Effect of Two-Dimensional Plasma Inhomogeneity in Magnetic Island on Threshold of Parametric Excitation of Trapped Upper Hybrid Waves and Level of Anomalous Absorption in ECRH Experiments

Abstract

The phenomenon of two-dimensional localization of the upper hybrid (UH) wave in a magnetic island was discovered. It was studied how this phenomenon affects the threshold and saturation level of the absolute parametric decay instability of the extraordinary wave, as a result of which a couple of two-dimensionally localized UH waves are excited.

APPENDIX

The set of equations describing the decay of the primary UH wave, ${{\varphi }_{n}} \propto \exp \left( { - i{{\omega }_{n}}t} \right)$, into the secondary IB wave, ${{\varphi }_{{{\text{IB}}}}} \propto \exp \left( { - i\left( {{{\omega }_{l}} - {{\omega }_{n}}} \right)t} \right)$, and the secondary UH wave, ${{\varphi }_{l}} \propto \exp \left( {i{{\omega }_{l}}t} \right)$, is as follows:

$$\left\{ \begin{gathered} {{{\hat {D}}}_{{{\text{UH}}}}}\left( {x,{{\omega }_{n}}} \right){{\varphi }_{n}} = - \hat {\chi }_{e}^{{nl}}{{\varphi }_{{{\text{IB}}}}}\varphi _{l}^{*} \hfill \\ {{{\hat {D}}}_{{{\text{UH}}}}}\left( {x,{{\omega }_{l}}} \right){{\varphi }_{l}} = - \hat {\chi }_{e}^{{nl}}{{\varphi }_{{{\text{IB}}}}}{{\varphi }_{n}}^{*} \hfill \\ {{{\hat {D}}}_{{{\text{IB}}}}}\left( {x,{{\omega }_{n}} - {{\omega }_{l}}} \right){{\varphi }_{{{\text{IB}}}}} = - \hat {\chi }_{e}^{{nl}}{{\varphi }_{n}}{{\varphi }_{l}}. \hfill \\ \end{gathered} \right.$$

(A1)

In weakly inhomogeneous plasma, the ${{\hat {D}}_{{{\text{UH,IB}}}}}$ ope-rators in the left-hand side of Eqs. (A1), acting on  arbitrary function F, can be represented in the follo-wing form: ${{\hat {D}}_{{{\text{UH,IB}}}}}F$ = $\frac{1}{{{{{(2\pi )}}^{3}}}}\int_{ - \infty }^\infty {F\left( {{\mathbf{r}}{\kern 1pt} '} \right)}$ × $\left( {\int_{ - \infty }^\infty {{{D}_{{{\text{UH,IB}}}}}\left( {{\mathbf{q}},\frac{{{\mathbf{r}} + {\mathbf{r}}{\kern 1pt} '}}{2}} \right)\exp \left( {i{\mathbf{q}}({\mathbf{r}} - {\mathbf{r}}{\kern 1pt} ')} \right)d{\mathbf{q}}} } \right)d{\mathbf{r}}{\kern 1pt} '$, where the kernel of integral operator ${{D}_{{{\text{UH}}}}} = 0$ is determined by Eq. (3), ${{D}_{{{\text{IB}}}}} = {{q}^{2}} + \chi = 0$ is the dispersion equation of the IB wave,

$$\chi = \sum\nolimits_{j = e,i} {\frac{{2\omega _{{pj}}^{2}}}{{\upsilon _{{tj}}^{2}}}} \left( {1 - \frac{{{{\omega }_{{{\text{IB}}}}}}}{{\left| {{{q}_{\xi }}} \right|{{\upsilon }_{{ti}}}}}\sum\nolimits_{m = - \infty }^\infty {Z\left( {\frac{{{{\omega }_{{{\text{IB}}}}} - m{{\omega }_{{cj}}}}}{{{{q}_{\xi }}{{\upsilon }_{{tj}}}}}} \right)} } \right. \times \left. {\exp \left( { - \frac{{q_{ \bot }^{2}\upsilon _{{tj}}^{2}}}{{2\omega _{{cj}}^{2}}}} \right){{I}_{m}}\left( {\frac{{q_{ \bot }^{2}\upsilon _{{tj}}^{2}}}{{2\omega _{{cj}}^{2}}}} \right)} \right)$$

is the linear plasma susceptibility, and $Z\left( \xi \right)$ = $2\exp \left( { - {{\xi }^{2}}} \right)\int_0^\xi {\exp \left( {{{t}^{2}}} \right)dt}$ – $i\sqrt \pi \exp \left( { - {{\xi }^{2}}} \right)$ is the plasma dispersion function. The operators in the right-hand side of Eqs. (A1), acting on the product of two functions U and V, have the following form:

$$\hat {\chi }_{e}^{{nl}}UV = \int\limits_{ - \infty }^\infty {U\left( {{\mathbf{r}}{\kern 1pt} '} \right)V\left( {{\mathbf{r}}{\kern 1pt} ''} \right)} \left( {\int\limits_{ - \infty }^\infty {\chi _{e}^{{nl}}\left( {{{{\mathbf{q}}}_{1}},{{{\mathbf{q}}}_{2}},{\mathbf{r}}} \right)\exp \left( {i{{{\mathbf{q}}}_{1}}({\mathbf{r}} - {\mathbf{r}}{\kern 1pt} ') + i{{{\mathbf{q}}}_{2}}({\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '')} \right)d{{{\mathbf{q}}}_{1}}d{{{\mathbf{q}}}_{2}}} } \right)\frac{{d{\mathbf{r}}{\kern 1pt} 'd{\mathbf{r}}{\kern 1pt} ''}}{{{{{(2\pi )}}^{6}}}}.$$

In homogeneous plasma, $\hat {\chi }_{e}^{{nl}}$ is the second-order nonlinear susceptibility of plasma [43–45]. Let us analyze Eqs. (A1) using the approach of perturbation theory. In the first stage of perturbation procedure, we neglect the nonlinear coupling of the waves. In this case, the unperturbed solutions corresponding to the two-dimensionally localized UH waves can be determined in the same way, as in Eq. (10). The potential of the IB wave propagating mainly along the direction of plasma inhomogeneity has the following form:

$$\begin{gathered} {{\varphi }_{{{\text{IB}}}}}\left( {{\mathbf{r}},t} \right) = \frac{{{{C}_{{{\text{IB}}}}}\left( {\mathbf{r}} \right)}}{{2\sqrt {\left| {\partial {{D}_{{{\text{IB}}}}}{\text{/}}\partial {{q}_{{Ix}}}} \right|} }} \\ \times \;\exp \left( {i\int\limits_{ - \infty }^x {{{q}_{{Ix}}}\left( {x{\kern 1pt} '} \right)} dx{\kern 1pt} ' - i{{\omega }_{{{\text{IB}}}}}t} \right){\text{ + c}}{\text{.c}}{\text{.,}} \\ \end{gathered} $$

(A2)

where the envelope amplitude obeys the following reduced equation:

$$\begin{gathered} i\frac{\partial }{{\partial x}}{{C}_{{{\text{IB}}}}} = - {{C}_{n}}\left( \xi \right){{\psi }_{{{{n}_{2}}}}}\left( \zeta \right){{C}_{l}}\left( \xi \right){{\psi }_{{{{l}_{2}}}}}\left( \zeta \right) \\ \times \;\frac{{\chi _{e}^{{nl}}{{\phi }_{{{{n}_{1}}}}}\left( x \right){{\phi }_{{{{l}_{1}}}}}\left( x \right)}}{{2{{{\left| {\partial {{D}_{{{\text{IB}}}}}{\text{/}}\partial {{q}_{{Ix}}}} \right|}}^{{1/2}}}}}\exp \left( { - i\int\limits_{ - \infty }^x {{{q}_{{Ix}}}\left( {x{\kern 1pt} '} \right)} dx{\kern 1pt} '} \right). \\ \end{gathered} $$

(A3)

In Eq. (A3), we omit the terms describing the negligible effect of IB wave diffraction in the $\zeta \times \xi$ plane. After integrating Eq. (A3), we obtain

$$\begin{gathered} {{\varphi }_{{{\text{IB}}}}}\left( {{\mathbf{r}},t} \right) = i\frac{{\chi _{e}^{{nl}}}}{4}\frac{{{{C}_{n}}\left( \xi \right){{\psi }_{{{{n}_{2}}}}}\left( \zeta \right){{C}_{l}}\left( \xi \right){{\psi }_{{{{l}_{2}}}}}\left( \zeta \right)}}{{{{{\left| {\partial {{D}_{{{\text{IB}}}}}\left( x \right){\text{/}}\partial {{q}_{{Ix}}}} \right|}}^{{1/2}}}}} \\ \times \;\int\limits_{ - \infty }^x {dx{\kern 1pt} '\frac{{{{\phi }_{{{{n}_{1}}}}}\left( {x{\kern 1pt} '} \right){{\phi }_{{{{l}_{1}}}}}\left( {x{\kern 1pt} '} \right)}}{{{{{\left| {\partial {{D}_{{{\text{IB}}}}}\left( {x{\kern 1pt} '} \right){\text{/}}\partial {{q}_{{Ix}}}} \right|}}^{{1/2}}}}}} \\ \times \;\exp \left( {i\int\limits_{x'}^x {{{q}_{{Ix}}}\left( {x{\kern 1pt} ''} \right)dx{\kern 1pt} ''\; - i{{\omega }_{{{\text{IB}}}}}t} } \right) + {\text{c}}{\text{.c}}. \\ \end{gathered} $$

(A4)

We substitute formula (A4) into the right-hand sides of the first two equations in set of Eqs. (A1) and multiply these equations by $\phi _{{{{n}_{1}}}}^{*}\left( x \right) \cdot \phi _{{{{n}_{2}}}}^{*}\left( \zeta \right)$ and $\phi _{{{{l}_{1}}}}^{*}\left( x \right) \cdot \phi _{{{{l}_{2}}}}^{*}\left( \zeta \right)$, respectively. Then we perform integration with respect to $x$ and $\zeta$ that results in the following equations:

$$\left\{ \begin{gathered} \frac{{\partial {{a}_{n}}}}{{\partial t}} + {{u}_{n}}\frac{{\partial {{a}_{n}}}}{{\partial \xi }} = - {{\nu }_{l}}\sqrt {\frac{{{{\omega }_{n}}}}{{{{\omega }_{p}}}}} {{\left| {{{a}_{l}}} \right|}^{2}}{{a}_{n}} \hfill \\ \frac{{\partial {{a}_{l}}}}{{\partial t}} - {{u}_{l}}\frac{{\partial {{a}_{l}}}}{{\partial \xi }} = \nu _{l}^{*}\sqrt {\frac{{{{\omega }_{p}}}}{{{{\omega }_{n}}}}} {{\left| {{{a}_{n}}} \right|}^{2}}{{a}_{l}}, \hfill \\ \end{gathered} \right.$$

(A5)

where ${{u}_{{n,l}}} = \left\langle {{{{\left| {\partial {{D}_{{{\text{UH}}}}}\left( x \right){\text{/}}\partial {{q}_{\xi }}} \right|}}_{{{{\omega }_{{n,l}}}}}}} \right\rangle {\text{/}}\left\langle {{{{\left| {\partial {{D}_{{{\text{UH}}}}}\left( x \right){\text{/}}\partial \omega } \right|}}_{{{{\omega }_{{n,l}}}}}}} \right\rangle$ are the group velocities of UH waves acquired due to the finite value of the parallel wave number, and

$$\begin{gathered} {{\nu }_{l}} = \frac{{4\pi {{{\left| {\chi _{e}^{{nl}}} \right|}}^{2}}}}{{\sqrt {{{\omega }_{n}}{{\omega }_{l}}} \left\langle {{{{\left| {\partial {{D}_{{{\text{UH}}}}}\left( x \right){\text{/}}\partial \omega } \right|}}_{{{{\omega }_{n}}}}}} \right\rangle \left\langle {{{{\left| {\partial {{D}_{{{\text{UH}}}}}\left( x \right){\text{/}}\partial \omega } \right|}}_{{{{\omega }_{l}}}}}} \right\rangle }} \\ \times \;\int\limits_{ - \infty }^\infty {d\zeta {{{\left| {{{\psi }_{{{{n}_{2}}}}}\left( \zeta \right)} \right|}}^{2}}{{{\left| {{{\psi }_{{{{l}_{2}}}}}\left( \zeta \right)} \right|}}^{2}}} \int\limits_{ - \infty }^\infty {dx\frac{{{{\phi }_{{{{n}_{1}}}}}\left( x \right){\kern 1pt} \text{*}{{\phi }_{{{{l}_{1}}}}}\left( x \right){\kern 1pt} \text{*}}}{{{{{\left| {\partial {{D}_{{{\text{IB}}}}}\left( x \right){\text{/}}\partial {{q}_{{Ix}}}} \right|}}^{{1/2}}}}}} \\ \times \;\int\limits_{ - \infty }^x {dx{\kern 1pt} '\frac{{{{\phi }_{{{{n}_{1}}}}}\left( {x{\kern 1pt} '} \right){{\phi }_{{{{l}_{1}}}}}\left( {x{\kern 1pt} '} \right)}}{{{{{\left| {\partial {{D}_{{{\text{IB}}}}}\left( {x{\kern 1pt} '} \right){\text{/}}\partial {{q}_{{Ix}}}} \right|}}^{{1/2}}}}}\exp \left( {i\int\limits_{x'}^x {{{q}_{{Ix}}}\left( {x{\kern 1pt} ''} \right)dx{\kern 1pt} ''} } \right)} \\ \end{gathered} $$

is the coefficient describing the secondary instability. The procedure for averaging the W arbitrary function over the modes localization region is defined as follows:

$$\left\langle {W\left( {x,\zeta } \right)} \right\rangle = \int\limits_{ - \infty }^\infty {dx{{{\left| {{{\phi }_{{{{n}_{1}}}}}\left( x \right)} \right|}}^{2}}} \int\limits_{ - \infty }^\infty {d\zeta {{{\left| {{{\psi }_{{{{n}_{2}}}}}\left( \zeta \right)} \right|}}^{2}}} W\left( {x,\zeta } \right).$$