1. Compressible MHD turbulence in molecular clouds Lucy Liuxuan Zhang Prof. Chris Matzner University of Toronto

2. Dynamics of molecular clouds - I Problem:  expected cloud collapse time ≤ 3x10 6 yrs  expected cloud lifetimes ≥ 3x10 7 yrs Environment:  n H2 =10 3 /cm 3, T=10K, ∂E/∂t=0.4L ☼  L=2pc, c s ≈0.2 km/s → t s ≡L/c s ≈10Myr  v a ≥ σ v » c s  isothermal approximation  existence of B field and turbulence motions

3. Dynamics of molecular clouds - II Possible solution (current opinions)  Turbulence as “turbulent pressure” to support the cloud from self-gravity  Magnetic fields as cushion to reduce dissipation rate  Supersonic, sub-alfvenic turbulence persists for more than flow crossing time over cloud size L

4. Intro hydrodynamics Lagrangian (SPH) Eulerian (grid-based)  Advantages large dynamical range in mass Computationally faster by several orders of magnitude Easy to implement and to parallelize  Basic principal: solve the integral Euler equations on a Cartesian grid by computing the flux of mass, momentum and energy across grid cell boundaries

5. Equations (no source term) 1.∂ t ρ+  (ρv)=0 2.∂ t (ρv)+  (ρvv+P  δ-bb)=0 3.∂ t e+  [(e+P  )v-bb·v]=0 4.e=ρv 2 /2+p/( γ -1)+b 2 /2 5.∂ t b=  x (v x b) 6.  ·b=0 7.P  =p+b 2 /2 P  total pressure, p gas pressure, є thermal

6. Our numerical model - ISOTHERMAL Adiabatic version:  “A Free, Fast, Simple and Efficient TVD MHD code” by Ue-Li Pen, Phil Arras, ShingKwong Wong (astro- ph/0305088 2003) Isothermal version ( γ=1) :  Eq(4) does not make sense!!  But then, we don’t have to solve for energy separately to update the pressure because p=ρc s 2 where c s is constant in space & time.  Eq(4) e=ρv 2 /2+p/( γ -1)+b 2 /2 and the quantity p drop out from the system  Eq(7) P  =ρc s 2 +b 2 /2 → P  =ρc s 2 +b 2 /2.

7. Energy dissipation in MHD turbulence Molecular clouds:  Isothermal, constant c s in space and time Initial conditions:  Cubic, periodic box of size L  Plasma of uniform density ρ 0  Uniform B field B 0 =(B 0,0,0) where b 0 =(ρ 0 c s 2 /β) 1/2 =B 0 /(4π) 1/2 Velocity perturbation δv:  Time intervals ∆t = 0.001 t s  Realization of Gaussian random field  Power spectrum |δv 2 |  k 6 exp(-8k/k pk ) 2, k pk =8(2π/L)   ·δv=0 divergenceless  ∫ρ δv=0 zero net momentum  ∂ t E =10 3 ρ 0 L 2 c s 3 → ∆E= ∂ t E · t energy normalization

8. Some results (partial) Comparison with “Dissipation in compressible magnetodydrodynamic turbulence” by Stone, Ostriker, Gammie Є=Є k +Є b +Є th Є th =p/( γ- 1), γ=1 → Є th =ρc s 2 log(ρ/ρ 0 )

9. Open questions Can molecular clouds be supported against gravitational collapse solely by magnetic turbulence? If not, how important a role MHD turbulence plays? What other mechanisms are realistic?