Numerical Study of Compressible Isothermal Magnetohydrodynamic Turbulence

Transcription

1 Numerical Study of Compressible Isothermal Magnetohydrodynamic Turbulence Junseong Park, Dongsu Ryu Dept. of Physics, Ulsan National Institute of Science and Technology. Ulsan, Korea 2016 KNAG meeting KASI, Daejeon, Korea

2 Purpose of Study Smith et al studied the distribution of Mach number in driven supersonic HD and MHD turbulence. However, they did not consider fast and slow Mach numbers distribution in molecular cloud conditions. In this study we will identify differences between the two kinds of shocks, because density and velocity structures in turbulence are associated with the fast and slow shocks. 1. The main goal is to identify the formulas for fast and slow Mach numbers and energy dissipation rate at shocks and understand the characteristic properties of fast and slow MHD shocks. 2. We studied MHD turbulence in a variety of astrophysical environments by performing numerical simulations of isothermal, compressible MHD turbulence with resolution up to Contents MHD equations of compressible, isothermal gas - simulation initial conditions Formula for Fast and Slow Mach numbers - fast and slow Mach number D structures - fast and slow Mach number distribution Formula for Energy dissipation rate at shocks - fast and slow energy dissipation rate Conclusion and Future work

3 MHD equations of compressible, isothermal gas ρ + ρv = 0 t v t + v v + c 2 s ρ ρ 1 ρ B v B = 0 t B B = 0 CONTINUITY MOMENTUM INDUCTION with the additional constraint for the absence of magnetic monopoles, B = 0 Simulation initial conditions Isothermal TVD Code (Kim, et al. 1999) Turbulence was driven by solenoidal ( v = 0) forcing (Stone et al. 1999; Mac Low 1999) Initial conditions: - computational domain L 0 = L x = L y = L z = 10 with periodic boundaries - uniform density ρ 0, magnetic field B 0 = 2/β, 0,0 where β = P g /P B = 2c s 2 /V A 2 = 0.1, 1, 10 - velocity forcing drawn from Gaussian random field, P k k 6 exp( 8k/k peak ), where k peak = 2k 0 (k 0 = 2π/L 0 ) - M s = v rms /c s = 0.5, 1,, 8 at saturation with time intervals t = 0.001L 0 /c s - flow length scale L d = 2π/k peak = L 0 /2, dynamical time t d = L d /v rms = L 0 / 2M s c s - resolution: 128, 256, grid cells - self-gravity was ignored.

4 MHD equations of compressible, isothermal gas ρ + ρv = 0 t v t + v v + c 2 s ρ ρ 1 ρ B v B = 0 t B B = 0 CONTINUITY MOMENTUM INDUCTION with the additional constraint for the absence of magnetic monopoles, B = 0 Simulation initial conditions Saturation was reached around ~2t d Comparison of observation & simulation : - cloud clump of size L 0 = 2pc, T = 10K, c s 0.2km/s, n H2 = 10 cm (observed MCs) - sound crossing time: t s = L 0 /c s ~10Myr, dynamical time t d = L d /v rms = L 0 / 2M s c s = 5Myr/M s - B 0 = 1.4μGβ 1/2 T/10K 1/2 n H2 /10 2 cm 1/2 ~ 14, 4.4, 1.4 μg for β = 0.1, 1, 10

5 Comparison of sliced density images (n = 512, t/t d = 9) M s = 0. 5 (subsonic) M s = 1 (transonic) M s = (supersonic) M s = 8 (supersonic) β = 0. 1 (sub-alfvénic) β = 1 (trans-alfvénic) B 0 β = 10 (super-alfvénic) Filaments and sheets with high density are formed in a flow with higher M s Supersonic turbulence Dense core formation

6 Formula for Fast and Slow Mach numbers Shock jump conditions of both sides of the shock front are as follows: ρ 1 v 1 = ρ 2 v 2 ρ 1 v c 2 s ρ B 1 2 = ρ 2 v c 2 s ρ B 2 2 (1) (2) plane-parallel shock pre-shock Post-shock ρ 1, v 1, v 1 ρ 2, v 2, v 2 B, B 1, c s B, B 2, c s ρ 1 v 1 v 1 B B 1 = ρ 2 v 2 v 2 B B 2 v 1 B 1 v 1 B = v 2 B 2 v 2 B () (4) n sh : parallel to shock normal : perpendicular to shock normal B, c s : same in pre-shock and Post-shock χ = ρ 2 /ρ 1 = v 1 /v 2 > 1 preferred frame for shock surface B 1 /B = v 1 /v 1, B 2 /B = v 2 /v 2 = v 2 χ/v 1 From 2, equations, we can rewrite as follows: ρ 1 v c 2 s ρ B 1 2 = ρ 2 1v 1 χ + χc s 2 ρ B 2 2 ρ 1 v 2 1 B 2 B 1 = ρ 1v 1 χ B 2 B 2 2 (5) (6) B 2 substituted into the equation 5, then 2ρ 1 v 6 1 4B B 1 ρ c 2 s ρ 1 χ + B 2 1 ρ 2 1 χ 2 v B 2 c 2 s ρ B 2 B B 4 ρ 1 χ 2 v 2 1 2B 4 c 2 s ρ 1 χ = 0

7 Formula for Fast and Slow Mach numbers Shock jump conditions of both sides of the shock front are as follows: ρ 1 v 1 = ρ 2 v 2 ρ 1 v c 2 s ρ B 1 2 = ρ 2 v c 2 s ρ B 2 2 ρ 1 v 1 v 1 B B 1 = ρ 2 v 2 v 2 B B 2 v 1 B 1 v 1 B = v 2 B 2 v 2 B (1) (2) () (4) divide by 2ρ 1 c s 6, and set M s = v 1 /c s, c A = B / ρ 1, c A = B 1 / ρ 1, α = c A /c s, α = c A /c s M s α α 2 χ α 2 χ 2 M s 4 + α α 2 + α 2 χ 2 M s 2 α 4 χ = 0 (7) the phase velocity of fast and slow modes is derived from equation 7 with χ = 1 as follows: 2 c fa,sl 2 = 1 c s α 2 + α 2 ± 1 + α α 2 2 4α Fast and Slow Mach numbers M fa = M s c s /c fa, M sl = M s c s /c sl (8)

8 Formula for Fast and Slow Mach numbers How to find M s from 6th order equation? let x = M s 2 x + C a x 2 + C b x 2 + C c = 0 where C a = 1 + 2α α 2 χ α 2 χ 2 C b = α α 2 + α 2 χ 2 C c = α 4 χ Compute Q = C a 2 C b /9, R = 2C a 9C a C b + 27C C /54 If Q 2 < R, then the cubic equation has three real roots (Press et al. 1986) as follows: x 1 = 2 Qcos θ C a x 2 = 2 Qcos θ + 2π C a x = 2 Qcos θ 2π C a where θ = arccos R/ Q Otherwise Q 2 > R, the cubic equation has only one real root. Compute A = sgn R R + R 2 Q 1, B = Q/A A 0 or 0 A = 0 and calculate the real root as follows: x 1 = A + B a/

9 Formula for Fast and Slow Mach numbers How to find M s from 6th order equation? let x = M s 2 x + C a x 2 + C b x 2 + C c = 0 where C a = 1 + 2α α 2 χ α 2 χ 2 C b = α α 2 + α 2 χ 2 C c = α 4 χ Compute Q = C a 2 C b /9, R = 2C a 9C a C b + 27C C /54 If Q 2 < R, then the cubic equation has three real roots (Press et al. 1986) as follows: x 1 = 2 Qcos θ slow mode C a x 2 = 2 Qcos θ + 2π fast mode C a x = 2 Qcos θ 2π C a where θ = arccos R/ Q Otherwise Q 2 > R, the cubic equation has only one real root. Compute A = sgn R R + R 2 Q 1, B = Q/A A 0 or 0 A = 0 and calculate the real root as follows: x 1 = A + B a/ fast mode 0 slow mode

10 Find shocks - Shock cells: ρ i+1 ρ i 1 ρ i+1 +ρ i 1 = χ 1 χ+1 > (i.e. χ = ) and v < 0 - Shock center: v is local minimum and labeled this center as part of a shock surface. - Shock quantities: max(ρ i+2, ρ i 2 ) is post-shock and min(ρ i+2, ρ i 2 ) is pre-shock Find Fast and Slow shocks - Fast shock B 2 /B 1 1, M fa 1.0, χ Slow shock 0 < B 2 /B 1 < 1, 1 < M sl < c A /c sl, 1 < χ < 2α 2 + α α 2 + α α 2 /4 ρ 1 v c 2 s ρ B 1 2 = ρ 2 1v 1 χ + χc s 2 ρ B 2 2 ρ 1 v 2 1 B 2 B 1 = ρ 2 1v 1 χ B 2 B 2 (5) (6) with B 2 = 0

11 Fast Mach number D structures (n = 512, t/t d = 9) M s = 0. 5 (subsonic) M s = 1 (transonic) M s = (supersonic) M s = 8 (supersonic) β = 0. 1 (sub-alfvénic) β = 1 (trans-alfvénic) β = 10 (super-alfvénic) x y B 0 z The locations of identified fast shocks along with the gas density, velocity and magnetic field distributions

12 Fast Mach number D structures (n = 512, t/t d = 9) M s = 0. 5 (subsonic) M s = 1 (transonic) M s = (supersonic) M s = 8 (supersonic) β = 0. 1 (sub-alfvénic) anisotropic β = 1 (trans-alfvénic) β = 10 (super-alfvénic) isotropic z x y B 0 The turbulence remains isotropic for M s = 8 and β = 10, but when it becomes M s = 0.5 and β = 0.1, the turbulence becomes highly anisotropic.

13 Slow Mach number D structures (n = 512, t/t d = 9) M s = 0. 5 (subsonic) M s = 1 (transonic) M s = (supersonic) M s = 8 (supersonic) β = 0. 1 (sub-alfvénic) β = 1 (trans-alfvénic) β = 10 (super-alfvénic) x y B 0 z The locations of identified slow shocks along with the gas density, velocity and magnetic field distributions

14 Slow Mach number D structures (n = 512, t/t d = 9) M s = 0. 5 (subsonic) M s = 1 (transonic) M s = (supersonic) M s = 8 (supersonic) β = 0. 1 (sub-alfvénic) anisotropic β = 1 (trans-alfvénic) β = 10 (super-alfvénic) isotropic z x y B 0 The turbulence remains isotropic for M s = 8 and β = 10, but when it becomes M s = 0.5 and β = 0.1, the turbulence becomes highly anisotropic.

15 Fast Mach number distribution with the number of shocks (t/t d = 4~9) β = 0.1 (sub-alfvénic) β = 1 (trans-alfvénic) β = 10 (super-alfvénic) The start in the power law is found to occur at M fa min given by M fa min ~2.5M s 0.2 /n 0.1 dn = na b M dm fa M fa where a~2.7 ± 0.2 s

16 Fast Mach number distribution with the number of shocks (t/t d = 4~9) β = 0.1 (sub-alfvénic) β = 1 (trans-alfvénic) β = 10 (super-alfvénic) The start in the power law is found to occur at M fa min given by M fa min ~2.5M s 0.2 /n 0.1 dn = na b M dm fa M fa where a~2.7 ± 0.2 s Power-law index b with M s b~m s 0.6 M s b As M s increases, b becomes smaller, but it becomes larger with increasing resolution Power law index is not significantly affected by plasma beta, and it follows 0.6 a power law b~m s

17 Slow Mach number distribution with the number of shocks (t/t d = 4~9) β = 0.1 (sub-alfvénic) β = 1 (trans-alfvénic) β = 10 (super-alfvénic) The start in the power law is found to occur at M sl min given by M sl min ~2.5M s 0.2 /n 0.1 dn = na b M dm sl M sl where a~2.7 ± 0.2 s

18 Slow Mach number distribution with the number of shocks (t/t d = 4~9) β = 0.1 (sub-alfvénic) β = 1 (trans-alfvénic) β = 10 (super-alfvénic) The start in the power law is found to occur at M sl min given by M sl min ~2.5M s 0.2 /n 0.1 dn = na b M dm sl M sl where a~2.7 ± 0.2 s Power-law index b with M s b~m s 0.49 M s b As M s increases, b becomes smaller, but it becomes larger with increasing resolution Power law index is not significantly affected by plasma beta, and it follows 0.49 a power law b~m s This relationship work very well for super-alfvénic case, but breaks down at sub/trans-alfvénic cases with M s =, 8

19 Slow Mach number distribution with the number of shocks (t/t d = 4~9) β = 0.1 (sub-alfvénic) β = 1 (trans-alfvénic) β = 10 (super-alfvénic) The start in the power law is found to occur at M sl min given by M sl min ~2.5M s 0.2 /n 0.1 dn = na b M dm sl M sl where a~2.7 ± 0.2 s Upstream Alfvénic Mach number M A (= ρ 1 v 1 /B 1 ) with M s fast shock slow shock When the turbulence becomes sub-alfvénic or trans-alfvénic, M A 0.2, anisotropies arise in the gas, because the back reaction of the magnetic field onto the flow is extremely strong for flows perpendicular to the magnetic field lines (Cho & Lazarian, 200; Esquivel & Lazarian, 2011) magnetic field and anisotropic become increasingly important

20 Formula for energy dissipation rate at shock When we adopt an isothermal equation of state (Mouschovias et al. 1974) that mimics the effect of radiative losses, the total energy is not conserved if shock waves are present in the system. E = 1 2 ρv2 + PlnP B2 dv From the conservation equation of energy, we obtain t 1 2 ρv2 + PlnP B ρv2 + PlnP + P v + B v B = 0 where v = v e + v e, B = B e + B e t 1 2 ρv2 + PlnP B2 + x 1 2 ρv2 + PlnP + P + B 2 v B B v + B v = 0 Some of the energy is dissipated at shocks with the isothermal conditions.

21 Formula for energy dissipation rate at shock When we adopt an isothermal equation of state (Mouschovias et al. 1974) that mimics the effect of radiative losses, the total energy is not conserved if shock waves are present in the system. E = 1 2 ρv2 + PlnP B2 dv From the conservation equation of energy, we obtain t 1 2 ρv2 + PlnP B ρv2 + PlnP + P v + B v B = 0 where v = v e + v e, B = B e + B e t 1 2 ρv2 + PlnP B2 + 1 x 2 ρv2 + PlnP + P + B 2 v B B v + B v = 0 The energy dissipation rate Q at shock discontinuity Q = 1 2 ρv2 + PlnP + P + B 2 v B B v + B v ρv2 + PlnP + P + B 2 v B B v + B v 2

22 Formula for energy dissipation rate at shock The energy dissipation rate Q at shock discontinuity Q = 1 2 ρv2 + PlnP + P + B 2 v B B v + B v ρv2 + PlnP + P + B 2 v B B v + B v 2 In the preferred frame for shock surface B /B = v /v then B 2 v B B v + B v = 0 and with ρ 1 v 1 = ρ 2 v 2 or P 1 v 1 = P 2 v 2 energy dissipation rate Q becomes Q = 1 2 ρ 1v ρ 1 lnρ 1 c s 2 v ρ 2v ρ 2 lnρ 2 c s 2 v 2 After some arithmetic manipulation Q = M s ρ 1 c 1 s 2 M s χ M s 2 α 2 2 α 1 M s 2 2 α 2 M 2 s α 2 χ 2 lnχ,for M s 1, most of the upstream flow energy is dissipated Q ~ M s

23 Fast and Slow Energy dissipation rate Q at shock (t/t d = 4~9) Q = 1 2 : 1 2 ρv2 + PlnP + P + B 2 v B B v + B v β = 0.1 (sub-alfvénic) β = 1 (trans-alfvénic) β = 10 (super-alfvénic) ~M fa Fast shock ~M sl Slow shock Q fa,sl increases with fast and slow Mach numbers, asymptotically follows a power law Q fa,sl ~M fa,sl The energy is dissipated by a small range of stronger shocks

24 Fast and Slow Energy dissipation rate Q at shock (t/t d = 4~9) Q = 1 2 : 1 2 ρv2 + PlnP + P + B 2 v B B v + B v β = 0.1 (sub-alfvénic) β = 1 (trans-alfvénic) β = 10 (super-alfvénic) ~M fa Fast shock ~M sl Slow shock How much percent of the turbulence energy is dissipated due to fast and slow shocks?

25 Conclusion We studied compressible isothermal MHD turbulence in a variety of M s and β, with resolution up to 1024 The power law for fast and slow Mach numbers distribution with the number of shocks - dn = na b M dm fa,sl M fa,sl s where a~2.7 ± as M s increases, b becomes smaller, but it becomes larger with increasing resolution Fast and Slow Energy dissipation rate at shock - as the plasma beta β increases, Q fa,sl becomes smaller. - Q fa,sl increases with fast and slow Mach numbers, asymptotically follows a power law Q fa,sl ~M fa,sl - energy is dissipated by a small range of stronger shocks Future work No physical dissipation terms are modelled, but numerical diffusion is unavoidable. In saturation Energy injection = Energy dissipation at shocks + Energy dissipation due to viscosity & resistivity how to quantify this value is the next project!!