Jet Stability: A computational survey

Transcription

1 Jet Stability Galway Jet Stability: A computational survey Rony Keppens Centre for Plasma-Astrophysics, K.U.Leuven (Belgium) & FOM-Institute for Plasma Physics Rijnhuizen & Astronomical Institute, Utrecht University Galway 2008, 8th-12th January 2008 With special thanks to collaborators: Bart van der Holst, Hubert Baty, Zakaria Meliani

2 Jet Stability Galway Jet Stability: a computational survey Overview of lectures Multi-D MHD evolutions: From planar shear flows to 3D magnetized jets Linear stability versus nonlinear interactions. Relativistic jets: Relativistic simulations and AMR Slowing down FR I jets: encountering density discontinuities Two-component jets: transverse stability Relativistic jets with Helical magnetic fields

3 Jet Stability Galway Magnetohydrodynamics or MHD IDEAL MHD: conserve MASS, MOMENTUM, ENERGY, MAGNETIC FLUX 8 non-linear PDE for density ρ, velocity v, energy e, and B add B = 0 no magnetic monopoles 7 wavespeeds entropy, ± slow, ± Alfvén, ± fast [anisotropic!] complicated shocks, Steady & Unsteady

4 Jet Stability Galway Multi-D MHD evolutions Kelvin-Helmholtz unstable v tanh yê x -profile only two dimensionless parameters: M = 1 M A = 10 (β = 120) linear stability analysis: growthrate/eigenfunction for exp(ik x x) choose box size in accord with λ x for maximal growthrate role of B in vortex flow: ρ at saturation B amplified at vortex perimeter: ρ depletions

5 Jet Stability compressible MHD studies of 2D KH unstable v = tanh ye x-profile role of B, without and with J-sheet induced island formation ( tearing ) when J-sheet Galway

6 Jet Stability Galway reversed case: initial J = B infinitely thin sheet limit b 0 with B x = tanh(y/b) [then p(y)] additional pinching mode (with reconnection) accessible current sheet gets amplified by vortex flow strong current sheet resistive dissipation important systematic study in resistive MHD for η 0 (finer grid) extra magnetic energy tapped

7 Jet Stability Galway anti-parallel field lines pushed together magnetic islands form (tearing unstable) reconnection plays role sooner than for uniform case turbulent state sets in fast, complicating saturation behaviour compressibility: density deviations up to 40 %

8 Jet Stability Galway Outline 2D planar Kelvin-Helmholtz with aligned B single billow evolution: exclude subharmonic modes wish to study vortex disruption and mode interactions extend domain size to multiple wavelengths of most unstable mode perform grid-adaptive (AMRVAC) studies Consider planar jets, or double shear layers (AMR) additional parameter: layer-layer separation Perform 3D studies 2.5D wake-current sheet: possible dominant 3D instabilities 3D cylindrical jet with uniform B current-carrying jets: Kelvin-Helmholtz & Current-Driven mode interaction

9 Jet Stability Galway Mach number M = V/c s, V total velocity jump M = 1 transonic layers Alfvén Mach number: initial B strength Weak B M A = 100 vortex pair/merge β = 12000

10 Jet Stability Galway stronger B: M = 1, Alfvén Mach M A = 7 (β = 58.8) reconnection disrupts single billows trend to large scale by pairing/merging 22 wavelengths of most unstable mode

11 Jet Stability Galway single layer: M = 1 M A = 7 or β = 58.8

12 Jet Stability Galway transition HD disruptive MHD MA = 30 or β 1000 locally amplified B survives > 1 roll-ups pairing/merging joins antiparallel field: tearing events at vortex periphery transit to turbulence + coalescence

13 Jet Stability Galway deterministic runs of two identical vortices exciting λ1 = Lx and λ2 = Lx/2 or only Lx study dependence on phase difference: Φ = 0 (top) or Φ = π/2 pair/merge when subharmonics excited, near-zero phase difference

14 Jet Stability Galway Follow-up study: planar jets or double layers additional parameter: jet width (radius) versus shear layer width a flow strength M, field strength M A, ratio R jet /a instability growth rates + eigenfunctions: linear MHD narrower jets: more inclined to sinuous deformations

15 Jet Stability Galway higher Mach flows: kink more unstable, non-surface like compressive perturbations into far surroundings

16 Jet Stability Galway M = 1 M A = 7 close layers R jet = 2.5a

17 Jet Stability Galway M = 1 M A = 50 close near HD layers (β = 3000) Batchelor coupling: counterrotating vortices pair (no merge), leave jet core

18 Jet Stability Galway M = 6 M A = 7 supersonic β = 1.63 layers Rapid shock-dominated transition

19 Jet Stability Galway Summary thus far Single shear layers: large-scale coalescence and vortex disruptions 2D Jets: trend to large-scale prevails layer-layer interactions eventually occur HD vortex pairing at (very) high β MHD coalescence at both jet boundaries Supersonic β 1 shock-dominated transitions

20 Jet Stability Galway Planar jet flows: idealized configurations, always uniform B(t = 0) similar effects also at play in coronal streamer belt planar wake flow (instead of jet) with cospatial current-sheet v = (1 cosh 1 y)ê x and force-free B(y) = (B x (y), 0, B z (y)) basic two parameters: flow and field strength take current sheet width to shear flow width (extra parameter) idealized local box: slow solar wind embedded amongst fast streams V x = V x (y) B x = B x (y) B z = B z (y) z y x SUN Computational domain

21 Jet Stability Galway Perform numerical quantification of linear ideal sinuous instability spectral MHD code: linearize about stationary equilibrium state maximal growth rate for varying sonic and Alfvénic Mach number possibility for dominant 3D instability! (k z 0 despite 2D equilibrium)

22 Jet Stability 2.5D and 3D simulations in super-alfve nic case fast magnetosonic shocks form, disturb far-field mechanism for in-situ shock formation in coronal streamer belt Galway

23 Jet Stability Galway wakes with sufficiently supersonic M s > 2.6 flows in β 1 conditions support dominant 3D sinuous instabilities! parameter regime charted for shock formation

24 Jet Stability Galway D jet configurations [astrophysical] jet of radius R jet : 3D Kelvin-Helmholtz case study cylindrical cross-section shear flow (width 2a) across its circumference t = 0 parallel uniform B, reference V 0 = 0.645, a = 0.05, B 0 = reference parameters M s = 0.5, M a = 5, β = 120 3D perturbation wavenumber n along jet, m about jet axis sideways kink perturbation m = 1 = n

25 Jet Stability Galway quasi-linear analytical prediction: (m, n) = (1, 1) excitation leads to (0, 2), (2, 2), and (2, 0) check results for t < 0.5: poloidal magnetic/kinetic energy evolution

26 Jet Stability Galway in horizontal cross-cut: doubled 2D result: ρ at t = 4

27 Jet Stability Galway high ρ isosurface and v x = 0 jet surface colored by p th

28 Jet Stability Galway p th gradient induces wavenumber doubling on top/bottom

29 Jet Stability Galway low ρ lanes: 3D fibril and sheet structures: cospatial with high B pol

30 Jet Stability Galway localized 3D high B pol regions control jet deformation

31 Jet Stability Galway Nonlinear evolution for varying initial 3D perturbation

32 Jet Stability Galway Kelvin-Helmholtz and Current-Driven modes 3D MHD simulations of cylindrical jets supersonic M = 1.26 jet segment with axial β = 32 unstable surface-type Kelvin-Helmholtz instabilities uniform B: nonlinear evolution as before astrophysical jet collimation azimuthal field components collimation by jet pinching (tension in B ϕ ) helical fields: current-carrying jets possibility for both KH and current-driven kink instabilities study interplay between KH and CD instabilities in jets

33 Jet Stability Galway Magnetic Lorentz force = tension + pressure Flux tube with twisted field lines: stabilizing axial B Z tension destabilizing B ϕ pressure in kink pure MHD kink instability when twist q = B ϕ /RB Z > q cr

34 Jet Stability Galway magnetized cylindrical jets axial flow profile V Z (R) tanh[(r j R)/a] 3 magnetic configurations: Uniform twisted fields For sufficiently twisted B fields: both KH and kink unstable what about their mutual interaction?

35 Jet Stability Galway current-carrying HEL2 (highest twist) case: eigenfunctions linear radial displacements show different character ξ m = 1 CD is localized centrally r m = ±1 KH is localized at jet radius R = R j = 1

36 Jet Stability saturation & disruption phase: compare UNI (a), HEL1 (b), HEL2 (c) density structure in jet cross-section at t = 14 development of fine structure in disruption: less when twisted decrease of axial kinetic energy: less in HEL2 case Galway

37 Jet Stability Galway jet coherency is maintained due to KH-CD interaction magnetic deformation due to centrally developing CD increases B ϕ & saturates KH vortices at jet surface 3D impression of jet after 14 transit times: UNI versus HEL2 increased (nonlinear) jet stability due to interacting instabilities!

38 Jet Stability Galway Selected references 2D planar Kelvin-Helmholtz J. Plasma Physics 61, 1 (1999) extend domain size to multiple wavelengths of most unstable mode Phys. of Plasmas 10(12), 2003, 4661 Consider planar jets, or double shear layers (AMR) Astron. & Astrophys. 447, (2006) 9-22 Perform 3D studies 2.5D wake-current sheet: possible dominant 3D instabilities Physics of Plasmas 10(11), 4478 (2003) 3D cylindrical jet with uniform B Physics of Plasmas 6(5), 1461 (1999) current-carrying jets: Kelvin-Helmholtz & Current-Driven mode interaction Astrophysical Journal 580, 800 (2002)