Magnetized ion collection by oblique surfaces including self-consistent drifts: Mach-probes of arbitrary shape.

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1 1 Magnetized ion collection by oblique surfaces including self-consistent drifts: Mach-probes of arbitrary shape I H Hutchinson Plasma Science and Fusion Center and and Engineering Department MIT APS DPP Dallas, Nov 2008

2 Abstract 2 A complete analytic theory for magnetized Mach-probes, when cross-field diffusion is neglected, is presented It is shown that the full self-consistent quasi-neutral fluid drift equations around an ion-collecting probe of arbitrary 3-D shape, in a magnetized isothermal plasma with background parallel and perpendicular flow, can be solved exactly The resulting flux to the probe (per unit area perpendicular to B) is n c s exp( 1 M + M cot θ), where θ is the angle between the surface and B in the plane of background-drift This exponential dependence is in good agreement with prior numerical fits of the diffusive case The (background) perpendicular Mach number, M, is that arising from the sum of ExB and, counter-intuitively, electron (not ion) diamagnetic drifts Fluid displacements in the magnetic presheath are important, and included in this expression, but give rise to small additional terms at some orientations Temperature-gradient diamagnetic drifts can be added, but only approximately: both electron and ion drifts contribute Corrections of order Larmor radius divided by electrode-dimensions are also evaluated They can bias the results for small probes

3 Probes in Tokamaks Used for many important studies of the plasma edge 3 Plasma flows, parallel and perpendicular to B, are important and can presumably be measured Represent generic problem of absorbing object in a flowing magnetized plasma My initial motivation for this longstanding work Layout of scanning probes in Alcator C-Mod

4 Magnetized Plasma Flow to Objects 4 Surface is a sink of ions and electrons Anisotropic collection with crossfield diffusion Ion acceleration along field Electron repulsion Self-consistent potential variation Local sheath structure is multi-layered Ions enter magnetic presheath at v = c s Enter Debye sheath at vk = c s v Magnetic Presheath B k θ Plasma: Quasineutral Drifts Thicknesses: Magnetic Presheath ρ S Solid Surface Debye Sheath Debye Sheath 4λ D

5 External Pre-existing Plasma Flow 5 Flow past the object is some combination of parallel and perpendicular ion drifts v v B Large flows in tokamak scrape-offs Plasma flow Object motion Plasma Flow v Object Space shuttle surface-charging Mach Probe Idea: Measure flow by ion flux density Electrodes facing in different directions should presumably collect different ion flux densities, depending on the external plasma flow θ Collectors B Insulator

6 Practical Mach probes are complicated 6 MacLatchy et al 1992 Gundestrup Peleman et al 2006 (Tokamak) Antoni et al 1996 (RFP) Smick and LaBombard 2008

7 Physics applies to some astrophysical situations 7 Interaction of Jupiter s moons with its magnetosphere is in the same regime: Larmor radius smaller than object (moon) smaller than scale-length

8 What s new about the treatment? 8 Old: Heuristic Diffusion v = D ln n + v Elliptic B v v External n = n, M = M New: Rigorous Convection: v (x) solved Hyperbolic v v v B Instead of invoking heuristic cross-field diffusion, ignore diffusion Treat pure imposed transverse-drift, solve self-consistently, no heuristics More satisfactory physics Rigorous mathematics Analytic solution! Result consistent Dependence on diamagnetic drifts discovered

9 Ion Fluid Drift Equations 9 (nv) = 0 mn(v )v = nze φ p + nze(v B) Uniform B Neglect friction, vicosity Zero diffusion Cross-field drift from (momentum equation) B replaces perpendicular momentum: [ ( Keep = Drift Approx v = φ + 1 ) Drop ] nze m p + Ze (v )v B }{{}}{{} B 2 Electric + Diamagnetic Polarization Drop the polarization drift term in the plasma region (not magnetic presheath) Use simplest possible closure: Uniform temperature T i =const (v ) ln n + v = 0 (v )v + Ze ( m φ + T ) i Ze ln n = 0 Electron parallel momentum φ = (T e /e) ln n e, with uniform T e (for now) Is used to eliminate potential φ by quasi-neutrality (Boltzmann electrons)

10 Hyperbolic system Analyze by Characteristics 10 Define c 2 s = (ZT e + T i )/m and M = v/c s : Mach Number Eliminate φ Then equations become M ln n + M = 0 M M + ln n = 0 which can be rearranged to display explicitly the characteristics (M + )(ln n + M ) = 0 (M )(ln n M ) = 0 Thus the quantities (ln n ± M ) are constant along their corresponding characteristics: dx = (M ± B/B)ds Integration along characteristics is the secret to solving hyperbolic systems

11 Solution by Characteristics 1 11 Assume M simply uniform (in y-direction) Take B along x 2-D problem Analyse the higher-x side of the object Two characteristics pass through any point Object Region P 3 P 4 θ 4 θ P 0 2 P P 1 Positive Characteristic θ m 2 y θ Perturbed Plasma Region M Unperturbed Plasma Region x B Negative Characteristic Negative characteristics start at y in unperturbed plasma: ln n M = ln n M If positive characteristic also starts at y then also: ln n + M = ln n + M Only solution then: n = n, M = M Point (P 0 ) in unperturbed plasma Unperturbed characteristics are straight lines with slope tanθ m = M /(M ± 1) If line from point to lowest part of object is shallower than M /(M + 1), then positive characteristic does not intersect object and point is unperturbed

12 2 The perturbed positive characteristics determine the solution 12 P 4 y M If positive characteristic intersects θ 4 P 2 θ x B object, ln n + M is still constant, but ln n + M Consequently Object Region P 3 θ P 0 2 P 1 Positive Characteristic θ m Perturbed Plasma Region Negative Characteristic Unperturbed Plasma Region M = const, ln n = const along straight +ve characteristics, on which M = M cot θ 1 Boundary condition is unconstrained outflow at object surface: ln n = ln n M + M M is most negative, consistent with overall solution θ is as large as possible This means the characteristic is tangent to surface eg P 1, P 2 Points such as P 3 in concave regions have a non-local tangency (M greater) Angles greater than π/2 are perfectly permissible (eg P 4 )

13 Uniform Velocity Solution* 13 For homogeneous perpendicular velocity M M h, M = M h cot θ 1, n = n exp( 1 M + M h cot θ) throughout the plasma region, where θ is the angle (to B) of the positive characteristic, which is the tangent of greatest possible angle At the plasma boundary with object, the outgoing flux per unit surface area is nc s (M h cos θ M sin θ) = nc s sin θ which is most conveniently written as flux per unit perpendicular area as Γ = nc s = n c s exp( 1 M + M h cot θ) with θ the local surface tangent (in convex regions) This is fine, but who says M is uniform? Generally it is not because of the perturbation to n and φ in the presheath For a 2-D probe (z an ignorable coordinate) the extra drifts are along z: no effect But for a 3-D object what happens? *I H Hutchinson Phys Rev Lett 101, (2008)

14 Now Comes the Magic! Consider a 3-D situation with the full, self-consistent transverse velocity Integrate the electron equation along the field to obtain φ φ (z) = (T e /e) ln(n/n ) Substitute into the drift equation to get transverse velocity in two parts uniform self consistent external v /c s = M = mc 2 s φ + }{{} Ze ln n }{{} M h M 1 Integrating the negative characteristic equation: B c s B 2 = M h + M 1 14 ln n = ln n + M M implies that n = n(m ), so M n Therefore a perpendicular velocity M 1 satisfying M 1 ln n = 0 (and ours does) gives zero contribution to the characteristics: ([M + M h + M 1 ] ± )(ln n ± M ) }{{} original characteristics = ([M + M h ] ± )(ln n ± M ) }{{} modified characteristics We can solve the problem using characterstics that contain just the homogeneous external transverse drift The uniform drift solution just discussed is correct for arbitrary 3-D probe shape = 0

15 Example 3-D Solution: Sphere 15 Contours at 01 spacing of M + 1 = 1 + M + ln(n/n ) Object: unity-radius sphere External perpendicular flow: M h = 024 External parallel flow: M = 02

16 Diamagnetic drift along contour lines 16 Sphere M = 0 M h = 03 M + 1 contours Coincide with ln n contours

17 Example 3-D Solution: Pyramid Probe 17 Contours at 01 spacing of M + 1 = 1 + M + ln(n/n ) Object: C-Mod-like pyramid probe External perpendicular flow: M h = 015 External parallel flow: M = 01

18 n External Diamagnetic Drift Suppose ln n (z) = ẑ/l n Integration along the negative characteristic gives 18 ln n M = ln(n (z )) M where z is the starting z-value of the characteristic The z-motion along the characteristic can t be ignored a priori ln(n/n (z)) = M + M δz/l n where δz z z is not constant But I ll show δz = δz(m ) In which case ln n/n is also function only of M, so write [ ( v /c s = M = φ + T ) ] i Ze ln n + mc2 s Ze ln(n/n ) }{{}}{{} M M h 1 B c s B 2, Write the characteristic equations to take advantage of this form d ( ) ln n ± M = d ( ) ln n/n ± M + Mz /L n = 0 ± ± c s dt c s dt where d c s dt M ± = (M ± 1) ± x + M y y + M z z, when acting on functions of ln n/n, can be d = (M ± 1) ± x + M h y c s dt

19 δz(m k ) analytical solution 19 Define d dm (δz/2l n ) r; so that d δz c s dt ± 2L n = c d s dt M d δz ± dm 2L n = r d c s dt M ± Eliminate M -derivatives from characteristics, eg d c s dt (2M δz + L n ) + M z L n = 0 to get d δz c s dt 2L n = M z 2L n and d δz c s dt + 2L n = M z r 2L n 1 r Then eliminate partial x-derivatives of ln n/n between the positive and negative characteristics, and partial y-derivatives using the z-velocity expression ln M z = ρ n ln n/n s y = ρ s y Substitute for d c s dt δz, and divide through by M z to arrive at the quadratic equation: ± [ ] 1 = M r +1 M r r (M 1) + 4L n M h /ρ s Solve, using the simplifying notation: u 0 2L n M h /ρ s, u M + u 0, to find r = (1/4)[u + 2 ± u 2 + 4u 0 ] Integrate to obtain the z-displacement δz = 2 rdm = 1 [ { u 2 + 4u u u L n u u 0 ln (±u + )} ] u u 2 + 4u 0 /2 u This solution is real only if 1/u o > 1/4, ie the ion diamagnetic drift magnitude M ni must be less than the E B drift magnitude M E

20 δz value proves unimportant I ve proved that the solution exists; that s critical for the presumptions Solution shows the δz values are small: 20 The values are nearly symmetric about M = 1: nearly even parity in M + 1 Perpendicular Mach number is derived from the odd-parity in M + 1 part of flux For practically all parameters, δz/l n is negligible < 2% contribution to M

21 Positive characteristics now curved 21 Schematic shapes of positive characteristics in x-y plane Curvature exaggerated Object Object θ m M y θ m M y 1 + M 1 + M L n positive M ni opposes M E L n negative M ni reinforces M E Notice that the curvature changes the definition of convex Plane surfaces are concave when the characteristics curve towards them We don t actually have to solve for the characteristic trajectories

22 Magnetic Presheath Displacement 22 Summary so far: the flux density from the plasma into the magnetic presheath has been shown, for an arbitrary (convex) 3-D object shape, to be with M h = M E + M ni and Γ = nc s = n (z )c s exp[ 1 (M M h cot θ)] n (z ) = n (z δz) = n (z) exp( δz/l n ) Although I ve solved for it, plasma displacement δz/l n can generally be ignored When n 0 (there s diamagnetic drift) the corresponding displacement in the MPS is not ignorable Must be calculated Consider general surface orientation normal: ˆk sin α = ˆkB/B other directions ˆl B ˆk, ĵ Integrate the ˆk-component of the full momentum equation including inertia to find the total displacement in the ˆl direction through the MPS After significant work, obtain δ l = ρ s [1 sin α] 2 sin α cos α

23 Displacement changes flux density 23 The displacement δ lˆl has a component along the gradient ln n The impact point samples flux entering MPS a distance δ lˆlẑ up the gradient z E 1 End B E E 2 y E B E B This has odd-parity (unlike δz) Effectively substracts a important fraction of M D from measurement x E δ lˆl θ 1 θ 2 δ lˆl E [Also causes potential end-effect] Stalk

24 Temperature Gradients 24 Accommodate the temperature-gradient part of Diamagnetic drifts by assuming T e and T i are functions only of z Two major changes occur First, c s (z) is spatially varying Characteristics become (v ± c s )[ln n/n ± v /c s ] = ±(v /c s )( v z /L c ) v z /L p where L c c s /dc s /dz is scale-length of c s -variation, and L p of ion-pressure Look for solutions that are of the form ln n/n M = g(m ) Reduce to a quadratic in r dg/2dm using same techniques as before (M 1) ( M L c + 1 L p ) ( r 1 + r + (M M + 1) L c + 1 L p ) = 2M h ρ s (1 + 1/2r) For this to be a consistent solution we require w M h L c /ρ s to be constant Solve for r Integrate to find g = 2 r dm

25 Characteristic-variation small 25 Solutions for g = ln n/n M Generally small and/or even-parity provided we avoid 0 < w < 3 [Mathematical Note: All this is necessary because these corrections are not small by ordering They are the same order in ρ s /L as M So we have to evaluate them before we can say if they are important] Incidentally, the geometric effect of g-variation is to make the positive characteristic curved The smallness of g amounts to saying the radius of curvature is bigger than the object

26 New temperature-gradient term looks like electron diamagnetic drift 26 Second T-effect New type of drift term [ ( ) ( Te M = ln( n n ) + φ + T ) i + T i e Ze Ze ln n + mc2 s Ze ln( n n ) }{{}}{{} M M h 1 Combines parallel density difference with perp T e -gradient The M h = ln(n/n )M Te + M Ti + M E + M ni is no longer homogeneous But, with boundary condition, M = M h cot θ 1 gives ln n/n = 1 + g + (M Ti + M E + M ni ) cot θ 1 + M Te cot θ ] B c s B 2 Adding the temperature gradient effects into the MPS analysis results in a similar term but full pressure-gradient effect [Skipping non-trivial algebra on MPS effects]

27 Final Full Flux Density Approximating g = M and writing to first order in M : 27 ln { Γ p nc s } = 1 M + [ (1 + M )M Te + M Di + M E ( ) ] 1 sin α M 1 + sin α D cot θ with M E = E B/c s B 2 ŷ Electric field drift, agreeing with intuition M Di = p B/(c s nzeb 2 )ŷ Total ion diamagnetic drift Intuitive M Te = T e B/(c s eb 2 )ŷ Electron diamagnetic due to T e gradient M D = M Di M De Difference between ion and electron diamag drifts α Angle between B and surface θ Angle in x-y-plane between B and surface The quantity in [ ] is what a transverse Mach probe measures Cf prior diffusive treatment ln { Γ p nc s } = 1 11(M M cot θ) * I H Hutchinson Phys Rev A, 37 (1988) 4358; IHH Principles of Plasma Diagnostics (2002)

28 Discussion 28 Solutions are exact Equations are approximate but good approximations Drift approximation probably better than diffusive for tokamaks Geometry is arbitrary (provided convex) Generality a gift from God! Results agree (within 10%) with standard diffusive M E formula which is based on ad hoc fit to numeric solutions But now given rigorous analytic justification Previously, the dependence on Diamagnetic drifts was simply unknown { } [ ( ) ] Γ p 1 sin α ln = 1 M + (1 + M )M nc Te + M Di + M E M s 1 + sin α D cot θ We now see that there is an extra apparent velocity in the measurements that is equal to M Te This may help to make sense of the Smick/LaBombard observations A more complete formula and fuller understanding of the effect of MPS displacement M D has been obtained The drift approximation is problematic when M D 1 because that requires ρ s /L 1 A local approximation is then impossible [Why solution by expansion impossible]

29 Summary 29 A complete analytic solution of the magnetized plasma drift equations surrounding an arbitrary 3-D absorbing object has been obtained It shows that the (appropriately normalized) flux-density depends only on surface orientation, not position on the object, for convex regions It shows why the (known) Mach probe calibration has the form it does, and what the (previously-unknown) diamagnetic drift dependence is

30 Supplemental 30

31 Eddy size equals or exceeds probe 31 C-Mod scanning probe head (schematic) compared in scale with Example of imaged turbulence eddies in SOL Probe body is comparable in size Probe electrodes are smaller