u(t)}^{2}_{L^{2}}+2\alpha\mathbb{E}\int_{0}^{t}\norm{\nabla u}^{2}_{L^{2}}\
ds+2\nu\mathbb{E}\int_{0}^{t}\norm{\Delta u}^{2}_{L^{2}}\
ds=\mathbb{E}\norm{\nabla u_{0}}^{2}_{L^{2}}+2\varepsilon_{1}t.$ (A.71)

Moreover, if $u$ is a stationary measure we have

$\displaystyle\alpha\mathbb{E}\norm{\nabla
u}^{2}_{L^{2}}+\nu\mathbb{E}\norm{\Delta u}^{2}_{L^{2}}=\varepsilon_{1},$
(A.72)

from which (A.67). ∎

###  A.4 Some Useful Results

The following theorem is a result from [58]:

######  Theorem A.6.

Let $\Omega$ be a bounded set in $\mathbb{R}^{n}$ and
$f_{n}:\Omega\to\mathbb{R}$ be functions whose $\nabla f_{n}$ are well defined
and converge uniformly (component by component) to a function
$g:\Omega\to\mathbb{R}^{n}$. Suppose $\exists\ x_{0}\in\Omega;\
\lim_{n\to\infty}f_{n}(x_{0})=L\in\mathbb{R}$. Then the functions $f_{n}$
converge uniformly to a differentiable function $f$ and $\nabla f$ exists and
it is equal to $g$.

###### Proof.

Since $\\{f_{n}(x_{0})\\}$ converges, for each $\varepsilon>0$ and $n,m$ large
enough we have

$\displaystyle|f_{n}(x)-f_{m}(x)|$
$\displaystyle\leq|f_{n}(x)-f_{m}(x)-f_{n}(x_{0})+f_{m}(x_{0})|+|f_{n}(x_{0})-f_{m}(x_{0})|$
$\displaystyle\leq\varepsilon\norm{x-x_{0}}+\varepsilon$
$\displaystyle\leq\varepsilon\text{ diam}(\Omega)+\varepsilon,$

where we used the mean value theorem. Hence $f_{n}$ converges uniformly on
$\Omega$ to a function $f$, moreover for each $\varepsilon>0$ and $m,n$ large
enough, the inequality

$\left|\frac{f_{n}(y)-f_{n}(x)}{y-x}-\frac{f_{m}(y)-f_{m}(x)}{y-x}\right|\leq\varepsilon$

holds for each $x\neq y\in\Omega$. The above relation implies that
$\frac{f_{n}(y)-f_{n}(x)}{y-x}$ converges uniformly to
$\frac{f(y)-f(x)}{y-x}$.

Now working, for instance, on the first component we can write

$\displaystyle\left|\frac{f(x^{1}+h,x^{2},...,x^{n})-f(x^{1},...,x^{n})}{h}-g^{1}(x)\right|\leq$
$\displaystyle\left|\frac{f(x^{1}+h,x^{2},...,x^{n})-f(x^{1},...,x^{n})}{h}-\frac{f_{n}(x^{1}+h,x^{2},...,x^{n})-f_{n}(x^{1},...,x^{n})}{h}\right|$
$\displaystyle+$
$\displaystyle\left|\frac{f_{n}(x^{1}+h,x^{2},...,x^{n})-f_{n}(x^{1},...,x^{n})}{h}-\partial_{1}f_{n}(x)\right|+\left|\partial_{1}f_{n}(x)-g^{1}(x)\right|.$

For each $\varepsilon>0$ and $n$ large enough we get

$\displaystyle\left|\frac{f(x^{1}+h,x^{2},...,x^{n})-f(x^{1},...,x^{n})}{h}-g^{1}(x)\right|\leq
2\frac{\varepsilon}{3}+\left|\frac{f_{n}(x^{1}+h,x^{2},...,x^{n})-f_{n}(x^{1},...,x^{n})}{h}-\partial_{1}f_{n}(x)\right|$

and for $h$ small enough

$\left|\frac{f(x^{1}+h,x^{2},...,x^{n})-f(x^{1},...,x^{n})}{h}-g^{1}(x)\right|\leq\varepsilon.$

By repeating this argument on each component, $\nabla f(x)$ exists and is
equal to $g(x)$. ∎

The following Lemma is derived in [5] that we rewrite here for the reader’s
convenience.

######  Lemma A.1.

Let $u\in L^{3}(\mathbb{T}\crossproduct I)$ be a divergence free vector field,
and let $T_{y}u(x)=u(x+y)$ denote its shift by $y\in\mathbb{R}^{2}$, then the
following identity holds in the sense of distributions

$\displaystyle\sum_{j=1,2}\partial_{y^{j}}\int_{\mathbb{T}\crossproduct
I}(u\otimes T_{y}u)u^{j}+(T_{y}u\otimes u)u^{j}-(T_{y}u\otimes
u)T_{y}u^{j}-(u\otimes T_{y}u)T_{y}u^{j}\ dx$ $\displaystyle=$
$\displaystyle\sum_{j=1,2}\partial_{y^{j}}\int_{\mathbb{T}\crossproduct
I}(\delta_{y}u\otimes\delta_{y}u)\delta_{y}u^{j}dx.$

###### Proof.

Expanding the difference we find,

$\displaystyle\int(\delta_{y}u\otimes\delta_{y}u)\delta_{y}u^{j}dx$
$\displaystyle=\int(T_{y}u\otimes T_{y}u)T_{y}u^{j}dx-\int(u\otimes
T_{y}u)T_{y}u^{j}dx-\int(T_{y}u\otimes T_{y}u)u^{j}dx$
$\displaystyle-\int(T_{y}u\otimes u)T_{y}u^{j}dx+\int(u\otimes
T_{y}u)u^{j}dx+\int(u\otimes u)T_{y}u^{j}dx+\int(T_{y}u\otimes u)u^{j}dx$
$\displaystyle-\int(u\otimes u)u^{j}dx.$

Using the fact that

$\int(T_{y}u\otimes T_{y}u)T_{y}u^{j}dx=\int(u\otimes u)u^{j}dx,$

and the divergence property of $u$,

$\displaystyle\sum_{i,j=1,2}\partial_{y^{j}}\int(T_{y}u\otimes T_{y}u)u^{j}dx$
$\displaystyle=\sum_{i,j=1,2}\partial_{y^{j}}\int(u\otimes u)T_{y}u^{j}dx$
$\displaystyle=\int(u\otimes u)\divergence u^{j}dx=0,$

we obtain

$\displaystyle\sum_{i,j=1,2}\partial_{y^{j}}\int(\delta_{y}u\otimes\delta_{y}u)\delta_{y}u^{j}dx=$
$\displaystyle-\sum_{i,j=1,2}\partial_{y^{j}}\int(u\otimes
T_{y}u)T_{y}u^{j}dx-\partial_{y^{j}}\int(T_{y}u\otimes u)T_{y}u^{j}dx$
$\displaystyle+\partial_{y^{j}}\int(u\otimes
T_{y}u)u^{j}dx+\partial_{y^{j}}\int(T_{y}u\otimes u)u^{j}dx$ $\displaystyle=$
$\displaystyle\sum_{j=1,2}\partial_{y^{j}}\int(u\otimes
T_{y}u)u^{j}+(T_{y}u\otimes u)u^{j}-(T_{y}u\otimes u)T_{y}u^{j}$
$\displaystyle-(u\otimes T_{y}u)T_{y}u^{j}dx.$

∎

## Notation

#### Vectors and sets

$\begin{array}[]{ll}\mathbb{R}^{2}&\text{2-dimensional real Euclidean
space}\\\ x&(x^{1},x^{2})\text{ a typical point in}\ \mathbb{R}^{2}\\\
\mathbb{T}_{x^{1}}&[0,L)\text{ torus of size }L\\\ I&[a,b]\text{ interval}\\\
\partial I&\text{boundary of }I\\\ D(\mathbb{T}\crossproduct(a,b))&\\{u\in
C^{\infty}(\mathbb{T}\crossproduct(a,b));u(0,x^{2})=u(L,x^{2})\text{ with
compact support in }x^{2}\\}\\\
D_{\sigma}(\mathbb{T}\crossproduct(a,b))&\\{u\in
D(\mathbb{T}\crossproduct(a,b)),\divergence u=0\\}\\\
H^{m}_{0}(\mathbb{T}\crossproduct(a,b))&\overline{D(\mathbb{T}\crossproduct(a,b))}^{H^{m}(\mathbb{T}\crossproduct(a,b))}\\\
H(\mathbb{T}\crossproduct(a,b))&\overline{D_{\sigma}(\mathbb{T}\crossproduct(a,b))}^{L^{2}(\mathbb{T}\crossproduct(a,b))}\\\
V(\mathbb{T}\crossproduct(a,b))&\overline{D_{\sigma}(\mathbb{T}\crossproduct(a,b))}^{H^{1}_{0}(\mathbb{T}\crossproduct(a,b))}\\\
\mathcal{X}&C\left(\mathbb{R}_{+};H\right)\cap
L^{2}_{loc}\left(\mathbb{R}_{+};V\right)\\\
\mathcal{X}_{T}&C\left([0,T];L^{2}_{\sigma}\right)\cap\
L^{2}\left([0,T];V\right)\\\ \mathcal{H}&\left\\{u\in
L^{2}\left([0,T];V^{\prime}\right);\partial_{t}u\in
L^{2}\left([0,T];H^{-1}_{\sigma}\right)\right\\}\\\
\norm{v}_{\mathcal{H}}&\norm{v}_{L^{2}_{[0,T]}H^{1}(\mathbb{T}\times
I;\mathbb{R}^{2})}+\norm{\partial_{t}v}_{L^{2}_{[0,T]}H^{-1}(\mathbb{T}\times
I;\mathbb{R}^{2})}\\\ \\{e_{j}\\}&\text{orthonormal basis in
}L^{2}_{\sigma}\\\ (\Omega,\mathcal{F},\mathbb{P})&\text{complete probability
space}\\\ \mathcal{G}_{t}&\text{filtration}\end{array}$

#### Operator

$\begin{array}[]{ll}\divergence{u}&\text{divergence of }u\\\
(u\cdot\nabla)&u^{1}\partial_{x^{1}}+u^{2}\partial_{x^{2}}\\\
B(u,v)&(u\cdot\nabla)v\\\ Fv&f\widehat{k}\times v\\\
A:B&\sum_{i,j}A_{ij}B_{ij}\\\ \hat{\cdot}&\text{Fourier transform}\\\
|A|&\sqrt{A:A}\\\ \delta_{h}u(\cdot)&u(\cdot+h)-u(\cdot)\\\
T_{y}u(\cdot)&u(\cdot+y)\\\ \Pi&\text{Leray projection}\\\ \end{array}$

#### Function notation

$\begin{array}[]{ll}u&\text{velocity vector field}\\\
\omega&\text{vorticity}\\\ u_{0}&\text{inital data}\\\ \varphi&\text{spatially
regular white noise in time stochastic process}\\\ \eta&\text{average
enstrophy per unit time per area}\\\ \varepsilon&\text{average energy per unit
time per area}\\\ \phi&\text{smooth test function}\\\
\\{\beta_{j}\\}&\text{sequence of independent standard Brownian motion}\\\
f&\text{Coriolis parameter}\\\ f_{0}+\beta x^{2}&\beta\text{-plane
approximation}\\\ f_{0}&\text{horizontal component of the rotation}\\\
\beta&\text{meridional gradient of }f\text{ at that fixed latitude}\\\
\rho_{\varepsilon}&\text{standard mollifier}\\\ E(k)&\text{ensemble average of
the energy spectrum}\\\ S_{n}&\text{ensemble average of the n point structure
function}\\\ S_{3}^{\|}&\text{third order longitudinal structure function}\\\
\wedge&\text{minimum}\\\ \Gamma&\text{two-point correlation of the
velocity}\\\ \varTheta&\text{two-point correlation of the Coriolis force}\\\
a&\text{two-point correlation of the noise}\\\ D^{j}&\text{velocity third
order structure function}\\\ \overline{\Gamma}&\text{spherically averaged two-
point correlation of the velocity}\\\ \overline{\varTheta}&\text{spherically
averaged two-point correlation of the Coriolis force}\\\
\overline{a}&\text{spherically averaged two-point correlation of the noise}\\\
\overline{D}&\text{spherically averaged third order structure function}\\\
\overline{\Gamma}_{\|}&\text{longitudinal spherically averaged two-point
correlation of the velocity}\\\ \overline{\varTheta}_{\|}&\text{longitudinal
spherically averaged two-point correlation of the Coriolis force}\\\
\overline{a}_{\|}&\text{longitudinal spherically averaged two-point
correlation of the noise}\\\ \overline{D}_{\|}&\text{longitudinal spherically
averaged third order structure function}\\\ \mathfrak{C}&\text{two point
correlation functions of the vorticity}\\\ \mathfrak{Q}&\text{two point
correlation functions of the curl of the Coriolis force}\\\
\mathfrak{a}&\text{two point correlation functions of the curl of the
noise}\\\ \mathfrak{D}&\text{vorticity third order structure function}\\\
\overline{\mathfrak{C}}&\text{spherically averaged two point correlation
functions of the vorticity}\\\ \overline{\mathfrak{Q}}&\text{spherically
averaged two point correlation functions of the curl of Coriolis force}\\\
\overline{\mathfrak{a}}&\text{spherically averaged two point correlation
functions of the curl of noise}\\\ \overline{\mathfrak{D}}&\text{spherically
averaged vorticity third order structure function}\end{array}$

#### Further symbols

$\begin{array}[]{ll}\alpha&\text{drag}\\\ \nu&\text{viscosity}\\\
b_{j}&\text{constant}\\\ \varTheta&\text{latitude}\\\ \Omega&\text{angular
velocity of the sphere}\\\ l_{\nu}&\text{dissipation scale}\\\
l_{I}&\text{injection scale}\\\ l_{\alpha}&\text{friction scale}\end{array}$

## References

  * [1] M.A.H. Al-Jaboori and D. Wirosoetisno.  Navier-Stokes equations on the $\beta$-plane.  Discrete and Continuous Dynamical Systems - B, 16 (3) : 687-701, 2011\.
  * [2] I. Arad, V.S. L’vov, and I. Procaccia.  Correlation functions in isotropic and anisotropic turbulence: The role of the symmetry group.  Phys. Rev. E (3) 59(6), 6753–6765, 1999.
  * [3] G.K. Batchelor.  Computation of the energy spectrum in homogeneous two-dimensional turbulence.  Phys. Fluids 12(12), II-233, 1969.
  * [4] G.K. Batchelor.  An Introduction to Fluid Dynamics.  Cambridge University Press, 2000.
  * [5] J. Bedrossian, M. Coti Zelati, S. Punshon-Smith, and F. Weber.  A Sufficient Condition for the Kolmogorov 4/5 Law for Stationary Martingale Solutions to the 3D Navier–Stokes Equations.  Equations. Commun. Math. Phys. 367, 1045–1075, 2019.
  * [6] J. Bedrossian, M. Coti Zelati, S. Punshon-Smith, and F. Weber.  Sufficient conditions for dual cascade flux laws in the stochastic 2d Navier-Stokes equations.  Arch Rational Mech Anal 237, 103–145, 2020.
  * [7] A. Bensoussan.  Stochastic Navier-Stokes equations.  Acta Appl. Mathematicae, 38, 267–304, 1995.
  * [8] D. Bernard.  Three-point velocity correlation functions in two-dimensional forced turbulence.  Physical Review E 60, no. 5, 6184, 1999.
  * [9] G. Boffetta.  Energy and enstrophy fluxes in the double cascade of two-dimensional turbulence.  J. Fluid Mech. 589:253–60, 2007.
  * [10] G. Boffetta and R.E. Ecke.  Two-Dimensional Turbulence.  Annual Review of Fluid Mechanics Vol. 44:427-451, 2012.
  * [11] Y. Cacchio’.  Global Well-Posedness for Eddy-Mean Vorticity Equations on $\mathbb{T}^{2}$.  arXiv:2303.00023, 2023.
  * [12] Y. Cacchio’, A. Hannani, and G. Staffilani.  In preparation.
  * [13] A. Chekhlov, S.A. Orszag, S. Sukoriansky, B. Galperin, and I. Staroselsky.  The effect of small-scale forcing on large-scale structures in two-dimensional flows.  Physica D: Nonlinear Phenomena, Volume 98, Issues 2–4, pp. 321-334, 1996\.
  * [14] N.C. Constantinou, B.F. Farrell, and P.J. Ioannou.  Emergence and Equilibration of Jets in Beta-Plane Turbulence: Applications of Stochastic Structural Stability Theory.  Journal of the Atmospheric Sciences, 71(5):1818–1842, 2014.
  * [15] N.C. Constantinou, B.F. Farrell, and P.J. Ioannou.  Statistical state dynamics of jet–wave coexistence in barotropic beta-plane turbulence.  Journal of the Atmospheric Sciences, 73(5):2229–2253, 2016.
  * [16] L. Cope.  The Dynamics of Geophysical and Astrophysical Turbulence.  University of Cambridge, 2020.
  * [17] G. Da Prato and A. Debussche.  Two-Dimensional Navier-Stokes Equations Driven by a Space-Time White Noise.  Journal of Functional Analysis 196, 180–210, 2002.
  * [18] S. Danilov and V. M. Gryanik.  Barotropic beta-plane turbulence in a regime with strong zonal jets revisited.  Journal of the Atmospheric Sciences, 61, 2283–2295, 2004.
  * [19] T. De Karman and L. Howarth.  On the statistical theory of isotropic turbulence.  Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, pp. 192–215, 1938.
  * [20] G.L. Eyink.  Exact results on stationary turbulence in 2D: consequences of vorticity conservation.  Physica D 91:97–142, 1996.
  * [21] R. Fjrtoft.  On the changes in the spectral distribution of kinetic energy for two dimensional, nondivergent flow.  Tellus 5(3), 225–230, 1953.
  * [22] F. Flandoli and D. Gatarek.  Martingale and stationary solutions for the stochastic Navier-Stokes equation.  Probab. Th. Rel. Fields, 102, 367–391, 1995.
  * [23] U. Frisch.  Turbulence: The Legacy of A.N. Kolmogorov.  Cambridge University Press, 1995.
  * [24] I. Gallagher and L. Saint-Raymond.  Mathematical study of the betaplane model: Equatorial waves and convergence results.  Mémoires de la Société Mathématique de France, no. 107, 122 p., 2006\.
  * [25] I. Gallagher and L. Saint-Raymond.  On the influence of the Earth’s rotation on geophysical flows.  Handbook of Mathematical Fluid Dynamics, Elsevier, (S. Friedlander and D. Serre, eds), 2006.
  * [26] B. Galperin, S. Sukoriansky, and N. Dikovskaya.  Zonostrophic turbulence.  Physica Scripta, T132:014034, 2008.
  * [27] B. Galperin, S. Sukoriansky, N. Dikovskaya, P.L. Read, Y.H. Yamazaki, and R. Wordsworth.  Anisotropic turbulence and zonal jets in rotating flows with a $\beta$-effect.  Nonlinear Processes in Geophysics, 13(1):83–98, 2006.
  * [28] G.R. Grimmett and D.R. Stirzaker.  Probability and Random Processes.  Oxford University Press. pp. 491–495, 2001.
  * [29] M. Hairer and J.C. Mattingly.  Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing.  Ann. of Math. 164(3), 993–1032, 2006.
  * [30] J.R. Holton.  An introduction to dynamical meteorology.  Academic Press, 2004.
  * [31] H. P. Huang and W. A. Robinson.  Two-dimensional turbulence and persistent zonal jets in a global barotropic model.  Nonlinear Processes Geophys., 55, 611–632, 1998.
  * [32] H.P. Huang, B. Galperin, and S. Sukoriansky.  Anisotropic spectra in two-dimensional turbulence on the surface of a rotating sphere.  Physics of Fluids, 13(1):225–240, 2001.
  * [33] A. Kolmogorov.  The local structure of turbulence in incompressible viscous fluid for very large Reynold’s numbers.  C. R. (Doklady) Acad. Sci. URSS (N.S.) 30, 301–305, 1941.
  * [34] A.N. Kolmogorov.  Dissipation of energy in the locally isotropic turbulence.  C. R. (Doklady) Acad. Sci. URSS (N.S.) 32, 16–18, 1941.
  * [35] A.N. Kolmogorov.  On degeneration of isotropic turbulence in an incompressible viscous liquid.  C. R. (Doklady) Acad. Sci. URSS (N. S.) 31, 538–540, 1941.
  * [36] R.H. Kraichnan.  Inertial ranges in two-dimensional turbulence.  Phys. Fluids 10(7), 1417–1423, 1967.
  * [37] R.H. Kraichnan and D. Montgomery.  Two-dimensional turbulence.  Rep. Prog. Phys. 43(5), 547, 1980.
  * [38] S. Kuksin and A. Shirikyan.  Mathematics of Two-Dimensional Turbulence.  Cambridge University Press, 2012.
  * [39] A. Kupiainen.  Ergodicity of two dimensional turbulence (after Hairer and Mattingly).  Astérisque 339, Exp. No. 1016, vii, 137–156. Séminaire Bourbaki. Vol. 2009/2010. Exposés 1012–1026, 2011.
  * [40] S. Kurien and K.R. Sreenivasan.  Measures of Anisotropy and the Universal Properties of Turbulence. In: Lesieur, M., Yaglom, A., David, F. (eds) New trends in turbulence Turbulence: nouveaux aspects.  Les Houches - Ecole d’Ete de Physique Theorique, vol 74. Springer, Berlin, Heidelberg, pp. 53–111, 2001.
  * [41] J. Leray.  Etude de diverses équations intégrales nonlinéaires e tde quelques problèmes que pose l’hydrodynamique.  J. Math. Pures Appl., 12, p. 1-82, 1933.
  * [42] J. Leray.  Essai sur le mouvement d’un liquide visquex emplissant l’espace.  Acta Math. 63, p. 193-248, 1934.
  * [43] J. Leray.  Essai sur les movementsplans d’un liquide visqueux que limitent des parois.  J Math. Pures et Appl., 13, p. 331-418, 1934.
  * [44] E. Lindborg.  Can the atmospheric kinetic energy spectrum be explained by two-dimensional turbulence? Journal of Fluid Mechanics 388, 259–288, 1999.
  * [45] J.C. McWilliams.  Fundamentals of Geophysical Fluid Dynamics.  Cambridge University Press, 2011.
  * [46] A. S. Monin and A. M. Yaglom.  Statistical fluid mechanics: mechanics of turbulence.  Vol. II, Dover Publications, Inc., Mineola, NY. Translated from the 1965 Russian original, Edited and with a preface by John L. Lumley, English edition updated, augmented and revised by the authors, Reprinted from the 1975 edition, 2007.
  * [47] S. Nazarenko.  Fluid Dynamics via Examples and Solutions.  Taylor and Francis Inc, 2014.
  * [48] Q. Nie and S. Tanveer.  A note on third-order structure functions in turbulence.  R.Soc.Lond.Proc.Ser.A Math. Phys. Eng. Sci. 455, 1615–1635, 1999.
  * [49] T. Nozawa and S. Yoden.  Formation of zonal-band structure in forced two-dimensional turbulence on a rotating sphere.  Physics of Fluids, 9, 2081-2093, 1997.
  * [50] S. Papathanasiou.  Sufficient conditions for local scaling laws for stationary martingale solutions to the 3D Navier–Stokes equations.  Nonlinearity 34(5):2937-2969, 2021.
  * [51] J. Pedlosky.  Geophysical fluid dynamics.  Springer-Verlag, 1992.
  * [52] P.B. Rhines.  Waves and turbulence on a beta-plane.  Journal of Fluid Mechanics, 69, 417-443, 1975.
  * [53] R. Salmon.  Lectures on Geophysical Fluid Dynamics.  Oxford University Press, 1998.
  * [54] R. K. Scott and L. M. Polvani.  Forced-Dissipative Shallow-Water Turbulence on the Sphere and the Atmospheric Circulation of the Giant Planets.  Journal of the Atmospheric Sciences, 64, 3158–3176, 2007.
  * [55] R.K. Scott and L.M. Polvani.  Equatorial super rotation in shallow atmospheres.  Geophysical Research Letters, 35(24):L24202, 2008.
  * [56] T.G. Shepherd.  Non-ergodicity of inviscid two-dimensional flow on a beta-plane and on the surface of a rotating sphere.  J. Fluid Mech., 184, pp. 289–302, 1987.
  * [57] K. Srinivasan and W.R. Young.  Zonostrophic Instability.  Journal of the Atmospheric Sciences 69(5):1633-1656, 2012.
  * [58] T. Tao.  Analysis II.  Hindustan Book Agency, 2009.
  * [59] R. Temam.  Navier-Stokes Equations: Theory and Numerical Analysis.  North-Holland Publishing Company, 1979.
  * [60] G. K. Vallis and M. E. Maltrud.  Generation of mean flow and jets on a beta plane and over topography.  J. Phys. Oceanogr., 23, 1346–1362, 1993.
  * [61] G.K. Vallis.  Atmospheric and Oceanic Fluid Dynamics.  Cambridge University Press, 2017.
  * [62] G.K. Vallis and M.E. Maltrud.  Energy spectra and coherent structures in forced two-dimensional and beta-plane turbulence.  Journal of Fluid Mechanics, 228, 321-342, 1991.
  * [63] J. Xie and O. Bühler.  Exact third-order structure functions for two-dimensional turbulence.  Journal of Fluid Mechanics, 851, 672-686, 2018.
  * [64] A. Yaglom.  On the local structure of a temperature field in a turbulent flow.  Dokl.akad. nauk sssr, pp. 743–746, 1949.
  * [65] V. Yakhot.  Two-dimensional turbulence in the inverse cascade range.  Physical Review E 60, no. 5, 5544, 1999.