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L–H Transition under Poloidal Nonuniform Ion Heating in Turbulent Tokamak Plasma

Abstract—

The turbulent dynamics of tokamak plasma in the case of poloidal nonuniform ion heating is studied on the basis of the numerical solution of the nonlinear equations of the reduced two-fluid Braginskii hydrodynamics. It is shown that, when the threshold in the heating intensity is exceeded, a dynamic bifurcation of the solution of the MHD system under consideration occurs, as a result of which the value of the average E × B drift velocity increases significantly. The interaction of this flow with turbulent fluctuations leads to the suppression of the latter and a decrease in radial transport, which leads to the transition of the plasma to the improved confinement mode called the L–H transition. It follows from the calculation results that the difference in the location of the heat source (e.g., in the upper or lower half-plane of the tokamak chamber) retains the L–H transition phenomenon, but leads to a difference in the efficiency of suppressing the turbulent heat flux from the electron component of the plasma. Numerical simulation showed that, under conditions of poloidal nonuniform heating, the main role in the generation of the poloidal rotation velocity is played not by the turbulent Reynolds stress force FRE, but by the geodesic force FSW and the neoclassical force arising from the longitudinal viscosity FNEO ~ (Ti0)5/2, the value of which significantly increases during ion heating.

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Correspondence to R. V. Shurygin.

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Translated by L. Mosina

APPENDIX

APPENDIX

As mentioned above, splitting Eq. (2) of field variables into axially symmetric and ballooning modes leads to the fact that the original system of MHD equations (1a)(1e) is split into two interacting subsystems of modes. Below is the system of evolutionary equations for ballooning modes \({{f}_{{S,C}}}(r,\theta ,t) = \) \(\left\{ {{{\phi }_{{S,C}}},{{p}_{{eS,C}}},{{p}_{i}}_{{S,C}}} \right\}\) derived from the original system (1a)–(1e) in the approximation m ≫ 1:

$$\begin{gathered} \frac{{\partial {{w}_{{S,C}}}}}{{\partial t}} \mp \frac{m}{r}\phi _{0}^{'}{{w}_{{C,S}}} \pm \frac{m}{r}w_{0}^{'}{{\phi }_{{C,S}}} \\ \, = \frac{1}{\nu }\frac{{{{\partial }^{2}}{{h}_{{S,C}}}}}{{\partial {{\theta }^{2}}}} + \frac{1}{{{{n}_{0}}}}{{R}_{{S,C}}} - {{\Pi }_{{{\text{||}}S,C}}} + {{\mu }_{ \bot }}\Delta {{w}_{{S,C}}}, \\ \end{gathered} $$
(A.1)
$$\begin{gathered} {{R}_{{S,C}}} = \pm {{p}_{{C,S}}}\frac{m}{r}\cos (\theta ) - C({{p}_{{S,C}}}); \\ {{p}_{{S,C}}} = {{p}_{{eS,C}}} + {{p}_{{iS,C}}}, \\ \end{gathered} $$
$$\begin{gathered} \frac{{\partial {{p}_{{eS,C}}}}}{{\partial t}} \mp \frac{m}{r}\phi _{0}^{'}{{p}_{{eC,S}}} \pm \frac{m}{r}\left( {p_{{e0}}^{'}} \right){{\phi }_{{C,S}}} \\ \, = \frac{5}{3}\sigma {{p}_{{e0}}}\frac{{{{\partial }^{2}}{{h}_{{S,C}}}}}{{\partial {{\theta }^{2}}}} + \frac{5}{3}g{{\Psi }_{{eS,C}}} \\ \, + {{{\hat {\chi }}}_{{||e}}}\frac{{{{\partial }^{2}}{{p}_{{eS,C}}}}}{{\partial {{\theta }^{2}}}} + {{\chi }_{ \bot }}\Delta {{p}_{{eS,C}}}, \\ \end{gathered} $$
(A.2)
$$\begin{gathered} {{\Psi }_{{eS,C}}} = \left\{ {{{M}_{{S,C}}} - \xi {{A}_{{S,C}}} \mp \frac{{m\cos (\theta )}}{r}} \right. \\ \, \times \left. {\mathop {[ - \xi {{T}_{{e0}}}{{p}_{e}}_{{C,S}} + {{p}_{{e0}}}{{\phi }_{{C,S}}}]}\limits_{} } \right\}, \\ \end{gathered} $$
$$\begin{gathered} {{A}_{{S,C}}} = {{p}_{{eS,C}}}\frac{{d{{T}_{{e0}}}}}{{dr}}\sin \theta + {{T}_{{e0}}}C({{p}_{{eS,C}}}), \\ {{M}_{{S,C}}} = {{p}_{{e0}}}C({{\phi }_{{S,C}}}) + {{p}_{{eS,C}}}C(\phi ), \\ \end{gathered} $$
$$\begin{gathered} \frac{{\partial {{p}_{{iS,C}}}}}{{\partial t}} \mp \frac{m}{r}\phi _{0}^{'}{{p}_{{iC,S}}} \pm \frac{m}{r}\left( {p_{{i0}}^{'}} \right){{\phi }_{{C,S}}} \\ \, = \frac{5}{3}\sigma {{p}_{{i0}}}\frac{{{{\partial }^{2}}{{h}_{{S,C}}}}}{{\partial {{\theta }^{2}}}} + \frac{5}{3}g{{\Psi }_{{iS,C}}} \\ \, + {{{\hat {\chi }}}_{{||i}}}\frac{{{{\partial }^{2}}{{p}_{{iS,C}}}}}{{\partial {{\theta }^{2}}}} + {{\chi }_{ \bot }}\Delta {{p}_{{iS,C}}}, \\ \end{gathered} $$
(A.3)
$$\begin{gathered} {{\Psi }_{{iS,C}}} = \left\{ {{{N}_{{S,C}}} - \xi {{B}_{{S,C}}}) \mp \frac{{m\cos (\theta )}}{r}} \right. \\ \, \times \left. {\mathop {[ - \xi {{T}_{{i0}}}{{p}_{{eC,S}}} + {{p}_{{i0}}}{{\phi }_{{C,S}}}]}\limits_{} } \right\} \\ \end{gathered} $$
$$\begin{gathered} {{B}_{{S,C}}} = {{p}_{{iS,C}}}\frac{{d{{T}_{{i0}}}}}{{dr}}\sin \theta + {{T}_{{i0}}}C({{p}_{{iS,C}}}), \\ {{N}_{{S,C}}} = {{p}_{{i0}}}C({{\phi }_{{S,C}}}) + {{p}_{{iS,C}}}C(\phi ), \\ \end{gathered} $$
$$\begin{gathered} {{w}_{{S,C}}} = \frac{{{{\partial }^{2}}{{Y}_{{S,C}}}}}{{\partial {{r}^{2}}}} - \frac{{{{m}^{2}}}}{{{{r}^{2}}}}{{Y}_{{S,C}}}, \\ {{h}_{{S,C}}} = \tau {{p}_{{eS,C}}} - {{\phi }_{{S,C}}}, \\ {{Y}_{{S,C}}} = \tau {{p}_{{iS,C}}} + {{\phi }_{{S,C}}}, \\ \end{gathered} $$
$$\begin{gathered} {{\Pi }_{{||S,C}}} = {{{{\nu }_{{||}}}T_{{i0}}^{{5/2}}} \mathord{\left/ {\vphantom {{{{\nu }_{{||}}}T_{{i0}}^{{5/2}}} {{{n}_{0}}}}} \right. \kern-0em} {{{n}_{0}}}}\left\{ {\left( {{{w}_{{S,C}}} + \frac{{5T{\kern 1pt} _{{i{\text{0}}}}^{'}}}{{2{{T}_{{i0}}}}}\frac{{\partial {{Y}_{{S,C}}}}}{{\partial r}}} \right)} \right. \\ \, \times \left. {\mathop {{{{\sin }}^{2}}\theta \mp \frac{m}{r}\sin 2\theta \frac{{\partial {{Y}_{{C,S}}}}}{{\partial r}} + \frac{{{{m}^{2}}}}{{{{r}^{2}}}}(1 - 2{{{\cos }}^{2}}\theta ){{Y}_{{S,C}}}}\limits_{} } \right\}. \\ \end{gathered} $$

The following boundary conditions were used: \(\frac{{\partial {{\phi }_{{S,C}}}(r = {{r}_{0}})}}{{\partial r}} = 0\), \({{\phi }_{{S,C}}}(r = a) = 0\).

Turbulent momentum, particle and heat fluxes associated with ballooning and axially symmetric modes have the following form:

$${{\Pi }_{B}} = - \frac{m}{r}{{\left\langle {\tilde {w}\frac{{\partial \tilde {\phi }}}{{\partial \lambda }}} \right\rangle }_{\lambda }} = \frac{m}{{2r}}\left( {{{\phi }_{C}}{{w}_{S}} - {{\phi }_{S}}{{w}_{C}}} \right),$$
$${{\Pi }_{{AX}}} = - \frac{1}{r}\left\langle {{{{\tilde {w}}}_{{AX}}}\frac{{\partial {{{\tilde {\phi }}}_{{AX}}}}}{{\partial \theta }}} \right\rangle ,$$
$${{Q}_{{e,iB}}} = - \frac{m}{r}{{\left\langle {{{{\tilde {p}}}_{{e,i}}}\frac{{\partial \tilde {\phi }}}{{\partial \lambda }}} \right\rangle }_{\lambda }} = \frac{m}{{2r}}\left( {{{\phi }_{C}}{{p}_{{e,iS}}} - {{\phi }_{S}}{{p}_{{e,iC}}}} \right),$$
$${{Q}_{{e,iAX}}} = - \frac{1}{r}\left\langle {{{{\tilde {p}}}_{{ei,AX}}}\frac{{\partial {{{\tilde {\phi }}}_{{AX}}}}}{{\partial \theta }}} \right\rangle .$$

We note that the main contribution to the transfers is made by the fluxes associated with the ballooning modes

$${{\Pi }_{B}} \gg {{\Pi }_{{AX}}},\quad {{Q}_{e}}_{{,iB}} \gg {{Q}_{e}}_{{,iAX}}.$$

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Shurygin, R.V. L–H Transition under Poloidal Nonuniform Ion Heating in Turbulent Tokamak Plasma. Plasma Phys. Rep. 47, 772–780 (2021). https://doi.org/10.1134/S1063780X21080080

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Keywords:

  • reduced MHD equations
  • turbulence in tokamak
  • L–H transition