This article introduces the derivation of dispersion relations of MHD waves in coronal cylindrical flux tubes.
A PDF version can be found here.
1. Introduction
We start with the ideal MHD equations, assuming small perturbations for each physical parameter, and combining the equations to get a differential equation containing the parameter to be solved. Then by utilizing the boundary conditions inside and outside of the cylindrical flux tube, the solution to the differential equation can be obtained.
In the following sections, we will treat the problem in cylindrical coordinate (the three base vectors are \(\mathbf{e_r},\ \mathbf{e_{\theta}}\) and \(\mathbf{e_z}\)). We assume the radius of the cylinder as \(r=a\), and the two typical speeds used during the derivation are local sound speed \(c_{s,i}\) and local Alfvén speed \(v_{A,i}\). The index \(i=o,\ e\) represents parameters in the interior and exterior regions, respectively.
2. Basic MHD Equations
The initial equations are ideal MHD equations (ignoring collisions and viscosity): \[\begin{equation} \frac{\partial\rho}{\partial t}+\nabla\cdot(\rho \mathbf{v})=0 \end{equation}\] \[\begin{equation} \rho\frac{\partial\mathbf{v}}{\partial t}+\mathbf{v}\cdot\nabla\mathbf{v}+\nabla p+\rho g+\frac{1}{\mu_0}[(\nabla\times \mathbf{B})\times\mathbf{B}]=0 \end{equation}\] \[\begin{equation} \nabla\times(\mathbf{v}\times\mathbf{B})-\frac{\partial\mathbf{B}}{\partial t}=0 \end{equation}\]
3. The Introduction of Small Perturbations
Assume a small perturbation in the parameters:
\(\rho=\rho_0+\rho_1,\quad p=p_0+p_1,\quad \mathbf{v}=\mathbf{v_1},\quad \mathbf{B}=\mathbf{B}_0+\mathbf{B}_1\),
We treat the perturbations as local phenomena, and neglect the large-scale gradients of macroscopic parameters: \(\rho g=0\), \(\nabla\rho_0=0\), \(\nabla p_0=0\), and \(\nabla\times B_0=0\). The magnetic field \(\mathbf{B}\) only has z-component: \(\mathbf{B}=(0,\ 0,\ B)\).
By introducing these small perturbations, the aforementioned equations (1)-(3) are written as: \[\begin{equation} \frac{\partial\rho_1}{\partial t}+\rho_0\nabla\cdot\mathbf{v}_1=0 \end{equation}\] \[\begin{equation} \rho_0\frac{\partial\mathbf{v}_1}{\partial t}+\nabla[p_1+\frac{1}{\mu_0}\nabla(\mathbf{B}_0\cdot\mathbf{B}_1)]-\frac{1}{\mu_0}(\mathbf{B}_0\cdot\nabla)\mathbf{B}_1=0 \end{equation}\] \[\begin{equation} \frac{\partial\mathbf{B}_1}{\partial t}+\mathbf{B}_0(\nabla\cdot\mathbf{v}_1)-(\mathbf{B}_0\cdot\nabla)\mathbf{v}_1=0 \end{equation}\]
In Eq. (5), we can write \(p_T=p_1+\frac{1}{\mu_0}\nabla(\mathbf{B}_0\cdot\mathbf{B}_1)\) as the total pressure perturbation (gas pressure plus magnetic pressure perturbations).
The above procedure converts Eq. (2) to Eq. (5) which is expressed by the total pressure perturbation \(p_T\), meanwhile, we can also change Eq. (2) to make it contain the density perturbation \(\rho_1\): \[\begin{equation} \rho_0\frac{\partial\mathbf{v}_1}{\partial t}+c_s^2\nabla\rho_1+\frac{1}{\mu_0}[\nabla(\mathbf{B}_0\cdot\mathbf{B}_1)-(\mathbf{B}_0\cdot\nabla)\mathbf{B}_1]=0 \end{equation}\]
4. Further Derivations
\(\dfrac{\partial}{\partial t}(5)\) gives \[\begin{equation} \rho_0\frac{\partial^2\mathbf{v}_1}{\partial t^2}+\nabla\cdot(\frac{\partial p_T}{\partial t})-\frac{1}{\mu_0}(\mathbf{B}_0\cdot\frac{\partial^2\mathbf{B}_1}{\partial z\partial t})=0 \end{equation}\] Combine Eq. (6) and Eq. (8) and know that local Alfvén speed \(v_A=\frac{B}{\sqrt{\mu_0\rho}}\): \[\begin{equation} \frac{\partial^2\mathbf{v}_1}{\partial t^2}+\frac{\nabla}{\rho_0}\cdot\frac{\partial p_T}{\partial t}-v_A^2(\frac{\partial^2\mathbf{v}_1}{\partial z^2}-\mathbf{e}_z\nabla\cdot\mathbf{v}_1)=0 \end{equation}\]
Take the divergence of the z-component of Eq. (8) (the last two terms in Eq. (8) will cancel): \[\begin{equation} \frac{\partial^2}{\partial t^2}(\nabla\cdot \mathbf{v}_1)+\frac{1}{\rho_0}\nabla^2(\frac{\partial p_T}{\partial t})=0 \end{equation}\]
From the r-component of Eq. (9) we get: \[\begin{equation} \rho_0\frac{\partial^2 v_r}{\partial t^2}-\rho_0v_A^2\frac{\partial^2v_r}{\partial z^2}+\frac{\partial^2}{\partial r^2}\frac{\partial p_T}{\partial t}=0 \end{equation}\]
\(\dfrac{\partial}{\partial t}(7)\) gives \[\begin{equation} \frac{\partial^2\mathbf{v}_1}{\partial t^2}=c_s^2\nabla(\nabla\cdot\mathbf{v}_1)+v_A^2[-\nabla(\frac{\partial v_z}{\partial z}-\nabla\mathbf{v}_1)+\frac{\partial}{\partial z}(\frac{\partial\mathbf{v}_1}{\partial z}-\mathbf{e}_z\nabla\cdot\mathbf{v}_1)] \end{equation}\]
Set \(\Delta=\nabla\cdot\mathbf{v}_1\) and \(\Gamma=\frac{\partial v_z}{\partial z}\), the z-component and divergence of Eq. (7) give: \[\begin{equation} \frac{\partial^2 v_z}{\partial t^2}=c_s^2\frac{\partial\Delta}{\partial z} \end{equation}\] \[\begin{equation} \frac{\partial^2\Delta}{\partial t^2}=(c_s^2+v_A^2)\nabla^2\Delta-v_A^2\nabla^2\Gamma=0 \end{equation}\]
Suppose both \(p_T\) and \(\Delta\) have the form of \(R(r)exp(i\omega t+in\theta+ikz)\): \[\begin{equation} p_T=-\frac{i\rho_0\omega}{k^2}\Delta+C \end{equation}\] \(C\) is a constant.
Eqs. (13) and (14) can be combined to be: \[\begin{equation} \frac{\partial^4\Delta}{\partial t^4}-(c_s^2+v_A^2)\frac{\partial^2}{\partial t^2}\nabla^2\Delta+c_s^2v_A^2=0 \end{equation}\]
Take the form \(\Delta=R(r)exp(i\omega t+in\theta+ikz)\) into Eq. (16): \[\begin{equation} \frac{d^2R(r)}{dr^2}+\frac{1}{r}\frac{dR(r)}{dr}-[\frac{n^2}{r^2}+k^2-\frac{\omega^4}{(c_s^2+v_A^2)\omega^2-k^2c_s^2v_A^2}]R(r)=0 \end{equation}\]
Set \(m^2=k^2-\dfrac{\omega^4}{(c_s^2+v_A^2)\omega^2-k^2c_s^2v_A^2}=\dfrac{(k^2c_s^2-\omega^2)(k^2v_A^2-\omega^2)}{(c_s^2+v_A^2)(k^2c_T^2-\omega^2)}\),
and \(c_T=\dfrac{c_s v_A}{\sqrt{c_s^2+v_A^2}}\) is the so-called tube speed.
5. Dispersion Relation of compressible waves
Eq. (17) is a modified Bessel equation, and its solution is modified Bessel functions.
5.1. Modified Bessel Equation
If we let \(x=ix\) in Bessel equation, we than get the modified Bessel equation: \(\dfrac{d^2y(x)}{dx^2}+\dfrac{1}{x}\dfrac{dy(x)}{dx}-(1+\dfrac{\nu^2}{x^2})y(x)=0\).
The solution of modified Bessel equation is \(y=AI_{\nu}(x)+BK_{\nu}(x)\). \(I_{\nu}(x)\) is the modified Bessel function of the first kind; when \(\nu>0\), \(I_{\nu}(0)=0\). \(K_{\nu}(x)\) is the modified Bessel function of the second kind; \(K_{\nu}(0)\rightarrow \infty\).
5.2. Solution to the Differential Equation
The solution to Eq. (17) is \(R(r)=AI_{n}(mr)+BK_{n}(mr)\) (the radius of the cylinder is \(a\)).
Inside the cylinder, because when \(r=0\), \(R(r)\) must be finite, \(B=0\Rightarrow R(r)=AI_{n}(mr)\ (r\le a)\); outside the cylinder, \(I_{n}(mr)\) is exponentially decreasing, so we take \(R(r)=BK_{n}(mr)\ (r>a)\).
Now we review Eqs. (11) and (15): \[\rho_0\frac{\partial^2 v_r}{\partial t^2}-\rho_0v_A^2\frac{\partial^2v_r}{\partial z^2}+\frac{\partial}{\partial r}\frac{\partial p_T}{\partial t}=0\quad (11)\] \[p_T=-\frac{i\rho_0\omega}{k^2}\Delta+C\quad (15)\] If we rewrite \(v_r\) in the form of \(v(r)exp(i\omega t+in\theta+ikz)\), then from Eq. (10) we have: \[\begin{equation} v_r=\frac{m\omega^2}{k^2(\omega^2-k^2v_A^2)}\frac{dR(r)}{dr} \end{equation}\]
Now we already have the solution of \(\Delta=\nabla\cdot \mathbf{v}_1\), and we know the total pressure perturbation \(p_T\) and radial velocity perturbation \(v_r\) are both related with \(\Delta\) (Eqs. (15) and (18), respectively).
The boundary conditions are: radial velocity perturbation \(v_r\) and total pressure \(p_T\) are continuous. Hence, we have \[\begin{equation} v_{r,o}(r=a-)=v_{r,e}(r=a+)\Rightarrow \frac{m_oI_n^{\prime}(m_o a)}{\omega^2-k^2v_{A,o}^2}=\frac{m_eK_n^{\prime}(m_e a)}{\omega^2-k^2v_{A,e}^2} \end{equation}\] \[\begin{equation} p_{T,o}(r=a-)=p_{T,e}(r=a+)\Rightarrow \rho_oI_n(m_oa)=\rho_eK_n(m_ea) \end{equation}\] Eq. (19)/ Eq. (20) gives the dispersion relation for compressible MHD waves: \[\begin{equation} \rho_e(\omega^2-k^2v_{A,e}^2)m_o\frac{I_n^{\prime}(m_o a)}{I_n(m_oa)}=\rho_o(\omega^2-k^2v_{A,o}^2)m_e\frac{K_n^{\prime}(m_e a)}{K_n(m_ea)} \end{equation}\]
Note here \(m_i^2=\dfrac{(k^2c_s^2-\omega^2)(k^2v_A^2-\omega^2)}{(c_s^2+v_A^2)(k^2c_T^2-\omega^2)}\).
\(n\) is an integer, the azimuthal modal structures of waves are determined by \(n\) in the dispersion relation.
\(n=0\): sausage modes
\(n=1\): kink modes
\(n>1\)(\(n\) is integer): flute or balloning modes
5.3. Dispersion Relation For Incompressible Situations
The above derivations are based on the fact that the plasma is compressible, which means we have \(p_T\ne 0\). If \(p_T=0\), meaning that the plasma is incompressible (the compressibility term \(\nabla\cdot\mathbf{v}_1=0\)), then Eq. (8) is reduced to \[\begin{equation} \rho_0\frac{\partial^2\mathbf{v}_1}{\partial t^2}-\frac{1}{\mu_0}(\mathbf{B}_0\cdot\frac{\partial^2\mathbf{B}_1}{\partial z\partial t})=0 \end{equation}\] And Eq. (6) is reduced to \[\begin{equation} \frac{\partial\mathbf{B}_1}{\partial t}-(\mathbf{B}_0\cdot\nabla)\mathbf{v}_1 =\\ \frac{\partial\mathbf{B}_1}{\partial t}-\mathbf{B}_0\cdot\frac{\partial\mathbf{v}_1}{\partial z}=0 \end{equation}\]
Substitute Eq. (23) into Eq. (22): \[\begin{equation} \frac{\partial^2\mathbf{v}_1}{\partial t^2}=\frac{B_0^2}{\mu_0\rho_0}\frac{\partial^2\mathbf{v}_1}{\partial z^2} \end{equation}\] Take the form \(\mathbf{v}_1=v_1(r)exp(i\omega t+in\theta+ikz)\) into Eq. (24): \[\begin{equation} \omega^2=v_A^2k^2 \end{equation}\] Eq. (25) describes a pure Alfvén wave propagating along the field lines, which is the torsional Alfvén wave in flux tubes.
6. Conclusion
In conclusion, we have two forms of dispersion relation:
For incompressible situation: \[\omega^2=v_A^2k^2\]
For compressible situations: \[\rho_e(\omega^2-k^2v_{A,e}^2)m_o\frac{I_n^{\prime}(m_o a)}{I_n(m_oa)}=\rho_o(\omega^2-k^2v_{A,o}^2)m_e\frac{K_n^{\prime}(m_e a)}{K_n(m_ea)}\] (\(I_n^{\prime}(m_o r)\) and \(K_n^{\prime}(m_e r)\) are the derivatives of \(I_n\) and \(K_n\), i.e., \(I_n^{\prime}=\dfrac{dI_n(m_o r)}{dr}\))
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