MHD Equilibrium and Kink Stability in Damavand Tokamak

Abstract

Magnetohydrodynamic (MHD) equilibrium is vulnerable to numerous destabilizing mechanisms. Instabilities introduce distortions to the plasma magnetic surfaces and its boundaries, their driving force being the radial gradient of plasma toroidal current density. For certain modal numbers, internal kink modes may develop, and their study is feasible according to the energy principle, in which the change in total potential energy due to the disturbance is evaluated. In this article, we present a totally new analysis of MHD equilibrium and stability, and apply it to Damavand tokamak which has a large aspect ratio. For this purpose, we combine perturbation and Green’s function methods to solve for the equilibrium configuration. At this stage, plasma profiles are found explicitly in terms of Bessel functions, and we present a simple expression for estimation of total toroidal plasma current. Then the rest of plasma profiles, including poloidal magnetic flux, safety factor, and toroidal current density, are obtained and plotted. In the next step, we turn to the stability calculations and show that Damavand plasma is resistant to most of the disturbances.

Appendices

Appendix

A-Poloidal Flux

We here study the tokamak plasma equilibrium using direct solution of Grad-Shafranov equation. For the first time, we show that the analytical solution of Grad-Shafranov equation within linear regime can be sought by combining perturbation and Green’s function methods. The results are particularly valid for a tokamak with low-β, near-circular cross-section and large aspect ratio; Damavand tokamak meets all of these requirements.

In the toroidal geometry with axial symmetry, the Grad-Shafranov equation in the cylindrical system of coordinates (R, z, ϕ) reads [4–6]

$$ \Updelta^{*} \psi = R\frac{\partial }{\partial R}\left( {\frac{1}{R}\frac{\partial \psi }{\partial R}} \right) + \frac{{\partial^{2} \psi }}{{\partial z^{2} }} = - \mu_{0} R^{2} \frac{\partial F\left( \psi \right)}{\partial \psi } - \mu_{0}^{2} I\left( \psi \right)\frac{\partial I\left( \psi \right)}{\partial \psi } $$

(13)

Where, F(ψ) is plasma pressure. Also, we have

$$ I\left( \psi \right) = \frac{{RB_{t} }}{{\mu_{0} }} $$

(14)

Here, B _t is the toroidal field. Now we employ the Solov’ev model [4, 6, 16] for large aspect ratio tokamaks. The pseudo-toroidal system of coordinates (r, θ, φ) and cylindrical system of coordinates (R, z, ϕ) are transformed as

$$ \begin{aligned} R = R_{0} + r\cos \theta \hfill \\ \varphi = \phi \hfill \\ z = r\sin \theta \hfill \\ \end{aligned} $$

(15)

Hence, (13) can be rewritten as

$$ \begin{gathered} \left[ {\frac{1}{r}\frac{\partial }{\partial r}\left( {r\frac{\partial }{\partial r}} \right) + \frac{1}{{r^{2} }}\frac{{\partial^{2} }}{{\partial \theta^{2} }}} \right]\psi - \frac{1}{{R_{0} + r\cos \theta }}\left( {\cos \theta \frac{\partial }{\partial r} - \frac{\sin \theta }{r}\frac{\partial }{\partial \theta }} \right)\psi = \hfill \\ - \mu_{0} \left( {R_{0} + r\cos \theta } \right)^{2} \frac{\partial F\left( \psi \right)}{\partial \psi } - \mu_{0}^{2} I\left( \psi \right)\frac{\partial I\left( \psi \right)}{\partial \psi } \hfill \\ \end{gathered} $$

(16)

In the Solov’ev equilibrium and pseudo-toroidal system of coordinates (r, θ, φ), magnetic surfaces are seen as concentric circles. As a good approximation one may write down

$$ \psi \left( {r,\theta } \right) = \psi_{0} \left( r \right) + \psi_{1} \left( {r,\theta } \right) $$

(17)

in which

$$ \left[ {\frac{1}{r}\frac{\partial }{\partial r}\left( {r\frac{\partial }{\partial r}} \right)} \right]\psi_{0} \left( r \right) = - \mu_{0} R_{0}^{2} \frac{{\partial F\left( {\psi_{0} } \right)}}{{\partial \psi_{0} }} - \mu_{0}^{2} I\left( {\psi_{0} } \right)\frac{{\partial I\left( {\psi_{0} } \right)}}{{\partial \psi_{0} }} $$

(18)

Hence for the perturbation function ψ ₁(r, θ) with the approximation $\psi_{1} \left( {r,\theta } \right) << \psi_{0} \left( r \right)$ we have

$$ \begin{gathered} \left[ {\frac{1}{r}\frac{\partial }{\partial r}\left( {r\frac{\partial }{\partial r}} \right) + \frac{1}{{r^{2} }}\frac{{\partial^{2} }}{{\partial \theta^{2} }}} \right]\psi_{1} \left( {r,\theta } \right) \approx \hfill \\ \frac{\cos \theta }{{R_{0} }}\frac{\partial }{\partial r}\psi_{0} - 2\mu_{0} R_{0} r\cos \theta \frac{{\partial F\left( {\psi_{0} } \right)}}{{\partial \psi_{0} }} - \frac{1}{(\partial \psi_{0}/ \partial r)}\frac{\partial }{\partial r}\left[ {\mu_{0} R_{0}^{2} \frac{{\partial F\left( {\psi_{0} } \right)}}{{\partial \psi_{0} }} - \mu_{0}^{2} I\left( {\psi_{0} } \right)\frac{{\partial I\left( {\psi_{0} } \right)}}{{\partial \psi_{0} }}} \right]\psi_{1} \hfill \\ \end{gathered} $$

(19)

Now we obtain the functions F(ψ) and I(ψ) from their polynomial expansions. Since in general these functions are not initially known, we may write down

$$ \begin{gathered} I\left( \psi \right)\frac{\partial }{\partial \psi }I\left( \psi \right) = \sum\limits_{n = 0}^{\infty } {I_{n} \psi^{n} } \approx I_{0} + I_{1} \psi + I_{2} \psi^{2} \hfill \\ \frac{\partial }{\partial \psi }F\left( \psi \right) = \sum\limits_{n = 0}^{\infty } {F_{n} \psi^{n} } \approx F_{0} + F_{1} \psi + F_{2} \psi^{2} \hfill \\ \end{gathered} $$

(20)

so that

$$ \frac{1}{r}\frac{\partial }{\partial r}\left( {r\frac{\partial }{\partial r}} \right)\psi_{0} \approx - A_{0} - A_{1} \psi_{0} - A_{2} \psi_{0}^{2} $$

(21a)

$$ \begin{gathered} \left[ {\frac{1}{r}\frac{\partial }{\partial r}\left( {r\frac{\partial }{\partial r}} \right) + \frac{1}{{r^{2} }}\frac{{\partial^{2} }}{{\partial \theta^{2} }}} \right]\psi_{1} \approx \hfill \\ \quad \frac{\cos \theta }{{R_{0} }}\frac{\partial }{\partial r}\psi_{0} - r\cos \theta \left( {B_{0} + B_{1} \psi_{0} + B_{2} \psi_{0}^{2} } \right) - \left( {A_{1} + 2A_{2} \psi_{0} } \right)\psi_{1} \hfill \\ \end{gathered} $$

(22b)

in which A _n  = μ ₀(R ²₀ F _n  + μ ₀ I _n ) and B _n  = 2μ ₀ R ₀ F _n . The initial conditions for (9) are

$$ \left. {\psi_{0} } \right|_{r = 0}\,= \left. {\frac{{\partial \psi_{0} }}{\partial r}} \right|_{r = 0} = 0 $$

(22a)

$$ \left. {\psi_{1} } \right|_{r = 0}\,= \left. {\frac{{\partial \psi_{1} }}{\partial r}} \right|_{r = 0} = 0 $$

(22b)

Therefore, as a linear approximation to the poloidal magnetic flux inside plasma we have

$$ \left[ {\frac{1}{r}\frac{\partial }{\partial r}\left( {r\frac{\partial }{\partial r}} \right) + A_{1} } \right]\psi_{0} = - A_{0} $$

(23a)

$$ \left[ {\frac{1}{r}\frac{\partial }{\partial r}\left( {r\frac{\partial }{\partial r}} \right) + \frac{1}{{r^{2} }}\frac{{\partial^{2} }}{{\partial \theta^{2} }} + A_{1} } \right]\psi_{1} \approx \frac{\cos \theta }{{R_{0} }}\frac{\partial }{\partial r}\psi_{0} - r\cos \theta \left( {B_{0} + B_{1} \psi_{0} } \right) $$

(23b)

Hence, the solution of (23a) can be found analytically using Bessel’s functions as

$$ \psi_{0} \left( r \right) = \psi_{c} \left[ {J_{0} \left( {kr} \right) - 1} \right] $$

(24)

Here, $k = \sqrt {A_{1} }$ and ψ _c  = A ₀/A ₁. In Fig. 2, a plot of (24) is shown which is in agreement with the general behavior of magnetic poloidal flux. Equation (24) is expected to be exact for large aspect ratio tokamaks and nearly circular cross-sections [13].

It is worth to point out that (23b) can be solved for tokamaks with non-circular cross section using (24) and Green’s function formalism [14, 17]. For this purpose, we define the Green’s function as

$$ \left[ {\frac{1}{r}\frac{\partial }{\partial r}\left( {r\frac{\partial }{\partial r}} \right) + \frac{1}{{r^{2} }}\frac{{\partial^{2} }}{{\partial \theta^{2} }} + k^{2} } \right]G\left( {{\mathbf{r}};{\mathbf{r}}^{\prime } } \right) = - \delta \left( {{\mathbf{r}} - {\mathbf{r}}^{\prime } } \right) = - \frac{1}{r}\delta \left( {r - r^{\prime } } \right)\delta \left( {\theta - \theta^{\prime } } \right) $$

(25)

where ${\mathbf{r}} = r\left( {\cos \theta \hat{r} + \sin \theta \hat{z}} \right)$ is the position vector measured from the plasma axis on the constant φ surface. Similarity of (13) with wave propagation in two-dimensional space results in the Green’s function as [14]

$$ G\left( {r;\;r^{\prime } } \right) = \text{Re} \left[ {\frac{i}{4}H_{0}^{\left( 1 \right)} \left( {k\left| {{\mathbf{r}} - {\mathbf{r}}^{\prime } } \right|} \right)} \right] $$

(26)

Here, H ⁽¹⁾₀ (·) is the Hankel’s function of the first kind and zeroth order. Hence

$$ \psi \left( {r,\;\theta } \right) = \int\limits_{0}^{\infty } {\int\limits_{0}^{2\pi } {G\left( {{\mathbf{r}};\;{\mathbf{r}}^{\prime } } \right)} f\left( {{\mathbf{r}}^{\prime } } \right)} r^{\prime } {\text{d}}\theta^{\prime } {\text{d}}r^{\prime } $$

(27a)

$$ f\left( {\mathbf{r}} \right) = \left\{ {\frac{{k\psi_{c} }}{{R_{0} }}J_{1} \left( {kr} \right) + r\left[ {\left( {B_{0} - B_{1} \psi_{c} } \right) + B_{1} \psi_{c} J_{0} \left( {kr} \right)} \right]} \right\}\cos \theta $$

(27b)

Therefore, the explicit solution of Grad-Shafranov equation within the linear regime can be accessed by combining perturbation and Green’s function methods. It is interesting to point out that we have recently used this combination for studying dissipative solitons in nonlinear plasma waves, which typically occur in Rayleigh-Taylor instabilities [18].

B-Safety Factor

In this section, we present the details of calculation of safety factor versus minor radius. For this purpose, it is sufficient that to find the toroidal magnetic flux according to the following definition for tokamaks with circular cross section

$$ \phi \left( r \right) = \int\limits_{0}^{r} {\int\limits_{0}^{2\pi } {B_{t} \left( {\mathbf{r}} \right)\rho \;{\text{d}}\theta \;{\text{d}}\rho } } $$

(28)

in which ${\mathbf{r}} = \left( {\rho ,\;\theta } \right)$ is the position vector measured from the plasma axis. Based on the Solov’ev equilibrium we have [16, 19]

$$ B_{t} \left( {\mathbf{r}} \right) = \frac{{B_{{t_{0} }} }}{{1 + ({\rho }/{{R_{0} }})\cos \theta }} $$

(29)

in which $B_{{t_{0} }}$ is the toroidal field on the plasma axis, and R ₀ is the major radius of plasma. For large aspect ratio approximation we have ρ << R ₀, and (29) can be rewritten as

$$ B_{t} \left( {\mathbf{r}} \right) = B_{{t_{0} }} \sum\limits_{n = 0}^{\infty } {\left( { - \frac{\rho }{{R_{0} }}\cos \theta } \right)^{n} } $$

(30)

By plugging (30) in (28) and making use of the identities

$$ \int\limits_{0}^{2\pi } {\left( {\cos \theta } \right)^{2n + 1} d\theta } = 0 $$

(31a)

$$ \int\limits_{0}^{2\pi } {\left( {\cos \theta } \right)^{2n} {\text{d}}\theta } = \frac{2\sqrt \pi }{n!}\Upgamma \left( {n + \frac{1}{2}} \right) $$

(31b)

where $\Upgamma \left( \cdot \right)$ is the Euler’s Gamma function we obtain

$$ \phi \left( r \right) = 2\sqrt \pi B_{{t_{0} }} \sum\limits_{n = 0}^{\infty } {\frac{{\Upgamma \left( {n + \tfrac{1}{2}} \right)}}{n!}\int\limits_{0}^{r} {\left( {\frac{\rho }{{R_{0} }}} \right)^{2n} \rho \;{\text{d}}\rho } } $$

(32)

After some algebra and simplification we get

$$ \phi \left( r \right) = \sqrt \pi B_{{t_{0} }} r^{2} \sum\limits_{n = 0}^{\infty } {\frac{{\Upgamma \left( {n + \tfrac{1}{2}} \right)}}{{\left( {n + 1} \right)!}}\left( {\frac{r}{{R_{0} }}} \right)^{2n} } $$

(33)

The latter equation can be rewritten as

$$ \phi \left( r \right) = \pi r^{2} B_{{t_{0} }} \left[ {1 + \frac{1}{4}\left( {\frac{r}{{R_{0} }}} \right)^{2} + \frac{3}{24}\left( {\frac{r}{{R_{0} }}} \right)^{4} + \cdots } \right] $$

(34)

As it can be seen here, within zeroth-order approximation we have $\phi \left( r \right) \approx \pi r^{2} B_{{t_{0} }}$, which shows that the toroidal magnetic flux is approximately equal to the product of cross-sectional area of the outermost magnetic surface and toroidal magnetic field on the plasma axis.

Now using (24) and (34) and the definition of safety factor as

$$ q\left( r \right) = \frac{1}{\mu \left( r \right)} = - \frac{{\phi^{\prime } \left( r \right)}}{{\psi^{\prime } \left( r \right)}} $$

(35)

we get

$$ q\left( r \right) = \frac{{2\sqrt \pi B_{{t_{0} }} }}{{k\psi_{c} J_{1} \left( {kr} \right)}}\sum\limits_{n = 0}^{\infty } {\frac{{\Upgamma \left( {n + \tfrac{1}{2}} \right)}}{n!}\frac{{r^{2n + 1} }}{{R_{0}^{2n} }}} $$

(36)

It is interesting if we take a look at safety factor on the plasma axis $r\sim 0$. Under these conditions, we have $J_{1} \left( {kr} \right)\sim \tfrac{1}{2}kr$ and higher order terms in (34) could be neglected. Thus

$$ q\left( {r \to 0} \right) = \frac{{4\pi B_{{t_{0} }} }}{{k^{2} \psi_{c} }} $$

(37)

But due to the sawtooth oscillations in tokamaks, the safety factor on the plasma axis is usually equal to unity [1, 4, 5], and we finally get

$$ \frac{{4\pi B_{{t_{0} }} }}{{k^{2} \psi_{c} }} = 1 $$

(38)

But according to the definition of k in Appendix A we had $k = \sqrt {A_{1} }$ _, and therefore the following equation for derivation of the constant A ₁ is obtained

$$ A_{1} = \frac{{4\pi B_{{t_{0} }} }}{{\psi_{c} }} $$

(39)