Simulating Two-Fluid MHD Dynamos And A Novel Paradigm for Geodesic Mesh MHD By Dinshaw Balsara, Sudip Garain (UND), Alex Lazarian, Siyao Xu (UWisc), Vladimir Florinski (UAH)
Turbulent Two-Fluid MHD Dynamos Dynamo action Amplifies strength of Magnetic field in a plasma and Increases the coherence length of the magnetic field. Small-scale dynamo has fastest growth; so we focus on that. Most astrophysical magnetic fields undergo dynamo action in partially ionized plasmas. Therefore, two-fluid, partially ionized plasmas constitute the focus on this study. Never studied before. Analytical theory for turbulent, small-scale dynamos in partially ionized plasmas makes two very important predictions (Xu & Lazarian 2016, 2017) – unique to partially ionized plasmas. We have verified those predictions via simulations. Applications to molecular clouds and early universe.
Single Fluid Small-scale Dynamos V/S Two-Fluid Small-scale Dynamos:- In both, the turbulent motions result in a stretch-twist-fold process which increases the field strength. Magnetic energy builds up fastest on the smallest scales. However, in order for the small scale magnetic fields to not quench the dynamo, there has also to be a small scale dissipation. For single fluid MHD (highly ionized plasma) turbulent diffusion provides small scale dissipation. For two-fluid MHD (partially ionized plasma) ion-neutral friction provides small scale dissipation. For very low ionization, the ions collide so infrequently with the neutrals that the KE of the neutrals is very inefficiently converted into magnetic energy. Small scale equipartition never reached.
Governing Equations for Partially-Ionized Fluids ¶ æ ö v 1 ( ) ( ) ( ) r + ×Ñ + Ñ + r ÑF + ´ Ñ´ = - ar r - i v v P B B v v ç ÷ i i i i i n i i n ¶ p è t ø 4 ¶ æ ö v ( ) ( ) r + ×Ñ + Ñ + r ÑF = - ar r - n v v P v v ç ÷ n n n n n n i n i ¶ è t ø V.V. I mp . = ar L V ~ 0.01 -- 0.05 pc for fiducial parameters AD A i ( ) ( ) r ¯ Trends: L as and L a s B AD i AD Þ ( pro tostellar cores also form on this length s c a le it is v. important) x - - 6 8 Recall: ~10 to 10 = pr In th e p a st: V B 4 was deemed too large for practical - A ion i computations -- The heavy ion approximation (HIA) was the compromise. HIA was found to discard essential physics -- HIA not used here .
First Prediction from theory:- linear exponential Magnetic field would initially Magnetic Field undergo exponential growth with time. Total Magnetic Field Once L AD reaches L driving , it undergoes linear growth with increasing time. Well-resolved 1024 3 zone, and upwards , simulations were needed o prove this. BW is/was unique machine for the task. Computations v.v. time-consuming!
Second Prediction from Theory:- The peak in the magnetic energy spectrum migrates initially to small scales. With increasing time, that peak migrates back to larger scales.
Geodesic Mesh MHD I) “On Being Round” Problem: Several Astrophysical systems are spherical; Codes for simulating them have been logically Cartesian. (r- q - f coordinates) Timestep and accuracy problems at poles! Example systems:- Accretion Disks and MRI – Done in Shearing Sheet boxes Jets propagating in pressure gradients around Galaxies Star and Planet Formation 7
Heliosphere Magnetospheres of planets Convection in the Sun Convection in AGB Stars Supernovae Possible uses in Galaxy formation Possible uses in NS-NS collisions Atmospheres of Proto- planets Global Weather Because of need for turbulence modeling, we need to learn how to do higher order MHD 8 optimally in spherical systems!
II) Geodesic Meshes and their Advantages – The Challenge of Meshing the Sphere :- The Computer Simulation of all such systems is hampered by the fact that spherical coordinate systems result in vanishingly small timesteps , and a loss of accuracy close to the poles . This is a coordinate singularity and should be removable. For General Relativistic systems, we want to go as close to the physical singularity at event horizon without blow-up. The Underlying Mesh should be free of these defects. It should give us the maximum possible angular isotropy . Icosahedron Spherical Icosahedron Level 4 zoning within each Level 1 sector division Level 0 sector level 1 sector. 9
Geodesic coordinate 2 Extrude the mesh in the radial direction to get a 3D mesh:- (Done here for a level 1 sector from Geodesic the previous page.) coordinate 1 Resulting zones have a shape called a fustrum. Radial direction 10
III) High Accuracy Divergence-Free MHD on Geodesic Meshes – Algorithmic Issues Built on the following four easy steps:- i) High order WENO Reconstruction on Unstructured Meshes. ii) Divergence-free reconstruction of magnetic fields. iii) Genuinely Multidimensional Riemann Solver. iv) High Order Temporal Update. Use Runge-Kutta or use ADER at high order. Let us address each of these very briefly in the next several transparencies and for the simplest case of second order accuracy. We have made all higher order extensions. Results shown in next section. This need for higher order accuracy is motivated by the fact that astrophysicists are beginning to face up to the presence of turbulence . Such problems have strong shocks ; we must handle shocks. Turbulence simulations always require the lowest possible numerical dissipation and dispersion . 11 High order accuracy is the only known way of beating down dissipation and dispersion.
III.1) High order WENO Reconstruction on Unstructured Meshes Central Stencil S 0 for target triangle T 0 E ach triangle starts with a single value for each ariable. v (Useful for smooth flow ; central stencil is most stable.) Our Goal is to use neighb or information to obtain the T s lopes in the target trian gle T :- T 6 0 5 ( ) x , y ( ) x , y 6 6 5 5 ( ) = + + ˆ ˆ u x y , u u x u y T S 0 0 S 0; x S 0; y 2 T ( ) T x y , 1 0 ( ) 2 2 x y , ( ) x , y Can be done b y atisfy s i ng the over-determined sys t e m :- T 1 1 ( ) 7 7 0,0 ( ) 7 T x , y 4 4 4 ( ) + = - ˆ ˆ x y , u x u y u u ; S 0; x 1 S 0; y 1 1 0 3 3 + = - ˆ ˆ u x u y u u ; S 0; x 2 S 0; y 2 2 0 T T ( ) x , y 8 3 + = - ˆ ˆ u x u y u u ; 8 8 ( ) S 0; x 3 S 0; y 3 3 0 x , y T 9 9 9 This is done in Least SQuare s sen e (LSQ s ) . 12
One-Sided Stencils S 1 , S 2 & S 3 for target triangle T 0 T T T T 6 6 (Useful at shocks are propagating from one or other 5 5 ( ) ( ) x , y ( ) x , y ( ) x , y x , y 6 6 6 6 side) 5 5 5 5 T T 2 2 T T ( ) ( ) T T x , y x y , 1 1 0 ( ) 0 ( ) 2 2 2 2 ( ) x y , ( ) x y , x , y x , y T T ( ) 1 1 1 1 ( ) 7 7 0,0 ( ) 7 7 7 T 0,0 ( ) 7 T x , y x , y 4 4 4 4 4 4 ( ) ( ) x y , x y , 3 3 3 3 T T ( ) T T ( ) 8 x , y 3 x , y 8 3 8 8 ( ) 8 8 ( ) x , y x , y T 9 9 T 9 9 9 9 Upwind biased stencil S 1 ; Flow features upwinded Upwind biased stencil S 2 ; Flow features upwinded towards left-lower corner of T 0 . towards right-lower corner of T 0 . T T 6 5 ( ) x , y ( ) x , y 6 6 5 5 The flow features can also be anisotropic, in which case a T 2 T ( ) T one-sided, upwind-biased stencil might be more appropriate. x , y 1 0 ( ) 2 2 ( ) x y , x , y T 1 1 ( ) We show three possible one-sided stencils shown by the three 7 7 ( ) 0,0 T 7 x , y 4 4 4 sets of triangles {T 0 , T 1 , T 4 , T 5 }, {T 0 , T 2 , T 6 , T 7 } and {T 0 , T 3 , ( ) x y , 3 3 T 8 , T 9 }. The stencils are shown by the solid lines. They T T ( ) x , y correspond to flow features that might need to be upwinded 8 3 8 8 ( ) x , y T 9 9 towards one of the three vertices of triangle T 0 . 9 Upwind biased stencil S 3 ; Flow features upwinded 13 towards upper corner of T 0 .
The elements form a 5-faced shape ca lled a frustrum . III.2) Divergence-free reconstruction of magnetic fields Within each frustrum we make a zone-centered WEN O top B reconstruction that is not divergence-free :- 1 r ( ) ( ) ( ) ( ) = + D + D + D B x y z , , B B x B y B z ; x x ;0 x x y x z x top 3 E qf ( ) ( ) ( ) ( ) top 2 E qf = + D + D + D B x y z , , B B x B y B z ; y y ;0 x y y y z y ( ) ( ) ( ) ( ) = + D + D + D B x y z , , B B x B y B z z z ;0 x z y z z z top 1 E qf 3 2 Our Goal is to obtain a magnetic field reconstruction 1 tha t is t he c l o s e st po ss ib l e t o the ne abo e o v , while E r a l s o rem i a n i n g divergenc e-free . W e pi c k : - 2 B 1 B qf qf 3 B ( ) ( ) 2 E = + + + + - 2 B x y z , , a a x a y a z a x C qf 3 E r x 0 x y z xx xx botm B r ( ) ( ) r + - + - a xy C a xz C xy xy xz xz ( ) ( ) = + + + + - 2 B x y z , , b b x b y b z b y C y 0 x y z yy yy ( ) ( ) botm 3 + - + - E qf b xy C b xz C xy xy xz xz botm 2 E qf ( ) ( ) = + + + + 2 - B x y z , , c c x c y c z c z C z 0 x y z zz zz ( ) ( ) 14 botm 1 E qf + - + - c xz C c yz C xz xz yz yz
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