dissipation range using a generalized Pao’s model [31].

The helical turbulence is also described using Craya-Herring and helical basis
[130, 174, 74, 175, 170]. In Craya-Herring basis [129, 176, 6, 10, 130, 19],
the unit vectors for a wavenumber k are

$\displaystyle\hat{e}_{3}({\bf k})=\hat{k};~{}~{}\hat{e}_{1}({\bf
k})=\frac{\hat{k}\times\hat{n}}{|\hat{k}\times\hat{n}|};~{}~{}\hat{e}_{2}({\bf
k})=\hat{e}_{3}({\bf k})\times\hat{e}_{1}({\bf k}),$ (182)

where $\hat{k}$ is the unit vector along wavenumber ${\bf k}$, and $\hat{n}$
could be along any direction, but it is typically chosen along the anisotropy
direction. We denote the velocity components along $\hat{e}_{1}({\bf k})$,
$\hat{e}_{2}({\bf k})$, $\hat{e}_{3}({\bf k})$ as $u_{1}({\bf k}),u_{2}({\bf
k}),u_{3}({\bf k})$ respectively. Among them $u_{3}({\bf k})=0$ due to
incompressibility, hence ${\bf u}({\bf k})=u_{1}({\bf
k})\hat{\textbf{e}}_{1}(\textbf{k})+u_{2}({\bf
k})\hat{\textbf{e}}_{2}(\textbf{k})$. In this basis, the mode-to-mode kinetic
energy transfer from wavenumber p to wavenumber k with the mediation of
wavenumber q is

$\displaystyle S^{uu}({\bf k^{\prime}|p|q})=S^{u_{1}u_{1}}({\bf
k^{\prime}|p|q})+S^{u_{2}u_{2}}({\bf k^{\prime}|p|q}),$ (183)

where

$\displaystyle S^{u_{1}u_{1}}({\bf k^{\prime}|p|q})$ $\displaystyle=$
$\displaystyle k^{\prime}\sin\bar{\beta}\cos\bar{\gamma}\Im\\{u_{1}({\bf
q})u_{1}({\bf p})u_{1}({\bf k^{\prime}})\\},$ (184) $\displaystyle
S^{u_{2}u_{2}}({\bf k^{\prime}|p|q})$ $\displaystyle=$
$\displaystyle-k^{\prime}\sin\bar{\beta}\Im\\{u_{1}({\bf q})u_{2}({\bf
p})u_{2}({\bf k^{\prime}})\\},$ (185)

with $\bar{\alpha},\bar{\beta},\bar{\gamma}$ as the internal angles across
$k,p,q$ of the triangle formed by the wavenumbers ${\bf(k^{\prime},p,q)}$ [5,
14].

Another useful basis called helical basis [130, 6, 10] is constructed using
the Craya-Herring vectors. In this basis, the unit vectors are

$\displaystyle\hat{e}_{s_{k}}({\bf k})=\frac{1}{\sqrt{2}}[\hat{e}_{2}({\bf
k})-is_{k}\hat{e}_{1}({\bf k})],$ (186)

where $s_{k}$ takes values $+1$ or $-1$. In terms of these unit vectors, the
velocity field is ${\bf u}({\bf k})=u_{+}({\bf k})\hat{e}_{+}({\bf
k})+u_{-}({\bf k})\hat{e}_{-}({\bf k})$, where

$\displaystyle u_{s_{k}}({\bf k})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}[u_{2}({\bf k})+is_{k}u_{1}({\bf k})].$ (187)

In the helical basis, the mode-to-mode kinetic energy transfer from ${\bf
u(p)}$ to ${\bf u(k^{\prime})}$ with the mediation of ${\bf u(q)}$ is

$\displaystyle S^{uu}({\bf k^{\prime}|p|q})$ $\displaystyle=$
$\displaystyle\sum_{s_{p},s_{k^{\prime}}}S^{uu}_{s_{k^{\prime}}s_{p}}({\bf
k^{\prime}|p|q}),$ (188)

where $S^{uu}_{s_{k^{\prime}}s_{p}}({\bf k^{\prime}|p|q})$, the kinetic energy
transfer from mode $u_{s_{p}}({\bf p})$ to $u_{s_{k^{\prime}}}({\bf
k^{\prime}})$ with the mediation of ${\bf u(q)}$, is [14]

$\displaystyle S^{uu}_{s_{k^{\prime}}s_{p}}({\bf k^{\prime}|p|q})$
$\displaystyle=$ $\displaystyle-\Im\left[{\bf\\{k^{\prime}\cdot
u(q)\\}}u_{s_{p}}({\bf p})u_{s_{k^{\prime}}}({\bf
k^{\prime}})\\{\hat{e}_{s_{p}}({\bf p})\cdot\hat{e}_{s_{k^{\prime}}}({\bf
k^{\prime}})\\}\right]$ (189) $\displaystyle=$
$\displaystyle-\frac{k^{\prime}}{2}\sin\bar{\beta}(1+s_{p}s_{k^{\prime}}\cos\bar{\gamma})\Im\\{u_{1}({\bf
q})u_{s_{p}}({\bf p})u_{s_{k^{\prime}}}({\bf k^{\prime}})\\}.$

Similarly, the mode-to-mode kinetic helicity transfer from wavenumber ${\bf
p}$ to wavenumber ${\bf k^{\prime}}$ with the mediation of wavenumber ${\bf
q}$ is [14]

$\displaystyle S^{H_{K}}({\bf k^{\prime}|p|q})$ $\displaystyle=$
$\displaystyle\sum_{s_{p},s_{k^{\prime}}}S^{H_{K}}_{s_{k^{\prime}}s_{p}}({\bf
k^{\prime}|p,q}),$ (190)

where $S^{H_{K}}_{s_{k^{\prime}}s_{p}}({\bf k^{\prime}|p,q})$, the elemental
helicity transfer from $u_{s_{p}}({\bf p})$ to $u_{s_{k^{\prime}}}({\bf
k^{\prime}})$ with the mediation of ${\bf u(q)}$, is

$\displaystyle S^{H_{K}}_{s_{k^{\prime}}s_{p}}({\bf k^{\prime}|p,q})$
$\displaystyle=$
$\displaystyle-\frac{1}{2}pk^{\prime}[s_{k^{\prime}}\sin\bar{\beta}+s_{p}\sin\bar{\alpha}]\Im\\{u_{1}({\bf
q})u_{s_{p}}({\bf p})u_{s_{k^{\prime}}}({\bf k^{\prime}})\\}$ (191)
$\displaystyle+\frac{1}{2}pk^{\prime}\sin\bar{\gamma}\Re\\{u_{2}({\bf
q})u_{s_{p}}({\bf p})u_{s_{k^{\prime}}}({\bf k^{\prime}})\\}.$

The kinetic energy flux from helical mode $u_{s_{g}}$ to mode $u_{s_{r}}$,
where $s_{g}$ and $s_{r}$ are the signs of giver and receiver modes
respectively, can be written as

$\displaystyle\Pi^{u_{s_{g}}<}_{u_{s_{r}}>}(k_{0})$ $\displaystyle=$
$\displaystyle\sum_{p\leq
k_{0}}\sum_{k^{\prime}>k_{0}}S^{uu}_{s_{g}s_{r}}({\bf k^{\prime}|p|q}).$ (192)

The corresponding kinetic helicity flux is

$\displaystyle\Pi^{H_{K}s_{g}}_{H_{K}s_{r}}(k_{0})=\sum_{p\leq
k_{0}}\sum_{k^{\prime}>k_{0}}S^{H_{K}}_{s_{r},s_{g}}({\bf k^{\prime}|p|q}).$
(193)

Alexakis and Biferale [74] and Sahoo et al. [175] constructed another set of
formulas for the energy flux, which is

$\displaystyle\Pi^{s_{1}s_{2}s_{3}}_{u}=\left\langle{\bf
u}^{<k}\cdot\mathcal{P}^{s_{1}}[{\bf
u}^{s_{2}}\times\bm{\omega}^{s_{3}}]\right\rangle,$ (194)

where $s_{i}$ takes values $\pm 1$ depending on the sign of kinetic helicity,
and $\mathcal{P}^{s_{1}}$ represents projection along $\hat{e}_{s_{1}}$. The
above flux formula is related to Eq. (189).

For large-scale external force, the energy and helicity spectra exhibit
$k^{-5/3}$ spectra and constant fluxes [6, 8]. However, the energy transfers
become more complex when the external force is employed at intermediate
scales, or when only homochiral modes (modes with same sign of kinetic
helicity) are present. Biferale et al. [177] showed that the nonlinear
interactions among homochiral modes yield an inverse cascade of kinetic energy
in 3D hydrodynamic turbulence. Sahoo et al. [178] varied the strengths of
different triadic interactions involving helical modes and observed a
discontinuous transition from inverse energy cascade to forward energy
cascade. Plunian et al. [179] performed numerical simulations with realistic
helical modes (rather than homochiral modes) and obtained similar results as
Sahoo et al. [178]. Detailed discussions on these issues are beyond the scope
of this review. For details, refer to references [174, 74, 175, 180, 181,
179].

We close this subsection with a comment that the secondary fields also induces
kinetic helicity. For example, bouyancy and Lorentz force induce kinetic
helicity. These topics, however, are too complex to be discussed here. Refer
to the references [6, 10, 130, 14] for further details.

###  8.3 Variable kinetic energy flux in decaying hydrodynamic turbulence

So far, we focussed on the steady-state behaviour of turbulence. In this
section, we will briefly discuss how variable energy flux plays an important
role in the decay of hydrodynamic turbulence for which ${\bf F}_{u}={\bf
F}_{\mathrm{LS}}=0$ [182, 183, 184, 185, 9, 6, 8]. For a viscous flow (flow
with zero nonlinearity), the kinetic energy flux is zero, while the total
kinetic energy decays as

$\displaystyle E_{u}(t)=\sum_{\bf k}E_{u}({\bf k},0)\exp(-\nu k^{2}t),$ (195)

where $E_{u}({\bf k},0)$ is the initial modal kinetic energy of wavenumber
${\bf k}$. The decay law for a turbulent flow is more complex, which will be
discussed below.

To analyse the evolution of decaying turbulence, we start with Eq. (27) and
integrate over small wavenumbers up to integral wavenumber $k_{\ell}\sim 1/l$
($l$ is the integral length scale). The wavenumber region $[0,k_{l}]$ contains
most of the kinetic energyof the turbulent flows. The above operation yields
[185, 9, 6, 8]

$\displaystyle\frac{dE}{dt}\approx\frac{\partial}{\partial
t}\int_{0}^{k_{\ell}}dkE_{u}(k,t)\approx-\Pi(k_{\ell})-\int_{0}^{k_{\ell}}dkD_{u}(k,t).$
(196)

Since the viscous dissipation is negligible in the wavenumber band
$[0,k_{\ell}]$, we obtain

$\displaystyle\frac{dE}{dt}=-\Pi(k_{\ell})=-\frac{U^{3}}{\ell},$ (197)

where $U$ is the large-scale velocity. Thus, the kinetic energy decays due to
the kinetic energy flux $\Pi(k_{\ell})$. In decaying turbulence, $U$ decreases
and $\ell$ increases, hence, $\Pi(k_{\ell})$ is a time-dependent quantity.

Equation (197) can be solved if we know the relationship between $U$ and
$\ell$. To derive this relation, Kolmogorov [185] employed conservation of
Loitsyansky integral, $\int r^{2}\left\langle{\bf u\cdot
u^{\prime}}\right\rangle d{\bf r}$ and obtained
$U^{2}\ell^{5}=\mathrm{constant}$ [3, 4]. This conservation law is related to
the spectrum $E_{u}(k)\sim k^{4}$ at small wavenumbers:

$\displaystyle E_{u}=U^{2}/2\sim\int_{0}^{k_{0}}E_{u}(k)dk\sim
k_{0}^{5}\sim\ell^{-5}.$ (198)

Using $U^{2}\ell^{5}=\mathrm{constant}$ we transform Eq. (197) to

$\displaystyle\frac{dE}{dt}=-\frac{U^{3}}{\ell}=-cU^{17/5}=-c^{\prime}E^{17/10},$
(199)

where $c$ and $c^{\prime}$ are constants. The above equation yields the
following decaying solution:

$\displaystyle E(t)\sim t^{-10/7};~{}~{}\ell(t)\sim t^{2/7}.$ (200)

Saffman [184] derived another decay law based on the assumption that
$E_{u}(k)\sim k^{2}$ at small wavenumbers. This infrared spectrum yields
$U^{2}\ell^{3}=\mathrm{constant}$, substitution of which in Eq. (197) yields

$\displaystyle E(t)\sim t^{-6/5};~{}~{}\ell(t)\sim t^{2/5}.$ (201)

Experiments on grid turbulence [186, 187, 188] tend to favour 6/5 decay law of
Saffman. However, numerical simulations exhibit one of the two laws depending
on initial $E_{u}(k)$ at small wavenumbers [189, 190]: 10/7 law for $k^{4}$
spectrum and 6/5 law for $k^{2}$ spectrum. The above experimental results tend
to suggest that small wavenumber spectrum in grid turbulence may follow
$E_{u}(k)\sim k^{2}$ for $k<k_{f}$, similar to that observed in Kolmogorov
flow [191] (see Sec. 10).

For MHD turbulence, Biskamp [150] derived decay laws based on conservation of
magnetic helicity. Verma [38] employed variations in the energy flux to derive
decay laws for MHD turbulence. Refer to the original papers for details.

##  9 Brief review of variable energy fluxes in 2D and quasi-2D turbulence

In this section, we present a brief review of variable energy fluxes in 2D and
quasi-two-dimensional (quasi-2D) hydrodynamic, buoyancy-driven, and MHD
turbulence. We start with a description of the energy and enstrophy fluxes of
2D hydrodynamic turbulence.

###  9.1 Fluxes of 2D hydrodynamic turbulence

For a 2D flow field ${\bf u}=u_{x}\hat{x}+u_{y}\hat{y}$, the vorticity field
${\bm{\omega}}=\omega\hat{z}=(\partial_{x}u_{y}-\partial_{y}u_{x})\hat{z}$ is
perpendicular to the plane of the flow. Hence, Eq. (161) yields the following
evolution equation for 2D vorticity:

$\displaystyle\frac{\partial{\omega}}{\partial t}+({\bf
u}\cdot\nabla){\omega}=F_{\omega,\mathrm{LS}}+\nu\nabla^{2}{\omega}.$ (202)

Note the absence of vortex stretching term in the above equation. For a 2D
field, the total enstrophy $E_{\omega}=\int d{\bf r}\frac{1}{2}\omega^{2}$ is
conserved [6, 8, 9], which is in addition to the conservation of total energy.
Note that in 2D hydrodynamics, the total kinetic helicity vanishes identically
because ${\bf u}$ and $\bm{\omega}$ are perpendicular to each other.

In 2D hydrodynamics, $\Pi^{u}_{\omega>}(k)=\Pi^{u}_{\omega<}(k)=0$ due to the
absence of ${\bm{\omega}}\cdot\nabla{\bf u}$ term in Eq. (202) (see Sec. 8.1).
Here, we assume that the external force supplies kinetic energy and enstrophy
at large scales (see Eq. (163)). Following Eq. (171), we deduce that
$\Pi_{\omega}(k)$ is constant in the inertial range and it equals the
enstrophy dissipation rate ($\epsilon_{\omega}$). This is in contrast to 3D
hydrodynamic turbulence where $\Pi_{\omega}(k)$ varies with $k$ due to the
variability of $\Pi^{u}_{\omega<}(k)$ and $\Pi^{u}_{\omega>}(k)$. Kraichnan
[192] showed that the above enstrophy flux yields
$E_{\omega}(k)\sim\epsilon_{\omega}^{2/3}k^{-1}$ in the inertial-range. Note
that $\Pi_{u}(k)$ is small in the inertial range. The above picture is a part
of Kraichnan [192]’s framework for 2D turbulence that will be described below.

Two-dimensional turbulence gets more complex when it is forced at an
intermediate wavenumber $k_{f}$. Using conservation laws and field-theoretic
tools, Kraichnan [192] constructed a phenomenology for such a scenario. In the
inertial range of $k<k_{f}$ region, $\Pi_{u}(k)=-\epsilon_{u}$ and
$E_{u}(k)=K_{2D}\epsilon_{u}^{2/3}k^{-5/3}$, where $\epsilon_{u}$ is the
magnitude of the kinetic energy flux, and $K_{2D}$ is a constant whose
numerical value is between 5.5 and 7.0. However, in the inertial range of
$k>k_{f}$ regime, $\Pi_{\omega}(k)=\epsilon_{\omega}>0$ and
$E_{u}(k)=K^{\prime}_{2D}\epsilon_{\omega}^{2/3}k^{-3}$, where
$K^{\prime}_{2D}$ is another constant whose numerical value is between 1.3 and
1.7. See Fig. 13 for an illustration. The results of many experiments (see
[193, 194, 195, 196] and references therein) and numerical simulations (see
[197, 196], and references therein) are consistent with the above predictions.
We do not detail these results here.

Figure 13: For 2D turbulence, (left panel) schematic illustrations of energy
and entropy spectra; (right panel) energy and enstrophy fluxes. Note that
$k_{\mathrm{in}}\equiv k_{f}$ and $k_{\nu}\equiv k_{d}$. From Alexakis and
Biferale [74]. Reprinted with permission from Elsevier.

Sharma et al. [45] and Gupta et al. [34] extended Kraichnan’s phenomenology
[192] beyond the inertial range by extrapolating Pao’s model [31] to 2D
turbulence. In the following we extend their calculations by including Ekman
friction. For steady state, away from the forcing band, the kinetic energy and
enstrophy fluxes obey the following equations:

$\displaystyle\frac{d}{dk}\Pi_{u}(k)$ $\displaystyle=$ $\displaystyle-2(\nu
k^{2}+\alpha)E_{u}(k),$ (203) $\displaystyle\frac{d}{dk}\Pi_{\omega}(k)$
$\displaystyle=$ $\displaystyle-2(\nu k^{2}+\alpha)E_{\omega}(k)=-2(\nu
k^{2}+\alpha)k^{2}E_{u}(k).$ (204)

For $k<k_{f}$, it is assumed that $E_{u}(k)/\Pi_{u}(k)$ depends only on the
kinetic energy dissipation rate ($\epsilon_{u}$) and $k$, but not on $\nu$.
Hence,

$\displaystyle\frac{E_{u}(k)}{\Pi_{u}(k)}=-K_{2D}\epsilon_{u}^{-1/3}k^{-5/3},$
(205)

where $\epsilon_{u}=-\Pi_{u}(k_{0})$ with $k_{0}$ being a wavenumber near
$k_{f}$. Substitution of the above in Eq. (203) and integration from $k$ to
$k_{0}$ yields

$\displaystyle\Pi_{u}(k)$ $\displaystyle=$
$\displaystyle-\epsilon_{u}\exp{\left(\frac{3}{2}K_{2D}[(k/k_{d})^{4/3}-(k_{0}/k_{d})^{4/3}]\right)}$
(206) $\displaystyle\times\exp{\left(-\frac{3\alpha K_{2D}}{\nu
k_{d}^{2}}[(k_{d}/k)^{2/3}-(k_{d}/k_{0})^{2/3}]\right)},$ $\displaystyle
E_{u}(k)$ $\displaystyle=$ $\displaystyle-
K_{2D}\Pi_{u}(k)\epsilon_{u}^{-1/3}k^{-5/3},$ (207)

where $k_{d}=(\epsilon_{u}/\nu^{3})^{1/4}$ is Kolmogorov’s wavenumber for 2D
turbulence. Interestingly, $\Pi_{u}(k)\rightarrow 0$ as $k\rightarrow 0$ due
to the Ekman friction. Substitution of the above $E_{u}(k)$ in Eq. (204)
yields the following enstrophy flux:

$\displaystyle\Pi_{\omega}(k)$ $\displaystyle=$
$\displaystyle\Pi_{\omega}(k_{0})+2K_{2D}\epsilon_{u}^{1/3}\int_{k_{0}}^{k}(\nu
k^{\prime 2}+\alpha)\Pi_{u}(k^{\prime})k^{\prime 1/3}dk^{\prime}.$ (208)

For $k>k_{f}$, following Pao [31], it has been argued that
$E_{\omega}(k)/\Pi_{\omega}(k)$ depends only on the enstrophy dissipation rate
($\epsilon_{\omega}$) and $k$, but not on $\nu$. Hence,

$\displaystyle\frac{E_{\omega}(k)}{\Pi_{\omega}(k)}$ $\displaystyle=$
$\displaystyle K^{\prime}_{2D}\epsilon_{\omega}^{-1/3}k^{-1}.$ (209)

Under this assumption, using Eq. (204) we obtain

$\displaystyle\Pi_{\omega}(k)=\Pi_{\omega}(k_{0})\left(\frac{k}{k_{0}}\right)^{-2\alpha
K^{\prime}_{2D}\epsilon_{\omega}^{-1/3}}\exp\left(-\frac{K^{\prime}_{2D}}{k_{d2D}^{2}}(k^{2}-k_{0}^{2})\right),$
(210)

where $k_{d2D}=\epsilon_{\omega}^{1/6}/\sqrt{\nu}$, and $\Pi_{\omega}(k_{0})$
is the reference value of the enstrophy flux at $k=k_{0}$ [25, 30]. We choose
$k_{0}\approx k_{f}$ and $\Pi_{\omega}(k_{f})\approx\epsilon_{\omega}$, where
$\epsilon_{\omega}$ is the enstrophy injection rate. Substitution of the above
in the equations for variable flux yields

$\displaystyle E_{u}(k)$ $\displaystyle=$ $\displaystyle
K^{\prime}_{2D}\epsilon_{\omega}^{2/3}k^{-3}\left(\frac{k}{k_{0}}\right)^{-2\alpha
K^{\prime}_{2D}\epsilon_{\omega}^{-1/3}}\exp\left(-\frac{K^{\prime}_{2D}}{k_{d2D}^{2}}(k^{2}-k_{0}^{2})\right),$
(211) $\displaystyle\Pi_{u}(k)$ $\displaystyle=$ $\displaystyle
K^{\prime}_{2D}\exp(x_{0})\frac{\epsilon_{\omega}}{k_{d2D}^{2}}\int_{x}^{\infty}\frac{1}{x^{\prime}}\left(1+\frac{\beta}{x^{\prime}}\right)\left(\frac{x^{\prime}}{x_{0}}\right)^{-\beta}\exp{(-x^{\prime})}dx^{\prime},$
(212)

where $x=K^{\prime}_{2D}(k/k_{d2D})^{2}$,
$x_{0}=K^{\prime}_{2D}(k_{0}/k_{d2D})^{2}$, and
$\beta=K^{\prime}_{2D}\alpha/\nu k_{d2D}^{2}$. Note that $E_{u}(k)$ is steeper
than $k^{-3}$ due to the Ekman friction and the viscous dissipation.
Asymptotically,

$\displaystyle\frac{\Pi_{u}(k)}{\epsilon_{\omega}}\approx\frac{K^{\prime}_{2D}}{k_{d2D}^{2}}E_{1}(K^{\prime}_{2D}(k/k_{d2D})^{2})\ll
1.$ (213)

That is, $\Pi_{u}(k)\ll\epsilon_{\omega}$ for $k\gg k_{f}$, and
$\Pi_{u}(k)\sim\log k$ for $\alpha=0$. The aforementioned scaling relations
are consistent with the analytical results of Gotoh [198], numerical results
of Gupta et al. [34] and Anas and Verma [25], as well as the experimental
results of Boffetta et al. [29].

Thus, the above phenomenology describes how the viscous and Ekman dissipation
steepens the 2D energy spectrum beyond the inertial range scaling.

###  9.2 Fluxes in quasi-2D turbulence

Strong rotation, gravity, magnetic field, and shear tend to make the flows
quasi-two-dimensional with dominant velocity field perpendicular to the
direction of the external field. Note that the above flows are stably
stratified. However, unstable stratification such as thermal convection tends
to strengthen the parallel component of the velocity field. In such flows,
pressure facilitates energy exchange between the parallel and perpendicular
components of the velocity fields (see Sec. 4.4). In this subsection, we
sketch some past work in this area without getting into complex details.

For fast rotation, researchers have reported strong two-dimensionalization of
the flow [199, 200, 45]. Such flows contain strong vortical structures. They
exhibit inverse energy cascade at small wavenumbers and forward energy cascade
at large wavenumbers [45, 201]. Similar energy transfers and fluxes are
observed in liquid-metal MHD turbulence under strong external magnetic field
[202, 22, 23, 107, 24]. Under strong external magnetic field, MHD turbulence
too exhibits similar anisotropic behaviour [203, 202, 128, 204]. Biferale et
al. [205] showed how flows transform from 2D to quasi-2D turbulence when the
helical modes are truncated. In contrast, buoyancy destabilizes convective
flows and generates plume structures; here, the velocity field parallel to
buoyancy is stronger than the perpendicular component.

Boffetta et al. [85] and Bartello [206] studied the energy fluxes of stably-
stratified turbulence; they observed an inverse cascade of kinetic energy and
a forward cascade of potential energy. They termed these forward and backward
fluxes as a flux-loop. Kumar et al. [137] simulated 2D stably-stratified
turbulence and observed variety of flow patterns. Falkovich and Kristsuk [86]
studied compressible turbulence and observed that planar structures and wave
turbulence exhibit inverse and forward energy cascades respectively.

These applications reveal importance of variable energy flux in the
description of quasi-2D turbulence. In all such fiows, pressure plays an
important role in the energy transfer between the parallel and perpendicular
components. In addition, there are interesting works on 2D turbulence with
buoyancy and MHD turbulence. However, these topics have been skipped in this
review.

##  10 Variable energy flux in dissipationless turbulence

###  10.1 Hydrodynamic systems

An inviscid or dissipationless fluid flow is described using Euler equation,
which is Navier-Stokes equation with $\nu=0$. Depending on the choice of
initial condition, Euler turbulence exhibits either zero kinetic energy flux
or a combination of positive and zero kinetic energy fluxes. Passive scalars
turbulence and MHD turbulence without dissipation too exhibit similar
properties. We describe these systems in the present section.

In the following discussion, we will focus on the properties of truncated
Euler turbulence where large-wavenumber Fourier modes are absent. Kraichnan
[57] and Lee [56] showed that the Liouville theorem is applicable to the phase
space formed by the Fourier modes of truncated Euler turbulence. Further, they
argued that the system is in equilibrium and ergodic in the available phase
space. Based on these observations, which are similar to those of equilibrium
thermodynamics, and using conservation of kinetic energy and kinetic helicity,
Kraichnan [57] and Lee [56] showed that $E_{u}(\textbf{k})$ and
$H_{K}(\textbf{k})$ are random variables with the following probability
distribution:

$P(E_{u}(\textbf{k}),H_{K}(\textbf{k}))=\frac{1}{Z}\exp[-\beta
E_{u}(\textbf{k})-\gamma H_{K}(\textbf{k})],$ (214)

where $E_{u}(\textbf{k})$ and $H_{K}(\textbf{k})$ are the modal kinetic energy
and kinetic helicity respectively, and $Z$ is a prefactor.

It is convenient to write down distribution functions for the helical energy
spectra, $E_{u\pm}(\textbf{k})$, where $u_{\pm}$ are the helical variables
defined in Sec. 8.2 [130, 14]. These distribution functions are

$P(E_{u_{+}}(\textbf{k}),E_{u_{-}}(\textbf{k}))=\beta_{+}\beta_{-}\exp[-\beta_{+}E_{u_{+}}(\textbf{k})-\beta_{-}E_{u_{-}}(\textbf{k})]$
(215)

that lead to $\left\langle E_{u_{\pm}}(\textbf{k})\right\rangle=1/\beta_{\pm}$
(because $u_{\pm}$ are independent variables). Using a change of variable,
$\beta_{\pm}=\beta\mp\gamma k$, we obtain [56, 57],

$\displaystyle\left\langle E_{u}(\textbf{k})\right\rangle=\left\langle
E_{u_{+}}\right\rangle+\left\langle
E_{u_{-}}\right\rangle=\frac{2\beta}{\beta^{2}-\gamma^{2}k^{2}},$ (216)
$\displaystyle\left\langle H_{K}(\textbf{k})\right\rangle=k(\left\langle
E_{u_{+}}\right\rangle-\left\langle E_{u_{-}}\right\rangle)=\frac{2\gamma
k^{2}}{\beta^{2}-\gamma^{2}k^{2}}.$ (217)

One-dimensional shell spectrum, $E_{u}(k)$, is a sum of the modes in a shell
of radius $k$. Therefore, $E_{u}(k)$ is proportional to $k^{2}$ for small and
moderate $k$’s, but it gets an upward bend for large $k$’s [56, 57, 74]. The
above forms of energy and kinetic helicity arise due to their conservation.

Detailed balance is an important property of systems under equilibrium.
Consequently, under equilibrium, we expect no energy exchange among the
Fourier modes and, hence, the energy flux $\Pi_{u}(k)=0$ (see Fig. 14(a)).
This observation is consistent with the fact that the phases of the Fourier
modes of a turbulent flow are random [59]222The mode-to-mode energy transfer
from mode u(p) to u(k) with the mediation of u(q) in Craya-Herring basis is
(see Sec. 8.2)  $\displaystyle\left\langle S^{u_{1}u_{1}}({\bf
k|p|q})\right\rangle$ $\displaystyle=$ $\displaystyle
k\sin\bar{\beta}\cos\bar{\gamma}\left\langle\Im[u_{1}(\textbf{p})u_{1}(\textbf{q})u^{*}_{1}(\textbf{k})]\right\rangle$
$\displaystyle=$ $\displaystyle
k\sin\bar{\beta}\cos\bar{\gamma}\left\langle|u_{1}(\textbf{p})u_{1}(\textbf{q})u^{*}_{1}(\textbf{k})|\sin(\phi_{1p}+\phi_{1q}-\phi_{1k})\right\rangle=0,$
$\displaystyle\left\langle S^{u_{2}u_{2}}({\bf k|p|q})\right\rangle$
$\displaystyle=$
$\displaystyle-k\sin\bar{\beta}\left\langle\Im[u_{1}(\textbf{p})u_{2}(\textbf{q})u^{*}_{2}(\textbf{k})]\right\rangle$
$\displaystyle=$
$\displaystyle-k\sin\bar{\beta}\left\langle|u_{1}(\textbf{p})u_{2}(\textbf{q})u^{*}_{2}(\textbf{k})|\sin(\phi_{2p}+\phi_{1q}-\phi_{2k})\right\rangle=0,$
where $\phi_{1k}$ and $\phi_{2k}$ are the phases of modes $u_{1}(\textbf{k})$
and $u_{2}(\textbf{k})$ respectively. The energy transfers are zero due to
random nature of the phases.. Contrast the above property with constant energy
flux for Kolmogorov’s theory of turbulence where the phases of the Fourier
modes are correlated (see Fig. 14(b)). Hence, the phase space for the
Kolmogorov’s scenario is not expected to be ergodic.

Figure 14: Schematic energy spectra and fluxes for various flows: (a) Euler-1:
Unforced Euler turbulence exhibiting $k^{2}$ for white noise as an initial
condition. (b) K41: Turbulent flow of Kolmogorov model. Such flows with small
viscosity are forced at large scales. (c) Euler-2: Unforced Euler turbulence
with large-scale Taylor–Green vortices as an initial condition. (d)
Kolmogorov-flow: A flow that is forced at an intermediate scale. The yellow,
green, and blue colours represent inertial, dissipative, and equilibrium
ranges respectively. The orange colour represents the thermal or microscopic
scales where particles move randomly.

A number of numerical simulations have been performed to verify the above
spectrum. In one such work, Cichowlas et al. [58] simulated Euler turbulence
with large-scale Taylor-Green vortices as initial condition and observed a
combination of Kolmogorov’s $k^{-5/3}$ spectrum in the intermediate range and
$k^{2}$ spectrum in the dissipative range, as shown in Fig. 14(c). The energy
flux is nonzero for the $k^{-5/3}$ regime due to the energy cascade from the
large-scale Taylor-Green vortices to the intermediate scales. Here, the
$k^{-5/3}$ regime is out of equilibrium, while the $k^{2}$ regime is
equilibrium. Over time, the $k^{-5/3}$ regime shrinks at the expense of
$k^{2}$ regime. This is the process of thermalization [58, 64, 207]. Note that
the predictions of Kraichnan [57] and Lee [56] are applicable only to the
equilibrium regime.

Interestingly, random uncorrelated signal, as in white noise, yields $k^{2}$
energy spectrum for the whole range of wavenumbers. Motivated by this
observation, Verma et al. [59, 60] simulated Euler turbulence using white
noise as an initial condition and observed an equilibrium configuration with
zero kinetic energy flux over the whole range of wavenumbers (also see [12,
63]). The numerical results are in agreement with the predictions of Kraichnan
[57] and Lee [56]; in particular, nonhelical flows yield $k^{2}$ kinetic
energy spectrum (see Fig. 14(a)). It appears that the random initial
condition, which has zero mode-to-mode energy transfer and zero kinetic energy
flux, manoeuvres the system towards equilibrium. Contrast this result with
those of Cichowlas et al. [58] who obtained a combination of $k^{-5/3}$ and
$k^{2}$ spectra for Taylor-Green vortex as an initial condition. Thus, an
initial condition plays a key role in the evolution of Euler turbulence; some
initial condition takes the system to equilibrium state, while some other to a
nonequilibrium state. We also remark that the spectral entropy,
$-\sum_{\textbf{k}}p(\textbf{k})\log(p(\textbf{k}))$ [208] where
$p(\textbf{k})=E(\textbf{k})/E$, could be employed to study how the entropy of
the system evolves during thermalization.

Kolmogorov flow, which is forced at intermediate length scales ($k\approx
k_{f}$), also exhibits equilibrium behaviour for $k\leq k_{f}$. For such
flows, Prasath et al. [209], Dallas et al. [191], and Alexakis and Biferale
[74] showed that $E_{u}(k)\sim k^{2}$ and $\Pi_{u}(k)\approx 0$ for $k<k_{f}$,
and $E_{u}(k)\sim k^{-5/3}$ and $\Pi_{u}(k)\approx\mathrm{const}>0$ for
$k>k_{f}$. These authors argue that the modes in $k<k_{f}$ regime are in
absolute equilibrium. See Fig. 14(d) for an illustration. Note that in
Kolmogorov flow, the large scales, rather than small scales, are in
equilibrium. This is in contrast to flows corresponding to K41 and Euler-2 of
Fig. 14 where the modes at small scales are in equilibrium.

Thus, Euler turbulence provides valuable insights into the thermalization
process, which is an important area of research in quantum and classical
physics. In Euler-2 and K41 flows, the energy at large scales cascades to
intermediate and dissipation ranges. The energy flux of the dissipative scale
is transferred to the thermal energy of the particles, who move randomly. In
Euler-2 flow, thermal energy of the particles appear as $k^{2}$ spectrum, but
thermal energy is not represented in K41 flow. We present these regimes in
Fig. 14 using different colors. The yellow, green, and blue colours represent
the inertial, dissipative, and equilibrium regimes respectively, while the
orange colour represents the thermodynamic regime or microscopic scales, where
the particles move randomly. Interestingly, $E_{u}(k)$ of Cichowlas et al.
[58] exhibits a small exponential transition regime (similar to that of
hydrodynamic turbulence) between the $k^{-5/3}$ and $k^{2}$ regimes.

Based on these examples, it has been argued that in 3D hydrodynamics,
thermalization occurs via a multiscale energy transfer from large and
intermediate scales (noequilibrium) to small scales (equilibrium). Here, the
coherent fluid energy of inertial-dissipative regime is converted to the
thermal energy of constituents molecules. This mechanism provides a scenario
for the emergence of friction or dissipation in generic systems, including
Hamiltonian and quantum systems [210]. The merging of rivers into the ocean,
as shown in Fig. 5(a), appears to provide an interesting analogy to the
aforementioned thermalization process. The river flow represents an out-of-
equilibrium system, while ocean represents a system in equilibrium.

The above formalism for Euler turbulence has been extended to secondary fields
advected by the velocity field. For diffusionless passive scalar turbulence,
conservation of $\sum_{\textbf{k}}E_{\zeta}(\textbf{k})$ leads to the
following distribution for $E_{\zeta}(\textbf{k})$:

$P(E_{\zeta}(\textbf{k}))=\beta_{\zeta}\exp[-\beta_{\zeta}E_{\zeta}(\textbf{k})]$
(218)

that leads to $\left\langle
E_{\zeta}(\textbf{k})\right\rangle=1/\beta_{\zeta}$ [6, 8]. The formulas for
MHD turbulence is more complex due the conservation of total energy, cross
helicity, and magnetic helicity. The reader is referred to the original papers
by Frisch et al. [61] and Stribling et al. [62]. For these systems too, the
equilibrium spectrum and fluxes are easier to obtain using white noise as an
initial condition. Burgers turbulence too exhibits similar properties during
thermalization [211].

The energy flux vanishes in the equilibrium regime of hydrodynamic turbulence.
It is important to contrast this effect from the suppression of the net energy
flux by two opposite fluxes, as observed in quasi-2D stably stratified
turbulence [85] and in compressible turbulence [86]. In the latter systems,
the detailed balance of energy transfers is broken by opposing energy fluxes,
called flux loops.

In addition to Euler turbulence and other dissipation-less turbulent systems,
many energy-conserving nonequilibrium systems exhibit strong fluctuations or
turbulence, which will be described below.

###  10.2 Thermalization in miscellaneous dissipation-less systems

In this section, we describe some generic properties of thermalization in
Hamiltonian systems and dissipation-less partial differential equations. But,
first we discuss the dynamics of dissipation-less Burgers equation.

For a wave (e.g., $\sin(x)$) as an initial condition, one-dimensional
dissipative Burgers equation yields shocks [212]. Such flows exhibit $k^{-2}$
spectrum followed by a dissipative spectrum [213]. In contrast, Frisch et al.
[65] and Ray et al. [211] studied thermalization of Burgers equation by
employing a minimal dissipation with small viscosity or hyperviscosity. In
these works, they report a mixed spectrum: $k^{-2}$ for intermediate
wavenumbers (nonequilibrium regime) and $k^{0}$ for large wavenumbers
(equilibrium regime), similar to the findings of Cichowlas et al. [58] for
Euler turbulence. Note, however, that a random initial condition yields an
equilibrium state with $E(k)\sim k^{0}$ and zero energy flux throughout the
wavenumber range.

For more than half a century, researchers have been studying thermalization in
Fermi-Pasta-Ulam-Tsingou model [214]. In this model, the energy cascades to
higher modes when a large-scale excitation is chosen as an initial condition.
The system eventually reaches a state where the energy is equipartitioned
among all the Fourier modes. However, after some time, the system returns to
the initial configuration, consistent with Poincaré recurrence theorem [215,
216, 217]. On the contrary, for random initial condition, the system reaches
an equilibrium state with energy equipartitioned among all Fourier modes [216,
217]. This observation again demonstrates how random initial condition aids in
taking a dynamical system towards thermalization. Similar behaviour has been
observed for KdV equation, which is related to the Fermi-Pasta-Ulam-Tsingou
model [217].

Thermalization processes in quantum systems, such as Bose-Einstein condensate
and superfluids, are more complex [63, 207]. Still, these systems have certain
similarities with Euler turbulence and other related systems. Thus, Euler
turbulence provides valuable insights into the thermalization process. More
work is being carried to the address the following questions among other: Do
Euler turbulence and related turbulent systems become ergodic under
thermalization? What are the consequences of Poincaré recurrence theorem on
thermalization? In Euler turbulence with Taylor-Green vortex as an initial
condition [58], would the system return to the initial configuration after
thermalization. Variable energy flux formalism may be a useful tool for
answering some of these interesting questions.

##  11 Variable energy fluxes in quantum turbulence and binary-mixtures
turbulence

Bose-Einstein condensates and superfluids, which are quantum systems, exhibit
turbulent behaviour for some set of parameters [49, 218, 219, 220]. Similarly,
turbulence is observed in binary-mixtures. As we show below, variable energy
flux formulation sheds important light into these complex systems.

###  11.1 Quantum turbulence

To describe quantum systems, we often employ macroscopic wavefunction
$\zeta(x,t)=\sqrt{n}\exp(iS)$, where $n$ is the number density of atoms and
$S$ is the phase. The time evolution of the wavefunction $\zeta(x,t)$ is
described using Gross-Pitaevskii (GP) equation [51, 218, 49, 220]:

$i\hbar\frac{\partial\zeta}{\partial
t}=\left[-\frac{\hbar^{2}}{2m}\nabla^{2}+V(\textbf{r})+g|\zeta|^{2}\right]\zeta,$
(219)

where $m$ is the mass of the quantum particle, $V(\textbf{r})$ is the external
potential, and $g$ is the proportionality constant for the interaction term.
The GP equation exhibits many interesting features, including entangled
quantum vortices, Kelvin waves, etc. However, in this subsection we focus on
the energy fluxes of quantum turbulence.

The velocity of a superfluid flow is given by the gradient of its
wavefunction, that is, $\textbf{u}_{s}=(\hbar/m)\nabla S$. In terms of $n$ and
$\textbf{u}_{s}$, the real and imaginary parts of the GP equation are [51]

$\displaystyle\frac{\partial n}{\partial t}+\nabla(n\textbf{u}_{s})=0,$ (220)
$\displaystyle\frac{\partial{\bf u}_{s}}{\partial
t}+{\bf(u_{s}\cdot\nabla)u_{s}}=-\frac{1}{mn}\nabla
p-\frac{1}{m}\nabla\left(\frac{\hbar^{2}}{2m\sqrt{n}}\nabla^{2}\sqrt{n}\right)-\frac{1}{m}\nabla
V.$ (221)

Note that $mn=\rho_{s}$ is the density of the superfluid. In Helium-4, the
superfluid component coexists with the normal fluid, whose velocity field is
denoted by $\textbf{u}_{n}$. The equations for the superfluid and normal-fluid
components are [46, 47]

$\displaystyle\frac{\partial{\bf u}_{s}}{\partial t}+({\bf
u}_{s}\cdot\nabla){\bf u}_{s}$ $\displaystyle=$
$\displaystyle-\frac{1}{\rho_{s}}\nabla p_{s}-\frac{\rho_{s}}{\rho}{\bf
F}_{ns},$ (222) $\displaystyle\frac{\partial{\bf u}_{n}}{\partial t}+({\bf
u}_{n}\cdot\nabla){\bf u}_{n}$ $\displaystyle=$
$\displaystyle-\frac{1}{\rho_{n}}\nabla p_{n}+\frac{\rho_{n}}{\rho}{\bf
F}_{ns}+\textbf{F}_{\mathrm{LS}}+\nu\nabla^{2}{\bf u}_{n},$ (223)

where $\rho_{n}$ is the density of the normal fluid, $\rho=\rho_{s}+\rho_{n}$
is the total density of the fluid, $\textbf{F}_{\mathrm{LS}}$ is the large-
scale force applied to the normal fluid, and ${\bf
F}_{ns}=(B/2)|\bm{\omega}|({\bf u}_{s}-{\bf u}_{n})$ is the mutual friction.
Here $\bm{\omega}$ is the superfluid vorticity, and $B$ is a constant. Note
that $p_{n}=(\rho_{n}/\rho)p+\rho_{s}ST$ and
$p_{s}=(\rho_{s}/\rho)p-\rho_{s}ST$ are partial pressures with $S,T,p$ as the
specific entropy, temperature, and pressure respectively. Also, the relative
density $\rho_{n}/\rho_{s}$ increases with temperature, and Helium-4 becomes a
normal fluid at the critical temperature of 2.17 K. In the following
discussion, for simplification, the fluid densities are assumed to be
constant.

Researchers have studied superfluid turbulence using experiments and numerical
simulations ([51, 218, 49, 220, 219] and references therein). These works show
that both superfluid and normal fluid exhibit Kolmogorov-like $k^{-5/3}$
spectra and nearly constant energy fluxes [49, 218, 219, 220]. At very low
temperatures, $\rho_{n}\approx 0$, which is observed in Helium-3; for such
flows, vortex reconnections and phonon coupling at small scales provide the
necessary dissipation to sustain the $k^{-5/3}$ spectrum. Also refer to [221]
for a recent work on energy flux in trapped Bose-Einstein condensate.

In this article, we focus on variable energy flux in quantum turbulence. In
the turbulent regime, the nonlinear terms of the above equations, $({\bf
u}_{s}\cdot\nabla){\bf u}_{s}$ and $({\bf u}_{n}\cdot\nabla){\bf u}_{n}$,
induce the respective energy cascades $\Pi_{u,s}(k)$ and $\Pi_{u,n}(k)$ for
the two components. These energy fluxes are affected by the mutual friction.
The energy injection rates by ${\bf F}_{ns}$ to the normal and superfluid
components are (see Sec. 3)

$\displaystyle\mathcal{F}_{u,s}({\bf k})=-\frac{\rho_{s}}{\rho}\Re[{\bf
F}_{ns}({\bf k})\cdot{\bf u}^{*}_{s}({\bf k})];~{}~{}~{}\mathcal{F}_{u,n}({\bf
k})=\frac{\rho_{n}}{\rho}\Re[{\bf F}_{ns}({\bf k})\cdot{\bf u}^{*}_{n}({\bf
k})].$ (224)

Therefore, the energy fluxes for the two fluids vary with $k$ as

$\displaystyle\frac{d}{dk}\Pi_{u,s}(k)=\mathcal{F}_{u,s}(k);~{}~{}~{}\frac{d}{dk}\Pi_{u,n}(k)=\mathcal{F}_{u,n}(k)-2\nu
k^{2}E_{n}(k),$ (225)

where $2\nu k^{2}E_{n}(k)$ is the dissipation rate for the normal fluid. Roche
et al. [47] and Wacks and Barenghi [48] analyzed the above energy transfers
using numerical simulations and observed that $\mathcal{F}_{u,s}<0$ and
$\mathcal{F}_{u,n}>0$. Note however that these quantities depend on the
temperature or $\rho_{s}/\rho_{n}$. Based on these observations, we expected
that $\Pi_{u,n}$ and $\Pi_{u,s}$ vary with $k$. These variations can induce
additional $k$ dependence in the energy spectra over Kolmogorov’s $k^{-5/3}$
spectrum. These predictions need to be verified using experiments and
numerical simulations.

Quantum systems are energy conserving. However, sustenance of $k^{-5/3}$
spectrum requires dissipation at small scales. It has been argued that the
compressible waves produced during vortex reconnections may provide the
required dissipation [49, 219, 220, 218, 210]. These issues, as well as the
energy spectrum and fluxes of GP equations, have been studied using numerical
simulations and experiments [63, 50, 49, 219, 220, 218, 222].

The above discussion illustrates the usefulness of variable energy flux in
quantum turbulence.

###  11.2 Variable energy fluxes in binary-mixture turbulence

In this section, we briefly describe the energy fluxes of binary fluid
mixtures [223, 224, 225, 226]. We consider a binary mixture with two
components whose relative densities are $\zeta({\bf r})$ and $1-\zeta({\bf
r})$. Researchers describe the dynamics of these fields using time-dependent
Ginzburg-Landau equation and Cahn-Hilliard equation [54, 83, 55]. Here we
illustrate the idea of variable energy flux for the above equations.

The time-dependent Ginzburg-Landau (TDGL) equation, also called model A, is
[55]:

$\frac{\partial\zeta}{\partial t}=\zeta-\zeta^{3}+\nabla^{2}\zeta.$ (226)

The corresponding equation for the spectral energy
$E_{\zeta}(\textbf{k})=\frac{1}{2}|\zeta(\textbf{k})|^{2}$ is

$\frac{d}{dt}E_{\zeta}({\bf k})=-\sum_{{\bf k}_{1},{\bf k}_{2}}\Re[\zeta({\bf
k}_{1})\zeta({\bf k}_{2})\zeta({\bf k}_{3})\zeta^{*}({\bf k})]+E_{\zeta}({\bf
k})-k^{2}E_{\zeta}({\bf k}),$ (227)

where ${\bf k}_{3}={\bf k}-{\bf k}_{1}-{\bf k}_{2}$. In Eq. (227), the second
term in RHS enhances $E_{\zeta}({\bf k})$, while the last term dissipates
$E_{\zeta}({\bf k})$. The nonlinear term, the first term in the RHS of above
equation, induces the following energy flux for a wavenumber sphere of radius
$k_{0}$:

$\Pi_{\zeta}(k_{0})=-\sum_{k\leq k_{0}}\sum_{{\bf k}_{1},{\bf
k}_{2}}\Re[\zeta({\bf k}_{1})\zeta({\bf k}_{2})\zeta({\bf
k}_{3})\zeta^{*}({\bf k})].$ (228)

Numerical simulation and analytical studies of TDGL equation reveal that
asymptotically ($t\rightarrow\infty$), the system exhibits a domain with
either $\zeta=1$ or $-1$ (see [55] and references therein). The energy flux
provides interesting inputs for understanding the above result. The arguments
for 1D TDGL are as follows. The scalar energy $E_{\zeta}$ is dissipated
strongly at small scales due to the $k^{2}$ factor in the dissipation rate.
Consequently, a forward cascade of $E_{\zeta}$ is set up that transfers the
energy of large and intermediate scales to small scales, where it is
dissipated. Note however that the mean energy, $E_{\zeta}({\bf k}=0)$, is not
dissipated (due to the structure of the dissipation term). Consequently, only
$k=0$ mode survives, while the rest of the modes vanish due to the forward
cascade and dissipation. Therefore, the final state is either $\zeta=1$ or
$-1$ with zero energy flux. The dynamics for 2D and 3D is expected to be
similar. Thus, the energy flux provides useful insights into the dynamics of
TDGL equation.

Similar analysis is applicable to model B or Cahn-Hilliard (CH) equation [55],
which is

$\frac{\partial\zeta}{\partial
t}=-\nabla^{2}(\zeta-\zeta^{3}+\nabla^{2}\zeta).$ (229)

The evolution equation for the modal energy of CH equation is

$\frac{d}{dt}E_{\zeta}({\bf k})=-k^{2}\sum_{{\bf k}_{1},{\bf
k}_{2}}\Re[\zeta({\bf k}_{1})\zeta({\bf k}_{2})\zeta({\bf
k}_{3})\zeta^{*}({\bf k})]+k^{2}E_{\zeta}({\bf k})-k^{4}E_{\zeta}({\bf k}),$
(230)

while the corresponding scalar energy flux is

$\Pi_{\zeta}(k_{0})=-k^{2}\sum_{k<k_{0}}\sum_{{\bf k}_{1},{\bf
k}_{2}}\Re[\zeta({\bf k}_{1})\zeta({\bf k}_{2})\zeta({\bf
k}_{3})\zeta^{*}({\bf k})].$ (231)

The energetics of CH equation is very similar to that of TDGL:
$k^{2}E_{\zeta}({\bf k})$ term feeds energy into the system, while
$k^{4}E_{\zeta}({\bf k})$ term dissipates the energy. The first term in the
RHS of Eq. (230) creates forward energy cascade. The cascaded energy gets
dissipated at small scales. Asymptotically, only $k=1$ mode survives because
the steady state solution has zero energy flux (or zero nonlinearity) and
$k^{2}-k^{4}=0$.

Inclusion of hydrodynamic effects into Cahn-Hilliard equation yields the
following equations for the velocity field and $\zeta$ [54]:

$\displaystyle\frac{\partial{\bf u}}{\partial t}+({\bf u}\cdot\nabla){\bf
u}=-\frac{1}{\rho}\nabla p+a\zeta\nabla\nabla^{2}\zeta+{\bf
F}_{\mathrm{LS}}+\nu\nabla^{2}{\bf u},$ (232)
$\displaystyle\frac{\partial\zeta}{\partial t}+({\bf
u}\cdot\nabla)\zeta=-\nabla^{2}(\zeta-\zeta^{3}+\nabla^{2}\zeta),$ (233)
$\displaystyle\nabla\cdot\mathbf{u}=0,$ (234)

with corresponding forces as ${\bf F}_{u}=a\zeta\nabla\nabla^{2}\zeta$ and
$F_{\zeta}=\nabla^{2}\zeta^{3}$. The energy spectra and fluxes of the above
system have been studied in detail by many researchers (see [225, 226] and
references therein).

The nonlinear terms $({\bf u}\cdot\nabla){\bf u}$ and $({\bf
u}\cdot\nabla)\zeta$ generate the respective fluxes $\Pi_{u}(k)$ and
$\Pi_{\zeta}(k)$ for the kinetic and scalar energies. The force ${\bf F}_{u}$
induces variations in $\Pi_{u}(k)$ due to the energy injection rate:

$\displaystyle\mathcal{F}_{u}({\bf k})$ $\displaystyle=$
$\displaystyle\Re[{\bf F}_{u}({\bf k})\cdot{\bf u^{*}(k)}]=\sum_{\bf
p}\Im[p^{2}\zeta({\bf k-p})\zeta({\bf p})\\{{\bf p\cdot u^{*}(k)}\\}].$ (235)

Refer to Eq. (230) for the expression of $\mathcal{F}_{\zeta}({\bf k})$.
Interestingly, hydrodynamic version of CH equation too exhibits coarsening
[225, 226]. Note that the scalar field appears to exhibit forward cascade.
Thus, the coarsening process in binary fluid is not due to any inverse cascade
(as in 2D turbulence), but it is due to the energy injection term of the
equation.

In summary, the scalar energy flux provides valuable inputs to the dynamics of
coarsening. However, more work is required for definitive conclusions.

##  12 Third-order structure function and energy flux

So far, we have discussed energy transfers and flux in Fourier space. Note,
however, that there are many interesting connections between energy flux and
field correlations in real space. In one of the path-breaking works,
Kolmogorov [1, 2] showed a relationship between the energy flux and the third-
order structure functions. Later, Kolmogorov’s theory was generalized to 2D
hydrodynamic turbulence, scalar turbulence, MHD turbulence, anisotropic
turbulence, and other related systems. We will briefly discuss these results
in this section.

Kolmogorov considered statistically homogeneous, isotropic, and steady 3D
turbulence, with forcing employed at large scales. In such a flow, he
considered two real-space points r and r+l where the velocities are u and
$\textbf{u}^{\prime}$ respectively. Under the limit of infinite Reynolds
number, for the inertial range, Kolmogorov [1, 2] showed that the third-order
longitudinal structure function $S_{3}(l)$ is

$\displaystyle\left\langle[{\bf(u^{\prime}-u)}\cdot\hat{\textbf{l}}]^{3}\right\rangle=-\frac{4}{5}\epsilon_{u}l,$
(236)

where $\hat{\textbf{l}}$ is the unit vector along vector ${\bf l}$, and
$\epsilon_{u}$ is the viscous dissipation rate that equals the kinetic energy
flux in the inertial range. A useful variant of the above relation is

$\displaystyle\left\langle|{\bf
u^{\prime}-u}|^{2}[{\bf(u^{\prime}-u)}\cdot\hat{\textbf{l}}]\right\rangle=-\frac{4}{3}\epsilon_{u}l.$
(237)

The above relations are borne out in many experiments and numerical
simulations, which are too numerous to be listed here. We refer the reader to
Frisch [8], Lesieur [6], and references therein for details.

For a passive secondary scalar field $\zeta$ advected in turbulent flow and
forced at large scales, in the inertial range [66, 3, 4],

$\displaystyle\left\langle{(\zeta^{\prime}-\zeta)}^{2}[{\bf(u^{\prime}-u)}\cdot\hat{\textbf{l}}]\right\rangle=-\frac{4}{3}\epsilon_{\zeta}l,$
(238)

where $\epsilon_{\zeta}$ is the dissipation rate or the energy flux of the
scalar field. The same relation holds for passive vector and tensorial fields
that are forced at large scales. However, the relationship of Eq. (238) does
not hold for the velocity and magnetic fields of MHD turbulence because MHD
equations have additional terms, $({\bf u\cdot\nabla})\bm{\zeta}$ and
$(\bm{\zeta}\cdot\nabla){\bf u}$, compared to passive vector (see Eqs. (117,
118)). However, the derivation of passive scalar can be generalized to
Elsässer variables ${\bf z}^{\pm}$ due to the similarities in the forms of the
equations (see Eq. (145)). Following this approach, Politano and Pouquet [227]
derived that

$\displaystyle S^{z^{+}}_{3}(l)$ $\displaystyle=$
$\displaystyle\left\langle|{\bf z^{+^{\prime}}-z^{+}}|^{2}[({\bf
z^{-^{\prime}}-z^{-}})\cdot\hat{\bf{l}}]\right\rangle=-\frac{4}{3}\epsilon_{z^{+}}l,$
(239) $\displaystyle S^{z^{-}}_{3}(l)$ $\displaystyle=$
$\displaystyle\left\langle|{\bf z^{-^{\prime}}-z^{-}}|^{2}[({\bf
z^{+^{\prime}}-z^{+}})\cdot\hat{\bf{l}}]\right\rangle=-\frac{4}{3}\epsilon_{z^{-}}l,$
(240)

where $\epsilon_{z^{\pm}}$ are the dissipation rates or the energy fluxes of
${\bf z^{\pm}}$ fields. Politano and Pouquet [227] also derived relations for
${\bf u}$ and $\bm{\zeta}$, but these relations are more complex than Eqs.
(239, 240).

The structure functions of 2D turbulence are more complex than above functions
due to dual scaling: the kinetic energy cascades backward in the wavenumber
band $k<k_{f}$, while the enstrophy cascades forward in the band $k>k_{f}$.
Here, $k_{f}$ is the forcing wavenumber band (see Sec. 9). For homogeneous and
isotropic 2D turbulence, in the inertial range of $k<k_{f}$ regime [228, 229,
228],

$\displaystyle\left\langle|{\bf
u^{\prime}-u}|^{2}[{\bf(u^{\prime}-u)}\cdot\hat{\textbf{l}}]\right\rangle=2\epsilon_{u}l;~{}~{}~{}\left\langle[{\bf(u^{\prime}-u)}\cdot\hat{\textbf{l}}]^{3}\right\rangle=\frac{3}{2}\epsilon_{u}l.$
(241)

However, in the inertial range of $k>k_{f}$ regime [228],

$\displaystyle\left\langle|\omega-\omega^{\prime}|^{2}[({\bf
u^{\prime}-u})\cdot\hat{l}]\right\rangle=-2\epsilon_{\omega}l;~{}\left\langle|{\bf
u^{\prime}-u}|^{2}[({\bf
u^{\prime}-u})\cdot\hat{l}]\right\rangle=-\frac{1}{8}\epsilon_{\omega}l,$
(242)

where $\epsilon_{\omega}$ is the enstrophy dissipation rate that equals the
enstrophy flux. These relations have been verified in several numerical
simulations (see [230, 197, 196] and references therein).

Isotropy is a major assumption in the derivation of Eq. (236). Researchers
have attempted to generalise Kolmogorov’s relation to anisotropic turbulence.
For example, Biferale and Procaccia [69] employed symmetry groups to decipher
anisotropic structure functions. Arad et al. [70] computed the anistropic
structure functions in atmospheric surface layer. Danaila et al. [71]
constructed anisotropic structure functions for axisymmetric anisotropic
turbulence by invoking scale-by-scale energy budget. Ching [68] and
Bhattacharya et al. [141] generalized Eq. (236) for turbulent thermal
convection. As described in Sec. 4.4, pressure plays an important role in
anisotropic spectral energy transfer. It will be interesting to relate the
Fourier-space anisotropic energy-transfer formulas to the anisotropic
structure functions. We also remark that the triple product in the left-hand-
side of Eq. (237) is related to the mode-to-mode energy transfer formula,
$S(\textbf{k}|\textbf{p}|\textbf{q})$, of Eq. (13). Refer to Verma [14] for
details.

The relations discussed so far in this section are related to the third-order
structure function. There have been valiant efforts to derive analytical
relations for higher-order structure functions, but these efforts have not
yielded the final results. These works are related to the intermittency
effects in turbulence, a topic which is beyond the scope of this review. We
refer the reader to Frisch [8], Dubrulle [231], Stolovitzky and Sreenivasan
[232], and references therein for details. Some of the notable field-theoretic
works regarding intermittency are [233, 234, 235] and references therein. It
has been argued that the higher-order structure functions are related to the
fluctuations in the energy flux.

##  13  Summary and conclusions

The energy flux is an important quantity in turbulence. In hydrodynamic
turbulence with large-scale forcing, the inertial-range energy flux is
constant. However, in the presence of inertial-range energy injection
($\mathcal{F}_{u}(k)$) and dissipation ($D_{u}(k)$), the kinetic energy flux
becomes scale-dependent and is described by
$d\Pi_{u}(k)/dk=\mathcal{F}_{u}(k)-D_{u}(k)$. In this review, we show how the
variable energy flux formalism provides valuable insights into the dynamics of
many turbulent systems, especially in determining their energy spectra and
fluxes. A summary of the results presented in the review is as follows.

  * •

Buoyancy driven turbulence: In stably stratified turbulence with moderate
stratification, the kinetic energy is transferred to the potential energy,
hence $\mathcal{F}_{u}(k)<0$. Therefore, $\Pi_{u}(k)$ decreases with $k$ in
the inertial range itself; in particular, $\Pi_{u}(k)\sim k^{-4/5}$ and
$E_{u}(k)\sim k^{-11/5}$, which is steeper than $k^{-5/3}$ spectrum [15, 16].
However, in turbulent thermal convection, $\mathcal{F}_{u}(k)>0$ for small and
moderate thermal Prandtl numbers (Pr). Hence, $E_{u}(k)$ for thermal
convection is expected to be shallower than $k^{-5/3}$. Yet, turbulent thermal
convection with $\mathrm{Pr}\lessapprox 1$ has behaviour similar to
hydrodynamic turbulence (nearly constant energy flux and $k^{-5/3}$ kinetic
energy spectrum in the inertial range). This is due to the fact that for
$\mathrm{Pr}\lessapprox 1$, $\mathcal{F}_{u}(k)$ decreases sharply with $k$
and is quite weak in the inertial range, similar to that in Kolmogorov’s model
for hydrodynamic turbulence.

  * •

MHD and Polymeric turbulence: In MHD turbulence, the nonlinear interactions
between the velocity and magnetic fields cause energy transfers from the
velocity field to the magnetic field. This conversion mechanism is responsible
for the growth of the magnetic field in astrophysical objects, and for making
$\Pi_{u}(k)$ a decreasing function of $k$. Similar energy transfers are
observed in turbulent flows with polymers. The suppression of $\Pi_{u}(k)$ or
the nonlinear term ${\bf(u\cdot\nabla)u}$ is one of the primary causes of drag
reduction in such flows. Using variable energy flux formalism we also derive
identities relating various fluxes.

  * •

Dissipation: In turbulence, viscous dissipation suppresses the energy flux in
the dissipation range. Using several assumptions, Pao [31, 125] showed that in
the inertial-dissipation range of hydrodynamic turbulence, the energy flux and
normalized energy spectrum vary as $\exp(-(k/k_{d})^{4/3})$, where $k_{d}$ is
Kolmogorov wavenumber. Ekman friction that acts at all scales steepens the
inertial-range energy spectrum further than the $k^{-5/3}$ power-law. In
quasi-static MHD turbulence, the Joule dissipation provides similar steepening
of the kinetic energy spectrum. Pao’s model has been generalized to 2D
hydrodynamic turbulence with Ekman friction.

  * •

Energy flux of a secondary flow $\zeta$: The advection term of a secondary
field $\zeta$, ${\bf(u\cdot\nabla)}\zeta$, has an associated secondary energy
flux $\Pi_{\zeta}$. This flux too exhibits variability:
$d\Pi_{\zeta}(k)/dk=\mathcal{F}_{\zeta}(k)-D_{\zeta}(k)$, where
$\mathcal{F}_{\zeta}(k)$ is the secondary energy injection rate, and
$D_{\zeta}(k)$ is the diffusion rate of the secondary field. Such flux
variations are present in MHD turbulence, stably stratified turbulence,
binary-mixture turbulence, and related complex flows.

Table 3: Table illustrating 3D turbulent systems along with their forces (${\bf F}_{u}({\bf k})$), the kinetic energy injection rates by the forces ($\mathcal{F}_{u}({\bf k})$), and the nature of kinetic energy fluxes in the inertial range. For the meaning of the symbols, refer to the discussion in the paper. IC stands for initial condition. System | ${\bf F}_{u}({\bf k})$ | $\mathcal{F}_{u}({\bf k})$ | Nature of $\Pi_{u}(k)$
---|---|---|---
Kolmogorov’s K41 law | 0 | 0 | Const
QS MHD turbulence | $-B_{0}^{2}\cos^{2}\theta{\bf u(k)}$ | $-2B_{0}^{2}\cos^{2}\theta E_{u}(k)<0$ | Decreases
Ekman friction (3D) | $-\alpha{\bf u}({\bf k})$ | $-2\alpha E_{u}({\bf k})<0$ | Decreases
Stably stratified turbulence | $-N\zeta({\bf k})$ | $-N\Re[\zeta({\bf k})u_{z}^{*}({\bf k})]<0$ | Decreases
Thermal convection | $\alpha g\zeta({\bf k})$ | $\alpha g\Re[\zeta({\bf k})u_{z}^{*}({\bf k})]>0$ | Marginally increases
Unstably stratified turbulence | $g\zeta({\bf k})$ | $g\Re[\zeta({\bf k})u_{z}^{*}({\bf k})]>0$ | Increases
MHD (Dynamo) | ${\bf[J\times B}]({\bf k})$ | $\Re\\{[{\bf J\times B}]({\bf k})\cdot{\bf u^{*}(k)}\\}<0$ | Decreases
Dilute polymer | $\frac{\mu}{\tau_{p}}\partial_{j}(f\zeta_{ij})$ | Complex convolution $<0$ | Decreases
Shear turbulence | Shear force | Positive | Increases
Euler turbulence | 0 | 0 | zero or mixed
|  |  | depending on IC

Variable energy flux is also useful for modelling shear turbulence, stably and
unstably stratified turbulence, Euler turbulence, quantum turbulence, etc.
Variable energy flux formalism provides valuable insights for understanding
quantum turbulence and the coarsening processes in time-dependent Ginzburg-
Landau and Cahn-Hilliard equations. Interestingly, the money supply in a free
market economy too exhibits a cascade across various income groups; this
cascade has similarities with the energy flux in turbulence [236]. It is also
important to note that the nature of variable energy flux depends on the space
dimensionality. In Table 3, we summarise the variable energy fluxes discussed
in this review.

For random (white noise) initial condition, truncated Euler turbulence
exhibits equilibrium behaviour with vanishing kinetic energy flux and nearly
$k^{2}$ energy spectrum. However, for orderly initial condition, such as
Taylor-Green vortex, Euler turbulence yields a mix of nonequilibrium and
equilibrium behaviour with a combination of $k^{-5/3}$ and $k^{2}$ spectra.
The $k^{-5/3}$ regime shrinks at the expense of $k^{2}$ regime. Dissipation-
less MHD turbulence and passive-scalar turbulence also exhibit similar
behaviour. These features of dissipation-less turbulence provide very valuable
insights for modelling thermalization of complex systems.

Thus, variable energy flux provides a unifying platform for modelling many
turbulent systems. Note, however, that there are many unresolved issues in
this framework, e.g., variable energy fluxes in anisotropic turbulence,
quantum turbulence, binary-mixture turbulence, etc. The present review does
not cover many important turbulent flows, such as electron
magnetohydrodynamics [237], shell model of turbulence [98], compressible
turbulence [238], weak turbulence [239], and plasma turbulence [240], where
variable energy fluxes are present. We have also omitted discussions on the
field-theoretic treatment of energy flux in turbulence. In summary, variable
energy flux is a very useful tool for understanding many turbulent systems.

## Acknowledgments

For writing this review I drew heavily from the discussions and idea exchanges
I had with my collaborators, namely, Franck Plunian, Rodion Stepanov, Ravi
Samtaney, Daniele Carati, Stephan Fauve, Jai Sukhatme, Sanjay Puri, K. R.
Sreenivasan, Alexandros Alexakis, Gaurav Dar, Vinayak Eswaran, and Bernard
Knaepen. I am very grateful to them for the same. In addition, I received
critical inputs and ideas from many past and present doctoral and master
students—Abhishek Kumar, Shashwat Bhattacharya, Roshan Samuel, Mohammad Anas,
Shadab Alam, Soumyadeep Chatterjee, Pankaj Mishra, Satyajit Barman, Manohar
Sharma, Shubhadeep Sadhukhan, Olivier Debliquy, Bogdan Teaca, Thomas
Lessinness, Valerii Titov, Ambrish Pandey, Sandeep Reddy, Anando Chatterjee,
Arvind Ayyer, and V. Avinash. I also thank J. K. Bhattacharjee, Peter Frick,
Annick Pouquet, Arnab Rai Choudhuri, Avinash Khare, P. K. Yeung, Diego Donzis,
Xavier Albets, Itamar Procaccia, Maurice Rossi, Andrei Teimurazov, Andrei
Sukhanovskii, Marc Brachet, Gregory Eyink, Luca Moriconi, Sagar Chakraborty,
Supratik Banerjee, Sonakshi Sachdev, Akanksha Gupta, and Luca Biferale for
useful discussions. I gratefully acknowledge the support of Indo-French
projects 4904-1 and 6104-1 from CEFIPRA, IFCAM project MA/IFCAM/19/90, Indo-
Russian project INT/RUS/ RSF/P-03 from Department of Science and Technology
India, and SERB project SERB/F/3279/2013-14 that made the collaborative work
and idea exchanges possible. Some of the results presented in the review have
been generated using SHAHEEN II of KAUST (project K1052) and HPC2013 of IIT
Kanpur.

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