«1 c Cambridge University Press 2015 J. Plasma Phys. (2015), vol. 81, 395810601 doi:10.1017/S0022377815001063 The electromotive force in multi-scale ...»
c Cambridge University Press 2015
J. Plasma Phys. (2015), vol. 81, 395810601
The electromotive force in multi-scale ﬂows at
high magnetic Reynolds number
Steven M. Tobias1, † and Fausto Cattaneo2
1 Department of Applied Mathematics, University of Leeds, Leeds LS8 1DS, UK
2 Department of Astronomy and Astrophysics, University of Chicago, 5640 South Ellis Avenue,
Chicago, IL 60637, USA
(Received 1 May 2015; revised 18 August 2015; accepted 19 August 2015) Recent advances in dynamo theory have been made by examining the competition between small- and large-scale dynamos at high magnetic Reynolds number Rm.
Small-scale dynamos rely on the presence of chaotic stretching whilst the generation of large-scale ﬁelds occurs in ﬂows lacking reﬂectional symmetry via a systematic electromotive force (EMF). In this paper we discuss how the statistics of the EMF (at high Rm) depend on the properties of the multi-scale velocity that is generating it.
In particular, we determine that different scales of ﬂow have different contributions to the statistics of the EMF, with smaller scales contributing to the mean without increasing the variance. Moreover, we determine when scales in such a ﬂow act independently in their contribution to the EMF. We further examine the role of large-scale shear in modifying the EMF. We conjecture that the distribution of the EMF, and not simply the mean, largely determines the dominant scale of the magnetic ﬁeld generated by the ﬂow.
1. Introduction It is a great pleasure and privilege to be invited to contribute to this volume in honour of the centenary of the birth of Professor Ya. B. Zel’dovich. Zel’dovich’s research interests were so wide ranging that it is possible to discuss almost any aspect of physics and describe the signiﬁcant and lasting impact that he had on that ﬁeld. We shall not even attempt to describe the breadth of the contributions and deep insight of Professor Zel’dovich’s research, since this has been noted repeatedly by both scientists and historians of science (Sunyaev 2004; Hargittai 2013), nor shall we review one of the many ﬁelds to which Zel’dovich made such telling contributions. Rather, we shall describe a new investigation that brings together two of Professor Zel’dovich’s research interests, random ﬂows in magnetohydrodynamics and dynamo theory.
An understanding and categorisation of the dynamo properties of turbulent ﬂows can only emerge with the recognition that turbulent ﬂows exist as a superposition of coherent and random structures. The ratio of the importance of each of these classes of ﬂow to the dynamo properties depends on the physical setting of the ﬂow. In general, for astrophysical and geophysical ﬂows, the interaction of rotation and stratiﬁcation leads to the enhanced importance of coherent structures (see e.g. Tobias & Cattaneo † Email address for correspondence: firstname.lastname@example.org 2 S. M. Tobias and F. Cattaneo 2008a). This typically involves long-lived structures – by which we mean structures with a coherence time longer than their turnover time – contributing signiﬁcantly to the generation of magnetic ﬁelds.
Dynamo theory has traditionally been separated into two distinct approaches.
The ﬁrst, often termed ‘small-scale dynamo theory’ or ‘ﬂuctuation dynamo theory’, examines whether and how ﬂuid ﬂows can act so as to sustain magnetic ﬁelds on scales smaller than or up to the typical scale of the energy-containing eddies. Despite the irritating presence of anti-dynamo theorems, which rule out the possibility of dynamo action if either the ﬂuid ﬂow or the magnetic ﬁeld possesses too much symmetry, it has now been established that sufﬁciently turbulent ﬂows at high enough magnetic Reynolds number (Rm) are almost guaranteed to act as small-scale dynamos (Vainshtein & Kichatinov 1986; Finn & Ott 1988; Galloway & Proctor 1992; Childress & Gilbert 1995). At this point it is worth noting that the two most famous and irritating theorems that dynamos must circumvent are Cowling’s theorem (Cowling 1933), which prescribes the possibility of an axisymmetric magnetic ﬁeld being generated by dynamo action, and Zel’dovich’s theorem (Zel’dovich 1957), which rules out two-dimensional ﬂows (i.e. ﬂows possessing only two components) as dynamos. Much ingenuity has been brought to bear in determining simple ﬂows that are able to circumvent the strictures of Zel’dovich’s theorem; in this paper we will be utilising a class of ﬂows (so-called 2.5-dimensional ﬂows) that are able to produce dynamos and are amenable to computation at high Rm.
The second approach, termed ‘large-scale dynamo theory’ (Steenbeck, Krause & Rädler 1966; Moffatt 1978; Krause & Raedler 1980; Brandenburg & Subramanian 2005), is utilised to describe how systematic magnetic ﬁelds can emerge on scales larger than the turbulent eddies. It is this theory that is often used to describe the dynamics of astrophysical magnetic ﬁelds such as those found in planets, stars and galaxies. Indeed the 11-year solar cycle, in which the global magnetic ﬁeld of the Sun waxes and wanes, with magnetic waves travelling from mid-latitudes towards the equator as the cycle progresses, is attributed to the action of a large-scale dynamo. Large-scale dynamos are subject to the same anti-dynamo theorems as small-scale dynamos (no assumption about the scale of the ﬁeld is made in either Cowling’s or Zel’dovich’s theorems), and so the utilisation of similar ingenious tricks as those brought to bear for small-scale dynamos (see e.g. Roberts 1972) may prove particularly useful.
The role of the turbulent cascade in both small- and large-scale dynamo theory has been extensively studied, and a complete review is well beyond the scope of this article. In many cases multi-scale ﬂows are driven by a forcing at moderate scales in a system at high ﬂuid Reynolds numbers Re. The importance of inertia at high Re usually leads to the formation of a turbulent cascade and the emergence of a statistically stationary ﬂow that exists on a large range of spatial scales (i.e. a multiscale ﬂow). This is a nice procedure, since the properties of the ﬂow can be changed by the addition of rotation or stratiﬁcation or modiﬁcation of the forcing. However, in this set-up it is extremely difﬁcult to retain precise control of the properties of the turbulent cascade (for example, the spectral slope of the ﬂow, the correlation time of the eddies and the (scale-dependent) degree of helicity of the ﬂow). Another popular setting for examining turbulent dynamo action is that where the ﬂow is driven by thermal driving leading to convection, in either plane layers or spherical shells (Tobias, Cattaneo & Brummell 2008; Käpylä, Korpi & Brandenburg 2010; Augustson et al.
2015). These are natural systems to study owing to their importance in geophysical and astrophysical ﬂuids. However, here energy input occurs on a range of spatial Multi-scale EMF at high Rm 3 and temporal scales, and control over the properties of the turbulent cascade is even more difﬁcult than for driven ﬂows. Categorising the properties of the spectra of such turbulent ﬂows (both forced and convective) is the focus of much ongoing research and we do not pursue this further in this article.
Rather, we take the view that, for the kinematic dynamo problem, more control can be exerted by prescribing the form of the ﬂow, rather than that of the driving.
This technique has been utilised successfully in determining what ﬂow properties are essential for dynamo action and indeed for dynamo action at high Rm. Statistical theories, such as those close to the heart of Zel’dovich, have also led to the characterisation of the dynamo properties of ﬂows with zero or short correlation times; for a review see Tobias, Cattaneo & Boldyrev (2013). It has been demonstrated, inter alia, that these random ﬂows can act as dynamos (even when the magnetic ﬁeld dissipates in the inertial range of the turbulence – the so-called low-Pm problem), and may produce large-scale ﬁelds if the ﬂow lacks reﬂectional symmetry (although these will be subdominant to ﬁelds generated on the resistive scale (Boldyrev, Cattaneo & Rosner 2005)).
The competition between coherent structures and random ﬂows in generating small-scale magnetic ﬁelds has been systematically studied in a series of papers (Cattaneo & Tobias 2005; Tobias & Cattaneo 2008b). It has been shown that the presence of coherence in space and time can overcome the randomness and lead to systematically enhanced small-scale dynamo action. Furthermore, the small-scale dynamo properties of a multi-scale ﬂow that is dominated by coherent (in time) structures has been elucidated; the slope of the spectrum was shown to play a key role in determining the velocity scale responsible for dynamo action. Here a competition between the local (i.e. at spatial scale 1/k) magnetic Reynolds number Rm(k) and turnover time τ (k) selects the eddy scale responsible for generating small-scale magnetic ﬁelds; hereinafter we term this scale the ‘dynamo scale’. The application of this theory to a multi-scale ﬂow enables the calculation of the expected growth time of the small-scale ﬁeld, as comparable with the turnover time of the ‘dynamo eddy’.
Of course, for ﬂows lacking reﬂectional symmetry, this small-scale dynamo must compete with that generating systematic large-scale ﬁelds; and compete it does – extremely effectively. In general, unless some process such as diffusion, shear suppression or nonlinearity acts so as to suppress the small-scale dynamo, it will outperform the large-scale dynamo.† At high Rm, diffusion is not really a viable mechanism for this suppression, and so shear and nonlinear effects remain as the prime candidates. In this paper, we shall not focus on this competition, although we shall return to this important consideration in the discussion. Rather, we shall examine the contribution to large-scale ﬁeld generation (via the electromotive force) of different scales in a multi-scale ﬂow. Once these contributions have been characterised, then a complete theory of the competition between large- and small-scale dynamos is possible.
In the next section we shall describe the general model problem of the calculation of the electromotive force (EMF) in 2.5-dimensional ﬂows and review previous ﬁndings for ﬂows on one scale; we shall conclude the section by generalising the set-up to include a ﬂow on a range of spatial scales. We shall argue that, although the mean EMF is important in determining the large-scale dynamo properties, higher moments of the distribution (for example, the variance of the EMF) may determine whether † This was termed the ‘suppression principle’ by Cattaneo & Tobias (2014).
4 S. M. Tobias and F. Cattaneo the large-scale mode is ever seen. In § 3.2 we determine under what circumstances scales in the ﬂow act independently of each other by determining the distributions of the EMF for ﬂows at different spatial scales. We then calculate the moments of the distribution of the EMF as a function of large-scale shear rate (for a variety of ﬂows with differing ranges of spatial scales and correlation times) and construct parametrisations of the effect of shear on the distribution of the EMF. We conclude in the discussion by postulating how our understanding of the factors controlling largeand small-scale dynamo action may be used to determine a priori whether a given ﬂow will lead kinematically to a large- or small-scale dynamo.
2. Formulation As for the dynamo calculations of Tobias & Cattaneo (2013) and Cattaneo & Tobias (2014), we consider a ﬂow for which the basic building block is a cellular ﬂow, with a well-deﬁned characteristic scale and turnover time. In addition, it is useful to consider ﬂows for which the EMF can be unambiguously measured. We therefore utilise the circularly polarised incompressible Galloway–Proctor ﬂow at scale k ﬁrst introduced by Cattaneo & Tobias (2005). We take Cartesian coordinates (x, y, z) on a 2π-periodic domain, and consider a ﬂow of the form uk = Ak (∂y ψk, −∂x ψk, kψk ), (2.1) where ψk (x, y, t) = sin k((x − ξk ) + cos ωk t) + cos k((y − ηk ) + sin ωk t). (2.2) This 2.5-dimensional ﬂow is maximally helical, taking the form of an inﬁnite array of clockwise and anticlockwise rotating helices such that the origin of the pattern itself rotates in a circle with frequency ωk. Here Ak is an amplitude, and ξk and ηk are offsets that can be varied so as to decorrelate the pattern, and therefore control the degree of helicity. Here they are random constants that are reset every τd, which can therefore be regarded as a decorrelation time. In this paper we only consider the case where ξk = ηk and the ﬂow remains maximally helical.
The dynamo properties of this type of ﬂow acting at one scale are well understood (Cattaneo & Tobias 2005), and so are the inductive properties as measured by the EMF (Roberts 1972; Plunian & Rädler 2002; Courvoisier, Hughes & Tobias 2006;
Courvoisier & Kim 2009). Because the velocity does not depend on the z coordinate, the EMF can easily be measured by applying a z-independent mean ﬁeld (B0, 0, 0) and measuring the resulting EMF E = u × b, where the angle brackets denote an average over horizontal planes (Roberts 1972).
In this paper we wish to calculate the large-scale induction of a superposition of these ﬂows and therefore set Ak and ωk at each scale k. We are free to choose Ak to mimic the properties of any spectrum of turbulence. Having chosen Ak, there is then a unique choice of ωk such that the associated dynamo action at scale k has the same asymptotic growth rate measured in units of the local turnover time (see e.g. Cattaneo & Tobias 2005). With this choice, all of these cells look the same at their own scale.
The combined cellular ﬂow takes the form of a superposition of these ﬂows on scales between kmin and kmax, i.e. we set kmax (2.3) uc = uk, kmin
We stress here that we have not attempted to model the scale dependence of the kinetic helicity for the multi-scale ﬂow. We believe that this is dependent not only on the form and scale of the driving mechanisms, but also on the prevailing conditions (i.e. rotation rate, stratiﬁcation and the presence or absence of large-scale shear). This will be investigated in a forthcoming paper.