We perform the first study of magnetohydrodynamic processes in the protolunar disk ( PLD ) .
With the use of published data on the chemical composition of the PLD , along with existing analytical models of the disk structure , we show that the high temperatures that were prevalent in the disk would have led to ionization of Na , K , SiO , Zn and , to a lesser extent , O _ { 2 } .
For simplicity , we assume that the disk has a vapor structure .
The resulting ionization fractions , together with a relatively weak magnetic field , possibly of planetary origin , would have been sufficient to trigger the magneto-rotational instability , or MRI , as demonstrated by the fact that the Elsasser criterion was met in the PLD : a magnetic field embedded in the flow would have diffused more slowly than the growth rate of the linear perturbations .
We calculate the intensity of the resulting magnetohydrodynamic turbulence , as parameterized by the dimensionless ratio \alpha of turbulent stresses to gas pressure , and obtain maximum values \alpha \sim 10 ^ { -2 } along most of the vertical extent of the disk , and at different orbital radii .
This indicates that , under these conditions , turbulent mixing within the PLD due to the MRI was likely capable of transporting isotopic and chemical species efficiently .
To test these results in a conservative manner , we carry out a numerical magnetohydrodynamic simulation of a small , rectangular patch of the PLD , located at 4 Earth radii ( r _ { E } ) from the center of the Earth , and assuming once again that the disk is completely gaseous .
We use a polytrope-like equation of state .
The rectangular patch is threaded initially by a vertical magnetic field with zero net magnetic flux .
This field configuration is known to produce relatively weak MRI turbulence in studies of astrophysical accretion disks .
We accordingly obtain turbulence with an average intensity \alpha \sim 7 \times 10 ^ { -6 } over the course of 280 orbital periods ( 133 days at 4 r _ { E } ) .
Despite this relatively low value of \alpha , the effective turbulent diffusivity D \sim 10 ^ { 10 } -10 ^ { 11 } cm ^ { 2 } s ^ { -1 } of a passive tracer introduced in the flow is large enough to allow the tracer to spread across a radial distance of 10 r _ { E } in \sim 13 - 129 yr , less than the estimated cooling time of the PLD of \sim 250 yr. Further improvements to our model will need to incorporate the energy balance in the disk , a complete two-phase structure , and a more realistic equation of state .