2. Standard model of plasma rotation in GDT

Consider the quasi-stationary state of the axially symmetric plasma discharge (on a time scale larger than the Alfvén time). In this case, the electric field can be considered electrostatic and described by means of the local plasma potential distributed on a flux surface $\psi$ along each field line, $\phi =\phi (\psi,z)$. Furthermore, due to the high electron temperature in this model, we will neglect the parallel resistivity of the discharge. Then the parallel electric field within the quasineutral bulk plasma is due entirely to the gradient of the electron pressure. Within the core plasma, the electric field ensures quasineutrality via the Boltzmann distribution of electrons. In the expanders (presheath), the distribution of electrons deviates from the equilibrium, but the potential profiles along a field line, including the non-neutral sheaths, can be calculated (Skovorodin Reference Skovorodin2019; Chapurin et al. Reference Chapurin, Jimenez, Smolyakov, Yushmanov and Dettrick2023).

The scheme of the potential distribution along internal field lines in GDT-type linear traps is presented in figure 1. The potential profile in the quasineutral region, including the core plasma and the expander volume, is essentially determined by the electron temperature and the density distribution, i.e. by the regime of the axial plasma confinement and outflow. As a result, the potential difference $\phi _{1}-\phi _{2}$ is a function of said parameters too. Meanwhile, the sheath potential difference $\Delta S=S_{1}-S_{2}$ depends on the parallel current densities to the endplates, $j_{1},j_{2}$. Thus, the sheath potentials balance the external ($A_{1}-A_{2}$) and the internal ($\phi _{1}-\phi _{2}$) voltages in the presence of the electron currents, making the plasma flow in expanders, in general, non-ambipolar.

Figure 1. Scheme of the potential distribution along a field line in GDT-type linear traps. Subscripts $1,2$ denote values relative to the west and east ends of the device, $M_{i}$ and $R_{i}$ are the axial positions of mirror throats and reflection points for sloshing hot ions, respectively, $A_{i}$ are the voltages applied to the endplates, $S_{i}$ and $P_{i}$ are the sheath and presheath (expander) potentials, $F_{i}$ are the heights of the ambipolar barriers due to the density peaks of hot ions at the reflection points.

The two plasma currents $j_{1}(S_{1})$ and $j_{2}(S_{2})$ have opposite directions due to the orientation of endplates, but let us set their signs the same (as measured by external circuits). Such convention is suitable for relating the currents to the volt–ampère characteristics of sheaths (ignoring their orientation). It is also convenient to measure the parallel current densities per unit flux, i.e. $j_{i}\equiv \partial I_{i}/\partial \psi \propto j_{\parallel }/B.$ Then, $j_{1},j_{2}$ can be rewritten in terms of the ‘net’ current through the trap $j_{n}=(\,j_{2}-j_{1})/2$ and the ‘biasing’ current $j_{b}=j_{1}+j_{2}$ (its closure is by the radial currents). For small currents, in the linear approximation, $\Delta S$ will depend only on the net current $j_{n}$, but in experiments, the logic is reversed, so $j_{n}$ is a function of $\Delta S$. For example, the difference in gas conditions and expansion ratio in the east and west ends of the GDT (due to positions of gas puffing and plasma gun placement) results in a significant net axial current (of the order of the ion current) on field lines with $A_{1}=A_{2}$. The net current can be used for ‘direct energy conversion’, but it results in enhanced energy losses of plasma electrons. Alternatively, the best axial electron confinement can be achieved for the minimum net axial current. This requires application of a suitable external voltage $A_{1}-A_{2}$ between opposite endplates. This is a feedback-type problem since such external voltage depends on plasma conditions and varies in time during the discharge. Alternatively, the endplates can be isolated, but in this case, the ability to set the biasing potential $\phi _{b}=(A_{1}+A_{2})/2$ is lost. A rough type of balancing can be achieved by setting a diode circuit on one and the voltage source on the opposite endplate. In the following, we will assume that some kind of balancing is in place, so that the current density to the endplate is less than the ion loss rate, while the functional dependence $j_{i}(S_{i})$ is approximately linear.

To relate figure 1 to the plasma rotation, we need to consider the dependence of shown potentials on the radial position of the field line, i.e. their dependence on the flux coordinate. One part of this functional dependence stems from the radial distribution of electron temperature and plasma density within the discharge, the ambipolar potential $\phi _{a}(\psi,z)$, the other is due to the radial dependence of the applied biasing voltage on the endplates, $\phi _{b}(\psi )$, while the third reflects the state of the angular momentum transfer due to radial currents, $\phi _{m}(\psi )$. The sheath potential $S_{i}$ contains contributions of the ambipolar and non-ambipolar nature, $S_{i}=S_{ia}+S_{in}$. This means that the sheath potential is calculated with the assumption of equal electron and ion loss rates, i.e. $j_{i}=0$, is $S_{ia}$. At small currents, $S_{i}\approx S_{ia}+j_{i}(\partial S_{i}/\partial j_{i})_{a}$, $S_{in}\approx j_{i}(\partial S_{i}/\partial j_{i})_{a}$. However, both $S_{ia},S_{in}$ result in $z$-independent contributions to the overall potential, so that the ambipolar part of the sheath potential can be included either in $\phi _{a}$ or in $\phi _{m}$. For the sake of brevity in terms, we will assume that $\phi _{m}$ is non-ambipolar, $\phi _{m}=(S_{1n}+S_{2n})/2\propto j_{b}$, while $\phi _{a}$ corresponds to the idealized axial equilibrium with $j_{b}=0$ and zero radial currents.

In summary, the plasma potential distribution can be written as a sum of the ambipolar potential, $\phi _{a}$, the biasing potential, $\phi _{b}$, and the non-ambipolar sheath potential, $\phi _{m}$:

(2.1)\begin{equation} \phi(\psi,z)=\phi_{a}(\psi,z)+\phi_{b}(\psi)+\phi_{m}(\,j_{b}(\psi)). \end{equation}

In the paraxial approximation, ${\rm d}\psi \approx 2{\rm \pi} rB\,{\rm d}r$, so that the ExB drift velocity can be described as

(2.2)\begin{equation} v_{E}=c\frac{-\phi'_{r}}{B}\approx2{\rm \pi} rc\frac{\partial\phi}{\partial\psi}. \end{equation}

The plasma rotation velocity can be found from the radial component of the equation of motion for the ion fluid (here we again assume a paraxial magnetic structure):

(2.3)\begin{equation} -nM\frac{v^{2}}{r}=-p'_{r}-qn\varphi'_{r}+\frac{qn}{c}vB, \end{equation}

where $v$ is the azimuthal velocity, $p'_{r}$ is the radial pressure gradient, $q$ is the ion charge and other notation is conventional. As a result,

(2.4)\begin{equation} \omega=\frac{v}{r}=\frac{\varOmega}{2}\left(\sqrt{1+4\frac{v_{E}}{\varOmega r}+\frac{4p'_{r}}{nM\varOmega^{2}r}}-1\right), \end{equation}

where $\varOmega =qB/Mc$ is the ion cyclotron frequency. In the limit of slow rotation, $2\omega /\varOmega \ll 1$, when the centrifugal effect is negligible,

(2.5)\begin{equation} \omega\approx v_{E}/r+p'_{r}/(nM\varOmega r)\equiv\omega_{E}+\omega_{d}, \end{equation}

where $\omega _{d}(\psi,z)=p'_{r}/(nM\varOmega r)$ is the diamagnetic drift frequency of ions. In this limit,

(2.6)\begin{equation} \omega(\psi,z)=\omega_{d}(\psi,z)+2{\rm \pi} c\frac{\partial\phi}{\partial\psi}\approx\omega_{da}(\psi,z)+\omega_{bm}(\psi). \end{equation}

Here, $\omega _{da}(\psi,z)=\omega _{d}(\psi,z)+2{\rm \pi} c(\partial /\partial \psi )\phi _{a}$ is fully determined by the equilibrium plasma parameters, while $\omega _{bm}(\psi )=2{\rm \pi} c(\partial /\partial \psi )(\phi _{b}+\phi _{m})$ is related to biasing and the conservation of the angular momentum. The important point here is that the rotation frequency $\omega _{bm}(\psi )$ is ‘rigid’ along the field lines (in the presheath and the plasma core), i.e. is constant on each flux surface. In particular, formally increasing the biasing potential far above the plasma temperature, $\omega _{da}(\psi,z)\ll \omega _{bm}(\psi )$, we should get a rigid rotation (along the field lines).

As mentioned above, $\phi _{m}$ and $\omega _{bm}$ are related to the radial currents, which, in turn, affect the angular velocity. These relationships can be described by the azimuthal component of the fluid equation for the plasma as a whole coupled with the current closure condition on a flux surface:

(2.7)\begin{gather} nM\frac{{\rm d}v}{{\rm d}t}=-\frac{1}{c}j_{\psi}B+f, \end{gather}

(2.8)\begin{gather} \left(\boldsymbol{b}\boldsymbol{\nabla}\right)\frac{j_{{\parallel}}}{B}+\frac{\partial}{\partial\psi}\left(2{\rm \pi} rj_{\psi}\right)=0. \end{gather}

Here, $j_{\psi }$ is the transverse (radial) current density and $f$ represents the azimuthal non-electromagnetic force per unit volume. In particular, $f$ can stand for the divergence of the viscous momentum flux, or for the collisional momentum transfer from the beam ions or by charge exchange with neutrals. For simplicity, we again used the paraxial approximation

The radial current density is very difficult to measure experimentally and thus it should be excluded from the system. We find

(2.9)\begin{equation} \frac{\partial}{\partial\psi}\left(2{\rm \pi} rj_{\psi}\right)=\frac{\partial}{\partial\psi}\frac{2{\rm \pi} rc}{B}\left(\,f-nM\frac{{\rm d}v}{{\rm d}t}\right)=-\left(\boldsymbol{b}\boldsymbol{\nabla}\right)\frac{j_{{\parallel}}}{B}, \end{equation}

(2.10)\begin{equation}\frac{\partial}{\partial\psi}\left(\frac{rnM}{B}\frac{{\rm d}}{{\rm d}t}r\omega\right)=\frac{1}{2{\rm \pi} c}\left(\boldsymbol{b}\boldsymbol{\nabla}\right)\frac{j_{{\parallel}}}{B}+\frac{\partial}{\partial\psi}\left(\frac{rf}{B}\right). \end{equation}

This equation describes the temporal evolution of the rotation frequency, $\omega$. However, as we found earlier, this frequency comprises two components $\omega =\omega _{da}(\psi,z)+\omega _{bm}(\psi ),$ of which the first one is essentially an external parameter, i.e. defined by equations of mass and energy transport outside of this model. Meanwhile, the variable part of the frequency $\omega _{bm}(\psi,t)$ is ‘rigid’, so that the equation for it can be reduced to the one-dimensional form by averaging along the field lines. After integrating (2.10) along field lines between the opposite endplates, we get

(2.11)\begin{equation} \frac{\partial}{\partial\psi}\left(\left\langle \frac{r^{2}nM}{B}\right\rangle \dot{\omega}_{bm}+U\omega_{bm}\right)=\frac{j_{b}}{2{\rm \pi} c}+\frac{\partial}{\partial\psi}\left\langle \frac{r}{B}\left(\,f-nM\frac{{\rm d}}{{\rm d}t}r\omega_{da}\right)\right\rangle .\end{equation}

Here, we assumed that the zero-order approximation $\boldsymbol {v}=v_{\parallel }\boldsymbol {b}+\boldsymbol {v}_{E}$ for the advection velocity in the inertial terms is sufficient, took into account that

(2.12)\begin{equation} \int\left(\boldsymbol{b}\boldsymbol{\nabla}\right)\frac{j_{{\parallel}}}{B}\,{\rm d}\ell=\left.\frac{j_{{\parallel}}}{B}\right|_{z_{1}}^{z_{2}}=j_{2}+j_{1}=j_{b}, \end{equation}

and introduced the notation

(2.13a,b)\begin{equation} \left\langle\ \right\rangle =\int_{z_{1}}^{z_{2}}\,{\rm d}\ell,\quad U(\psi)=\left\langle \frac{rnM}{B}v_{{\parallel}}\left(\boldsymbol{b}\boldsymbol{\nabla}\right)r\right\rangle . \end{equation}

Coefficient $U$ accounts for the angular momentum transfer with the axial plasma flow. Strictly speaking, this is a non-paraxial effect, but it can be important in the case of intense flows in expanders. Namely, taking into account the rigid rotation frequency, the loss rate of the angular momentum with the outflow in expanders will be larger due to the increasing radius of rotation at the endplates.

Equation (2.11) can describe the evolution of $\omega _{bm}(\psi,t)$ if its coefficients and the source terms on the right-hand side are known. However, the line-tying current density $j_{b}$ depends on the non-ambipolar sheath potential $\phi _{m}$, which contributes to $\omega _{bm}$ itself:

(2.14)\begin{equation} \omega_{bm}=2{\rm \pi} c\frac{\partial}{\partial\psi}\left(\phi_{b}+\phi_{m}\right), \end{equation}

(2.15)\begin{equation} j_{b}=j_{b}(\phi_{m})=j_{b}\left(\frac{1}{2{\rm \pi} c}\int\omega_{bm}\,{\rm d}\psi-\phi_{b}\right). \end{equation}

The explicit form of relation (2.15) depends on the IV characteristic of the sheath layer. For low-emission endplates and low net axial currents (below the ion saturation threshold), (2.15) can be linearized and rewritten as

(2.16)\begin{equation} j_{b}=j_{{\rm ion}}\frac{e\phi_{m}}{T_{e}}=j_{{\rm ion}}\frac{e}{T_{e}}\left(\frac{1}{2{\rm \pi} c}\int\omega_{bm}\,{\rm d}\psi-\phi_{b}\right), \end{equation}

where $j_{\textrm {ion}}=J_{i}/B$ is the ion current density (per flux tube) to the endplate. In particular, such a model was used earlier by Beklemishev et al. (Reference Beklemishev, Bagryansky, Chaschin and Soldatkina2010).

There is at least one more term on the right-hand side of (2.11) that depends on $\omega _{bm}$. It is the corresponding viscous force

(2.17)\begin{equation} f_{\eta b}=\frac{\eta}{r^{2}}\frac{\partial}{\partial r}\left(r^{3}\frac{\partial\omega_{bm}}{\partial r}\right),\end{equation}

where $\eta$ is the hydrodynamic viscosity. In fusion plasmas, it is usually considered to be very small, but can be anomalous in the presence of turbulence. This term contains the highest radial derivative.

Taking into account the above relations, (2.11) can be rewritten in terms of the plasma potential induced by biasing, $\phi =\phi _{b}+\phi _{m}$:

(2.18)\begin{equation} \frac{\partial}{\partial\psi}\left(I\frac{\partial\dot{\phi}}{\partial\psi}+U\frac{\partial\phi}{\partial\psi}\right)-H\left(\phi-\phi_{b}(\psi)\right)-\frac{\partial^{2}}{\partial\psi^{2}}\left(\varUpsilon\frac{\partial^{2}\phi}{\partial\psi^{2}}\right)=F,\end{equation}

where

(2.19)\begin{equation} \left. \begin{gathered} I=\left\langle \frac{r^{2}nM}{B}\right\rangle ,\quad F=\frac{\partial}{\partial\psi}\left\langle \frac{r}{2{\rm \pi} cB}\left(\hat{f}-nM\frac{{\rm d}}{{\rm d}t}r\omega_{da}\right)\right\rangle ,\\ \varUpsilon=4{\rm \pi}^{2}\left\langle \eta Br^{4}\right\rangle ,\quad H=j_{{\rm ion}}\frac{e}{4{\rm \pi}^{2}c^{2}T_{e}}, \end{gathered} \right\} \end{equation}

$H$ corresponds to the sheath-adjacent plasmas and angular brackets denote integration along the field lines. Note that the biasing potential $\phi _{b}$ enters through the line-tying term $H(\phi -\phi _{b}(\psi ))$ only.

As an example, let us consider rotation of the plasma column with radially uniform density and electron temperature, driven by biasing adjacent concentric rings of endplates (as in the ‘vortex confinement’ (Beklemishev Reference Beklemishev2008; Beklemishev et al. Reference Beklemishev, Bagryansky, Chaschin and Soldatkina2010)). Then, $\phi _{b}(\psi )$ is discontinuous like a Heaviside function at $\psi =\psi _{0}$. Further assume the plasma to be in equilibrium with low axial losses and no momentum injection with beams, $F=0,U=0.$ Then, (2.18) describes how the rotation potential $\phi$ tries to relax to the applied biasing potential $\phi _{b}$, but is hindered by viscosity. If $\eta$ is small, the rotation layer in equilibrium is in a small area around $\psi _{0}$, and the solution can be found analytically. It was earlier published in Beklemishev (Reference Beklemishev2008) and is shown in figure 2. The main point here is the weak dependence of the solution width $\Delta$ around $\psi _{0}$ on viscosity

(2.20)\begin{equation} \varDelta=(\varUpsilon/4H)^{1/4}. \end{equation}

In GDT, the solution width is of the order of 20 % of the plasma radius (by estimates and by probe measurements). If this width were comparable to the discharge radius, we would need to solve the boundary problem and study the boundary conditions for (2.18) (in $\psi$) in detail.

Figure 2. Equilibrium profile of the rotation potential driven by biasing adjacent endplates and regularized by viscosity. Here, $x=\psi -\psi _{0}$, $\kappa =(4H/\varUpsilon )^{1/4}.$

In summary, the frequency of the plasma rotation in GDT-like devices can be described as having two distinct components: (1) the ‘ambipolar’ frequency $\omega _{da}(\psi,z)$, which is a function of distributions of density and temperature in the discharge; and (2) the ‘driven’ frequency $\omega _{bm}$, which reflects the radial distribution of the angular momentum. If the axial ion confinement is independent of the ‘driven’ rotation, then $\omega _{bm}$ is constant on each flux surface and its temporal evolution can be found from the equation of the angular momentum transport, (2.18).

3. Rotation of low-temperature plasmas

Plasma rotation in the SMOLA device (Sudnikov et al. Reference Sudnikov, Ivanov, Inzhevatkina, Larichkin, Postupaev, Sklyarov, Tolkachev and Ustyuzhanin2022) exhibits a rigid-rotor type of behaviour in the radius up to the SOL boundary for all types of the biasing distribution. It can be seen from the recently published probe measurements of the radial electric field (Inzhevatkina et al. Reference Inzhevatkina, Ivanov, Postupaev, Sudnikov, Tolkachev and Ustyuzhanin2024) shown in figure 3.

Figure 3. Radial distribution of the radial electric field in SMOLA at $Z = 2.04\ \textrm {m}$ in different magnetic field configurations (Inzhevatkina et al. Reference Inzhevatkina, Ivanov, Postupaev, Sudnikov, Tolkachev and Ustyuzhanin2024). Approximately linear profile in the plasma core ($r<3.8\ \textrm {cm}$) corresponds to rigid rotation, while in the SOL ($r>3.8\ \textrm {cm}$), the rotation velocity decays to zero.

The radially rigid rotation indicates that, unlike in GDT, the effective viscous transport of momentum in radius is dominant. It should be noted that in these experiments the electron temperature is less than 10 % of the biasing potential, while the diamagnetic drift frequency should also be small. This means that in terms of the previous section, the ambipolar frequency $\omega _{da}$ can be neglected. There are also no obvious non-electromagnetic sources of the angular momentum within the discharge core, $\hat {f}=0.$

As compared with the above GDT-relevant model, the theoretical description of SMOLA discharges should include effects of finite conductivity and fast axial plasma flow. Indeed, the rotation and its potential in SMOLA vary along the discharge. In conjunction with low electron temperature, it means that the parallel gradient of the plasma potential is ohmic. In addition, the plasma gun of SMOLA generates a constant sub-sonic axial flow of plasma through the device. This flow is sure to play an important role in the axial transport of the angular momentum.

Let us construct a simplified model relevant for low-temperature viscous plasmas. Assume that a cylindrical plasma column flows along the uniform magnetic field $B$ and rotates in crossed fields. The ambipolar potential is zero. Assume the rotation to be rigid in radius with frequency $\varOmega (z)$ up to the ‘edge’ at $r=a$ (in the SOL, at $r>a$, the rotation is no longer rigid.) Assumption of the rigid rotor replaces the equation of the momentum transport in radius, while the feature of the model will be construction of the equation for the axial transport of the angular momentum.

For a rigid $E\times B$ rotation, the plasma potential is parabolic ($v(r)\propto r$ and the edge potential is $\phi (a)=\varphi _{*}(z)$):

(3.1)\begin{equation} \phi(r,z)=\frac{B}{2c}\varOmega(z)\left(r^{2}-a^{2}\right)+\varphi_{*}(z). \end{equation}

The axial plasma current density can then be found from Ohm's law:

(3.2)\begin{equation} j_{z}(r,z)=\sigma E_{z}=\frac{\sigma B}{2c}\varOmega'(z)\left(a^{2}-r^{2}\right)-\sigma\varphi_{*}'(z).\end{equation}

The radial current density can be found from the current closure condition. In the case of radially uniform conductivity $\sigma (r)=const.$, it is also parabolic:

(3.3)\begin{equation} j_{r}(r,z)=-\frac{1}{r}\int_{0}^{r}\frac{\partial j_{z}}{\partial z}r\,{\rm d}r=\sigma\frac{Br}{8c}\varOmega''(z)\left(r^{2}-2a^{2}\right)+\sigma\frac{r}{2}\varphi_{*}''(z). \end{equation}

In our model, there are two possible sources of torque: Ampère's force, $j_{r}B_{z}$, and the edge friction. Then the balance of the angular momentum looks like

(3.4)\begin{equation} \frac{\partial\varPhi(z)}{\partial z}=M_{A}+M_{\eta}. \end{equation}

Here, $\varPhi$ is the axial flux of the angular momentum,

(3.5)\begin{gather} \varPhi(z)=2{\rm \pi}\varOmega(z)\int_{0}^{a}\rho v_{z}r^{3}\,{\rm d}r\equiv G\varOmega, \end{gather}

(3.6)\begin{gather} M_{A}=-\frac{2{\rm \pi}}{c}\int_{0}^{a}j_{r}Br^{2}\,{\rm d}r=\frac{{\rm \pi}\sigma B^{2}}{12c^{2}}a^{6}\varOmega''(z)-\frac{{\rm \pi}\sigma B}{4c}a^{4}\varphi_{*}''(z) \end{gather}

is the torque of Ampère's force, while the torque of the edge friction, $M_{\eta }$, should be found from the boundary conditions in radius.

The structure of the model discharge as well as the realistic boundaries are shown in figure 4. Along the magnetic field $\boldsymbol {B}$, the discharge is in electrical contact with the cathode and the biased endplate. There is a plasma source at the cathode and a sink at the endplate. At $r=a$, the discharge is in contact with the SOL plasma. The SOL plasma is in contact with the anode, limiters and floating shielding. Some current may flow to the inner faces of the anode and limiters, or to their sides across the SOL.

Figure 4. Scheme of the structure and the boundary conditions of the model discharge.

In general, areas on top of limiters are rather small and should not exert too much of an influence on plasma dynamics (though this may not be the case for the anode). These areas should exhibit the sheath property of shielding the external electric potential of nearby electrodes by higher density of electrons. However, the radial current density there should be in electrons mostly (in the absence of surface ion sources) and thus be small, unless there is a significant azimuthal friction force on electrons. Such a force can be due to the neutral gas atoms or the plasma ions. Both these contributing forces should be present on the SOL boundary away from the limiters as well. The SOL boundary has a much larger area than the limiter tops, so that the radial current flows mostly through this boundary and then to the sides of the limiters. The anode top may be different due to higher gas density there and its higher area in some trap designs.

Discussing the role of limiters, it should also be noted that their side surfaces act as electrodes with line-tying properties, i.e. provide an effective friction for the plasma rotation due to the sheath dissipation. This friction in principle can be mitigated by splitting the side surfaces into radially isolated rings. Biasing the limiters should be effective only if the plasma density in the SOL is sufficiently high, since the biasing current is limited by the ion current from the SOL to the limiter side. Shielding the sides of a limiter by floating electrodes reduces the biasing current limit and ultimately becomes equivalent to setting the whole limiter at the floating potential.

The SOL plasma experiences azimuthal radially differential flow, while the axial flow can be neglected. Then, in the slab approximation (the SOL width is usually smaller than its radius):

(3.7a,b)\begin{equation} \frac{\partial\varPi_{xy}}{\partial x}=-\frac{1}{c}j_{x}B-qv_{y},\quad\varPi_{xy}=\rho v_{x}v_{y}-\eta\frac{\partial v_{y}}{\partial x}, \end{equation}

where $x$ is in the radial and $y$ is in the azimuthal directions, $\varPi _{xy}$ is the radial flux of the azimuthal momentum and $qv_{y}$ is the momentum transfer to neutrals. Thus, the radial flux of momentum in the SOL can decay either due to Ampère's force via currents to the limiters, or by recycling.

Using again the current closure and Ohm's law,

(3.8)\begin{equation} \frac{\partial j_{x}}{\partial x}=-\frac{\partial j_{z}}{\partial z}=\sigma_{s}\frac{\partial^{2}\phi}{\partial z^{2}}, \end{equation}

and in terms of the electrostatic potential, we get ($\rho v_{x}\rightarrow 0$, $v_{y}=c/B(\partial \varphi /\partial x)$):

(3.9)\begin{equation} \left(\frac{B^{2}\sigma_{s}}{c^{2}}\right)\frac{\partial^{2}\phi}{\partial z^{2}}=\eta\frac{\partial^{4}\phi}{\partial x^{4}}-q\frac{\partial^{2}\phi}{\partial x^{2}}.\end{equation}

Though (3.9) is comprehensive, it is too complicated to use, especially since the plasma parameters in the SOL are usually known rather poorly.

Consider the line-tying regime in the SOL: neglect recycling ($q=0$), integrate between limiters ($L_{m}$ is the line length) and apply boundary conditions on them ($j_{z}=\pm J_{\textrm {lim}}$):

(3.10)\begin{equation} \frac{\partial^{4}\phi}{\partial x^{4}}=\frac{2B^{2}J_{{\rm lim}}}{\eta c^{2}L_{m}}. \end{equation}

Assuming that the limiters are biased to a fixed potential $\phi _{\textrm {lim}}$ and the sheath current density is far from saturation, the current density to the limiters is

(3.11)\begin{equation} J_{{\rm lim}}\approx J_{i}\left(1-\exp\left(e(\phi_{{\rm lim}}-\phi)/T_{e}\right)\right)\approx eJ_{i}/T_{e}\left(\phi-\phi_{{\rm lim}}\right), \end{equation}

where $J_{i}$ is the ion flux density. Then,

(3.12)\begin{equation} d_{\eta}^{4}\phi^{\prime\prime\prime\prime}+\phi-\phi_{{\rm lim}}=0, \end{equation}

with the characteristic radial SOL width

(3.13)\begin{equation} d_{\eta}=\left(\frac{\eta c^{2}L_{m}T_{e}}{2J_{i}eB^{2}}\right)^{1/4}. \end{equation}

This width can be estimated for SMOLA parameters as $d_{\eta }\sim 0.6\ \textrm {cm}$ (with the classic viscosity $\eta$ Braginskii Reference Braginskii1965). However, this estimate is not really reliable and can be regarded as a rough lower limit. In the presence of turbulence, $\eta$ and $d_{\eta }$ should be larger, while the turbulence is in fact observed at the SOL boundary. Another possible cause for the increase in $d_{\eta }$ is the saturation of the electron sheath current. It occurs due to the fact that the transverse current is mostly in ions, while the non-ambipolar current to the limiters is in electrons. Thus, the number of electrons that can be extracted from each flux tube is limited. In other terms, the presheath length can become larger than $L_{m}$. In this case, the line-tying effect weakens and the SOL width is enhanced.

Another possible plasma behaviour in the SOL can be denoted as the recycling regime: neglect Ampère's force (the left-hand side) in (3.9), then it yields

(3.14)\begin{equation} \phi'''-\frac{q}{\eta}\phi'=0,\quad\phi=\left(\varphi_{*}-\varphi_{0}\right)\exp\left(-x/d_{q}\right)+\varphi_{0}, \end{equation}

where $d_{q}=\sqrt {\eta /q}$ and $\varphi _{0}$ is the potential of the outer shell. For SMOLA, $d_{q}\sim 0.3{-}3\ \textrm {cm}$, depending on estimates of viscosity and the density of neutrals, which is comparable to $d_{\eta }$. This means that both the recycling and the line-tying can play a role.

The matching conditions for the plasma core to the boundary of the SOL are as follows: the potential $\phi$ is continuous,

(3.15)\begin{gather} \varOmega=\frac{v_{y}(a)}{a}=\frac{c}{aB}\left.\frac{\partial\phi}{\partial x}\right|_{a}, \end{gather}

(3.16)\begin{gather} M_{\nu}=2{\rm \pi} a^{2}\eta\left.\frac{\partial v_{y}}{\partial x}\right|_{a} \end{gather}

and $j_{\perp }(a)=j_{x}$. The last condition (of the current continuity) is relevant for the line-tying regimes in the SOL, while in the recycling regime the current density plays no role in the plasma dynamics.

It is very difficult to measure the transport coefficients in the SOL to evaluate the matching conditions. However, the problem can be reversed: we can obtain information about conditions in the SOL by measuring the radial profile of potential there. Assume that we can measure the gradient length $d$ of the radial electric field at the inner edge of the SOL (${\partial v_{y}}/{\partial x}|_{a}=-({a\varOmega }/{d})$) and the asymptotic value of the electrostatic potential at the outer edge of the SOL, $\varphi _{0}$. Then the matching conditions are

(3.17)\begin{equation} M_{\nu}=-2{\rm \pi} a^{3}\frac{\eta}{d}\varOmega, \end{equation}

and, assuming an approximately exponential profile of $\phi$ (like in the recycling regime of the SOL),

(3.18)\begin{equation} \varphi_{*}\approx\varphi_{0}-\frac{Bda}{c}\varOmega. \end{equation}

The effective viscosity, $\eta$, is also difficult to measure. However, it can be found from the value of $M_{\nu }$ via measurements of $\varOmega (z)$ profiles. This relationship is discussed below.

The scheme of matching the rigidly rotating discharge core to the SOL is shown in figure 5. The $\varOmega (r)$ profile in the matching zone $r\approx a$ satisfies the necessary condition for the Kelvin–Helmholtz instability. Indeed, its smoothed version contains the inflection point. In SMOLA, the core edge appears to be the source of fluctuations (Tolkachev et al. Reference Tolkachev, Inzhevatkina, Sudnikov and Chernoshtanov2024). As a result, viscosity and resistivity there are likely anomalous. As shown in the graph for the axial current density, $j_{z}$, its absolute value in the edge zone is enhanced, while the axial ohmic field can cause the axial dependence of $\varphi _{*}(z)=\phi (a,z)$ and hence, $\varOmega (z)$.

Figure 5. Matching the discharge core ($r< a$) and the SOL layer.

Axial dependence of $\varOmega$ can be deduced from the torque balance equation, $\partial \varPhi (z)/\partial z=M_{A}+M_{\nu }$. Using the ‘experimental’ matching conditions, in terms of $\varOmega (z)$, it looks as

(3.19)\begin{equation} \frac{{\rm \pi}\sigma B^{2}}{12c^{2}}a^{5}(a+d)\frac{\partial^{2}\varOmega}{\partial z^{2}}-\frac{\partial}{\partial z}\left(G\varOmega\right)-\frac{2{\rm \pi} a^{3}}{d}\eta\varOmega=\frac{{\rm \pi}\sigma B}{4c}a^{4}\varphi_{0}''(z). \end{equation}

The physical meaning of the equation terms here is as follows.

1. (i) The torque via Ampère's force due to the biasing of end electrodes. In ideally conducting plasma ($\sigma \rightarrow \infty$), it is dominant and ensures $\varOmega =\textrm {const.}$ in $z$.

2. (ii) The divergence of the axial flow of the angular momentum with the plasma flow ensures its continuity.

3. (iii) The frictional torque caused by interaction with the SOL plasma at the core edge.

4. (iv) The effect of the axially distributed SOL biasing, if present (via the limiters in the line-tying regime or via the conducting shell).

The above equation for $\varOmega (z)$ can be rewritten in the normalized form as

(3.20)\begin{equation} L^{2}\frac{\partial^{2}\varOmega}{\partial z^{2}}-\varOmega=\frac{\partial}{\partial z}\left(\hat{G}\varOmega\right)+D.\end{equation}

Here, $L$ is the characteristic axial variation length,

(3.21)\begin{equation} L=c_{A}\sqrt{\tau_{\sigma}\tau_{\eta}/24}\propto Ba^{3/2}\sigma^{1/2}\eta^{-1/4}, \end{equation}

where $c_{A}$ is the Alfvén velocity, $\tau _{\sigma }$ is the resistive time and $\tau _{\eta }$ is the viscous momentum transport time:

(3.22a,b)\begin{gather} \tau_{\sigma}=\frac{4{\rm \pi}\sigma}{c^{2}}a^{2},\quad\tau_{\eta}=\frac{\rho d(a+d)}{\eta}, \end{gather}

(3.23)\begin{gather} \hat{G}=\int_{0}^{1}v_{z}\tau_{\eta}\left(\frac{\rho r^{3}}{\rho_{*}a^{3}}\right)\mathrm{d}\left(\frac{r}{a}\right) \end{gather}

is the normalized axial plasma flux and

(3.24)\begin{equation} D=\frac{\sigma Bad}{8c\eta}\varphi_{0}''(z). \end{equation}

Equation (3.20) can be solved with suitable boundary conditions (modelling the cathode and the endplates) along the discharge, if its coefficients are defined. Unfortunately, they are not, as it is very difficult to measure or calculate the anomalous plasma viscosity. In these circumstances, it would be more effective to use (3.20) to find $L$ and then $\eta$ via measurements of $\varOmega (z)$.

The axial distribution of the electric field depends on the plasma resistivity as well. It is easier to measure than the viscosity, but there may exist another complication. Namely, Ohm's law for discharges with emissive cathodes (like in SMOLA) may need modification to include the electron inertia. Indeed, the parabolic form of the plasma potential in the radius suggests that the sheath potential at the cathode edge in the radius may be large, of the order of the full biasing potential. Then, at least part of the axial electron current may be sustained by the beam effect, which is not included in the present model.