Analytic structure of solutions of the one-dimensional Burgers equation with modified dissipation
aa r X i v : . [ n li n . C D ] A ug Analytic structure of solutions of the one-dimensionalBurgers equation with modified dissipation
Walter Pauls , and Samriddhi Sankar Ray ‡ Max-Planck-Institut für Dynamik und Selbstorganisation, Am Faßberg 11, 37073 Göttingen,Germany Mühlweg 4, 73460 Hüttlingen, Germany International Centre for Theoretical Sciences, Tata Institute of Fundamental Research,Bangalore 560089, India
Abstract.
We use the one-dimensional Burgers equation to illustrate the effect of replacing thestandard Laplacian dissipation term by a more general function of the Laplacian – of whichhyperviscosity is the best known example – in equations of hydrodynamics. We analyzethe asymptotic structure of solutions in the Fourier space at very high wave-numbers byintroducing an approach applicable to a wide class of hydrodynamical equations whosesolutions are calculated in the limit of vanishing Reynolds numbers from algebraic recursionrelations involving iterated integrations. We give a detailed analysis of their analytic structurefor two different types of dissipation: a hyperviscous and an exponentially growing dissipationterm. Our results, obtained in the limit of vanishing Reynolds numbers, are validated by high-precision numerical simulations at non-zero Reynolds numbers. We then study the bottleneckproblem, an intermediate asymptotics phenomenon, which in the case of the Burgers equationarises when ones uses dissipation terms (such as hyperviscosity) growing faster at high wave-numbers than the standard Laplacian dissipation term. A linearized solution of the well-knownboundary layer limit of the Burgers equation involving two numerically determined parametersgives a good description of the bottleneck region.PACS numbers: 47.27.-i, 82.20.-w, 47.51.+a, 47.55.df
1. Introduction
The physics of a fluid in a turbulent state is multiscale. Hence, it is convenient to studyturbulence by separating the scales into energy injection L , inertial r , and dissipation η ranges [1]. Such a classification has proved useful, both theoretically and numerically, todevelop models which mimic such scales. These models have the advantage of being lesscomplex than the original system and hence, more tractable. Indeed, we owe much of ourunderstanding of the physics and mathematics of turbulent flows, validated by experiments,observations and detailed simulations, to such reduced models. The most celebratedexample of this is the tremendous advance made within the framework of three-dimensional,homogeneous, isotropic turbulence (in the limit of vanishing kinematic viscosity ν ). Such a ‡ Author to whom all correspondence should be addressed. nalytic structure of solutions of the one-dimensional Burgers equation with modified dissipation ν → and suitable initial conditions,provides a complete description of the velocity field u in space and time, the pitfalls of atheoretical treatment of such an equation, at small scales in particular, is best illustrated by thefollowing: At small scales, the properties of the solutions of the Navier-Stokes equation canbe conveniently studied by neglecting the nonlinear convection term yielding ˆ u k ∼ e − ( k/k d ) ,where k d is the energy dissipation wavenumber. However, more refined theoretical arguments,based on the analytic properties of velocity fields at small scales [17] or on some estimatesof velocity field correlations [18] suggest an exponential decay as k → ∞ . Indeed, directnumerical simulations suggest that the energy spectrum, at large k , is consistent with thefunctional form ( k/k d ) γ e − δ ( k/k d ) [19, 20]. The constant δ is believed to be a Reynolds numberdependent quantity, whereas γ is expected to be universal. The exact numerical value of γ isunknown and the only prediction so far is based on Kraichnan’s DIA equations [21] giving γ = 3 .The large- k asymptotic discussed above is relevant, of course, to the deep dissipationrange and have their roots in issues of regularity and finite-time blow-up of solutions of theincompressible Navier-Stokes and Euler equations. For moderate values of k , in the so-calledinertial range where the ideas of Kolmogorov hold, the energy spectrum E ( k ) ∼ k − / (upto intermittency corrections). Between this intermediate and the large- k asymptotics, lies thebottleneck region. The bottleneck is defined as a bump in the turbulence spectrum which leadsto a non-monotonic behaviour in a narrow range of scales between the inertial and dissipationrange (an instance of a bottleneck in a numerical simulation can be found in [22]). It hasbeen argued that this pile-up is due to suppression of energy transfer to smaller scales by the nalytic structure of solutions of the one-dimensional Burgers equation with modified dissipation ν ∆ u by amore general function f ( √− ∆) (or in the Fourier space f ( | k | ) ). An instance of such adissipation term is the well-known hyperviscous dissipation term ν ( − ∆) α which is frequentlyused in numerical simulations. Clearly, the constants γ and δ determining the behavior ofthe energy spectrum in the dissipation range change with α , in particular the fall-off of thespectrum becomes steeper with growing α . The bottleneck has been shown to become morepronounced with increasing α , see, e.g., Ref. [24, 25, 26, 27] as well as Ref. [28] for a review,allowing for theoretical calculations, in the large α limit to be checked against numericalsimulations. It is also important to remember that the use of hyperviscosity has shed light onthe problem of finite-time blow-up: It was shown [12, 29] that there is no finite-time blow-upfor α > / despite the existence of complex singularities. For more general dissipativefunctions, there is one example that we know of where the dissipative term of the form f ( | k | ) = exp( | k | ) leads to entire solutions [30].In this paper, by using a generalised dissipative term f ( √− ∆) , we revisit the problem ofthe nature of the velocity field in the far dissipation range as well as derive analytical resultsin the bottleneck region which connects the intermediate asymptotics of the inertial rangeto the true asymptotics of the far dissipation range. However, the use of such a generaliseddissipation is not completely amenable to a rigorous theoretical treatment. This is because, asis well-known, since Euler’s discovery of the equations for ideal fluids more than 250 yearsago, we are still far from having a complete analytical handle of the nature of the velocityfield in viscous and idealised fluids. We therefore resort to a simpler model, namely that ofthe one-dimensional Burgers equation [31], which, while retaining the same structure of thenon-linearity in the Euler and Navier-Stokes equations, allow for a more rigorous analyticaltreatment [16, 32, 33]. Most of our results are easily generalisable to higher dimensionalequations of hydrodynamics; we shall comment on these later.Our paper is organised as follows. We begin our investigations in Section 2 byconsidering solutions of the one-dimensional (compressible) Burgers equation with modifieddissipation ∂ t u + u∂ x u = − f ( p − ∂ x ) u. (1)The approach that we use can be easily generalized to other hydrodynamical equationssuch as the Navier–Stokes equations. We show that the leading order contribution to suchsolutions can be calculated recursively from an algebraic recursion relation involving iteratedintegrations. Furthermore, in the limit of large times this recursion relation can be transformedto a simple algebraic recursion relation. All of these considerations apply not only tohydrodynamical equation with the standard viscous term but also allow for more generalviscous terms such as hyperviscous or exponentially growing dissipation terms [30]. nalytic structure of solutions of the one-dimensional Burgers equation with modified dissipation α > . Motivated by the recent work of Frisch, et al. ,[26] we investigate how the presence of the bottleneck is related to Gibbs-type oscillationsin the velocity field arising in the neighbourhood of strongly dissipating structures. In theframework of the one-dimensional Burgers equation the method of matched asymptotics canbe used to derive a simplified equation for such structures which in this case are shocks [34]. Itis known that one can determine the asymptotics of the (oscillatory) solutions of this simplifiedequation. However, since neither the amplitude nor the phase of these oscillations is known,we determine them numerically and show that the asymptotic solution indeed gives the rightdescription of the bottleneck. We also derive analytical relations which allow us to estimatefor what kind of dissipation term the simplified boundary layer Burgers equation will exhibita bottleneck.In Section 4 we compare the analytical and semi-analytical results of Sections 2 and3 with state-of-the-art direct numerical simulations. By using high-precision simulations andasymptotic extrapolation of sequences [36] we determine the asymptotic structure of solutionsin the dissipation range and compare it with theoretical results. We also show that asymptoticsolutions of the boundary layer Burgers equation obtained in Section 3 are in agreement withthe numerical solutions in the bottleneck region.In the last section, we discuss the implications of the results proved in the earlier sectionsand make concluding remarks.
2. Solutions of the Burgers equation with modified dissipation
The solution of the d -dimensional Burgers equation with standard (Laplacian) dissipation,in the limit of vanishing viscosity, has been studied extensively, and successfully, by usingvarious techniques [16, 34]. However, these established analytical approaches are limited inscope when applied to the present problem of the Burgers equation with a dissipation termwhich is not necessarily a Laplacian. This is because in the dissipation range, such methodsrarely allow us to determine beyond the leading order asymptotics. A second drawback– which holds even for the usual Laplacian dissipation – is the reliance of conventionaltechniques on properties peculiar to the Burgers equation. Consequently, generalising forthe higher dimensional Euler and Navier-Stokes equations have proved formidable.Given this, we present an approach which does not rely on the specific properties ofthe one-dimensional Burgers equation and hence can be generalized to the multi-dimensionalNavier–Stokes equation. This approach is in the spirit of recent studies by Lee and Sinaiof several hydrodynamic equations with complex-valued initial conditions as well as for thecase of a bounded domain with periodic boundary conditions. Of course, we should note, thatif one were to be interested only in the one-dimensional Burgers equation, other theoreticalmethods are more efficient; the advantage of our approach lies in it being easily adapted tomultidimensional equations of hydrodynamics.We will consider two cases: (i) initial conditions which are real-valued in the physical nalytic structure of solutions of the one-dimensional Burgers equation with modified dissipation By using the semi-group e − tf ( √ − ∂ x ) , generated by − f ( p − ∂ x ) , and the Duhamelprinciple [35], Eq. (1) can be written as u ( x, t ) = − Z t e − ( t − s ) f ( √ − ∂ x ) u ( x, s ) ∂ x u ( x, s ) ds, (2)or in the Fourier space representation ˆ u ( k, t ) = ˆ u ( k, e − f ( k ) t − ik X l + l ′ = k e − f ( k ) t Z t e f ( k ) s ˆ u ( l, s ) ˆ u ( l ′ , s ) ds. (3)Here, and in the following, we assume that the generalised dissipation function f ( k ) is apositive, non-decreasing, strictly convex even function of k with f (0) = 0 .We now introduce an explicit dependence on the amplitude of the initial condition via u ( x, t ) | t =0 = Au ( x ) . (4)Here, when all other parameters are fixed, A plays the role of the Reynolds number. We nowexpand the solution corresponding to the initial condition (4) in a formal power series in Au ( x, t ) = ∞ X n =1 u ( n ) ( x, t ) A n ; (5)for suitable initial conditions, this series has a non-vanishing radius of convergence [37].Furthermore, u ( n ) satisfy the recurrence relations u (1) = exp h − tf ( p − ∂ x ) i u , (6)for n = 1 and u ( n ) = − Z t exp h − ( t − s ) f ( p − ∂ x ) i n − X m =1 u ( m ) ( s ) ∂ x u ( n − m ) ( s ) ds, (7)for n > . In the Fourier space representation we use, for convenience, ˆ u ( k, t ) = i ˆ v ( k, t ) , sothat the respective recursion relations become ˆ v (1) ( k, t ) = ˆ v ( k ) e − f ( k ) t (8)and ˆ v ( n ) ( k, t ) = k X l + l ′ = k e − f ( k ) t Z t e f ( k ) s n − X m =1 ˆ v ( m ) ( l, s )ˆ v ( n − m ) ( l ′ , s ) ds. (9)By using these recursive formulas it is easy to make the following observation: Prop. 2.1
Suppose that ˆ v ( k ) is supported only by finitely many modes in the Fourier space,i.e., k ∈ Σ = {− K, ..., K } , where K is a positive integer. Then for any n ≥ the function ˆ v ( n ) ( k, t ) also has a finite support in the Fourier space n Σ = {− nK, ..., nK } and every ˆ v ( n ) ( k, t ) can be calculated by finitely many operations.nalytic structure of solutions of the one-dimensional Burgers equation with modified dissipation u ( x ) = sin x . Then the first two terms of the expansion are u (1) ( x, t ) = sin x e − tf (1) , (10)and u (2) ( x, t ) = −
12 1 f (2) − f (1) sin 2 x (cid:16) e − f (1) t − e − f (2) t (cid:17) . (11)It is not difficult to verify that the higher order terms are of the form u ( n ) ( x, t ) = g ( n )0 ( t ) sin nx + g ( n )2 ( t ) sin( n − x + ... . (12)This representation can be easily transferred into the Fourier representation as v ( n ) ( n, t ) = − g ( n )0 ( t ) , v ( n ) ( n − , t ) = − g ( n )2 ( t ) , ... (13) v ( n ) ( − n, t ) = 12 g ( n )0 ( t ) , v ( n ) ( − n + 2 , t ) = 12 g ( n )2 ( t ) , ... (14)We now note that although for a fixed wavenumber k obtaining ˆ v ( k, t ) requires summationof the whole series in A , to obtain small- A asymptotics it suffices to consider only the lowestpower of A for a given k . For the initial condition sin x this lowest power for wavenumber k is A k and we obtain the following small A asymptotics ˆ v ( k, t ) ∼ ˆ v as ( k, t ) = A − k ˆ v ( − k ) ( k, t ) k < (15) ˆ v ( k, t ) ∼ ˆ v as ( k, t ) = A k ˆ v ( k ) ( k, t ) k > . (16)The function ˆ v as ( k, t ) can be represented as a sum of two functions ˆ v +as ( k, t ) and ˆ v − as ( k, t ) with support on the positive and negative half-axis, respectively. The function ˆ v +as ( k, t ) is thesolution of the recurrent equation (for k > ) ˆ v +as ( k, t ) = ˆ v ( k, e − f ( k ) t + k k − X l =1 e − f ( k ) t Z t e f ( k ) s ˆ v +as ( l, s ) ˆ v +as ( k − l, s ) ds. (17)The function ˆ v − as ( k, t ) satisfies an analogous equation with k < . Let us stress that thefunctions ˆ v +as ( k, t ) and ˆ v − as ( k, t ) are also solutions of the Burgers equation, however, withone-mode complex-valued initial conditions. The function ˆ v +as ( k, t ) corresponds to the initialcondition ˆ v +as (1 ,
0) = A ˆ v (1) and the function ˆ v − as ( k, t ) to the initial conditions ˆ v − as ( − ,
0) = A ˆ v ( − . The asymptotic solution in the physical space u as ( x, t ) can be written in terms ofthe coefficients g ( n )0 ( t ) u as ( x, t ) = ∞ X n =1 g ( n )0 ( t ) sin nx A n , (18)where g ( n )0 ( t ) = X α +2 α ... + nα n = n n − G ( n ; α , α , ..., α n ) e − t ( α f (1)+ α f (2)+ ... + α n f ( n )) . (19)The sum is taken over all combinations of n non-negative integers α i , i = 1 , ..., n such that α +2 α + ... + nα n = n . We note that these combinations correspond to partitions of integers.For example, for n from to , we obtain the following combinations: nalytic structure of solutions of the one-dimensional Burgers equation with modified dissipation • n = 1 : { ( α ) } = { (1) } , • n = 2 : { ( α , α ) } = { (2 , , (0 , } , • n = 3 : { ( α , α , α ) } = { (3 , , , (1 , , , (0 , , } , • n = 4 : { ( α , α , α , α ) } = { (4 , , , , (2 , , , , (0 , , , , (1 , , , , (0 , , , } .The recurrence relation for coefficients G ( n ; ( n, , ..., , n ≥ corresponding tocombinations ( n, , ..., is G ( n ; ( n, , ..., − n f ( n ) − nf (1) n − X m =1 G ( m ; ( m, , ..., G ( n − m ; ( n − m, , ..., . Note that these coefficients contribute to terms with decay rate e − ntf (1) , which is theslowest decay rate possible for g ( n )0 ( t ) . For coefficients G ( n ; ( n − , , , ..., , n ≥ ofcombinations ( n − , , , ..., , we obtain G ( n ; ( n − , , , ..., − nf ( n ) − ( n − f (1) − f (2) G ( n − , ( n − , , , ..., − n f ( n ) − ( n − f (1) − f (2) n − X m =2 h G ( m ; ( m, , ..., G ( n − m ; ( n − m − , , , ..., G ( m ; ( m − , , , ..., G ( n − m ; ( n − m, , ..., i . (20)Generally, for coefficients G ( n ; ( α , ..., α n − , of combinations of the type ( α , ..., α n − , ,where the last entry vanishes, we obtain the relation G ( n ; ( α , ..., α n − , − n f ( n ) − α f (1) − ... − α n − f ( n − n − X m =1 X β + γ = α G ( m ; ( β , ..., β m ) G ( n − m ; ( γ , ..., γ n − m )) . (21)Here we denote α = ( α , ..., α n − ) ∈ N n − , β = ( β , ..., βm, , ..., ∈ N n − and γ = ( γ , ..., γ n − m , , ..., ∈ N n − , where β + 2 β + ... + mβ m = m and γ + 2 γ + ... + ( n − m ) γ n − m = n − m . Coefficients G ( n ; (0 , , ..., , can be calculated from thecoefficients corresponding to combinations with α n = 0 G ( n ; (0 , , ..., , n − X m =1 ( n − m ) X α +2 α ... + mα m = m X β +2 β ... +( n − m ) β n − m = n − m G ( m ; α , α , ..., α m ) G ( n − m ; β , β , ..., β n − m ) f ( n ) − α f (1) − α f (2) − ... − α m f ( m ) − β f (1) − β f (2) − ... − β n − m f ( n − m ) . (22)We note that in the case, when the function f ( · ) describe the standard dissipation − ν∂ x thecoefficients G ( · ; · ) can be found explicitly by means of the Hopf-Cole transformation whichyields the familiar expression u ( x, t ) = − ν ln (cid:16) e tν∂ x e − A ν u (cid:17) . (23) nalytic structure of solutions of the one-dimensional Burgers equation with modified dissipation As has been noted in Ref. [30], for initial conditions supported on the positive half-line, i.e., ˆ v ( k,
0) = 0 for k ≤ , the Fourier coefficient of the solution at a fixed wavenumber can becalculated iteratively by finitely many operations via Eq. (17). Thus, e,g., we obtain ˆ v (1 , t ) = ˆ v (1) e − f (1) t , (24) ˆ v (2 , t ) = h ˆ v (2) − f (2) − f (1) ˆ v (1) i e − f (2) t + ˆ v (1) e − f (1) t f (2) − f (1) (25)and ˆ v (3 , t ) = (cid:26) ˆ v (3) − v (1)ˆ v (2) f (3) − f (1) − f (2)+ 3ˆ v (1) f (2) − f (1) h f (3) − f (1) − f (2) − f (3) − f (1) i(cid:27) e − f (3) t + 3 h ˆ v (1)ˆ v (2) − ˆ v (1) f (2) − f (1) i e − [ f (1)+ f (2)] t f (3) − f (1) − f (2)+ 3ˆ v (1) e − f (1) t [ f (2) − f (1)][ f (3) − f (1)] . (26)In general the Fourier coefficients of the solution will have a form which is similar to Eq. (19) ˆ v ( k, t ) = X α +2 α + ... + kα k = k F ( k ; ( α , α , ..., α k )) e − t ( α f (1)+ α f (2)+ ...α k f ( k )) . (27)It is instructive to compare this form with the explicit expression found in the case of f ( p − ∂ x ) = − ν∂ x and u ( x ) = Ae ix obtained by using Faà di Bruno’s formula ˆ u ( k, t ) = − ν ( − k k ! A k k ν k k X l =1 ( − l − ( l − X j ,j ,...,j k − l +1 k ! j ! j ! ...j k − l +1 ! × (cid:18) (cid:19) j (cid:18) (cid:19) j ... (cid:18) k − l + 1)! (cid:19) j k − l +1 e − νt ( j + j + ...j k − l +1 ( k − l +1) ) , (28)where the second sum is taken over k − l + 1 nonnegative integers j , ..., j k − l +1 such that j + j + ... + j k − l +1 = l, and j + 2 j + ... + ( k − l + 1) j k − l +1 = k. In the explicit solution, the dependence on the amplitude of the initial condition A manifestsitself by the term A k , in agreement with the observation made previously. The timedependence is also clearly exhibited by the terms e − νt ( j + j + ...j k − l +1 ( k − l +1) ) , which in themore general case become e − t ( α f (1)+ α f (2)+ ...α k f ( k )) . Finally, the exact solution also givesexplicit expressions for the coefficients F ( k ; ( α , α , ..., α k )) .In general, calculating solutions of the Burgers equation with modified dissipation byrecursive determination of coefficients F ( k ; ( α , α , ..., α k )) is quite cumbersome. We nowtake advantage of the fact that in the limit of large times the terms with exponential decay e − kf (1) dominate over the other terms. nalytic structure of solutions of the one-dimensional Burgers equation with modified dissipation Prop. 2.2
From the assumptions on f ( · ) , it follows that f ( · ) is a super-additive function. Theterm in the sum on the right-hand-side of Eq. (27) with the slowest decay in t correspondsto ( α , α , ..., α k ) = ( k, , ..., , with the rate of temporal decay e − kf (1) . The coefficient F ( k, ( k, , ..., h ( k ) satisfies the following recursion relation h ( k ) = k f ( k ) − kf (1) k − X l =1 h ( l ) h ( k − l ) , (29) with h (1) = ˆ v (1) . Thus, for a fixed k and t → + ∞ ˆ v ( k, t ) ∼ F ( k, ( k, , ..., e − kf (1) t . (30)Note that the high wavenumber contributions to the initial conditions are suppressed when t → ∞ and the solution therefore becomes independent of the initial condition since the k thmode is proportional to ˆ v k (1) . Thus, the behaviour of solutions of Eq. (3) is universal at largetimes. To study the solutions of the recursion relation Eq. (29), we introduce the generating function h ( x ) of h ( k ) h ( x ) = ∞ X k =1 h ( k ) e kx , (31)so that Eq. (29) becomes an ordinary pseudo-differential equation f ( ∂ x ) h − f (1) ∂ x h = 12 ∂ x h , (32)with boundary conditions h ( x ) ∼ ˆ v (1) e x for x → −∞ . It is well-known that the asymptoticproperties of h ( k ) can be deduced from the analytic properties of h ( ξ ) . Here we will considertwo cases: (i) hyperviscosity f ( k ) = k α , (33)and (ii) exponentially growing dissipation f ( k ) = e k . (34)In case (i) the solution h ( x ) has a singularity at some point x . We know that the solutionof Eq. (32) in the neighborhood of the singularity behaves as ( x − x ) α − h ( x ) = 1( x − x ) α − g ( x − x ) . (35)To determine higher-order contributions, we assume that the function g ( x − x ) can be writtenas g ( x − x ) = g (1) ( x − x ) + ( x − x ) γ g (2) ( x − x ) + h . o . t ., (36)where the functions g (1) ( x − x ) and g (2) ( x − x ) are analytic with Taylor expansions g (1) ( ξ ) = ∞ X l =0 g (1) l ξ l , g (2) ( ξ ) = ∞ X l =0 g (2) l ξ l ; (37) nalytic structure of solutions of the one-dimensional Burgers equation with modified dissipation γ α − X m =0 ( − m (cid:18) α − m (cid:19) (2 α − m )!(2 α − γ ) α − − m + (4 α − α − . (38)Here ( γ ) α − m − is the Pochhammer symbol. One solution is γ = − ; the other solutions arecomplex and we denote them by γ ± i , i = 1 , ..., α − , with Re ( γ ± i ) > and ( γ + i ) ∗ = γ − i . Theterms ( x − x ) γ ± i imply that the asymptotic expansion of h ( k ) for k → ∞ has the form h ( k ) ≃ Ck α − e − δk (cid:16) b k + ... + α − X i =1 c i k − γ + i + α − X i =1 ( c i ) ∗ k − γ − i + ... (cid:17) (39)For case (ii), Eq. (32) becomes a difference-differential equation h ( x + 1) = ∂ x n h ( x ) + e h ( x ) o . (40)Solutions of this equation are entire functions [30]; therefore we concentrate on theirbehaviour for x → ∞ . Assuming that h ( x ) → ∞ for x → ∞ we write h ( x ) = exp[ S ( x )] obtaining e S ( x +1) = ∂ x n e S ( x ) + e S ( x )+1 o . (41)The dominant behaviour can be deduced from the relation S ( x + 1) = 2 S ( x ) , (42)which is solved by S ( x ) = β ( x ) e x ln 2 . (43)Here β ( x ) is a periodic function with period : β ( x + 1) = β ( x ) . Thus, to the leading orderthe solution is given by H ( x ) = exp (cid:2) β ( x ) e x ln 2 (cid:3) . (44)From this representation it is easy to determine the behaviour of h ( k ) as follows: Introducinga new variable ξ = e x we see that h ( k ) are the Taylor coefficients of ˜ h ( ξ ) = h (ln e x ) and thatfor ξ → ∞ ˜ h ( ξ ) ∼ exp (cid:2) β (ln ξ ) ξ ln 2 (cid:3) . (45)The function ˜ h ( ξ ) is thus an entire function of order ln 2 . It is well-known that the growthrate of entire functions at infinity determines the behavior of their Taylor coefficients for k → ∞ [39] so that h ( k ) ∼ e − k ln k . (46)Actually, the asymptotic behaviour of h ( k ) can be determined directly from the recursionrelation for h ( k ) along with sub-dominant terms h ( k ) ≃ √ π ln 2 k − e ( δ + g (ln k )) k e − k ln k , (47) nalytic structure of solutions of the one-dimensional Burgers equation with modified dissipation g ( · ) is periodic with period ln 2 . The presence of the function g (ln k ) in theasymptotic expansion of h ( k ) is related to the presence of the function β ( x ) in the expansionof h ( x ) at infinity.Finally, we remark that the estimate (46) of the dominant part in the high wavenumberasymptotics of solutions of the Burgers equation with exponentially growing dissipation canbe proved rigorously [37].
3. Bottleneck effect in the boundary layer of the one-dimensional Burgers equation
The analysis presented in the previous section applies only to small Reynolds numbers andcan thus be relevant only for the dissipation range. To study the transition zone betweenthe dissipation range and the inertial range we have to take recourse to asymptotic matchingwhich so far is known to work only for the Burgers equation. We write the Burgers equationwith modified dissipation in the form ∂ t u + u∂ x u = − ν f ( ν p − ∂ x ) u. (48)In the limit of vanishing viscosity ν → the outer solution, which is the entropic solution ofthe inviscid Burgers equation, is matched against the inner solution of the equation f r − d dX ! u (in) + u (in) ddX u (in) = 0 , (49)satisfying the boundary conditions lim X →−∞ u (in) ( X ) = 1 and lim X → + ∞ u (in) ( X ) = − [26, 40]. In this section we shall study various aspects of solutions of the inner equation(49), in particular, with an eye on the bottleneck effect. Equation (49), for the case of the hyperviscous dissipation term f ( k ) = k α , has been studiedby asymptotic and numerical methods in Refs. [26, 34]. The same methods are easily extendedto Eq. (49) for more general dissipation terms. Thus, in the case of a more general f ( k ) (here,we only assume that it grows faster than linearly) we can use asymptotic expansion of thesolution at ±∞ : neglecting nonlinear contributions Eq. (49) is written as f r − d dX ! u (in)as + ddX u (in)as = 0 , X → −∞ , (50) f r − d dX ! u (in)as − ddX u (in)as = 0 , X → + ∞ . (51)By using the ansätz u (in)as = e − iζx and u (in)as = e iζx we obtain an equation for ζ ζ f ( ζ ) = i. (52) nalytic structure of solutions of the one-dimensional Burgers equation with modified dissipation ζ i of Eq. (52) for which Im ζ i > . Let us consider the solution ζ min of Eq. (52) with the smallest imaginary part Im ζ min . The leading order asymptotics for X → ±∞ are u (in) ( X ) ≃ u (in)as ( X ) = 1 − Ae λX sin( ωX + φ ) , X → −∞ ; u (in) ( X ) ≃ u (in)as ( X ) = − − Ae − λX sin( ωX − φ ) , X → + ∞ ; (53)where λ = Im ζ and ω = Re ζ and A is chosen to be positive. In the case when Re λ min = 0 ,the solution oscillates around ± . However, neither the amplitude of the oscillation A , northe phase φ can be determined from a linear analysis.The Fourier transform of the linearized solution is ˆ u (in)as ( k ) = − i r π k − i r π A k k sin φ + λ sin φ − λ ω cos φ − ω sin φ ( λ + ω − ω k + k ) ( λ + ω + 2 ω k + k ) . (54)At high k the linearized asymptotic solution ˆ u (in)as ( k ) decays as − i p /π (1 + A sin φ ) k − contrary to the actual solutions of Eq. (49) which decay exponentially or faster thanexponentially. Nevertheless, we expect that the dissipation range is being mimicked alsofor the linearized asymptotic solution: For high k the asymptotic solution has to decreasefaster than the small k solution − i p /π k − and this is possible only when A sin φ < asconfirmed by numerical simulations.In Fig. (1), we compare numerical solutions and linearized asymptotic solutions for thehyperviscous dissipation term f ( k ) = k α with α = 4 , , . The agreement between the two is α = 4Spectrum of the linear asymptotic approximation for α = 4-2.5-2-1.5-1-0.5 0 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 -2.5-2-1.5-1-0.5 0 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 Fourier coefficients for α = 4Asymptotic approximation for α = 4 -2.5-2-1.5-1-0.5 0 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4k Fourier coefficients for α = 4Linear asymptotic approximation for α = 4Fourier coefficients for α = 5Linear asymptotic approximation for α = 5Fourier coefficients for α = 6Linear asymptotic approximation for α = 6 Figure 1.
Comparison between the imaginary part of numerical solution of Eq. (49) in theFourier space and the imaginary part of the linearized asymptotic solution Eq. (53) in theFourier space for f ( k ) = k (solid lines), f ( k ) = k (dashed lines) and f ( k ) = k (dottedlines). Note that in the bottleneck region the numerical and the asymptotic solutions have thesame shape with the asymptotic solution shifted down compared to the numerical solution. remarkably good, in particular in the bottleneck region they seem to have similar shapes. Wedo note, however, that the linear asymptotic solution is shifted with respect to the completesolution. Thus, although the expression (54) is an excellent model for the solution in thebottleneck region, there is a drawback: The amplitude A and the phase shift φ cannot bedetermined analytically and has to be extracted numerically. nalytic structure of solutions of the one-dimensional Burgers equation with modified dissipation G ( u (in) ) and integrate over X , obtaining Z R G ( u (in) ) f r − d dX ! u (in) dX = g (1) − g ( − , (55)where the functions G ( · ) and g ( · ) are related by ddu g ( u ) = uG ( u ) . (56)For G ( u ) = u , we obtain the relation Z R f ( k ) | ˆ u (in) ( k ) | dk = 23 . (57)We divide the Fourier space into three ranges: the small wave-number range ( − k i , k i ) ,corresponding to the inertial range, the intermediate wave-number range ( − k d , − k i ) ∪ ( k i , k d ) and the high wave-number range ( −∞ , − k d ) ∪ ( k d , + ∞ ) which corresponds to the dissipationrange. Because of the exponential decay of the Fourier coefficients in the dissipation rangethe contribution to the integral (57) from the high wave-number range is negligible.To a first approximation, we estimate the width of the small wave-number range ( − k i , k i ) by assuming that the entire contribution to (57) comes from this range π Z k i − k i f ( k ) k dk = 23 . (58)Obviously, the solution of Eq. (58) gives an upper bound for the higher end of the inertialrange. By setting f ( k d ) = 1 at the lower end of the dissipation range, we estimate thebeginning of the dissipation range. For the definitions of k i and k d to be consistent we requirethat k i < k d . However, for f ( k ) such that f ( k ) /k are small for small k this consistencycondition is violated.Consider for example a dissipation term given by f ( k ) = k + ak . Then k i and k d canbe calculated explicitly k i ( a ) = 12 q π + 2 √ a + π − a p π + 2 √ a + π k d ( a ) = 12 q − a + 2 √ a + 4 . (59)It follows that for a < a ⋆ ≈ . , where a ⋆ is the solution of k i ( a ⋆ ) = k d ( a ⋆ ) , theconsistency condition is violated and a significant contribution to the integral (57) has tocome from the intermediate (bottleneck) range.We perform detailed numerical simulations toconfirm this result. In Fig. (2) solutions of (49) are represented for a = 0 , / , / , and abottleneck is observed only for a = 1 / and a = 0 . For a = 1 there is clearly no bottleneckand there is practically no bottleneck in the case a = 1 / either.We remark that for a ∈ (0 , , that is for all values of a that we analyzed above,Eq. (52) has complex solutions. Thus, the corresponding solutions of Eq. (49) oscillate around ± for X → ±∞ . But, as can be easily seen in Fig. (2) the amplitude of the oscillationsdecreases with increasing a . Thus, we view the oscillations appearing in the solutions when nalytic structure of solutions of the one-dimensional Burgers equation with modified dissipation f ( k ) falls off too fast with k → as another manifestation of the bottleneck phenomenon.The mere possibility of oscillations in solutions of Eq. (49) does not necessarily lead to abump in the spectrum. A special case in which the bottleneck effect can be analyzed analytically is Eq. (49) withdissipation given by the function f ( k ) = k α , when α is close to unity. We write α = 1 + ε and use ε as a small parameter. Noting that k α = k k ε = k ∞ X n =0 n ! (2 ln k ) n ε n , (60)and assuming that u in has an expansion in powers of εu in = ∞ X n =0 u ( n ) ε n , (61)with u (0) = − tanh x being the exact solution of Eq. (49) in the case f ( k ) = k , we obtainthe following system of equations for the functions u ( n ) , n ≥ : at the leading order n = 1(2 ln p − ∂ x )( − ∂ x ) u + ( − ∂ x ) u (1) + u ∂ x u (1) + u (1) ∂ x u = 0 , (62)and for n > − ∂ x ) u ( n ) + u ∂ x u ( n ) + u ( n ) ∂ x u + n X m =1 (2 ln p − ∂ x ) m m ! ( − ∂ x ) u ( n − m ) + n − X m =1 u ( m ) ∂ x u ( n − m ) = 0 . (63)Now we show that at every fixed n , in particular at n = 1 , the function u ( n ) can be explicitlywritten in terms of u ( m ) , with ≤ m < n and their Fourier transforms. Indeed, uponintegrating the above equations we can rewrite them as ∂ x u ( n ) = u u ( n ) − g ( n ) , (64)where, for n > g ( n ) = 2 n n ! ∂ x (ln p − ∂ x ) n u + n − X m =1 m m ! ∂ x (ln p − ∂ x ) m u ( n − m ) − n − X m =1 u ( m ) u ( n − m ) (65)and g (1) = 2 ∂ x ln p − ∂ x u , (66)for n = 1 . Since for all n ≥ functions u ( n ) are odd, we can write the solutions of the linearinhomogeneous equations (64) as u ( n ) ( x ) = − x Z x g ( n ) ( x ′ ) cosh x ′ dx ′ , (67)or, inserting the expressions for g ( n ) , as u ( n ) = − n X m =1 m m ! 1cosh x Z x (cid:16) ∂ x (ln p − ∂ x ) m u ( n − m ) (cid:17) cosh x ′ dx ′ + 12 n − X m =1 x Z x u ( m ) u ( n − m ) cosh x ′ dx ′ . (68) nalytic structure of solutions of the one-dimensional Burgers equation with modified dissipation + k f(k) = k + 0.5 k f(k) = k + 0.25 k f(k) = k + k f(k) = k + 0.5 k f(k) = k + 0.25 k f(k) = k -1.2-1-0.8-0.6-0.4-0.2 0 0 2 4 6 8 10 12 14x f(k) = k + k f(k) = k + 0.5 k f(k) = k + 0.25 k f(k) = k Figure 2.
Numerical simulations of Eq. (49) with the dissipation term f ( k ) = k + ak ,with resolution N = 1024 and domain size L = 400 π for (a) and L = 100 π for (b). In (a)we represent the spectrum of the inner solution | ˆ u (in) | for a = 1 (solid line), a = (dashedline), a = (dash-dotted line) and a = 0 (dotted line). In the inset we represent f ( k ) | ˆ u (in) | for different values of a : a = 1 (solid line), a = (dashed line), a = (dash-dotted line)and a = 0 (dotted line). In (b) we represent the solutions u (in) in the physical space for a = 1 (solid line), a = (dashed line), a = (dash-dotted line) and a = 0 (dotted line). Theexponentially decaying oscillations around − become stronger for smaller a . Finally, by using explicit representation for the action of the pseudo-differential operators (ln p − ∂ x ) m we obtain the following expression for the function u ( n ) u ( n ) = n X m =1 m m ! a ( m ) n ( x ) + 12 n − X m =1 b ( m ) n ( x ) , (69)where a ( m ) n = 1 √ πi hZ R k (ln | k | ) m k ( F u ( n − m ) )( k ) sin kx dk + ddx tanh x Z R (ln | k | ) m ( F u ( n − m ) )( k ) sin kx k dk i (70)and b ( m ) n = 1cosh x Z x u ( m ) ( x ′ ) u ( n − m ) ( x ′ ) cosh x ′ dx ′ . (71)To study the bottleneck effect, it is enough to consider the first order term in Eq. (61), whichgives u (1) ( x ) = 2 Z R k ln | k | sinh( πk ) sin kx k dk + ddx x Z R ln | k | sinh( πk ) sin kx k dk (72)or, in the Fourier space, (cid:0) F u (1) (cid:1) ( k ) = 2 √ πi k ln | k | k πk + 2 √ πi k Z R k ′ ) + 1 ln | k ′ | sinh πk ′ π ( k − k ′ ) dk ′ , where the integral has to be regularized in a suitable sense. nalytic structure of solutions of the one-dimensional Burgers equation with modified dissipation The arguments presented in the previous section imply that the best way to generate abottleneck is to take for f ( k ) a function which vanishes for k smaller than a certain cut-off(which, without any loss of generality, we take to be ) and is infinite for k above the cut-off f tr ( k ) = ( | k | < , + ∞ for | k | > . (73)However, it is not clear how to implement such a dissipation term in Eq. (49). We approximatesuch a cut-off dissipation term by considering a function f ( k ) which depends on a certainparameter in such a way that when the parameter tends to infinity, f ( k ) tends to f tr ( k ) . Herewe consider two examples of such functions: (i) hyperviscosity f ( k ) = k α in the limit α → ∞ , a problem which has also been studied in [34] and (ii) a cosh-dissipation termexponentially growing for | k | → ∞ f ( k ) = e − µ (cosh µk − , (74)introduced in [30] and studied further in [41]. Both functions tend to f tr ( k ) for α → ∞ and µ → ∞ but behave differently in the dissipation range, as we have seen in Section 2.3.For both types of dissipation we found that the solutions in the Fourier space seem totend to a well-defined limit for | k | < and tend to zero for | k | > ; this is illustrated inFig. (3). The latter observation follows immediately from Eq. (57). The former follows fromthe numerical results for the hyperviscous and the cosh-dissipation terms, with representativeplots shown in Fig. (3), which also suggest that the limiting function, which we denote by u ∞ ,does not depend on the precise form of f ( k ) . Numerically, the high α and µ solutions in theneighborhood of ± are well described by the functional form ˆ u ∞ ( k ) = ( a (1 + k ) − ∆ + b, − < k < , k ∼ − , − a (1 − k ) − ∆ − b, < k < k ∼ . (75)A good agreement of the numerical data with the functional form (75) is achieved for ∆ ≈ / and a ≈ . , b ≈ . .Unfortunately, we did not manage to establish an equation for u ∞ , and thus, we donot have a theory which would explain the exponent in its Fourier space representation.The main difficulty in establishing such an equation consists in determining the α → ∞ or µ → ∞ limit of the right hand side of Eq. (49) which we denote by R ( u ∞ ) . Clearly, whereasthe support of the limiting function itself is supp u ∞ = [ − , , the support of R ( u ∞ ) iscontained in ( −∞ , − ∪ [1 , + ∞ ) . More precisely, since on the left hand side of Eq. (49)we have a quadratic term, the support is equal to [ − , − ∪ [1 , . Some information about R ( u ∞ ) can be obtained by using Eq. (55) for g ( u ) = u | u | and G ( u ) = sign( u ) , so that Z R sign( u (in) ) f r − d dX ! u (in) dX = 1 . (76) nalytic structure of solutions of the one-dimensional Burgers equation with modified dissipation -9-8-7-6-5-4-3-2-1 0 0.2 0.4 0.6 0.8 1 1.2k α = 10 α = 20 α = 30 α = 40 α = 50 α = 60 α = 70 α = 80 α = 90 α = 100 α = 110 α = 120 α = 130 α = 140 α = 150 α = 160 α = 170 α = 180 α = 190 α = 200-9-8-7-6-5-4-3-2-1 0 0.96 0.97 0.98 0.99 1 1.01 1.02 -6-5-4-3-2-1 0.5 0.6 0.7 0.8 0.9 1k Dissipation term k α Dissipation term e - µ (cosh µ k - 1)-6-5-4-3-2-1 0.96 0.97 0.98 0.99 1 1.01 Figure 3.
Numerical simulations of Eq. (49) with hyperviscous dissipation terms and cosh-dissipation terms. In (a) we represent solutions for hyperviscous dissipation terms with α =10 , , ..., . With increasing α the solution tends to zero for k > and seems to acquirea well-defined limit for k < . In (b) we compare solutions for hyperviscous dissipationterms with α = 10 , , ..., (solid lines) and cosh-dissipation terms with µ = 20 , , ..., (dashed lines). Based on numerical results we assume that sign( u (in) ) = − sign( X ) , which gives us thefollowing relation for the term on the right hand side of Eq. (49) Z R sign( X ) f r − d dX ! u (in) ( X ) dX = − , (77)from which follows, via Parseval’s theorem, Z R f ( k ) k ˆ u (in) ( k ) dk = − i r π . (78)Relations (57) and (78), combined with numerical results, suggest that R ( u ∞ ) is a functionand not a distribution. However, we did not succeed in determining the functional form of thisfunction.From the numerically obtained functional form of ˆ u ∞ we deduce the asymptotic form of u ∞ in the physical space for X → ∞ u ∞ ( X ) ≃ a q π Γ (cid:16) (cid:17) ( − X ) − sin (cid:16) X + π (cid:17) for X → −∞ , − a q π Γ (cid:16) (cid:17) X − sin (cid:16) X − π (cid:17) for X → + ∞ . (79)
4. Spectrum of the one-dimensional Burgers equation with modified dissipation
In Sections 2 and 3 we have studied simplified models derived for solutions of the Burgersequation. Whereas the results of Section 2 concern dissipation scales only, Section 3 dealswith the intermediate range between the inertial range and the dissipation range. In this sectionwe shall see how far the results of the previous two sections can be used to analyze numericalsolutions of the Burgers equation with modified dissipation in the Fourier space. We employthe following strategy: we solve the Burgers equation by using high-precision pseudo-spectral nalytic structure of solutions of the one-dimensional Burgers equation with modified dissipation k . We employ the Exponential Time Differencing Runge-Kutta scheme [45, 44].We concentrate essentially on the behavior of solutions in two ranges: the dissipationrange (or the high wavenumber range) and the bottleneck range (or the transition range fromthe inertial to the dissipation range).The functional form of solutions in the Fourier space in the dissipation range is studiedby using the example of two different kinds of dissipation: the hyperviscous dissipationand the cosh-dissipation. Here we have the advantage that our numerical investigations ofthe | k | → ∞ asymptotics can be checked against the theoretical predictions of Eqns. (35)and (47). This is important in particular with regard to numerical studies of more generalequations, such as the Navier–Stokes equations, for which analytical results concerning theform of the dissipation range are few.To analyze the asymptotics in the dissipation range numerically, we apply the asymptoticextrapolation procedure of van der Hoeven [36]. This procedure can be viewed as a sequenceof transformation techniques in which the main idea is to remove the higher leading-orderterms by applying a suitable sequence of transformation and then, knowing the sub-leadingorder terms, to obtain the leading-order terms. The choice of the order and the type oftransformations depends on the functional form of the analyzed sequence. In our case weessentially take the sequence used in [46] and [30].For the hyperviscous dissipation term ν α − k α the dissipation range begins roughly at /ν . Taking ν to be of order one gives us a solution which lies entirely in the dissipationrange. For such a solution the small Reynolds number results of Section 2 apply in the firstplace and thus give us a means to check the validity of the small-Reynolds number expansionof Section 2. Unfortunately, numerical analysis in the case of ν ∼ turns out to be difficult,because for such values of ν the solution in the Fourier space falls off very quickly, so thatvery high precision and extremely small time steps are required: the higher are the modeswhose Fourier coefficients we calculate, the higher the precision and computational accuracyis needed.As a consequence, in the expansion (35) only the leading and the two sub-leadingterms can be reliably determined. For example, the exponent of the algebraic prefactor isobtained with a relative precision of order − whereas for the rate of exponential decaywe get a precision of − as shown in Fig. (4). We did not succeed in determining anyfurther sub-leading terms, such as complex powers of k , because of several reasons related toinsufficiently small time steps and a lack of sufficient number of modes for extrapolation.Simulations employing cosh-dissipation give results similar to the hyperviscous case.Hence for the dissipation term f ( k ) = (cosh k − the leading-order term exp( − Ck ln k ) can be clearly identified. In particular, the numerical value of the constant C = − / ln 2 conjectured in [30] and predicted by Eq. (47) can be confirmed with certainty as shown inFig. (4b). Unfortunately, the determination of higher order term in the asymptotic expansionis hampered by the logarithmic-scale oscillations present in the next-order correction exp( kg (ln k )) (function g ( · ) is periodic with period ln 2 ) giving rise to the logarithmic scale nalytic structure of solutions of the one-dimensional Burgers equation with modified dissipation A sy m p t o t i c e x t r apo l a t i on r e s u l t s k Prefactor exponent for ∆ t = 10 -5 Prefactor exponent for ∆ t = 10 -6 -5.86248-5.86248-5.86248-5.86247-5.86247-5.86247-5.86247-5.86247-5.86247-5.86247-5.86247-5.86247 10 15 20 25 30 35Rate of exp. decay for ∆ t = 10 -5 Rate of exp. decay for ∆ t = 10 -6 Figure 4. (a) Results of asymptotic extrapolation procedure applied to the Fourier coefficientsof the solution of Eq. (2) with ν = 1 and α = 2 , initial condition sin x , calculated with 200digits and time step − . At the fourth stage of asymptotic extrapolation the sequence tendsto the constant value − /α . The deviations from this value is of the order − . oscillations in Fig. (4b).What happens when the dissipation term starts acting at wavenumbers much higher thanone, so that a substantial inertial range can be developed? As we shall see now, although thefunctional form predicted by the small Reynolds number expansion can be identified in thedissipation range, for dissipation terms producing large bottlenecks this becomes increasinglydifficult, since one has to go to higher and higher wavenumbers to recover the asymptoticbehavior of the Fourier coefficients of solutions. At the same time the parts of the bottleneckregion adjacent to the inertial range are satisfactorily described by the linear asymptoticapproximation based on Eq. (54).We have calculated solutions of the Burgers equation with hyperviscous dissipation termsof the type ν α − k α for small ν numerically by using high-precision for several values of α .For α = 2 the functional form of the dissipation range is identified quite accurately: Forexample, for the numerical solution using ν = 10 − , the exponent of the algebraic prefactoris determined with the relative precision of the order − , for the rate of exponential decaythe relative precision is of the order − as shown in Fig. (5). Remarkably, to obtain thefunctional form of the solution in the dissipation range accurately even for α = 2 we have togo quite far beyond the wavenumber /ν ≈ , i.e. the wavenumber at which dissipationsets in. For example, as can be seen in Figure 5, the relative error in the determination of theprepfactor exponent drops below − only for k > /ν ≈ . For α = 3 and ν = 10 − we would have to go even farther beyond the wavenumber /ν ≈ : for k > /ν ≈ and up to N/ the asymptotic extrapolation procedure for the algebraic prefactorexponent does not converge to any value, displaying oscillations similar to those in Fig. (5a),but much stronger. For the rate of exponential decay the error is of the order − if weassume that the algebraic prefactor is k . Even worse convergence to asymptotic behavior isobserved for the exponentially growing dissipation terms for which even the identification ofthe leading order term requires resolutions much higher than /ν .For the bottleneck region we use the results of Section 3.1, in particular the numerical nalytic structure of solutions of the one-dimensional Burgers equation with modified dissipation α = 22 1.94 1.96 1.98 2 2.02 2.04 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000 -0.014-0.0135-0.013-0.0125-0.012-0.0115-0.011 500 1000 1500 2000k Rate of exponential decay for α = 2-0.01348853-0.013489-0.0134888-0.0134886-0.0134884-0.0134882-0.013488 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 Figure 5.
Results of asymptotic extrapolation procedure applied to Fourier coefficients ofthe solution of Eq. (2)with ν = 10 − and α = 2 , initial condition sin x , calculated with 54digits, time step ∆ t = 10 − and resolution N = 2 at time t = 1 . . Panel (a) are theresults for the algebraic prefactor exponent for which the analytical value is α − . Thedeviations from the theoretical value are of order − for wavenumbers between and . Panel (b) shows the results of asymptotic extrapolation for the rate of exponential decayin the leading-order term. values of the amplitude A and of the phase shift φ . The functional form of solutions in thebottleneck region (Eq. (54)) is approximated by the linearized asymptotic solution ˆ u (in)as ( k ) ofthe boundary layer Burgers equation (49) as ˆ u ( k ) ≃ √ π Jk e ˆ u (in)as ( k/k e ) , (80)where J is the jump in the entropic solution at the shock, by Fast-Legendre transforms, and k e = α − √ J /ν is the effective dissipation wavenumber. As can be seen in Fig. (6) on theexample of the hyperviscous Burgers equation with α = 2 and ν = 10 − , the agreement ofthe approximative solution with the actual solution is extremely good.
5. Conclusions
In this article we have seen by using the example of the one-dimensional Burgers equationwith a modified dissipation term how the structure of solutions of a hydrodynamical equationcan be described by simplified models which can be obtained from the original equation bysystematic reduction. We have concentrated on the far dissipation region and the transitionregion from the inertial range to the dissipation range.To study the far dissipation region we have presented a method which allows us to studysolutions of hydrodynamical equations at small Reynolds numbers in domains with periodicinitial conditions. This method takes advantage of the fact that for initial conditions withsuitably restricted modes the interaction between modes is restricted and solutions can beobtained recursively without any errors due to truncation or time-stepping. It is applicableto more general hydrodynamical equations such as the Navier–Stokes equations which wasone of the reasons to present it here. For the one-dimensional Burgers equation in the limit nalytic structure of solutions of the one-dimensional Burgers equation with modified dissipation k Hyperviscous Burgers equation for α = 2Boundary layer hyperviscous Burgers equation for α = 2Linear asymptotic approximation for α = 2Inertial range 1e-04 0.001 100 Hyperviscous Burgers equation for α = 2Linear asymptotic approximation for α = 2Dissipation range asymptotics for α = 2Inertial range Figure 6.
Log-log scale representation of the Fourier coefficients of the solution of Eq. (2)with ν = 10 − and α = 2 , initial condition sin x , calculated with 54 digits, time step ∆ t = 10 − and resolution N = 2 at time t = 1 . . We compare this solution with thesolution of the boundary layer Burgers equation, the rescaled linearized asymptotic solutionand the inertial range scaling ∼ k − . In the inset the numerical solution of Eq. (2) is comparedwith the functional form in the dissipation range, the rescaled linearized asymptotic solutionand the inertial range scaling. of long times the problem can be simplified even further, so that the problem reduces to anon-linear difference-differential equation. By using this equation we have studied the highwavenumber asymptotics in detail and verified the results by using high-precision pseudo-spectral numerical simulations.We have seen that the transition range from the inertial range to the dissipation range inthe case of the Burgers equation can be described quite well by a linearized solution of theboundary layer problem in the neighborhood of shocks. However, in contrast to the studyof the far dissipation range where the analysis has been done by a method which a prioridoes not use any special properties of the one-dimensional Burgers equation, in the study ofthe intermediate range we had to rely on a very special property of the Burgers equation. Afurther drawback is that we were not able to determine analytically the amplitude and thephase of oscillations near the shocks and had to use numerics to determine them.How far the analysis presented in this article applicable to the Navier–Stokes equations?As we stated above, the method for the analysis of the far dissipation range presented herecan be extended to the Navier–Stokes equations in arbitrary dimensions. The main differenceto the Burgers case is that the corresponding recursion relations are hard to deal withanalytically and have to be studied numerically using high-precision arithmetics, analogouslyto singularities of the Euler equation [48, 49, 50]. The results of this ongoing work will bepublished elsewhere.The treatment of the bottleneck problem seems to be more difficult because the Burgerstype analysis does not apply to incompressible flows. It is known that the bump in the energy nalytic structure of solutions of the one-dimensional Burgers equation with modified dissipation [1] L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon Press, 1987), Volume 6 of Course ofTheoretical Physics, Second English Edition, Revised.[2] U. Frisch,
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