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Nonlinear superposition of direct and inverse cascades in two-dimensional turbulence forced at large and small scales

We inquire about the properties of 2d Navier-Stokes turbulence simultaneously forced at small and large scales. The background motivation comes by observational results on atmospheric turbulence. We show that the velocity field is amenable to the sum of two auxiliary velocity fields forced at large and small scale and exhibiting a direct-enstrophy and an inverse-energy cascade, respectively. Remarkably, the two auxiliary fields reconcile universal properties of fluxes with positive statistical correlation in the inertial range.

eral explanations for the observed spectra have been proposed. Interpreting the synoptic −3 spectrum as an enstrophy cascade, forced by instabilities of the horizontal motion at the planetary scale (∼10 4 Km) [11], the −5/3 mesoscales spectrum may result: from a 2d-inverse energy cascade forced by convection driven by thermal gradients in the troposphere [13][14][15]; or, less likely [10,16], from gravity waves [17,18]. Interpreting the −5/3 spectrum as a convection-driven 2d-inverse energy cascade, the steeper synoptical spectrum may result from largescale coherent structures due to forcing [19] or to energy condensation at the planetary scale [19,20]. Interestingly, such coherent motions may mask the inverse cascade inducing a positive energy-flux [19], which could explain observations [12]. Other proposed mechanisms (not discussed here) consider 3d or quasi-2d scenarios accounting for stratification and other effects [21][22][23].
In Ref. [16], to explain the observed universality of atmospheric spectra, it was conjectured that the two sources may not be independent. However, large and small scale excitations originate from different physical processes [11] characterized by separate timescales (small-scale convection being faster), bearing their independence. Within the independent-source model, spectral universality can be ascribed to that of the inversecascade [4]. In [14,16] it was also hypothesized that oppositely directed cascades could not coexist without a sink between the forcing scales. Lilly [15], using closure theories, showed that there is no need of such sink. Maltrud and Vallis [24] made, as far as we know, the unique numerical study of Eq. (1), providing evidence of two overlapping cascades. This is confirmed by Fig. 1 which shows that the energy spectrum E v (k) displays the basic features of the atmospheric one. Moreover, enstrophy Π ω (k) and energy Π v (k) fluxes are constant with opposite signs (Π ω (k) > 0 and Π v (k) < 0) meaning that direct enstrophy and inverse energy cascades superimpose, apparently undisturbed, in the same range [k L : k ℓ ]. We show below that, remarkably, the superposition is realized maintaining non-trivial correlations between the degrees of freedom associated with the two cascades.
To scrutinize this superposition we propose a decomposition able to disentangle the two cascades, by evolving in parallel with Eq. (1) two auxiliary equations where v is the same as in Eq. (1). Since f L and f ℓ in (2-3) are the same realizations of the forcings in (1), ω L ,ℓ are two "active" pseudo-scalar fields [25], such that  [27]). Grey symbols in (b) refer to passive scalar flux Π θ , see text (Cfr. Eq. (10)). The solid curves show energy and enstrophy fluxes obtained by integrating (2) with fL = 0 (a) or f ℓ = 0 (b), their superposition to Π v 0;00 and Π ω 0;00 , resp. further demonstrates the cascades overlap when both forcings are present. Notice that Π v 0;22 ≈ Π v 0;00 < 0 and Π v 0;11 ≈ 0 making evident that v2 are the degrees of freedom associated with the inverse cascade. Similar considerations apply to vorticity, though Π ω 0;11 > Π ω 0;00 , which will be discussed later in relation to Fig. 4.
The behavior of the energy spectra ( Fig. 1) and the fluxes (Fig. 2) associated with the auxiliary fields suggest the identification of the components v 1,2 as the carriers of the degrees of freedom mainly associated with, respectively, the direct and the inverse cascade. This observation can be substantiated by means of the Kármán-Howarth-Monin (KHM) equation 0jk (r, t) (as customary, we assumed translation, rotation and parity invariance). The KHM equation links the 3 rd order structure-tensor, Kraichnan's theory is equivalent to the following three hypotheses [28]: existence of steady state for Galilean invariant statistical indicators; smoothness at finite viscosity; absence of velocity dissipative anomaly. Using these hypotheses, a careful analysis along the lines of [10,28,30] justifies in the range ℓ ≪ r ≪ L, the expansion with shorthand notation for summation over cyclic permutations, Eq. (5) holds true strictly for i = 0 in the ideal limit of infinite volume at vanishing hypofriction (α q = 0), with, furthermore, the explicit prediction for the coefficients:  (c) C ω 12 (r) =≺ ω1(0)ω2(r) ≻ vs r. The behaviors of Z0;12(r) (which should vanish in the ideal limit) and of C ω 12 (r) are evidences of the sensitivity of directly cascading degrees of freedom to the hypofriction, see text (Cfr. Eq. (9)).  (5) can still be justified using parity invariance and the incompressibility of the fields v i . The quantities S 0jk are thus associated with dynamical fluxes, while S 1jk and S 2jk only provide statistical information, and will be dubbed "statistical" fluxes. Eqs. (5-7) recover the 3 rd -order longitudinal structure function derived in Ref. [10]:

Out of the ideal limit, the constants
S L 000 (r) ≡ r α1 r α2 r α3 S α1α2α3 000 (r)/r 3 = 3 2 ǫ r + 1 8 η r 3 , (8) which is numerically very well satisfied (Fig. 3a). Interestingly, other quantities deviate from the ideal limit prediction (e.g S L 012 (r) = 0 is not reproduced by data, see Fig. 3a). As discussed below, similar issues are present for 3 rd -order quantities associated with ω i (i = 0, 1, 2). The deviations from ideal-limit predictions should be ascribed to the finiteness of the simulation domain, where the presence of hypofriction stymies the derivation of neat relations such as (7). The A {···:1} 's become coupled to the infra-red component of the kinetic energy hinting at a more non-local and less universal behavior of the enstrophy cascade. This fact relates to non-trivial correlations existing between the degrees of freedom associated with the two cascades. The importance of such correlations can be quantified by comparing the "statistical" fluxes Π v,ω 1;jj (k) and Π v,ω 2;jj (k) (with j = 0, 1, 2), defined in [27], with the "dynamical" fluxes Π v,ω 0;jj 's (for j = 0, 1, 2). As shown in Fig. 4, two phenomena stand out. Whilst Π v 0;00 ≃ Π v 0;22 < 0 (Fig. 2a) validates the identification of v 2 as the degrees of freedom associated with the inverse cascade, the left panels of Fig. 4 show that Π v 1;00 < 0 (Π v 1;22 < 0) with intensity comparable to that of Π v 2;00 (Π v 2;22 ) and thus indicates that v 1 contributes to the inverse cascade of the total field v 0 . This may appear, at first glance, surprising in consideration of the observed absence of flux of kinetic energy of v 1 (Π v i;11 ≃ 0 for any i). The non-intuitive and, possibly, non-universal behavior of the statistical energy fluxes Π v i;jj (i = 1, 2) is, however, a consequence of their dependence on the full statistics. Similar considerations apply to enstrophy fluxes, with the roles of v 1 and v 2 exchanged (Fig. 4 right).
The second phenomenon appertains to the relative intensity of enstrophy fluxes. The ideal-limit energybalance predictions (6-7) suggest Π ω 0;00 ≃ Π ω 0;11 > 0 with Π ω 0;22 ≃ 0. The right panels of Fig. 4 confirm the latter prediction coherently with the interpretation of v 2 as the carrier of the inverse cascade degrees of freedom. They also show that the "dynamical" enstrophy flux Π ω 0;11 is significantly enhanced with respect to Π ω 0;00 (see also Fig. 2b). The phenomenon can be regarded as a consequence of the alleged fact that the main contribution to v 1 comes from the degrees of freedom of v 0 undergoing the direct cascade and of the stronger sensitivity of this latter to non-local effects. In order to substantiate the claim we inspected the equations governing the correlation functions C ω ij (r) =≺ω i (0)ω j (r)≻. From the energy balance in the presence of hypofriction together with the assumption of inverse cascading v 2 (i.e. Π ω for some γ > 0, as enstrophy is bounded [28]. Eq. (9) evinces the non-local nature of the enstrophy transfer. Its positive value implies that the correlation C ω 12 (r) must be also positive in the inertial range, as confirmed in Fig. 3c. In terms of the vorticity structure functions Z i;jk (r) ≡≺ δv i (r)δω j (r)δω k (r) ≻, Eq. (9) translates into Z 0;12 (r) = r · Z 0;12 (r)/r > 0 with linear dependence on r (as shown in Fig. 3b), while it should be 0 in the ideal limit. To further elucidate the enhancement of Π ω 0;11 with respect to Π ω 0;00 , it is instructive to liken the ultra-violet behavior of the vorticity ω 1 to that of a passive scalar θ evolving in the same velocity field, as in Ref. [29]: Here, the large-scale forcing f ′ L has the same statistics of that used in (2) but with independent realizations, making θ a passive scalar [25]. Flow incompressibility brings about a scalar energy ≺ θ 2 ≻/2 cascade to smaller scales, and similarly to (9), by energy balance, we have for some positive γ ′ , where C θ (r) =≺ θ(r, t)θ(0, t) ≻. We expect the difference in (11) to be positive as the infra-red content of C ω 00 is "fattened" by the inverse energy transfer at k < k L . As shown in Fig. 2b, the comparison between the passive scalar energy flux, Π θ , with the enstrophy flux, Π ω 0;00 validates this prediction. The inference is that the same phenomenon underlies the positive sign of (9).
Summarizing, we showed that 2d-turbulence sustained by a large and a small scale forcing gives rise in between the sources to an inertial range where a direct and an inverse cascade co-exist and overlap. We also showed that there exists a natural decomposition of the Navier-Stokes field compartmentalizing the degrees of freedom associated with the direct and inverse cascade in two auxiliary velocity fields, obtained considering a single large and small scale source, respectively. Although these auxiliary fields satisfy energy balance relations as if they were independent, actually they are not and exhibit nontrivial correlations pinpointed by the inspection of third order statistics of "statistical" objects, evading the energy balance relations. In contrast to the settings used here, realistic forcings in the atmosphere should be timecorrelated, with the large-scale excitation being slower than the small-scale one. Provided the forcings are independent with separate spatial and temporal scale the picture presented here should remain essentially unaltered. However, a slow large-scale forcing may induce coherent motions that, as argued in Ref. [19], can change the sign of the (total) energy flux and thus mask the inverse cascade process, which may explain observations [12].
We conclude with a theoretical remark. The decomposition in terms of auxiliary velocity fields here proposed can be straightforwardly generalized to 2d Navier-Stokes equations with an energy input distributed over different scales. In this perspective, the cascade overlap of the two sources model, here investigated, evinces the physical mechanism for why Kraichnan theory applies also in the presence of power-law sources and, consequently, for the inability of renormalization group approach to correctly predict Navier-Stokes energy spectra [30], even in what may seem a priori a perturbative regime.
MC and AV acknowledge support from MIUR PRIN2009 "Nonequilibrium fluctuations: theory and applications". PMG acknowledges support from the Finnish Academy CoE "Analysis and Dynamics" and from KITP (grant No. NSF PHY05-51164).