7th Asia-Pacific Conference on Plasma Physics, 12-17 Nov, 2023 at Port Messe Nagoya

Scaling law of the plasma rotation in visco-resistive magnetohydrodynamicsystems
A. Krupka1, M.-C. Firpo1
1LPP, CNRS, Ecole Polytechnique, Francee-mail (speaker): anna.krupka@lpp.polytechnique.frThe rotation of plasma has been identified as a possible
key factor in the heat and particle confinement properties
in tokamaks. Numerous experimental observations havesupported the existence of intrinsic plasma rotation inconnection with the high-confinement mode [1]. One
option to avoid instabilities could be to make plasma
rotate to stabilize it. The understanding and control ofplasma rotation are of utmost importance for achieving
enhanced plasma performance and sustainable fusion
reactions in tokamak devices.
The usual approach begins by using the axisymmetricGrad-Shafranov equation at zeroth order to reconstructmagnetic flux surfaces in real-time. In this context, any
existing velocity field is considered to be primarily theresult of “turbulence”, including its axisymmetric part. Itmeans that in the usual approach there is no steady-state
plasma velocity: an equilibrium plasma is a non-rotation
plasma.
The present approach follows Montgomery's works [2].
To address this matter, we conduct numericalcomputations of the steady-state Navier-Stokes equation,
which includes the non-linear (v·grad)v term [3, 4, 5]. To
obtain a closed system of partial differential equations wesolve it on the cross-section plasma domain Ω together
with Maxwell equations, Ohm's law and boundary
conditions [6, 7]. We numerically compute theaxisymmetric steady states of the visco-resistivemagnetohydrodynamic equations using the finite element
method through the open-source platform FreeFem++ forsolving partial differential equations [8].
By considering a visco-resistive magnetohydrodynamic
model for a tokamak plasma with a given toroidal current
drive, we predict and numerically check a scaling law of
the toroidal velocity as the function of the resistivity, η,and the Hartmann number H ≡ (ην)-1/2.

This scaling law is {vφ}rms ≃ ηf(H) for some function fand should be valid as long as the inertial term remains
negligible. Then, we reconsider this in light of thepossible anisotropy of plasma resistivity and argue thatthe perpendicular resistivity can be retained in thisscaling.
References[1] J. Rice et al., Inter-machine comparison of intrinsic
toroidal rotation in tokamaks, Nuclear Fusion 47, 1618
(2007).
[2] L. P. Kamp and D. C. Montgomery, Toroidal flows inresistive magnetohydrodynamic steady states, Physics ofPlasmas 10, 157–167 (2003).
[3] J. A. Morales, W.J. Bos, K. Schneider and D.C.
Montgomery, Intrinsic rotation of toroidally confinedmagnetohydrodynamics, Phys. Rev. Lett. 109, 175002(2012).
[4] H. Oueslati, T. Bonnet, N. Minesi, M.-C. Firpo andA. Salhi, Numerical derivation of steady flows in viscoresistive magnetohydrodynamics for JET and ITER-likegeometries with no symmetry breaking, AIP Conf.
Proceedings 2179, 020009 (2019).
[5] M. Holst, V. Kungurtsev and S. Mukherjee, A Noteon Optimal Tokamak Control for Fusion PowerSimulation, arXiv:2211.08984 (2022).
[6] H. Oueslati, M.-C. Firpo, Breaking up-down
symmetry with magnetic perturbations in tokamak
plasmas: Increase of axisymmetric steady-state
velocities, Phys. Plasmas 27, 102501 (2020).
[7] E. Roverc'h, H. Oueslati and M.-C. Firpo, Steadystate flows in a visco-resistive magnetohydrodynamic
model of tokamak plasmas with inhomogeneous heating,
Journal of Plasma Physics 87, 905870217 (2021).
[8] F. Hecht, New development in FreeFem++,J. Numer. Math. 20 (3-4), 251-265 (2012).