Modelling Axial Flow Generation and Profile Evolution in the CSDX Linear Device R. J. Hajjar, P. H. Diamond, G. R. Tynan, R. Hong, S.

Transcription

1 Modelling Axial Flow Generation and Profile Evolution in the CSDX Linear Device R. J. Hajjar, P. H. Diamond, G. R. Tnan, R. Hong, S. Thakur This material is based upon work supported b the U.S. Department of Energ, Office of Science, Office of Fusion Energ Sciences, under Award Number DE-FG0-04ER54738

2 What are we doing? Main Goals: 1. Investigate the relation between the parallel and perpendicular flow dnamics in a coupled drift-ion acoustic wave plasma.. Model the profiles evolution and the fluctuations measurements in the CSDX linear device. 3. Investigate how the CSDX axial flows are generated and their relation to azimuthal shear and densit gradient.

3 Along the side: 1. Relate the mean profile evolution and the variations of the transport fluxes to variations of the mean plasma densit, via the adiabaticit parameter.. Interpretation of the corresponding variations in view of a zonal flow production/turbulence enhancement perspective. 3

4 How is this related to tokamaks? 1. Intrinsic rotation is essential for plasma stabilit, speciall in large scale devices where methods of external axial momentum input are not efficient.. The relation between the axial flow shear and the edge pressure gradients in CSDX, reminds us of the well known Rice scaling in tokamaks. 3. Tokamak L-H transitions are associated with a nonlinear energ transfer to the mean flows via the Renolds stress. The current model captures the energ transfer between flows and fluctuations, in both parallel and perpendicular directions. It can thus be used to understand tokamak phsics. 4

5 Wh do we care? 1. The reduced model presents an opportunit to stud profile evolution in terms of kinetic energies v z,v (Rice scaling) and turbulent energ ε.. The model relates variations of the parallel and perpendicular Renolds stresses: v ~ v~ and v ~ v~ x z x, to the gradients of n, v z and v. All of these quantities are experimentall measurable in CSDX. 3. It adds to what we alread know about DWs and ZFs, b introducing a similar phsical relation between DWs and axial flows. 4. The model allows us to understand the coupling/competition relation between the parallel and the perpendicular flow dnamics. 5

6 Experimental Results (R. Hong in preparation) 6

7 Experimental Results (R. Hong in preparation) As B is raised: 1. Densit gradient increases.. Parallel stress increases in magnitude. 3. Residual parallel stress and Renolds force increase significantl. 4. Radial shear of v z increases. 5. Axial flow shear increases with the densit gradient (proportional to azimuthal shearing) coupling between parallel and perpendicular flow shear. 7

8 Wh another reduced model? 1. CSDX plasma is a multiscale sstem a reduced model is a necessar and appropriate intermediar between the experiments and the full blown simulations.. Reduced models are essential to formulate the right questions that guide the experiment and interpret the experimentall acting brute forces. 3. Reduced models facilitate understanding the phsics behind the experimental feedback loops between the mean profiles and the fluctuation intensities. 4. Reduced models have a relativel low computational cost. 8

9 What is new in this model? 1. Self-consistent treatment of the coupling relation between the parallel and perpendicular flow dnamics.. The acoustic coupling breaks conservation of PV Need to formulate a new conserved energ. This new energ is formulated in terms of both parallel and perpendicular kinetic energies, as well as the plasma internal energ. 3. The model introduces an experimental measure of the correlator k k m z that relates the parallel residual stress proportional to the densit gradient. 9

10 Formulation of the model Hasegawa-Wakatani model + Parallel compression term Parallel compression PV conservation is broken define a new conserved energ ε: 10

11 Model Equations Diffusion Coupling terms Production t x ( 1/ l mix ) x dn dx v~ v~ x dv dx v~ v~ x z dvz dx l 3/ mix v th ei [ z ( ~ n~ )] nv ~~ z P Dissipation We need expressions for: 1. Particle Flux. Vorticit flux and Renolds work 3. Parallel Renolds stress 4. Mixing length 11

12 1- The Quasi Linear form of the Particle Flux: * r In the adiabatic limit,, v C / L and the particle flux is: 1 ks d s s n with k ei Here the adiabaticit parameter is: v z th 1 1

13 - Vorticit Flux: Renolds power rate: v~ x v~ x fl d mix v dx d dx v res xz fl mix ci L n v~ v~ x v~ v~ x dv dx dv dx v v [ [ fl d v dx d v dx mix res x ] fl mix ci L n ] Suppress ZF DW Drive 13

14 3- Parallel Renolds Stress a) Quasi Linear Expression: v~ v~ x z fl mix dv dx z k m k z C s 3 s l [ mix k s ] b) Empirical form: In analog with pipe flows, we write 14

15 The parallel Renolds stress is then: σ VT is an empiricall testable constant of the correlator <k m k z >, since both L // and <k > can be determined experimentall: 15

16 res xz VTC s lmix / L// L n 1) Densit gradient is responsible of generating the axial flow (the parallel residual stress) via the relation: ) Since v also depends on n, we can write: Parallel and Perpendicular Coupling 16

17 4-Mixing length Mixing length is inversel proportional to the shear: v and v z. This allows for a closed feedback loop to form for the energ, thereb enhancing the mean energ. k m =1/l 0 and k z =1/L // τ c = l 0 /<δv x > 1/ =l 0 /(fl mix ) 1/ In CSDX, azimuthal shear is greater than the axial shear 17

18 Simplification b slaving For fast correlation time τ c <τ conf, and when parallel energ transfer is much smaller than perpendicular transfer, we simplif the model b dropping v z equation and slaving ε to the densit and the azimuthal shear: Improved model from Hinton et al. (1993) where fluctuations are treated as an ad hoc constant. Where Δ=f(n, v ) 18

19 Transition from adiabatic to hdrodnamic response General Expression for the particle flux: General Expression for the vorticit flux: 19

20 Adiabatic limit 1 * 1/ 0 Plasma densit increases Hdrodnamic limit 1 * Fluxes are: T e decreases Adiabaticit parameter α decreases Fluxes are: Turbulence enhancement Zonal flow production decreases 0

21 Future Work 1. Numerical investigation of the 3-field model and the corresponding paralle to perpendicular coupling.. Numerical investigation of the -field model. 3. Numerical investigation of the transition from an adiabatic to a hdrodnamic plasma regime 1