Chapter 5 Angular momentum transport by sheared flows
5.8 Kinetic electrons
We next examine the effect of kinetic electrons on the E×B driven momentum transport. In Fig.5.7 (top) the momentum fluxes for (u′∥,sˆ)=(0,1) are shown to be comparable with the adiabatic case once the increased turbulence due to the trapped electron drive is factored out. Comparing the parallel potential structure Fig.5.7 (bottom), we see an similar asymmetric shift for the kinetic and adiabatic cases. This differs from the Coriolis pinch [45] in which the Coriolis drift generates an asymmetry in the parallel velocity structure, and the potential shift in response compensates (completely in the case of adiabatic electrons) the symmetry breaking of the Coriolis drift [155]. Here, no such ‘compensation effect’ occurs since the asymmetry in the solution is generated directly through the convection and effective shift of the ballooning angle.
Finally, we also investigated theE×B momentum transport in pure trapped electron mode turbulence in nonlinear simulations with(u′∥,sˆ)=(0,1). This case,
Figure 5.7: (Top) Comparison ofMRfor thega-stdcase with kinetic electrons and
adiabatic electrons. (Bottom) Parallel perturbed potential structures compared: The trapped electron drive is localised on the outboard side, but the asymmetric shift is equivalent. 0 1 2 3 4 −1 −0.5 0 γE/ γmax M || / χ e 0 1 2 3 4 0 0.5 1 1.5 γE / γ max χ e / (c s ρ s 2 /a ) 0 0.5 1 0 0.05 0.1 k θρs γ (a/c s ) 0 0.5 1 10−2 10−1 100 101 k θρs χ e / (c s ∆ k ρ s 3 /a) pure TEM GA−STD a) b) c) d)
Figure 5.8: Trapped electron mode turbulence,pure-tem case: a) E×B induced momentum momentum transport dimensionless ratio, b) E×B suppression of elec- tron heat flux, c) growth rate spectrum, d) nonlinear electron heat flux spectrum compared toga-std kinetic case.
which we label pure-tem has the same geometry parameters as the ga-std case, but the electron temperature gradient is reduced toR/LTe =7, the ion temperature
gradient reduced toR/LTi =0 and the ion temperature reduced to Ti/Te=3/10 to
eliminate the smallest scale electron modes [34] and ensure a peaked growth rate spectrum (Fig.5.8c). The modes all propagate in the electron diamagnetic direction, but the sign of theE×B driven momentum flux is unchanged (Fig.5.8a), indicating no dependence on the mode propagation direction (a possibility suggested in Ref. [160]). For this case, the thermal transport is dominated by the electronsχe≫χi,
and normalisingM∥withχE gives a dimensionless number of magnitude comparable
toMRfor thega-stdcase (which is dominated by ion thermal transport with both
kinetic and adiabatic electrons). The shift in the turbulence maximum is in the same direction as in Fig.5.2a, and the sign and form of the Γ∥ spectrum in kψ is
the same as the ˆs=1 case in Fig. 5.4. Investigation of the symmetry breaking in the drive for TEM turbulence is more difficult since we find no stable result for this TEM case for ˆs=0.
TheE×B suppression of turbulence still occurs for the pure TEM case, but at a higher normalised shearing rateγE/γmax. This is explained by the shift in the
turbulence spectrum to smaller scales (Fig.5.8d), since theE×B shear acts first to quench transport at the largest scales, as discussed above.
5.9
Summary
The nonlinear local gyrokinetic simulations presented in this chapter have confirmed and quantified the quasilinear slab prediction of parallel momentum transport due to backgroundE×B shear [151]. The shearedE×B flow both quenches turbulence and drives a parallel asymmetry that produces anomalous parallel momentum transport. Given these competing effects, the turbulent momentum transport due to this term in isolation exhibits an extrema in the shearing rateγE (as predicted in Ref. [151]).
The simulations show that theE×B contribution to the momentum transport (M∥γE) has a stronger quenching withγE than that observed for the heat diffusivity.
The magnitude of the effect relative to the overall turbulence level is quantified by the dimensionless number MR = M∥/χi, which is found to be a strong function
of γE. This quantity is analogous to the turbulent Prandtl number (the ratio of
momentum diffusivityχ∥ to thermal diffusivityχi), but unlike the Prandtl number
shows significant sensitivity to plasma parameters.
The results presented in this chapter show that theE×B driven momentum flux interacts with the diffusive transport such that MR also depends on u′∥. It is
The symmetry breaking mechanisms responsible for momentum transport therefore cannot be considered independent in the turbulent regime with background E×B shearing, which sets limits on the applicability of linearised decompositions with independent terms (Eq. (5.3)). When all the mechanisms are included together, nonlinear simulations are required for each point in parameter space to accurately quantify momentum transport, even in the local model.
The direction of the E×B induced momentum transport reverses with the sign of∇Er and ˆs, such that for positive magnetic shear, the tendency is to trans-
port momentum inwards, enhancing any existing rotation gradient (for both ITG and TEM turbulence). The sign of the flux reverses for negative magnetic shear, and we also find that a weaker symmetry breaking mechanism generates momentum transport at zero magnetic shear. This mechanism will transport toroidal momen- tum towards a minimum in the q-profile when there is an existing peaked rotation profile. This local ‘spin up’ could play a role in the formation of some internal transport barriers, especially if (as indicated in Ref. [145]) the poloidal flows in the region of a transport barrier are elevated well above the neoclassical level.
For low parallel flow gradients, the size of the E×B contribution can be a significant correction to the diffusive momentum transport, under certain conditions resulting in null flow sustaining equilibrium rotation gradients. In the case of purely toroidal rotation, the effective momentum diffusivity can be significantly reduced at lower toroidal rotation gradients. Simulations of ITG turbulence with kinetic electrons have comparable MR to those with adiabatic electrons, with no parallel
mode structure compensation effect (as seen for the Coriolis pinch [155]) observed. Whilst the Coriolis pinch [45] requires a seed rotation, for theE×Bdriven momentum transport effect an initial gradient in the rotation must be present.
In summary, whilst the diffusive parallel momentum Prandtl number is roughly constant over a range of parameter space, the results of this chapter show that the equivalent dimensionless ratio for E×B induced momentum transport is a strong function of shearing rate, parallel flow gradient, and magnetic shear.