Kinetic theory of ions in the magnetic presheath

Transcription

1 Kinetic theory of ions in the magnetic presheath Alessandro Geraldini 1,2, Felix I. Parra 1,2, Fulvio Militello 2 1. Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford, Oxford OX1 3NP, UK 2. CCFE, Culham Science Centre, Abingdon, OX14 3DB, UK 9 th Plasma kinetic working group meeting (Vienna, 2016) 1

2 Introduction Boundary conditions for fluid codes used to simulate the SOL plasma Ones currently used* are obtained using fluid equations: aim is to obtain boundary conditions using a kinetic treatment Could be used with future drift kinetic codes of SOL Interesting problem from a purely theoretical point of view: generalizing gyrokinetics to strongly distorted orbits in the magnetic presheath geometry * J. Loizu, P. Ricci, F.D. Halpern and S. Jolliet, Phys. Plasmas 19, (2012). 2

3 Geometry flux surface ζ divertor target through the looking glass Figure (left) from: P.C. Stangeby, The plasma boundary of magnetic fusion devices (2000) 3

4 Boundary layers Width Estimate Collisional layer αλ mfp 100 mm Magnetic presheath ρ i 0.7 mm Debye sheath λ D 0.02 mm λ D << ρ i << αλ mfp Estimates using data from: F. Militello and W. Fundamenski, Plasma Phys. Control. Fusion, 53, (2011) N.B. wall = divertor target 4

5 Assump+ons Focus on ion gyroradius scale ρ i => derive magnetic presheath equations λ D << ρ i ~ αλ mfp => include collisions but keep quasineutral Angle α satisfies (1º=) 0.02 ~ (m e /m i ) ½ << α << 1 Magnetic field constant and presheath electrostatic Turbulent gradients parallel to the wall inherited from outside magnetic presheath neglect / y~1/l~δ/ρ i and / z~δ/l~δ 2 /ρ i Order / t~δ 2 % << presheath timescale δ% Maximal ordering α~δ<<1 α<<1 and δ=ρ i /l <<1 N.B. l~1cm => δ~0.1 5

6 Assumption: (not too) small angle Q: When does it take an ion and an electron the same time to reach the wall? Ion intersects wall after a single orbit: time~ρ i /v t,i Electrons travel along field at v t,e reaching wall after time d/v t,e Times equal when α ρ i /d= (m e /m i ) In order to assume negatively charged wall, require electrons to reach wall much faster than ions Corresponds to α>> (m e /m i ) 6

7 The zeroth order problem: α=δ=0 Equations of motion of single particle are v x = v y = v z =0 Constants of motion: Ze d (x) m i dx v x + v y Orbit position Perpendicular energy x = x + (1/*)v y U = ½v x 2 + ½v y 2 + Zeφ/m i Total energy U = U + ½v z 2 See also: R.H. Cohen and D.D. Ryutov, Phys. Plasmas 5, 808 (1998) 7

8 Gyrophase The zeroth order motion periodic when magnetic force large enough to make the ion turn Can write v x = σ x [2(U - χ(x))] ½ where σ x =±1 Both x and v x are periodic if particle is trapped around a minimum of the effective potential χ(x) = ½* 2 (x-x ) 2 + Zeφ(x)/m i Modified ion gyrofrequency % mod 2 mod =2 Gyrophase ϕ Z xt x b dx v x =2 Z xt x b p 2(U? dx ' = + mod t = x mod Z x (x)) 1 p x b 2(U? (x 0 )) dx0 8

9 Change of variables Can describe particle motion entirely using new set of variables (x, y, z, v x, v y, v z ) à (x, y, z, U, ϕ, U, σ ) x, U and U are constant and y, z are symmetry directions Define gyroaverage of a quantity as an average over ϕ while holding all other variables fixed h...i ' 1 2 I (...) d' = mod Z xt (...) p x b 2(U? (x)) dx 9

10 Single particle motion in system v x = v y = v z = with α~δ<<1 The exact equations of motion are m (x, y, z, neglect t)+ v 1 y cos Ze m (x, y, z, t) v 1 x cos v z sin Ze small m (x, y, z, t)+ v y sin α Changes in the orbit parameters: ExB drift parallel to wall Orbit drift normal to wall Perpendicular energy change Total energy change D U? E hẏi ' = 1 B ' 1 = v k B D UE h xi ' = v k 1 ' ' 10

11 First order constants: y * and 0 Frame exists in which zeroth order is periodic à => An adiabatic invariant * exists, with <d0/dt> = O(α 2 0) when α~δ<<1 Quantity y * proportional to z- canonical momentum derived by integrating dv z /dt = α*v y to obtain v z = α*(y y * ), with dy * /dt = O(α 2 y * ) Adiabatic invariant* y-star µ = 1 Z xt x b p 2(U? (x))dx y? = y h µi ' =0 1 v k hẏ? i ' =0 replaces U replaces y new change of variables: (x, y, z, U, ϕ, U, σ ) à (x, y *, z, *, ϕ, U, σ ) * R.H. Cohen and D.D. Ryutov, Phys. Plasmas 5, 808 (1998) D.L. Holland, B.D. Fried and G.J. Morales, Phys Fluids B, 5, 1723 (1993) 11

12 Gyrokine+c x h x + hẏi ' F <F> ϕ first order = C[F + D = hc[f ]i ' To solve it, require a form for the potential φ(x,y) (that we will determine) and boundary conditions: 1. F = 0 at yà± (corresponds to outside SOL) 2. F = 0 at U à 3. F = F (y, U, U) for <dx /dt> ϕ < 0 at x à 4. F = 0 for open orbits (ones that intersect wall) 12

13 Geometry flux surface ζ divertor target through the looking glass Figure (left) from: P.C. Stangeby, The plasma boundary of magnetic fusion devices (2000) 13

14 Gyrokine+c x +ẏ? F <F> ϕ first order @µ + '@F = C[F ] small small Boundary conditions are: F 0 = F 0 (y *, 0, U) for <dx /dt> ϕ < 0 at x à F 0 = 0 for open orbits 14

15 Quasineutrality The assumption α>>(m e /m i ) ½ => Boltzmann electrons n e =n e, exp[e(φ-φ )/T e ] λ D << ρ i => quasineutrality holds throughout magnetic presheath To obtain ion density, integrate F 0 in velocity space using (x, U, U) instead of (v x, v y, v z )* e ( (x, y) 1 (y)) n e1 (y)exp = Z X k=±1 Z 1 x m (x,y) Z d x T e M ( x,y) (x, x,y) 2 du p? 2(U? (x, x)) Z 1 F ( x, y, U?,U, k) p du U? 2(U U? ) Quasineutrality (above) and the gyrokinetic equation allow to solve for F and φ self consistently in the magnetic presheath * M.J. Gerver, S.E. Parker and K. Theilhaber, Phys. Fluids B, 2, 1069 (1990) 15

16 Collisionless magnetic presheath If ρ i << αλ mfp, gyrokinetic equation becomes (using x =0 The solution is ( therefore F 1 (y?,µ,u) for closed orbits, F = 0 for open orbits. An iteration scheme with quasineutrality would allow to determine the self-consistent φ(x,y) e ( (x, y) 1 (y)) n e1 (y)exp = Z X k=±1 Z 1 x m (x,y) Z d x T e M ( x,y) (x, x,y) will improve on this! 2 du p? 2(U? (x, x)) Z 1 U? F 1 (y?,µ,u)) p 2(U U? ) du See also: R.H. Cohen and D.D. Ryutov, Phys. Plasmas 5, 808 (1998). 16

17 Open orbits Orbit opening 17

18 Why are open orbits important close to the wall? At exit of quasineutral presheath, electric field diverges (breakdown of quasineutrality) but potential does not (φ ~ x)* Expect potential of this form à Electric force always overcomes magnetic force (=Zev y B) close to x=0 =>Effective potential always has maximum near wall à No closed orbits => need open orbit density * K.-U. Riemann, Journal of Physics D: Applied Physics 24, 493 (1991) 18

19 Quasineutrality without open orbits Closed orbit density n i,closed goes to zero at wall Electron density n e ~ n exp[e(φφ )/T e ] ~ n exp[e φ MPS /T e ] Gives φ MPS =- Density of ion open orbits required to obtain finite potential jump Zn i,open n e near wall n i,closed/e / n x / ρ i 19

20 χ(x) Open orbit density Conservation of distribution function F in phase space f open v x U? = M dv x dv y dv z = Density is Z 1 n i,open = 3 du F (U, U? = M, x) v xi v k 0 Z 1 0 d x Z x M x t U? U? = M dx i dxdu x M dx i (x i x M ) v x v xi F 1 ( x, U? = M,U) x U =χ M x i v i x M x t x x i Confirmed by calculation of velocity corrections near X-point Divergence at v x =0: calculate correction at X point v x = Z qv 2M +2( M (x)) with xm V 2M =2 2 s x v k p M x i 2( M x M y (s)) ds 20

21 Preliminary numerical results n i/e /n Green: electron density Blue: ion density Without With open orbits n i/e /n x/ρ i Analytically, n i,open ~ αn => φ MPS ~ (T/e) ln α Numerical results shown in plots obtained using: α=0.1 F ~v 2 exp(-m(v 2 +2*B)/2T i ) φ=1.05*ln(1/α)*exp(-0.19 x-0.49x) 21 x/ρ i

22 Conclusions Derived gyrokinetic equations of ions in magnetic presheath Assumed small magnetic field to wall angle α, small gradients parallel to the wall (δ<<1) and constant B field Assumed electron repelling wall => Boltzmann electrons Derived form of quasineutrality Proposed and currently applying iteration scheme that could solve for collisionless magnetic presheath Solution valid to lowest order in α and δ Open orbits are important close to the wall In progress: quantifying effects of open orbits near the wall + including open orbit density in numerical work Future work: study the purely collisional layer αλ mfp wide 22

23 BACKUP SLIDES 23

24 Assumption: turbulent gradients We include weak gradients parallel to the wall of the electrostatic potential φ and the ion and electron densities, due to the width of turbulent structures l=ρ i /δ with δ~α<<1 These gradients are in the y direction, across the magnetic field The direction (almost) parallel to the magnetic field is associated with even smaller gradients because turbulent structures are elongated along this direction: v z / z~α 2 * 0 Time derivatives are ordered very small: / t~α 2 * 0 neglect Figure: The elongation of a turbulent structure along the magnetic field causes the gradients in the z direction to be very small. The characteristic length scale in the y direction (out of the page) is l. 24

25 Assumption: (not too) small angle When does it take the same time for an ion and an electron to reach the wall? Suppose we are very close to the wall It takes an ion a single orbit to intersect the wall ~ρ i /v t,i Electrons have a much smaller Larmor radius, so they travel parallel to the field at v t,e and reach the wall after a time d/v t,e Times equal when α ρ i /d= (m e /m i ) 25 Figure: A schematic which shows that, when very close to the wall, ions intersect the wall during their gyromotion while electrons are tied much more closely to the field line and have to drift along B with speed ~v t,e much faster than the characteristic ion speed.. At small enough angle α, the ions reach the wall more quickly than thefaster-moving electrons, which travel almost parallel to the wall.

26 Assumption: (not too) small angle (2) In order to assume a negatively charged wall, we require electrons to reach it much faster than ions Corresponds to α>> (m e /m i ) Provided this is satisfied, almost all of the electrons are repelled by the wall We can therefore assume that electrons are in equilibrium and are Boltzmann distributed n e =n e, exp[e(φ-φ )/T e ] Figure: A schematic which shows that, when very close to the wall, ions intersect the wall during their gyromotion while electrons are tied much more closely to the field line and have to drift along B with speed ~v t,e much faster than the characteristic ion speed.. At small enough angle α, the ions reach the wall more quickly than thefaster-moving electrons, which travel almost parallel to the wall. 26

27 dv y /dt = - %v x => v y the orbit position Orbit parameters = %(x - x) where x is Perpendicular energy U and total energy U are conserved Orbit position Perpendicular energy x = x + (1/*)v y U = ½v x 2 + ½v y 2 + Zeφ/m i Total energy U = U + ½v z 2 27

28 Kine+c equa+on change variables small small small zeroth order ~%F expand F = F 0 + F 1 + O(α 2 F) to get => F 0 = <F 0 > ϕ 28

29 Changes in the orbit parameters ExB drift parallel to wall Orbit drift normal to wall Perpendicular energy change Total energy change hẏi ' = 1 B h xi ' = v k 1 B D U? E ' 1 = v k B D UE @y ' ' 29

30 Gyrokine+c equa+on expand F and recover F 0 = <F 0 > first order ~α%f and gyroaverage small small Boundary conditions are: F 0 = F 0 (y *, 0, U) for <dx /dt> ϕ < 0 at x à F 0 = 0 for open orbits 30