2 Basic equations and background equilibria

The dynamics of a perfectly conducting plasma with anisotropic pressure is determined by the following set of equations:

(2.1) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\unicode[STIX]{x1D71A}^{\ast }(D_{t}\boldsymbol{V}^{\ast })=\boldsymbol{J}^{\ast }\times \boldsymbol{B}^{\ast }-\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D64B}^{\ast },\quad \unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D71A}^{\ast }+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D71A}^{\ast }\boldsymbol{V}^{\ast })=0,\\ \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{B}^{\ast }=0,\quad \unicode[STIX]{x1D735}\times \boldsymbol{B}^{\ast }=\unicode[STIX]{x1D707}_{0}\boldsymbol{J}^{\ast },\quad \unicode[STIX]{x2202}_{t}\boldsymbol{B}^{\ast }=\unicode[STIX]{x1D735}\times (\boldsymbol{V}^{\ast }\times \boldsymbol{B}^{\ast }).\end{array}\right\} & & \displaystyle\end{eqnarray}$$

Here, $\boldsymbol{B}^{\ast }(\boldsymbol{r},t)$ is the magnetic field, $\boldsymbol{J}^{\ast }(\boldsymbol{r},t)$ the current density, $\unicode[STIX]{x1D707}_{0}$ the magnetic constant, $\boldsymbol{V}^{\ast }(\boldsymbol{r},t)$ the plasma velocity, $\unicode[STIX]{x1D71A}^{\ast }(\boldsymbol{r},t)$ the mass density and $\unicode[STIX]{x1D64B}^{\ast }$ the pressure tensor defined as

(2.2) $$\begin{eqnarray}\unicode[STIX]{x1D64B}^{\ast }:=P_{\bot }^{\ast }\unicode[STIX]{x1D644}+\frac{\unicode[STIX]{x1D70E}_{d}^{\ast }}{\unicode[STIX]{x1D707}_{0}}\boldsymbol{B}^{\ast }\boldsymbol{B}^{\ast },\end{eqnarray}$$

where $\unicode[STIX]{x1D644}$ is the unit tensor; $\unicode[STIX]{x2202}_{t}$ denotes the time derivative, while the Lagrangian derivative is defined as $D_{t}\equiv \unicode[STIX]{x2202}_{t}+(\boldsymbol{V}^{\ast }\boldsymbol{\cdot }\unicode[STIX]{x1D735})$ . The pressure tensor is diagonal in a local rectangular system with respect to $\boldsymbol{B}^{\ast }$ , and consists of the scalar pressure elements $P_{\Vert }^{\ast }(\boldsymbol{r},t)$ and $P_{\bot }^{\ast }(\boldsymbol{r},t)$ along and across $\boldsymbol{B}^{\ast }(\boldsymbol{r},t)$ , respectively, while the dimensionless function

(2.3) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{d}^{\ast }:=\frac{\unicode[STIX]{x1D707}_{0}(P_{\Vert }^{\ast }-P_{\bot }^{\ast })}{|\boldsymbol{B}^{\ast }|^{2}}\end{eqnarray}$$

is a measure of the pressure anisotropy. Particle collisions equilibrating parallel and perpendicular energies lower the value of $\unicode[STIX]{x1D70E}_{d}^{\ast }$ and, therefore a highly collisional plasma is described accurately by a single scalar pressure, $P_{\text{isotropic}}^{\ast }$ . In view of this fact, when pressure anisotropy is present it is useful to introduce an effective isotropic pressure,

(2.4) $$\begin{eqnarray}{\mathcal{P}}^{\ast }(\boldsymbol{r},t):=\frac{P_{\Vert }^{\ast }+P_{\bot }^{\ast }}{2},\end{eqnarray}$$

that reduces to $P_{\text{isotropic}}^{\ast }$ in the absence of anisotropy. By substituting (2.3) and (2.4) into the momentum equation of set (2.1) we obtain

(2.5) $$\begin{eqnarray}\unicode[STIX]{x1D71A}^{\ast }(D_{t}\boldsymbol{V}^{\ast })=(1-\unicode[STIX]{x1D70E}_{d}^{\ast })\boldsymbol{J}^{\ast }\times \boldsymbol{B}^{\ast }-\unicode[STIX]{x1D735}{\mathcal{P}}^{\ast }-\frac{\boldsymbol{B}^{\ast }}{\unicode[STIX]{x1D707}_{0}}(\boldsymbol{B}^{\ast }\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D70E}_{d}^{\ast })+\frac{|\boldsymbol{B}^{\ast }|^{2}}{2\unicode[STIX]{x1D707}_{0}}\unicode[STIX]{x1D735}\unicode[STIX]{x1D70E}_{d}^{\ast },\end{eqnarray}$$

in which the scalar pressures $P_{\Vert }^{\ast }$ and $P_{\bot }^{\ast }$ do not appear explicitly. This is very useful for the stability analysis to follow in the next sections.

The counterparts to set (2.1) equilibrium equations are

(2.6) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\unicode[STIX]{x1D71A}(\boldsymbol{V}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{V}=\boldsymbol{J}\times \boldsymbol{B}-\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D64B},\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D71A}\boldsymbol{V})=0,\\ \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{B}=0,\quad \unicode[STIX]{x1D735}\times \boldsymbol{B}=\unicode[STIX]{x1D707}_{0}\boldsymbol{J},\quad \unicode[STIX]{x1D735}\times (\boldsymbol{V}\times \boldsymbol{B})=0,\end{array}\right\} & & \displaystyle\end{eqnarray}$$

where the absence of the superscript $\ast$ denotes equilibrium and therefore no dependence on time.

In the special case of collinear velocity and magnetic fields related through

(2.7) $$\begin{eqnarray}\boldsymbol{V}=\frac{\unicode[STIX]{x1D706}(\boldsymbol{r})}{\sqrt{\unicode[STIX]{x1D707}_{0}\unicode[STIX]{x1D71A}}}\boldsymbol{B},\end{eqnarray}$$

with $\unicode[STIX]{x1D706}$ an arbitrary dimensionless scalar function, the continuity equation of the set (2.6) takes the form $(\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D71A})/\unicode[STIX]{x1D71A}=-2(\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D706})/\unicode[STIX]{x1D706}$ ; substitution of the latter relation in the counterpart to (2.5), the equilibrium momentum equation, yields

(2.8) $$\begin{eqnarray}\displaystyle (1-\unicode[STIX]{x1D70E}_{d}-\unicode[STIX]{x1D706}^{2})\boldsymbol{J}\times \boldsymbol{B} & = & \displaystyle \unicode[STIX]{x1D735}\left({\mathcal{P}}+\unicode[STIX]{x1D706}^{2}\frac{\boldsymbol{B}^{2}}{2\unicode[STIX]{x1D707}_{0}}\right)\nonumber\\ \displaystyle & & \displaystyle +\,\frac{\boldsymbol{B}^{2}}{2\unicode[STIX]{x1D707}_{0}}\unicode[STIX]{x1D735}(1-\unicode[STIX]{x1D70E}_{d}-\unicode[STIX]{x1D706}^{2})-\frac{\boldsymbol{B}}{\unicode[STIX]{x1D707}_{0}}[\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}(1-\unicode[STIX]{x1D70E}_{d}-\unicode[STIX]{x1D706}^{2})].\end{eqnarray}$$

Equation (2.8) is valid for arbitrary functions $\unicode[STIX]{x1D71A}$ and $\unicode[STIX]{x1D70E}_{d}$ , and therefore for compressible flows.

Furthermore, we assume that the mass density and the anisotropy function remain constant everywhere inside a volume ${\mathcal{D}}$ : $\unicode[STIX]{x1D71A},\unicode[STIX]{x1D70E}_{d}=\text{const.}$ Consequently, the corresponding field-aligned equilibrium flow becomes incompressible and the function $\unicode[STIX]{x1D706}$ is constant on magnetic surfaces $\unicode[STIX]{x1D713}=\text{const.}$ , whenever such surfaces exist. Also, both the magnetic and velocity fields lie on those surfaces, $\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D713})=\boldsymbol{V}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D713})=0$ . It is known that the existence of three-dimensional equilibria with nested toroidal magnetic surfaces is not guaranteed and, in general, irrespective of the existence of magnetic surfaces, the function $\unicode[STIX]{x1D706}$ is constant on both the magnetic field lines and the velocity streamlines. Henceforth, we will presume the existence of well-defined equilibrium magnetic surfaces. Under these assumptions, equation (2.8) takes the simpler form

(2.9) $$\begin{eqnarray}(1-\unicode[STIX]{x1D70E}_{d}-\unicode[STIX]{x1D706}^{2})\boldsymbol{J}\times \boldsymbol{B}=\unicode[STIX]{x1D735}{\mathcal{P}}_{s}-(\unicode[STIX]{x1D706}^{2})^{\prime }\frac{\boldsymbol{B}^{2}}{2\unicode[STIX]{x1D707}_{0}}\unicode[STIX]{x1D735}\unicode[STIX]{x1D713},\end{eqnarray}$$

where the prime denotes differentiation with respect to $\unicode[STIX]{x1D713}$ and ${\mathcal{P}}_{s}$ is the total effective pressure in the absence of flow $(\unicode[STIX]{x1D706}=0)$ , defined as

(2.10) $$\begin{eqnarray}{\mathcal{P}}_{s}:={\mathcal{P}}+\unicode[STIX]{x1D706}^{2}\frac{\boldsymbol{B}^{2}}{2\unicode[STIX]{x1D707}_{0}}.\end{eqnarray}$$

Moreover, projection of (2.9) along $\boldsymbol{B}$ implies that the static total effective pressure is a surface quantity, ${\mathcal{P}}_{s}={\mathcal{P}}_{s}(\unicode[STIX]{x1D713})$ , and therefore, the momentum equation (2.9) can be cast into the useful form

(2.11) $$\begin{eqnarray}\boldsymbol{J}\times \boldsymbol{B}=g(\unicode[STIX]{x1D713},\boldsymbol{B}^{2})\unicode[STIX]{x1D735}\unicode[STIX]{x1D713},\end{eqnarray}$$

where

(2.12) $$\begin{eqnarray}g(\unicode[STIX]{x1D713},\boldsymbol{B}^{2}):=(1-\unicode[STIX]{x1D70E}_{d}-\unicode[STIX]{x1D706}^{2})^{-1}\left[{\mathcal{P}}_{s}^{\prime }-(\unicode[STIX]{x1D706}^{2})^{\prime }\frac{\boldsymbol{B}^{2}}{2\unicode[STIX]{x1D707}_{0}}\right].\end{eqnarray}$$

Equation (2.11) implies that, for equilibria with field-aligned incompressible flows, constant density and constant anisotropy function, the current surfaces coincide with the magnetic surfaces, that is, the vectors $\boldsymbol{J}$ , $\boldsymbol{B}$ lie on common surfaces $\unicode[STIX]{x1D713}=\text{const.}$ In the subsequent sections we examine the stability of the aforementioned equilibria to small three-dimensional perturbations. To this end, we define the following quantities:

(2.13a,b ) $$\begin{eqnarray}\boldsymbol{N}:=\boldsymbol{J}\times \boldsymbol{B},\quad \boldsymbol{M}:=\unicode[STIX]{x1D735}\times \boldsymbol{N}=\unicode[STIX]{x1D735}g\times \unicode[STIX]{x1D735}\unicode[STIX]{x1D713},\end{eqnarray}$$

from which it follows that

(2.14) $$\begin{eqnarray}\boldsymbol{N}\boldsymbol{\cdot }\boldsymbol{M}=0.\end{eqnarray}$$

3 Energy principle and perturbation potential energy

In order to examine the linear stability of a given equilibrium with anisotropic pressure and incompressible flow, $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{V}=0$ , we assume that the equilibrium position, $\boldsymbol{r}$ , is perturbed to a position $\boldsymbol{r}^{\ast }(\boldsymbol{r},t)$ , through the usual Lagrangian displacement vector $\unicode[STIX]{x1D743}(\boldsymbol{r},t)\equiv \boldsymbol{r}^{\ast }-\boldsymbol{r}$ , so that

(3.1) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\boldsymbol{B}^{\ast }=\boldsymbol{B}(\boldsymbol{r})+\boldsymbol{b}(\boldsymbol{r},t),\quad \boldsymbol{V}^{\ast }=\boldsymbol{V}(\boldsymbol{r})+\boldsymbol{v}(\boldsymbol{r},t),\quad \boldsymbol{J}^{\ast }=\boldsymbol{J}(\boldsymbol{r})+\boldsymbol{j}(\boldsymbol{r},t),\\ {\mathcal{P}}^{\ast }={\mathcal{P}}(\boldsymbol{r})+p(\boldsymbol{r},t),\quad \unicode[STIX]{x1D71A}^{\ast }=\unicode[STIX]{x1D71A}(\boldsymbol{r})+\unicode[STIX]{x1D6FF}(\boldsymbol{r},t),\quad {\unicode[STIX]{x1D70E}_{d}}^{\ast }=\unicode[STIX]{x1D70E}_{d}(\boldsymbol{r})+\unicode[STIX]{x1D716}(\boldsymbol{r},t).\end{array}\right\} & & \displaystyle\end{eqnarray}$$

Here, $\boldsymbol{b},\boldsymbol{j},p,\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D716}$ and

(3.2) $$\begin{eqnarray}\boldsymbol{v}=\boldsymbol{u}(\boldsymbol{r},t)+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D743}}{\unicode[STIX]{x2202}t}\end{eqnarray}$$

correspond to small perturbations of the respective equilibrium quantities. Note that we have assumed perturbations of the effective pressure, ${\mathcal{P}}$ , and the anisotropy function, $\unicode[STIX]{x1D70E}_{d}$ , instead of explicit perturbations of the scalar pressures $P_{\Vert }$ and $P_{\bot }$ . Also, on the fixed boundary $\unicode[STIX]{x2202}{\mathcal{D}}$ surrounding a plasma domain ${\mathcal{D}}$ of interest, we adopt the following conditions:

(3.3) $$\begin{eqnarray}\boldsymbol{b}\boldsymbol{\cdot }\hat{\boldsymbol{n}}=\boldsymbol{u}\boldsymbol{\cdot }\hat{\boldsymbol{n}}=0,\end{eqnarray}$$

where $\hat{\boldsymbol{n}}$ is the perpendicular outward unit vector on the boundary. Introducing perturbations (3.1) into the dynamical equations (2.1) and employing the equilibrium equations (2.6), we obtain the following linearised equations:

(3.4) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{b}=0,\quad \boldsymbol{j}=\frac{1}{\unicode[STIX]{x1D707}_{0}}\unicode[STIX]{x1D735}\times \boldsymbol{b},\quad \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x2202}t}+\boldsymbol{V}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D71A}\boldsymbol{v})=0,\\ \displaystyle \unicode[STIX]{x1D71A}\frac{\unicode[STIX]{x2202}\boldsymbol{v}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D71A}[(\boldsymbol{V}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{v}+(\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{V}]-(1-\unicode[STIX]{x1D70E}_{d})(\boldsymbol{J}\times \boldsymbol{b}+\boldsymbol{j}\times \boldsymbol{B})+\unicode[STIX]{x1D735}p=G(\boldsymbol{r},t),\end{array}\right\} & & \displaystyle\end{eqnarray}$$

where

(3.5) $$\begin{eqnarray}\displaystyle G & := & \displaystyle -\unicode[STIX]{x1D6FF}(\boldsymbol{V}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{V}-\unicode[STIX]{x1D716}\boldsymbol{J}\times \boldsymbol{B}+\frac{1}{\unicode[STIX]{x1D707}_{0}}\left[\vphantom{\left(\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{b}+\frac{\boldsymbol{b}^{2}}{2}\right)}\frac{\boldsymbol{B}^{2}}{2}\unicode[STIX]{x1D735}\unicode[STIX]{x1D716}-(\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D716}+\boldsymbol{b}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D70E}_{d})\boldsymbol{B}\right.\nonumber\\ \displaystyle & & \displaystyle \left.+\left(\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{b}+\frac{\boldsymbol{b}^{2}}{2}\right)\unicode[STIX]{x1D735}\unicode[STIX]{x1D70E}_{d}-(\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D716})\boldsymbol{b}\right].\end{eqnarray}$$

At this point we note that, in order that the set of dynamical equations (2.1) be closed, one needs, in connection with the pressures $P_{\Vert }^{\ast }$ and $P_{\bot }^{\ast }$ , a couple of energy equations or equations of state, e.g. the double adiabatic equations Chew et al. (Reference Chew, Goldberger and Low1956) associated with the known CGL model. In the present study, one such equation of state corresponds to incompressibility in connection with an evolution with constant mass density, $\unicode[STIX]{x1D71A}^{\ast }=\unicode[STIX]{x1D71A}=\text{const.}$ , $(\unicode[STIX]{x1D6FF}=0)$ . Also, the fact that the momentum equation can be cast in the form (2.5) involving the pressures $P_{\Vert }^{\ast }$ and $P_{\bot }^{\ast }$ only implicitly through the effective pressure, ${\mathcal{P}}^{\ast }$ , and the anisotropy function, $\unicode[STIX]{x1D70E}_{d}$ , motivated us to adopt as second equation of state the constrain that the latter function remains constant, $\unicode[STIX]{x1D70E}_{d}^{\ast }=\unicode[STIX]{x1D70E}_{d}=\text{const.}$ , $(\unicode[STIX]{x1D716}=0)$ . This implies that $P_{\Vert }^{\ast }$ and $P_{\bot }^{\ast }$ (precisely their difference) evolve in such a way that they keep proportional to the magnetic pressure $B^{2}/(2\unicode[STIX]{x1D707}_{0})$ , which, on physical grounds, is an acceptable approximation. Consequently, equation (3.5) implies that $G=0$ , while the third of the linearised equations (3.4) reduces to

(3.6) $$\begin{eqnarray}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{v}=0.\end{eqnarray}$$

In view of this relation, we consider incompressible perturbations, $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D743}=0$ , such that the condition $\unicode[STIX]{x1D743}\boldsymbol{\cdot }\hat{\boldsymbol{n}}=0$ is satisfied on the boundary. Then, equation (3.6) implies that $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0$ . Subsequently, the perturbations of the magnetic and velocity fields can be expressed in terms of $\unicode[STIX]{x1D743}$ , as

(3.7a,b ) $$\begin{eqnarray}\boldsymbol{b}=\unicode[STIX]{x1D735}\times (\unicode[STIX]{x1D743}\times \boldsymbol{B}),\quad \boldsymbol{u}=\unicode[STIX]{x1D735}\times (\unicode[STIX]{x1D743}\times \boldsymbol{V}),\end{eqnarray}$$

while the linearised force balance equation of the set (3.4) is put in the form

(3.8) $$\begin{eqnarray}\unicode[STIX]{x1D71A}\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D743}}{\unicode[STIX]{x2202}t^{2}}+2\unicode[STIX]{x1D71A}(\boldsymbol{V}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D743}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}f=\hat{\boldsymbol{F}}\unicode[STIX]{x1D743}.\end{eqnarray}$$

Here,

(3.9) $$\begin{eqnarray}\hat{\boldsymbol{F}}:=\unicode[STIX]{x1D71A}\boldsymbol{u}\times (\unicode[STIX]{x1D735}\times \boldsymbol{V})+\unicode[STIX]{x1D71A}\boldsymbol{V}\times (\unicode[STIX]{x1D735}\times \boldsymbol{u})+(1-\unicode[STIX]{x1D70E}_{d})\boldsymbol{J}\times \boldsymbol{b}-\frac{1}{\unicode[STIX]{x1D707}_{0}}\boldsymbol{B}\times (\unicode[STIX]{x1D735}\times \boldsymbol{b}),\end{eqnarray}$$

is the symmetric force operator and the scalar function $f$ is defined as

(3.10) $$\begin{eqnarray}f:=\unicode[STIX]{x1D71A}\boldsymbol{V}\boldsymbol{\cdot }\boldsymbol{u}+p.\end{eqnarray}$$

The energy principle is based of the fact that the total perturbation energy

(3.11) $$\begin{eqnarray}E=K+W=\frac{1}{2}\int _{{\mathcal{D}}}\unicode[STIX]{x1D71A}\dot{\unicode[STIX]{x1D743}}^{2}\,\text{d}^{3}r-\frac{1}{2}\int _{{\mathcal{D}}}\unicode[STIX]{x1D743}\boldsymbol{\cdot }\hat{\boldsymbol{F}}\unicode[STIX]{x1D743}\,\text{d}^{3}r\end{eqnarray}$$

is conserved, where $K$ is the kinetic energy and $W$ the potential energy. Stability is related to the sign of $E$ . Since $K$ is quadratic in velocity, and therefore non-negative–definite, a sufficient condition for a given equilibrium to be linearly stable is $W\geqslant 0$ . On account of (3.3)–(3.10), $W$ is expected to depend only on the physical quantities of the background equilibrium and the displacement vector, $\unicode[STIX]{x1D743}$ , with the exception of the perturbation, $p$ , of the effective pressure, appearing in the gradient of the quantity $f$ . However, the contribution of the latter term to $W$ vanishes due to the implied boundary conditions on $\unicode[STIX]{x2202}{\mathcal{D}}$ . Thus, one finds for the perturbation potential energy

(3.12) $$\begin{eqnarray}W=\frac{1}{2}\int _{{\mathcal{D}}}\left\{\unicode[STIX]{x1D71A}\boldsymbol{u}\boldsymbol{\cdot }[\unicode[STIX]{x1D743}\times (\unicode[STIX]{x1D735}\times \boldsymbol{V})]-\unicode[STIX]{x1D71A}\boldsymbol{u}^{2}+(1-\unicode[STIX]{x1D70E}_{d})\boldsymbol{b}\boldsymbol{\cdot }(\boldsymbol{J}\times \unicode[STIX]{x1D743})+\frac{(1-\unicode[STIX]{x1D70E}_{d})}{\unicode[STIX]{x1D707}_{0}}\boldsymbol{b}^{2}\right\}\text{d}^{3}\boldsymbol{r}.\end{eqnarray}$$

Now we focus our study on the investigation of the linear stability for the equilibria with field-aligned incompressible flows, defined by (2.7). In this case (3.12) becomes

(3.13) $$\begin{eqnarray}W=\frac{1}{2\unicode[STIX]{x1D707}_{0}}\int _{{\mathcal{D}}}\{(1-\unicode[STIX]{x1D70E}_{d}-\unicode[STIX]{x1D706}^{2})[\boldsymbol{b}^{2}+\boldsymbol{b}\boldsymbol{\cdot }(\unicode[STIX]{x1D707}_{0}\boldsymbol{J}\times \unicode[STIX]{x1D743})]-2\unicode[STIX]{x1D706}(\unicode[STIX]{x1D743}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D706})(\unicode[STIX]{x1D743}\boldsymbol{\cdot }[(\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{B}])\}\,\text{d}^{3}\boldsymbol{r}.\end{eqnarray}$$

In the next section, we employ the form of $W$ given in (3.13) to derive a sufficient condition for the linear stability of the respective kinds of equilibria.

4 Sufficient condition for linear stability

It is recalled that for equilibria with field-aligned incompressible flows, constant mass density and constant pressure anisotropy function, $\unicode[STIX]{x1D70E}_{d}$ , defined by (2.7)–(2.13), the current density stays on magnetic surfaces, $\unicode[STIX]{x1D713}=\text{const.}$ and, thus, the vectors $\boldsymbol{B}$ , $\boldsymbol{J}$ and $\boldsymbol{N}=\boldsymbol{B}\times \boldsymbol{J}$ , form a basis in $\mathbb{R}^{3}$ space. Accordingly, following the analysis in Throumoulopoulos & Tasso (Reference Throumoulopoulos and Tasso2007) we expand the displacement vector in this basis, as

(4.1) $$\begin{eqnarray}\unicode[STIX]{x1D743}=\unicode[STIX]{x1D6FC}(\boldsymbol{r},t)\boldsymbol{N}+\unicode[STIX]{x1D6FD}(\boldsymbol{r},t)\boldsymbol{J}+\unicode[STIX]{x1D6FE}(\boldsymbol{r},t)\boldsymbol{B},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}$ , $\unicode[STIX]{x1D6FD}$ and $\unicode[STIX]{x1D6FE}$ are arbitrary, appropriately dimensional scalar functions. We note that in Throumoulopoulos & Tasso (Reference Throumoulopoulos and Tasso2007) a sufficient condition was derived for the linear stability of equilibria with field-aligned incompressible flows, isotropic pressure and constant density; in fact, the constant density and the vacuum magnetic permeability constant were set to unity therein. These equilibria are recovered from the respective anisotropic pressure equilibria described in § 2 for $\unicode[STIX]{x1D70E}_{d}=0$ . Also, we observe that the form (3.13) of the potential energy for $\unicode[STIX]{x1D70E}_{d}=0$ (and  $\unicode[STIX]{x1D71A}=\unicode[STIX]{x1D707}_{0}=1$ ) recovers the one obtained in equation (13) of Throumoulopoulos & Tasso (Reference Throumoulopoulos and Tasso2007) for respective isotropic equilibria. Thus, it is straightforward to derive, along the same lines as in Throumoulopoulos & Tasso (Reference Throumoulopoulos and Tasso2007), a sufficient condition for the linear stability of the present anisotropic equilibria. Specifically, we decompose the potential energy of (3.13) into two integrals as

(4.2) $$\begin{eqnarray}W=W_{1}+W_{2},\end{eqnarray}$$

and, following step by step the procedure in the Appendix of Throumoulopoulos & Tasso (Reference Throumoulopoulos and Tasso2007), we obtain

(4.3) $$\begin{eqnarray}W_{1}=\frac{1}{2\unicode[STIX]{x1D707}_{0}}\int _{{\mathcal{D}}}(1-\unicode[STIX]{x1D70E}_{d}-\unicode[STIX]{x1D706}^{2})(\boldsymbol{b}+\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D707}_{0}\boldsymbol{J}\times \boldsymbol{N})^{2}\,\text{d}^{3}\boldsymbol{r},\end{eqnarray}$$

and

(4.4) $$\begin{eqnarray}W_{2}=\frac{1}{2\unicode[STIX]{x1D707}_{0}}\int _{{\mathcal{D}}}{\mathcal{A}}(\sqrt{2}g\unicode[STIX]{x1D6FC})^{2}\,\text{d}^{3}\boldsymbol{r},\end{eqnarray}$$

where

(4.5) $$\begin{eqnarray}\displaystyle {\mathcal{A}} & := & \displaystyle -(1-\unicode[STIX]{x1D70E}_{d}-\unicode[STIX]{x1D706}^{2})\{|\unicode[STIX]{x1D707}_{0}\boldsymbol{J}\times \unicode[STIX]{x1D735}\unicode[STIX]{x1D713}|^{2}-(\unicode[STIX]{x1D707}_{0}\boldsymbol{J}\times \unicode[STIX]{x1D735}\unicode[STIX]{x1D713})\boldsymbol{\cdot }[(\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{B}]\}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{\unicode[STIX]{x1D707}_{0}}{2}[\ln (1-\unicode[STIX]{x1D70E}_{d}-\unicode[STIX]{x1D706}^{2})]^{\prime }|\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}|^{2}\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\left(P_{\bot }+\frac{\boldsymbol{B}^{2}}{2\unicode[STIX]{x1D707}_{0}}\right).\end{eqnarray}$$

Equations (4.3) and (4.4) imply that $W$ is non-negative if both quantities $1-\unicode[STIX]{x1D70E}_{d}-\unicode[STIX]{x1D706}^{2}$ and ${\mathcal{A}}$ are also non-negative–definite in ${\mathcal{D}}$ . Thus, we conclude with the following statement: an equilibrium with anisotropic pressure with  $\unicode[STIX]{x1D70E}_{d}=\text{const.}$ , of a field-aligned incompressible flow in connection with a constant plasma density is linearly stable if both of the following conditions are satisfied:

(4.6) $$\begin{eqnarray}\displaystyle & \displaystyle 1-\unicode[STIX]{x1D70E}_{d}-\unicode[STIX]{x1D706}^{2}\geqslant 0, & \displaystyle\end{eqnarray}$$

(4.7) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{A}}\geqslant 0. & \displaystyle\end{eqnarray}$$

Conditions (4.6) and (4.7) can be applied to any steady state without geometrical restriction. They generalise the ones derived in Throumoulopoulos & Tasso (Reference Throumoulopoulos and Tasso2007) for isotropic pressure, since for $\unicode[STIX]{x1D70E}_{d}=0$ condition (4.6) reduces to sub-Afvénic flows, $\unicode[STIX]{x1D706}^{2}<1$ , while (4.7) reduces to equation (22) in Throumoulopoulos & Tasso (Reference Throumoulopoulos and Tasso2007). In fact, the expression (4.5) is more compact because it consists of three terms, instead of the four terms in equation (22) of Throumoulopoulos & Tasso (Reference Throumoulopoulos and Tasso2007). The first of these terms,

(4.8) $$\begin{eqnarray}A_{1}=-(1-\unicode[STIX]{x1D70E}_{d}-\unicode[STIX]{x1D706}^{2})|\unicode[STIX]{x1D707}_{0}\boldsymbol{J}\times \unicode[STIX]{x1D735}\unicode[STIX]{x1D713}|^{2},\end{eqnarray}$$

is negative and therefore always destabilising, in potential connection with current-driven modes. The second term,

(4.9) $$\begin{eqnarray}A_{2}=(1-\unicode[STIX]{x1D70E}_{d}-\unicode[STIX]{x1D706}^{2})(\unicode[STIX]{x1D707}_{0}\boldsymbol{J}\times \unicode[STIX]{x1D735}\unicode[STIX]{x1D713})\boldsymbol{\cdot }[(\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{B}],\end{eqnarray}$$

is related to the magnetic shear (i.e. it depends on the variation of $\boldsymbol{B}$ across the magnetic surfaces), and can be either stabilising or destabilising. The third term,

(4.10) $$\begin{eqnarray}A_{3}=\frac{\unicode[STIX]{x1D707}_{0}}{2}[\ln (1-\unicode[STIX]{x1D70E}_{d}-\unicode[STIX]{x1D706}^{2})]^{\prime }|\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}|^{2}\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\left(P_{\bot }+\frac{\boldsymbol{B}^{2}}{2\unicode[STIX]{x1D707}_{0}}\right),\end{eqnarray}$$

can be regarded as a flow term, although it is affected by anisotropy, since it vanishes in the absence of flow ( $\unicode[STIX]{x1D706}=0$ ). Note that it relates to the variation of the total pressure perpendicular to the magnetic surfaces; indeed, on account of (2.7) and (2.10) one finds

(4.11) $$\begin{eqnarray}P_{\bot }+\frac{\boldsymbol{B}^{2}}{2\unicode[STIX]{x1D707}_{0}}=\underbrace{{\mathcal{P}}_{s}(\unicode[STIX]{x1D713})}_{\text{static}}-\underbrace{\frac{1}{2}\unicode[STIX]{x1D71A}\boldsymbol{V}^{2}}_{\text{flow}}+\underbrace{(1-\unicode[STIX]{x1D70E}_{d})\frac{\boldsymbol{B}^{2}}{2\unicode[STIX]{x1D707}_{0}}}_{\text{magnetic}},\end{eqnarray}$$

which involves all three pressures, static effective, flow and magnetic, the latter being influenced by the pressure anisotropy through the factor ( $1-\unicode[STIX]{x1D70E}_{d}$ ). In addition, satisfaction of condition (4.6) in the absence of flow, $\unicode[STIX]{x1D70E}_{d}\leqslant 1$ , implies that the corresponding static anisotropic equilibria are stable under the firehose instability (see Cheviakov & Bogoyavlenskij Reference Cheviakov and Bogoyavlenskij2004).

Before closing this section we note that every equilibrium state with incompressible flow and an anisotropy function that is uniform on the magnetic surfaces, $\unicode[STIX]{x1D70E}_{d}=\unicode[STIX]{x1D70E}_{d}(\unicode[STIX]{x1D713})$ having certain continuous geometrical symmetry, is governed by a generalised Grad–Shafranov (GS) equation for the flux function $\unicode[STIX]{x1D713}$ , e.g. equation (28) in Evangelias, Kuiroukidis & Throumoulopoulos (Reference Evangelias, Kuiroukidis and Throumoulopoulos2018) governing helically symmetric equilibria. That equation contains a quadratic $|\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}|^{2}$ -term. For this reason we have introduced the integral transformation,

(4.12) $$\begin{eqnarray}U(\unicode[STIX]{x1D713})=\int _{0}^{\unicode[STIX]{x1D713}}\sqrt{1-\unicode[STIX]{x1D70E}_{d}(x)-\unicode[STIX]{x1D706}^{2}(x)}\,\text{d}x,\quad 1-\unicode[STIX]{x1D70E}_{d}-\unicode[STIX]{x1D706}^{2}\geqslant 0,\end{eqnarray}$$

under which the respective transformed GS equation, becomes free of a quadratic term as $|\unicode[STIX]{x1D735}U|^{2}$ , and can be solved by analytical techniques in the $U$ -space. Transformation (4.12) does not change the topology of the magnetic surfaces, but just relabels them by the flux function $U$ , and consists of a generalisation of the transformation in Clemente (Reference Clemente1993) for static anisotropic equilibria $(\unicode[STIX]{x1D706}=0)$ and that in Simintzis, Throumoulopoulos & Pantis (Reference Simintzis, Throumoulopoulos and Pantis2001) for respective stationary, isotropic equilibria $(\unicode[STIX]{x1D70E}_{d}=0)$ . Under transformation (4.12), condition (4.6) is trivially satisfied, while condition (4.7), valid for $\unicode[STIX]{x1D70E}_{d}=\text{const.}$ , is expressed in $U$ -space as

(4.13) $$\begin{eqnarray}\displaystyle {\mathcal{A}} & = & \displaystyle -|\unicode[STIX]{x1D707}_{0}\boldsymbol{J}\times \unicode[STIX]{x1D735}U|^{2}+(\unicode[STIX]{x1D707}_{0}\boldsymbol{J}\times \unicode[STIX]{x1D735}U)\boldsymbol{\cdot }[(\unicode[STIX]{x1D735}U\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{B}]\nonumber\\ \displaystyle & & \displaystyle +\,\frac{\unicode[STIX]{x1D707}_{0}}{2(1-\unicode[STIX]{x1D70E}_{d}-\unicode[STIX]{x1D706}^{2})}\frac{\text{d}\ln (1-\unicode[STIX]{x1D70E}_{d}-\unicode[STIX]{x1D706}^{2})}{\text{d}U}|\unicode[STIX]{x1D735}U|^{2}\unicode[STIX]{x1D735}U\boldsymbol{\cdot }\unicode[STIX]{x1D735}\left(P_{\bot }+\frac{\boldsymbol{B}^{2}}{2\unicode[STIX]{x1D707}_{0}}\right).\end{eqnarray}$$

Condition ${\mathcal{A}}\geqslant 0$ in connection with form (4.13) will be employed in § 6 to examine the stability of specific helically symmetric equilibria.

5 Stability under symmetry transformations

In a recent work (Evangelias & Throumoulopoulos Reference Evangelias and Throumoulopoulos2019) a set of symmetry transformations that map anisotropic plasma equilibria with field-aligned incompressible flows was introduced; specifically, these transformations, when applied to a given equilibrium with parallel incompressible flow and anisotropy function, $\unicode[STIX]{x1D70E}_{d}$ , uniform on the magnetic surfaces labelled by the function $\unicode[STIX]{x1D713}$ , $\{\boldsymbol{B},\unicode[STIX]{x1D71A}(\unicode[STIX]{x1D713}),\boldsymbol{V}=\unicode[STIX]{x1D706}\boldsymbol{B}/(\sqrt{\unicode[STIX]{x1D707}_{0}\unicode[STIX]{x1D71A}}),{\mathcal{P}},\unicode[STIX]{x1D70E}_{d}(\unicode[STIX]{x1D713})\}$ , produce an infinite family of respective equilibria with field-aligned incompressible flows, but a density and anisotropy function that may vary on the magnetic surfaces, $\{\boldsymbol{B}_{1},\unicode[STIX]{x1D71A}_{1},\boldsymbol{V}_{1},{\mathcal{P}}_{1},\unicode[STIX]{x1D70E}_{d_{1}}\}$ . These transformations are defined by

(5.1) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \boldsymbol{B}_{1}=\frac{b_{1}(\boldsymbol{r})}{n_{1}(\boldsymbol{r})}\boldsymbol{B},\quad \boldsymbol{V}_{1}=\frac{c_{1}(\boldsymbol{r})\sqrt{1-\unicode[STIX]{x1D70E}_{d}}}{a_{1}(\boldsymbol{r})\sqrt{\unicode[STIX]{x1D707}_{0}\unicode[STIX]{x1D71A}}}\boldsymbol{B},\\ \displaystyle \unicode[STIX]{x1D71A}_{1}(\boldsymbol{r})=a_{1}^{2}(\boldsymbol{r})\unicode[STIX]{x1D71A},\quad {\mathcal{P}}_{1}=C{\mathcal{P}}+(1-\unicode[STIX]{x1D70E}_{d})(C-b_{1}^{2}(\boldsymbol{r}))\frac{\boldsymbol{B}^{2}}{2\unicode[STIX]{x1D707}_{0}},\\ \displaystyle \unicode[STIX]{x1D70E}_{d_{1}}=1-n_{1}^{2}(\boldsymbol{r})(1-\unicode[STIX]{x1D70E}_{d}),\quad C=\frac{[b_{1}^{2}(\boldsymbol{r})-c_{1}^{2}(\boldsymbol{r})](1-\unicode[STIX]{x1D70E}_{d})}{1-\unicode[STIX]{x1D70E}_{d}-\unicode[STIX]{x1D706}^{2}}=\text{const.}\neq 0,\end{array}\right\} & & \displaystyle\end{eqnarray}$$

where $a_{1},b_{1},c_{1}$ and $n_{1}$ are scalar functions, which must satisfy the relations

(5.2a,b ) $$\begin{eqnarray}\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\left(\frac{b_{1}}{n_{1}}\right)=0,\quad \boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}(a_{1}c_{1})=0.\end{eqnarray}$$

Transformations (5.1), consisting a generalisation of those introduced in Bogoyavlenskij (Reference Bogoyavlenskij2001) for respective equilibria with isotropic pressure, preserve the topology of the magnetic surfaces, $\unicode[STIX]{x1D713}\equiv \unicode[STIX]{x1D713}_{1}=\text{const.}$ , between the original and the transformed equilibrium, i.e. $\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}=\boldsymbol{B}_{1}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}$ . It was also proved in Evangelias & Throumoulopoulos (Reference Evangelias and Throumoulopoulos2019) that these transformations can break the geometrical symmetry of a given equilibrium only when the fields $\boldsymbol{B}$ and $\boldsymbol{V}$ are purely poloidal.

From (5.1) it readily follows that the transformed collinear velocity and magnetic fields are related through

(5.3a,b ) $$\begin{eqnarray}\boldsymbol{V}_{1}=\frac{\unicode[STIX]{x1D706}_{1}}{\sqrt{\unicode[STIX]{x1D707}_{0}\unicode[STIX]{x1D71A}_{1}}}\boldsymbol{B}_{1},\quad \unicode[STIX]{x1D706}_{1}=\frac{c_{1}n_{1}}{b_{1}}\sqrt{1-\unicode[STIX]{x1D70E}_{d}},\end{eqnarray}$$

and thus, the transformed equilibria satisfy a force balance equation analogous to (2.8)

(5.4) $$\begin{eqnarray}\displaystyle & & \displaystyle (1-\unicode[STIX]{x1D70E}_{d_{1}}-\unicode[STIX]{x1D706}_{1}^{2})\boldsymbol{J}_{1}\times \boldsymbol{B}_{1}\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D735}\left({\mathcal{P}}_{1}+\unicode[STIX]{x1D706}_{1}^{2}\frac{\boldsymbol{B}_{1}^{2}}{2\unicode[STIX]{x1D707}_{0}}\right)+\frac{\boldsymbol{B}_{1}^{2}}{2\unicode[STIX]{x1D707}_{0}}\unicode[STIX]{x1D735}(1-\unicode[STIX]{x1D70E}_{d_{1}}-\unicode[STIX]{x1D706}_{1}^{2})-\frac{\boldsymbol{B}_{1}}{\unicode[STIX]{x1D707}_{0}}[\boldsymbol{B}_{1}\boldsymbol{\cdot }\unicode[STIX]{x1D735}(1-\unicode[STIX]{x1D70E}_{d_{1}}-\unicode[STIX]{x1D706}_{1}^{2})].\qquad\end{eqnarray}$$

With the aid of (5.1) and (5.3) we obtain the useful relations

(5.5) $$\begin{eqnarray}\displaystyle & \displaystyle 1-\unicode[STIX]{x1D70E}_{d_{1}}-\unicode[STIX]{x1D706}_{1}^{2}=C\left(\frac{n_{1}}{b_{1}}\right)^{2}(1-\unicode[STIX]{x1D70E}_{d}-\unicode[STIX]{x1D706}^{2}), & \displaystyle\end{eqnarray}$$

(5.6) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{P}}_{1}+\unicode[STIX]{x1D706}_{1}^{2}\frac{\boldsymbol{B}_{1}^{2}}{2\unicode[STIX]{x1D707}_{0}}=C\left({\mathcal{P}}+\unicode[STIX]{x1D706}^{2}\frac{\boldsymbol{B}^{2}}{2\unicode[STIX]{x1D707}_{0}}\right)=C{\mathcal{P}}_{s}, & \displaystyle\end{eqnarray}$$

under which equation (5.4) reduces to

(5.7) $$\begin{eqnarray}\left(\frac{n_{1}}{b_{1}}\right)^{2}\boldsymbol{J}_{1}\times \boldsymbol{B}_{1}=\boldsymbol{J}\times \boldsymbol{B}+\frac{\boldsymbol{B}_{1}^{2}}{2\unicode[STIX]{x1D707}_{0}}\unicode[STIX]{x1D735}\left(\frac{n_{1}}{b_{1}}\right)^{2}.\end{eqnarray}$$

Now, presume that the original equilibrium belongs to the family described in § 2 for which (2.11) holds; then (5.7) becomes

(5.8) $$\begin{eqnarray}\left(\frac{n_{1}}{b_{1}}\right)^{2}\boldsymbol{J}_{1}\times \boldsymbol{B}_{1}=g(\unicode[STIX]{x1D713},\boldsymbol{B}^{2})\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}+\frac{\boldsymbol{B}_{1}^{2}}{2\unicode[STIX]{x1D707}_{0}}\unicode[STIX]{x1D735}\left(\frac{n_{1}}{b_{1}}\right)^{2}.\end{eqnarray}$$

Projection of (5.8) along $\boldsymbol{J}_{1}$ yields

(5.9) $$\begin{eqnarray}\boldsymbol{J}_{1}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}=-\frac{\boldsymbol{B}_{1}^{2}}{2\unicode[STIX]{x1D707}_{0}}\left[\boldsymbol{J}_{1}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\left(\frac{n_{1}}{b_{1}}\right)^{2}\right],\end{eqnarray}$$

and thus, it turns out that the transformed current density, $\boldsymbol{J}_{1}$ , remains on the magnetic surfaces if and only if the ratio $n_{1}/b_{1}$ is uniform on those surfaces

(5.10) $$\begin{eqnarray}\frac{n_{1}}{b_{1}}:=y(\unicode[STIX]{x1D713}).\end{eqnarray}$$

In this case, the transformed vectors, $\boldsymbol{J}_{1},\boldsymbol{B}_{1}$ , and $\boldsymbol{N}_{1}\equiv \boldsymbol{J}_{1}\times \boldsymbol{B}_{1}$ , form a basis in $\mathbb{R}^{3}$ , and thus, for constant $\unicode[STIX]{x1D71A}_{1}$ and $\unicode[STIX]{x1D70E}_{d_{1}}$ , sufficient conditions for the linear stability of the transformed equilibria, satisfying (5.1), (5.3) and (5.10), analogous to (4.6) and (4.7) can be derived

(5.11) $$\begin{eqnarray}1-\unicode[STIX]{x1D70E}_{d_{1}}-\unicode[STIX]{x1D706}_{1}^{2}\geqslant 0,\end{eqnarray}$$

(5.12) $$\begin{eqnarray}{\mathcal{A}}_{1}\geqslant 0,\end{eqnarray}$$

where

(5.13) $$\begin{eqnarray}\displaystyle {\mathcal{A}}_{1} & = & \displaystyle -(1-\unicode[STIX]{x1D70E}_{d_{1}}-\unicode[STIX]{x1D706}_{1}^{2})\{|\unicode[STIX]{x1D707}_{0}\boldsymbol{J}_{1}\times \unicode[STIX]{x1D735}\unicode[STIX]{x1D713}|^{2}+(\unicode[STIX]{x1D707}_{0}\boldsymbol{J}_{1}\times \unicode[STIX]{x1D735}\unicode[STIX]{x1D713})\boldsymbol{\cdot }[(\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{B}_{1}]\}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{\unicode[STIX]{x1D707}_{0}}{2}[\ln (1-\unicode[STIX]{x1D70E}_{d_{1}}-\unicode[STIX]{x1D706}_{1}^{2})]^{\prime }|\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}|^{2}\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\left({\mathcal{P}}_{\bot _{1}}+\frac{\boldsymbol{B}_{1}^{2}}{2\unicode[STIX]{x1D707}_{0}}\right).\end{eqnarray}$$

In order to investigate the stability of the aforementioned transformed equilibria, we presume that the original equilibrium has constant density and anisotropy functions ( $\unicode[STIX]{x1D71A}=\text{const.}$ , $\unicode[STIX]{x1D70E}_{d}=\text{const.}$ ) and is stable under small three-dimensional perturbations; therefore, conditions (4.6) and (4.7) are satisfied in ${\mathcal{D}}$ . Note that in this case the scalar functions of transformation (5.1) must have a structure of the form: $a_{1}=\text{const.}$ , $n_{1}=\text{const.}$ , $b_{1}=b_{1}(\unicode[STIX]{x1D713})$ , $c_{1}=c_{1}(\unicode[STIX]{x1D713})$ ; as a result, breaking potential geometrical symmetry of the original equilibrium is not possible by the transformation even for purely poloidal $\boldsymbol{B}$ and $\boldsymbol{V}$ fields. In Ilin & Vladimirov (Reference Ilin and Vladimirov2004) the stability of respective isotropic equilibria was examined; in particular, it was stated therein that all equilibrium families resulting from the application of the respective isotropic transformations introduced by Bogoyavlenskij to given equilibria of field-aligned incompressible flows, with isotropic pressure and constant mass density, are linearly stable, if either the original equilibria is stable and $C>0$ , or the original equilibria is unstable and $C<0$ . A straightforward calculation shows that

(5.14) $$\begin{eqnarray}{\mathcal{A}}_{1}=C{\mathcal{A}},\end{eqnarray}$$

so that conditions (5.11) and (5.12) assume the form

(5.15) $$\begin{eqnarray}\displaystyle & \displaystyle C\left(\frac{n_{1}}{b_{1}}\right)^{2}(1-\unicode[STIX]{x1D70E}_{d}-\unicode[STIX]{x1D706}^{2})\geqslant 0, & \displaystyle\end{eqnarray}$$

(5.16) $$\begin{eqnarray}\displaystyle & \displaystyle C{\mathcal{A}}\geqslant 0. & \displaystyle\end{eqnarray}$$

By inspection of the later relations, we come to the conclusion that the transformed equilibrium is linearly stable if, either (i) the original one is linearly stable and $C>0$ , or (ii) neither of the conditions (4.6), (4.7) are satisfied and $C<0$ . However, when $C$ is negative equation (5.6) in the absence of flow $(\unicode[STIX]{x1D706}=0)$ yields the physically unacceptable relation

(5.17) $$\begin{eqnarray}\frac{{\mathcal{P}}_{1}}{{\mathcal{P}}}<0.\end{eqnarray}$$

Thus, we finally conclude with the formulation of the following statement:

The infinite class of equilibria, obtained from the application of the symmetry transformations (5.1) for the case  $\unicode[STIX]{x1D71A}_{1}=\text{const.}$  and  $\unicode[STIX]{x1D70E}_{d_{1}}=\text{const.}$ to given respective equilibria which are linearly stable (by satisfying the sufficient conditions (4.6)–(4.7)), are also linearly stable if the transformation constant $C$ is positive definite.

This statement corrects and generalises the respective statement in Ilin & Vladimirov (Reference Ilin and Vladimirov2004) for isotropic equilibria $(\unicode[STIX]{x1D70E}_{d}=\unicode[STIX]{x1D70E}_{d_{1}}=0,n_{1}=1)$ .

6 Linear stability of helically symmetric equilibria

In this section we consider helically symmetric equilibria with field-aligned incompressible flows and the anisotropic pressure derived in Evangelias et al. (Reference Evangelias, Kuiroukidis and Throumoulopoulos2018), for which the following relations hold:

(6.1) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \boldsymbol{B}=I(U)\boldsymbol{h}+(1-\unicode[STIX]{x1D70E}_{d}-M_{p}(U)^{2})^{-1/2}\boldsymbol{h}\times \unicode[STIX]{x1D735}U(r,u),\quad \boldsymbol{V}=\frac{M_{p}(U)}{\unicode[STIX]{x1D707}_{0}\unicode[STIX]{x1D71A}}\boldsymbol{B},\\ \displaystyle \unicode[STIX]{x1D71A}=\text{const.},\quad \unicode[STIX]{x1D70E}_{d}=\text{const.},\quad {\mathcal{P}}={\mathcal{P}}_{s}(U)-M_{p}^{2}(U)\frac{\boldsymbol{B}^{2}}{2\unicode[STIX]{x1D707}_{0}}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

Here, the function $U(r,u)$ labels the magnetic surfaces, where $(r,u,z)$ are helical coordinates defined through the usual cylindrical ones $(\unicode[STIX]{x1D70C},\unicode[STIX]{x1D719},\unicode[STIX]{x1D701})$ as $r=\unicode[STIX]{x1D70C}$ , $u=z-\unicode[STIX]{x1D702}\unicode[STIX]{x1D719}$ , $z=\unicode[STIX]{x1D701}$ ; $I$ relates to the helicoidal magnetic field; ${\mathcal{P}}_{s}$ is the static effective pressure; $M_{p}$ is the Alfvén Mach function for the poloidal field, defined as $M_{p}:=(\sqrt{\unicode[STIX]{x1D707}_{0}\unicode[STIX]{x1D71A}}|\boldsymbol{V}_{pol}|)/\boldsymbol{B}_{pol}$ , which for parallel flows equals to the total Mach function $(M=\unicode[STIX]{x1D706}=\sqrt{\unicode[STIX]{x1D707}_{0}\unicode[STIX]{x1D71A}}|\boldsymbol{V}|/\boldsymbol{B})$ . All these functions are uniform on the magnetic surfaces. In addition, the vector $\boldsymbol{h}:=(-\unicode[STIX]{x1D702}/(k^{2}r^{2}+\unicode[STIX]{x1D702}^{2}))\boldsymbol{g}_{z}$ points along the helical direction, where $\unicode[STIX]{x1D702}$ is an arbitrary constant, and $\boldsymbol{g}_{i}=\unicode[STIX]{x2202}\boldsymbol{r}/\unicode[STIX]{x2202}i~(i=r,u,z)$ are the covariant helical basis vectors. According to Evangelias et al. (Reference Evangelias, Kuiroukidis and Throumoulopoulos2018), equilibria (6.1) are governed by the following generalised GS equation

(6.2) $$\begin{eqnarray}{\mathcal{L}}U+\frac{1}{2}\frac{\text{d}}{\text{d}U}[(1-\unicode[STIX]{x1D70E}_{d}-M_{p}^{2})I^{2}]+\unicode[STIX]{x1D707}_{0}(k^{2}r^{2}+\unicode[STIX]{x1D702}^{2})\frac{\text{d}{\mathcal{P}}_{s}}{\text{d}U}+\frac{2\unicode[STIX]{x1D702}}{k^{2}r^{2}+\unicode[STIX]{x1D702}^{2}}(1-\unicode[STIX]{x1D70E}_{d}-M_{p}^{2})^{1/2}I=0,\end{eqnarray}$$

where the elliptic operator involved is defined as ${\mathcal{L}}:=(r^{2}+\unicode[STIX]{x1D702}^{2})[\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}/(r^{2}+\unicode[STIX]{x1D702}^{2}))]$ . In fact, equation (6.2) constitutes a reduced form of the more general one (Evangelias et al. (Reference Evangelias, Kuiroukidis and Throumoulopoulos2018), equation (30)) valid in general for non-collinear $\boldsymbol{V}$ and $\boldsymbol{B}$ , with the replacement of the parameters $m$ and $k$ therein by $-\unicode[STIX]{x1D702}$ and $-1$ , respectively, here. Under the following linearising ansatz for the free function terms,

(6.3a,b ) $$\begin{eqnarray}(1-\unicode[STIX]{x1D70E}_{d}-M_{p}^{2})^{1/2}I=w_{1}U,\quad {\mathcal{P}}_{s}=w_{2}-2w_{3}^{2}\frac{U^{2}}{\unicode[STIX]{x1D707}_{0}},\end{eqnarray}$$

the GS equation (6.2) reduces to the form

(6.4) $$\begin{eqnarray}\frac{1}{r^{2}}\frac{\unicode[STIX]{x2202}^{2}U}{\unicode[STIX]{x2202}u^{2}}+\frac{1}{r}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}\left(\frac{r}{r^{2}+\unicode[STIX]{x1D702}^{2}}\frac{\unicode[STIX]{x2202}U}{\unicode[STIX]{x2202}r}\right)+\left(\frac{w_{1}^{2}}{r^{2}+\unicode[STIX]{x1D702}^{2}}+\frac{2w_{1}\unicode[STIX]{x1D702}}{(r^{2}+\unicode[STIX]{x1D702}^{2})^{2}}-4w_{3}^{2}\right)U=0,\end{eqnarray}$$

where $w_{1},w_{2}$ and $w_{3}$ are arbitrary constants. Partial differential equation (PDE) (6.4) was solved analytically in Bogoyavlenskij (Reference Bogoyavlenskij2000) and the exact solution obtained therein is

(6.5) $$\begin{eqnarray}U_{Nql}=\text{e}^{-w_{3}r^{2}}\{f_{N}R_{0N}(s)+r^{q}R_{ql}(s)[c_{ql}\cos (qu/\unicode[STIX]{x1D702})+d_{ql}\sin (qu/\unicode[STIX]{x1D702})]\},\end{eqnarray}$$

where $N,q,l$ are arbitrary integers ${\geqslant}0$ satisfying the condition $2N>2l+q$ ; $f_{N},c_{ql},d_{ql}$ are arbitrary coefficients, and $s=2w_{3}r^{2}$ ; the form of the polynomial functions $R_{ql}(s)$ involves derivatives of the Laguerre polynomials $L_{q+l}(s)$ (see Bogoyavlenskij (Reference Bogoyavlenskij2000), equations (3.7)–(3.8)), and $R_{0N}$ is the respective polynomial for $q=0$ , $l=N\neq 0$ . On account of solution (6.5), different classes of exact helically symmetric MHD equilibria describing astrophysical jets with isotropic pressure $(\unicode[STIX]{x1D70E}_{d}=0)$ were constructed in Bogoyavlenskij (Reference Bogoyavlenskij2000). For such equilibria, the magnetic field, the flow velocity and the current density fall rapidly to zero at $r\rightarrow \infty$ , while the pertinent isotropic pressure takes a limiting constant value therein. We have to note that solution (6.5) can also accurately describe helically symmetric CGL anisotropic pressure equilibria, with $\unicode[STIX]{x1D70E}_{d}$ being a surface quantity (or constant), and incompressible flow. This is due to the fact that although the MHD and CGL models are established through different physical assumptions for the particle collisions, the form of the generalised GS equations governing them are identical, such that passing from the one equilibrium equation to the other is mathematically convenient.

Figure 1. Poloidal cut of helicoidal magnetic surfaces $U(x,y,z=0)=\text{const}$ . for the helically symmetric equilibrium solution (6.6).

Figure 2. Profile of the function $U(x,y=0,z=0)=\text{const.}$ for the helically symmetric equilibrium solution (6.6).

Figure 3. The profiles of the scalar pressures (a) $P_{\bot }(x,y=0,z=0)$ and (b) $P_{\Vert }(x,y=0,z=0)$ for the constructed stationary equilibria, for $M_{p_{0}}^{2}=10^{-3}$ . The blue dashed curve corresponds to $\unicode[STIX]{x1D70E}_{d}=0.1$ , while the red dotted one to $\unicode[STIX]{x1D70E}_{d}=-0.1$ .

In order to construct specific equilibria, we restrict our analysis to solution (6.5) for $N=2,q=1,l=0$ and $w_{1}=7/(6\unicode[STIX]{x1D702})$ , which assumes the simpler form (see Bogoyavlenskij (Reference Bogoyavlenskij2000), equation (4.2))

(6.6) $$\begin{eqnarray}U(r,u)=\text{e}^{-w_{3}r^{2}}[1-10w_{3}r^{2}+8w_{3}^{2}r^{4}+c_{10}\cos (u/\unicode[STIX]{x1D702})].\end{eqnarray}$$

Solution (6.6) of the GS equation (6.2), associated with the relations (6.1) and the profiles (6.3), defines a special class of exact helically symmetric equilibria with incompressible flows with constant density and anisotropy functions, valid for any functional dependence of $M_{p}^{2}(U)$ . Therefore, to completely determine an equilibrium, we employ the following profile for the Mach function:

(6.7) $$\begin{eqnarray}M_{p}^{2}(U)=M_{p_{0}}^{2}U^{2},\end{eqnarray}$$

where $M_{p_{0}}$ is an arbitrary parameter. The magnetic surfaces of the above constructed equilibria on the Cartesian plane $z=0$ , for $w_{3}=0.01,c_{10}=0.5,w_{2}=2$ and $\unicode[STIX]{x1D702}=2.318$ , are shown in figure 1; all dimensional quantities present in this section are measured in appropriate SI units. Also, the profile of the flux function $U$ along the $x$ -axis is given in figure 2. In figures 1 and 2 it can be seen that the plasma domain consists of two sub-domains: an outer one consisting of magnetic surfaces with circular poloidal cross-sections extending up to infinity ( $r\rightarrow \infty$ ), and an inner sub-domain containing three lobes and a couple of saddle points (X-points). The inner X-point, corresponding to a maximum of $U$ , is located at $x=-17.6$ . Then on the right-hand side of this first X-point are located, successively, two lobes, then the second X-point and farther outwards the third lob. The respective magnetic axes of the lobes are located at $x=-7.07$ , $x=2.31$ and $x=15.3$ while the second X-point is located at $x=7.07$ . Each helix composing such a helically symmetric configuration is characterised by a pitch length, equal to $2\unicode[STIX]{x03C0}\unicode[STIX]{x1D702}$ , and a torsion given by

(6.8) $$\begin{eqnarray}\unicode[STIX]{x1D70F}=\frac{\unicode[STIX]{x1D702}}{r^{2}+\unicode[STIX]{x1D702}^{2}}.\end{eqnarray}$$

The above equilibrium can model helically symmetric jets with anisotropic pressure, tending to become isotropic at very long distances ( $r\rightarrow \infty$ ). In more detail, for any $\unicode[STIX]{x1D70E}_{d}=\text{const.}\neq 0$ inside ${\mathcal{D}}$ , the scalar pressures parallel and perpendicular to $\boldsymbol{B}$ are given by the relations

(6.9a,b ) $$\begin{eqnarray}P_{\bot }={\mathcal{P}}-\unicode[STIX]{x1D70E}_{d}\frac{\boldsymbol{B}^{2}}{2\unicode[STIX]{x1D707}_{0}},\quad P_{\Vert }={\mathcal{P}}+\unicode[STIX]{x1D70E}_{d}\frac{\boldsymbol{B}^{2}}{2\unicode[STIX]{x1D707}_{0}},\end{eqnarray}$$

indicating that when $\unicode[STIX]{x1D70E}_{d}>0$ , its increase results in an enhancement of $P_{\bot }$ while $P_{\Vert }$ decreases, and vice versa for $\unicode[STIX]{x1D70E}_{d}<0$ , as can be seen at the profiles of $P_{\Vert }$ and $P_{\bot }$ shown in figure 3. In the limit of $r\rightarrow \infty$ the magnetic field, current density and velocity vanish and therefore the scalar pressures become equal to each other, i.e. $P_{\bot }=P_{\Vert }=w_{2}=\text{const.}$ because of the second of (6.3) and (6.9). Thus, the configuration becomes, in this limit, isotropic. Note that this is compatible with a non-zero value of $\unicode[STIX]{x1D70E}_{d}$ on account of the definition (2.3), which makes $\unicode[STIX]{x1D70E}_{d}$ indefinite in the limit of $r\rightarrow \infty$ . Profiles of the magnetic field magnitude, $B$ , and the helicoidal component of the current density, $J_{h}$ , are shown in figure 4. The values of $B$ become greater (lower) for $\unicode[STIX]{x1D70E}_{d}>0$ ( $\unicode[STIX]{x1D70E}_{d}<0$ ), in connection with a diamagnetic (paramagnetic) behaviour. The helicoidal current density, $J_{h}$ , reverses near the origin and becomes more peaked (hollow) for $\unicode[STIX]{x1D70E}_{d}>0$ ( $\unicode[STIX]{x1D70E}_{d}<0$ ). In addition, profiles of the helicoidal velocity component, $V_{h}$ and the Mach function, $M_{p}^{2}$ , are provided in figure 5. It is noted that $V_{h}$ reverses in the region of the left lobe, where $U$ becomes negative, and that the flow in terms of the Mach function, $M_{p}^{2}$ , strengthens the impact of pressure anisotropy for $\unicode[STIX]{x1D70E}_{d}>0$ , due to the factor $1-\unicode[STIX]{x1D70E}_{d}-M_{p}^{2}$ (cf. (6.2)). In this respect, it is expected that the increase of the parameter $M_{p_{0}}^{2}$ has the same impact on $V_{h}$ as that shown in figure 5(a) for $\unicode[STIX]{x1D70E}_{d}>0$ .

Figure 4. Variation of (a) $|\boldsymbol{B}(x,y=0,z=0)|$ and (b) $J_{h}(x,y=0,z=0)$ , for the stationary equilibria constructed here, for $M_{p_{0}}^{2}=10^{-3}$ and the impact of pressure anisotropy on them for positive and negative values of $\unicode[STIX]{x1D70E}_{d}$ .

Figure 5. Profiles of (a) the helicoidal velocity component for $M_{p_{0}}^{2}=0.06$ , and the impact of pressure anisotropy through $\unicode[STIX]{x1D70E}_{d}$ , and (b) the Mach function along the $x$ -axis for different values of the flow parameter $M_{p_{0}}^{2}$ .

In what follows, we employ the derived suffiecient condition by calculating the quantity ${\mathcal{A}}=A_{1}+A_{2}+A_{3}$ of (4.13) (in connection with (4.5), (4.8)–(4.10) in the $\unicode[STIX]{x1D713}$ -space) for the helically symmetric equilibria, defined by the relations (6.1), (6.3) and (6.6)–(6.9), in order to examine the impact of the pressure anisotropy, flow and torsion on their linear stability, through the variation of the parameters $\unicode[STIX]{x1D70E}_{d},M_{p_{0}}$ and $\unicode[STIX]{x1D702}$ , respectively. We recall that the aforementioned relations were obtained by applying the integral transformation (4.12) (see Evangelias et al. Reference Evangelias, Kuiroukidis and Throumoulopoulos2018); in this respect the condition (4.6) is trivially satisfied. Figure 6(a) shows that the condition ${\mathcal{A}}\geqslant 0$ in the absence of pressure anisotropy and flow, $\unicode[STIX]{x1D70E}_{d}=M_{p_{0}}^{2}=0$ , is satisfied in a broad region including the outer domain and the two magnetic axes located on the left and right sides of the origin $(x=y=0)$ . In these regions, the term $A_{2}$ , being stabilising, surpasses the destabilising term $A_{1}$ , as shown in figure 6(b) (while in this case the flow term $A_{3}$ vanishes). However, the condition is satisfied neither near the central magnetic axis, where $J_{h}$ reverses and $U$ is negative, nor in a small region located on the left side of the magnetic axis of the left lobe, in which $V_{h}$ reverses; in this respect it should be noted that, since the condition is only sufficient, the white coloured regions in figure 6(a) and in the figures to follow, where ${\mathcal{A}}<0$ , do not necessarily imply instability. Thus, we will consider only regions in which the condition ${\mathcal{A}}\geqslant 0$ is satisfied.

Figure 6. (a) For the static anisotropic helically symmetric equilibrium ( $\unicode[STIX]{x1D70E}_{d}=M_{p_{0}}^{2}=0$ ) the stability condition ${\mathcal{A}}\geqslant 0$ is satisfied in the orange coloured regions. (b) The term $A_{2}$ has a stabilising effect (red-dotted curve) which counteracts the destabilising one of $A_{1}$ (blue-dashed curve), so that the quantity ${\mathcal{A}}=A_{1}+A_{2}$ , indicated by the black-straight curve, becomes positive in the aforementioned orange coloured regions.

The presence of pressure anisotropy does not affect the isotropic stability map of figure 6(a), as is clearly indicated in the profiles of figure 7. However, it affects the values of  ${\mathcal{A}}$ . Specifically, in the regions where ${\mathcal{A}}\geqslant 0$ , for $P_{\Vert }>P_{\bot }$ ( $\unicode[STIX]{x1D70E}_{d}>0$ ) the anisotropy has a stabilising impact, in the sense that the maximum values of ${\mathcal{A}}$ become larger than the respective isotropic ones, and a destabilising effect for $\unicode[STIX]{x1D70E}_{d}<0$ occurs. These characteristics are illustrated in figure 7. In addition, it is found that the flow in terms of $M_{p}^{2}$ has a peculiar effect on stability. Specifically, on the one hand, it results in shrinking of the orange coloured area located on the left-hand side of the first lobe, where the helicoidal velocity reverses, as can be seen in figure 8(a,b). This shrinking is connected with a destabilising effect of $M_{p_{0}}^{2}$ on both terms $A_{1}$ and $A_{2}$ , as shown in figure 9(a,b). On the other hand, the flow has a stabilising effect, similar to that of $\unicode[STIX]{x1D70E}_{d}>0$ , because the respective maximum values of ${\mathcal{A}}$ get larger in this area as $M_{p}^{2}$ increases, as can be seen in figure 8(c). Also, the flow gives rise to a stabilising contribution via the term $A_{3}$ . However, this contribution in the region of interest is an order of magnitude lower than the destabilising impact of $M_{p}^{2}$ on $A_{1}$ and $A_{2}$ , as can be seen in figure 9(c). Because of the stronger impact of the pressure anisotropy on ${\mathcal{A}}$ than the destabilising effect of the flow, the presence of both anisotropy and flow has an overall stabilising effect in terms of the region where the condition ${\mathcal{A}}\geqslant 0$ is satisfied and the maximum values of ${\mathcal{A}}$ . This is shown in figure 10.

Figure 7. The impact of pressure anisotropy on the quantities (a) ${\mathcal{A}}$ , (b) $A_{1}$ and (c) $A_{2}$ in the absence of flow ( $M_{p_{0}}=0$ ) for $\unicode[STIX]{x1D70E}_{d}=0.2$ (blue-dashed curves) and $\unicode[STIX]{x1D70E}_{d}=-0.2$ (red-dotted curve). For comparison, also given are the respective isotropic black continuous curves. In the regions where ${\mathcal{A}}\geqslant 0$ , this impact is stabilising for $\unicode[STIX]{x1D70E}_{d}>0$ and destabilising for $\unicode[STIX]{x1D70E}_{d}<0$ .

Figure 8. Impact of the flow through $M_{p_{0}}^{2}$ in the central orange coloured region where the stability condition ${\mathcal{A}}\geqslant 0$ is satisfied in comparison with the respective static isotropic equilibrium. The maximum used value of the parameter $M_{p_{0}}$ , for which all the pressures remain positive, is 0.08.

Figure 9. The impact of the flow parameter $M_{p_{0}}^{2}$ on the terms (a) $A_{1}$ , (b) $A_{2}$ and (c)  $A_{3}$ , for $\unicode[STIX]{x1D70E}_{d}=0$ .

Figure 10. The overall stabilising impact of pressure anisotropy in combination with flow on the stability of the constructed helically symmetric equilibria. The pertinent parametric values employed are as follows: (straight black curve) $\unicode[STIX]{x1D70E}_{d}=M_{p_{0}}^{2}=0$ , (blue dashed curve) $\unicode[STIX]{x1D70E}_{d}=0$ , $M_{p_{0}}^{2}=0.07$ , (red dotted curve) $\unicode[STIX]{x1D70E}_{d}=0.07$ , $M_{p_{0}}^{2}=0$ and (green dot-dashed curve) $\unicode[STIX]{x1D70E}_{d}=M_{p_{0}}^{2}=0.07$ .

Figure 11. The impact of the torsion of the magnetic axis and of the pitch on the linear stability of helically symmetric equilibria. Each pair of figures illustrates configurations that have same torsion, but different pitch values, as explained in the text.

Finally, as concerns the impact of the torsion, $\unicode[STIX]{x1D70F}$ , on the quantity ${\mathcal{A}}$ , for a specific helix, defined by the equations $r=r_{c}=\text{const.}$ , $u=u_{c}=\text{const.}$ (in helical coordinates), equation (6.8) implies that $\unicode[STIX]{x1D70F}$ depends only on the parameter $\unicode[STIX]{x1D702}$ , which characterises the pitch of that helix. Inspection of (6.8) implies that $\unicode[STIX]{x1D70F}$ has an extremum for $\unicode[STIX]{x1D702}=r_{c}$ , corresponding to the maximum torsion, $\unicode[STIX]{x1D70F}_{\max }=1/2r_{c}$ . For example, for the static, isotropic equilibrium of figure 1 the helical magnetic axis of the central lobe intersects the plane $z=0$ at the position $x_{c}=2.318,y_{c}=0$ , corresponding to $r_{c}=2.318$ , and has maximum torsion, $\unicode[STIX]{x1D70F}_{\max }=0.2157$ . Therefore, there can exist two different helically symmetric configurations with the same torsion but different pitches, one for $\unicode[STIX]{x1D702}<r_{c}$ and the other $\unicode[STIX]{x1D702}>r_{c}$ . However, the same torsion does not imply that these configurations have necessarily the same stability properties. The region in which the condition ${\mathcal{A}}\geqslant 0$ is satisfied for the equilibrium of figure 1 is shown in figure 6. We found the respective stability maps for the three pairs of equilibria, shown in figure 11. Each pair corresponds to the same torsion, $\unicode[STIX]{x1D70F}<\unicode[STIX]{x1D70F}_{\max }$ but different pitch lengths, $\unicode[STIX]{x1D702}<r_{c}$ and $\unicode[STIX]{x1D702}>r_{c}$ . Specifically, the torsion and pitch values we employed in connection with these equilibrium pairs are the following: (upper pair consisting of figure 11 a,b) $\unicode[STIX]{x1D70F}_{(a),(b)}=0.207,\unicode[STIX]{x1D702}_{(a)}=1.75,\unicode[STIX]{x1D702}_{(b)}=3.07$ , (middle pair consisting of the figure 11 c,d) $\unicode[STIX]{x1D70F}_{(c),(d)}=0.157,\unicode[STIX]{x1D702}_{(c)}=1,\unicode[STIX]{x1D702}_{(d)}=5.37$ and (lower pair consisting of the figure 11 e,f) $\unicode[STIX]{x1D70F}_{(e),(f)}=0.089,\unicode[STIX]{x1D702}_{(e)}=0.5,\unicode[STIX]{x1D702}_{(f)}=10.75$ . The stability maps indicate that the condition ${\mathcal{A}}\geqslant 0$ is satisfied in a wider region as the torsion decreases from its maximum value and for a given torsion ${\mathcal{A}}$ it gets larger as the pitch length $\unicode[STIX]{x1D702}>r_{c}$ increases. Thus, we conclude that helical configurations with smaller torsion and bigger pitch lengths may have improved stability characteristics. This result is reasonable if one considers the limit of zero torsion and infinite pitch length in which a helically symmetric plasma column degenerates into an one-dimensional, cylindrical $\unicode[STIX]{x1D703}$ -pinch. It is well known that such a configuration has favourable stability properties, since the safety factor approaches infinity. However, confinement in a $\unicode[STIX]{x1D703}$ -pinch is not possible in the presence of toroidicity because the axial magnetic field becomes purely toroidal.