4.1 Compactification and boundary conditions

For every correlation scalar field of isotropic turbulence, we introduce an abstract scalar field satisfying all formal relations satisfied by the correlation scalar field locally in its natural domain. In particular, we introduce an abstract field $F$ , which is an even function of a coordinate $r$ on an infinite line, and an abstract field $H$ , which is an odd function on that line, satisfying all relations from (2.2) to (2.7). Predominantly, the Karman–Howarth equation (2.4) is satisfied. The relation (2.7) is regarded essentially as a definition of the kinetic energy $K$ , and (2.2) and (2.6) are equivalent definitions of the energy dissipation rate $\unicode[STIX]{x1D700}$ .

A priori, the natural domains of the abstract scalar fields are lines, planes and hyperplanes, subspaces of an infinite-dimensional hyperplane. Then, we compactify, in a way that preserves flatness: we impose that infinite lines are topologically ‘wound’ to circles, planes to two-tori and hyperplanes to $n$ -tori, subspaces of an infinite-dimensional torus. Explicitly, this means that we impose periodicity on all abstract scalar fields, a multidimensional analogue of the periodicity condition (3.1). As discussed in § 2, our mathematical set-up inherits the so-called closure problem, meaning that the interactions are non-trivially entangled, and we probably need the full set of the infinite scalar fields in order to answer fundamental questions from first principles. Hence, we restrict ourselves to the lowest order, i.e. the one-dimensional components of the full theory, and study its compactification to a circle.

The infinite line $-\infty <r<\infty$ is compactified to a circle of circumference $2\unicode[STIX]{x03C0}$ . We shall work in the interval $-\unicode[STIX]{x03C0}\leqslant r\leqslant \unicode[STIX]{x03C0}$ . The circumference or period $l=2\unicode[STIX]{x03C0}$ simply reflects the usual choice in DNS flows; it can be changed to any value by rescaling $r$ . The dynamical equation in the problem, the Karman–Howarth equation, is second order in the field $F$ and first order in the field $H$ . Therefore, the boundary conditions encoding periodicity on the fields $F$ and $H$ in this interval are

The field $F$ is an even function, which means that $\unicode[STIX]{x2202}_{r}F$ is odd, and the field $H$ is odd. Thus, the first condition on $F$ is trivially satisfied and the remaining two conditions in (4.1a ) read equivalently

A most fundamental fact which will be realized in the course of our discussion is that although the fields $F$ and $H$ must be infinitely differentiable everywhere in $-\unicode[STIX]{x03C0}\leqslant r\leqslant \unicode[STIX]{x03C0}$ , they are only twice and once differentiable respectively, if one indeed views this interval as a parameterization of the circle where the point $r=-\unicode[STIX]{x03C0}$ is identified with the point $r=+\unicode[STIX]{x03C0}$ : higher derivatives of the fields become discontinuous as one crosses the point $-\unicode[STIX]{x03C0}\sim +\unicode[STIX]{x03C0}$ . Employing the notion of $C^{k}$ differentiability class (that is, $k$ times continuously differentiable), we may say that the field $F$ is $C^{2}$ class periodic while $H$ is $C^{1}$ class periodic. Such functions are not unusual. They occur every time one creates a periodic function by piecing together copies of the same segment of the curve of, e.g., a polynomial, because the matching is only finitely smooth at the joining points; that is, as one joins one boundary point of the curve segment to the other. In our case, the ‘failure’ of higher-order periodicity is encoded in a sequence of relations on the fields $F$ and $H$ which are fundamental for the completion of our theory. These relations follow once one develops the spectral version of the theory, to which we now turn.

4.2 The spectral space and the dynamical degrees of freedom

The abstract field $F(r)$ should give rise to an abstract energy spectrum $E(k)$ . Its construction with all of the associated details is the main subject of this subsection. Of course, we need first to find out what $k$ is.

The spectral space is associated with the normal modes of the theory of $F$ in the compact space, determined by the non-interacting part of the Karman–Howarth equation, i.e. the part of it which is linear in $F$ . This is a neat answer which may sound either perfectly natural or ad hoc depending on whether one doubts the relevance to the problem of the splitting of the field equation into linear (free) and interacting parts. For that reason, we prefer to show that it is indeed the correct answer.

We may put our thoughts in order by returning to the ideal isotropic flows. The relation between the autocorrelation function $F(r)$ and the energy spectrum $E(k)$ in isotropic turbulence is

where $k$ is the magnitude of the wavenumber vector $k_{i}$ . The domain of the ideal flows is the infinite space; therefore, the wavenumber space is continuous and infinite itself.

Use of Fourier transforms in the tensorial quantities in three dimensions means that the normal modes are the usual plane waves $\exp (\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{r})$ , leaving no mystery as to what spectral space means. On the other hand, at the level of scalar quantities which solely interest us, one is left with (4.2). Is expression (4.2) itself an expansion of the field $F$ on normal modes? It may be so if the functions

are the normal modes of some fundamental dynamical equation in the problem. It should be noted that the $f_{k}(r)$ are normalized so that $f_{k}(r)\rightarrow 1$ for $r\rightarrow 0$ .

Indeed, the $f_{k}(r)$ are eigenfunctions of the linear operator $r^{-4}\unicode[STIX]{x2202}_{r}\,r^{4}\unicode[STIX]{x2202}_{r}$ that acts on the field $F$ in the Karman–Howarth equation (2.4), which we may rewrite as

The eigenvalue equation for this linear operator is

The eigenvalues $-k^{2}$ fix the time dependence of the respective modes $f_{k}(r)$ , $\exp (-2\unicode[STIX]{x1D708}k^{2}t)$ . As anticipated above, the normal modes are indeed specified most naturally by the free part of the field equations.

Equation (4.5) is a Sturm–Liouville eigenvalue equation (Arfken & Weber Reference Arfken and Weber2005), and $f_{k}(r)$ are its solutions that are analytic around the origin. No special conditions need to be imposed at the ‘boundary’ $r=\pm \infty$ ; thus, there are no restrictions on the values of $k$ . The spectral space is continuous. Equation (4.2) is an expansion of $F(r)$ on the modes $f_{k}(r)$ for all $k$ . The coefficients form the energy spectrum, $E(k)$ . Equation (4.3) is consistent with (2.7) by $\int _{0}^{\infty }\text{d}k\,E(k)=K$ .

Compactifying the infinite line to a circle of circumference $2\unicode[STIX]{x03C0}$ means that the modes $f_{k}(r)$ satisfy the eigenvalue equation (4.5) everywhere in $-\unicode[STIX]{x03C0}\leqslant r\leqslant \unicode[STIX]{x03C0}$ and satisfy the boundary conditions on $F$ given by (4.1a ) or equivalently (4.1b,c ),

bearing in mind that the modes (4.3) are even functions the same as the field $F$ . We should explicitly include the zero mode $f_{0}(r)=1$ , whose existence is downplayed in the infinite-line case. All modes are square integrable in the interval $-\unicode[STIX]{x03C0}\leqslant r\leqslant \unicode[STIX]{x03C0}$ . The conditions (4.6) specify the eigenvalues $-k^{2}$ . In other words, they specify the spectral space of the compactified theory.

The spectral space is discrete, a result of the compactness of the domain. Explicitly, equation (4.6) gives that $k$ is given by the solutions of the transcendental equation

The first few solutions of (4.7) are $k=$ 0 and

One may in fact show (see appendix A) that the $n$ th non-zero wavenumber $k$ is the $n$ th zero of the Bessel function of the first kind of order $5/2$ : $J_{5/2}(\unicode[STIX]{x03C0}k)=0$ .

One observes that there is no value in the vicinity of $k=1$ . Moreover, there is the $k=0$ mode. Both phenomena admit interesting interpretations that we will discuss below. One also observes that values of $k$ are not integers, but they are in the vicinity of integers and approach them asymptotically. The large values of $k$ are given by

for large integers $n$ . The properties of $k$ are discussed in more detail in appendix A.

We have therefore completely determined the normal modes of the compactified theory. We may finally write down the mode expansion of the field $F$ in the compact space,

where the sum is taken over the non-negative solutions $k$ of (4.7). The energy spectrum $E(k)$ is defined by this expansion as the abstract energy spectrum. By the relations (2.6) and (2.7), we obtain

These relations have the form expected by the interpretation of the quantities involved.

The modes $f_{k}(r)$ are eigenfunctions of a Sturm–Liouville problem and therefore are orthogonal; orthogonality of course holds in the $\unicode[STIX]{x1D6FF}$ -function sense in the infinite-line case as well. We make orthogonality explicit in order to invert the expansion (4.10). For any positive $k$ and $k^{\prime }$ , we find (appendix B)

For the zero mode $f_{0}(r)=1$ and any positive $k$ mode $f_{k}(r)$ , orthogonality reads

These equations are an immediate implication of the eigenvalue equation (4.5) and the boundary conditions (4.6).

We are now able to deduce an explicit expression for the abstract energy spectrum as a transform of $F$ ,

for any positive $k$ , and

for $k=0$ . We learn most convincingly that the zero mode not only cannot be flippantly discarded within the discrete spectrum of the compactified theory but also its energy is proportional to what one would reasonably call the Loitsyansky integral in this context. We also learn that the zero mode is the sole contributor to the Loitsyansky integral, as it is most clearly expressed by (4.13). We will demystify these highly important properties of the compactified theory below. We shall also drop the adjective ‘abstract’ from now on for the fields $F$ , $H$ and $E(k)$ , understanding that we refer to the compact space analogues of the autocorrelation function, triple correlation function and energy spectrum.

Inserting the mode expansion (4.10) of the field $F$ into the Karman–Howarth equation (2.4), we obtain a set of dynamical equations for the spectrum $E(k)$ . At this point, the fact that the $f_{k}(r)$ are the normal modes of the free (non-interacting) part of the Karman–Howarth comes into play and ‘diagonalizes’ the free part of the equations for $E(k)$ . For all positive $k$ modes, one finds

where the energy transfer $T(k)$ is given by

The orthogonality relation for $f_{k}^{\prime }(r)$ is similar to (4.12) times a factor $k^{2}$ (relation (B 5)). This allows us to invert (4.17) to write the field $H(r)$ as an expansion of the modes $f_{k}^{\prime }(r)$ ,

Here, $H(r)$ is an odd function, the same as the modes $f_{k}^{\prime }(r)$ . By the condition (4.6), the field $H$ automatically satisfies the boundary condition (4.1b,c ). The conservation of energy transferred between the modes, expressed by $\sum _{k}T(k)=0$ , is equivalent to $H^{\prime }(0)=0$ . The latter condition is a well-known fact regarding the triple correlation function in isotropic turbulence (Monin & Yaglom Reference Monin and Yaglom1971), which follows explicitly from the incompressibility condition on the velocity field. The sum in (4.18) excludes the zero mode, to which we now turn.

The energy $E(k)$ of all positive $k$ modes decays through dissipation at a rate $2\unicode[STIX]{x1D708}k^{2}E(k)$ , and it is transferred to other modes through $T(k)$ which expresses the cascade. For the zero mode, we have

The zero-mode energy does not decay through dissipation, as the associated rate $2\unicode[STIX]{x1D708}k^{2}E(k)$ vanishes, and it is not transferred to other modes either, unless we can allow for a non-zero value for $T(0)$ . We will be explicitly prove (4.19) by a different route below.

With the aid of the last result, we may rewrite the mode expansion (4.10) as follows:

Therefore, the ‘vacuum’ configurations, $F=\text{const.}$ and $H=0$ , are associated with the zero mode. Indeed, if we proceed inversely and put $F=\text{const.}$ into (4.14), the orthogonality relation (4.13) tells us that $E(k)=0$ for all $k>0$ .

The transformation

where $c_{1}$ is a constant, leaves the Karman–Howarth equation and the boundary conditions unchanged. That is, this transformation is a symmetry of the theory, mapping a solution of the equations to another solution. The entirety of the ‘vacua’ $F=\text{const.}$ and $H=0$ is generated from the solution $F=0$ and $H=0$ by the transformation (4.21). What is more, the transformation (4.21) acts on the general configuration (4.19) by simply moving the ‘vacuum’,

The integral on the right-hand side of (4.14) cannot ‘see’ additive constants in $F$ . In other words, by varying the zero-mode energy $E(0)$ of an arbitrary configuration, we obtain another configuration that differs only by an additive constant in the field $F$ . We conclude then that the zero mode is not only a ‘vacuum’ mode, but also is completely decoupled from the other modes of the system: $E(0)$ cannot affect the evolution of the system in any way and sets only a lower value for the total energy $K$ . One loses nothing by setting that lower value to zero, $E(0)=0$ .

What is a little strange is that $E(0)$ is also a proper integral of motion. Formula (4.15) reminds us that $E(0)$ is essentially the Loitsyansky integral of the theory; the proportionality constant involves the fixed circumference $l$ , chosen here to be equal to $2\unicode[STIX]{x03C0}$ . Therefore, the Loitsyansky integral is also constant throughout motion. This result clearly expresses a conservation law of the theory that it is derivable directly at the level of the field equations for the given boundary conditions. Indeed, by multiplying the Karman–Howarth equation (2.4) by $r^{4}$ and integrating, we have that

and by the periodicity conditions (4.1), the right-hand side vanishes identically, proving explicitly equation (4.19).

Now, our previous conclusions that the specific value of $E(0)$ is of no particular consequence and we may choose to work with fields $F$ such that $E(0)=0$ suggest that the specific value of the conserved Loitsyansky integral is of no particular consequence either, and we may choose initial conditions such that

The dynamical and interacting positive $k$ modes preserve (4.24) through time as they have the necessary property built into them, expressed by (4.13). Generic initial conditions can be generated through the transformation (4.21) and give rise to solutions that differ from the solutions (4.24) by an ignorable additive constant in the field F (which equals $2/3$ times the energy of the decoupled zero mode).

4.3 Loitsyansky integral and scaling

The conclusion that (4.24) identifies the entire class of dynamically distinct solutions – modulo irrelevant additive constants – calls for some further discussion. It is well known that the Loitsyansky integral $\int _{0}^{\infty }r^{4}F(r)\,\text{d}r$ is a quantity with a special significance in the phenomenology of isotropic turbulence; see, e.g., Monin & Yaglom (Reference Monin and Yaglom1971). It would be at least instructive if we paused momentarily the formal development of the theory in order to contrast the status of the Loitsyansky integral between the compactified theory and the ideal isotropic flows.

In ideal isotropic flows, the physical space is infinite and the spectral space is continuous all the way to $k=0$ . The existence of the Loitsyansky integral depends on the behaviour of the spectrum $E(k)$ in the infinitesimal neighbourhood of $k=0$ . When it exists, its (possibly approximate) conservation has quite strong dynamical implications. By neat symmetry arguments involving the freedom of rescaling lengths and times at large Reynolds numbers, see, e.g., Oberlack (Reference Oberlack2002) and references therein, one may deduce that the Loitsyansky ‘invariant’, with dimensions $[L]^{7}[T]^{-2}$ , shapes the evolution of the system such that, for example, the integral scale $L$ evolves by the power law $L\sim t^{2/7}$ at times adequately distant from the initial conditions. Power laws for other characteristic quantities of turbulence, such as the total energy or the Reynolds number, are then also specified.

In the compactified theory, there cannot be a problem of existence for the Loitsyansky integral, and it is exactly conserved. Moreover, there is no such thing as an infinitesimal neighbourhood of $k=0$ , and hence the integral is solely determined by the value of the energy at $k=0$ . The scaling symmetry considerations apply of course in the compactified theory as well, but the result is different. In this case, there is a constant length in the problem, the fixed circumference $l$ of the compact space. This breaks the symmetry of rescaling lengths that held in the infinite space, and implies that any large length scale $L$ should eventually approach a constant of order $l$ – always for large enough Reynolds numbers.

On the other hand, we consider times such that $L$ is not as large as $l$ , so that rescaling lengths are still a good symmetry and the scaling arguments we mentioned above apply. We observe also that the very presence of the fixed length $l$ allows one to multiply the Loitsyansky integral with an arbitrary power of $l$ and get an infinity of conserved quantities of dimension $[L]^{any~number}[T]^{-2}$ , one of them being the energy $E(0)$ given by (4.15). In other words, we seem to have a wealth of invariants available for scaling arguments. It should be noted also that these additional invariants have come for free. The reason why they came for free is that they mean nothing scalewise: the Loitsyansky integral is zero, as far as dynamics is concerned. Scaling symmetry arguments applied to any of these invariants produce an empty statement, $L^{any~number}t^{-2}\times 0=0$ . What predominantly shapes the evolution scalewise in the long run is the compactness of the space itself. On the other hand, other invariants may very well arise during the intermediate stages of the evolution. Nonetheless, it seems to be difficult to single out any non-trivial potential invariant in compact space as one can do with the Loitsyansky or Saffman invariants (Monin & Yaglom Reference Monin and Yaglom1971) in the ideal isotropic flows.

4.4 Spectrum regularity conditions

A physically realizable spectrum $E(k)$ of developed turbulence is positive and falls off exponentially for large wavenumbers. We shall call such a spectrum regular or physical; when this notion is applied for finite wavenumbers, it will mean that the spectrum is positive and decreasing for large wavenumbers. The compactness of the domain where the field $F$ lives has non-trivial implications when combined with the condition that the associated energy spectrum must be regular.

We start by observing that all of the energy moments exist; that is, the quantities

are finite for all $i=1,2,3,\ldots .$ The moments of the spectrum are related to the derivatives of the field $F$ at $r=0$ . Indeed, the eigenvalue equation (4.5) tells us that if we apply $i$ times the operator $-r^{-4}\unicode[STIX]{x2202}_{r}r^{4}\unicode[STIX]{x2202}_{r}$ on an eigenfunction, we obtain

Then, multiplying both sides by $E(k)$ , summing over all $k$ and recalling the mode expansion of the field $F$ , we find

Taking the limit $r\rightarrow 0$ of both sides, we have that the right-hand side goes to the $i$ th moment given in (4.25) by the normalization of the eigenfunctions, $f_{k}(r)\rightarrow 1$ for $r\rightarrow 0$ . The left-hand side involves derivatives of the field $F$ up to order $2i$ at $r=0$ . The general relation between derivatives of $F$ and the energy moments given by the limit of (4.27) is a useful formal fact to bear in mind.

In order to make this relation more explicit, we expand the eigenfunctions $f_{k}(r)$ in power series around $r=0$ , to find the relations

The mode expansion (4.10) of the field $F$ implies then that

The overall conclusion is that the differentiability of $F$ , understood through its mode expansion, rests on the regularity of the spectrum: the left-hand side of the relation (4.27) or (4.29) makes sense as much as right-hand side makes sense. The last statement will be revisited and refined in an interesting way below (§ 4.5.4), dictated by the very properties of the energy spectrum in compact space.

We consider now the ‘boundary’ point $r=\unicode[STIX]{x03C0}$ . We differentiate (4.26) once and evaluate the result at that point,

where the right-hand side vanishes by the boundary condition (4.6). We therefore obtain a sequence of relations for $i=1,2,\ldots ,\infty$ , the first few of which can be written as

Either by (4.30) or by the explicit relations (4.31), one observes that the values of the odd derivatives of the modes at $\unicode[STIX]{x03C0}$ are related to the values of lower even derivatives at $\unicode[STIX]{x03C0}$ through equations with constant ( $k$ -independent) coefficients. Explicit relations such as (4.31) require the use of the lower derivative relations to eliminate all of the odd derivative terms that arise in (4.30) for a given $i$ , but we shall not try to give a general simplified result here; the relations (4.31) will be more than adequate and it is sufficient to know that the general relation is given by (4.30).

An alternative and somewhat instructive derivation of (4.31) comes from differentiating the eigenfunctions as given by (4.3) explicitly. Indeed, the first few derivatives of the eigenfunctions at $r=\unicode[STIX]{x03C0}$ read

where, after differentiating at the desired order, we have the substitution

everywhere, which is nothing but a rearrangement of (4.7) that defines $k$ , which itself is nothing but the explicit form of the boundary condition (4.6). Then, the relations (4.31) follow by a little algebra.

With the help of the large- $k$ identity (A 7), the relations (4.32) allow us to see that for large $k$ ,

a result that can be established in general by (4.26), (4.32a ) which allows us to deduce the large- $k$ behaviour of $f_{k}^{(2n)}(\unicode[STIX]{x03C0})$ , and (4.30) which shows that $f_{k}^{(2n+1)}(\unicode[STIX]{x03C0})\sim f_{k}^{(2n)}(\unicode[STIX]{x03C0})$ for large $k$ (modulo constant coefficients).

The importance of the relations (4.30) can hardly be overemphasized. Clearly, they show that the modes are only $C^{2}$ class periodic functions: $f_{k}(r)$ are even functions, whose odd derivatives (third and higher) do not vanish at $r=\pm \unicode[STIX]{x03C0}$ and therefore are not continuous. In infinite space, where the ‘boundary’ lies at infinity, relations such as (4.30) are void of content, as all quantities involved vanish. In compact space, these relations imply non-trivial constraints on the fields $F$ and $H$ when combined with the regularity of the energy spectrum and energy transfer function.

Indeed, we consider ideally regular energy spectra and energy transfer functions; that is, they are positive and exponentially decreasing for infinitely large $k$ . Then, through the mode expansions (4.10) and (4.18) of the fields $F$ and $H$ respectively, the relations (4.30) imply that

for any value of $i$ , as by (4.34) any derivative of the mode expansions converges for such spectral functions. Applying (4.30), we have used the fact that $\unicode[STIX]{x2202}_{r}(r^{-4}\unicode[STIX]{x2202}_{r}r^{4}\unicode[STIX]{x2202}_{r})^{i}=(\unicode[STIX]{x2202}_{r}r^{-4}\unicode[STIX]{x2202}_{r}r^{4})^{i}\unicode[STIX]{x2202}_{r}$ , which one may very easily verify by writing down the consecutive operators explicitly. We shall call these relations the regularity conditions or relations, and the set of relations for $i\leqslant n$ form the order- $n$ , or the $n$ th-order, set of regularity conditions.

For $n=\infty$ , by (4.34) and the mode expansions of the fields $F$ and $H$ , if the sum of $(\pm )k^{2i-2}E(k)$ and $(\pm )k^{2i-4}T(k)$ over all $k$ is finite for all $i$ , then the regularity relations (4.35) hold for all $i$ . If the spectral functions are ideally regular (positive and exponentially decreasing for large $k$ ), then the finiteness of these sums for any $i$ follows, and hence so do the relations (4.35). Thus, (4.35) are a set of necessary relations for ideal regularity. There are then two intriguing facts. First, (4.35) are not an adequate set of conditions for really well-behaved energy spectrum and energy transfer functions. Second, in practice, only a finite number of these conditions can be employed in order to actually solve a problem in compact space. The point is though, that by imposing the regularity relations in practice, we actually enforce regularity to an adequate degree so that these conditions are practically adequate: it turns out that by increasing the order of regularity, the spectral functions remain physical for an increasingly larger regime of wavenumbers. Moreover, we shall see that the order of regularity is related to the ‘well-resolved’ scales, or equivalently and more concretely, to the number of well-defined energy moments that we need to have available in order to reconstruct in this language an actual case of DNS flows; these are always very few. These comments will become very clear in what follows.

We may give (4.35) an explicit form for the first few orders of regularity. Either through the relations (4.31) and the mode expansions of the fields, or by direct evaluation of (4.35), we obtain for $i=1,2,3$

for the field $F$ and for the field $H$ . One should note that all regularity conditions, on both $F$ and $H$ , express values of odd functions at $r=\unicode[STIX]{x03C0}$ in terms of even functions at that point. This is already clear at the level of the general relations (4.35).

We may now complete the discussion we started at the end of § 4.1 regarding the periodicity of the fields $F$ and $H$ . Although the fields are $C^{\infty }$ differentiability class as functions over the interval  $-\unicode[STIX]{x03C0}\leqslant r\leqslant \unicode[STIX]{x03C0}$ , they can also be viewed as $C^{2}$ and $C^{1}$ periodic respectively over the circle parameterized by $-\unicode[STIX]{x03C0}\leqslant r\leqslant \unicode[STIX]{x03C0}$ . Presumably, the field corresponding to the transverse autocorrelation function, which is related to $F$ by the first derivative relation $(r^{2}F)^{\prime }/2r$ (Monin & Yaglom Reference Monin and Yaglom1971), is also $C^{1}$ periodic. The ‘failure’ of having smoother periodicity is due to those derivatives of the fields that are odd functions and do not vanish at $r=\pm \unicode[STIX]{x03C0}$ , hence becoming discontinuous as we cross the point $-\unicode[STIX]{x03C0}\sim +\unicode[STIX]{x03C0}$ . That ‘failure’, or, better, obstruction, is encoded fundamentally in the modes of the theory, expressed through the relations (4.30). The regularity conditions (4.35) are a formal statement of that obstruction at the level of the fields $F$ and $H$ . The link between the modes and the fields is provided by the latter being admissible to physical interpretation.

4.5 Interactions and renormalization

Up to this point, we have worked out the free-field theory; that is, we have constructed the modes of the Karman–Howarth equation in compact space, and explored the host of interesting facts about the nature of the fields $F$ and $H$ to which one is led by the properties of those modes. However, closing off the hierarchy at the level of one-dimensional fields, as discussed in § 2, means that the field $H$ must be regarded as a functional of $F$ , expressing its self-interaction. What makes things non-trivial is the regularity conditions (4.35). It is by no means obvious how and why any reasonable relationship between the fields should or could conform to these necessary conditions. It turns out that this difficulty resolves itself in a rather beautiful way.

5.1 The spectral space of the DNS flows

The spectral space of the DNS flows consists of vectors that are triplets of integers, $\boldsymbol{k}=(n_{1},n_{2},n_{3})$ . The spectral space is organized in shells of wavenumbers labelled by integers $n$ such that $n-0.5<|\boldsymbol{k}|<n+0.5$ . By averaging functions of all wavenumbers in each shell, one obtains direction-independent quantities that are regarded as functions of $n$ . The larger the (shell) number $n$ is, the better the approximation becomes. Then, one may think of each shell labelled by $n$ as a single wavenumber $n$ . We have discussed the fact that the $k$ values of the compact space, given by the solutions of (4.7), approach integer values as they become large. Thus, for large wavenumbers, the two spectral spaces become similar, $k\simeq n$ . The dimensionful compact wavenumbers, $2\unicode[STIX]{x03C0}k/l$ , also coincide if we choose a common length $l$ entering in the boundary conditions (3.1) and (4.1). This is the reason why we have chosen the circumference $l$ of the compact space to coincide with the periodicity $2\unicode[STIX]{x03C0}$ of the DNS.

For relatively small $n$ , things become more intricate. Especially as we approach the smaller values that correspond to the increasingly non-isotropic modes, the shell approximation of the DNS flow spectral space becomes a very crude one. There are too few wavenumber vectors in each shell, and the use of shells serves more to complete the bookkeeping of each spectral function in question in this one-dimensional reorganization of spectral space. On the compactified theory side, the spectral space is inherently one-dimensional, and every single mode is as isotropic as any other, as they all obey the equations of the ideal isotropic flows. What is different in the relatively small values of $k$ is that they lie slightly away from the nearest integer, as we see in (4.8). There is a decoupled $k=$ 0 mode that contributes pure constants in the field $F$ , and the lowest non-zero wavenumber $k$ is near the integer 2 and not near 1. One may interpret near-integer wavenumbers as a more carefully weighted labelling of the shells in the spectral space of the DNS flows, which become increasingly sparse for longer waves. On the downside, at the lowest end of the two spectra, the correspondence between them becomes nebulous. A good reason for this is that the lowest modes, and especially the $|\boldsymbol{k}|=1$ ones, are the most anisotropic modes of the DNS flows.

5.2 Large-scale cutoff

The vanishing (compact space) Loitsyansky integral arose in our discussion above as a fundamental property of the compactified theory. In § 4.3, we learned that a major implication of this fact is that the conserved Loitsyansky integral of the theory does not operate in the same way as in ideal isotropic flows: scalewise our system has only to conform to the compactness of the domain. In this respect, it resembles the DNS flows which have no other obvious constraint scalewise except the compactness of the space in which they live. Additionally, the Loitsyansky integral of the compact space is essentially the energy of the zero mode of the flow, which is not dynamical, and serves only as a constant lower value of the energy $K$ which may as well be set to zero. The seemingly peculiar implication of zero Loitsyansky condition (4.24) is that $F$ , which is understood as an analogue to the longitudinal autocorrelation functions, must become negative somewhere in its domain. We argue that this is a natural byproduct of the behaviour of actual flows that are bounded, explicitly or effectively, in space.

We start with ideal isotropic flows that live in infinite space. Formula (4.2) relates the spectrum $E(k)$ with the longitudinal autocorrelation function $F(r)$ . We consider a simple spectrum with an infrared cutoff: $E(k)=0$ for $k<k_{0}$ for some wavenumber $k_{0}$ , while $E(k)$ assumes any reasonable form for the higher wavenumbers. To be concrete, let us take, for example, that $E(k)=k^{4}\text{e}^{-k}$ for $k>k_{0}$ , in suitable units. The introduction of a sharp cutoff is a legitimate way to allow modes only up to a certain size within the infinite space; information of this kind is stored more consistently in spectral space where the positivity of the spectrum is a clear cut condition that needs to be satisfied with no analogue in physical space. It is then straightforward to show that the asymptotic behaviour of $F(r)$ at large distances is

Clearly, this $F(r)$ vanishes, becomes negative and starts to oscillate around zero for distances beyond $r=O(1/k_{0})$ . The same happens when the sharp infrared cutoff is implemented in a smooth way by some exponential function: the result exhibits the same behaviour, except that the calculations are a little more difficult.

In the brief discussion that follows, we shall describe $F(r)$ such as (5.1) as asymptotically oscillating. The trademark properties are the existence of extrema that are both positive and negative, i.e. the existence of nodes between them. The usual longitudinal functions of isotropic turbulence do not exhibit such behaviour. There is a close relationship between the field $F$ of the compactified theory and these asymptotically oscillating functions. By the boundary conditions (4.1b,c ), each of the endpoints of the interval $-\unicode[STIX]{x03C0}\leqslant r\leqslant \unicode[STIX]{x03C0}$ is a stationary point of the field $F$ (and, as usual, we use the evenness of $F$ to restrict ourselves to half of that interval). In the simplest configurations, the field $F$ is continuously decreasing and reaches its first minimum at the end of the interval. Now, the field $F$ , modulo an ignorable additive constant, satisfies (4.24). This means that in some neighbourhood of its minimum, the field $F$ must have passed to the negative values. That is, there must be a node between $r=0$ and $r=\unicode[STIX]{x03C0}$ . These configurations of the field $F$ can be regarded as appropriate asymptotically oscillating configurations truncated at their first minimum. This is the form of the first mode; that is, the eigenfunction (4.3) that corresponds to the lowest non-zero wavenumber $k=$ 1.835 (given in (4.8)).

Similarly, configurations that possess multiple nodes, that is, two or more nodes in the interval $0\leqslant r\leqslant \unicode[STIX]{x03C0}$ , can be regarded as asymptotically oscillating ones truncated at an appropriate minimum or maximum. Now, the form of the configurations is only half of the story. The discrete spectral space of the compactified theory introduces an a priori cutoff on the energy spectrum of any configuration of the field $F$ . That cutoff can be regarded as a counterpart of the sharp spectral cutoff that gives rise to the asymptotically oscillating configurations. However, there is an important difference between them: the cutoff of the compactified theory is permanent as compactness is permanent. The single- and multinode configurations can be distinguished on the basis of this difference.

Single-node configurations of the field $F$ arise naturally. Plausible positive-definite spectra that one uses to set up initial conditions for the field $F$ are most likely to lead to configurations with a single node. As noted already above, the node may move but not disappear, courtesy of (4.24). From the spectral space point of view, the permanence of these configurations rests on the naturalness of their energy spectra. The spectra possess a single cutoff which is the a priori and permanent spectral cutoff of the compactified theory. Multinode configurations arise when we consider spectra with an additional low-wavenumber cutoff. That is, by repeating in the discrete spectral space the procedure we applied in the continuous spectral space of the ideal isotropic flows that led to the asymptotically oscillating functions. In a sense, the multinode configurations are a counterpart of the asymptotically oscillating configurations in compact space. In their respective domains, both kinds of configurations have at least one node too many, or, equivalently, they have one sharp cutoff too many in the size of their modes, rendering them unstable structures. The reason is that the sharp cutoff in their spectra is unsustainable. The nonlinear interactions are continuously redistributing energy into wavenumbers in the vicinity of the cutoff, softening its sharpness. Gradually, the nodes are untangled and the configurations settle into the most elementary form they assume in their respective domains. In compact space, that form is the single-node configuration. In conclusion, thinking in analogy with ideal isotropic flows, we acquired an intuitive basis for the necessary appearance of nodes in the configurations of $F$ , a fact that is elegantly encoded in (4.24), and clarified their function.

6.1 Forcing in compact space

We assume that we feed each mode $k$ with energy at a rate $\unicode[STIX]{x1D700}_{k}$ . This means that the (total) energy balance (2.2) reads

while the spectral form of the energy balance (4.16) reads

which, in $r$ -space, takes the form of a Karman–Howarth equation (2.4) with forcing terms

The numerical coefficient $2/3$ , present also in the mode expansion (4.10), reflects the normalization of the field $F$ with respect to the total energy. Each mode $k$ is represented in $r$ -space simply and directly by the corresponding eigenfunction  $f_{k}(r)$ . This allows the simple representation of forcing we see in (6.3). The formulae (4.10), (4.12) and (4.14) allow the back and forth transformation between (6.2) and (6.3).

In the DNS, forcing is usually applied in the first lowest shells, that is, bunches of DNS turbulence modes centred around $k\sim 1$ and $k\sim 2$ , as we explained in § 5.1, such that the total energy $K$ is kept constant. Here, these two shells are represented by the first wavenumber given in (4.8), $k_{1}=1.835$ . Therefore, we feed energy only to the first mode, so that the total energy is kept constant. Then, (6.1) tells us that the forcing rate on the first mode $k_{1}$ must be equal to the dissipation, $\unicode[STIX]{x1D700}_{k_{1}}=\unicode[STIX]{x1D700}$ , where $\unicode[STIX]{x1D700}$ is given at any instant of time by (2.6). All in all, we have

Given some initial configuration of the field $F$ that corresponds to a regular spectrum, a closure scheme $H[F]$ , as we explained in §§ 4.4 and 4.5, and the boundary conditions on $F$ given by (4.1) we have a well-defined problem to solve.

We should also make sure that the initial configuration of $F$ respects the condition (4.24) of zero Loitsyansky integral in compact space. Indeed, as discussed in § 4.2, the Loitsyansky integral in compact space is nothing but the energy $E(0)$ of the zero mode of the spectrum, given explicitly by (4.15). As discussed in § 4.2, the zero mode is entirely decoupled from all of the $k>0$ modes, and its energy $E(0)$ is a constant of motion, acting merely as an arbitrary lower value of the total energy without physical relevance. The existence and the non-dynamical nature of the zero mode are implications of the formal symmetry (4.21) of the Karman–Howarth equation with the boundary conditions (4.1); this symmetry is respected by the dressed closure scheme (4.45), as discussed in § 4.5.3, and hence by (6.3). Thus, if we want the sum of the $E(k)$ to correspond to the actual total energy $K$ of the dynamical degrees of freedom, that is, if we want to be able to identify $F(0)$ with $(2/3)K$ according to (2.7), then we must choose initial configurations such that $E(0)=0$ , which will be preserved throughout the evolution of the system by the constancy of $E(0)$ . The condition $E(0)=0$ is equivalent to the integral condition (4.24), which we rewrite in the half-interval, utilizing the evenness of the field $F$ as usual, requiring that

Once the initial configuration of $F$ is consistent with (6.5), we have a complete formulation of the problem.

6.2 Steady state

As in the DNS, we are primarily interested in the asymptotic state of the system, which is a steady state: there is no time dependence on any quantity. Then,

We may multiply this equation by $r^{4}$ and integrate from $r=0$ to $r=r$ . As both $\unicode[STIX]{x2202}_{r}F$ and $H$ vanish at $r=0$ , we obtain the first-order equation

where we denote differentiation with respect to $r$ by a prime. The integral that arose in the forcing term can be understood as an incomplete form of the compact space Loitsyansky integral of the mode $k_{1}$ . By (C 2), it is related to the derivative of the corresponding mode $k_{1}$ . Thus, we obtain the rather elegant steady-state equation

At $r=\unicode[STIX]{x03C0}$ , (6.8) is satisfied due to the boundary conditions (4.1) on the fields $F$ and $H$ and (4.6) for the eigenfunction, i.e. each term vanishes independently. Clearly, the same procedure can also be applied to the general equation (6.3). Thus, a DNS type of forcing can be nicely accommodated in our all-isotropic formalism, leading to a first-order differential equation for the steady state (as long as the closure is also first order in $F$ ).

An important implication is that if the field $F$ is regular up to a certain order, then (6.8) implies that we must have a closure $H[F]$ such that $H$ is also regular at that order. Indeed, we assume, e.g., that $F$ is third-order regular, that is, conditions (4.36) hold. As these follow from the relations (4.31) of the eigenfunctions, similar relations hold for the forcing term in (6.8). Therefore, equation (6.5) requires us to have a field $H$ that is also third-order regular. In § 4.5.3, we discussed how this is accomplished by renormalizing the selected closure at the desired order.

Let us write down (6.8) in an explicit form, using also the renormalized form of the closure given in (4.45),

We may immediately observe a few things. First of all, the forcing term is proportional to an eigenfunction and hence vanishes at $r=\unicode[STIX]{x03C0}$ by the boundary condition that is obeyed by the eigenfunctions and the field $F$ , relations (4.6) and (4.1) respectively. Then, by the form of the cascade process term, the derivative of the field $F$ is a common factor in the first two terms of (6.8). Thus, the boundary condition $F^{\prime }(\unicode[STIX]{x03C0})=0$ is actually implied by (6.6) and need not be imposed independently. This makes sense, as all we are left with in the steady state is the first-order equation (6.8), whose only necessary ‘initial data’ along the line of $r$ are exhausted by the value of the field $F$ at $r=0$ , that is, by the value of $u^{\prime 2}=(2/3)K$ , which is set once and for all by the initial condition and preserved by the forcing.

Second, in order to actually be able to identify $F(0)=u^{\prime 2}$ , (6.9) must be supplemented by the zero Loitsyansky integral condition (6.5). This condition is imposed at the initial configurations on $F$ and is carried over time by the conservation of the compact space Loitsyansky integral all the way to the steady state. Working formally directly with the steady state, we must impose it independently in order to obtain the unique solution of (6.9) that corresponds to initial configurations of zero Loitsyansky integral.

In order to immediately see the role of this condition, we imagine that somehow the cascade process is switched off. Then, (6.9) is solved trivially, to give

modulo an additive integration constant. However, from the zero Loitsyansky integral condition (6.5) and the fact that the mode already satisfies that condition, equation (4.13), we have that the additive constant must be zero. Then, using the fact that $F(0)=u^{\prime 2}$ and  $f_{k1}(0)=1$ , we have that

That is, we obtain the resulting value of the dissipation, given the viscosity, the forcing mode and the initial energy. Although the steady-state value of the dissipation of the full problem (6.9) is not given by (6.11), the latter does emphasize the correspondence between the initially given energy and the steady-state value of the dissipation. Clearly, without employing the condition (6.5), we would not be able to identify the actual total energy of the system and derive a unique energy balance relation such as (6.11).

It is convenient to introduce the normalized (and dimensionless) field

which satisfies $f(0)=1$ . We rewrite (6.9) in the form

This equation involves two dimensionless constants. The quantity $u^{\prime }/\unicode[STIX]{x1D708}$ is a dimensionless quantity, in the arbitrary length units we use to measure $r$ . It is essentially a Reynolds number of the general form $LU/\unicode[STIX]{x1D708}$ , where $L$ and $U$ are associated with the size and the total energy of the system respectively. Given that the forcing maintains the energy $(3/2)u^{\prime 2}$ constant at its initial value, this quantity is fixed by the initial conditions. Moreover, recalling the Taylor microscale Reynolds number (Monin & Yaglom Reference Monin and Yaglom1971)

we obtain a dimensionless quantity associated with the dissipation of the steady state. Therefore, this quantity characterizes the steady state we are led to, for the given initial conditions. It is easy to see that the coefficient of the forcing term reads $5(u^{\prime }/\unicode[STIX]{x1D708})k_{1}^{-2}Re_{\unicode[STIX]{x1D706}}^{-2}$ ; that is, the coefficients of (6.13) are indeed expressed in terms of these two Reynolds numbers. Solution of the problem means that we determine $\unicode[STIX]{x1D700}$ for the given $u^{\prime }$ . This means, in turn, that we determine $Re_{\unicode[STIX]{x1D706}}$ for the given $u^{\prime }/\unicode[STIX]{x1D708}$ . We should perhaps note that for a given initial configuration, say $F_{0}(r)$ , there is also an initial value of the dissipation, say $\unicode[STIX]{x1D700}_{0}$ , given by $-15\unicode[STIX]{x1D708}F_{0}^{\prime \prime }(0)$ . However, the dissipation is not preserved by the forcing, therefore $\unicode[STIX]{x1D700}_{0}$ is not a characteristic of the initial state that remains relevant in the steady state.

6.3 Renormalization

The function $w(r)$ is in general a functional of the field $f$ , which we constructed in § 4.5.3 for the first three orders of regularity. Indeed, beyond the first order, the function $w(r)$ depends on the derivatives of the field $F$ and therefore of $f$ at $r=\unicode[STIX]{x03C0}$ . This is slightly unsettling, as higher-order derivatives of the unknown function enter the (first-order) differential equation (6.13). On the other hand, $w(r)$ is a local functional of $f$ at the point $r=\unicode[STIX]{x03C0}$ . Therefore, we may use the information contained in (6.13) in the neighbourhood of $r=\unicode[STIX]{x03C0}$ to rewrite $w(r)$ in an equivalent but more agreeable way such that it depends only on the value of the field $f$ at $r=\unicode[STIX]{x03C0}$ . The function $\unicode[STIX]{x1D711}(r)$ , defined in (4.52), reads equivalently

We consider the $w(r)$ imposed by the second-order regularity conditions. Its coefficients, given by (4.51), involve the second derivative of $f$ at $r=\unicode[STIX]{x03C0}$ . Differentiating (6.13) once and evaluating the result at $r=\unicode[STIX]{x03C0}$ , using also the boundary condition $f^{\prime }(\unicode[STIX]{x03C0})=0$ , we obtain a quadratic equation for $f^{\prime \prime }(\unicode[STIX]{x03C0})=0$ which is easily solved to obtain

where

It should be noted that we write $f_{\unicode[STIX]{x03C0}}$ in place of $f(\unicode[STIX]{x03C0})$ . We also used (4.32b ) to replace the quantity $f_{k_{1}}^{\prime \prime }(\unicode[STIX]{x03C0})$ that arises. It should also be noted that both constants $A$ and $B$ are positive. Using the result (6.16) in (4.51), we find

Thus, $w(r)$ is written in such a way that it involves only the value of $f$ at $r=\unicode[STIX]{x03C0}$ . This is quite convenient for the numerical solution of the problem.

As a matter a consistency, we note that, working at the level of second-order regularity, the condition (4.36a ) relating the third to the second derivative of $F$ is assumed to hold. By differentiating (6.13) twice and evaluating the result at $r=\unicode[STIX]{x03C0}$ , the third derivative of $F$ at the point arises. However, that third derivative term can be replaced via (4.36a ) with a second derivative term, giving again an equation for $f^{\prime \prime }(\unicode[STIX]{x03C0})$ . The result is identical to (6.16). This is rather expected, given that at each given order of regularity, a number of odd derivatives of $F$ depend on lower even-order derivatives at $r=\unicode[STIX]{x03C0}$ , information already encoded in the Karman–Howarth equation. Hence, relations involving these odd derivatives should not contain any new information.

For the third-order regularity, the function $w(r)$ depends on the second and fourth derivatives of $F$ at $r=\unicode[STIX]{x03C0}$ , as can be seen from (4.54). The first derivative of (6.3) at $r=\unicode[STIX]{x03C0}$ gives one relation for $f^{\prime \prime }(\unicode[STIX]{x03C0})$ and $f^{(4)}(\unicode[STIX]{x03C0})$ . The third derivative of (6.13) at $r=\unicode[STIX]{x03C0}$ gives, using also the first-order regularity condition (4.36a ), a second relation for the quantities $f^{\prime \prime }(\unicode[STIX]{x03C0})$ and $f^{(4)}(\unicode[STIX]{x03C0})$ . These two relations define $f^{(4)}(\unicode[STIX]{x03C0})$ in a simple manner in terms of $f^{\prime \prime }(\unicode[STIX]{x03C0})$ . The quantity $f^{\prime \prime }(\unicode[STIX]{x03C0})$ is defined now through a cubic equation. The result is explicit, but it is rather too involved to quote here. In the application that follows, it will become clear that we need not use such a high-order regularity; even for illustration purposes, the application of regularity up to the second order makes all basic characteristics of renormalization manifest.

6.4 Application

We have to solve equation (6.13) using the condition (6.5) of zero Loitsyansky integral, for different orders of regularity. For concreteness, we shall use the DNS data of Ishihara et al. (Reference Ishihara, Kaneda, Yokokawa, Itakura and Uno2005, Reference Ishihara, Kaneda, Yokokawa, Itakura and Uno2007) as a basis for input values. We consider the lowest-Reynolds-number case $(Re_{\unicode[STIX]{x1D706}}\sim 100)$ , which allows us to go quite far into the dissipation range by only a few hundred wavenumbers. We set the viscosity at the value $\unicode[STIX]{x1D708}=2/1000$ and the kinetic energy at $K=5/10$ . The dissipation $\unicode[STIX]{x1D700}$ is to be found as part of the solution: the solution of the problem is a function $f(r)$ that solves (6.13) for the specific value of $\unicode[STIX]{x1D700}$ for which $f(r)$ satisfies (6.5). As explained earlier, in § 4.2, there is a correspondence between the initially set (and preserved by the forcing) value of $K$ and the final (steady-state) value of $\unicode[STIX]{x1D700}$ . The only other numerical input required is the value of the closure parameter $\unicode[STIX]{x1D6FC}$ introduced in (4.41) relating it to the inertial range Kolmogorov constant $C_{2}$ . The constant $\unicode[STIX]{x1D6FC}$ can also be expressed in terms of the skewness $S$ of the velocity-gradient distribution: $\unicode[STIX]{x1D6FC}=(\surd 2/9)(-S)$ , as we discuss in appendix D. For our purposes, we shall use the nominal value $-S=50/100$ , rounding the quoted value in Ishihara et al. (Reference Ishihara, Kaneda, Yokokawa, Itakura and Uno2007).

We start with the zeroth order of regularity. That is, we use the bare form of the closure for which $w(r)=r$ . We need to find the value of $\unicode[STIX]{x1D700}$ so that the associated solution $f(r)$ is consistent with the zero Loitsyansky integral condition (6.5). This can be done iteratively through a midpoint procedure. From the data of Ishihara et al. (Reference Ishihara, Kaneda, Yokokawa, Itakura and Uno2005), the dissipation $\unicode[STIX]{x1D700}$ is expected around 0.09. We start with a pair of values of $\unicode[STIX]{x1D700}$ around 0.09 so that the Loitsyansky integral associated with each solution has a different sign, say 0.08 and 0.10, and calculate also the Loitsyansky integral associated with the solution for the midpoint, 0.09, of that interval. The half-interval whose endpoints correspond to Loitsyansky integrals of different sign is picked as the new interval. We proceed in this way until the Loitsyansky integral is small enough ( ${\sim}10^{-20}$ appears to be adequate for our purposes). This determines $\unicode[STIX]{x1D700}$ also to some 20 significant figures.

We proceed to the first order of regularity. The function $w(r)$ is given by (4.47). The coefficient of $w(r)$ is a constant number, and the method of solution is the same as with the zeroth order.

At the second order, the coefficients $a_{1}$ and $a_{2}$ of $w(r)$ are given by (4.51), or, equivalently in the present problem, by the simplified equations (6.18). They depend on the value of the field $f$ at $r=\unicode[STIX]{x03C0}$ . The only modification we need to make here is to insert an additional set of iterations for each value of $\unicode[STIX]{x1D700}$ . For each value of $\unicode[STIX]{x1D700}$ in the midpoint procedure, we start with $f_{\unicode[STIX]{x03C0}}=0$ in the expressions (6.18) and find the associated solution $f(r)$ and the value of $f(\unicode[STIX]{x03C0})$ for that solution. We set $f_{\unicode[STIX]{x03C0}}=f(\unicode[STIX]{x03C0})$ in the expressions (6.18) and find a new solution. The new value of $f(\unicode[STIX]{x03C0})$ provides a new value $f_{\unicode[STIX]{x03C0}}$ for the next step. We proceed iteratively until the input $f_{\unicode[STIX]{x03C0}}$ and the resulting $f(\unicode[STIX]{x03C0})$ differ adequately little (an error of $10^{-25}$ arises after 20 iterations). This provides the second-order regularity solution associated with the given value of $\unicode[STIX]{x1D700}$ . By applying the procedure at each step of the midpoint procedure for $\unicode[STIX]{x1D700}$ , we find the desired value of $\unicode[STIX]{x1D700}$ and the associated solution. Thus, the second-order solution is characterized by the value of $\unicode[STIX]{x1D700}$ , the same as the lower-order solutions, as well as the value of $f_{\unicode[STIX]{x03C0}}$ . The latter is required to specify the coefficients $a_{1}$ and $a_{2}$ , which are functionals of the field $f$ at this order and are determined simultaneously with the solution. The obtained values for all of these quantities are given in table 1, where we include the values corresponding to the lower orders for completeness. The dissipation $\unicode[STIX]{x1D700}$ is given to many decimal places, which will be convenient below. The quoted decimal places of the results for the other quantities are indicative, serving only to illustrate their size and/or variation from order to order.

Table 1.

Solution parameters derived for each order of regularity.

For each order of regularity, once the solution is obtained, the spectrum $E(k)$ is calculated by the formula (4.14). The spectra of energy, dissipation, palinstrophy and $k^{6}$ -moment are given in figure 3 for the zeroth, first and second orders of regularity; the dissipation, palinstrophy and $k^{6}$ -moment spectra are given in the dimensionless form (E 1). The wavenumbers are quoted characteristically in the dimensionless form $k\unicode[STIX]{x1D702}$ , where $\unicode[STIX]{x1D702}$ is the dissipation range Kolmogorov scale $\unicode[STIX]{x1D702}=(\unicode[STIX]{x1D708}^{3}/\unicode[STIX]{x1D700})^{1/4}$ . We plot the absolute values of the spectra; beyond a certain wavenumber, each $E(k)$ starts to oscillate in the sign-changing fashion we encounter in § 4.5.3. The results realize fairly clearly the claims and conjectures we were led to in § 4.5.3 through simple numerical and analytical examples. First, the spectra exhibit asymptotic irregularities in the form of oscillatory (sign-changing) power laws, and, second, these irregularities are gradually removed by applying higher-order regularity conditions. The large- $k$ behaviour of the spectrum at the $n\text{th}$ order of regularity is

This becomes clear in the figure 3(b–d), where the graph of the spectrum of the $k^{2(n+1)}$ -moment, that is, the quantity $k^{2n+2}E(k)$ , becomes a straight horizontal line at large $k$ , for $n=0,1,2$ . This fact is also verified by the numbers: in the regimes shown, the absolute value of the spectra $k^{2n+2}E(k)$ is constant to some six significant figures.

Figure 3.

Plots of the absolute value of the spectrum $E(k)$ , the dimensionless dissipation spectrum $(\unicode[STIX]{x1D700}\unicode[STIX]{x1D708})^{-1/2}(k\unicode[STIX]{x1D702})^{2}E(k)$ , the dimensionless palinstrophy spectrum $(\unicode[STIX]{x1D700}\unicode[STIX]{x1D708})^{-1/2}(k\unicode[STIX]{x1D702})^{4}E(k)$ and the dimensionless $k^{6}$ -moment spectrum $(\unicode[STIX]{x1D700}\unicode[STIX]{x1D708})^{-1/2}(k\unicode[STIX]{x1D702})^{6}E(k)$ against $k\unicode[STIX]{x1D702}$ are shown in (a–d) respectively for the zeroth, first and second order of regularity, indicated by different tones of grey dotted lines. The order increases downwards, showing the advancement of the physical regime deeper into the dissipation subrange. The softening of the irregularities in the large- $k$ regime, formalized by (4.58), is also visible.

Recalling also the discussion of § 4.5.4, we observe that the $k^{2i}$ -moment is well defined in the strong sense for all $i<n+1$ . Moreover, in accordance with our reasoning there, by increasing the order of regularity, the regime of wavenumbers where the spectrum remains physical (i.e. positive definite and decreasing for large $k$ ) is extended deeper into the dissipation subrange. Indeed, this is clearly visible in figure 3, and we find specifically that at the zeroth order of regularity, the spectrum is physical up to $k\unicode[STIX]{x1D702}=0.35$ , at the first order up to $k\unicode[STIX]{x1D702}=2.2$ and at the second order up to $k\unicode[STIX]{x1D702}=3.7$ . This is a most important result, exhibiting the effect of ‘renormalization’, as we have called it. An empirically postulated closure, such (4.41), is ‘unaware’ of the subtleties of the compact space, leading to irregular behaviour of the spectrum in the dissipation subrange. The subtleties are a series of constraints imposed by the compact space on the fields $F$ and $H$ , involving increasingly higher derivatives of the fields, given by (4.35). The problem is also the key to its solution: by adding suitable corrective terms to the original closure, any specific number of such constraints can be satisfied. This turns the spectrum physical up to a large wavenumber, which increases with the number of satisfied constraints. These results also illustrate the following idea. There is nothing wrong with a spectrum becoming unphysical at large wavenumbers, as long as the asymptotic behaviour does not lead to greater inconsistencies, apart from the spectrum being essentially undefined beyond a certain wavenumber, and, also, as long as that wavenumber can be pushed as deep into the dissipation subrange as we want. The point is that, from a practical point of view, we are interested in finite wavenumbers, and from the theoretical point of view, this kind of spectrum is still manageable. One could put forward a working definition that such spectra are the ones for which all energy moments are at least weakly defined.

We now turn to the consistent calculation of these moments and – through them – the quantification of the information carried by the spectrum in its physical regime. For each order of regularity, the results for the first three energy moments, i.e. the dissipation $\unicode[STIX]{x1D700}$ , the (dimensionless) palinstrophy and the (dimensionless) $k^{6}$ -moment, are given in table 2. The palinstrophy and the $k^{6}$ -moment in dimensionless form are defined by the left-hand side of (E 1) in appendix E for $i=2,3$ , and they are calculated by the right-hand sides of the relations (E 2) and (E 3) respectively, once the value of the dissipation $\unicode[STIX]{x1D700}$ has been found. The results are quoted to a relatively large number of decimals. In fact, given that $\unicode[STIX]{x1D700}$ has been determined by even more decimals, the moments can be determined through the right-hand sides of (E 2) and (E 3) with even higher accuracy. The number of significant figures we choose to quote is explained below. The spectral (i.e. left-hand) sides of equalities (E 2) and (E 3) are a specific realization of equation (4.68) and must be understood in a regularized sense, as implied by (4.68). In practice, where we use finite sums, regularization is implemented very effectively by the resummation formulae of appendix F. Equalities (E 2) and (E 3), as well as the relations

are verified to the quoted decimals by including wavenumbers up to $k\unicode[STIX]{x1D702}\sim 5$ (the total energy is calculated even more very accurately; thus, we give it as an exact fraction value). To the second order of regularity, the simplest resummation ( $m=1$ case of formulae (F 2) or (F 3)) is adequate to obtain the $k^{6}$ -moment to the quoted decimals, while the dissipation and palinstrophy relations are verified to even higher accuracy. To the first order of regularity, the dissipation is obtained by the $m=1$ of formulae (F 2) or (F 3) essentially to the quoted decimals, while the higher moments require the $m=2$ resummation formulae. The zeroth order requires consistently higher $m$ in order to verify the relations to the given decimals. All in all, using a suitable regularization, one may check and verify to high accuracy the consistency of the solution in $k$ - and $r$ -space.

Table 2.

Energy moment values derived for each order of regularity.

In table 3, we give the results for the moments calculated from the physical regime alone. We quote the result to the number of significant figures for which they agree with the accurate values (in the given decimals) of the moments shown in table 2. The zeroth-order results from the physical regime are quite poor, due to the fact that the physical regime at that order does not even reach the peak of the palinstrophy spectrum. Specifically, the total energy and the dissipation turn out to be 0.49 and 0.057 respectively. For the first and second orders, one observes the high accuracy with which the moments are obtained from the physical regime of the spectrum. Clearly, this is possible due to the depth of the physical regime already at the first order. The number of figures we use in table 2 is a merely uniform choice which is required for quoting the correct significant figures of the most accurate result of table 3; after all, the moments have been uniformly determined by almost twice as many decimals for all of these cases. One should also note the rather significant level of accuracy with which palinstrophy is determined in the first order of regularity in the physical regime. We consider this to be an implication of the mild divergence of the palinstrophy spectrum at that order: $k^{4}E(k)$ approaches $(-1)^{[k]}$ for large $k$ , therefore palinstrophy is what we called in § 4.5.4 marginally ill defined (in the strong sense). The same can be noted for the value of the $k^{6}$ -moment at the second order of regularity. On the other hand, one may note that at the first order of regularity, the spectrum $k^{6}E(k)$ diverges as $(-1)^{[k]}k^{2}$ , which is a strong divergence. The implication is that the associated moment, although it is represented quite decently in the physical regime by three correct significant figures, is significantly less accurately represented than the palinstrophy.

Table 3.

Physical regime values of the energy moments.

We may note at this point that the value $\unicode[STIX]{x1D700}\sim 0.09$ we find as part of the solution is consistent with the DNS value for the $Re_{\unicode[STIX]{x1D706}}\sim 100$ case of Ishihara et al. (Reference Ishihara, Kaneda, Yokokawa, Itakura and Uno2005) which we used as a reference. This, in turn, means that other global characteristics, that is, the Reynolds number $Re_{\unicode[STIX]{x1D706}}$ and the dissipation length scale $L_{\unicode[STIX]{x1D700}}=(2K)^{3/2}/\unicode[STIX]{x1D700}=1/\unicode[STIX]{x1D700}$ , which are specified by $\unicode[STIX]{x1D700}$ , are also consistent with Ishihara et al. (Reference Ishihara, Kaneda, Yokokawa, Itakura and Uno2005). The values of these quantities for the different orders of regularity are given in table 4. Therefore, we understand that the toy model (4.41), although it does not encapsulate finer phenomena of turbulence such as the bottleneck effect or anomalous scaling (Ishihara et al. Reference Ishihara, Gotoh and Kaneda2009; Donzis & Sreenivasan Reference Donzis and Sreenivasan2010), does capture well the relation between $K$ and $\unicode[STIX]{x1D700}$ and hence the global characteristics of the flow. We may also note that other more intrinsic characteristics are also reproduced: the peaks of the dissipation and the palinstrophy spectrum arise roughly at $k\unicode[STIX]{x1D702}\sim 0.2$ and $k\unicode[STIX]{x1D702}\sim 0.5$ respectively, consistently with the behaviour of the DNS flows (Ishihara et al. Reference Ishihara, Kaneda, Yokokawa, Itakura and Uno2005, Reference Ishihara, Kaneda, Yokokawa, Itakura and Uno2007). Hence, the model (4.41) can be the basis for more elaborate models, as we discuss in the next section. In table 4, we also include the values of the integral scale $L_{11}$ for each order of regularity. The integral scale is naturally defined here as the integral of $f(r)$ from $r=0$ to $r=\unicode[STIX]{x03C0}$ . In appendix G, we rewrite this definition also in spectral space, deriving a formula quite similar to the standard spectral formula for $L_{11}$ in isotropic turbulence. The data of Ishihara et al. (Reference Ishihara, Kaneda, Yokokawa, Itakura and Uno2005) show that $L_{11}\sim 1.1$ in the DNS, in contrast to $L_{11}\sim 0.9$ obtained here. The integral scale is very strongly affected by the very first few wavenumbers. The discrepancy may be attributed essentially to the existence of the $k=1$ modes in the DNS, as we discussed in § 5.1. This could be corrected by organizing and labelling the shells, especially the first one, of the DNS differently. Hence, we regard the discrepancy more as a matter of definition than as an essential one. At any rate, one should bear in mind that the integral scale is strongly affected by the most anisotropic modes of the DNS, and its value is by nature an imprecise characteristic of the flow.

The results given in table 3 can be read from another point of view, that is, as illustrating the degree of convergence of the results according to the order of regularity. One observes that the change of these quantities, which are essentially determined through the dissipation, is of the order of one part in a thousand, as we go from the first to the second order of regularity. The dissipation, being the lowest energy moment, is affected the most by renormalization, which changes the spectrum slightly all the way to the smallest wavenumbers. As we see in table 3, the palinstrophy changes by of the order of one part in ten thousand, from the first and second order of regularity. The convergence of the moments quantifies the convergence of the spectrum towards a limiting spectrum that remains physical for all wavenumbers. The latter statement is clearly an important assumption underlying everything we do here, but it is also the most difficult to handle mathematically. The fact that the spectrum does not change significantly as the physical regime is extended at larger wavenumbers by regularity is already visible in the graphs of figure 3. The moments give a more specific measure of convergence. One may note the relatively slow convergence of the $k^{6}$ -moment at these orders. This can be regarded as another implication of the fact that at the first order of regularity, the spectrum $k^{6}E(k)$ diverges strongly, as we noted above, discussing the relatively low accuracy of the representation of this moment in the physical regime of the spectrum. All in all, we may say, taking now convergence into account explicitly, that the values of the dissipation, dimensionless palinstrophy and dimensionless $k^{6}$ -moment are 0.0840(2), 0.07583(1) and 0.0367(1) respectively, with the estimated error in the last digit given in the brackets.

Table 4.

The Reynolds number, the dissipation length scale and the integral scale derived for each order of regularity.

Regarding the energy transfer $T(k)$ , the energy balance equation (6.2) implies that at the steady state we have

which encodes the conservation of energy by the cascade processes

and also verifies the relation (4.61) between the energy spectrum and the energy transfer for large $k$ that came from our discussion in § 4.5.3. At the steady state, the energy transfer function differs by a factor of $k^{2}$ from the energy spectrum for the wavenumbers that are not forced or, in general, if one goes to large enough wavenumbers, as is implied by the asymptotic formulae we have derived here in various ways.

Finally, the effect of the Reynolds number on renormalization can be understood in the following way. The coefficients of the function $w(r)$ of the renormalized cascade are of order one, in units of the size of the compact space. Introducing a dissipation scale dimensionless distance $x$ by $r=\unicode[STIX]{x1D702}x$ , we have that $w=\unicode[STIX]{x1D702}x(1-O(Re_{\unicode[STIX]{x1D706}}^{-3})x^{2}+O(Re_{\unicode[STIX]{x1D706}}^{-6})x^{4}+\cdots \,)$ . We may then say that the counterterms are increasingly lighter with regard to the large-wavenumber effects for higher Reynolds numbers. That is, any given counterterm is increasingly less necessary for removing the irregularities from the spectrum for any given wavenumber $k\unicode[STIX]{x1D702}$ . Put differently, the physical regime associated with any given order of regularity becomes larger and larger as we increase the Reynolds number. Indeed, for the first order of regularity, we mentioned above that for $Re_{\unicode[STIX]{x1D706}}\sim 100$ , the physical regime extends to the wavenumber $k\unicode[STIX]{x1D702}=2.2$ . Using again data from Ishihara et al. (Reference Ishihara, Kaneda, Yokokawa, Itakura and Uno2005, Reference Ishihara, Kaneda, Yokokawa, Itakura and Uno2007) as sensible input, we find that for $Re_{\unicode[STIX]{x1D706}}\sim 200$ , the physical regime extends to $k\unicode[STIX]{x1D702}\sim 2.5$ , and for $Re_{\unicode[STIX]{x1D706}}\sim 400$ to $k\unicode[STIX]{x1D702}\sim 2.8$ , for the same order of regularity. These results tie in well with the interpretation of our renormalization as a perturbative procedure with the inverse Reynolds number as the small parameter (§ 4.5.3).

On the other hand, the rescaled distance argument we used above should be carefully applied. In compact space, the effects of large and small distances are not cleanly separated, as is very much the case in infinite space. We have experienced this fact throughout this work, where the – figuratively – large-distance differentiability problems of the field $F$ at $r=\unicode[STIX]{x03C0}$ affect the large- $k$ behaviour of the spectrum. Hence, the naïve conclusion deriving from the above argument, that for very large Reynolds numbers one may not need even the first regularity condition, is wrong, as the regularity conditions that operate through $r=\unicode[STIX]{x03C0}$ are inescapable. In fact, for the truth of the matter, one finds that, e.g., for $Re_{\unicode[STIX]{x1D706}}\sim 400$ , the first negative values of the spectrum arise at somewhat larger wavenumbers, $k\unicode[STIX]{x1D702}\sim 1.1$ as opposed to $k\unicode[STIX]{x1D702}\sim 0.94$ of the $Re_{\unicode[STIX]{x1D706}}\sim 100$ case, but the uniform monotonicity of the spectrum is ruined in the strict sense due to small-amplitude oscillations that start already from the smallest wavenumbers. Hence, strictly speaking, the spectrum is not physical anywhere at the zeroth order of regularity for Reynolds numbers in the range of hundreds or higher. Due to such phenomena, we have refrained from giving too rigorous a definition of what is physical, because, after all, whether or not small oscillations can be tolerated in a positive-definite spectrum is a matter of realistic assessment. Moreover, these types of irregularities in the regime of small wavenumbers, e.g. on the left of the bottleneck effect peak, are very much present in the DNS spectra.