# Cascades transition in generalised two-dimensional turbulence

Vibhuti Bhushan Jha1,2<EMAIL_ADDRESS>Kannabiran Seshasayanan3
<EMAIL_ADDRESS>
Vassilios Dallas4<EMAIL_ADDRESS>1Space Applications Centre,
Indian Space Research Organisation, Ahmedabad, Gujarat, India 2Department of
Physics, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal,
India 3Department of Applied Mechanics, Indian Institute of Technology Madras,
Chennai, India 4Mathematical Institute, University of Oxford, Oxford, OX2 6GG,
UK

###### Abstract

The generalized two-dimensional (2D) fluid is characterised by a relationship
between a scalar field $q$, called generalised vorticity, and the stream
function $\psi$, namely $q=-(-\nabla^{2})^{\frac{\alpha}{2}}\psi$. We study
the transition of cascades in generalised 2D turbulence by systematically
varying the parameter $\alpha$ and investigating its influential role in
determining the directionality (inverse, forward, or bidirectional) of these
cascades. We identify a finite critical threshold $\alpha_{c}$, where a
transition occurs in cascade directionality, emphasising that $\alpha=0$
signifies a laminar flow state with a vanishing nonlinear term. At a critical
value of $\alpha_{c}$ a small wavelength instability is induced in the laminar
flow. Using perturbation analysis we obtain that $\alpha_{c}$ is inversely
proportional to the square of the Reynolds number – a scaling validated
through numerical simulations. We derive upper bounds for the dimensionless
dissipation rates of generalised energy $E_{G}$ and enstrophy $\Omega_{G}$ as
the Reynolds number tends to infinity. These findings corroborate numerical
simulations, illustrating the inverse cascade of $E_{G}$ and forward cascade
of $\Omega_{G}$ for $\alpha>0$, contrasting with the reverse behaviour for
$\alpha<0$. The dependence of dissipation rates on system parameters
reinforces these observed transitions, substantiated by spectral fluxes and
energy spectra, which hint at Kolmogorov-like scalings at large scales but
discrepancies at smaller scales. These discrepancies are possibly due to
nonlocal transfers, which dominate the dynamics as we go from positive to
negative values of $\alpha$. Intriguingly, the forward cascade of $E_{G}$ for
$\alpha<0$ reveals similarities to three-dimensional turbulence, notably the
emergence of vortex filaments within a 2D framework, marking a unique feature
of this generalised model.

###### keywords:

Keywords

##  1 Introduction

The presence of nonlinear interactions allow energy to be redistributed across
different length scales in turbulent flows. While in three-dimensional (3D)
turbulence, energy is distributed to small scales (forward cascade), energy is
transferred to large scales in two-dimensional (2D) turbulence (inverse
cascade). This, along with the existence of other quadratic invariants make 2D
turbulence an exciting system. Not entirely limited to theoretical or
numerical excursions, such behaviour has been observed in atmospheric (Lilly,
1969; Rhines, 1979; Nastrom et al., 1984; Read, 2001) and planetary flows
(Siegelman et al., 2022). The presence of external mechanisms like rotation,
stratification, presence of magnetic field or compactification of a dimension
can make the prediction of the cascade direction challenging (Celani et al.,
2010; Alexakis, 2011; Sen et al., 2012; Marino et al., 2013; Deusebio et al.,
2014) due to competing behaviour between the forward and inverse cascades,
often dependent on some control parameter like anisotropy, rotation rate, etc.
This results in bidirectional cascades, which depend on the control parameter.
Such bidirectional cascades have been observed in numerical (Smith et al.,
1996; Alexakis, 2011; Seshasayanan et al., 2014a; Sozza et al., 2015) and
experimental settings (Shats et al., 2010; Xia et al., 2011; Campagne et al.,
2014). Furthermore, there is evidence for bidirectional cascade in atmospheric
(Byrne & Zhang, 2013; King et al., 2015; Shao et al., 2023), oceanic (Scott &
Wang, 2005; Arbic et al., 2013; Balwada et al., 2016; Khatri et al., 2018) and
planetary flows (Lesur & Longaretti, 2011; Young & Read, 2017).

The transition from one type of cascade to another can be either smooth or at
a critical point (Alexakis & Biferale, 2022). The presence of a critical
dimension at which the cascade direction changes has been demonstrated by
Frisch et al. (1976). Benavides & Alexakis (2017) found the transition from a
forward to a bidirectional cascade in thin layers of fluid turbulence to be
critical (Ecke, 2017). In 2D magnetohydrodynamic (MHD) turbulence, the
variation in magnetic field above a critical value leads to cascade transition
(Seshasayanan et al., 2014a), and the energy flux scales as a power law.
Similarly under certain regimes, MHD flows can behave like a 2D flow at large
scales and 3D like at smaller scales (Alexakis, 2011). In this case, the
inverse cascade of energy is highly sensitive to the strength of the magnetic
field.

Pierrehumbert et al. (1994a) considered a generalised model of 2D turbulence
characterised by a parameter $\alpha$ which links the streamfunction to a
scalar field called generalised vorticity. Certain values of $\alpha$ lead to
equations relevant in the context of geophysical flows. For $\alpha=2$ the
generalised model gives the familiar barotropic vorticity equation from 2D
Navier-Stokes (Tabeling, 2002; Boffetta & Ecke, 2012). For $\alpha=1$ it gives
the surface quasi-geostrophic (SQG) equation, which describes the motion of a
rotating stratified fluid (Held et al., 1995; Lapeyre, 2017). For $\alpha=-2$
it gives a rescaled shallow-water quasi-geostrophic equation in the asymptotic
limit of length scales large compared to the deformation scale (Larichev &
McWilliams, 1991; Smith et al., 2002). In generalised 2D turbulence, changing
the value of $\alpha$ leads to varying degrees of forward and inverse
cascades. There are various studies using different values of $\alpha$ and
analysing the associated energy spectra (Tran, 2004; Watanabe & Iwayama,
2004a; Iwayama et al., 2015).

In the presence of an inverse cascade, strain rate arguments for spectral
locality are based on negative eddy viscosity. In other words, the notion that
strain thins smaller scale vortices and so energy is transferred toward large
scales (Kraichnan, 1976). The cascades of 2D Navier-Stokes turbulence
($\alpha=2$) has been predicted that are less local than of the 3D turbulence
(Kraichnan, 1971; Boffetta & Ecke, 2012). Pierrehumbert et al. (1994a) focus
on positive values of $\alpha$ only and argue that $\alpha=2$ is at the
transition between local to nonlocal enstrophy transfers, in the sense that
dominant straining of small scales comes from the largest scales when
$\alpha>2$. This is confirmed by numerical simulations (Watanabe & Iwayama,
2004b). On the other hand, it is argued that the energy transfers become
nonlocal when $\alpha>4$ (Pierrehumbert et al., 1994a). However, depending on
the dynamics generating an inverse cascade, spectral locality of strain rate
may not indicate local transfers (Burgess & Shepherd, 2013). On the other
hand, Watanabe & Iwayama (2007); Foussard et al. (2017) found nonlocal
interactions in the generalised enstrophy transfer for $1\leq\alpha\leq 2$.

The value of $\alpha$ decides the nature of cascade. For positive $\alpha$,
energy (enstrophy) cascades to large (small) scales whereas energy (enstrophy)
cascades to small (large) scales for negative $\alpha$. Thus, depending on the
sign and value of $\alpha$, the cascade would either be inverse, forward or
bidirectional. This suggests that there is some critical $\alpha_{c}$ at which
there is a transition in the direction of the cascades. However, the
aforementioned studies did not study systematically the $\alpha<0$ regime. In
this work we focus on the theoretical and numerical study of the cascades in
generalised 2D turbulence by systematically varying $\alpha$ from positive to
negative values. We use mathematical inequalities to derive bounds on
generalised energy and enstrophy dissipation rates and perform DNS to
determine the critical point $\alpha_{c}$. Finally, our present study deals
with the degree of non-locality of triadic interactions for different values
of $\alpha$.

The paper is organised as follows. In section 2 we discuss the problem
formulation and details of the model. In section 3 we derive theoretical
bounds on the generalised energy and enstrophy dissipation rates. We present
numerical results in section 4 and finally we summarise our concluding remarks
in section 5.

##  2 Problem Formulation

Consider the 2D evolution equation of the generalised vorticity $q(x,y,t)$ in
a periodic domain

$\partial_{t}q+\mathcal{J}(\psi,q)=\nu_{+}\nabla^{2n}q+\nu_{-}\nabla^{-2m}q+f_{q}$
(1)

where $\psi(x,y,t)$ is the streamfunction, the nonlinear term given by the
Jacobian
$\mathcal{J}(\psi,q)=\partial_{x}\psi\partial_{y}q-\partial_{y}\psi\partial_{x}q$,
$\nu_{+}$ is the small scale diffusion coefficient, $\nu_{-}$ is the large
scale diffusion coefficient and
$f_{q}=-(-\nabla^{2})^{\frac{\alpha}{2}}f_{\psi}$ is the external forcing,
where $f_{\psi}=f_{0}\sin(k_{f}x)\sin(k_{f}y)$ with $f_{0}$ denoting the
amplitude of the forcing. The fluid velocity is given by
$\mathbb{u}=\nabla\times(\psi\hat{z})$. The generalised relationship between
$q$ and $\psi$ in 2D fluid dynamics is,

$q=-(-\nabla^{2})^{\frac{\alpha}{2}}\psi$ (2)

as discussed by Pierrehumbert et al. (1994b).

In the presence of an inverse cascade the energy of the large-scale modes grow
to extreme values forming a condensate, the growth saturates when the viscous
dissipation at the largest scale balances the energy injection (Chertkov et
al., 2007; Boffetta & Ecke, 2012). To prevent the formation of a very large
condensate and to reach a turbulent stationary regime we supplement our system
with a large scale dissipative term $\nu_{-}\nabla^{-2m}q$ that is responsible
for saturating the inverse cascade. Here we consider hypo-viscosity by raising
the inverse Laplacian to the power of $m=2$ and hyper-viscosity by raising the
Laplacian of the viscous term $\nu_{+}\nabla^{2n}q$ to the power of $n=2$. The
hyper-viscosity models very large Reynolds number flows giving a wider
inertial range, as the viscous term kicks in at much smaller scales compared
to the normal viscosity case. The hypo-viscosity does the same by extending
the inertial range for scales larger than the forcing scale.

We denote the dimensions of a quantity $g$ by using the square bracket $[g]$.
Considering the fact that the variables $x,y$ have units of length $[L]$, and
$t$ has units of time $[T]$, then $[\psi]=[L]^{2}[T]^{-1}$ and
$[q]=[L]^{2-\alpha}[T]^{-1}$. So, the two dimensionless parameters are the
small scale Reynolds number

$Re_{+}=\sqrt{f_{o}k_{f}}/(\nu_{+}k_{f}^{2n-1/2}),$ (3)

and the large scale Reynolds number

$Re_{-}=\sqrt{f_{o}k_{f}}/(\nu_{-}k_{f}^{-2m-1/2}),$ (4)

with $[f_{0}]=[L]^{2}[T]^{-2}$, $[\nu_{+}]=[L]^{2n}[T]^{-1}$ and
$[\nu_{-}]=[L]^{-2m}[T]^{-1}$.

In the limit of $\nu_{+}\to 0$, $\nu_{-}\to 0$ and $f_{q}=0$ the integral over
a periodic domain of any function of the scalar field $q$ is conserved.
Therefore, there are infinite number of invariants (Smith et al., 2002).
Following the Kolmogorov-Kraichnan phenomenology of 2D Navier-Stokes
turbulence, the two quadratic invariants that determine the cascade directions
in generalised 2D turbulence are:

$E_{G}=\langle{\psi q}\rangle,\quad\Omega_{G}=\langle{q^{2}}\rangle,$ (5)

where we refer to $E_{G}$ as the generalised energy and $\Omega_{G}$ as the
generalised enstrophy with units $[E_{G}]=[L]^{4-\alpha}[T]^{-2}$ and
$[\Omega_{G}]\sim[L]^{4-2\alpha}[T]^{-2}$. The angle brackets
$\langle{.}\rangle$ denote spatiotemporal averaging. Multiplying (1) by $\psi$
and integrating over space and time, we can derive the evolution equation of
the generalised energy, yielding

$0=-\epsilon_{+}-\epsilon_{-}+\epsilon,$ (6)

where the left hand side $d_{t}E_{G}=0$ due to statistical stationarity,
$\epsilon_{+}=-\nu_{+}\langle{\psi\nabla^{2n}q}\rangle$ is the small scale
dissipation rate, $\epsilon_{-}=-\nu_{-}\langle{\psi\nabla^{-2m}q}\rangle$ is
the large scale dissipation rate and $\epsilon=\langle{\psi f_{q}}\rangle$ is
the injection rate of the generalised energy. Similarly, if we multiply (1)
with $q$ and integrate over space and time, the evolution equation of the
generalised enstrophy reads as

$0=-\xi_{+}-\xi_{-}+\xi,$ (7)

where the left hand side $d_{t}\Omega_{G}=0$ due to statistical stationarity,
$\xi_{+}=-\nu_{+}\langle{q\nabla^{2n}q}\rangle$ is the small scale dissipation
rate, $\xi_{-}=-\nu_{-}\langle{q\nabla^{-2m}q}\rangle$ is the large scale
dissipation rate and $\xi=\langle{qf_{q}}\rangle$ is the injection rate of the
generalised enstrophy.

The equivalent expression of (2) in Fourier space is

$\widehat{q}({\bf k},t)=-k^{\alpha}\widehat{\psi}({\bf k},t)$ (8)

where $k=\sqrt{k_{x}^{2}+k_{y}^{2}}$ is the isotropic two-dimensional
wavenumber. The notation $\widehat{.}$ denotes the Fourier transform
coefficients. The spectra of the two quadratic invariants can be connected
using the relationship (8) together with Parseval’s theorem to get

$\Omega_{G}(k)=k^{\alpha}E_{G}(k).$ (9)

Now, the spectra of the two quadratic invariants in the inertial range can be
derived following Kolmogorov (1941) scaling arguments, where the spectral flux
is assumed to be constant in the inertial range and the spectral densities
$E_{G}(k)$ and $\Omega_{G}(k)$ are only functions of the local scale and
spectral flux. Hence, using dimensional arguments for the generalised energy
and enstrophy flux we get

$\displaystyle\frac{kE_{G}(k)}{\tau_{E}(k)}=\epsilon=const.,\quad\tau_{E}(k)=[k^{5-\alpha}E_{G}(k)]^{-1/2}$
(10)
$\displaystyle\frac{k\Omega_{G}(k)}{\tau_{\Omega}(k)}=\xi=const.,\quad\tau_{\Omega}(k)=[k^{5-2\alpha}\Omega_{G}(k)]^{-1/2}$
(11)

where $\tau_{E}(k)$ is the local timescale that takes $E_{G}$ to be
transferred across wavenumbers. Then, by combining (10) and (11) with (9) we
obtain the following power laws that depend on the value of $\alpha$

$\displaystyle E_{G}(k)\propto\epsilon^{2/3}k^{(\alpha-7)/3},$
$\displaystyle\Omega_{G}(k)\propto\epsilon^{2/3}k^{(4\alpha-7)/3},$ (12)
$\displaystyle E_{G}(k)\propto\xi^{2/3}k^{(-\alpha-7)/3},$
$\displaystyle\Omega_{G}(k)\propto\xi^{2/3}k^{(2\alpha-7)/3},$ (13)

for wavenumbers in the inertial ranges above and below the intermediate
forcing wavenumber $k_{f}$.

The flux $\Pi$ is a measure of the nonlinear cascade of a conserved quantity
in turbulence (Alexakis & Biferale, 2018). The energy flux for a circle of
radius $k$ in the 2D wavenumber space is the total energy transferred from the
modes within the circle to the modes outside the circle. Consequently, we
define the flux of generalised energy $\Pi_{E}(k,t)$ and enstrophy
$\Pi_{\Omega}(k,t)$ as

$\displaystyle\Pi_{E}(k,t)$ $\displaystyle=\sum_{k^{\prime}\leq
k}T_{E}(k^{\prime},t),$ (14a) $\displaystyle\Pi_{\Omega}(k,t)$
$\displaystyle=\sum_{k^{\prime}\leq k}T_{\Omega}(k^{\prime},t),$ (14b)

where $T_{E}(k,t)$ and $T_{\Omega}(k,t)$ are the non-linear generalised energy
and enstrophy transfers across $k$

$\displaystyle T_{E}(k,t)$ $\displaystyle=\sum_{k\leq|{\bf k}|<k+\Delta
k}\widehat{\psi}^{*}({\bf k},t)\widehat{\mathcal{J}(\psi,q)}({\bf k},t),$
(15a) $\displaystyle T_{\Omega}(k,t)$ $\displaystyle=\sum_{k\leq|{\bf
k}|<k+\Delta k}\widehat{q}^{*}({\bf k},t)\widehat{\mathcal{J}(\psi,q)}({\bf
k},t),$ (15b)

where the sum is performed over the Fourier modes with wavenumber amplitude
$k$ in a shell of width $\Delta k=2\pi/L$.

In general we cannot determine the direction of cascade of the two invariants
unless there are special relations between the two like Eq. (9). A generalised
Fjørtoft (1953) argument on the directions of the cascades of two quadratic
invariants has been reformulated by Alexakis & Biferale (2022), which
essentially states that

$\text{if}\;|E_{G}(k)|\leq
ck^{-\alpha}\Omega_{G}(k)\;\text{with}\;c>0\;\text{and}\;\alpha>0\;\text{then}\;E_{G}\;\text{cannot
cascade forward},$ (16)

and

$\text{if}\;\Omega_{G}(k)\leq
ck^{\alpha}|E_{G}(k)|\;\text{with}\;c>0\;\text{and}\;\alpha>0\;\text{then}\;\Omega_{G}\;\text{cannot
cascade inversely}.$ (17)

In other words, when $\alpha>0$ then $E_{G}$ is transferred towards large
scales while $\Omega_{G}$ is transferred towards small scales and the opposite
is true when $\alpha<0$. In the next section we demonstrate this rigorously
using mathematical inequalities on the injection and dissipation rates of the
generalised energy and enstrophy.

##  3 Bounds

Let us start with Eq. (1) and consider the case of small scale dissipation
with the power of the Laplace operator $n=1$ and the large scale diffusion
coefficient $\nu_{-}=0$. We then have

$\partial_{t}q+\mathcal{J}(\psi,q)=\nu_{+}\nabla^{2}q+f_{q},$ (18)

where $f_{q}$ is time independent. The corresponding streamfunction equation
can be found using Eq. (2), i.e.

$\partial_{t}\psi+\mathcal{J^{\prime}}(\psi,q)=\nu_{+}\nabla^{2}\psi+f_{\psi},$
(19)

where
$\mathcal{J}^{\prime}(\psi,q)=-(-\nabla^{2})^{-\alpha/2}\mathcal{J}(\psi,q)$.
Using the rms velocity
$U=\langle{|\mathbb{u}|^{2}}\rangle^{1/2}=\langle{|{\nabla}\psi|^{2}}\rangle^{1/2}$,
we define the dimensionless measures of generalized energy and enstrophy
dissipation as:

$c_{\epsilon}=\frac{\epsilon_{+}}{U^{3}k_{f}^{\alpha-1}},\quad\quad
c_{\xi}=\frac{\xi_{+}}{U^{3}k_{f}^{2\alpha-1}}.$ (20)

The Reynolds number based on the rms velocity field is defined as

$\real=\frac{U}{k_{f}\nu_{+}},$ (21)

where the forcing wavenumber
$k_{f}=\langle{|\nabla^{2}f_{\psi}|^{2}}\rangle^{1/2}/\langle{|f_{\psi}|^{2}}\rangle^{1/2}$.
Now let us consider separately the cases for positive and negative $\alpha$.

###  3.1 Positive $\alpha$

In statistically stationary state, we have the balance between the dissipation
and the injection of generalised enstrophy and so from Eqn. (7) we have
$\xi_{+}=\nu_{+}\langle|\nabla q|^{2}\rangle=\langle qf_{q}\rangle$. Using
equation (8) we can write $\xi_{+}=\langle qf_{q}\rangle\leq
k_{f}^{2\alpha}\langle\psi f_{\psi}\rangle$ and with the Cauchy-Schwarz
inequality we get

$\displaystyle\xi_{+}\leq
k_{f}^{2\alpha-1}\langle|\psi|^{2}\rangle^{1/2}\langle|f_{\psi}|^{2}\rangle^{1/2}=k_{f}^{2\alpha-1}UF_{\psi},$
(22)

where $F_{\psi}=\langle|f_{\psi}|^{2}\rangle^{1/2}$ and

$\langle|\psi|^{2}\rangle^{1/2}\leq
Ck_{f}^{-1}\langle{|\mathbb{u}|^{2}}\rangle^{1/2}=Ck_{f}^{-1}U,$ (23)

using the Poincare inequality as $\psi$ is a lower derivative than the
velocity field ${\bf u}$, where $C$ is a constant that depends on the domain
geometry. Thus, Eq. (22) yields a bound on the generalised enstrophy
dissipation rate at the small scales in terms of forcing.

Now we multiply Eq. (19) by a smoothly varying doubly differentiable function
$\phi$,

$\displaystyle\phi\partial_{t}\psi+\phi\mathcal{J^{\prime}}(\psi,q)$
$\displaystyle=\nu_{+}\phi\nabla^{2}\psi+\phi f_{\psi}.$

Performing a spatio-temporal averaging and changing the order of
differentiation during the integration by parts, we get

$\langle(\mathbb{u}\cdot\nabla\phi)\psi\rangle=\nu_{+}\langle\psi\nabla^{2}\phi\rangle+\langle{\phi
f_{\psi}}\rangle.$ (24)

For the left hand side term of (24), we use the Hölder’s and Cauchy-Schwartz
inequalities to get

$\displaystyle\langle(\mathbb{u}\cdot\nabla\phi)\psi\rangle$
$\displaystyle\leq\|\nabla\phi\|_{\infty}\langle{|\psi|^{2}}\rangle^{1/2}\langle{|{\bf
u}|^{2}}\rangle^{1/2}$ $\displaystyle\leq
C\|\nabla\phi\|_{\infty}k_{f}^{-1}U^{2}.$ (25)

Similarly for the first term on the right hand side of (24), we can write

$\nu_{+}\langle\psi\nabla^{2}\phi\rangle\leq\nu_{+}\langle{|\psi|^{2}}\rangle^{1/2}\langle{|\nabla^{2}\phi|^{2}}\rangle^{1/2}.$
(26)

Putting equations (25) and (26) back in (24), we get

$\langle\phi f_{\psi}\rangle\leq
C||\nabla\phi||_{\infty}k_{f}^{-1}U^{2}+\nu_{+}\langle{|\psi|^{2}}\rangle^{1/2}\langle{|\nabla^{2}\phi|^{2}}\rangle^{1/2}.$
(27)

Now, by letting $\phi=f_{\psi}/F_{\psi}$ we can simplify the expression and
obtain a bound on the forcing, namely

$F_{\psi}\leq C_{1}U^{2}+C_{2}\nu_{+}k_{f}U,$ (28)

where $C_{1},C_{2}$ are constants that depend on the form of the forcing
function and the domain geometry. Substituting (28) in the expression for the
generalised enstrophy dissipation rate (22) we obtain the following bound,

$\displaystyle\xi_{+}$ $\displaystyle\leq
k_{f}^{2\alpha-1}U(C_{1}U^{2}+C_{2}\nu_{+}k_{f}U),$ (29)

which in non-dimensional form can be written as,

$\displaystyle c_{\xi}\leq C_{1}+C_{2}Re^{-1},$ (30)

dividing by $k_{f}^{2\alpha-1}U^{3}$ and using the definitions (20) and (21).
So, as $Re\rightarrow\infty$ the generalised enstrophy dissipation rate
$c_{\xi}\rightarrow$ constant, which is independent of the viscosity $\nu_{+}$
and amplitude of the velocity field $U$. This result is similar to 2D Navier-
Stokes turbulence (i.e. $\alpha=2$), where the enstrophy dissipation rate goes
to a constant as $Re\rightarrow\infty$. Thus, for any $\alpha>0$ we find that
a generalised 2D turbulent flow will lead to a constant $c_{\xi}$ as
$Re\rightarrow\infty$.

We then look to bound the generalised energy dissipation rate given by
$\epsilon_{+}=\nu_{+}\langle\psi\nabla^{2}q\rangle$. Using the Cauchy-Schwarz
inequality, we get

$\displaystyle\epsilon_{+}$
$\displaystyle\leq\nu_{+}\langle|\nabla\psi|^{2}\rangle^{1/2}\langle|\nabla
q|^{2}\rangle^{1/2}$ $\displaystyle\leq\nu_{+}^{1/2}U\xi_{+}^{1/2}.$ (31)

Using the bound for $\xi_{+}$ from Eq. (29), this inequality simplifies to

$\displaystyle c_{\epsilon}\leq Re^{-1/2}(C_{1}+C_{2}Re^{-1})^{1/2},$ (32)

dividing by $U^{3}k_{f}^{\alpha-1}$ and using the definitions (20) and (21).
Thus, $c_{\epsilon}\rightarrow 0$ as $\real\rightarrow\infty$, implying that
there is no forward cascade of generalised energy for $\alpha>0$ in the limit
of infinite Reynolds number.

###  3.2 Negative $\alpha$

In a similar fashion to positive $\alpha$, in statistically stationary state
we have the balance between the dissipation and the injection rate of
generalised energy. So from Eq. (6) we have
$\epsilon_{+}=\nu_{+}\langle{\psi\nabla^{2}q}\rangle=\langle{\psi
f_{q}}\rangle$. Using the Cauchy-Schwarz and Poincare inequalities we get

$\displaystyle\epsilon_{+}$
$\displaystyle\leq\langle{|\psi|^{2}}\rangle^{1/2}\langle{|f_{q}|^{2}}\rangle^{1/2}$
$\displaystyle\leq Ck_{f}^{-1}UF_{q},$ (33)

where $F_{q}=\langle{|f_{q}|^{2}}\rangle^{1/2}$. Now, we multiply Eq. (18) by
a smoothly varying doubly differentiable function $\phi$,

$\phi\partial_{t}q+\phi\mathcal{J}(\psi,q)=\nu_{+}\phi\nabla^{2}q+\phi f_{q}.$
(34)

By performing spatiotemporal averaging and changing the order of
differentiation during the integration by parts, we get

$\langle{(\mathbb{u}\cdot\nabla\phi)q}\rangle=\nu_{+}\langle{q\nabla^{2}\phi}\rangle+\langle{\phi
f_{q}}\rangle$ (35)

For the left hand side term of (35), we use the Hölder’s and Cauchy-Schwartz
inequalities to get

$\displaystyle\langle{(\mathbb{u}\cdot\nabla\phi)q}\rangle$
$\displaystyle\leq||\nabla\phi||_{\infty}\langle{|q|^{2}}\rangle^{1/2}\langle{|\mathbb{u}|^{2}}\rangle^{1/2}$
$\displaystyle\leq\widetilde{C}||\nabla\phi||_{\infty}k_{f}^{\alpha-1}U^{2},$
(36)

using also the Poincare inequality as $q$ is a lower derivative of the
velocity field $\mathbb{u}$ for $\alpha<0$ and so

$\langle{|q|^{2}}\rangle^{1/2}\leq\widetilde{C}k_{f}^{\alpha-1}\langle{|\mathbb{u}|^{2}}\rangle^{1/2}=\widetilde{C}k_{f}^{\alpha-1}U,$
(37)

where $\widetilde{C}$ is a constant that depends on the domain geometry.
Similarly for the first term of the right hand side of Eq. (35), we can write

$\displaystyle\nu_{+}\langle q\nabla^{2}\phi\rangle$
$\displaystyle\leq\nu_{+}\langle{|q|^{2}}\rangle^{1/2}\langle{|\nabla^{2}\phi|^{2}}\rangle^{1/2}$
$\displaystyle\leq\widetilde{C}\nu_{+}k_{f}^{\alpha-1}U\langle{|\nabla^{2}\phi|^{2}}\rangle^{1/2}$
(38)

Putting equations (36) and (38) back in (35), we get

$\langle\phi
f_{q}\rangle\leq||\nabla\phi||_{\infty}k_{f}^{\alpha-1}U^{2}+\widetilde{C}\nu_{+}k_{f}^{\alpha-1}U\langle{|\nabla^{2}\phi|^{2}}\rangle^{1/2}$
(39)

Now, by letting $\phi=f_{q}/F_{q}$ we can simplify the expression and obtain a
bound on the forcing, namely

$F_{q}\leq k_{f}^{\alpha}(C_{3}U^{2}+C_{4}\nu_{+}k_{f}U),$ (40)

where $C_{3}$, $C_{4}$ are constants that depend on the form of the forcing
function and the domain geometry. Substituting (40) in the expression for the
generalised energy dissipation rate (33) we obtain the following bound

$\displaystyle\epsilon_{+}$ $\displaystyle\leq
k_{f}^{\alpha-1}U(C_{3}U^{2}+C_{4}\nu_{+}k_{f}U)$ (41)

which in non-dimensional form can be written as,

$\displaystyle c_{\epsilon}$ $\displaystyle\leq C_{3}+C_{4}Re^{-1}$ (42)

dividing by $k_{f}^{\alpha-1}U^{3}$ and using the definitions (20) and (21).
Thus, as $Re\rightarrow\infty$ we find that a generalised 2D turbulent flow
will lead to a constant generalised energy dissipation rate $c_{\epsilon}$,
independent of the viscosity $\nu_{+}$ and amplitude of the velocity field $U$
for any $\alpha<0$.

We then look to bound the generalised enstrophy dissipation rate
$\xi_{+}=\nu_{+}\langle{|\nabla q|^{2}}\rangle$ which can be written in the
spectral space as

$\displaystyle\xi_{+}$ $\displaystyle=\nu_{+}\sum_{\bf
k}k^{1+\frac{\alpha}{2}}k^{1-\frac{\alpha}{2}}|\widehat{q}(\mathbb{k},t)|^{2}.$
(43)

Using the Cauchy-Schwarz inequality we get the expression

$\displaystyle\xi_{+}$ $\displaystyle\leq\nu_{+}\left(\sum_{\bf
k}k^{2+\alpha}|\widehat{q}(\mathbb{k},t)|^{2}\right)^{\frac{1}{2}}\left(\sum_{\bf
k}k^{2-\alpha}|\widehat{q}(\mathbb{k},t)|^{2}\right)^{\frac{1}{2}}.$ (44)

For the first term in the brackets we can use the Poincare inequality to get

$\displaystyle\left(\sum_{\bf
k}k^{2+\alpha}|\widehat{q}(\mathbb{k},t)|^{2}\right)^{\frac{1}{2}}=\left(\sum_{\bf
k}k^{2+3\alpha}|\widehat{\psi}(\mathbb{k},t)|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\leq C_{5}k_{f}^{3\alpha/2}U,$ (45)

where $C_{5}$ is a constant that depends on the form of the forcing function
and the domain geometry. Now, the second term in the brackets can be written
as

$\displaystyle\left(\sum_{\bf
k}k^{2-\alpha}|\widehat{q}(\mathbb{k},t)|^{2}\right)^{\frac{1}{2}}=\langle{q\nabla^{2}\psi}\rangle^{1/2}=(\epsilon_{+}/\nu_{+})^{1/2}.$
(46)

Putting equations (45) and (46) back in (44), we get

$\displaystyle\xi_{+}$ $\displaystyle\leq
C_{5}\nu_{+}^{1/2}k_{f}^{3\alpha/2}U\epsilon_{+}^{1/2}.$ (47)

Using the bound for $\epsilon_{+}$ from Eq. (41), this inequality simplifies
to

$\displaystyle c_{\xi}\leq Re^{-1/2}(C_{1}+C_{2}Re^{-1})^{1/2}$ (48)

dividing by $U^{3}k_{f}^{2\alpha-1}$ and using the definitions (20) and (21).
Thus, $c_{\xi}\to 0$ as $\real\to\infty$, implying that there is no forward
cascade of generalised enstrophy for $\alpha<0$ in the limit of infinite
Reynolds number.

In summary, the bounds we derived for positive and negative $\alpha$
demonstrate the transition of the cascades between the generalised energy and
enstrophy. In other words, for $\alpha>0$ the bounds suggest that in the limit
of infinite Reynolds number $\epsilon_{+}$ tends to zero due to the inverse
cascade, and $\xi_{+}$ remains finite due to the dissipation anomaly of the
forward cascade, while the opposite is true for $\alpha<0$.

##  4 Numerical Results

###  4.1 Transition of generalised energy and enstrophy cascades

To investigate the transition of the cascades, we numerically simulate the
generalised vorticity equations (1)-(2) in the limit of large Reynolds numbers
by varying $\alpha$ systematically. The details of the numerical method and
simulation parameters used are mentioned in Appendix A. We quantify the
variation in the generalized energy and enstrophy fluxes to large scales using
the large scale dissipation rates $\epsilon_{-}$ and $\xi_{-}$ defined in Eqs.
(6), (7). For the parameters explored, $\epsilon_{-}$ and $\xi_{-}$ are
localised at large length scales giving an estimate of the flux to large
scales occurring through an inverse cascade or transfer.

(a)

(b)

Figure 1:  Normalised large scale dissipation rate of generalised energy
$\epsilon_{-}/\epsilon$ and enstrophy $\xi_{-}/\xi$ as a function of $\alpha$.
Plot a) shows the bifurcation diagram for fixed $Re_{+}=1381$ and increasing
values of $\real_{-}$, where $Re_{-}^{(1)}=10^{6}$, $Re_{-}^{(2)}=4\times
10^{7}$ and $Re_{-}^{(3)}=2\times 10^{9}$. Plot b) shows the bifurcation
diagram for fixed $Re_{-}=4\times 10^{7}$ and increasing value of $Re_{+}$,
where $Re_{+}^{(1)}=345$, $Re_{+}^{(2)}=1381$ and $Re_{+}^{(3)}=3450$.

In Fig. 1 we show the large scale dissipation rates normalised by their
respective injection rates, $\epsilon_{-}/\epsilon$ and $\xi_{-}/\xi$, as a
function of $\alpha$. In Fig. 1(a) we consider different values of $Re_{-}$
with fixed $Re_{+}$, while in Fig. 1(b) we consider different values of
$Re_{+}$ with fixed $Re_{-}$. We find that for $\alpha=2$ the energy
dissipation is predominantly at large length scales due to the inverse cascade
of energy in 2D Navier-Stokes turbulence. On the other hand, for $\alpha=-1$
we observe the transition of the cascades with the energy to be predominantly
dissipated at small scales suggesting a forward cascade of energy. For
intermediate values of $\alpha$ we find that energy is dissipated at both
large and small scales implying the presence of a dual cascade.

As seen from Fig. 1(a), for a given $Re_{+}$ increasing $Re_{-}$ leads to
smaller inverse cascade of both the generalised energy and enstrophy. In
Figure 1(b) for a fixed $Re_{-}$ increasing $Re_{+}$ leads to smaller forward
cascade of both the generalised energy and enstrophy. Thus, we find that the
behaviour close to $\alpha=0$ depends strongly on the control parameters
$\real_{+}$ and $\real_{-}$ as the nonlinearity, which leads to the cascade,
vanishes as $\alpha\rightarrow 0$. At $\alpha=0$ there is no cascade as the
nonlinearity vanishes, i.e. $\mathcal{J}(\psi,\psi)=0$ by definition. This
leads both the forcing and dissipation to be localised at the wavenumber
$k_{f}$. We refer to $\alpha=0$ as the threshold where the cascades vanish for
any value of the dissipation coefficients. For the simulation parameters that
are explored here, the small scale dissipation dominates over the large scale
dissipation near $\alpha\approx 0$ implying that the dissipation due to large
scale friction is negligible, namely $\epsilon_{-}\ll 1$. The exact behaviour
close to $\alpha=0$ will be discussed in section 4.2.

In Fig. 2 we show the generalised energy spectrum for different values of
$\alpha$. The dashed lines show the phenomenological predictions from Eqs.
(12), (13). The red dashed lines show the prediction for $\alpha=2$ and the
blue dashed lines show the prediction for $\alpha=-1$. We see that the
generalised energy spectra in the range $k_{min}<k<k_{f}$ have exponents close
to the predictions (12), while in the range $k_{f}<k<k_{max}$ the exponents
are far from the phenomenological predictions. This is also seen in 2D Navier-
Stokes turbulence with $\alpha=2$, where the exponent of the range
$k_{f}<k<k_{max}$ depends on the small scale Reynolds number (Boffetta & Ecke,
2012). Simulations at very high resolutions are required to shed light on the
differences in the spectral exponents for the different values of $\alpha$ and
the validity of Kolmogorov type arguments to determine the exponents.

Figure 2:  Energy spectra for different $\alpha$ along with their theoretical
exponents in red ($\alpha=2$) and blue ($\alpha=-1$) are shown. The spectra
are obtained from DNS with $Re_{+}=1381$, $Re_{-}=4\times 10^{7}$ and
$k_{f}=16\sqrt{2}$.

Figure 3 shows the fluxes of generalised energy and enstrophy, normalized with
their respective injection rates, for different values of $\alpha$. For the
Navier-Stokes model ($\alpha=2$) we see that the energy cascades mostly to
large scales (or low $k$) while the enstrophy cascades mostly to small scales
(or high $k$). For the $\alpha=-1$ we find the opposite scenario where the
energy mostly cascades to small scales while the generalised enstrophy
cascades to large scales. Intermediate values of $\alpha$ shows dual cascade
where the generalised energy and enstrophy cascade to both large and small
scales. This is similar to the cascade transitions seen in other turbulent
systems such as in Celani et al. (2010), Seshasayanan et al. (2014b),
Benavides & Alexakis (2017). Close to $\alpha=0$ we find that the cascade
vanishes with energy injection and dissipation occurring at the same length
scale.

(a)

(b)

Figure 3:  a) Normalized energy flux and b) normalised enstrophy flux for
different $\alpha$. The fluxes are obtained from DNS with $Re_{+}=1381$,
$Re_{-}=4\times 10^{7}$ and $k_{f}=16\sqrt{2}$.

The top row of Fig. 4 shows the generalised vorticity distribution for
different values of $\alpha$. For $\alpha=2$, we get the two-dimensional large
scale vortices which are generated by the inverse cascade of energy. As
$\alpha$ is reduced we see that the vortices become diffused with the sizes of
the vortices becoming large. For $\alpha=1$, Carton et al. (2016) found that
the distance of vortex merging was generally smaller than for $\alpha=2$. It
was explained that for a point vortex of the form $q(r)=\delta(r)$, the
velocity field decreases like $1/r^{2}$ for the case SQG model ($\alpha=1$)
while for the standard Navier-Stokes model ($\alpha=2$) the velocity reduces
like $1/r$ from the core of the vortices. Hence, the region of influence of
the vortices decreases as we reduce $\alpha$. To distinguish the behaviour of
the classical vorticity from the generalised vorticity, we show in the bottom
row of Fig. 4 the contour plot of the classical vorticity defined as
$\omega=\nabla^{2}\psi$ for all values of $\alpha$. As seen from the figures
the classical vorticity goes to smaller length scales as $\alpha$ is reduced.
For $\alpha<0$ we observe a few vortices along with many filamentary
structures that resemble the thin vortex filaments observed in 3D homogeneous
isotropic turbulence.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 4:  Generalised vorticity field for a) $\alpha=2.0$ b) $\alpha=1.0$ c)
$\alpha=-0.5$ d) $\alpha=-1.0$. The corresponding classical vorticity fields
are shown in e) $\alpha=2.0$ f) $\alpha=1.0$ g) $\alpha=-0.5$ and h)
$\alpha=-1.0$. For $\alpha=2,1$ and $-0.5$ the colorbar has been capped at
$50\%$ of the maximum value and at $10\%$ for $\alpha=-1$ to emphasise the
structures of the fields. The fields are obtained from DNS with $Re_{+}=1381$,
$Re_{-}=4\times 10^{7}$ and $k_{f}=16\sqrt{2}$.

###  4.2 Near the threshold $\alpha=0$

We investigate the behaviour of the system close to the threshold $\alpha=0$.
At $\alpha=0$ the nonlinearity vanishes and the flow becomes laminar. As one
moves away from the threshold, the flow remains laminar until it undergoes a
linear instability. Here the instability is found to occur at smaller length
scales than the laminar flow, unlike the negative viscosity instability that
is observed in many 2D flows. The threshold of the instability is found to be
the same for both positive and negative values of $\alpha$. For simplicity we
report on the critical point of the instability $\alpha_{c}$ as we increase
$\alpha$ away from zero to positive values. In Fig. 5 we show the critical
point of the instability $\alpha_{c}$ as a function of $\real_{-}$ and
$\real_{+}$. For the parameters of $\real_{+}=1381$, $\real_{-}=4\times
10^{7}$ at $\alpha=0$ the large scale friction is negligible, thus we find
that the instability threshold $\alpha_{c}$ is almost independent of
$\real_{-}$ (see Fig. 5(a)). We also find that the threshold decreases like a
power law as $\real_{+}$ increases (see Fig. 5(b)), thus neglecting the effect
of large scale friction we expect that $\alpha_{c}\rightarrow 0$ as
$\real_{+}\rightarrow\infty$. The dependence on $\real_{+}$ indicates that the
behaviour very close to the critical point is sensitive to the dissipation
coefficients.

(a)

(b)

Figure 5:  Figure shows the variation of $\alpha_{c}$ as a function of a)
$\real_{-}$ and b) $\real_{+}$. The dashed line in figure b) denotes the
scaling $\alpha_{c}\propto\real^{-2}_{+}$.

The scaling of $\alpha_{c}$ with $\real_{+}$ can be found by doing a
perturbation around $\alpha=0$. We write the fields as a sum of the base flow
and a perturbation, $q=q_{b}+\widetilde{q},\psi=\psi_{b}+\widetilde{\psi}$,
where the terms with the subscript b denote the base flow component and the
terms with $\widetilde{\;}$ denote the perturbation fields. Substituting this
decomposition into the nonlinear term of equation (1) we get,
$\mathcal{J}(\psi,q)=\mathcal{J}(\psi_{b},q_{b})+\mathcal{J}(\widetilde{\psi},q_{b})+\mathcal{J}(\psi_{b},\widetilde{q})+\mathcal{J}(\widetilde{\psi},\widetilde{q})$.
The term $\mathcal{J}(\psi_{b},q_{b})$ is zero for the laminar solution and
the term $\mathcal{J}(\widetilde{\psi},\widetilde{q})$ can be neglected as we
are looking at the linear onset of the instability. Thus we get,

$\mathcal{J}(\psi,q)\approx\mathcal{J}(\widetilde{\psi},q_{b})+\mathcal{J}(\psi_{b},\widetilde{q}).$
(49)

We express the terms in the Fourier space using the relation (8) and then
expand the above expression in powers of $\alpha$. In the limit of
$\alpha\rightarrow 0$ we can expand
$k^{\alpha}=k_{f}^{\alpha}(1+\alpha\log(\frac{k}{k_{f}})+O(\alpha^{2}))$.
Then, equation (49) at leading order in $\alpha$ will take the form

$\mathcal{J}(\psi,q)\approx\alpha\left[\mathcal{J}(\widetilde{\psi},\phi_{b})+\mathcal{J}(\psi_{b},\widetilde{\phi})\right]+O(\alpha^{2}),$
(50)

where the Fourier coefficients of the fields $\phi_{b}$ and $\widetilde{\phi}$
are defined as

$\phi_{b}({\bf k},t)=-k_{f}^{\alpha}\log(\frac{k}{k_{f}})\psi_{b}({\bf
k},t),\quad\quad\widetilde{\phi}({\bf
k},t)=-k_{f}^{\alpha}\log(\frac{k}{k_{f}})\psi({\bf k},t).$ (51)

Using Eq. (50) we can derive a scaling of $\alpha_{c}$ with $Re_{+}$ from the
balance between the non-linear term $J(\psi,q)$ and the dissipation term
$\nu_{+}\nabla^{2n}\widetilde{\psi}$ where $n=2$. At the threshold
$\alpha_{c}$ of the linear instability we use Eqs. (50) and (51) to find the
following leading order scaling

$\displaystyle J(\psi,q)$
$\displaystyle\sim\alpha_{c}\frac{f_{0}}{\nu_{+}k_{f}^{2}}\;g_{1}\left(\frac{k_{i}}{k_{f}}\right)\widetilde{\psi},$
(52) $\displaystyle\nu_{+}\nabla^{4}\widetilde{\psi}$
$\displaystyle\sim\nu_{+}k_{f}^{4}\;g_{2}\left(\frac{k_{i}}{k_{f}}\right)\widetilde{\psi},$
(53)

where $k_{i}$ denotes the typical wavenumber of the unstable mode
$\widetilde{\psi}$ and $g_{1},g_{2}$ are two functions of $k_{i}/k_{f}$.
Equating (52) with (53) we get,

$\alpha_{c}\frac{f_{0}}{\nu_{+}k_{f}^{2}}\sim\nu_{+}k_{f}^{4}\;g_{3}\left(\frac{k_{i}}{k_{f}}\right),$
(54)

where $g_{3}$ is a function of $k_{i}/k_{f}$. We find that the ratio
$k_{i}/k_{f}$ remains almost constant as we vary $\real_{+}$ over a few
decades, which leads to the following scaling law for the threshold of the
instability

$\alpha_{c}\sim Re_{+}^{-2}.$ (55)

This scaling law is denoted by the dashed line in Fig. 5(b) and has a good
agreement with the thresholds found numerically for different $Re_{+}$.

###  4.3 Locality of cascades - $\alpha$ dependence

We quantify the locality of the nonlinear triadic interactions by analysing
the transfer rate of generalised energy transferred via the generalised
vorticity advection term from one shell of wavenumbers $Q<k<Q+\Delta k$ to
another shell of wavenumbers $K<k<K+\Delta k$. We define this shell to shell
transfer function as

$T_{E}(K,Q,t)=\langle{\psi_{K}(\mathbb{u}\cdot\nabla q_{Q})}\rangle,$ (56)

where $\psi_{K}$ and $q_{Q}$ are the streafunction and generalised vorticity
field filtered such that only the wavenumbers at shell $K$ and $Q$ are kept,
respectively. The transfer term $T_{E}(K,Q,t)$ conserves the generalised
energy, i.e. it does not generate or destroy $E_{G}$ but it is responsible for
the redistribution of the generalised energy across different scales. This is
expressed by the fact that $T_{E}(K,Q,t)$ is antisymmetric under the exchange
of $K$ and $Q$, i.e. $T_{E}(K,Q,t)=-T_{E}(Q,K,t)$. Similarly, this is true for
the transfer rate of generalised enstrophy from the mode $K$ into the mode
$Q$, which is defined as

$T_{\Omega}(K,Q,t)=\langle{q_{K}(\mathbb{u}\cdot\nabla q_{Q})}\rangle.$ (57)

The contour plots in the top row of Fig. 6 show in logarithmic scale the
absolute value of the time averaged shell to shell transfer function
$T_{E}(K,Q)$ normalised by the energy injection rate $\epsilon$ for a)
$\alpha=2.0$ b) $\alpha=1.0$, c) $\alpha=-0.5$ and d) $\alpha=-1.0$.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 6:  The top row shows shell-to-shell transfers of generalised energy as
$\log(|T_{E}(K,Q)/\epsilon|)$ for (a) $\alpha=2.0$, (b) $\alpha=1.0$, (c)
$\alpha=-0.5$, and (d) $\alpha=-1.0$. The bottom row shows shell-to-shell
transfers of generalised enstrophy as $\log(|T_{\Omega}(K,Q)/\xi|)$ for (e)
$\alpha=2.0$, (f) $\alpha=1.0$, (g) $\alpha=-0.5$, and (h) $\alpha=-1.0$. The
colorbars are the same for all the plots and are shown only in plots d) and h)
for clarity. The shell-to-shell transfers are obtained from DNS with
$Re_{+}=1381$, $Re_{-}=10^{6}$ and $k_{f}=8\sqrt{2}$.

For $\alpha=2$ the shell to shell transfer $T_{E}(K,Q)$ is mostly concentrated
at $K/k_{f}<1$, $Q/k_{f}<1$ and close to the diagonal $K=Q$, indicating that
generalised energy cascades to large scales and mostly via local interactions.
For values of $\alpha<2$, we observe that the transfer becomes more
significant at $K/k_{f}>1$ and $Q/k_{f}>1$ and this is because the cascade
transitions more and more to small scales. Moreover, for $\alpha=1$ and -0.5
the shell to shell transfer $T_{E}(K,Q)$ happens not only locally close to
$K=Q$ but there are also two non-local branches, which become more and more
significant as $\alpha\to-0.5$. These are the vertical branch, which is at the
wavenumber range $K/k_{f}<1$ along the wavenumbers $Q/k_{f}$ and the
horizontal branch, which is at the wavenumber range $Q/k_{f}<1$ along the
wavenumbers $K/k_{f}$. At $\alpha=-1$, we find that $T_{E}(K,Q)$ is fully non-
local spanning the whole range of wavenumbers.

In a similar fashion the contour plots in the bottom row of Fig. 6 show in
logarithmic scale the absolute value of the time averaged shell to shell
transfer function $T_{\Omega}(K,Q)$ normalised by the energy injection rate
$\xi$ for e) $\alpha=2.0$ f) $\alpha=1.0$, g) $\alpha=-0.5$ and h)
$\alpha=-1.0$. For $\alpha=2$ the shell to shell transfer $T_{\Omega}(K,Q)$
occurs predominantly at $K/k_{f}>1$, $Q/k_{f}>1$ and close to the diagonal
$K=Q$, indicating that generalised enstrophy cascades to small scales though
local interactions in wavenumber. For values of $\alpha<2$, we observe that
$T_{\Omega}(K,Q)$ turns gradually to low wavenumber, i.e. at $K/k_{f}<1$ and
$Q/k_{f}<1$ and this happens since the cascade transitions to large scales. In
addition, for $\alpha=1$ and -0.5 the shell to shell transfer happens via
three branches. One local branch at $K=Q$ and two non-local branches, which
become more and more significant as $\alpha\to-0.5$. Again these branches are
the vertical and horizontal branches that have similar pattern to the non-
local branches of $T_{E}(K,Q)$. Finally, at $\alpha=-1$, we also find that
$T_{\Omega}(K,Q)$ is fully non-local with a wide spread across the
wavenumbers.

The fact that the cascades of the generalised energy and enstrophy become
gradually non-local as we go from positive to negative values of $\alpha$ is a
notion clearly at variance with the typical locality assumption of the
Kolmogorov phenomenology. This might a reason why the energy spectra in Fig. 2
do not agree with the scaling predictions from Eqs. (12) and (13), which are
drawn in the plot for comparison.

##  5 Conclusion

In this paper we focus on the theoretical and numerical analysis of the
cascades in generalised 2D turbulence by systematically varying the parameter
$\alpha$ of the model. The value and sign of $\alpha$ determines if the
cascade would be inverse, forward or bidirectional. This suggests that there
is some critical $\alpha_{c}$ at which a transition happens in the direction
of the cascades. At the threshold $\alpha=0$ the nonlinear term vanishes and
the flow is laminar. We find that as we move away from $\alpha=0$ the laminar
flow undergoes a linear small wavelength instability at a critical value
$\alpha_{c}$ which is the same both for positive and negative $\alpha$. We
observe that $\alpha_{c}\approx const.$ as we increase $Re_{-}$, however
$\alpha\propto Re_{+}^{-2}$ as $Re_{+}$ increases. This scaling is also
verified theoretically by doing a perturbation analysis.

For $\alpha>0$ using mathematical inequalities we are able to bound the
dimensionless dissipation rates of generalised energy
$c_{\epsilon}\equiv\epsilon/(U^{3}k_{f}^{\alpha-1})\leq
Re^{-1/2}(C_{1}+C_{2}Re^{-1})^{1/2}$ and enstrophy
$c_{\xi}\equiv\xi/(U^{3}k_{f}^{2\alpha-1})\leq C_{1}+C_{2}Re^{-1}$. These
bounds state that as $Re\to\infty$ then $c_{\epsilon}$ tends to zero due to
the inverse cascade, while $c_{\xi}$ remains finite due to the dissipation
anomaly of the forward cascade. On the other hand, for $\alpha<0$ we obtain
the bounds $c_{\epsilon}\leq C_{1}+C_{2}Re^{-1}$ and $c_{\xi}\leq
Re^{-1/2}(C_{1}+C_{2}Re^{-1})^{1/2}$. This behaviour is due to the transition
of the cascades, where $c_{\epsilon}$ remains finite due to the dissipation
anomaly of the forward cascade, while $c_{\xi}$ tends to zero due to the
inverse cascade.

By varying $\alpha$ in numerical simulations we confirm this picture; meaning
that when $\alpha>0$ then generalised energy $E_{G}$ cascades inversely while
generalised enstrophy $\Omega_{G}$ cascades forward and the opposite happens
when $\alpha<0$ (see Fig. 1). Moreover, we find that the amount of dissipation
rates of $E_{G}$ and $\Omega_{G}$ depend on the system parameters $Re_{-}$ and
$Re_{+}$. This transition from positive to negative alpha is also clear from
the spectral fluxes (see Fig. 3), which determine the spectral exponents of
the energy spectra. The energy spectra at large scales, i.e. at length scales
larger than the forcing length scale, seem to agree with Kolmogorov type
scalings of the generalised 2D turbulence, however, this is not true for
length scales smaller than the forcing length scale. A reason of this
discrepancy might be that higher resolution computations are required to shed
light or the nonlocal transfers of generalised energy and enstrophy might be
the cause of this disagreement. Finally, it is interesting to mention that due
to the forward cascade of $E_{G}$ for $\alpha<0$ the classical vorticity
field, which is dominated by vortex filaments, is reminiscent of the filaments
observed in 3D Navier-Stokes turbulence.

This study shows that the generalised model of 2D turbulence is another nice
set-up to study the transition of turbulent cascades. However, this model is
unique as this is the first system that shows features that appear in 3D
turbulence, like the vortex filaments, emerging in two dimensions. So, further
numerical studies are required to understand deeper and relate the dynamics of
the vortex filaments in two dimensions to the dissipation anomaly that is
present for negative values of $\alpha$. Further mathematical analysis of this
model dissipation anomaly for negative values of $\alpha$ might provide new
insights to the regularity of solutions of the 3D Navier-Stokes equations. The
study of locality of cascades using the shell to shell transfers from
observational and experimental data of geophysical relevance would complement
our findings, e.g. for the SQG ($\alpha=1$) model, and it gives a warning to
the community that we should be thinking beyond the Kolmogorov type of
arguments.

##  Appendix A Numerical Setup

We perform direct numerical simulations (DNSs) in a periodic square domain by
numerically integrating using the pseudospectral method (Gottlieb & Orszag,
1977). We decompose the streamfunction $\psi(x,y,t)$ into basis functions of
Fourier modes, viz.

$\displaystyle\psi({\bf x},t)$ $\displaystyle=\sum^{N/2}_{{\bf
k}=-N/2}\widehat{\psi}_{\bf k}(t)e^{i{\bf k}\cdot\mathbf{x}},$ (58)

where $\widehat{\psi}_{\bf k}$ is the amplitude of the ${\bf k}=(k_{x},k_{y})$
mode of $\psi$, and $N$ denotes the number of aliased modes in the $x$ and $y$
directions. A third-order Runge-Kutta scheme is used for time advancement. The
aliasing errors are removed with the 2/3 rule, which implies that the maximum
wavenumber $k_{max}=N/3$. The computations were performed on graphics cards
(GPUs), which provided three times speedup in contrast to computations on
processors (CPUs). Time-averaged quantities are computed once the system has
reached a statistically stationary regime. The numerical parameters we
considered are listed in Table 1.

a)$Re_{+}=1381$ b)$Re_{-}=4\times 10^{7}$ $Re_{-}^{(1)}=10^{6}$
$Re_{-}^{(2)}=4\times 10^{7}$ $Re_{-}^{(3)}=2\times 10^{9}$ $Re_{+}^{(1)}=345$
$Re_{+}^{(2)}=1381$ $Re_{+}^{(3)}=3450$ $Re_{+}^{(4)}=6900$ N 512 1024 2048
1024 1024 1024 1024 $k_{f}$ $8\sqrt{2}$ $16\sqrt{2}$ $32\sqrt{2}$ $16\sqrt{2}$
$16\sqrt{2}$ $16\sqrt{2}$ $16\sqrt{2}$

Table 1: Numerical parameters for the two set of runs corresponding to a)
fixed $Re_{+}$ with varying $Re_{-}$ and b) fixed $Re_{-}$ with varying
$Re_{+}$.

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