Source: https://www.groundai.com/project/dissipative-structures-in-a-nonlinear-dynamo/
Timestamp: 2019-04-19 22:12:30+00:00

Document:
and Mathematical Geophysics, Moscow, Russia.
This paper considers magnetic field generation by a fluid flow in a system referred to as the Archontis dynamo: a steady nonlinear magnetohydrodynamic (MHD) state is driven by a prescribed body force. The field and flow become almost equal and dissipation is concentrated in cigar-like structures centred on straight-line separatrices. Numerical scaling laws for energy and dissipation are given that extend previous calculations to smaller diffusivities. The symmetries of the dynamo are set out, together with their implications for the structure of field and flow along the separatrices. The scaling of the cigar-like dissipative regions, as the square root of the diffusivities, is explained by approximations near the separatrices. Rigorous results on the existence and smoothness of solutions to the steady, forced MHD equations are given.
Much is known about fast dynamo action: the rapid growth of magnetic fields at high magnetic Reynolds number in fluid flows with chaotic streamlines, but the mechanisms for the dynamical saturation of such fields remain poorly understood. In many cases when the growing field equilibrates by modifying the fluid motion, the effect is to switch off the chaotic stretching in the flow, as measured for example by a reduction in the finite-time Liapunov exponents (e.g., Cattaneo, Hughes & Kim, 1996; Zienicke, Politano & Pouquet, 1998) . What is left is a fluid threaded by a magnetic field which resists stretching and so suppresses overturning fluid motions, but supports elastic wave-like motions, essentially Alfvén waves with coupled field and flow (e.g., Courvoisier, Hughes & Proctor, 2010) . The final state of many simulations shows apparently chaotic behaviour in space and time, suggestive of an attractor of moderate or high dimension, although because of the three-dimensionality of MHD systems little can be done to explore its properties, for example the fractal dimension or spectrum of Liapunov exponents.
Although this appears to be the outcome of many simulations, as far as they can be run, there are some intriguing examples where a further phase of evolution takes place: the magnetic field and flow align, depleting the nonlinear terms, and both fields evolve to a steady (or very slowly evolving) state. The key point is that in unforced, ideal magnetohydrodynamics (see equations (2.1–2.3) below with ν=η=0 and f=0) any state with u=±b is an exact steady solution. The remarkable fact that simulations of forced, non-ideal MHD turbulence could evolve to something very close to such a state was first observed by Archontis (2000) in his thesis, and published in Dorch & Archontis (2004) (hence referred to as DA), and Archontis, Dorch & Nordlund (2007) . These simulations use a compressible code with a Kolmogorov forcing function, (2.4) below, first used as the form of a flow for simulations of fast, kinematic dynamo action by Galloway & Proctor (1992) . Subsequently Cameron & Galloway (2006a) undertook incompressible simulations of the same system as Archontis, and pushed up the fluid and magnetic Reynolds numbers; our work is linked closely to this paper, which we refer to as CG in what follows.
What these authors found was that, starting with a forced fluid flow and a seed magnetic field, the growing magnetic field initially equilibrates in rough equipartition with the velocity field, in a messy, chaotic time-dependent state. However during this state, there is a slow but persistent exponential growth in the average alignment of the u and b vectors, as measured by the cross-helicity. This process of alignment continues until there takes place a sudden increase in the fluid and magnetic energies, and both fields tend to a steady state of almost perfect alignment, discrepancies being controlled by the weak dissipation and the forcing. In fact since any solution u=±b is a neutrally stable solution of the ideal problem (Friedlander & Vishik, 1995) , the solution that is selected must depend delicately on balances involving these subdominant diffusive and forcing effects. We note that some alignment of field and flow has been noted in many other MHD flows, for example see Dobrowolny, Mangeney & Veltri (1980) , Pouquet, Meneguzzi & Frisch (1986) , Mason, Cattaneo & Boldyrev (2006) and references therein, but of a less dramatic nature.
This observation of dynamo saturation in a steady state with such a high degree of alignment was a new phenomenon: CG refer to the saturated state as the ‘Archontis dynamo’, though we prefer the term ‘Archontis saturation mechanism’. CG observed this aligned state as a solution branch over a wide range of magnetic and fluid Reynolds numbers (taking the magnetic Prandtl number to be unity in much of their work). Further developments include the development of bursts of rapid time dependence after some time in the steady state, in the study Archontis, Dorch & Nordlund (2007) . However this appears only to occur in the compressible case, as it has not been seen by CG nor in our simulations; we will therefore not discuss this further. Cameron & Galloway (2006b) also find slow time-dependent evolution of the saturated state for the Kolmogorov forcing with magnetic Prandtl number Pr=ν/η not equal to unity, and for more general spatially periodic steady forcings. In all cases though, the field and flow settle into a state of very close alignment, even if they then evolve on a slow time scale.
The focus of the present paper is to understand more about the structure of the steady saturated state for the Kolmogorov forcing and unit magnetic Prandtl number Pr, with a particular focus on the regions where dissipation occurs and on rigorous results on existence and smoothness. DA and CG find a complex geometrical picture for the field and flow and identify these regions of high dissipation: they are localised along straight-line separatrices that join a family of stagnation points; similar structures are found in the 1:1:1 ABC flow (Dombre et al., 1986) . These are found to have a width scaling as √ε where ε is a dimensionless measure of the diffusivity, and one of our aims is to understand this power law.
We set up the governing equations in §2 and extend the solution branch to yet smaller values of the diffusivity ε by means of large scale simulations in §3. In §4 we then classify the symmetries of the Kolmogorov forcing, which are preserved by the nonlinear, saturated field and flow. These symmetries are the reason for the presence of the non-generic straight line separatrices that join stagnation points in the flow and field, and they constrain the local flow: it is in these regions that dissipation is strongest. We plot the local structure of fields along the separatrix from (0,0,0) to (π,π,π) in §5. We determine the effects of diffusion by setting up PDEs for the advection of field as it enters the dissipative regions in §6 and use these to justify the order √ε scaling for the cigar widths found in CG. We then proceed with a formal mathematical investigation of the existence of steady-state solutions to the MHD problem at hand and bounds for them in various function spaces in §§7–9. The reader should note that these sections use functional analysis and so have a different flavour from the earlier ones. Finally §10 offers concluding discussion.
We undertook a number of runs to investigate the structure of the steady, equilibrated Archontis dynamo for Pr=1 and values of ε down to 10−4 in the (2π)3 periodic domain T3. The steady solutions were found by following the solution branch: that is taking the output from a run with a given value of ε and using it as the initial condition for a run with a reduced value of ε. This establishes the Archontis dynamo as a robust local attractor, in the range of ε used, in agreement with DA and CG. Whether it is a global attractor over some or all sufficiently small values of ε remains unknown, and extremely difficult to address in view of the long transients that may occur. Our runs were undertaken with a pseudo-spectral code using N3 modes with N=128 for ε=0.02 and 0.01, N=256 for ε=10−3, and N=512 for ε=10−4. There were other, less well resolved runs with N=128 for ε=10−3 and N=256 for ε=10−4, which we refer to below as our ‘testing simulations’. For comparison, CG go down to ε=1.25×10−3 in their study, with resolution 1283. Our results thus extend theirs by a little over a decade, and in this section we present measures of the magnetic field and flow in the equilibrated state.
Figure 1: Numerical results plotted against log10ε−1. Plotted are (a) kinetic energy EK (triangles) and magnetic energy EM (squares), (b) enstrophy ΩK (triangles) and squared current ΩM (squares), (c) normalised cross helicity HX/(2EKEM), (d) energy log10E− of Λ− (dotted line gives ε2 dependence).
Figure 2: Numerical results for the case ε=10−3 with N=256. Plotted are (a) 10EK (upper) and 10EM (lower), (b) 10ΩK (lower) and 10ΩM (upper), against t/10. (c,d) are the same but plotted against 104/(t−1400). In each panel dotted lines are linear fits.
We then use an estimate of the limiting value as A0; for example see figure 2 (c,d). This was done for all the results in table 1.
Figure 3: Cross sections showing |Λ−| in the (x,y)-plane for z=π/2. In (a) ε=10−2, (b) 10−3 and (c) 10−4 and the colour scale shown runs from zero (bottom) to (a) 0.031, (b) 0.0041 and (c) 0.00058 (top).
One of the aims of this paper is to focus on dissipative regions in the system: these occur along a series of straight line separatrices (DA/CG) and in figure 3, we show colour plots of |Λ−| for a range of diffusivities. Clearly seen in each case, but especially in (c) at the smallest ε, are cross sections of spiralling field, centred on the separatrices, where small scales are generated with consequently enhanced diffusion.
These also form a subgroup of the group of 24 symmetries of the 1:1:1 ABC flow (Arnold & Korkina, 1983; Dombre et al., 1986) , and the above follows the notation in Gilbert (1992) . The symmetries all commute with the inversion symmetry j(r)=(−x,−y,−z) and so the full symmetry group of the forcing f is the direct product A4×Z2.
with ζ along the separatrix. From there we may further define cylindrical polar coordinates (ρ,θ,ζ), whose axis is ζ along the separatrix with μ=ρcosθ and χ=ρsinθ.
Trajectories spiral in for ζ≃0 and spiral out for ζ≃√3π. On the separatrix itself u=(0,0,2s(ζ)) and ∇×u=(0,0,2Ω(ζ)), directed along the axis.
Figure 4: Plot of components of fields against ζ for runs with ε=0.02, 0.01, 10−3 and 10−4 for (a) the field Λ+, reading down the curves (separated by adding 0, −0.1, −0.2, etc.), (b) the field ∇×Λ+, reading down the curves (c) the field ε−1/2∇×Λ−, reading down the curves (separated by adding 0, −0.25, −0.5, etc.) (d,e,f) the field ε−1Λ− for ε equal to (d) 0.01, (e) 10−3 and (f) 10−4.
There is only slight steepening at (π,π,π) as ϵ is reduced. Panel 4(b) shows traces of the components of ∇×Λ+ with clear cosine form, in keeping with (5.6) and (5.5) but of somewhat enhanced amplitude, and with evidence of some finer scale structure near (π,π,π). These indicate that the approximation (5.6) is reasonable for the leading order fields on the separatrices.
The picture is naturally more complicated for the Λ− field, which tends to zero in the limit of small ε. Panels 4(d,e,f) plot the components of ε−1Λ− along the separatrix: there is clear evidence of finer scale oscillations emerging in the limit, but the nature of the limiting distribution is unclear. Panel 4(c) shows the fields ε−1/2∇×Λ− (separated by constants). These show the development of a cusp at (π,π,π), the stagnation point where the two separatrices converge. In conclusion, the field Λ− on the separatrix scales as O(ε), but its curl scales as O(ε1/2), giving a natural O(ε1/2) cigar width length scale, confirming results in CG and to be explored further below.
We now have some knowledge of the local structure of the flow and field on the separatrices, in terms of both the general form it must take, namely (5.4), and the actual behaviour for small values of ε in figure 4. The aim of the present section is to derive the dissipative lengthscale of √ε noted by CG. Of course we are not able to put together a solution that is complete: the dissipative, cigar-like regions process field that is drawn in, in a spiralling fashion, and then churn it out again. A complete picture would involve matching to the outer region, which is a highly three-dimensional problem, beyond what we can do; nonetheless a local picture gives some information.
Note that a straightforward estimate of the width of a diffusive layer based on (6.3) would suggest an order ε scaling from balancing Λ⋅∇λ∼ε∇2λ, but this is too small, as it does not take into account the different scales of variation of λ along and across the characteristics of Λ, and the following, more delicate argument is needed.
from requiring that ∇⋅λ=0. Everything is exact up to this point but we note that this representation will generally break down at isolated points where Λ=0.
Thus singularities of c can arise, where the helicity type term involving (Λ⋅∇)Λ (i.e., the denominator in (6.17)) vanishes.
To understand the diffusive O(ε1/2) scaling in the cigars and to determine something of the local structure of the fine-scaled λ field the following strategy is adopted: solve the equations (6.10) and (6.11) by integrating along characteristics of Λ given locally by (6.18) and reconstruct λ via (6.13). As the characteristics of Λ approach the origin and are squeezed along the outgoing separatrix, given by ζ=O(1), ρ=0, high gradients build up and the terms in ε that were earlier neglected increase: when these come into balance with the terms we have retained, we reach the scale at which diffusive effects become important, fixing the width of the dissipative regions.
Figure 5: Schematic figure showing the flow in the (ρ,θ,ζ) coordinates.
Our strategy now is to solve the outer, diffusionless equations (6.10–6.11) for transport of a and c for the simplified form (6.19) of Λ. This is done exactly, but then to see how large the neglected, diffusion terms are, we approximate by taking ζ=O(1) but ρ≪1, so our results are valid in the outer region, near to the origin, on the outward-going separatrix, as depicted schematically in figure 5. Of course, by the time ζ=O(1) we are, strictly speaking, away from the stagnation point at the origin and the form (6.19) that we are using no longer applies. However the above form is sufficient to obtain the overall structure of the outer solution as the separatrix is approached, together with the scaling of the diffusive layer width.
Here A gives the form of the field being carried in from the outer region, and we do not know much about it, except that it has 3-fold rotational symmetry (see figure 3 and figure 13 of CG). It is perhaps helpful to think of A as being some function of order unity with the appropriate symmetry, for example A=A0+A3cos3θ.
as ρ→0, where Aθ denotes the derivative of A with respect to its first argument. Here we have obtained a component growing as ρ−1 which arises because of the incoming values of A on different characteristics being squeezed together.
This now has a ρ−2 growth, by virtue of differentiating A again. In equation (6.7) it is clear that the final εΛ⋅∇×λa term with diffusion will be the same order as the term Λ⋅∇×Λ=4ωσζ we originally included, when ερ−2=O(1). This gives the ρ=O(√ε) scaling of the diffusive cigar width. Similarly at these values of ρ, in (6.8) and (6.9) the terms ε∇×λa become of similar magnitude to ∇×Λ (though note here that the ∇a terms are actually larger in magnitude at this point).
where C(θ0,ζ0) gives the incoming values of c on the surface ρ=ρ0, as before and the particular integral CPI involves A but is of order unity as ρ→0.
as ρ→0 on the outgoing separatrix. As a by-product of our calculations we observe that the small-scale field λ will show components λa perpendicular to streamlines that diverge as ρ−1 as the separatrix is approached from the outer solution. These will peak at levels λ=O(ε−1/2) when diffusive suppression begins to occur at scales ρ=√ε. This is in keeping with the scalings seen by CG, who note that Λ−=ελ=O(√ε) near the separatrices (their section 3.2.1, figures 13 and 16). In view of the cos3θ dependence of the leading field identified here, this component must go to zero on the axis itself and is presumably strongly suppressed by diffusion. Thus we cannot make a detailed link with figure 4: the field here originates with the mean component of A, independent of θ, for which the onset of diffusion will be delayed until smaller values of ρ. This also presumably explains the structure seen in figure 3 (most clearly in (b)) or figure 13 of CG, with three incoming sheets of field merging in an axisymmetric ‘collar’ at smaller values of ρ. In this way, there could be several nested boundary layers along the separatrices in the limit ε→0.
In this section we define weak solutions to these equations and formally prove their existence, adapting the approach of Ladyzhenskaya (1969) . In the next two sections we will show that the weak solutions are classical smooth functions, satisfying the equations at any point in space.
Embedding theorem (see Bergh & Löfström (1976) , Taylor (1981) and references therein).
(ii) For 0<s<N/p and q=Np/(N−ps), Wsp(T3)⊂Lq(T3) (in particular, ∥Φ∥q≤Cs,p∥Φ∥s,p).
We will show in the remainder of this section that for any space-periodic forcing f from the Lebesgue space L2(T3), the system of equations (6.2), (6.3) and (7.1) has at least one weak space-periodic solution from the Sobolev space W12(T3). The assumption that the box of periodicity is the cube T3≡[0,2π]3 is technical: our arguments can be repeated almost literally for the case of an arbitrary parallelepiped of periodicity. Note that in this and the following sections we do not restrict ourselves to the Kolmogorov forcing (2.4); higher regularity of f will be required in §9.
Thus the integrals are well-defined.
respectively, understood as equalities in H.
is compact, i.e. B(Λn,λn) is a strongly converging sequence in H⊗H for any sequence (Λn,λn) weakly converging in H⊗H.
belongs to a ball in H⊗H of a radius independent of μ for 0≤μ≤1.

References: §2
 §3
 §4
 §5
 §6
 §10
 §9