Jet Stability Galway 2008-1 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
Jet Stability Galway 2008-2 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
Jet Stability Galway 2008-3 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
Jet Stability Galway 2008-4 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
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 2008-5
Jet Stability Galway 2008-6 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
Jet Stability Galway 2008-7 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 %
Jet Stability Galway 2008-8 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
Jet Stability Galway 2008-9 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
Jet Stability Galway 2008-10 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
Jet Stability Galway 2008-11 single layer: M = 1 M A = 7 or β = 58.8
Jet Stability Galway 2008-12 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
Jet Stability Galway 2008-13 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
Jet Stability Galway 2008-14 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
Jet Stability Galway 2008-15 higher Mach flows: kink more unstable, non-surface like compressive perturbations into far surroundings
Jet Stability Galway 2008-16 M = 1 M A = 7 close layers R jet = 2.5a
Jet Stability Galway 2008-17 M = 1 M A = 50 close near HD layers (β = 3000) Batchelor coupling: counterrotating vortices pair (no merge), leave jet core
Jet Stability Galway 2008-18 M = 6 M A = 7 supersonic β = 1.63 layers Rapid shock-dominated transition
Jet Stability Galway 2008-19 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
Jet Stability Galway 2008-20 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
Jet Stability Galway 2008-21 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)
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 2008-22
Jet Stability Galway 2008-23 wakes with sufficiently supersonic M s > 2.6 flows in β 1 conditions support dominant 3D sinuous instabilities! parameter regime charted for shock formation
Jet Stability Galway 2008-24 3D 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 = 0.129 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
Jet Stability Galway 2008-25 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
Jet Stability Galway 2008-26 in horizontal cross-cut: doubled 2D result: ρ at t = 4
Jet Stability Galway 2008-27 high ρ isosurface and v x = 0 jet surface colored by p th
Jet Stability Galway 2008-28 p th gradient induces wavenumber doubling on top/bottom
Jet Stability Galway 2008-29 low ρ lanes: 3D fibril and sheet structures: cospatial with high B pol
Jet Stability Galway 2008-30 localized 3D high B pol regions control jet deformation
Jet Stability Galway 2008-31 Nonlinear evolution for varying initial 3D perturbation
Jet Stability Galway 2008-32 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
Jet Stability Galway 2008-33 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
Jet Stability Galway 2008-34 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?
Jet Stability Galway 2008-35 current-carrying HEL2 (highest twist) case: eigenfunctions linear radial displacements show different character 0.10 0.05 ξ 0.00 0.05 m = 1 CD is localized centrally 0.10 0.0 0.5 1.0 1.5 2.0 r m = ±1 KH is localized at jet radius R = R j = 1
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 2008-36
Jet Stability Galway 2008-37 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!
Jet Stability Galway 2008-38 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)