.

Kolmogorov Theory of Turbulence: The Turbulent Cascade

Kolmogorov turbulence
Eddies 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 contain energy input thermal energy dissipation cascade log(l) log(k)

Kolmogorov turbulence
Kolmogorov turbulent cascade (must be a powerlaw!) energy input (turbulent driving) log(E(k)) energy dissipation (molecular viscosity) at the „Kolmogorov scale“ log(k)

Kolmogorov turbulence
Driving turbulence with energy input ε [erg/gram.s] For scales i.e. 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 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 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 scales.

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 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 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 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 numbers

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 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 μ=2.3)

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 % dust. Just about enough to form Earth. Seem thus to be ok! Mass within 1 large turbulent eddy:

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 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) driven:
The Magnetorotational Instability (ref: Book by Phil Armitage)

Magnetorotational Instability
Highly 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! A B 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. A B Causes turbulence in the disk

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 Instability
However, in a rotating system the rotation can stabilize the Kelvin-Helmholtz instability. The Rayleigh criterion says: instability A Keplerian disk has: Keplerian disks are Rayleigh- Stable

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 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 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 Instability
Now define a local (x,y) coordinate system: Inserting this into the previous equations, and discarding quadratic terms, yields (after some calculation):

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 Instability
These produce a magnetic tension force (from ): Alfven velocity

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

Magnetorotational Instability
Unstable Stable Most unstable mode Most 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. Let‘s assume magnetic equipartition: Then the instability occurs at: Scale larger than disk thickness: Equipartition disk=stable

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 today.