# Variable energy flux in turbulence

Mahendra K. Verma  Department of Physics, Indian Institute of Technology
Kanpur, Kanpur 208016, India<EMAIL_ADDRESS>
###### Abstract

In three-dimensional hydrodynamic turbulence forced at large length scales, a
constant energy flux $\Pi_{u}$ flows from large scales to intermediate scales,
and then to small scales. It is well known that for multiscale energy
injection and dissipation, the energy flux $\Pi_{u}$ varies with scales. In
this review we describe this principle and show how this general framework is
useful for describing a variety of turbulent phenomena. Compared to
Kolmogorov’s spectrum, the energy spectrum steepens in turbulence involving
quasi-static magnetofluid, Ekman friction, stable stratification,
magnetohydrodynamics, and solution with dilute polymer. However, in turbulent
thermal convection, in unstably stratified turbulence such as Rayleigh-Taylor
turbulence, and in shear turbulence, the energy spectrum has an opposite
behaviour due to an increase of energy flux with wavenumber. In addition, we
briefly describe the role of variable energy flux in quantum turbulence, in
binary-fluid turbulence including time-dependent Landau-Ginzburg and Cahn-
Hillianrd equations, and in Euler turbulence.

  * May 2020

##  1 Introduction

Turbulence is observed in most natural flows, for example, in atmospheres and
interiors of planets and stars, in oceanic flows, and in stellar and galactic
winds. Many engineering flows, as in air conditioners and combustion engines,
as well as most kitchen flows are turbulent. These complex flows have multiple
components that could be a combination of velocity, temperature, density, and
magnetic fields. The complexities of above turbulent flows appear daunting,
yet, the mathematical models and tools developed over the last two centuries
provide reasonable understanding of such flows. In this review article we will
describe one such tool called variable energy flux.

The nonlinear interactions among the Fourier modes of a turbulent flow cause
energy exchange among the modes [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,
14]. In one of the pioneering works, Kolmogorov [1, 2] showed that when
hydrodynamic turbulence is forced at large scales, the large-scale kinetic
energy is transferred to intermediate scales, called inertial range, and then
to small scales. The flow is homogeneous and isotropic in the inertial range.
For hydrodynamic turbulence, we define kinetic energy flux $\Pi_{u}(k_{0})$,
which is the net energy transfer from the Fourier modes of a wavenumber sphere
of radius $k_{0}$ to the Fourier modes outside the sphere. Kolmogorov [1, 2]
argued that in the inertial range, an absence of forcing and weak dissipation
leads to a constant $\Pi_{u}(k_{0})$. In addition, the inertial-range kinetic
energy spectrum, $E_{u}(k)$, varies as $k^{-5/3}$, also called Kolmogorov’s
energy spectrum [5, 6, 7].

Kolmogorov’s theory of turbulence describes the energy spectrum and flux of
hydrodynamic turbulence without any external force in the intermediate scales.
However this assumption is not valid for many complex flows where the forcing
and/or dissipation are active in the intermediate scales. For example, gravity
acts at all scales in buoyant flows (stably stratified turbulence and thermal
convection); the Lorentz force acts at all scales in MHD turbulence; Ekman
friction dissipates kinetic energy at all scales. Due to the additional
forcing and/or dissipation, the inertial-range energy flux of the such flows
varies with wavenumber. In addition, the inertial-range kinetic energy
spectrum differs from Kolmogorov’s $k^{-5/3}$ spectrum. Interestingly, complex
variations in energy spectrum and flux can be quantified using an equation for
the variable energy flux: $d\Pi_{u}(k)/dk=\mathcal{F}_{u}(k)-D_{u}(k)$, where
$\mathcal{F}_{u}(k)$ is the kinetic energy injection rate by the external
force at wavenumber $k$, and $D_{u}(k)$ is the dissipation rate at $k$ [8, 9,
11, 14]. The framework of variable energy flux, which is the theme of this
review, helps us understand a wide range of turbulent phenomena. We introduce
these topics in this section and detail them in subsequent chapters.

Figure 1:  A figure illustrating variable energy flux. The blue and red arrows
depict $\mathcal{F}_{u}(k)$ and $D_{u}(k)$ respectively. The red curve
represents Kolmogorov’s model for which $\Pi_{u}(k)=$ constant in the inertial
range because $\mathcal{F}_{u}(k)=0$ and $D_{u}(k)=0$. The green curve
represents the case when $\mathcal{F}_{u}(k)>0$ and $d\Pi_{u}(k)/dk>0$, while
the blue curve represents flows with $\mathcal{F}_{u}(k)<0$ and
$d\Pi_{u}(k)/dk<0$.

Gravity acts at all scales in stably stratified turbulence and in thermal
convection and generates variable kinetic energy flux in these systems. For
the stably stratified turbulence with moderate stratification, Bolgiano [15]
and Obukhov [16] showed that $\mathcal{F}_{u}(k)<0$ and $d\Pi_{u}(k)/dk<0$ due
to a conversion of kinetic energy to potential energy by buoyancy. In
particular, $\Pi_{u}(k)\sim k^{-4/5}$ and $E_{u}(k)\sim k^{-11/5}$ (different
from Kolmogorov’s $k^{-5/3}$ spectrum). See Fig. 1 for an illustration.

The physics of thermal convection, however, is quite different from the stably
stratified turbulence even though the equations for the two systems are the
same. This is because thermal convection is unstable, while stably stratified
turbulence is stable. In the inertial range of turbulent convection, thermal
plumes drive the velocity field. Hence, $\mathcal{F}_{u}(k)>0$ leading to
$d\Pi_{u}(k)/dk>0$. See Fig. 1 for an illustration. Detailed studies, however,
show that for small and moderate Prandtl numbers, $\mathcal{F}_{u}(k)$ is
primarily concentrated at small wavenumbers, as in Kolmogorov’s theory of
hydrodynamic turbulence. Consequently, the kinetic energy spectrum and flux of
turbulent convection are similar to those predicted by Kolmogorov’s turbulence
theory [17, 18, 19]. These observations indicate the usefulness of variable
energy flux in modelling buoyant flows.

Variable energy flux is also useful for describing magnetohydrodynamic (MHD)
turbulence and dynamo. Here, the Lorentz force, which is active at all scales,
transfers energy from the velocity field to the magnetic field. These energy
transfers are responsible for the enhancement of magnetic field in
astrophysical objects via dynamo mechanism [20, 21, 14]. These transfers also
lead to a reduction in kinetic energy flux and an enhancement of magnetic
energy flux with wavenumber [14]. In quasi-static MHD turbulence, Joule
dissipation is significant at all scales. Consequently, the inertial-range
kinetic energy flux decreases with $k$, and $E_{u}(k)$ is steeper than
Kolmogorov’s $k^{-5/3}$ spectrum [22, 23, 24]. Anas and Verma [25] showed that
the variable energy formalism successfully explains the spectral steepening
observed in numerical simulations and experiments [22, 23, 24]. For very
strong magnetic field, the Joule dissipation steepens $E_{u}(k)$ even further
and yields an exponential spectrum, which is $\exp(-ck)$, where $c$ is a
constant.

In shear turbulence, shear acts at small wavenumbers and injects energy to the
flow. This injected energy leads to an increase in the kinetic energy flux
with $k$. This variation in the kinetic energy flux may be responsible for the
$1/f$ noise reported for many systems [26, 27, 28]. On the other hand, Ekman
friction acts at all scales and depletes the kinetic energy flux in the
inertial range leading to a steeper kinetic energy spectrum than $k^{-5/3}$
[29, 30].

In the viscous range of hydrodynamic turbulence, $d\Pi_{u}(k)/dk=-2\nu
k^{2}E_{u}(k)$, where $\nu$ is the kinematic viscosity. The viscous
dissipation leads to steep decline in $\Pi_{u}(k)$ and $E_{u}(k)$. Reseachers
[31, 32, 33, 11] have attempted to model the energy spectrum in this range. In
particular, Pao [31] derived that in the inertial-dissipation range of 3D
hydrodynamic turbulence, $\Pi_{u}(k)$ and $E_{u}(k)k^{5/3}$ vary as
$\exp\\{-c(k/k_{d})^{4/3}\\}$, where $c$ is a constant, and $k_{d}$ is the
Kolmogorov wavenumber. Pao’s model for the 3D inertial-dissipation range has
been extended to 2D hydrodynamic turbulence [14, 34]. Falkovich [35] and Verma
and Donzis [36] showed that the energy flux plays an important role in the
bottleneck effect. In this review we argue that the bottleneck effect may
possibly be due to a sudden suppression of the energy flux in the dissipation
range.

In most complex flows, one or several secondary fields are coupled to the
velocity field. Some examples of secondary fields: the density field in
buoyant flows, the temperature field in thermal convection, the magnetic field
in MHD turbulence, and the conformation tensor of polymers in polymeric flows.
The nonlinear term associated with the secondary field also induces scalar
energy transfer or scalar energy flux. For example, the potential energy
fluxes of stably stratified turbulence and turbulent thermal convection are
constant. In addition, a coupling between the velocity field with the
secondary field often yields energy exchange between the velocity field and
the secondary field, as well as variability in the secondary energy flux [37,
38, 39]. This phenomena is related to the turbulent drag reduction in
polymeric turbulence [40, 41, 42] and in MHD turbulence [43].

The enstrophy ($\int d{\bf r}\frac{1}{2}|\bm{\omega}|^{2}$, where
$\bm{\omega}$ is the vorticity field) and kinetic helicity ($\int d{\bf
r}\frac{1}{2}|[{\bf u}\cdot\bm{\omega}]$) are important quantities of
hydrodynamic turbulence. The fluxes of these quantities exhibit interesting
properties. For example, the enstrophy flux has a similar structure as those
of kinematic dynamo [20, 14]. In MHD turbulence, the flux of magnetic helicity
too exhibit interesting properties [21].

Many turbulent systems, including buoyancy-driven turbulence, MHD turbulence,
and rotating turbulence are anisotropic [44, 14]. Under strong external field,
stably stratified turbulence, MHD turbulence, and rotating turbulence become
quasi-two-dimensional with $|{\bf u}_{\perp}|\gg u_{\parallel}$, where ${\bf
u}_{\perp}$ and $u_{\parallel}$ are the perpendicular and parallel components
of the velocity field in relation to the anisotropy direction. On the other
hand, in thermal convection, $|{\bf u}_{\perp}|\ll u_{\parallel}$. In these
flows, the pressure acts as a mediator for the energy exchange between the
parallel and perpendicular components of the velocity field [24, 14, 45]. The
energy fluxes of ${\bf u}_{\perp}$ and $u_{\parallel}$ provide useful insights
into the anisotropic nature of such flows. The secondary fields too have
similar anisotropic fluxes, but these quantities have not been computed so
far.

Quantum systems, such as superfluids and Bose-Einstein condensate, too exhibit
turbulent behaviour for a parameter range. Researchers have studied energy
spectra and fluxes of these systems. For example, in Helium-4, normal and
superfluid components interact with each other that leads to variability in
their energy fluxes (e.g., [46, 47, 48, 49] and references therein). The
scenario is more complicated in Helium-3 that lacks normal component; here
phonon coupling at small scales is expected to provide the dissipation [50,
51].

The energy flux is a useful diagnostic tool for other nonequilibrium systems
as well. For example, in binary-mixture turbulence, the energy flux provides
valuable insights into the field dynamics, especially phase separation and
coarsening [52, 53]. Researchers have also employed time-dependent Ginzburg-
Landau and Cahn-Hilliard equations to model the coarsening process [54, 55],
where the energy flux is proving to be a very useful tool [52, 53].

Lee [56] and Kraichnan [57] showed that inviscid hydrodynamic turbulence
$(\nu=0)$ exhibits equilibrium behaviour (also called absolute equilibrium).
The energy flux for this case vanishes due to the detailed energy balance
among the Fourier modes [8]. The evolution of such systems depends quite
critically on the initial condition. For example, Cichowlas et al. [58] showed
that Taylor-Green vortex as an initial condition yields a mixture of
$k^{-5/3}$ and $k^{2}$ spectra. On the contrary, for white noise as initial
condition, the system exhibits $k^{2}$ spectrum for the whole range of
wavenumbers [59, 60]. The former system exhibits variable energy flux, but the
latter system (equilibrium configuration) has no energy flux. The absolute
equilibrium theory of hydrodynamics has been extended to MHD turbulence [61,
62], quantum turbulence [63, 64], Burger turbulence [65], and other forms of
turbulence [8, 6].

So far, the energy flux has been defined for the wavenumber space. Note,
however, that Kolmogorov [1, 2] formulated a relationship between the energy
flux and the real-space third-order structure function, which is related to
the velocity difference between two points. This theory of Kolmogorov has been
generalized to passive scalar turbulence [66], MHD turbulence [67], rotating
turbulence, thermal convection [68], etc. Biferale and Procaccia [69], Arad et
al. [70], and Danaila et al. [71] have attempted to generalize the above
Kolmogorov’s theory to anisotropic systems. Since structure functions and
associated energy flux are covered in great detail in many books [8, 6] and
reviews [72, 73], they are not covered in detail in this review.

The aforementioned turbulent systems have been studied in great detail in the
past, including in books [3, 4, 5, 6, 7, 8, 9, 10, 11, 14] and review articles
[74]. The equation of variable energy flux,
$d\Pi_{u}(k)/dk=\mathcal{F}_{u}(k)-D_{u}(k)$, too appears in several
textbooks, for example [6, 7, 8, 9]. In this review article, we illustrate how
various turbulent phenomena can be connected via variable energy flux. We also
present the scaling laws and energy fluxes of many turbulent systems
thematically in the framework of variable energy flux. This perspective
provides valuable and unique insights. For example, using variable energy
flux, it has been shown that the dynamics of turbulent thermal convection is
very different from that of stably stratified turbulence, contrary to a
popular view that the Bolgiano-Obukhov scaling for stably stratified
turbulence [15, 16] also works for turbulent thermal convection [75, 76, 77].
The contrast in the energy fluxes of the two systems played a key role in the
resolution of this critical puzzle.

The structure of the review is as follows. Section 2 introduces the energy
transfers and energy flux in hydrodynamic turbulence. The formalism of
variable energy flux is presented in Sec. 3. In this section, we present
various examples, including variable energy fluxes in the dissipation range of
hydrodynamic turbulence, in quasi-static MHD turbulence, and in shear
turbulence. In Sec. 4 we derive the energy flux for the secondary field that
is advected by velocity field and show how it could become variable when a
multiscale force is applied to the secondary field. Here, we also describe the
energy exchange between the secondary field and the velocity field, as well as
those between the field components in anisotropic turbulence. Section 5
contains discussions on turbulence in stably stratified flows and thermal
convection. Sections 6 describes variable energy fluxes in MHD turbulence and
in a turbulent flow with dilute polymer. Here, we discuss several exact
relations among the fluxes of MHD turbulence. Sections 8 describes the fluxes
associated with enstrophy and kinetic helicity, while Sec. 9 contains
discussions on 2D turbulence. Variable energy fluxes in dissipation-less
systems are discussed in Sec. 10, while those in quantum turbulence and
binary-mixture turbulence are discussed in Sec. 11. Section 12 summarises
Kolmogorov’s four-fifth law for hydrodynamic turbulence, as well as the laws
for passive-scalar turbulence and MHD turbulence. We conclude in Sec. 13.

##  2 Energy flux in hydrodynamics

In this section, we introduce the kinetic energy flux of hydrodynamics. We
start with the basic equations of hydrodynamics in real and Fourier spaces.

###  2.1 Basic equations of hydrodynamics

The Navier-Stokes (NS) equations given below describe an incompressible flow
[78]:

$\displaystyle\frac{\partial{\bf u}}{\partial t}+({\bf u}\cdot\nabla){\bf
u}=-\frac{1}{\rho}\nabla p+{\bf F}_{u}+{\bf F}_{\mathrm{LS}}+\nu\nabla^{2}{\bf
u},$ (1) $\displaystyle\nabla\cdot\mathbf{u}=0,$ (2)

where ${\bf u}({\bf r},t)$ is the velocity field, $p({\bf r},t)$ is the
pressure field, ${\bf F}_{\mathrm{LS}}$ is the large-scale external force,
${\bf F}_{u}$ is the force field such as buoyancy, and $\nu$ is the kinematic
viscosity. Under incompressible limit, the fluid density $\rho$ can be treated
as a constant. In this review, without loss of generality, $\rho$ is taken to
be unity. We distinguish ${\bf F}_{\mathrm{LS}}$ and ${\bf F}_{u}$ to clearly
demarcate the energy transfers from these forces. Note that ${\bf F}_{u}$
could be a function of the velocity field; for example, Ekman friction is
$-\alpha{\bf u}$, where $\alpha$ is a constant. For an incompressible flow,
the pressure field is determined using

$\displaystyle
p=-\nabla^{-2}\nabla\cdot\left[(\mathbf{u}\cdot\nabla)\mathbf{u}-{\bf
F}_{u}-{\bf F}_{\mathrm{LS}}\right].$ (3)

The ratio of the nonlinear term $({\bf u}\cdot\nabla){\bf u}$ and the viscous
term $\nu\nabla^{2}{\bf u}$ is called Reynolds number $\mathrm{Re}$, which is
${UL}/{\nu}$, where $U,L$ are the large-scale velocity and length
respectively.

In three-dimensional (3D) inviscid ($\nu=0$) hydrodynamics, for a periodic or
vanishing boundary condition, the total kinetic energy, $E_{u}=\frac{1}{2}\int
d{\bf r}u^{2}$, and the total kinetic helicity, $H_{K}=\frac{1}{2}\int d{\bf
r}({\bf u}\cdot\bm{\omega})$, are conserved [6, 8, 9]. Here,
$\bm{\omega}=\nabla\times\bf{u}$ is the vorticity field. In two-dimensional
(2D) hydrodynamics, the total kinetic energy and the total enstrophy
$\frac{1}{2}\int d{\bf r}|\bm{\omega}|^{2}$ are conserved. Note that the
physics of turbulence in 2D and 3D are quite different. In this review, we
will focus on the fluxes of kinetic energy and associated secondary energy in
3D flows. The fluxes of other quantities, such as enstrophy and kinetic
helicity, will be discussed briefly.

The multiscale energy transfers and fluxes are conveniently described using
the velocity Fourier modes. For compactness, we denote the Fourier transform
of ${\bf u(r)}$ using ${\bf u(k)}$; here ${\bf r}$ and ${\bf k}$ denote the
real and Fourier space coordinates respectively. In a Fourier space
convolution, the other wavenumbers are denoted by ${\bf p}$ and ${\bf q}$. The
wavenumbers are discrete for a flow confined in a finite box, but they form a
continuum for a flow in an infinite box.

The Navier-Stokes equations are transformed in Fourier space as [6, 8, 9, 19]

$\displaystyle\frac{d}{dt}{\bf u}(\mathbf{k})+{\bf N}_{u}({\bf k})=-i{\bf
k}p(\mathbf{k})+{\bf F}_{u}({\bf k})+{\bf F}_{\mathrm{LS}}({\bf k})-\nu
k^{2}{\bf u}(\mathbf{k}),$ (4) $\displaystyle{\bf k\cdot u}(\mathbf{k})=0,$
(5)

where

$\displaystyle{\bf N}_{u}({\bf k})$ $\displaystyle=$ $\displaystyle i\sum_{\bf
p}{\bf\\{k\cdot u(q)\\}u(p)}$ (6)

is the Fourier transform of the nonlinear term ${\bf(u\cdot\nabla)u}$. Here
${\bf q=k-p}$. The equation for the pressure mode $p(\mathbf{k})$ is

$\displaystyle p(\mathbf{k})=\frac{i}{k^{2}}{\bf k}\cdot\\{{\bf N}_{u}({\bf
k})-{\bf F}_{u}({\bf k})-{\bf F}_{\mathrm{LS}}({\bf k})\\}.$ (7)

We define modal kinetic energy for wavenumber ${\bf k}$ as $E_{u}({\bf
k})=\frac{1}{2}|{\bf u}({\bf k})|^{2}$. Note that Parseval’s theorem yields
the following relation for the total kinetic energy:

$\displaystyle E_{u}=\frac{1}{2}\langle|{\bf u}({\bf
r})|^{2}\rangle=\frac{1}{\mathrm{Vol}}\int d{\bf r}\frac{1}{2}|{\bf u}({\bf
r})|^{2}=\sum_{\bf k}\frac{1}{2}|{\bf u}({\bf k})|^{2},$ (8)

where ${\mathrm{Vol}}$ is the volume of the box. We derive the following
dynamical equation for $E_{u}({\bf k})$ by performing a scalar product of Eq.
(4) with ${\bf u^{*}(k)}$ and adding the resulting equation with its complex
conjugate [6, 8, 9, 19]:

$\displaystyle\frac{d}{dt}E_{u}(\mathbf{k})$ $\displaystyle=$ $\displaystyle
T_{u}({\bf k})+\mathcal{F}_{u}{\bf k})+\mathcal{F}_{\mathrm{LS}}{\bf
k})+D_{u}({\bf k})$ (9) $\displaystyle=$ $\displaystyle\sum_{\bf
p}\Im\left[{\bf\\{k\cdot u(q)\\}\\{u(p)\cdot u^{*}(k)\\}}\right]+\Re[{\bf
F}_{u}({\bf k})\cdot{\bf u^{*}(k)}]$ $\displaystyle+\Re[{\bf
F}_{\mathrm{LS}}({\bf k})\cdot{\bf u^{*}(k)}]-2\nu k^{2}E_{u}({\mathbf{k}}),$

where ${\bf q=k-p}$, and $\Re[.]$, $\Im[.]$ stand respectively for the real
and imaginary parts of the argument. In the above equation, $T_{u}({\bf k})$
is the nonlinear energy transfer from all the Fourier modes to ${\bf u(k)}$;
$\mathcal{F}_{u}({\bf k}),\mathcal{F}_{\mathrm{LS}}{\bf k})$ are the
respective energy supply rates from ${\bf F}_{u}$ and ${\bf F}_{\mathrm{LS}}$
to ${\bf u(k)}$; and $D_{u}({\bf k})$ is the viscous dissipation rate of ${\bf
u(k)}$.

The nonlinear interactions of Eq. (9) induce complex energy transfers among
the Fourier modes. However, a peep into a single wavenumber triad provides
interesting insights into the nature of nonlinear interactions, which will be
described below.

###  2.2 Triadic energy transfers and energy flux in hydrodynamics

Kraichnan [79] focussed on a pair of interacting wavenumber triads,
${\bf(k,p,q)}$ and ${\bf(-k,-p,-q)}$ with a condition that ${\bf k=p+q}$. The
corresponding Fourier modes are ${\bf u(k),u(p),u(q)}$ and their complex
conjugates: ${\bf u^{*}(k),u^{*}(p),u^{*}(q)}$. For convenience, we set
$\nu=0$ to suppress the viscous dissipation rate, which is a trivial linear
term of Eq. (9). In addition, we assume that ${\bf F}_{u}=0$ and ${\bf
F}_{\mathrm{LS}}=0$.

The wavenumbers ${\bf k,p,q}$ of Eq. (9) are not symmetric (note ${\bf
k=p+q}$). The formulas for the energy transfers are best expressed using a
symmetric set $({\bf k^{\prime},p,q})$ obeying a constraint, ${\bf
k^{\prime}+p+q}=0$. Note that ${\bf k^{\prime}=-k}$. For this triad, the
dynamical equation for the modal energy $E_{u}({\bf k^{\prime}})$ is

$\displaystyle\frac{d}{dt}E_{u}({\bf k^{\prime}})$ $\displaystyle=$
$\displaystyle-\Im\left[{\bf\\{k^{\prime}\cdot u(q)\\}\\{u(p)\cdot
u(k^{\prime})\\}+\\{k^{\prime}\cdot u(p)\\}\\{u(q)\cdot
u(k^{\prime})\\}}\right]$ (10) $\displaystyle=$ $\displaystyle S^{uu}({\bf
k^{\prime}|p,q}).$

The equations for $E_{u}({\bf p})$ and $E_{u}({\bf q})$ are written in a
similar manner.

The function $S^{uu}({\bf k^{\prime}|p,q})$, called the combined energy
transfer, represents the net kinetic energy transfer from modes ${\bf u(p)}$
and ${\bf u(q)}$ to ${\bf u(k^{\prime})}$. Note that the energy of the mode
${\bf u(k^{\prime})}$ grows at a rate of $dE_{u}({\bf k^{\prime}})/dt$. Using
Eq. (10) and the incompressibility condition, ${\bf k^{\prime}\cdot
u(k^{\prime})}=0$, we derive that [79, 6]

$\displaystyle S^{uu}({\bf k^{\prime}|p,q})+S^{uu}({\bf
p|q,k^{\prime}})+S^{uu}({\bf q|k^{\prime},p})=0$ (11)

that leads to the conservation of the total energy within a triad.

Even though Kraichnan’s combined energy transfer formula has been widely used,
it does not provide individual energy transfers among the Fourier modes. This
task was first achieved by Dar et al. [37] who derived the mode-to-mode energy
transfer from mode ${\bf u(p)}$ to mode ${\bf u(k^{\prime})}$ with the
mediation of mode ${\bf u(q)}$ as [38].

$\displaystyle S^{uu}({\bf k^{\prime}|p|q})=-\Im\left[{\bf\\{k^{\prime}\cdot
u(q)\\}\\{u(p)\cdot u(k^{\prime})\\}}\right].$ (12)

The formula contains a scalar product between the receiver mode ${\bf
u(k^{\prime})}$ and the giver mode ${\bf u(p)}$, and another scalar product
between the receiver wavenumber ${\bf k^{\prime}}$ and the mediator mode ${\bf
u(q)}$. The successive arguments of $S^{uu}$ are receiver, giver, and mediator
wavenumbers respectively. In terms of wavenumbers ${\bf k,p,q}$ with ${\bf
k=p+q}$, the above formula is written as

$\displaystyle S^{uu}({\bf k|p|q})=\Im\left[{\bf\\{k\cdot u(q)\\}\\{u(p)\cdot
u^{*}(k)\\}}\right].$ (13)

Note that the mode-to-mode energy transfer functions satisfy the following
properties:

$\displaystyle S^{uu}({\bf k^{\prime}|p|q})+S^{uu}({\bf
k^{\prime}|q|p})=S^{uu}({\bf k^{\prime}|p,q}),$ (14) $\displaystyle
S^{uu}({\bf k^{\prime}|p|q})=-S^{uu}({\bf p|k^{\prime}|q}).$ (15)

The latter property follows from the incompressibility condition: ${\bf
k^{\prime}\cdot u(k^{\prime})}=0$. Dar et al. [37] and Verma [38] showed that
Eq. (12) satisfies Eqs. (14, 15), but it is not a unique solution to Eqs. (14,
15). A circulating energy transfer that traverses along wavenumbers ${\bf
p\rightarrow k^{\prime}\rightarrow q\rightarrow p}$ could be added to the
respective mode-to-mode energy transfers of Eq. (12) without violating Eqs.
(14, 15). The circulating transfer enters and leaves a mode, hence they do not
alter the energy flux, which is a measurable quantity. Therefore, the
circulating transfer could be safely ignored. Later, using tensor analysis and
symmetry arguments, Verma [14] showed that Eq. (12) provides a unique formula
for the mode-to-mode energy transfer.

A fluid flow has many Fourier modes. Hence, the net energy transfer to the
mode ${\bf u(k)}$ is a sum of energy transfers from all other modes. In terms
of mode-to-mode energy transfer, the net energy transfer to ${\bf u(k)}$ is
[37, 38]

$\displaystyle\frac{d}{dt}E_{u}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\sum_{\bf p}S^{uu}({\bf k|p|q})+\mathcal{F}_{u}({\bf
k})+\mathcal{F}_{\mathrm{LS}}({\bf k})-D_{u}({\bf k}),$ (16)

where ${\bf q=k-p}$. A comparison of Eq. (16) and Eq. (9) shows that

$\displaystyle T_{u}({\bf k})=\sum_{\bf p}S^{uu}({\bf k|p|q}),$ (17)

which is the net energy transfer to mode ${\bf u(k)}$ by nonlinearity. Also
note that for any wavenumber region $A$,

$\displaystyle\sum_{{\bf p}\in A}\sum_{{\bf k}\in A}S^{uu}({\bf k|p|q})=0.$
(18)

The above relation hinges on Eq. (15).

A very useful quantity in turbulence is the energy flux $\Pi_{u}(k_{0})$,
which is the total energy transfer from all the modes inside a wavenumber
sphere of radius $k_{0}$ to all the modes outside the sphere. Starting from
Kraichnan [79], researchers have derived a number of formulas for
$\Pi_{u}(k_{0})$ [1, 2, 79, 6, 9, 37, 38, 8, 80]. Here we present the flux
formula in terms of mode-to-mode energy transfer, which is

$\displaystyle\Pi_{u}(k_{0})=\sum_{p\leq k_{0}}\sum_{k>k_{0}}S^{uu}({\bf
k|p|q})=\sum_{p\leq k_{0}}\sum_{k>k_{0}}\Im\left[{\bf\\{k\cdot
u(q)\\}\\{u(p)\cdot u^{*}(k)\\}}\right]$ (19)

with ${\bf q=k-p}$. In the sum, the giver wavenumber ${\bf p}$ is inside the
wavenumber sphere, while the receiver wavenumber ${\bf k}$ is outside the
sphere. For the above formula, it does not matter whether the mediator
wavenumber ${\bf q}$ is inside or outside the sphere. This liberty does not
exist for the combined energy transfer formula in which both ${\bf u(p)}$ and
${\bf u(q)}$ supply energy to ${\bf u(k^{\prime})}$. The ambiguity of ${\bf
u(q)}$ in the combined energy transfer poses a serious challenge for computing
shell-to-shell energy transfer [37, 38].

We derive another set of formulas for the energy flux using $T_{u}({\bf k})$.
We sum the terms of Eq. (16) over all the modes inside the sphere that yields
[79, 6, 8, 9]

$\displaystyle\frac{d}{dt}\sum_{k\leq k_{0}}E_{u}({\bf k})=\sum_{k\leq
k_{0}}T_{u}({\bf k})+\sum_{k\leq k_{0}}\mathcal{F}_{u}({\bf k})+\sum_{k\leq
k_{0}}\mathcal{F}_{\mathrm{LS}}({\bf k})-\sum_{k\leq k_{0}}D_{u}({\bf k}).$
(20)

To the wavenumber sphere of radius $k_{0}$, $\sum_{k\leq
k_{0}}\mathcal{F}_{u}({\bf k})$ and $\sum_{k\leq
k_{0}}\mathcal{F}_{\mathrm{LS}}({\bf k})$ are the net energy supply rates by
${\bf F}_{u}$ and ${\bf F}_{\mathrm{LS}}$ respectively; and $\sum_{k\leq
k_{0}}D_{u}({\bf k})$ is the net viscous dissipation rate in the sphere (see
Fig. 2). The sum $\sum_{k\leq k_{0}}T_{u}({\bf k})$ is the net energy transfer
due to the nonlinear term from the modes outside the sphere to the modes
inside the sphere. Hence, by definition,

$\displaystyle\Pi_{u}(k_{0})=-\sum_{k\leq k_{0}}T_{u}({\bf k}).$ (21)

Using Eqs. (20, 21) we derive

$\displaystyle\Pi_{u}(k_{0})=-\frac{d}{dt}\sum_{k\leq k_{0}}E_{u}({\bf
k})+\sum_{k\leq k_{0}}\mathcal{F}_{u}({\bf k})+\sum_{k\leq
k_{0}}\mathcal{F}_{\mathrm{LS}}({\bf k})-\sum_{k\leq k_{0}}D_{u}({\bf k}).$
(22)

Similarly, using the following equation,

$\displaystyle\frac{d}{dt}\sum_{k>k_{0}}E_{u}({\bf
k})=\sum_{k>k_{0}}T_{u}({\bf k})+\sum_{k>k_{0}}\mathcal{F}_{u}({\bf
k})+\sum_{k>k_{0}}\mathcal{F}_{\mathrm{LS}}({\bf k})-\sum_{k>k_{0}}D_{u}({\bf
k}),$ (23)

one obtains $\Pi_{u}(k_{0})=\sum_{k>k_{0}}T_{u}({\bf k})$ [6, 14]. Verma et
al. [81] employed Eq. (22) to compute the energy fluxes of magnetohydrodynamic
turbulence.

Figure 2: Illustration of the energy flux $\Pi_{u}(k_{0})$, the kinetic energy
injection rates by $\mathcal{F}_{u}({\bf k})$ and
$\mathcal{F}_{\mathrm{LS}}({\bf k})$ (wheels), and the local viscous
dissipation rate $D_{u}({\bf k})$ (wavy lines). The viscous dissipation is
negligible at small wavenumbers.

Interestingly, $T_{u}(k)$-based energy flux formula can be used to systems
whose nonlinear interactions differ from those of incompressible
hydrodynamics. For example, for dissipation-less Burgers equation,
$\partial_{t}u=-\partial_{x}u^{2}/2$, the energy equation is

$\frac{d}{dt}\frac{1}{2}|u(k)|^{2}=\sum_{p}\Im[ku(k-p)u(p)u^{*}(k)]=T(k).$
(24)

Hence, the energy flux $\Pi(k_{0})=-\sum_{k<k_{0}}T(k)$. We can also use the
above procedure to compute the energy flux for $\phi^{4}$ theory where the
interaction is quartic and $T(k)\sim\phi({\bf k}_{1})\phi({\bf
k}_{2})\phi({\bf k}_{3})\phi({\bf k}_{4})$ with ${\bf k}_{1}+{\bf k}_{2}+{\bf
k}_{3}+{\bf k}_{4}=0$ [82, 83]. Such schemes are useful for modelling energy
transfers in binary fluids, and time-dependent Ginzburg-Landau and Cahn-
Hilliard equations. (see Sec. 11).

Kraichnan [79], Frisch [8], Alexakis et al. [80, 39] and others have derived
formulas for the energy flux based on nonlinear interactions in Fourier space.
These formulas will not be discussed here due to lack of space and due to
similarities with those discussed above. Kolmogorov’s formula for the energy
flux [1, 2], which is based on the third-order structure function, will be
discussed in Sec. 12.

An important point to note: the formulas of Eqs. (19, 21) are applicable to
isotropic as well as anisotropic flows. For anisotropic flows, the spectrum
$E_{u}(\textbf{k})$ depends on the angle between k and the anisotropic axis.
However, the energy flux, which involves a sum over the modes of a sphere, is
well-defined for such flows. Due to these reasons, energy flux remains an
important quantity for buoyant and MHD turbulence, even though such flows
could be strongly anisotropic.

The formalism of kinetic energy flux can be generalized to other energy
transfers. For example, the net energy transfer from wavenumber region $A$ to
wavenumber region $B$ is

$T^{u,A}_{u,B}=\sum_{{\bf p}\in A}\sum_{{\bf k}\in B}\Im\left[{\bf\\{k\cdot
u(q)\\}\\{u(p)\cdot u^{*}(k)\\}}\right],$ (25)

which can be computed numerically using Fast Fourier Transform (FFT) [37]. In
addition to the energy flux, the other two popular measures of energy
transfers are shell-to-shell energy transfer and ring-to-ring energy transfer
[37, 84, 14, 10], but these topics are beyond the scope of this review.

In the next section, we will show how the energy flux varies with wavenumbers
due to multiscale external force and dissipation.

##  3 Variable energy flux

The kinetic energy flux varies with wavenumbers in the presence of multiscale
forcing and dissipation [6, 8, 9, 11, 19]. In this review we summarise how
past works employed this observation to deduce interesting properties of
turbulence. We start this section with an equation for the variable energy
flux. In this section we assume the wavenumber $k$ to be a continuous
variable.

###  3.1 Formalism

Rewriting Eq. (20) for spheres of radii $k$ and $k+dk$, and then taking a
difference between the two equations yields

$\displaystyle\frac{d}{dt}\sum_{k<k^{\prime}\leq k+dk}E_{u}({\bf k^{\prime}})$
$\displaystyle=$ $\displaystyle\sum_{k<k^{\prime}\leq k+dk}T_{u}({\bf
k^{\prime}})+\sum_{k<k^{\prime}\leq k+dk}\mathcal{F}_{u}({\bf k^{\prime}})$
(26) $\displaystyle+\sum_{k<k^{\prime}\leq k+dk}\mathcal{F}_{\mathrm{LS}}({\bf
k^{\prime}})-\sum_{k<k^{\prime}\leq k+dk}D_{u}({\bf k^{\prime}}).$

Using $\sum_{k<k^{\prime}\leq k+dk}T_{u}({\bf
k^{\prime}})=[-\Pi_{u}(k+dk)+\Pi_{u}(k)]$ and taking the limit $dk\rightarrow
0$, we obtain the following evolution equation for one-dimensional energy
spectrum $E_{u}(k)$ [6, 8, 9, 11, 19]:

$\displaystyle\frac{\partial}{\partial t}E_{u}(k,t)$ $\displaystyle=$
$\displaystyle-\frac{\partial}{\partial
k}\Pi_{u}(k,t)+\mathcal{F}_{u}(k,t)+\mathcal{F}_{\mathrm{LS}}(k,t)-D_{u}(k,t),$
(27)

where

$\displaystyle E_{u}(k)dk$ $\displaystyle=$
$\displaystyle\sum_{k<k^{\prime}\leq k+dk}E_{u}({\bf k^{\prime}}),$ (28)
$\displaystyle\mathcal{F}_{u}(k)dk$ $\displaystyle=$
$\displaystyle\sum_{k<k^{\prime}\leq k+dk}\Re[{\bf F}_{u}({\bf
k^{\prime}})\cdot{\bf u^{*}(k^{\prime})}],$ (29)
$\displaystyle\mathcal{F}_{\mathrm{LS}}(k)dk$ $\displaystyle=$
$\displaystyle\sum_{k<k^{\prime}\leq k+dk}\Re[{\bf F}_{\mathrm{LS}}({\bf
k^{\prime}})\cdot{\bf u^{*}(k^{\prime})}],$ (30) $\displaystyle D_{u}(k)dk$
$\displaystyle=$ $\displaystyle 2\nu\sum_{k<k^{\prime}\leq k+dk}k^{\prime
2}E_{u}({\mathbf{k}}^{\prime}).$ (31)

Figure 3 illustrates the above quantities for a wavenumber shell whose inner
and outer radii are $k$ and $k+dk$ respectively. Also note that $T_{u}(k,t)$
is defined as [5, 6]

$\displaystyle T_{u}(k,t)=-\frac{\partial}{\partial k}\Pi_{u}(k,t).$ (32)

Equation (27) describes how $\Pi_{u}(k)$ varies with $k$ in an unsteady flow.

Figure 3:  A figure illustrating the energy flux difference,
$\Pi_{u}(k+dk)-\Pi_{u}(k)$; the energy injection rates $\mathcal{F}_{u}(k)dk$
and $\mathcal{F}_{\mathrm{LS}}(k)dk$ represented by wheels; and the viscous
dissipation rate $D_{u}(k)dk$ represented by wavy lines. Refer to Eq. (27).

In this review we study the behaviour of the kinetic energy flux under a
steady state. Setting $\partial E_{u}(k,t)/\partial t=0$ in Eq. (26) yields
the following equation for the wavenumber shell of radius $k$:

$\frac{d}{dk}\Pi_{u}(k)=\mathcal{F}_{u}(k)+\mathcal{F}_{\mathrm{LS}}(k)-D_{u}(k).$
(33)

That is, the energy flux $\Pi_{u}(k)$ varies with $k$ due to the energy
injection rates $\mathcal{F}_{u}(k)$ and $\mathcal{F}_{\mathrm{LS}}(k)$, and
the viscous dissipation rate $D_{u}(k)$. We remark that Eq. (33), derived
using the energy conservation, is an exact relation in a statistical sense.
That is, the quantities $\Pi_{u}(k)$, $\mathcal{F}_{u}(k)$,
$\mathcal{F}_{\mathrm{LS}}(k)$, and $D_{u}(k)$ may fluctuate around their
mean, but Eq. (33) holds on an average. This relation is analogous to
Kolmogorov’s four-fifth law, which is also related to the energy conservation
[8, 14]. Note however that Eq. (33) is valid for anisotropic flows as well,
which is the not the case for Kolmogorov’s four-fifth law.

Figure 4: Four categories of $\Pi_{u}(k)$ depending on the inertial-range
forcing $\mathcal{F}_{\mathrm{in}}(k)$: (a) $\mathcal{F}_{\mathrm{in}}(k)=0$
and $\Pi_{u}(k)=0$ (equilibrium); (b) $\mathcal{F}_{\mathrm{in}}(k)=0$ and
$\Pi_{u}(k)=C>0$ (nonequilibrium); (c)
$\mathcal{F}_{\mathrm{in}}(k)<0\Rightarrow d\Pi_{u}(k)/dk<0$; (d)
$\mathcal{F}_{\mathrm{in}}(k)>0\Rightarrow d\Pi_{u}(k)/dk>0$. Viscous
dissipation $D_{u}(k)$ dominates the nonlinear term beyond wavenumber
$k_{\mathrm{DI}}$.

In the intermediate wavenumber range, the nonlinear term dominates the viscous
term, a reason for which this is called the inertial range. In addition,
$\mathcal{F}_{\mathrm{LS}}(k)=0$ and $D_{u}(k)\approx 0$ in this range.
Therefore,

$\frac{d}{dk}\Pi_{u}(k)=\mathcal{F}_{u,\mathrm{in}}(k),$ (34)

where the subscript “in” stands for the inertial range. Thus, the energy flux
varies due to $\mathcal{F}_{u,\mathrm{in}}(k)$. Behaviour of $\Pi_{u}(k)$ can
be classified into the following four categories:

  1. 1.

$\mathcal{F}_{u,\mathrm{in}}(k)=0$ and $\Pi_{u}(k)=0$: This case corresponds
to the absolute equilibrium scenario of Euler’s equation (Navier Stokes
equation with $\nu=0$) [56, 57]. For this case, the average energy exchange
between any two Fourier modes is zero. That is, $\left\langle S^{uu}({\bf
k^{\prime}|p|q})\right\rangle=0$ for any triad, thus satisfying detailed
balance of kinetic energy transfer. Therefore, as in thermodynamics, the modal
kinetic energy spectrum $E_{u}({\bf k})=$ constant, leading to $E_{u}(k)\sim
k^{2}$ and $\sim k$ for 3D and 2D hydrodynamic turbulence respectively [56,
57, 12, 6, 74, 38]. We will discuss these cases in more detail in Sec. 10. A
related phenomenon to this cases is flux loop cascade [85, 86], which will be
discussed in Sec. 9.

  2. 2.

$\mathcal{F}_{u,\mathrm{in}}(k)=0$ and $\Pi_{u}(k)=C>0$: This regime is
described by Kolmogorov’s theory of turbulence. For this nonequilibrium case,
$\left\langle S^{uu}({\bf k^{\prime}|p|q})\right\rangle>0$ for $k^{\prime}>p$
and vice versa. We describe this case in more detail in Sec. 3.2.

  3. 3.

$\mathcal{F}_{u,\mathrm{in}}(k)<0$ and $d\Pi_{u}(k)/dk<0$: For
$\mathcal{F}_{\mathrm{in}}(k)<0$, the kinetic energy flux decreases with $k$.
Notable examples in this category are Quasi-static MHD turbulence, flows with
Ekman friction, and stably-stratified turbulence. These flows are described in
subsequent sections.

  4. 4.

$\mathcal{F}_{u,\mathrm{in}}(k)>0$ and $d\Pi_{u}(k)/dk>0$: The kinetic energy
flux increases in this case. Leading examples for this case are turbulent
thermal convection and shear turbulence that will be described in later
sections.

The above four cases are illustrated in Fig. 4.

In the following discussion, we provide several examples of variable energy
flux in hydrodynamic turbulence. However, first, we begin with a brief
description of Kolmogorov’s theory of turbulence that forms a basis for many
turbulence phenomenologies.

###  3.2 Kolmogorov’s theory of turbulence: constant energy flux

In this section we briefly describe Kolmogorov’s theory of turbulence, which
corresponds to case (ii) described above. For a more detailed derivations,
refer to [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. Kolmogorov [1, 2]
considered a steady turbulent flow that is driven at large length scales or at
a wavenumber band near $k_{f}\approx 1/L$, where $L$ is the system size. The
energy thus injected at the large scales cascades to the inertial range and
then to the dissipative scales. In the inertial range,
$\mathcal{F}_{\mathrm{in}}(k)=0$ and $D_{u}(k)=0$. Therefore, under a steady
state, Eq. (33) yields $\Pi_{u}(k)=\Pi_{u}=\mathrm{const}$, as shown in Fig.
4(b). To be precise, for any $k>k_{f}$, an integration of Eq. (33) yields

$\displaystyle\Pi_{u}(k)=\int_{k}^{\infty}D_{u}(k^{\prime})dk^{\prime}.$ (35)

The viscous dissipation is negligible in the forcing band $(0,k_{f})$. Hence,
for a wavenumber $k$ in the inertial range,

$\displaystyle\Pi_{u}(k)\approx\int_{0}^{\infty}D_{u}(k^{\prime})dk^{\prime}=\epsilon_{u},$
(36)

where $\epsilon_{u}$ is the total viscous dissipation rate. Thus, under steady
state, $\epsilon_{u}$ is approximately equal to the energy injection rate by
the large-scale forcing. Note however that Eqs. (35, 36) imply that
$\Pi_{u}(k)\lessapprox\epsilon_{u}$ in the inertial range.

Another important assumption of Kolmogorov’s theory of turbulence is that the
physics in the inertial range is independent of the forcing and dissipative
mechanisms [5, 6, 8]. Hence, the energy spectrum $E_{u}(k)$ is isotropic, and
it depends only on $\epsilon_{u}$ and local wavenumber $k$. Absence of any
external length scale implies that $E_{u}(k)$ is a power law in $k$. This
assumption is related to the locality of interactions in hydrodynamic
turbulence [87, 88, 89], but this discussion will take us beyond the scope of
this review.

Using the above inputs and dimensional analysis, we derive that

$E_{u}(k)=K_{\mathrm{Ko}}\epsilon_{u}^{2/3}k^{-5/3},$ (37)

where $K_{\mathrm{Ko}}$ is Kolmogorov’s constant. The above theory of
turbulence is universal because Eq. (37) is independent of forcing and
dissipative mechanisms, initial condition, and fluid properties such as
viscosity and density. The above law has been observed in many laboratory
experiments, in natural flows such terrestrial atmosphere, and in numerical
simulations [6, 7, 8, 9, 10, 11]. However, Kolmogorov’s theory of turbulence
is inapplicable in the presence of multiscale external force and/or
dissipation. As described below, the formalism of variable energy flux is
useful for modelling such flows.

###  3.3 Variable energy flux in the inertial-dissipation range and bottleneck
effect

Kolmogorov’s $k^{-5/3}$ energy spectrum is observed in the inertial range of
hydrodynamic turbulence, but not in the dissipative range. Using the data from
numerical simulations, Chen et al. [32] and Martinez et al. [33] proposed that
in the far-dissipation range, the energy spectrum varies as
$k^{\alpha}\exp\left(-ck/k_{d}\right)$, where $\alpha,c$ are constants, and
$k_{d}=(\epsilon_{u}/\nu^{3})^{1/4}$ is Kolmogorov’s wavenumber. Pao [31]
argued in favour of $k^{-5/3}\exp(-c(k/k_{d})^{4/3})$ spectrum. Pope [11]
showed that the following energy spectrum is a good fit to many experimental
observations (see e.g. [90]):

$\displaystyle
E_{u}(k)=K_{\mathrm{Ko}}\epsilon^{2/3}k^{-5/3}f_{L}(kL)f_{\eta}(k/k_{d}),$
(38)

where $f_{L}(kL),f_{\eta}(\tilde{k})$ represent the large-scale and
dissipative-scale components respectively.

Here we present Pao’s model [31] for the inertial-dissipation range of
hydrodynamic turbulence. This formula is based on variable energy flux and the
following ansatz. In the inertial-dissipation range, where
$\mathcal{F}_{u}=\mathcal{F}_{\mathrm{LS}}=0$, Eq. (33) yields

$\displaystyle\frac{d}{dk}\Pi_{u}(k)=-2\nu k^{2}E_{u}(k).$ (39)

In addition, Pao [31] assumed that in the inertial-dissipative range,
$\Pi_{u}(k)/E_{u}(k)$ is independent of $\nu$ and is function only of $k$ and
$\epsilon_{u}$ (also see [5]). Under these assumptions, dimensional analysis
with an aid of Eq. (37) yields

$\displaystyle\frac{E_{u}(k)}{\Pi_{u}(k)}=K_{\mathrm{Ko}}\epsilon_{u}^{-1/3}k^{-5/3}.$
(40)

The above conjecture assumes locality of turbulent interactions because
$\Pi_{u}(k)$ depends on local $k$ and $E_{u}(k)$. Substitution of Eq. (40) in
Eq. (39) yields

$\displaystyle\Pi_{u}(k)$ $\displaystyle=$
$\displaystyle\epsilon_{u}\exp{\left(-\frac{3}{2}K_{\mathrm{Ko}}(k/k_{d})^{4/3}\right)},$
(41) $\displaystyle E_{u}(k)$ $\displaystyle=$ $\displaystyle
K_{\mathrm{Ko}}\epsilon_{u}^{2/3}k^{-5/3}\exp{\left(-\frac{3}{2}K_{\mathrm{Ko}}(k/k_{d})^{4/3}\right)}.$
(42)

Pao [31] showed that the above energy spectrum describes many experimental
observations [91] reasonably well111Using Eq. (39), Verma et al. [92] argue
that the energy flux and spectrum vary as $\exp(-k)$ for laminar flows (small
Reynolds number). Note that the energy flux is nonzero for flows with small
but nonzero Reynolds number.. Recently, Verma et al. [92] showed that the
above spectrum and flux describe the results of high-resolution numerical
simulations, except for a small discrepancy in the dissipation range, where
Pao’s model overpredicts the energy flux and spectrum (see Fig. 5). This
feature appears to be related to the bottleneck effect, which will be
discussed below. Interestingly, Eqs. (41, 42) describe the energy spectrum and
flux of the shell model of turbulence [93] quite well. (see Fig. 5(c,d)).

Figure 5:  (a,b) The energy spectrum and flux of hydrodynamic turbulence: for
a $4096^{3}$ DNS (red curve), Pao’s model (green curve), Pope’s model (dashed
curve), for a flow with hyperviscosity (schematic diagram only). (c,d) The
energy spectrum and flux of a shell model (red curves) are in good agreement
with the predictions of Pao’s model (green curves). Figures (a,b) adopted from
a figure of Verma et al. [92].

In hydrodynamic turbulence, the normalised spectrum $E_{u}(k)k^{5/3}$ exhibits
a hump between the inertial range and the dissipation range [35, 90, 36, 94,
95] (see Fig. 5(a)). This phenomena, called bottleneck effect, is not yet
fully understood. Falkovich [35] related the bottleneck effect to the
nonlinear energy transfer $T_{u}(k)$ of Eq. (32). Verma and Donzis [36] argue
that the observed bottleneck effect in most experiments and simulations is due
to the insufficient inertial range available for facilitating the energy
transfer from the large scales to small scales. Kuchler et al. [94] performed
an experimental study of bottleneck effect and observed general agreement
between the experimental results and model predictions [35, 36]. Spyksma et
al. [96] showed that the bottleneck effect gets enhanced due to
hyperviscosity, as illustrated schematically in Fig. 5(a). Interestingly, as
shown Fig. 5(c), Pao’s model and the shell model do not exhibit bottleneck
effect.

In the following discussion we model the bottleneck effect by relating the
kinetic energy flux to the mass flux in a river front when the front meets the
ocean (see Fig. 6(a)). The mass flux of the water flow can be approximated by
$\rho h{\bf u}$, where $\rho$ and ${\bf u}$ are respectively the density and
velocity of the water, and $h$ is the water level height of the river (we
ignore the effects of boundary layers, waves, etc.). A river moves with a
certain velocity, thus carrying a mass flux. However, the river slows down or
${\bf u}$ decreases as the river approaches the ocean, as shown Fig. 6(b).
Note however that the mass flux should remain constant. Therefore, $h$ tends
to increase at the front in order to conserve the mass flux. Note however that
${\bf u}=0$ when the river has attained equilibrium with the ocean (somewhat
deep inside the ocean). We also remark that the increase in $h$ is small or
insignificant if the river has slowed down considerably before meeting the
ocean. The hump in $h$ is observed only when the river front has significant
mass flux when it meets the ocean. Similar phenomena is observed in a traffic
jam. The traffic velocity decreases considerably in a traffic jam leading to a
pileup of vehicles at the bottleneck. We also observe similar accumulation of
people when a large crowd attempts to exit from a narrow gate.

Figure 6:  (a) River front meeting the ocean. The height of the front
increases when the river front is signficantly fast. (b) A schematic diagram
of diminishing velocity u and increasing of the water level height h. (c) In
the bottleneck zone of hydrodynamic turbulence, the normalized spectrum,
$E_{u}(k)k^{5/3}$ increases due to the suppression of $\Pi_{u}(k)$ by viscous
dissipation. Figure (a) is an adoption of a video from
https://www.youtube.com/watch?v=y0L5-owOr9U.

Now, let us connect the bottleneck effect to the aforementioned examples of
river front. The turbulent energy flux is analogous to the mass flux of the
river or the traffic, while the energy spectrum to the local water volume. The
constant energy flux in the inertial range corresponds to the smooth flow of
the river or traffic. The turbulent energy flux decreases in the dissipation
range, similar to the mass flux in the river front during its merger with the
ocean. In Pao’s model, the energy flux decreases gradually, hence it does not
exhibit any bottleneck effect. This feature is akin to a slow merger of a
river front with the ocean, during which $h$ does not increase. However, as
shown in Fig. 5(a), the energy flux in the DNS decreases more rapidly than
Pao’s model. Therefore, $E_{u}(k)k^{5/3}$ of the DNS exhibits a hump or
bottleneck effect. The hyperviscous dissipation suppresses the energy flux
even further than the DNS, which leads to stronger bottleneck effect. Thus,
the bottleneck effect is related to the extent of suppression of the turbulent
energy flux in the dissipation range.

Another interesting observation is that the numerical results of most shell
models do not exhibit any bottleneck effect (see Fig. 5(c,d)). These figures
also illustrate that the energy spectrum and flux of the shell model are
described reasonably well by Pao’s model, which is possibly because most shell
models only incorporate local energy transfers (among neighbouring shells). In
contrast, the nonlocal shell models [97, 98] may exhibit bottleneck effect.
The aforementioned connections between locality, the bottleneck effect, and
Pao’s model need to be explored further.

The above discussion on Pao’s model and the bottleneck effect illustrates the
usefulness of variable energy flux.

###  3.4 Quasi-static MHD turbulence

In quasi-static (QS) MHD, which is applicable to liquid metals, the nonlinear
term of the induction equation is ignored beause $\mathrm{RePm}\rightarrow 0$,
where $\mathrm{Pm}$ is the magnetic Prandtl number [23, 24]. For such flows,
under a strong mean magnetic field ${\bf B}_{0}$, the Lorentz force is
approximated by the following expression [23, 24]:

$\displaystyle{\bf F}_{u}({\bf k})=-[N(U/L)\cos^{2}\theta]{\bf u}({\bf k}),$
(43)

where $\theta$ is the angle between wavenumber ${\bf k}$ and ${\bf B}_{0}$;
$U,L$ are the large-scale velocity and length respectively; and
$N=B_{0}^{2}L/(U\kappa)$ is the interaction parameter with $\kappa$ as the
magnetic diffusivity.

The effects of the Lorentz force in QS MHD turbulence is an important topic of
research. Laboratory experiments [99, 100, 101, 102, 103] and numerical
simulations [104, 105, 106, 107] of QS-MHD turbulence reveal that for small
interaction parameters ($N\lessapprox 1$), the Lorentz force has a weak effect
on the flow, thus yielding a weak steepening of the energy spectrum compared
to $k^{-5/3}$ spectrum. However, with the increase of $N$, $E_{u}(k)$ steepens
significantly with the spectral index approaching as low as $-5$ for $N\approx
30$. For very large $N$, $E_{u}(k)\sim\exp(-bk)$ with $b$ as a positive
constant.

Several models have been constructed to explain the steepening of the energy
spectrum in QS MHD turbulence [99, 101, 102, 108, 109, 22, 110, 23, 25]. In
particular, many researchers invoked two-dimensionalization of the flow to
explain the steepened $E_{u}(k)$ [102, 108, 109, 22]. However, in the
following discussion we present how variable energy flux can explain the
steepening of the spectrum [110, 24, 25]. In QS-MHD turbulence, the energy
injected by the Lorentz force gets converted to heat by the Joule dissipation
$D_{J}({\bf k})$ (see Eq. (43)) as shown below:

$\displaystyle\mathcal{F}_{u}({\bf k})=\Re[{\bf F}_{u}({\bf k})\cdot{\bf
u^{*}(k)}]=-[2N(U/L)\cos^{2}\theta]E_{u}({\bf k})=-D_{J}({\bf k}).$ (44)

In addition, the modal energy spectrum is not isotropic. However, the shell
spectrum and energy flux, which is averaged over polar angle, obey the
following equation [110, 24]:

$\displaystyle\frac{d}{dk}\Pi_{u}(k)=-[2\nu k^{2}+2c_{2}N(U/L)]E_{u}(k),$ (45)

where $c_{2}$ is a constant.

Anas and Verma [25] solved Eq. (45) by making an assumption similar to that by
Pao [31] for 3D hydrodynamic turbulence and derived the following formulas for
the energy flux and spectrum. For $N\lessapprox 1$,

$\displaystyle\log\left(\frac{\Pi_{u}(k)}{\Pi_{u}(k_{0})}\right)$
$\displaystyle=$
$\displaystyle-\frac{3}{2}K_{\mathrm{Ko}}[(k/k_{d})^{4/3}-(k_{0}/k_{d})^{4/3}]$
(46)
$\displaystyle+3c_{2}K_{\mathrm{Ko}}\left[(k/k_{d2})^{-2/3}-(k_{0}/k_{d2})^{-2/3}\right],$
$E_{u}(k)=K_{\mathrm{Ko}}\Pi_{u}(k)\epsilon^{-1/3}k^{-5/3},$ (47)

and for $N\gg 1$,

$\displaystyle E_{u}(k)$ $\displaystyle=$ $\displaystyle
A\exp(-k/\bar{k}_{d}),$ (48) $\displaystyle\Pi_{u}(k)$ $\displaystyle=$
$\displaystyle
A\left[2\nu\bar{k}_{d}(k^{2}+2k\bar{k}_{d}+2\bar{k}_{d}^{2})+2(NU/L)c_{2}\bar{k}_{d}\right]\exp(-k/\bar{k}_{d}),$
(49)

where $k_{d2},A,\bar{k}_{d}$ are constants. See Anas and Verma [25] for
further details. The above formulas provide a good description to past
experimental and numerical results, e.g., [99, 100, 106]. Thus, variable
energy flux helps describe the QS MHD turbulence quite well.

###  3.5 $k^{-1}$ spectrum in shear turbulence

Many turbulent systems exhibit $k^{-1}$ energy spectrum at small wavenumbers
or $1/f$ frequency spectrum at small frequencies. For example, Tichen [26] and
Pereira et al. [28] observed $k^{-1}$ energy spectrum in shear-driven
turbulence. In the solar wind, Matthaeus and Goldstein [27] reported $1/f$
spectrum for small frequencies where solar wind jets may create shear
turbulence. Recently, Duguid et al. [111] reported $1/f$ spectrum for the
total kinetic energy of thermal convection at large time scales. To explain
the $k^{-1}$ spectrum at small $k$’s, Tchen [26] modelled shear turbulence at
large scales using Heisenberg’s turbulence model [112]; he argued that the
shear induces strong resonance in the flow, which in turn yields
$u_{k}\sim\mathrm{constant}$. Hence, $E_{u}(k)\sim u_{k}^{2}/k\sim k^{-1}$.

In the following discussion we derive $k^{-1}$ spectrum using variable energy
flux. The velocity shear injects kinetic energy to the small wavenumber modes
of the flow. Hence, we expect that $\mathcal{F}_{\mathrm{LS}}(k)>C$ that leads
to an increase in the energy flux $\Pi_{u}(k)$ with $k$, as in case (iv)
discussed in Sec. 3.1. Consequently, the energy spectrum
$E_{u}(k)\sim[\Pi_{u}(k)]^{2/3}k^{-5/3}$ is expected to be shallower than
$k^{-5/3}$. In particular, for $\mathcal{F}_{\mathrm{LS}}(k)=C$, Eq. (34)
yields
$\Pi_{u}(k)=\int^{k}\mathcal{F}_{\mathrm{LS}}(k^{\prime})dk^{\prime}=Ck$. Now,
using dimensional analysis we derive that

$\displaystyle E_{u}(k)\sim C^{2/3}k^{-1}.$ (50)

As argued in Sec. 2, the energy flux can be defined for shear turbulence even
though it is anisotropic. The above wavenumber-dependent
$\mathcal{F}_{\mathrm{LS}}(k)$ is tune with the earlier works by Yakhot and
Orszag [113] and Sain et al. [114].

The above mechanism provides a plausible explanation for the $k^{-1}$ energy
spectrum in shear turbulence [26, 28, 27]. The low-frequency $1/f$ spectrum in
the solar wind may be due to the shear experienced by wind jets [27], while
that in thermal convection may be due to shear among the large-scale thermal
plumes [111]. Note that we convert the wavenumber spectrum to the frequency
spectrum using Taylor’s hypothesis, which is applicable to the solar wind
because it is much faster than spacecrafts [115]. In thermal convection,
Taylor’s hypothesis is expected to work under certain conditions [116, 117,
118].

The above theory for $k^{-1}$ spectrum in shear turbulence hinges on the
assumption that $\mathcal{F}_{u}({\bf k})=C$, which needs to be tested using
experiments and/or numerical simulations. Interestingly, $1/f$ noise has been
reported in a large number of physical systems—electric currents, ion-channel
currents, music, earthquakes, etc. (see [119, 120, 121], and references
therein). It is possible that $1/f$ spectra in the electric and ion-channel
currents are connected to shear in the electron flow. This conjecture,
however, needs to be tested.

In addition to the above, there are many more examples of variable energy
flux. In stably-stratified turbulence, $\Pi_{u}(k)$ decreases as $k^{-4/5}$
due to buoyancy [15, 16]. On the contrary, $\Pi_{u}(k)$ increases marginally
in turbulent thermal convection. We will describe these fluxes in Sec. 5. In
MHD turbulence, the kinetic energy flux varies in the inertial range itself
due to the Lorentz force. Similar variations are observed in solvents with
polymers, quantum turbulence, binary-mixture turbulence, etc. We will discuss
these systems in Sections 6 and 11. In addition, the energy flux variations
can be generalised to other quantities such as enstrophy and kinetic helicity;
these issues will be discussed in Sec. 8.

With this, we close our brief discussion on variable energy flux in
hydrodynamic turbulence. In the next section, we describe such variations in
flows with a secondary field, e.g., temperature, magnetic field.

##  4 Variable energy flux in flows with a secondary field: Formalism

Consider a secondary field $\zeta$ advected by the velocity field ${\bf u}$.
This secondary field could be a scalar, a vector, or a tensor. Leading
examples of a scalar field are density and temperature of a fluid; that of a
vector field are magnetic field, dipolar field, and flock velocity; and that
of a tensor are the configuration tensor of a polymer and stress tensor of an
elastic fluid. In this section, we present energy fluxes associated with a
secondary field and those arising due to interactions between the velocity and
secondary fields. As we describe below, many features of energy transfers are
common among the scalar, vector, and tensor secondary fields.

Table 1: Energy ($E_{\zeta}$) and modal energy ($E_{\zeta}({\bf k})$) of a secondary field | scalar | vector | tensor
---|---|---|---
$E_{\zeta}$ | $\frac{1}{2}\left\langle\zeta^{2}\right\rangle$ | $\frac{1}{2}\left\langle\bm{\zeta}\cdot\bm{\zeta}\right\rangle$ | $\frac{1}{2}\left\langle\zeta_{ij}\zeta_{ij}\right\rangle$
$E_{\zeta}({\bf k})$ | $\frac{1}{2}|\zeta({\bf k})|^{2}$ | $\frac{1}{2}|\bm{\zeta}({\bf k})|^{2}$ | $\frac{1}{2}\zeta_{ij}({\bf k})\zeta^{*}_{ij}({\bf k})$

###  4.1 Variable energy flux associated with a secondary field

The equations for the velocity field ${\bf u}$ are same as those covered in
Sec. 2, except that the force field for the velocity field (${\bf F}_{u}$)
could be a function of ${\bf u}$, $\zeta$, ${\bf r}$, and $t$. The equations
for the secondary field are

$\displaystyle\mathrm{Scalar}:\frac{\partial{\zeta}}{\partial t}+({\bf
u}\cdot\nabla){\zeta}$ $\displaystyle=$ $\displaystyle F_{\zeta}({\bf
u},\zeta)+\kappa\nabla^{2}{\zeta},$ (51)
$\displaystyle\mathrm{Vector}:\frac{\partial{\bm{\zeta}}}{\partial t}+({\bf
u}\cdot\nabla){\bm{\zeta}}$ $\displaystyle=$ $\displaystyle{\bf
F}_{\zeta}({\bf u},\bm{\zeta})+\kappa\nabla^{2}{\bm{\zeta}},$ (52)
$\displaystyle\mathrm{Tensor}:\frac{\partial{{\zeta}_{ij}}}{\partial t}+({\bf
u}\cdot\nabla){{\zeta}_{ij}}$ $\displaystyle=$ $\displaystyle
F_{{\zeta},ij}({\bf u},\zeta)+\kappa\nabla^{2}{{\zeta}_{ij}},$ (53)

where $\kappa$ is the diffusion coefficient of the secondary field, and
$F_{\zeta}$ is the force field for the secondary field. Two important
nondimensional parameters are Prandtl number $\mathrm{Pr}=\nu/\kappa$ and
Pélet number $\mathrm{Pe}$, which is the ratio of the nonlinear term and the
diffusion term in the equation for the secondary field, that is,

$\displaystyle\mathrm{Pe}=\frac{({\bf
u}\cdot\nabla){\zeta}}{\kappa\nabla^{2}{\zeta}}=\frac{UL}{\kappa},$ (54)

where $U,L$ are the large-scale velocity and length respectively.

Similar to the kinetic energy, we define secondary energy and associated modal
energy, as listed in Table 1. For discrete wavenumbers, one-dimensional
secondary energy spectrum is defined as $E_{\zeta}(k)=\sum_{k-1<k^{\prime}\leq
k}E_{\zeta}({\bf k^{\prime}})$, but for continuum wavenumbers,
$E_{\zeta}(k)dk=\sum_{k<k^{\prime}\leq k+dk}E_{\zeta}({\bf k^{\prime}})$. The
evolution equation for the modal secondary energy of a scalar is

$\displaystyle\frac{d}{dt}E_{\zeta}(\mathbf{k})$ $\displaystyle=$
$\displaystyle T_{\zeta}({\bf k})+\mathcal{F}_{\zeta}({\bf k})-D_{\zeta}({\bf
k})$ $\displaystyle=$ $\displaystyle\sum_{\bf p}\Im\left[{\bf\\{k\cdot
u(q)\\}\\{\zeta(p)\zeta^{*}(k)\\}}\right]+\Re[F_{\zeta}({\bf k})\zeta^{*}({\bf
k})]-2\kappa k^{2}E_{\zeta}({\mathbf{k}}).$

In the above equation, $T_{\zeta}({\bf k})$ is the nonlinear transfer of
secondary energy to $\zeta({\bf k})$, $\mathcal{F}_{\zeta}({\bf k})$ is the
secondary energy transfer to $\zeta({\bf k})$ by $F_{\zeta}({\bf k})$, and
$D_{\zeta}({\bf k})$ is the diffusion or dissipation rate of $\zeta({\bf k})$.
The equations for the vector and tensor fields are very similar to the above
equations, except that the field multiplication is performed appropriately
(scalar product or tensor product).

Several important points regarding the secondary field are

  1. 1.

When $F_{\zeta}$ is a linear function of ${\bf u}$, the scalar energy
injection rate $\mathcal{F}_{\zeta}({\bf k})=\Re[F_{\zeta}({\bf
k})\zeta^{*}({\bf k})]$ is a function of ${\bf k}$, ${\bf u(k)}$, and
$\zeta({\bf k})$. We encounter such forms of $F_{\zeta}({\bf k})$ in stably
stratified turbulence and in thermal convection where $F_{\zeta}\propto
u_{z}$. Similar properties hold for $\mathcal{F}_{u}({\bf k})$ when ${\bf
F}_{u}$ is a linear function of $\zeta$.

  2. 2.

When $F_{\zeta}$ is a nonlinear function of ${\bf u}$ and/or $\zeta$, the
scalar energy injection rate $\mathcal{F}_{\zeta}({\bf k})$ is a convolution.
Therefore, $\mathcal{F}_{\zeta}({\bf k})$ involves wavenumbers other than
${\bf k}$. For example, in MHD turbulence, where ${\bf
F}_{\bm{\zeta}}=\bm{\zeta}\cdot\nabla{\bf u}$ with $\bm{\zeta}$ as the
magnetic field,

$\displaystyle\mathcal{F}_{\zeta}({\bf k})$ $\displaystyle=$
$\displaystyle\sum_{\bf
p}-\Im\left[{\bf\\{k\cdot\bm{\zeta}(k-p)\\}\\{u(p)\cdot\bm{\zeta}^{*}(k)\\}}\right]$
(56)

is a convolution. The nonlinear $F_{u}$ yields a similar convolution.

The nonlinear term $({\bf u}\cdot\nabla)\zeta$ (and similar ones for vector
and tensor) for the secondary field facilitates scalar energy transfer. For a
wavenumber triad $({\bf k^{\prime},p,q})$, the mode-to-mode secondary energy
transfer from wavenumber p to wavenumber k with the mediation of wavenumber q
is [122, 18, 19]:

$\displaystyle\mathrm{Scalar}:S^{\zeta\zeta}({\bf k^{\prime}|p|q})$
$\displaystyle=$ $\displaystyle-\Im\left[{\bf\\{k^{\prime}\cdot
u(q)\\}\\{{\zeta}({\bf p}){\zeta}({\bf k^{\prime}})\\}}\right],$ (57)
$\displaystyle\mathrm{Vector}:S^{\zeta\zeta}({\bf k^{\prime}|p|q})$
$\displaystyle=$ $\displaystyle-\Im\left[{\bf\\{k^{\prime}\cdot
u(q)\\}\\{\bm{\zeta}({\bf p})\cdot\bm{\zeta}({\bf k^{\prime}})\\}}\right],$
(58) $\displaystyle\mathrm{Tensor}:S^{\zeta\zeta}({\bf k^{\prime}|p|q})$
$\displaystyle=$ $\displaystyle-\Im\left[{\bf\\{k^{\prime}\cdot
u(q)\\}}\\{{\zeta}_{ij}({\bf p}){\zeta}_{ij}({\bf k^{\prime}})\\}\right].$
(59)

In the above equations, the giver and receiver modes are from the secondary
field, while a velocity mode acts as a mediator for the secondary energy
transfer. The aforementioned form of energy transfer also follows from the
structure of nonlinear term $({\bf u}\cdot\nabla)\zeta$ where ${\bf u}$
advects the scalar field $\zeta$. Also, the superscript of $S^{\zeta\zeta}$
refer to the receiver and giver fields, both being $\zeta$.

Using incompressibility condition, we can show that the mode-to-mode secondary
energy transfer functions satisfy the following property:

$\displaystyle S^{\zeta\zeta}({\bf k^{\prime}|p|q})=-S^{\zeta\zeta}({\bf
p|k^{\prime}|q}).$ (60)

Using Eq. (60) we deduce that for a wavenumber region $A$ (including a triad),

$\displaystyle\sum_{{\bf k^{\prime}}\in A}\sum_{{\bf p}\in
A}S^{\zeta\zeta}({\bf k^{\prime}|p|q})=0.$ (61)

This relation also implies that $E_{\zeta}$ is conserved when $F_{\zeta}=0$
and $\kappa=0$. Using the formula for the mode-to-mode energy transfers, we
define the secondary energy flux for a wavenumber sphere of radius $k_{0}$ as
[5, 6, 122, 8, 18, 19]:

$\displaystyle\Pi_{\zeta}(k_{0})=\sum_{k^{\prime}>k_{0}}\sum_{p\leq
k_{0}}S^{\zeta\zeta}(\mathbf{k^{\prime}|p|q}).$ (62)

Here, $\Pi_{\zeta}(k_{0})$ is the net secondary energy transfer from all the
modes inside the sphere to all the modes outside the sphere.

Following similar lines of arguments as in Sec. 3, we derive the following
evolution equation for the scalar energy spectrum $E_{\zeta}(k)$ [6, 14]:

$\displaystyle\frac{\partial}{\partial
t}E_{\zeta}(k,t)=-\frac{\partial}{\partial
k}\Pi_{\zeta}(k,t)+\mathcal{F}_{\zeta}(k,t)-D_{\zeta}(k,t),$ (63)

where

$\displaystyle E_{\zeta}(k)dk$ $\displaystyle=$
$\displaystyle\sum_{k<k^{\prime}\leq k+dk}E_{\zeta}({\bf k^{\prime}}),$ (64)
$\displaystyle\mathcal{F}_{\zeta}(k)dk$ $\displaystyle=$
$\displaystyle\sum_{k<k^{\prime}\leq k+dk}\Re[F_{\zeta}({\bf
k^{\prime}})\zeta^{*}({\bf k^{\prime}})],$ (65) $\displaystyle D_{\zeta}(k)dk$
$\displaystyle=$ $\displaystyle\sum_{k<k^{\prime}\leq k+dk}2\kappa k^{\prime
2}E_{\zeta}({\bf k^{\prime}}).$ (66)

Here, $\mathcal{F}_{\zeta}(k)$ represents the scalar energy supply rate by
$F_{\zeta}$ to shell $k$, and $D_{\zeta}(k)$ represents the diffusion or
dissipation rate of the scalar energy in shell $k$. For the vector and tensor
secondary fields, Eq. (65) involves vector and tensor products respectively.
For a steady state ($\partial E_{\zeta}(k)/\partial t=0$), Eq. (63) yields

$\displaystyle\frac{d}{dk}\Pi_{\zeta}(k)=\mathcal{F}_{\zeta}(k)-D_{\zeta}(k).$
(67)

Thus, the energy flux of a secondary field is affected by
$\mathcal{F}_{\zeta}(k)$ and $D_{\zeta}(k)$. We may obtain a steady state when
$\zeta$ field is forced at large scales.

In the inertial range, $D_{\zeta}(k)\approx 0$, hence

$\displaystyle\frac{d}{dk}\Pi_{\zeta}=\mathcal{F}_{\zeta,\mathrm{in}}(k),$
(68)

where the subscript “in” of $\mathcal{F}_{\zeta,in}$ represents the inertial
range. Therefore, in the inertial range, similar to the description of Sec. 3,
variations of $\Pi_{\zeta}(k)$ can be classified into four categories:

  1. 1.

$\mathcal{F}_{\zeta,\mathrm{in}}(k)=0$ and $\Pi_{\zeta}(k)=0$,

  2. 2.

$\mathcal{F}_{\zeta,\mathrm{in}}(k)=0$ and $\Pi_{\zeta}(k)=C>0$ ,

  3. 3.

$\mathcal{F}_{\zeta,\mathrm{in}}(k)<0$ and $d\Pi_{\zeta}(k)/dk<0$ ,

  4. 4.

$\mathcal{F}_{\zeta,\mathrm{in}}(k)>0$ and $d\Pi_{\zeta}(k)/dk>0$ .

The interpretation of the above four cases are very similar to those for the
kinetic energy flux discussed in Sec. 3 and exhibited in Fig. 4. Note that the
first case corresponds to the equilibrium configuration for the secondary
field. We will describe these cases in the subsequent sections.

###  4.2 Cross energy transfers between the velocity and secondary fields

In this subsection we describe the energy transfers from the velocity field to
the secondary field and vice versa. We consider $k_{0}$ beyond $k_{f}$ and
rewrite Eq. (20) as [6, 14]

$\displaystyle\frac{d}{dt}\sum_{k\leq k_{0}}E_{u}({\bf k},t)$ $\displaystyle=$
$\displaystyle\sum_{k\leq k_{0}}T_{u}+\sum_{k\leq k_{0}}\mathcal{F}_{u}({\bf
k})+\sum_{k\leq k_{0}}\mathcal{F}_{\mathrm{LS}}({\bf k})-\sum_{k\leq
k_{0}}D_{u}({\bf k})$ (69) $\displaystyle=$
$\displaystyle-\Pi_{u}(k_{0})+\Pi^{\zeta}_{u<}(k_{0})+\varepsilon_{\mathrm{inj}}-\sum_{k\leq
k_{0}}D_{u}({\bf k}),$

where $\varepsilon_{\mathrm{inj}}$ is the kinetic energy injection rate by the
large-scale force, and $\Pi^{\zeta}_{u<}(k_{0})$ is the net energy transfer
from all the $\bm{\zeta}$ modes to the velocity modes within the sphere of
radius $k_{0}$. In $\Pi^{\zeta}_{u<}(k_{0})$, the superscript and subscripts
denote the giver and receiver field variables respectively, while $<$ denotes
the modes within the sphere. In the same vein, we define

$\displaystyle\Pi^{\zeta}_{u>}(k_{0})=\sum_{k>k_{0}}\mathcal{F}_{u}({\bf k})$
(70)

as the net energy transfer from all the $\bm{\zeta}$ modes to the velocity
modes outside (represented by the symbol $>$) the sphere of radius $k_{0}$.
Using the evolution equation for $E_{\zeta}(\mathbf{k})$ we deduce that

$\displaystyle\Pi^{u}_{\zeta<}(k_{0})=\sum_{k\leq
k_{0}}\mathcal{F}_{\zeta}({\bf
k});~{}~{}~{}\Pi^{u}_{\zeta>}(k_{0})=\sum_{k>k_{0}}\mathcal{F}_{\zeta}({\bf
k})$ (71)

are the respective energy transfers from all the velocity modes to the $\zeta$
modes inside and outside of the sphere of radius $k_{0}$. Figure 7 illustrates
these fluxes.

Figure 7: (a) The flux $\Pi^{\zeta}_{u<}(k_{0})$ ($\Pi^{\zeta}_{u>}(k_{0})$)
represents the energy transfer from $\zeta$ modes to the velocity modes inside
(outside) the sphere. (b) The flux $\Pi^{u}_{\zeta<}(k_{0})$
($\Pi^{u}_{\zeta>}(k_{0})$) represents the reverse energy transfers, i.e.,
from all the velocity modes to $\zeta$ modes inside (outside) the sphere.

The net energy transfer from $\zeta$ to ${\bf u}$ is

$\displaystyle\mathcal{F}_{u}=\sum_{\bf k}\mathcal{F}_{u}({\bf
k})=\Pi^{\zeta}_{u<}(k)+\Pi^{\zeta}_{u>}(k).$ (72)

Since $\mathcal{F}_{u}$ is a fixed number, the sum
$\Pi^{\zeta}_{u<}(k)+\Pi^{\zeta}_{u>}(k)$ is constant in $k$. Similarly, the
net energy transfer from ${\bf u}$ to $\zeta$ is

$\displaystyle\mathcal{F}_{\zeta}=\sum_{\bf k}\mathcal{F}_{\zeta}({\bf
k})=\Pi^{u}_{\zeta<}(k)+\Pi^{u}_{\zeta>}(k).$ (73)

with the sum $\Pi^{u}_{\zeta<}(k)+\Pi^{u}_{\zeta>}(k)$ as a constant in $k$.
Note however that the individual fluxes (e.g., $\Pi^{u}_{\zeta<}(k)$) may vary
with $k$. For some systems, $\int d{\bf r}(u^{2}+\zeta^{2})/2$ represents the
total energy and

$\displaystyle\mathcal{F}_{u}+\mathcal{F}_{\zeta}=0~{}~{}~{}\mathrm{or}~{}~{}~{}\Pi^{\zeta}_{u<}(k)+\Pi^{\zeta}_{u>}(k)+\Pi^{u}_{\zeta<}(k)+\Pi^{u}_{\zeta>}(k)=0.$
(74)

MHD turbulence and stably stratified turbulence are examples of such systems;
they will be discussed later in this review.

Using the definitions of the above fluxes, we derive the following relations
[14]:

$\displaystyle\frac{d}{dk}\Pi^{\zeta}_{u>}(k)=-\mathcal{F}_{u}(k);~{}~{}~{}\frac{d}{dk}\Pi^{\zeta}_{u<}(k)=\mathcal{F}_{u}(k);$
$\displaystyle\frac{d}{dk}\Pi^{u}_{\zeta>}(k)=-\mathcal{F}_{\zeta}(k);~{}~{}~{}\frac{d}{dk}\Pi^{u}_{\zeta<}(k)=\mathcal{F}_{\zeta}(k).$
(75)

If we assume an absence of large scale forcing for $\zeta$, then the fluxes
obey the following properties under steady state:

$\displaystyle\Pi_{u}(k_{0})+\Pi^{\zeta}_{u>}(k_{0})=\int_{k_{0}}^{\infty}dkD_{u}(k^{\prime},t),$
(76)
$\displaystyle\Pi_{\zeta}(k_{0})+\Pi^{u}_{\zeta>}(k_{0})=\int_{k_{0}}^{\infty}dkD_{\zeta}(k),$
(77)
$\displaystyle\Pi_{u}(k_{0})=\Pi_{u<}^{\zeta}(k_{0})+\varepsilon_{\mathrm{inj}},$
(78) $\displaystyle\Pi_{\zeta}(k_{0})=\Pi^{u}_{\zeta<}(k_{0}).$ (79)

For $k_{0}$ in the inertial range,
$\int_{k_{0}}^{\infty}dkD_{u,\zeta}(k^{\prime},t)\approx\epsilon_{u,\zeta}$
respectively. The above identities follow from the energy conservation. See
Figs. 8 and 9 for illustrations.

Figure 8: Schematic diagram illustrating the identities (a)
$\Pi_{u}(k_{0})+\Pi^{\zeta}_{u>}(k_{0})=\int_{k_{0}}^{\infty}dk^{\prime}D_{u}(k^{\prime})$;
(b)
$\Pi_{\zeta}(k_{0})+\Pi^{u}_{\zeta>}(k_{0})=\int_{k_{0}}^{\infty}dk^{\prime}D_{\zeta}(k^{\prime})$.
The wheel in the centre of (a) represents kinetic energy injection rate
$\varepsilon_{\mathrm{inj}}$.  Figure 9:  Schematic diagram illustrating
identities for a steady state: (a)
$\Pi_{u}(k_{0})\approx\Pi_{u<}^{\zeta}(k_{0})+\varepsilon_{\mathrm{inj}}$; (b)
$\Pi_{\zeta}(k_{0})\approx\Pi^{u}_{\zeta<}(k_{0})$. The wheel in the centre of
(a) represents kinetic energy injection rate $\varepsilon_{\mathrm{inj}}$.

In subsequent sections, we will describe the role of variable energy flux in
several turbulent flows with secondary fields, namely, passive scalar flow,
buoyancy driven flows, magnetohydrodynamics, and polymer solution. Note that
in the turbulent limit, the nonlinear terms ${\bf(u\cdot\nabla)u}$ and
${\bf(u\cdot\nabla)\bm{\zeta}}$ dominate the respective dissipation terms, and
hence $\mathrm{Re}\gg 1$ and $\mathrm{Pe}\gg 1$.

In the next subsection we describe the scaling of passive secondary
turbulence.

###  4.3 Turbulence with a passive secondary field

Consider a flow whose $\mathcal{F}_{u}$ is independent of the secondary field
$\zeta$ and, hence, the velocity field is unaffected by the secondary field.
However, the secondary field $\zeta$ is affected by ${\bf u}$. This is the
reason why such a $\zeta$ is called a passive secondary field [3, 4, 6, 5, 8,
7]. We will discuss the properties of such flows in this subsection.

If the forces on the velocity and the passive secondary fields are active only
at large scales, then $\mathcal{F}_{u,\mathrm{in}}(k)=0$ and
$\mathcal{F}_{\zeta,\mathrm{in}}(k)=0$. For such field configurations, using
Eqs. (34, 68) we deduce that $\Pi_{u}(k)$ and $\Pi_{\zeta}(k)$ are constant in
the inertial range. Since the velocity field is unaffected by $\zeta$, the
kinetic energy is described by Kolmogorov’s theory of turbulence (see Sec.
3.2). Therefore, the kinetic energy spectrum is given by Eq. (37).

In the inertial range, $\Pi_{\zeta}(k)=\mathrm{const}=\epsilon_{\zeta}$, where
$\epsilon_{\zeta}$ is the energy dissipation rate of the secondary field.
Using dimension analysis and similar arguments as in Sec. 3.2, one obtains [6,
7, 8]

$\displaystyle
E_{\zeta}(k)=K_{\mathrm{OC}}\epsilon_{\zeta}(\epsilon_{u})^{-1/3}k^{-5/3},$
(80)

where $K_{\mathrm{OC}}$ is the Obukhov-Corrsin constant. The above scaling has
been verified using several numerical simulations and experiments [6, 7, 9, 8,
123, 124]. For the inertial-dissipation range, using arguments similar to
those in Sec. 3.3, Pao [125] derived that

$\displaystyle\Pi_{\zeta}(k)$ $\displaystyle=$
$\displaystyle\epsilon_{\zeta}\exp{\left(-\frac{3}{2}K_{\mathrm{OC}}(k/k_{c})^{4/3}\right)},$
(81) $\displaystyle E_{\zeta}(k)$ $\displaystyle=$ $\displaystyle
K_{\mathrm{OC}}\epsilon_{\zeta}\epsilon_{u}^{-1/3}k^{-5/3}\exp{\left(-\frac{3}{2}K_{\mathrm{OC}}(k/k_{c})^{4/3}\right)},$
(82)

where $k_{c}=\left(\epsilon_{u}/\kappa^{3}\right)^{1/4}$ is Kolmogorov’s
diffusion wavenumber. Hence, $k_{c}/k_{d}=\mathrm{Pr}^{3/4}$.

The above arguments are valid when $\mathrm{Re}\gg 1$ and $\mathrm{Pe}\gg 1$.
The spectral properties are quite different for other regimes [126, 127, 19],
which are not discussed in this review.

###  4.4 Variable energy flux in anisotropic turbulence

Typically, a fluid flow becomes anisotropic in the presence of a strong
external field (e.g., mean magnetic field, buoyancy, external rotation field).
In such flows, we denote the velocity components perpendicular and parallel to
the external field as ${\bf u}_{\perp}$ and $u_{\parallel}$, and the
corresponding energy spectra as $E_{u,\parallel}({\bf
k})=\frac{1}{2}|u_{\parallel}({\bf k})|^{2}$ and $E_{u,\perp}({\bf
k})=\frac{1}{2}|{\bf u_{\perp}(k)}|^{2}$ respectively. The corresponding
energy fluxes are [24, 19]:

$\displaystyle\Pi_{u,\parallel}(k_{0})$ $\displaystyle=$
$\displaystyle\sum_{k^{\prime}>k_{0}}\sum_{p\leq k_{0}}-\Im\left\\{\left[{\bf
k}^{\prime}\cdot{\bf{u}}({\bf q})\right]\left[{u}_{\parallel}({\bf
k}^{\prime}){u}_{\parallel}({\bf p})\right]\right\\},$ (83)
$\displaystyle\Pi_{u,\perp}(k_{0})$ $\displaystyle=$
$\displaystyle\sum_{k^{\prime}>k_{0}}\sum_{p\leq k_{0}}-\Im\left\\{\left[{\bf
k}^{\prime}\cdot{\bf{u}}({\bf q})\right]\left[{\bf{u}}_{\perp}({\bf
k}^{\prime})\cdot{\bf{u}}_{\perp}({\bf p})\right]\right\\},$ (84)

where ${\bf k=p+q}$ and ${\bf k^{\prime}}=-{\bf k}$.

The evolution equations for the one-dimensional spectra $E_{u,\perp}(k)$ and
$E_{u,\parallel}(k)$ are

$\displaystyle\frac{\partial}{\partial t}E_{u,\parallel}(k,t)$
$\displaystyle=$ $\displaystyle-\frac{\partial}{\partial
k}\Pi_{u,\parallel}(k,t)+\mathcal{P}(k,t)-2\nu
k^{2}E_{u,\parallel}(k,t)+\mathcal{F}_{u,\parallel}(k)+\mathcal{F}_{\mathrm{LS,\parallel}}(k),$
$\displaystyle\frac{\partial}{\partial t}E_{u,\perp}(k,t)$ $\displaystyle=$
$\displaystyle-\frac{\partial}{\partial
k}\Pi_{u,\perp}(k,t)-\mathcal{P}(k,t)-2\nu
k^{2}E_{u,\perp}(k,t)+\mathcal{F}_{u,\perp}(k)+\mathcal{F}_{\mathrm{LS,\perp}}(k),$

where

$\displaystyle\mathcal{P}(k)dk=\sum_{k<k^{\prime}\leq
k+dk}\Im\left\\{k_{\parallel}{p}({\bf k})u^{*}_{\parallel}({\bf k})\right\\},$
(87)

and $\mathcal{F}_{u,\parallel}(k),\mathcal{F}_{\mathrm{LS,\parallel}}(k)$,
$\mathcal{F}_{u,\perp}(k)$, and $\mathcal{F}_{\mathrm{LS,\perp}}(k)$ are the
energy injection rates to $u_{\parallel}$ and ${\bf u}_{\perp}$ by the
parallel and perpendicular components of ${\bf F}_{u}$ and ${\bf
F}_{\mathrm{LS}}$. Under a steady state and in the inertial range with ${\bf
F}_{u}=0$ and ${\bf F}_{\mathrm{LS}}=0$,

$\displaystyle\frac{d}{dk}\Pi_{u,\parallel}(k)=-\frac{d}{dk}\Pi_{u,\perp}(k)=\mathcal{P}(k).$
(88)

Thus, $\Pi_{u,\perp}(k)$ and $\Pi_{u,\parallel}(k)$ vary with $k$. However,

$\displaystyle\frac{d}{dk}\Pi_{u}(k)=\frac{d}{dk}[\Pi_{\parallel}(k)+\Pi_{u,\perp}(k)]=0.$
(89)

That is, the energy flux $\Pi_{u}(k)$ is constant in inertial range, as
expected from Kolmogorov’s theory of turbulence. Note that $\mathcal{P}(k)$
facilitates energy transfers between the perpendicular and parallel components
of the velocity field.

For QS MHD turbulence with a strong mean magnetic field, Reddy et al. [107]
showed that $\mathcal{P}(k)$ is positive, and hence there is an energy
transfer from $u_{\perp}$ to $u_{\parallel}$. In such flows, $u_{\perp}$ is
stronger than $u_{\parallel}$. MHD turbulence exhibits a similar behaviour
[128]. The situation however is reversed in thermal convection where
$\mathcal{P}(k)<0$ that leads to an energy transfer from $u_{\parallel}$ to
$u_{\perp}$.

The anisotropic energy transfers are more complex in rotating turbulence.
Here, $\mathcal{F}_{u}({\bf k})=0$ because the velocity field is perpendicular
to the Coriolis force ${\bf u}\times\bm{\Omega}$. However, the Coriolis force
and the pressure field induce asymmetry between $u_{\parallel}$ and ${\bf
u}_{\perp}$. The pressure field due by the Coriolis force ($p_{\mathrm{Co}}$)
generates the following $\mathcal{P}({\bf k})$:

$\displaystyle\mathcal{P}({\bf k})$ $\displaystyle=$ $\displaystyle
k_{\parallel}\Im[p_{\mathrm{Co}}({\bf k})u^{*}_{\parallel}({\bf k})]$ (90)
$\displaystyle=$ $\displaystyle 2k_{\parallel}\Im[i\frac{\Omega}{k}\sin\zeta
u_{1}({\bf k})u_{2}^{*}({\bf k})\sin\zeta]$ $\displaystyle=$
$\displaystyle\frac{2\Omega k_{\parallel}}{k}\sin^{2}\zeta\Re[u_{1}({\bf
k})u_{2}^{*}({\bf k})],$

where $u_{1},u_{2}$ are components of the velocity field in Craya-Herring
basis [129, 130, 14]. The corresponding contribution by the nonlinear term
$({\bf u\cdot\nabla}){\bf u}$ is

$\displaystyle\mathcal{P}({\bf k})$ $\displaystyle=$ $\displaystyle
k_{\parallel}\Im[p_{\mathrm{nlin}}({\bf k})u^{*}_{\parallel}({\bf k})]$ (91)
$\displaystyle=$ $\displaystyle-\frac{k_{\parallel}}{k^{2}}\sum_{\bf
p}\Im[u_{\parallel}^{*}({\bf k}){\bf\\{k\cdot u(k-p)\\}\\{k\cdot u(p)\\}}].$

The aforementioned quantities induce energy exchange between $u_{\parallel}$
and ${\bf u}_{\perp}$, and they need to be quantified in future. In Sec. 9.2
we show how the above energy transfers take an active part in quasi-2D
turbulence generated by strong rotation, external magnetic field, or gravity.
We refer the reader to earlier works [45, 131, 130, 74] for further details.
We also remark that similar formulas need to be derived for the secondary
fields as well. For example, it will be interesting to investigate how the
parallel and perpendicular components of the magnetic field exchange energy
among themselves.

In the next section we describe how the ideas of variable energy flux yields
interesting insights into the physics of buoyancy-driven turbulence.

##  5 Variable energy flux in buoyancy-driven turbulence

Buoyancy-driven flows can be broadly classified into two categories: stable
and unstable [132, 44, 10, 19]. We will show below that the properties of
these two categories of flows are very different. For brevity, our focus would
be on flows with linear stratification, which is a good approximation for a
small region of planetary or stellar atmospheres.

In the next two subsections we will describe turbulence phenomenologies of
stably stratified and unstably stratified flows. Since gravity affects the
velocity field, the secondary fields in such flows are called active fields.
The buoyant flows are typically anisotropic due to external gravity. Still,
one-dimensional energy spectrum and flux are often employed to characterize
such flows because they provide cumulative effects over the polar angles
(angle between the buoyancy direction and wavenumber k).

###  5.1 Stably stratified turbulence

A flow is said to be stably stratified when the density of a fluid under
gravity decreases with height. See Fig. 10(a) for an illustration. The
background density profile is

$\displaystyle\bar{\zeta}(z)=\zeta_{b}+\frac{d\bar{\zeta}}{dz}z=\zeta_{b}+\frac{\zeta_{t}-\zeta_{b}}{d}z,$
(92)

where $\bar{\zeta}(z)$ is the vertical density profile, which is assumed to be
linear; gravity is along $-\hat{z}$; and $\zeta_{b},\zeta_{t}$ are respective
densities at the bottom and top layers of the flow that is confined within a
vertical distance $d$. Stably stratified environment supports internal gravity
waves with Brunt-Väisälä frequency, which is given by [132, 44]

$\displaystyle
N=\sqrt{\frac{g}{\zeta_{m}}\left|\frac{d\bar{\zeta}}{dz}\right|},$ (93)

where $\zeta_{m}$ is the mean density of the whole fluid, and $g$ is the
acceleration due to gravity. Another useful nondimensional number is
Richardson number, which is the ratio of buoyancy and nonlinear term.

The governing equations for the system are [132, 44, 10, 19]

$\displaystyle\frac{\partial{\mathbf{u}}}{\partial{t}}+(\mathbf{u}\cdot\nabla)\mathbf{u}$
$\displaystyle=$
$\displaystyle-\nabla\sigma-N\zeta\hat{z}+\nu\nabla^{2}\mathbf{u}+{\bf
F}_{\mathrm{LS}},$ (94)
$\displaystyle\frac{\partial{\zeta}}{\partial{t}}+(\mathbf{u}\cdot\nabla)\zeta$
$\displaystyle=$ $\displaystyle Nu_{z}+\kappa\nabla^{2}\zeta,$ (95)

where $\sigma$ is the pressure, $\zeta\rightarrow(g\zeta)/(N\zeta_{m})$ is the
density fluctuation in velocity units, and $-N\zeta\hat{z}$ is buoyancy. For
periodic or vanishing boundary condition and in the absence of dissipative
terms, the total energy,

$\displaystyle E_{u}+E_{\zeta}=\int d{\bf r}\frac{1}{2}u^{2}+\int d{\bf
r}\frac{1}{2}\zeta^{2},$ (96)

is conserved [132, 133, 44, 10, 19]. In the above expression, $E_{\zeta}$ is
the total potential anergy.

The forces related to buoyancy are

$\displaystyle{\bf F}_{u}=-N\zeta\hat{z};~{}~{}~{}F_{\zeta}=Nu_{z},$ (97)

which are linear functions of the field variables. Clearly, the energy
injection rates by these two forces are

$\displaystyle\mathcal{F}_{u}({\bf k})$ $\displaystyle=$
$\displaystyle-N\Re[\zeta({\bf k})u_{z}^{*}({\bf k})],$ (98)
$\displaystyle\mathcal{F}_{\zeta}({\bf k})$ $\displaystyle=$ $\displaystyle
N\Re[\zeta({\bf k})u_{z}^{*}({\bf k})].$ (99)

Hence, $\mathcal{F}_{\zeta}({\bf k})+\mathcal{F}_{u}({\bf k})=0$. Therefore,
in the inertial range,

$\displaystyle\frac{d}{dk}[\Pi_{u}(k)+\Pi_{\zeta}(k)]=\mathcal{F}_{u}+\mathcal{F}_{\zeta}=0$
(100)

leading to constancy of $\Pi_{u}(k)+\Pi_{\zeta}(k)$ in the inertial range. The
above two equations follow from the conservation of total energy in the
inviscid limit (see Eq. (96)). Due to the local interactions in
$\mathcal{F}_{u}({\bf k})$ and $\mathcal{F}_{\zeta}({\bf k})$, the cross
fluxes $\Pi^{u<}_{\zeta>}(k)$, $\Pi^{u>}_{\zeta<}(k)$, $\Pi_{u>}^{\zeta<}(k)$,
and $\Pi_{u<}^{\zeta>}(k)$ are zeros.

Figure 10: For stably stratified turbulence: (a) a schematic diagram depicting
the density $\zeta(z)$ that decreases with height; (b) Kinetic energy flux,
$\Pi_{u}(k)$, decreases for $k<k_{B}$, and is constant for
$k_{B}<k<k_{\mathrm{DI}}$, where $k_{B}$ is the Bolgiano wavenumber, and
$k_{\mathrm{DI}}$ is the wavenumber beyond which the dissipation range starts.
(c) Secondary energy flux, $\Pi_{\zeta}(k)$, increases marginally for
$k<k_{B}$, and is constant for $k_{B}<k<k_{\mathrm{DI}}$. For turbulent
thermal convection: (d) a schematic diagram depicting a fluid between two
plates whose temperatures are $T_{b}$ and $T_{t}$ ($T_{b}>T_{t}$); (e, f)
$\Pi_{u}(k)$ and $\Pi_{\zeta}(k)$ are approximately constant in the inertial
range.

Stably stratified turbulence has complex properties, which are discussed in
detail in many books and papers, for example, Lindborg [133], Davidson [44],
and references therein. In this review, to present applications of variable
energy flux in buoyant flows, we describe the spectral analysis for moderately
stratified flows where $|{\bf u\cdot\nabla u}|\approx N\zeta$. For such flows,
Richardson number is of the order of unity.

For moderately stratified flows, Bolgiano [15] and Obukhov [16] provided the
first phenomenological model. They assumed that for a wavenumber band in the
inertial range,

$\displaystyle
ku_{k}^{2}=N\zeta_{k};~{}~{}~{}\Pi_{\zeta}=k\zeta_{k}^{2}u_{k}=\epsilon_{\zeta}$
(101)

that yield the following fluxes and spectra for the velocity and density
fields:

$\displaystyle
E_{u}(k)=c_{1}\epsilon_{\zeta}^{2/5}N^{4/5}k^{-11/5},~{}~{}~{}\Pi_{u}(k)$
$\displaystyle=$ $\displaystyle c_{3}\epsilon_{\zeta}^{3/5}N^{6/5}k^{-4/5},$
(102) $\displaystyle
E_{\zeta}(k)=c_{2}\epsilon_{\zeta}^{4/5}N^{-2/5}k^{-7/5},~{}~{}~{}\Pi_{\zeta}(k)$
$\displaystyle=$ $\displaystyle\epsilon_{\zeta}.$ (103)

Clearly, the kinetic energy flux $\Pi_{u}(k)$ decreases with $k$, in contrast
to constant $\Pi_{u}$ in the inertial range of 3D hydrodynamic turbulence. The
reduction of $\Pi_{u}(k)$ occurs due to the conversion of kinetic energy to
potential energy, and it leads to a steepening of $E_{u}(k)$ [15, 16, 133, 17,
44, 19]. See Fig. 10(b,c) for an illustration.

In addition, Obukhov [15] and Obukhov [16] predicted that buoyancy becomes
weak for $k>k_{B}$, where $k_{B}\approx
N^{3/2}\epsilon_{u}^{-5/4}\epsilon_{\zeta}^{3/4}$ is Bolgiano wavenumber. Due
to weak buoyancy, Bolgiano and Obukhov predicted that both kinetic and
secondary energies exhibit $k^{-5/3}$ spectrum for $k_{B}<k<k_{\mathrm{DI}}$,
where $k_{\mathrm{DI}}$ is the wavenumber beyond which dissipation becomes
significant. See Fig. 10(b) for an illustration. Using the constraint
$k_{B}\ll k_{d}$, where $k_{d}$ is Kolmogorov’s wavenumber, Alam et al. [134]
showed that simultaneous presence of both the scaling regimes ($k^{-11/5}$ and
$k^{-5/3}$) requires very large Reynolds number. Hence, the $k^{-5/3}$ regime
of Bolgiano-Obukhov phenomenology is quite difficult to reproduce in numerical
simulations.

Kimura and Herring [135] reported Bolgiano-Obukhov scaling for a narrow
wavenumber band in their decaying simulation on a $128^{3}$ grid. Kumar et al.
[17] performed a numerical simulation of stably stratified turbulence on a
$1024^{3}$ grid for Richardson number around unity and observed a good
agreement between numerical results and the predictions of Eqs. (102, 103).
Rosenberg et al. [136] reported Bolgiano-Obukhov scaling in their simulation
of rotating stratified turbulence.

Strong buoyancy (large Richardson number) makes the flow anisotropic, hence
Bolgiano-Obukhov scaling is inapplicable to such flows. Lindborg [133] and
Davidson [44] argued that the longitudinal and traverse velocity components
exhibit $k^{-3}$ and $k^{-5/3}$ spectra respectively. Variable energy flux may
provide interesting clues for this regime as well. For two-dimensional stably
stratified turbulence, Kumar et al. [137] derived several interesting
relations using variable energy flux. Also note that buoyancy is weak for
flows with small Richardson number. Hence, such flows exhibit Kolmogorov’s
spectrum [17].

In the next subsection we will employ the ideas of variable energy flux to
turbulent thermal convection.

###  5.2 Turbulent thermal convection

Thermal convection too is driven by buoyancy. However, in contrast to the
stably stratified flows, the fluid density increases with height that makes
the flow unstable. A setup, exhibited in Fig. 10(d), consists of a thin fluid
layer confined between two thermally conducting plates separated by a distance
$d$. The temperatures of the bottom and top plates are $T_{b}$ and $T_{t}$
respectively.

In thermal convection, the temperature is a sum of externally-imposed linearly
varying temperature $\bar{T}(z)$ and fluctuation $\zeta(x,y,z)$:

$\displaystyle T(x,y,z)=\bar{T}(z)+\zeta(x,y,z),$ (104)

where

$\displaystyle\bar{T}(z)=T_{b}+\frac{d\bar{T}}{dz}z=T_{b}-\frac{T_{b}-T_{t}}{d}z.$
(105)

The equations for thermal convection under Boussinesq approximation are [138]

$\displaystyle\frac{\partial{\bf u}}{\partial t}+({\bf u}\cdot\nabla){\bf u}$
$\displaystyle=$ $\displaystyle-\frac{1}{\zeta_{m}}\nabla\sigma+\alpha
g\zeta\hat{z}+\nu\nabla^{2}{\bf u},$ (106)
$\displaystyle\frac{\partial\zeta}{\partial t}+({\bf u}\cdot\nabla)\zeta$
$\displaystyle=$ $\displaystyle\frac{\Delta}{d}u_{z}+\kappa\nabla^{2}\zeta,$
(107) $\displaystyle\nabla\cdot{\bf u}=0,$ (108)

where $\alpha,\kappa$ are respectively the thermal expansion coefficient and
thermal diffusivity of the fluid, $g$ is the acceleration due to gravity, and
$\Delta=T_{b}-T_{t}$. The two important parameters of turbulent thermal
convection are Prandtl number, $\mathrm{Pr}=\nu/\kappa$, and Rayleigh number,

$\displaystyle\mathrm{Ra}=\frac{\alpha gd^{3}\Delta}{\nu\kappa}.$ (109)

Note that the forces

$\displaystyle{\bf F}_{u}=\alpha
g\zeta\hat{z},~{}~{}~{}F_{\zeta}=\frac{\Delta}{d}u_{z}$ (110)

are linear functions of the field variables.

In thermal convection, hot plumes ascend and cold ones descend. Hence,
$\langle\zeta({\bf r})u_{z}({\bf r})\rangle>0$. Therefore, using Parceval’s
theorem we deduce that

$\displaystyle\sum_{\bf k}\Re\left[\left\langle\zeta({\bf k})u_{z}^{*}({\bf
k})\right\rangle\right]>0.$ (111)

Further, numerical simulations reveals that $\Re\left[\left\langle\zeta({\bf
k})u_{z}^{*}({\bf k})\right\rangle\right]>0$ for most Fourier modes of thermal
convection. Hence,

$\displaystyle\mathcal{F}_{u}({\bf k})$ $\displaystyle=$ $\displaystyle\alpha
g\Re[\zeta({\bf k})u_{z}^{*}({\bf k})]>0$ (112)
$\displaystyle\mathcal{F}_{\zeta}({\bf k})$ $\displaystyle=$
$\displaystyle\frac{\Delta}{d}\Re[\zeta({\bf k})u_{z}^{*}({\bf k})]>0$ (113)

Therefore, in the inertial range,

$\displaystyle\frac{d}{dk}[\Pi_{u}(k)-\frac{\alpha
gd}{\Delta}\Pi_{\zeta}(k)]=0$ (114)

leading to

$\displaystyle\Pi_{u}(k)-\frac{\alpha
gd}{\Delta}\Pi_{\zeta}(k)=\mathrm{const}.$ (115)

In the dissipationless limit, $\int d{\bf r}\frac{1}{2}[u^{2}-(\alpha
gd/\Delta)\zeta^{2}]$, a sum of kinetic energy and potential energy, is
conserved. Here, the potential energy, $-\frac{1}{2}(\alpha
gd/\Delta)\zeta^{2}$, is negative because the system is unstable. Since this
potential energy can be converted to kinetic energy, it is also called
available potential energy. Contrast this potential energy with that for
stably stratified turbulence.

Equation (112) implies that $\Pi_{u}(k)$ should increase with $k$. However,
for $\mathrm{Pr}\sim 1$, Verma et al. [18] showed that
$\mathcal{F}_{u}(k)\sim(kL)^{-5/3}$, hence, $\mathcal{F}_{u}(k)$ is small in
the inertial range because $kL\gg 1$. The function $\mathcal{F}_{u}(k)$ is
even steeper for small Pr [19]. These observations indicate that large-scale
thermal plumes drive the flow, similar to the forcing in Kolmogorov’s theory
for hydrodynamic turbulence. These results falsify the popular statements that
thin thermal plumes drive thermal convection.

Based on similarities between the forcing in thermal convection and
Kolmogorov’s model of turbulence, Kumar et al. [17] and Verma et al. [18]
argued that for $\mathrm{Pr}\lessapprox 1$, $\Pi_{u}(k)$ remains an
approximate constant in the inertial range and $E_{u}(k)\sim k^{-5/3}$. Verma
et al. [17, 18] verified the above phenomenology using high-resolution
numerical simulations. The above flux-based arguments resolve the long impasse
in the field regarding the energy spectrum. Note that several past works [139,
75, 76, 77] projected that Bolgiano-Obukhov’s scaling for stably stratified
turbulence ($E_{u}(k)\sim k^{-11/5}$) holds for turbulent thermal convection
as well, while experimental and numerical works were inconclusive. For large
Prandtl numbers, the flow is dissipative and $E_{u}(k)\sim k^{-13/3}$ [140,
19].

However, there is a major difference between hydrodynamic turbulence and
turbulent convection. In turbulent convection, inertial-range kinetic energy
flux is a fraction of the total viscous dissipation rate. This is because
thermal plumes drive the flow at all scales. In particular, under steady
state, for a $k$ in the inertial range,

$\displaystyle\Pi_{u}(k)$ $\displaystyle\approx$
$\displaystyle\int_{0}^{k}dk^{\prime}\mathcal{F}_{u}(k^{\prime})=\int_{0}^{\infty}dk^{\prime}\mathcal{F}_{u}(k^{\prime})-\int_{k}^{\infty}dk^{\prime}\mathcal{F}_{u}(k^{\prime})$
(116) $\displaystyle=$
$\displaystyle\epsilon_{u}-\int_{k}^{\infty}dk^{\prime}\mathcal{F}_{u}(k^{\prime}),$

where $\epsilon_{u}$ is the total viscous dissipation rate. Using Eq. (116) we
deduce that the inertial-range kinetic energy flux $\Pi_{u}(k)<\epsilon_{u}$
due to the presence of buoyancy at all scales. Using numerical simulations,
Bhattacharya et al. [141] showed that for $\mathrm{Pr}=1$, the inertial-range
$\Pi_{u}(k)$ is around one-third of $\epsilon_{u}$.

There are many turbulent flows with unstable stratification, notably Rayleigh-
Taylor turbulence [142], bubbly turbulence [143], Taylor-Couette turbulence
[144, 145], etc. Based on the arguments of this subsection, we expect that the
turbulence properties of unstable stable stratification are similar to those
of hydrodynamic turbulence, e.g., $E_{u}(k)\sim k^{-5/3}$ [18, 19]. The
results on Rayleigh-Taylor turbulence [146, 147, 142] and Taylor-Couette
turbulence [144] are consistent with the above observations.

In the aforementioned buoyant flows, ${\bf F}_{u}$ and ${\bf F}_{\zeta}$ are
linear functions of $\zeta$ and ${\bf u}$ respectively. Hence the cross energy
transfer occurs among the modes with same wavenumbers. Therefore,
$\mathcal{F}_{u}({\bf k})$ and $\mathcal{F}_{\zeta}({\bf k})$ are functions
only of local wavenumber ${\bf k}$. However, when ${\bf F}_{u}$ and ${\bf
F}_{\zeta}$ are nonlinear functions of ${\bf u}$ and/or $\zeta$, the cross
energy transfers are convolutions of ${\bf u}$ and $\zeta$ Fourier modes. In
the following section, we illustrate such energy transfers in MHD turbulence.

##  6 Variable energy fluxes in magnetohydrodynamic turbulence

Magnetohydrodynamic (MHD) turbulence is a vast area of research with many
astrophysical and engineering applications. It is covered in several books,
e.g., [148, 149, 150], review articles, e.g., [38, 21, 151, 74], and
references therein. The present section does not attempt to summarise vast
number of results of MHD turbulence, but focusses on variable energy fluxes of
MHD turbulence.

A magnetofluid, which is a quasi-neutral and electrically-conducting
collisional plasma, is described by velocity field and magnetic field, which
is denoted by $\bm{\zeta}$. In the following subsection, we describe the
governing equations for MHD turbulence.

###  6.1 MHD turbulence: Governing equations and cross transfers

The dynamical equations for the velocity and magnetic fields of MHD are [148,
149, 150]

$\displaystyle\frac{\partial{\mathbf{u}}}{\partial{t}}+({\bf
u}\cdot\nabla){\bf u}$ $\displaystyle=$ $\displaystyle-\nabla
p+(\bm{\zeta}\cdot\nabla)\bm{\zeta}+{\bf F}_{\mathrm{LS}}+\nu\nabla^{2}{\bf
u},$ (117) $\displaystyle\frac{\partial{\bm{\zeta}}}{\partial{t}}+({\bf
u}\cdot\nabla)\bm{\zeta}$ $\displaystyle=$
$\displaystyle(\bm{\zeta}\cdot\nabla){\bf u}+\kappa\nabla^{2}\bm{\zeta},$
(118) $\displaystyle\nabla\cdot{\bf u}=\nabla\cdot\bm{\zeta}$ $\displaystyle=$
$\displaystyle 0,$ (119)

where $p$ is the total (thermodynamic + magnetic) pressure, ${\bf
F}_{\mathrm{LS}}$ is the large-scale external force, and $\nu,\kappa$ are the
kinematic viscosity and magnetic diffusivity respectively. In the above
equation, the magnetic field is in Alfvénic units, which has same dimension as
the velocity field. In the above equations, the forces on ${\bf u}$ and
$\bm{\zeta}$ fields are

$\displaystyle{\bf F}_{u}=(\bm{\zeta}\cdot\nabla)\bm{\zeta};~{}~{}~{}{\bf
F}_{\zeta}=(\bm{\zeta}\cdot\nabla){\bf u}.$ (120)

For inviscid flow ($\nu=\kappa=0$), under periodic or vanishing boundary
condition, the total energy

$\displaystyle E_{u}+E_{\zeta}=\int d{\bf r}\frac{1}{2}u^{2}+\int d{\bf
r}\frac{1}{2}\zeta^{2}$ (121)

is conserved [148, 149, 150]. Here, $E_{\zeta}$ is the total magnetic anergy.
The other conserved quantities of inviscid MHD are cross helicity, $\int d{\bf
r}\frac{1}{2}({\bf u}\cdot\bm{\zeta})$, and magnetic helicity, $\int d{\bf
r}\frac{1}{2}({\bf A\cdot\bm{\zeta}})$, where ${\bf A}$ is the vector
potential. In this review, we focus on kinetic and magnetic energies only.

In Fourier space, the forces of Eq. (120) are

$\displaystyle{\bf F}_{u}({\bf k})=i\sum_{\bf p}\\{{\bf k}\cdot\bm{\zeta}({\bf
q})\\}\bm{\zeta}({\bf p});~{}~{}~{}{\bf F}_{\zeta}({\bf k})=i\sum_{\bf
p}\\{{\bf k}\cdot\bm{\zeta}({\bf q})\\}{\bf u}({\bf p}),$ (122)

where ${\bf q=k-p}$. The equation for the kinetic and magnetic modal energies
are

$\displaystyle\frac{d}{dt}E_{u}(\mathbf{k})$ $\displaystyle=$ $\displaystyle
T_{u}({\bf k})+\mathcal{F}_{u}({\bf k})+{\bf F}_{\mathrm{LS}}({\bf
k})-D_{u}({\bf k})$ $\displaystyle=$ $\displaystyle\sum_{\bf
p}\Im\left[{\bf\\{k\cdot u(q)\\}\\{{\bf u(p)\cdot
u^{*}(k)}\\}}\right]+\mathcal{F}_{u}({\bf k})+{\bf F}_{\mathrm{LS}}({\bf
k})-2\nu k^{2}E_{u}({\mathbf{k}}),$
$\displaystyle\frac{d}{dt}E_{\zeta}(\mathbf{k})$ $\displaystyle=$
$\displaystyle T_{\zeta}({\bf k})+\mathcal{F}_{\zeta}({\bf k})-D_{\zeta}({\bf
k})$ (124) $\displaystyle=$ $\displaystyle\sum_{\bf p}\Im\left[{\bf\\{k\cdot
u(q)\\}\\{\bm{\zeta}(p)\cdot\bm{\zeta}^{*}(k)\\}}\right]+\mathcal{F}_{\zeta}({\bf
k})-2\kappa k^{2}E_{\zeta}({\mathbf{k}}),$

where the kinetic and magnetic energy injection rates by the forces ${\bf
F}_{u}({\bf k})$ and ${\bf F}_{\zeta}({\bf k})$ are

$\displaystyle\mathcal{F}_{u}({\bf k})$ $\displaystyle=$
$\displaystyle\sum_{\bf
p}-\Im\left[{\bf\\{k\cdot\bm{\zeta}(q)\\}\\{\bm{\zeta}(p)\cdot
u^{*}(k)\\}}\right],$ (125) $\displaystyle\mathcal{F}_{\zeta}({\bf k})$
$\displaystyle=$ $\displaystyle\sum_{\bf
p}-\Im\left[{\bf\\{k\cdot\bm{\zeta}(q)\\}\\{u(p)\cdot\bm{\zeta}^{*}(k)\\}}\right].$
(126)

Note that ${\bf F}_{u}({\bf k})$ facilitates energy transfer from $\bm{\zeta}$
to ${\bf u}$, while ${\bf F}_{\zeta}({\bf k})$ yields energy transfer from
${\bf u}$ to $\bm{\zeta}$. As remarked earlier, $\mathcal{F}_{u}({\bf
k}),\mathcal{F}_{\zeta}({\bf k})$ are convolutions because ${\bf F}_{u},{\bf
F}_{\zeta}$ are nonlinear functions of ${\bf u}$ and $\bm{\zeta}$. In
contrast, the corresponding transfers for the buoyant flows are functions of
fields at local wavenumber (see Sec. 5).

The nonlinear structure of the cross transfers between ${\bf u}$ and
$\bm{\zeta}$ can be formulated in terms of mode-to-mode energy transfers. For
a triad ${\bf(k^{\prime},p,q)}$ satisfying ${\bf k^{\prime}+p+q}=0$, Dar et
al. [37] and Verma [38] derived the following formulas for the mode-to-mode
energy transfers from ${\bf u}$ to $\bm{\zeta}$ and vice versa:

$\displaystyle S^{u\zeta}({\bf k^{\prime}|p|q})$ $\displaystyle=$
$\displaystyle\Im\left[{\bf\\{k^{\prime}\cdot\bm{\zeta}(q)\\}\\{\bm{\zeta}(p)\cdot
u(k^{\prime})\\}}\right]=-\Im\left[{\bf\\{k\cdot\bm{\zeta}(q)\\}\\{\bm{\zeta}(p)\cdot
u^{*}(k)\\}}\right],$ $\displaystyle S^{\zeta u}({\bf k^{\prime}|p|q})$
$\displaystyle=$
$\displaystyle\Im\left[{\bf\\{k^{\prime}\cdot\bm{\zeta}(q)\\}\\{u(p)\cdot\zeta(k^{\prime})\\}}\right]=-\Im\left[{\bf\\{k\cdot\bm{\zeta}(q)\\}\\{u(p)\cdot\zeta^{*}(k)\\}}\right].$

The former is the mode-to-mode energy transfer from $\bm{\zeta}({\bf p})$ to
${\bf u}({\bf k^{\prime}})$ with the mediation of $\bm{\zeta}({\bf q})$, while
the latter provides the energy transfer from ${\bf u}({\bf p})$ to
$\bm{\zeta}({\bf k^{\prime}})$ with the mediation of $\bm{\zeta}({\bf q})$. In
$S^{ab}({\bf k^{\prime}|p|q})$, the superscript $ab$ refer to receiver field
$a$ and giver field $b$.

These transfers satisfy the property:

$\displaystyle S^{u\zeta}({\bf k^{\prime}|p|q})=-S^{\zeta u}({\bf
p|k^{\prime}|q}),$ (129)

that is, the energy gained by ${\bf u}({\bf k^{\prime}})$ from
$\bm{\zeta}({\bf p})$ is negative of the energy gained by $\bm{\zeta}({\bf
p})$ from ${\bf u}({\bf k^{\prime}})$. This is a property of energy exchange.
Using the above property, we can show that for any region $A$ of Fourier
space, including a triad,

$\displaystyle\sum_{{\bf k^{\prime}}\in A}\sum_{{\bf p}\in A}[S^{u\zeta}({\bf
k^{\prime}|p|q})+S^{\zeta u}({\bf p|k^{\prime}|q})]=0.$ (130)

By summing over all the Fourier modes, we deduce that
$\mathcal{F}_{u}+\mathcal{F}_{\zeta}=0$, where $\mathcal{F}_{u}$ and
$\mathcal{F}_{\zeta}$ are respectively the total energy gained by the velocity
and secondary fields via cross energy transfers. Note that stably-stratified
turbulence too has $\mathcal{F}_{u}+\mathcal{F}_{\zeta}=0$.

In the next subsection we define the energy fluxes of MHD turbulence.

###  6.2 Various energy fluxes of MHD turbulence

Using the mode-to-mode energy transfers of Eqs. (6.1, 6.1), Dar et al. [37]
and Verma [38] derived formulas for the energy fluxes of MHD turbulence for a
wavenumber sphere of radius $k_{0}$. Note that $\Pi_{u}(k_{0})$ and
$\Pi_{\zeta}(k_{0})$ are respective fluxes for the kinetic and magnetic
energies, while the energy flux $\Pi^{u<}_{\zeta>}(k_{0})$ represents the net
energy transfer from all the velocity modes inside the sphere to all the
magnetic modes outside the sphere, that is,

$\displaystyle\Pi^{u<}_{\zeta>}(k_{0})=\sum_{p\leq
k_{0}}\sum_{k^{\prime}>k_{0}}S^{\zeta u}({\bf k^{\prime}|p|q}).$ (131)

The other fluxes, $\Pi^{\zeta<}_{u>}(k_{0})$, $\Pi^{u<}_{\zeta<}(k_{0})$, and
$\Pi^{u>}_{\zeta>}(k_{0})$ are defined similarly. Note that an application of
Eq. (129) yields

$\displaystyle\Pi^{ua}_{\zeta b}(k_{0})=-\Pi^{\zeta b}_{ua}(k_{0}),$ (132)

where $a,b$ represent $<$ or $>$. The above energy fluxes of MHD turbulence,
depicted in Fig. 11, have been studied in great detail [37, 152, 39].
Interestingly, $\Pi^{u<}_{\zeta>}(k)$ and $\Pi^{\zeta<}_{u>}(k)$ are absent in
buoyant flows due to the products of field variables with the same wavenumber
(see Sec. 5).

Figure 11: The six energy fluxes of MHD turbulence.
$\epsilon_{u},\epsilon_{\zeta}$ are the dissipation rates of the velocity and
magnetic fields respectively. The wheel in the centre of the velocity sphere
represents the external forcing at large scales.

Now we relate the above quantities to the fluxes $\Pi^{u}_{\zeta}$ and
$\Pi^{\zeta}_{u}$ discussed in Sec. 4.2. Clearly,

$\displaystyle\Pi^{\zeta}_{u<}(k_{0})=\sum_{k\leq k_{0}}\mathcal{F}_{u}({\bf
k})$ $\displaystyle=$
$\displaystyle\Pi^{\zeta<}_{u<}(k_{0})+\Pi^{\zeta>}_{u<}(k_{0}),$ (133)
$\displaystyle\Pi^{\zeta}_{u>}(k_{0})=\sum_{k>k_{0}}\mathcal{F}_{u}({\bf k})$
$\displaystyle=$
$\displaystyle\Pi^{\zeta<}_{u>}(k_{0})+\Pi^{\zeta>}_{u>}(k_{0}),$ (134)
$\displaystyle\Pi^{u}_{\zeta<}(k_{0})=\sum_{k\leq
k_{0}}\mathcal{F}_{\zeta}({\bf k})$ $\displaystyle=$
$\displaystyle\Pi^{u<}_{\zeta<}(k_{0})+\Pi^{u>}_{\zeta<}(k_{0}),$ (135)
$\displaystyle\Pi^{u}_{\zeta>}(k_{0})=\sum_{k\leq
k_{0}}\mathcal{F}_{\zeta}({\bf k})$ $\displaystyle=$
$\displaystyle\Pi^{u<}_{\zeta>}(k_{0})+\Pi^{u>}_{\zeta>}(k_{0}).$ (136)

The identities of Eqs. (72, 73) are translated to the following identities for
MHD turbulence:

$\displaystyle\mathcal{F}_{u}$ $\displaystyle=$
$\displaystyle\Pi^{\zeta<}_{u<}(k)+\Pi^{\zeta<}_{u>}(k)+\Pi^{\zeta>}_{u<}(k)+\Pi^{\zeta>}_{u>}(k)=C_{1},$
(137) $\displaystyle\mathcal{F}_{\zeta}$ $\displaystyle=$
$\displaystyle\Pi^{u<}_{\zeta<}(k)+\Pi^{u<}_{\zeta>}(k)+\Pi^{u>}_{\zeta<}(k)+\Pi^{u>}_{\zeta>}(k)=C_{2},$
(138)

where $C_{1}$ and $C_{2}$ are constants. The above two sums are constant in
$k$ even though the individual flux in the sum can vary with $k$. Note that
$C_{1}$ is the net energy transfer from $\bm{\zeta}$ to u, while $C_{2}$ is
the net energy transfer from u to $\bm{\zeta}$. Interestingly, the vorticity
field has similar properties as the magnetic field (with some important
deviations) [153]. In Sec. 8.1 we will describe various fluxes associated with
the vorticity field.

Under a steady state with no external forcing for $\bm{\zeta}$, the net
nonlinear energy transfer to the magnetic field, $\mathcal{F}_{\zeta}$,
balances the magnetic diffusion or dissipation rate, $\epsilon_{\zeta}$. That
is,

$\displaystyle\mathcal{F}_{\zeta}=\sum_{\bf k}2\eta
k^{2}E_{\zeta}(k)=\epsilon_{\zeta}>0.$ (139)

Note however that an imbalance in $\mathcal{F}_{\zeta}$ and $\epsilon_{\zeta}$
makes the flow unsteady. For the velocity field,
$\mathcal{F}_{u}=-\mathcal{F}_{\zeta}<0$. In addition, the viscous force
dissipates the kinetic energy. Hence, for the velocity field, a large-scale
force, ${\bf F}_{\mathrm{LS}}$, is required to maintain a steady state. Using
energy balance we obtain

$\displaystyle\varepsilon_{\mathrm{inj}}+\mathcal{F}_{u}=\sum_{\bf k}2\nu
k^{2}E_{u}(k)=\epsilon_{u},$ (140)

where $\varepsilon_{\mathrm{inj}}$ is the total energy injection rate by ${\bf
F}_{\mathrm{LS}}$. In astrophysics, a supernova is an example of one such
energy source. Note that under a steady state, the net energy injection rate
by ${\bf F}_{\mathrm{LS}}$ equals the sum of viscous dissipation and Joule
dissipation, i.e., $\varepsilon_{\mathrm{inj}}=\epsilon_{u}+\epsilon_{\zeta}$.

Equations (76-79) are applicable to MHD turbulence as well. When $k$ is in the
inertial range, the identities of Eqs. (76-79) get simplified to

$\displaystyle\Pi_{u}(k)+\Pi^{\zeta}_{u>}(k)=\epsilon_{u};~{}~{}~{}\Pi_{\zeta}(k)+\Pi^{u}_{\zeta>}(k)=\epsilon_{\zeta},$
(141)
$\displaystyle\Pi_{u}(k)+\Pi^{u<}_{\zeta}(k)=\varepsilon_{\mathrm{inj}},~{}~{}~{}\Pi_{\zeta}(k)-\Pi^{u}_{\zeta<}(k)=0.$
(142)

The above equations follow from the energetics. For example, a physical
interpretation of the equation,
$\Pi_{u}(k)+\Pi^{u<}_{\zeta}(k)=\varepsilon_{\mathrm{inj}}$, is as follows.
Here, the velocity field is forced at large scales by ${\bf F}_{\mathrm{LS}}$.
A part of the injected kinetic energy by ${\bf F}_{\mathrm{LS}}$ cascades to
the inertial range as $\Pi_{u}$, but a fraction of it is transferred to the
magnetic energy as $\Pi^{u<}_{\zeta}(k)>0$ [37, 38, 21, 39]. This cross
transfer from ${\bf u}$ to $\bm{\zeta}$ amplifies the magnetic field; this
mechanism is responsible for the generation of large-scale magnetic field in
planets, stars, and galaxies [20, 154, 21, 38]. Note that this cross energy
transfer in MHD turbulence makes both $\Pi_{u}(k)$ and $\Pi^{u<}_{\zeta}(k)$
functions of $k$, in contrast to constant $\Pi_{u}(k)$ in the inertial range
of hydrodynamic turbulence. Using similar arguments we can show that the other
MHD fluxes also vary with $k$. However, the total energy flux is constant
under a steady state. That is,

$\displaystyle\varepsilon_{\mathrm{inj}}$ $\displaystyle=$
$\displaystyle\Pi_{u}(k)+\Pi^{u<}_{\zeta}(k)=\Pi_{u}(k)+\Pi^{u<}_{\zeta<}(k)+\Pi^{u<}_{\zeta>}(k)$
(143) $\displaystyle=$
$\displaystyle\Pi_{u}(k)+\Pi_{\zeta}(k)+\Pi^{\zeta<}_{u>}(k)+\Pi^{u<}_{\zeta>}(k)=\Pi_{\mathrm{tot}}(k)=\epsilon_{u}+\epsilon_{\zeta}.$

Here we employ the identity
$\Pi^{u<}_{\zeta<}=\Pi_{\zeta}+\Pi^{\zeta<}_{u>}(k)$. These relations do not
hold for unsteady flows. For example, in decaying MHD turbulence, magnetic
field feeds energy to the velocity field when $b^{2}\gg u^{2}$ and vice versa
[152].

The cross energy transfer $\Pi^{u<}_{\zeta}(k)$ is also responsible for the
drag reduction in MHD turbulence [43]. Since $\Pi^{u<}_{\zeta}(k)>0$, using
Eq. (142) we deduce that $\Pi_{u}(k)<\varepsilon_{\mathrm{inj}}$. Note that in
hydrodynamic turbulence, $\Pi_{u}(k)\approx\varepsilon_{\mathrm{inj}}$.
Therefore, for the same $\varepsilon_{\mathrm{inj}}$, $\Pi_{u}(k)$ in MHD
turbulence is lower than that for hydrodynamic turbulence. Therefore, the
turbulent drag, $F_{D}\approx\Pi_{u}/U$, will be lower for MHD turbulence
compared to its hydrodynamic counterpart [43]. In the next section we will
show that a similar dynamics is a work in turbulent flows with dilute polymer.

Variable energy flux also provides valuable insights into the dynamics and
spectral laws of MHD turbulence and dynamo. For example, the kinetic and
magnetic energy spectra are often modelled as [155, 156, 150, 151, 157, 158,
38]

$\displaystyle
E_{u,\zeta}(k)\sim\epsilon_{u,\zeta}^{2/3}k^{-5/3},~{}~{}\mathrm{or}~{}~{}~{}(B_{0}\epsilon_{u,\zeta})^{1/2}k^{-3/2},$
(144)

where $B_{0}$ is the magnitude of the mean magnetic field. The above equations
need proper interpretations because $\Pi_{u}(k)$ and $\Pi_{\zeta}(k)$, which
are variables of $k$, cannot be simply replaced by $\epsilon_{u}$ and
$\epsilon_{\zeta}$ respectively. For the spectral studies, it is more
appropriate to employ the total energy flux or the fluxes of Elsäser variables
(topic of the next subsection) because they are constant in $k$.

###  6.3 Energy fluxes associated with Elsäser variables

An alternative formulation of MHD turbulence is in terms of Elsäser variables,
${\bf z}^{\pm}={\bf u}\pm\bm{\zeta}$. The MHD equations in terms ${\bf
z}^{\pm}$ and a mean magnetic field ${\bf B}_{0}$ are [148, 150]

$\displaystyle\frac{\partial{\mathbf{z}^{\pm}}}{\partial{t}}\mp({\bf
B}_{0}\cdot\nabla){\bf z}^{\pm}+({\bf z}^{\mp}\cdot\nabla){\bf
z}^{\pm}=-\nabla p+\nu_{+}\nabla^{2}{\bf z}^{\pm}+\nu_{-}\nabla^{2}{\bf
z}^{\mp},$ (145) $\displaystyle\nabla\cdot{\bf z}^{\pm}=0,$ (146)

where $\nu_{\pm}=\frac{1}{2}(\nu\pm\eta)$. For ${\bf z}^{+}$, in a triad
$({\bf k^{\prime},p,q})$, the mode-to-mode energy transfer from ${\bf
z^{+}(p)}$ to ${\bf z^{+}(k^{\prime})}$ with the mediation of ${\bf z^{-}(q)}$
is [38]

$\displaystyle S^{z^{+}z^{+}}({\bf k^{\prime}|p|q})$ $\displaystyle=$
$\displaystyle-\Im\left[{\bf\\{k^{\prime}\cdot z^{-}(q)\\}\\{z^{+}(p)\cdot
z^{+}(k^{\prime})\\}}\right].\ $ (147)

For ${\bf z}^{-}$, the mode-to-mode energy transfer from ${\bf z^{-}(p)}$ to
${\bf z^{-}(k^{\prime})}$ with the mediation of ${\bf z^{+}(q)}$ is

$\displaystyle S^{z^{-}z^{-}}({\bf k^{\prime}|p|q})$ $\displaystyle=$
$\displaystyle-\Im\left[{\bf\\{k^{\prime}\cdot z^{+}(q)\\}\\{z^{-}(p)\cdot
z^{-}(k^{\prime})\\}}\right].$ (148)

Note that there is no cross transfer from ${\bf z^{+}}$ to ${\bf z^{-}}$ and
vice versa. The corresponding energy fluxes are

$\displaystyle\Pi_{z^{+}}(k_{0})=\sum_{p\leq
k_{0}}\sum_{k>k_{0}}S^{z^{+}z^{+}}({\bf
k|p|q});~{}\Pi_{z^{-}}(k_{0})=\sum_{p\leq
k_{0}}\sum_{k>k_{0}}S^{z^{-}z^{-}}({\bf k|p|q}).$ (149)

Due to the absence of cross transfers between ${\bf z}^{+}$ and ${\bf z}^{-}$,
$\Pi_{z^{\pm}}(k_{0})$ are constant in the inertial range of MHD turbulence.
In addition, in the inertial range, $\Pi_{z^{\pm}}(k)=\epsilon_{z^{\pm}}$,
where $\epsilon_{z^{\pm}}$ are the total dissipation rates of ${\bf z}^{\pm}$.
Constancy of $\Pi_{z^{\pm}}(k)$ in the inertial range makes them suitable
candidates for modelling the energy spectrum of MHD turbulence. For example,
Marsch [159] argued that

$\displaystyle
E_{z^{\pm}}(k)=K_{z^{\pm}}(k)\epsilon_{z^{\pm}}^{4/3}\epsilon_{z^{\mp}}^{-2/3}k^{-5/3},$
(150)

where $K_{z^{\pm}}(k)$ are nondimensional constants. Refer to Bismamp [150]
and Verma [38] for further details.

In the next section we will briefly describe the energy fluxes for a solvent
with dilute polymers.

##  7 Variable energy fluxes in a turbulent flow with dilute polymers

In this section, we discuss the energy transfers and drag reduction in a
solution of dilute polymers [160, 161, 162, 163, 164, 165]. In such flows, the
polymer is often described by finitely extensible nonlinear elastic-Peterlin
(FENE-P) model. The equations for the velocity field and polymer-conformation
tensor $\zeta$ of FENE-P model are [166, 167]:

$\displaystyle\frac{\partial{u_{i}}}{\partial
t}+u_{j}\partial_{j}u_{i}=-\frac{1}{\rho}\partial_{i}p+\nu\partial_{jj}u_{i}+\frac{\mu}{\tau_{p}}\partial_{j}(f\zeta_{ij})+F_{\mathrm{LS},i},$
(151) $\displaystyle\frac{\partial{\zeta_{ij}}}{\partial
t}+u_{l}\partial_{l}\zeta_{ij}=\zeta_{il}\partial_{l}u_{j}+\zeta_{il}\partial_{j}u_{l}+\frac{1}{\tau_{p}}[f\zeta_{ij}-\delta_{ij}],$
(152) $\displaystyle\partial_{i}u_{i}=0,$ (153)

where $p$ is the pressure, $\rho$ is the mean density of the solvent, $\nu$ is
the kinematic viscosity, $\mu$ is an additional viscosity parameter,
$\tau_{p}$ is the polymer relaxation time, and $f$ is the renormalized
Peterlin’s function. We also remark that energetics of polymer turbulence has
many similarities with those for MHD turbulence. As we describe below, there
is a preferential energy transfer from the velocity field to the polymer, just
like the energy transfer from the velocity field to the magnetic field in
dynamos. Refer to deGennes [40], Fouxon and Lebdev [166], and references
therein for details.

In the above equations, the following forces (apart from constants) associated
with ${\bf u}$ and $\zeta$ induce cross energy transfers:

$\displaystyle
F_{u,i}=\partial_{j}(f\zeta_{ij}),~{}~{}~{}F_{\zeta,ij}=\zeta_{il}\partial_{l}u_{j}+\zeta_{il}\partial_{j}u_{l}.$
(154)

In Fourier space, the respective energy feed by these forces to the kinetic
energy and the tensor energy are

$\displaystyle\mathcal{F}_{u}({\bf k})$ $\displaystyle=$
$\displaystyle-\sum_{\bf p}\Im\left[k_{j}f({\bf q})\zeta_{ij}({\bf
p})u^{*}_{i}({\bf k})\right],$ (155) $\displaystyle\mathcal{F}_{\zeta}({\bf
k})$ $\displaystyle=$ $\displaystyle-\sum_{\bf p}\Im\left[\zeta_{il}({\bf
q})p_{l}u_{j}({\bf p})\zeta^{*}_{ij}({\bf k})+\zeta_{il}({\bf
q})p_{j}u_{l}({\bf p})\zeta^{*}_{ij}({\bf k})\right],$ (156)

where ${\bf q=k-p}$. Both, $\mathcal{F}_{u}({\bf k})$ and
$\mathcal{F}_{\zeta}({\bf k})$ are convolutions similar to those in MHD
turbulence. However, the structure of the nonlinear terms for the polymers is
more complex than that for MHD turbulence. Till date, there are no formulas
for the mode-to-mode energy transfers from ${\bf u}$ to $\zeta$ and vice
versa. Yet, the following equations can be used to describe the energy fluxes
from the velocity field to the polymer field.

$\displaystyle\Pi^{u}_{\zeta<}(k_{0})=\sum_{k\leq
k_{0}}\mathcal{F}_{\zeta}({\bf
k});~{}~{}~{}\Pi^{u}_{\zeta>}(k_{0})=\sum_{k>k_{0}}\mathcal{F}_{\zeta}({\bf
k}).$ (157)

In fact, Eqs. (141, 142) too are applicable to turbulent flows with dilute
polymers.

From Eq. (142), under a steady state and in the inertial range,

$\displaystyle\Pi_{u}(k)=\varepsilon_{\mathrm{inj}}-\Pi^{u<}_{\zeta}(k)$ (158)

where $\varepsilon_{\mathrm{inj}}$ is the energy injection rate by large-scale
forcing, and $\Pi^{u<}_{\zeta}(k)$ is the energy transfer from the large-scale
velocity field to $\zeta$. See Fig. 9(a) for an illustration. In polymeric
flows, it has been shown that the velocity field stretches the polymers [160,
161], similar to the stretching of the magnetic field in MHD turbulence and
dynamo. Therefore, we expect that

$\displaystyle\Pi^{u<}_{\zeta}(k)>0\Rightarrow\Pi_{u}(k)<\varepsilon_{\mathrm{inj}}.$
(159)

That is, for the same $\varepsilon_{\mathrm{inj}}$, the kinetic energy flux in
the polymer solution will be reduced compared to hydrodynamic turbulence.
Therefore, the turbulent drag, $F_{D}\approx\Pi_{u}/U$, for a polymeric flow
will be lower than its hydrodynamic counterpart [43]. Several numerical
simulations exhibit the aforementioned reduction in $\Pi_{u}(k)$ [41, 42].
Thus, variable energy flux provides valuable insights into the mechanism of
turbulent drag reduction in polymer solution and in MHD turbulence.

In the next section we describe the flux associated with enstrophy and kinetic
helicity.

##  8 Variable enstrophy and helicity fluxes

In this section we focus on the enstrophy and kinetic-helicity fluxes of 3D
hydrodynamics. Note that $\textbf{F}_{u}=0$ for pure hydrodynamics. However,
we will retain the large-scale forcing (${\bf F}_{\mathrm{LS}}$).

The vorticity field, $\bm{\omega}=\nabla\times{\bf u}$, plays an important
role in hydrodynamic turbulence. The dynamical equation of $\bm{\omega}$ is
[6, 8, 9]

$\displaystyle\frac{\partial\bm{\omega}}{\partial t}=\nabla\times({\bf
u}\times\bm{\omega})+{\bf F}_{\omega,\mathrm{LS}}+\nu\nabla^{2}\bm{\omega},$
(160)

or

$\displaystyle\frac{\partial\bm{\omega}}{\partial t}+({\bf
u}\cdot\nabla)\bm{\omega}=(\bm{\omega}\cdot\nabla){\bf u}+{\bf
F}_{\omega,\mathrm{LS}}+\nu\nabla^{2}\bm{\omega},$ (161)

where ${\bf F}_{\omega,\mathrm{LS}}=\nabla\times{\bf F}_{\mathrm{LS}}$. The
total enstrophy, $E_{\omega}=\frac{1}{2}\int d{\bf r}\omega^{2}$, and the
modal enstrophy, $E_{\omega}({\bf k})=\frac{1}{2}|\bm{\omega}({\bf k})|^{2}$,
are important quantities of hydrodynamics. The evolution equation for the
latter is

$\displaystyle\frac{d}{dt}E_{\omega}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\sum_{\bf p}\left\\{\Im\left[{\bf\\{k\cdot
u(q)\\}\\{\bm{\omega}(p)\cdot\bm{\omega}^{*}(k)\\}}\right]-\Im\left[\\{{\bf
k\cdot\bm{\omega}(q)\\}\\{u(p)\cdot\bm{\omega}^{*}(k)\\}}\right]\right\\}$
(162) $\displaystyle+\mathcal{F}_{\omega,\mathrm{LS}}(\mathbf{k})-2\nu
k^{2}E_{\omega}({\mathbf{k}}),$

where

$\displaystyle\mathcal{F}_{\omega,\mathrm{LS}}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\Re[i{\bf k\times F}_{u,\mathrm{LS}}({\bf
k})\cdot\bm{\omega}^{*}({\bf k})]=k^{2}\Re[{\bf u}^{*}({\bf k})\cdot{\bf
F}_{\mathrm{LS}}({\bf k})]$ (163)

is the enstrophy injection rate by ${\bf F}_{\mathrm{LS}}$. In Eq. (162), the
first term in the right-hand-side represents the advection of the vorticity
field by the velocity field, while the second term represents vortex
stretching. For inviscid and force-free 3D hydrodynamics, $E_{\omega}$ is not
conserved due to the vortex stretching by the velocity field [6, 8, 9]. Note
however that $E_{\omega}$ is conserved in 2D hydrodynamics; this issue will be
discussed in Sec. 9.

###  8.1 Variable enstrophy flux

When we compare Eq. (161) with Eq. (52) for the secondary vector $\bm{\zeta}$,
we obtain the following correspondence

$\displaystyle\bm{\zeta}\rightarrow\bm{\omega};~{}~{}~{}{\bf
F}_{\zeta}\rightarrow{\bf F}_{\omega}=(\bm{\omega}\cdot\nabla){\bf u},$ (164)

and

$\displaystyle\mathcal{F}_{\omega}({\bf k})$ $\displaystyle=$
$\displaystyle\sum_{\bf
p}-\Im\left[{\bf\\{k\cdot\bm{\omega}(q)\\}\\{u(p)\cdot\bm{\omega}^{*}(k)\\}}\right];~{}~{}~{}\mathcal{F}_{u}({\bf
k})=0.$ (165)

Here, $\mathcal{F}_{\omega}({\bf k})$ induces enstrophy enhancement via vortex
stretching by the velocity field. This process is similar to the magnetic
field stretching by the velocity field in MHD turbulence (see Sec. 6), first
proposed by Batchelor [153]. Note however that vorticity field does not back-
react on the velocity field because $\mathcal{F}_{u}({\bf k})=0$.

By making an analogy with MHD turbulence, we define the following enstrophy
fluxes for a wavenumber sphere of radius $k_{0}$ (see Sec. 6):

$\displaystyle\Pi^{u}_{\omega<}(k_{0})=\sum_{k\leq
k_{0}}\mathcal{F}_{\omega}({\bf
k});~{}~{}~{}\Pi^{u}_{\omega>}(k_{0})=\sum_{k>k_{0}}\mathcal{F}_{\omega}({\bf
k}).$ (166)

Here, $\Pi^{u}_{\omega<}(k_{0})$ ($\Pi^{u}_{\omega>}(k_{0})$) represents the
net enstrophy gain by the vorticity modes within (outside) the sphere due to
the nonlinear interactions with the velocity modes. Following Eq. (73), we
deduce that the net enstrophy enhancement rate due to the vortex stretching is

$\displaystyle\Pi^{u}_{\omega<}(k)+\Pi^{u}_{\omega>}(k)=\mathcal{F}_{\omega}=\mathrm{const}.$
(167)

The sum in the above equation is independent of $k$ even though its
constituents, $\Pi^{u}_{\omega<}(k)$ and $\Pi^{u}_{\omega>}(k)$, may vary with
$k$.

Since $\mathcal{F}_{u}({\bf k})=0$ (no back reaction from the vorticity to the
velocity field), both $\Pi^{\omega}_{u<}(k)$ and $\Pi^{\omega}_{u>}(k)$ are
zeros. Hence, the vortex dynamics is similar to that of kinematic dynamo where
the magnetic field does not back-react on the velocity field [20]. Note
however that in MHD turbulence, the magnetic field back-reacts on the velocity
field. Also note that the enstrophy and kinetic energy have different
dimensions.

The term $({\bf u}\cdot\nabla)\bm{\omega}$ provides advection to the vorticity
field, analogous to the advection of the secondary vector by the term $({\bf
u}\cdot\nabla)\bm{\zeta}$ (see Sec. 4). Consequently, following Eq. (58), we
define the mode-to-mode enstrophy transfer from $\bm{\omega}({\bf p})$ to
$\bm{\omega}({\bf k^{\prime}})$ with the mediation of ${\bf u(q)}$ as [14,
168]

$\displaystyle S^{\omega\omega}({\bf k^{\prime}|p|q})$ $\displaystyle=$
$\displaystyle-\Im\left[{\bf\\{k^{\prime}\cdot u(q)\\}\\{\bm{\omega}({\bf
p})\cdot\bm{\omega}({\bf k^{\prime}})\\}}\right].$ (168)

Hence, the enstrophy flux is (see Eq. (62))

$\displaystyle\Pi_{\omega}(k_{0})=\sum_{k^{\prime}>k_{0}}\sum_{p\leq
k_{0}}S^{\omega\omega}(\mathbf{k^{\prime}|p|q}).$ (169)

Now, following Eqs. (67, 75), we deduce that during a steady state, in the
inertial range,

$\displaystyle\frac{d}{dk}\Pi_{\omega}(k)=\mathcal{F}_{\omega}(k)=-\frac{d}{dk}\Pi^{u}_{\omega>}(k)=\frac{d}{dk}\Pi^{u}_{\omega<}(k).$
(170)

Therefore,

$\displaystyle\Pi_{\omega}(k)+\Pi^{u}_{\omega>}(k)=C_{3};~{}~{}~{}\Pi_{\omega}(k)-\Pi^{u}_{\omega<}(k)=C_{4},$
(171)

where $C_{3},C_{4}$ are constants. Since $\Pi^{u}_{\omega>}(k)$ and
$\Pi^{u}_{\omega<}(k)$ are nonzero, we conclude that $\Pi_{\omega}(k)$ varies
with $k$, unlike $\Pi_{u}(k)$, which is constant in the inertial range. The
above arguments and Eq. (167) yields $C_{3}-C_{4}=\mathcal{F}_{\omega}$.

We derive several new formulas by making an analogy between the vorticity
field and the magnetic field. For example, a comparison of vorticity dynamics
with Eqs. (6.1, 6.1) yields the following formulas for the mode-to-mode
enstrophy transfers:

$\displaystyle S^{u\omega}({\bf k^{\prime}|p|q})$ $\displaystyle=$
$\displaystyle 0,$ (172) $\displaystyle S^{\omega u}({\bf k^{\prime}|p|q})$
$\displaystyle=$
$\displaystyle\Im\left[{\bf\\{k^{\prime}\cdot\bm{\omega}(q)\\}\\{u(p)\cdot\bm{\omega}(k^{\prime})\\}}\right].$
(173)

Consequently, analogous to MHD fluxes, we can define four enstrophy fluxes:
$\Pi^{u<}_{\omega<}$, $\Pi^{u<}_{\omega>}$, $\Pi^{u>}_{\omega<}$,
$\Pi^{u>}_{\omega>}$. See Fig. 11 of Sec. 6 for an illustration. Note that

$\displaystyle\Pi^{u}_{\omega<}(k)=\Pi^{u<}_{\omega<}(k)+\Pi^{u>}_{\omega<}(k);~{}~{}~{}\Pi^{u}_{\omega>}(k)=\Pi^{u<}_{\omega>}(k)+\Pi^{u>}_{\omega>}(k).$
(174)

Substitutions of these relations in the identity of Eqs. (167) yields the
following relation [168]:

$\displaystyle\Pi^{u<}_{\omega<}(k)+\Pi^{u>}_{\omega<}(k)+\Pi^{u<}_{\omega>}(k)+\Pi^{u>}_{\omega>}(k)=\mathcal{F}_{\omega}.$
(175)

The above sum is constant in $k$ (both in inertial and dissipation range),
akin to Eq. (138) for MHD turbulence.

Sadhukhan et al. [168] performed numerical simulations of hydrodynamic
turbulence and computed the aforementioned enstrophy fluxes, as well as the
conserved quantity of Eq. (175). These quantities are exhibited in Fig. 12.
Note that the individual fluxes may vary with $k$ due to cross transfers, but
the sum of Eq. (175) is a constant. Interestingly, $\Pi_{\omega}(k)\sim k^{2}$
and $\Pi^{u<}_{\omega>}(k)\sim k$. Verma [14] argued that $\Pi_{\omega}(k)\sim
k^{2}$ due to the term $\bm{\omega}({\bf p})\cdot\bm{\omega}({\bf
k^{\prime}})$ of $S^{\omega\omega}({\bf k^{\prime}|p|q})$. At small
wavenumbers, $\Pi^{u>}_{\omega>}(k)$ is the most dominant among all the fluxes
implying that the intermediate and small-scale vortices are stretched most
significantly.

Figure 12:  For the numerical simulations by Sadhukhan et al. [168], plots of
enstrophy fluxes and the conserved flux of Eq. (175). From Sadhukhan et al.
[168]. Reprinted with permission from APS.

We summarise the similarities and dissimilarities between the vorticity field
and magnetic field in Table 2. A major difference between the two fields is
related to the back-reaction on the velocity field—vorticity does not back-
react, but magnetic field does. Hence, enstrophy dynamics is similar to that
of kinematic dynamo where the magnetic field does not affect the velocity
field. Note that enstrophy fluxes do not have a relation equivalent to the
conservation of total energy flux in MHD turbulence (Eq. (143)).

Table 2:  Similarities (first nine rows) and dissimilarities (last two rows) between the magnetic and vorticity fields.  Magnetic field | Vorticity
---|---
$S^{\zeta\zeta}$ | $S^{\omega\omega}$
$S^{\zeta u}$ | $S^{\omega u}$
$\Pi_{\zeta}$ | $\Pi_{\omega}$
$\Pi^{u}_{\zeta}$ | $\Pi^{u}_{\omega}$
Stretching of magnetic field lines | Vortex stretching
$u$-to-$\zeta$ transfer |  $u$-to-$\omega$ transfer
Growth of magnetic field | Enhancement of enstrophy
$\zeta$-to-$\zeta$ transfer (forward for nonhelical) |  $\omega$-to-$\omega$ transfer (forward)
No dynamo in 2D | No vortex stretching in 2D
$S^{u\zeta}\neq 0$ | $S^{u\omega}=0$
$\zeta$-to-$u$ transfer (back-reaction), | No $\omega$-to-$u$ transfer (no back-reaction)
except for kinematic dynamo where $S^{u\zeta}=0$ |

In the next subsection, we will the describe variable kinetic-helicity flux.

###  8.2 Flux of kinetic helicity

Kinetic helicity, which is defined as $H_{K}=\frac{1}{2}\int d{\bf r}({\bf
u}\cdot\bm{\omega})$, plays a major role in the growth of vorticity and
magnetic field [6, 9, 20]. Using Eqs. (1, 160) we can show that for $\nu=0$,
${\bf F}_{u}=0,{\bf F}_{\mathrm{LS}}=0$, and periodic or vanishing boundary
condition, the total kinetic helicity is conserved in 3D hydrodynamics [6, 8,
9]; this is in addition to the conservation of total kinetic energy.

The evolution equation for the modal kinetic helicity, $H_{K}({\bf
k})=\frac{1}{2}\Re[{\bf u}({\bf k})\cdot\bm{\omega}^{*}({\bf k})]$, is [6, 8,
9, 14]

$\frac{d}{dt}H_{K}({\bf k})=\sum_{\bf p}\Re[{\bf u(q)}\cdot\\{\bm{\omega}({\bf
p})\times\bm{\omega}^{*}({\bf k})\\}]+\mathcal{F}_{H_{K},\mathrm{LS}}({\bf
k})-2\nu k^{2}H_{K}({\bf k}),$ (176)

where $\mathcal{F}_{H_{K},\mathrm{LS}}(\mathbf{k})=\Re[\bm{\omega}^{*}({\bf
k})\cdot{\bf F}_{\mathrm{LS}}({\bf k})]$ is the kinetic helicity injection
rate by ${\bf F}_{\mathrm{LS}}$. Based on Eq. (176), researchers have derived
various formulas for the kinetic helicity flux. For example, Müller et al.
[169] argued that the kinetic flux is given by

$\displaystyle\Pi_{H_{K}}(k_{0})=\sum_{k>k_{0}}\\{i{\bf
k\times[u\times\bm{\omega}}]({\bf k})\\}\cdot{\bf u^{*}(k)}+c.c.,$ (177)

where $c.c.$ stands for the complex conjugate. In the following discussion we
will describe more flux formulas for the kinetic helicity.

Following the structure of the nonlinear term of Eq. (176), Verma [14],
Sadhukhan et al. [168], and Plunian et al. [170] showed that the mode-to-mode
kinetic helicity from wavenumber ${\bf p}$ to wavenumber ${\bf k^{\prime}}$
with the mediation of wavenumber ${\bf q}$ is

$\displaystyle S^{H_{K}}(\mathbf{k^{\prime}|p|q})=\Re[{\bf
u(q)}\cdot\\{\bm{\omega}({\bf p})\times\bm{\omega}({\bf k^{\prime}})\\}].$
(178)

In terms of $S^{H_{K}}(\mathbf{k^{\prime}|p|q})$, the kinetic helicity flux
for a wavenumber sphere of radius $k_{0}$ is

$\displaystyle\Pi_{H_{K}}(k_{0})=\sum_{p\leq
k_{0}}\sum_{k^{\prime}>k_{0}}S^{H_{K}}(\mathbf{k^{\prime}|p|q}).$ (179)

Following the same lines of arguments as in Sec. 3, we obtain

$\displaystyle\frac{d}{dk}\Pi_{H_{K}}(k)=\mathcal{F}_{H_{K},\mathrm{LS}}(k)-D_{H_{K}}(k),$
(180)

where $D_{H_{K}}(k)=2\nu k^{2}H_{K}(k)$ is the dissipation rate of kinetic
helicity in shell $k$, and $\mathcal{F}_{H_{K},\mathrm{LS}}(k)$ is the kinetic
helicity injection rate in the shell due to the force ${\bf F}_{\mathrm{LS}}$.
In the inertial range, $\mathcal{F}_{H_{K},\mathrm{LS}}(k)=0$ and
$D_{H_{K}}(k)=0$, hence $\Pi_{H_{K}}(k)=\epsilon_{H_{K}}=\mathrm{constant}$,
where $\epsilon_{H_{K}}$ is the total dissipation rate of kinetic helicity.
Using the constancy of $\Pi_{H_{K}}(k)$, dimensional analysis and field-
theoretic arguments, the following inertial-range spectrum for kinetic
helicity has been derived [6, 171, 172]:

$\displaystyle H_{K}(k)=K_{H}\epsilon_{H_{K}}(\epsilon_{u})^{-1/3}k^{-5/3},$
(181)

where $K_{H}$ is a nondimensional constant, whose value has been estimated to
be of the order of unity. The above scaling has been verified in many
numerical simulations [6, 173, 168]. Interestingly, Sadhukhan et al. [168]
also modelled the kinetic helicity spectrum and flux in the inerital-