Dissipative structures in a nonlinear dynamo

Dissipative structures in a nonlinear dynamo

Abstract

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.

keywords:
fast dynamo, Archontis dynamo, dissipation, symmetry

1 Introduction

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 (2.12.3) below with and ) any state with 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 and 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 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 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 , 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 where 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 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 to 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 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 §§79. 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.

2 Governing equations

We begin with the dimensional equations for incompressible MHD, in the form

(2.1)
(2.2)
(2.3)

where and are the kinematic viscosity and magnetic diffusivity. We take to be a steady force of magnitude acting on a length scale . We will consider the Kolmogorov forcing , whose dimensionless form is given by

(2.4)

In non-dimensionalising we have only the parameters , together with the form (2.4) of the forcing function. From these we can define a magnetic Prandtl number and a Grashof number as in similar forced flow problems (see, e.g., Childress, Kerswell & Gilbert, 2001) by

(2.5)

We have as diagnostics the Reynolds number and magnetic Reynolds number given by

(2.6)

where is a measure of the fluid velocity at a given time, for example the norm, taken as the root-mean-square value, averaged over the periodicity box. We rescale as

(2.7)

with the choice of velocity scale

(2.8)

This yields the non-dimensional formulation, dropping the stars, as

(2.9)
(2.10)
(2.11)

with given in (2.4) and the only parameters specified are . The corresponding Reynolds and magnetic Reynolds numbers are

(2.12)

We refer to as the Grashof number and will be interested in the inviscid limit . The Reynolds number and magnetic Reynolds number are diagnostics depending on the flow regime realised.1 Indeed, they change greatly during the saturation process, when the fields align and , increase significantly. As in CG, the governing equations may be written in a more symmetrical form in terms of Elsasser variables

(2.13)

which gives, for ,

(2.14)
(2.15)
(2.16)

3 Numerical results

We undertook a number of runs to investigate the structure of the steady, equilibrated Archontis dynamo for and values of down to in the periodic domain . The steady solutions were found by following the solution branch: that is taking the output from a run with a given value of and using it as the initial condition for a run with a reduced value of . This establishes the Archontis dynamo as a robust local attractor, in the range of used, in agreement with DA and CG. Whether it is a global attractor over some or all sufficiently small values of 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 modes with for and , for , and for . There were other, less well resolved runs with for and for , which we refer to below as our ‘testing simulations’. For comparison, CG go down to in their study, with resolution . 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 . Plotted are (a) kinetic energy (triangles) and magnetic energy (squares), (b) enstrophy (triangles) and squared current (squares), (c) normalised cross helicity , (d) energy of (dotted line gives dependence).
0.02 64 0.1797 0.1745 0.1849 0.1791 0.3532
0.01 64 0.1781 0.1765 0.1825 0.1816 0.3543
0.001 256 0.1717 0.1717 0.1786 0.1789 0.3435
0.0001 512 0.1722 0.1722 0.1787 0.1787 0.3443
Table 1: Numerical results.

Numerical values are given in table 1 and plotted in figure 1. Panel 1(a) shows the kinetic and magnetic energies in the equilibrated state, given by

(3.1)

These show an initial decrease with (as in CG) but then a slight increase from to : this is quite small bearing in mind the scale on the vertical axis, but appears to be real as it is borne out in our test simulations. In all these runs though this is not apparent from the numbers in table 1 nor in panel 1(a). Panel 1(b) shows the enstrophy and integrated squared current, defined by

(3.2)

The total dissipation is given by and this tends to zero as , as does the input of mechanical energy. Panel 1(c) shows the cross helicity

(3.3)

in normalised form, which rapidly tends to its theoretical upper bound of unity, within the accuracy of our simulations, indicating the strong alignment of field as . Finally panel 1(d) shows the energy in the Elsasser variable, where

(3.4)

This shows a rapid decrease to zero as consistent with the scaling (dotted line) in agreement with the discussion in CG and below.2

Figure 2: Numerical results for the case with . Plotted are (a) (upper) and (lower), (b) (lower) and (upper), against . (c,d) are the same but plotted against . In each panel dotted lines are linear fits.

In panel 1(b) it is notable that the two curves, for enstrophy and total current squared, cross between and . The enstrophy is a little smaller than for and in fact is also for and in our test simulations, making us confident that this is a real effect. This opens up the question of how we measure these quantities, since the rate of evolution of the state becomes extremely slow for small . Figure 2(a,b) shows , , and as functions of time for the case and : comparison with linear fits (dotted) shows clear curvature, as expected, but also highlights the slow evolution. This suggests neutral stability of the final state, and an expansion for any quantity in the form

(3.5)

Although asymptotically the origin of time does not matter, we found it helpful to choose an origin of time (once per run) so as to obtain the best linear fit for quantities in the form

(3.6)

We then use an estimate of the limiting value as ; for example see figure 2 (c,d). This was done for all the results in table 1.

(a) (b) (c)

Figure 3: Cross sections showing in the -plane for . In (a) , (b) and (c) 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 for a range of diffusivities. Clearly seen in each case, but especially in (c) at the smallest , are cross sections of spiralling field, centred on the separatrices, where small scales are generated with consequently enhanced diffusion.

4 Symmetries

We have seen that the dissipation tends to concentrate in cigar shaped regions, with one extending from to the stagnation points at . The reason these straight line separatrices are robust structures is linked to the symmetries of the forcing (2.4) and also applies to the kinematic dynamo study by Galloway & Proctor (1992) of the Kolmogorov flow

(4.1)

These symmetries turn out to be preserved by the solution in the nonlinear regime: there is no symmetry breaking. The forcing (2.4) is -periodic in each coordinate: we only consider symmetries up to this periodicity (and that do not reverse time). Note first that any map maps a vector field according to

(4.2)

It is then easily checked that the forcing in (2.4) is preserved by the following 12 orientation-preserving symmetries, with , which form the group of even permutations of 4 objects, or the symmetry group of the tetrahedron,

(4.3)

These also form a subgroup of the group of 24 symmetries of the :: 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 and so the full symmetry group of the forcing is the direct product .

5 Flow and field on the separatrices

The above symmetries constrain the behaviour of the magnetic field and flow on the separatrices. Take, for definiteness, the separatrix joining to and call this the ‘main separatrix’ for brevity. Because of the symmetries and in (4.3) there is a three-fold rotational symmetry about this separatrix, as seen in DA/CG, and any vector field on the separatrix can only point along the separatrix. We may introduce rotated Cartesian coordinates via

(5.1)

with along the separatrix. From there we may further define cylindrical polar coordinates , whose axis is along the separatrix with and .

Our aim now is to investigate more of the behaviour of the flow near to the separatrix, in the saturated regime. However to fix ideas and establish a benchmark, we consider first the Kolmogorov flow in (4.1). For this flow it can be shown that on the main separatrix motion is governed by

(5.2)

with solution

(5.3)

Here tends to zero as and to as . Near to the separatrix, the radial coordinate and the flow field may be expanded in powers of . In view of the three-fold rotational symmetry, the flow is axisymmetric about the main separatrix at leading order and streamlines are given by

(5.4)

with

(5.5)

Trajectories spiral in for and spiral out for . On the separatrix itself and , directed along the axis.

Figure 4: Plot of components of fields against for runs with , , and for (a) the field , reading down the curves (separated by adding , , , etc.), (b) the field , reading down the curves (c) the field , reading down the curves (separated by adding , , , etc.) (d,e,f) the field for equal to (d) , (e) and (f) .

Now in the nonlinear, equilibrated regime, the symmetries of the system are observed to be preserved and so the motion near and along the separatrix is given by (5.4) for some functions and . These functions characterise aspects of the nonlinear saturation on the separatrices and so of the spiral dissipative structures that form there, visible in figure 3. We can measure the equivalent functions for any field, and in our runs we find that the traces for , and are identical to graphical accuracy. In figure 4(a) we show the components of along the separatrix (separated by constants). This figure in fact depicts two separatrices, the main separatrix from to and the next one that continues to , with . The components of show a sinusoidal form in keeping with the property of the equilibrated fields noted by CG, namely3

(5.6)

There is only slight steepening at as is reduced. Panel 4(b) shows traces of the components of 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 . 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 field, which tends to zero in the limit of small . Panels 4(d,e,f) plot the components of 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 (separated by constants). These show the development of a cusp at , the stagnation point where the two separatrices converge. In conclusion, the field on the separatrix scales as , but its curl scales as , giving a natural cigar width length scale, confirming results in CG and to be explored further below.

6 Local behaviour and scaling in the cigars

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 in figure 4. The aim of the present section is to derive the dissipative lengthscale of 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.

6.1 Uncurling the induction equation

We start with the formulation in Elsasser variables (2.132.16) and for brevity set

(6.1)

We assume the key scaling of CG that , at least in the outer region, which means away from the stagnation points and the separatrices. Without approximation, the steady equations are

(6.2)
(6.3)

Note that a straightforward estimate of the width of a diffusive layer based on (6.3) would suggest an order scaling from balancing , but this is too small, as it does not take into account the different scales of variation of along and across the characteristics of , and the following, more delicate argument is needed.

Subtracting (6.3) from (6.2) gives an equation equivalent to the induction equation (2.10),

(6.4)

which may be uncurled as

(6.5)

where is a scalar field. Taking the divergence gives an elliptic equation for ,

(6.6)

This development can be pursued further, to obtain a general closed but complicated system of scalar PDEs that link the field and flow to the external forcing, as in Zheligovsky (2009). However our present aims are more limited: we only need that (6.5) is equivalent to two equations,

(6.7)

and

(6.8)

where is another scalar field which obeys

(6.9)

from requiring that . Everything is exact up to this point but we note that this representation will generally break down at isolated points where .

Now we approximate: first consider an ‘outer’ region, well away from the dissipative, cigar-like structures that lie on the separatrices joining stagnation points.We neglect diffusion in the outer region, the fields having a greater length-scale. The leading order outer problem is obtained by simply setting in the above equations (6.76.9), leaving a pair of quasi-linear equations for and giving transport along characteristics of , namely

(6.10)
(6.11)

together with an equation that then reconstructs , from (6.8), which we write as a sum of three terms,

(6.12)

with

(6.13)

Finally for this section, we note that in the outer region, can be calculated explicitly in terms of and . Substitution of (6.13) into (6.3), where the diffusive term involving is neglected, yields

(6.14)

By virtue of (6.11), this equation takes the form

(6.15)

where

(6.16)

Scalar multiplication of (6.15) by yields

(6.17)

Thus singularities of can arise, where the helicity type term involving (i.e., the denominator in (6.17)) vanishes.

6.2 Field in the outer region, near the main separatrix

In the outer region, as the main separatrix is approached, it is observed that the field is relatively smooth, as seen from the numerical simulations of CG, and also in view of the leading order Laplacian in (6.2), whereas develops fine scales. Using the formulation in (5.4) we from now on define and by

(6.18)

in cylindrical polar coordinates defined in (5.1) and below. Here the functions and defined for are shown in figure 4(a,b) and are not known analytically; nonetheless their functional form is similar to that of the Kolmogorov flow (5.5)

To understand the diffusive scaling in the cigars and to determine something of the local structure of the fine-scaled field the following strategy is adopted: solve the equations (6.10) and (6.11) by integrating along characteristics of given locally by (6.18) and reconstruct via (6.13). As the characteristics of approach the origin and are squeezed along the outgoing separatrix, given by , , high gradients build up and the terms in 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 coordinates.

There are two problems with this approach: first that the incoming values of and are determined by the outer solution and links to other cigars: as this is beyond what can be addressed analytically, unknown functions have to be introduced. Secondly, even with the simplified, general local form (6.18), analytical calculations rapidly become unwieldy. The first problem will remain with us, but to ameliorate the second problem we simplify further and consider only the motion near to the origin, in which will simply take the field to be, exactly,

(6.19)

in the local cylindrical polar coordinate system . Here and are taken as constants, which we may identify as

(6.20)

We also note from (6.19) that

(6.21)

Our strategy now is to solve the outer, diffusionless equations (6.106.11) for transport of and for the simplified form (6.19) of . This is done exactly, but then to see how large the neglected, diffusion terms are, we approximate by taking but , 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 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.

Equation (6.10) becomes

(6.22)

and letting be a time parameter along characteristics, we integrate this in the standard way, with

(6.23)

and the solution in terms of initial conditions on a characteristic,

(6.24)

If we suppose that we specify the incoming values of on a surface (see figure 5) with

(6.25)

at , then we have the solution:

(6.26)

Here 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 as being some function of order unity with the appropriate symmetry, for example .

Given we can now reconstruct the appropriate part of in (6.13). We have

(6.27)

and so

(6.28)

as , where denotes the derivative of with respect to its first argument. Here we have obtained a component growing as which arises because of the incoming values of on different characteristics being squeezed together.

6.3 The effect of diffusive terms

With this component of in hand, we can now revisit the diffusive equation (6.7). We calculate

(6.29)

and

(6.30)

This now has a growth, by virtue of differentiating again. In equation (6.7) it is clear that the final term with diffusion will be the same order as the term we originally included, when . This gives the scaling of the diffusive cigar width. Similarly at these values of , in (6.8) and (6.9) the terms become of similar magnitude to (though note here that the terms are actually larger in magnitude at this point).

This is the main part of the argument: although we have simplified by focusing solely on in (6.13), consideration of the scalar and component does not affect the discussion, nor does , given straightforwardly by

(6.31)

and so is negligible. To check this we now look at the equation (6.11) for and the corresponding component . After a straightforward calculation (6.11) becomes

(6.32)

The key point is that the right-hand side is of order unity as , as was the case for in (6.22). Thus without solving the equation in detail, it is clear that the solution analogous to (6.26) for will take the form

(6.33)

where gives the incoming values of on the surface , as before and the particular integral involves but is of order unity as .

Now when we reconstruct via (6.13), the component along streamlines will be subdominant to the component , the inverse power of arising from taking the gradient. Thus our focus on in the above discussion of the diffusive breakdown of the outer solution is justified and we have

(6.34)

as on the outgoing separatrix. As a by-product of our calculations we observe that the small-scale field will show components perpendicular to streamlines that diverge as as the separatrix is approached from the outer solution. These will peak at levels when diffusive suppression begins to occur at scales . This is in keeping with the scalings seen by CG, who note that near the separatrices (their section 3.2.1, figures 13 and 16). In view of the 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 , independent of , for which the onset of diffusion will be delayed until smaller values of . 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 . In this way, there could be several nested boundary layers along the separatrices in the limit .

7 Existence of weak steady-state solutions

We consider now the system of equations (6.2) and (6.3) in Elsasser variables, together with the solenoidality conditions

(7.1)

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.

We start by recalling some definitions. Consider the class of functions whose domain is the periodicity cell . The norm in the Lebesgue space is defined, for , as

(7.2)

Since in the above-mentioned class is a positively defined self-adjoint operator (where is the identity), whose eigenfunctions are Fourier harmonics, we can define in the usual way the powers for an arbitrary real , by considering Fourier series. For and any

(7.3)
(7.4)

The Sobolev space is defined for , as the closure in the norm

(7.5)

of the set of infinitely smooth periodic functions, whose domain is . (Evidently, .) We will work in the subspace of zero-mean vector fields, in which the operator can be used instead of in these definitions. In particular, we define (without introducing a new notation) a norm, equivalent to (7.5), in the subspace of zero-mean fields in as

(7.6)

Since the Laplacian is a self-adjoint operator, in the important particular case this implies

(7.7)

We will employ the following:

Embedding theorem (see Bergh & Löfström (1976), Taylor (1981) and references therein).

(i) For , .

(ii) For and , (in particular, ).

We will show in the remainder of this section that for any space-periodic forcing from the Lebesgue space , the system of equations (6.2), (6.3) and (7.1) has at least one weak space-periodic solution from the Sobolev space . The assumption that the box of periodicity is the cube 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 will be required in §9.

Consider then, the set of infinitely smooth solenoidal zero-mean periodic functions, whose domain is the periodicity cell , and denote by its closure in the Sobolev space . A pair of vector fields is a weak solution to the system (6.2), (6.3) and (7.1), if the integral identities

(7.8)

and

(7.9)

hold true for any vector field . (If and are smooth, these identities immediately follow from (6.2) and (6.3).) By Hölder’s inequality and the embedding theorem, for any function ,

(7.10)

where is a constant independent of . Consequently, the Cauchy–Bunyakowsky–Schwarz inequality implies that the integrals involving nonlinear terms admit the bounds

(7.11)

being a constant independent of and , and similarly

(7.12)

Thus the integrals are well-defined.

Consider the scalar product in

(7.13)

Integrating by parts we recast the identities (7.8) and (7.9) in an alternative form involving the scalar product (7.13):

(7.14)

and

(7.15)

Here

(7.16)

denoting, as usual, the inverse Laplacian, and

(7.17)

is a bilinear operator, where is the projection onto the subspace of solenoidal vector fields. (In fact, for the Kolmogorov forcing , but in what follows we do not employ this equality.)

Using (7.7) for , we find

(7.18)

For any

(7.19)
(7.20)

and therefore

(7.21)

Now we develop (7.18), using Hölder’s inequality and the embedding theorem,

(7.22)

which shows that .

Thus, we have shown that for and the first factors in the scalar products in the right-hand sides of (7.14) and (7.15) belong to . Since smooth vector fields are dense in in the norm induced by the scalar product , (7.14) and (7.15) are equivalent to equations

(7.23)

and

(7.24)

respectively, understood as equalities in .

The existence of solutions to the system (7.23), (7.24) is guaranteed by the Leray–Schauder principle (see Leray & Schauder (1934) and Ladyzhenskaya (1969)) under two conditions:

(i) The operator defined as

(7.25)

is compact, i.e. is a strongly converging sequence in for any sequence weakly converging in .

(ii) Any solution to the set of equations

(7.26)

belongs to a ball in of a radius independent of for .

The proof of (i) relies on the embedding theorem for Sobolev spaces, whereby the embedding is compact for , i.e., for , for any sequence weakly converging in