FIELD THEORY MODEL FOR TWO-DIMENSIONAL TURBULENCE: VORTICITY-BASED APPROACH

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1 to appear in Acta Physica Slovaca 6 FIELD THEORY MODEL FOR TWO-DIMENSIONAL TURBULENCE: VORTICITY-BASED APPROACH M.V. Altaisky Joint Institute for Nuclear Research, Dubna, 4980, Russia; Space Research Institute, Profsoyuznaya 84/3, Moscow, 7997, Russia John C. Bowan Departent of Matheatical and Statistical Sciences, University of Alberta, Edonton, Alberta, T6G G, Canada Subitted May 5, 00 Renoralization group analysis is applied to the two-diensional Navier Stokes vorticity equation driven by a Gaussian rando stirring. The energy-range spectru C Kε /3 k 5/3 obtained in the one-loop approxiation coincides with earlier double epsilon expansion results, with C K = This result is in good agreeent with the value C K = 3.35 obtained by direct nuerical siulation of the two-diensional turbulent energy cascade using the pseudospectral ethod. PACS: 47.7.Gs, 47.7.Eq,.0.Gh Statistical hydrodynaics constitutes one of the ost proising applications of quantu field theory ethods to the physics of classical systes. The statistical description of twodiensional hydrodynaic turbulence is in turn one of the ost challenging probles in hydrodynaics. In addition to having intrinsic atheatical beauty, the renoralization group (RG) odel of turbulence in three diensions has provided realistic values of the Kologorov constant in the range.4 to.7 [,, 3]. Along with the energy, which is conserved by the nonlinear ters of the incopressible Navier Stokes equation in any diension; there exists in two diensions an additional positive-seidefinite invariant, the ean-squared vorticity. This enstrophy conservation law forbids universal scaling in the whole range of scales fro energy injection to energy dissipation, aking the dynaical RG technique [4, ] hardly applicable in two diensions [5, 6]. The goal of applying the RG technique to hydrodynaic turbulence is to decrease the nuber of odes required to describe the syste, retaining at the sae tie all basic syetries and properties of the syste [4]. The sae idea underlies the spectral reduction ethod, where the effect of discarded odes is taken into account by enhancing the nonlinear interaction coefficients [7]. This is accoplished by coarse-graining the vorticity equation in Fourier space in E-ail address: altaisky@x.iki.rssi.ru E-ail address: bowan@ath.ualberta.ca Institute of Physics, SAS, Bratislava, Slovakia

2 M.V. Altaisky, J.C. Bowan such a way that the averaged equations autoatically respect the energy and enstrophy invariants. For the two-diensional enstrophy cascade, spectral reduction has been shown to provide an efficient nuerical approxiation to the turbulent statistics. In the RG approach, the effect of the deleted sall scale odes is taken into account by integrating over these odes in the presence of a rando stirring force added to the right-hand side of the Navier Stokes equation. Technically, two-diensional turbulence RG calculations are ore coplicated than threediensional ones because of extra one-loop divergences that require a counterter having no counterpart in the original action [8]. A significant breakthrough in this proble was achieved when the double paraetric renoralization group [6] was applied to the field theory foralis of two-diensional turbulence to cancel the extra one-loop divergence [9]. However, it has been difficult to estiate by other theoretical eans if the obtained value of C K = is relevant and to confir it with appropriate nuerical siulations. In this paper we apply the double paraetric expansion RG ethod to essentially the sae vorticity equation in Fourier space that is used for pseudospectral siulations of two-diensional turbulence. Our analytical calculations atch the result C K = obtained by Honkonen [9] and M. Hnatich et al. [3]. The nuerical siulation perfored with the pseudospectral code gives C K The RG ethod for two-diensional turbulence is based on a two-ter force correlator that odels the phase transition between lainar flow and developed turbulence: the first ter atches the (asyetric) turbulent regie, while the second ter atches the syetric (with respect to (3)) lainar regie. We briefly review the field theory approach to stochastic hydrodynaics [4, 8], starting fro the two-diensional Navier Stokes equation t v + (v )v = ν v P + f, where the velocity field v is incopressible ( v = 0), ν is the viscosity, P is the pressure, and f denotes an external stirring force. The curl of this equation describes the evolution of the vorticity vector ζ(t, x) = v(t, x) = ζẑ, where ẑ is the noral to the plane of flow [0], t ζ ν ζ = [ ( ζ) ζ ] + ξẑ; () the incopressible velocity field being the inverse Laplacian of the curl of the vorticity field: v(t, x) = ζ(t, x). The rando stirring force ξẑ = f is supposed to be Gaussian: ( ξ(x)ξ(x ) = D(x x ), P [ξ] = exp ) ξ(x)d (x x )ξ(x ) dx dx. () Here we use the + d notation x (x 0, x), where x 0 = t and d = is the diension. Consider a stochastic process u(t, x), governed by the stochastic integro-differential equation F [u] = ξ ˆη, where F is a nonlinear integro-differential operator, ˆη(x) is a regular force, and ξ(x) is a Gaussian rando force satisfying (). The field theory approach is based on the characteristic functional Z[η, ˆη] = ( exp i ) dx η(x)u(x), u n = δn i n δη n. η=0 By prescribing to each trajectory u = u(t, x) the weight P [u, ξ] = δ(f [u] ξ + ˆη)P [ξ] and representing the δ-function as a functional integral over an auxiliary field û, one finds that Z[η, ˆη] = DuDû exp (is[û, u] + iηu + iˆηû), where the action S[û, u] can be expressed in ters of the rando force correlation function D as S[û, u] = ûf [u] + iûdû/.

3 Field theory for D turbulence 3 In the absence of a rando force, the vorticity equation () is invariant under the Lie group G of scale transforations: t = e γ t, x = e γ x, ζ = e γ ζ, (3) where γ is the infinitesial transforation paraeter. The corresponding Lie algebra generators are ˆΓ t = t t, ˆΓ x = x x, ˆΓ ζ = ζ ζ. In ters of the characteristic functional, invariance under a Lie group iplies []: [ ] δ ˆΓ iδη, iη, x, x Z[η, ˆη] = 0 (4) for all generators ˆΓ[u, u, x, x ] of the syetry group of the equation F [u] = 0. Following [], we can easily find the constraints on the correlators that underly the syetry (3) even in the case of a rando force (D 0). Since ˆζ is just an auxiliary field with no dynaical constraints iposed on it, we can fix the transforation law of ˆζ with respect to scale transforations to satisfy the invariance of ˆζ(t, x)f [ζ(t, x)]dt d d x identically with respect to (3). This eans that ˆζ = e γdˆζ ˆζ, where Dˆζ = d. Now the invariance of the action S[ζ, ˆζ] with respect to scale transforations is copletely deterined by the transforation properties of ˆζDˆζ with respect to (3). To keep it invariant, the identity ˆζ D ˆζ d d x dt d d x dt = ˆζ Dˆζ d d x dt d d x dt should hold. Siple power counting gives D = e γdd D, with D D = 8, or D = 0. The second possibility corresponds to the free equation, i.e. decaying turbulence. The first case corresponds to a special power-type stirring, say D(x, t) δ(t) x 6 or D(x, t) t α x β with α + β = 8. (5) Thus, for a white-noise rando forcing, a correlator that is invariant under (3) ust have a Fourier transfor proportional to k 4. Now, assuing the invariance of the characteristic functional with respect to scale transforations (3), let us derive the scaling equations for the correlation and response functions. Here we follow the ethod already used to derive the governing statistical equations for the Navier Stokes velocity field in a siilar setting []. The correlation and response functions are C() = δ ln Z[ˆη, η] δη(x )δη(x ), G() = i δ ln Z[ˆη, η] iδη(x )iδˆη(x ). (6) Since the action should be identically invariant under (3), we have to account only for the variation of the generating part dx dt (ηζ + ˆη ˆζ) in the exponent of the generating functional, so that the invariance of the characteristic functional under the group of scale transforations δ G dx dt (ηζ + ˆη ˆζ) = 0, where δg eans variation with respect to (3), will hold. After straightforward calculations, siilar to those presented in [], we obtain the syetries of the correlation and Green function, which hold when the scaling syetry (3) is observed, i.e. for the class (5) of rando force correlators, [x x + t t + 4] C(t, x) = 0, [k k + ω ω 4] C(ω, k) = 0, (7) [x x + t t + d] G(t, x) = 0, [k k + ω ω d] G(ω, k) = 0. (8)

4 4 M.V. Altaisky, J.C. Bowan The vorticity equation () gives rise to the field theory Z[J] = e is[φ]+ij Φ DΦ, S[Φ] = i ˆζDˆζ + ˆζ [ t ζ ν ζ U[ζ] ]. (9) The notation Φ = (ζ, ˆζ), J = (η, ˆη) is used for book-keeping, where the nonlinear interaction in two diensions is given by the nonlocal vertex ( ) δ(k p q) U[ζ] = U(k p, q)ζ(p)ζ(q) dp dq, U(k p, q) = (π) 3 (p q) z q p. The force correlation operator D is chosen in accord with the double paraetric expansion: ξ(k )ξ(k ) = (π) d+ δ(k + k )D(k ) and D(k) = ν 3 (g µ ɛ k 4 ɛ + g k 4 ), where µ is the renoralization ass [6, 9, 3]. The coupling constants g and g can be ade diensionless by rescaling both vorticity and tie (): t = tν 0 Λ, ζ = ζ ν 0 /D 0, η = η/(λ ν 0 D 0 ), where D 0 = g,0 ν0 3 is the unrenoralized strength of the vorticity forcing and Λ is the ultraviolet oentu cutoff. The renoralized viscosity is given by a one-loop contribution to the Green function ˆζζ, [ ( )] G (k) = 4G 0 (k) (q k) z D(q) G 0 (q) [ ( ( q k) z q k q k q )] G 0 (k q) dd+ q (π) d+, where G 0 (q) = ( iq 0 + ν 0 q ). In the long-wave liit k 0 this results in a turbulent dressing of the viscosity, [ ν = ν 0 + g,0 3π ɛ + g,0 3π ln Λ ], (0) (for ɛ > 0), where and Λ are the lower (infrared) and the upper (ultraviolet) liits of integration and g,0 and g,0 are the bare constants of the vorticity force correlation. The positive turbulent dressing of the viscosity (0) can be absorbed into the renoralization of the viscosity ν 0 = νz ν, [ Z ν = A g ɛ + g ln Λ ], A = 3π. () Siilarly, the one-loop divergent contribution to the correlation function ζζ is D (k) = ν [ ] 6 Λ qdq (π) dθ 4 q k sin k θ +kq cos θ q (k +q kq cos θ) (g q 4 ɛ +g q 4 )(g k q 4 ɛ +g k q 4 ) ν 3 (q +k kq cos θ)q (k +q kq cos θ). In the liit k 0 the one-loop contribution to the correlator is D (k) = ν3 k ( g,0 3π 4ɛ + g,0g,0 ɛ + g,0 ln Λ ), ()

5 Field theory for D turbulence 5 which can be absorbed into the renoralization of the coupling constant g, ( g Z = A g 4ɛ + g ɛ + g ln Λ ). (3) The renoralization of g is related to Z ν in the usual way: g,0 = µ ɛ g Z, g,0 = g Z Z, Z = Zν 3. The corresponding anoalous diensions γ i = µ µ ln Z i g,0,g,0,ν 0=const and the functions β i = µ µ g i g,0,g,0,ν 0=const are evaluated as a series in the coupling constants g and g. Retaining up to second-order ters in g and g, we find β = g ( ɛ + 3Ag + 3Ag ) and β = A( g + g g + g ). This yields the nontrivial infrared fixed point (g, g ), where [6] g = 4ɛ 9A, g = ɛ 9A. (4) Our results () and (3), obtained in the one-loop approxiation, exactly atch the results obtained in two diensions for the forced strea function [9] and velocity [3] equations. Aong a variety of controversial theoretical and nuerical results for two-diensional turbulence obtained by different eans, it sees ost natural to copare the value of the Kologorov constant obtained by a two paraeter renoralization group (Z ν, Z ) with a quasistationary nuerical result obtained by pseudospectral siulation of the sae vorticity equation used to derive the values Z ν and Z. Referring the reader to [9, 3] for details of the derivation of the energy spectru deterined by the fixed point (4), we just state the final results. The ean energy dissipation rate ε is related to the field theory odel paraeters: ε = ν0 3g,0k 4 ɛ d /(6π), where k d is the dissipation wavenuber and ɛ =. We now evaluate the constant C K in the Kologorov law for the energy cascade, E(κ) = C K ε /3 κ 5/3, where κ = k and the two-diensional energy spectru E(κ) is defined by E(κ) = Tr This yields [9, 3] C K = πκ dκ dω v(k)v( k) π (π) = π κ dκ dω ζ(k)ζ( k) π (πκ). ( ) /3 g + g π g = (4ɛ) /3, (5) /3 so that C K for ɛ = and d =. Euclidean field theory (9) describing two-diensional turbulence is siilar to the Landau- Ginzburg phase transition theory. In the absence of a rando force or a syetry preserving force (ɛ = 0), two-diensional turbulence in an unbounded doain is invariant under the Lie group of scale transforations (3), which forally corresponds to the vorticity correlator ζ(k)ζ( k) k 4. The Kologorov regie requires that the ean energy dissipation rate ε is exactly copensated by energy injection. This state (ɛ = ) corresponds to the breaking of the original Lie group (3) syetry down to a new state of the syste deterined by the anoalous diensions γ i = γ i (ɛ, g, g ). For a given value of ɛ, the existence of the stable fixed point β i (ɛ, g,ɛ, g,ɛ) = 0 iplies, in the language of phase transitions, the existence of a stable phase. The trivial fixed point g = g = 0 corresponds to the syetric phase (lainar flow); the infrared stable fixed point (4) at ɛ = corresponds to the asyetric phase, the Kologorov

6 6 M.V. Altaisky, J.C. Bowan (a) (b) 0 Fig.. (a) Transient spectral energy density E(κ) obtained fro a dealiased pseudospectral siulation. The dashed line indicates the fitted Kologorov spectru 3.35ε /3 κ 5/3. (b) Variation of the estiated value of Kologorov constant for the inverse energy cascade with wavenuber. regie. The diffeoorphis between phases is controlled by a single paraeter ɛ that deterines the syetry of the forcing. If other fields are incorporated (e.g. passive scalar advection) the phase diagra of the cobined syste becoes ultidiensional [3]. In Fig., we depict the transient energy spectru and estiated Kologorov constant C K = 3.35 obtained with a dealiased pseudospectral siulation of the two-diensional energy cascade driven by a white-noise rando forcing restricted to κ [98, 0], taking ε = and ν = To iniize the required resolution, we replaced the viscous ter νκ ζ by νκ H(κ 0)ζ, where H(κ) denotes the Heaviside unit step function. An adaptive fifth order Cash Karp Fehlberg Runge Kutta integrator was used to advance () and its second oent. We would like to thank Drs. N. Antonov, M. Hnatich, and R. Seancik for useful discussions. References [] L.Ts. Adzheyan, A.N. Vasil ev, Yu. M. Pis ak: Theor. and Math. Physics 57 (983) 68 [] V. Yakhot, S. Orszag: Phys. Rev. Lett. 57 (986) 7 [3] M. Hnatich et al.: Phys. Rev. E 59 (999) 4 [4] D. Forster, D.R. Nelson, M.J. Stephen: Phys. Rev. A 6 (977) 73 [5] P. Olla: Int. J. Mod. Phys. B 8 (994) 58 [6] J. Honkonen, M.Yu. Naliov: Z.Phys.B 99 (996) 58 [7] J.C. Bowan, B.A. Shadwick, P.J. Morrison: Phys. Rev. Lett. 83 (999) 549 [8] L.Ts. Adzheyan, N.V. Antonov, A.N. Vasiliev: Field Theoretic Renoralization Group in Fully Developed Turbulence (Gordon and Breach, 999) [9] J. Honkonen: Phys. Rev. E 58 (998) 453 [0] J. G. Charney: Geof. Publ. XVII (948) 3 [] M.V. Altaisky, S.S. Moiseev: J. Phys. I France (99) 079