2. Model equations

We consider a cylindrical plasma bounded radially and immersed in a homogeneous magnetic field such that $\boldsymbol {B}=B \hat {e}_z$ (see figure 1). The model is based on the continuity and momentum equations for electrons and ions

(2.1)\begin{gather} \frac{\partial n_j}{\partial t}+{\boldsymbol{\nabla}}\boldsymbol{\cdot}(n_j\boldsymbol{v}_j)=0, \end{gather}

(2.2)\begin{gather}n_im_i\left(\frac{\partial\boldsymbol{v}_{\boldsymbol{i}} }{\partial t}+ \boldsymbol{v}_{\boldsymbol{i}} \boldsymbol{\cdot}{\boldsymbol{\nabla}}\boldsymbol{v}_{\boldsymbol{i}} \right)= n_i e(-\boldsymbol{\nabla} \phi+\boldsymbol{v}_{\boldsymbol{i}} \times\boldsymbol{B}) -T_i{\boldsymbol{\nabla}}n_i-m_in_i\nu_{\rm in}\boldsymbol{v}_i, \end{gather}

(2.3)\begin{gather}0={-}en_e(-\boldsymbol{\nabla} \phi+\boldsymbol{v}_e\times\boldsymbol{B})-{\boldsymbol{\nabla}}(n_eT_e), \end{gather}

with

(2.4)\begin{equation} \boldsymbol{\nabla}=\frac{\partial}{\partial r}\hat{e}_r+\frac{1}{r} \frac{\partial}{\partial \theta}\hat{e}_\theta+ \frac{\partial}{\partial z}\hat{e}_z,\end{equation}

where $\hat {e}_r$, $\hat {e}_\theta$ and $\hat {e}_z$ are the unit vectors in $r$, $\theta$ and $z$ directions, respectively. Here, $j=i,e$ denotes either ions or electrons, $n_{j}$ is the number density, $\boldsymbol {v}_{j}$ is the velocity of the species, $m_j$ is the mass of the species, $T_{i,e}$ is the species temperature, $\phi$ is the electric potential and $\nu _{\rm in}$ is the ion–neutral collision frequency. The following assumptions are used:

• • Electrostatic approximation, ${\partial \boldsymbol {B}}/{\partial t}=0$.

• • Quasi-neutrality, $n_i=n_e$.

• • No particle source.

• • No variations in the axial direction $(k_{\|}=0)$.

• • Radially uniform ion temperature i.e. $T_i\approx 0.1$ eV.

• • No Gyroviscosity ($\boldsymbol {\nabla } \boldsymbol {\cdot }{\rm \pi} _i=0$, $\boldsymbol {\nabla } \boldsymbol {\cdot }{\rm \pi} _e=0$).

• • Electron inertia neglected as a consequence of the small mass of electrons i.e. $m_e/m_i\ll 1$.

• • Electron collisions with ions and neutrals neglected. This is, for instance, relevant for the regimes met in MISTRAL (Annaratone et al. Reference Annaratone, Escarguel, Lefevre, Rebont, Claire and Doveil2011) where $\nu _{ei},\nu _{en}/\omega _{ce}\ll 1$.

The plasma density, flow and electric potential are written as the sum of a time-independent equilibrium part denoted by subscript 0 and a fluctuating part denoted by superscript $\sim$ as $n=n_0+\tilde {n}$, $\boldsymbol {v}=\boldsymbol {v}_0+\tilde {\boldsymbol {v}}$ and $\phi =\phi _0+\tilde {\phi }$, where the fluctuating part has the following form:

(2.5)\begin{equation} \left.\begin{array}{c@{}} \tilde{n}= n_1(r) \exp[{\rm i}(m\theta-\omega t)]\\ \tilde{\boldsymbol{v}}=\boldsymbol{v}_1(r) \exp[{\rm i}(m\theta-\omega t)]\\ \tilde{\phi}= \phi_1(r) \exp[{\rm i}(m\theta-\omega t)] \end{array}\right\}. \end{equation}

Here, $n_0$ is the equilibrium density of ions or electrons, $\phi _0$ is the equilibrium electric potential and $\boldsymbol {v_0}$ is the equilibrium flow. For the fluctuating part, $n_1$ and $\phi _1$ give the perturbation amplitudes of the density and potential, respectively, $\boldsymbol {v}_1(r)=v_{r_1}\hat {e}_r+v_{\theta _1}\hat {e}_\theta$, $v_{r1}$ and $v_{\theta 1}$ are the radial and azimuthal components of the perturbed velocity, respectively, $m$ is the azimuthal mode number and ${\omega }={\omega }_r+i{\gamma }$, where ${\omega }_{r}$ is the mode frequency and ${\gamma }$ is the growth rate.

Figure 1. Cylindrical coordinate system and direction of rotation for ion-cyclotron frequency $\omega _\textrm {ci}$, positive $E\times B$ frequency ($\omega _{E0}>0$) and positive diamagnetic frequency ($\omega _{*0}>0$).

The equilibrium density ($n_0$) and plasma potential $(\phi _0)$ are assumed to have Gaussian and parabolic profiles, respectively. This is compatible with typical profiles measured in MISTRAL (see the Appendix)

(2.6a,b)\begin{equation} n_0(r)=n_{00} \exp\left(-\frac{r^2}{r_0^2}\right); \quad \phi_0=p_1r^2+p_2,\end{equation}

where $n_{00}$, $p_1$ and $p_2$ are constants. Here, $r$ is the radial coordinate and $r_0$ is the width of the Gaussian used to parametrize the density profile; $r_0$ characterizes how fast the plasma density decays to zero when moving radially outward. These equilibrium profiles are consistent with the rigid body rotation assumption used for the equilibrium, see § 3.

3. Equilibrium flow

In this section, we derive the expression for the ion equilibrium flow as a function of the $E\times B$ flow and the diamagnetic flow under the assumption of rigid body rotation. The equilibrium flow velocity for ions is first split into radial and azimuthal components

(3.1)\begin{equation} \boldsymbol{v}_{\boldsymbol{i0}}(r)=v_{{ir}_0}\hat{e}_r + v_{i\theta_0} \hat{e}_{\theta},\end{equation}

and rigid body rotation is assumed such that $v_{i\theta _0}=r\omega _0$ with $\omega _0'=\omega _0''=0$, where $'$ represents ${\partial }/{\partial r}$ and $''$ represents ${\partial ^2}/{\partial r^2}$. The ion inertial term $\boldsymbol {v}_{\boldsymbol {i0}}\boldsymbol {\cdot }{\boldsymbol {\nabla }} \boldsymbol {v}_{\boldsymbol {i0}}$ entering (2.2) becomes

(3.2)\begin{equation} (\boldsymbol{v}_{i0}\boldsymbol{\cdot}{{\boldsymbol{\nabla}}}) \boldsymbol{v}_{i0}=\left(v_{{ir}_0}\frac{\partial v_{{ir}_0}}{\partial r} -r\omega_0^2\right)\hat{e}_{r}+2\omega_0 v_{{ir}_0}\hat{e}_{\theta}.\end{equation}

Substituting equations (3.1) and (3.2) into the ion momentum equation ((2.2)), assuming ${\partial \boldsymbol {v}_{i0}}/{\partial t}=0$, taking the cross-product with $\boldsymbol {B}$ and then projecting along $\hat {e}_r$, one gets

(3.3)\begin{equation} v_{{ir}_0}=\frac{-\nu_{\rm in}r\omega_0}{\omega_{\rm ci}+2\omega_0}.\end{equation}

The equation above is normalized by dividing $v_{{ir}_0}$ with $v_{\textrm {th}_i}=\sqrt {{T_i}/{m_i}}$; $r$ with $\rho _i={mv_{\textrm {th}_i}}/{eB}$; $\nu _\textrm {in}$, $\omega _{0}$ with $\omega _\textrm {ci}={eB}/{m_i}$ where the normalized quantities are noted with an overbar, see table 1,

(3.4)\begin{equation} \bar{v}_{{ir}_0}=\frac{-\bar{\nu}_{\rm in} \bar{r}\bar{\omega}_0}{1+2\bar{\omega}_0},\end{equation}

which is the equilibrium flow of ions in the radial direction.

Table 1. Normalized parameters and their definitions. Here, $T_{{e0}_\textrm {ref}}$ is the reference value of the electron temperature.

Using (3.3) in (2.2) and projecting along $\hat {e}_\theta$, the azimuthal flow frequency $\omega _0=v_{i\theta _0}/r$ is given by

(3.5)\begin{equation} \left(\frac{\nu_{\rm in}\omega_0}{2\omega_0+ \omega_{\rm ci}}\right)^2-\omega_0^2={-}\omega_{\rm ci}\omega_{E0}+\omega_{\rm ci} \omega_0-\omega_{\rm ci}\omega_{*0} +\left(\frac{\nu_{\rm in}^2\omega_0}{2\omega_0+\omega_{\rm ci}}\right).\end{equation}

Here, $\omega _{E0}$ is the $E\times B$ drift frequency

(3.6)\begin{equation} \omega_{E0}=\frac{\boldsymbol{B}\times \boldsymbol{\nabla} \phi_0}{rB^2}\boldsymbol{\cdot}\hat{e}_\theta=\frac{\phi_0'}{rB}, \end{equation}

and $\omega _{*0}$ is the ion diamagnetic drift frequency

(3.7)\begin{align} \omega_{*0}& =\frac{T_i}{en_0B}\frac{\boldsymbol{B}\times \boldsymbol{\nabla} n_0}{rB}\boldsymbol{\cdot}\hat{e}_\theta=\frac{T_i}{erB}\frac{n_0'}{n_0} \end{align}

(3.8)\begin{align} & ={-}\frac{T_i}{erB}\frac{1}{L_n}, \end{align}

where ${1}/{L_n}=-{n_0'}/{n_0}={2r}/{r_0^2}$ is the logarithmic density gradient. It should be noted that $\omega _{E0}$ and $\omega _{*0}$ are independent of $r$ because of the choice of $n_0$ and $\phi _0$ given by (2.6a,b).

The rotation direction for positive $E\times B$ and diamagnetic frequency is illustrated in figure 1. Azimuthal flows are counted positive in the direction of increasing $\theta$.

The normalized form of (3.5) is

(3.9)\begin{equation} 4\left(\bar{\omega}_0+\frac{1}{2}\right)^4- (1-\bar{\nu}_{\rm in}^2+4(\bar{\omega}_{E0}+ \bar{\omega}_{*0}))\left(\bar{\omega}_0+\frac{1}{2}\right)^2 -\frac{\bar{\nu}_{\rm in}^2}{4} = 0,\end{equation}

which is a fourth-order polynomial in $\bar {\omega }_0$ whose solutions are given by

(3.10)\begin{equation} \bar{\omega}_0={\pm}\tfrac{1}{2}\sqrt{\tfrac{1}{2} [b+\sqrt{b^2 +4\bar{\nu}_{\rm in}^2}]}-\tfrac{1}{2},\end{equation}

where $b=1+4(\bar {\omega }_{*0}+\bar {\omega }_{E0})-\bar {\nu }_\textrm {in}^2$. Equation (3.9) has four roots. Only two roots will be considered since the other two are imaginary and the equilibrium flow is undefined. Equation (3.10) gives the remaining two roots. The branch for which $\bar {\omega }_0$ increases with increasing $\bar {\omega }_{E0}+\bar {\omega }_{*0}$ is the slow rotation mode and the one that decreases with increasing $\bar {\omega }_{E0}+\bar {\omega }_{*0}$ is the fast rotation mode (Rax et al. Reference Rax, Fruchtman, Gueroult and Fisch2015).

The normalized equilibrium flow $\bar {\omega }_0$ is shown in figure 2 as a function of the sum of the normalized $E\times B$ and diamagnetic flows, $\bar {\omega }_{E0}+\bar {\omega }_{*0}$, for different values of $\bar {\nu }_\textrm {in}$.

Figure 2. Normalized equilibrium flow frequency ($\omega _0/\omega _\textrm {ci}$) as a function of normalized $E \times B$ drift frequency ($\omega _{E0}/\omega _\textrm {ci}$) and ion diamagnetic drift frequency ($\omega _{*0}/\omega _\textrm {ci}$) for different values of normalized ion–neutral collision frequency ($\nu _\textrm {in}/\omega _\textrm {ci}$). The black dashed line presents the stability limit for $\bar {\nu }_\textrm {in}=0$. The red dashed line is the diagonal.

For zero collisionality i.e. $\bar{\nu} _\textrm {in} = 0$, (3.9) reduces to

(3.11)\begin{equation} \bar{\omega}_0^2+\bar{\omega}_0-(\bar{\omega}_{E0}+\bar{\omega}_{*0})=0,\end{equation}

and the equilibrium flow $\bar {\omega }_0$ is given by (Chen Reference Chen1966; Jassby Reference Jassby1972; Gueroult et al. Reference Gueroult, Rax and Fisch2017)

(3.12)\begin{equation} \bar{\omega}_0=\frac{-1\pm\sqrt{1+4(\bar{\omega}_{E0} +\bar{\omega}_{*0})}}{2}.\end{equation}

Equation (3.12) shows that, for the equilibrium to exist at $\bar {\nu }_\textrm {in}=0$, the following condition should be satisfied:

(3.13)\begin{equation} \bar{\omega}_{E0}+\bar{\omega}_{*0}>{-}\tfrac{1}{4}. \end{equation}

For finite collisionality, $\bar {\omega }_{E0}+\bar {\omega }_{*0}>-1/4$ is no longer required for the equilibrium to exist. From figure 2, it is seen that collisions increase the angular frequency of the fast rotation mode and decrease the angular frequency of the slow rotation mode. A more detailed discussion of collisional and non-collisional equilibrium flow can be found in Rax et al. (Reference Rax, Fruchtman, Gueroult and Fisch2015). Turning now to electrons and writing the equilibrium flow velocity as

(3.14)\begin{equation} \boldsymbol{v}_{\boldsymbol{e0}}=v_{{er}_0}\hat{e}_r + v_{{e\theta}_0}\hat{e}_{\theta},\end{equation}

(2.3) is solved directly to get

(3.15a–c)\begin{equation} \bar{v}_{{er}_0}=0 ; \quad \bar{v}_{{e\theta}_0}= \bar{r}\bar{\omega}_{0e}\quad \textrm{and}\quad \bar{\omega}_{0e}=\bar{\omega}_{E0}+\bar{\omega}_{*e},\end{equation}

where $\bar {\omega }_{0e}$ is the electron equilibrium flow frequency and $\bar {\omega }_{*e}$ is the electron diamagnetic drift frequency ${\omega }_{*e}$ normalized to the ion-cyclotron frequency, with

(3.16)\begin{equation} \omega_{*e}={-}\frac{1}{en_0B}\frac{\boldsymbol{B}\times \boldsymbol{\nabla} (n_0T_{e0})}{rB}\boldsymbol{\cdot}\hat{e}_\theta. \end{equation}

After deriving the equilibrium flow, the next section will focus on the linear stability of the plasma. Finite ion–neutral collisions (and ionization sources) result in a finite radial equilibrium flow, which adds many contributions to the dispersion relation. In the following, we will focus on the collisionless case and assume $\bar{\nu} _\textrm {in}=0$.

4. Dispersion relation

To proceed with the derivation of the dispersion relation, we first linearize the model equations and then use the momentum equations, (2.2) and (2.3), to express the ion and electron flow in the continuity equation. The system is closed by invoking quasi-neutrality. From the electron momentum equation, (2.3), the electron flow is written in the customary form

(4.1)\begin{equation} \boldsymbol{v}_e = \frac{\boldsymbol{b}\times\boldsymbol{\nabla} \phi}{B} + \frac{1}{en_e}\frac{\boldsymbol{b}\times\boldsymbol{\nabla} (n_eT_e)}{B}. \end{equation}

When $\boldsymbol {B}$ is homogeneous and straight (linear plasma column), for any function $A$, we have

(4.2)\begin{equation} \boldsymbol{\nabla} \boldsymbol{\cdot} ({\boldsymbol{b}\times\boldsymbol{\nabla} A}) = 0.\end{equation}

Therefore, on multiplying equation (4.1) with $n_e$ and taking the divergence on both sides

(4.3)\begin{equation} \boldsymbol{\nabla}\boldsymbol{\cdot}(n_e\boldsymbol{v}_e)= \frac{\boldsymbol{b}\times\boldsymbol{\nabla} \phi}{B}\boldsymbol{\cdot}\boldsymbol{\nabla} n_e,\end{equation}

which upon linearization yields

(4.4)\begin{align} \boldsymbol{\nabla}\boldsymbol{\cdot}(n_e\boldsymbol{v}_e)|_1 & = \frac{im}{rB} [- \phi_1 n_0' + \phi_0'n_1] \end{align}

(4.5)\begin{align} & ={-}im\frac{\phi_1}{rB} n_0' + im\omega_{E0}n_1. \end{align}

Combining equation (4.5) and the electron continuity equation ((2.1)), one obtains the relationship between perturbed density $(n_1)$ and perturbed potential $(\phi _1)$

(4.6)\begin{equation} \frac{n_1}{n_0}=\frac{m}{rL_n}\frac{1}{\omega- m\omega_{E0}}\frac{\phi_1}{B}.\end{equation}

Normalizing length to ion Larmor radius ($\rho _i$) and frequencies to the ion-cyclotron frequency ($\omega _\textrm {ci}$), the normalized form of the above equation is

(4.7)\begin{equation} \bar{n}_1=\frac{m}{\bar{r}\bar{L}_n} \frac{1}{\bar{\omega}-m\bar{\omega}_{E0}}\tau\bar{\phi}_1,\end{equation}

where $\tau =T_{{e0}_\textrm {ref}}/T_{io}$ with $T_{{e0}_\textrm {ref}}$, the reference value of the electron temperature. It should be noted that the radial variation of the electron temperature is retained here but,since the diamagnetic flux is divergence free ((4.1)–(4.3)), it does not enter the continuity equation. The relation between the perturbed density of electrons and perturbed potential given by (4.7) is therefore identical to that of Chen (Reference Chen1966), Rognlien (Reference Rognlien1973) and Gueroult et al. (Reference Gueroult, Rax and Fisch2017), where the electron diamagnetic flow was neglected.

Turning now to ions, the linearized momentum equation writes

(4.8)\begin{equation} -{\rm i}\omega\boldsymbol{v}_{i1} +(\boldsymbol{v}_{i0}\boldsymbol{\cdot}\boldsymbol{\nabla})\boldsymbol{v}_{i1} + (\boldsymbol{v}_{i1}\boldsymbol{\cdot}\boldsymbol{\nabla})\boldsymbol{v}_{i0} = \frac{e}{m_i}[-\boldsymbol{\nabla} \phi +\boldsymbol{v}_{i1}\times\boldsymbol{B}] -\frac{T_i}{m_i}\boldsymbol{\nabla}\frac{n_1}{n_o}.\end{equation}

For a background rigid body rotation, $\boldsymbol {v}_{i0}=r\omega _0\hat {e}_\theta$, the inertial terms can be written as

(4.9)\begin{equation} (\boldsymbol{v}_{i0}\boldsymbol{\cdot}\boldsymbol{\nabla})\boldsymbol{v}_{i1} = im\omega_0 \boldsymbol{v}_{i1} -\omega_0 \boldsymbol{v}_{i1}\times \boldsymbol{b}, \end{equation}

and

(4.10)\begin{equation} (\boldsymbol{v}_{i1}\boldsymbol{\cdot}\boldsymbol{\nabla})\boldsymbol{v}_{i0} = v_{ir1}\frac{\partial \boldsymbol{v}_{i0}}{\partial r} + \frac{v_{i\theta 1}}{r}\frac{\partial \boldsymbol{v}_{i0}}{\partial \theta}={-}\omega_0 \boldsymbol{v}_{i1}\times \boldsymbol{b}. \end{equation}

When included in the linearized ion momentum equation, (4.8), it yields

(4.11)\begin{equation} -{\rm i}(\omega-m\omega_0)\boldsymbol{v}_{i1} ={-}\frac{\omega_{\rm ci}}{B}\boldsymbol{\nabla} \phi + (\omega_{\rm ci}+2\omega_0)\boldsymbol{v}_{i1}\times \boldsymbol{b}-\frac{\omega_{\rm ci}T_i}{eB}\boldsymbol{\nabla}\frac{n_1}{n_o}.\end{equation}

The background flow enters in the Doppler shifted frequency, $\omega -m\omega _0$, on the left-hand side and in the linearized Coriolis force, $F_\textrm {co}=2m_i\boldsymbol {v}_{i1}\times \omega _0\boldsymbol {b}$, on the right-hand side.

Upon normalization, we get

(4.12)\begin{equation} -{\rm i}(\bar{\omega}-m\bar{\omega}_0)\bar{\boldsymbol{v}}_{i1} = (1+2\bar{\omega}_0)\bar{\boldsymbol{v}}_{i1}\times \boldsymbol{b}-\boldsymbol{\nabla}\varPhi_1. \end{equation}

Writing, $C=1+2\bar {\omega }_0$, the factor by which the Laplace force is modified due to the inertial force, $\bar {\omega }_\textrm {ph}=\bar {\omega }-m\bar {\omega }_0$, and the normalized Doppler shifted frequency and combining the perturbed density and potential terms into $\varPhi _1=\bar {n}_1+\tau \bar {\phi }_1$, the linearized ion momentum equation then writes

(4.13)\begin{equation} -{\rm i}\bar{\omega}_{\rm ph}\bar{\boldsymbol{v}}_{i1} ={-}\boldsymbol{\nabla}\varPhi_1 + C\bar{\boldsymbol{v}}_{i1}\times \boldsymbol{b}.\end{equation}

Taking first the cross-product of (4.13) with $\boldsymbol {b}$ and using again (4.13) to replace $\boldsymbol {v}_{i1}\times \boldsymbol {b}$ in that new equation, we get

(4.14)\begin{equation} \bar{\boldsymbol{v}}_{i1} = \frac{C}{C^2 - \bar{\omega}_{\rm ph}^2}\left[\boldsymbol{b}\times\boldsymbol{\nabla} \varPhi_1 + i\frac{\bar{\omega}_{\rm ph}}{C}\boldsymbol{\nabla} \varPhi_1\right]. \end{equation}

The first term in the brackets is the combination of the perturbed $E\times B$ and diamagnetic flows. The second one is the polarization flow. Inertial effects are included in the factor $C$. The polarization flow matters when the mode frequency $\omega$ is comparable to $\omega _\textrm {ci}$, which is precisely the regime of interest here. Note that the polarization flow makes the plasma incompressible form $(\boldsymbol {\nabla }\boldsymbol {\cdot }\bar {\boldsymbol {v}}_{i1}=0)$.

The final step needed before obtaining the dispersion relation is to compute the linearized divergence of the ion particle flux

(4.15)\begin{equation} \boldsymbol{\nabla} \boldsymbol{\cdot} (n_i \bar{v}_i)|_1 = n_0\boldsymbol{\nabla} \boldsymbol{\cdot} \bar{\boldsymbol{v}}_{i1} + \bar{\boldsymbol{v}}_{i1} \boldsymbol{\cdot} \boldsymbol{\nabla} n_0 + \bar{\boldsymbol{v}}_0 \boldsymbol{\cdot} \boldsymbol{\nabla} n_1. \end{equation}

These terms are given by

(4.16)\begin{gather} n_0\boldsymbol{\nabla} \boldsymbol{\cdot} \bar{\boldsymbol{v}}_{i1} = n_0\frac{i\bar{\omega}_{\rm ph}}{C^2 - \bar{\omega}_{\rm ph}^2} \nabla^2 \varPhi_1 , \end{gather}

(4.17)\begin{gather}\bar{\boldsymbol{v}}_{i1} \boldsymbol{\cdot} \boldsymbol{\nabla} n_0 = \frac{C}{C^2 - \bar{\omega}_{\rm ph}^2}\left[-\frac{im}{\bar{r}}\varPhi_1n_0' + i \frac{\bar{\omega}_{\rm ph}}{C}\varPhi_1'n_0'\right], \end{gather}

(4.18)\begin{gather}\bar{\boldsymbol{v}}_0 \boldsymbol{\cdot} \boldsymbol{\nabla} n_1 = im\bar{\omega}_0 n_1, \end{gather}

where $\nabla ^2 \varPhi _1 = \varPhi _1''+\varPhi _1'/r-m^2/r^2\varPhi _1$.

Combining quasi-neutrality, $n_e=n_i$, and the continuity equations yields

(4.19)\begin{equation} \boldsymbol{\nabla} \boldsymbol{\cdot} (n_e v_e)|_1 = \boldsymbol{\nabla} \boldsymbol{\cdot} (n_i v_i)|_1, \end{equation}

which implies

(4.20)\begin{equation} \frac{m}{\bar{r}}\frac{1}{\bar{L}_n}\tau\bar{\phi}_1 + m\bar{\omega}_{E0}\bar{n}_1= \frac{C}{C^2-\bar{\omega}_{\rm ph}^2}\left[\frac{m}{\bar{r}}\frac{1}{\bar{L}_n}\varPhi_1 - \frac{\bar{\omega}_{\rm ph}}{C}\frac{1}{\bar{L}_n}\varPhi_1' + \frac{\bar{\omega}_{\rm ph}}{C}\nabla^2\varPhi_1\right]+m\bar{\omega}_0\bar{n}_1.\end{equation}

Now, using the electron continuity equation, (4.7), to express $\bar {n}_1$ as a function of $\bar {\phi }_1$ in $\varPhi _1=\bar {n}_1+\tau \bar {\phi }_1$, we get

(4.21)\begin{equation} \varPhi_1 = (1+\alpha_*)\tau\bar{\phi}_1,\end{equation}

with

(4.22)\begin{equation} \alpha_*={-}\frac{m\bar{\omega}_{*0}}{\bar{\omega}_{\rm ph}-m(\bar{\omega}_{E0}-\bar{\omega}_{0})}.\end{equation}

Note that, when the ion pressure gradient is neglected in the ion momentum equation ($T_i=0$), we get $\alpha _*=0$ and $\varPhi _1=\tau \bar {\phi }_1$.

Using (4.21) to express $\tau \bar {\phi }_1$ and $\bar {n}_1$ as a function of $\varPhi _1$ and recalling that, from the equation dictating the equilibrium flow, $\bar {\omega }_0+\bar {\omega }_0^2=\bar {\omega }_{*0}+\bar {\omega }_{E0}$, (4.20) can be written as

(4.23)\begin{equation} \varPhi_1'' +\left[ \frac{1}{\bar{r}} - \frac{1}{\bar{L}_n}\right]\varPhi_1' -\frac{m^2}{\bar{r}^2}\varPhi_1 + \frac{1}{\bar{r}\bar{L}_n}N\varPhi_1=0,\end{equation}

where

(4.24)\begin{equation} N = m\left[\frac{C}{\bar{\omega}_{\rm ph}}-\frac{C^2-\bar{\omega}_{\rm ph}^2}{\bar{\omega}_{\rm ph}-m\bar{\omega}_0^2}\right].\end{equation}

Equations (4.23) and (4.24) provide an extension of the model derived in Chen (Reference Chen1966) for arbitrary frequency values but in the limit of vanishing gyro-viscosity. The low-frequency expansion involved in Chen (Reference Chen1966) consists in approximating $C^2-\bar {\omega }_\textrm {ph}^2\sim C^2$. In this limit

(4.25)\begin{equation} N = m\left[\frac{C}{\bar{\omega}_{\rm ph}}-\frac{C^2}{\bar{\omega}_{\rm ph}-m\bar{\omega}_0^2}\right],\end{equation}

and one exactly recovers (25) in Chen (Reference Chen1966) for $\boldsymbol {\nabla }\boldsymbol {\cdot } \boldsymbol {{\rm \pi} }_i=0$.

Note that $N$ is radially constant because of the assumption of rigid body rotation. The differential equation (4.23) can be solved by the method used in Rosenbluth et al. (Reference Rosenbluth, Krall and Rostoker1962), Chen (Reference Chen1966) and Rognlien (Reference Rognlien1973) by exploiting the change of variables

(4.26)\begin{equation} z=r^2/r_0^2, \end{equation}

where $r_0$ is the width of the Gaussian used to parametrize the density profile defined by (2.6a,b), and enters (4.23) through ${1}/{L_n}={2r}/{r_0^2}$, combined with

(4.27)\begin{equation} \varPhi_1=z^{-{1}/{2}}\, {\rm e}^{{z}/{2}}W(z), \end{equation}

to obtain Whittaker's equation (Whittaker & Watson Reference Whittaker and Watson1966)

(4.28)\begin{equation} \frac{{\rm d}^2W}{{\rm d}z^2}+\biggl\{-\frac{1}{4}+ \frac{N+1}{2z}+\frac{1-m^2}{4z^2}\biggr\}W=0. \end{equation}

The non-singular solution of this equation is given by,

(4.29)\begin{equation} W_{N,m}(z)=z^{({m+1})/{2}}\, {\rm e}^{-{z}/{2}} F\left(\frac{m-N}{2},1+2m;z\right), \end{equation}

where $F(({m-N})/{2},1+2m;z)$ is the confluent hyper-geometric function of the first kind known as Kummer's function. Imposing the boundary condition $\varPhi (Z)=0$, with $Z=r_b^2/r_0^2$ (where $r_b$ is the outer radial boundary of the cylindrical vessel), fully determines the possible values of $N$, which for different mode numbers $m$ are evaluated from the zeros of the Kummer function $F(({m-N})/{2},1+2m;Z)$. These zeros can be evaluated numerically. Alternatively, the asymptotic values of $N$ $(Z\rightarrow \infty )$ are $N=m+2n$, where $n=0, 1, 2, 3,\ldots$ is the radial mode number (Rosenbluth et al. Reference Rosenbluth, Krall and Rostoker1962; Chen Reference Chen1966). The radial mode number $n$ simply indicates which zero of $F$ we are referring to; e.g. $n=0$ implies the first value of $N$ at which the function $F$ goes to zero, $n=1$ implies the second value of $N$ at which the function $F$ goes to zero and so on. In figure 3, the numerical solutions for $N$, obtained using the whitm function in the Python library mpmath, are compared with the asymptotic solutions for $n=0$. Convergence is reached at $Z>6$ for $m=1$, but higher $Z$ values are required at high $m$. For practical applications, such as in MISTRAL where $r_b=10 \textrm {cm}$ and $r_0\approx 3\ \textrm {cm}$, it is, therefore, preferable to use the numerical solution. In the following discussion, we will only use the values of $N$ evaluated numerically.

Figure 3. Values of $N$ corresponding to the first zero of Kummer's function for different values of $Z=r_b^2/r_0^2$. The solid line denotes the values evaluated numerically and the dashed line $(--)$ denotes the asymptotic values.

Note that, for a given radial mode number $n$, the value of $N$ and the eigenfunction shape only depend on the azimuthal mode number $m$ and the value of $Z$, which represents the ratio of the cylinder radius to the plasma radius. The eigenfunctions are therefore independent of the background flow $\bar {\omega }_0$. Eigenfunctions obtained for $m=1,2,5$ and $10$ for different $Z$ values, including the one of MISTRAL ($Z\approx 10.8$), are shown in figure 4. The solutions of (4.24) are purely real, therefore, there is no radial variation of the phase of the eigenfunctions.

Figure 4. Eigenfunction $\varPhi$ as a function of $\bar {r}/\bar {r}_b$ for $m=1,2,5$ and $10$. The solid lines represent the case when $Z=10.8$ (for MISTRAL) and the dashed lines represent the case when $Z=6.9$.

Once $N$ is known, rearranging (4.24) gives the cubic dispersion relation

(4.30)\begin{equation} \bar{\omega}_{\rm ph}^3-\frac{N}{m}\bar{\omega}_{\rm ph}^2+(N\bar{\omega}_0^2-2C\bar{\omega}_0)\bar{\omega}_{\rm ph}-mC\bar{\omega}_0^2=0,\end{equation}

from which the mode growth rate and frequency can be computed.

If the LFA is applied i.e. if $C^2-\bar {\omega }_\textrm {ph}^2\approx C^2$, the equation above becomes

(4.31)\begin{equation} \frac{N}{m}\bar{\omega}_{\rm ph}^2-(N\bar{\omega}_0^2-2C\bar{\omega}_0)\bar{\omega}_{\rm ph}+mC\bar{\omega}_0^2=0,\end{equation}

which is exactly equivalent to (30) in Chen (Reference Chen1966) if the terms with $1/r_0^2$ entering because of the gyro-viscosity tensor are dropped.

4.1. Local limit

To make the link with previous work, e.g. Chen (Reference Chen1966) and Gueroult et al. (Reference Gueroult, Rax and Fisch2017), the local limit is obtained by assuming $\varPhi _1'=0$, $\varPhi _1''=0$ in (4.23)

(4.32)\begin{equation} \bar{\omega}_{\rm ph}^3-\frac{m\bar{r}_0^2}{2\bar{r}^2}\bar{\omega}_{\rm ph}^2+\left({-}2C\bar{\omega}_0+m^2\bar{\omega}_0^2 \frac{\bar{r}_0^2}{2\bar{r}^2}\right)\bar{\omega}_{\rm ph}-mC\bar{\omega}_0^2=0.\end{equation}

This is the same as the dispersion relation obtained by equating (17a) and (17b) in Gueroult et al. (Reference Gueroult, Rax and Fisch2017). Note that, in Gueroult et al. (Reference Gueroult, Rax and Fisch2017), the diamagnetic drift of the ions was neglected. It is kept here but only enters the equation by modifying the equilibrium azimuthal flow $\bar {\omega }_0$. Various asymptotic regimes and stability limits regarding the dispersion relation ((4.32)) have been discussed in Gueroult et al. (Reference Gueroult, Rax and Fisch2017) for $-0.25\leq \omega _{E0}\leq 0$. We extend this discussion to $-0.5\leq \omega _{0}\leq 1.5$ in § 5.

Using the LFA in (4.32), leads to the following dispersion relation:

(4.33)\begin{equation} \bar{\omega}_{\rm ph}^2+\left(\frac{4\bar{r}^2C \bar{\omega}_0}{m\bar{r}_0^2}-m\bar{\omega}_0^2\right) \bar{\omega}_{\rm ph}+\frac{2\bar{r}^2C \bar{\omega}_0^2}{\bar{r}_0^2}=0.\end{equation}

5. Linear stability

In this section, the linear stability and the parametric dependency of the growth rate are discussed for the global dispersion relation derived in § 4. We first examine the role of the radial mode number $n$ in linear stability.

Figure 5 represents the radial mode number $n$ which yields the largest growth rate evaluated using the global dispersion relation ((4.30)) for a given mode number $m$ as a function of $Z$ and $\omega _0/\omega _\textrm {ci}$. For $m=1, 2$ and for the given range of $\omega _0/\omega _\textrm {ci}$, the radial mode number $n=0$ has the largest growth rate for $Z<3$. For $Z>3$, higher radial mode numbers are progressively dominant as $\bar {\omega }_0$ and $Z$ increases. For $m=10$, the lowest radial mode number $n=0$ corresponds to the largest growth rate when $-0.3\leq \omega _0/\omega _\textrm {ci}\leq 0.3$. For large values of $|\bar {\omega }_0|$, the radial mode number $n$ that gives the largest growth rate also increases with $|\bar {\omega }_0|$ and $Z$. When the growth rate is evaluated using the global dispersion relation with LFA ((4.31)), the lowest radial mode number $n=0$ has the largest growth rate. In the following discussion, we will focus on the radial mode number, $n=0$.

Figure 5. Radial mode number $n$ corresponding to the largest growth rate $\bar {\gamma }$ as a function of $Z=\bar {r}_b^2/\bar {r}_0^2$ and $\omega _0/\omega _\textrm {ci}$. The colour bar represents the radial mode number $n$.

The most unstable mode obtained from the global dispersion relation without ((4.30)) and with LFA ((4.31)) for mode number $n=0$ and $m = 1, 2 \textrm {and} 10$ is shown in figure 6 as a function of $\omega _0/\omega _\textrm {ci}$ and $Z$. The two models predict the growth rate to increase with $|\bar {\omega }_0|$, with an asymmetry with respect to $\bar {\omega }_0$, originating from the inertial term in the effective magnetization factor $C$. The difference between two models increases with increasing $m$ and equilibrium flow frequency $\bar {\omega }_0$. For $m = 10$, the region of higher growth rate, as well as the stability region, are radically different with and without LFA. Without the LFA, the largest growth rate is obtained at low $Z$ and large $\bar {\omega }_0$, whereas this becomes a stable region and the growth rate is maximum at large $Z$ with the LFA. The difference in the stability region stems from the neglect of the terms of the order of $\bar {\omega }_\textrm {ph}^3$. For frequencies satisfying ${\omega }-m\omega _0\ll \omega _\textrm {ci}$, the LFA is valid, and hence the dispersion relation with LFA ((4.31)) yields correct results, but as we move towards regimes with high-frequency values, the LFA ordering fails. There is a common region that is stable $(\bar {\gamma } = 0)$ for both the cases and that region corresponds to $\bar {\omega }_0=0$.

Figure 6. Normalized growth rate $\gamma /\omega _\textrm {ci}$ as a function of normalized equilibrium flow frequency ($\omega _{0}/\omega _\textrm {ci}$) and $Z=\bar {r}_b^2/\bar {r}_0^2$, where $\bar {r}_b$ is the radial boundary and $\bar {r}_0$ is the width of the Gaussian normalized to the Larmor radius $\rho _i$ for the global dispersion relation given by (4.30) (a–c) and (4.31) (d–f). The colour bar represents the normalized growth rate $(\bar {\gamma }=\gamma /\omega _\textrm {ci})$.

5.1. Effect of LFA

The validity domain of the LFA as a function of $\omega _0/\omega _\textrm {ci}$ is emphasized in figure 7, where the solution without the LFA ((4.30)) with the red curve, is compared with the solution with the LFA valid when ${\omega }_\textrm {ph}/\omega _\textrm {ci}=O(\rho ^2)$ (green curve, (4.31)) and to another solution with the LFA but valid at higher frequency i.e. ${\omega }_\textrm {ph}/\omega _\textrm {ci}=O(\rho )$ (blue curve, (38) in Chen (Reference Chen1966) with $\boldsymbol {\nabla }\boldsymbol {\cdot } \boldsymbol {{\rm \pi} }_i=0$).

Figure 7. Normalized growth rate $\gamma /\omega _\textrm {ci}$ and normalized Doppler shifted frequency $(\omega _r-m\omega _{0})/\omega _\textrm {ci}$ as a function of normalized equilibrium flow frequency ($\omega _{0}/\omega _\textrm {ci}$) for (a,c) $m=2$ and (b,d) $m=20$ for $Z=10.78$.

All three dispersion relations predict the same growth rate $\bar {\gamma }$ and the real part of Doppler shifted frequency $\bar {\omega }_r-m\bar {\omega }_0$ when the values of $\bar {\omega }_0$ are close to zero. As $\bar {\omega }_0$ increases, the model predictions deviate, especially for higher mode numbers. This accounts from the fact that terms involving higher orders of $\bar {\omega }_\textrm {ph}=\bar {\omega }-m\bar {\omega }_0$, i.e. the Doppler shifted frequency, have been neglected in evaluating the dispersion relations in Chen (Reference Chen1966) and as the factor $m\bar {\omega }_0$ increases, the assumption is no longer valid.

5.2. Impact of radial boundary on growth rate

The position of the boundary also has a strong influence on the mode growth rate for a given value of $\bar {\omega }_0$. The growth rate and real part of the Doppler shifted frequency at different radial boundary positions $\bar {r}_b$ keeping $\bar {r}_0$ fixed, for various values of $m$ evaluated by the global dispersion relation ((4.30)) areshown in figure 8. At fixed plasma size, $\bar {r}_0$, increasing the cylinder radius $\bar {r}_b$, for which $\varPhi (Z)=0$ is imposed, first destabilizes all modes and then has limited-to-no impact on the growth rate once $\bar {r}_b\sim 3\bar {r}_0$ ($Z\sim 9$). Note that $m=1$ has a different behaviour and gets fully stabilized when the bounding cylinder radius is increased.

Figure 8. Normalized growth rate $\bar {\gamma } = \gamma /\omega _\textrm {ci}$ and normalized Doppler shifted frequency $(\omega _r-m\omega _{0})/\omega _\textrm {ci}$ as a function of $Z=\bar {r}_b^2/\bar {r}_0^2$ for various mode numbers $m$ using dispersion relation ((4.30)); (a,c) for $\bar {\omega }_0=-0.4$ and (b,d) for $\bar {\omega }_0=0.4$.

For the real part of the normalized Doppler shifted frequency $\bar {\omega }_r-m\bar {\omega }_0$, for all mode numbers, the frequency is maximum for small values of $Z$ and then decreases as $Z$ increases except for $m=1$ when $\bar {\omega }_0=-0.4$. The sign of $\bar {\omega }_0$ plays a critical role in determining the sign of Doppler shifted frequency ($\bar {\omega }_r-m\bar {\omega }_0$).

5.3. Eigenfunction and phase difference

The expression of the eigenfunctions for the normalized perturbed density $n_1/n_0$ and perturbed potential $e\phi _1/T_{{e0}_\textrm {ref}}$ is obtained by using (4.7) and (4.21)

(5.1)\begin{gather} \bar{n}_1=\frac{m}{\bar{r}\bar{L}_n(\bar{\omega} -m\bar{\omega}_{E0})}\frac{\varPhi_1}{(1+\alpha_*)}, \end{gather}

(5.2)\begin{gather}\bar{\phi}_1=\frac{\varPhi_1}{(1+\alpha_*)\tau}, \end{gather}

where $\alpha _*$ is given by (4.22). Using these expressions, the eigenfunctions $\bar {n}_1$ for $m=1$ and 10 are shown in figure 9 for $\tau =1$. The perturbations in density and potential are more spread out for $m=1$ than for $m=10$ or, in other words, modes with higher azimuthal mode numbers are more localized towards the boundary region, as already discussed in figure 4.

Figure 9. Normalized perturbed density $n_1/n_0$ for (a) $m=1$ and (b) $m=10$ as a function $\bar {r}/\bar {r}_b$ using $Z=10.8$. The parameters used to obtain these eigenfunctions are $\bar {\omega }_{E0}=0.95$, $\bar {\omega }_{*0}=-0.35$, $\bar {\omega }_0=0.42$ and $\tau =1$.

Another essential information regarding the mode structure of the instability is the phase difference between density and potential fluctuations. This is a quantity that can be measured experimentally and which determines the level of particle flux driven by the fluctuations $\bar {n}_1\bar {\boldsymbol {v}}_{i1}$. To calculate the phase difference between $\bar {\phi }_1$ and $\bar {n}_1$, we divide (5.2) by (5.1)

(5.3)\begin{equation} \frac{\bar{\phi}_1}{\bar{n}_1}=(\bar{\omega}- m\bar{\omega}_{E0})\frac{\bar{r}\bar{L}_n}{m\tau}.\end{equation}

Writing ${\bar {\phi }_1}/{\bar {n}_1}=A\, \textrm {e}^{\textrm {i}\phi _p}$, the phase difference $\phi _p$ is,

(5.4)\begin{equation} \phi_p=\tan^{{-}1}\left(\frac{\bar{\gamma}}{\bar{\omega}_r- m\bar{\omega}_{E0}}\right).\end{equation}

Figure 10 shows the phase difference between $\bar {\phi }_1$ and $\bar {n}_1$ as a function of $\omega _{E0}/\omega _\textrm {ci}$ and $-2/\bar {r}_0^2$ for $m=1$ and $m=2$ with $\tau =1$. Note that $\bar {r}_0$ does not appear explicitly in (5.4) but comes in the expression for $Z=\bar {r}_b^2/\bar {r}_0^2$ and $\bar {\omega }_0$ which determines $\bar {\gamma }$ and $\bar {\omega }_r$. Therefore, by varying $\bar {r}_0$, the combined effect of $Z$ as well as $\bar {\omega }_0$ on the phase difference can be observed. The phase shift is close to zero, except in a narrow region where $\bar {\omega }_r-m\bar {\omega }_{E0}$ is approaching zero. In this region, the phase shift becomes large, $|\phi _p|\sim 90^{\circ }$ and changes sign. Furthermore, the critical value of $\bar {\omega }_{E0}$ at which the phase shift changes from negative to positive increases with decreasing $\bar {r}_0$.

Figure 10. Phase difference between $\bar {\phi }_1$ and $\bar {n}_1$ for (a) $m=1$ and (b) $m=2$ as a function of normalized $E\times B$ flow frequency ($\omega _{E0}/\omega _\textrm {ci}$) and $-2/\bar {r}_0^2$, where $\bar {r}_0$ is the normalized plasma size. The colour bar represents the phase difference in degrees and the constant lines on the contour represent $\bar {\omega }_0$.

5.4. Azimuthal mode number spectra

In figure 11, the normalized growth rate $\bar {\gamma }$ and normalized real frequency $\bar {\omega }_r$, computed numerically by solving the dispersion relation ((4.30)) are shown as a function of $m$. The growth rate is increasing with the mode number $m$ irrespective of the sign of $\bar {\omega }_0$. At high $m$, finite Larmor radius (FLR) effects are strongly stabilizing (Hoh Reference Hoh1963) and should be taken into account. In a fluid description they enter in the gyroviscosity tensor, neglected here, but have been shown to stabilize high $m$ numbers in Chen (Reference Chen1966). In other words, FLR effects are important when $k_\theta \rho _i\sim 1$ where $k_\theta =m/r$ is the azimuthal wavenumber. This corresponds to, $m\sim r/\rho _i$, which implies that the FLR stabilization ($\gamma \rightarrow 0$) comes into effect when $m>r/\rho _i$.

Figure 11. (a) Normalized growth rate $\bar {\gamma } = \gamma /\omega _\textrm {ci}$, (b) normalized Doppler shifted frequency $(\omega _r-m\omega _{0})/\omega _\textrm {ci}$ and (c) normalized frequency $\bar {\omega }_r = \omega _r/\omega _\textrm {ci}$ as a function of azimuthal mode number $m$ for various values of normalized equilibrium flow frequency $\bar {\omega }_0 = \omega _0/\omega _\textrm {ci}$ used in the global dispersion relation ((4.30)).

The growth rate is zero for $\bar {\omega }_0=0$, which is consistent with the linear stability diagram (figure 6). For $m=1$, $\bar {\gamma }$ is of the order of $10^{-2}\omega _\textrm {ci}$ for positive values of $\bar {\omega }_0$ and for $\bar {\omega }_0=-0.2$, and zero for $\bar {\omega }_0=-0.4$ and $0$. For similar values of $\bar {\omega }_0$ but in opposite directions there is a small difference in the growth rate up to $m = 5$ and this difference in the growth rate escalates with increasing mode number $m$. Overall, the growth rate increases with the increase in $\bar {\omega }_0$. The Doppler shifted frequency ($\bar {\omega }_{r}-m\bar {\omega }_0$) has the sign opposite to that of $\bar {\omega }_0$ for $\bar {\omega }_0<0$. For $\bar {\omega }_0>0$, the Doppler shifted frequency has the sign opposite to $\bar {\omega }_0$ until $m<20$. The real part of the frequency $\bar {\omega }_r$ has also been shown in figure 11(c) to show the dominance of the factor $m\bar {\omega }_0$.

6. Comparison of local and global dispersion relations

In this section, the impact of the local approximation ((4.32) and (4.33)) is discussed. In figure 12, the mode growth rate obtained in the local approximation with and without LFA is shown as a function of $\bar {\omega }_0$ and $\bar {r}^2/\bar {r}_0^2$ for $m=1,2\textrm { and }10$. Similarly to the radially global results, the LFA assumption is shown to have a validity domain restricted to low $\bar {\omega }_0$ values and low $m$. Relaxing the LFA opens up new instability regions, in particular at low $m$, where an unstable zone is obtained at $\bar {\omega }_0<0$. For $\bar {\omega }_0=0$, no instability exists and stable anti-drift modes with a propagation frequency $\bar {\omega }_r=m\bar {r}_0^2/2\bar {r}^2$ are predicted without the LFA (Fridman Reference Fridman1964).

Figure 12. Normalized growth rate $\gamma /\omega _\textrm {ci}$ as a function of normalized equilibrium flow frequency ($\omega _{0}/\omega _\textrm {ci}$) and $\bar {r}^2/\bar {r}_0^2$ for the local dispersion relation given by (4.32) (a–c) and (4.33) (d–f). The colour bar represents the normalized growth rate $(\bar {\gamma }=\gamma /\omega _\textrm {ci})$.

In contrast to the local dispersion relation, which evaluates the growth rate at each radial position, the global dispersion relation describes the growth rate of an eigenmode extending over the whole cylinder radius. To compare the local and global model predictions, we show in figure 13, the maximum growth rate, $\bar {\gamma }_{\max }$, obtained with the local model over the interval $0\leq \bar {r}\leq \bar {r}_b$ as a function of $\bar {\omega }_0$ and $Z=\bar {r}_b^2/\bar {r}_0^2$. This quantity is compared with the global model predictions in figure 13(d–f). All results are shown without the LFA. In figure 13(a–c), we see that, for $\bar {\omega }_0>0$, the value of $\bar {\gamma }_{\max }$ is largely independent of $Z=\bar {r}_b^2/\bar {r}_0^2$. This is because the radial position at which the maximum growth rate is obtained in the local model is close to zero, see figure 12(a–c). The situation is different for $\bar {\omega }_0<0$ and low $m$, where the local growth rate increases with $\bar {r}$ (see figure 12a). This is reflected by an increase of $\bar {\gamma }_{\max }$ with $\bar {r}_b^2/\bar {r}_0^2$.

Figure 13. Normalized maximum growth rate $\gamma _{\max }/\omega _\textrm {ci}$ as a function of normalized equilibrium flow frequency ($\omega _{0}/\omega _\textrm {ci}$) and $\bar {r}_b^2/\bar {r}_0^2$ for the local dispersion relation given by (4.32) (a–c). Normalized relative growth rate $\gamma _{\textrm {rel}}/\omega _\textrm {ci}$ as a function of normalized equilibrium flow frequency ($\omega _{0}/\omega _\textrm {ci}$) and $\bar {r}^2/\bar {r}_0^2$, where $\bar {\gamma }_{\textrm {rel}}=2(\bar {\gamma }_{\max }-\bar {\gamma }_{\textrm {global}})/(\bar {\gamma }_{\max }+\bar {\gamma }_{\textrm {global}})$ and $\bar {\gamma }_{\textrm {global}}$ is evaluated using (4.30) (d–f). The constant lines on (d–f) represents the difference between $\bar {\gamma }_{\max }$ and $\bar {\gamma }_{\textrm {global}}$.

In both cases, $\bar {\gamma }_{\max }$ is obtained close to the radial boundaries, either $\bar {r}=0$ or $\bar {r}=\bar {r}_b$, where global effects are non-negligible. This is why the relative difference between the $\bar {\gamma }_{\max }$ and $\bar {\gamma }_{\textrm {global}}$, shown in figure 13, is always significant, except perhaps when the growth rate is closer to zero. The dark blue region in figure 13(d–f) where $\bar {\gamma }_{\textrm {rel}}$ is maximum, corresponds to the region where $\bar {\gamma }_{\max }=0$ but $\bar {\gamma }_{\textrm {global}}$ remains finite, leading to large value of $\bar {\gamma }_{\max }-\bar {\gamma }_{\textrm {global}}$. The white region in figure 13(d–f) corresponds to the region where both $\bar {\gamma }_{\max }$ and $\bar {\gamma }_{\textrm {global}}$ correspond to zero. From the comparison, it is evident that the local dispersion relation cannot be used to study the global behaviour of weakly magnetized rotating plasma systems having frequencies comparable to the ion-cyclotron frequency.