Electric Field and Poloidal Rotation in the Turbulent Edge Plasma of the T-10 Tokamak

Abstract

The dynamics of turbulent edge plasma in the T-10 tokamak is simulated numerically by solving reduced nonlinear MHD equations of Braginskii’s two-fluid hydrodynamics. It is shown that the poloidal plasma velocity is determined by the combined effect of two forces: the turbulent Reynolds force F_R and the Stringer–Winsor geodesic force F_SW, which is associated with the geodesic acoustic mode of the total plasma pressure $\left\langle {p{\text{sin}}\theta } \right\rangle$. It follows from the simulation results that the F_R and F_SW forces are directed oppositely and partially balance one another. It is shown that, as the electron temperature increases, the resulting balance of these forces changes in such a way that the amplitude of the poloidal ion flow velocity and, accordingly, the electrostatic potential ${{\phi }_{0}}(r,t)$ decrease. As the plasma density increases, the “driving forces” of turbulence (the dn₀/dr and dp₀/dr gradients) also increase, while dissipation due to the longitudinal current decreases, which results in an increase in the amplitude of turbulent fluctuations and the Reynolds force F_R. On one hand, the force F_SW increases with increasing plasma density due to an increase in the pressure $\left\langle {p{\text{sin}}\theta } \right\rangle$; however, on the other hand, it decreases in view of the factor 1/n₀. As a result, the net force driving poloidal rotation increases, which leads to the growth of the plasma potential. Both under electron-cyclotron resonance heating and in regimes with evolving plasma density, the results of numerical simulations qualitatively agree with experimental data on the electrostatic potential of the T-10 plasma.

APPENDIX

Most experimental data concern plasma parameters averaged over the poloidal angle, ${{f}_{0}}(r,t) =$${{\left\langle {f(r,\theta ,t)} \right\rangle }_{\theta }}$. It is useful to derive separate equations for the time evolution of the variables f₀ = {${{V}_{0}},{{n}_{0}},{{p}_{{e,i\,0}}},{{V}_{{||0}}}$} by using the expansion $f(r,\theta ,t) = {{f}_{0}}(r,t) + \tilde {f}(r,\theta ,t)$, where ${{f}_{0}} = \left\langle f \right\rangle = {{\left( {2\pi } \right)}^{{ - 1}}}\int_0^{2\pi } {fd\theta }$. After averaging the equations presented in Section 2, we obtain the following equations for the averaged variables:

$$\begin{gathered} \frac{{\partial {{n}_{0}}}}{{\partial t}} + {{\nabla }_{r}}{{\Gamma }_{n}} = {{\Psi }_{n}} + {{D}_{ \bot }}\frac{{{{d}^{2}}{{n}_{0}}}}{{d{{r}^{2}}}}, \\ {{\Gamma }_{n}} = \left\langle {{{{\tilde {V}}}_{{Er}}}\tilde {n}} \right\rangle ,\quad {{\Psi }_{n}} = \left\langle {g\,[nC(\phi ) - \alpha C({{p}_{e}})]} \right\rangle ; \\ \end{gathered} $$

((A.1))

$$\begin{gathered} \frac{{\partial {{p}_{{e0}}}}}{{\partial t}} + {{\nabla }_{r}}{{Q}_{e}} = {{\Psi }_{{pe}}} - \frac{2}{3}\sigma \left\langle {\frac{{\partial {{{\tilde {p}}}_{e}}}}{{\partial \theta }}\frac{{\partial \tilde {H}}}{{\partial \theta }}} \right\rangle \\ - \;\frac{2}{3}\gamma \left\langle {{{{\tilde {p}}}_{e}}\frac{{\partial {{{\tilde {V}}}_{{||i}}}}}{{\partial \theta }}} \right\rangle + {{\chi }_{{ \bot e}}}\frac{{{{d}^{2}}{{p}_{{e0}}}}}{{d{{r}^{2}}}} - \left\langle {{{W}_{{ei}}}} \right\rangle , \\ {{Q}_{e}} = \left\langle {{{{\tilde {V}}}_{{Er}}}{{{\tilde {p}}}_{e}}} \right\rangle , \\ {{\Psi }_{{pe}}} = \left\langle {\frac{5}{3}g\left\{ {[{{p}_{e}}C(\phi ) - \xi C({{p}_{e}}{{T}_{e}})]} \right\}} \right\rangle ; \\ \end{gathered} $$

((A.2))

$$\begin{gathered} \frac{{\partial {{p}_{{i0}}}}}{{\partial t}} + {{\nabla }_{r}}{{Q}_{i}} = {{\Psi }_{{pi}}} - \frac{5}{3}\sigma \left\langle {\frac{{\partial {{{\tilde {p}}}_{i}}}}{{\partial \theta }}\frac{{\partial \tilde {H}}}{{\partial \theta }}} \right\rangle \\ - \;\frac{2}{3}\gamma \left\langle {{{{\tilde {p}}}_{i}}\frac{{\partial {{{\tilde {V}}}_{{||i}}}}}{{\partial \theta }}} \right\rangle + {{\chi }_{{ \bot i}}}\frac{{{{d}^{2}}{{p}_{{i0}}}}}{{d{{r}^{2}}}} + \left\langle {{{W}_{{ei}}}} \right\rangle , \\ {{Q}_{i}} = \left\langle {{{{\tilde {V}}}_{{Er}}}{{{\tilde {p}}}_{i}}} \right\rangle , \\ {{\Psi }_{{pi}}} = \left\langle {\frac{5}{3}g} \right.\{ [{{p}_{i}}C(\phi ) + \xi C({{p}_{i}}{{T}_{i}}) \\ \left. { - _{{}}^{{}}\xi {{T}_{i}}C({{p}_{e}} + {{p}_{i}})]\} } \right\rangle ; \\ \end{gathered} $$

((A.3))

$$\begin{gathered} \frac{{\partial {{V}_{{||0}}}}}{{\partial t}} + {{\nabla }_{r}}{{\Pi }_{{r||}}} = {{\mu }_{ \bot }}\frac{{{{d}^{2}}{{V}_{{||0}}}}}{{d{{r}^{2}}}} + \xi g\frac{{d\left\langle {{{{\tilde {V}}}_{{i||}}}\sin \theta } \right\rangle }}{{dr}}, \\ {{\Pi }_{{r||}}} = \gamma \left\langle {{{{\tilde {V}}}_{{Er}}}{{{\tilde {V}}}_{{i||}}}} \right\rangle . \\ \end{gathered} $$

((A.4))

Set of equations (A.1)–(A.4) should be supplemented with Eq. (8) for the poloidal velocity V₀(r, t). When solving set of equations (A.1)–(A.4), the following boundary conditions were used:

$$\begin{gathered} {{n}_{{\text{0}}}}({{r}_{0}}) = {{n}_{b}},\quad {{n}_{{\text{0}}}}(a) = {{n}_{W}},\quad {{p}_{{{\text{0}}e}}}({{r}_{0}}) = {{p}_{{be}}}, \\ {{p}_{{{\text{0}}e}}}(a) = {{p}_{{We}}},\quad {{p}_{{{\text{0}}i}}}({{r}_{0}}) = {{p}_{{bi}}},\quad {{p}_{{{\text{0}}i}}}(a) = {{p}_{{Wi}}}, \\ d{{V}_{{||0}}}({{r}_{0}}){\text{/}}dr = 0,\quad {{V}_{{||0}}}(a) = {{C}_{S}}{\text{/}}{{V}_{{Ti}}},\quad {{V}_{{Ti}}} = \sqrt {\frac{{{{T}_{*}}}}{{{{m}_{i}}}}} . \\ \end{gathered} $$

Turbulent fluxes of the momentum, particles, and heat are calculated as sums of the fluxes of axisymmetric (f_A) and ballooning (f_B) modes,

$$\begin{gathered} \Pi = \Pi {}_{A}\; + {{\Pi }_{B}},\quad {{\Gamma }_{i}} = {{\Gamma }_{A}} + {{\Gamma }_{B}},\quad {{q}_{{e,i}}} = {{Q}_{{Ae,i}}} + {{Q}_{{Be,i}}}, \\ {{\Pi }_{A}} = \left\langle {{{{\tilde {V}}}_{{Er}}}\tilde {w}} \right\rangle ,\quad {{\Gamma }_{{AX}}} = \left\langle {{{{\tilde {V}}}_{{Er}}}\tilde {n}} \right\rangle ,\quad {{Q}_{{AXe,i}}} = \left\langle {{{{\tilde {V}}}_{{Er}}}{{{\tilde {p}}}_{{e,i}}}} \right\rangle , \\ {{{\tilde {V}}}_{{Er}}} = - \frac{{\text{1}}}{r}\frac{{\partial \tilde {\phi }}}{{\partial \theta }}, \\ \end{gathered} $$

$${{\Pi }_{B}} = - \frac{m}{r}{{\left\langle {\tilde {\omega }\frac{{\partial \tilde {\phi }}}{{\partial \lambda }}} \right\rangle }_{\lambda }} = \frac{m}{{2r}}\left( {{{\phi }_{{BC}}}{{\omega }_{{BS}}} - {{\phi }_{{BS}}}{{\omega }_{{BC}}}} \right),$$

$${{\Gamma }_{B}} = - \frac{m}{r}{{\left\langle {\tilde {n}\frac{{\partial \tilde {\phi }}}{{\partial \lambda }}} \right\rangle }_{\lambda }} = \frac{m}{{2r}}\left( {{{\phi }_{{BC}}}{{n}_{{BS}}} - {{\phi }_{{BS}}}{{n}_{{BC}}}} \right),$$

$$\begin{gathered} {{Q}_{{Bj}}} = - \frac{m}{r}{{\left\langle {{{{\tilde {p}}}_{j}}\frac{{\partial \tilde {\phi }}}{{\partial \lambda }}} \right\rangle }_{\lambda }} = \frac{m}{{2r}}\left( {{{\phi }_{{BC}}}{{p}_{{BSj}}} - {{\phi }_{{BS}}}{{p}_{{BCj}}}} \right), \\ j = e,i. \\ \end{gathered} $$

We note that the main contribution to the transport is made by the fluxes ${{\Pi }_{B}}$, ${{\Gamma }_{B}}$, and ${{Q}_{{Bj}}}$ ($j = e,i$), which are associated with ballooning modes. The set of equations for the ballooning modes  f_BS,BC = {W_BS,BC, ${{n}_{{BS,BC}}}$, ${{p}_{{e\,BS,BC}}}$} in the case of p_i = τp_e is presented in [12].