Synergy of Turbulent Momentum Drive and Magnetic Braking

In absence of external torque, plasma rotation in tokamaks results from a balance between collisional magnetic braking and turbulent drive. The outcome of this competition and cooperation is essential to determine the plasma flow. A reduced model, supported by gyrokinetic simulations, is first used to explain and quantify the competition only. The ripple amplitude above which magnetic drag overcomes turbulent viscosity is obtained. The synergetic impact of ripple on the turbulent toroidal Reynolds stress is explored. Simulations show that the main effect comes from an enhancement of the radial electric field shear by the ripple, which in turn impacts the residual stress.

Figure 1

Sketch of the modeled ripple-turbulence competition on the equilibrium toroidal velocity estimated with local momentum conservation in the case of cocurrent $V_{\mathrm{turb}}$. The synergistic effects are not accounted for here, but are detailed further below.

Figure 2

Radial profiles of the toroidal velocity $V_T$, its shear $V_T^{\prime }$, and the stress tensor $\mathrm{\Pi }$ taken at turbulent saturation for simulations without ripple and with different initial toroidal velocity profiles $V_T(t=0)=V_{T0}\exp [-32(r/a-0.5{)}^2]$ with $a$ the minor radius. Velocities are normalized to the ion thermal velocity and lengths to ion Larmor radius ${\rho }_i$.

Figure 3

Radial profile of magnetic drag ${\nu }_{\phi }$ for different ripple amplitudes and the turbulent viscous contribution $\chi {\lambda }_v$. Orange zone represents $\chi |{\lambda }_v|$ for $a/2\le |{\lambda }_v{|}^{-1/2}\le a$. Time is normalized to the cyclotron period ${\omega }_{c0}^{-1}$.

Figure 4

Time trace of the (a) toroidal velocity $V_T$ and (b) the radial electric field $E_r$ for different ripple amplitudes in $0.45<r/a<0.55$, shaded areas, and radially averaged in this same interval, solid lines.

Figure 5

Solid lines, radial profile of coarse-grained (temporally and spatially) (a) radial electric field and (b) its shear, as well as (c) residual stress and (d) the opposite of its divergence for different ripple amplitudes. Dashed lines, time average only.