4.1 The energy budget in the full MHD equations

Let us set $B_{0}=1$ in (2.1) (the result is also valid for $B_{0}=-1$ by the symmetry of the MHD equations). Using the Elsässer variables $\boldsymbol{\unicode[STIX]{x1D702}}_{\pm }=\boldsymbol{v}\pm \boldsymbol{b}=[\unicode[STIX]{x1D709}_{\pm },\boldsymbol{\unicode[STIX]{x1D702}}_{\pm },\unicode[STIX]{x1D701}_{\pm }]$ , equations (2.1) can be transformed into

Note that the Elsässer variables are divergence free from (2.4c ). The ‘energies’ for those variables can be defined as

where the angle brackets denote the spatial average over the computational domain $[0,2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FC}]\times [-1,1]\times [0,2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FD}]$ . Averaging $\boldsymbol{\unicode[STIX]{x1D702}}_{-}\boldsymbol{\cdot }(4.1a)$ and $\boldsymbol{\unicode[STIX]{x1D702}}_{+}\boldsymbol{\cdot }(4.1b)$ , we can derive two energy budget equations

Here

are the diffusion term and the energy input term due to the base flow, respectively.

The boundary terms

may also act as an energy input mechanism (recall that the double overline is the $x$ – $z$ average). Those terms can be simplified by using the magnetic boundary conditions, as summarised in table 1. The assumption (iii) mentioned in § 1 essentially requires that all the boundary terms are negligibly small; this case may occur when the coherent structure is localised at the mid-channel (as seen in Deguchi (Reference Deguchi2015)), or the problem is considered in an unbounded domain. For the perfectly insulating case (2.3), all the boundary terms are non-zero. At perfectly conducting walls, $a_{y},c_{y}$ and $b$ must vanish from (2.2), and hence $\unicode[STIX]{x1D6E4}$ and $\unicode[STIX]{x1D6EC}$ must be zero, while $\unicode[STIX]{x1D706}$ may not be zero.

Table 1.

The properties of the boundary terms in (4.3) and (4.14).

Under Chandrasekhar’s assumptions (i)–(iii), all the terms on the right-hand side of (4.3) can be dropped except for the energy input term $2I$ from the base flow. In the linear framework, a time derivative can be replaced by the exponential growth rate of the flow amplitude. From the first equation (4.3a ), clearly the velocity and magnetic field eigenfunctions must be identical to have a non-zero growth rate. However, from the second equation (4.3b ), this class of eigenfunctions must also have a zero growth rate, because $I\equiv 0$ . Therefore the Chandrasekhar theorem follows. Note that the only boundary term that survives in the diffusionless approximation is $\unicode[STIX]{x1D6E4}$ ; therefore, as widely accepted, the theorem is valid for perfectly conducting conditions (see table 1).

We further remark here that for unidirectional flows the theorem can be used whatever the magnetic boundary conditions. From the linearised ideal induction equations, we can deduce that $(U_{b}-s)b=U_{b}v$ must hold, where $s$ is the complex phase speed of the wave. Therefore, except for the unlikely situation that the wave is neutral and its travelling speed coincides with the speed of the base flow on one of the walls, $b$ must vanish on the walls (and hence $\unicode[STIX]{x1D6E4}=0$ ) as long as the impermeability conditions are satisfied there.

The mechanism of the linear instability seen in Rüdiger et al. (Reference Rüdiger, Schultz, Stefani and Mond2015) can be explained using (4.3). When $P_{m}=1$ , the stability result is consistent with the Chandrasekhar theorem, and hence the stability property can be explained by the ideal mechanism. However, the existence of the instability for $P_{m}\neq 1$ suggests that the terms multiplied by $(1-P_{m}^{-1})$ in (4.3) have some destabilisation effects. Those terms indeed play a role when a singularity occurs in the ideal problem, even when the instability is predominantly driven by some ideal mechanism almost everywhere (e.g. Bogdanova-Ryzhova & Ryzhov Reference Bogdanova-Ryzhova and Ryzhov1994; Sakurai et al. Reference Sakurai, Goossens and Hollweg1991). The absorption of the wave at the singularity occurs in a small scale, and hence the derivative of the flow appearing in the corresponding terms in (4.3) may not be small.

Now we turn our attention to the nonlinear regime. The stability of the Chandrasekhar state can actually be shown even when nonlinearity and diffusivity are present in the flow. More precisely, the assumptions (i) and (ii) are not necessary to prove the stability, as long as $P_{m}=B_{0}=1$ and the assumption (iii) is satisfied. The proof of this new result follows from (4.3a ) and the total energy equation, which can be obtained for example by adding (4.3a ) and (4.3b ):

where $E=(\langle |\boldsymbol{v}|^{2}+|\boldsymbol{b}|^{2}\rangle /2)+{\mathcal{E}}\geqslant 0$ and $D=\langle |\unicode[STIX]{x1D735}\boldsymbol{v}|^{2}+|\unicode[STIX]{x1D735}\boldsymbol{b}|^{2}\rangle +{\mathcal{D}}\geqslant 0$ . The functions ${\mathcal{E}}$ and ${\mathcal{D}}$ are defined in appendix A; they have come from the boundary term $\unicode[STIX]{x1D6EC}$ , and $D=E=0$ occurs only for the trivial solution. The other boundary terms $\unicode[STIX]{x1D6E4}$ and $\unicode[STIX]{x1D706}$ are cancelled out. When (iii) is satisfied, there is no energy input term in (4.3a ) and thus $E_{-}$ must decay monotonically; thus the equipartitioned state $\boldsymbol{v}=\boldsymbol{b}$ is realised ultimately. Approaching the equipartition state the shear energy input term behaves like $I\rightarrow 0$ . If $I=0$ , from (4.6) we can show that $E$ is a Lyapunov function of the base state in the entire phase space.

The extended version of the Chandrasekhar theorem above implies that if a non-trivial nonlinear state exists in the Chandrasekhar state, the magnetic perturbation generated by it should strongly interact with the walls. Unlike the original Chandrasekhar theorem, the extended result is not valid when (iii) is replaced by perfectly conducting conditions, because of the non-vanishing viscous boundary terms $\unicode[STIX]{x1D706}$ and $\unicode[STIX]{x1D6EC}$ . Also, for the perfectly insulating problem, the boundary terms hinder further theoretical progresses.

The next step we can take towards a theoretical understanding of the numerical result in § 3 is to apply some rational reduction to the governing equations. In § 4.3, we use the vortex–Alfvén wave interaction theory derived at the large-Reynolds-number limit for this purpose. Before the asymptotic analysis, in the next section, we give a detailed look at the nonlinear solutions found in the Chandrasekhar state using the perfectly conducting conditions. The examination of the flow structure computed at moderately large Reynolds number may also help the reader’s understanding of the asymptotic theory.

4.2 Nonlinear states at $P_{m}=B_{0}=1$ for the perfectly conducting conditions

For the perfectly conducting case, non-trivial solutions can be found even when $P_{m}=B_{0}=1$ as seen in § 3. Therefore, the best we can do in (4.6) for this case is to find the critical energy Reynolds number $R_{E}$ below which the laminar flow is globally stable. To compute $R_{E}$ , first the maximum of $I/D$ over all $\boldsymbol{b}$ and divergence-free $\boldsymbol{v}$ must be computed. The maximum $M$ can be found by solving the Euler–Lagrange equations

where $u=v=w=b=a_{y}=0$ at $y=\pm 1$ . The optimum perturbation found at $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})=(1.79,0)$ gives the energy Reynolds number $R_{E}=M^{-1}=72.58$ . The value found is remarkably higher than that for purely hydrodynamic plane Couette flow ( $(R_{E},\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})=(20.66,0,1.56)$ ; see Joseph (Reference Joseph1966)), implying that the magnetic field causes a stabilisation effect to the flow.

Figure 5.

The flow visualisation of the mirror-symmetric nonlinear solutions seen in figure 4. The Chandrasekhar state at $R=1000$ . The (a,c) left- and (b,d) right-hand panels show the hydrodynamic and magnetic fields, respectively. Yellow and blue represent isosurface of 50 % streamwise vorticity (current) which are dominated by the wave component excited by the resonance occurring at the middle of the channel. The grey surface is the zero of the streamwise velocity (magnetic field). The contour of the hydrodynamic (magnetic) streak is also shown at the front of the computational box ( $[0,2\unicode[STIX]{x03C0}]\times [-1,1]\times [0,\unicode[STIX]{x03C0}]$ ).

All non-trivial equilibrium solutions should exist for $R>R_{E}$ . This can be checked by varying $R$ from the solutions obtained in figure 3. In figure 4 the bifurcation diagram for $B_{0}=P_{m}=1$ is shown. The two points marked by the open and filled circles are those seen in figure 3. From the upper point, we can trace the branch towards the saddle-node point at $R=575$ , but the turned branch does not return to the lower point. The branch continued from the lower point revealed that it is originated from the bifurcation point $R=802$ , at which the solid curve connects to the dashed curve. The solution on the dashed curve has the mirror symmetry

in addition to (2.13) (the hydrodynamic part of the symmetry is the same as EQ7/EQ8 in Gibson, Halcrow & Cvitanovic (Reference Gibson, Halcrow and Cvitanovic2009) and Itano & Generalis (Reference Itano and Generalis2009)). In figure 4, blue solid curves are used for the sinuous mode that has the symmetry (2.13) but does not satisfy (4.8). This mirror-symmetric mode has a saddle-node point sitting at a lower Reynolds number ( ${\approx}273$ ) than that of the sinuous mode. Nevertheless, this value is much higher than that for the purely hydrodynamic plane Couette flow, $R\approx 162$ (see Deguchi Reference Deguchi2019a ).

Deguchi (Reference Deguchi2019b ) found that there are two typical types of asymptotic behaviours of the solutions at large Reynolds numbers. The upper and lower branches of the mirror-symmetric solution seen in figure 4 best illustrate the difference of those two.

For the lower branch, the shear $\unicode[STIX]{x1D6E5}$ seems to converge towards some constant with increasing $R$ . This is because the flow asymptotes to the vortex–Alfvén wave interaction type of state studied in Deguchi (Reference Deguchi2019a ,Reference Deguchi b ); the sinuous mode solutions we have seen in figures 2 and 3 are also of this type. The lower panels of figure 5 display the structure of the lower-branch solution at $R=1000$ . The mechanism sustaining the solution can be explained by the asymptotic theory as follows. First we note in the figure that the grey surfaces showing the isosurface of the zero streamwise velocity field are almost flat (recall that the streamwise direction is  $x$ ). In fact, the velocity and magnetic fields are dominated by the $x$ average of the streamwise component, called the streak. The origin of the less dominant three-dimensional structure can be explained by the ideal linear neutral stability of that streak; therefore, only one streamwise Fourier mode is necessary to describe the leading-order structures. The structure of the lower-branch states is similar to the so-called alpha–omega dynamo, where the strong shear generates fields in the $x$ direction from transverse fields; meanwhile waves on the flow can give an alpha effect so that diffusion can sustain the transverse fields. The idea of using mean-fluctuation decomposition can be seen in various mean-field dynamo theories (Braginsky Reference Braginsky1964; Moffatt Reference Moffatt1978; Krause & Rädler Reference Krause and Rädler1980).

However, the vortex–Alfvén wave interaction theory is more than the mean-field theory, because the closure problem inevitably associated with the decomposition is rationally resolved by only assuming the largeness of the Reynolds number. The key to the resolution was the use of the matched asymptotic expansions. The stability problem possesses a singularity at the resonant position, where the velocity relative to the phase speed of the wave coincides with the Alfvén velocity. For the mirror-symmetric mode, by symmetry, the resonant position is simply $y=0$ , where the grey surface sits. The absorption of the Alfvén wave there creates the strong streamwise vortex and current shown by the blue and yellow surfaces. The vorticity/current is trapped within a thin diffusive layer, whose thickness gets thinner with increasing $R$ . The strong fluctuation emerging within the diffusive layer causes a nonlinear feedback effect to the $x$ -averaged field through the Reynolds stress. Thus the wave causes a strongly nonlinear feedback effect to the mean field, despite its rather simple structure in the streamwise direction. The entire mechanism outlined here is very similar to the purely hydrodynamic counterpart studied in Hall & Sherwin (Reference Hall and Sherwin2010), which can be found on the edge of laminar and turbulent attractors, as shown by Wang et al. (Reference Wang, Gibson and Waleffe2007).

For the upper branch, by contrast, the value of $\unicode[STIX]{x1D6E5}$ increases indefinitely for large $R$ (see figure 4). This asymptotic behaviour of $\unicode[STIX]{x1D6E5}$ is typical in turbulence simulations, due to the presence of a near-wall boundary-layer structure in the flow. In the upper panels of figure 5, where the upper-branch solution at $R=1000$ is visualised, the presence of the near-wall boundary layer structure can indeed be seen in the contour of the streamwise field shown at the front of the computational box. The streamwise velocity and magnetic fields are no longer dominated by the $x$ average, as can be seen in the strongly modulated wavy grey surfaces. The streamwise vorticity and current fields have very complicated structures spread out in the entire channel. The existence of a formal asymptotic theory for that complicated flow remains an open question.

4.3 Non-existence of the vortex–Alfvén wave interaction type of nonlinear states for perfectly insulating conditions

The large-Reynolds-number limit corresponds to the singular perturbation problem, and thus the limit of the diffusive problem may not coincide with the ideal results. Deguchi (Reference Deguchi2019a ) derived a set of reduced nonlinear equations by taking the large-Reynolds-number limit of the viscous–resistive MHD equations. We shall shortly see that the diffusivity is in fact not negligible in the limiting reduced leading-order problem. The aim of this section is to show that there is no nontrivial solution in the reduced problem when $B_{0}=P_{m}=1$ and the perfectly insulating conditions are used. That theoretical result may explain the numerical result obtained in figure 2.

The vortex–Alfvén wave interaction theory derived in Deguchi (Reference Deguchi2019a ) is valid for travelling-wave solutions propagating in $x$ with speed $s$ . The leading-order flow has the form

Here the variables with overline are the $x$ -averaged components called the ‘vortex’, or ‘roll-streak’, part; the streamwise and cross-streamwise components are referred to as the streak and roll, respectively. The ‘wave’ part is the component fluctuating in $x$ ; the amplitude $\unicode[STIX]{x1D716}$ is to be specified later. Only one Fourier mode is needed in this component because it is determined by a linear stability of the streak field.

Substituting (4.9) into (2.1), and then applying the large-Reynolds-number asymptotic reduction, Deguchi (Reference Deguchi2019a ) found a closure for the leading-order solution. If $B_{0}$ and $P_{m}$ are both unity, the $x$ -average parts of the leading-order equations are

whilst the fluctuation part becomes

Equation (4.11) comes from the ideal linear stability problem of the hydrodynamic streak $U=U_{b}+\overline{u}$ , and the magnetic streak $A=U_{b}+\overline{a}$ . Having determined $\widetilde{q}$ , we can easily relate it to $\widetilde{\boldsymbol{v}}$ and $\widetilde{\boldsymbol{b}}$ . In particular,

and the ‘frozen’ condition

holds.

On a curve where either $\overline{u}-s=\overline{a}$ or $\overline{u}-s=-\overline{a}$ is satisfied, $(U-s)^{2}-A^{2}$ vanishes and hence equation (4.11) becomes singular. Around the critical curves at which the singularity occurs, we must retain the diffusive effects to regularise the flow. The thickness of the diffusion layer is $O(R^{-1/3})$ , which is well known in resonant absorption studies (Sakurai et al. Reference Sakurai, Goossens and Hollweg1991; Goossens et al. Reference Goossens, Ruderman and Hollweg1995). Two types of diffusion-layer structures were determined in Deguchi (Reference Deguchi2019a ) together with the appropriate wave amplitude $\unicode[STIX]{x1D716}$ . Type 1 occurs when the two diffusive layers are well separated; the amplitude in this case is $\unicode[STIX]{x1D716}=1$ . In type 2, the two diffusive layers merge so that the resonance there becomes stronger; as a consequence, a smaller size of the wave amplitude $\unicode[STIX]{x1D716}=R^{-1/6}$ must be used.

The sharp structure within the thin diffusive layer creates a jump in the derivative of the outer hydrodynamic roll components. The appropriate forms of the jump for types 1 and 2 were found in Deguchi (Reference Deguchi2019a ). Equations (4.10) and (4.11) form a closure valid at the large-Reynolds-number limit together with the jump condition.

Although the reduced equations may look complicated, the important point here is twofold; first, the wave fields $\widetilde{\boldsymbol{v}},\widetilde{\boldsymbol{b}}$ are generated by the ideal linear stability problem of the streak fields (4.11); second, there is no direct feedback from the wave to the streak as one can easily confirm in (4.10). Hence, the perturbation streak fields $\overline{u},\overline{a}$ are maintained indirectly through the roll fields driven by the wave field, which is induced by the instability of the streak. The three-point cyclic coupling between the wave, roll and streak fields is similar to that seen in the hydrodynamic theory (Hall & Smith Reference Hall and Smith1991; Wang et al. Reference Wang, Gibson and Waleffe2007; Hall & Sherwin Reference Hall and Sherwin2010).

Now, let us show that there is no solution if the wall is perfectly insulating. The core of the theoretical result are the following cross-energy equations similar to (4.3) but for the perturbation Elsässer streak field $\overline{\unicode[STIX]{x1D709}}_{\pm }=\overline{u}\pm \overline{a}$ :

where ${\mathcal{I}}=-\langle U_{b}^{\prime }\overline{v}\overline{\unicode[STIX]{x1D709}}_{+}\rangle$ and

Those equations can be derived by the $x$ components of (4.10). As remarked earlier, there is no energy input from the wave activity, because of the asymptotic reduction. Thus, the diffusion terms on the left-hand side of (4.14) are maintained by the roll energy input term ${\mathcal{I}}$ and the boundary terms $\unicode[STIX]{x1D6FE}_{\pm }$ .

For the insulating case, the magnetic streak $\overline{a}=\overline{\overline{a}}+\overline{g}_{z}$ must vanish on the walls, from (2.5) and (2.9). Thus there is no contribution from the the boundary terms $\unicode[STIX]{x1D6FE}_{\pm }$ . From the first equation in (4.14), we see that the streak components must satisfy the equipartition condition $\overline{u}=\overline{a}$ . We then recall that the wave component is generated through the ideal linear stability problem of the streak. Thus, we can show by the generalised version of Chandrasekhar’s theorem that there is no wave component generated from the equipartitioned streaks (see § 4.1). Without the wave, the roll component cannot be maintained, and thus ${\mathcal{I}}=0$ . Therefore, using (4.14) again, we arrive at the conclusion that the perturbation streaks $\overline{u}$ and $\overline{a}$ must be identically zero (more technical details of the discussion here can be found in appendix B).

The discussion above critically relies on the behaviour of the boundary terms in the streak energy equations (4.14). For perfectly conducting walls, $\overline{a}$ is non-vanishing on the walls, and hence the equipartitioned streaky fields are not realised, as indeed can be seen in figure 5. This may be the reason why we can compute nonlinear solutions even when $B_{0}=P_{m}=1$ .

We close this section by noting that the mechanism behind the equipartitioning of the streaky fields is somewhat similar to the Reynolds analogy seen in stratified shear flows (i.e. the mean temperature is approximately proportional to the mean streamwise velocity when the Prandtl number is close to unity). In fact, the analysis in § 3 is very similar to that in Deguchi (Reference Deguchi2017), where the vortex–wave interaction theory was applied for stratified plane Couette flow and it was found that the Reynolds analogy exactly holds. This result and the outcome in this section are good examples showing the usefulness of the asymptotic reduction in deducing further theoretical results unreachable in the full equations.