Consider a rotating tornado with a straight magnetic field (Ω0 ≠ 0,J0 = 0). In this case, the dispersion relation (14) reduces to (15)\begin{eqnarray} \lefteqn{(\omega^{2}-C_{\mathrm {A}}^{2}k^{2})(\omega^{2}-C_{\rm T}^{2}k^{2})}\nonumber\\ &amp; &amp;+\frac{a^2}{4(C_{\mathrm {A}}^{2}+C_{\rm s}^{2})}\left[4\Omega_{0}^{2}\omega^{4}-2\Omega_0^2k^2C_{\rm s}^2\omega^{2} -2\Omega_0^2k^{4}C_{\rm s}^{2}C_{\mathrm {A}}^{2})\right]\nonumber\\ &amp; &amp;-\frac{a^2}{4(C_{\mathrm {A}}^{2}+C_{\rm s}^{2})}\left[(\omega^{2}-C_{\mathrm {A}}^{2}k^{2})^{2}(\omega^{2}-C_{\rm s}^{2}k^{2})\right]=\left(\frac{\rho_{0e}}{2\rho_0}\right)\nonumber\\ &amp; &amp; \times \frac{(\omega^2-k^2C_\mathrm{A}^2)(\omega^2-k^2C_{\rm s}^2)(\omega^2-k^2C_{\mathrm{Ae}}^2)a}{(C_\mathrm{A}^2+C_{\rm s}^2)m_{\rm e}}\frac{K_0(m_{\rm e}a)}{K_1(m_{\rm e}a)}\cdot \label{rotationext} \end{eqnarray}(ω2−CA2k2)(ω2−CT2k2)+a24(CA2+Cs2)[4Ω02ω4−2Ω02k2Cs2ω2−2Ω02k4Cs2CA2)]−a24(CA2+Cs2)[(ω2−CA2k2)2(ω2−Cs2k2)]=ρ0e2ρ0The second term on the left-hand side of Eq. (15) shows the effect of the internal rotation where the third term indicates the confinement of the tube. The term on the right-hand side of Eq. (15) shows the effect of the perturbations of the external medium. If we write the dispersion relation in terms of the fast and slow speeds we would have (16)\begin{eqnarray} &amp;&amp;\left(C_{\mathrm {A}}^{2}+C_{\rm s}^{2}+2{\cal R}C_{\mathrm {A}}^{2}\right)\left(\omega^{2}-C_{+}^{2}k^{2}\right)\left(\omega^{2}- C_{-}^{2}k^{2}\right)\nonumber\\ &amp;&amp;\quad-\frac{a^2}{4}\left(\omega^{2}-C_{\mathrm {A}}^{2}k^{2}\right)^{2}\left(\omega^{2}-C_{\rm s}^{2}k^{2}\right)= \left(\frac{\rho_{0e}}{2\rho_0}\right)\nonumber\\&amp;&amp;\qquad\times \frac{\left(\omega^2-k^2C_\mathrm{A}^2\right)\left(\omega^2-k^2C_{\rm s}^2\right)\left(\omega^2-k^2C_{\mathrm{Ae}}^2\right)a}{m_{\rm e}}\frac{K_0(m_{\rm e}a)}{K_1(m_{\rm e}a)}, \end{eqnarray}(CA2+Cs2+2ℛCA2)(ω2−C+2k2)(ω2−C−2k2) −a24(ω2−CA2k2)2(ω2−Cs2k2)=ρ0e2ρ0  ×(ω2−k2CA2)(ω2−k2Cs2)(ω2−k2CAe2)ameK0(mea)K1(mea),where (17)\begin{eqnarray} \label{PhaseRotation} C_{\pm}^{2}&amp;=&amp;C_{\mathrm {A}}^{2}\frac{(C_{\mathrm {A}}^{2}+2C_{\mathrm {s}}^{2})+{\cal R}C_{\mathrm {s}}^{2}\pm\sqrt{{\cal P}}}{2(C_{\mathrm {A}}^{2}+C_{\mathrm {s}}^{2})+4C_{\mathrm {A}}^{2}{\cal R}},\nonumber\\ {\cal P}&amp;=&amp;C_{\mathrm {A}}^{4}-2{\cal R}C_{\mathrm {s}}^{2}(C_{\mathrm {A}}^{2}-4C_{\mathrm {s}}^2)+{\cal R}^{2}C_{\mathrm {s}}^{2}(8C_{\mathrm {A}}^{2}+C_{\mathrm {s}}^{2}), \end{eqnarray}C±2=CA2(CA2+2Cs2)+ℛCs2±𝒫2(CA2+Cs2)+4CA2ℛ,𝒫=CA4−2ℛCs2(CA2−4Cs2)+ℛ2Cs2(8CA2+Cs2),and the dimensionless parameter, ℛ, representing the rotation is (18)\begin{equation} {\cal R}=\frac{a^2\Omega^{2}_{0}}{2\,C_{\mathrm {A}}^{2}}\cdot \end{equation}ℛ=a2Ω022 CA2·Equation (17) indicates the modification of the Alfvén speed and the tube speed by the equilibrium rotation to the fast (C+) and slow (C−) magnetosonic speeds, respectively (see also Zhugzhda & Nakariakov 1999; Vasheghani Farahani et al. 2010). As such the torsional wave would propagate with the speed C+, and the longitudinal wave would propagate with the speed C−. If we consider the dispersion to be weak (k2A0 ≪ 1), and take \hbox{$\omega^{2}\approx\,C_{\pm}^{2}k^{2}$}ω2≈ C±2k2, an explicit expression for the dispersion relation could be obtained (19)\begin{eqnarray} \label{PhaseR} \omega^{2}&amp;\approx&amp;\,C_{\pm}^{2}k^{2}\pm\,\frac{a^2}{4}\frac{(C_{\pm}^{2}-C_{\mathrm {A}}^{2})^{2}(C_{\pm}^{2}-C_{\mathrm {s}}^{2})}{C_{\mathrm {A}}^{2}\,\sqrt{{\cal P}}}k^{4}+ \left(\frac{\rho_{0e}}{2\rho_0}\right)\nonumber\\&amp;&amp;\quad\times \frac{(C_{\pm}^2-C_\mathrm{A}^2)(C_{\pm}^2-C_{\rm s}^2)(C_{\pm}^2-C_{\mathrm{Ae}}^2)a}{GC_{\mathrm {A}}^{2}\,\sqrt{{\cal P}}}\nonumber\\&amp;&amp;\quad\times\frac{K_0(Ga)}{K_1(Ga)}k^4, \end{eqnarray}ω2≈ C±2k2± a24(C±2−CA2)2(C±2−Cs2)CA2 𝒫k4+ρ0e2ρ0 ×(C±2−CA2)(C±2−Cs2)(C±2−CAe2)aGCA2 𝒫 ×K0(Ga)K1(Ga)k4,where (20)\begin{eqnarray} G^2=\frac{\left(k^2C^2_{\mathrm{Ae}}-C_{\pm}^2\right)\left(k^2C^2_{\rm se}-C_{\pm}^2\right)} {\left(C^2_{\mathrm{Ae}}+C^2_{\rm se}\right)\left(k^2C^2_{\rm Te}-C_{\pm}^2\right)}, \,\mbox{ \ } C_{\rm Te}^{2}=\frac{C_{\mathrm {Ae}}^{2}C_{\rm se}^{2}}{C_{\mathrm {A}e}^{2}+C_{\rm se}^{2}}\cdot \label{me} \end{eqnarray}G2=(k2CAe2−C±2)(k2Cse2−C±2)(CAe2+Cse2)(k2CTe2−C±2), CTe2=CAe2Cse2CAe2+Cse2·The second term on the right-hand side of Eq. (19) is the dispersive correction term, while the third term on the right-hand side represents the effects of the perturbation of the external medium. Equation (19) clearly shows the interplay of the equilibrium rotation and the density contrast of the internal and external media. The modification of the speeds at k = 0 is clearly exhibited by both the rotation and the density contrast. Hence, the modification of the tube and Alfvén speeds to the slow and fast magnetoacoustic speeds is clearly observed, see also Zhugzhda & Nakariakov (1999). In the long wavelength limit where the tube radius is much smaller than the wavelength, the dispersion would be week, therefore the arguments of the modified Bessel function of the second kind K would be small. In such a regime we are able to use the first terms of the expansions of the modified Bessel function of the second kind for small arguments (Abramowitz et al. 1988) which are (21)\begin{eqnarray} K_{0}(Ga)=-\mathrm{ln}\left(\frac{1}{2}Ga\right),\,\,\,\,\,\,\,\,\, K_{1}(Ga)=\frac{1}{Ga}\cdot \label{K} \end{eqnarray}K0(Ga)=−ln12Ga,         K1(Ga)=1Ga·By substituting the terms of Eq. (21) in Eq. (19) we obtain an explicit expression that is easier to read: (22)\begin{eqnarray} \omega^{2}\approx\,C_{\pm}^{2}k^{2}\pm\,\frac{a^2}{4}\frac{(C_{\pm}^{2}-C_{\mathrm {A}}^{2})^{2}(C_{\pm}^{2}-C_{\mathrm {s}}^{2})}{C_{\mathrm {A}}^{2}\,\sqrt{{\cal P}}}k^{4}- \left(\frac{\rho_{0e}}{\rho_0}\right)\nonumber\\\times \frac{a^2}{2}\frac{(C_{\pm}^2-C_\mathrm{A}^2)(C_{\pm}^2-C_{\rm s}^2)(C_{\pm}^2-C_{\mathrm{Ae}}^2)}{ C_{\mathrm {A}}^{2}\,\sqrt{{\cal P}}}\mathrm{ln}\left(\frac{1}{2}Ga\right)k^4. \label{PhaseR2} \end{eqnarray}ω2≈ C±2k2± a24(C±2−CA2)2(C±2−Cs2)CA2 𝒫k4−ρ0eρ0×a22(C±2−CA2)(C±2−Cs2)(C±2−CAe2)CA2 𝒫ln12Gak4.Equation (22) represents the dispersion relation for magnetoacoustic waves propagating in a steadily rotating magnetic flux tube (e.g. a solar tornado), which is more convenient for observation purposes and coronal seismology.
Citation: Vasheghani Farahani et al. 2010
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