Hartmann Flow Physics at Plasma-Insulator Boundary in the Maryland Centrifugal Experiment (MCX)

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1 Hartmann Flow Physics at Plasma-Insulator Boundary in the Maryland Centrifugal Experiment (MCX) Sheung-Wah Ng, A. B. Hassam IREAP, University of Maryland, College Park ICC 2006, Austin, TX

2 Maryland Centrifugal Experiment (MCX) E A rotating mirror system. Shear rotational flow stabilizes instabilities and provides centrifugal force confining the plasma towards the mid-plane.

3 Frictions on Rotational Flow? For the MCX idea to be practical for generating energy, the energy required to maintain the supersonic rotational flow should be small. Considerations: Neutrals Drag? Hartmann Friction? Parameter of concern: Momentum confinement time, τ mom

4 The Role of Neutrals Limited existing works for centrifugal confined plasma: Open field lines and existence of centrifugal force. We would like to know what role neutrals are playing τ mom in the momentum confinement time,. We will begin our study by considering the neutrals distributions in the system with ionization, charge exchange with prefect recycling.

5 The Geometry x z N w Side Wall End Wall n core B Mid-Plane N w Side Wall

6 Normalized Model Equations ( nu)' = Nn nu = n nn u U + NnU 2 ( )' γ ' γ ( ) ( NU )' = Nn 2 ( )' γ ' γ ( ) NU = N + nn u U NnU +ng (1) Normalization: n = n, t = 1/ αn, where n = (n+n)dv /V o cx, i g u γ = α α T = T = T ave l = c n 2 ( / 2)', CX / I, i 0 and e N o ave ave 2 1/2 s /( αiαcx ) ~ 3cm to 5c m Geometrical mean of λ and λ CX ion

7 Wall Neutral Density vs. Confinement Force 1D numerical result N w e g / γ g / γ ( 2 ) In MCX, g / γ ~ M / 2 s Rotation Mach number

8 Cross-field 1D Solution It can also be shown that [1], Neutral density at Wall Plasma classical cross-field 2 diffusion coefficient, ηnt/b Plasma core density N D w n 2D core N 1 Neutral penetration depth is also ~ l cx, i Neutrals diffusion coefficient, v 2 t,i / α CX n [1] R J Goldston and P H Rutherford 1995, Introduction to Plasma Physics, Institute of Physics Publishing, Philadelphia.

9 Numerical Calculations in 2D system $ L x = L = 1, artifical confinement z force acting towards right in z. g=0, no confinement g=1, with confinement End wall End wall

10 2D numerical results: Ratio of Neutrals densities at different walls

11 Remarks on Neutrals Distributions N << w 1 and N w ~ O(1) in 1D separately. N N 2 w exp( M s / 2) in 1D if confinement (i.e. rotation) is considered w / N increases as better confinement w is achieved in 2D.

12 Hartmann Physics Insulating Hartmann Wall Main Flow Across the Strong Mirror Field Top and Bottom Hartmann layers Prevent Supersonic Flow? Conducting side wall B Flow

13 Classical 1D Hartmann Results % B o Given, driving force density,system L µ η size,viscosity and resistivity. z Hartmann Number: Momentum confinement time: H ~ B L / µη Hartmann Layer width: lh ~ Lz / H a Maximum flow : u a 0 2 FLz, ~ 2 µh y core z H nmu % J. D. Jackson, "Classical Electrodynamics", 2nd ed., (Wiley, New York, 1975). τ a F F y, core / 2 ~ τ µ H a

14 Large Current Sheet J x ~ l H x z y B u y Applied Force, F

15 H Hartmann numbers in MCX If we substitute the MCX parameters* into the -5 6 l 4 H formula, we have H ~ 10, ~ 10 cm and τ ~10 sec. Therefore by considering the conventional Hartmann physics alone, the idea of MCX is not feasible f or a generating energy because too much energy is required to support a supersonic flow (smalless of τ ). H

16 Neutrals Effects on Hartmann One of the main physical parameters in conventional Hartmann problem is the resistivity η. The effect of neutrals on η can be estimated from the ratio of e-n and e-i collision frequencies: 3 ( ) ν / ν ~ 3 10 N / n en ei which might not be significant because the confinement limits the neutrals densities at the wall.

17 Boundary Plasma Density Effect In the classical Hartmann calculation, the plasma density is a constant. Yet, a simple analysis shows m that, by taking µ = n ɶ µ (i.e. µ is Braginskii's η for m = 2), we have u y, core ~ 1 m n / 2 wall m m FL B Therefore, mechanism that lowers the boundary η µɶ plasma density could increase the core flow and thus the momentum confinement time, τ mom. o z 1

18 Such mechanisms could be (i) plasma outflow (recycling) or (ii) confinement. (i) For the recycling effects, we can exam. the change in τ mon/ τ H as α i changes. Redefine 2 p dz τ mom F µ = nɶ µ 1 y dz ɶ µ 1 = 0.05 η = 0.05 *Note that part of the total momentum goes to the neutrals. τ τ mom H α i

19 (ii) Putting an artificial confinement force along z towards the mid-plane, the τ mom is found to be increased with g as expected. τ τ mom H µ = nɶ µ 1 ɶ µ 1 = 0.05 η = 0.05 g

20 Selected Profiles along z with different confinement strengths ( ) g=1 g=0.5 g=0 µ = nɶ µ 1 ɶ µ 1 = 0.05 η = 0.05 n n = ave (n+n)dv/v = 1 N Force acting to the left (mid-plane) 2 p / 2 u / F y y F

21 Hall Effects The existence of the thin current sheet inside the Hartmann layers leads to the consideration of adding Hall terms to the Ohm s Law: E = u B + η j + j B p ne e It can be shown analytically that the Hartmann layers are broadened and the flow is increased with a fixed driving force. The contribution of the hall effect can be measured from the paramet er ε c / ω L. p, i z

22 * Since for MCX, ~ ε >> η ~ 10 after some normalizations, the hall effect could be significant in the Hartmann layers and it can be shown that τ H, hall ~ τ µ τ µ >> = τ H a hall H a, However, secondary flow in the x-direction is being generated too. That might be a concern for MCX. H

23 Profiles with Hall effects It can be shown that τ ε / µ hall whe n ε >> η. 2 uy / F z η = µ = ε = ε = 0.05 ε = ux / F z

24 Conclusions Supersonic azimuthal flow is required in MCX. Neutrals density is small at the Hartmann wall when confinement is good (and recycling is the only source). Thus drag from neutrals should be limited. Classical Hartmann layer limits core flow speed. Small plasma density at the wall loose this restriction. Good confinement helps. Hall effects around Hartmann layer help further. But generate secondary flows.

25 Future Works Analysis in full geometry is required especially for the Hall effects secondary flows. Since both plasma and neutrals densities are small around the walls, kinetic effects (e.g. in η and µ ) might need for further analysis.

26 Appendix * MCX Parameters B T n L a 20-3 o = 1 T, = 10 ev, core = 10 m, z = 1 m, = 0.3 m $ Numerical simulations parameters The 2D simulation parameters are normalized based on the MCX parameters above. β g α α I C X l c x, i 0.3 ~ ' 2 p / B ~ D simulations Real System (MCX) 1 ~ ~ 1 η, µ 0.0 1, ~ 2 1 0, u /

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