2.1. Nonlinear equations

The standard set of nonlinear equations of dissipative incompressible magnetohydrodynamics consists of the Navier–Stokes equation for the fluid velocity $\boldsymbol{u}$ and of the induction equation for the magnetic field $\boldsymbol{B}$ :

(2.1a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\partial \boldsymbol{u}}{\partial t}+\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{u}-\displaystyle \frac{1}{{\it\mu}_{0}{\it\rho}}\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{B}+\frac{1}{{\it\rho}}\boldsymbol{{\rm\nabla}}P-{\it\nu}{\rm\nabla}^{2}\boldsymbol{u}=0, & \displaystyle\end{eqnarray}$$

(2.1b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\partial \boldsymbol{B}}{\partial t}+\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{B}-\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{u}-{\it\eta}{\rm\nabla}^{2}\boldsymbol{B}=0, & \displaystyle\end{eqnarray}$$

$P=p+\boldsymbol{B}^{2}/2{\it\mu}_{0}$

$p$

${\it\rho}=\text{const.}$

${\it\nu}=\text{const.}$

${\it\eta}=({\it\mu}_{0}{\it\sigma})^{-1}$

${\it\sigma}$

${\it\mu}_{0}$

(2.2a,b ) $$\begin{eqnarray}\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{u}=0,\quad \boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{B}=0.\end{eqnarray}$$

2.2. Steady state

We consider the rotational fluid flow in the gap between the radii $R_{1}$ and $R_{2}>R_{1}$ , with an imposed magnetic field sustained by electrical currents. Introducing the cylindrical coordinates $(R,{\it\phi},z)$ we consider the stability of a steady-state background liquid flow with the angular velocity profile ${\rm\Omega}(R)$ in a helical background magnetic field (a magnetized Taylor–Couette (TC) flow):

(2.3a−c ) $$\begin{eqnarray}\boldsymbol{u}_{0}(R)=R\,{\rm\Omega}(R)\,\boldsymbol{e}_{{\it\phi}},\quad p=p_{0}(R),\quad \boldsymbol{B}_{0}(R)=B_{{\it\phi}}^{0}(R)\boldsymbol{e}_{{\it\phi}}+B_{z}^{0}\boldsymbol{e}_{z}.\end{eqnarray}$$

Note that if the azimuthal component is produced by an axial current $I$ confined to $R<R_{1}$ , then

(2.4) $$\begin{eqnarray}B_{{\it\phi}}^{0}(R)=\frac{{\it\mu}_{0}I}{2{\it\pi}R}.\end{eqnarray}$$

Introducing the hydrodynamic Rossby number $(\mathit{Ro})$ by means of the relation

(2.5) $$\begin{eqnarray}\mathit{Ro}:=\frac{R}{2{\rm\Omega}}\partial _{R}{\rm\Omega},\end{eqnarray}$$

where $\partial _{R}=\partial /\partial R$ , we find that the solid body rotation corresponds to $\mathit{Ro}=0$ and the Keplerian rotation to $\mathit{Ro}=-\frac{3}{4}$ , whereas the velocity profile ${\rm\Omega}(R)\sim R^{-2}$ corresponds to $\mathit{Ro}=-1$ . Note that although Keplerian rotation is not possible globally in TC-flow, an approximate Keplerian profile can locally be achieved.

Similarly, following Kirillov & Stefani (Reference Kirillov and Stefani2013) we define the magnetic Rossby number

(2.6) $$\begin{eqnarray}\mathit{Rb}:=\frac{R}{2B_{{\it\phi}}^{0}R^{-1}}\partial _{R}(B_{{\it\phi}}^{0}R^{-1}).\end{eqnarray}$$

$\mathit{Rb}=0$ results from a linear dependence of the magnetic field on the radius, $B_{{\it\phi}}^{0}(R)\propto R$ , as would be produced by a homogeneous axial current in the fluid. $\mathit{Rb}=-1$ corresponds to the radial dependence given by (2.4). Note that the logarithmic derivatives ${\it\alpha}$ and ${\it\beta}$ introduced in the work by Ogilvie & Pringle (Reference Ogilvie and Pringle1996) are nothing else but the doubled hydrodynamic and magnetic Rossby numbers: ${\it\alpha}=2\mathit{Ro}$ , ${\it\beta}=2\mathit{Rb}$ .

2.3. Linearization with respect to non-axisymmetric perturbations

To describe natural oscillations in the neighbourhood of the magnetized Taylor–Couette flow we linearize equations (2.1)–(2.2) in the vicinity of the stationary solution (2.3) assuming general perturbations $\boldsymbol{u}=\boldsymbol{u}_{0}+\boldsymbol{u}^{\prime }$ , $p=p_{0}+p^{\prime }$ , and $\boldsymbol{B}=\boldsymbol{B}_{0}+\boldsymbol{B}^{\prime }$ and leaving only the terms of first order with respect to the primed quantities:

(2.7a ) $$\begin{eqnarray}\displaystyle & & \displaystyle \partial _{t}\boldsymbol{u}^{\prime }+\boldsymbol{u}_{0}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{u}^{\prime }+\boldsymbol{u}^{\prime }\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{u}_{0}-\frac{1}{{\it\rho}{\it\mu}_{0}}\left(\boldsymbol{B}_{0}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{B}^{\prime }+\boldsymbol{B}^{\prime }\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{B}_{0}\right)-{\it\nu}{\rm\nabla}^{2}\boldsymbol{u}^{\prime }\nonumber\\ \displaystyle & & \displaystyle \quad =-\frac{1}{{\it\rho}}\boldsymbol{{\rm\nabla}}p^{\prime }-\frac{1}{{\it\rho}{\it\mu}_{0}}\boldsymbol{{\rm\nabla}}(\boldsymbol{B}_{0}\boldsymbol{\cdot }\boldsymbol{B}^{\prime }),\end{eqnarray}$$

(2.7b ) $$\begin{eqnarray}\displaystyle & & \displaystyle \partial _{t}\boldsymbol{B}^{\prime }+\boldsymbol{u}_{0}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{B}^{\prime }+\boldsymbol{u}^{\prime }\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{B}_{0}-\boldsymbol{B}_{0}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{u}^{\prime }-\boldsymbol{B}^{\prime }\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{u}_{0}-{\it\eta}{\rm\nabla}^{2}\boldsymbol{B}^{\prime }=0,\end{eqnarray}$$

(2.8a,b ) $$\begin{eqnarray}\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }=0,\quad \boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{B}^{\prime }=0.\end{eqnarray}$$

Note that by adding/subtracting (2.7b ) divided by $\sqrt{{\it\rho}{\it\mu}_{0}}$ to/from (2.7a ) yields the linearized equations in a more symmetrical Elsasser form; in the ideal magnetohydrodynamic (MHD) case they were derived e.g. by Friedlander & Vishik (Reference Friedlander and Vishik1995).

Introducing the gradients of the background fields represented by the two $3\times 3$ matrices

(2.9a ) $$\begin{eqnarray}\displaystyle & {\mathcal{U}}(R)=\boldsymbol{{\rm\nabla}}\boldsymbol{u}_{0}={\rm\Omega}\left(\begin{array}{@{}ccc@{}}0 & -1 & 0\\ 1+2\mathit{Ro} & 0 & 0\\ 0 & 0 & 0\\ \end{array}\right), & \displaystyle\end{eqnarray}$$

(2.9b ) $$\begin{eqnarray}\displaystyle & {\mathcal{B}}(R)=\boldsymbol{{\rm\nabla}}\boldsymbol{B}_{0}=\displaystyle \frac{B_{{\it\phi}}^{0}}{R}\left(\begin{array}{@{}ccc@{}}0 & -1 & 0\\ 1+2\mathit{Rb} & 0 & 0\\ 0 & 0 & 0\\ \end{array}\right), & \displaystyle\end{eqnarray}$$

(2.10a ) $$\begin{eqnarray}\displaystyle & & \displaystyle (\partial _{t}+{\mathcal{U}}+\boldsymbol{u}_{0}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}})\boldsymbol{u}^{\prime }-\displaystyle \frac{1}{{\it\rho}{\it\mu}_{0}}\left({\mathcal{B}}+\boldsymbol{B}_{0}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\right)\boldsymbol{B}^{\prime }\nonumber\\ \displaystyle & & \displaystyle \quad -\,{\it\nu}{\rm\nabla}^{2}\boldsymbol{u}^{\prime }+\displaystyle \frac{1}{{\it\rho}}\boldsymbol{{\rm\nabla}}p^{\prime }+\displaystyle \frac{1}{{\it\rho}{\it\mu}_{0}}\boldsymbol{{\rm\nabla}}(\boldsymbol{B}_{0}\boldsymbol{\cdot }\boldsymbol{B}^{\prime })=0,\end{eqnarray}$$

(2.10b ) $$\begin{eqnarray}\displaystyle & & \displaystyle (\partial _{t}-{\mathcal{U}}+\boldsymbol{u}_{0}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}})\boldsymbol{B}^{\prime }+({\mathcal{B}}-\boldsymbol{B}_{0}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}})\boldsymbol{u}^{\prime }-{\it\eta}{\rm\nabla}^{2}\boldsymbol{B}^{\prime }=0.\end{eqnarray}$$

8.1. The threshold of instability

Consider the dispersion relation (7.3). Substitute its coefficients (B 2) to the matrix $\unicode[STIX]{x1D63D}$ in (6.1) and then calculate the determinants $m_{i}$ according to (6.2). Set $\mathit{Pm}=0$ in the resulting expressions for $m_{i}$ . For the coefficient $m_{4}=\det \unicode[STIX]{x1D63D}$ this leads to a great simplification and yields the instability threshold in a compact and closed form:

(8.1) $$\begin{eqnarray}\frac{\mathit{Ha}^{4}{\it\beta}^{2}(n{\it\beta}+1)^{2}\mathit{Ro}^{2}-\left(C^{2}-4\mathit{ Ha}^{4}{\it\beta}^{2}(n{\it\beta}+1)^{2}\right)(\mathit{Ro}+1)}{(1+\mathit{ Ha}^{2}(n{\it\beta}+1)^{2})^{2}-4\mathit{Ha}^{2}{\it\beta}^{2}(1+\mathit{Ha}^{2}(n{\it\beta}+1)^{2})\mathit{ Rb}-4\mathit{Ha}^{4}{\it\beta}^{2}(n{\it\beta}+1)^{2}}=\frac{C^{2}}{4\mathit{Re}^{2}},\end{eqnarray}$$

where $C=1+\mathit{Ha}^{2}((n{\it\beta}+1)^{2}-2{\it\beta}^{2}\mathit{Rb})$ . Further on in the text we will also derive this threshold from the explicit expressions for the growth rate of the perturbation.

In the limit $\mathit{Re}\rightarrow \infty$ (8.1) reduces to a quadratic equation with respect to $\mathit{Ro}$ in the numerator of its left-hand side. Dividing this equation by $\mathit{Ha}^{4}$ and then letting $\mathit{Ha}\rightarrow \infty$ yields

(8.2) $$\begin{eqnarray}{\it\beta}^{2}(n{\it\beta}+1)^{2}\mathit{Ro}^{2}-(((n{\it\beta}+1)^{2}-2{\it\beta}^{2}\mathit{Rb})^{2}-4{\it\beta}^{2}(n{\it\beta}+1)^{2})(\mathit{Ro}+1)=0\end{eqnarray}$$

with the roots

(8.3) $$\begin{eqnarray}\displaystyle \mathit{Ro}^{\pm } & = & \displaystyle -2+\frac{(n{\it\beta}+1)^{2}-2{\it\beta}^{2}\mathit{ Rb}\pm \sqrt{((n{\it\beta}+1)^{2}-2{\it\beta}^{2}\mathit{ Rb })^{2}-4{\it\beta}^{2}(n{\it\beta}+1)^{2}}}{2{\it\beta}^{2}(n{\it\beta}+1)^{2}((n{\it\beta}+1)^{2}-2{\it\beta}^{2}\mathit{Rb})^{-1}}.\end{eqnarray}$$

At $n=0$ and $\mathit{Rb}=-1$ the expression (8.3) reduces to that of Kirillov & Stefani (Reference Kirillov and Stefani2011)

(8.4) $$\begin{eqnarray}\mathit{Ro}^{\pm }=\frac{4{\it\beta}^{4}+1\pm (2{\it\beta}^{2}+1)\sqrt{4{\it\beta}^{4}+1}}{2{\it\beta}^{2}}.\end{eqnarray}$$

From (8.4) one can find ${\it\beta}$ as a function of $\mathit{Ro}$ to recover the results of Priede (Reference Priede2011).

Figure 1.

(a) The function $\mathit{Ro}^{-}({\it\beta},n)$ given by (8.3) at $\mathit{Rb}=-\frac{25}{32}$ . (b) Its extrema (8.6) along which the critical hydrodynamic Rossby number attains the maximal value $\mathit{ Ro}^{-}=-\frac{3}{4}$ . The cross-section of the function at (c) $n=0$ (HMRI) and (d) as ${\it\beta}\rightarrow +\infty$ (AMRI).

8.2. Extremal properties of the critical hydrodynamic Rossby number

For a given magnetic Rossby number, $\mathit{Rb}$ , the functions $\mathit{R}o^{\pm }(n,{\it\beta})$ take their extrema

(8.5) $$\begin{eqnarray}\mathit{Ro}_{extr}^{\pm }=-2-4\mathit{Rb}\pm 2\sqrt{2\mathit{Rb }(2\mathit{Rb }+1)}\end{eqnarray}$$

at the following extremal curves in the $(n,{\it\beta})$ -plane (cf. Kirillov et al. Reference Kirillov, Stefani and Fukumoto2012):

(8.6a,b ) $$\begin{eqnarray}{\it\beta}_{extr}^{+}=\frac{-1}{n+\sqrt{-2\mathit{Rb}}},\quad {\it\beta}_{extr}^{-}=\frac{-1}{n-\sqrt{-2\mathit{Rb}}},\end{eqnarray}$$

see figure 1(b). The branch $\mathit{Ro}^{-}({\it\beta},n)$ is shown in figure 1(a) for $\mathit{Rb}=-\frac{25}{32}$ (this particular value has been chosen since it is the minimum value which leads to destabilization of Keplerian profiles, as we will see below). The maximal value of the hydrodynamic Rossby number is constant along the curves (8.6), see figure 1(a,b).

The cross-section of the function $\mathit{Ro}^{-}({\it\beta},n)$ at $n=0$ corresponding to the helical magnetorotational instability (HMRI) is plotted in figure 1(c). In contrast, figure 1(d) shows the limit ${\it\beta}\rightarrow +\infty$

(8.7) $$\begin{eqnarray}\mathit{Ro}^{-}(n,\mathit{Rb})=-2+(n^{2}-2\mathit{Rb})\frac{n^{2}-2\mathit{Rb}-\sqrt{(n^{2}-2\mathit{Rb })^{2}-4n^{2}}}{2n^{2}}\end{eqnarray}$$

corresponding to the azimuthal magnetorotational instability (AMRI).

The function (8.7) attains its maximal value $\mathit{Ro}_{extr}^{-}$ given by (8.5) at

(8.8) $$\begin{eqnarray}n=\pm \sqrt{-2\mathit{Rb}}.\end{eqnarray}$$

Since by definition (7.2) $n=m/{\it\alpha}$ where $k_{z}/\sqrt{k_{R}^{2}+k_{z}^{2}}=:{\it\alpha}\in [0,1]$ , then for $\mathit{ Rb}\in [-1,-0.5]$ the condition (8.8) yields the only possible integer values of the azimuthal wavenumber:

(8.9) $$\begin{eqnarray}\displaystyle m=\pm 1. & & \displaystyle\end{eqnarray}$$

This means that the effect of the purely azimuthal magnetic field is most pronounced at the maximal possible range of variation of the hydrodynamic Rossby number for the lowest azimuthal modes with $m=\pm 1$ (Hollerbach et al. Reference Hollerbach, Teeluck and Rüdiger2010; Kirillov, Stefani & Fukumoto Reference Kirillov, Stefani and Fukumoto2014).

8.3. A band structure periodic in ${\it\alpha}$

What is the general dependence of the mode number $m$ on ${\it\alpha}$ , $\mathit{ Rb}$ and the ratio ${\it\omega}_{A_{{\it\phi}}}/{\it\omega}_{A_{z}}$ of the azimuthal to the axial magnetic fields? Let us express ${\it\alpha}$ from (8.6) as

(8.10) $$\begin{eqnarray}{\it\alpha}=\pm \left(m+\frac{{\it\omega}_{A_{z}}}{{\it\omega}_{A_{{\it\phi}}}}\right)\frac{1}{\sqrt{-2\mathit{Rb}}}.\end{eqnarray}$$

The ratio ${\it\omega}_{A_{{\it\phi}}}/{\it\omega}_{A_{z}}$ when plotted against ${\it\alpha}$ at different $m$ and $\mathit{Rb}$ uncovers a regular pattern shown in figure 2. The pattern is periodic in ${\it\alpha}$ with the period $1/\sqrt{-2\mathit{Rb}}$ . On the other hand, the dependence of ${\it\alpha}$ on the ratio of the magnetic fields is a continuous piecewise smooth function inside every vertical ‘band’ with the width $1/(2\sqrt{-2\mathit{ Rb}})$ in ${\it\alpha}$ , see figure 2(b). The vertical bands in figure 2 are further separated into cells with the boundaries at

(8.11) $$\begin{eqnarray}\frac{{\it\omega}_{A_{{\it\phi}}}}{{\it\omega}_{A_{z}}}=\frac{2}{1},\frac{2}{2},\frac{2}{3},\frac{2}{4},\frac{2}{5},\frac{2}{6},\dots .\end{eqnarray}$$

Every particular cell in figure 2 corresponds to a unique integer azimuthal wavenumber $m$ . Note however that the cells with the same $m$ are grouped along the same hyperbolic curve given by (8.10).

Figure 2.

The periodic-in- ${\it\alpha}$ band structure at (a) $\mathit{Rb}=-1$ (with grey cells corresponding to $m=-2$ ) and (b) $\mathit{Rb}=-0.5$ (with the grey stripe marking the first band).

The curves (8.10) with slopes of the same sign do not cross. The crossings of two hyperbolic curves (8.10) with the integer indices $m_{1}$ and $m_{2}$ and slopes of different sign happen at

(8.12a,b ) $$\begin{eqnarray}{\it\alpha}=\frac{m_{1}-m_{2}}{2\sqrt{-2\mathit{Rb}}}\quad \text{and}\quad \frac{{\it\omega}_{A_{{\it\phi}}}}{{\it\omega}_{A_{z}}}=\frac{-2}{m_{1}+m_{2}},\end{eqnarray}$$

which explains the sequence (8.11). Within one and the same band with ${\it\alpha}\geqslant 0$ the cells are encoded by two interlacing series of integers, e.g. for the third band in figure 2(a)

(8.13) $$\begin{eqnarray}m=+1,-2,0,-3,-1,-4,-2,-5,-3,-6,\dots ,\end{eqnarray}$$

which is a mixture of the sequences

(8.14) $$\begin{eqnarray}\displaystyle +1,0,-1,-2,-3,\dots \quad \text{and}\quad -2,-3,-4,-5,-6,\dots , & & \displaystyle\end{eqnarray}$$

whereas for the second band with ${\it\alpha}\geqslant 0$ in figure 2(a) we have

(8.15) $$\begin{eqnarray}m=-1,0,-2,-1,-3,-2,-4,-3,-5,-4,\dots ,\end{eqnarray}$$

which corresponds to the two interlacing sequences

(8.16) $$\begin{eqnarray}-1,-2,-3,-4,-5,\dots \quad \text{and}\quad 0,-1,-2,-3,-4,\dots .\end{eqnarray}$$

Therefore, at the crossings (8.12) with ${\it\alpha}\neq 0$ the quadruplet of cells consists of two pairs; each pair corresponding to a different value of the azimuthal wavenumber $m$ . At the crossings with ${\it\alpha}=0$ all four cells in the quadruplet have the same $m$ .

Figure 3.

(a) The lower (LLL) and the upper (ULL) Liu limits existing at $\mathit{Rb}=-1$ are points on the quasi-hyperbolic curve (8.20) in the $\mathit{Ro}{-}\mathit{Rb}$ plane. (b) A scaled fragment of the limiting curve (8.18) demonstrating that the inductionless forms of the magnetorotational instability exist above the lower Liu limit $\mathit{Ro}_{extr}^{-}=2-2\sqrt{2}$ when $\mathit{Rb}>-\frac{25}{32}$ . The open circle marks the Keplerian value for the Rossby number $\text{Ro}=-\frac{3}{4}$ at $\mathit{Rb}=-\frac{25}{32}$ whereas the black circle corresponds to $\mathit{Ro}=\mathit{Rb}=-\frac{2}{3}$ . The dashed diagonal is given by the equation $\mathit{Ro}=\mathit{Rb}$ .

8.4. Continuation of the Liu limits to arbitrary $\mathit{Rb}$

The extrema (8.5) can be represented in the form

(8.17) $$\begin{eqnarray}\mathit{Ro}_{extr}^{\pm }=-1-\left(\sqrt{2\mathit{Rb}}\mp \sqrt{1+2\mathit{Rb}}\right)^{2},\end{eqnarray}$$

which is equivalent to the expression

(8.18) $$\begin{eqnarray}\mathit{Rb}=-\frac{1}{8}\frac{(\mathit{Ro}+2)^{2}}{\mathit{Ro}+1}\end{eqnarray}$$

that is universal in the sense that it contains only logarithmic derivatives of the radial profiles of the flow and the azimuthal field.

The particular case of (8.18) at $\mathit{Rb}=-1$ yields the result of Liu et al. (Reference Liu, Goodman, Herron and Ji2006), also reproduced by Kirillov & Stefani (Reference Kirillov and Stefani2011) and Priede (Reference Priede2011). Solving (8.18) at $\mathit{Rb}=-1$ , we find that the critical Rossby numbers $\mathit{Ro}(\mathit{Ha},\mathit{Re},n,{\it\beta})$ given by (8.1), and thus the instability domains, lie at $\mathit{Pm}=0$ and $\mathit{Rb}=-1$ outside the stratum

(8.19) $$\begin{eqnarray}\displaystyle 2-2\sqrt{2}=:\mathit{Ro}_{LLL}<\mathit{Ro}(\mathit{Ha},\mathit{Re},n,{\it\beta})<\mathit{Ro}_{ULL}:=2+2\sqrt{2}, & & \displaystyle\end{eqnarray}$$

where $\mathit{Ro}_{LLL}$ is the value of $\mathit{Ro}_{extr}^{-}$ at the lower Liu limit (LLL) and $\mathit{Ro}_{ULL}$ is the value of $\mathit{Ro}_{extr}^{+}$ at the upper Liu limit (ULL) corresponding to the critical values of ${\it\beta}$ given by (8.6).

Figure 3(a) shows how the LLL and ULL continue to the values of $\mathit{Rb}\neq -1$ . In fact, the LLL and the ULL are points on the quasi-hyperbolic curve

(8.20) $$\begin{eqnarray}(\mathit{Rb}-\mathit{Ro}-1)^{2}-(\mathit{Rb}+\mathit{Ro}+1)^{2}={\textstyle \frac{1}{2}}(\mathit{Ro}+2)^{2},\end{eqnarray}$$

which is another representation of (8.18). At $\mathit{ Rb}=-\frac{1}{2}$ the branches $\mathit{Ro}_{extr}^{-}(\mathit{Rb})$ and $\mathit{ Ro}_{extr}^{+}(\mathit{Rb})$ meet each other. Therefore, the inductionless magnetorotational instability at negative $\mathit{Rb}$ also exists at positive $\mathit{Ro}$ when $\mathit{Ro}>\mathit{Ro}_{extr}^{+}(\mathit{Rb})$ , figure 3(a). Notice also the second stability domain at $\mathit{Rb}>0$ and $\mathit{Ro}<-1$ .

We see that in the inductionless case $\mathit{Pm}=0$ when the Reynolds and Hartmann numbers subsequently tend to infinity as described in § 8.1 and ${\it\beta}$ and $n$ are under the constraints (8.6), the maximal possible critical Rossby number $\mathit{Ro}_{extr}^{-}$ increases with the increase of $\mathit{Rb}$ . At

(8.21) $$\begin{eqnarray}\mathit{Rb}\geqslant -{\textstyle \frac{25}{32}}=-0.78125\end{eqnarray}$$

$\mathit{Ro}_{extr}^{-}(\mathit{Rb})$ exceeds the critical value for the Keplerian flow: $\mathit{Ro}_{extr}^{-}\geqslant -\frac{3}{4}$ .

Therefore, the possibility for $B_{{\it\phi}}(R)$ to depart from the profile $B_{{\it\phi}}(R)\propto R^{-1}$ allows us to break the conventional lower Liu limit and extend the inductionless versions of MRI to the velocity profiles ${\rm\Omega}(R)$ as flat as the Keplerian one and even to less steep profiles, including that of the solid body rotation, i.e. $\mathit{ Ro}=0$ , for $\mathit{Rb}\geqslant -\frac{1}{2}$ , figure 3(b).

Figure 4.

Domains of the inductionless HMRI for $\mathit{Pm}=0$ , $n=0$ , and $\mathit{Rb}=-0.74$ when the Hartmann and Reynolds numbers follow the scaling law (8.22) with ${\it\beta}$ satisfying the restraints (8.6): (a) $\mathit{Ha}=5$ and $\mathit{Re}\approx 92$ ; (b) $\mathit{Ha}=100$ and $\mathit{ Re}\approx 736\,158$ . With the increase in $\mathit{Ha}$ and $\mathit{Re}$ the HMRI domain easily breaks the conventional Liu limits as well as the Kepler line $\mathit{Ro}=-0.75$ and its boundary tends to the curves (8.3) that touch the new limits for the critical Rossby number: $\mathit{Ro}_{extr}^{-}(-0.74)\approx -0.726$ and $\mathit{Ro}_{extr}^{+}(-0.74)\approx 2.646$ .

8.5. Scaling law of the inductionless MRI

What asymptotic behaviour of the Reynolds and Hartmann numbers at infinity leads to maximization of negative (and simultaneously to minimization of positive) critical Rossby numbers? To get an idea, we investigate extrema of $\mathit{Ro}$ as a solution to (8.1) subject to the constraints (8.6). Taking e.g. ${\it\beta}={\it\beta}_{extr}^{-}$ in (8.1), differentiating the result with respect to $\mathit{Ha}$ , equating it to zero and solving the equation for $\mathit{Re}$ , we find the following asymptotic relation between $\mathit{Ha}$ and $\mathit{Re}$ when $\mathit{Ha}\rightarrow \infty$ :

(8.22) $$\begin{eqnarray}\mathit{Re}=2\mathit{Rb}\sqrt{3\mathit{Rb }+2}(\sqrt{1+2\mathit{Rb}}+\sqrt{2\mathit{Rb}}){\it\beta}^{3}\mathit{Ha}^{3}+O(\mathit{Ha}).\end{eqnarray}$$

For example, at $\mathit{Rb}=-1$ and $n=0$ we have ${\it\beta}={\it\beta}_{extr}^{-}=\frac{1}{\sqrt{2}}$ . After taking this into account in (8.22) we obtain the scaling law of HMRI found in Kirillov & Stefani (Reference Kirillov and Stefani2010):

(8.23) $$\begin{eqnarray}\mathit{Re}=\frac{2+\sqrt{2}}{2}\mathit{Ha}^{3}+O(\mathit{Ha}).\end{eqnarray}$$

Figure 4 shows the domains of the inductionless helical MRI at $\mathit{Rb}=-0.74$ when the Hartmann and the Reynolds numbers are increasing in accordance with the scaling law (8.22). The instability thresholds easily penetrate the LLL and ULL as well as the Keplerian line and tend to the curves (8.3) that touch the new limits for the critical Rossby number: $\mathit{Ro}_{extr}^{-}(-0.74)\approx -0.726$ and $\mathit{Ro}_{extr}^{+}(-0.74)\approx 2.646$ . Note that the curves (8.3) also correspond to the limit of vanishing Elsasser number $\mathit{N}$ , because according to the scaling law (8.22) we have $\mathit{ N}\propto 1/\mathit{Ha}$ as $\mathit{Ha}\rightarrow \infty$ . This observation makes the dispersion relation (7.6) advantageous for investigation of the inductionless versions of MRI.

8.6. Growth rates of HMRI and AMRI and the critical Reynolds number

We will calculate the growth rates of the inductionless MRI with the use of the dispersion relation (7.6). Assuming in (7.6) $\mathit{ Rm}:=\mathit{Re}\mathit{Pm}=0$ , we find the roots explicitly:

(8.24) $$\begin{eqnarray}{\it\lambda}_{1,2}=-\text{i}n+\mathit{N}\left(2{\it\beta}^{2}\mathit{Rb}-(n{\it\beta}+1)^{2}\right)-\frac{1}{\mathit{Re}}\pm 2\sqrt{X+\text{i}Y},\end{eqnarray}$$

where

(8.25a,b ) $$\begin{eqnarray}X=\mathit{N}^{2}{\it\beta}^{2}\left({\it\beta}^{2}\mathit{Rb}^{2}+(n{\it\beta}+1)^{2}\right)-\mathit{Ro}-1,\quad Y=\mathit{N}{\it\beta}(\mathit{Ro}+2)(n{\it\beta}+1).\end{eqnarray}$$

Separating the real and imaginary parts of the roots, we find the growth rates of the inductionless MRI in closed form:

(8.26) $$\begin{eqnarray}({\it\lambda}_{1,2})_{r}=\mathit{N}\left(2{\it\beta}^{2}\mathit{Rb}-(n{\it\beta}+1)^{2}\right)-\frac{1}{\mathit{Re}}\pm \sqrt{2X+2\sqrt{X^{2}+Y^{2}}}.\end{eqnarray}$$

Particularly, at $n=0$ we obtain the growth rates of the axisymmetric helical magnetorotational instability in the inductionless case.

Introducing the Elsasser number of the azimuthal field as

(8.27) $$\begin{eqnarray}\mathit{N}_{A}:={\it\beta}^{2}\mathit{N}\end{eqnarray}$$

and then taking the limit of ${\it\beta}\rightarrow \infty$ we obtain from (8.26) the growth rates of the inductionless azimuthal MRI:

(8.28) $$\begin{eqnarray}\displaystyle & & \displaystyle ({\it\lambda}_{1,2})_{r}=\mathit{N}_{A}(2\mathit{Rb}-n^{2})-\frac{1}{\mathit{Re}}\pm \sqrt{2}\nonumber\\ \displaystyle & & \displaystyle \quad \times \,\sqrt{\mathit{N }_{A}^{2}(\mathit{Rb }^{2}+n^{2})-\mathit{ Ro }-1+\sqrt{\left(\mathit{N }_{A}^{2}(\mathit{Rb }^{2}+n^{2})-\mathit{ Ro }-1\right)^{2}+\mathit{ N }_{A}^{2}(\mathit{Ro }+2)^{2}n^{2}}}.\end{eqnarray}$$

In the inviscid limit $\mathit{Re}\rightarrow \infty$ the term $1/\mathit{Re}$ vanishes in the expressions (8.26) and (8.28).

Figure 5.

Domain of the viscous inductionless MRI (8.29) of a Keplerian flow ( $\mathit{Ro}=-0.75$ ) at $\mathit{Rb}=-0.75$ when (a) $n=0$ (axisymmetric HMRI) and (b) ${\it\beta}\rightarrow \infty$ (non-axisymmetric AMRI). Open circles mark the location of the minima of the critical hydrodynamic Reynolds number: (a) ${\it\beta}\approx 0.848$ , $\mathit{N}\approx 0.193$ , $\mathit{Re}\approx 198.5$ ; (b) $n\approx 1.180$ , $\mathit{ N}_{A}\approx 0.139$ , $\mathit{Re}\approx 198.5$ .

On the other hand, in the viscous case the condition $({\it\lambda}_{1,2})_{r}>0$ yields the critical hydrodynamic Reynolds number beyond which inductionless MRI appears. For example, from (8.26) we find the following criterion for the onset of the instability:

(8.29) $$\begin{eqnarray}\mathit{Re}>\frac{1}{\mathit{N}\left(2{\it\beta}^{2}\mathit{Rb}-(n{\it\beta}+1)^{2}\right)+\sqrt{2X+2\sqrt{X^{2}+Y^{2}}}}.\end{eqnarray}$$

With $n=0$ the criterion (8.29) corresponds to the axisymmetric HMRI. A similar criterion for the onset of AMRI follows from (8.28).

The critical hydrodynamic Reynolds number for a Keplerian flow at $\mathit{ Rb}=-0.75$ is plotted in figure 5 as a function of the Elsasser number and (figure 5 a) ${\it\beta}$ in case of the axisymmetric HMRI and (figure 5 b) $n$ in case of the non-axisymmetric AMRI. In both cases the minimal $\mathit{Re}$ at the onset of instability is about $200$ .

Figure 6.

$\mathit{Ro}=-0.75$ . (a) The growth rates ${\it\lambda}_{r}\geqslant 0$ of viscous ( $\mathit{Re}=800$ ) HMRI ( $n=0$ ) in the inductionless $(\mathit{Pm}=0)$ limit in projection onto the $\mathit{N}{-}{\it\beta}$ plane. (b) The same in the inviscid $(\mathit{Re}\rightarrow \infty )$ limit. (c) The growth rates ${\it\lambda}_{r}\geqslant 0$ of AMRI ( ${\it\beta}\rightarrow \infty$ ) in the inductionless and inviscid limit in projection onto the $\mathit{N}_{A}{-}n$ plane. The open circles mark the maxima of ${\it\lambda}_{r}$ . Row (a) (left to right): ${\it\beta}\approx 0.830$ , $\mathit{N}\approx 0.153$ , ${\it\lambda}_{r}\approx 0.002$ ; ${\it\beta}\approx 0.848$ , $\mathit{N}\approx 0.193$ , ${\it\lambda}_{r}\approx 0.004$ ; ${\it\beta}\approx 0.869$ , $\mathit{N}\approx 0.236$ , ${\it\lambda}_{r}\approx 0.007$ . Row (b): ${\it\beta}\approx 0.830$ , $\mathit{ N}\approx 0.153$ , ${\it\lambda}_{r}\approx 0.003$ ; ${\it\beta}\approx 0.848$ , $\mathit{N}\approx 0.193$ , ${\it\lambda}_{r}\approx 0.005$ ; ${\it\beta}\approx 0.869$ , $\mathit{N}\approx 0.236$ , ${\it\lambda}_{r}\approx 0.008$ . Row (c): $n\approx 1.204$ , $\mathit{N}_{A}\approx 0.105$ , ${\it\lambda}_{r}\approx 0.003$ ; $n\approx 1.180$ , $\mathit{N}_{A}\approx 0.139$ , ${\it\lambda}_{r}\approx 0.005$ ; $n\approx 1.151$ , $\mathit{N}_{A}\approx 0.178$ , ${\it\lambda}_{r}\approx 0.008$ .

8.7. HMRI and AMRI as magnetically destabilized inertial waves

Consider the Taylor expansion of the eigenvalues (8.24) with respect to the Elsasser number $\mathit{N}$ in the vicinity of $\mathit{N}=0$ :

(8.30) $$\begin{eqnarray}\displaystyle \displaystyle {\it\lambda}_{1,2} & = & \displaystyle -\text{i}(n\mp 2\sqrt{\mathit{Ro }+1})-\frac{1}{\mathit{Re}}\nonumber\\ \displaystyle & & \displaystyle +\,\displaystyle \mathit{N}\left(-(n{\it\beta}+1)^{2}+2{\it\beta}^{2}\mathit{Rb}\pm {\it\beta}\frac{(\mathit{Ro}+2)}{\sqrt{\mathit{Ro }+1}}(n{\it\beta}+1)\right)+O(\mathit{N}^{2}).\end{eqnarray}$$

Expressions (8.24) and (8.30) generalize the result of Priede (Reference Priede2011) to the case of arbitrary $n$ , $\mathit{Re}$ and $\mathit{Rb}$ and exactly coincide with it at $n=0$ , $\mathit{Re}\rightarrow \infty$ and $\mathit{Rb}=-1$ .

In the absence of the magnetic field ( $\mathit{N}=0$ ) the eigenvalues ${\it\lambda}_{1,2}$ correspond to damped inertial (or Kelvin) waves. According to (8.29) at finite $\mathit{Re}$ there exists a critical finite $\mathit{N}>0$ that is necessary to trigger destabilization of the inertial waves by the magnetic field, see figures 5 and 6(a). However, as the expansion (8.30) demonstrates, in the limit $\mathit{Re}\rightarrow \infty$ the inductionless magnetorotational instability occurs when the effect of the magnetic field is much weaker than that of the flow – even when the Elsasser number is infinitesimally small, figure 6(b,c).

In the inviscid case the boundary of the domain of instability (8.29) takes the form

(8.31) $$\begin{eqnarray}\mathit{N}=\pm 2\sqrt{\frac{{\it\beta}^{2}(\mathit{Ro }+2)^{2}(n{\it\beta}+1)^{2}-((n{\it\beta}+1)^{2}-2{\it\beta}^{2}\mathit{ Rb })^{2}(\mathit{Ro }+1)}{((n{\it\beta}+1)^{2}-4{\it\beta}^{2}(\mathit{Rb}+1))((n{\it\beta}+1)^{2}-2{\it\beta}^{2}\mathit{Rb})^{2}(n{\it\beta}+1)^{2}}}.\end{eqnarray}$$

The lines (8.31) bound the domain of non-negative growth rates of HMRI in figure 6(b).

When $\mathit{Ro}=\mathit{Rb}$ , the stability boundary (8.31) has a self-intersection at

(8.32a,b ) $$\begin{eqnarray}n=-\frac{1}{{\it\beta}}\pm 2\sqrt{\mathit{Rb }+1},\quad \mathit{N}=\frac{\pm 1}{2{\it\beta}^{2}}\sqrt{\frac{-(3\mathit{Rb }+2)}{(\mathit{Rb}+1)(\mathit{Rb}+2)}}.\end{eqnarray}$$

For example, when $\mathit{Rb}=\mathit{Ro}=-0.75$ and $n=0$ , the intersection happens at ${\it\beta}=1$ and $\mathit{N}=\frac{\sqrt{5}}{5}$ , figure 6(b). If $\mathit{Rb}=\mathit{Ro}=-0.75$ and ${\it\beta}\rightarrow \infty$ , the intersection point is at $n=1$ and $\mathit{N}_{A}=\frac{\sqrt{5}}{5}$ , figure 6(c). In general, the intersection exists at $\mathit{N}\neq 0$ for

(8.33) $$\begin{eqnarray}\mathit{Ro}=\mathit{Rb}<-{\textstyle \frac{2}{3}}.\end{eqnarray}$$

At $\mathit{Ro}=\mathit{Rb}=-\frac{2}{3}$ the intersection occurs at $\mathit{N}=0$ .

In figure 6 we see that when $\mathit{Ro}<-\frac{2}{3}$ and $\mathit{ Ro}>\mathit{Rb}$ , the instability domain consists of two separate regions. In the case when $\mathit{Ro}<-\frac{2}{3}$ and $\mathit{Ro}<\mathit{Rb}$ , the two regions merge into one. When the condition (8.33) is fulfilled, the two sub-domains touch each other at the point (8.32). At $\mathit{ Ro}=\mathit{Rb}=-\frac{2}{3}$ the lower region shrinks to a single point which simultaneously is the intersection point (8.32) with $\mathit{ N}=0$ .

On the other hand, given $\mathit{Ro}<-\frac{2}{3}$ and decreasing $\mathit{Rb}$ we find that the single instability domain tends to split into two independent regions after crossing the line $\mathit{Ro}=\mathit{Rb}$ . The further decrease in $\mathit{Rb}$ yields diminishing size of the lower instability region, see figure 6. At which $\mathit{Rb}$ does the lower instability region completely disappear?

Clearly, the lower region disappears when the roots of the equation $\mathit{N}(n)=0$ become complex. From (8.31) we derive

(8.34) $$\begin{eqnarray}(n{\it\beta}+1)^{2}\pm {\it\beta}\frac{\mathit{Ro}+2}{\sqrt{\mathit{Ro }+1}}(n{\it\beta}+1)-2{\it\beta}^{2}\mathit{Rb}=0.\end{eqnarray}$$

The equations (8.34) have the roots $n$ complex if and only if their discriminant is negative:

(8.35) $$\begin{eqnarray}\frac{(\mathit{Ro}+2)^{2}}{\mathit{Ro}+1}+8\mathit{Rb}<0,\end{eqnarray}$$

which is simply the domain with the boundary given by the curve connecting the two Liu limits (8.18) that is shown in figure 3. Note that $\mathit{Rb}=-\frac{2}{3}$ and $\mathit{Ro}=-\frac{2}{3}$ satisfy (8.18), which indicates that the line $\mathit{Ro}=\mathit{Rb}$ is tangent to the curve (8.18) at the point $(-\frac{2}{3},-\frac{2}{3})$ in the $\mathit{ Ro}$ – $\mathit{Rb}$ plane, see figure 3(b).

Finally, we notice that the left-hand side of (8.34) is precisely the coefficient at $\mathit{N}$ in the expansion (8.30). In the inviscid case it determines the limit of the stability boundary as $\mathit{N}\rightarrow 0$ , quite in accordance with the scaling law (8.22). Solving (8.34) for $\mathit{Ro}$ , we exactly reproduce the formula (8.3). At $n=0$ and $\mathit{Rb}=-1$ equation (8.34) exactly coincides with that obtained by Priede (Reference Priede2011).

9.1. Islands of HMRI at various integer $n$ and their reconnection

Consider the dispersion relation (7.3) and substitute its coefficients (B 2) into the Bilharz matrix (6.1). Applying the Bilharz criterion (6.2) to the result, we find that it is the condition of vanishing the determinant of the Bilharz matrix that determines the instability threshold. Fixing the Rossby number at the Keplerian value $\mathit{Ro}=-0.75$ and assuming some reasonable values for the Hartmann and Reynolds numbers, e.g. $\mathit{Ha}=30$ and $\mathit{ Re}=40\,000$ , we choose the magnetic Rossby number slightly to the right of the line $\mathit{Ro}=\mathit{Rb}$ , which according to the criterion (8.21) should result in instability, at least in the inductionless case. For various integer modified azimuthal wavenumbers $n$ we plot the instability domains in the ${\it\beta}{-}\mathit{Pm}$ plane, figure 7(a,b).

Figure 7.

$\mathit{Ha}=30$ , $\mathit{Ro}=-0.75$ , $\mathit{Re}=40\,000$ , $\mathit{ Rb}=-0.755$ . (a,b) Islands of low- $\mathit{Pm}$ magnetorotational instability (axisymmetric HMRI with $n=0$ , non-axisymmetric HMRI with $n\neq 0,+1$ at ${\it\beta}\leqslant 2$ and AMRI with $n=+1$ at ${\it\beta}>2$ ) corresponding to different modified azimuthal wavenumbers $n$ . (c,d) The growth rates ${\it\lambda}_{r}$ of the perturbation: (c) as functions of ${\it\beta}$ at $\mathit{Pm}=4\times 10^{-6}$ and different $n$ and (d) as functions of $\mathit{Pm}$ at different $n$ and ${\it\beta}$ .

Figure 8.

Subsequently reconnecting instability islands at $\mathit{Ha}=30$ , $\mathit{Ro}=-0.75$ , $\mathit{Re}=40\,000$ and growing $\mathit{Rb}>-0.75$ : (a) $\mathit{Rb}=-0.74933$ , (b) $\mathit{Rb}=-0.747$ , and (c) $\mathit{Rb}=-0.742$ .

We see that at every integer $n$ there are several instability islands (curiously resembling those originating in a pure hydrodynamical Taylor–Couette problem Altmeyer et al. (Reference Altmeyer, Hoffmann and Lücke2011)). As is visible in figure 7(b) they tend to group into two clusters.

The first cluster consists of the islands situated at $\mathit{ Pm}<10^{-5}$ and contains intervals of the positive ${\it\beta}$ -axis at all $n$ but $n=+1$ , figure 7(a). Note that the modified azimuthal wavenumber $n$ corresponding to these islands follows exactly the sequence (8.13) that we have found in the inductionless case! The growth rates, i.e. the real parts of the roots ${\it\lambda}$ defined via relations (7.5), are plotted in figure 7(c) for a fixed value of $\mathit{Pm}$ that cuts this cluster along a vertical line. Figure 7(d) demonstrates the growth rates at various ${\it\beta}$ corresponding to particular values of $n$ . We see that for all $n$ but $n=+1$ the growth rates tend to some positive values as $\mathit{ Pm}\rightarrow 0$ indicating a smooth transition to the inductionless magnetorotational instabilities that occur both for axisymmetric ( $n=0$ ) and non-axisymmetric ( $n\neq 0$ ) perturbations. This seems to be the main reason for the manifestation of the sequence (8.13) in the pattern of the instability islands of figure 7(c).

The second cluster of instability islands occupies the region at $\mathit{Pm}>10^{-5}$ and is encoded by the sequence (8.15), see figure 7(b). We see that the islands of the two clusters tend to form quadruplets. Each quadruplet consists of the two pairs of islands corresponding to the indices $n$ that differ by 2, for example: 0 and $-2$ , $-1$ and $-3$ , $-2$ and $-4$ etc. Each quadruplet whose pairs are labelled with the indices $n_{i},n_{j}\leqslant 0$ tends to be centred at ${\it\beta}={\it\beta}_{ij}$ , where

(9.1) $$\begin{eqnarray}{\it\beta}_{ij}=-\frac{2}{n_{i}+n_{j}},\end{eqnarray}$$

which is exactly (8.12b ). Moreover, the whole pattern of the instability islands in figure 7(b) repeats the pattern of cells in the second and third bands shown in figure 2(a).

It is natural to ask what are the conditions for reconnection of the islands in the pairs that constitute each particular quadruplet. To get an idea we play with the two Rossby numbers in figure 8. We fix $\mathit{Ro}=-0.75$ and slightly increase $\mathit{Rb}$ . As a result, at $\mathit{Rb}=-0.74933$ the islands with the indices $n=-2$ and $n=0$ reconnect at ${\it\beta}_{-2,0}=1$ , figure 8(a). At $\mathit{Rb}=-0.747$ these islands overlap whereas the islands in the next quadruplet with $n=-3$ and $n=-1$ reconnect at ${\it\beta}_{-3,-1}=\frac{1}{2}$ , figure 8(b). At $\mathit{Rb}=-0.742$ the reconnection happens in the third quadruplet at ${\it\beta}_{-4,-2}=\frac{1}{3}$ , and so on, figure 8(c).

This sequence of reconnections indicates the special role of the line $\mathit{Ro}=\mathit{Rb}$ which seems to be even more pronounced in the inviscid limit $(\mathit{Re}\rightarrow \infty )$ . In the following we check these hypotheses when the magnetic field has only the azimuthal component which corresponds to the limit ${\it\beta}\rightarrow \infty$ .

9.2. AMRI as a dissipation-induced instability of Chandrasekhar’s equipartition solution

In the matrix (7.7) let us replace via the relation (8.27) the Elsasser number $\mathit{N}$ of the axial field with the Elsasser number of the azimuthal field $\mathit{N}_{A}$ and then let ${\it\beta}\rightarrow \infty$ . If, additionally

(9.2) $$\begin{eqnarray}\mathit{N}_{A}=\mathit{Rm}\end{eqnarray}$$

and

(9.3) $$\begin{eqnarray}\mathit{Ro}=Rb,\end{eqnarray}$$

then in the ideal limit ( $\mathit{Re}\rightarrow \infty$ , $\mathit{Rm}\rightarrow \infty$ ) the roots of the dispersion relation (7.6) are

(9.4a,b ) $$\begin{eqnarray}{\it\lambda}_{1,2}=0,\quad {\it\lambda}_{3,4}=-2\text{i}(n\pm 1).\end{eqnarray}$$

With the use of the relations (7.2) it is straightforward to verify that the condition (9.2) requires that ${\rm\Omega}={\it\omega}_{A_{{\it\phi}}}$ . Thus, the conditions (9.2) and (9.3) are equivalent to (6.15), which at $\mathit{Rb}=\mathit{Ro}=-1$ define the Chandrasekhar equipartition solution (Kirillov et al. Reference Kirillov, Stefani and Fukumoto2014) belonging to a wide class of exact stationary solutions of MHD equations for the case of ideal incompressible infinitely conducting fluid with total constant pressure that includes even flows with knotted magnetic surfaces (Golovin & Krutikov Reference Golovin and Krutikov2012). It is well-known that the Chandrasekhar equipartition solution is marginally stable (Chandrasekhar Reference Chandrasekhar1956, Reference Chandrasekhar1961; Bogoyavlenskij Reference Bogoyavlenskij2004). According to (9.4) the marginal stability is also preserved in the ideal case when $\mathit{Rb}=\mathit{Ro}\neq -1$ . Will the roots (9.4) acquire only negative real parts with the addition of electrical resistivity?

Figure 9.

(a) The threshold of instability (9.5) at $\mathit{N}_{A}=\mathit{ Rm}$ and $\mathit{Re}\rightarrow \infty$ in the $(n,\mathit{Rb},\mathit{Rm})$ space and (b) its projection onto the $\mathit{Rb}{-}n$ plane. The increase in $\mathit{Rm}$ makes the instability domain more narrow so that in the limit $\mathit{Rm}\rightarrow \infty$ it degenerates into a ray (dashed) on the straight line (9.6) that emerges from the point (open circle) with the coordinates $n=\frac{2\sqrt{3}}{3}$ and $\mathit{Rb}=-\frac{2}{3}$ and passes through the point with $n=1$ and $\mathit{Rb}=-1$ .

In general, the answer is no. Indeed, under the constraints (9.2) and (9.3) in the limit of vanishing viscosity ( $\mathit{Re}\rightarrow \infty$ ) the Bilharz criterion applied to the dispersion relation (7.6) gives the following threshold of instability:

(9.5) $$\begin{eqnarray}\displaystyle & & \displaystyle 16(n^{2}-\mathit{Rb}^{2})(n^{2}-\mathit{Rb}-2)^{2}\mathit{Rm}^{4}\nonumber\\ \displaystyle & & \displaystyle \quad +\,(n^{6}-12n^{2}\mathit{Rb}^{2}+32n^{2}(\mathit{Rb}+1)-16\mathit{Rb}^{2}(\mathit{Rb}+2))\mathit{Rm}^{2}\nonumber\\ \displaystyle & & \displaystyle \quad -\,4\mathit{Rb}^{2}+4n^{2}(\mathit{Rb}+1)=0.\end{eqnarray}$$

In the $(n,\mathit{Rb},\mathit{Rm})$ space the domain of instability is below the surface specified by (9.5), see figure 9(a). When the electrical resistivity is vanishing, the cross-sections of the instability domain in the $\mathit{Rb}{-}n$ plane become smaller and in the limit $\mathit{Rm}\rightarrow \infty$ they tend to the ray on the line

(9.6) $$\begin{eqnarray}\mathit{Rb}=-\frac{2}{3}+\frac{n\sqrt{3}-2}{3(2-\sqrt{3})}\end{eqnarray}$$

that starts at the point with the coordinates $n=\frac{2\sqrt{3}}{3}$ and $\mathit{Rb}=-\frac{2}{3}$ and passes through the point with $n=1$ and $\mathit{Rb}=-1$ , figure 9(b). For example, at $\mathit{ Ro}=Rb=-\frac{3}{4}$ (9.6) yields

(9.7) $$\begin{eqnarray}n=\frac{1}{4}+\frac{\sqrt{3}}{2}.\end{eqnarray}$$

On the other hand, when the magnetic Reynolds number $\mathit{Rm}$ decreases, the instability domain widens and in the inductionless limit at $\mathit{Rm}=0$ it is bounded by the curve

(9.8) $$\begin{eqnarray}\mathit{Rb}=\frac{n(n-\sqrt{n^{2}+4})}{2}.\end{eqnarray}$$

The wide part of the instability domain shown in figure 9(a) that exists at small $\mathit{Rm}$ belongs to the realm of the azimuthal magnetorotational instability (AMRI). We see that this instability quickly disappears with the increase of $\mathit{Rm}$ or $\mathit{Rb}$ . On the other hand, the ideal solution with the roots (9.4) that corresponds to the limit $\mathit{Rm}\rightarrow \infty$ is destabilized by the electrical resistivity. For $n$ given by (9.7) we have, for example, an unstable root ${\it\lambda}\approx 0.00026-\text{i}0.00493$ at $\mathit{Rm}=100$ . In general, if $\mathit{Rb}$ and $n$ satisfy (9.6), then already an infinitesimally weak electrical resistivity has destabilized the solution specified by the constraints (9.2) and (9.3) at vanishing kinematic viscosity that includes Chandrasekhar’s equipartition solution as a special case. This dissipation-induced instability (Kirillov Reference Kirillov2009; Kirillov & Verhulst Reference Kirillov and Verhulst2010; Kirillov Reference Kirillov2013) further develops into the AMRI with $\mathit{Rm}$ decreasing to zero.

Figure 10.

At $n\approx 1.27842$ (a) the domain of the inductionless AMRI bounded by the curve (10.2) with $\mathit{Ro}=-1$ and $\mathit{ Rb}=-1$ that gradually transforms into the domain that includes the Tayler instability (TI) as $\mathit{Rb}$ increases from $-1$ to 0; (b) Tayler instability corresponding to $\mathit{Ro}=0$ and $\mathit{Re}=0$ . The open circle marks the onset of the standard TI at $\mathit{Rb}=0$ with the critical $\mathit{Hb}\approx 1.04117$ .