Three-dimensional kink modes in solar coronal slabs: group velocities and their implications for impulsively excited waves

Abstract

Little attention has been paid to group velocities of three-dimensional (3D) MHD waves in solar coronal seismology. This study aims to present a rather comprehensive examination on the group velocities of trapped 3D kink modes in coronal slabs, emphasizing the connection of mode analysis to both mode characterization and impulsively excited 3D kink waves. We work in linear, ideal, pressureless MHD, and take the equilibrium slab to be symmetrically structured only in one transverse direction. The dispersion relation is numerically solved, with the results understood by making in-depth analytical progress. We address both the transverse fundamental and its first overtone. We develop a three-subgroup scheme for categorizing 3D kink modes on the plane spanned by the axial and out-of-plane wavenumbers. The group ($\vec{v}_{\rm gr}$) and phase velocities ($\vec{v}_{\rm ph}$) sit on the same side of the equilibrium magnetic field ($\vec{B}_0$) for the ``$\vec{B}_0$-same-side A'' and ``$\vec{B}_0$-same-side F'' subgroups, which are further discriminated by the directional similarity of $\vec{v}_{\rm gr}$ and $\vec{B}_0$. The ``$\vec{B}_0$-straddling'' subgroup is peculiar in that $\vec{v}_{\rm gr}$ and $\vec{v}_{\rm ph}$ lie astride $\vec{B}_0$, a feature that cannot be found for waves in unbounded uniform media in pressureless MHD. This ``$\vec{B}_0$-straddling'' subgroup pertains to both the fundamental and its overtones. We further place our results in the context of impulsive waves, employing the method of stationary phase to predict the large-time wavefront morphology in the plane of symmetry of the equilibrium slab. Wavefronts directed toward $\vec{B}_0$ derive exclusively from ``$\vec{B}_0$-straddling'' modes, and are confined to narrow sectors.

Figures (8)

• Figure 1: Characterization of trapped 3D kink modes in solar coronal slabs. A density contrast of $\rho_{\rm i}/\rho_{\rm e}=3$ is chosen for illustrative purposes. Plotted in each panel are the dependencies on the out-of-plane wavenumber ($k_y$) of the mode frequency of the transverse fundamental (with transverse order $j=1$, the black solid curve) and the first overtone ($j=2$, black dash-dotted). The hatched portions in any $k_y-\omega$ plane represent where trapped modes are forbidden, with four curves serving as the relevant borders (labeled in panel c1). Curves $0$, $1$, $2_0$, and $3$ correspond to where $\kappa_{\rm i}^2 \coloneqq k_z^2 - \omega^2/v_{\rm Ai}^2=0$, $\kappa_{\rm e}^2 \coloneqq k_z^2 - \omega^2/v_{\rm Ae}^2=0$, $n_{\rm i}^2 \coloneqq \omega^2/v_{\rm Ai}^2 - (k_y^2+k_z^2) =0$, and $m_{\rm e}^2 \coloneqq k_y^2+k_z^2 - \omega^2/v_{\rm Ae}^2=0$, respectively. The curves labeled $2_{J\pi}$ ($J=1/2, 1, 3/2$) further correspond to where $n_{\rm i} = J\pi$. These curves pertain to the characterization of the $j=1$ and $j=2$ modes, whose dispersive behavior is typified by different panels where the dimensionless axial wavenumber ($k_z d$) increases sequentially. See text for more details.
• Figure 2: Transverse fundamental kink mode in a coronal slab with a density contrast $\rho_{\rm i}/\rho_{\rm e}=3$. The dependence on $[k_y, k_z]$ of the phase speed $v_{\rm ph}=\omega/\sqrt{k_y^2+k_z^2}$ is shown by the equally spaced contours. Trapped modes are allowed for all $[k_y \ge 0, k_z >0]$, with both body ($n_{\rm i}^2 > 0$) and surface ($n_{\rm i}^2 < 0$) modes being relevant. The color map represents the parameter $n_{\rm i}$, and is restricted to body modes.
• Figure 3: Transverse fundamental kink mode in a coronal slab with a density contrast $\rho_{\rm i}/\rho_{\rm e}=3$. Shown are the dependencies on $[k_y, k_z]$ of the $y$-component (the black contours) and the $z$-component (red) of the group velocity. All contours are equally spaced.
• Figure 4: Transverse fundamental kink mode in a coronal slab with a density contrast $\rho_{\rm i}/\rho_{\rm e}=3$. Shown are the dependencies on $[k_y, k_z]$ of (a) the angle between the group velocity $\boldsymbol{v}_{\rm gr}$ and the $z$-direction, and (b) the angle between the group velocity $\boldsymbol{v}_{\rm gr}$ and the phase velocity $\boldsymbol{v}_{\rm ph}$. The angles are in degrees and measured counterclockwise from $\boldsymbol{v}_{\rm gr}$ to $\hat{\boldsymbol{e}}_z$ or $\boldsymbol{v}_{\rm ph}$. Superimposed are the filled contours of $\Lambda_{\rm i}$, the ratio of the magnetic pressure gradient force to the magnetic tension force. All contours are equally spaced. The angle information allows the fundamental to be classified into two regimes as labeled in panel c. See text for details.
• Figure 5: Similar to Fig. \ref{['fig_fund_vph']} but for the first transverse overtone. Trapped 3D modes are forbidden within the circle $d\sqrt{k_y^2 + k_z^2} = k_z^{(1/2)} d = (\pi/2)/\sqrt{\rho_{\rm i}/\rho_{\rm e} -1}$, which reads $1.11$ for the chosen $\rho_{\rm i}/\rho_{\rm e}=3$. Trapped 2D modes are further prohibited for $k_z^{(1/2)} < k < k_z^{(1)} = 2k_z^{(1/2)}$ along $k_y=0$. Trapped modes are exclusively of the body type ($n_{\rm i}^2 >0$).