Intermittency of Burgers' Turbulence

Viscous Instanton for Burgers’ Turbulence E. Balkovskya, G. Falkovicha , I. Kolokolovb and V. Lebedeva,c arXiv:chao-dyn/9603015v1 31 Mar 1996 a Physics Department, Weizmann Institute of Science, Rehovot 76100, Israel b Budker Institute of Nuclear Physics, Novosibirsk 630090, Russia c Landau Inst. for Theor. Physics, Moscow, Kosygina 2, 117940, Russia (February 5, 2008) We consider the tails of probability density functions (PDF) for different characteristics of velocity that satisfies Burgers equation driven by a large-scale force. The saddle-point approximation is employed in the path integral so that the calculation of the PDF tails boils down to finding the special field-force configuration (instanton) that realizes the extremum of probability. We calculate high moments of the velocity gradient ∂x u and find out that they correspond to the PDF with ln[P(∂x u)] ∝ −(−∂x u/Re)3/2 where Re is the Reynolds number. That stretched exponential form is valid for negative ∂x u with the modulus much larger than its root-mean-square (rms) value. The respective tail of PDF for negative velocity differences w is steeper than Gaussian, ln P(w) ∼ −(w/urms )3 , as well as single-point velocity PDF ln P(u) ∼ −(|u|/urms )3 . For high velocity derivatives u(k) = ∂xk u, the general formula is found: ln P(|u(k) |) ∝ −(|u(k) |/Rek )3/(k+1) . 1. INTRODUCTION The forced Burgers equation ∂t u + u∂x u − ν∂x2 u = φ (1.1) describes the evolution of weak one-dimensional acoustic perturbations in the reference frame moving with the sound velocity [1]. It is natural to assume the external force φ to be δ-correlated in time in that frame: hφ(t1 , x1 )φ(t2 , x2 )i = δ(t1 − t2 )χ(x1 − x2 ). (1.2) Then the statistics of φ is Gaussian and is entirely characterized by (1.2). We are interested in turbulence excited by a large-scale pumping with the typical correlation length L so that χ does not essentially change at x < ∼ L and goes to zero where x > L. Besides L, the correlation function χ is characterized by the parameter ω = [−(1/2)χ′′ (0)]1/3 having the dimensionality of frequency. Then e.g. χ(0) ∼ L2 ω 3 . Developed turbulence corresponds to the large value of Reynolds number Re = L2 ω/ν. The physical picture of Burgers turbulence is quite clear: arbitrary localized perturbation evolves into shock wave with the viscous width of the front, that gives k −2 for the energy spectrum at Re ≫ kL ≫ 1 [1,2]. The presence of shocks leads to a strong intermittency, PDF of velocity gradients is substantially non-Gaussian [3] and there is an extreme anomalous scaling for the moments of velocity differences w = u(ρ) − u(−ρ): hwn i ∝ (ρ/L) for n > 1 [4]. Simplicity of the equation and transparency of underlying physics make it reasonable to hope that consistent formalism for the description of intermittency could be developed starting from Burgers equation [5–7] as well as from the problem of passive scalar advection [8–11]. That would be an appropriate way to mark centenary of Burgers’ birth. The most striking manifestation of intermittency is the strong deviation of the PDF tails from Gaussian or the behavior of high moments. The PDF’s P(∂x u) and P(w) are not symmetric, the asymmetry is due to the simple fact that positive gradients are smeared while the steepening of negative gradients could be stopped only by viscosity. It has been realized recently that those tails can be found by considering the saddle-point configurations (we call them instantons) in the path integral determining the PDF [12]. The instanton formalism has been applied to the Burgers equation by Gurarie and Migdal [13] who found the right tail ln P(w) ∼ −[w/(ωρ)]3 determined by practically inviscid behavior of smooth ramps between the shocks. That cubic right tail has been first predicted by Polyakov from the conjecture on the operator product expansion [5], it corresponds to the same right tail for the gradients ln P(∂x u) ∼ −(∂x u/ω)3 which also has been derived from the mapping closure by Gotoh and Kraichnan [14]. Here we find the viscous instantons that give the left tails of PDF’s and single-point velocity PDF. Even though some calculations are lengthy, the simple picture appears as a result. Since white forcing pumps velocity by the law w2 ∝ φ2 t while the typical time of growth is restricted by the breaking time t ∼ L/w then the Gaussianity of the forcing ln P(φ) ∝ −φ2 /χ(0) leads to ln P(w) ∼ −[|w|/(Lω)]3 . At a shock, w2 ≃ −ν∂x u so that ln P(∂x u) ∼ −[−∂x u/(ωRe)]3/2 . These simple estimates are confirmed below by consistent calculations. 1 2. SADDLE-POINT APPROXIMATION A general instanton formalism for the description of the high-order correlation functions and of the tails of probability density functions has been developed in [12]. Here, we will be interested in the high-order moments of the velocity gradient ∂x u, difference w = u(x = ρ) − u(x = −ρ) and the velocity itself: Z −1 n h[∂x u] i = Z0 DuDp exp {iI + n ln[∂x u(0, 0)]} , (2.1) Z hwn i = Z0−1 DuDp exp {iI + n ln[u(0, ρ) − u(0, −ρ)]} , (2.2) Z hun i = Z0−1 DuDp exp {iI + n ln u(0, 0)} . (2.3) R Here, Z0 is the normalization constant and the effective action I = dt L(u, p). The form of the Lagrangian L for (1.1,1.2) is determined by the dynamical equations [15–17]: Z Z i dx1 dx2 p1 χ12 p2 . (2.4) L = dx (p∂t u + pu∂x u − νp∂x2 u) + 2 The expressions (2.1,2.2) with (2.4) are formally exact relations. A. Saddle-Point Equations The main idea implemented here is that the high-order correlation functions (for n ≫ 1) are determined by the saddle-point configurations of the path integrals (2.1,2.2). The corresponding saddle-point equations are Z 2 ∂t u + u∂x u − ν∂x u = −i dx′ χ(x − x′ )p(x′ ), (2.5) ∂t p + u∂x p + ν∂x2 p = δ(t)λ(x). (2.6) We should substitute here λ ⇒ [in/a(0)]δ ′ (x) a(0) = ∂x u(0, 0) for gradients, λ ⇒ −(in/w)[δ(x − ρ) − δ(x + ρ)] for differences. λ ⇒ −[in/u(0, 0)]δ(x) for velocity. (2.7) (2.8) (2.9) A solution of the equations (2.5,2.6) determines the saddle-point value of the arguments of the exponent in (2.1,2.2) and consequently the corresponding correlation functions in the steepest descend approximation. The equation (2.5) can be solved with arbitrary initial condition at remote negative time. The condition for the equation (2.6) should be formulated for the final time [12]. It is p = 0 as it follows from the variation of the action R R (2.4) over u at the final time, since the corresponding contribution to the action originating from dt dx p∂t u is dx pδu. That means that p = 0 at t > 0. Thus we can solve the system of equations (2.5) and ∂t p + u∂x p + ν∂x2 p = 0, (2.10) at negative time t only. The role of the last term in (2.6) is then reduced to the initial (or more precisely final) condition p(x) = −λ(x) imposed on p at t = −0. The initial condition imposed on u at a remote negative time is forgotten after a finite time and does not influence the answer. From the other hand the final condition on the field p is also forgotten at a finite negative time if to move backwards in time. Thus our solution is characterized by a long stationary stage where both fields u and p are zero. We will refer to that stage as vacuum since it is realized at the absence of the source λ in the right-hand side of (2.6). In the following we will treat the vacuum value u = 0 as the initial condition for the field u. As to the field p it grows from a small perturbation at moving in the positive direction in time due to the negative sign at the viscous term in (2.10). We can say that our instanton solution describes the instability inherent to the vacuum. The equations (2.5,2.10) are invariant under x → −x, u → −u, p → −p. Thus for gradients and differences, the antisymmetry of the boundary term λ leads to an antisymmetric solution u(x) = −u(−x), p(x) = −p(−x). For velocity, the solution will be asymmetric. 2 In the saddle-point approximation h[∂x u]n i ∼ exp(iIextr )[a(0)]n , hwn i ∼ exp(iIextr )[w(0)]n , hun i ∼ exp(iIextr )[u(0, 0)]n , (2.11) where we should substitute the corresponding values found as a solution of (2.5,2.6) and the saddle-point value of the effective action. Of course, besides the saddle-point contributions, there exist fluctuation corrections to the path integrals (2.1–2.3). The corrections at the vacuum stage are compensated by the normalization constant Z0 since the vacuum just corresponds to n = 0 that is to λ = 0. The contribution of the fluctuations around the instanton (associated with nonzero saddle-point values of the fields u and p) is finite and small in comparison with the saddlepoint one if n ≫ 1. B. Conservation Laws The equations (2.5,2.10) are Lagrangian equations and the Lagrangian (2.4) does not explicitly depend on time. Then the “energy” E is conserved which can be found by the standard canonical transformation: Z iE = dt p∂t u − L. (2.12) The explicit expression for E can be obtained from (2.4): Z Z E = i dx (pu∂x u + ν∂x p∂x u) − (1/2) dx1 dx2 p1 χ12 p2 . (2.13) Since the Lagrangian (2.4) does not explicitly depend on coordinates then the “momentum” J is conserved as well: Z iJ = dx p∂x u. (2.14) Because of the conservation laws we should treat solutions of (2.5,2.6) with zero values of both E and J since they are zero in the vacuum. Using E = 0 we conclude from (2.4,2.13,2.12) that the saddle-point value of the effective action figuring in (2.11) is Z Iextr = dt dx p∂t u. (2.15) Any conservation law is an important heuristic tool to be used at the beginning to understand general properties of the solutions. Let us consider the final instant t = −0. Here p(x) = −λ(x). Now, substituting that into the energy, we can analyze the balance of different terms. For the gradient, E = −na(0)+ω 3 n2 /a2 (0)−nν∂x3 u(0, 0)/a(0) = 0. Striking difference between the cases a(0) > 0 and a(0) < 0 is now clearly seen from the energy conservation. Indeed, for the case of positive gradient, the viscous contribution to the energy is unessential and two first terms can compensate each other. Then one gets a(0) = ωn1/3 , which corresponds to the right PDF tail ln P(∂x u) ∼ −(∂x u/ω)3 given by an inviscid instanton (see the next subsection). The instanton that gives a(0) < 0 cannot exist without viscosity as it is seen from energy conservation. The same consideration can be developed for velocity differences with the same result that positive w comes from an inviscid instanton while negative from a viscous one. For the velocity, we use both conservation laws. Requiring J to be zero we obtain ∂x u(0, 0) = 0. That leads to the following energy at t = −0: E = χ(0)n2 /u2 (0) − ν∂x2 u(0, 0). Without viscous term, energy conservation cannot be satisfied. Let us emphasize that the answer we shall obtain for the tails of velocity PDF (as well as that for velocity differences) does not contain viscosity while it’s consistent derivation requires the account of the viscous terms in the equations. C. Right Tail Let us first describe the instantons producing the right tails of the PDFs for gradients and differences [5,13,14]. The tails correspond to the positive slope of the velocity u near the origin. Then one can expect that the field u is smooth enough and the viscosity plays a minor role in its dynamics. Due to the initial condition p = −λ at t = 0 the field p is localized near the origin. At moving backward in time the viscosity will spread the field p. Nevertheless, if u > 0 at x > 0 and u < 0 at x < 0 then the sweeping will tend to “compress” the field p. In this situation one 3 can expect that the width of p remains much smaller than L. Then, it is possible to formulate the closed system of equations for the quantities a(t) = ∂x u(t, 0) and Z c(t) = −i dx x p(t, x), (2.16) regarding that the width of p is so small that the velocity u can be replaced by it’s linear expansion term ax. The equation for c can be obtained from (2.10) if to multiply it by x and integrate over x. The equation for a can be R obtained from (2.5) where for narrow p we can put dx′ χ(x − x′ )p(t, x′ ) → −i∂x χ(x)c(t) and for small x we can substitute ∂x χ(x) ≈ −2ω 3 x. We thus find ∂t c = 2ac, ∂t a = −a2 + 2ω 3 c. (2.17) The energy conserved by (2.17) can be derived from (2.13) at the same approximation: E = −ca2 + ω 3 c2 . The condition E = 0 means that the instanton is a separatrix solution of (2.17). For gradients p(−0, x) = −[in/a(0)]δ ′ (x) i.e. a(0)c(0) = n. From the energy conservation ω 3 c2 = ca2 , one gets a(0) = ω 3 c2 (0)/n = ωn1/3 . The same solution describes also the instanton for differences since in this case p(−0, x) is determined by (2.8) what leads to w = 2a(0)ρ and then to the same initial condition a(0)c(0) = n. Note that the life time of the instanton is T ∼ n−1/3 ω −1 . The main term in (2.11) is [a(0)]n , that immediately leads to h(∂x u)n i ∼ [a(0)]n ∼ ω n nn/3 which gives the right cubic tail of the PDF ln P(∂x u) ∼ −(∂x u/ω)3 . Analogously one can obtain ln P(w) ∼ −[w/(ρω)]3 found previously in [5,13,14]. Then one can check that the extremum value (2.15) is negligible in comparison with n ln[a(0)]: Z Iextr = i dt c∂t a ∼ a(0)c(0) = n. One can incorporate viscosity and show that it has negligible influence on the answer and that the width of p is much less than L through the relevant time of evolution giving the main contribution into the action [13]. The right tails of P(∂x u) and P(w) are thus universal i.e. independent of the large-scale properties of the pumping. D. Left Tail The main subject of this paper is the analysis of the instantons that give the tails of P(u) and the left tails of P(∂x u) and P(w) corresponding to negative values of a(0), w. Although the field p is narrow at t = 0, we cannot use the simple system of equations (2.17) to describe this instanton as it has been seen from the conservation laws. In terms of the function u, the reason is in the shock forming near the origin which cannot be described in terms of the inviscid equations. In addition, sweeping by negative velocity slope provides for stretching (rather than compression) of the field p at moving backwards in time. Thus we should use the system of full differential equations (2.5,2.6) to describe the instantons which can be called viscous instantons. Before presenting analytical consideration, let us qualitatively describe the instantons and make simple estimates of their parameters. We shall see that, apart from a narrow front near x = 0, the velocity field has L as the only characteristic scale of change. The life time |T | of the instanton is then determined by the moment when the width of p reaches L due to sweeping by the velocity u0 : |T | ∼ L/u0 . Such a velocity u0 itself has been created during the life time |T | by the forcing so that u0 ∼ |c|max T Lω 3. To estimate the maximal value of |c(t)|, let us consider the backward evolution from t = 0. We first notice that the width of p (which was zero at t = 0) is getting larger than the width of the velocity front ≃ u0 /a already after the short time ≃ a−1 . After that time, the values of c and a are of order of their values at t = 0. Then, one may in time) in the almost homogeneous velocity field R R consider thatRp(t, x) propagates (backwards u0 so that ∂t c = −i dx xupx ∼ u0 dx p. The integral dx p can be estimated by it’s value at t = 0 which is n/u0 . Therefore, we get cmax ≃ nT so that |T | ≃ n−1/3 ω −1 and u0 ≃ Lωn1/3 . Since at the viscosity-balanced shock, the velocity u0 and the gradient a are related by u20 ≃ νa then a(0) ≃ ωRe n2/3 . The main contribution to the saddle-point value (2.11) is again related to the term [∂x u(0, 0)]n and we find h(∂x u)n i ≃ [a(0)]n ≃ (ωRe)n n2n/3 , which corresponds to the following left tail of PDF at ∂x u ≫ ωRe P(∂x u) ∝ exp[−C(−∂x u/ωRe)3/2 ] . (2.18) The velocity difference w(ρ) is u0 for L ≫ ρ ≫ x0 where the width of the shock x0 ≃ u0 /a(0) ≃ n−1/3 Re−1 . We thus have hwn i ≃ (Lω)n nn/3 which corresponds to the cubic left tail P(w) ∝ exp{−B[w/(Lω)]3 } . 4 The product Lω plays the role of urms . The instanton describing the velocity statistics is asymmetric since the velocity maximum should reach x = 0 at t = 0 as it has been seen from the momentum conservation. Nevertheless, all above estimates concerning w are valid also for u and we obtain P(u) ∝ exp{−D[u/(Lω)]3 } . The numerical factors C, B and D are of order unity, they are determined by the evolution at t ≃ T i.e. by the behavior of pumping correlation function χ(x) at x ≃ L. The left tails are thus nonuniversal; in addition to L and ω, they are determined by the large-scale properties of the pumping. 3. ANALYTIC DESCRIPTION OF THE VISCOUS INSTANTON The formation of the shock near the origin occurs at time |t| < ∼ |T | and at small x. At moderate x we can use the inviscid version of the equations (2.5,2.10) Z ∂t u + u∂x u = −i dx′ χ(x − x′ )p(x′ ), (3.1) ∂t p + u∂x p = 0 . (3.2) That system obeys a rescaling symmetry which admits the following form of solutions u = N f1 (t/T, x/L), p = N 2 f2 (t/T, x/L), T = −N −1 ω −1 , (3.3) where f1 , f2 satisfy corresponding equations. We will demonstrate that a unique family with N related to n is realized for our viscous instanton. To prove that, we should find a solution of the viscous equations (2.5,2.10) at x ≪ L, |t| ≪ |T | and to match the solution with (3.3) at intermediate scales and time. We shall construct this short-scale solution at this section. Here, we want to note only that the shock formed at small time near the origin can be in the main approximation described by the stationary equation u∂x u − ν∂x2 u = 0 which can be derived from (2.5) where the time derivative and the pumping term can be neglected in comparison with large space derivatives. Then we find the well known expression u= 2ν tanh(x/x0 ), x0 (3.4) describing the field u at t = 0 and small enough x, the width of the shock x0 should be much less than L. At scales x ≫ x0 we return to the behavior (3.3) what gives the estimate LωN for the amplitude of (3.4). Then we find from (3.4) x0 ∼ LRe−1 N −1 , a(0) ∼ ωN 2 Re, w ∼ LωN, (3.5) where w = u(0, ρ) − u(0, −ρ) for x0 ≪ ρ ≪ L. Since the instanton describing the velocity statistics is asymmetric the shock is not formed near the origin. It can be described by the expression u = (2ν/x0 ) tanh[(x − ρ)/x0 ] + const obtaining from (3.4) by shifting. Thus the general conclusions made above are valid for the instanton but the velocity statistics needs a separate consideration. The next subsections are devoted to the instantons describing the statistics of the derivatives and of the differences and the last subsection presents the instanton for the velocity statistics. A. Cole-Hopf Representation It is known that the equation (1.1) can be rewritten in the linear form by using Cole-Hopf substitution [1]. To proceed further we shall introduce the substitution for our instanton. Namely we reformulate the problem for the new variables Ψ and P : u ∂x Ψ = − Ψ, 2ν∂x p = P Ψ. (3.6) 2ν The action (2.4) is rewritten as 5 I= Z dt dx P (∂t Ψ − ν∂x2 Ψ) + i 2 Z dt dx1 dx2 χ12 p1 p2 . (3.7) The energy (2.13) is now Z Z iν E = i dx ν∂x P ∂x Ψ + dx F P Ψ 2 Z Z iν = −i dx P ∂t Ψ − dx F P Ψ , where 2 Z i ∂x F = − 2 dx′ χ(x − x′ )p(x′ ). 2ν (3.8) (3.9) We will fix F by the condition F (x = 0) = 0. Note that for the antisymmetric solution (describing the gradients and the differences) Z Z i F (x = +∞) − F (x = −∞) = − 2 dx χ(x) dx′ p(x′ ) = 0. 2ν The saddle-point equations for the functions Ψ and P read ∂t Ψ − ν∂x2 Ψ + νF Ψ = 0, ∂t P + ν∂x2 P − νF P − 2νλ′ (x)δ(t)Ψ−1 = 0. (3.10) (3.11) Here, we have introduced the term from (2.6), producing the final condition for P , into the corresponding equation. One concludes from (3.10,3.11) that ∂t (P Ψ) + ν∂x (∂x P Ψ − P ∂x Ψ) = 2νδ(t)λ′ (x), (3.12) R and the quantity dx P Ψ is conserved. Actually, the integral is equal to zeroRsince P (x) = 0 at t = +0. That means R that we can treat functions p(x) tending to zero at x → ±∞ since dx P Ψ ∝ dx ∂x p in accordance with (3.6). Note that the saddle-point value of the action (2.15) can be rewritten also in terms of the fields Ψ and P : Z Iextr = dt dx P ∂t Ψ. (3.13) As we already pointed out, the spatial width of p(t, x) is small at small |t|. It enables one to go further in formulating the equations for the fields P and Ψ. That allows us to expand χ(x − x′ ) in (3.9) in the series over x′ . The leading term gives F = c [χ(0) − χ(x)]. 2ν 2 Here c is defined by (2.16), c can be also rewritten in terms of the fields Ψ and P Z i c= dx x2 P Ψ. 4ν (3.14) (3.15) For x ≪ L (3.14) is reduced to F (x) = [ω 3 /(2ν 2 )]cx2 . Then, the system (3.10,3.11) can be rewritten as follows ω3 2 cx Ψ = 0, 2ν ω3 2 ∂t P + ν∂x2 P − cx P − 2νδ(t)λ′ (x)Ψ−1 = 0, 2ν ∂t Ψ − ν∂x2 Ψ + (3.16) (3.17) and the energy (3.8) is E = −i Z dx P ∂t Ψ − ω 3 c2 . 6 (3.18) B. Generating Function The temporal evolution of p for time where it’s width is less than L can be effectively described in terms of the generating function for the moments Z Y (η) = −i dx exp(−ηx2 /2)∂x p . (3.19) Note that Y (0) = 0 and ∂η Y (0) = c where c is determined by the expression (2.16). We shall express the behavior of the moments of p via some derivatives of u taken at the initial time t = 0. Since the field p is narrow enough at small time we can use (3.16,3.17) at x ≪ L to calculate the generating function (3.19). It is useful to pass to the new variables  ϕ  ϕ  1 1 (3.20) x2 P̃ , Ψ = √ exp − x2 Ψ̃, P = √ exp 4ν 4ν ∆ ∆ with the parameters that satisfy ordinary differential equations ∂t ϕ + ϕ2 − 2ω 3 c = 0. ∂t ∆ = ϕ∆, (3.21) Since the quantities ∆ and ϕ are introduced by the differential equations where initial conditions are not fixed, then we can impose two arbitrary conditions without changing any observable quantity. Note that the equation (3.21) for ϕ describes the evolution of the linear profile u = ϕx of the velocity which the pumping c would produce at x ≪ L in accordance with the inviscid equations. Let us introduce the dimensionless variables y = x/∆, ds = −2ν∆−2 dt. (3.22) Then the system (3.16,3.17) leads to the following equations for the functions Ψ̃ and P̃ 1 ∂s Ψ̃ + ∂y2 Ψ̃ = 0, 2 1 ∂s P̃ − ∂y2 P̃ = −2νδ(s)∂y λ Ψ̃−1 . 2 (3.23) (3.24) Now, we see the essence of the transformation (3.20,3.21): the equations (3.23,3.24) are separated and do not contain the pumping explicitly. The expression (3.19) is now rewritten as Z i Y =− dy P̃ (s, y)Ψ̃(s, y) exp(−η̃y 2 /2), (3.25) 2ν where η̃ = ∆2 η. The energy (3.18) in terms of the new variables takes the form Z Z 2iν dy P̃ ∂s Ψ̃ + iϕ dy y P̃ ∂y Ψ̃ + ω 3 c2 − cϕ2 . E= 2 ∆ (3.26) The formal solution of the equations (3.23,3.24) can be presented as Ψ̃(s, y) = exp[−s∂y2 /2]Ψ̃(0, y), P̃ (s, y) = −2ν (3.27) exp[s∂y2 /2]Ψ̃−1 (0, y)∂y λ. (3.28) R The expression (3.25) becomes Y = i dy Ψ̃(s, y) exp(−η̃y 2 /2) exp[s∂y2 /2]Ψ̃−1 (0, y)∂y λ. Integrating over dy by parts and substituting (3.27) we obtain   Z η̃ −1 2 Y = i dy Ψ̃ (0, y)∂y λ exp − [y + s∂y ] Ψ̃(0, y). (3.29) 2 Here we used the following relation exp(s∂y2 /2) exp(−η̃y 2 /2) exp(−s∂y2 /2) 7  η̃ 2 = exp − [y + s∂y ] . 2  (3.30) At t = 0 near the origin there is the shock (3.4). Taking into account also the linear profile we find at x ≪ L     1 x 2 (3.31) exp (ϕ0 + ϕ1 )x , Ψ̃(0, x) ∝ cosh x0 4ν where ϕ0 is the value of ϕ at t = 0. We have introduced also the parameter ϕ1 which accounts for the deviation of the actual solution from tanh(x/x0 ) at t = 0, ϕ1 x is the linear profile to be added to (3.4). Starting from (3.31) one can find the explicit expression for the generating function Y via (3.29). First we use the Hubbard-Stratonovich trick to convert the second-order operator at the exponent into the first-order one:   Z   dξ η̃ 1 2 2 √ exp − [y + s∂y ] = exp − ξ + iξ[y + s∂y ] 2 2η̃ 2π η̃     Z dξ 1 2 1 2 √ exp − ξ exp − ξ s exp (iξy) exp (iξs∂y ) . = (3.32) 2η̃ 2 2π η̃ We used the relation exp(t1 y + t2 ∂y ) = exp(t1 t2 /2) exp(t1 y) exp(t2 ∂y ). Substituting (3.31,3.32) into (3.29) we obtain after integrating over ξ     Z y y 1 η̃ (1 + ϕ̃2 s)2 y 2 + s2 /y02 1 p cosh cosh−1 , (3.33) exp − Y = i dy ∂y λ 2 1 + η̃s(1 + ϕ̃ s) 1 + η̃s(1 + ϕ̃ s) y y 2 2 0 0 1 + η̃s(1 + ϕ̃2 s) where y0 = x0 /∆0 and ϕ̃2 = ∆20 (ϕ0 + ϕ1 )/(2ν) C. A solution To analyze the t-dependence of (3.33) we should know a solution of the system (3.21) to restore s(t) from (3.22). For this, it is convenient to use another set of dimensionless variables 2ν 2ω ϕ̃ = ϕ̃, ˜2 ∆2 ∆ Then the equations (3.21) are rewritten as ϕ= a= 2ν 2ω ã = ã, ˜2 ∆2 ∆ ˜ ∆ = L(Re)−1/2 ∆. 1 ˜6 ∂s ϕ̃ + ϕ̃2 + ∆ c̃ = 0, 2 Let us write also the equation for the time t following from (3.22) ˜ = −ϕ̃∆, ˜ ∂s ∆ ∂s t = − ˜ 2 c̃. ωc = ∆ 1 ˜2 ∆ . 2ω (3.34) (3.35) (3.36) One derives from (3.20,3.23,3.31) the following relations 1 (ã − ϕ̃) ⇒ ã0 = −1/y02 − ϕ̃1 , 2 where ϕ̃1 = ∆2 ϕ1 /2ν. After expanding (3.33) we find   Z iν y y 2 c̃ = − (1 + ϕ̃2 s) dy ∂y λ y (1 + ϕ̃2 s) + 2s tanh , 2 y0 y0 ∂s ln Ψ̃(s, y = 0) = (3.37) (3.38) We see that c̃ is the function of the second order over s. Substituting now the formal solution (3.27,3.28) into (3.26) we find after some calculations using the same tricks as above E ˜ 4 c̃2 − 4c̃ϕ̃2 /∆ ˜ 2 + 4(ϕ̃/∆ ˜ 2 )∂s c̃ − (2/∆ ˜ 2 )∂ 2 c̃. =∆ (3.39) s ω One can directly check that the energy (3.39) is conserved as a consequence of (3.35,3.36). Now, to establish the t-dependence of c we should solve the system (3.35) and then reconstruct the function s(t) from (3.36). We will see that the last term in the equation (3.35) for ϕ̃ can be neglected in the wide interval of changing s. Inside that interval (to be specified below), the equation (3.35) has a solution ϕ̃ = z −1 , ˜ = A0 z −1 , ∆ z = s + ϕ̃0−1 , ωt = − A20 sϕ̃0 , 2z (3.40) ˜ grows with decreasing z and consequently the behavior (3.40) is implying that ϕ̃0 < 0. We see from (3.40) that ∆ destroyed for small enough z. 8 D. Gradients For the gradients, λ is determined by (2.7). Thus one should substitute λ = in(2νã0 )−1 δ ′ (y) into (3.33):     1 (1 + sϕ̃2 )η̃ s2 η̃ n 1 [2s + s2 (1 + sϕ̃2 )η̃] exp − . + 1 + s ϕ̃ Y = 2 2νã0 [1 + s(1 + sϕ̃2 )η̃]3/2 y02 [1 + s(1 + sϕ̃2 )η̃] 2y02 1 + sη̃(1 + ϕ̃2 s) (3.41) Expanding (3.41) one finds c̃ =   n 2s (1 + sϕ̃2 ) 2 + (1 + sϕ̃2 ) . 2ã0 y0 (3.42) ˜ is very small and It is time to recall that the energy should be equal to zero at our solution. At t = 0 the value of ∆ it is possible to neglect the first term in the right-hand side of (3.39). Then, equating the residue to zero (and taking s = 0) we find ϕ̃2 ≈ ϕ̃0 that is we can believe ϕ̃1 = 0. That means that with our accuracy it is possible to neglect the linear profile in the the velocity at t = 0 in comparison with the shock contribution. Neglecting ϕ̃1 in (3.37) we find also ã0 = −1/y02 . Substituting the relations into (3.42) one obtains   1 c̃ = nϕ̃0 z −s + ϕ̃0 z , (3.43) 2ã0 ˜ with decreasing where z is determined by (3.40). We see that c tends to zero where z → 0. It is related to growing ∆ |z| since the first term in the expression (3.39) for the energy E cannot tend to infinity. For z ≪ 1 we can disregard the second term in (3.43) and write c̃ = −nsϕ̃0 z. At z ≪ 1, it follows from the definition 1/3 2 of z that s ≈ −ϕ̃−1 A0 since 0 and we conclude that c̃ ≈ nz. The behavior (3.40) is observed if |z| ≫ z1 where z1 ∼ n ˜ 6 in (3.35) is getting substantial. In the region |z| < z1 , both ∆ ˜ and ϕ̃ do not at z ∼ −z1 the term proportional to ∆ ∼ vary essentially and can be estimated as follows ˜ ∼ n−1/3 A−1 , ∆ 0 ϕ̃ ∼ n−1/3 A−2 0 . (3.44) Using (3.35,3.40) one obtains for z1 ≪ |z| ≪ 1 c = 2nt. (3.45) This result is correct, in particular, for the time where the width of p is much larger than the width of the shock. Then (3.45) has to be compared with c(t) = N 2 f3 (t/T ) what is the consequence of (3.3) and of the definition (2.16). We thus conclude N = n1/3 . Then we find from (3.5) a(0) ∼ ωn2/3 Re, T = −ωn−1/3 . (3.46) ˜ = (Re)1/2 It will be convenient for us to normalize ∆ such that ∆ = L at z = 0, then from (3.34) it follows that ∆ there. Comparing this value with (3.44) we conclude A0 ∼ Re−1/2 n−1/3 z1 ∼ n−1/3 Re−1 . (3.47) ˜ 2 Ren2/3 ∼ ϕ̃2 . It can be rewritten as y0 ϕ0 ∼ 1. Then the second contribution Then we find from (3.34,3.40) ã0 ∼ ∆ 0 0 2 to c̃ in (3.43) is ∼ nz and is consequently much smaller than the first contribution at small z what justifies its neglecting. Now we can find the width of the field p at t = T . We see that at z → 0 all coefficients at η̃ in (3.41) tends to zero excepting for the coefficient in the argument of the exponent which for small z is exp[−s2 η̃/(2y02 )]. Just the term determines the width of the field p at small z, the width is s∆/y0 ∼ ∆ since at small z s ≈ −ϕ̃−1 and 0 y0 ϕ̃0 ∼ −1. In accordance with (3.34,3.40) the width of the field p increases with |t|: ∆ ∼ n1/3 Lωt at |t| ≪ |T | and reaches the value ∆ ∼ L at t ∼ T . Now we can estimate corrections associated with neglecting the first term in the expression for the energy (3.39) at t = 0. Using (3.42) we obtain from (3.39) at t = 0   E n2 ˜ 4 2n 2 2 ϕ̃ + ϕ̃ = 2∆ − 1 , ˜ 2 ã0 y02 1 ω 4ã0 0 ∆ 0 where ϕ̃1 = ϕ̃2 − ϕ̃0 determines the linear profile of the velocity. The term with ϕ̃21 can be neglected here and we find ϕ̃1 ∼ nA60 ϕ̃20 . Let us estimate the correction to the expression (3.43) for c̃ following at nonzero ϕ̃1 from (3.42). First, 9 due to the presence of ϕ̃1 , 1/y02 differs from −ã0 , see (3.37). The ratio ϕ̃1 /a0 ∼ A60 is negligible, see (3.47). Second, the term 1 + sϕ̃2 is now ϕ̃0 (z + ϕ̃1 ϕ̃0−2 ). The question is in the value of the correction at z = z1 , it is estimated −2 −2/3 as ϕ̃1 ϕ̃−2 n , see (3.47), the correction is negligible. Thus we proved the correctness of the expression 0 /z1 ∼ Re (3.43). Note that the anomalously small value of the linear profile ϕ1 shows that the formation of the shock is finished at t = 0 since ϕ1 directly determines the time derivative of the width of the shock: ∂t ln x0 = ϕ1 . This is not surprising taking into account extremal properties of the instanton. E. Velocity differences Let us apply the same scheme for the differences. We introduce the dimensionless quantities w̃ = ∆0 w , 2ν ρ̃ = ρ ∆0 (3.48) so that we find from (2.8) λ = in[δ(y + ρ̃) − δ(y − ρ̃)]/(2ν w̃). Substituting this expression into (3.33) we obtain   n [(1 + sϕ̃2 )ρ̃ + s/y0 ] η̃[(1 + sϕ̃2 )ρ̃ + s/y0 ]2 Y = . (3.49) η̃(1 + sϕ̃2 ) exp − ν w̃ 2[1 + s(1 + sϕ̃2 )η̃] [1 + s(1 + sϕ̃2 )η̃]3/2 At calculating (3.49) we substituted cosh by exp because of the inequality ρ̃ ≫ y0 that means that the separation ρ between the points is much larger than the (viscous) width of the front. Expanding (3.49) one finds   n s c̃ = (1 + sϕ̃2 ) (1 + sϕ̃2 )ρ̃ + . (3.50) w̃ y0 Again, using the relation E = 0 with the energy (3.39) we find that ϕ̃1 = 0. Then disregarding ϕ̃1 also in (3.31) we derive the relation w̃ = −2/y0. Finally, c̃ = nϕ̃0 z (−2w̃s + ρ̃ϕ̃0 z) . w̃ (3.51) Now, we should solve the equations (3.35) with c̃ given by (3.51). Again, we have the behavior (3.40) stopped by the term with c̃ in (3.35). To estimate corresponding z1 we should use the asymptotic of (3.51) at small z. As we shall see, the leading term is the linear one so that c̃ ∼ nz. Then we return to the same situation as for gradients. That gives w ∼ Lωn1/3 , w̃ ∼ Re1/2 A0 n1/3 , (3.52) where we used the condition ρ ≫ x0 to be satisfied. To justify the above assertions we should check that at z ∼ −z1 the second term in (3.51) is small in comparison with the first one. Utilizing ρ̃ ∼ Re1/2 ρ/(LA0 ) and (3.47) we obtain the estimates for the ratios of the noted terms ρ̃ ρ ρ ωT z1 ∼ n−1/3 ≪ 1, z1 ∼ w̃ LA20 L (3.53) Thus the term is small since ρ/L ≪ 1 is implied. All the other estimates are the same as for the case of gradients. F. Higher derivatives The above formalism allows us also to find the PDF tails for the odd derivatives u(k) of the velocity. The average value h[u(k) ]n i can be written like (2.1,2.2) what leads to the saddle-point expression of the (2.11) type. The corresponding instanton is determined by the equations (2.5,2.6) where instead of (2.7,2.8) one should take λ to be λ= in δ (k) (x). u(k) (0, 0) All above expressions containing λ will be correct for odd k. Substituting (3.54) into (3.38) one finds 10 (3.54) c̃ = − k+1 nsϕ̃0 z 2 (3.55) Performing the analysis similar to that of subsection 3 D we obtain that N ∼ [(k + 1)n]1/3 and therefore the life time T of the instanton is T ∼ ω −1 [n(k + 1)]−1/3 . It becomes smaller for derivatives of higher order and therefore our approximation works better for larger k. Then from (3.4,3.5) one finds the characteristic value u(k) (0, 0) ∼ N k+1 L1−k ωRek leading to E D (3.56) [u(k) ]n ∼ ωRek L1−k n(k+1)/3 . The result (3.56) can be rewritten in terms of PDF: "  (k) k−1 3/(k+1) #   |u |L (k) . P |u | ∝ exp −Ck ωRek (3.57) Note that the non-Gaussianity increases with increasing k. On the other hand, the higher k the more distant is the (k) validity region of (3.57): u(k) ≫ urms ∼ L1−k ωRek . G. Extremum action −1 Let us now calculate the extremum value of the action (3.13). For s < ∼ |ϕ̃0 | one can rewrite it using (3.20,3.21) Z Z Z ˜ 6 ) − i dsdyy P̃ ∂y Ψ̃ϕ̃. (3.58) iIextr = −i ds dy P̃ ∂s Ψ̃ + ds c̃(2ϕ̃2 − c̃∆ Using now the expression (3.26) for the energy E we can rewrite (3.58) as ! Z ˜2 ∆ 1 2 ˜6 iIextr = −i ds − E − c̃ ∆ + ã . 2ω 2 (3.59) Let us recall that the energy E in (3.59) is 0. Now estimate all the terms in (3.59) utilizing asymptotics R we can R the 1/3 2 2 −1/3 2 ˜6 ˜ (3.40) valid at z > z where z ∼ n A . Then ds ∆ ∼ n . Substituting (3.50) we find ds c̃ ∆ ∼ n. The 1 0 ∼ 1 last contribution in (3.59) can be rewritten as Z Z ds ã = dta ≈ −2 ln[Ψ(0, 0)/Ψ(T, 0)], the last relation is the consequence of (3.6,3.10). We see that only one contribution to iIextr is proportional to n, it is ∼ n. Thus with our accuracy we can neglect the contribution. It is natural that ρ-dependence of P(w) cannot be found in a saddle-point approximation; as a predexponent, it can be obtained only at the next step by calculating the contribution of fluctuations around the instanton solution. This is consistent with the known fact that the scaling exponent is n-independent for n > 1: hwn (ρ)i ∝ ρ. All said above is applicable also for high derivatives. H. Statistics of velocities. Let us now consider the statistics of velocity u. Equations for fields P and Ψ are the same as for odd derivatives: ∂t Ψ − ν∂x2 Ψ + νF Ψ = 0, ∂t P + ν∂x2 P − νF P − 2νλ′ (x)δ(t)Ψ−1 = 0. while λ = −inδ(x)/u(0, 0) is not an odd function of x so that F is different now. Since linear (rather than quadratic) function of x for narrow p: Z i χ(0) F (x) = 2 bx, b = − dxp(x) ν 2 11 (3.60) (3.61) R p dx is nonzero then F is a (3.62) This substantially simplifies the calculations. Substitution which separates variables Ψ= 1 exp(ϕx)Ψ̃(t, y), ∆ P = ∆ exp(−ϕx)P̃ (t, y), (3.63) includes y = x + ρ, ∆ and ϕ which satisfy the following equations ∂t ϕ + χ(0) b = 0, ν ∂t ρ − 2νϕ = 0, ∂t ∆ + νϕ2 ∆ = 0. (3.64) The equation for ∆ is separated and since ∆ does not enter any observable we can forget about it. The equations on Ψ̃ and P̃ are ∂t Ψ̃ − ν∂y2 Ψ̃ = 0, ∂t P̃ − ν∂y2 P̃ = − 2inν ′ δ (y − ρ0 )δ(t)Ψ̃−1 u0 (3.65) Formal solutions of (3.65) are 2iνn exp(−νt∂y2 )δ ′ (y − ρ0 )Ψ̃−1 (0, y), Ψ̃ = exp(νt∂y2 )Ψ̃(0, y) u0 R Substituting (3.66) into b = i dyy Ψ̃P̃ /4ν we obtain Z n b=− dyδ ′ (y − ρ0 )Ψ̃−1 (0, y) exp(−νt∂y2 )y Ψ̃(t, y) 2u0 # " Z n ∂y Ψ̃(0, y) ′ =− dyδ (y − ρ0 ) y − 2νt 2u0 Ψ̃(0, y) P̃ = (3.66) (3.67) The conservation of the momentum (2.14) gives iJ = Z dxP ∂x Ψ = Z 2iνn dy P̃ ∂y Ψ̃ = u0 Z in ∂y Ψ̃(0, y) =− dyδ (y − ρ0 ) u Ψ̃(0, y) 0 ′ Z dxδ ′ (x) [u(0, x) + ϕ] = 0, (3.68) what leads to ∂x u(0, 0) = 0. That naturally means that the maximum of u reaches the point x = 0 at t = 0. We see that J is the same integral which enters (3.67). Therefore, b is a constant (given by it’s value at t = −0): Z n n (3.69) dyδ(y − ρ0 ) = b= 2u0 2u(0, 0) Similarly to the consideration at Sec. 2 D, we can now estimate the value of u(0, 0). Velocity stretches the field p so that the width of p reaches L at T ≃ L/u(0, 0) while the velocity itself is produced by the pumping during the same time: u(0, 0) ≃ χ(0)bT = χ(0)nT /2u(0, 0) ≃ nχ(0)L/u(0, 0). That gives u(0, 0) ≃ Lωn1/3 . As well as in the preceding subsection, we can neglect the term with Iextr in (2.11) and conclude that P(u) ∼ exp[−D(u/Lω)3 ] . Here D is some numerical factor of order unity which is determined by the evolution at t ≃ T i.e. by the large-scale behavior of the pumping. 4. CONCLUSION At smooth almost inviscid ramps, velocity differences and gradients are positive and linearly related w(ρ) ≡ u(ρ) − u(−ρ) ≈ 2∂x uρ so that the right tails of PDF have the same form P(∂x u) ∼ exp[−(∂x u/ω)3 ] , P(w) ∼ exp[−(w/2ρω)3 ] . Those tails are universal i.e. they are determined by a single characteristics of the pumping correlation function χ(r), namely, by it’s second derivative at zero ω = [−(1/2)χ′′ (0)]1/3 . Contrary, the left tails contain nonuniversal constant 12 which depends on a large-scale behavior of the pumping. The left tails come from shock fronts where w2 ≃ −ν∂x u so that cubic tail for velocity differences corresponds to semi-cubic tail for gradients: P(∂x u) ∼ exp[−C(−∂x u/ωRe)3/2 , P(w) ∼ exp[−B(w/Lω)3 ] . The formula for P(w) is valid for w ≫ urms where urms ∼ Lω. It is natural to assume that the probability is small for both u(ρ) and u(−ρ) being large simultaneously. Therefore, P(w) should coincide there with a single-point P(u). Indeed, we saw that at u ≫ urms P(u) ∼ exp[−D(u/Lω)3 ] . Here, B, C and D are (related) nonuniversal constants dependent upon the behavior of χ(r) at r ≃ L. Note that the cubic tail for the single-point PDF has been obtained for decaying turbulence by a similar method employing the saddle-point approximation in the path integral with time as large parameter [19]. Also, our formula for the derivatives P(|u(k) |) ∝ exp[−Ck (|u(k) |Lk−1 /ωRek )3/(k+1) ] coincides with that of [19] for the particular case of white (in space) initial conditions. That, probably, means that white-in-time forcing corresponds to white-in-space initial conditions. One should be cautious, however, comparing the results for forced and decaying turbulence. Another important restriction is our assumption on delta-correlated pumping. If the pumping has a finite correlation time τ then our results, strictly speaking, are valid for u, w ≪ L/τ . ACKNOWLEDGMENTS We are grateful to M. Chertkov, D. Khmelnitskii and R. Kraichnan for useful discussions. This work was partially supported by the Rashi Foundation (G. F.), by the Minerva Center for Nonlinear Physics (I. K. and V. L.), and by the Minerva Einstein Center (V. L.). [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] J.M. Burgers, The Nonlinear Diffusion Equation (Reidel, Dordrecht 1974). R. Kraichnan, Phys. Fluids 11, 265 (1968). T. Gotoh and R. Kraichnan, Phys. Fluids 5, 445 (1993). T. Gotoh, Phys. Fluids 6, 3985 (1994). A.M. Polyakov, Phys. Rev. E 52, 6183 (1995). J. Bouchaud, M. Mézard and G. Parisi, Phys. Rev. E 52, 3656 (1995). A. Chekhlov and V. Yakhot, Phys. Rev. E 52, 5681 (1995). R. Kraichnan, V. Yakhot and S. Chen, Phys. Rev. Lett. 75, 240 (1995). M. Chertkov, G. Falkovich, I. Kolokolov, V. Lebedev, Phys. Rev. E 52, 4294 (1995). K. Gawȩdzki and A. Kupiainen, Phys. Rev. Lett. 75, 3834 (1995) B. Shraiman and E. Siggia, C.R. Acad. Sci. Paris, 321, 279 (1995). G. Falkovich, I. Kolokolov, V. Lebedev and A. Migdal, Phys. Rev. E (submitted), chao-dyn/9512006. V. Gurarie and A. Migdal, Phys. Rev. E (submitted), chao-dyn/9512012. T. Gotoh and R. Kraichnan, (in preparation) P. C. Martin, E. D. Siggia, and H. A. Rose, Phys. Rev. A 8, 423 (1973). C. de Dominicis, J. Physique (Paris) 37, c01-247 (1976). H. Janssen, Z. Phys. B 23, 377 (1976). C. de Dominicis and L. Peliti, Phys. Rev. B 18, 353 (1978). M. Avellaneda, R. Ryan and E. Weinan, Phys. Fluids 7, 3067 (1995). APPENDIX A: Consistent account of fluctuations and calculation of predexponent in PDF will be the subject of future work. Yet one may worry if it is possible that fluctuations destroy instanton i.e. give some divergent contribution into the action. That may happen due to some hidden symmetries and particular fluctuations that are deformations along symmetry 13 group. A special feature of our problem is associated with δ-correlated character of the pumping which leads to a quasi-symmetry of the gauge-invariance type. Namely, under transformations p(t, x) → p(t, x − r), u(t, x) → u(t, x − r) + ∂t r, r = r(t), (A1) R the action I = dt L determined by Lagrangian (2.4) is transformed as I → I + dt dx p ∂t2 r. This additional term is equal to zero for functions p satisfying Z 2 ∂t dx p = 0. (A2) R For antisymmetric objects [gradients Ru′ and differences w with λ being δ ′ (x) and δ(x + ρ) − δ(x − ρ) respectively], λ(x) = −λ(−x). Then the integral dt dx λu entering (2.1,2.2) is invariant under the transformation (A1) if we require r(t = 0) = 0 which will be implied below. We thus have to do with the quasi-gauge transformation which left the effective action invariant on a wide subclass of functions. That means that the integration over the quasi-gauge degree of freedom should be performed exactly. The conventional way to integrate over the gauge degree of freedom is to pass to the action with a fixed gauge, excluding the volume of the corresponding gauge group. To perform the program for the transformation (A1) we include into the integrand in (2.1,2.2) the additional factor Z δ[u(t, r) − ∂t r] , (A3) Dr δ[u(t, r) − ∂t r]J , J = det δr where the integration is performed over functions r(t). The factor (A3) is equal to R R unity by its construction and therefore does not influence Z. Let us now change the order of integration to Dr DuDp, and perform the transformation (A1). Then the effective action I is shifted and δ[u(t, r) − ∂t r] → δ[u(t, 0)]. The Jacobian J introduced by (A3) is reduced to J → det[−∂t + ∂x u(t, x)|x=0 ]. Then the integration over r gives the functional δ-function  Z  Z  Z Dr exp i dt dx ∂t2 r p ⇒ δ dx ∂t2 p(t, x) , (A4) (A5) imposing the constraint on the field p. Now, we should find the Jacobian (A4) which depends on the regularization of the transformation (A1). Really, this regularization is fixed by the regularization of the integral (2.1,2.2). The point is that at deriving (2.4) from the equation (1.1) the retarded regularization was implied (otherwise, some additional u-dependent term appears originating from the corresponding Jacobian [18]). The retarded regularization means that at discretizing time we should substitute ∂t u + u∂x u ⇒ un − un−1 + un−1 u′n−1 , ǫ (A6) where ǫ is the step in time and u′ = ∂x u. The discretized version of the transformation (A1) is un (x) → un (x − rn ) + ṙn , (A7) where ṙ = ∂t r and ṙn is the expression to be determined. Substituting (A7) into the right-hand side of (A6) we obtain 1 un − un−1 + un−1 u′n−1 → [un (x − rn ) − un−1 (x − rn−1 )] + un−1 u′n−1 + ṙn−1 u′n−1 + r̈ ǫ ǫ 1 1 + [un (x − rn ) − un−1 (x − rn−1 )] − (rn − rn−1 )u′n−1 + ṙn−1 u′n−1 + un−1 u′n−1 + r̈. ǫ ǫ (A8) We see that the left-hand side of (A8) is preserved (up to terms r̈) if ṙn−1 = (rn − rn−1 )/ǫ or ṙn = (rn+1 − rn )/ǫ. (A9) Note that because of the condition r(0) = 0 the contribution to the Jacobian will be different depending on whether we consider positive or negative time. At t < 0 with the regularization rule (A9), Jacobian (A4) equals to 14 ...  ... J = det  ... ...  ... ... ... ... ... 0 1/ǫ + u′n −1/ǫ 0 0 0 0 1/ǫ + u′n+1 −1/ǫ 0 ... ... ... ... ...  ... Z  ...  ⇒ exp dt ∂x u(t, 0) . ...  ... (A10) At deriving (A10) we have not interested in factors which do not depend on the field u. All the factors should be absorbed into the normalization constant Z0 introduced in (2.1,2.2). At t > 0 we should go in the positive direction in time stating from r(0) = 0. Then we deal with the retarded regularization which does not produce the contribution to the Jacobian (A10). With the Jacobian, the action acquires the form Z Z I ≡ dt L, L = dx (p∂t u + pu∂x u − νp∂x2 u) Z i dx1 dx2 p1 χ12 p2 − iθ(−t)∂x u(t, 0), (A11) + 2 where θ is the step function. Now, at calculating the path integral (2.1,2.2) one should use only functions satisfying (A2) and u(t, 0) = 0. The instanton equation for p will acquire the additional term ∂t p + u∂x p + ν∂x2 p − iθ(−t)δ ′ (x) = δ(t)λ(x) which leads to a nontrivial vacuum: u∂x u − ν∂x2 u = −c∂x χ(x), u∂x p + ν∂x2 p = iδ ′ (x). (A12) (A13) The requirements that u(x) is a bounded function and p(x) tends to zero where x → ∞ (to provide the finite value for c), together with the gauge u(t, 0) = 0, determine the unique solution of (A12,A13). The equation (A12) has the first integral: u2 /2 − ν(∂x u − a) + c[χ(x) − χ(0)] = 0 . (A14) p If x → ±∞ then u → ±u∞ = ± 2cχ(0) − 2νa. The second term under the square root is a small correction of the order of Re−1 . For small x, we have u = ax and we find from (A14) the relation a2 = 2cω 3 . Substituting u → ax into (A13), multiplying it by x and integrating over x, we find 2ac = 1. Then c = 1/(2ω) and a = ω, that means particularly u∞ ∼ Lω which is urms . To obtain the form of p(x), we substitute u → ax into (A13) and solve the equation: p=i r a sign(x) 2πν 3 Z∞ |x|  a  dξ exp − ξ 2 . 2ν (A15) The width of p(x) is L/Re i.e. it is small indeed. The energy of the vacuum is finite E = 3ω/4. It is important to realize that it is the “gauge” term in the right-hand side of the equation for p that makes vacuum non-trivial i.e. not given just by u = 0 and p = 0. In our formalism, the breakdown of Galilean invariance, discussed by Polyakov [5], manifests itself due to fixing gauge and making an explicit account of fluctuations of gauge degrees of freedom. Now, we can repeat all the above instanton consideration with a new vacuum and the additional term in the equation for p. That gives no contribution at the main order in large n. 15 View publication stats