2The idea behind the α-formula Viscosity is length times velocity:Maximum height of an eddy:Maximum velicity of an eddy:
3The idea behind the α-formula Time scales:Maximum height of an eddy:Maximum velicity of an eddy:Simulations show:
4Reynolds Number Reynolds Number: Typically Re>>1 Turbulent eddies cannot have smaller LV than:i.e.because such eddies are quickly viscously dissipated.
5Kolmogorov Theory ofTurbulence:The Turbulent Cascade
6Kolmogorov 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 containenergy inputthermal energy dissipationcascadelog(l)log(k)
7Kolmogorov 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)
8Kolmogorov turbulence Driving turbulence with energy input ε [erg/gram.s]For scalesi.e.we canuse dimensional analysis to get the powerlaw slope. The questionis: What combination of k and ε gives E? Dimensions (using erg=gram cm2 s-2):Only possible combination with the right dimensions:
9Kolmogorov turbulence Now a similar dimensional analysis for the typical velocity vof turbulent eddies at each scale l=2π/k:Only possible combination with the right dimensions:
10Kolmogorov 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 timescales ~ tkepler, the smaller eddies have shorter turn-over timescales.
11Kolmogorov turbulence Contribution 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.
12Kolmogorov turbulence At which scale does the turbulence dissipate (i.e. what is thevalue 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:
13Kolmogorov turbulence Back to the energy input ε: 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 α-turbulence theory:
14Kolmogorov 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!)
16Estimates 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 toThe 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.
17Estimates for disks & turbulence @ 1 AU We can now calculate at 1 AU:Large eddy size:Large eddy velocity:
18Estimates 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)
19Estimates 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 enoughto form Earth. Seem thus to be ok!Mass within 1 large turbulent eddy:
20Estimates for disks & turbulence @ 1 AU Molecular cross section of H2 = 2x10-15 cm2Mean free path for gas is:The molecular viscosity is then:The Reynolds number of the turbulence is thus:
21Estimates for disks & turbulence @ 1 AU Now calculate Komogorov scale. Remember:At the largest eddies we haveAt the smallest (Kolmogorov) scale (l=lη) we have Re=1. So:
22How turbulence is (presumably) driven: The Magnetorotational Instability(ref: Book by Phil Armitage)
23Magnetorotational 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!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 disk
24Kelvin-Helmholtz Instability Now let‘s do this a bit better. We follow a discussion from thebook of Armitage.Kelvin-Helmholtz instability (shear instability):Photo credit: Beverly Shannon (1999)
25Kelvin-Helmholtz Instability However, in a rotating system the rotation can stabilize theKelvin-Helmholtz instability. The Rayleigh criterion says:instabilityA Keplerian disk has:Keplerian disksare Rayleigh-Stable
26Magnetorotational Instability Let 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:
27Magnetorotational Instability Now let‘s transform this to cylindrical coordinates. This is nottrivial. But let‘s 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:
28Magnetorotational Instability Taking our equations at ϕ=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 law
29Magnetorotational Instability Now define a local (x,y) coordinate system:Inserting this into the previous equations, and discarding quadraticterms, yields (after some calculation):
30Magnetorotational Instability Now let‘s look at an (x,y) displacement varying with height zand time t:Remember now that gas displacements carry along the B-field.Let‘s assume a weak vertical initial B-field. Then the displacementscreate x- and y- components of this B-field:
31Magnetorotational Instability These produce a magnetic tension force (from ):Alfvenvelocity
32Magnetorotational Instability The equation of motion then becomes:Combining them yields the following dispersion relation:
34Magnetorotational Instability Stable for:Conclusion: If the field is too strong, the disk is stable. SoMRI 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 acertain scale where the driving is the strongest.Let‘s assume magnetic equipartition:Then the instability occurs at:Scale larger thandisk thickness:Equipartition disk=stable
35Magnetorotational Instability Note: 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.This leads to so called „dead zones“ in disks.The debate for what causes turbulence in disks remains wide opentoday.