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Scientific paper
Dec 2008
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2008apjs..179..553j&link_type=abstract
The Astrophysical Journal Supplement Series, Volume 179, Issue 2, pp. 553-555.
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1
Scientific paper
The descriptions of some of the numerical tests in our original paper are incomplete, making reproduction of the results difficult. We provide the missing details here. The relevant tests are described in § 4 of the original paper (Figs. 8-11).
We use the analytical solutions outlined by B. M. Johnson (ApJ, 660, 1375 [2007]) as the initial conditions for the linear tests in B. M. Johnson et al. (ApJS, 177, 373, [2008]). The incompressive solution is given by the real parts of expressions (80)-(82) of that paper. For imaginary ω and ω˜ and a Keplerian rotation profile, these areδv=δv˜cos(k˙x+π/4) (1)andδvA=δv˜Acos(k˙x-π/4), (2)withδv˜=Ai(k2x-k2,kxky-(k2)/2α,kxkz+(k2ky)/(2αkz)) (3)andδv˜A=-(vA˙k)/|ω|Ai(k2x-k2,kxky+2αk2z,kxkz-2αkykz), (4)where Ai=ɛcsH|ω˜|/Ωsqrt(|ω|Ω/(2|ω˜2|k2+Ω2k2z)) (5)andα=Ω|ω|/(|ω˜2|). (6)Here, H=cs/Ω is the disk scale height, k is the initial wavenumber of the perturbation, and ɛ is an arbitrary perturbation amplitude; other symbols have their usual meanings. These solutions have been normalized to the correct dimensional units.2 The density perturbation is given byδρ/(ρ0)=[-(vA)/(cs)˙(δv˜A)/(cs)+2Ω/(csk)((kx)/kŷ+(ky)/2kx̂)˙(δv˜)/(cs)cos(k˙x-π/4). (7)
The unstable branch of the incompressive dispersion relation is|ω˜2|=((kzΩ)/k)2[sqrt(1+((4kvA˙k)/(kzΩ))2)-1 (8)and|ω|=sqrt(|ω˜2|-(vA˙k)2). (9)For our choice of initial parameters, vA=sqrt(15/16)(Ω/kz)ẑ and Hk=2π(-2/10,1/10,1), these become|ω˜2|=Ω25/21(sqrt(67)-2)~=1.47Ω2 (10)and|ω|=Ωsqrt(5/21)(sqrt(67)-95/16)1/2~=0.732Ω. (11)The perturbations in this limit are given byδv˜=-(Ai)/(H2)(2π)2(101/100,1/50+21/40α,1/5-21/400α) (12)andδv˜A=sqrt(15/16)Ω/|ω|(Ai)/(H2)(2π)2(101/100,1/50-2α,1/5+α/5), (13)with(Ai)/(H2)=ɛcs|ω|/2πΩ(2/αsqrt(67))1/2 (14)andα=(sqrt(21)(sqrt(67)-95/16)1/2)/(sqrt(5)(sqrt(67)-2))~=0.497. (15)Dividing through by an overall factor of H2(k2x-k2)=-(2π)2(101/100) gives the initial conditions quoted in our original paper (with ɛ=10-6 and cs=Ω=ρ0=1).
We make comparisons based upon the amplitude of the solution, i.e. δv˜ and δv˜A rather than δv and δvA. In Figures 8-10 of our original paper, then, the quantity that is being plotted is δv˜2+δv˜2A. To extract these quantities from the code, we perform spatial sine and cosine Fourier transforms in shearing coordinates on each of the velocity and magnetic field components, and sum the squares of the transforms. As a concrete example, the cosine transform of the radial velocity component is given by(δṽx[tn])numerical=2/(NxNyNz)Σi=1NxΣj=1NyΣk=1Nzvxnijkcos(k[tn˙xijk), (16)where k(t)=k(0)+qΩkytx̂. Since the solution as expressed above breaks down as ω transitions from imaginary to real, we calculate the analytical amplitudes for the incompressive tests based upon an integration of the full set of linear equations.3
The compressive solution is given by the real part of expressions (83)-(85) of B. M. Johnson (ApJ, 660, 1375 [2007]):(δv,δvA,δρ)=(δv˜,δv˜A,δρ˜)cos(k˙x), (17)withδv˜=ω/ω˜Ac((ω2)/(k2)k-vA˙kvA), (18)δv˜A=(ω2)/ω˜Ac(vA-(vA˙k)/(k2)k), (19)andδρ˜/(ρ0)=ω˜Ac, (20)where Ac=ɛHksqrt(ωΩ/(ω4-(vA˙k)2c2sk2)). (21)
Our choice of initial parameters for this test, vA=cs(0.1,0.2,0.0) and Hk=4π(-2,1,1), gives vA˙k=0, so that the nonzero solution to the compressive dispersion relation isω2=(c2s+v2A)k2. (22)The perturbations in this limit are given by(δv˜,δv˜A,δρ˜/ρ0)=ɛ(vAsqrt(1+β)k̂,vA,1)(Hksqrt(β/1+β))1/2, (23)where β=c2s/v2A. For our initial conditions (β=20), this is(δv˜,δv˜A,δρ˜/ρ0)=ɛ((cs)/2sqrt(7/10)(Hk)/4π,vA,1)(8πsqrt(10/7))1/2, (24)which matches the numbers given in our original paper (with ɛ=10-6 and cs=Ω=ρ0=1).
Figure 11 of our original paper shows the evolution of the azimuthal component of δv˜A. The numerical results are(δṽAy[tn])numerical=2/(NxNyNz)Σi=1NxΣj=1NyΣk=1Nz((bynijk)/(sqrt(4πρ0))-vAy[tn])cos(k[tn]˙xijk). (25)and the analytical results are calculated within the code using the time dependent version of expression (24), i.e.(δṽAy[tn])analytical=vAy[tn](8πsqrt(10/7))1/2cos(Σn'=0nω[tn']dtn'), (26)with dt0=0.
As a final practical consideration, implementing the solutions as described above can introduce divergence into the initial conditions. To avoid this, we calculate the vector potential in the Coulomb gauge (k˙δA=0) for the above solutions and numerically calculate its curl to obtain the initial magnetic field perturbation. The perturbed vector potential is(δA)/(sqrt(4πρ0))=-(vA˙k)/|ω|Aikz(2α[(k2x)/(k2)-1,2α(kxky)/(k2)-1,2α(kxkz)/(k2)+(ky)/(kz))cos(k˙x+π/4) (27)for the incompressive solution and(δA)/(sqrt(4πρ0))=(ω2vAXk)/(ω˜k2)Acsin(k˙x) (28)for the compressive solution. For our initial conditions, these reduce to(δA)/(sqrt(4πρ0))=ɛ(csH)/14(1/30αsqrt(67))1/2(202α,4α+105,40α-21/2)cos(k˙x+π/4) (29)for the incompressive solution (with α given by expression [15]), and(δA)/(sqrt(4πρ0))=ɛ(csH)/60(1/πsqrt(5/14))1/2(2,-1,5)sin(k˙x) (30)for the compressive solution.
This work was performed under the auspices of Lawrence Livermore National Security, LLC, (LLNS) under Contract DE-AC52-07NA27344.
Gammie Charles F.
Guan Xiaoyue
Johnson Bryan M.
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