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1. Introduction
Certain
hydrodynamical systems exhibit steady-state flow patterns,
while others oscillate in a regular periodic fashion. Still
others vary in 3-D irregular, seemingly haphazard manner,
and, even when observed for long periods of time, do not appear
to repeat their previous history.
These modes of behavior may all be observed
in the familiar rotating-basin experiments, described by Fultz,
et al. (1959) and Hide (1958). In these experiments, a cylindrical
vessel containing water is rotated about its axis, and is
heated near its rim and cooled near its center in a steady
symmetrical fashion. Under certain conditions the resulting
flow is as symmetric and steady as the heating which gives
rise to it. Under different conditions a system of regularly
spaced waves develops, and progresses at a uniform speed without
changing its shape. Under still different conditions an irregular
flow pattern forms, and moves and changes its shape in an
irregular nonperiodic manner.
Lack of periodicity is very common in natural
systems, and is one of the distinguishing features of turbulent
flow. Because instantaneous turbulent flow patterns are so
irregular, attention is often confined to the statistics of
turbulence, which, in contrast to the details of turbulence,
often behave in a regular well-organized manner. The short-range
weather forecaster, however, is forced willy-nilly to predict
the details of the large scale turbulent eddies – the
cyclones and anticyclones – which continually arrange
themselves into new patterns.
1The research
reported in this work has been sponsored by the Geophysics
Research Directorate of the Air Force Cambridge Research Center,
under Contract No. AF 19(604)-4969.
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Thus there are occasions
when more than the statistics of irregular flow are of very
real concern.
In this study we shall work with systems
of deterministic equations which are idealizations of hydrodynamical
systems. We shall be interested principally in nonperiodic
solutions, i.e., solutions which never repeat their past history
exactly, and where all approximate repetitions are of finite
duration. Thus we shall be involved with the ultimate behavior
of the solutions, as opposed to the transient behavior associated
with arbitrary initial conditions.
A closed hydrodynamical system of finite
mass may ostensibly be treated mathematically as a finite
collection of molecules – usually a very large finite
collection – in which case the governing laws are expressible
as a finite set of ordinary differential equations. These
equations are generally highly intractable, and the set of
molecules is usually approximated by a continuous distribution
of mass. The governing laws are then expressed as a set of
partial differential equations, containing such quantities
as velocity, density, and pressure as dependent variables.
It is sometimes possible to obtain particular
solutions of these equations analytically, especially when
the solutions are periodic or invariant with time, and, indeed,
much work has been devoted to obtaining such solutions by
one scheme or another. Ordinarily, however, nonperiodic solutions
cannot readily be determined except by numerical procedures.
Such procedures involve replacing the continuous variables
by a new finite set of functions of time, which may perhaps
be the values of the continuous variables at a chosen grid
of points, or the coefficients in the expansions of these
variables in series of orthogonal functions. The governing
laws then become a finite set of ordinary differential
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