Small-Scale Dynamo and the Magnetic Prandtl Number
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1 MRI Turbulence Workshop, IAS, Princeton, Small-Scale Dynamo and the Magnetic Prandtl Number Alexander Schekochihin (Imperial College) with Steve Cowley (Culham & Imperial) Greg Hammett (Princeton) Alexey Iskakov (UCLA) Russell Kulsrud (Princeton) Leonid Malyshkin (Chicago) Jason Maron (AMNH) Jim McWilliams (UCLA) François Rincon (Toulouse) Sam Taylor (Princeton) Tarek Yousef (Imperial) [Pm>1: ApJ 612, 276 (2004); Pm<1: New J. Phys 9, 300 (2007)]
2 The Magnetic Prandtl Number Pm = kinematic viscosity magnetic diffusivity ~ T 4 /n (ionised plasma)
3 The Magnetic Prandtl Number Pm = kinematic viscosity magnetic diffusivity ~ T 4 /n (ionised plasma) Pm large: hot thin plasmas galaxies (10 14 )
4 The Magnetic Prandtl Number Pm = kinematic viscosity magnetic diffusivity ~ T 4 /n (ionised plasma) Pm large: hot thin plasmas galaxies (10 14 ) clusters (10 29 )
5 The Magnetic Prandtl Number Pm = kinematic viscosity magnetic diffusivity ~ T 4 /n (ionised plasma) Pm large: hot thin plasmas galaxies (10 14 ) clusters (10 29 ) fusion devices
6 The Magnetic Prandtl Number Pm = kinematic viscosity magnetic diffusivity ~ T 4 /n (ionised plasma) Pm large: hot thin plasmas galaxies (10 14 ) clusters (10 29 ) fusion devices Pm small: liquid metals ( ) (planetary, laboratory dynamos)
7 The Magnetic Prandtl Number Pm = kinematic viscosity magnetic diffusivity ~ T 4 /n (ionised plasma) Pm large: hot thin plasmas galaxies (10 14 ) clusters (10 29 ) fusion devices Pm small: liquid metals ( ) stars ( )
8 The Magnetic Prandtl Number Pm = kinematic viscosity magnetic diffusivity ~ T 4 /n (ionised plasma) Pm large: hot thin plasmas galaxies (10 14 ) clusters (10 29 ) fusion devices Pm small: liquid metals ( ) stars ( ) discs
9 The Magnetic Prandtl Number Pm = kinematic viscosity magnetic diffusivity ~ T 4 /n (ionised plasma) Pm large: hot thin plasmas galaxies (10 14 ) clusters (10 29 ) fusion devices Pm small: liquid metals ( ) stars ( ) discs [Balbus & Henri, arxiv: ]
10 Fundamental Problem Pm = kinematic viscosity magnetic diffusivity ~ T 4 /n (ionised plasma) Pm large: hot thin plasmas galaxies (10 14 ) clusters (10 29 ) fusion devices Pm small: liquid metals ( ) stars ( ) discs [Balbus & Henri, arxiv: ]
11 Fundamental Problem
12 Fundamental Problem
13 Fundamental Problem Set B = 0, start with small seed field Will magnetic energy B 2 grow? (fluctuation, or small-scale, dynamo) At what scales? How does it saturate? (fully developed isotropic MHD turbulence) except at the end of this talk NB: no mean flux no helicity no shear no rotation {no mean-field (large-scale) dynamo
14 Fundamental Problem Set B = 0, start with small seed field Will magnetic energy B 2 grow? YES (fluctuation, or small-scale, dynamo) At what scales? } DEPENDS How does it saturate? ON Pm (fully developed isotropic MHD turbulence) except at the end of this talk NB: no mean flux no helicity no shear no rotation {no mean-field (large-scale) dynamo
15 The Stability Curve Pm > 1: Rm c ~ 60 Pm<<1: Rm c < 200 [Iskakov et al. 2007, PRL 98, ; Schekochihin et al. 2007, New J. Phys 9, 300]
16 The Stability Curve Pm > 1: Rm c ~ 60 Pm<<1: Rm c < 200 MRI Turbulence [Fromang et al. 2007, A&A 476, 1123]
17 The Stability Curve Pm > 1: Rm c ~ 60 Pm<<1: Rm c < 200 [Iskakov et al. 2007, PRL 98, ; Schekochihin et al. 2007, New J. Phys 9, 300]
18 Kinetic energy Pm >> 1: An Easy Dynamo k -5/3 Magnetic energy grows: fluctuation dynamo forcing k 0 k ν ~ Re 3/4 k 0 k η ~ Pm 1/2 k ν k Pm >> 1 Rm >> Re >> 1 Most known small-scale dynamos are of this sort! Chaotic dynamo flows (ABC, etc.) Numerical dynamos with Pm 1
19 Kinetic energy Pm >> 1: An Easy Dynamo k -5/3 Magnetic energy grows: fluctuation dynamo forcing k 0 k ν ~ Re 3/4 k 0 k η ~ Pm 1/2 k ν k Pm >> 1 Rm >> Re >> 1 Lagrangian stretching of field lines by the chaotic but smooth velocity field of the viscous-scale eddies [see Schekochihin et al. 2004, ApJ 612, 276; Schekochihin & Cowley, astro-ph/ for an account of theory and simulations]
20 Kinetic energy Pm >> 1: An Easy Dynamo k -5/3 Magnetic energy grows: fluctuation dynamo forcing k 0 k ν ~ Re 3/4 k 0 k η ~ Pm 1/2 k ν k Pm >> 1 Rm >> Re >> 1 Pm 1: it is well established numerically that nonhelical fluctuation dynamo exists provided Rm > Rm c ~ 60; easy to get already at 64 3 [Meneguzzi, Frisch & Pouquet 1981, PRL 47, 1060]
21 Pm << 1: A Harder Dynamo Kinetic energy k -5/3 This regime is fundamentally different! No scale separation between field and flow Flow not smooth (real turbulence!) forcing Dynamo k 0 k η ~ Rm 3/4 k 0 k ν ~ Re 3/4 k 0 Pm << 1 Re >> Rm >> 1 Pm << 1: eluded numerical detection until recently because of higher threshold Rm > Rm c ~ 200; need at least k [Iskakov et al. 2007, PRL 98, ; Schekochihin et al. 2007, New J. Phys 9, 300]
22 Kinetic energy Pm >> 1: An Easy Dynamo k -5/3 Magnetic energy grows: fluctuation dynamo forcing k 0 k ν ~ Re 3/4 k 0 k η ~ Pm 1/2 k ν k Pm >> 1 Rm >> Re >> 1 Lagrangian stretching of field lines by the chaotic but smooth velocity field of the viscous-scale eddies [see Schekochihin et al. 2004, ApJ 612, 276; Schekochihin & Cowley, astro-ph/ for an account of theory and simulations]
23 Numerical Dynamo (Pm >> 1, Re ~ 1) B 2 grows exponentially, then saturates Field at the resistive scale (k η ~ Pm 1/2 k ν ) Quantitative analytical theory possible [Schekochihin et al. 2004, ApJ 612, 276 and references therein]
24 Pm >> 1: An Easy Dynamo Stretch/shear Direction reversals at the resistive scale: k ~ k η Field varies slowly along itself: k ~ k flow Lagrangian stretching of field lines by the chaotic but smooth velocity field of the viscous-scale eddies [see Schekochihin et al. 2004, ApJ 612, 276; Schekochihin & Cowley, astro-ph/ for an account of theory and simulations]
25 Pm >> 1: An Easy Dynamo u B Lagrangian stretching of field lines by the chaotic but smooth velocity field of the viscous-scale eddies [see Schekochihin et al. 2004, ApJ 612, 276; Schekochihin & Cowley, astro-ph/ for an account of theory and simulations]
26 Kinetic energy Onset of Back Reaction k -5/3 Folds provide a direction in space that is locally coherent at the scale of the flow forcing k 0 k ν ~ Re 3/4 k 0 k η ~ Pm 1/2 k ν k Kinematic growth continues until B B ~ u u k B 2 ~ k ν u 2 i.e., B 2 ~ u 2 Mag. energy ~ visc. eddies energy [Schekochihin et al. 2002, PRE 65, ]
27 Intermediate Nonlinear Growth (Pm=1) Slower than exponential growth [Schekochihin et al. 2004, ApJ 612, 276]
28 Kinetic energy Intermediate Nonlinear Growth k -5/3 forcing k 0 k s (t) k ν ~ Re 3/4 k 0 k η ~ Pm 1/2 k ν k Define stretching scale l s (t) : [Schekochihin et al. 2002, NJP 4, 84]
29 Kinetic energy Intermediate Nonlinear Growth Supported by DNS [Schekochihin et al. 2004, ApJ 612, 276] k -5/3 Magnetic energy grows linearly in time forcing k 0 k s (t) k ν ~ Re 3/4 k 0 k η ~ Pm 1/2 k ν k Define stretching scale l s (t) : [Schekochihin et al. 2002, NJP 4, 84]
30 Kinetic energy Intermediate Nonlinear Growth Supported by DNS [Schekochihin et al. 2004, ApJ 612, 276] k -5/3 Magnetic energy grows linearly in time forcing k 0 k s (t) k ν ~ Re 3/4 k 0 k η ~ Pm 1/2 k ν k Define stretching scale l s (t) : [Schekochihin et al. 2002, NJP 4, 84]
31 Kinetic energy Intermediate Nonlinear Growth Supported by DNS [Schekochihin et al. 2004, ApJ 612, 276] k -5/3 Magnetic energy grows linearly in time forcing k 0 k s (t) k ν ~ Re 3/4 k 0 k η ~ Pm 1/2 k ν k selective decay It is possible to construct a Fokker-Planck model of this self-similar intermediate growth stage [Schekochihin et al. 2002, NJP 4, 84]
32 Kinetic energy Intermediate Nonlinear Growth Supported by DNS [Schekochihin et al. 2004, ApJ 612, 276] k -5/3 Magnetic energy grows linearly in time forcing k 0 k s (t) k ν ~ Re 3/4 k 0 k η k selective decay It is possible to construct a Fokker-Planck model of this self-similar intermediate growth stage [Schekochihin et al. 2002, NJP 4, 84]
33 Kinetic energy k -5/3 Intermediate Nonlinear Growth Supported by DNS [Schekochihin et al. 2004, ApJ 612, 276] Magnetic energy grows linearly in time forcing k 0 k s (t) k ν ~ Re 3/4 k 0 k η k selective decay It is possible to construct a Fokker-Planck model of this self-similar intermediate growth stage [Schekochihin et al. 2002, NJP 4, 84]
34 Kinetic energy Saturation Supported by DNS [Schekochihin et al. 2004, ApJ 612, 276] Magnetic energy saturates forcing k 0 k ν ~ Re 3/4 k 0 k η ~ Re 1/4 Pm 1/2 k ν k Nonlinear growth/selective decay/fold elongation continue until l s ~ l 0 B 2 ~ u 2 and l η ~ Rm 1/2 l 0 ~ Re 1/4 Pm 1/2 l ν [Schekochihin et al. 2002, NJP 4, 84]
35 Intermediate Nonlinear Growth (Pm=1) Slower than exponential growth Selective decay and fold elongation [Schekochihin et al. 2004, ApJ 612, 276]
36 Kinetic energy Saturation Supported by DNS [Schekochihin et al. 2004, ApJ 612, 276] Magnetic energy saturates forcing k 0 k ν ~ Re 3/4 k 0 k η ~ Re 1/4 Pm 1/2 k ν k Nonlinear growth/selective decay/fold elongation continue until l s ~ l 0 B 2 ~ u 2 and l η ~ Rm 1/2 l 0 ~ Re 1/4 Pm 1/2 l ν NB: l η and l ν distinguishable only if Pm >> Re 1/2 >> 1!!! [Schekochihin et al. 2002, NJP 4, 84]
37 Kinetic energy Saturation Supported by DNS [Schekochihin et al. 2004, ApJ 612, 276] Magnetic energy saturates forcing k 0 k ν ~ Re 3/4 k 0 k η ~ Re 1/4 Pm 1/2 k ν k NB: l η and l ν distinguishable only if Pm >> Re 1/2 >> 1!!! [Schekochihin et al. 2002, NJP 4, 84]
38 Kinetic energy Saturation Supported by DNS [Schekochihin et al. 2004, ApJ 612, 276] Magnetic energy saturates forcing k 0 k η ~ Rm 1/2 k 0 k ν ~ Re 3/4 k 0 k NB: If Pm < Re 1/2, l η > l ν All current numerical simulations with Re >> 1 are in this regime! [Schekochihin et al. 2002, NJP 4, 84]
39 What Is the Saturated State at 1 Pm Re 1/2? KINEMATIC, Pm = 1 [New simulations by Alexey Iskakov]
40 What Is the Saturated State at 1 Pm Re 1/2? SATURATED, Pm = 1 [New simulations by Alexey Iskakov]
41 What Is the Saturated State at 1 Pm Re 1/2? SATURATED, Pm = 10 [New simulations by Alexey Iskakov]
42 What Is the Saturated State at 1 Pm Re 1/2? SATURATED, Pm = 10 Cf. Sebastien Fromang s shallowish velocity spectrum at largest resolution [New simulations by Alexey Iskakov]
43 Folded Structure at 1 Pm Re 1/2 u B Pm = 1, Re = Rm ~ 1300 [New simulations by Alexey Iskakov]
44 Folded Structure at 1 Pm Re 1/2 u B Pm = 1, Re = Rm ~ 1300 [New simulations by Alexey Iskakov]
45 Spontaneous Current Sheets at 1 Pm Re 1/2 Reconnection requires inflows and outflows that have gradients on the resistive scale ~ L Rm 1/2 This is not possible when Pm
46 Saturated State at Pm > Re 1/2 u B Pm = 10, Re ~ 80, Pm > Re 1/2 [Schekochihin et al. 2004, ApJ 612, 276]
47 Saturated State at Pm > Re 1/2 Pm > Re 1/2 Pm < Re 1/2 u B Pm = 10, Re ~ 80, Pm > Re 1/2 [Schekochihin et al. 2004, ApJ 612, 276]
48 What Is the Saturated Spectrum at Pm >> 1? Pm >> 1, Re ~1 [Schekochihin et al. 2004, ApJ 612, 276]
49 What Is the Saturated Spectrum at Pm >> 1? [Yousef, Rincon & Schekochihin 2006, JFM 575, 111] with prob. 1/2
50 What Is the Saturated Spectrum at Pm >> 1? k 1 [Yousef, Rincon & Schekochihin 2006, JFM 575, 111] [Schekochihin et al. 2004, ApJ 612, 276] with prob. 1/2
51 What Is the Saturated Spectrum at Pm >> 1? This is probably too simplistic a model k? [Schekochihin et al. 2004, ApJ 612, 276]
52 Saturated State of Isotropic MHD Turbulence Kinetic energy NON-LOCAL Magnetic energy forcing Folds k 0 k ν ~ Re 3/4 k 0 k η ~ Re 1/4 Pm 1/2 k ν k Interactions are NONLOCAL in k space! [Schekochihin et al. 2004, ApJ 612, 276]
53 Pm << 1: A Harder Dynamo Kinetic energy k -5/3 This regime is fundamentally different! No scale separation between field and flow Flow not smooth (real turbulence!) forcing Dynamo k 0 k η ~ Rm 3/4 k 0 k ν ~ Re 3/4 k 0 Pm << 1 Re >> Rm >> 1 Pm << 1: eluded numerical detection until recently because of higher threshold Rm > Rm c ~ 200; need at least k [Iskakov et al. 2007, PRL 98, ; Schekochihin et al. 2007, New J. Phys 9, 300]
54 Is There a Dynamo at Pm << 1? [Schekochihin et al. 2004, PRL 92, ; 2005, ApJ 625, L115]
55 Is There a Dynamo at Pm << 1? [Schekochihin et al. 2004, PRL 92, ; 2005, ApJ 625, L115]
56 Porterhouse Dynamo Keep η fixed (Rm ~ const) and decrease ν (increase Re) Dynamo! forcing k 0 k ν k η k This has to be done at sufficient resolution, so that Rm > Rm c ~ 10 2 [Iskakov et al. 2007, PRL 98, ; Schekochihin et al. 2007, New J. Phys 9, 300]
57 Porterhouse Dynamo Keep η fixed (Rm ~ const) and decrease ν (increase Re) Dynamo! forcing k 0 k ν k η k This has to be done at sufficient resolution, so that Rm > Rm c ~ 10 2 [Iskakov et al. 2007, PRL 98, ; Schekochihin et al. 2007, New J. Phys 9, 300]
58 Porterhouse Dynamo Keep η fixed (Rm ~ const) and decrease ν (increase Re) Dynamo? forcing k 0 k η k ν k This has to be done at sufficient resolution, so that Rm > Rm c ~ 10 2 [Iskakov et al. 2007, PRL 98, ; Schekochihin et al. 2007, New J. Phys 9, 300]
59 Porterhouse Dynamo Keep η fixed (Rm ~ const) and decrease ν (increase Re) Dynamo?? forcing k 0 k η k ν k This has to be done at sufficient resolution, so that Rm > Rm c ~ 10 2 [Iskakov et al. 2007, PRL 98, ; Schekochihin et al. 2007, New J. Phys 9, 300]
60 Porterhouse Dynamo Keep η fixed (Rm ~ const) and decrease ν (increase Re) Dynamo??? forcing k 0 k η k ν k This has to be done at sufficient resolution, so that Rm > Rm c ~ 10 2 [Iskakov et al. 2007, PRL 98, ; Schekochihin et al. 2007, New J. Phys 9, 300]
61 Numerical Experiment: Results γ (Re, Rm) const(rm) as Re [Iskakov et al. 2007, PRL 98, ; Schekochihin et al. 2007, New J. Phys 9, 300]
62 The Stability Curve Rm c (Re) const as Re Pm > 1: Rm c ~ 60 Pm<<1: Rm c < 200 [Iskakov et al. 2007, PRL 98, ; Schekochihin et al. 2007, New J. Phys 9, 300]
63 The Stability Curve Rm c (Re) const as Re Pm > 1: Rm c ~ 60 Pm<<1: Rm c < 200 [Iskakov et al. 2007, PRL 98, ; Schekochihin et al. 2007, New J. Phys 9, 300]
64 Rm c (Re) const as Re What Kind of Dynamo? Pm > 1: Rm c ~ 60 Pm<<1: Rm c < 200 Inertial-range dynamo? [Rogachevskii & Kleeorin 1997, PRE 56, 417 Boldyrev & Cattaneo 2004, PRL 92, ] Forcing-scale dynamo? (mean field + magnetic induction)
65 What Kind of Dynamo? Kinetic energy k 5/3 forcing Inertial-range dynamo? [Rogachevskii & Kleeorin 1997, PRE 56, 417 Boldyrev & Cattaneo 2004, PRL 92, ] Forcing-scale dynamo? (mean field + magnetic induction) Magnetic energy k 11/3 k 0 k k ν ~ Re 3/4 η ~ Rm 3/4 k 0 k 0 Very fast: would overwhelm any mean-flow or mean-field dynamo k
66 What Kind of Dynamo? Kinetic energy k 5/3 forcing Inertial-range dynamo? [Rogachevskii & Kleeorin 1997, PRE 56, 417 Boldyrev & Cattaneo 2004, PRL 92, ] Forcing-scale dynamo? (mean field + magnetic induction) Magnetic energy k 0 k k ν ~ Re 3/4 η ~ Rm 3/4 k 0 k 0 Very fast: k 11/3 would overwhelm any mean-flow or mean-field dynamo Less fast: k
67 What Kind of Dynamo? An example of dynamo with saturating growth rate [Mininni 2007, physics/ ] Forcing-scale dynamo? (mean field + magnetic induction) Less fast:
68 Dynamo at Pm 1 u B Rm = 440, Re = 440 [Schekochihin et al. 2007, New J. Phys 9, 300]
69 Dynamo at Pm << 1 u B Rm = 430, Re = 6200 [Schekochihin et al. 2007, New J. Phys 9, 300]
70 Practical Point: Hyperviscosity is OK [Schekochihin et al. 2007, New J. Phys 9, 300]
71 Practical Point: Hyperviscosity is OK u 8th-order hyperviscosity: Re = 4000, Rm = 230 B [Schekochihin et al. 2007, New J. Phys 9, 300]
72 Practical Point: Hyperviscosity is OK u Laplacian viscosity: Re = 3600, Rm = 220 B [Schekochihin et al. 2007, New J. Phys 9, 300]
73 How Universal Is the Stability Curve? Existence of dynamo universal Asymptotic threshold probably robust Exact shape of stability curve not universal [Iskakov et al. 2007, PRL 98, ; Schekochihin et al. 2007, New J. Phys 9, 300]
74 How Universal Is the Stability Curve? Existence of dynamo universal Asymptotic threshold probably robust Exact shape of stability curve not universal [Iskakov et al. 2007, PRL 98, ; Schekochihin et al. 2007, New J. Phys 9, 300]
75 How Universal Is the Stability Curve? Existence of dynamo universal Asymptotic threshold probably robust Exact shape of stability curve not universal [Iskakov et al. 2007, PRL 98, ; Schekochihin et al. 2007, New J. Phys 9, 300]
76 Spectra at const Pm: (same colour = same Rm) Where Is the Energy? [Schekochihin et al. 2007, New J. Phys 9, 300]
77 Spectra at const Pm: (same colour = same Rm) Where Is the Energy? [Schekochihin et al. 2007, New J. Phys 9, 300]
78 Spectra at const Pm: (same colour = same Rm) Where Is the Energy? [Schekochihin et al. 2007, New J. Phys 9, 300]
79 Spectra at const Pm: (same colour = same Rm) Where Is the Energy? [Schekochihin et al. 2007, New J. Phys 9, 300]
80 Spectra at const Pm: (same colour = same Rm) Where Is the Energy? [Schekochihin et al. 2007, New J. Phys 9, 300]
81 Spectra at const Pm: (same colour = same Rm) Where Is the Energy? [Schekochihin et al. 2007, New J. Phys 9, 300]
82 So How Do We Get Large-Scale Fields?
83 So How Do We Get Large-Scale Fields? A topical example Large-scale field MRI Small-scale MHD fluctuations Mean-field dynamo feedback? Geoffroy Lesur (his talk yesterday) has a model for this Francois Rincon thinks it s his subcritial dynamo [see his paper in Astron. Nachr., Catania proceedings] Axel Brandeburg thinks it s incoherent alpha effect
84 Shear Dynamo? Keplerian rotation and shear No rotation, no shear Numerical experiments by Tarek Yousef and Tobi Heinemann SEE POSTER [Yousef, Heinemann et al. 2008, PRL 100, ]
85 Shear Dynamo? Keplerian rotation and shear No rotation, no shear Numerical experiments by Tarek Yousef and Tobi Heinemann SEE POSTER [Yousef, Heinemann et al. 2008, PRL 100, ]
86 Conclusions Pm 1: [ApJ 612, 276] Fluctuation dynamo well understood: random stretching folded fields Progress in understanding saturation as well Must distinguish between Pm < Re 1/2 and Pm >> Re 1/2 Pm << 1: [New J. Phys. 9, 300] Fluctuation dynamo exists, but is hard to get (Rm c is high) No mechanistic model of it yet Not known if dynamo is driven by inertial range Saturation has not been studied Turbulence + shear = large-scale field [PRL. 100, ] In fact, must model astrophysical plasmas kinetically!!! [PoP 13, ; arxiv: ]
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