7892014201401XXXYY

11institutetext:  Leibniz-Institut für Astrophysik, An der Sternwarte 16,
D-14482 Potsdam, Germany

later

# Mean-field dynamos:
the old concept and some recent developments

K.-H. Rädler
<EMAIL_ADDRESS>
(XX January 2014; YY ZZZ 2014)

###### Abstract

This article reproduces the Karl Schwarzschild lecture 2013. Some of the basic
ideas of electrodynamics and magnetohydrodynamics of mean fields in
turbulently moving conducting fluids are explained. It is stressed that the
connection of the mean electromotive force with the mean magnetic field and
its first spatial derivatives is in general neither local nor instantaneous
and that quite a few claims concerning pretended failures of the mean-field
concept result from ignoring this aspect. In addition to the mean-field dynamo
mechanisms of $\alpha^{2}$ and $\alpha\Omega$ type several others are
considered. Much progress in mean-field electrodynamics and
magnetohydrodynamics results from the test-field method for calculating the
coefficients that determine the connection of the mean electromotive force
with the mean magnetic field. As an important example the memory effect in
homogeneous isotropic turbulence is explained. In magnetohydrodynamic
turbulence there is the possibility of a mean electromotive force that is
primarily independent of the mean magnetic field and labeled as Yoshizawa
effect. Despite of many efforts there is so far no convincing comprehensive
theory of $\alpha$ quenching, that is, the reduction of the $\alpha$ effect
with growing mean magnetic field, and of the saturation of mean-field dynamos.
Steps toward such a theory are explained. Finally, some remarks on laboratory
experiments with dynamos are made.

###### keywords:

Cosmic magnetic fields – mean-field electrodynamics – mean-field
magnetohydrodynamics – mean-field dynamos

##  1 Introduction

At the beginning of the last century mankind knew the magnetic field of the
Earth, but nothing about magnetic fields at other celestial bodies. In 1908
George Ellery Hale proposed to interpret line splittings in the spectrum of
the light coming from sunspots, which were not understood at this time, as a
consequence of strong magnetic fields (of a few kilogauss) within them. Eleven
years later, in 1919, Sir Joseph Larmor came up with the idea that magnetic
fields at the Sun could be generated by self-exciting dynamos just as
introduced in engineering for instance by Ernst Werner von Siemens in 1867
111The idea of the self-exciting dynamo has been stated several years before
by Anyos Jedlik, by Søren Hjorth and by Samuel Alfred Varley, in 1867 also by
Charles Wheatstone. Von Siemens is known for having recognized the practical
importance of the dynamo principle.. Of course, Larmor’s proposal was not
readily accepted and there were many attempts to check it or to rule it out. A
dynamo in a homogeneous fluid is quite different from its technical version
built up with insolated wires. More mathematically spoken, a dynamo working in
a singly-connected conducting region is different from that in a multiply-
connected region. In 1934 Thomas George Cowling proved a theorem which we may
now (after some generalizations) formulate by saying that a dynamo can never
work with an axisymmetric magnetic field. Another important theorem traces
back to investigations by Walter M. Elsasser in 1946 and by Edward C. Bullard
and H. Gellman in 1954 and excludes a dynamo in a sphere due to motions
without radial components. Quite a few modifications of such theorems have
been proven in the course of time showing the impossibility of dynamos with
some simple geometrical structures of the magnetic field or the fluid flow.

In 1947 Horace W. Babcock discovered a star with a strong magnetic field (of
34 kilogauss), and in 1958 he published a catalogue of magnetic stars. Later
we learned about a large number of various objects which exhibit magnetic
fields, including galaxies with rather weak but very extended fields (of the
order of $10^{-5}$ gauss) or neutron stars with extremely strong ones (up to
the order of $10^{15}$ gauss).

A rigorous existence proof for a homogeneous dynamo has been delivered by A.
Herzenberg in 1958. The velocity distribution he assumed was, however, far
away from from that expected in the Earth’s interior, in the Sun or in stars.

Already before Herzenberg’s proof, 1955 and 1957, Eugene N. Parker designed a
model for the Sun in which “cyclonic convection” together with rotational
shear, that is differential rotation, produce a magnetic cycle. In 1964
Stanislaus I. Braginsky published his model of the “nearly symmetric dynamo”
which reflects features of the Earth’s magnetic field.

In the early sixties Max Steenbeck in Jena pushed Fritz Krause and myself to
think about the question of how the Sun or the Earth could generate their
magnetic fields. Many conceivable mechanisms were discussed and investigated
in the course of time. At the end the mean-field electrodynamics of
electrically conducting turbulently moving fluids was established. A central
issue of this theory is the $\alpha$ effect, the occurrence of an
electromotive force with a part parallel (or antiparallel) to the mean
magnetic field as a consequence of induction processes caused by irregular
motions. The $\alpha$ effect allows dynamo action. The first paper on this
topic has been published by Steenbeck, Krause and Rädler in 1966
(unfortunately only in German language). Since then mean-field electrodynamics
and, more general, mean-field magnetohydrodynamics have been elaborated in
great detail and dynamo models have been proposed for the Sun, planets,
several types of stars and for galaxies. In this lecture I would like to
explain a few results of this research field. (For comprehensive presentations
see, e.g., Moffatt 1978, Krause and Rädler 1980 or Brandenburg and Subramanian
2005.)

Before doing so, however, I would like to say: It is a great honor for me to
receive the Karl Schwarzschild Medal. I am very grateful to the Board of the
Astronomische Gesellschaft for this distinction. It is also a great honor to
deliver this Schwarzschild lecture.

Let me start with a few remarks about Max Steenbeck (1904-81) and the place,
Jena, where mean-field electrodynamics was born. Max Steenbeck was no geo- or
astrophysicist. He was one of the great pioneers of plasma physics, worked
until the end of the Second World War in the Siemens Company in Berlin, dealt
there with heavy current technology, for example rectifiers, constructed the
first working betatron etc. At the end of the war he has been interned in the
Soviet Union. He spent there (involuntarily) eleven years, dealing in
particular with the separation of Uranium isotopes in the framework of the
Soviet Atomic program. After his return to the G.D.R. he dealt there with
magnetic materials, with nuclear power stations, and in 1959 he established
the Institute for Magnetohydrodynamics in Jena with the idea to deliver
contributions to nuclear fusion research, which looked at that time very
promising. Sometimes, on his frequent rides between Jena and occasionally on
Saturdays and Sundays, he thought about possibilities of processes in the Sun
or in the Earth’s interior that might produce the observed magnetic fields,
and then attacked Fritz Krause and me with his ideas. Since the Institute for
Magnetohydrodynamics was not a place with astrophysical or geophysical
tradition there was, at least at the beginning, no contact to leading
scientists in these fields.

##  2 Mean-field electrodynamics

###  2.1 The basic idea

In what follows we deal with electromagnetic processes in an electrically
conducting moving fluid. Adopting the magnetohydrodynamic approximation we
assume that the electromagnetic fields obey the pre-Maxwell equations

${\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$E$}}=-\partial_{t}{\mbox{\boldmath$B$}}\,,\quad{\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$B$}}=\mu{\mbox{\boldmath$J$}}\,,\quad{\mbox{\boldmath$\nabla$}}\cdot{\mbox{\boldmath$B$}}=0$
(1)

and Ohm’s law for moving matter in the form

${\mbox{\boldmath$J$}}=\sigma({\mbox{\boldmath$E$}}+{\mbox{\boldmath$U$}}\times{\mbox{\boldmath$B$}})\,.$
(2)

As usual, $E$ denotes the electric field, $B$ the magnetic field, $J$ the
electric current density, and $U$ the fluid velocity; further $\mu$ means the
magnetic permeability of free space and $\sigma$ the electric conductivity of
the fluid. From (1) and (2) we may derive the induction equation

$\eta{\mbox{\boldmath$\nabla$}}^{2}{\mbox{\boldmath$B$}}+{\mbox{\boldmath$\nabla$}}\times({\mbox{\boldmath$U$}}\times{\mbox{\boldmath$B$}})-\partial_{t}{\mbox{\boldmath$B$}}={\bf
0}\,,\quad{\mbox{\boldmath$\nabla$}}\cdot{\mbox{\boldmath$B$}}=0\,,$ (3)

with $\eta=1/\mu\sigma$ being the magnetic diffusivity. For simplicity we
ignore here any electromotive force independent of electromagnetic fields,
which would act as a battery.

Until further notice we consider the fluid velocity $U$ as given. If the
induction equation is solved and so the magnetic field $B$ is known, we may
calculate the electric field $E$ and the electric current density $J$ without
further integrations.

Thinking of the situation in many astrophysical objects, we assume further
that the fluid velocity $U$ and so also the electromagnetic fields $B$, $E$
and $J$ show irregular fluctuations in space and time. Considering these
fluctuations we simply speak of turbulence (without having a specific
definition of turbulence in mind). We then focus attention on mean fields
defined as averages of these fields and showing simpler dependencies on space
and time coordinates. Space or time or statistical averages or combinations of
them are admitted. It is merely important that the Reynolds averaging rules,
already known from hydrodynamic turbulence theory, are (exactly or
approximately) satisfied. We denote averages by overbars. Let $F$ and $G$ be
quantities showing irregular behavior, that is fluctuations, and put
$F=\overline{F}+f$ and $G=\overline{G}+g$. Then these rules read

$\displaystyle\overline{F+G}=\overline{F}+\overline{G}$ (4)
$\displaystyle\overline{\overline{F}}=\overline{F}\;\mbox{or, what is
equivalent,}\;\overline{f}=0$ (5)
$\displaystyle\overline{F\,G}=\overline{F}\,\overline{G}+\overline{f\,g}$ (6)
$\displaystyle\overline{\partial F/\partial x}=\partial\overline{F}/\partial
x\,,\quad\overline{\partial F/\partial t}=\partial\overline{F}/\partial t\,.$
(7)

We stress that the average of the product of two fluctuating quantities is not
equal to the product of the corresponding mean quantities, but there is an
additional term determined by the fluctuations.

Returning to electrodynamics we subject equations (1) and (2) to averaging. We
find then their mean-field versions

${\mbox{\boldmath$\nabla$}}\times\overline{{\mbox{\boldmath$E$}}}=-\partial_{t}\overline{{\mbox{\boldmath$B$}}}\,,\quad{\mbox{\boldmath$\nabla$}}\times\overline{{\mbox{\boldmath$B$}}}=\mu\overline{{\mbox{\boldmath$J$}}}\,,\quad{\mbox{\boldmath$\nabla$}}\cdot\overline{{\mbox{\boldmath$B$}}}=0$
(8)

and

$\overline{{\mbox{\boldmath$J$}}}=\sigma(\overline{{\mbox{\boldmath$E$}}}+\overline{{\mbox{\boldmath$U$}}}\times\overline{{\mbox{\boldmath$B$}}}+{\mbox{\boldmath$\cal{E}$}})\,.$
(9)

When averaging (3) in the same way we obtain

$\displaystyle\\!\\!\\!\\!\eta{\mbox{\boldmath$\nabla$}}^{2}\overline{{\mbox{\boldmath$B$}}}+{\mbox{\boldmath$\nabla$}}\times(\overline{{\mbox{\boldmath$U$}}}\times\overline{{\mbox{\boldmath$B$}}}+{\mbox{\boldmath$\cal{E}$}})-\partial_{t}\overline{{\mbox{\boldmath$B$}}}$
$\displaystyle={\bf 0}\,,$ (10)
$\displaystyle\qquad\qquad\qquad\qquad\qquad{\mbox{\boldmath$\nabla$}}\cdot\overline{{\mbox{\boldmath$B$}}}$
$\displaystyle=0\,.$

$\cal{E}$ is the mean electromotive force due to the fluctuations of the fluid
velocity and the magnetic field,
${\mbox{\boldmath$u$}}={\mbox{\boldmath$U$}}-\overline{{\mbox{\boldmath$U$}}}$
and
${\mbox{\boldmath$b$}}={\mbox{\boldmath$B$}}-\overline{{\mbox{\boldmath$B$}}}$,
that is,

${\mbox{\boldmath$\cal{E}$}}=\overline{{\mbox{\boldmath$u$}}\times{\mbox{\boldmath$b$}}}\,.$
(11)

The form of the equations (8) - (10) agrees widely with that of the original,
not averaged equations (1) - (3). The only, but decisive deviation consists in
the occurrence of the new electromotive force $\cal{E}$.

The crucial point in the elaboration of mean-field electrodynamics is the
determination of that mean electromotive force $\cal{E}$. We first consider
$u$ as given. As for $b$ we may derive from (3) and (10) that

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\eta{\mbox{\boldmath$\nabla$}}^{2}{\mbox{\boldmath$b$}}+{\mbox{\boldmath$\nabla$}}\times(\overline{{\mbox{\boldmath$U$}}}\times{\mbox{\boldmath$b$}}+{\mbox{\boldmath$\epsilon$}})-\partial_{t}{\mbox{\boldmath$b$}}=-{\mbox{\boldmath$\nabla$}}\times({\mbox{\boldmath$u$}}\times\overline{{\mbox{\boldmath$B$}}})$
$\displaystyle\qquad\qquad{\mbox{\boldmath$\epsilon$}}={\mbox{\boldmath$u$}}\times{\mbox{\boldmath$b$}}-\overline{{\mbox{\boldmath$u$}}\times{\mbox{\boldmath$b$}}}\,,\quad{\mbox{\boldmath$\nabla$}}\cdot{\mbox{\boldmath$b$}}=0\,.$
(12)

Clearly $\epsilon$ is the fluctuating part of
${\mbox{\boldmath$u$}}\times{\mbox{\boldmath$b$}}$. Equation (12) tells us
that $b$ is a functional of $u$, $\overline{{\mbox{\boldmath$U$}}}$ and
$\overline{{\mbox{\boldmath$B$}}}$, which is linear (not necessarily linear
and homogeneous) in $\overline{{\mbox{\boldmath$B$}}}$. Consequently,
$\cal{E}$ depends also on these quantities and may be represented as a sum

${\mbox{\boldmath$\cal{E}$}}={\mbox{\boldmath$\cal{E}$}}^{(0)}+{\mbox{\boldmath$\cal{E}$}}^{(B)}$
(13)

of a part ${\mbox{\boldmath$\cal{E}$}}^{(0)}$ independent of
$\overline{{\mbox{\boldmath$B$}}}$ and another part
${\mbox{\boldmath$\cal{E}$}}^{(B)}$ which is is linear and homogeneous in
$\overline{{\mbox{\boldmath$B$}}}$.

Let us assume here that $b$ decays to zero if
$\overline{{\mbox{\boldmath$B$}}}$ vanishes. This implies also the absence of
small-scale dynamos (see section 2.4). Under this assumption
${\mbox{\boldmath$\cal{E}$}}^{(0)}$ decays to zero, too. As a result,
$\cal{E}$ agrees with ${\mbox{\boldmath$\cal{E}$}}^{(B)}$ and it must allow a
representation in the form of the convolution

$\\!{\cal{E}}_{i}({\mbox{\boldmath$x$}},t)=\\!\\!\int_{0}^{\infty}\\!\\!\\!\\!\int_{\infty}\\!\\!\\!\\!{\cal
K}_{ij}({\mbox{\boldmath$x$}},t;{\mbox{\boldmath$\xi$}},\tau)\overline{B}_{j}({\mbox{\boldmath$x$}}+{\mbox{\boldmath$\xi$}},t-\tau)\,\mbox{d}^{3}\xi\,\mbox{d}\tau$
(14)

with some tensorial kernel ${\cal K}_{ij}$, which depends on $u$ and
$\overline{{\mbox{\boldmath$U$}}}$. We know the explicit dependence of ${\cal
K}_{ij}$ on $u$ and $\overline{{\mbox{\boldmath$U$}}}$ only for very special
cases, but conclude from the turbulent nature of the velocity fluctuations
that ${\cal K}_{ij}$ vanishes for sufficiently large
$|{\mbox{\boldmath$\xi$}}|$ and $\tau$. As a consequence, $\cal{E}$ in a given
point in space and time depends only on the behavior of
$\overline{{\mbox{\boldmath$B$}}}$ in a certain surroundings of this point,
the extent of which is determined by the correlation length and the
correlation time of $u$.

It is appropriate to split the kernel ${\cal K}_{ij}$ in (14) into two parts,
one symmetric and the other one antisymmetric in $\xi$, and to express the
last one by a derivative of a tensor symmetric in $\xi$. Doing so and
subjecting then (14) to an integration by parts we arrive easily at the
equivalent representation

$\displaystyle\\!\\!\\!\\!{\cal{E}}_{i}({\mbox{\boldmath$x$}},t)$
$\displaystyle=\int_{0}^{\infty}\\!\\!\\!\int_{\infty}\\!\\!\Big{(}{\cal
A}_{ij}({\mbox{\boldmath$x$}},t;{\mbox{\boldmath$\xi$}},\tau)\overline{B}_{j}({\mbox{\boldmath$x$}}+{\mbox{\boldmath$\xi$}},t-\tau)$
(15) $\displaystyle\;+{\cal
B}_{ijk}({\mbox{\boldmath$x$}},t;{\mbox{\boldmath$\xi$}},\tau)\frac{\partial\overline{B}_{j}({\mbox{\boldmath$x$}}+{\mbox{\boldmath$\xi$}},t-\tau)}{\partial
x_{k}}\Big{)}\mbox{d}^{3}\xi\mbox{d}\tau$

with two tensors ${\cal A}_{ij}$ and ${\cal B}_{ijk}$ which are both symmetric
in $\xi$.

Assume for a moment that $\overline{{\mbox{\boldmath$B$}}}$ varies only weakly
in space and time, that is, there are distinct gaps in the spectra of the
length and time scales of the total magnetic field,
$\overline{{\mbox{\boldmath$B$}}}+{\mbox{\boldmath$b$}}$, separating large and
small scales. We speak then of ideal scale separation. In this case relation
(15) turns into a simpler one,

${\cal{E}}_{i}=a_{ij}\overline{B}_{j}+b_{ijk}\frac{\partial\overline{B}_{j}}{\partial
x_{k}}$ (16)

with

$a_{ij}=\int_{0}^{\infty}\\!\\!\int_{\infty}\\!\\!{\cal
A}_{ij}({\mbox{\boldmath$x$}},t;{\mbox{\boldmath$\xi$}},\tau)\,\mbox{d}^{3}\xi\,\mbox{d}\tau$
(17)

and an analogous connection between $b_{ijk}$ and ${\cal B}_{ijk}$.

While relation (15) connects $\cal{E}$ at a given point in space and time with
$\overline{{\mbox{\boldmath$B$}}}$ in an arbitrary spatial surroundings of
this point and at this time and arbitrary past times, relation (16) describes
a local and instantaneous connection of $\cal{E}$ with
$\overline{{\mbox{\boldmath$B$}}}$ and its first spatial derivatives. The
latter relation, which has to be understood as application of the former one
to idealized cases, explains a large number of phenomena in turbulent fluids,
in particular some types of dynamos. It is, however, unable to capture, e.g.,
memory effects, that is, the dependence of the evolution of a mean field not
only on its current values but also on its history (see section 2.6). In what
follows we will, as long as it is appropriate, refer to (16) but switch to
(15) where necessary. We want to stress that many statements on pretended
failures of mean-field electrodynamics or on allegedly narrow limits of its
applicability result from ignoring (15) and considering instead (16) as the
most general relation for the mean electromotive force $\cal{E}$.

Let us finally mention the technical issue that convolutions like (15) turn
under a proper Fourier transformation (or, concerning the time, also a Laplace
transformation) into simpler algebraic relations. Ignore for simplicity any
dependence of ${\cal A}_{ij}$ and ${\cal B}_{ijl}$ on $x$ and $t$. With a
transformation

$\displaystyle F({\mbox{\boldmath$x$}},t)$
$\displaystyle=(2\pi)^{-4}\int\\!\\!\int\hat{F}({\mbox{\boldmath$k$}},\omega)$
(18)
$\displaystyle\qquad\qquad\qquad\exp\big{(}\mbox{i}({\mbox{\boldmath$k$}}\cdot{\mbox{\boldmath$x$}}-\omega
t)\big{)}\mbox{d}^{3}k\,\mbox{d}\omega$

relation (15) turns then into

$\displaystyle\hat{\cal{E}}_{i}({\mbox{\boldmath$k$}},\omega)$
$\displaystyle=\hat{\cal
A}_{ij}({\mbox{\boldmath$k$}},\omega)\,\hat{\overline{B}}_{j}({\mbox{\boldmath$k$}},\omega)$
(19) $\displaystyle\qquad\qquad\qquad+\mbox{i}\,\hat{\cal
B}_{ijk}({\mbox{\boldmath$k$}},\omega)\,\hat{\overline{B}}_{j}({\mbox{\boldmath$k$}},\omega)k_{k}\,.$

###  2.2 A simple example

We consider first the case in which no mean flow exists,
$\overline{{\mbox{\boldmath$U$}}}={\bf 0}$, and the velocity fluctuations $u$
correspond to a homogeneous isotropic turbulence. As for the mean magnetic
field $\overline{{\mbox{\boldmath$B$}}}$ we assume ideal scale separation in
the sense explained above so that (16) applies.

We define homogeneity of the turbulence by the invariance of all averaged
quantities depending on $u$ under arbitrary translations of the $u$ field, and
isotropy by the invariance of all such quantities under arbitrary rotations of
this field about arbitrary axes. We may also fix the $u$ field and subject the
coordinate system in which we describe it to translations or rotations. Then
homogeneity and isotropy occur as invariance of all averaged quantities under
arbitrary translations and arbitrary rotations of the coordinate system. In
particular the tensors $a_{ij}$ and $b_{ijk}$ that occur in (16) have to show
these properties, that is, their components have to be independent of space
coordinates and independent of rotations of the coordinate system. This
qualifies them as space-independent isotropic tensors, that is, they differ
only by simple factors, say $\alpha$ and $\beta$, from the Kronecker tensor
$\delta_{ij}$ and the Levi-Civita tensor $\epsilon_{ijk}$, that is,
$a_{ij}=\alpha\,\delta_{ij}$ and $b_{ijk}=\beta\epsilon_{ijk}$. In this way we
arrive at

${\mbox{\boldmath$\cal{E}$}}=\alpha\,\overline{{\mbox{\boldmath$B$}}}-\beta\,{\mbox{\boldmath$\nabla$}}\times\overline{{\mbox{\boldmath$B$}}}\,,$
(20)

and the mean-field version (9) of Ohm’s law takes the form

$\overline{{\mbox{\boldmath$J$}}}=\sigma_{\rm
m}(\overline{{\mbox{\boldmath$E$}}}+\alpha\overline{{\mbox{\boldmath$B$}}})$
(21)

with the mean-field conductivity $\sigma_{\rm m}$ given by

$\sigma_{\rm m}=\sigma/(1+\mu\sigma\beta)\,.$ (22)

While $\alpha$ is a pseudoscalar, $\beta$ is a scalar.

Homogeneity and isotropy of turbulence do not exclude reflexional symmetry. We
define it by the invariance of all averaged quantities depending on $u$ under
reflexion of the $u$ field at a point. In the case of homogeneity and isotropy
this is equivalent to reflexions at any plane. A reflexion turns a right-
handed structure in the flow field in a left-handed one and vice versa.
Reflexional symmetry in the above sense implies therefore an equipartition of
right-handed and left-handed structures in a fluid flow, that is, the absence
of any preferred handedness. A simple consequence is, e.g., that the mean
kinetic helicity
$\overline{{\mbox{\boldmath$u$}}\cdot({\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$u$}})}$
vanishes. In the case of homogeneous isotropic ref lexionally symmetric
turbulence we may easily show that the pseudoscalar $\alpha$ in (20) and (21)
has to be zero. There is, however, no restriction to the scalar $\beta$.

As long as there are no deviations of the underlying homogeneous isotropic
turbulence from reflexional symmetry the mean-field version of Ohm’s law reads
simply $\overline{{\mbox{\boldmath$J$}}}=\sigma_{\rm
m}\overline{{\mbox{\boldmath$E$}}}$ with $\sigma_{\rm m}$ as given by (22).
The insight, that for mean fields a conductivity different from that relevant
for the original fields applies, can already be found in papers by Sweet
(1950) and Csada (1951). As we will see later (section 2.5) the ratio
$\sigma_{\rm m}/\sigma$ can be much bigger than unity. In the solar convection
zone, e.g., it may take values of the order of $10^{4}$, what explains in
particular the observed life times of sunspots.

Turbulence in rotating bodies, on which we want to focus our attention later,
is subject to the Coriolis force. It deviates therefore not only from isotropy
but also from reflexional symmetry. (This corresponds to the fact that the
Coriolis force is determined, e.g., by a right-hand rule.) Preparing
investigations of this complex situation, we consider here first the more or
less academic case of homogeneous isotropic but not reflexionally symmetric
turbulence, in which (20) and (21) with $\alpha\not=0$ apply. The occurrence
of the electromotive force $\alpha\overline{{\mbox{\boldmath$B$}}}$ is called
“$\alpha$ effect”. It has been first considered by Steenbeck, Krause and
Rädler (1966) in a slightly different context (see section 2.3).

The $\alpha$ effect allows growing mean magnetic fields, that is, dynamo
action. In order to show this we consider the mean-field induction equation
(10) with $\overline{{\mbox{\boldmath$U$}}}={\bf 0}$ and $\cal{E}$ specified
according to (20), that is,

$\eta_{\rm
m}{\mbox{\boldmath$\nabla$}}^{2}\overline{{\mbox{\boldmath$B$}}}+\alpha{\mbox{\boldmath$\nabla$}}\times\overline{{\mbox{\boldmath$B$}}}-\partial_{t}\overline{{\mbox{\boldmath$B$}}}={\bf
0}\,,\quad{\mbox{\boldmath$\nabla$}}\cdot\overline{{\mbox{\boldmath$B$}}}=0\,,$
(23)

where $\eta_{\rm m}$ is the magnetic mean-field diffusivity,

$\eta_{\rm m}=\eta+\beta\,.$ (24)

Simple particular solutions $\overline{{\mbox{\boldmath$B$}}}$ of (23) read

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\overline{{\mbox{\boldmath$B$}}}$
$\displaystyle=B_{0}(\cos kz,\pm\sin kz,0)\exp(\lambda t)\,,$ (25)
$\displaystyle\qquad\qquad\qquad\quad\lambda=-(\eta+\beta)k^{2}\pm\alpha k\,,$

with a wave number $k$, which we consider as positive, and a growth rate
$\lambda$. Growing solutions, i.e., such with $\lambda>0$, are possible as
soon as $|\alpha|>(\eta+\beta)\,k$. We will see later (section 2.5) that this
condition can well be satisfied.

###  2.3 A more realistic example

In a next, somewhat more realistic case, we consider turbulence on a rotating
body. We assume that for a co-rotating observer no mean flow exists,
$\overline{{\mbox{\boldmath$U$}}}={\bf 0}$, but admit slight deviations of the
turbulence, $u$, from homogeneity and isotropy. We further assume that the
inhomogeneity can be described by a vector $g$, e.g., the intensity gradient
${\mbox{\boldmath$\nabla$}}\overline{u^{2}}$ of the turbulence. The anisotropy
depends, of course, apart from $g$ also on the angular velocity $\Omega$ which
defines the Coriolis force. Again, we restrict ourselves to sufficiently small
variations of the mean magnetic field $\overline{{\mbox{\boldmath$B$}}}$ in
space and time so that (16) applies. Considering the deviations of the
turbulence from homogeneity and isotropy as sufficiently small, we assume that
$a_{ij}$ and $b_{ijk}$ are linear in both $g$ and $\Omega$. Studying then the
possible tensorial structures of $a_{ij}$ and $b_{ijk}$ we find

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\mbox{\boldmath$\cal{E}$}}=\alpha_{1}({\mbox{\boldmath$g$}}\cdot{\mbox{\boldmath$\Omega$}})\overline{{\mbox{\boldmath$B$}}}+\alpha_{2}{\mbox{\boldmath$g$}}({\mbox{\boldmath$\Omega$}}\cdot\overline{{\mbox{\boldmath$B$}}})+\alpha_{3}{\mbox{\boldmath$\Omega$}}({\mbox{\boldmath$g$}}\cdot\overline{{\mbox{\boldmath$B$}}})+\gamma{\mbox{\boldmath$g$}}\times\overline{{\mbox{\boldmath$B$}}}$
$\displaystyle\\!\\!-\beta{\mbox{\boldmath$\nabla$}}\times\overline{{\mbox{\boldmath$B$}}}-\delta{\mbox{\boldmath$\Omega$}}\times({\mbox{\boldmath$\nabla$}}\times\overline{{\mbox{\boldmath$B$}}})-\delta_{*}{\mbox{\boldmath$\nabla$}}({\mbox{\boldmath$\Omega$}}\cdot\overline{{\mbox{\boldmath$B$}}})$
(26)

with scalar coefficients $\alpha_{1}$, $\alpha_{2}$, … $\delta_{*}$
independent of $g$ and $\Omega$.

The first line of (26) reproduces the momentous result by Steenbeck, Krause
and Rädler (1966). Due to the Coriolis force there is now (at least locally) a
preference of either right-handed or left-handed helical patterns in the
turbulent flow. As a consequence the three contributions to $\cal{E}$ with the
coefficients $\alpha_{1}$, $\alpha_{2}$ and $\alpha_{3}$ occur in (26). The
term
$\alpha_{1}({\mbox{\boldmath$g$}}\cdot{\mbox{\boldmath$\Omega$}})\overline{{\mbox{\boldmath$B$}}}$
corresponds to $\alpha\overline{{\mbox{\boldmath$B$}}}$ in relation (20), that
is, in the case of homogeneous isotropic turbulence lacking reflexional
symmetry. However, the $\alpha_{1}$ term is now accompanied by two others, the
$\alpha_{2}$ and $\alpha_{3}$ terms. We speak here again of an $\alpha$
effect, more precisely, if all three terms are considered, of an anisotropic
$\alpha$ effect. On a spherical body, on which $g$ points in radial direction,
$\alpha_{1}({\mbox{\boldmath$g$}}\cdot{\mbox{\boldmath$\Omega$}})$ changes its
sign when moving from the northern hemisphere to the southern one. The
$\alpha$ effect as considered here is crucial for special types of mean-field
dynamo mechanisms (see section 2.4).

The $\gamma$ term in (26) describes the transport of mean magnetic flux by
inhomogeneous turbulence. This effect has been first, for a two-dimensional
turbulence, considered by Zeldovich (1956), later in a more general context by
Rädler (1968a,b). It has been discussed as “pumping of mean magnetic flux” or
(since the mean magnetic flux is expelled from regions of high turbulence
intensity) as “turbulent diamagnetism”.

The $\beta$ term corresponds to that which occurred already in the case of
homogeneous isotropic turbulence, that is in (20), and gives rise to introduce
the mean-field conductivity in the mean-field version of Ohm’s law or the
mean-field diffusivity in the mean-field induction equation.

The effect described by the $\delta$ term in (26) has been first considered by
Rädler (1969a,b). It is often called
“${\mbox{\boldmath$\Omega$}}\times{\mbox{\boldmath$J$}}$ effect”, in what
follows also simply “$\delta$ effect”. Combined with mean shear it is capable
of dynamo action (see section 2.4). Other than the $\alpha$ effect, the
$\delta$ effect requires spatial variations of the mean magnetic field; it
does not occur with a homogeneous field. Apart from this it is in a sense
simpler than the $\alpha$ effect. It needs no gradient of the turbulence
intensity but occurs already with homogeneous turbulence. The sum of the
$\beta$ and $\delta$ terms can also be described by an anisotropic mean-field
conductivity with a non-symmetric conductivity tensor. The $\delta_{*}$ term
is of minor importance. It does not influence the mean-field induction
equation as long as $\delta_{*}$ is spatially constant.

###  2.4 Mean-field dynamo mechanisms

When discussing dynamo mechanisms due to turbulent motions we focus attention
on mean-field dynamos. They are characterized by the ability to generate
magnetic fields with length scales much larger than the typical length scales
of the turbulent motions. Therefore we call them also “large-scale dynamos”.
In this context we should have in mind the finding by Kazantsev (1968)
according to which a homogenous isotropic turbulence, which needs not to
deviate from reflexional symmetry, may under certain conditions maintain
irregular magnetic fields with length scales smaller than or equal to those of
the turbulent motion, which therefore do not contribute to a mean magnetic
field. We speak then of a “small-scale dynamo”. The influence of large-scale
on small-scale dynamos and vice versa is an interesting subject (see, e.g.,
Brandenburg et al. 2012), which we however do not want to discuss here.

Let us consider dynamos due to turbulence on an axisymmetric rotating fluid
body. We restrict attention first to mean magnetic fields that are symmetric
about the axis of rotation. Each such field can be split into a poloidal part
that lies completely in meridional planes, and a toroidal part perpendicular
to them. Within this frame, dynamos can always be understood as an interplay
between the poloidal and the toroidal part of the mean magnetic field. We
admit here a mean motion in the form of differential rotation, that is, a
dependence of the corresponding angular velocity $\Omega$ on radius or
latitude. In addition to induction effects of turbulent motions described,
e.g., by $\alpha$ or $\delta$ effects, we have then also the effect of
rotational shear, which we call “$\Omega$ effect”. While $\alpha$ and $\delta$
effect generate poloidal magnetic fields from toroidal ones and vice versa,
the $\Omega$ effect generates only a toroidal field from a poloidal one.

The simplest mean-field dynamo mechanism is that of $\alpha^{2}$ type, in
which both the generation of the poloidal field from the toroidal one as well
as that of the toroidal field from the poloidal one is due the $\alpha$
effect. The first spherical models of this type were proposed by Steenbeck and
Krause (1969b), and were on the basis of numerical simulations discussed in
view of the Earth and the planets. If we admit in addition the $\Omega$ effect
and assume that it dominates the generation of the toroidal field, we arrive
at dynamos of the $\alpha\Omega$ type. Models of this type have been
investigated also by Steenbeck and Krause (1969a) and applied to explain
essential features of the magnetic solar cycle. The case in which both the
$\alpha$ and the $\Omega$ effect contribute markedly to the generation of the
toroidal field is often labeled as mechanism of $\alpha^{2}\Omega$ type. In
the course of time, a large number of dynamo models of the mentioned types
have been investigated (see, e.g., Rädler 1986).

Mean-field dynamos require not necessarily the $\alpha$ effect. It is easy to
see that there is no dynamo due to the $\delta$ effect alone. However, the
combination of the $\delta$ and $\Omega$ effects is, as demonstrated by Rädler
(1969b, 1976), capable of dynamo action. Meanwhile there are several results
for dynamo models of that type (see, e.g, Rädler 1974, 1986). Also the
combination of another effect which can be described as an anisotropy of the
mean-field conductivity with the $\Omega$ effect can lead to a dynamo (Rädler
1986).

So far we focussed attention on dynamos with axisymmetric mean fields, which
allow a simple description. The mechanisms mentioned here work, however, also
with non-axisymmetric mean fields, and there are quite a few cases in which
such fields are easier to excite than axisymmetric ones (see, e.g., Rädler
1986).

Rogachevskii and Kleorin (2004) claimed that the induction effects that occur
in a turbulent fluid under the influence of a global shear, which are similar
to those in the $\delta\Omega$ mechanism, are also capable of dynamo action,
and they spoke of a “shear-current dynamo”. They calculated the relevant mean-
field coefficients however in a defeasible approximation. Several
investigations on this ”shear-current dynamo” have been carried out, but no
reliable proof for its existence has been given so far.

It has been shown for a wide range of assumptions that the magnetic mean-field
diffusivity $\eta_{\rm m}=\eta+\beta$ is positive so that a growth of a mean
magnetic field due to negative mean diffusivity can be excluded. There is
however a very recent result which might provoke scruples in this respect.
Some properties of mean-field dynamos are reflected by mean-field models
derived from the dynamos investigated by Roberts (1972), working not with
turbulence but with regular three-dimensional flows periodic in, say, $x$ and
$y$ and independent of $z$. It has been shown very recently by Devlen et al.
(2013) that in one of these mean-field models growing mean magnetic fields are
generated with no other mean-field induction effect than a negative mean-field
diffusivity. It remained open whether this result can be extended to mean-
field dynamos working with real turbulence.

###  2.5 Calculation of mean-field coefficients

####  2.5.1 Second-order correlation approximation

So far we have have considered connections of the mean electromotive force
$\cal{E}$ with the mean magnetic field $\overline{{\mbox{\boldmath$B$}}}$ and
its spatial derivatives that are defined by coefficients like $\alpha$,
$\beta$, $\alpha_{1}$ or $\delta$, but nothing has been said about their
actual values or their dependence on the magnitude or other properties of the
fluctuating motions. In the early days of mean-field electrodynamics many
calculations of $\cal{E}$ were carried out on the basis of equation (12) for
the magnetic fluctuations $b$ but with the $\epsilon$ term canceled. This is
an approximation that can be justified for sufficiently small velocity
fluctuations $u$ only, often called “second-order correlation approximation”
(SOCA) or “first-order smoothing approximation” (FOSA) or “quasilinear
approximation”.

Consider as an example again the case in which the mean velocity
$\overline{{\mbox{\boldmath$U$}}}$ vanishes and the fluctuating velocity $u$
corresponds to a homogeneous isotropic turbulence. Restrict attention further
to small variations of the mean magnetic field
$\overline{{\mbox{\boldmath$B$}}}$ in space and time, that is, ideal scale
separation, and to the in applications most important high-conductivity limit,
defined by $\eta\tau_{\rm c}/\lambda_{\rm c}^{2}\ll 1$, where $\tau_{\rm c}$
and $\lambda_{\rm c}$ are correlation time and correlation length of the
turbulent motions. Within the frame of SOCA we find then

$\displaystyle\alpha$
$\displaystyle=-{\textstyle\frac{1}{3}}\int_{0}^{\infty}\overline{{\mbox{\boldmath$u$}}(t)\cdot({\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$u$}}(t-\tau))}\,\mbox{d}\tau$
(27) $\displaystyle\beta$
$\displaystyle={\textstyle\frac{1}{3}}\int_{0}^{\infty}\overline{{\mbox{\boldmath$u$}}(t)\cdot{\mbox{\boldmath$u$}}(t-\tau)}\,\mbox{d}\tau\,.$

We may write this also in the form

$\alpha=-{\textstyle{1\over
3}}\overline{{\mbox{\boldmath$u$}}\cdot({\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$u$}})}\,\tau^{(\alpha)}\,,\quad\beta={\textstyle{1\over
3}}\overline{u^{2}}\,\tau^{(\beta)}$ (28)

with $\tau^{(\alpha)}$ and $\tau^{(\beta)}$ defined by equating the
corresponding expressions in (27) and (28). Under reasonable assumptions both
$\tau^{(\alpha)}$ and $\tau^{(\beta)}$ are approximately equal to $\tau_{\rm
c}$. These results are in many respects instructive. In the high-conductivity
limit considered here, however, the application of SOCA can only readily be
justified if the Strouhal number $St=u_{\rm c}\tau_{\rm c}/\lambda_{\rm c}$,
with $u_{\rm c}$ being a characteristic value of $u$, is small compared with
unity. In realistic cases of turbulence it is close to unity.

It is well possible to proceed from the second-order approximation to a third-
order one with $\epsilon$ expressed by second-order results, then to a fourth-
order one with $\epsilon$ expressed by third-order results etc., and it has
been proven that this procedure converges (Krause 1968). Analytic calculations
of that kind are however very tedious and, apart from a few fourth-order
results, no results of practical interest have been gained in this way.

####  2.5.2 Test-field method

Several other techniques for obtaining results for mean-field coefficients
have been proposed, using assumptions which look to a certain extent plausible
but cannot be justified in a clean way (for a critical review see, e.g.,
Rädler and Rheinhardt 2007). In recent years, with growing possibilities of
numerical calculations, the “test-field method”, established immediately on
the basic equations, brought much progress in the reliable determination of
mean-field coefficients.

The method was developed by Schrinner et al. (2005, 2007) in the context of
this task: Consider a simple geodynamo model, with the magnetic field
maintained by convection. Define mean fields by averaging over the azimuthal
coordinate; they are then axisymmetric. Extract the mean-field coefficients
from the numerical results for this model. Construct a mean-field model with
these coefficients. Compare then the mean fields obtained from the original
model by direct numerical simulations with those obtained from the mean-field
model. In the ideal case they should agree with each other.

Let us sketch the idea of the test-field method for the case of the simple
connection of the mean electromotive force $\cal{E}$ with
$\overline{{\mbox{\boldmath$B$}}}$ and its first spatial derivatives as given
by (16). We choose a set of test fields $\overline{{\mbox{\boldmath$B$}}}^{\rm
T}$ and replace $\overline{{\mbox{\boldmath$B$}}}$ in (12) consecutively by
each of its elements, calculate the corresponding ${\mbox{\boldmath$b$}}^{\rm
T}$ and finally ${\mbox{\boldmath$\cal{E}$}}^{\rm
T}=\overline{{\mbox{\boldmath$u$}}\times{\mbox{\boldmath$b$}}^{\rm T}}$. These
${\mbox{\boldmath$\cal{E}$}}^{\rm T}$ have to obey

$a_{ij}\overline{B}_{j}^{\rm T}+b_{ijk}\partial\overline{B}_{j}^{\rm
T}/\partial x_{k}={\cal{E}}_{i}^{\rm T}\,.$ (29)

With a sufficient number of independent $\overline{{\mbox{\boldmath$B$}}}^{\rm
T}$ we obtain a system of equations which allows us the determination of the
$a_{ij}$ and $b_{ijk}$ from the ${\cal{E}}_{i}^{\rm T}$ calculated for the
chosen set of $\overline{{\mbox{\boldmath$B$}}}^{\rm T}$. It turned out that
there is a high degree of freedom in the choice of the test fields. They need
not to be solenoidal and have not to satisfy specific boundary conditions.

Let us return once more to the coefficients $\alpha$ and $\beta$ for
homogeneous isotropic turbulence. Referring to numerical simulations of
hydrodynamic turbulence in a weakly compressible fluid, Sur et al. (2008) used
the test-field method for the determination of these coefficients. The
turbulence was specified to have an energy input at a wavenumber $k_{\rm f}$,
and to be maximally helical, that is,
$\overline{({\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$u$}})^{2}}/\overline{{\mbox{\boldmath$u$}}^{2}}=k_{\rm
f}^{2}$. Calculations with different values of the hydrodynamic Reynolds
number $Re=u_{\rm rms}/\nu k_{\rm f}$, where $\nu$ means the kinematic
viscosity, were carried out. In Figs.1 and 2 some results for
$\alpha/\alpha_{0}$ and $\beta/\beta_{0}$ with $\alpha_{0}=-{\textstyle{1\over
3}}u_{\rm rms}$ and $\beta_{0}={\textstyle{1\over 3}}u_{\rm rms}/k_{\rm f}$
are shown in dependence on the magnetic Reynolds number $Rm=u_{\rm rms}/\eta
k_{\rm f}$. In the turbulence considered here the Strouhal number $St$ turned
out to be of the order of unity. So the reported results confirm that (27) and
(28), which were derived for $St\ll 1$ only, apply also with realistic values
of $St$.

The test-field method for the determination of the mean-field coefficients
brought much progress in mean-field electrodynamics and beyond. It has been
extended to a very broad range of assumptions, is in particular not limited to
cases with scale separation (see, e.g., Brandenburg et al. 2008, Rheinhardt
and Brandenburg 2010, 2012).

Figure 1:  Normalized mean-field coefficients $\alpha/\alpha_{0}$ and
$\beta/\beta_{0}$ as functions of $Rm$, obtained in test-field calculations by
Sur et al. (2009) based on turbulence simulations with $Re=2.2$ Figure 2:
Same as Fig.1 but simulations with $Re=10\,Rm$

###  2.6 Imperfect scale separation

####  2.6.1 Apparent discrepancies

In the examples considered so far we have reduced the general representations
(14) or (15) of the mean electromotive force $\cal{E}$ as a convolution
depending on the mean magnetic field $\overline{{\mbox{\boldmath$B$}}}$ in all
space and at the current time and all past times, to the simple local and
instantaneous connection (16) of $\cal{E}$ with
$\overline{{\mbox{\boldmath$B$}}}$ and its first spatial derivatives. On this
level the theory may deliver incomplete or even wrong statements.

One example for that is the aforementioned incomplete agreement of the mean-
field geodynamo models derived immediately from the basic equations and those
constructed with mean-field coefficients determined by the simple version of
the test-field method, which considers only local and instantaneous
connections of $\cal{E}$ with $\overline{{\mbox{\boldmath$B$}}}$ and its first
derivatives (section 2.5.2).

####  2.6.2 Memory effect

Another interesting example concerns the growth of a mean magnetic field in a
turbulence showing $\alpha$ effect. As Hubbard and Brandenburg (2009) pointed
out, the growth rates obtained in direct numerical simulations clearly differ
from those derived from a dispersion relation with mean-field coefficients
gained in a static approximation, that is, assuming an instantaneous
connection of $\cal{E}$ and $\overline{{\mbox{\boldmath$B$}}}$ as in (16). The
difference disappears if a proper connection of $\cal{E}$ at a given time with
$\overline{{\mbox{\boldmath$B$}}}$ at former times, that is, some memory of
the turbulent system, is taken into account. We know meanwhile many examples
in which such memory effects play an important role and can even be crucial
for the existence of dynamos (Rheinhardt et al. 2014).

For an illustration of the memory effect, Hubbard and Brandenburg (2009)
considered a Roberts flow instead of a real turbulence. They assumed
${\mbox{\boldmath$u$}}\\!=\\!-{\mbox{\boldmath$e$}}\times{\mbox{\boldmath$\nabla$}}\psi+k_{\rm
f}\psi\,{\mbox{\boldmath$e$}}$ and $\psi=(u_{0}/k_{0})\cos k_{0}x\cos k_{0}y$,
where $e$ means the unit vector in $z$ direction and $u_{0}$, $k_{\rm f}$ and
$k_{0}$ are constants, and they restricted attention on the case of a maximal
modulus of the relative helicity
$\overline{{\mbox{\boldmath$u$}}\cdot({\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$u$}})}/\overline{{\mbox{\boldmath$u$}}^{2}}k_{\rm
f}$, which occurs with $k_{\rm f}=\sqrt{2}k_{0}$. They further defined mean
fields by averaging over all $x$ and $y$. Fig 3 shows the normalized growth
rates $\lambda/\lambda_{0}$, with $\lambda_{0}=u_{\rm rms}k_{\rm f}$, as
functions of $Rm$, obtained (i) in direct numerical simulations and (ii) from
the dispersion relation with mean-field coefficients determined in a static
approximation. Note the substantial deviations of the two results for large
$Rm$.

Figure 3:  Normalized growth rates $\lambda/\lambda_{0}$ of a mean magnetic
field in a Roberts flow as functions of $Rm$, (i) obtained in direct numerical
simulations and (ii) calculated from a dispersion relation with mean-field
coefficients determined in a static approximation, according to Hubbard and
Brandenburg (2009)

##  3 Mean-field magnetohydrodynamics

###  3.1 Momentum balance and consequences

So far the fluid velocity has been considered as prescribed. We now relax this
assumption and use in addition to the electromagnetic equations (1) and (2),
or the induction equation (3), also the momentum balance. For the sake of
simplicity we restrict ourselves to an incompressible fluid. Admitting a
rotating frame of reference we have then

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\varrho(\partial_{t}{\mbox{\boldmath$U$}}+({\mbox{\boldmath$U$}}\cdot{\mbox{\boldmath$\nabla$}}){\mbox{\boldmath$U$}})=-{\mbox{\boldmath$\nabla$}}P+\varrho\nu{\mbox{\boldmath$\nabla$}}^{2}{\mbox{\boldmath$U$}}-2\varrho{\mbox{\boldmath$\Omega$}}\times{\mbox{\boldmath$U$}}$
$\displaystyle\quad+(1/\mu)({\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$B$}})\times{\mbox{\boldmath$B$}}+{\mbox{\boldmath$F$}}\,,\quad{\mbox{\boldmath$\nabla$}}\cdot{\mbox{\boldmath$U$}}=0\,.$
(30)

Here $\varrho$ means the mass density, $\nu$ again the kinematic viscosity of
the fluid, and $P$ the hydrodynamic pressure. The angular velocity $\Omega$
defines the rotation of the frame and so the Coriolis force, and $F$ stands
for any external force. The inertial term in (30) is balanced by the pressure
gradient, the viscous force, the Coriolis force, the Lorentz force and
possibly some external force.

Let us focus attention again on turbulent situations. Taking then the average
not only of the induction equation (3) but also of the momentum balance (30),
we find in addition to the mean-field induction equation (10) with the mean
electromotive force $\cal{E}$ given by (11) the mean-field version of the
momentum balance,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\varrho(\partial_{t}\overline{{\mbox{\boldmath$U$}}}+(\overline{{\mbox{\boldmath$U$}}}\cdot{\mbox{\boldmath$\nabla$}})\overline{{\mbox{\boldmath$U$}}})=-{\mbox{\boldmath$\nabla$}}\overline{P}+\varrho\nu{\mbox{\boldmath$\nabla$}}^{2}\overline{{\mbox{\boldmath$U$}}}-2\varrho{\mbox{\boldmath$\Omega$}}\times\overline{{\mbox{\boldmath$U$}}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!+(1/\mu)({\mbox{\boldmath$\nabla$}}\times\overline{{\mbox{\boldmath$B$}}})\times\overline{{\mbox{\boldmath$B$}}}+\overline{{\mbox{\boldmath$F$}}}+{\mbox{\boldmath$\cal{F}$}}\,,\quad{\mbox{\boldmath$\nabla$}}\cdot\overline{{\mbox{\boldmath$U$}}}=0\,,$
(31)

with a mean ponderomotive force $\cal{F}$,

${\mbox{\boldmath$\cal{F}$}}=-\varrho\overline{({\mbox{\boldmath$u$}}\cdot{\mbox{\boldmath$\nabla$}}){\mbox{\boldmath$u$}}}+(1/\mu)\overline{({\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$b$}})\times{\mbox{\boldmath$b$}}}\,.$
(32)

If we ignore the magnetic field we return to pure hydrodynamics. The mean
ponderomotive force $\cal{F}$ covers then, for example, the contribution of
the turbulence to the mean-field viscosity, often discussed as eddy viscosity,
further the $\Lambda$ effect, which, on a rotating body, may drive
differential rotation (Rüdiger 1989), or the anisotropic kinetic $\alpha$
effect (AKA effect, Frisch et al. 1987). We do not want to discuss these
subjects here but focus attention on cases with magnetic field.

Generalizing the considerations on the mean electromotive force $\cal{E}$
explained above, we see that both the electromotive force $\cal{E}$ and the
ponderomotive force $\cal{F}$ may be considered as functionals of fluctuations
${\mbox{\boldmath$u$}}^{(0)}$ and ${\mbox{\boldmath$b$}}^{(0)}$, for example
relating to the case of vanishing mean motion and mean magnetic field, of the
mean velocity $\overline{{\mbox{\boldmath$U$}}}$, the mean magnetic field
$\overline{{\mbox{\boldmath$B$}}}$ and their first spatial derivatives and
also of the angular velocity $\Omega$ that determines the Coriolis force.
These functionals are not necessarily linear in
$\overline{{\mbox{\boldmath$U$}}}$, $\overline{{\mbox{\boldmath$B$}}}$ or
$\Omega$.

Focussing attention on the mean electromotive force $\cal{E}$ we write once
again
${\mbox{\boldmath$\cal{E}$}}={\mbox{\boldmath$\cal{E}$}}^{(0)}+{\mbox{\boldmath$\cal{E}$}}^{(B)}$,
with a part ${\mbox{\boldmath$\cal{E}$}}^{(0)}$ independent of the mean
magnetic field $\overline{{\mbox{\boldmath$B$}}}$. In our short presentation
of mean-field electrodynamics we argued, considering purely hydrodynamic
background turbulence, that the part ${\mbox{\boldmath$\cal{E}$}}^{(0)}$ will
always decay to zero. Now we may no longer exclude magnetohydrodynamic
turbulence, and then this is no longer necessarily the case. A non-zero part
${\mbox{\boldmath$\cal{E}$}}^{(0)}$ of $\cal{E}$ corresponds to a battery. If
such a part exists, the mean-field induction equation is no longer homogeneous
in the mean magnetic field $\overline{{\mbox{\boldmath$B$}}}$ and has always
non-decaying solutions. In the absence of conditions that allow a mean-field
dynamo the magnitude of the corresponding mean magnetic fields is determined
by ${\mbox{\boldmath$\cal{E}$}}^{(0)}$. If a dynamo is possible, they may act
as seed fields.

###  3.2 Yoshizawa effect

An interesting example of a contribution to the mean electromotive force
independent of the mean magnetic field, ${\mbox{\boldmath$\cal{E}$}}^{(0)}$,
has been given by Yoshizawa in 1990. Let us consider magnetohydrodynamic
turbulence in the presence of a mean flow or on a rotating body, that is,
under the influence of the Coriolis force. Assuming originally homogeneous
isotropic turbulence and ideal scale separation in space and time, we may
expect that

${\mbox{\boldmath$\cal{E}$}}^{(0)}=c_{U}\overline{{\mbox{\boldmath$U$}}}+c_{W}{\mbox{\boldmath$\nabla$}}\times\overline{{\mbox{\boldmath$U$}}}+c_{\Omega}{\mbox{\boldmath$\Omega$}}$
(33)

with three coefficients $c_{U}$, $c_{W}$ and $c_{\Omega}$. If the turbulence
shows Galilean invariance, $c_{U}$ has to be zero. The two coefficients
$c_{W}$ and $c_{\Omega}$ must be pseudo scalars, and it turns out that they
are closely connected with the cross-helicity
$\overline{{\mbox{\boldmath$u$}}\cdot{\mbox{\boldmath$b$}}}$. The Yoshizawa
effect, that is, an electromotive force like (33) with nonzero $c_{W}$ or
$c_{\Omega}$, is capable of building up and maintaining a mean magnetic field.
Its strength depends on $c_{W}$ and $c_{\Omega}$. As explained above, it may
act as a seed field if there are conditions which allow a further growth of
mean magnetic fields. Of course, the production of cross-helicity in general
depends also on the mean magnetic field, and in that sense the mean
electromotive force under consideration, too, may depend on this mean magnetic
field.

###  3.3 $\alpha$ effect and $\alpha$ quenching

Let us return to the situation as considered in section 2.2, that is, no mean
motion, no Coriolis force and only small variations of
$\overline{{\mbox{\boldmath$B$}}}$ in space and time. Instead of purely
hydrodynamic turbulence we assume however now homogeneous isotropic
magnetohydrodynamic turbulence, for which $b$, like $u$, remains non-zero if
$\overline{{\mbox{\boldmath$B$}}}\to{\bf 0}$. Then $\cal{E}$ has to satisfy
again (20), that is
${\mbox{\boldmath$\cal{E}$}}=\alpha\,\overline{{\mbox{\boldmath$B$}}}-\beta\,({\mbox{\boldmath$\nabla$}}\times\overline{{\mbox{\boldmath$B$}}})$.
Solving the equation governing $b$ under SOCA and that for $u$ under an
analogous approximation, further restricting ourselves to the high-
conductivity limit, $\eta\tau_{\rm c}/\lambda_{\rm c}^{2}\ll 1$, and the
analogous low-viscosity limit, $\nu\tau_{\rm c}/\lambda_{\rm c}^{2}\ll 1$, we
find

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle\alpha_{\rm
K}+\alpha_{\rm M}$ $\displaystyle\alpha_{\rm K}$ $\displaystyle=$
$\displaystyle-{\textstyle{1\over
3}}\overline{{\mbox{\boldmath$u$}}^{(0)}\cdot({\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$u$}}^{(0)})}\,\tau^{(\alpha{\rm
K})}$ (34) $\displaystyle\alpha_{\rm M}$ $\displaystyle=$
$\displaystyle{\textstyle{1\over
3\mu\varrho}}\overline{{\mbox{\boldmath$b$}}^{(0)}\cdot({\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$b$}}^{(0)})}\,\tau^{(\alpha{\rm
M})}$

and

$\beta=\beta_{\rm K}={\textstyle{1\over
3}}\overline{{{\mbox{\boldmath$u$}}^{(0)}}^{2}}\,\tau^{(\beta)}\,.$ (35)

(see, e.g., Rädler and Rheinhardt 2007). Here ${\mbox{\boldmath$u$}}^{(0)}$
and ${\mbox{\boldmath$b$}}^{(0)}$ stand for $u$ and $b$ in the limit
$\overline{{\mbox{\boldmath$B$}}}\to{\bf 0}$, and $\tau^{(\alpha{\rm K})}$,
$\tau^{(\alpha{\rm M})}$ and $\tau^{(\beta)}$ are quantities approximately
equal to the correlation time $\tau_{\rm c}$. Within this framework the
$\alpha$ effect has in addition to the kinetic part, which is connected with
the mean kinetic helicity
$\overline{{\mbox{\boldmath$u$}}\cdot({\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$u$}})}$,
a magnetic part connected with the mean current helicity
$\overline{{\mbox{\boldmath$b$}}\cdot({\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$b$}})}=\overline{\mu{\mbox{\boldmath$j$}}\cdot{\mbox{\boldmath$b$}}}$.
Such a magnetic part has been first considered by Pouquet et al. in 1976.
Remarkably the coefficient $\beta$, which determines the mean-field
diffusivity, has no such magnetic contribution. If we put
${\mbox{\boldmath$b$}}^{(0)}=0$ we return to our old result for purely
hydrodynamic turbulence.

Let us now admit an arbitrarily strong mean magnetic field. It causes an
anisotropy of the turbulence such that the tensor $a_{ij}$ in (16) has the
structure $\alpha_{1}\delta_{ij}+\alpha_{2}e_{i}e_{j}$, where $\alpha_{1}$ and
$\alpha_{2}$ may depend on $|\overline{{\mbox{\boldmath$B$}}}|$, and $e$
stands for the unit vector in the direction of
$\overline{{\mbox{\boldmath$B$}}}$. Considering then (16) but ignoring, for
simplicity, the terms with derivatives of $\overline{{\mbox{\boldmath$B$}}}$,
we find again
${\mbox{\boldmath$\cal{E}$}}=\alpha\,\overline{{\mbox{\boldmath$B$}}}$ with
$\alpha=\alpha_{1}+\alpha_{2}$ being a function of
$|\overline{{\mbox{\boldmath$B$}}}|$. In general we expect a reduction of the
modulus of $\alpha$ with growing $|\overline{{\mbox{\boldmath$B$}}}|$. In this
case we speak of “$\alpha$ quenching”. It limits the growth of the mean
magnetic field and defines a saturation field strength.

The determination of the dependence of $\alpha$ on
$|\overline{{\mbox{\boldmath$B$}}}|$ is a complex task. A simple ansatz that
has been frequently discussed in the past reads

$\alpha=\frac{\alpha_{0}}{1+c\,\overline{B}^{2}/B^{2}_{eq}}\,,$ (36)

where $\alpha_{0}$ is the value of $\alpha$ in the limit
$\overline{{\mbox{\boldmath$B$}}}\to{\bf 0}$, further $c$ a dimensionless
positive constant and $B_{eq}$ the equipartition field strength defined such
that the energies stored in the fluctuating velocity field and in the mean
magnetic field are equal to each other, that is,
$B^{2}_{eq}=\mu\varrho\,\overline{u^{2}}$.

In 1992 Vainshtein and Cattaneo suggested on the basis of analytical
considerations and numerical calculations with an imposed magnetic field a
relation like (36) with $c\approx Rm$, where $Rm$ means again the magnetic
Reynolds number. In the solar convection zone, for example, $Rm$ takes values
of $10^{6}$ or even $10^{9}$, and $\overline{B}/B_{eq}$ values of the order of
unity. Then $\alpha$ would be very close to zero and we could not expect any
dynamo. Therefore this kind of quenching has been called “catastrophic
quenching”.

This finding has initiated many discussions and investigations. Considerable
progress has been made by investigating the simplest possible dynamo systems
with $\alpha$ effect in the nonlinear regime. A fully satisfactory theory of
this subject is, however, still missing.

One important issue in the recent investigations on $\alpha$ quenching are the
hypotheses that $\alpha$ is always a sum of a kinetic part $\alpha_{\rm K}$
and a magnetic part $\alpha_{\rm M}$ and that the latter is determined by the
part of the mean current helicity due to the electric current and magnetic
field fluctuations,
$\overline{{\mbox{\boldmath$j$}}\cdot{\mbox{\boldmath$b$}}}$. The other
important issue is the role of the magnetic helicity in a dynamo. We recall
here that the magnetic helicity, say $H$, is defined as a volume integral over
the magnetic helicity density
$h={\mbox{\boldmath$A$}}\cdot{\mbox{\boldmath$B$}}$, where $A$ is a vector
potential of the magnetic field $B$, that is
${\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$A$}}={\mbox{\boldmath$B$}}$.
If the electromagnetic fields satisfy specific conditions at the surface of
this volume, in particular the magnetic field does not intersect this surface,
$H$ is independent of the special choice of the vector potential $A$, that is,
under gauge transformations of $A$. Then the basic equations imply further
that, in the limit of infinite conductivity, $H$ is a conserved quantity, that
is, does not change in time. Within the mean-field concept the magnetic
helicity density $h$ is the sum of two parts, one originating from the mean
magnetic field $\overline{{\mbox{\boldmath$B$}}}$ and the other from the
fluctuating part of the magnetic field, $b$. The mean part of the latter,
$\overline{{\mbox{\boldmath$a$}}\cdot{\mbox{\boldmath$b$}}}$ with
${\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$a$}}={\mbox{\boldmath$b$}}$,
is closely related to the magnetic contribution $\alpha_{\rm M}$ to $\alpha$,
which is, as explained above, determined by the part
$\overline{{\mbox{\boldmath$j$}}\cdot{\mbox{\boldmath$b$}}}=(1/\mu)\overline{{\mbox{\boldmath$b$}}\cdot({\mbox{\boldmath$\nabla$}}\times{\mbox{\boldmath$b$}})}$
of the mean current helicity. If, for example, in the limit of infinite
conductivity $H$ is initially equal to zero and the mean magnetic field grows,
the modulus of $\alpha_{\rm M}$ must grow, too.

With the hypothesis $\alpha=\alpha_{\rm K}+\alpha_{\rm M}$, further the
evolution equation of the mean magnetic helicity density due to fluctuations,
and a few plausible assumptions an evolution equation for $\alpha_{\rm M}$,

$\partial_{t}\alpha_{\rm M}=-2\eta_{\rm t}k_{\rm
f}^{2}\Big{(}\frac{{\mbox{\boldmath$\cal{E}$}}\cdot\overline{{\mbox{\boldmath$B$}}}}{B_{eq}^{2}}+\frac{\alpha_{\rm
M}}{Rm}\Big{)}-{\mbox{\boldmath$\nabla$}}\cdot{\mbox{\boldmath$F$}}\,,$ (37)

has been derived (see, e.g., Hubbard and Brandenburg 2011). As usual in this
context, we write here $\eta_{\rm t}$ instead of $\beta$, and $k_{\rm f}$
denotes again the wavenumber of the energy-carrying scale in the turbulence.
$\cal{E}$ should be specified to be equal to $(\alpha_{\rm K}+\alpha_{\rm
M})\overline{{\mbox{\boldmath$B$}}}-\eta_{t}{\mbox{\boldmath$\nabla$}}\times\overline{{\mbox{\boldmath$B$}}}$.
As above, $B_{eq}$ is the equipartition field strength, and $F$ means a mean
magnetic helicity flux. In simple models with periodic boundary conditions the
term ${\mbox{\boldmath$\nabla$}}\cdot{\mbox{\boldmath$F$}}$ does not change
the total mean magnetic helicity inside a dynamo volume. In general, however,
the mean magnetic helicity flux plays a crucial role, and expressions for $F$
have been elaborated which depend, for example, on differential rotation.
Models incorporating such results reflect indeed many properties of dynamos in
the non-linear regime including saturation field strengths (see, e.g.,
Brandenburg and Subramanian 2005, Hubbard and Brandenburg 2011,2012, Del Sordo
et al. 2013).

##  4 Laboratory experiments

The development of dynamo theory was accompanied and supported by several
laboratory experiments. As early as in 1967, one year after the first paper
about this subject, the $\alpha$ effect has been demonstrated in a liquid
sodium flow in the Institute of Physics in Riga (Steenbeck et al. 1967). The
measurements were carried out at the so-called “$\alpha$ box”, in which a
proper flow geometry has been organized by baffles.

Already at this time there were many discussions on the realization of a
dynamo in a conducting fluid. It was clear from the very beginning that such
an experiment requires a large fluid volume and high flow rates. Only in the
last days of the last century, in December 1999, after expensive preparations,
two dynamos ran successfully with liquid sodium flows, one in Riga (Gailitis
et al. 2000) and one in Karlsruhe (Müller and Stieglitz 2000, 2002). The first
one (Riga) is clearly different from a mean-field dynamo, but the second one
(Karlsruhe) can be well understood as a mean-field dynamo of $\alpha^{2}$ type
(see Rädler et al. 2002).

I do not want to go into the details of these experiments but add a more
general remark on the sometimes underestimated practical value of basic
research. We have learned in geophysically or astrophysically motivated
studies that the self-excitation of magnetic fields in moving electrically
conducting fluids is possible as soon as the magnetic Reynolds number
$Rm=UL/\eta$, with $U$ and $L$ being typical values of fluid velocity and
linear dimensions of the considered device, exceeds a critical value, which
depends on the flow geometry and lies in all investigated cases above unity.
For a long time situations of that kind did not appear in laboratories or in
industrial devices. In the sixties and seventies of the last century, however,
big fast breeder reactors were built with huge circuits of liquid sodium,
which transport the heat produced in the active zone to the places where it is
transformed into electric power. Such devices imply indeed the possibility of
self-excitation of magnetic fields, what constitutes a big danger. These
fields could quickly grow, heavily hamper the sodium flow and so seriously
disturb the whole reactor regime or even cause a catastrophy. At first the
reactor engineers were not aware of that. It was the people thinking about
cosmic dynamos who pointed out this danger. Max Steenbeck, as a Foreign Member
of the Soviet Academy, presented there in 1971 a corresponding memorandum and
initiated so investigations in this field and measures to avoid this danger.
Some years later, independent of that, two papers of British scientists about
this topic have appeared (Bevir 1973, Pierson 1975). This development fostered
the willingness of authorities to support theoretical and experimental
research on dynamos and related topics.

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# Cauchy–Schwarz-type inequalities for additive functions

Zsolt Páles  Institute of Mathematics, University of Debrecen, H-4002
Debrecen, Pf. 400, Hungary<EMAIL_ADDRESS>and  Mahmood Kamil Shihab
Doctoral School of Mathematical and Computational Sciences, University of
Debrecen, Hungary; Department of Mathematics, College of Education for Pure
Sciences, University of Kirkuk, Iraq<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The main goal of this paper is to show that if a real valued function defined
on a groupoid satisfies a certain Levi–Civita-type functional equation, then
it also fulfills a Cauchy–Schwarz-type functional inequality. In particular,
if the groupoid is the multiplicative structure of commutative ring, then we
can establish the existence of nontrivial additive functions satisfying
inequalities connected to the multiplicative structure.

###### Key words and phrases:

Levi–Civita-type functional equation, Cauchy–Schwarz-type functional
inequality, additive function

###### 2000 Mathematics Subject Classification:

Primary 39B22, 39B52, 39B62, 39B72

The research of the first author was supported by the K-134191 NKFIH Grant.

##  1\. Introduction

In the theory of real and additive functions (see the monograph [8] of Kuczma)
there are several results which establish the existence of a discontinuous
additive function which satisfies further algebraic conditions. One of the
first problems of this kind was posed by Szabó [10] motivated by a question of
Benz [2] and solved by Kominek, Reich, and Schwaiger [7]. They proved that if
$A:\mathbb{R}\to\mathbb{R}$ is an additive function which satisfies the
equality $A(x)A(y)=0$ for all $(x,y)\in C$, where $C$ is a the unit circle, or
is a hyperbola, or is an algebraic curve given by polynomials, then $A$ has to
be equal to zero identically. Boros and Fechner [4] and Boros, Fechner and
Kutas [5] extended these results to sets defined via generalized polynomials
and to quadratic functions instead of additive ones, respectively, and they
also examined the stability versions of such problems.

In [5], the case when $C$ is the graph of the hyperbola $xy=1$ was left open.
Kanappan [6, Chapter 1] proved that if, for some positive constant $a$, an
additive function $A$ satisfies the condition $A(x)A(1/x)=a$ for all $x\neq
0$, then $A$ has to be continuous. On the other hand, according to the remarks
[1] and [3], there exist discontinuous additive functions which fulfill the
inequality $A(x)A(1/x)>0$ for all $x\neq 0$. On the other hand, using the
theory of valuations of fields, Kutas [9, Theorem 24] proved that there exists
a nonzero (henceforth discontinuous) additive function which satisfies the
equality $A(x)A(1/x)=0$ for all $x\neq 0$.

The above results motivated us to construct discontinuous additive real
functions that enjoy properties that are connected to the multiplicative
structure. It turned out that such properties could be possessed if the
additive function satisfies Levi–Civita-type functional equations with respect
to the multiplicative structure.

More generally, let $(G,*)$ be a groupoid. (Recall that a pair $(G,*)$ is said
to be a groupoid if $\cdot$ is a binary operation on $G$, i.e., $\cdot:G\times
G\to G$.) Let $A:G\to\mathbb{R}$ be a function such that there exist functions
$f_{1},\dots,f_{n},g_{1},\dots,g_{n}:G\to\mathbb{R}$ such that the functional
equation

$\displaystyle A(x*y)=f_{1}(x)g_{1}(y)+\dots+f_{n}(x)g_{n}(y)\qquad(x,y\in G)$

is fulfilled. Under certain assumptions on $n$ and on the functions
$f_{1},\dots,f_{n},g_{1},\dots,g_{n}$, we are going to prove that $A$ will
satisfy either the inequality $A(x*y)^{2}\leq A(x*x)A(y*y)$ or the reversed
one $A(x*x)A(y*y)\leq A(x*y)^{2}$. In the important particular case when the
groupid is the multiplicative structure of a commutative ring and $A$ is
additive, we will establish the existence of nontrivial additive functions
which satisfy one of the above mentioned inequalities.

##  2\. The inequality $A(x*y)^{2}\leq A(x*x)A(y*y)$

In our first result we assume that the function $A$ satisfies a Levi–Civita-
type functional equation over a groupoid.

######  Theorem 2.1.

Let $(G,*)$ be a groupoid and let $A:G\to\mathbb{R}$ be a function. Assume
that there exist $n\in\mathbb{N}$ and functions
$f_{1},\dots,f_{n}:G\to\mathbb{R}$ such that $A$ the Levi–Civita-type
functional equation

(1) $\displaystyle A(x*y)=f_{1}(x)f_{1}(y)+\dots+f_{n}(x)f_{n}(y)$

holds for all $x,y\in G$. Then, $A$ satisfies the functional inequality

(2) $\displaystyle A(x*y)^{2}\leq A(x*x)A(y*y)$

for all $x,y\in G$.

###### Proof.

Let $x,y\in G$. In view of the functional equation (1), the inequality (2) can
be rewritten as

$\displaystyle\big{(}f_{1}(x)f_{1}(y)+\dots+f_{n}(x)f_{n}(y)\big{)}^{2}\leq\big{(}f_{1}(x)^{2}+\dots+f_{n}(x)^{2}\big{)}\big{(}f_{1}(y)^{2}+\dots+f_{n}(y)^{2}\big{)},$

which follows from the Cauchy–Schwarz inequality when we apply it to the
$n$-dimensional vectors $(f_{1}(x),\dots,f_{n}(x))$ and
$(f_{1}(y),\dots,f_{n}(y))$. ∎

If the groupoid is the multiplicative semigroup of a commutative ring
$(R,+,\cdot)$ and $A$ is additive, then we can establish a characterization of
the corresponding inequality. Recall that in a ring, the product $x\cdot y$ of
the elements $x,y\in R$ is simply denoted by $xy$, and $x^{2}$ is defined to
be the product $x\cdot x$.

######  Theorem 2.2.

Let $(R,+,\cdot)$ be a commutative ring and let $A:R\to\mathbb{R}$ be an
additive function. If one of the following conditions

  1. (i)

$A=0$,

  2. (ii)

$A(x^{2})\geq 0$ for all $x\in R$,

  3. (iii)

$A(x^{2})\leq 0$ for all $x\in R$

hold, then $A$ satisfies the inequality

(3) $\displaystyle A(xy)^{2}\leq A(x^{2})A(y^{2})$

for all $x,y\in R$. Conversely, if $R$ has a multiplicative unit element and
$A$ satisfies the inequality (3) for all $x,y\in R$, then one of the
conditions (i), (ii), and (iii) must be satisfied.

###### Proof.

The inequality (3) is obvious if $A=0$, i.e., if condition (i) holds.

Now assume that $A$ satisfies condition (ii) and let $x,y\in R$ be fixed.
Then, for all $n\in\mathbb{N}$ and $k\in\mathbb{Z}$, we get that

$\displaystyle 0\leq
A((nx+ky)^{2})=A(n^{2}x^{2}+2nkxy+k^{2}y^{2})=n^{2}A(x^{2})+2nkA(xy)+k^{2}A(y^{2}).$

Dividing this inequality by $n^{2}$, we can conclude that

$\displaystyle 0\leq A(x^{2})+2\frac{k}{n}A(xy)+\frac{k^{2}}{n^{2}}A(y^{2}).$

Because $n\in\mathbb{N}$ and $k\in\mathbb{Z}$ were arbitrary, we obtain that

$\displaystyle 0\leq A(x^{2})+2rA(xy)+r^{2}A(y^{2})$

is valid for all rational number $r$. By the density of rational numbers, it
follows that the above inequality is true for all real number $r$. The
polynomial on the right hand side cannot have two distinct real roots,
therefore, its discriminant has to be non positive, i.e.,

$\displaystyle(2A(xy))^{2}-4A(x^{2})A(y^{2})\leq 0.$

This inequality reduces to (3).

In the case when condition (iii) holds, then the additive function $(-A)$
satisfies condition (ii) and hence the inequality (3) holds with $(-A)$
instead of $A$, which again shows that (3) is valid.

To verify the reversed implication, assume that $R$ possesses a multiplicative
unit element which will be denoted by $e$ and assume that $A$ satisfies
inequality (3) for all $x,y\in R$. Substituting $y:=e$ in (3), it follows that

(4) $\displaystyle 0\leq A(x)^{2}\leq A(x^{2})A(e)$

holds for all $x\in R$. We can now distinguish three cases according to the
possibilities $A(e)=0$, $A(e)>0$, and $A(e)<0$.

If $A(e)=0$, then (4) yields that $A(x)=0$ for all $x\in R$, i.e., condition
(i) is valid.

In the case when $A(e)>0$, it follows that $A$ satisfies condition (ii), while
in the case when $A(e)<0$, we can see that $A$ satisfies condition (iii). ∎

##  3\. The inequality $A(x*x)A(y*y)\leq A(x*y)^{2}$

In the subsequent two theorems, we present to Levi–Civita-type functional
equations which imply the inequality in the title of this section.

######  Theorem 3.1.

Let $(G,*)$ be a groupoid and $A:G\to\mathbb{R}$ be a function. Assume that
there exist two functions $f,g:G\to\mathbb{R}$ such that the Levi–Civita-type
functional equation

(5) $\displaystyle A(x*y)=f(x)f(y)-g(x)g(y)$

holds for all $x,y\in G$. Then $A$ satisfies the functional inequality

(6) $\displaystyle A(x*x)A(y*y)\leq A(x*y)^{2}$

for all $x,y\in G$.

###### Proof.

Let $x,y\in G$. According to the functional equation (5), the inequality (6)
can be rewritten as

$\displaystyle(f(x)^{2}-g(x)^{2})(f(y)^{2}-g(y)^{2})\leq(f(x)f(y)-g(x)g(y))^{2}.$

Observe that this inequality is equivalent to

$\displaystyle 0\leq(g(x)f(y)-f(x)g(y))^{2},$

which is obviously valid. ∎

######  Theorem 3.2.

Let $(G,*)$ be a groupoid. Let $A:R\to\mathbb{R}$ be a function. Assume that
there exist $f,g:R\to\mathbb{R}$ such that the Levi–Civita-type functional
equation

(7) $\displaystyle A(x*y)=f(x)g(y)+g(x)f(y)$

holds for all $x,y\in G$. Then, for all $x,y\in G$, $A$ satisfies the
functional inequality (6).

###### Proof.

Let $x,y\in G$. According to the functional equation (7), the inequality (6)
can be rewritten as

$\displaystyle 4f(x)g(x)f(y)g(y)\leq(f(x)g(y)+g(x)f(y))^{2}.$

Observe that this inequality is equivalent to

$\displaystyle 0\leq(g(x)f(y)-f(x)g(y))^{2},$

which is obviously valid. ∎

######  Corollary 3.3.

Assume that $A:\mathbb{R}\to\mathbb{R}$ satisfies the Leibniz Rule, i.e.,

$\displaystyle A(xy)=xA(y)+A(x)y\qquad(x,y\in\mathbb{R}).$

Then, for all $x,y\in\mathbb{R}$, the inequality

(8) $\displaystyle A(x^{2})A(y^{2})\leq A(xy)^{2}$

holds.

###### Proof.

Observe that with groupoid $(G,*):=(\mathbb{R},\cdot)$ and with the notations
$g:=A$ and $f(x):=x$, $(x\in G)$, the equality (7) of Theorem 3.2 holds.
Therefore, $A$ satisfies inequality (6) for all $x,y\in\mathbb{R}$, hence (8)
is also satisfied. ∎

In particular, if $A:\mathbb{R}\to\mathbb{R}$ is a derivation (i.e., $A$ is
additive and satisfies the Leibniz Rule with respect, then the above corollary
implies that it fulfills the inequality (8).

If the groupoid is the multiplicative semigroup of a commutative ring
$(R,+,\cdot)$ and $A$ is additive, then we can establish a characterization of
the inequality (6) over a particular subset of the ring.

######  Theorem 3.4.

Let $(R,+,\cdot)$ be a commutative ring with a multiplicative unit element $e$
and $A:R\to\mathbb{R}$ be an additive function with $A(e)\neq 0$. Let the
subset $R_{A}\subseteq R$ be defined by

$\displaystyle R_{A}:=\\{x\in R\mid 0\leq A(x^{2})A(e)\\}$

Then $e\in R_{A}$ and $A$ satisfies the following functional inequality

(9) $\displaystyle A(x^{2})A(y^{2})\leq A(xy)^{2}$

for all $x,y\in R_{A}$ if and only if

(10) $\displaystyle A(x^{2})A(e)\leq A(x)^{2}$

for all $x\in R_{A}$.

###### Proof.

The inclusion $e\in R_{A}$ is obvious. Now, putting $y:=e$, we can see that
the inequality (9) implies (10).

To prove the reversed implication, assume that (10) is valid for all $x\in
R_{A}$. Then it is also valid for all $x\in R$, since, for $x\in R\setminus
R_{A}$, the left hand side of the inequality is negative, while the right hand
side is nonnegative. Introduce the function $A_{0}:=A/A(e)$. Then, $A_{0}$ is
additive, $A_{0}(e)=1$ and, dividing (10) by $A(e)^{2}>0$ side by side, for
all $x\in R$, we get that

(11) $\displaystyle A_{0}(x^{2})\leq A_{0}(x)^{2}$

Let $x,y\in R_{A}$ be fixed and $n\in\mathbb{N}$, $k\in\mathbb{Z}$ be
arbitrary. Then, (11) yields

$\displaystyle A_{0}((nx+ky)^{2})\leq A_{0}(nx+ky)^{2}.$

Using the additivity of $A_{0}$, we get

$\displaystyle n^{2}A_{0}(x^{2})+2nkA_{0}(xy)+k^{2}A_{0}(y^{2})\leq
n^{2}A_{0}(x)^{2}+2nkA_{0}(x)A_{0}(y)+k^{2}A_{0}(y)^{2}.$

Dividing this inequality by $n^{2}$, we obtain

$\displaystyle
A_{0}(x^{2})+2\tfrac{k}{n}A_{0}(xy)+\big{(}\tfrac{k}{n}\big{)}^{2}A_{0}(y^{2})\leq
A_{0}(x)^{2}+2\tfrac{k}{n}A_{0}(x)A_{0}(y)+\big{(}\tfrac{k}{n}\big{)}^{2}A_{0}(y)^{2}.$

Therefore, for any rational number $r\in\mathbb{Q}$,

$\displaystyle
0\leq(A_{0}(x)^{2}-A_{0}(x^{2}))+2r(A_{0}(x)A_{0}(y)-A_{0}(xy))+r^{2}(A_{0}(y)^{2}-A_{0}(y^{2})).$

Using the continuity of both sides as a function of $r$, it follows that the
same inequality is valid for all $r\in\mathbb{R}$. Thus, the discriminant of
this quadratic polynomial has to be nonpositive, i.e.,

(12)
$\displaystyle(A_{0}(x)A_{0}(y)-A_{0}(xy))^{2}\leq(A_{0}(x)^{2}-A_{0}(x^{2}))(A_{0}(y)^{2}-A_{0}(y^{2}))$

and hence

$\displaystyle|A_{0}(x)A_{0}(y)-A_{0}(xy)|\leq\sqrt{(A_{0}(x)^{2}-A_{0}(x^{2}))(A_{0}(y)^{2}-A_{0}(y^{2}))}=Q(x)Q(y),$

thus

$\displaystyle\big{|}|A_{0}(x)A_{0}(y)|-|A_{0}(xy)|\big{|}\leq|A_{0}(x)A_{0}(y)-A_{0}(xy)|\leq
Q(x)Q(y),$

where $Q(u):=\sqrt{A_{0}(u)^{2}-A_{0}(u^{2})}\geq 0$ ($u\in R$). Then, for all
$u\in R$,

(13) $\displaystyle A_{0}(u)^{2}=Q(u)^{2}+A_{0}(u^{2}).$

Therefore $|A_{0}(xy)|$ satisfies the inequality

(14)
$\displaystyle|A_{0}(x)A_{0}(y)|-Q(x)Q(y)\leq|A_{0}(xy)|\leq|A_{0}(x)A_{0}(y)|+Q(x)Q(y).$

We are going to show that

(15) $\displaystyle A_{0}(x^{2})A_{0}(y^{2})\leq A_{0}(xy)^{2}.$

To see this inequality, we will prove that

(16) $\displaystyle Q(x)^{2}Q(y)^{2}\leq A_{0}(x)^{2}A_{0}(y)^{2}.$

and

(17) $\displaystyle
A_{0}(x^{2})A_{0}(y^{2})\leq\Big{(}|A_{0}(x)A_{0}(y)|-Q(x)Q(y)\Big{)}^{2}.$

Since $x$ and $y$ belong to $R_{A}$, therefore, we have that $A_{0}(x^{2})\geq
0$ and $A_{0}(y^{2})\geq 0$, then, with $u\in\\{x,y\\}$, the equality (13)
implies that

$\displaystyle Q(x)^{2}\leq A_{0}(x)^{2}\qquad\mbox{and}\qquad Q(y)^{2}\leq
A_{0}(y)^{2}.$

Multiplying these inequalities side by side, we get that (16) holds.
Therefore, we can conclude that

(18) $\displaystyle Q(x)Q(y)\leq|A_{0}(x)A_{0}(y)|$

which is equivalent to (16).

By the obvious inequality

$\displaystyle\big{(}|A_{0}(x)|Q(y)-|A_{0}(y)|Q(x)\big{)}^{2}\geq 0,$

we have that

(19) $\displaystyle 2|A_{0}(x)A_{0}(y)|Q(x)Q(y)\leq
A_{0}(x)^{2}Q(y)^{2}+A_{0}(y)^{2}Q(x)^{2}.$

Therefore using the equality (13) with $u\in\\{x,y\\}$ and the inequality
(19), we obtain

$\displaystyle A_{0}(x^{2})A_{0}(y^{2})$
$\displaystyle=A_{0}(x)^{2}A_{0}(y)^{2}-A_{0}(x)^{2}Q(y)^{2}-Q(x)^{2}A_{0}(y)^{2}+Q(x)^{2}Q(y)^{2}$
$\displaystyle\leq
A_{0}(x)^{2}A_{0}(y)^{2}-2|A_{0}(x)A_{0}(y)|Q(x)Q(y)+Q(x)^{2}Q(y)^{2}$
$\displaystyle=(|A_{0}(x)A_{0}(y)|-Q(x)Q(y))^{2}.$

This shows that the inequality (17) holds.

In view of (18), the first inequality in (14) implies that

$\displaystyle(|A_{0}(x)A_{0}(y)|-Q(x)Q(y))^{2}\leq A_{0}(xy)^{2}.$

This, combined with the inequality (17) yields that (15) is valid, indeed.
Therefore,

$\displaystyle A(x^{2})A(y^{2})=A_{0}(x^{2})A_{0}(y^{2})A(e)^{2}\leq
A_{0}^{2}(xy)A(e)^{2}=A(xy)^{2},$

which completes the proof of the inequality (9) for $x,y\in R_{A}$. ∎

In the following example we show that the additivity of the function $A$ in
Theorem 3.4 is necessary.

######  Example 3.5.

Let $q\in(0,1)$ and let $f:\mathbb{R}\to\mathbb{R}$ be a function defined by

$\displaystyle f(x)=\begin{cases}x&x\neq 1,\\\ q&x=1.\end{cases}$

Clearly, $f$ is not additive. Therefore, for $x\not\in\\{1,-1\\}$, we have
that

$\displaystyle f(x^{2})f(1)=qx^{2}\leq x^{2}=f(x)^{2}$

for $x=\pm 1$,

$\displaystyle f(1^{2})f(1)=q^{2}=f(1)^{2},\qquad
f((-1)^{2})f(1)=q^{2}<1=f(-1)^{2},$

which shows that (10) is satisfied for all $x\in\mathbb{R}$. On the other
hand, for $x,y\in\mathbb{R}\setminus\\{1,-1\\}$ with $xy=1$, we can conclude
that

$\displaystyle f(x^{2})f(y^{2})=x^{2}y^{2}=1>q^{2}=f(xy)^{2},$

which shows that (9) is not satisfied.

The next example shows that if the function $A$ in Theorem 3.4 is non-
additive, continuous and satisfies $A(e)=0$, then the conclusion of Theorem
3.4 may not be valid.

######  Example 3.6.

Let $A:\mathbb{R}\to\mathbb{R}$ be defined as $A(x)=|x-1|$. Note that $A$ is
continuous and not additive. Since $A(1)=0$ this implies that

$\displaystyle A(x^{2})A(1)=0\leq(x-1)^{2}=A(x)^{2}.$

Thus the inequality (10) holds for all $x\in\mathbb{R}$. On the other hand we
have that

$\displaystyle A(x^{2})A(y^{2})=|x^{2}-1||y^{2}-1|\qquad\mbox{and}\qquad
A(xy)^{2}=(xy-1)^{2}.$

Hence for $x=2$ and $y=\frac{1}{2}$ we have that
$A(x^{2})A(y^{2})=\frac{9}{4}$ but $A(xy)^{2}=0$, therefore the inequality (9)
does not hold.

##  4\. Consequences of systems of Levi–Civita-type functional equations

######  Theorem 4.1.

Let $(G,*)$ be a groupoid and $A,B:G\to\mathbb{R}$ be functions. Assume that
there exist $f,g:G\to\mathbb{R}$ such that $A$ and $B$ satisfy the following
system of Levi–Civita-type functional equations

(20) $\displaystyle A(x*y)$ $\displaystyle=f(x)f(y)-g(x)g(y)\qquad\mbox{and}$
$\displaystyle B(x*y)$ $\displaystyle=f(x)g(y)+g(x)f(y)$

for all $x,y\in G$. Then the inequalities

(21) $\displaystyle-B(x*y)^{2}\leq A(x*x)A(y*y)\leq A(x*y)^{2}$

and

(22) $\displaystyle-A(x*y)^{2}\leq B(x*x)B(y*y)\leq B(x*y)^{2}$

hold for all $x,y\in G$.

###### Proof.

In view of two functional equations in (20), for $x,y\in G$, we have that

$\displaystyle B(x*y)^{2}$ $\displaystyle+A(x*x)A(y*y)$
$\displaystyle=f(x)^{2}g(y)^{2}+2f(x)g(y)g(x)f(y)+g(x)^{2}f(y)^{2}+(f(x)^{2}-g(x)^{2})(f(y)^{2}-g(y)^{2})$
$\displaystyle=(f(x)f(y)+g(x)g(y))^{2}\geq 0,$

which proves the left hand side inequality in (21). The right hand side
inequality in (21) is a direct consequence of Theorem 3.1.

Again, in view of two equations in (20), for $x,y\in G$ we have that

$\displaystyle A(x*y)^{2}$ $\displaystyle+B(x*x)B(y*y)$
$\displaystyle=f(x)^{2}f(y)^{2}-2f(x)f(y)g(x)g(y)+g(x)^{2}g(y)^{2}+4f(x)g(x)f(y)g(y)$
$\displaystyle=(f(x)f(y)+g(x)g(y))^{2}\geq 0.$

This implies the left hand side inequality in (22). On the other hand,
applying Theorem 3.2 for the function $B$ instead of $A$, we obtain that

$\displaystyle B(x*x)B(y*y)\leq B(x*y)^{2}.$

This shows that the second inequality of (22) holds for $x,y\in G$. ∎

An interesting consequence of the functional equations in (20) is that $A$ and
$B$ satisfy the following identity:

$\displaystyle B(x*y)^{2}+A(x*x)A(y*y)=A(x*y)^{2}+B(x*x)B(y*y)\qquad(x,y\in
G).$

Therefore, the inequalities (21) and (22) can be expressed as the following
chain of inequalities

$\displaystyle 0$ $\displaystyle\leq A(x*x)A(y*y)+(B(x*y))^{2}$
$\displaystyle=B(x*x)B(y*y)+A(x*y)^{2}\leq A(x*y)^{2}+B(x*y)^{2}\qquad(x,y\in
G).$

######  Corollary 4.2.

For all $x,y\in\mathbb{R}$, we have

$\displaystyle-\sin(x+y)^{2}$
$\displaystyle\leq\cos(2x)\cos(2y)\leq\cos(x+y)^{2}\qquad\mbox{and}$
$\displaystyle-\cos(x+y)^{2}$
$\displaystyle\leq\sin(2x)\sin(2y)\leq\sin(x+y)^{2}.$

###### Proof.

Observe that the trigonometric functions $\cos:\mathbb{R}\to\mathbb{R}$ and
$\sin:\mathbb{R}\to\mathbb{R}$ satisfy the functional equations

$\displaystyle\cos(x+y)$
$\displaystyle=\cos(x)\cos(y)-\sin(x)\sin(y)\qquad\mbox{and}$
$\displaystyle\sin(x+y)$ $\displaystyle=\sin(x)\cos(y)+\cos(x)\sin(y)$

for all $x,y\in\mathbb{R}$. Therefore, (20) holds with $A:=f:=\cos$ and
$B:=g:=\sin$ over the groupoid $(\mathbb{R},+)$. Consequently, (21) and (22)
are satisfied for all $x,y\in\mathbb{R}$, which imply the assertion. ∎

######  Corollary 4.3.

Let $(G,*)$ be a groupoid and let $\varphi:G\to\mathbb{C}$ be a homomorphism
into the multiplicative semigroup of complex numbers. Define $A:=\Re\varphi$
and $B:=\Im\varphi$. Then, for all $x,y\in G$, the inequalities (21) and (22)
hold.

###### Proof.

Using the multiplicativity of $\varphi$, for all $x,y\in G$, we get that

$\displaystyle A(x*y)$
$\displaystyle=\Re(\varphi(x*y))=\Re(\varphi(x)\varphi(y))$
$\displaystyle=\Re((A(x)+iB(x))(A(y)+iB(y)))=A(x)A(y)-B(x)B(y),$
$\displaystyle B(x*y)$
$\displaystyle=\Im(\varphi(x*y))=\Im(\varphi(x)\varphi(y))$
$\displaystyle=\Im((A(x)+iB(x))(A(y)+iB(y)))=A(x)B(y)+B(x)A(y).$

Therefore, the functional equations in (20) are satisfied with $f:=A$ and
$g:=B$. Thus, according to Theorem 4.1, we obtain that the inequalities (21)
and (22) hold for all $x,y\in G$, which was to be shown. ∎

######  Corollary 4.4.

Let $\varphi:\mathbb{C}\to\mathbb{C}$ be an automorphism of the field
$\mathbb{C}$. Define $A:=\Re\varphi$ and $B:=\Im\varphi$. Then
$A:\mathbb{C}\to\mathbb{R}$ and $B:\mathbb{C}\to\mathbb{R}$ are additive
mappings, furthermore, for all $x,y\in\mathbb{C}$,

(23) $\displaystyle-B(xy)^{2}\leq A(x^{2})A(y^{2})\leq
A(xy)^{2}\quad\mbox{and}\quad-A(xy)^{2}\leq B(x^{2})B(y^{2})\leq B(xy)^{2}.$

###### Proof.

Using the additivity of $\varphi$, for all $x,y\in\mathbb{C}$, we obtain that

$\displaystyle A(x+y)$
$\displaystyle=\Re(\varphi(x+y))=\Re(\varphi(x)+\varphi(y))$
$\displaystyle=\Re((A(x)+iB(x))+(A(y)+iB(y)))=A(x)+A(y),$ $\displaystyle
B(x+y)$ $\displaystyle=\Im(\varphi(x+y))=\Im(\varphi(x)+\varphi(y))$
$\displaystyle=\Im((A(x)+iB(x))+(A(y)+iB(y)))=B(x)+B(y).$

These equalities show that $A:\mathbb{C}\to\mathbb{R}$ and
$B:\mathbb{C}\to\mathbb{R}$ are additive mappings.

By the multiplicativity of $\varphi$, it maps the groupoid
$(G,*):=(\mathbb{C},\cdot)$ into itself. Thus, according to Corollary 4.3, we
obtain that the inequalities (21) and (22) hold for all $x,y\in\mathbb{C}$.
This yields the assertion. ∎

The following result is a counterpart of Theorem 4.1.

######  Theorem 4.5.

Let $(G,*)$ be a groupoid and $A,B:G\to\mathbb{R}$ be functions. Assume that
there exist $f,g:G\to\mathbb{R}$ such that $A$ and $B$ satisfy the following
Levi–Civita-type functional equations

(24) $\displaystyle A(x*y)$ $\displaystyle=f(x)f(y)+g(x)g(y)\qquad\mbox{and}$
$\displaystyle B(x*y)$ $\displaystyle=f(x)g(y)+g(x)f(y)$

for all $x,y\in G$. Then the inequalities

(25) $\displaystyle B(x*x)B(y*y)\leq A(x*y)^{2}\leq A(x*x)A(y*y)$

and

(26) $\displaystyle B(x*x)B(y*y)\leq B(x*y)^{2}\leq A(x*x)A(y*y).$

hold for all $x,y\in G$.

###### Proof.

In view of two functional equations in (24), for $x,y\in G$, we have that

$\displaystyle A(x*y)^{2}$ $\displaystyle-B(x*x)B(y*y)$
$\displaystyle=f(x)^{2}f(y)^{2}+2f(x)f(y)g(x)g(y)+g(x)^{2}g(y)^{2}-4f(x)g(x)f(y)g(y)$
$\displaystyle=(f(x)f(y)-g(x)g(y))^{2}\geq 0.$

This implies the left hand side inequality in (25). On the other hand,
applying Theorem 2.1 for the function $A$ and $n=2$, $f_{1}:=f$, $f_{2}:=g$,
we obtain that the second inequality of (25) holds for $x,y\in G$.

Again, in view of two equations in (24), for $x,y\in G$, we have that

$\displaystyle A(x*x)$ $\displaystyle A(y*y)-B(x*y)^{2}$
$\displaystyle=(f(x)^{2}+g(x)^{2})(f(y)^{2}+g(y)^{2})-f(x)^{2}g(y)^{2}-2f(x)g(y)g(x)f(y)-g(x)^{2}f(y)^{2}$
$\displaystyle=(f(x)f(y)-g(x)g(y))^{2}\geq 0,$

which proves the right hand side inequality in (26). The left hand side
inequality in (26) is a direct consequence of Theorem 3.2 (applied to $B$
instead of $A$). ∎

An interesting consequence of the functional equations in (20) is that $A$ and
$B$ satisfy the following identity:

$\displaystyle B(x*x)B(y*y)+A(x*x)A(y*y)=A(x*y)^{2}+B(x*y)^{2}\qquad(x,y\in
G).$

######  Corollary 4.6.

For all $x,y\in\mathbb{R}$, we have

(27)
$\displaystyle\sinh(2x)\sinh(2y)\leq\sinh(x+y)^{2}<\cosh(x+y)^{2}\leq\cosh(2x)\cosh(2y)$

###### Proof.

Observe that the hyperbolic functions $\cosh:\mathbb{R}\to\mathbb{R}$ and
$\sinh:\mathbb{R}\to\mathbb{R}$ satisfy the functional equations

$\displaystyle\cosh(x+y)$
$\displaystyle=\cosh(x)\cosh(y)+\sinh(x)\sinh(y)\qquad\mbox{and}$
$\displaystyle\sinh(x+y)$ $\displaystyle=\sinh(x)\cosh(y)+\cosh(x)\sinh(y)$

for all $x,y\in\mathbb{R}$. Therefore, (24) holds with $A:=f:=\cosh$ and
$B:=g:=\sinh$ over the groupoid $(\mathbb{R},+)$. Consequently, (25) and (26)
are satisfied for all $x,y\in\mathbb{R}$, which imply the first and last
inequalities in (27). The central inequality follows from the identity
$\cosh^{2}-\sinh^{2}=1$. ∎

To formulate the next result, let $p$ be a square free positive integer and
let $\mathbb{Q}(\sqrt{p})$ denote the subfield of $\mathbb{R}$ generated by
$\sqrt{p}$. Then, one can see that
$\mathbb{Q}(\sqrt{p})=\\{a+b\sqrt{p}:a,b\in\mathbb{Q}\\}$.

######  Theorem 4.7.

Let $p$ be a square free positive integer. Then there exist two discontinuous
additive functions $A:\mathbb{Q}(\sqrt{p})\to\mathbb{R}$ and
$B:\mathbb{Q}(\sqrt{p})\to\mathbb{R}$ such that the functions $A$ and $B$
fulfill the inequalities

(28) $\displaystyle B(x^{2})B(y^{2})$ $\displaystyle\leq A(xy)^{2}\leq
A(x^{2})A(y^{2}),$ $\displaystyle B(x^{2})B(y^{2})$ $\displaystyle\leq
B(xy)^{2}\leq A(x^{2})A(y^{2})$

for all $x,y\in\mathbb{Q}(\sqrt{p})$.

###### Proof.

Define the functions $A:\mathbb{Q}(\sqrt{p})\to\mathbb{R}$ and
$B:\mathbb{Q}(\sqrt{p})\to\mathbb{R}$ by

$\displaystyle A(a+b\sqrt{p}):=a\qquad\mbox{and}\qquad
B(a+b\sqrt{p}):=b\sqrt{p}\qquad(a,b\in\mathbb{Q}).$

We show that $A$ and $B$ are additive. Indeed, let $x=a_{1}+b_{1}\sqrt{p}$ and
$y=a_{2}+b_{2}\sqrt{p}$ be two arbitrary points of $\mathbb{Q}(\sqrt{p})$,
where $a_{1},a_{2},b_{1},b_{2}\in\mathbb{Q}$. According to the definition of
$A$ and $B$, we have that

$\displaystyle A(x+y)$
$\displaystyle=A(a_{1}+a_{2}+(b_{1}+b_{2})\sqrt{p})=a_{1}+a_{2}=A(x)+A(y)$

and

$\displaystyle B(x+y)$
$\displaystyle=B(a_{1}+a_{2}+(b_{1}+b_{2})\sqrt{p})=(b_{1}+b_{2})\sqrt{p}=B(x)+B(y).$

This proves that $A$ and $B$ are additive, indeed.

Next we prove that $A$ and $B$ satisfy the functional equations in (24) with
$f:=A$, and $g:=B$, where the groupoid $(G,*)$ is equal to
$(\mathbb{Q}(\sqrt{p}),\cdot)$. Indeed, let $x=a_{1}+b_{1}\sqrt{p}$ and
$y=a_{2}+b_{2}\sqrt{p}$ be two arbitrary points of $\mathbb{Q}(\sqrt{p})$,
where $a_{1},a_{2},b_{1},b_{2}\in\mathbb{Q}$. Then

$\displaystyle
A(xy)=A((a_{1}a_{2}+b_{1}b_{2}p)+(a_{1}b_{2}+a_{2}b_{1})\sqrt{p})=a_{1}a_{2}+b_{1}b_{2}p=A(x)A(y)+B(x)B(y)$

and similarly,

$\displaystyle
B(xy)=B((a_{1}a_{2}+pb_{1}b_{2})+(a_{1}b_{2}+a_{2}b_{1})\sqrt{p})=(a_{1}b_{2}+a_{2}b_{1})\sqrt{p}=A(x)B(y)+B(x)A(y).$

Therefore, according to Theorem 4.5, the inequalities (25) and (26) holds,
which prove that the inequalities in (28) are also valid.

Finally, we show that the functions $A$ and $B$ are discontinuous at the point
$u:=1+\sqrt{p}$. By the density of the set $\mathbb{Q}$ in $\mathbb{R}$, there
exists a sequence $(x_{n})$ of rational numbers converging to $u/\sqrt{p}$.
Then the sequence $(x_{n}\sqrt{p})$ converges to $u$. We have that $A(u)=1$
but, for all $n\in\mathbb{N}$, $A(x_{n}\sqrt{p})=0$, therefore $A$ is
discontinuous at $u$. Furthermore, there exists a sequence $(y_{n})$ of
rational numbers, which converges to $u$. Since $B(u)=\sqrt{p}$ and
$B(y_{n})=0$ for all $n\in\mathbb{N}$, thus we can conclude that the function
$B$ is also discontinuous at $u$. ∎

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