We expose
a survey of unpublished research on the evolution of structure in turbulence, in
whatever form that it presents, or developes itself in the world but, formally,
understood as the problem of studying the ensemble of solutions to an arbitrary,
deliberately unknown system of differential equations.
Real world examples of turbulence come, traditionally, from fluids described by the Navier-Stokes equations [32,40,60,73,91], although one may recognize the same, problematic situation when studying the magnetic confinement of plasmas, the fault dynamics in the earth's crust, or even the behavior of markets in an economy [10,33,46,50,65,86,90,92]. Now, facing such a generic sort of phenomena, our intuition is impressed by the prevalence of scale invariance that it is, concomitantly, exhibited. And the idea follows that something, like structure, should be conserved, as such a symmetry prevails; an idea that, as it is well known, has its roots in the work of Noether [69], in 1918. But, there is no exact scale invariance in any real world phenomenon, whatsoever, and so, we are naturally led to consider the question of determining the flow of that something, like structure, that, in fact, it is not conserved.
Aiming for
objectiveness, we concentrate on the emergence and the evolution of coherent
structures [40,66,87,88], as they are exhibited, for example, by vortices in a
turbulent flow. The scope of our analysis unfolds, therefore, for us to face
both the many-scale problem with strong coupling between the constituent scales
and the problem of the connection between scaling and structure [91]. We observe
such a task to be a feasible one, as we manage to develope a conceptual
framework where both these two problems, and the main challenge that originated
them, can all be treated within the rigorous terms of a mathematical
formalization that generates models encompassing their respective, possible
phenomenological manifestations, whatever they may be. And some consequences
follow, allowing the efectiveness of this framework to be put to the trial of
experience. For such an efectiveness to obtain, a few options have to be made,
at the expense of a rather elementary use of some, quite demanding mathematics,
notwithstanding the fact that, in what concerns modelization methodology, we do
not depart significantly from current practices.
We start
with the common observation that many turbulent flows display coherent
structures across several orders of spatial or temporal magnitude . We take this
fact as evidence for the concomitant involution of something quite likely to be
expressed, by analogy, as a renormalization flow [17], of which the observed
coherent structures constitute the snapshots. Consequently, we identify these
snapshots as the scale-dependent state representations of what we call Sigma
structures, and we study their evolution under a certain, convenient law of
action, Y,
having scale as its parameter. It is fundamental to acknowledge that we must
leave the (physico-mathematical) reference, or the (ontological) status of these
Sigma structures undetermined [22,59].
Following
these guidelines, we consider the specifics of a particular turbulent fluid flow
and the associated Sigma structure. As we stated, the display of coherent
structures at a given order of, say, spatial magnitude is to be interpreted as a
scale-dependent state representation of the Sigma structure. Now, we postulate
that such a state representation is faithfully modelized by a convergent power
series, with integer coefficients, restricted to be defined on some subset of
the maximal ideal of a complete, algebraically closed, local field containing
the rational numbers [11,56,77,81]. The modeling procedure is acomplished in
three, interwoven steps: to begin with, the topological structure of the
vorticity field associated with the turbulent fluid flow must be obtained
[73,79]; then, an euclidean model is given to the p-adic integer ring contained
in the ground number field [5,20,76,77]; finally, an adjustment is to be sought
for, between the topological structure of the physical field and the phase
portrait of an adequate power series, on the given euclidean model.
Thereafter,
a certain, convenient law of action, or operation on the set, Zp[x],
defined by these power series is taken as the central object of our study.
Starting from the postulate that it must satisfy an extremum principle, in fact,
a minimum one, we will prove, first, that it is unique, if it exists and if it
is subjected to the proper, formal constraints; afterwards, it will be shown
that it does exist and, finally, that it factors in a very special, also unique
way.
The
demonstration of these three main results constitutes the fulcrum of our
research. In particular, the strategy that we adopt in order to achieve them,
exploring the various levels of mathematical structure that the chosen power
series veil, reveals itself as a sucessful one, as it authorizes us on the
far-reaching assertion that the evolution
of structure in turbulence phenomena follows some well determined paths, that
are geodesic, in a precise, experimentally testable way.
After
these introductory lines, our survey is organized in the following way:
In section
2, there is a preliminary discussion, aiming to set the context and the
motivation for the achievement of our first main result; then, we show that, if
there exists a law of action, or operation on the set Zp[x],
and if it is restricted by some proper, bottom-up, scale-dependent formal
constraints, then, the result of that operation upon a given, initial power
series, u(x), which itself is restricted to be defined only up to a certain
order of magnitude, is a unique one.
In section
3, we deal with the problem of existence for the law of action that was
previously dealt with, showing, in particular, that it can be qualified as one
of analytic extension through T-filters, between power series defined on
distinct subsets of a totally disconnected space; this is our second main
result.
In section
4, after another small, preliminary discussion, we show that such an action
admits a unique, non-trivial factorization along morphisms in the category of
rigid analytic varieties, exposing, thereby, the third and last of our main,
mathematical results.
Sections 5
and 6 will debate the apparently innocuous character of our factorization
result; their presentations are somewhat terse, as we wish not to dwell on the
further mathematics called upon by the analysis of the strategy of proof that
was adopted before, in section 4. Instead of that, in section 5, we offer an
informal illustration of some connection between our work and, both, number
theory and dynamical systems theory, and afterwords, in section 6, we elaborate
on the inference of the claim that we submit to an eventual, possible test in
the laboratory, that is to say, the proposition that the development of
turbulences follows the proper geodesics that we define in an auxiliary space.
In section
7, this survey is driven to its end, with a few, final considerations.
For the
convenience of an eventual reader not familiar with the variety of mathematical
theories that are summoned along our survey, and/or the physics to which they
are applied, we have chosen to be somewhat redundant on the references (section
8) that are given along the text. In what concerns notation, we have followed
standards, whenever not explicitly stated otherwise.
Finally,
we observe that the proofs of theorems 1 and 2 may be skipped, on a first
reading of this survey, without losing grasp of the core of its physical
content; on the contrary, the proof of theorem 3 is more important than the
theorem itself.
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On the Evolution of Structure in Turbulence Phenomena