Clément Surville, Lucio Mayer. Insitute for Computational Science, University of Zürich. Yann Alibert. Institute of Physics, University of Bern

Transcription

1 The code a multi-fluid method for the formation of primordial bodies in protoplanetary disks Carving through the Codes : Challenges in Computational Astrophysics Clément Surville, Lucio Mayer Insitute for Computational Science, University of Zürich Yann Alibert Institute of Physics, University of Bern Davos, February th 7

2 The code Protoplanetary systems are composed of a proto-star, a gas disk (due to angular momentum) and solid components embeded (dust, pebbles, planetesimals, planets). The level of complexity of the dynamics can be high : instabilities, accretion, self-gravity and spiral waves, etc... In the lowest level of complexity, the system looks like as steady with : Spherical gravitational field (star), Axisymetric gas flow, with no radial accretion (V r = ) This system is not necessary easy to evolve numerically!

3 The code Finite Volume of the code RoSSBi I designed the code RoSSBi from scratch to address the dynamics of protoplanetary disks. The main properties are : D polar coordinates, or D spherical, to fit the main symmetry of the system. Finite Volume Method approach (FVM), on a fixed grid. At least second order in time and space. I try to be as close as possible to the Euler equations of conservation : t+ t t+ t t Euler dv dt = Flux ds dt t V t S t+ t () + Source dv dt t V To be second order in time, we need to approximate for example the sources as : Source = Source(t) + t Source(t+ t/) Source(t) t/

4 The code Toward a The of is due to the radial variation of the steady low complexity solution of the system, which can be the MMSN model. The radial velocity is null, and the azimuthal velocity of this solution is : V (r) = r Ω K (r) + rσ (r) r P (r). () The steady axisymmetric Euler equations reduce to : r r rp P (r) = σ Vθ σ (r) V (r) σ σ (r) V K (r) + P P (r), () r r r Integrating in space with the conservative form of the FVM gives : [ ] P r + r + P (r)r r θ = σ Vθ σ (r)v r (r)dr θ r + r + σ σ (r)v r K (r)dr θ + P P (r)dr θ r ()

5 The code Toward a To obtain a, we use the following relations : r + r σ (r)v (r)dr θ = r + r σ (r)v K (r)dr θ + = [P (r)r] r+ r θ r + r + r r + r r P (r)dr θ + σ (r)v r K (r)dr θ P (r)dr θ r () The two last integrals can be approximated at second order, as long as the same values are used for the Keplerian and pressure source terms.

6 The code The RoSSBi : results x 8 x Radial Velocity Error = rot = rot t = rot Radius [AU] Radial Velocity Error = rot = rot t = rot Radius [AU] x x. Density Error.. Density Error.8 = rot = rot t = rot Radius [AU] t = rot t = rot t = rot Radius [AU] Unbalanced Scheme Balanced Scheme

7 The code Parallel approach and performances We use OpenMP paralellism on local blocks, and MPI parallelism for the whole grid, to perform global D simulations. π,,,,,,,,,,,, R in,, Radius,, R out

8 The code Parallel approach and performances Production runs on the Piz Daint Swiss system scale linearly for blocks of 8 : up to nodes (9 threads) for (N r, N θ ) = (8, 9) up to nodes (8 threads) for (N r, Nθ ) = (9, 89) Relative speedup (8 9) (9 89) Linear scaling Number of nodes (MPI tasks) Scaling as function of node numbers Efficiency..8. (8 9) (9 89). Number of nodes (MPI tasks) Efficiency to linear scaling

9 The code

10 The code The motion of a solid grain embeded in protoplanetary disks is driven by different forces : m a(grain) = G(star) + F(friction) + F(collisions) () As most of the acceleration is compensed by the star gravity, the dominant force is the drag friction for small grains. As long as the distance of efficiency of the drag is shorter than the mean distance between grains, collisions are unlikely. The grains will have a collective motion imposed by the gas, and thus can be treated as a fluid. The for coupled fluids, is when the friction is on short timescale ( small grains). Resolving the friction explicitely will impose timesteps shorter than for advection, and make the solution slow and inaccurate.

11 Implicit solution of the We consider the impulsion of the gas and particle fluids, P g = σgvg and P p = σpvp, respectively, fully coupled by the. ( ) tpg = A g + Ω k (r)s t P p σp Pg, (7) σ g ( ) tpp = Ap Ωk (r)s t σ p P p Pg. (8) σ g If we introduce A = A p ɛag, P = P p ɛpg, and the drag frequency ω p = Ωk (r)s t, one obtains t P = A ( + ɛ) ω p P. (9) A solution over the timestep, t [t, t + t] is P(t ) = P(t) exp[ ( + ɛ)ω pt ] + A(t) ( + ɛ)ω p ( exp[ ( + ɛ)ωpt ] ), ()

12 Implicit solution of the To keep the second order accuracy of the advection and not couple advection with friction, we can get rid of the A term because : (i) ( + ɛ)ω p t <, third order term (superposition of solutions) (ii) ( + ɛ)ω p t >, short friction timescale limit Error of the term in A (+ǫ)ω p t

13 Implicit solution of the Then the time integration of the motion equations becomes : t+ t t P g(t + t) Pg(t) = A gdt + ω p P(t) exp[ ( + ɛ)ωpt ]dt, t t+ t t P p(t + t) Pp(t) = A pdt ω p P(t) exp[ ( + ɛ)ωpt ]dt, t () We can thus compute implicitely the time integral of the friction, and obtain : t ω p P(t) exp[ ( + ɛ)ω pt ]dt = P(t) ( + ɛ) () [ exp[ ( + ɛ)ω p t]]. Keeping in mind the integral form of the FVM helps to find accurate solutions of the equations for critical conditions.

14 Mutli-fluid case The method can be expanded to two dust fluids, and we obtain : t Pg = +ωp ( Pp ɛ Pg ), () +ω p ( Pp ɛ Pg ) () t Pp = ω p ( Pp ɛ Pg ) () t Pp = ω p ( Pp ɛ Pg ) () (7) In the same notation system as before we obtain after replacing the gas equation : t P p = ( + ɛ ) ω p P p ɛ ω p P p (8) t P p = ɛ ω p P p ( + ɛ ) ω p P p. (9) The solution of the system is obtained with the exponential of the matrix defining the linear system of differential equations. We have implemented the dust fluids case, and are working on the dust fluids solution (exponential of a matrix...)

15 The code The following results are taken from : Surville, C., Mayer, L., Lin, D. N. C. "Dust capture and long-lived density enhancements triggered by vortices in D protoplanetary disks", The Astrophysical Journal, 8, 8 Surville, C., and Mayer, L., "Effect of small grains on the evolution of vortices in D PPDisks" 7, The Astrophysical Journal, under review Surville, C., Mayer, L., and Alibert, Y., "Dust rings triggered by super Earths" 7, The Astrophysical Journal, in preparation

16 Dust capture in vortices and ring formation The code Dust density e+ e+ e+ e Radius [r ].. S t = S t = S t = S t = Model for S t = Model for S t = Model for S t = Model for S t = Time [πω (r )].. Capture of solids inside the vortex. creates an exponential growth of the dust density in the vortex. e Rossby number.. Maximum of σp e+ e Radius [r ]... The linear model fits the numerical results, even for frictions in the critical regime ( + )ωp t

17 Dust capture in vortices and ring formation e+ e+ e+ e+ e+ e+ e+ e+ e e Radius [r] e Radius [r] e Dust density for S t = Dust density for S t = Larger grains tend to produce norrow rings with dense eddies, while smaller grains produce wide rings with smoother structures (resolution issue?). The structure of dust rings produced by vortices are different as function of the grain size

18 Radius [r] Two dust population capture Rossby number of the gas e+ e+ e+ e+ e+ e Radius [r] e Radius [r] e Density of large grains (S t =.) Density of small grains (S t =.)

19 Planet-disk interaction : dust ring formation We performed a study with Yann Alibert of a typical setup of a Earth masses planet orbiting at AU with dust of S t = and S t = (. and. cm) We observe gap opening and accumulation of solids at the outer edge.. e+. e+. e e e.. Radius [r]... Radius [r] e 8 Gas density at t = rot Dust density for S t =

20 Planet-disk interaction : dust ring formation The edge becomes unstable to the drag/vorticity instability (in competition with the RWI) e+ e+ e+ e+ e e e e e e.. Radius [r] e 8.. Radius [r] e 8 Dust density at t = rot Dust density at t = rot

21 Planet-disk interaction : dust ring formation As time goes, and as the planet migrates, the gap widens, and the dust ring forms with eddies of dust-to-gas ratio larger than unity.. e+. e+. e e e.. Radius [r]... Radius [r] e 8 Gas density at t = 8 rot Dust density for S t =

22 Planet-disk interaction : dust ring formation The code Second ring formation : the gap moves with the planet, triggering a second instability and ring formation.. e e. e+. e+.. Radius [r ] Gas vorticity at t = 8 rot... e. Radius [r ] Dust density for St = e 8

23 Planet-disk interaction : dust ring formation The code The process continues and additional dust rings can form, which are decorrelated from the initial gap.. e e. e+. e+.. Radius [r ] Gas vorticity at t = rot... e. Radius [r ] Dust density for St = e 8

24 Planet-disk interaction : dust ring formation The code The planet has cleaned the solid component inside the gap. But a wide dusty region with rings and eddies is left in the outer parts.. e e. e+. e+.. Radius [r ] Gas vorticity at t = rot... e. Radius [r ] Dust density for St = e 8

25 The code The code RoSSBi is designed to solve the complexity of protoplanetary disks dynamics thanks to a high order FFM and a well balanced.

26 The code The code RoSSBi is designed to solve the complexity of protoplanetary disks dynamics thanks to a high order FFM and a well balanced. To stay close to the conservative form of the equations helps to find implicit or accurate solutions of the source terms.

27 The code The code RoSSBi is designed to solve the complexity of protoplanetary disks dynamics thanks to a high order FFM and a well balanced. To stay close to the conservative form of the equations helps to find implicit or accurate solutions of the source terms. The high resolution achievable with multi-fluids ( dust populations) is an innovation to understand the dynamics of primordial small size solids.

28 The code The code RoSSBi is designed to solve the complexity of protoplanetary disks dynamics thanks to a high order FFM and a well balanced. To stay close to the conservative form of the equations helps to find implicit or accurate solutions of the source terms. The high resolution achievable with multi-fluids ( dust populations) is an innovation to understand the dynamics of primordial small size solids. We discovered a new drag instability, the drag/vorticity instability, that destroys vortices, but sustains dust rings under many conditions.

29 The code The code RoSSBi is designed to solve the complexity of protoplanetary disks dynamics thanks to a high order FFM and a well balanced. To stay close to the conservative form of the equations helps to find implicit or accurate solutions of the source terms. The high resolution achievable with multi-fluids ( dust populations) is an innovation to understand the dynamics of primordial small size solids. We discovered a new drag instability, the drag/vorticity instability, that destroys vortices, but sustains dust rings under many conditions. (Close-)Future improvements, like disk self-gravity, D spherical, will help to understand the complex dynamics of solids in PPDisks, and to upgrade planet formation models, in particular the core accretion scenario.

30 Thank you for your attention!