Helical inward orbits of high-$Z$ impurities

TL;DR

This paper investigates how helical orbits, driven by plasma rotation mechanisms, can cause high-$Z$ impurity ions to flow inward and accumulate centrally in tokamaks, impacting plasma purity.

Contribution

It introduces a new understanding of impurity transport via helical orbits caused by rotation and baroclinic effects in tokamaks.

Findings

Helical orbits connect edge and core regions in tokamaks.

Rotation mechanisms can induce inward impurity flow.

Implications for impurity accumulation and plasma purity.

Abstract

In a tokamak the edge and the central region may become connected by helical orbits resulting from the combination of global poloidal rotation and localized radial shifts. The rotation sustained by the Stringer mechanism and the baroclinic effect are able to generate such helical orbit. This can be a contribution to the inward flow and central accumulation of high-$Z$ impurity ions.

Equations

$$\frac{\epsilon }{q}\left(1+2q^2\right)\left(\frac{\partial V_{\theta }}{\partial t}+{\gamma }_{MP}{V}_{\theta }\right)+q{V}_{\theta }\frac{1}{nr}\frac{\partial }{\partial r}\left(nr⟨2\cos \theta v_r⟩\right)=0$$

$$\frac{\partial }{\partial t}\omega +\nabla \times \left[\left(\mathbf{v}\cdot \nabla \right)\mathbf{v}\right]=\frac{1}{{\rho }^2}\nabla \rho \times \nabla p-{\gamma }_{\omega }\omega {\hat{\mathbf{e}}}_{\phi }$$

$$\omega \sim \frac{c_s^2}{{\gamma }_{\omega }}\frac{1}{n_0}\frac{\partial \ln \tilde{n}}{r\partial \theta }\frac{\partial }{\partial r}\ln n_0$$

$$\frac{\partial \omega }{\partial t}{\hat{\mathbf{e}}}_{\phi }\sim c_s^2\frac{1}{n_0^2}\frac{\partial \tilde{n}\left(\theta \right)}{r\partial \theta }{\hat{\mathbf{e}}}_{\theta }\times \frac{\partial n_0}{\partial r}{\hat{\mathbf{e}}}_r$$

$$\mathrm{a}ngle\sim \omega \Delta t\sim \frac{1}{n_0^2}\frac{\partial \tilde{n}}{r\partial \theta }\frac{d{n}_0}{dr}\times \Delta t$$

$$\delta r^{inward}$$

Full text

The high $Z$ impurity ions that enter in the tokamak plasma from the first wall are a source of energy loss via radiation and can be a danger for the reactor regime. The experiments show that the density of heavy ion impurity contamination has a substantial poloidal nonuniformity, with maximum in the Low Field Side, and that in the $H$ mode there is an accumulation of ions in the center of the discharge. The high-$Z$ ions have a fast displacement toward the plasma core (possibly faster than diffusion).

The physical picture of the access and the accumulation of heavy impurity ions in the center of the discharge is rather complex and have received several explanations.

There is a collisional momentum transfer from the background ions to the high-$Z$ ion. The equilibrium gradient of the background ions, with decrease of density from the center toward the edge, may suggest that to an impurity ion it is collisionally imposed an outward diffusion. However, the argument of Longmire and Rosenbluth [1] shows that the impurity ions actually have opposite diffusion, i.e. radially inward (”counter-gradient”). In an alternative picture, the impurity ions must have inward diffusion in order to preserve ambipolarity: the background ions have outward diffusion (fixed by the gradient of their density) while the electrons, that are tied to the magnetic surface, cannot follow. This would produce a non-ambipolar radial flux and the impurities are requested to respond by an inflow that compensate for the radial outflow of the basic ions, $\Gamma_{i}-\Gamma_{imp}\times Z=0$. In the presence of ICRH, the minority with large perpendicular energy have large banana orbits and their ourward half traverses a small region at the outboard edge. It then results an accumulation of charge in this area with generation of an electric field. The poloidal projection of this electric field, combined with the confining magnetic field, generates a radially inward flow of heavy ions.

There are mechanisms that can contribute to the faster-than-diffusion access and accumulation of heavy ions in the center, like the Turbulence Equipartition density pinch and the transient ballistic flows when the discharge is divided into a toroidal dipolar symmetric vortex.

We here examine a new possiblity: the existence of radially inward helical orbits connecting the plasma edge with the center.

The basic elements that make possible an inward spiraling motion for an element of plasma, are a global poloidal rotation of plasma and the baroclinic effect that produces local radial deviations from a purely poloidal orbit.

For the poloidal rotation we consider the mechanism of Stringer, able to produce a poloidal torque when there is a poloidal nonuniform radial flux of particles.

The Stringer spontaneous (*i.e. *without externally applied torque) rotation results from the conservation laws [2], [3], [4]. The equation for the plasma poloidal velocity is

[TABLE]

Here $\gamma_{MP}$ is the rate of damping of poloidal rotation by the magnetic pumping [5]. The radial flux is $\Gamma_{r}\approx\left\langle 2\cos\theta\ n\widetilde{v}_{r}\left(\theta\right)\right\rangle$ where the surface averaging operation is $\left\langle f\right\rangle\equiv\oint\frac{d\theta}{2\pi}\left(1+\varepsilon\cos\theta\right)\;f$. The Stringer mechanism is a basic neoclassical effect. The variation on surface of the density $\widetilde{n}$ and of the radial diffusive flow $\widetilde{v}_{r}$ produces a local flux $\widetilde{\Gamma}_{r}\left(\theta\right)$ that, due to the tokamak geometry, has non-zero divergence. A poloidal flow must exist to compensate and $V_{\theta}$ would grow indefinitely (a shock line is eventually created at the inner side of plasma), The magnetic pumping damping $\gamma_{MP}$ is very efficient in suppressing the purely neoclassical Stringer torque. However if the radial flux $\widetilde{\Gamma}\left(\theta\right)$ with variation on poloidal angle $\theta$ is due to anomalous transport, the torque can be higher than $\gamma_{MP}$ and the plasma has poloidal rotation. This is a mechanism independent to the Reynolds stress, which is usually considered the source of poloidal rotation in the $H$ mode and in Internal Transport Barrier. Several analyses on the Stringer effect induced by the poloidally non-uniform anomalous transport fluxes [2], [3] show that it can be much higher than the Reynolds stress-induced torque and this permit us to consider in the following that the plasma has a global poloidal rotation.

Now we discuss the effect of the poloidal nonuniformities of the density from a different angle. We consider $\widetilde{n}\left(r,\theta\right)$ the perturbation to the density $n_{0}\left(r\right)$ uniform on surface. It has the maximum on the equatorial plane and is symmetric above and below the plane, with identical spatial (*i.e. *$\sim\theta$) decay. This means that $\widetilde{n}\left(r,\theta\right)$ decreases on each surface $r$, from the maximum on the equator, along the arc distances $\pm r\left|\theta\right|$. The particular function is not essential now and we only remark that there are gradiants $\mathbf{\nabla}_{\theta}\widetilde{n}\left(r,\theta\right)$ tangent to the surface $r$ with orientation $-\widehat{\mathbf{e}}_{\theta}$ above and $\widehat{\mathbf{e}}_{\theta}$ below the plane ($\widehat{\mathbf{e}}_{\theta}=r\mathbf{\nabla}_{\theta}\theta$ is the versor of $\theta$-variation). Such gradients induce a baroclinic effect. For simplification we consider the momentum balance equation and apply the $\mathit{curl}$ operator.

[TABLE]

where $\gamma_{\omega}$ is a schematic representation of any dynamical damping of the vorticity, i.e. limiting the development of higher shear of the velocity. Projecting along the toroidal direction $\widehat{\mathbf{e}}_{\varphi}$, a stationary state of plasma rotation induced by only this term should be

[TABLE]

but we note that the gradients of $\widetilde{n}\left(r,\theta\right)$ have opposite sign above and below the equatorial plane and the effect is cancelled. We are interested in the effect of the baroclinic term for an element of plasma that, in the global poloidal rotation, traverses the two regions where the gradients of $\widetilde{n}\left(r,\theta\right)$ are substantial. The baroclinic effect is a source of vorticity

[TABLE]

($\widehat{\mathbf{e}}_{\varphi}=\widehat{\mathbf{e}}_{\theta}\times\widehat{\mathbf{e}}_{r}$) and this means the generation of a localized effect of rotation. The approximate measure of the rotational effect which acts for a short time interval $\Delta t$ is an angular displacement

[TABLE]

In the Figure we represent the spatial deviation induced by this angular displacement. An element of plasma initially located in $O$ is displaced, by the baroclinic effect, radially inward to a new position $A$, at a smaller radius $r_{A}<r_{O}$. It continues the global rotation and arrives in the region where it is acted upon again by the baroclinic term but this time in the opposite direction. However the two small, baroclinic-induced angular displacements are not exactly equal and opposite since the element of plasma is now at a radius $r_{A}$ which is different of the initial magnetic surface $r_{O}$. The displacement will be to a new point $B$ which is not exactly on the initial surface $r_{O}$ . A small effective radial displacement results. At every turn in the global poloidal rotation, the element of plasma receives a small radial displacement. Its orbit is helically inward, a spiral that may connect the edge to the central region. An estimation

[TABLE]

where $L_{Z\theta}$ is the gradient length of the poloidal nonuniformity of the density $\rho_{Z}$ of high-$Z$ impurities, $\Omega$ is the global (Stringer) rotation of plasma, $p_{0}\left(r\right)$ is the equilibrium plasma pressure. The condition for the existence of a helical orbit is $\frac{\partial}{\partial r}\frac{L_{Z\theta}}{V_{\theta}^{2}}>0$. Inserting some physical data one obtains an order of magnitude of few millimeters for the effective inward radial displacement at every poloidal turn and a velocity of inward advancement of about $1\ \left(m/s\right)$.