#  Universal constants and
equations of turbulent motion

by Helmut Z. Baumert  Institute for Applied Marine and Limnic Studies (IAMARIS
e.V.), Hamburg, Germany

###### Abstract

In the spirit of Prandtl’s [1926] conjecture, for turbulence at
$Re\rightarrow\infty$ we present an analogy with the kinetic theory of gases,
with dipoles made of quasi-rigid and “dressed” vortex tubes as frictionless,
incompressible but deformable quasi-particles. Their movements are governed by
Helmholtz’ elementary vortex rules applied locally. A contact interaction or
“collision” leads either to random scatter of a trajectory or to the formation
of two likewise rotating, fundamentally unstable whirls forming a dissipative
patch slowly rotating around its center of mass which is almost at rest. This
approach predicts von Karman’s constant as $\kappa=1/\sqrt{2\pi}\simeq 0.399$
and the spatio-temporal dynamics of energy-containing time and length scales
controlling turbulent mixing [[]]baumert09 in agreement with observations. A
link to turbulence spectra was missing so far. In the present paper it is
shown that the above image of random vortex-dipole movements is compatible
with Kolmogorov’s turbulence spectra if dissipative patches, beginning as two
$likewise$ rotating eddies, evolve locally into a space-filling bearing in the
sense of [Herrmann (1990)], i.e. into an “Apollonian gear” consisting of
incompressible and flexibly deformable vortex tubes which are frictionless,
excepting the dissipative scale of size zero.

For steady and locally homogeneous conditions our approach predicts the pre-
factor in the three-dimensional Eulerian wavenumber spectrum,
$E(k)=\alpha_{1}\,\varepsilon^{2/3}\,k^{-5/3}$, as
$\alpha_{1}=\frac{1}{3}(4\,\pi)^{2/3}\simeq 1.802$, and in the Lagrangian
frequency spectrum, $E(\omega)=\beta_{1}\,\varepsilon\,\omega^{-2}$, as
$\beta_{1}=2$. The unique values for $\alpha_{1},\beta_{1}$ and $\kappa$ are
situated well within the broad scatter range of observational, experimental
and approximative results. Our derivations rest on geometry, methods from
many-particle physics, and on elementary conservation laws.

Baumert Universal constants and equations H. Z. Baumert, IAMARIS, Bei den
Mühren 69A,
D-20457 Hamburg, Germany<EMAIL_ADDRESS>Submitted as v2 to arXiv.org

##  1 Introduction

In the present paper we show that fluid turbulence can be understood in an
idealised sense as a statistical many-body ensemble – a tangle of vortex tubes
taken as discrete particles. We follow an early conjecture by Ludwig [Prandtl
(1926)] who discussed an analogy between molecular diffusion and turbulence.
He related his mixing length111According to [Hinze (1953)] also G. I. Taylor
made early use of the notion mixing length. or Mischungsweg with the mean-free
path of kinetic gas theory and considered his fluid elements or fluid lumps
(Flüssigkeitsballen) of locally nearly same size as relatives of gas
molecules. He further assumed his mixing length to scale with the “diameter”
of his fluid elements.

Although purely heuristic, his concept became popular in the years before WW2.
The question how to compute the details could not be answered without
reference to measurements. After WW2 the gas-kinetic analogy found thus strong
criticism from the continuous-image side [[, see Introduction in]]batchelor53.
This was plausible because the gas analogy represents a discrete concept that,
if useful, would label a Copernicanian turn or a paradigm shift in our grown
view of turbulence as an exclusive matter of continuum mechanics.

In this sitution Werner [Albring (1981)], in the footsteps of Prandtl,
explicitely challenged the continuous paradigm of [Reynolds (1895)], [Keller
and Friedmann (1924)], [Taylor (1935)], [Batchelor (1953)] and their many
followers, when he posed the fundamental question:

> Can the Navier-Stokes equation be used to calculate turbulent flows?

His doubts were based on the feeling that the continuous image of RANS does
possibly override features of most elementary vortex interactions at small
scales.

In analogy to Albring one might ask: Can we deduce a rose’s blossom from the
periodic system of chemical elements? Intuitively we answer with no. However,
this answer is justified because generally “more is different” [Anderson
(1972)].

Looking in this sense at Albring’s above question we may add that the
prediction horizon of the Navier-Stokes equation (NSE) is strongly limited by
a series of higher-order non-equilibrium phase transitions when a growing
Reynolds number goes through a number of critical values. In these super-
critical regions the sensitivity against initial conditions becomes relevant
and leads to the famous butterfly effect [[]]lorenz63. This is closely related
with irreversibility. At least in the limit of vanishing viscosity the
initial-value problem for NSE looks like a reversible one. But we know that
turbulence at $Re=\infty$ is an irreversible process.

In principle this problem was already known by Poincare and others as an
aspect of the three-body problem of celestial mechanics, but in a fluid-
mechanical context it has been demonstrated only after WW2 by [Lorenz (1960),
Lorenz (1963)]. Without going into details of the onset of turbulence and non-
equilibrium phase transitions in hydrodynamics and their reflections in
various branches of pure mathematics, together with Prandtl and Albring we
hypothesize that NSE is not sufficient to understand the secrets of turbulence
or to even predict turbulent flows, in particular not in the limit
$Re\rightarrow\infty$. Some readers might even go a step further and argue
that higher-order elements of the Friedman-Keller expansions of NSE, e.g. the
second and third-order turbulence closures discussed in [Voropayeva (2007)]
and the literature quoted therein, as well as the various “corrections” of
those closures populating turbulence theory, are modern analogues of the
epicycles of geocentric times. But this would exceed the limits of the present
report. In this respect we better refer to the agreeable picture of today’s
status of turbulence science drawn by [Davidson (2004)] in the preface of his
book where he mentions even “religious wars…between the different camps” of
turbulence theory.

There are not many physical phenomena resisting theoreticians so long like
turbulence. In the past it was mainly supra-conductivity which took about 30
years [[]]feynman63b. Today similarly obstinate problems are dark matter, dark
energy, and super-symmetry which have also now an age of about up to 40 years.
But turbulence waits still much longer for redemption and remained
particularly as the very last unsolved enigma of classical physics.

Below we develop an asymptotically invariant alternative to RANS and higher-
order closures, which is not primarily based on NSE but does not violate NSE
either.

Although the theory of many-particle physics offers a huge reservoir of
potentially helpful methods and tools, a closer inspection reveals that the
number of directly applicable tools is less impressive. Neither Liouville
theorem, ergodic hypothesis or Hamilton formalism nor other concepts for
thermodynamic equilibrium are applicable in their classical forms. Turbulence
is essentially an open-system, thermodymically non-equilibrium phenomenon. In
best cases we have a steady state or Fließgleichgewicht in the sense of
[Bertalanffy (1953)], [Glansdorff and Prigogine (1971)] and [Haken (1978),
Haken (1983)]. However, the concepts of Ising about particle dressing, quasi-
particles and renormalization – in the broader sense of [Dresden (1993)] and
[McComb (2004)] – appeared as essential guidelines in the slow evolution of
our thoughts.

Surely, turbulence can successfully be attacked from more than one
side222Theories generally compromise portrait and design aspects and in some
case mathematically different images of turbulence will show up eventually as
fully equivalent in their physical predictions, like Heisenberg’s matrix
mechanics and Schrödinger’s wave mechanics.. Whereas the continuous image has
led to a number of important results, for instance RANS and the broad spectrum
of different heuristic closure schemes discussed e.g. by [Wilcox (2006)],
without additional phenomenological input it gave neither unique values of the
universal constants of turbulent motion like von-Karman’s and Kolmogorov’s
spectral constants nor a closed image of turbulent flows. This has led to the
following view shared by many theoreticians [[]p. 173]landaulifshitz_eng87:

> …and $\kappa$ (is) a numerical constant, the von Karman constant, whose
> value cannot be calculated theoretically and must be determined
> experimentally. It is found to be $\kappa=0.4$.

In this respect one can actually be more optimistic. Frictionless turbulence
is governed in a weak sense by the Euler equation, a special case of NSE. The
Euler equation together with the conservation of mass represent pure “inert
geometry”. We can thus expect that the universal constants of turbulent motion
are ruled by certain geometrical constants, e.g. by irrational numbers like
$\sqrt{2}$ and/or transcendental numbers like $\pi$. Indeed, this is possible.
In a precursor study based on the particle picture used also below, von
Karman’s constant could be derived as $\kappa=1\left/\sqrt{2\pi}\right.\simeq
0.399$ [Baumert (2009)]. This interesting result led to the conjecture that
the particle concept can give also insight into spectral aspects of
turbulence.

Below we make explicit use of theoretico-physical thinking, which differs from
other scientific activities like computer modelling and mathemtical physics,
from applied mathematics, and from the measuring and observational
disciplines.

Computer modelling and measurements have in common that they deal with really
existing, finite systems and data, in particular with real and integer
numbers, with variables of finite size.

Theoretical physics, however, deals with “things” which never did, which do
not and never will exist: e.g. with point masses, homogeneous continua,
idealized vortex tubes, linear waves, frictionless fluids, plane and
impenetrable walls, infinitely small or large variables. A further building
block of theoretical physics is an often unspoken principle which we apply
also here: it “dictates that, all things being equal, one goes for the
simplest possibility–a rule that has worked remarkably well.” [[]p. 85, see
also Chandrasekhar, 1979]zee86. Sometimes this principle is called Occam’s
razor.

Mathematics also deals with similar non-existing objects, but their relations
to the real world are outside its responsibility. It is the responsibility of
theoretical physics to relate idealized thought systems with the real world.
The following report concentrates solely on the latter aspects.

With one exception, the following text contains no “fancy shmancy
mathematics”. It is the relation between equations (7), (8) and (9) that would
need some time to be derived from scratch. Furtunately, this has already been
done by other authors many years ago and went into the textbooks on
stochastic-dynamic systems [[,
e.g.]]kraichnan68,haken78,haken1983,stratonovich92,stratonovich94 to which we
refer the theoretically interested reader. We further suggest to begin first
with [Baumert (2009)] because it is actually the basis of the present report.

Everywhere the pronomen “we” is used in this text, it means the two of us, the
dear reader and the author.

##  2 Particles

If we take Prandtl’s discrete particle concept seriously but not literally,
then a number of questions need to be posed – and answered. A first question
to be addressed is the following.

#### Where do the particles come from?

With respect to their substance our particles are indistinguishable from their
fluid or gaseous environment. It is only their state of motion which makes the
difference to the sourrounding fluid so that we better should talk about
quasi-particles, but keep for brevity the term particles in the following.
They are characterized by their geometry, their specific kinetic energy, by
their vectors of linear and angular momentum.

Our particles are generated as representatives of turbulent fluctuations such
that their energy has to be drawn from the mean flow field via so-called shear
production333Convective turbulence and internal-wave breaking will be treated
in another study. to keep the total kinetic energy of the flow in balance.
I.e. if there is a shear in the mean flow and non-vanishing turbulent
viscosity, then the mean flow looses kinetic energy which reappears in form of
TKE444In the following the word turbulent kinetic energy or TKE is often used.
In the given context it means actually a kinetic energy density, i.e. energy
per unit of mass. Therefore we measure TKE most comfortably in the units m2
s-2., i.e. as energy of vortex motions. Hereby an exact process description of
vortex generation is not needed as long as we know how much energy is lost and
how much vorticity is generated so that we can place vortices of corresponding
properties into the flow. In a way we may talk about an emergence of the
particles.

#### What do the particles look like?

Due to Helmholtz’ principle of conservation of circulation, a circulation-free
fluid volume should exhibit zero circulation over the course of time. I.e.
particle generation cannot add any circulation to the volume. This implies
that turbulence production can take place only in form of the generation of
couples of counter-rotating vortices with zero total circulation. Consequently
we specify Prandtl’s fluid elements as dipoles where the simplest form is
locally fully symmetric, which is to be understood in a statistical sense.

This assumption is central for our theory but deserves a comment. It is well-
known that in laminar and turbulent flows wall friction generates velocity
gradients or current shear, which is identical with circulaton induction into
the mean flow. This process is governed by RANS, more specifically by its
turbulent friction term containing correlators like $\langle
u^{\prime}\,w^{\prime}\rangle$, i.e. by the Reynolds tensor. This tensor is
calculated further below on the basis of a particle theory. If a particle
would have own circulation on the local level, it would add extra circulation
to the flow, which would violate RANS. Our particles can therefore have no own
circulation.

Thus we have the challenging theory situation that a mean flow exhibits
turbulent friction and circulation while the turbulent vortices responsible
for fluctuations and fricition have no circulation and are essentially
frictionless, excepting the singular dissipation scale of size zero, see
further below.

With respect to the specific form of vortices in our dipoles we refer to the
classical notion of vortex tubes wherein vorticity is confined to the interior
of a tube of smaller or larger cross section [[, see further below and]who
quote papers by Kuo & Corrsin, 1972, and Brown & Roshko, 1974, showing tubes
as dominating characteristic structures]pullinsaffman98. A nice and more
recent study of vortex tubes has been presented by [Wilczek (2011)] who also
published on the internet a number of valuable animations of vortex ensembles
in motion555http://pauli.uni-muenster.de/tp/menu/forschen/ ag-
friedrich/mitarbeiter/wilczek-michael.html. We note that, as always, the
centerlines of our vortex tubes form either closed loops or they are attached
to boundaries.

Ordinary linear momentum of a dipole is known from classical theory in local
approximation and can be imagined in analogy to the motions of smoke rings. It
is conserved in our many-particle image if each particle, generated in a
locally homogeneous volume element, exhibits no preferred direction of motion.
The linear momentums of two dipoles with opposite directions compensate each
other. Assuming a locally “sufficiently high” and even number of dipoles thus
guarantees that the particle-induced linear momentum and circulation are zero
and total momentum of the flow exclusively governed by the mean-flow momentum
balance including the Reynolds stress terms.

#### How do the particles move?

The motion of a single isolated dipole [[, a “naked” particle in the
terminology of many-particle physics, see e.g.]]dresden93,mccomb2004 can be
given by classical rules [[, see]]lamb32, albring81,saffman92,
baumert05b,baumert09. However, if the selected dipole is embedded in a dense
tangle and thus surrounded by a cloud of similar dipoles, its properties are
“screened” by the cloud and thus modified: it appears to be “dressed” –
without violation of governing conservation laws. The solution is discussed in
the next section.

##  3 Vortices, energy, and scales

### Traditional vortex models

The hydrodynamic literature presents a larger number of elementary analytic
models for single isolated vortices under different conditions. Whereas the
so-called potential vortex is less realistic, the Oseen vortex, the Rankine
and the Taylor vortex, the Burgers, the Lundgren, and the Long, the Sullivan
and the spherical Hill vortex are more realistic models of isolated vortices
far from others and from boundaries [[, for overviews
see]]lamb32,lugt79,albring81,saffman92,pullinsaffman98,davidson04.

Excepting the potential vortex, all other vortex models are principally
realistic and show first a core with radially increasing tangential velocity,
then a saddle, and then a tail wherein the tangential velocity decreases
radially down to zero. However, in our context these models are not applicable
because they hold for conditions of isolation only. They cannot be transferred
to conditions of a dense vortex tangle where the distance between vortex
dipoles is small and the surrounding cloud of similar vortices screens the
effects from the rest. Therefore a new approach is needed, without violating
governing conservation laws, neither on a global nor on the scale of the
locally homogeneous and isotropic fluid volume.

I.e. an ideally “dressed” vortex dipole moving frictionless within a tangle of
similar objects should necessarily be characterized by a finite effective
radius, $r$, within which all the kinetic energy, $\cal K$, and vorticity of
the vortex is concentrated so that we may talk of an “energy-containing
radius”. For simplicity, in a statistical sense $r$ and $\omega$ are taken
identical for the two vortices forming the internally symmetric dipole.

We summarize as follows:

  * •

The effective scales $\omega$ and $r$ are governed by local conservation of
energy and angular momentum.

  * •

The radius $r$ defines a boundary within which kinetic energy and vorticity of
a vortex are confined.

  * •

The effective tangential velocity of a vortex is $u=\omega\,r$ and equals the
propagation velocity of the dipole. The resulting kinetic energy of the dipole
is thus ${\cal K}=2\times u^{2}/2=2\times
r^{2}\,\omega^{2}/2=(r\,\omega)^{2}$.

  * •

The vortices are incompressible but deformable quasi-solid bodies which move
frictionless in the vortex tangle. I.e. vorticity is uniformly distributed
within the cross-section of a vortex tube and has the value $2\times\omega$.

  * •

At solid boundaries vortices perform frictionless roll motions. Dissipation
happens exclusively $within$ the fluid at scale zero. (In reality the
dissipation interacts with boundaries through heat and sound generation.

These assumptions describe our image of vortices in a turbulent vortex tangle
without violating the conservation laws of kinetic energy, momentum, angular
momentum and circulation. Compared with the above-mentioned vortex models our
approach may be interpreted as a renormalization procedure leading to a finite
kinetic energy and a finite spatial extension.

While on a first glance this procedure looks somewhat arbitrary, exactly this
image is well established since long in physical oceanography, which is a
world of highest Reynolds numbers. In physical oceanography, meteorology and
physical limnology the a.m. characteristic radius, $r$, is traditionally
related with the “energy-containing” scale, which is a direct observable under
conditions of stable stratification and related with the so-called overturning
and Thorpe scales. Central aspects are discussed in detail in the following
section.

### Vortex tubes and rigid-body rotation

In stable stratification turbulent vortices are directly recognizable in
vertical profiles of density, $\rho$, and other scalar variables, such as
temperature and salinity, as “overturning” of density profiles. Within certain
energetic limits they move heavier over lighter fluid and thus produce a
direct imprint of the vortices in instantaneous depth profiles of density,
$\rho(z)$. This imprint has a unique character in stratified flows because the
buoyancy force implies that there should be a stably stratified “reference”
state, $\bar{\rho}$, a state in the absence of turbulent overturning.

This imprint appears typically as a Z-pattern, see Fig. 1, which obviously
stands for a solid-body rotation wherein the tangential (azimutal) velocity
grows linearly with the radius coordinate up to a maximum at its outer radius
and then (outside the body) drops rapidly down to zero. We interprete a
Z-pattern as a realisation of a vortex embedded in a dense vortex ensemble.
i.e. as a so-called rigid vortex as discussed by [Lugt (1979)] and further
below666 The idea of a rigid vortex is compatible with fluid mechanics because
Helmholtz’ laws also apply here. But the connection with continuum mechanics
is not trivial because the velocity distribution in a rigid vortex exhibits a
singularity at the outer radius where it drops from a finite value down to
zero. Further, the isolated rigid vortex is not stable..

To our knowledge, the use of overturning deviations from the stable state to
derive vortex radii was first introduced and explored by Thorpe (1977) in an
analysis of temperature profiles from Loch Ness, Scotland. Thorpe showed how
the reference state can be reconstructed from measurements and how overturns
and overturning scales can be detected. His approach is a solid basis to
measure the radii (or diameters) of vortices in stably stratified fluids
directly.

Thorpe (1977) described the general case of a monotonous reference density
distribution and the overturning fluid body as frozen for the time of the
overturn. This is justified as long as the length achieved through molecular
diffusion during the time of rotation, $l_{d}=\sqrt{2\,\nu_{m}\,t_{ot}}$, is
small compared with the energy-containing radius $r$ of the overturn777Here
$\nu_{m}$ is molecular viscosity and $t_{ot}$ the duration of an overturn at
the energy-containing scale $r\propto L$.. In water this condition is always
satisfied as $l_{d}$ remains here in the range of centimeters and less.

Later authors like Imberger and Boashash (1986) considered a subset of the
monotonous case, the linear reference-density distribution as depicted in our
Fig. 1. Itsweire et al. (1986) in their Fig. 1 were the first to discribe the
overturning motion explicitely as the effect of a rigid-vortex motion (cf.
also Pullin and Saffman, 1998).

Figure 1: Overturning and overturning scales in the sense of (Imberger and
Boashash, 1986, Fig. 6 there). The cartoon depicts a $180\deg$ overturning of
a 20 m thick layer that brings relatively heavy water up and lighter water
down. Note the characteristic Z-pattern of the vertical displacement $\zeta$.
The original stable density profile is the blue line, the unstable density in
the overturn is red and dashed. By courtesy of Dr. Hartmut Peters, Earth and
Space Research, Seattle, USA.

### Thorpe’s method in oceanography

Background. A short digression on the background of Thorpe’s method might be
in place. Geophysical fluid dynamics is a domain of pioneering turbulence
studies at high Reynolds numbers because atmosphere and oceans offer the
necessary conditions for free. The first massive demonstration of measured
Kolmogorov spectra was carried out by Grant et al. (1959) at a Reynolds number
of about $Re\approx 10^{8}$ in a 100 m deep tidal channel888Today’s DNS of
turbulent flows do not exceed $Re\approx 10^{5}$ and even the European
superpipe CiCLOPE will in the best case not significantly exceed $Re\approx
10^{6}$ Rüedi et al. (2009).. They demonstrated the validity of the 5/3 law
over an intervall of 3.5 orders of magnitude (about 12 octaves). Together with
the logarithmic law of the wall and the decay of TKE in homogeneous isotropic
turbulence according to $t^{-1}$ (or $(x/U)^{-1}$ in the wind tunnel; cf.
Batchelor, 1953, Chapter VII) the 5/3 law belongs until now to the most
prominent universal features of high-$Re$ turbulence.

Many observational techniques and dynamical concepts of geophysical fluid
dynamics were first developed in meteorology and later adopted in physical
oceanography and limnology. That this was not the case with turbulent
overturning is likely related to the fact that the troposphere, the most
accessible part of the atmosphere, does not show consistent mean
stratification. In contrast, oceans and lakes are almost everywhere stably
stratified outside of boundary layers. It further makes sense that turbulent
overturning was first explored in a lake where temperature, $T$, is the sole
stratifying agent. In contrast, ocean stratification depends on temperature as
well as salinity, $S$, and density $\rho=\rho(p,T,S)$ cannot be measured with
the same resolution as $T$ alone. We note in passing that real fluids are
compressible such that analyzes of overturning scales have to be based on
potential density and potential temperature.

The imprint of turbulence in vertical density profiles allows defining and
extracting overturning scales such as the Thorpe scale, $L_{Th}$ Thorpe
(1977). The cartoon of Fig. 1 illustrates the overturning of a $L_{ot}=20$ m
thick layer of the water column by a single vortex in solid body rotation of
diameter $L_{ot}$. According to our above discussion of vortex dipoles as
quasi-particles we conclude that $L_{ot}=2\,r$.

The graph depicts the moment after a 180$\deg$ rotation that brings heavy
water up by vertical displacements with a range of $\zeta=0\,...+L_{ot}$ and
moves lighter water down by $\zeta=0\,...-L_{ot}$. The Thorpe scale is defined
as the r.m.s. of $\zeta$. The linear original density profile of the cartoon
implies

$L_{Th}=L_{ot}/\sqrt{3}\;=\frac{2}{\sqrt{3}}\,r\;=\;1.15\;r\;.$ (1)

We define $\zeta$ by the path from its original depth to a displaced depth.
That is, $\zeta$ carries the same sign as the vertical turbulent velocity
$w^{\prime}$ that has caused it.

Figure 2: A big overturn in the Pacific Equatorial Undercurrent at
$0\deg,140\deg$W during the Tropic Heat II cruise (Peters et al., 1995,
adapted from ): (a) potential temperature ($\Theta$, red) and potential
density ($\sigma_{\Theta}$, black), Thorpe-sorted $\sigma_{\Theta}$ (blue),
and Thorpe scale $L_{Th}$ (shaded); (b) turbulent vertical displacement
$\zeta$. By courtesy of Dr. Hartmut Peters.

Fig. 1 may seem simplistic. It can easily be made more realistic by adding
that, in the course of the overturning, and owing to the unstable
stratification within overturns, flow instability will occur and generate
turbulence and a range of smaller scales than $L_{ot}$, inside the big
overturn.

The ocean is full of overturns that look just like this scenario. Fig. 2
depicts a big overturn in the Pacific Equatorial Undercurrent, EUC, on the
equator at $140\deg$W from Peters et al. (1995). Note the Z-shape of the big
overturn and its sharp upper and lower edges.

In oceanography and limnology, Thorpe’s concept of vortices overturning parts
of the water column are applied to observations as in Fig. 2. Measured
potential density ($\sigma_{\Theta}$; by convention 1000 kg m-3 is subtracted)
or potential temperature ($\Theta$) data points are “Thorpe-sorted” into
monotonically rising or falling sequences corresponding to stable density
stratification. The sorted profiles are taken as a proxy for the reference, or
“mean” profile that gave rise to the observed turbulence and overturning. The
vertical distance over which data points have to be moved to make the profile
monotonic is $-\zeta$. I.e. Thorpe-sorting “undoes” the overturning.
Individual overturns are defined by $\sum_{i}\,\zeta_{i}=0$ Dillon (1982).

The shaded bars in Fig. 2(a) show $L_{Th}$ averaged over individual overturns.
This definition is highly robust. A squared buoyancy frequency $N^{2}$
computed from the sorted $\sigma_{\Theta}$ or $\Theta$ is non-negative999 Here
we may restrict our considerations to incompressible fluids where
$N^{2}=-g/\langle\rho\rangle d\langle\rho\rangle/dz$ with $g$ as gravitational
acceleration, $g=9,81$ m s-2., $N^{2}\geq 0$.

Turbulent length scales related to overturning scales are unaffected by the
presence of internal gravity waves (IGWs). IGWs are ubiquitous in stratified
geophysical flows and dominate velocity and scalar spectra at vertical scales
of the order of O(100 m) and smaller (e.g. Peters et al., 1995). The TKE
accounts only for a tiny fraction of the integral of these “red” spectra.
Owing to this property, length scales commonly used in laboratory experiments,
such as the Ellison (1957) scale,
$L_{E}=\overline{\rho^{\prime}}/\langle\partial\langle\rho\rangle/\partial
z\rangle$, are unsuitable for a characterization of turbulent length scales in
geophysical flows because density fluctuations
$\rho^{\prime}=\rho-\langle\rho\rangle$ inevitably are dominated by IGW
signals so that $L_{E}$ can no longer be interpreted as a characteristic
length scale of turbulent fluctuations. Here, angular brackets denote ensemble
averages $\overline{\rho^{\prime}}=\langle\rho^{\prime 2}\rangle^{1/2}$ is an
r.m.s. value.

Thorpe scale and Taylor scaling. The great power of Thorpe’s [1977] concept
can be demonstrated by considering the so-called Taylor scaling, with $q$ as
an r.m.s. measure of turbulent velocity fluctuations. Dimensional analysis
strictly gives for $l$ as an energy-related length scale the following:

$l\propto\,{q^{3}}/{\varepsilon}\;.$ (2)

Here $l$ is obviously related with TKE because of $\bar{\cal K}\propto q^{2}$,
and $\varepsilon$ is the TKE dissipation rate101010We use the convention that
the symbol ${\cal K}$ denotes the energy of one selected dipole whereas
$\bar{\cal K}$ is the local average of the same variable for a whole ensemble.

If the Thorpe scale $is$ indeed a measure of the size of the energy-containing
vortices and thus related with the energy-containing scale, i.e.
$L_{Th}\propto l$, then we can write (2) as follows,

$\varepsilon\times L_{Th}\propto\,{q^{3}}\;,$ (3)

or

$q\;\propto\,(\varepsilon\times L_{Th})^{1/3}\;.$ (4)

Instead of $q$ one can alternatively use $\bar{\cal K}^{1/2}$ with turbulent
kinetic or mean vortex kinetic energy, $\bar{\cal K}$. Fig. 3 shows
measurements in the ocean demonstrating that relation (4) is true, in a
statistical sense. The Thorpe scale characterizes indeed the size of the
energy-containing eddies.

8.2cm

Figure 3: Turbulent velocity $q_{\epsilon}\\!=\\!(\epsilon\,l)^{1/3}$ derived
from measured $\epsilon$ via Taylor scaling, (4), versus measured turbulent
velocity fluctuation $q$. Adapted from Peters et al. (1995) where
$l\\!=\\!1.6\,L_{th}$ and where $q$ is the spectral velocity variance at
vertical wavenumbers $\geq\,1/l$. Shown are data for individual overturns at
least 1 m thick from depths 60–350 m in the Equatorial Undercurrent at
$0\deg,\,140\deg$W. The gray line indicates the median ratio of $q_{\epsilon}$
over $q$. By courtesy of Dr. Hartmut Peters.

However, this statement needs a comment. Thorpe-sorting as a technique to
measure the radius or diameter of energy-containing turbulent vortices rests
on the existence, and hence works only, in stratified flows. Under geophysical
conditions this automatically implies that a certain coexistence of IGWs and
turbulence is inescapable. The $q$ in (2) must only reflect TKE and must not
be contaminated by the much larger IGW energy.

While the separation of waves and turbulence is beyond the scope of this
note111111The reader is referred to Peters et al. (1995) and D’Asaro and Lien
(2000)., a workaround is to consider spectra and to study only the ultra-
violett or short-wavelengths part where IGW existence is excluded through
their dispersion relation. Peters et al. (1995) did exactly this and extracted
$q$ from oceanic observations as the velocity variance at scales of $l$ and
smaller. On this basis, Taylor scaling clearly emerges from their observations
(Fig. 3).

Thorpe’s [1977] concept of turbulent vortices and their imprint through
vertical overturning on density profiles allows extracting energy-scale
variables even in conditions where velocity and scalar spectra are heavily
dominated by internal gravity waves. It allows relating energy-scale- to
dissipation scale variables through the turbulent length scale that carries
Thorpe’s name. These  directly observable energy-containing length scales
stand for the radius or diameter of our renormalized vortices discussed
further above.

### Dressing a rigid-vortex tube

As we learned from the considerations and examples above, turbulence may be
imagined as vortex tubes resembling a dense “local cloud” of dipoles in
chaotic motion, the cloud having zero angular and linear momentum. So far the
rigid-vortex tubes have been considered as somewhat arbitrary idealizations of
real-world vortices.

Translated into the language of home cooking, a snapshot of rigid-vortex
turbulence may be imagined as a dense, entangled heap of hot spaghetti
arrabiata. The spaghetti rotate around their inner centerlines and move
frictionless within the (inviscid) sauce arrabiata. Therefore the only
interactions between individual spaghetti occurs when they touch each other
randomly.

It arises the question whether the rigid-vortex tube and a corresponding
dipole is a stable solution of the Euler equations. We leave it to the
interested reader to derive from these equations that an isolated rigid vortex
indeed solves these equations, but, as long as it exists, generates the
following pressure head as a consequence of inertial (centrifugal) forces:

$p=p_{0}+\frac{\rho}{2}\times\omega^{2}r^{2}.$ (5)

Here $p_{0}$ is the background pressure of a laminar reference flow, e.g. in
the ocean the depth-depending hydrostatic pressure. If the pressure outside
the vortex would simply be $p_{0}$ then, due to the action of the outwards-
directed pressure head of the vortex motion given by the second term in (5),
the vortex would loose stability.

The stability of quasi-rigid vortices as observed in real-world turbulence can
thus be guaranteed only by the help of a compensating force of equal strength.
Here a concept of many-particle physics comes into play: dressing. As far as
we will embed our initially isolated rigid vortex into a locally homogeneous
and isotropic large ensemble (“cloud”) of similar vortices (more precisely:
dipols made of rigid vortices), the members of the cloud generate more or less
exactly the counter pressure needed to compensate (5) and keep the vortex
“stable enough”.

According to (5) and with $u=\omega\,r$ the pressure deviation can be written
as follows,

$\delta p=p-p_{0}=\frac{\rho}{2}\times u^{2}=\rho\times\cal K\,,$ (6)

where $\cal K$ is the kinetic energy density in a vortex. Obviously the
measurement of turbulent pressure fluctuations can help estimating turbulent
kinetic energy. This corresponds to an early classical but approximate result
of continuum theory (eq. (8.3.21) on p. 182 in Batchelor, 1953; Hinze, 1953,
p. 242, last eq.). We will not explore the potential of (6) further and leave
it to the interested reader.

##  4 Turbulence and kinetic gas theory

### Similarities

About 150 years ago, James Clerk Maxwell presented his kinetic theory of
gases. Even if the details of molecular interaction forces in vacuum would
have been known that time, it would have not been helpful. An integration of
Newtons law of motion in terms of ordinary differential equations for each one
of the billion particles was practically excluded. But the creative use of
symmetries and conservation laws radically simplified the situation and led
eventually to a closed description of the most important macroscopic
properties of gases. Main elements are the following:

  * (i)

The particles in a gas are perfectly elastic points with non-zero mass.

  * (ii)

They are in permanent random motion which is to be described in terms of
statistical moments and sometimes called “molecular chaos”. The scatter
motions exhibit no preferred dirctions.

  * (iii)

The collision results depend only on the local angular orientation of the
collision partners.

  * (iv)

Between collisions they move uniformly and independently, without preferred
direction.

  * (v)

Due to chemical neutrality, collisions lead only to the scatter of
trajectories.

The corresponding “turbulent relatives” of the above are:

  1. (i’)

The particles are locally symmetric vortex-dipole tubes with finite cross-
sectional area, with vorticity and kinetic energy confined in the tubes, and
with zero circulation.

  2. (ii’)

Their random translatory motions prefer no directions and may be termed ‘dipol
chaos’ Marmanis (1998).

  3. (iii’)

The result of collisions depends only on the local angular orientation of the
colliding dipole elements.

  4. (iv’)

Between collisions the particles move along complex trajectories which may be
curved.

  5. (v’)

For symmetry reasons, 50 % of all collisions occur between two counter-
rotating dipole elements leading to dipole recombinations (or reconnections,
like those reported in a turbulent superfluid by Paoletti et al., 2010) and at
the end to quasi-elastic, random scatter motions resembling turbulent
diffusion and mixing, see left branch in Figure 14.

We underline that these rules shall apply only locally in space and time. Of
course, the prediction of the global pathway of a vortex dipole is impossible.

### Differences

Major differences between ideal gases and turbulence are the following:

  * (a)

The number of chemically neutral gas particles in a countainer is constant in
time. They form a closed thermodynamic system. There is no particle
annihilation.

  * (b)

The particles of turbulence are excited energy states, their number decreases
via collisions and subsequent energy dissipation. They need to be replaced by
shear generation of new particles if their number shall be kept more or less
in balance.

  * (c)

In a gas, trajectories of point masses are simple lines in space, trajectories
of vortex-dipole tubes produce curved areas.

While gases evolve towards static thermodynamic equilibria, turbulence evolves
either towards $dynamic$ equilibria or steady states121212This neither implies
homogeneity, equifinality nor uniqueness of steady states., which may have
periodic character, or turbulence dies off.

Figure 4: Two possible collision results of two dipoles: the left branch is
“diffusive” because it leads to a recombination of the dipole elements and
scatter of trajectories which is known as turbulent diffusion. The right
branch is “dissipative” because it leads to an unstable vortex configuration
which decays “somehow” into heat141414The details of this decay process are
discussed later below.. For symmetry reasons both branches have identical
probabilities of 0.5. Note that the circular form of the vortex cross sections
presented here are chosen for reasons of clarity. They hold as an statistical
average picture only. Real vortices have elliptic or even strongly deformatted
cross sections.

To specify this important difference we have to supplement our above property
list for “turbulence particles” as follows:

  1. (d)

For the same symmetry reasons like in (v’) of section 4 above, the remaining
50 % of all dipole collisions occur between two likewise rotating dipole
elements. Each case generates a fundamentally unstable vortex couple forming a
slowly rotating dissipative patch with its center of mass more or less at
rest. This patch decays through turbulent dissipation in the special form of a
“devil’s gear” or Kolmogorov spectrum. For details see further below and the
right branch in Fig. 14.

##  5 Equations of turbulent motion

### Brownian and turbulent motions

The turbulence properties (iii’) and (v’) of section 4 above establish an
analogy of turbulent dipole movements with the Brownian motion of particles
suspended in a fluid at rest (in the sense of Einstein, 1905). In the present
turbulent case it is not the kinetic theory of heat which governs the motions,
it rather is Helmholtz’ theory of dipol motions.

The (local) temporal path increments $\delta\vec{Y}_{j}$ of a vortex dipole
$j$ may be found by integration of the following Langevin equation over the
time increment $\delta t$,

$\frac{d{\vec{Y}_{j}}}{dt}=\vec{V}_{j}\;\;,\;j=1\dots{\cal N},$ (7)

where $\vec{V}_{j}$ is the random center-of-mass velocity of the selected
dipole $j$.

To specify the stochastic process $\vec{V}_{j}=\vec{V}_{j}(t)$ as simple as
possible, we choose a zero-mean, white-noise Gaussian process. Its strength is
controlled by the locally averaged dipole properties $\bar{\cal K}$ and
$\bar{\omega}$. More elaborate random processes like the Ornstein-Uhlenbeck or
the Kraichnan model Kraichnan (1968) and others are beyond the focus of the
present paper.

We now need to make a bigger jump over the broad river of stochastic dynamic
systems theory where the probability density function for the solution of a
dynamic system151515A dynamic system is here understood as a set of possibly
non-linear ordinary differential equations driven at their right-hand sides by
stochastic processes. may easily be taken from applied textbooks like those of
Haken (1978, 1983) in form of solutions of the Fokker-Planck equation (FPE)
corresponding to the extremely simple stochastic-dynamic system (7). In the
present case the FPE reads as follows,

$\frac{\partial{\cal N}}{\partial
t}-\frac{\partial}{\partial\vec{x}}\left(\nu\;\frac{\partial{\cal
N}}{\partial\vec{x}}\right)=0\,,$ (8)

where $\cal N$ is the probability density mentioned above, an equivalent of
the particle number density itself. $\nu$ characterizes the strength or
intensity of the noise $V_{j}(t)$ in (7). As already mentioned it is governed
by the control variables ($\bar{\cal K}$, $\bar{\omega}$) and appears later as
coefficient of eddy diffusivity of momentum, i.e. as eddy viscosity.

In the transition from (7) to (8) it has implicitely provided that all
gradients exhibit sufficiently smooth and slow behavior. Those local quasi-
equilibrium conditions are typically assumed in non-equilibrium thermodynamics
and many-particle physics. This clearly excludes shocks and steep fronts from
our considerations.

A further comment concerns an assumption used implicitely above. It is an
analogue of the so-called Ising assumption known from the theory of magnetism:
we treat triple or higher interactions between vortex filaments as
negligible161616This is not fully trivial. Under specific conditions of Bose-
Einstein condensates stable configurations consisting of one vortex and two
anti-vortices have been observed in the laboratory, either in linear setups or
equilateral triangles Seman et al. (2009). The latter is most symmetric and
called a $tripole$. Here we assume that in our very dense “vortex gas”
tripole-tripole interactions are controlled by dipole-dipole interactions of
their subsets. Note that tripoles might violate the principle of zero
circulation on the level of elementary interactions needed to keep the mean-
flow momentum balance in correct balance..

Figure 5: Local cross section through the first developmental stage of an
unstable pair of likewise rotating vortices resulting from a dipole-dipole
collision (right branch in Fig. 14). The green circles represent the primary
energy-containing vortices with identical radii $r$. Note that the green
circles do not touch each other! They are separated by the red circles who
symbolize secondary vortices in the beginning phase of a whole vortex cascade.
The dipole cloud surrounding the above structure not only generates the
necessary pressure head to keep individual vortices stable but it also acts as
a source of perturbations initiating roll-up instabilities and thus tertiary
and higher-order vortices and eventually a fully developed dissipative patch,
see text. The broken blue line and the arrows symbolize the slow rotation of
the whole patch around the common center of mass.

### Generation and annihilation of particles

Property (d) in Section 4 above allows to supplement the right-hand side of
equation (8) with sink and source terms, leading to a reaction-diffusion type
equation:

$\frac{\partial{\cal N}}{\partial
t}-\frac{\partial}{\partial\vec{x}}\left(\nu\;\frac{\partial{\cal
N}}{\partial\vec{x}}\right)=\Pi-\beta\;{\cal N}^{2}\;.$ (9)

Here $\Pi$ is the rate of quasi-particle generation. It is to be expressed in
terms of kinetic energy per unit time and is thus necessarily proportional to
the energy loss of the mean flow.

The second term corresponds to the energy-dissipation rate of TKE,
$\varepsilon$, i.e. TKE conversion into heat and/or sound, with $\beta$ being
a constant (for details see Baumert, 2009). $\varepsilon$ scales with the rate
of particle annihilation resulting from collisions. As we learn from chemical
kinetics (see Haken, 1978, 1983, for details), the annihilation term is
quadratic in the particle number, $\cal N$, because $two$ particles need to
collide to generate $one$ unstable couple which is then converted into heat
and/or sound. This is discussed in the next Section.

##  6 Dissipative patches

### Formation of vortex spectra

Our Fig. 14 and the concept behind it may seem simplistic. It demonstrates the
only two possible results of a dipole collision. While the left half of the
Figure shows the recombination of counter-rotating vortices from counter-
rotating vortices, the right half shows the the formation of a couple of
likewise rotating vortices from counter-rotating vortices. If isolated or
naked, the likewise rotating couple revolves around a common center of mass
and remais thus nearly at rest. This couple is known to be unstable, to form
stationary dissipative patches Sommerfeld (1948)171717It somehow resembles the
“dissipative elements” discussed by the group around Norbert Peters (see
Schäfer et al., 2010)..

This picture with the dissipative patch almost at rest implies that
dissipation is a spatially patchy phenomenon called intermittency and studied
extensively by various authors (for an overview see e.g. Frisch, 1995). We do
not go into details because intermittency is outside the focus of this study.
We discuss instead the way how the unstable configurations at the right half
of Fig. 14 and in Fig. 5 could be transformed into a Kolmogorov-Richardson
spectrum. This problem has attracted early attention by Taylor (1937) and
Kolmogorov (1941). The latter found on dimensional grounds that for a steady
energy flux from large to small scales the kinetic energy spectrum as function
of wavenumber may be presented as follows181818Actually, the original
arguments in Kolmogorov’s derivation were more subtle, but the use of
arguments in the sense of Rayleigh’s method of dimensional analysis or,
stricter, Buckingham’s $\pi$ theorem, is “strict enought” for the present
discussion.:

$d{\mathcal{K}}=\alpha_{1}\,\varepsilon^{\alpha_{2}}\,k^{-\alpha_{3}}\,dk\;,$
(10)

where here $k=2\pi/\lambda$ is the wave number and $\lambda$ the wavelength.
$\varepsilon$ is the dissipation rate of TKE, $\cal K$. Based on strict
dimensional arguments, Kolmogorov (1941) proved that $\alpha_{2}=2/3$ and
$\alpha_{3}=5/3$, in excellent agreement with the famous observations by Grant
et al. (1959) in a tidal inlet with $Re\approx 10^{8}$ and a depth of about
100 m. A theoretically sound value of $\alpha_{1}$ was open until today and
given now below.

### Devil’s gear

Our view of the Kolmogorov-Richardson cascade has been changed through a study
by Herrmann (1990) who has shown that Kolmogorov’s value for $\alpha_{3}$
corresponds numerically to the data of a space-filling bearing (see also
Herrmann et al., 1990). The latter is the densest non-overlapping (Apollonian)
circle packing in the plane, with side condition that the circles are
pointwise in contact but able to rotate freely, without friction or slipping.
One may call it a “devil’s gear” Pöppe (2004).

The contact condition for two different “wheels” with indizes 1 and 2 of the
gear reads

$u=\omega_{1}\;r_{1}=\omega_{2}\;r_{2}\;,$ (11)

where $u$ is necessarily constant throughout the gear and governed by the
energy of the decaying vortex pair as $u=\sqrt{2\,\cal K}$. It follows that

$\omega_{2}=\omega_{1}\;\frac{r_{1}}{r_{2}}\;,$ (12)

so that for very small $r_{2}$ the frequency $\omega_{2}$ may become
acustically relevant.

If the above gear is frictionless then the next question arises $where$ –
within this picture – energy could be dissipated. Clearly, in a fluid with
vanishing but non-zero viscosity, dissipation happens at scales where velocity
gradients are high enough, here: at a scale of measure zero. Our dissipative
patch (Fig. 5 shows the first stage of its formation) is thus “almost
frictionless” and therefore a Hamiltonian system, excepting scales of size
zero.

The formation of a fully developed spectrum of “wheels” from Fig. 5 deserves
certain perturbations “from the sides”, a condition which is guaranteed by the
random reconnection/recombination and scatter processes sketched in the left
half of Fig. 14 and also by the incomplete mutual pressure compensation of the
vortices in our vortex ensemble.

Without speculating too much we may expect that in a quasi-steady state
patches like in Fig. 5 are formed via roll-up instabilities at the “surface”
of the respective larger vortices. They steadily evolve into a fully developed
gear. Its energy, dissipated at the smallest radii, will decrease the energy
content of the primary (initial) vortex pair unless it is fed by mean flow.

The outer limits of such a patch are sketched in Fig. 6 for the begin of the
cascade process. The most important message which we gain from this figure is
that the $longest$ or energy-containing wavelength of the dissipative patch
equals $\lambda_{0}=2\,r$. The wavelength in a dipole is $4\,r$ and it forms
no patch or spectrum. This difference is essential. Further below we use
$\lambda_{0}$ as a lower integration limit for the spectral energy
distribution. It is important to underline that $\lambda_{0}$ labels the upper
wavelength limit in a dissipative patch. This limit is actually not influenced
by the formation details of the spectrum.

We finally notice that the transformation of the unstable configuration of two
likewise rotating vortices deserves time to set the greater masses of the
smaller scales into motion. This inertia effect might play a role in highly
dynamic scenarios.

Figure 6: Outer limits of a dissipative patch ($c.f.$ Fig. 5). The maximum
wavelength is obviously equal to $\lambda_{0}=2\,r$.

##  7 Universal constants

In a precursor study Baumert (2009) it has been shown that the geometric-
mechanical concept given in the present paper allows the derivation of von-
Karman’s constant as $\kappa=1/\sqrt{2\,\pi}$. It fueled hopes that other
universal constants of turbulent motion might also be derived from that
apparatus.

### Kolmogorov constant $\alpha_{1}$
in the wavenumber spectrum

The TKE can be calculated by integrating (10) over the dissipatve patch in the
sense sketched in Figures 5 and 6 yielding

${\cal
K}=\alpha_{1}\,\varepsilon^{2/3}\,\int_{k_{0}}^{\infty}k^{-5/3}\;dk=\alpha_{1}\,\frac{3}{2}\,\left(\frac{\varepsilon}{k_{0}}\right)^{2/3},$
(13)

where $k_{0}=2\,\pi/\lambda_{0}$ characterizes the lower end of the
$turbulence$ spectrum in the wavenumber space. We loosely assign the
wavenumber range $k=0\dots k_{0}$ to the mean flow which may basically be
resolved in numerical models.

The dissipation rate $\varepsilon$ in (13) can be expressed as follows,

$\varepsilon={\cal K}/\tau\,,$ (14)

with $\tau$ being the lifetime of a dissipative patch. Inserting (14) in (13)
and rearranging gives the following:

$\alpha_{1}=\frac{2}{3}\,\left(2\,\pi\right)^{2/3}{\cal
K}^{1/3}\left(\frac{\tau}{2\,r}\right)^{2/3}\,.$ (15)

Here we took from Fig. 6 that the energy-containing initial or primary
wavelength of a dissipative patch is given by $\lambda_{0}=2\,r$.

Now we make a local quasi-equilibrium assumption for conditions of extremely
dense vortex packing: Our marching dipoles can occupy only those places which
are simultaneously “emptied” from dissipative patches by decay. This means
that the life time of a dissipative patch, $\tau={\cal K}/\varepsilon$, should
equal the time of “free flight” of a dipole over a distance $2\,r$ (cf.
Baumert, 2009):

$\tau={\cal K}/\varepsilon=2\,r/u\;.$ (16)

Here we used the scalar dipole velocity $u$,

$u=\omega\,r=\sqrt{2\,{\cal K}}.$ (17)

After some algebra, the dimensionless pre-factor of the three-dimensional
wavenumber spectrum reads as follows:

$\alpha_{1}\;=\;\frac{1}{3}\,(4\,\pi)^{2/3}\;=\;1.802\,.$ (18)

The corresponding value of an ideal one-dimensional spectrum is one third of
the above, i.e. 0.60.

### Kolmogorov constant $\beta_{1}$
in the frequency spectrum

The generalized form of the Lagrangian frequency spectrum of fluid turbulence
is the following,

$d{\mathcal{K}}=\beta_{1}\,\varepsilon^{\beta_{2}}\,\omega^{-\beta_{3}}\,d\omega\;,$
(19)

where $\omega$ is the angular frequency. Tennekes and Lumley (1972) derived
$\beta_{2}=1$ and $\beta_{3}=2$ in a similar fashion like Kolmogorov (1941)
derived $\alpha_{2}$ and $\alpha_{3}$ (see also McComb, 2004). The integration
of (19) from $\omega=\omega_{0}$ to $\omega=\infty$ gives after rearrangement
and using (14) the following,

$\beta_{1}\;=\;\omega_{0}\;\tau\,.$ (20)

Now we take from the right part of (16) $\tau=2\,r/u$, from the left part of
(17) $\omega_{0}=u/r$ and insert both in (20) to get finally

$\beta_{1}\;=\;\omega_{0}\;\tau\,=\;\frac{u}{r}\,\frac{2\,r}{u}\;=\;2\,.$ (21)

### Pre-factor in the velocity
autocorrelation function

The spatial autocorrelation function $B(\rho)$ of fluctuating velocities is a
special second-order case of a structure function191919For details see §34 in
Landau and Lifshitz (1987).. With the abbreviation
$\rho=\left|\vec{\rho}\right|$, it is defined as follows,

$B(\rho)=\langle u_{1}(\vec{x})\times u_{2}(\vec{x}+\vec{\rho})\rangle.$ (22)

Here $u_{1},u_{2}$ are the (scalar) velocity components of the flow velocity
$\vec{u}$ in the direction $\vec{\rho}$ connecting the points $\vec{x}$ and
$\vec{x}+\vec{\rho}$ where the velocities $u_{1}$ and $u_{2}$ are taken
respectively:

$\displaystyle u_{1}$ $\displaystyle=$
$\displaystyle\vec{u}(\vec{x})\cdot{\vec{\rho}}/{\rho}\;,$ (23) $\displaystyle
u_{2}$ $\displaystyle=$
$\displaystyle\vec{u}(\vec{x}+\vec{\rho})\cdot{\vec{\rho}}/{\rho}\;.$ (24)

The central dot denotes the scalar product (or dot product). Notice that,
rather than a density, $\rho$ is here a spatial distance.

In their §34 on p. 145 Landau and Lifshitz (1987) have shown that, based on
early results by Kolmogorov, $B(r)$ in (22) may be written as follows,

$B(r)=C\times(\varepsilon\,\rho)^{2/3},$ (25)

where $C$ is a dimensionless numerical constant which is related with
$\alpha_{1}$ from the universal wavenumber spectrum (10) as follows,

$\displaystyle C$ $\displaystyle=$
$\displaystyle\alpha_{1}\,\frac{27}{55}\,\Gamma(1/3)$ (26) $\displaystyle=$
$\displaystyle\frac{1}{3}\,(4\,\pi)^{2/3}\,\frac{27}{55}\,\Gamma(1/3)$ (27)
$\displaystyle\approx$ $\displaystyle 2.37\;.$ (28)

Here $\Gamma(z)$ is the Euler gamma function.

In contrast to our derivations of $\kappa$, $\alpha_{1}$ and $\beta_{1}$ the
value (28) for $C$ should be taken with care because the derivation of its
relation with $\alpha_{1}$ by Landau and Lifshitz uses an approximation and is
thus valid for small values of the distance variable $\rho$ only, i.e. for
$\ll\lambda_{0}=2\,\bar{r}$.

##  8 Discussion

### Comparison with observations

The rounded numerical values $\kappa=0.4$, $\alpha_{1}=1.8$,
$\alpha_{1}/3=0.6$, and $\beta_{1}=2$ for von Karman’s and Kolmogorov’s
universal constants predicted by our theory are situated well within the error
bars of many high-$Re$ number observations, NSE and RG based analytical
approximations, laboratory and DNS experiments. Based on observations,
Tennekes and Lumley (1972) gave the values $\alpha_{1}=1.62$ and
$\beta_{1}=2.02$, but with greater uncertainty.

Later works have been analysed in an important study by Sreenivasan (1995) who
possibly gave the most comprehensive literature review of experimental and
observational values for the number $\alpha_{1}/3$ until now. Later Yeung and
Zhou (1997) reported a value of $\alpha_{1}=1.62$ based on high-resolution DNS
studies with up to 5123 grid points.

Figure 7: Experimental and observational results for $\alpha_{1}/3$ measured,
collected from the literature, and analysed by Sreenivasan (1995). The solid
green line follows our somewhat arbitrary approximation
$0.6\times{\sqrt{Re_{\lambda}}}/{\left(\sqrt{Re_{*}}+\sqrt{Re_{\lambda}}\right)}$
wherein $\alpha_{1}/3=0.6$ is the theoretically derived asymptotic value. Here
we took $Re_{*}=10$. A similar presentation has been chosen by Sreenivasan
(1995) for $\alpha_{1}/3$ in his Fig. 3, and by Yeung et al. (2006) for
$\pi\beta_{1}$ in their Fig. 2.

The results of their later efforts suggest, again with DNS but based on a grid
of 20483 points, the value $\beta_{1}=2.1$ (Yeung et al., 2006, their Fig. 3).
In a recent study by Donzis and Sreenivasan (2010) a DNS grid of $4096^{3}$
has led to $\alpha_{1}\approx 1.58$.

Based on their oceanic measurements (with much higher Reynolds numbers
compared with DNS) Lien and D’Asaro (2002) found that $\beta_{1}=1.75\dots
2.04$. They state that

> …since the present uncertainty is comparable to that between high quality
> estimates of the Eulerian one-dimensional longitudinal Kolmogorov constant
> measured by many dozen investigators over the last 50 years, large
> improvements in the accuracy of the estimate of $\beta_{1}$ seem unlikely.

Beginning with an initiating work by Forster et al. (1977), systematic
analytical approximations using RG methods and related techniques for NSE
became further sources of estimates for the universal constants. E.g. Yakhot
and Orszag (1986b, a) reported $\alpha_{1}\approx 1.62$ whereas McComb and
Watt (1992) derived $\alpha_{1}=1.60\pm 0.01$ and Park and Deem (2003)
obtained $\alpha_{1}=1.68$. We note that these approximations are technically
extremely complex and neither unique nor part of an integrated descriptive
concept for turbulence.

Turbulence plays a crucial role in almost all fields of engineering, including
medical applications, and in geophysical fluid dynamics up to climate-change
studies. It plays an essential part in our everyday life. When we leave our
house or ride our bike, when we jump into our pool – we always are literally
embedded in a turbulent fluid. Compared with most other fields of modern
physics the present scatter in the values of the universal constants is
uniquely high and therefore actually not longer acceptable. However, our
theoretical results containing irrational and transcendental numbers suggest
that the universal constants of turbulence can principally not be measured in
real-world fluids. Or would measuring $\pi$ be a reasonable task?

A possible solution of this dilemma is the asymptotic analysis of measurements
at higher and higher Reynolds numbers so that an extrapolation to
$Re\rightarrow\infty$ becomes feasible with some certainty. The data analyses
by Sreenivasan (1995) and the report by Yeung et al. (2006) could serve as a
methodical model. Fig. 7 shows a re-plot of Sreenivasan’s data with $Re$ on
the abscissa and measured $\alpha_{1}/3$ on the ordinate. The data exhibit a
visible tendency to grow with increasing $Re$ to a saturation value which is
statistically indistinguishable from our 0.6.

### Universality and fundamentality;
turbulence, physics, and geometry

Besides problems and tasks posed in engineering and geophysical fluid dynamics
where prediction accuracy and extrapolability matter most, there is a question
posed by natural philosophy: how are universal constants of turbulent motion,
fundamental physics constants like the electron’s elementary charge (see e.g.
Fritzsch, 2009), and mathematical constants like $\pi$ related with each
other?

First, the measurement accuracy of “turbulent numbers” is extremely poor.

Second, “turbulent constants” characterize universal properties of a specific
(turbulent) form of dynamic and self-similar motions rather than more “static”
properties like the elementary charge. They are thus closer to the vacuum
speed of light or the Hubble constant describing the (accelerated) expansion
motion of the universe.

Third, like mathematical constants, “turbulent numbers” are dimensionless
whereas fundamental physics motion constants like the Hubble or the vacuum
speed of light are given in kilometers per second. The latter are absolute
values. “Our” constants characterize self-similar motions202020 If we exclude
from our consideration physical cosmology and arbitrarily chosen ratios
between masses of the various atoms and molecules then physics has actually
only one dimensionless exception: Sommerfeld’s fine-structure constant, with a
value of about $137$. Wolfgang Pauli has long been preoccupied with the
question of why, and Richard Feynman (1985) even speculated about a relation
with $\pi$. Here we have shown that $\pi$ is at least related with constants
of turbulent motion..

This provocates the question whether geometry is part of physics or vice
versa, as discussed for instance by Palais (1981). One might argue that the
discovery of the constant angle sum in triangles was historically the first
discovery of a physical conservation law by man, made and explicitly
formulated much earlier than the conservation laws of volume, mass etc.212121
These thoughts are actually outside the scope of the present report. Some
patience is needed as a “theory of everything” is not in sight Laughlin
(2005), and just-answered questions typically give birth to new conundrums.

### The completed image of turbulent flows

For the sake of clarity we concentrate on the most simple non-trivial
situation, a spatially one-dimensional channel flow with velocity component
$U$ in horizontal ($x$) direction, with vertical variation along $z$, and with
horizontal, $\tilde{u}$, and vertical velocity fluctuations, $\tilde{w}$. An
example of this situation has been presented earlier (see eq. (4.20) and
(4.21) in Baumert, 2005a, where stratification is already considered through
the squared buoyancy frequency, $N^{2}$). The Reynolds decomposition of our
flow field then reads as follows:

$\displaystyle U(z,t)$ $\displaystyle=$ $\displaystyle\langle
U\rangle+\tilde{u}(z,t)\,,$ (29) $\displaystyle W(z,t)$ $\displaystyle=$
$\displaystyle\langle W\rangle+\tilde{w}(z,t)\,.$ (30)

Mean flow: RANS. One may now insert (29, 30) into the corresponding two-
dimensional Euler equation to find together with mass conservation the
following,

$\frac{\partial\langle U\rangle}{\partial
t}+\frac{\partial\langle\tilde{u}\,\tilde{w}\rangle}{\partial
z}=-\frac{\partial\langle p\rangle}{\partial x}\,,$ (31)

where $\langle p\rangle$ is the pressure. Equation (31) is usually called a
Reynolds-averaged Navier-Stokes equation (RANS). Here we take the case of
vanishing molecular viscosity.

Turbulent mixing: downgradient flux.  Equation (31) is not yet closed because
the correlator describing a diffusive flux,
$\langle\tilde{u}\,\tilde{w}\rangle$, needs to be specified, a task which
consumed substantial efforts over the last 60 years. The following quasi-
linear flux-gradient relation is theoretically well established in many
branches of many-particle physics and reads in our case as follows:

$-\,\langle\tilde{u}\,\tilde{w}\rangle\;=\;\nu\,\frac{\partial}{\partial
z}\,\langle U\rangle\,.$ (32)

Kolmogorov-Prandtl relation. However, also (32) does not yet close the problem
because now the so-called turbulent viscosity, $\nu$, needs to be specified.
In our picture the latter is again presented in a many-particle format in
analogy to Einstein’s theory of Brownian motion as follows (for details see
Baumert, 2009):

$\nu=L^{2}/\tau={\cal K}/(\pi\,\Omega)\,.$ (33)

Here $L=r/\kappa$ and $\tau=1/\Omega$ are locally averaged222222For reasons of
transparency the overbars are omitted. space and time scales which are
expressed within the framework of our mechanistic dipol-chaos model in terms
of $\cal K$ and $\omega=2\,\pi\,\Omega$.

Equations of turbulent motion, neutral stratification. We skip here the
technical details of derivations given in Baumert (2009) and quote only the
result that the variables $\cal K$ and $\Omega$ are governed by a specific
system of nonlinear partial differential equations as follows:

$\displaystyle\frac{\partial\cal K}{\partial t}-\frac{\partial}{\partial
z}\left(\nu\frac{\partial\cal K}{\partial z}\right)$ $\displaystyle=$
$\displaystyle\nu\left[\left(\frac{\partial\langle U\rangle}{\partial
z}\right)^{2}-\Omega^{2}\right],$ (34)
$\displaystyle\frac{\partial\Omega}{\partial t}-\frac{\partial}{\partial
z}\left(\nu\,\frac{\partial\Omega}{\partial z}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\left[\frac{1}{2}\left(\frac{\partial\langle
U\rangle}{\partial z}\right)^{2}-\Omega^{2}\right]$ (35)

These equations resemble at least the structure of the so-called $\cal
K$-$\Omega$ closure model used by many authors (loc. cit. Wilcox, 2006). They
use differing empirically gained sets of prefactors of the terms and cannot
identify the physical nature of $\Omega$. Our derivation of (34, 35) should
not be confused with a new “scheme” or closure of this sort because – as we
have shown – our equations are founded solidly on the most simple principles
of hydrodynamics and many-particle physics, without use of phenomenological
data and, for the first time, giving even the universal constants of turbulent
motion. On the other hand, the experience of Wilcox (2006) and colleagues
shows that practical experiments and application needs guide the creative
engineer very close to the physically correct solution, of course, without
guiding further to the universal constants.

Turbulence spectra. With the dissipation rate $\varepsilon$,

$\varepsilon={\cal K}\,\Omega/\pi\,,$ (36)

we may express the spectra in the admissible turbulent ranges
$k=k_{0}\dots\infty$ and $\omega=\omega_{0}\dots\infty$ as follows:

$\displaystyle\frac{d{\cal K}}{dk}$ $\displaystyle=$
$\displaystyle\alpha_{1}\,\varepsilon^{2/3}\,k^{-5/3}\,\;,$ (37)
$\displaystyle\frac{d{\mathcal{K}}}{d\omega}$ $\displaystyle=$
$\displaystyle\beta_{1}\,\varepsilon\;\omega^{-2}\,\;,$ (38)

where turbulence begins at

$\displaystyle k_{0}$ $\displaystyle=$
$\displaystyle\pi/\kappa\,L\;=\;\pi^{2}\,\Omega\sqrt{2/{\cal K}}\;,$ (39)
$\displaystyle\omega_{0}$ $\displaystyle=$
$\displaystyle\Omega/\kappa^{2}\;=\;2\pi\,\Omega\,,$ (40)

respectively. Longer scales or slower motions are possibly turbulent, but not
in the sense of a fully developed spectrum. In the case of slow variations of
the driving forces the motions outside the validity range belong to the mean
flow. Notice that the lower limits of the turbulent spectral ranges are
$dynamic$ quantities which are controlled by the dynamic variables $\cal K$
and $\Omega$.

Universal constants.  The values of our theoretically derived universal
constants of turbulent motion are

$\displaystyle\alpha_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\,(4\,\pi)^{2/3}\;=\;1.802\,,$ (41)
$\displaystyle\beta_{1}$ $\displaystyle=$ $\displaystyle 2\,,$ (42)
$\displaystyle\kappa$ $\displaystyle=$ $\displaystyle
1/\sqrt{2\,\pi}=0.399\,.$ (43)

Solid boundaries. TKE cannot penetrate solid boundaries like walls, which is
sometimes called an ‘adiabatic boundary condition’, in analogy to heat
conduction. Assuming the wall at $z=z_{*}$ this means that the following
condition has to be satisfied:

$\left.\frac{\partial{\cal K}}{\partial z}\right|_{z=z_{*}}=0\,.$ (44)

This is already sufficient to solve (34, 35) at solid walls and produce the
logarithmic law of the wall. A second condition for $\Omega$ is not needed due
to the a.m. nonlinearities of the equations (for details see Baumert, 2005a).

Stable stratification. Stable stratification may be described by the following
modification of (34):

$\frac{\partial\cal K}{\partial t}-\frac{\partial}{\partial
z}\left(\nu\frac{\partial\cal K}{\partial
z}\right)=\nu\left[\left(\frac{\partial\langle U\rangle}{\partial
z}\right)^{2}-2N^{2}-\Omega^{2}\right].$ (45)

This description is valid as long as $\Omega\geq N$. In the case of free decay
and linear stratification, $\Omega$ approaches $N$ “from above”. Due to the
dispersion relation for internal waves, slow disordered turbulent motions at
$\Omega\approx N$ and slower can only exist as waves so that turbulence is
converted into internal waves when $\Omega\rightarrow N$. The stratification
aspects are discussed by Baumert and Peters (2004, 2005) in greater detail and
in relation to observations and measurements.

Limitations. A trivial limitation for (45) is $N^{2}\geq 0$. For unstable or
convective situations these equations are possibly not applicable, but we did
not yet test this case.

Equations (29 – 44) form a closed, complete description of mean and turbulent
motions, including their wavenumber and frequency spectra with their universal
spectral constants, applicable under neutrally stratified conditions. For
stably stratified flows (34) needs to be replaced with (45), provided that
there are $no$ external sources of internal-wave energy. This means that, due
to the ubiquituous presence of internal waves in most geophysical flows, the
latter are $not$ covered by (45). Some modifications are necessary and a
certain knowledge of the sources of internal-wave energy is needed. The
necessary modifications are not part of this report.

The difference between (34, 35) or (45, 35) on the one hand, and traditional
phenomenologically based closure schemes discussed e.g. by Wilcox (2006) on
the other, is mathematically minimal but physically relevant. Those schemes
use for instance quantities like $\varepsilon$ or $\tau$ as primary variables
in balance equations although conservation laws for those quantities do not
exist. The differences are becoming generally relevant and visible at solid
boundaries and, in the stratified case, also within the turbulent fluid
volume.

An important limitation has already been mentioned above. Further, variations
in the mean flow field should be relatively slow and spatial gradients not too
strong so that we may talk about time-dependent but quasi-steady behavior with
sufficiently homogeneous state variables on local scales and sufficiently
developed Kolmogorov spectra. This excludes shock-waves from our
considerations. But alos this limitation has not been tested yet.

The free decay of turbulence might be such a case. Our theory predicts the
decay of TKE to scale with $t^{-1}$ [or with $(x/U)^{-1}$ in the wind tunnel].
This is also supported through a group-theoretical analysis by Oberlack
(2002). Whereas wind-tunnel data Batchelor (1953) and selected free-decay
measurements Dickey and Mellor (1980) agree surprisingly well with our theory,
in some experiments if rapid free decay the exponent is somewhat greater than
unity. This situation is challenging and needs a further study.

We may summarize as follows:

  * •

The present theory is based on conservation laws and geometry only, without
use of empirical data.

  * •

Besides spectrally integrated parameters like turbulent viscosity, dissipation
rate etc., it gives full turbulent spectra, even under non-stationary
conditions.

  * •

The theory predicts fundamental constants of turbulent motion.

  * •

The approach is limited to slow and smooth mean flows under neutral or stable
stratification.

  * •

Internal gravity waves are taken into account only so far as they are
intrinsically coupled with local current shear.

Our methodology differs from past concepts. The latter use specific series
expansions of the Navier-Stokes equation into a system of partial differential
equations for higher and higher moments in the perturbations, with the zeroth
hierarchy element being the Reynolds equation (RANS). The $k$-$\varepsilon$,
the Mellor-Yamada, $k$-$\tau$, the $k$-$\omega$ and the many other turbulence-
closure schemes populating the literature are examples (e.g. Baumert et al.,
2005; Wilcox, 2006).

The present picture includes a perturbation step, too, but it stops at the
zeroth level, at RANS. Here the turbulent fluxes still remain unknown. We then
took into account that each fluid element experiences an effective total force
field consisting additively of the external mean field $\bar{\mathcal{H}}$
acting in the total fluid volume, and of the field $\tilde{\mathcal{H}}$
controlled by nearest neighboring vortex dipoles (“Ising assumption”),

${\mathcal{H}}=\bar{\mathcal{H}}+\tilde{\mathcal{H}}.$ (46)

Our results show that $\tilde{\mathcal{H}}$ is locally best described by a
simple white-noise stochastic force. This approach has successfully been used
about 100 years ago by Langevin, Smoluchowsky and Einstein in theoretical
analyses of Brownian motion. This assumption directly delivers Fokker-Planck
equations for the expectation-value distribution functions of particle number
and of particle properties $\cal K$ and $\Omega$.

The theory applies exclusively to locally homogeneous, locally isotropic and
weakly unsteady turbulent flows. To extreme non-stationarities and/or sharp
spatial gradients like in shockwaves it is possibly not applicable. As a rule,
temporal changes of the mean flow should take place on time scales
sufficiently greater than $1/\Omega$ because otherwise the Kolmogorov spectra
(37, 38) are not yet well enough established.

Coda. The theory completed here will hopefully not form a Procrustean bed in
Saffman’s (1977) sense (loc. cit. Davidson, 2004, p. 107) who even feared that
“in searching for a theory of turbulence, perhaps we are looking for a chimera
…”. This view has just recently been enforced by Hunt (2011) as follows: “But
there are good reasons why the answer to the big question that Landau and
Batchelor raised about whether there is a general theory of turbulence is
probably ‘no’. ”

As the above smacks of defeatism, we pass the word to Sir Winston Churchill
who demanded in a critical situation “We shall never surrender!” He surely
would encourage a new generation of men and women to follow their own fresh
ideas towards a general theory of turbulence at asymptotically high Reynolds
numbers.

###### Acknowledgements.

This report profited substantially from the scientific cooperation between the
author and Hartmut Peters at Earth and Space Research in Seattle, WA, who even
contributed Figures 1 to 3, and major paragraphs of Chapter 3 of the present
report. The cooperation took place within the framework of the Department of
the Navy Grant N62909-10-1-7050 issued by the Office of Naval Research Global.
The United States Government has a royalty-free license throughout the world
in all copyrightable material contained herein. The author is thankful to
Eckhard Kleine who directed the attention to the remarkable works of Hans J.
Herrmann and helped as a robust sparring partner in turbulent debates. Special
thanks are due to Stephen A. Thorpe who critically commented the precursor
version of this report. Thanks are further due to David I. Benjamin, Peter
Braun, Eric D’Asaro, Bruno Eckhardt, Philippe Fraunier, Harald Fritzsch, Boris
Galperin, Rupert Klein, Ren-Chieh Lien, Jim Riley, Gisbert Stoyan, John
Simpson, Jürgen Sündermann, Oleg F. Vasiliev, Michael Wilczek, and Sergej S.
Zilitinkevich. The author thanks particularly the conference organizers of
Turbulent Mixing and Beyond 2011 at the Abdus Salam International Centre of
Theoretical Physics, Trieste/Italy, where the above text has been presented
for the first time. Special thanks are due to Snezhana Abarzhi and Joseph J.
Niemela who managed to create a fair, open and creative scientific atmosphere.

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