Cylindrical geometry limit

Consider the form of matrix $E$ in the cylindrical geometry limit, in which the equilibrium quantities are independent of poloidal angle. Equation () indicates that the geodesic curvature $\kappa _s$ is zero in this case. Thus, the matrix elements $E_{12}$ and $E_{21}$ are zero. Next, consider the matrix elements $E_{11}$ and $E_{22}$. Because all equilibrium quantities are independent of the poloidal angle, different poloidal harmonics of the perturbation are decoupled. Therefore, we can consider a perturbation with a single poloidal mode number . For a poloidal harmonic with poloidal mode number $m$, matrix element $E_{11}$ is written

$$E_{11} = - \frac{\omega^2 \rho_0 \vert \nabla \Psi \vert^2}{B_0^2} -{\text... ...left( \frac{\vert \nabla \Psi \vert^2}{B^2_0} \mathbf{B}_0 \cdot \nabla \right)$$

$$= - \frac{\omega^2 \rho_0 \vert \nabla \Psi \vert^2}{B_0^2} +{\text... ...frac{\vert \nabla \Psi \vert^2}{B^2_0} (\Psi' \mathcal{J}^{- 1})^2 (m - n q)^2,$$ (216)

and matrix element $E_{22}$ is written

$$E_{22} = \frac{{\textmu}_0^{- 1} B_0^2 + \gamma p_0}{B_0^2} +{\textmu}_0^{... ...mathbf{B}_0 \cdot \nabla \left( \frac{\mathbf{B}_0 \cdot \nabla}{B^2_0} \right)$$

$$= \frac{{\textmu}_0^{- 1} B_0^2 + \gamma p_0}{B_0^2} -{\textmu}_0^{... ...p_0}{\omega^2 \rho_0} \frac{1}{B_0^2} (\Psi' \mathcal{J}^{- 1})^2 (m - n q)^2 .$$ (217)

The continua are the roots of the equation $\det (E) = 0$, which, in the cylindrical geometry limit, reduces to

$$E_{11} E_{22} = 0.$$ (218)

Two branches of the roots of Eq. (218) are given by $E_{11} = 0$ and $E_{22} = 0$, respectively. The equation $E_{11} = 0$ is written

$$- \omega^2 \rho_0 +{\textmu}_0^{- 1} (\Psi' \mathcal{J}^{- 1})^2 (m - n q)^2 = 0,$$ (219)

which gives

$$\omega^2 = \frac{(\Psi' \mathcal{J}^{- 1})^2 (m - n q)^2}{{\textmu}_0 \rho_0},$$ (220)

which is the Alfvén branch of the continua. Figure 1a plots the results of Eq. (220). The equation $E_{22} = 0$ is written

$$\omega^2 = \frac{\gamma p_0}{{\textmu}_0^{- 1} B_0^2 + \gamma p_0} \frac{(\Psi' \mathcal{J}^{- 1})^2}{{\textmu}_0 \rho_0} (m - n q)^2,$$ (221)

which is the sound branch of the continua. Figure 1b plots the results of Eq. (221).

Figure: $n = 1$ Alfven continua (left) and sound continua (right) in the cylindrical geometry limit for $m = 0, 1, 2, 3, 4$, and $5$ (calculated by using Eqs. (220) and (221)). The equilibrium used for this calculation is for EAST discharge #38300@3.9s (G-eqdsk filename g038300.03900, which was provided by Dr. Guoqiang Li). The number density of ions is given in Fig. 12. Because the Jacobian $\mathcal {J}$ in toroidal geometry depends on the poloidal angle, the average value of $\mathcal {J}$ on a magnetic surface is used in evaluating the right-hand side of (220).

Figures compares the Alfven continua in the cylindrical limit with those in the toroidal geometry. The results indicate that the Alfven continua in the toroidal geometry reconnect, forming gaps near the locations where the Alfven continua in the cylindrical limit intersect each other.

Figure 2: Comparision of the the Alfven continua in toroidal geometry (black solid lines) and Alfven continua in the cylindrical limit (other lines). The Alfven continua in toroidal geometry are obtained by using the slow-sound-approximation. The equilibrium is EAST discharge #38300 at 3.9s.

The result in Eq. (220) is not clear from the view of physics since it involves the Jacobian, which is a mathematical factor due to the freedom in the choice of coordinates. Next, we try to write the right-hand side of Eq. (220) in more physical form. In cylindrical geometry limit, magnetic surfaces are circular. Thus the radial coordinate can be chosen to be the geometrical radius of the circular magnetic surface, and the usual poloidal angle (i.e., equal-arc angle) can be used as the poloidal coordinate. Then the poloidal magnetic flux is written as

$$\Psi_p = \int_0^r B_p (r) 2 \pi R_0 d r,$$ (222)

where $2 \pi R_0$ is the length of the cylinder. We know that $\Psi$ used in the Grad-Shafranov equation is related to $\Psi_p$ by

$$\Psi = \pm \frac{\Psi_p}{2 \pi} + C.$$ (223)

Using Eqs. (222) and (223), we obtain

$$\Psi' \equiv \frac{d \Psi}{d r} = \pm B_p R_0 .$$ (224)

Next, we calculate the Jacobian $\mathcal {J}$, which is defined by

$$\mathcal{J}^{- 1} = \nabla \psi \times \nabla \theta \cdot \nabla \zeta .$$ (225)

Since we choose $\psi = r$ and $\zeta = \phi$ (the positive direction of $\theta$ is count clockwise when observers view along the positive direction of $\phi$), the above equation is written

$$\mathcal{J}^{- 1} = \nabla r \times \nabla \theta \cdot \frac{\hat{\ensuremath{\boldsymbol{\phi}}}}{R_0} .$$

$$= - \frac{1}{r R_0} .$$ (226)

Using Eqs. (224) and (226) , $(\Psi' \mathcal{J}^{- 1})^2$ is written

$$(\Psi' \mathcal{J}^{- 1})^2 = \left( B_p R_0 \frac{1}{r R_0} \right)^2 = \frac{B_p^2}{r^2}$$ (227)

Using these, Eq. (220) is written

$$\omega^2 = \frac{B_p^2 (m - n q)^2}{r^2 {\textmu}_0 \rho_0} .$$ (228)

Using the definition of safety factor in the cylindrical geometry

$$q = \frac{B_{\phi} r}{R_0 B_p},$$ (229)

equation (228) is written

$$\omega^2 = \frac{B_{\phi}^2 (m - n q)^2}{{\textmu}_0 \rho_0 q^2 R_0^2} .$$ (230)

In the cylindrical geometry, the parallel (to equilibrium magnetic field) wave-number is given by

$$k_{\parallel} = \frac{m - n q}{q R_0} .$$ (231)

Using this, Eq. (230) is written

$$\omega^2 = k_{\parallel}^2 \frac{B_{\phi}^2}{{\textmu}_0 \rho_0} .$$ (232)

Using the definition of Alfven speed $V_{A \phi}^2 \equiv B_{\phi}^2 / ({\textmu}_0 \rho_0)$, the above equation is written as

$$\omega^2 = k_{\parallel}^2 V_{A \phi}^2,$$ (233)

which gives the well known Alfven resonance condition. For later use, define

$$\omega_a^2 = \frac{B_p^2 (m - n q)^2}{r^2 {\textmu}_0 \rho_0},$$ (234)

then Eq. (228) is written as $\omega^2 = \omega_a^2$.

Similarly, by using Eq. (227), equation (217) for $E_{22}$ is written as

$$E_{22} = \frac{{\textmu}_0^{- 1} B_0^2 + \gamma p_0}{B_0^2} -{\te... ...{\gamma p_0}{\omega^2 \rho_0} \frac{1}{B_0^2} \frac{B_p^2}{r^2} (m - n q)^2 .$$

Then equation $E_{22} = 0$ reduces to

$$\frac{{\textmu}_0^{- 1} B_0^2 + \gamma p_0}{B_0^2} -{\textmu}_0^{... ...{\gamma p_0}{\omega^2 \rho_0} \frac{1}{B_0^2} \frac{B_p^2}{r^2} (m - n q)^2 = 0$$ (235)

$$\Rightarrow \omega^2 = \frac{{\textmu}_0^{- 1} \frac{\gamma p_0}{... ...{B_p^2}{r^2} (m - n q)^2}{\frac{{\textmu}_0^{- 1} B_0^2 + \gamma p_0}{B_0^2}}$$

$$\Rightarrow \omega^2 = \frac{{\textmu}_0^{- 1} \frac{\gamma p_0}{\rho_0} }{{\textmu}_0^{- 1} B_0^2 + \gamma p_0} \frac{B_p^2}{r^2} (m - n q)^2$$

$$\Rightarrow \omega^2 = \frac{\frac{\gamma p_0}{\rho_0} }{\frac{B... ... \frac{\gamma p_0}{\rho_0}} \frac{B_p^2}{{\textmu}_0 \rho_0 r^2} (m - n q)^2$$

$$\Rightarrow \omega^2 = \frac{C_s^2}{V_A^2 + C_s^2} \omega_a^2,$$ (236)

where $C_s^2 = \gamma p_0 / \rho_0$, $V_A^2 = B_0^2 / ({\textmu}_0 \rho_0)$. Equation () gives the sound branch of the continua. For present tokamak plasma parameters, $C_s$ is usually one order smaller than $V_A$. Thus, equation (236) indicates the sound continua are much smaller than the Alfven continua for the same $m$ and $n$.