3.1. Large $H$ asymptotic analysis

Here we study the large $H$ fate of the solutions. The results of increasing $H$ from the green and blue branches in figure 1(b) are shown in figures 1(c) and 1(d), respectively. As can be seen in the figures, the generation of the magnetic field is suppressed as $H$ increases. More specifically, when $H$ is sufficiently large, the strength of the magnetic field seems to scale like $\boldsymbol {b}= O(H^{-1})$. It is not so difficult to investigate this asymptotic behaviour by a regular perturbation analysis employing $H^{-1}$ as a small parameter. We assume that all other parameters are $O(H^0)$ quantities, and use the regular asymptotic expansion for the velocity, pressure and magnetic field. The leading-order terms in the velocity and pressure expansions should be $O(H^0)$ but from the asymptotic balance of the induction equation the magnetic field must expand as $\boldsymbol {b}=H^{-1}\boldsymbol {b}_0+\cdots$.

The leading-order system can be obtained by simply setting $H=1$ in the induction equations and dropping the Lorentz force term from the momentum equations (see (2.5)). There is no feedback from the magnetic field to the velocity field – in this sense, when $H$ is large, the dynamo mechanism is mostly kinematic, though the induction equations are still nonlinear with respect to the magnetic field. The solution of the leading-order system can be found easily by applying some minor modifications to the numerical code. The asymptotic result is shown by the black solid curves in figure 1(c,d) and compared with the finite $H$ results.

In both results, we see that the magnetic field decays in agreement with the asymptotic result, but the magnetic field generated in figure 1(d) is one order magnitude larger than that in figure 1(c). This difference is also reflected in the structure of the magnetic field, as shown in figure 4. Figures 4(a) and 4(b) show the streamwise current of the small amplitude solution (figure 1c) and the large amplitude solution (figure 1d), respectively. In figure 4(a) we can see that the structure of the solution is relatively simple, mainly because the perturbation energy is concentrated in the zeroth and first Fourier modes in the $x$-direction. In contrast, the flow shown in figure 4(b) clearly has a more complicated structure in the $x$ direction.

Figure 4.

The same plots as figure 2 but for the large $H$ asymptotic solutions. Panels (a,b) correspond to the black solid branches seen in figures 1(c) and 1(d), respectively.

The structure of the former type of solution is very similar to the purely hydrodynamic solutions typically found on the edge of laminar and turbulent attractors (Wang et al. Reference Wang, Gibson and Waleffe2007). As in the hydrodynamic case, the precise dynamo mechanism of the edge-state type solutions can be understood by a large-Reynolds-number asymptotic analysis, as we shall see in the next section. The asymptotic behaviour of solutions with large amplitudes is still unclear even for the hydrodynamic case and so no in-depth discussions will be presented in this paper (see Deguchi Reference Deguchi2019b also).

4.1. Large-Reynolds-number limit of the Couette flow solutions

In this section we study the large-Reynolds-number asymptotic property of the solutions, fixing the magnetic Prandtl number and the periodic box to those used in the previous section. This means that we are considering the behaviour of vortices of about the channel width in a situation where both the hydrodynamic and magnetic Reynolds numbers are large. Note that, although the underlying philosophy of the asymptotic analysis is somewhat similar to the usual scaling arguments used in the turbulent physics community, the main question asked in the analysis is that if the assumed scaling is really mathematically consistent with the governing equations.

We consider asymptotic expansions of the physical quantities choosing the inverse of the Reynolds number as a small perturbation parameter. This is a singular perturbation problem and, in general, careful consideration is needed to find the appropriate expansions. In the current case, the expansions are similar to those obtained by Deguchi (Reference Deguchi2019a) for $H = 0$. The main question here is how large the Hall parameter $H$ should be to balance the Hall term in the asymptotic structure. We shall see shortly that the appropriate size is $H=O(R^{-2/3})$; hence, hereafter we use the rescaled Hall parameter $H_0=R^{2/3}H$.

The asymptotic theory by Deguchi (Reference Deguchi2019a) is strongly motivated by the widely accepted sustainment mechanism of coherent structures in purely hydrodynamic shear flows; the streamwise mean flow (or roll–streak) produces unstable waves, which induce feedback to the mean field through the Reynolds stresses. This loop interaction between the mean flow and the wave is called the vortex-wave interaction (Hall & Smith Reference Hall and Smith1991) or self-sustaining process (Waleffe Reference Waleffe1997) in the shear flow community, and has many similarities with the mean-field dynamo theory. The unstable wave in MHD flows involves the Alfven wave which formally becomes singular in ideal approximation. As a result of the resonant absorption at the singular point, current sheets become prominent in the flow field, as frequently seen in many MHD simulations.

The form of the outer asymptotic expansions used by Deguchi (Reference Deguchi2019a) are

(4.1a)\begin{gather} \left[ \begin{array}{c} u\\ v\\ w\\q \end{array} \right]= \left[ \begin{array}{c} \bar{u}(y,Z)\\ R^{{-}1}\bar{v}(y,Z)\\ R^{{-}1}\bar{w}(y,Z)\\ R^{{-}2}\bar{q}(y,Z) \end{array} \right] +R^{{-}7/6} \left \{ \exp({\textrm{i} \alpha (x-s_x t)}) \left[ \begin{array}{c} \tilde{u}(y,Z)\\ \tilde{v}(y,Z)\\ \tilde{w}(y,Z)\\ \tilde{q}(y,Z)\end{array} \right] +\text{c.c.} \right \}+\cdots, \end{gather}

(4.1b)\begin{gather}\left[ \begin{array}{c} a\\b\\c \end{array} \right]= \left[ \begin{array}{c} R^{{-}1/3}\bar{a}(y,Z)\\ R^{{-}4/3}\bar{b}(y,Z)\\ R^{{-}4/3}\bar{c} (y,Z)\end{array} \right] +R^{{-}3/2} \left \{ \exp({\textrm{i} \alpha (x-s_xt)}) \left[ \begin{array}{c} \tilde{a}(y,Z)\\ \tilde{b}(y,Z)\\ \tilde{c}(y,Z)\\ \end{array} \right] +\text{c.c.} \right \}+\cdots, \end{gather}

where $\alpha =2{\rm \pi} /L_x$ is the streamwise wavenumber, $Z=z-s_zt$, $s_z=R^{-1}s$ and c.c. stands for complex conjugate. The reason why those expansions are called ‘outer’ expansions is that different asymptotic expansions must be used within the resonant (or critical) layer.

In the expansions above we leave only the terms necessary to obtain the leading-order system. The first terms on the right-hand side are the leading-order part of the streamwise averaged field. Here and hereafter $\bar {u}$ and $(\bar {v},\bar {w})$ are called streak and roll, respectively, and we adopt the same terminologies for the magnetic field components $\bar {a}$ and $(\bar {b},\bar {c})$. The second terms on the right-hand side are the leading-order part of the field fluctuating in the $x$ direction. There is only the fundamental Fourier mode in the $x$-direction; the asymptotic analysis can be used to verify that all the harmonics must be higher-order terms and do not participate in the leading-order system.

A feature of the roll–streak field is that it is $x$-independent and, hence, is advected only by the rolls. The $R^{-1}$ scaling of the roll components was chosen to balance the viscous term and the advection term in the $x$-averaged equations. Similarly, the magnetic roll should be $R$ times smaller than the magnetic streak. However, the absolute magnitude of the magnetic roll–streak field cannot be determined from the outer problem, because outside the resonant layer the Lorentz force is negligible and, hence, the equations are linear with respect to the magnetic field. The analysis of the resonant layer is therefore necessary to complete the derivation of the asymptotic scaling, but before doing this we shall analyse the outer problem. Substituting the outer expansions into the governing equations (2.5) and only retaining the leading-order terms, it is easy to show that the roll–streak equations are viscous resistive as expected, i.e.

(4.2a)\begin{gather} {}[\bar{v}\partial_y +(\bar{w}-s)\partial_Z-(\partial_y^2+\partial_Z^2)] \left[ \begin{array}{c} \bar{u}\\ \bar{v}\\ \bar{w} \end{array} \right] ={-}\left[ \begin{array}{c} 0\\ \partial_y\bar{q}\\ \partial_Z\bar{q} \end{array} \right],\end{gather}

(4.2b)\begin{gather} {}[\bar{v}\partial_y +(\bar{w}-s)\partial_Z-P_m^{{-}1}(\partial_y^2+\partial_Z^2)] \left[ \begin{array}{c} \bar{a}\\ \bar{b}\\ \bar{c} \end{array} \right] -[\bar{b}\partial_y +\bar{c}\partial_Z] \left[ \begin{array}{c} \bar{u}\\ \bar{v}\\ \bar{w} \end{array} \right] \nonumber\\ =H_0 \left[ \begin{array}{c} 0\\\partial_Z(\bar{b}\partial_y\bar{a}+\bar{c}\partial_Z\bar{a}) \\ -\partial_y(\bar{b}\partial_y\bar{a}+\bar{c}\partial_Z\bar{a}) \end{array} \right], \end{gather}

(4.2c)\begin{gather}\partial_y\bar{v}+\partial_Z\bar{w}=0, \quad \partial_y\bar{b}+\partial_Z\bar{c}=0. \end{gather}

On the other hand, all the dissipative effects are negligible in the wave part. After some manipulation, they can be combined into a single equation for the pressure wave

(4.3)\begin{equation} \partial_y \left ( \frac{\partial_y\tilde{q}}{(\bar{u}-s_x)^2} \right ) +\partial_Z \left ( \frac{\partial_Z\tilde{q}}{(\bar{u}-s_x)^2} \right ) -\alpha^2\frac{\tilde{q}}{(\bar{u}-s_x)^2} =0.\end{equation}

This equation becomes singular when $\bar {u}-s_x$ vanishes; hereafter we denote this location as $y=f(Z)$. This singularity appears when the flow is viewed on the outer scale, but a close-up shows that it is actually regular due to the dissipative effects.

In order to analyse the behaviour of the outer solution around the singularity, it is convenient to introduce the coordinates $(n,l)$ attached to the curve $y=f(Z)$ in the $y$–$Z$ plane; here we denote the signed length measured along straight lines that are normal to this curve as $n$, and the arc length measured along the curve as $l$. The usual regular singular point analysis for the outer wave equations tells us that as the singularity is approached the velocity wave amplitude is increased like $n^{-1}$, whilst the magnetic wave amplitude grows at an even faster rate proportional to $n^{-2}$. Those singular behaviours of the waves are, of course, resolved by the dissipative effects acting on a smaller scale. The thin dissipative layer surrounding the singularity is exactly what we called the resonant layer, where we must use the inner expansion. The layer thickness can be found by the viscous-convective balance for the wave equations. Let us assume that the dissipation effect in the wave equations is non-negligible in a layer of thickness $\delta$ around the curve $y=f(Z)$. By balancing the advection operator $(\bar {u}-s_x)\partial _x = O(\delta )$ and the viscous operator $R^{-1}\partial _y^2=O( R^{-1}\delta ^{-2})$, we arrive at $\delta =R^{-1/3}$ which is nothing but the well-known classical critical layer thickness.

The next thing we need to do is to see how the nonlinear self-interaction of the waves amplified around the resonant layer drive the roll components. First, we note that there are two kinds of nonlinear terms in the $x$-averaged momentum equations, namely the Reynolds stress term and the Lorentz force term. The analysis of the former term is similar to the asymptotic structure derived in Hall & Sherwin (Reference Hall and Sherwin2010) for purely hydrodynamic shear flows. Let us denote the size of the velocity wave outside the resonant layer as $\rho$. Then the size of the wave inside the layer is found as $\rho \delta ^{-1}$ taking into account the effect of the singularity. Since the Reynolds stress only occurs inside the layer with thickness $\delta$, its size can be estimated to be $O((\rho \delta ^{-1})^2 \delta )$. From the requirement that this forcing should balance the viscous stress of the velocity roll of size $O(R^{-2})$, we have $\rho =\delta ^{1/2}R^{-1}=R^{-7/6}$ as seen in (4.1). Likewise, Deguchi (Reference Deguchi2019a) showed that within the resonant layer the Lorentz force affects the motion of the fluid, and, hence, the dynamo is not kinematic. The structure of this feedback effect is somewhat similar to that of the Reynolds stress term. However, as we have already seen, the magnetic wave component grows more rapidly around the singularity than the velocity counterpart. As a result, the amplitude of the magnetic wave outside the resonant layer must be $O(\delta \rho )=O(R^{-3/2})$, which is $O(\delta )$ times smaller than that for the velocity field (see (4.1)). The velocity roll field driven by those two nonlinear terms induces the streak due to the lift-up effect. The inhomogeneity that appeared in the streak field plays a crucial role in generating the waves as without it an inviscid instability is not possible.

The analysis of the $x$-averaged induction equations is similar to what we saw in the momentum equations; the waves drive the magnetic roll (poloidal) component through the electromotive force in the resonant layer, and then the omega effect induces the magnetic streak (toroidal) component, as seen in Deguchi (Reference Deguchi2019a). Furthermore, we have to take account of the Hall effect as an additional wave forcing term. To balance the electromotive force by the wave with the Ohmic dissipation by the magnetic roll within the resonant layer, the magnetic roll–streak field must be $O(\delta )$ smaller than that of the velocity field, and, hence, the magnetic streak and roll in the outer scale must be $O(R^{-1/3})$ and $O(R^{-4/3})$, respectively. The magnitude of the Hall term is now completely determined by the analysis so far, because $H=O(R^{-2/3})$ must be chosen by the ratio of the hydrodynamic roll size $O(R^{-1})$ to the streak magnetic field size $O(R^{-1/3})$ in view of the outer roll–streak equation (4.2a). When this size of $H$ is adopted, the wave forcing due to the Hall effect in the roll system is in fact non-negligible within the resonant layer. Since the effect within the resonant layer is not trivial, a detailed matched asymptotic expansion analysis is needed here.

To this end, it is convenient to convert the $y,z$ components of the fields into the $n,l$ components by introducing the unit vectors $\boldsymbol {e}_n$ and $\boldsymbol {e}_l$. The appropriate inner expansions for $u$, $a$, $\{(v,w)\cdot \boldsymbol {e}_n\}$, $\{(b,c)\cdot \boldsymbol {e}_n\}$, $\{(v,w)\cdot \boldsymbol {e}_l\}$, $\{(b,c)\cdot \boldsymbol {e}_l\}$ and $q$ are as follows: for the wave components,

(4.4a)\begin{gather} \delta^{{-}1}R^{{-}7/6}\tilde{U}_0(N,l)+\cdots,\quad \delta^{{-}2}R^{{-}3/2}\tilde{A}_0(N,l)+\cdots, \end{gather}

(4.4b)\begin{gather}R^{{-}7/6}\tilde{V}_0(N,l)+\cdots,\quad \delta^{{-}1}R^{{-}3/2}\tilde{B}_0(N,l)+\cdots, \end{gather}

(4.4c)\begin{gather}\delta^{{-}1}R^{{-}7/6}\tilde{W}_0(N,l)+\cdots,\quad \delta^{{-}2}R^{{-}3/2}\tilde{C}_0(N,l)+\cdots, \end{gather}

and $R^{-7/6}\tilde {Q}_0(l)+\cdots$, respectively, and for the roll–streak components,

(4.5a)\begin{gather} s_x+\delta \lambda(l)N+\cdots,\quad R^{{-}1/3}\gamma(l)+\cdots,\hspace{20mm} \end{gather}

(4.5b)\begin{gather}R^{{-}1}\{\bar{V}_0(l)+\delta \bar{V}_1(N,l)\}+\cdots,\quad R^{{-}4/3}\{\bar{B}_0(l)+\delta \bar{B}_1(N,l)\}+\cdots, \end{gather}

(4.5c)\begin{gather}R^{{-}1}\{\bar{W}_0(l)+\delta \bar{W}_1(N,l)\}+\cdots,\quad R^{{-}4/3}\{\bar{C}_0(l)+\delta \bar{C}_1(N,l)\}+\cdots \end{gather}

and $R^{-2}\bar {Q}_0(N,l)+\cdots$, respectively. Here $N=n/\delta$ is the stretched normal coordinate appropriate to the inner scale. The leading-order wave equations are found as

(4.6a)\begin{gather} \textrm{i}\alpha (\lambda N \tilde{U}_0 -\gamma \tilde{A}_0)+\lambda \tilde{V}_0+\lambda'N\tilde{W}_0-\gamma '\tilde{C}_0={-}\textrm{i}\alpha \tilde{Q}_0 +\partial_N^2\tilde{U}_{0}, \end{gather}

(4.6b)\begin{gather}\textrm{i}\alpha (\lambda N \tilde{W}_0 -\gamma \tilde{C}_0)={-}\partial_l\tilde{Q}_{0}+\partial_N^2\tilde{W}_{0}, \end{gather}

(4.6c)\begin{gather}\textrm{i}\alpha (\lambda N \tilde{A}_0 -\gamma \tilde{U}_0)-\lambda \tilde{B}_0-\lambda'N \tilde{C}_0+\gamma '\tilde{W}_0=P_m^{{-}1}\partial_N^2\tilde{A}_{0}, \end{gather}

(4.6d)\begin{gather}\textrm{i}\alpha (\lambda N \tilde{B}_0 -\gamma \tilde{V}_0)=P_m^{{-}1}\partial_N^2\tilde{B}_{0}, \end{gather}

(4.6e)\begin{gather}\textrm{i}\alpha (\lambda N \tilde{C}_0 -\gamma \tilde{W}_0)=P_m^{{-}1}\partial_N^2\tilde{C}_{0}, \end{gather}

(4.6f)\begin{gather}\textrm{i}\alpha \tilde{U}_0+\partial_N\tilde{V}_{0}+\partial_l\tilde{W}_{0}=0, \end{gather}

(4.6g)\begin{gather}\textrm{i}\alpha \tilde{A}_0+\partial_N\tilde{B}_{0}+\partial_l\tilde{C}_{0}=0. \end{gather}

Of the leading-order roll–streak equations, the important ones are the $N$ and $l$ components of the momentum equations, and the $l$ component of the induction equations. Integrating them in the $N$-direction, we have the conditions

(4.7a)\begin{gather} {}[\bar{Q}_0]^{\infty}_{-\infty}=\chi \int^{\infty}_{-\infty}(|\tilde{W}_0|^2-|\tilde{C}_0|^2)\,\textrm{d}N+\text{c.c.}, \end{gather}

(4.7b)\begin{gather}{}[\bar{W}_{1N}]^{\infty}_{-\infty}=\frac{\partial}{\partial l}\int^{\infty}_{-\infty}(|\tilde{W}_0|^2-|\tilde{C}_0|^2)\,\textrm{d}N+\text{c.c.}, \end{gather}

(4.7c)\begin{gather}{}[\bar{C}_{1N}]^{\infty}_{-\infty}=P_m\chi \int^{\infty}_{-\infty} (\tilde{B}_0\tilde{W}_0^*-\tilde{V}_0\tilde{C}_0^*)+P_mH_0\left (\tilde{B}_0\frac{\partial \tilde{A}_{0}^*}{\partial N}+\tilde{C}_0^*\frac{\partial \tilde{A}_{0}}{\partial l}\right )\,\textrm{d}N +\text{c.c.}, \end{gather}

where the asterisk represents complex conjugation, and $\chi (l)=-f''/(1+f'^2)^{3/2}$ is the curvature of the resonant layer.

In principle, we can find the solutions of the wave equations (4.6) by matching them to the outer solution, and use them to work out the right-hand sides of (4.7). Therefore, the upshot of the inner analysis is that we can express the jumps $[\bar {q}]^{0_+}_{n=0_-}$, $[\{(\bar {v},\bar {w})\cdot \boldsymbol {e}_l\}_n]^{0_+}_{n=0_-}$ and $[\{(\bar {b},\bar {c})\cdot \boldsymbol {e}_l\}_n]^{0_+}_{n=0_-}$ in terms of $\chi$ and the values of $\bar {u}_n$, $\bar {a}$, $\tilde {q}$ at $y=f(Z)$; in particular, when $P_m=1$, we can derive an analytic expression of the jumps as shown in Appendix A. The important point here is that the outer roll–streak equations (4.2), the outer pressure wave equation (4.3) and the jumps form a closure for the leading-order flow. Physically, the jumps are energy injections from the waves to the rolls, of which the Hall effect cannot be neglected in those resulting from the induction equation. The jump in the induction equations essentially corresponds to those empirically modelled by the alpha effect in the mean-field dynamo theory.

The scaling described above can be confirmed in the numerical solution. It has already been shown by Deguchi (Reference Deguchi2019b) for pure MHD flows that the edge-state type solutions behave in a manner consistent with the asymptotic theory. The solution at point A in figure 1(a) was used in this study. Using the same solution, here we shall check if the scaling of the Hall effect derived above can be observed. In order to obtain a sufficiently accurate asymptotic scaling in the numerical computation, the Reynolds number of the solution is increased up to $30\,000$. The computation requires higher resolution in the $y$ and $z$ directions because, as the theoretical result tells, the resonant layer gets thinner. Figure 5 shows the result of changing $H$ from point A for $R=20\,000$ and 30 000. The quantity $I_z$ defined in (2.11a,b) is one of the appropriate quantities to measure the magnitude of the magnetic roll. This component is associated with the roll field and, thus, should be scaled by $R^{4/3}$ because of the expansion (4.1). Indeed, in figure 5(a) the results for the two different $R$ collapse onto a single curve after rescaling $H$ by $R^{2/3}$, and $I_z$ by $R^{4/3}$. To check this collapse is not a coincidence, a similar comparison is made for $I_x$, as shown in figure 5(b). In this figure $I_x$ is rescaled by $R^{1/3}$ because it measures the magnitude of the magnetic streak; see (4.1). We also checked that no inconsistency with the theory was found using other physical quantities such as the energy of each component.

Figure 5.

The large $R$ convergence of the solution branch. The green dashed and blue dotted curves are computed at $R=20\,000$ and $R=30\,000$, respectively. In order to compute the high-Reynolds-number solutions we have increased the number of the Chebychev polynomials and the Fourier harmonics in the $z$ direction to 100 and 28, respectively. While thanks to the asymptotic property of the edge-state type solutions, we only need three or four Fourier harmonics in the $x$ direction.

4.2. Vertically localised states

In this section we will discuss what would happen if the sides of the periodic box $L_x$ and $L_z$, which were fixed in the previous section, are reduced in dependence on the Reynolds number. For the sake of simplicity, we fix the ratio $L_x/L_z$ and consider a situation where both $L_x$ and $L_z$ are scaled by $h\ll 1$. This is an interesting question as it is well known that fine eddies typically appear at high-Reynolds-number turbulent flows. For the purely hydrodynamic cases, a similar computation has been independently performed by Blackburn et al. (Reference Blackburn, Hall and Sherwin2013) and Deguchi (Reference Deguchi2015), where the results showed that all nonlinear effects are concentrated within a layer of depth $h$, which we hereafter call the production layer.

For the MHD problem, similar localisation can be observed as shown in figure 6. The asymptotic property of the localised solution can be easily found. Outside the production layer there is merely a linear shear which is almost the laminar Couette flow. We require that the shear $\partial _y u$ is $O(1)$ there, so $u$ must be scaled as $O(h)$ in the production layer. After rescaling $\boldsymbol {v}$, $\boldsymbol {b}$, $p$ and $H$ by $h,h,h^2$ and $h$, respectively, it is easy to see that the new Reynolds number becomes $Rh^2$. As long as this ‘effective Reynolds number’ is large, the matched asymptotic expansion analysis, as in the previous section, is possible for the nonlinear structures in the production layer. The structure of this asymptotic regime analogues to that found in Blackburn et al. (Reference Blackburn, Hall and Sherwin2013). When $h$ is $O(R^{-1/2})$, i.e. at the Kolmogorov microscale, the effective Reynolds number becomes $O(1)$, and, hence, all the terms in the governing equations must be retained, similar to the hydrodynamic version by Deguchi (Reference Deguchi2015). The maximum Hall parameter we can take is $H=O(R^{-1/2})$, which is much larger than $H=O(R^{-2/3})$ found in the previous section.

Figure 6.

The solutions computed for small $L_x$ and $L_z$. Point $D$ shown in figure 7(b) is used. All the fluctuations are localised in the production layer, as shown by the isosurfaces of the streamwise current (see figure 2 for the definition) and the streamwise magnetic field (the colour map on the back). In the upper and lower outer regions, the flow is merely a linear shear to leading order. Results are shown for (a) $R=10\,000$, (b) $R=30\,000$.

The above order-to-magnitude estimate can be refined a bit more, using the fact that the flow outside of the production layer is not completely the laminar Couette flow. Let us consider the Kolmogorov microscale states localised in the production layer, and assume that it is sandwiched by the upper and lower outer regions (see figure 3 of Deguchi Reference Deguchi2017). We denote the streamwise and spanwise drag on the wall as $D_x$ and $D_z$, respectively. Then to leading order the velocity $\boldsymbol {v}$ in the upper and lower outer regions must be written in the form

(4.8a,b)\begin{equation} (D_xy- D_x+ 1,0,D_z y - D_z)\quad \text{and}\quad (D_xy+ D_x- 1,0,D_z y + D_z), \end{equation}

respectively. Note that the mean shear should not change across the production layer, and the outer flows should satisfy the boundary conditions. Likewise using $I_x$ and $I_z$ defined in (2.11a,b), the leading-order magnetic fields $\boldsymbol {b}$ in the upper and lower outer regions are written as

(4.9a,b)\begin{equation} (I_x y - I_x, 0, I_z y - I_z)\quad \text{and}\quad (I_x y + I_x, 0, I_z y + I_z), \end{equation}

respectively.

The equations satisfied by the canonical structure within the production layer should be found using the rescaled variable $(X,Y,Z)=h^{-1}(x,y,z)$, assuming that the centre of the layer is at $y=0$ (it is easy to extend the theory to general cases). Requiring that the rescaled equations must have a unit effective Reynolds number, the appropriate scaling can be found as

(4.10a–c)\begin{equation} \boldsymbol{v}=\frac{1}{Rh}\boldsymbol{V}(X,Y,Z,T),\quad \boldsymbol{b}=\frac{1}{Rh}\boldsymbol{B}(X,Y,Z,T),\quad p=\frac{1}{R^2h^2}P(X,Y,Z,T), \end{equation}

where $T=t/(Rh^2)$. In fact, by writing $H=h\hat {H}$, the governing equations become

(4.11a)\begin{gather} (\partial_T+\boldsymbol{V}\boldsymbol{\cdot} \widehat{\boldsymbol{\nabla}})\boldsymbol{V} - (\boldsymbol{B}\boldsymbol{\cdot} \widehat{\boldsymbol{\nabla}})\boldsymbol{B} ={-}\boldsymbol{\nabla} P + \widehat{\nabla}^2 \boldsymbol{V}, \end{gather}

(4.11b)\begin{gather}\partial_T\boldsymbol{b} = \widehat{\boldsymbol{\nabla}} \times [(\boldsymbol{V}-\hat{H}\widehat{\boldsymbol{\nabla}} \times \boldsymbol{B})\times \boldsymbol{B}]+\frac{1}{P_m} \widehat{\nabla}^2 \boldsymbol{B}, \end{gather}

(4.11c)\begin{gather}\widehat{\boldsymbol{\nabla}} \boldsymbol{\cdot} \boldsymbol{V} =0,\quad \widehat{\boldsymbol{\nabla}} \boldsymbol{\cdot} \boldsymbol{B}=0, \end{gather}

where $\widehat {\boldsymbol {\nabla }}=(\partial _X,\partial _Y,\partial _Z)$. Here the periodicities of the flow in $X,Z$ become

(4.11d)\begin{equation} [\boldsymbol{V},\boldsymbol{B}](X,Y,Z,T)=[\boldsymbol{V},\boldsymbol{B}](X+L_{x0},Y,Z,T)=[\boldsymbol{V},\boldsymbol{B}](X,Y,Z+L_{z0},T), \end{equation}

with the normalised periodicity $L_{x0}=L_x/h$, $L_{z0}=L_z/h$. For the normal direction, we require that on exiting the layer the rescaled velocity tends to $Y$ to leading order so that

(4.11e)\begin{equation} \boldsymbol{V}\rightarrow (Y\mp D_{x0},0,\mp D_{z0}),\quad \boldsymbol{B}\rightarrow ({\mp} I_{x0}, 0,\mp I_{z0}) \end{equation}

as $Y\rightarrow \pm \infty$. Here $D_{x0}, D_{z0}, I_{x0}$ and $I_{z0}$ are constants determined by solving the canonical problem (4.11), and, hence, they are functions of $L_{x0}, L_{z0}, P_m$ and $\hat {H}$.

The scaling factor $h$ must be fixed by matching the production layer solutions with the outer solutions. Let us compare the outer solution (4.8a,b) rewritten in terms of the production layer variable $Y$ and the limiting behaviour of the inner solution (4.11e),

(4.12)\begin{equation} Rh(1-D_x+D_xhY)\approx Y-D_{x0}. \end{equation}

From the coefficients of $Y^0$ and $Y^1$, we have

(4.13a,b)\begin{equation} D_x\approx 1+\frac{D_{x0}}{Rh},\quad D_x\approx \frac{1}{Rh^2}, \end{equation}

respectively. Thus, to leading order the drag $D_x$ is unity, and $h=R^{-1/2}$ as we obtained earlier. The formal asymptotic analysis yields that within the production layer the leading-order flow satisfies (4.11), and creates the small jump in the streamwise velocity across the layer. This jump is the origin of the small next-order correction ${D_{x0}}/{R^{1/2}}$ to the drag.

However, in order to observe the asymptotic convergence of the finite-Reynolds-number result, it is more convenient to use the scaling factor

(4.14)\begin{equation} h=\frac{1}{\sqrt{D_{x}R}}\end{equation}

directly deduced from the second equation in (4.13a,b). The difference to the previous one is that some higher-order terms seen in the approximation of $D_x$ are incorporated into the scaling. This is beneficial in observing a better asymptotic convergence in the numerical result, because some of the next-order correction terms in the asymptotic analysis can be retained. Figure 7(a) is the numerical results obtained by reducing $L_x$ and $L_z$ from the solution seen in figure 5, and figure 7(b) is their rescaled version. Here, we consider the asymptotic convergence of $I_x$. The scaling is motivated by the fact that $I_{0x}$ appeared in (4.11e) is a function of $L_{z0}=L_z/h$. Comparing the outer solution (4.9a,b) and the limiting inner solution (4.11e), we have

(4.15)\begin{equation} Rh({-}I_x+I_xhY) \approx{-}I_{0x}, \end{equation}

from which to leading order we have $Rh I_x=I_{0x}$. In figure 7(b) the horizontal and vertical axes are $1/L_{0x}$ and $I_{0x}$, and, hence, as expected, we can observe the convergence towards the canonical result with increasing the Reynolds number. In this figure, $h$ is the one defined in (4.14). If we use the simpler $h=R^{-1/2}$, the convergence will be worse and we will need to use a larger Reynolds number.

Figure 7.

(a) The behaviour of the solutions for small $L_x$ and $L_z$. The ratio $L_x/L_z$ is fixed to 2 in this figure. The solution branch is continued from that shown in figure 5. The red solid, green dashed and blue dotted curves are the result for $R=10\,000$, 20 000 and 30 000, respectively. Panel (b) is the same result but with the rescaling obtained by the theory. Typically 180 Chebyshev polynomials are used in $y$, while in the $x$ and $z$ directions up to 6th and 24th harmonics are retained, respectively.

4.3. The high-Reynolds-number solution in a shearing box

The fact that the nonlinear interaction is maintained even away from the wall suggests that the theoretical result so far can be extended to more general situations. For example, the localised canonical structure can be embedded for any $y$ by using the Galilean transform to make the streamwise phase velocity almost the same as the laminar velocity. Moreover, we can even embed multiple nonlinear solutions in the flow, as long as they are far enough apart from each other.

In the case the coherent structures are placed closely, we need to include some interaction between them. For effective Reynolds number of order unity, the simplest way to incorporate such an interaction is to use a framework called the shearing box, which is a common way to study distributed coherent structures (Schumann Reference Schumann1985; Gerz, Schumann & Elgobashi Reference Gerz, Schumann and Elgobashi1989; Brandenburg et al. Reference Brandenburg, Nordlund, Stein and Torkelsson1995; Hawley, Gammie & Balbus Reference Hawley, Gammie and Balbus1995). The flow configuration considered in this framework is similar to plane Couette flow but the walls are replaced by the shear-periodic condition – namely the flow is required to be invariant by moving the flow in the $y$ direction for one period and then Galilean transforming it in the $x$ direction for the same period. Finding nonlinear solutions in the shearing box at high Reynolds numbers is not an easy task (Sekimoto & Jiménez Reference Sekimoto and Jiménez2017). However, when the effective Reynolds number is sufficiently large, Blackburn et al. (Reference Blackburn, Deguchi and Hall2021) recently showed that the asymptotic theory by Hall (Reference Hall2018) can be used to compute the solutions more easily. Here we shall briefly see that such a computation is also possible for MHD flows.

Figure 8 shows the MHD dynamo solution found by this method. The computation can be done by applying a minor modification to the existing numerical code as follows. First, as we are assuming a high effective Reynolds number of the flow, we can apply the asymptotic theory seen in § 4.1. We treat the roll–streak and wave components of the flow separately. For the shearing box problem, the roll–streak component ($x$-independent part) should be described by the sum of the laminar homogeneous shear and the perturbation field which is periodic in the $y$ direction. Therefore, in the numerical computation, we simply need to replace the Chebyshev polynomials with Fourier modes for this component. In figure 8 we plotted 1.5 periods of the solution in $y$. As can be seen in the figure, there are two symmetrically arranged waves in one period. Looking at the central half-period box, we can see that the basic structure of the solution is very similar to that found in the Couette flow (see figure 2a). Recall that the wave is induced by the instability of the streak field. The streak field is not periodic due to the inclusion of the laminar homogeneous shear flow, and, thus, the generated wave component is not periodic in $y$. However, the shear-periodic condition is nevertheless satisfied by the periodic arrangement of the waves with different speeds appropriate to each half-periodic cell (see Blackburn et al. Reference Blackburn, Deguchi and Hall2021 for detail). Here we computed only the central wave and used its copies to calculate the periodic feedback to the roll component. The amplitude of the central wave decreases rapidly as moving away from the half-periodic box at the centre. Practically, it is sufficient to use the computational domain about three times larger than the roll–streak period, so again the modification to the numerical code is not too difficult.

Figure 8.

The dynamo solution in the shearing box at $R=10\,000$. See figures 2 and 6 for the definition of the surfaces and the colour map. Translucent surfaces are used except for the central wave. The dimension of the box shown is $[0, 2{\rm \pi} ]\times [-3,3]\times [0,{\rm \pi} ]$. The wave is resolved by using 260 Chebyshev polynomials in $y$, and taking up to 20th Fourier harmonics in $z$. For the roll–streak component up to 60th Fourier harmonics are used in $y$. Note that in this figure, the size of the central half-period box is the same as that used in figures 2 and 4.

4.4. Induction through the free-stream coherent structure mechanism

When a localised solution such as the one discussed in § 4.2 occurs in a boundary layer flow, an interesting phenomenon occurs. The results in this section are an extension of the purely hydrodynamic study by Deguchi & Hall (Reference Deguchi and Hall2014b), where it is shown that a small three-dimensional equilibrium solution placed in the vicinity of the free stream has the ability to produce even larger disturbances in the boundary layer (see figure 2 of Deguchi & Hall Reference Deguchi and Hall2017). This amplification mechanism is due to the lift-up effect where the roll drives the streak. Here we shall see in the MHD version that the growth also occurs in the magnetic field through the omega effect.

For the sake of simplicity, here we consider the asymptotic suction boundary layer flow with the base flow profile $\boldsymbol {v}=(1-\textrm {e}^{-y},-R^{-1},0)$, where $y$ is the distance from the wall. For large $y$, the flow approaches a uniform free stream. We impose periodicity in the $x$ and $z$ directions with the fixed periods $L_x$ and $L_z$, respectively. The key point of the theory is that the unit effective Reynolds number nonlinear states are operational without changing the length scale, if the velocity is scaled like $O(R^{-1})$ in the Navier–Stokes equations. Of course the question is whether the base flow will allow this scaling. Deguchi & Hall (Reference Deguchi and Hall2014b) showed for the asymptotic suction boundary layer flow that such a situation indeed occurs when the base shear $\textrm {e}^{-y}$ becomes $O(R^{-1})$, i.e. when $y$ is $O(\ln R)$ (note that a Galilean transformation can be used to eliminate the constant part so we only need to check the size of the base shear). This is the location of the production layer, which in this case coincides with what is known as the edge layer in the boundary layer transition studies (Duck, Ruban & Zhikharev Reference Duck, Ruban and Zhikharev1996). Some readers may think that the theory is probably only applicable to the simple model flow, but note that for any boundary layer flows, the effective Reynolds number decreases as it approaches the free stream. The truth is that, somewhat surprisingly, the same production layer solution in the asymptotic suction boundary layer is generic to many non-parallel boundary layer flows (Deguchi & Hall Reference Deguchi and Hall2015) and jet flows (Deguchi & Hall Reference Deguchi and Hall2018).

Let us consider a pure MHD flow ($H = 0$) in the asymptotic suction boundary layer flow. We assume that the wall is insulating as in the previous sections. The production layer problem (4.11) with $\hat {H}=0$ can be found by applying the coordinate transform $(X,Y,Z,T)=(x-t,y-\ln R,z,R^{-1}t)$, and writing

(4.16a)\begin{gather} \boldsymbol{v}=(1,0,0)+R^{{-}1}\boldsymbol{V}(X,Y,Z,T)+\cdots, \end{gather}

(4.16b)\begin{gather}p=R^{{-}2}P(X,Y,Z,T)+\cdots,\quad \boldsymbol{b}=R^{{-}1}\boldsymbol{B}(X,Y,Z,T)+ \cdots \end{gather}

in the governing equations (2.5). The boundary conditions for the production layer problem are the periodic conditions

(4.17)\begin{equation} [\boldsymbol{V},\boldsymbol{B}](X,Y,Z,T)=[\boldsymbol{V},\boldsymbol{B}](X+L_{x},Y,Z,T)=[\boldsymbol{V},\boldsymbol{B}](X,Y,Z+L_{z},T), \end{equation}

and the far-field conditions

(4.18a)\begin{gather} (U,V,W,A,B,C)\rightarrow (0,-1,0,0,0,0)\quad \text{as} \ Y\rightarrow \infty, \end{gather}

(4.18b)\begin{gather}(Ue^Y,V,W,Ae^Y,B,C)\rightarrow ({-}1,-1,0,0,0,0)\quad \text{as} \ Y\rightarrow -\infty. \end{gather}

The latter conditions require that, for large negative $Y$, the flow goes back to the laminar flow to leading order. However, note that (4.18b) allows some exponential growth of perturbation at slower rate than that of the base flow.

As the nonlinearity is assumed to be localised within the production layer, the growth of the perturbation as $Y\rightarrow -\infty$ can be seen by analysing the linearised equations. As shown in Deguchi & Hall (Reference Deguchi and Hall2014b), the $x$ dependent part must be more rapidly damped than the roll–streak component in this limit. Furthermore, among the roll components the harmonics decay more rapidly than the fundamental mode as they feel the viscous effect more strongly. Let us write the wavenumber of the fundamental mode as $\beta$ (note that it does not necessarily equal to ${2{\rm \pi} }/{L_z}$, see Deguchi & Hall Reference Deguchi and Hall2014b, Reference Deguchi and Hall2017). Then the limiting behaviour of $(V,W)$ and $(B,C)$ as $Y\rightarrow -\infty$ may be written as

(4.19a,b)\begin{equation} (\mathcal{V}(Y)\cos \beta z,\mathcal{W}(Y)\sin \beta z)+\cdots \quad \text{and}\quad (\mathcal{B}(Y)\cos \beta z,\mathcal{C}(Y)\sin \beta z)+\cdots, \end{equation}

respectively. Here the functions $\mathcal {V}(Y)$ and $\mathcal {B}(Y)$ must satisfy

(4.20a)\begin{gather} (\partial_Y+\partial_Y^2-\beta^2)(\partial_Y^2-\beta^2)\mathcal{V}=0, \end{gather}

(4.20b)\begin{gather}(\partial_Y+P_m^{{-}1}(\partial_Y^2-\beta^2))\mathcal{B}=0, \end{gather}

from the linearised roll equations, and $\mathcal {W}(Y)=-\beta ^{-1}\mathcal {V}'$, $\mathcal {C}(Y)=-\beta ^{-1}\mathcal {B}'$ from the continuity and Gauss's law. Solving (4.20) and looking for the slowest decaying mode we find that the leading-order behaviour of the roll components

(4.21a,b)\begin{equation} \mathcal{V}(Y)\sim K\textrm{e}^{\omega Y}, \quad \mathcal{B}(Y)\sim J\textrm{e}^{\gamma Y} \end{equation}

as $Y\rightarrow -\infty$. Here

(4.22a,b)\begin{equation} \omega=\frac{1}{2}\left(\sqrt{1+4\beta^2}-1\right)>0,\quad \gamma=\frac{1}{2}\left(\sqrt{P_m^2+4\beta^2}-P_m\right)>0 \end{equation}

and $J$ and $K$ are constants determined by the production layer solution. The reason for discarding the exponentially increasing mode in (4.20a) is that if they were present, nonlinearities would appear, violating the assumption. The roll component is hence attenuated, but it interacts with the base flow through the lift-up term to promote the streak growth. Importantly, this growth does not produce any nonlinear effect. Let us write the leading-order behaviour of $U$ and $A$ as

(4.23a,b)\begin{equation} \mathcal{U}(Y)\cos \beta z+\cdots \quad \text{and}\quad \mathcal{A}(Y)\cos \beta z+\cdots, \end{equation}

respectively. The behaviour of the functions $\mathcal {U}(Y)$ and $\mathcal {A}(Y)$ can be found from the linearised streak equations

(4.24)\begin{align} (\partial_Y+\partial_Y^2-\beta^2)\mathcal{U}=\textrm{e}^{{-}Y}\mathcal{V}, \end{align}

(4.25)\begin{align} (\partial_Y+P_m^{{-}1}(\partial_Y^2-\beta^2))\mathcal{A}={-}\textrm{e}^{{-}Y}\mathcal{B}, \end{align}

from which we can find

(4.26a,b)\begin{equation} \mathcal{U}(Y)\sim{-}\frac{K\textrm{e}^{(\omega-1) Y}}{2\omega},\quad \mathcal{A}(Y)\sim \frac{P_m J \textrm{e}^{(\gamma-1) Y}}{P_m+2\gamma -1}, \end{equation}

as $Y\rightarrow -\infty$. Therefore, the growth of the hydrodynamic streak occurs when $\omega -1<0$, i.e. $\beta <\sqrt {2}$ as seen in Deguchi & Hall (Reference Deguchi and Hall2014b), while the condition for the magnetic field growth can be found as

(4.27)\begin{equation} \beta<\sqrt{1+P_m}, \end{equation}

which ensures $\gamma -1<0$. Moreover, $({\partial }/{\partial P_m})(\gamma -\omega )<0$ implies that

(4.28)\begin{align} \gamma > \omega \quad \text{if} \ P_m<1, \end{align}

(4.29)\begin{align} \gamma \leq \omega \quad \text{if} \ P_m\geq 1. \end{align}

Therefore, when $P_m>1$ ($P_m< 1$), the magnetic streak grows faster (slower) than the hydrodynamic part.

The leading-order field in the boundary layer can be found by seeking a linearised roll–streak solution that matches the above far-field behaviour of the production layer solution when $y$ is large and satisfies the boundary conditions at $y=0$. The hydrodynamic part of the leading-order boundary layer solution can be seen in Deguchi & Hall (Reference Deguchi and Hall2014b). Similarly, after some algebra, the leading-order magnetic field can be found as

(4.30)\begin{equation} a=R^{-\gamma}F(y)\cos \beta z+\cdots,\end{equation}

where

(4.31)\begin{equation} F(y)=P_mJ\left \{ \frac{(\textrm{e}^{(\gamma-1)y}-\textrm{e}^{-(\gamma+P_m)y})}{P_m+2\gamma-1} +\frac{(\beta-\gamma)(\textrm{e}^{-(P_m+\gamma+1)y}-\textrm{e}^{-(\gamma+P_m)y})}{(P_m+1)(P_m+\gamma+\beta)(2P_m+2\gamma+1)}\right \}.~~~~~~ \end{equation}

The function $F(y)$ is shown in figure 9(a) for $\beta =1$ and various $P_m$. For this value of $\beta$, consistent with (4.27), no matter how small $P_m$ is the growth of the magnetic streak occurs towards the wall. Moreover, the larger the value of $P_m$ is, the more rapidly the magnetic field increases. Note that the behaviour of the hydrodynamic streak is similar to the $P_m=1$ result shown in this picture, as highlighted by the thick green dashed curve. When the value of $\beta$ is increased to $2$, as shown in figure 9(b), the growth of the hydrodynamic streak is suppressed because the growth condition $\beta <\sqrt {2}$ is violated. Nevertheless, when $P_m>3$, the growth of the magnetic streak occurs; see (4.27).

Figure 9.

The near-wall perturbation driven by a small nonlinear perturbation in the production layer at $y=\ln R$. The amplitude $F(y)$ of the magnetic field in the boundary layer for $P_m=0.2$, 0.6, 1.0, 1.4, 1.8, 2.2, 2.6, 3.0, 3.4, 3.8 and 4.2. The result for $P_m=1$ is highlighted by the green dashed curve. Results are shown for (a) $\beta =1$, (b) $\beta =2$.

Now we include the Hall effect in the flow, considering $H>0$. A simple order-to-magnitude analysis shows that as long as $H$ is smaller than $O(1)$, the leading-order structure of the production layer remains unchanged. However, even if $H$ is small, the structure of the amplified magnetic field below the layer will be affected. To see this, we treat $H$ as a perturbation to the asymptotic structure obtained for the pure MHD. For example, the magnetic field can be expanded as

(4.32)\begin{equation} \boldsymbol{b}=R^{{-}1}\{\boldsymbol{B}(X,Y,z)+H\boldsymbol{B}_1(X,Y,z)+\cdots\}. \end{equation}

In order to analyse the region beneath the production layer, it is convenient to introduce the new stretched variable $\xi =y/(\ln R)$, where $\xi =1$ and 0 correspond to the locations of the production layer and the boundary layer, respectively. The solution we developed earlier is valid for $\xi \in (\xi _c,1)$, where $\xi _c$ is the position at which the leading-order structure breaks down due to the Hall effect. The scenario of the breakdown is as follows. (i) First we note that a growing $y$-$z$ mean magnetic field can be generated by the electromotive force. The size of the forcing can be found by the interaction of the magnetic streak and the hydrodynamic roll. (ii) The magnetic roll component in $\boldsymbol {B}_1$ is driven by the Hall term, whose leading-order part comes from the interaction of the mean magnetic field and the magnetic roll in $\boldsymbol {B}$. This forcing can grow beneath the production layer, and, thus, the next-order magnetic roll can also grow towards the wall. (iii) In summary, the magnetic roll component in $\boldsymbol {B}$ should decay, but that in $\boldsymbol {B}_1$ can grow. The breakdown of the asymptotic expansion occurs when $\boldsymbol {B}$ and $H\boldsymbol {B}_1$ become the same size. At this breakdown point, we need to solve a nonlinear induction equation, similar to what we saw in § 3.1. The nonlinearity suppresses the growth of the magnetic field for $\xi <\xi _c$.

To determine $x_c$, the size of each component appeared in the above argument needs to be identified as follows.

1. (i) We already know the behaviour of the leading-order roll–streak as seen in (4.21a,b) and (4.26a,b). Using $Y=(\ln R)(\xi -1)$ and recalling (4.16), the size of them in terms of the original variables $\boldsymbol {v},\boldsymbol {b}$ can be found as shown in table 2. For example, the size of the magnetic streak is $O(R^{-1}\textrm {e}^{(\gamma -1)Y})=O(R^{(\xi -1)(\gamma -1)-1})$. At the boundary layer ($\xi =0$), the size is $O(R^{-\gamma })$ which is consistent with our earlier result (4.30).

2. (ii) Next, we estimate the size of the $x$–$z$ averaged part of the perturbation. The mean magnetic field is driven by the electromotive force. The behaviour of this forcing term can be found as $\textrm {e}^{(\omega +\gamma -1)Y}$ by multiplying the magnetic streak and the hydrodynamic roll. The mean field behaves like this forcing as $Y\rightarrow -\infty$. The behaviour of the hydrodynamic part can be found in a similar manner by noting that it is driven by the Reynolds stress and the Lorentz force.

3. (iii) Finally, we consider the roll component in $\boldsymbol {B}_1$. This component satisfies an equation similar to (4.25), but forced by the Hall term. As the leading-order part of the forcing comes from the interaction of the magnetic mean field and the magnetic roll component, it behaves like $\textrm {e}^{(2\gamma +\omega -1)Y}$. The size of the next-order roll in terms of the original variable is $O(R^{-1}He^{(2\gamma +\omega -1)Y})=O(HR^{(\xi -1)(2\gamma +\omega -1)-1})$.

Table 2.

The size of the perturbation beneath the production layer. Here $\xi =1$ is the location of the production layer, and $\xi =0$ corresponds to the boundary layer. Note that the hydrodynamic components do not include the base flow. The roll, streak and mean components refer to the poloidal, toroidal, and $x$–$z$ averaged fields, respectively.

The asymptotic expansion breaks down when $O(R^{(\xi -1)\gamma -1})=O(HR^{(\xi -1)(2\gamma +\omega -1)-1})$. The critical value $\xi _c$ can be found by solving $R^{(\xi _c-1)\gamma -1}=HR^{(\xi _c-1)(2\gamma +\omega -1)-1}$, i.e.

(4.33)\begin{equation} \xi_c=\frac{\ln (HR^{1-\gamma-\omega})}{\ln (R^{1-\gamma-\omega})}. \end{equation}

The above discussion holds even when the effective Reynolds number in the production layer is large, but the scaling becomes a bit more complicated.