PHYSICAL REVIEW E 90, 032924 (2014)
Nonlinear decay of random waves described by an integrodifferential equation Sergey N. Gurbatov* Lobachevsky State University of Nizhni Novgorod, 23 Gagarin Ave., Nizhni Novgorod 603950, Russia
Oleg V. Rudenko† Lomonosov Moscow State University, Moscow 119991, Leninskie Gory, Moscow, Russia (Received 19 March 2014; published 29 September 2014) The evolution of random nonlinear waves (high-intensity noise) in a dissipative and dispersive media is studied. To describe wave processes, the mathematical model in the form of a nonlinear integrodifferential equation is used. The concrete integrand kernels are determined by both frequency-dependent absorption and velocity of the wave. The nonlinear energy loss of broadband noise is considered in two limiting cases: (i) at the initial stage of propagation, when the wave profile contains a small number of shock fronts, and (ii) at the later stage when the wave reshapes to a sawtoothlike one with randomly located shocks. DOI: 10.1103/PhysRevE.90.032924
PACS number(s): 05.45.−a, 43.25.−x
I. INTRODUCTION
Nonlinear wave propagation in media with complicated frequency-dependent properties is governed by integrodifferential equations. Such models are suggested for waves in geophysical structures, soft biological tissues, and other systems whose internal dynamics affects the wave. One of the typical forms written in dimensionless variables is [1] ∂p ∂p ˆ −p = L(p), ∂z ∂θ ∞ ∂2 ˆ L(p) =D 2 K(s)p(θ − s,z)ds. ∂θ 0
(1) (2)
Commonly, θ is the normalized time and z is the normalized distance. For acoustic waves p is the pressure. For a noisy wave, the problem is reduced to the search for the statistical characteristics of the field p(θ,z) at an arbitrary distance z. We assume that the initial statistics of the field at the source located at z = 0 is known. Physical properties of the medium are defined by the kernel K(s). Equation (1) is widely used for relaxing media where the kernel has the exponential form (see formula (7) of Ref. [2]). Energy exchange between the translational degree of freedom and vibrational and rotational degrees of freedom in gases, internal molecular processes in liquids, reversible chemical reactions such as dissociation recombination can serve as examples of relaxation processes. Waves in soft biological tissues are also described by this model but here commonly K(s) ∼ s −σ , 0 < σ < 1 (see (23), (25) of Ref. [3]). Waves in wet porous soils are described by the same model with σ = 0.5 (see (23), Ref. [4]). Nonlinear integrodifferential equations with a proper kernel can describe wave energy loss in the given medium due to three physical mechanisms: (i) viscosity and thermal conductivity,
*
[email protected] † [email protected]; Also at Lobachevsky State University of Nizhni Novgorod, 23 Gagarin Ave., Nizhni Novgorod 603950, Russia; Prokhorov General Physics Institute of Russian Academy of Sciences. 1539-3755/2014/90(3)/032924(7)
(ii) nonequilibrium relaxation processes, and (iii) nonlinear loss at shock fronts depending on the intensity of a wave. These processes go on simultaneously and interact with each other. To calculate correctly the structure and spectrum of a nonlinear field, it is necessary to take into account all kinds of energy loss. Correct calculation of the absorption of the nonlinear wave is necessary to evaluate the impact of the wave on the medium. In particular, the impact can be in the form of heating, radiation pressure, and acoustic streaming. These phenomena are used in modern technologies [5] and in medical applications [6]. In the limiting cases of short or long memory, Eq. (1) transforms to the Riemann-Hopf or Burgers equations. The well known Burgers equation describes a variety of nonlinear wave phenomena arising in the theory of wave propagation, acoustics, plasma physics, and so on (see, e.g., [7–13]). In the classical form ∂v ∂ 2v ∂v +v = ν 2. (3) ∂t ∂x ∂x This equation was originally introduced by Burgers as a model of hydrodynamical turbulence [14,15]. It shares a number of properties with the Navier-Stokes equation: the same type of nonlinearity of invariance groups and of energy-dissipation relation, the existence of a multidimensional version [16]. However, the Burgers equation is known to be integrable and therefore lacks the property of sensitive dependence on the initial conditions. Nevertheless, the differences between the Burgers and Navier-Stokes equations are as interesting as the similarities [17] and this is also true for the multidimensional Burgers equation. Here we will use the integrodifferential equations (1) in the form similar to the classical Burgers equation ∞ ∂v ∂v ∂2 K(s)v(x − s,t)ds. (4) +v =ν 2 ∂t ∂x ∂x 0 The description by model (4) uses the same approach as for the Burgers equation (3). We assume the initial random field v(x,t = 0) = v0 (x) at the moment t = 0 is given. The goal of the present paper is to investigate the evolution of random nonlinear waves described by the integrodifferential equation (4), when the dissipation plays the leading role. The
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SERGEY N. GURBATOV AND OLEG V. RUDENKO
PHYSICAL REVIEW E 90, 032924 (2014)
behavior of waves in such media is similar to the behavior of waves in the classical Burgers equation. The paper is organized as follows. In Sec. II, the problem is formulated. Some connected results about the decay at the initial stage of evolution of nonlinear random waves in the limit of zero viscosity (Riemann waves) are listed. The decay of nonlinear random Riemann waves exists due to the artificial attenuation when we use the double Fourier transformation from Eulerian to the Lagrangian coordinates in the solution for simple waves. In Sec. III, the influence of dissipation and dispersion described by the integral term of (4) is taken into account. A surprising phenomenon is discovered for broadband noise. Namely, even weak noise at small acoustic Reynolds numbers can transform to a strong nonlinear signal. In Sec. IV, the last stage of evolution is considered when the noise is a sequence of sawtooth pulses with the random position of shocks. The high-frequency asymptotic of noise spectrum at large times of evolution is determined by the kernel K(s) of integrodifferential equation (4) and by fluctuations of the width of shock fronts. II. DECAY OF RANDOM RIEMANN WAVES
We shall be concerned with the initial value problem for the Burgers equation (3) with zero viscosity, when this equation transforms to the simple wave (Riemann-Hopf) equation: ∂v ∂v +v = 0. ∂t ∂x
(5)
Equation (5) is a Eulerian representation of the flow of particles uniformly moving along the x axis. Let us assume that the particle at the point y at the initial moment of time t = 0 has the velocity v0 (y). Then in Lagrangian representation the particle’s motion is given by the following equations: X(y,t) = y + v0 (y)t, V (y,t) = v0 (y).
we have the implicit form of solution for the simple wave v(x,t) = v0 [x − tv(x,t)].
(10)
The problems of statistical description of the random simple (Riemann) wave in Eulerian representation are connected with the implicit form of this solution. Nevertheless, in the Lagrangian representation the statistical description is trivial. Based on the relation between statistical description in Eulerian and Lagrangian representation [9], it is possible to find exact expressions for both the spectra and probability distributions of a random field [8–10,13]. In particular, it was shown that, despite the strong distortions of the profile, the one-point probability distribution of a Riemann wave is conserved, while the higher probability distributions are distorted. Expressions for the energy spectra and correlation functions of Riemann (simple) waves were obtained in Refs. [18–20]. In the case of a statistically homogeneous Riemann field v(x,t) its spatial energy spectrum ∞ 1 E(k,t) = B(s,t)e−iks ds, (11) 2π −∞ where B(s,t) = v(x,t)v(x + s,t)
(12)
is the correlation function of the Eulerian field v(x,t), is equal to ∞ 1 E(k,t) = e−iks 2π (kt)2 −∞ × [θ2 (kt, − kt,s) − θ1 (kt)θ1 (−kt)]ds.
(13)
Here
θ2 (k1 ,k2 ,s) = ei[k1 v0 (x)+k2 v0 (x+s)] , θ1 (k) = eikv0 (x) (14)
(8)
are the two-point and one-point characteristic functions of the initially statistically homogeneous field v0 (x). Let us discuss some characteristic features of evolution of the spectral density of the statistically homogeneous Riemann field v(x,t) by using the initially Gaussian field v0 (x) with a zero mean value and a given correlation function B0 (s), B0 (0) = σ02 . In this case, the spectral density (13) takes on the form [18–21] ∞ B0 (s)k2 t 2 1 −(σ0 kt)2 E(k,t) = e e − 1 e−iks ds. (15) 2 2π (kt) −∞
Let the field v(x,t) of particle velocities in a flow be given as a function of the Eulerian coordinate x and time t. If, in addition to that, the mapping (7) of the Lagrangian to Eulerian coordinates is also known, then the dependence of the velocity field on the Lagrangian coordinates are given by the following equation:
Let us also note that the rate of nonlinear self-action and generation of new harmonics of the field v(x,t) depends on the spatial frequency k in the following way: the smaller k, the slower these processes. Therefore, at σ0 kt 1, it is possible to expand the exponents in (15) into a Taylor series and retain several first terms, e.g.:
V (y,t) = v[X(y,t),t].
E(k,t) = E0 (k) + 12 (kt)2 [E0 (k) ⊗ E0 (k)
(6)
By varying y, we obtain the laws of motion of other particles in the flow. The mapping from the Lagrangian into Eulerian coordinates is described by the following equation: x = X(y,t).
(7)
In the case of uniformly moving particles, this equation has the following form: x = y + v0 (y)t.
(9)
In what follows, the fields describing the behavior of particles in the Lagrangian coordinate system will be called the Lagrangian fields, and the fields in the Eulerian coordinate system will be referred to as the Eulerian fields. So v(x,t) is the Eulerian particle-velocity field, and X(y,t) is the Lagrangian field of the Eulerian coordinates of the particles. So, from (9)
− 2E0 (0)E0 (k)] + · · · ,
(16)
where E0 (k) is the spectrum of the initial field v0 (x), the sign ⊗ denotes the convolution operation. Taking only the first term on the right-hand side into account corresponds to neglecting nonlinear effects; two first terms take nonlinear interaction of pairs of initial field harmonics into account, which leads to the
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NONLINEAR DECAY OF RANDOM WAVES DESCRIBED BY . . .
appearance of spectral components with difference and sum wave numbers (single interaction), etc. Let us now consider the behavior of the spectral density of the field v(x,t) at large wave numbers k. In this case, to evaluate the integral (15), it is possible to use the method of steepest descent. By retaining in Eq. (15) the first few terms of the Taylor expansion of the correlation function B0 (s) with respect to powers of s, k12 s 2 k24 s 4 2 + − ··· , (17) B0 (s) = σ0 1 − 2! 4! we obtain: σ02
1 E(k,t) = √ exp − 2 3 2τ k1 2π (κτ ) τ = σ0 k1 t = t/tnl , tnl = 1/σ0 k1 ,
,
(18)
v (x , t ) + − −6
−4
−2
2
4
kx
+ FIG. 1. Field v± (x,t) found by summing 50 terms of the series (24) at τ = kat = 2 [13]. The dashed line shows the multistream velocity field of uniformly moving particles. Pluses and minuses indicate the sign, with which each stream is added to the total sum. Characteristically pointed peaks at the boundaries of the intervals of multistreamness are due to nondifferentiability of the velocity field at these points.
κ = k/k1 . (19)
where tnl is a nonlinear time of evolution. From here, it follows that the spectral density of the Riemann field decays at k → ∞ according to the universal power law E(k,t) ∼ k −3 .
PHYSICAL REVIEW E 90, 032924 (2014)
velocities vi (x,t) of the multistream Riemann solution: n n i−1 v± (x,t) = (−1) vi (x,t) = (−1)i−1 v0 [yi (x,t)]. i=1
i=1
(20)
(23)
Nevertheless, the physical interpretation of these results is not trivial. For a Riemann wave, the energy should be conserved; however, from the expressions obtained for the spectra (16) it follows that the energy, when calculated as the integral over the spectrum, decreases with distance from the input [9]. This is related to the fact that the passage from the Eulerian description to the Lagrangian one in the spectral representation is equivalent to the replacement of the multivalued solution by a single-flux one, which is obtained by an alternating-sign summation of the branches of the Riemann solution [9,13,19–21]. In the case of Gaussian statistics, the regions of ambiguity of the Riemann wave appear at arbitrarily small distances, so we have the decay of energy in (16) from the very beginning. It is also true for the expression for the asymptotic of the spectra (18), where the universal power law (20) is due to the appearance of singularities of the velocity field v(x,t) ∼ (x − x∗ )1/2 , and the factor exp(−1/2τ 2 ) in (18) is proportional to the average number of singularities in the random Riemann solution. These phenomena will be clear if we use the transformation from Eulerian coordinates to Lagrangian in the Fourier transform of the velocity ∞ 1 v(κ,t) ˜ = v(x,t) e−iκx dx 2π −∞ ∞ 1 ∂X(y,t) dy. (21) v(κ,t) ˜ = e−iκX(y,t) v0 (y) 2π −∞ ∂y
Figure 1 shows the field (23) for a sinusoidal initial condition, constructed by partial summation of the series
where X(y,t) = y + v0 (y)t (6). The inverse Fourier integral of (21) ∞ v∗ (x,t) = v(κ,t) ˜ eiκx dκ
v± (x,t) = 2a
∞
(−1)n+1
n=1
Jn (nτ ) sin(nkx). nτ
Thus, it means that we have an artificial attenuation of random multiflux Riemann waves. This decay goes on somewhat faster than that for the shocked solution. Nevertheless, based on these equations, we can estimate the decay of high-intensity multiflux noise on the initial stage, and moreover, we can estimate the influence of nonlinear distortion on the decay of noise governed by integrodifferential equation (4). We will first consider the decay of the artificial random Riemann wave. After the inverse Fourier transform of (13) for the energy of the artificial Riemann wave, we have y 1 ∞ ∂ 2 B0 (y)dy. (25) g σ (t) = 2 −∞ ∂y 2t(σ02 − B0 (y))1/2 z 2 2 e−s ds. (26) g(z) = Erf(z) ≡ √ π 0 Assuming that the initial correlation function has the form B0 (y) =
σ02 1 + k∗2 y 2
(22)
is a single-valued function. It can easily be shown that the field defined in this way is an alternating-sign sum of stream
(27)
from (25), (27) for the energy of the artificial Riemann wave we have σ 2 (t) = σ02 [1 − (t)]
−∞
(24)
(28)
√ 2
(τ ) = (1 + τ −2 )[1 − Erf(1/ 2τ )] − τ −1 e−1/2τ 2/π (29) √ where τ = 2σ0 k∗ t is dimensionless time. From (29) we have that even at τ = 1 the additional nonlinear decay is relatively
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Assuming that nonlinear distortion of the wave is relatively small for the correlation function of the noise described by the integrodifferential equation we have [23] 1 ∞ ∂ g σ 2 (t) = 2 −∞ ∂y
y × 2 1/2 Blin (y,t)dy. (34) 2t σlin (t) − Blin (y,t)
FIG. 2. The additional nonlinear decay (τ ) as function of dimensionless time τ (29).
small— (τ = 1) ≈ 0.15 (see Fig. 2). At (τ 1) we have from (29) 4 3 1 (τ ) = √ τ exp − 2 . (30) 2τ 2π In the general case at the initial stage of evolution (τ = σ0 k1 t 1) we can use in (25) the first few terms of the Taylor expansion of the correlation function B0 (s) with respect to powers of s (17) and for the nonlinear decay of artificial Riemann we obtain 2 t (31) (t) = Cart k14 k2−4 (t/tnl )3 exp − nl2 , 2t √ where Cart = 24/ 2π , tnl = 1/σ0 k1 is a nonlinear time (19). For Gaussian statistics, the mean number of singularities in random Riemann solutions is proportional to exp(−1/2τ 2 ) [9,21]. Consequently, the additional nonlinear decay in (25) is directly caused by the appearance of artificial shocks. We have the same equation for the decay of a random shock wave at the initial stage [22], (see also Eq. (5.17) in Ref. √ [9]), but with a different numerical constant Cshock = 54/π . So it means that the equation for the energy of the Riemann wave (25) qualitatively describes the decay of the shock waves at the initial stage. In the next section we consider the influence of nonlinear decay on the propagation of random waves in media with dissipation and weak dispersion. III. EVOLUTION OF NONLINEAR RANDOM WAVES IN MEDIA WITH DISSIPATION AND WEAK DISPERSION
Let us consider the decay of the wave described by integrodifferential equations (4). In the linear case, for the correlation function of the velocity field we have [1] ∞ Elin (k,t) exp(−ikx)dk, (32) Blin (x,t) = −∞
where the linear spectrum is 2 Elin (k,t) = E0 (k) exp −2νk 0
∞
K(y) cos(ky)dy .
(33)
To get this expression we introduce in (4) a new variable that is equal to the transformation from Eulerian to Lagrangian coordinates, and use the double Fourier transform and statistical averaging, similar to that for the Riemann equation. As for the Riemann wave, we can use in (34) the first few terms of the Taylor expansion of the correlation function Blin (y) with respect to powers of y: k24 (t)y 4 k12 (t)y 2 2 + − · · · , (35) Blin (y) = σlin (t) 1 − 2! 4! where the parameters of expansion are determined by the first moments of the linear spectrum ∞ 2 2n σlin (t)kn (t) = k 2n Elin (k,t)dk. (36) −∞
And then from (34) we have 2 (t)[1 − (t)]. σ 2 (t) = σlin
2 t (t) (t) = Cart k14 (t)k2−4 (t)[t/tnl (t)]3 exp − nl 2 . 2t
(37) (38)
Here the local nonlinear time tnl (t) is determined as tnl (t) = 1/σlin (t)k1 (t).
(39)
We will now consider the evolution of noise in media, when the dissipation is proportional exp(−2tνk m ). For the first moments of the linear spectrum, which determined the nonlinear decay, we have ∞ 2 σlin (t)kn2n (t) = 2 k 2n E0 (k) exp(−2tνk m )dk, (40) 0
where E0 (k) is the initial spectrum of the noise. Let us consider the evolution of truncated white noise E0 (k) ∼ exp[−(k/k0 )2 ] described by the Burgers equation (m = 2). From (40), (38) for the additional nonlinear decay we have 2 t (t) = C2 (1 + t/tlin )−9/4 (t/tnl )3 exp − nl2 (1 + t/tlin )3/2 . 2t (41) 2 Here C2 = Cart /3, √ tlin = 1/2νk0 is the time of linear decay and tnl = 1/σ0 k1 = 2/σ0 k0 is the nonlinear time of evolution. The ratio of linear and nonlinear times is an initial Reynolds number of Re0 = σ0 /k0 ν = tlin /tnl . From (41) we see that linear attenuation leads to a reduction of nonlinear effects. Nevertheless, even in case when Re0 1, at rather large times the linear stage of evolution will be transformed to the nonlinear stage. From (41) we have that at t tlin the
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NONLINEAR DECAY OF RANDOM WAVES DESCRIBED BY . . .
additional nonlinear decay may be written as
tnl2 3/4 (t) ∼ t exp − . 3/2 2t 1/2 tlin
where ψ0 (x) is the initial potential of the velocity field. Accordingly, for the velocity field we have (42)
From this equation we see, that the nonlinear effects become important at time tnl∗ = tlin (tnl /tlin )4 = tlin Re−4 0 . This effect has a simple interpretation. Assuming that we have only a linear attenuation for the local Reynolds number Re(t) we have Re(t) =
σlin (t) = Re0 (1 + t/tlin )1/4 . νk1 (t)
(43)
The dissipation leads to a decrease in the amplitude of the wave σlin (t) and an increase of spatial scale L(t) = 1/k1 (t), and for the truncated white noise the increase in the scale dominates over the decrease in the amplitude. Due to these phenomena, the Reynolds number increases with time and the linear stage of evolution is followed by the nonlinear one. Moreover, it is possible to show from the exact Hopf-Cole solution of the Burgers equation, that for E0 (k) ∼ k n and n < 1 a strongly nonlinear regime of sawtooth waves is actually set up, independently of the Re0 value, and that the field statistical properties become self-similar [9]. Finally consider the general case when the dissipation is proportional to exp(−2tνk m ) and the initial spectrum has the form E0 (k) ∼ k n exp[−(k/k0 )2 ].
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(44)
In this case, we have that at t tlin the additional nonlinear decay may be written as
tnl2 (t) ∼ exp − . (45) (3+n)/m 2t (2m−n−3)/m tlin
v(x,t) = −ψx (x,t) =
(47)
where y(x,t) is the coordinate of the maximum of (46) for given x and t. At long time, the parabola (x − y)2 /2t in Eq. (46) becomes a smooth function at the scale of the initial potential ψ0 (y), and the coordinates of the global maxima in (46) practically coincide with the coordinates yk of some local maxima of ψ0 (y), which are zeros of the velocity field with a positive gradient. Hence, at long time, a continuous random field is transformed into a random sawtooth wave—a sequence of cells with the universal behavior within each cell v(x,t) = (x − yk )/t, but with random positions of discontinuities separating the cells. Here yk are the zeros of the sawtooth wave. Coalescence of the cells leads to the growth of the overall scale of the turbulence L(t), therefore the energy of the random field E(t) ∼ L2 (t)/t 2 decreases slower than the energy of a periodic signal. The rate of coalescence of discontinuities is determined by the statistical characteristics of the discontinuity velocities, which, in their turn, depend on the initial potential ψ0 (x). At long time, after multiple coalescence of discontinuities, the properties of the turbulence are determined by the statistical properties of the increments ψ0 (x + L) − ψ0 (x) at large separations L, and at Gaussian statistics of the initial field—by the asymptotic behavior of the structure function of the initial potential. Thus, the turbulence-development scenario is determined by the behavior of the large-scale part of the initial energy spectrum E0 (k) of the velocity field [9–13]. If the Burgers turbulence is indeed characterized by a single scale, then, assuming self-similarity, its spectrum may be represented in the form [9–13,24,25]:
Consequently, we have that if m > m∗ = (3 + n)/2 the linear stage of evolution will be transformed to the nonlinear stage.
E(k,t) =
L3 (t) ˜ E[kL(t)]. t2
(48)
For sufficiently large wave numbers, the behavior of the spectrum is always determined by discontinuities. Therefore
IV. LATE STAGE OF EVOLUTION
In the last part of the paper we consider the evolution of the wave at the late stage, when the wave is a sequence of sawtooth waves with random positions of the shocks. At the late stage, the high-frequency asymptotics of noise will be determined by the kernel K(s) of the integrodifferential equation and fluctuations in the width of discontinuities. Let us first discuss the properties of the random solutions of (3), (4) in the limit of vanishing viscosity, assuming that the spectral density at large scales (small wave numbers k) is in the form of a power law: E0 (k) ∼ |k|n . This situation is equal to the case of free decaying of the Burgers turbulence. In the theory of the Burgers turbulence, it is customary to introduce a potential ψ(x,t) of the velocity field v(x,t) = −ψx (x,t). In the limit ν → 0, the solution for the potential is representable in the following form (see, e.g., [9–13]): (x − y)2 , ψ(x,t) = max ψ0 (y) − y 2t
x − y(x,t) , t
(46)
E(k,t) ∼
B(t) k2
(large k),
(49)
and, from Eq. (48), it follows that B(t) ∼ L(t)/t 2 . Thus a general property of the decaying Burgers turbulence is the establishment of self-similarity at long times and, in particular, the self-similar behavior of the spectrum (48). In the case of finite viscosity, the shocks have the structure determined by properties of the kernel K(s) in (1), (4). Let us assume that we know the stationary solution of Eq. (4) ∞ ∂2 ∂F (x) =ν 2 U F (x) K(s)F (x − s)ds (50) ∂x ∂x 0 with the boundary condition for the dimensionless function F (x): F (−∞) = 1, F (∞) = −1, and U is the amplitude of the stationary solution. We assume that the structure of the shocks of sawtooth waves is the same as the structure of the stationary wave and its amplitude U (t) is determined by the zero viscosity
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solution U = ηk /2t = (yk+1 − yk )/t.
(51)
It is convenient to consider first the spectrum of the gradient of the velocity field, which is a sequence of the pulses situated at the position of the shocks 1 1 ηi ∂v = − f (x − xk ). (52) q(x,t) = ∂x t t 2t Here ∂F (x) f (x) = . (53) ∂x It is obvious that Ev (k,t) = Eq (k,t)/k 2 . If cq (k) is the Fourier transform of the function f (x) ∞ cq (k) = f (x) exp(ikx)dx (54) −∞
for the energy spectrum of the pulse field q(x,t) at large k [k 1/L(t)], when one can neglect interpulse interference of Fourier components of the velocity gradient pulses, we have [9] ¯ ηi 2 2 n(t) Eq (k,t) = cq (k,t) . (55) 2π 2t
In the viscosity interval, the behavior of the spectrum depends on the statistical properties of probability distribution Wη ( ). Here, we discuss the evolution of the Burgers turbulence at n > 1, when the initial field has Gaussian statistics, and the velocity and potential are statistically homogeneous. The statistics of the velocity field is determined by the statistical characteristics of the absolute maximum coordinates. At long times, the parabola in (46) becomes a smooth function at the scale of the initial action ψ0 (x), and for the right to be the absolute maximum in (46) competes a large number of local maxima of ψ0 (x). This made it possible to use for an analysis of the Burgers turbulence at long time [9,20,27,28] the statistical theory of large overshoots [29]. In the limit of vanishing viscosity, when the time t tends to infinity, the statistical characteristics of the Burgers turbulence become self-similar and, in particular, the energy spectrum assumes the form (48). The integral scale of the turbulence L(t) is given by the following expression t 1/2 −1/4 , (59) L(t) ≈ (tσψ ) ln 2π tnl where
¯ ∼ 1/L(t) is the average number of the shocks per Here n(t) unit length, and the angle brackets mean the averaging over random values ηi . We need to stress that the width of the shock F (x) and consequently, the function cq (k,t) depends on random amplitude of the shock ηi /2t. In the general case we can introduce the inner scale of turbulence as δ ∼ tν/L(t). If in the sequent space interval the stationary front has the singularities of the type F (x − x∗ ) = A(U )|x − x∗ |α , (α = 0 is equal to the discontinuities of the velocity field and α = 1— to the discontinuities of the velocity gradient). Then for the asymptotic of the energy spectrum we have ηi ¯ A2 (56) Ev (k,t) ∼ n(t) k −2(α+1) . 2t
are the nonlinearity time and the initial integral scale of the turbulence respectively. In this case, for the probability distribution Wη ( ) we have [27] 2 exp − . (61) Wη ( ) = 2L2 (t) 4L2 (t) √ ¯ = 1/ π L(t). In the internal interval k > 1/δ we and n(t) can use in (58) the asymptotic sinh(x) ≈ exp(x)/2 and in this interval for the spectrum we have
From this equation we see that the fluctuation of the shock width does not change the shape of the spectrum. Thus for the classical Burgers turbulence we have in the inertial interval of the spectrum Ev (k,t) ∼ k −2 . The propagation of the intense acoustics in the tube can be described√ by Eq. (1), where in the spectral form the operator is ˆ L(iω) ∼ ω(1 − i). In this case we have the discontinuities of the field of velocity derivative. It leads to the asymptotic E(ω,z) ∼ ω−4 , which was observed experimentally [26]. In the case of the Burgers turbulence, when we have the analytical solution for the stationary front and in (53) (57) f (x) = sec2 (x/δi ).
V. DISCUSSION AND CONCLUSION
here δi = 4νt/ηi is the width of the shock. We shall assume that we know the probability distribution Wη ( ) of the distance between the zeros ηi = yi+1 − yi of the sawtooth wave. Then from (55), (57), we have for the asymptotic of energy spectrum ∞ Wη ( )d 2 ¯ Ev (k,t) = 2π ν n(t) . (58) sinh2 (2π νkt/ ) 0 In the inertial interval of the spectrum 1/L k 1/δ, we have from the spectrum the universal behavior Ev (k,t) ∼ k −2 (49).
tnl ≡ L20 /σψ = L0 /σv , L0 ≡ σψ /σv
Ev (k,t) ∼ exp{−3[π δ(t)k]2/3 }.
(60)
(62)
Here δ = νt/L(t) is the internal scale of turbulence. We need to stress that for the turbulence the spectrum in the internal interval falls off slower then for the periodic signals where E(k,t) ∼ exp[−kδ(t)] and δ = νt/L0 .
Here we have considered the evolution of random nonlinear waves in media with dissipation and weak dispersion. The behavior of waves in such media is similar to the behavior of the waves described by the Burgers equation. We have also considered here the influence of nonlinearity and dissipation on the decay of broadband noise in two limiting cases: at the initial stage, when the number of the shocks is relatively small, and at the late stage, when the wave is a sequence of sawtooth waves with the random positions of the shocks. It is well known that for the periodic waves at large Reynolds numbers due to the increasing width of the shock, the nonlinear stage of evolution is replaced by the linear stage of decay. For the broadband noise the nontrivial behavior may exist: even for small initial Reynolds numbers, the linear stage of decay may change to the nonlinear stage of evolution. In the limiting case of vanishing viscosity, the continuous wave transforms
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NONLINEAR DECAY OF RANDOM WAVES DESCRIBED BY . . .
to the sequence of sawtooth waves. For the complex media with dissipation and weak dispersion, the shocks have a finite width, and the structure of the shocks will be determined by the kernel K(s) of the integrodifferential equation.
[1] O. V. Rudenko and S. N. Gurbatov, Acoust. Phys. 58, 243 (2012). [2] O. V. Rudenko and S. I. Soluyan, Sov. Phys. Acoust. USSR 18, 352 (1973). [3] A. Sarvazyan, O. V. Rudenko, S. D. Swanson, J. B. Folwkes, and S. Y. Emelianov, Ultrasound Med. Biol. 24, 1419 (1998). [4] O. V. Rudenko, A. L. Sobisevich, L. E. Sobisevich, C. M. Hedberg, and N. V. Shamaev, Acoust. Phys. 58, 99 (2012). [5] O. V. Rudenko, Phys.-Usp., Adv. Phys. Sci. 49, 69 (2006). [6] O. V. Rudenko, Phys.-Usp., Adv. Phys. Sci. 50, 359 (2007). [7] G. B. Whitham, Linear and Nonlinear Waves (John Wiley & Sons Inc., New York, 1974). [8] O. V. Rudenko and S. I. Soluyan, Theoretical Foundations of Nonlinear Acoustics (Plenum Press, Consultants Bureau, New York, 1977). [9] S. N. Gurbatov, A. N. Malakhov, and A. I. Saichev, Nonlinear Random Waves and Turbulence in Nondispersive Media: Waves, Rays, Particles (Manchester University Press, Manchester, 1991). [10] W. A. Woyczynski, Burgers–KPZ Turbulence. Gottingen Lectures (Springer-Verlag, Berlin, 1998). [11] U. Frisch and J. Bec, Burgulence. Les Houches 2000: New Trends in Turbulence (Springer EDP-Sciences, Berlin, 2001). [12] J. Bec and K. Khanin, Phys. Rep. 447, 1 (2007). [13] S. N. Gurbatov, O. V. Rudenko, and A. I. Saichev, Waves and Structures in Nonlinear Nondispersive Media. General Theory and Applications to Nonlinear Acoustics (Springer-Verlag and HEP, Berlin, 2012).
PHYSICAL REVIEW E 90, 032924 (2014) ACKNOWLEDGMENTS
We have benefited from discussions with A. Saichev and I. Demin. The work is supported by the Russian Science Foundation, Grant No. 14-12-00882.
[14] J. M. Burgers, Kon. Ned. Akad. Wet. Verh. 17, 1 (1939). [15] J. M. Burgers, The Nonlinear Diffusion Equation (Reidel, Dordrecht, 1974). [16] U. Frisch. Turbulence: The Legacy of A. N. Kolmogorov (Cambridge University Press, Cambridge, 1995). [17] R. Kraichnan, Phys. Fluids Mech. 11, 265 (1968). [18] O. V. Rudenko and A. S. Chirkin, Sov. Phys. Dokl. 19, 64 (1974). [19] A. I. Saichev, Radiophys. Quantum Electron. 17, 781 (1974). [20] J. D. Fournier and U. Frisch, J. Mec. Theor. Appl. (France) 2, 699 (1983). [21] S. N. Gurbatov and A. I. Saichev, Chaos 3, 333 (1993). [22] S. N. Gurbatov, Radiophys. Quantum Electron. (Engl. Transl.) 20, 73 (1977). [23] O. V. Rudenko, S. N. Gurbatov, and I. Yu. Demin, Acoust. Phys. 59, 584 (2013). [24] S. N. Gurbatov, S. I. Simdyankin, E. Aurel, U. Frisch, and G. T. T´oth, J. Fluid Mech. 344, 339 (1997). [25] A. Noullez, S. N. Gurbatov, E. Aurel, and S. I. Simdyankin, Phys. Rev. E 71, 056305 (2005). [26] L. Bj¨orn¨o and S. N. Gurbatov, Sov. Phys. Acoust. 31, 179 (1985). [27] S. N. Gurbatov and A. I. Saichev, Sov. Phys. JETP 53, 347 (1981). [28] S. N. Gurbatov and A. I. Saichev, Radiophys. Quantum Electr. (Engl. Transl.) 27, 303 (1984). [29] M. R. Leadbetter, G. Lindgren, and H. Rootzen, Extremes and Related Properties of Random Sequences and Processes (Springer, Berlin, 1983).
032924-7
Nonlinear decay of random waves described by an integrodifferential equation Sergey N. Gurbatov* Lobachevsky State University of Nizhni Novgorod, 23 Gagarin Ave., Nizhni Novgorod 603950, Russia
Oleg V. Rudenko† Lomonosov Moscow State University, Moscow 119991, Leninskie Gory, Moscow, Russia (Received 19 March 2014; published 29 September 2014) The evolution of random nonlinear waves (high-intensity noise) in a dissipative and dispersive media is studied. To describe wave processes, the mathematical model in the form of a nonlinear integrodifferential equation is used. The concrete integrand kernels are determined by both frequency-dependent absorption and velocity of the wave. The nonlinear energy loss of broadband noise is considered in two limiting cases: (i) at the initial stage of propagation, when the wave profile contains a small number of shock fronts, and (ii) at the later stage when the wave reshapes to a sawtoothlike one with randomly located shocks. DOI: 10.1103/PhysRevE.90.032924
PACS number(s): 05.45.−a, 43.25.−x
I. INTRODUCTION
Nonlinear wave propagation in media with complicated frequency-dependent properties is governed by integrodifferential equations. Such models are suggested for waves in geophysical structures, soft biological tissues, and other systems whose internal dynamics affects the wave. One of the typical forms written in dimensionless variables is [1] ∂p ∂p ˆ −p = L(p), ∂z ∂θ ∞ ∂2 ˆ L(p) =D 2 K(s)p(θ − s,z)ds. ∂θ 0
(1) (2)
Commonly, θ is the normalized time and z is the normalized distance. For acoustic waves p is the pressure. For a noisy wave, the problem is reduced to the search for the statistical characteristics of the field p(θ,z) at an arbitrary distance z. We assume that the initial statistics of the field at the source located at z = 0 is known. Physical properties of the medium are defined by the kernel K(s). Equation (1) is widely used for relaxing media where the kernel has the exponential form (see formula (7) of Ref. [2]). Energy exchange between the translational degree of freedom and vibrational and rotational degrees of freedom in gases, internal molecular processes in liquids, reversible chemical reactions such as dissociation recombination can serve as examples of relaxation processes. Waves in soft biological tissues are also described by this model but here commonly K(s) ∼ s −σ , 0 < σ < 1 (see (23), (25) of Ref. [3]). Waves in wet porous soils are described by the same model with σ = 0.5 (see (23), Ref. [4]). Nonlinear integrodifferential equations with a proper kernel can describe wave energy loss in the given medium due to three physical mechanisms: (i) viscosity and thermal conductivity,
*
[email protected] † [email protected]; Also at Lobachevsky State University of Nizhni Novgorod, 23 Gagarin Ave., Nizhni Novgorod 603950, Russia; Prokhorov General Physics Institute of Russian Academy of Sciences. 1539-3755/2014/90(3)/032924(7)
(ii) nonequilibrium relaxation processes, and (iii) nonlinear loss at shock fronts depending on the intensity of a wave. These processes go on simultaneously and interact with each other. To calculate correctly the structure and spectrum of a nonlinear field, it is necessary to take into account all kinds of energy loss. Correct calculation of the absorption of the nonlinear wave is necessary to evaluate the impact of the wave on the medium. In particular, the impact can be in the form of heating, radiation pressure, and acoustic streaming. These phenomena are used in modern technologies [5] and in medical applications [6]. In the limiting cases of short or long memory, Eq. (1) transforms to the Riemann-Hopf or Burgers equations. The well known Burgers equation describes a variety of nonlinear wave phenomena arising in the theory of wave propagation, acoustics, plasma physics, and so on (see, e.g., [7–13]). In the classical form ∂v ∂ 2v ∂v +v = ν 2. (3) ∂t ∂x ∂x This equation was originally introduced by Burgers as a model of hydrodynamical turbulence [14,15]. It shares a number of properties with the Navier-Stokes equation: the same type of nonlinearity of invariance groups and of energy-dissipation relation, the existence of a multidimensional version [16]. However, the Burgers equation is known to be integrable and therefore lacks the property of sensitive dependence on the initial conditions. Nevertheless, the differences between the Burgers and Navier-Stokes equations are as interesting as the similarities [17] and this is also true for the multidimensional Burgers equation. Here we will use the integrodifferential equations (1) in the form similar to the classical Burgers equation ∞ ∂v ∂v ∂2 K(s)v(x − s,t)ds. (4) +v =ν 2 ∂t ∂x ∂x 0 The description by model (4) uses the same approach as for the Burgers equation (3). We assume the initial random field v(x,t = 0) = v0 (x) at the moment t = 0 is given. The goal of the present paper is to investigate the evolution of random nonlinear waves described by the integrodifferential equation (4), when the dissipation plays the leading role. The
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behavior of waves in such media is similar to the behavior of waves in the classical Burgers equation. The paper is organized as follows. In Sec. II, the problem is formulated. Some connected results about the decay at the initial stage of evolution of nonlinear random waves in the limit of zero viscosity (Riemann waves) are listed. The decay of nonlinear random Riemann waves exists due to the artificial attenuation when we use the double Fourier transformation from Eulerian to the Lagrangian coordinates in the solution for simple waves. In Sec. III, the influence of dissipation and dispersion described by the integral term of (4) is taken into account. A surprising phenomenon is discovered for broadband noise. Namely, even weak noise at small acoustic Reynolds numbers can transform to a strong nonlinear signal. In Sec. IV, the last stage of evolution is considered when the noise is a sequence of sawtooth pulses with the random position of shocks. The high-frequency asymptotic of noise spectrum at large times of evolution is determined by the kernel K(s) of integrodifferential equation (4) and by fluctuations of the width of shock fronts. II. DECAY OF RANDOM RIEMANN WAVES
We shall be concerned with the initial value problem for the Burgers equation (3) with zero viscosity, when this equation transforms to the simple wave (Riemann-Hopf) equation: ∂v ∂v +v = 0. ∂t ∂x
(5)
Equation (5) is a Eulerian representation of the flow of particles uniformly moving along the x axis. Let us assume that the particle at the point y at the initial moment of time t = 0 has the velocity v0 (y). Then in Lagrangian representation the particle’s motion is given by the following equations: X(y,t) = y + v0 (y)t, V (y,t) = v0 (y).
we have the implicit form of solution for the simple wave v(x,t) = v0 [x − tv(x,t)].
(10)
The problems of statistical description of the random simple (Riemann) wave in Eulerian representation are connected with the implicit form of this solution. Nevertheless, in the Lagrangian representation the statistical description is trivial. Based on the relation between statistical description in Eulerian and Lagrangian representation [9], it is possible to find exact expressions for both the spectra and probability distributions of a random field [8–10,13]. In particular, it was shown that, despite the strong distortions of the profile, the one-point probability distribution of a Riemann wave is conserved, while the higher probability distributions are distorted. Expressions for the energy spectra and correlation functions of Riemann (simple) waves were obtained in Refs. [18–20]. In the case of a statistically homogeneous Riemann field v(x,t) its spatial energy spectrum ∞ 1 E(k,t) = B(s,t)e−iks ds, (11) 2π −∞ where B(s,t) = v(x,t)v(x + s,t)
(12)
is the correlation function of the Eulerian field v(x,t), is equal to ∞ 1 E(k,t) = e−iks 2π (kt)2 −∞ × [θ2 (kt, − kt,s) − θ1 (kt)θ1 (−kt)]ds.
(13)
Here
θ2 (k1 ,k2 ,s) = ei[k1 v0 (x)+k2 v0 (x+s)] , θ1 (k) = eikv0 (x) (14)
(8)
are the two-point and one-point characteristic functions of the initially statistically homogeneous field v0 (x). Let us discuss some characteristic features of evolution of the spectral density of the statistically homogeneous Riemann field v(x,t) by using the initially Gaussian field v0 (x) with a zero mean value and a given correlation function B0 (s), B0 (0) = σ02 . In this case, the spectral density (13) takes on the form [18–21] ∞ B0 (s)k2 t 2 1 −(σ0 kt)2 E(k,t) = e e − 1 e−iks ds. (15) 2 2π (kt) −∞
Let the field v(x,t) of particle velocities in a flow be given as a function of the Eulerian coordinate x and time t. If, in addition to that, the mapping (7) of the Lagrangian to Eulerian coordinates is also known, then the dependence of the velocity field on the Lagrangian coordinates are given by the following equation:
Let us also note that the rate of nonlinear self-action and generation of new harmonics of the field v(x,t) depends on the spatial frequency k in the following way: the smaller k, the slower these processes. Therefore, at σ0 kt 1, it is possible to expand the exponents in (15) into a Taylor series and retain several first terms, e.g.:
V (y,t) = v[X(y,t),t].
E(k,t) = E0 (k) + 12 (kt)2 [E0 (k) ⊗ E0 (k)
(6)
By varying y, we obtain the laws of motion of other particles in the flow. The mapping from the Lagrangian into Eulerian coordinates is described by the following equation: x = X(y,t).
(7)
In the case of uniformly moving particles, this equation has the following form: x = y + v0 (y)t.
(9)
In what follows, the fields describing the behavior of particles in the Lagrangian coordinate system will be called the Lagrangian fields, and the fields in the Eulerian coordinate system will be referred to as the Eulerian fields. So v(x,t) is the Eulerian particle-velocity field, and X(y,t) is the Lagrangian field of the Eulerian coordinates of the particles. So, from (9)
− 2E0 (0)E0 (k)] + · · · ,
(16)
where E0 (k) is the spectrum of the initial field v0 (x), the sign ⊗ denotes the convolution operation. Taking only the first term on the right-hand side into account corresponds to neglecting nonlinear effects; two first terms take nonlinear interaction of pairs of initial field harmonics into account, which leads to the
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appearance of spectral components with difference and sum wave numbers (single interaction), etc. Let us now consider the behavior of the spectral density of the field v(x,t) at large wave numbers k. In this case, to evaluate the integral (15), it is possible to use the method of steepest descent. By retaining in Eq. (15) the first few terms of the Taylor expansion of the correlation function B0 (s) with respect to powers of s, k12 s 2 k24 s 4 2 + − ··· , (17) B0 (s) = σ0 1 − 2! 4! we obtain: σ02
1 E(k,t) = √ exp − 2 3 2τ k1 2π (κτ ) τ = σ0 k1 t = t/tnl , tnl = 1/σ0 k1 ,
,
(18)
v (x , t ) + − −6
−4
−2
2
4
kx
+ FIG. 1. Field v± (x,t) found by summing 50 terms of the series (24) at τ = kat = 2 [13]. The dashed line shows the multistream velocity field of uniformly moving particles. Pluses and minuses indicate the sign, with which each stream is added to the total sum. Characteristically pointed peaks at the boundaries of the intervals of multistreamness are due to nondifferentiability of the velocity field at these points.
κ = k/k1 . (19)
where tnl is a nonlinear time of evolution. From here, it follows that the spectral density of the Riemann field decays at k → ∞ according to the universal power law E(k,t) ∼ k −3 .
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velocities vi (x,t) of the multistream Riemann solution: n n i−1 v± (x,t) = (−1) vi (x,t) = (−1)i−1 v0 [yi (x,t)]. i=1
i=1
(20)
(23)
Nevertheless, the physical interpretation of these results is not trivial. For a Riemann wave, the energy should be conserved; however, from the expressions obtained for the spectra (16) it follows that the energy, when calculated as the integral over the spectrum, decreases with distance from the input [9]. This is related to the fact that the passage from the Eulerian description to the Lagrangian one in the spectral representation is equivalent to the replacement of the multivalued solution by a single-flux one, which is obtained by an alternating-sign summation of the branches of the Riemann solution [9,13,19–21]. In the case of Gaussian statistics, the regions of ambiguity of the Riemann wave appear at arbitrarily small distances, so we have the decay of energy in (16) from the very beginning. It is also true for the expression for the asymptotic of the spectra (18), where the universal power law (20) is due to the appearance of singularities of the velocity field v(x,t) ∼ (x − x∗ )1/2 , and the factor exp(−1/2τ 2 ) in (18) is proportional to the average number of singularities in the random Riemann solution. These phenomena will be clear if we use the transformation from Eulerian coordinates to Lagrangian in the Fourier transform of the velocity ∞ 1 v(κ,t) ˜ = v(x,t) e−iκx dx 2π −∞ ∞ 1 ∂X(y,t) dy. (21) v(κ,t) ˜ = e−iκX(y,t) v0 (y) 2π −∞ ∂y
Figure 1 shows the field (23) for a sinusoidal initial condition, constructed by partial summation of the series
where X(y,t) = y + v0 (y)t (6). The inverse Fourier integral of (21) ∞ v∗ (x,t) = v(κ,t) ˜ eiκx dκ
v± (x,t) = 2a
∞
(−1)n+1
n=1
Jn (nτ ) sin(nkx). nτ
Thus, it means that we have an artificial attenuation of random multiflux Riemann waves. This decay goes on somewhat faster than that for the shocked solution. Nevertheless, based on these equations, we can estimate the decay of high-intensity multiflux noise on the initial stage, and moreover, we can estimate the influence of nonlinear distortion on the decay of noise governed by integrodifferential equation (4). We will first consider the decay of the artificial random Riemann wave. After the inverse Fourier transform of (13) for the energy of the artificial Riemann wave, we have y 1 ∞ ∂ 2 B0 (y)dy. (25) g σ (t) = 2 −∞ ∂y 2t(σ02 − B0 (y))1/2 z 2 2 e−s ds. (26) g(z) = Erf(z) ≡ √ π 0 Assuming that the initial correlation function has the form B0 (y) =
σ02 1 + k∗2 y 2
(22)
is a single-valued function. It can easily be shown that the field defined in this way is an alternating-sign sum of stream
(27)
from (25), (27) for the energy of the artificial Riemann wave we have σ 2 (t) = σ02 [1 − (t)]
−∞
(24)
(28)
√ 2
(τ ) = (1 + τ −2 )[1 − Erf(1/ 2τ )] − τ −1 e−1/2τ 2/π (29) √ where τ = 2σ0 k∗ t is dimensionless time. From (29) we have that even at τ = 1 the additional nonlinear decay is relatively
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Assuming that nonlinear distortion of the wave is relatively small for the correlation function of the noise described by the integrodifferential equation we have [23] 1 ∞ ∂ g σ 2 (t) = 2 −∞ ∂y
y × 2 1/2 Blin (y,t)dy. (34) 2t σlin (t) − Blin (y,t)
FIG. 2. The additional nonlinear decay (τ ) as function of dimensionless time τ (29).
small— (τ = 1) ≈ 0.15 (see Fig. 2). At (τ 1) we have from (29) 4 3 1 (τ ) = √ τ exp − 2 . (30) 2τ 2π In the general case at the initial stage of evolution (τ = σ0 k1 t 1) we can use in (25) the first few terms of the Taylor expansion of the correlation function B0 (s) with respect to powers of s (17) and for the nonlinear decay of artificial Riemann we obtain 2 t (31) (t) = Cart k14 k2−4 (t/tnl )3 exp − nl2 , 2t √ where Cart = 24/ 2π , tnl = 1/σ0 k1 is a nonlinear time (19). For Gaussian statistics, the mean number of singularities in random Riemann solutions is proportional to exp(−1/2τ 2 ) [9,21]. Consequently, the additional nonlinear decay in (25) is directly caused by the appearance of artificial shocks. We have the same equation for the decay of a random shock wave at the initial stage [22], (see also Eq. (5.17) in Ref. √ [9]), but with a different numerical constant Cshock = 54/π . So it means that the equation for the energy of the Riemann wave (25) qualitatively describes the decay of the shock waves at the initial stage. In the next section we consider the influence of nonlinear decay on the propagation of random waves in media with dissipation and weak dispersion. III. EVOLUTION OF NONLINEAR RANDOM WAVES IN MEDIA WITH DISSIPATION AND WEAK DISPERSION
Let us consider the decay of the wave described by integrodifferential equations (4). In the linear case, for the correlation function of the velocity field we have [1] ∞ Elin (k,t) exp(−ikx)dk, (32) Blin (x,t) = −∞
where the linear spectrum is 2 Elin (k,t) = E0 (k) exp −2νk 0
∞
K(y) cos(ky)dy .
(33)
To get this expression we introduce in (4) a new variable that is equal to the transformation from Eulerian to Lagrangian coordinates, and use the double Fourier transform and statistical averaging, similar to that for the Riemann equation. As for the Riemann wave, we can use in (34) the first few terms of the Taylor expansion of the correlation function Blin (y) with respect to powers of y: k24 (t)y 4 k12 (t)y 2 2 + − · · · , (35) Blin (y) = σlin (t) 1 − 2! 4! where the parameters of expansion are determined by the first moments of the linear spectrum ∞ 2 2n σlin (t)kn (t) = k 2n Elin (k,t)dk. (36) −∞
And then from (34) we have 2 (t)[1 − (t)]. σ 2 (t) = σlin
2 t (t) (t) = Cart k14 (t)k2−4 (t)[t/tnl (t)]3 exp − nl 2 . 2t
(37) (38)
Here the local nonlinear time tnl (t) is determined as tnl (t) = 1/σlin (t)k1 (t).
(39)
We will now consider the evolution of noise in media, when the dissipation is proportional exp(−2tνk m ). For the first moments of the linear spectrum, which determined the nonlinear decay, we have ∞ 2 σlin (t)kn2n (t) = 2 k 2n E0 (k) exp(−2tνk m )dk, (40) 0
where E0 (k) is the initial spectrum of the noise. Let us consider the evolution of truncated white noise E0 (k) ∼ exp[−(k/k0 )2 ] described by the Burgers equation (m = 2). From (40), (38) for the additional nonlinear decay we have 2 t (t) = C2 (1 + t/tlin )−9/4 (t/tnl )3 exp − nl2 (1 + t/tlin )3/2 . 2t (41) 2 Here C2 = Cart /3, √ tlin = 1/2νk0 is the time of linear decay and tnl = 1/σ0 k1 = 2/σ0 k0 is the nonlinear time of evolution. The ratio of linear and nonlinear times is an initial Reynolds number of Re0 = σ0 /k0 ν = tlin /tnl . From (41) we see that linear attenuation leads to a reduction of nonlinear effects. Nevertheless, even in case when Re0 1, at rather large times the linear stage of evolution will be transformed to the nonlinear stage. From (41) we have that at t tlin the
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NONLINEAR DECAY OF RANDOM WAVES DESCRIBED BY . . .
additional nonlinear decay may be written as
tnl2 3/4 (t) ∼ t exp − . 3/2 2t 1/2 tlin
where ψ0 (x) is the initial potential of the velocity field. Accordingly, for the velocity field we have (42)
From this equation we see, that the nonlinear effects become important at time tnl∗ = tlin (tnl /tlin )4 = tlin Re−4 0 . This effect has a simple interpretation. Assuming that we have only a linear attenuation for the local Reynolds number Re(t) we have Re(t) =
σlin (t) = Re0 (1 + t/tlin )1/4 . νk1 (t)
(43)
The dissipation leads to a decrease in the amplitude of the wave σlin (t) and an increase of spatial scale L(t) = 1/k1 (t), and for the truncated white noise the increase in the scale dominates over the decrease in the amplitude. Due to these phenomena, the Reynolds number increases with time and the linear stage of evolution is followed by the nonlinear one. Moreover, it is possible to show from the exact Hopf-Cole solution of the Burgers equation, that for E0 (k) ∼ k n and n < 1 a strongly nonlinear regime of sawtooth waves is actually set up, independently of the Re0 value, and that the field statistical properties become self-similar [9]. Finally consider the general case when the dissipation is proportional to exp(−2tνk m ) and the initial spectrum has the form E0 (k) ∼ k n exp[−(k/k0 )2 ].
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(44)
In this case, we have that at t tlin the additional nonlinear decay may be written as
tnl2 (t) ∼ exp − . (45) (3+n)/m 2t (2m−n−3)/m tlin
v(x,t) = −ψx (x,t) =
(47)
where y(x,t) is the coordinate of the maximum of (46) for given x and t. At long time, the parabola (x − y)2 /2t in Eq. (46) becomes a smooth function at the scale of the initial potential ψ0 (y), and the coordinates of the global maxima in (46) practically coincide with the coordinates yk of some local maxima of ψ0 (y), which are zeros of the velocity field with a positive gradient. Hence, at long time, a continuous random field is transformed into a random sawtooth wave—a sequence of cells with the universal behavior within each cell v(x,t) = (x − yk )/t, but with random positions of discontinuities separating the cells. Here yk are the zeros of the sawtooth wave. Coalescence of the cells leads to the growth of the overall scale of the turbulence L(t), therefore the energy of the random field E(t) ∼ L2 (t)/t 2 decreases slower than the energy of a periodic signal. The rate of coalescence of discontinuities is determined by the statistical characteristics of the discontinuity velocities, which, in their turn, depend on the initial potential ψ0 (x). At long time, after multiple coalescence of discontinuities, the properties of the turbulence are determined by the statistical properties of the increments ψ0 (x + L) − ψ0 (x) at large separations L, and at Gaussian statistics of the initial field—by the asymptotic behavior of the structure function of the initial potential. Thus, the turbulence-development scenario is determined by the behavior of the large-scale part of the initial energy spectrum E0 (k) of the velocity field [9–13]. If the Burgers turbulence is indeed characterized by a single scale, then, assuming self-similarity, its spectrum may be represented in the form [9–13,24,25]:
Consequently, we have that if m > m∗ = (3 + n)/2 the linear stage of evolution will be transformed to the nonlinear stage.
E(k,t) =
L3 (t) ˜ E[kL(t)]. t2
(48)
For sufficiently large wave numbers, the behavior of the spectrum is always determined by discontinuities. Therefore
IV. LATE STAGE OF EVOLUTION
In the last part of the paper we consider the evolution of the wave at the late stage, when the wave is a sequence of sawtooth waves with random positions of the shocks. At the late stage, the high-frequency asymptotics of noise will be determined by the kernel K(s) of the integrodifferential equation and fluctuations in the width of discontinuities. Let us first discuss the properties of the random solutions of (3), (4) in the limit of vanishing viscosity, assuming that the spectral density at large scales (small wave numbers k) is in the form of a power law: E0 (k) ∼ |k|n . This situation is equal to the case of free decaying of the Burgers turbulence. In the theory of the Burgers turbulence, it is customary to introduce a potential ψ(x,t) of the velocity field v(x,t) = −ψx (x,t). In the limit ν → 0, the solution for the potential is representable in the following form (see, e.g., [9–13]): (x − y)2 , ψ(x,t) = max ψ0 (y) − y 2t
x − y(x,t) , t
(46)
E(k,t) ∼
B(t) k2
(large k),
(49)
and, from Eq. (48), it follows that B(t) ∼ L(t)/t 2 . Thus a general property of the decaying Burgers turbulence is the establishment of self-similarity at long times and, in particular, the self-similar behavior of the spectrum (48). In the case of finite viscosity, the shocks have the structure determined by properties of the kernel K(s) in (1), (4). Let us assume that we know the stationary solution of Eq. (4) ∞ ∂2 ∂F (x) =ν 2 U F (x) K(s)F (x − s)ds (50) ∂x ∂x 0 with the boundary condition for the dimensionless function F (x): F (−∞) = 1, F (∞) = −1, and U is the amplitude of the stationary solution. We assume that the structure of the shocks of sawtooth waves is the same as the structure of the stationary wave and its amplitude U (t) is determined by the zero viscosity
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PHYSICAL REVIEW E 90, 032924 (2014)
solution U = ηk /2t = (yk+1 − yk )/t.
(51)
It is convenient to consider first the spectrum of the gradient of the velocity field, which is a sequence of the pulses situated at the position of the shocks 1 1 ηi ∂v = − f (x − xk ). (52) q(x,t) = ∂x t t 2t Here ∂F (x) f (x) = . (53) ∂x It is obvious that Ev (k,t) = Eq (k,t)/k 2 . If cq (k) is the Fourier transform of the function f (x) ∞ cq (k) = f (x) exp(ikx)dx (54) −∞
for the energy spectrum of the pulse field q(x,t) at large k [k 1/L(t)], when one can neglect interpulse interference of Fourier components of the velocity gradient pulses, we have [9] ¯ ηi 2 2 n(t) Eq (k,t) = cq (k,t) . (55) 2π 2t
In the viscosity interval, the behavior of the spectrum depends on the statistical properties of probability distribution Wη ( ). Here, we discuss the evolution of the Burgers turbulence at n > 1, when the initial field has Gaussian statistics, and the velocity and potential are statistically homogeneous. The statistics of the velocity field is determined by the statistical characteristics of the absolute maximum coordinates. At long times, the parabola in (46) becomes a smooth function at the scale of the initial action ψ0 (x), and for the right to be the absolute maximum in (46) competes a large number of local maxima of ψ0 (x). This made it possible to use for an analysis of the Burgers turbulence at long time [9,20,27,28] the statistical theory of large overshoots [29]. In the limit of vanishing viscosity, when the time t tends to infinity, the statistical characteristics of the Burgers turbulence become self-similar and, in particular, the energy spectrum assumes the form (48). The integral scale of the turbulence L(t) is given by the following expression t 1/2 −1/4 , (59) L(t) ≈ (tσψ ) ln 2π tnl where
¯ ∼ 1/L(t) is the average number of the shocks per Here n(t) unit length, and the angle brackets mean the averaging over random values ηi . We need to stress that the width of the shock F (x) and consequently, the function cq (k,t) depends on random amplitude of the shock ηi /2t. In the general case we can introduce the inner scale of turbulence as δ ∼ tν/L(t). If in the sequent space interval the stationary front has the singularities of the type F (x − x∗ ) = A(U )|x − x∗ |α , (α = 0 is equal to the discontinuities of the velocity field and α = 1— to the discontinuities of the velocity gradient). Then for the asymptotic of the energy spectrum we have ηi ¯ A2 (56) Ev (k,t) ∼ n(t) k −2(α+1) . 2t
are the nonlinearity time and the initial integral scale of the turbulence respectively. In this case, for the probability distribution Wη ( ) we have [27] 2 exp − . (61) Wη ( ) = 2L2 (t) 4L2 (t) √ ¯ = 1/ π L(t). In the internal interval k > 1/δ we and n(t) can use in (58) the asymptotic sinh(x) ≈ exp(x)/2 and in this interval for the spectrum we have
From this equation we see that the fluctuation of the shock width does not change the shape of the spectrum. Thus for the classical Burgers turbulence we have in the inertial interval of the spectrum Ev (k,t) ∼ k −2 . The propagation of the intense acoustics in the tube can be described√ by Eq. (1), where in the spectral form the operator is ˆ L(iω) ∼ ω(1 − i). In this case we have the discontinuities of the field of velocity derivative. It leads to the asymptotic E(ω,z) ∼ ω−4 , which was observed experimentally [26]. In the case of the Burgers turbulence, when we have the analytical solution for the stationary front and in (53) (57) f (x) = sec2 (x/δi ).
V. DISCUSSION AND CONCLUSION
here δi = 4νt/ηi is the width of the shock. We shall assume that we know the probability distribution Wη ( ) of the distance between the zeros ηi = yi+1 − yi of the sawtooth wave. Then from (55), (57), we have for the asymptotic of energy spectrum ∞ Wη ( )d 2 ¯ Ev (k,t) = 2π ν n(t) . (58) sinh2 (2π νkt/ ) 0 In the inertial interval of the spectrum 1/L k 1/δ, we have from the spectrum the universal behavior Ev (k,t) ∼ k −2 (49).
tnl ≡ L20 /σψ = L0 /σv , L0 ≡ σψ /σv
Ev (k,t) ∼ exp{−3[π δ(t)k]2/3 }.
(60)
(62)
Here δ = νt/L(t) is the internal scale of turbulence. We need to stress that for the turbulence the spectrum in the internal interval falls off slower then for the periodic signals where E(k,t) ∼ exp[−kδ(t)] and δ = νt/L0 .
Here we have considered the evolution of random nonlinear waves in media with dissipation and weak dispersion. The behavior of waves in such media is similar to the behavior of the waves described by the Burgers equation. We have also considered here the influence of nonlinearity and dissipation on the decay of broadband noise in two limiting cases: at the initial stage, when the number of the shocks is relatively small, and at the late stage, when the wave is a sequence of sawtooth waves with the random positions of the shocks. It is well known that for the periodic waves at large Reynolds numbers due to the increasing width of the shock, the nonlinear stage of evolution is replaced by the linear stage of decay. For the broadband noise the nontrivial behavior may exist: even for small initial Reynolds numbers, the linear stage of decay may change to the nonlinear stage of evolution. In the limiting case of vanishing viscosity, the continuous wave transforms
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NONLINEAR DECAY OF RANDOM WAVES DESCRIBED BY . . .
to the sequence of sawtooth waves. For the complex media with dissipation and weak dispersion, the shocks have a finite width, and the structure of the shocks will be determined by the kernel K(s) of the integrodifferential equation.
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PHYSICAL REVIEW E 90, 032924 (2014) ACKNOWLEDGMENTS
We have benefited from discussions with A. Saichev and I. Demin. The work is supported by the Russian Science Foundation, Grant No. 14-12-00882.
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