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Kengo Deguchi
Abstract

Rational large Reynolds number matched asymptotic analyses of three-dimensional magneto-hydrodynamic dynamo states are concerned. The dynamos, here assumed to be predominantly driven by a unidirectional shear, are fully nonlinear and can be sustained without any linear instability of the base flow. Two classes of dynamos are found. The first class of dynamos is emerged out of a nice combination of the purely hydrodynamic vortex/wave interaction theory by Hall & Smith (1991) and the resonant absorption theories for Alfvén waves, developed in the solar physics community. Similar to the hydrodynamic theory, the mechanism of the dynamos can be explained by the successive interaction of the roll, streak, and wave fields, which are now defined both for hydrodynamic and magnetic fields. The derivation of this ‘vortex/Alfvén wave interaction’ state is rather straightforward as the scalings for both of the hydrodynamic and magnetic fields are identical. However, it turns out that the leading order magnetic field cannot be supported without a small external magnetic field. Without the leading order magnetic field at first glance dynamos are not possible. Nevertheless, the second class of ‘self-sustained shear driven dynamo theory’ shows the magnetic generation of slightly smaller size in the absence of any external field. Despite the small size of the magnetic field it causes the novel feedback mechanism to the velocity field through the absorption, where the magnetic wave becomes more strongly amplified than the hydrodynamic counterpart.

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High-speed shear driven dynamos. Part 1. Asymptotic analysis]High-speed shear driven dynamos.
Part 1. Asymptotic analysis

1 Introduction

We are concerned with mathematical descriptions of high-speed nonlinear magneto-hydrodynamic (MHD) flows driven by simple subcritical shear flows such as plane Couette flow. MHD turbulence states occur in electrically conductive fluid flows (e.g. plasma or liquid metal) and are made of various complicated vortex structures. The interaction of the velocity and magnetic fields in the flow have attracted much attention, as it could produce Ôdynamo statesÕ, which serve an efficient way to convert kinetic energy to electricity. The dynamo states are of central importance in future power engineering and in astrophysics, where such energy conversion mechanism is common. However, most of the mathematical dynamo studies are limited to the kinematic situation where the magnetic field is so small that it does not affect to the hydrodynamic field.

The strong nonlinear coupling of the velocity and magnetic fields in the MHD turbulence implies that we must greatly rely on numerical analysis, from which it is difficult to infer general properties. The numerical dynamo studies aiming to compute the saturated magnetic field from the kinematic dynamos use meticulously designed complicated flow geometries in order to avoid various anti-dynamo theorems (see the review by Brandenburg & Subramanian (2005) and references therein) or otherwise assume some linear instability of the flow (e.g. magneto-convection). Numerical studies of dynamos in linearly stable flows are rather rare (Rincon et al. 2007, Rincon et al. 2008; Riols et al. 2013; Neumann & Blackman 2017).

Another difficulty of fully computational study lies in the fact that the high Reynolds number of practical relevance hinders the simulation that requires very fine grid points. In order to analyse high Reynolds number MHD turbulence locally, a number of isotropic homogeneous simulations are performed with some artificial external forcing pouring energy to the hydrodynamic field (see Kida et al. 1991, Brandenburg et al. 2012, for example). However, it is not clear how those local MHD turbulent states are sustained in physically realisable flow configurations. Elsewhere modelling approaches are used to simplify or modify the governing equations to perform global high Reynolds number simulations, but in order to close the model system, introduction of some artificial assumptions is inevitable (e.g. large eddy simulations, turbulent models, mean-field models).

To make some theoretical progresses towards the rational high-speed dynamo theory using minimal assumptions, in this paper we employ the large Reynolds number matched asymptotic expansion of viscous-resistive MHD equations. The large hydrodynamic and magnetic Reynolds numbers limit is known as the singular limit of the governing equations and thus the leading-order solution may possess singularities. The small-scale flow induced near the singular point must behave a consistent manner as the large-scale flows, so this is exactly the place where we need the matching. That matching condition is the crux to close the asymptotically reduced system, where unlike heuristic model approaches there is no need of artificial assumption or tuning parameters. In fact the method on which our analysis is based is the cornerstone of progress with understanding high Reynolds number solutions of the Navier-Stokes and resistive MHD equations, although nowadays only a few researchers working in this field due to its technical complexity associated with the singularity. In purely hydrodynamic study such singular point is known as ‘critical layer’, which occurs whenever the speed of the background flow coincides with the phase speed of the instability wave. For the simplest linear case, the delicate interaction of the small- and large-scale flows through the matching conditions was firstly established by Lin (1945, 1955).

In solar physics community, the similar singular behaviour of wave like instabilities has been studied in the resonant absorption theory. Since Alfvén (1942) a number of researchers studied instability waves riding on an uniform background velocity and magnetic fields. However solar atmosphere is highly non-uniform in reality so the uniform flow assumption used by Alfvén (1942) is actually not valid. The consideration of non-uniform background magnetic fields led to the finding of the Alfvén resonant point, where the wave behaves a singular manner at the limit of large hydrodynamic and magnetic Reynolds numbers. The asymptotic structure of linear disturbances excited by a background magnetic field was derived in Sakurai et al. (1991), Goossens et al. (1995). The theory and its extended results to steady non-zero background shear flows (Goossens et al. (1992), Erdélyi et al. (1995), Erdélyi (1997)) successfully describe the inner structure of the thin dissipative layer surrounding the resonant point, and how it should be matched to the outer Alfvén wave solution. Despite the similarity, the nature of the singularities was found to be mathematically different to the purely hydrodynamic critical layer.

To date, most of the existing nonlinear hydrodynamic/MHD asymptotic theories use the relatively small instability waves so that they do not have an ability to modify the leading order background flow (Smith & Bodonyi (1982); Clack & Ballai (2009); Goossens & Erdélyi (2011); Deguchi & Walton (2018)). The assumed background flows in those studies vary only in one direction and hence the wave-like perturbation is essentially two-dimensional. In this case the feedback effect from the wave to the background flow is difficult to implement theoretically, as it destroys the basic assumption that the singularity only occurs at the critical or resonant layers. This means that most of the interactions previously considered are merely one way from the background flow to the induced wave, although in turbulent flows of course the background flow profile can be largely distorted and it may cause some feedback effect to the wave instability.

Figure 1: The nonlinear steady solution of plane Couette flow at Reynolds number (streamwise wavenumber 1.2, spanwise wavenumber , upper branch; see Deguchi & Hall (2014) for definition). Here is taken to be the direction of the base flow; see section 2. The wave field is visualised by 30% maximum of absolute streamwise vorticity (red is positive, blue is negative). The roll-streak field corresponds to the streamwise average of the total flow field. The streak field is the zero locus of the streamwise velocity, whilst the roll field is the isosurfaces of the stream function for the cross-stream components of the averaged velocity (red is positive, blue is negative).

It has been pointed out by Hall and Smith in late 80’s to early 90’s that the three-dimensionality of the flow allows us to incorporate that feedback effect to the theory without destroying the critical layer structure. Their theory, called the vortex/wave interaction theory (Hall & Smith 1989), was originally found for boundary layer flows by considering the mutual interaction between the streamwise vortices developing slowly and much rapid three-dimensional viscous (Tollmien-Shlichting) waves; see Hall (1983) and Smith (1979) also. Subsequently those authors noticed that the similar interaction is possible for inviscid (Rayleigh) waves (Hall & Smith 1991). What is remarkable in the latter type of the theory is that it rationally describes the mechanism by which the small wave significantly modifies the background flow to leading order. This means that the large Reynolds number approximation has a potential to interpret the mysterious mechanism of nonlinear subcritical instability that can destabilise the linearly stable flows, e.g. plane Couette flow. It was Waleffe (1997) who firstly recognised that possibility, although what he independently proposed was rather heuristic model approach coined ‘self-sustaining process’. His theory has rapidly spreaded among fluid dynamisists as it concisely explains the sustaining mechanism of the nonlinear states thorough the successive interaction of the roll, streak and wave. Here the streak and roll constitutes the streamwise and the cross-stream components of the vortex component in the vortex/wave interaction; see figure 1 where the roll, streak and wave components computed in Deguchi & Hall (2014) are shown. Despite the great success of the self-sustaining process, there had been an important ingredient overlooked until Wang, Gibson & Waleffe (2007) - the critical layer structure. Subsequently Hall & Sherwin (2010) showed that the self-sustaining process is actually equivalent to mathematically rational framework of vortex/wave interaction if the former theory is supplemented by the critical layer analysis. Two independently developed research streams over the last two decades, both of which greatly inspired by earlier Benney (1984), Benney & Chow (1989), have now been integrated together.

The primal aim of this paper is to develop a nonlinear three-dimensional asymptotic dynamo theory by replacing the Rayleigh wave in Hall & Smith (1991) by the Alfvén wave. The formulation of this ‘vortex/Alfvén wave interaction theory’ would be rather straightforward, because the particular form of the viscous resistive MHD equations suggests that we should be able to apply the roll-streak-wave decomposition to the magnetic fields as well as the velocity fields. Indeed, in Rincon et al. (2007), Rincon et al. (2008), Riols et al. (2013) some similarities of the nonlinear dynamo theory and the self-sustaining process have been pointed out. (However, it should be noted that those works treat dynamos driven by the magneto-rotational instability, while the effect of system rotation is not considered here. We shall comment on implication of our work on the rotational case in section 5.)

After formulating our problem in the next section, we begin the asymptotic analysis in section 3 assuming the largeness of the Reynolds number. We shall see that the Alfvén wave produced in inhomogeneous mean fields, namely hydrodynamic and magnetic streaks, can be analysed using the resonant absorption theory. Interestingly, the size of the wave in general differ from the purely hydrodynamic case, because the interaction from the wave to the vortex now occurs through the Alfvén resonant layer rather than the critical layer. In the same section the sustained mechanism of the dynamo will be examined to show that unless small magnetic field is applied, the vortex/Alfvén wave interaction state is not possible. However, it does not mean that the dynamo states cannot be realised in the zero external magnetic field limit. In fact, the numerical study in part 2 of this paper (Deguchi 2018) shall show the production of the magnetic field at that limit. Motivated by that numerical result, section 4 concerns the mathematical description of such self-sustained shear driven dynamos, and show that the induced magnetic field is indeed slightly smaller than the vortex/Alfvén wave interaction states. Finally in section 5 we draw some conclusions.

2 Formulation of the problem

Throughout the paper we use the Cartesian coordinates . Consider incompressible viscous resistive MHD equations for the velocity field , the magnetic field , the current density , and the pressure :

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

where , is the fluid density , is the fluid kinematic viscosity, is the fluid magnetic diffusivity, and is the fluid magnetic permeability.

We assume that the flow is predominantly driven by some unidirectional hydrodynamic basic flow, and take to be the typical size and the spatial scale of it. Using the length scale and the velocity scale , we define the non-dimensional variables as

(2.0a)
(2.0b)

to get the non-dimensional version of the governing equations

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

where , and

(2.0d)

is the total pressure. From the induction equation (2) we can show that if the solenoidal condition holds at certain instant, it should be always satisfied for all . In the absence of magnetic monopole this is indeed the case so we assume the solenoidality throughout the paper.

The flow is governed by the hydrodynamic Reynolds number and the magnetic Reynolds number

(2.0)

The ratio of those parameters is known as the magnetic Prandtl number

(2.0)

The theory to be developed can be applied for quite wide range of shear flows but here for the sake of simplicity and definiteness we consider plane Couette flow. Following the convention of shear flow study we take directions to be streamwise, wall-normal, and spanwise directions, respectively, assuming the periodicity of the flow in and the presence of the no-slip and perfectly insulating walls at . The system has a laminar basic flow solution with , while our main interest is other nonlinear solutions.

If the no-slip conditions are replaced by certain generalised periodic conditions, the system becomes ‘shearing box’ frequently used in local analyses of astrophysical flows. However, note that our notation is different from the common notation used in astrophysics community, where the azimuthal (streamwise), radial (vertical), and axial (spanwise) directions of the computational box are denoted by , respectively (e.g. Riols et al. 2013).

3 The vortex/Alfvén wave interaction theory

Now let us construct the large asymptotic theory for the nonlinear three-dimensional MHD dynamos. We assume that the nonlinear state is a travelling wave propagating in with a phase speed . Once the theory for the travelling wave is found, it can then be easily extended to describe more general time dependent solutions. However the formulation becomes much more complicated (see Deguchi & Hall (2016) for purely hydrodynamic cases) and thus that possibility is omitted here.

Here and hereafter we apply the Galilean transform to convert the travelling wave to the steady state. Redefining the coordinate , the governing equations (2) can be written in component form

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

where . In the following large asymptotic analysis, we assume that or larger. The analyses for small cases are given in Appendix B.

The use of analogy from the Hall-Smith theory yields the asymptotic expansion

(3.0)

Here the coefficients are real functions of and we call them as the vortex components. The coefficients depend on as well but they are complex and are called the wave components. The c.c. stands for complex conjugate. Note that in (3) we only display the terms relevant to derive the leading order equations. The full asymptotic expansion must of course contain the higher-order vortex or wave terms with higher streamwise harmonics but after we derive the leading order system we can check that the presence of those higher order terms does not modify the leading order system. The parameter is the wave amplitude to be fixed in terms of later and is meanwhile assumed to be of or smaller.

Substituting (3) to (3) and then neglecting some small terms, we have the vortex equations

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

and the wave equations

(3.0d)
(3.0e)
(3.0f)

where , and

(3.0)

is the doppler-shifted hydrodynamic streak. In obtaining the above equations we retain the diffusion terms in (3.0d), (3.0e) because those terms become important within the thin ‘dissipative layers’ surrounding the singularity as we shall explain shortly. Outside the layers we can formally neglect those diffusion terms.

In order to elucidate the motivation of the scaling (3), it is convenient to adopt the terminology introduced by Waleffe (1997), namely we call the - and -components of the vortex part as the streak and roll, respectively. The roll scale is chosen so that the viscous-convective balance in the roll-streak equations is achieved. On the other hand, the wave part is predominantly convected by the streak component (including the basic flow), and therefore this part can be approximated by ideal flows almost everywhere. The wave equations can be regarded as the linear stability problem of the streak, and thus only the monochromatic wave neutral to the streak can be excited to leading order (this is the reason why there in only one Fourier mode in (3)). From the form of the roll-streak equations the mechanism by which the wave field causes the feedback to the roll field. In the momentum equations the feedback terms are produced by the advection terms and the Lorentz force terms, hereafter called the wave Reynolds stress and the Maxwell stress, respectively. The similar feedback terms in the induction equations are called the wave electromotive force terms.

If the diffusion terms are neglected the wave equations (3.0d)–(3.0f) are combined to yield the single pressure equation

(3.0)

The pressure equation is the natural generalisation of that derived by Sakurai et al. (1991), Goossens (1992) for one-dimensional background flows. (In those works compressible flows are concerned so incompressible limit must be taken. See the review by Goossens & Erdélyi (2011) also). Given the hydrodynamic and magnetic streaks, the equation (3.0) constitutes the linear eigenvalue problem for the eigenvalue , and thus can in principle be solved with the boundary conditions at which ensures that the walls are impermeable. (Near wall boundary layers are necessary to satisfy other boundary conditions but the dynamics in the layer is passive so the analysis of it is not necessary.) This singularity occurring when vanishes corresponds to the Alfvén resonant point discussed in the resonant absorption theory. At this point the hydrodynamic and magnetic streak energies are equipartitioned (i.e. ). Hereinafter we denote the singular positions of the resonant layers as and assume that is satisfied there.

Once is solved, we can express the other wave components as

(3.0a)
(3.0b)

Here we remark that the behaviour of the cross-stream components of the magnetic wave is simply related to the hydrodynamic counterpart as

(3.0)

This is due to the absence of the diffusion in the wave induction equations and analogues to the Alfvén’s frozen-in theorem. Hereinafter we say that the magnetic wave is ‘frozen’ to the hydrodynamic wave when (3.0) holds.

On making use of those inviscid wave expressions to (3), we have the outer roll-streak equations

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

Now we can find that there is actually no wave electromotive force terms in the majority of the flow, because the frozen wave (3.0) ensures the cancellation of the two wave nonlinear terms.

3.1 Body-fitted coordinate

Figure 2: Schematic of type 1 interaction studied in section 3.2. The body-fitted coordinate is shown for the upper resonant curve . Type 2 interaction occurs when the distance between the two resonant curves becomes ; see section 3.3.

One of the major aims in the resonant absorption theory is to find the connection formulae, that tells us how to connect the outer solutions across the resonant point. Here the similar formulae must be found from the inner analysis to find the jump, which completes the nonlinear system at large with the outer problem. However, the singularity now occurs on a curve in plane rather than a point so it is convenient to use ‘body-fitted’ coordinate attached to the resonant curve.

Let us consider the curve and write the length measured along straight lines that are normal to this curve as , and the arc length measured along the curve as ; see figure 2. Then we write the components of the velocity and magnetic field vectors as

(3.0)
(3.0)

where are the unit vectors. The curvilinear coordinate expression of the governing equations (3) can be found as (see Slattery (1999) for example)

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

where

(3.0d)
(3.0e)
(3.0f)

The Lamé coefficient

(3.0)

where is the curvature of the critical curve

(3.0)

is fixed by the definition of . Then for we can also apply the vortex-wave decomposition

(3.0)

to find the body-fitted coordinate version of the vortex and wave equations. The local behaviour of the outer solution near the resonant curve gives the matching conditions, which are needed to solve the solutions within the dissipative layer. Near the resonant curve the wave behaves a singular way so that its amplitude must be increased. It is this amplified inner wave that creates derivative jumps in the outer roll components. The aim of the next two subsections is to derive the analytic form of the jump, by the generalisation of the connection formulae obtained in the previous resonant point analysis. The wave Reynolds-Maxwell stress and the roll stress within the dissipative layer must be in balance and that condition gives the appropriate size of the wave amplitude in terms of our intrinsic small parameter . We shall show that the magnitude of changes depending on the distance between the two resonant layers.

3.2 Type 1: the connection formulae for the case

Figure 3: The upper panel shows the inner wave solution for type 1 interaction (3.0). The representative parameters are chosen for . The lower panel is the forcing from the inner wave to the hydrodynamic roll component (3.0).

In this section we assume the distance between the two resonant curves is ; this case is called type 1 interaction and schematic of it is shown in figure 2. Using the body-fitted coordinate near the resonant curve , we can Taylor expand the streak as

(3.0)

This means that we can write

(3.0)

where we assume for all . Of course is the resonant curve and here vanishes as required. The singularity there must be resolved within the thin dissipative layers surrounding , where we retain some diffusion effects. The flows inside and outside of the dissipative layer analysed separately are then connected through the matching conditions.

Now let us analyse the singular behaviour of the outer wave solution at the edge of the dissipative layer. In the body-fitted coordinates, the inviscid wave equation (3.0) becomes

(3.0)

Since when is small, the above equation suggests the Frobenius expansion

(3.0)

where

(3.0)

The quantity appeared in (3.0) is the jump associated with the logarithmic singularity and physically represents the phase shift of the wave across the resonant curve. The jump is induced by the diffusion so the value of should be fixed by matching to the solution within the dissipative layer. We can fix the coefficients from the boundary conditions, and all the other coefficients in the expansion can be fixed in terms of those two coefficients. In particular, substituting (3.0) to (3.0) we can find

(3.0)

From the body-fitted coordinate expression of (3) and (3.0) we can find the small asymptotic behaviours of the other wave components

(3.0)

More precisely, the cross-stream components expand

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

Using the stretched normal coordinate with the dissipative layer thickness , the inner expansions consistent to (3.0), (3.0) can be found as

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

The dissipative layer thickness is chosen in such a way that the diffusion terms influence the inner flow.

The inner wave equations are obtained by substituting (3.0), (3.2) to the body-fitted coordinate version of the wave equations. We must analyse the equations up to the first order. From the -components we can find that

(3.0a)
(3.0b)

whilst the -components yield

(3.0c)
(3.0d)
(3.0e)

From (3.0b) and (3.0c) we see that the zeroth order components remain frozen within the dissipative layer.

It is readily shown from equations (3.0c)-(3.0e) that they can be combined to yield the single equation for

(3.0)

Here is a function of only from (3.0a) and the prime denotes a ordinary differentiation. The equation is linear and the solution can be found analytically. Note that the -component of the wave is the most singular and it is the only component necessary to estimate the jump in the roll components.

We note in passing that in the above derivation we can safely assume that there are no significant effects by the nonlinear terms with respect to the waves. In the essentially two-dimensional cases, the resonant absorption theories have been extended to include the wave nonlinearity within the dissipation layer. The theories treat mostly for the compressible waves and in that case the singularity also occurs at the cusp wave resonant point. For example Ruderman et al. (1997), Ballai & Erdelyi (1998), Clack & Ballai (2008) showed that the nonlinear version of the dissipative layer may appear around the cusp resonant point if the wave amplitude is tuned to be sufficiently large, whilst Clack et al. (2009) concluded that the corresponding nonlinearity is always negligible for the Alfvén resonant point; see the review by Ballai & Ruderman (2011) also. However, in our case the appropriate wave amplitude must be fixed in order to drive the roll component, as we shall see shortly. Therefore we cannot vary it to balance the nonlinear terms in the dissipative layer wave equations as did in the previous studies. The similar conclusion was obtained in the purely hydrodynamic studies. Hall & Smith (1991) found that the inner wave equations in their theory should be always linear by the same reason, although if the flow is essentially two-dimensional so-called nonlinear critical layer is possible (Smith & Bodonyi 1982; Deguchi & Walton 2018).

Using the new variable

(3.0)

the equation (3.0) can be transformed into

(3.0)

The solution of this equation matching to the outer solution can be found as

(3.0)

where

(3.0)

satisfies and in the far-field limit ().

The integral of with respect to

(3.0)

produces a logarithmic function with a constant jump for large

(3.0)

where is a constant (see Appendix A.1). The logarithm and jump are exactly what we saw in the pressure expansion (3.0). Thus in order to match the solution the value of the logarithmic phase shift in (3.0) must be

(3.0)

The analogues result was obtained by Sakurai et al. (1991) for the first time in solar physics community, although it has long been recognised in the hydrodynamic study; see Haberman (1972) for example.

The other wave components can be explicitly solved but since they are not necessary to find the jumps in the roll components we omit further inner wave analysis here.

The inner roll fields expand

(3.0a)
(3.0b)

The outer roll velocity and magnetic fields should be continuous across the dissipative layer, but the normal derivative of the tangential components, namely the leading order vorticity and current, might suffer jumps.

Here note that there is no jump in the normal derivative of the normal components because from the solenoidal conditions we have . Note that the condition (where ) can be written in the outer variable expression (where ).

The vorticity and current jumps can be found from the roll equations. Substituting the inner expansions (3.2) and (3.2) to the body-fitted coordinate forms of the roll equations and integrate them over ,

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

where the superscript asterisks are used to express the complex conjugate. The first two equations are derived from the - and -components of the hydrodynamic roll equations, and the third equation comes from the -component of the magnetic roll equations (analysis of other equations are not necessary to find the jumps). In order to derive the above equations it is important to note that the terms like