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Kengo Deguchi
Abstract

This paper aims to numerically verify the large Reynolds number asymptotic theory of magneto-hydrodynamic flows proposed in part 1 (Deguchi 2018). In order to avoid complexity associated with chaotic nature of turbulence and flow geometry, nonlinear steady solutions of the viscous-resistive magneto-hydrodynamic equations in plane Couette flow are concerned. Two classes of dynamo states, which convert kinematic energy to magnetic energy effectively, are found. The first class of dynamos can be obtained when a small spanwise uniform magnetic field is applied to the known hydrodynamic solution branch of plane Couette flow. The dynamos are characterised by the hydrodynamic/magnetic roll-streak and the resonant layer at which strong vorticity and current sheets are observed. Those flow features, and the induced strong streamwise magnetic field, are fully consistent with the vortex/Alfvén wave interaction theory proposed in part 1. When the spanwise uniform magnetic field is switched off, the solutions become purely hydrodynamic. However, it turns out that dynamos can actually be found at the zero-external magnetic field limit using different form of external field. That second class of ‘self-sustained shear driven dynamos’ can be found by the homotopy via the forced states subject to a spanwise uniform current field. The discovery of purely shear driven dynamos motivated the corresponding matched asymptotic analysis in part 1. Here the reduced equations due to the asymptotic theory are solved numerically. The asymptotic solution is found to give remarkably good predictions for the finite Reynolds number dynamo solutions.

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High-speed shear driven dynamos. Part 2. Numerical analysis]High-speed shear driven dynamos.
Part 2. Numerical analysis

1 Introduction

Our concern is with the large Reynolds number development of unstable invariant solutions underpinning turbulent magneto-hydrodynamic (MHD) flows. In recent years dynamical systems theory picture of turbulence has been much studied among fluid dynamics researchers. The Naver-Stokes equations become a dynamical system after spatial discretisation, and it was uncovered by many researchers that unstable invariant solutions of the equations can be regarded as ÔskeletonsÕ on which the chaotic dynamics of turbulence hang. Indeed some solutions reproduce remarkable statistical properties of turbulent flows, although such unstable solutions themselves are never realised in the flow dynamics obtained by direct numerical simulations or experiments; see the review by Kawahara et al. (2012) and the references therein. This is exactly what can be expected from famous Lorenz’s toy model computation of atmospheric convection (Lorenz 1963), but the corresponding computations of the Navier-Stokes equations are not as easy as this.

In practice, unstable invariant solutions can be captured by the multi-dimensional Newton method, and thus exactly satisfy the full fluid dynamic equations within numerical accuracy. Early years of the computations were motivated to explain finite amplitude (subcritical) transition in plane Couette flow. Laminar plane Couette flow is always linearly stable but transition to turbulence occurs at moderate Reynolds numbers by some perturbations characterised by streamwise roll-streak. This apparent contradiction has been resolved by relating the nonlinear three-dimensional solutions and the subcritical transition using the dynamical systems theory picture. Finding solutions in linearly stable system is a tricky task, as we cannot use the bifurcation analysis from a linear critical point to find the first solution. In the latter case we can consider the continuous (homotopic) deformation from the augmented system where a nonlinear solution is known to the target system, wishing that the solution branch could be continued throughout the deformation. The challenge to find nontrivial solutions in plane Couette flow began in Nagata’s PhD project with his supervisor Busse, who proposed to use buoyancy as a homotopy parameter. After some unsuccessful attempts, the first nonlinear solutions were found in Nagata (1990) where Colioris force is chosen as a homotopy parameter, while their original approach has been accomplished in Clever & Busse (1992); nowadays the solutions they found are called the Nagata-Busse-Clever (NBC) solutions. Subsequently existence of myriad of solutions in the flow has been uncovered; see Schmiegel (1999), Waleffe (2003), Itano & Generalis (2009), Gibson et al. (2009). As remarked earlier, the importance of those solutions and some nearby dynamics in fully developed turbulence has been pointed out repeatedly by Kawahara & Kida (2001), Gibson et al. (2008), van Veen & Kawahara (2011), Kreilos & Eckhardt (2012). Moreover in bistable systems such as plane Couette flow, some solutions play a gatekeeper role at the edge of the laminar and turbulent attractors; see Itano & Toh (2001), Skufca, Yorke & Eckhardt (2006), Wang, Gibson & Waleffe (2007), Deguchi & Hall (2016).

The numerical studies also unveiled the physical interpretation of the sustainment mechanism of roll-streak, called self-sustaining process (Waleffe 1997, Wang, Gibson & Waleffe 2007). Before the numerical computation of the three-dimensional solutions, the appearance of streamwise roll in plane Couette flow was rather puzzling, because it is easy to show by the energy balance that two-dimensional streamwise independent roll cannot be sustained by its own right. This means that in order to support the roll component some three dimensionarity of the flow is necessary. The self-sustaining process explains the origin of the three-dimensional component by the wave-like instability of the streak, which is the mean flow modulated by the roll via the lift-up mechanism.

There is another similar but more mathematical research stream, called the vortex/wave interaction theory (Hall & Smith 1991), originally formulated for boundary layer flows rather than plane Couette flow. While both the vortex/wave interaction and self-sustaining process theories concern the Reynolds number scaling of streamwise vortices when that number is large, there is nonetheless an important difference. Formally, the wave stability problem is inviscid when the large Reynolds number limit is taken and thus produces a singularity at the critical layer, where the wave speed coincides with the streak speed. The vortex/wave interaction approach use the method of matched asymptotic expansion to analyse the flow around the singularity, and found that the amplified wave there causes the major feedback effect to the roll. On the other hand, the self-sustaining process retains the viscous effect in the wave equations so the importance of the critical layer had been overlooked until Wang et al. (2007) where the critical layer structure was numerically found. Motivated by the latter work, Hall & Sherwin (2010) computed the reduced system obtained by the vortex/wave interaction theory for the first time for plane Couette flow, and found that the extrapolated vortex/wave interaction result from infinite Reynolds number reproduces the Navier-Stokes result by Wang et al. (2007) even at moderate Reynolds numbers. In order to solve the singular problem Hall & Sherwin (2010) used certain technical regularisation method. Subsequently, much simpler regularisation based on the ‘fictitious viscosity’ has been adopted (Blackburn et al. 2013; Deguchi & Hall 2014b, 2016). That hybrid approach, carried forward by the earlier work Hall & Horseman (1991), led identical formulation to the self-sustaining process as a result, but an important point to note is that now that formulation is backed up by the fully rational asymptotic analysis due to Hall & Smith (1991).

Asymptotic development of invariant solutions are perhaps of enormous importance as the scaling of the flow dynamics in terms of the Reynolds number has been the central interest in fluid dynamic study. An innovative combination of two techniques, asymptotic analysis and unstable invariant solutions, was then employed in Deguchi et al. (2013), Deguchi & Walton (2013a, 2013b, 2018), Deguchi & Hall (2014a, 2014b, 2014c, 2015), Deguchi (2015, 2017), Dempsey et al. (2016), Ozcakir et al. (2016), sparked by the excellent agreement seen in Hall & Sherwin (2010). The advantage to use the dual approach is that it is particularly useful for confirming or finding new asymptotic theories, because the simple structure of unstable invariant solutions enable a clean quantitative comparison of the theories with the full numerical results. Moreover, some invariant solutions can be found at very high Reynolds numbers and can produce remarkably accurate Reynolds number asymptotic scaling. For example, in a channel flow the nonlinear solutions computed in Dempsey et al. (2016) reach Reynolds number of order , which is much higher than the maximum Reynolds number available by direct numerical simulations or experiments.

In this paper, we shall extend those above purely hydrodynamic studies to MHD flows. So far invariant solutions in MHD equations have been studied in the flow under Kepler rotation, which is linearly stable as well. There is much astrophysical interests in that flow because it may model flows around certain celestial bodies. Balbus & Hawley (1991) introduced an external magnetic field to bring a linear instability mechanism, known as magneto-rotational instability, in the flow. However the unknown precise origin of that external magnetic effect at the beginning of the astrophysical flow formation posed the unanswered question of whether the subcritical transition plays an important role in the instability mechanism. To this end, the first invariant nonlinear dynamo solution without any external magnetic effect was found in MHD rotating plane Couette flow by Rincon et al. (2007), and their results were subsequently extended to rotating shearing box; see Riols et al. (2013). The dynamo solutions found by those authors are all three-dimensional consistent to Cowling’s anti-dynamo theorem (Cowling 1934), which states that there is no two-dimensional dynamo states. Inspired by the self-sustaining process, the origin of three-dimensionality was explained by the magneto-rotational instability triggered by the induced magnetic streak (Rincon et al. 2008). However, unfortunately their solutions are found to be very difficult to continue large Reynolds number regime of astrophysical importance, opposed to the purely hydrodynamic plane Couette flow counterparts.

In part 1 of this paper, it was shown that the straightforward extension of the vortex/wave interaction theory to MHD flows is possible, but the asymptotic state should be destroyed when a strong rotating effect presents. Therefore, here we only focus on the dynamo states driven in non-rotating shear flows as we are interested in high Reynolds number regime. To date, numerical computation of such non-rotating shear driven dynamos have received surprisingly less attention (see recent numerical study by Neumann & Blackman 2017 and references therein), although such energy conversion mechanism via three-dimensional magnetic field is common in astrophysics and solar physics. The key idea used in part 1 is to use the Alfvén wave instability as a driving mechanism of the roll-streak field now defined for both velocity and magnetic fields. The Alfvén wave is stimulated by inhomogeneous hydrodynamic and magnetic streaky fields, and possesses the singularity known as Alfvén resonant point; see Sakurai et al. (1991), Goossens et al. (1992) for example. Near the resonant point the wave is amplified to produce strong vortex and current sheets, which in turn produce a feedback effect to the hydrodynamic and magnetic roll-streak. However, an important caveat found in the asymptotic analysis of part 1 is that the vortex/Alfvén interaction cycle above is not realised unless there is an weak externally applied magnetic field.

After formulating our problem in the next section selecting MHD plane Couette flow as a canonical flow configuration, we begin our computation in section 3. The excitation of three-dimensional magnetic field can be found by applying a uniform unidirectional magnetic or current field to the NBC solutions. In particular, the small uniform spanwise magnetic field is one of the most appropriate choices to drive the vortex/Alfvén interaction states. The invariant solutions weakly forced in this way indeed produce much stronger streamwise magnetic field, consistent to the theoretical analysis in part 1. In section 4 we use an external uniform spanwise current field as a homotopy parameter to show that we can compute the invariant dynamo solution branch even at the zero external magnetic field limit. Motivated by this rather surprising result, another asymptotic theory was formulated in part 1 to explain the presence of that self-sustained shear driven dynamos, S dynamos for short. In the same section the corresponding asymptotic problem is to be numerically solved using the hybrid approach. Finally, in section 5, we draw some conclusion.

2 Computational method

Consider electricaly conducting fluid flow between the perfectly insulating walls of infinite extent placed at . For directions we assume periodicity with the wavenumbers , respectively. Within that computational box, we numerically solve incompressible viscous resistive MHD equations non-dimensionalised similar manner as part 1

(2.0a)
(2.0b)
(2.0c)
(2.0d)

where , and is the total pressure. The flow is driven by the given base flow , which represents the laminar flow solution of the system and is function of only. More specifically, we consider plane Couette flow , forced by some external magnetic field. In that case the hydrodynamic Reynolds number and the magnetic Reynolds number are defined using the half channel height and the wall speed as the length and velocity scales, respectively.

The perturbation to the base flow must satisfy the no-slip and insulating conditions on the walls. In order to satisfy the latter conditions, the magnetic perturbation must match to the magnetic field outside of the computational domain. There should be no current in the outer region so the outer magnetic field must have a potential so that . Hence from the solenoidality we must solve the Laplace equation for the potential for and requiting that the potential decays in the far-field .

For the sake of simplicity here we only consider perturbations of travelling wave form with the streamwise and spanwise wave speed , , respectively. It is convenient to use the coordinate attached to the travelling wave so that we have a steady problem. In the transformed coordinate the operator in (2) must be replaced by ; note that this means that here and hereafter the coordinates are redefined.

The divergence free fields in the periodic box can be written by the toroidal-poloidal potentials as

(2.0)

Here the double overline represents the average

(2.0)

Thus the components are functions of only and correspond to the mean components, which can further be decomposed into the base flow and the mean perturbation as

(2.0a)
(2.0b)

The equations for the potentials and the mean flows can be obtained from the fluctuations parts of the equations , , , , and the mean parts of the equations , , , . Now we substitute the Fourier expansions with the basis

(2.0a)
(2.0b)

(note that there is no mean component by definition) to those equations and operate

(2.0)

to discretise the equations in and . For the fluctuating parts the discretised equations are obtained as

(2.0a)
(2.0b)
(2.0c)
(2.0d)

where the summations are taken for , with the nonlinear terms shown in Appendix A. Here we have used the shorthand notations

(2.0a)
(2.0b)

and the Fourier transformed variables are abbreviated like

(2.0)

Likewise, the mean part of the Fourier transformed equations are obtained as

(2.0a)
(2.0b)
(2.0c)
(2.0d)

where the summations are taken for , .

In order to further discretise the equations in , we expand the Fourier coefficients by modified Chebyshev polynomials. No-slip conditions on the walls

(2.0)

are satisfied by using the basis functions

In order to find the appropriate basis for the magnetic part we need some analysis of the Fourier transformed outer magnetic potential

(2.0)

The potential satisfying the Laplace equation with the required far-field conditions can be found as

(2.0)

where are constants and . As mentioned above the magnetic perturbation of the flow must match to this outer field on the walls. Thus the mean part should vanish and the fluctuating parts satisfy

(2.0a)
(2.0b)
(2.0c)

at . Therefore, we finally get the boundary conditions

(2.0)

which is exactly the narrow-gap limit version of the insulating conditions used for Taylor-Couette flow; see Roberts (1964), Willis & Barenghi (2002), Rudiger et al. (2003). The choice of the basis functions

ensures (2.0).

For the numerical purpose we truncate the expansion and only use the coefficients , , . Evaluating (2) and (2) at the collocation points

(2.0)

the resultant equations with the Fourier discretisation parameters and constitutes the algebraic equations for the spectral coefficients and the phase speeds . Note that about half of the coefficients/equations are redundant because of the reality conditions , and that the two extra equations needed for the phase speeds are some conditions that can eliminate the freedom of the solution associated with the arbitrary shift in and .

The algebraic equations are quadratic so the exact expression of the Jacobean matrix can be computed straightforwardly. The range of numerical resolutions we used are . The resolutions and the convergence in the iteration are checked as Deguchi, Hall & Walton (2013).

3 Dynamos with external magnetic fields

We use plane Couette flow with throughout the computation of this paper. In the absence of any magnetic effect, the MHD equations become the Navier-Stokes equations. Therefore the hydrodynamic solutions obtained in the previous studies constitute the solutions of the MHD plane Couette flow if there is no external magnetic field. They serve a good starting point of the Newton continuation to find the dynamo solution branch, as the imposed external magnetic fields stimulate three-dimensional magnetic fields.

3.1 Symmetry of the MHD solutions

The symmetries of the dynamo solutions are important because we fully use them to reduce the computational costs. Here we shall clarify how the form of external magnetic fields affects the symmetry of the solutions.

We begin the symmetry analysis by the purely hydrodynamic cases. The results to be obtained below are similar to Gibson et al. (2009) but here we approach the problem from slightly different direction. We first aim to derive all possible symmetry classes of the solutions assuming the invariance of the equations under the operations

(3.0a)
(3.0b)
(3.0c)
(3.0d)
(3.0e)

The above operations are motivated by the invariance of the Navier-Stokes equations under the reflection and shift in or directions. Here we do not consider the shift in direction because the fluid motion is limited by the walls. For directions we only focus on half-shift for the sake of simplicity (if we allow general shifts it just produces a few special cases). We use the abbreviation used in Gibson et al. (2009) for the combined operations; namely , etc.

In order to classify the solutions in terms of their symmetry, we check whether they are invariant under the operations in (3.1) and/or their combinations. However in order to classify the solutions in terms of their symmetry some operations should be excluded from the consideration from the reasons below.

  1. It is desirable to choose the periodic box so that there is no identical flow copy within the box. So any solution invariant under or must be excluded.

  2. Whether a solution is invariant under a reflection operator or not actually depends on the choice of the origin. In order to develop the theory independent of that choice, hereafter we say that a solution is invariant under an operation when we can choose at least one coordinates where this invariance appears. This means that for example the operation like can be omitted because the invariance of it is now equivalent to that of .

  3. The operations, which are not consistent with the basic flow, should be excluded. For example, if the flow is invariant under , this means that , and for the basic part (recall that we are now using the travelling wave coordinates). So whenever , we must exclude from the possible operations. The relationship of the operations and the expected restriction for the base flow when the flow is invariant to the corresponding operator is summarised in table 1. (All magnetic effects in the table should be omitted at this stage. Also, the operator is to be defined shortly for the MHD equations).

Given the extended meaning of the invariance and the set of operations under consideration, we now define the classes of the solutions by the curly bracket within which the invariant operators are written. For plane Couette flow, there are five possible operations we must consider, , in view of (a)-(c) above and table 1. The NBC solution belongs to the class , because the solution is invariant under , and not invariant under . Note that the solutions in this class must be steady, because the shifted basic flow is an odd function, and should be zero.

All possible classes for plane Couette flow solutions can be found fairly easily. The least symmetric class is of course , where the solutions belong to this class do not have any symmetry. The next less symmetric 5 classes can be found straightforwardly: , , , , . If the solutions are invariant under two operations they must also be invariant under the combined operation of those two. Thus the next 6 higher symmetric classes are made of three elements: , , , , , (Note that in the latter class the origin used for the operations , must differ by a half shift in the direction). Furthermore we can find one highest symmetric class of six elements .

Operators
zero any zero any
even even even even
any zero any zero
even odd even odd
zero zero zero zero
odd even odd even
odd odd odd odd
zero any any zero
even even odd odd
any zero zero any
even odd odd even
zero zero any any
odd even even odd
odd odd even even
Table 1: The required symmetry for the base flows when the solution is invariant under the corresponding operator.
Figure 1: The -averaged field of the NBC solutions modified by the external magnetic fields. The left panels use with (see figure 8), the middle panels use with (see figure 13), and the right panels use with (see figure 3). Upper panels are the hydrodynamic roll-streak ( for all three figures). Lower panels are the magnetic roll-streak ( from left to right).
Figure 2: The same visualisation as figure 1 but for isosurface of 50% streamwise vorticity (upper panels) and current (lower panels) Red: positive, blue: negative.

In order to extend the above symmetry argument to the MHD flows, we must decide the action of the operators on the magnetic components. The most natural choice would be to transform the magnetic parts as the same manner as the hydrodynamic parts . In fact the MHD equations are invariant under the operations (3.1) extended in this way. However, the MHD equations are also invariant under the flip of the polarity

(3.0)

which brings some more complication. The above operator does not affect to the hydrodynamic part. Therefore, for example, all the extended classes given below preserve the hydrodynamic part of the symmetries of the NBC class, but for the magnetic part we have completely different symmetry:

(3.0a)
(3.0b)
(3.0c)
(3.0d)

The symmetry of the mean flow inferred from the operators in the above classes should be consistent to the applied base magnetic field , as summarised in table 1.

For the first case (3.0a), it is easy to see from table 1 that should be an odd function, and . This is expected result because for this class the hydrodynamic and magnetic fields must have identical symmetries. The left panels in figure 1 and 2 show the flow field of this class computed by imposing the base magnetic field corresponding to the uniform spanwise current. Here the constant represents the intensity of the external field. Figure 1 shows the roll-streak field and , which is defined by the streamwise average

(Note that the definition of the overline is different from that in part 1.) Figure 2 is the total field visualised by the isosurfaces of the streamwise vorticity and current. In the latter figure the emergence of the strong vortex and current sheets can be seen, and they are in fact the signatures of the amplified Alfvén wave due to the resonance.

Table 1 suggests also for the second case (3.0b) but now should be an even function. Imposing the uniform streamwise magnetic field, , we can indeed generate the three dimensional magnetic field of different symmetry, as shown in the middle panels of figures 1,2.

In the third case (3.0c) we must turn off the streamwise magnetic field so that but instead can apply a spanwise magnetic field of an even function . In the computation we use the spanwise uniform magnetic field . The symmetry of the solution visualised in the right columns of figure 1,2 is actually the same as that observed in the MHD rotating plane Couette flow computation by Rincon et al. (2007).

There is the fourth case (3.0d), which can be generated by the spanwise magnetic field of odd function, but we omit the computation of that case here.

The external magnetic fields mentioned above are the only possible cases to preserve the NBC symmetry for the hydrodynamic field. In fact, it is easy to theoretically/numerically confirm that if for example the base magnetic field is applied, then the NBC symmetry seen in the hydrodynamic part is destroyed for .

3.2 Asymptotic scaling of the vortex/Alfvén wave interaction states

Streak Roll Outer wave Inner wave Wave amplitude
Type 1
Type 2
Table 2: The scaling of the vortex/Alfvén interaction states.

Our aim in this section is to check one of the nonlinear three dimensional dynamo theories proposed in part 1, called the vortex/Alfvén wave interaction, using the MHD solutions supported by the uniform spanwise magnetic field (corresponding to the third case (3.0c)). The theory concerns the interaction of the roll-streak (i.e. and ) components and the wave components defined by and . The leading order scaling of those components are summarised in table 2. The streak and roll fields are typical for vortex/wave interaction and self-sustaining process theories.

The singularity occurs in the inviscid wave problem whenever the resonant conditions or are satisfied. Near those curves the wave is amplified singular manner and thus the dissipative layers of thickness surrounding the singularity is necessary to regularise the flow there. In table 2 we also show the theoretical wave scaling for the inside and outside of the dissipative layer. As shown in part 1 two types of the inner/outer wave scaling are possible, depending on the two resonant layers are well-separated (type 1) or degenerated (type 2).

Figure 3: The bifurcation diagram for the external field . The solid curve is the sinuous mode branch continued from the NBC solution (the magenta triangle). The shear on the upper wall () is used to measure the solution. The visualisations in the right panels of figures 1,2 correspond to the filled circle. The dashed curve is the mirror-symmetric mode branch. The open circle corresponds to the visualisation in figure 4.
Figure 4: The same plot as figures 1 and 2 but for the mirror-symmetric mode computed in figure 3 (). The upper panels are the -averaged fields (left panel: , right panel: ). The lower panels are the 50 % streamwise vorticity and current.

For the vortex/Alfvén wave interaction the leading order magnetic field cannot be maintained without some external magnetic field because of the lack of energy input to the magnetic roll equations. Here we apply the small uniform external magnetic field with expecting that it will stimulate larger magnetic field in the flow through the dynamo mechanism. Figure 3 shows the bifurcation diagram obtained by applying that base magnetic field to the NBC solutions (magenta triangle). The vertical axis of this diagram is the shear on the upper wall (). The continued MHD solutions, hereafter called the sinuous mode, generate the three-dimensional magnetic field as seen in figures 1, 2. Further increasing , the solution branch experiences a turning point at , and on the way back it connects to another solution branch shown by the dashed curve. The solutions on the latter branch, which we call the mirror-symmetric mode, belong to the higher symmetry class ; see the visualisation shown in figure 4. At the hydrodynamic limit () the solution reduces to the known hydrodynamic solution branch (Itano & Generalis (2009); Gibson et al. (2009)), having a flat critical layer at (Deguchi & Hall 2014a).

Figure 5 compares the vorticity of the sinuous and mirror-symmetric mode solutions at . There are two resonant curves for the sinuous mode so this is formally type 1, whilst for the mirror-symmetric modes the two resonant positions must be degenerated at by their symmetry so it could be regarded as a good example of type 2. The vorticity is dominated by the amplified wave component near the resonant curves, consistent to the theory. Similar amplification of the magnetic wave component by visualising the current; see figure 2.

Figure 5: The streamwise vorticity at . The left half is the sinuous mode (, the same solution as the right column of figure 2), whilst the right half is the mirror-symmetric mode (, the same solution as figure 4). The resonant curves are indicated by the black solid curves.
Figure 6: The amplitudes of the solutions subject to the uniform spanwise magnetic field with . Upper panel: the sinuous mode, lower panel: the mirror-symmetric mode. The open symbols are the hydrodynamic component, the filled symbols are the magnetic component. The red triangles, blue circles and green squares are the roll, streak, and wave amplitudes, respectively. The slope of the blue dotted, red solid, and green dashed lines are , and , respectively.

For those two types of solutions we can check the asymptotic scaling changing the Reynolds number. In order to measure the size of the roll-streak-wave component, it is convenient to define the amplitude by the square root of the energies defined below:

where

(3.0)

represents the volumetric average over the computational domain. Note that only the inner wave occurring within the thickness contributes to the wave amplitude to leading order. For type 1, the scaling of the amplitude can be estimated as . Likewise the amplitude of type 2 is , as shown in table 2.

Figure 6 shows the dependence of the amplitudes on the Reynolds number. The upper and lower panels correspond to the sinuous and mirror-symmetric modes, respectively. The spanwise uniform magnetic field with is used to support the roll components shown by the red triangles. As predicted by the theory, the streak parts of the solution shown by the blue circles tend to constants for large Reynolds numbers, thereby proving the magnetic field generation by much smaller external magnetic field. The scaling of the wave amplitude, shown by the green squares, is yet less clear at this range of . The wave amplitudes of the mirror-symmetric mode, formally being type 2, drops slightly faster than the sinuous mode of type 1; this is certainly consistent with what has been shown in part 1. However, the slopes of the numerical solutions do not perfectly match to the theory. For the mirror-symmetric mode it is known that the asymptotic convergence is not very good even for the purely hydrodynamic case (Deguchi & Hall 2014). That slow convergence of the flow typically appears when there are two or more possible asymptotic regimes (the boundary-region equations type asymptotic structure of Deguchi, Hall & Walton 2013 can also occur locally for the present case; see Deguchi & Hall 2015, Ozcakir 2016 also). On the other hand, the slow convergence of the sinuous mode seems to be due to the different mechanism that the distance between the two resonant layers (shown in figure 5) is so close in numerical sense that the two dissipative layers are not completely separated. This means that the Reynolds number must be further increased to observe better scaling.

4 Dynamos without any external magnetic field

4.1 Homotopy using the streamwise magnetic field

Figure 7: The bifurcation diagram for the external field . The lower two panels are the magnified plot of the upper panel. There are three solutions at : the magenta triangle is the NBC solution, whilst the other two are the S solutions. The solution at the filled circle corresponds to the visualisation in the left panels of figures 1,2.
Figure 8: The 50% streamwise vorticity and current of the S dynamo solution in figure 7. Red: positive, blue: negative.
Figure 9: The streamwise vorticity (left half, ) and the current (right half, ) of the S dynamo at . The same solution as figure 8. The resonant curve is indicated by the solid black curve.

In the previous section, the nonlinear solutions lose all the magnetic effects as , and become purely hydrodynamic solutions. This is certainly consistent with the vortex/Alfvén wave interaction theory where the leading order magnetic field cannot survive at this limit. However, further numerical investigations reveal that the induced magnetic field does not always vanish at the zero external magnetic field limits. This section is devoted to the analysis of such self-sustained shear-driven dynamos, S dynamos for short.

In order to find the S dynamos, we consider a homotopy continuation of the solution branch using a different type of external magnetic field. Figure 7 shows the continuation of the solution branch when the streamwise magnetic field of odd function, , is used (corresponding to the first case (3.0a)). Again we begin the calculation from the NBC solution indicated by the magenta triangle. The solution branch can be continued by gradually increasing until it reaches the turning point around . The blue filled circle on the branch corresponds to the visualisation in the left panels in figures 1,2. Then after the turning point the branch returns to the limit as the previous case, but slightly different manner.

Here we note that by symmetry the bifurcation diagram must be symmetric with respect to . Thus the NBC solution branch continued towards negative must also return to the same point, but the branch does not coincides with the former branch unless . The appearance of the two degenerated solutions at the unforced limit is in fact the signature of the magnetic field production there. At the symmetry (3.0) ensures that if a S dynamo exists then there should be another S dynamo of the same velocity but the opposite polarity. Therefore, if that branch is continued from S dynamos for finite the symmetry of the magnetic field becomes imperfect and hence two distinct branches should appear. On the other hand, such imperfection does not occur when the branch is continued from purely hydrodynamic solutions. After the crossing point the branch behaves rather complicated way so the computation is terminated in figure 7.

The three-dimensional structure of one of the S dynamos obtained is shown in figure 8. The structure of the solution is similar to the externally forced solution seen in the left columns in figure 2, and the symmetry (3.0a) is preserved. The emergence of the vortex and current sheets in the flow is due to the resonant absorption but it is slightly different to the previous cases, as we shall explain shortly.

Varying the S dynamo solution can then be traced back to its origin at the saddle-node as the NBC branch as shown in figure 10. It turns out that the solutions we have seen in figure 7 at are the lower branch states. The saddle-node point of the S dynamo solution, shown by the solid curve, sits at slightly larger Reynolds number () than that of the NBC branch (). It has been repeatedly shown in the hydrodynamic community that the unstable invariant solutions are the precursor of turbulence (Gibson et al. (2009), Kreilos & Eckhardt (2012)). Thus the bifurcation diagram shown in figure 10 suggests that in order to observe the self magnetic field generation we need larger Reynolds number than the critical Reynolds number for the hydrodynamic turbulence. This seems to be consistent with the recent numerical simulation by Neumann & Blackman (2017).

Figure 10: The bifurcation diagram of plane Couette flow with no external magnetic field. The S dynamo solution branch is the solid curve, whilst the NBC solution branch is the dashed curve.

4.2 Asymptotic scaling of the self-sustained shear driven dynamos

Now let us examine the large Reynolds number asymptotic fate of the S dynamos. We use the self-consistent matched asymptotic theory proposed in part 1, which was entirely motivated by the numerical results in the last section. The ability of the theory to predict the finite Reynolds numbers results will be tested against the numerical invariant solutions.

In view of the results in section 3 one might think the generation of the magnetic field at large Reynolds numbers is contradicting to the vortex/Alfvén wave interaction theory. However, the fact is that the theory only inhibits the presence of the magnetic field of the order shown in table 2, and hence actually smaller magnetic fields could be maintained. The difficulty in formulating the new asymptotic theory at the S limit choosing the smaller leading order magnetic field was that the dynamos tend to become kinematic dynamos, which is linear with respect to the magnetic field. In that case the magnetic field amplitude is undetermined so we are unsure if there is saturated magnetic field generation.

The asymptotic theory proposed in part 1 has overcome this difficulty by showing the presence of the nonlinear magnetic effects due to the resonant absorption, now formally occurring when . There is only one resonant layer in the flow and in fact the structure is similar to type 2 of the vortex/Alfvén wave interaction theory. To leading order the instability wave is purely hydrodynamic and its amplitude behaves like near the resonant layer, where is the distance to the resonant location. However it turned out that the magnetic wave produced at the next order behaves like which is much singular. That discovery implies that the hydrodynamic and magnetic waves within the dissipative layer of thickness becomes comparable in size if the magnetic field size is chosen to be smaller than amount compared with the hydrodynamic counterpart; see table 3. With that choice of the scaling the wave Maxwell stress governing the nonlinear magnetic feedback effect to the hydrodynamic flow should appear within the dissipative layer because it now has the same magnitude as the wave Reynolds stress there. The vortex and current sheets seen in figure 8 are due to that amplification; see figure 9 also where the structure at is shown with the resonant curve.

Streak Roll Outer wave Inner wave Wave amplitude
Hydrodynamic
Magnetic