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Trans-resonant evolution of wave singularities and vortices
Physics Letters A 311 (2003) 192–199 www.elsevier.com/locate/pla
Trans-resonant evolution of wave singularities and vortices Sh.U. Galiev Department of Mechanical Engineering, The University of Auckland, Private Bag 92019, Auckland 1, New Zealand Received 12 June 2000; received in revised form 9 March 2003; accepted 13 March 2003 Communicated by A.R. Bishop
Abstract Nonlinear resonant wave phenomena are treated. It is assumed that near and at the resonance first-order linear terms in perturbed wave equations annihilate each other. As a result, the perturbed wave equations reduce to basic highly nonlinear ordinary differential equation or the basic algebraic equation for traveling waves. These equations determine the evolution of smooth waves into shock waves. Then the jump curls and eventually breaks, nucleating drops, bubbles or vortices. 2003 Elsevier Science B.V. All rights reserved. PACS: 03.40.Kf; 43.25; 91.30.Mv Keywords: Wave universality; Wave singularity; Nonlinearity; Stability
1. Introduction Wave nonlinearity is an important element of nature. This nonlinearity is often focused near and at critical points where greatly different physical systems exhibit a strong similarity. These points were called ‘resonance’ [1]. In the resonant band the nonlinear effects can increase strongly and the first-order linear effects drop to zero. In particular, harmonic waves can be transformed into anomalous waves and wave structures (breakers, mushroom-like waves, jets and vortices). The traditional methods of nonlinear physics fail at the resonance because of the singularity. Therefore a development of the theory is required to describe nonlinear trans-resonant wave phenomena. This development was presented in [1–4], where the trans-resonant wave processes in different nonlinE-mail address: [email protected] (Sh.U. Galiev).
ear, dispersive–dissipative systems were considered. It is assumed that near and at the resonance linear terms in perturbed wave equations can annihilate each other (ut t ≈ a02 ∇ 2 u). As a result, the perturbed wave equations reduce to a following basic highly nonlinear ordinary differential equation for traveling waves [1,4] kJ + µ∗ J + D2 + D1 J + D3 (J )2 + D5 (J )3 + D6 (J )4 J + D∗ (J )2 + s2 J |J | = X ,
(1)
where J is the one-sided traveling wave and J = dJ /dr, X = dX/dr. For one-dimensional waves r = c0 t + x. If k = µ∗ = D∗ = s2 = 0, then (1) yields the basic resonant algebraic equation D2 J + 0.5D1 (J )2 + 0.333D3(J )3 + 0.25D5(J )4 + 0.2D6 (J )5 = X + C.
0375-9601/03/$ – see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0375-9601(03)00492-4
(2)
Sh.U. Galiev / Physics Letters A 311 (2003) 192–199
Coefficient D2 in (1) and (2) changes from negative to positive value within the resonant band. It was emphasized in [1,4] that the model equations (1) and (2) describe the trans-resonant evolution of waves in different systems ranging from microresonators to the early Universe. Indeed, waves in these systems are described by similar perturbed wave equations [1]. Although the above equations can have quite different intrinsic characteristics, coefficients and different solutions they also have a class of similar resonant solutions. Some of them were constructed in [1–4]. These solutions demonstrate the analogies between surface waves, nonlinear and atom optics, field theories and acoustics of the early Universe. The solutions describe forced, parametric-excited and free waves, and wave patterns, which have both classical and quantummechanical features [1]. Here we study the following particular case of Eq. (1) kJ + µ∗ J + D2 + D1 J + D3 (J )2 + D5 (J )3 + D6 (J )4 J = X . (3) Eq. (3) determines the evolution of smooth waves into shock waves and mushroom-like waves. According to (3) the shock-jump curls as a result of the instability and eventually breaks, nucleating drops, bubbles and vortices. Here high nonlinear effects are considered. Transresonant evolution of quadratic nonlinear waves was treated in [4,5].
2. Governing equations and solutions for traveling resonant wave phenomena As an example of the trans-resonant evolution we consider here one-dimensional waves in strings, lattices and surface layers. The following equation is valid for these waves [1,4,6] ut t − a02 uxx = βux uxx + β1 u2x uxx + β2 u3x uxx + β3 u4x uxx + µut xx + kuxxxx + 0.5gηx+ ,
(4)
where u is the longitudinal displacement, a0 is the sound velocity, t and x are time and coordinate, respectively. Here and below letter subscripts denote
193
differentiation with respect to the corresponding value. In (4) there are quadratic, cubic, fourth and fifth order terms with regard to u. The dissipative (µut xx ) and dispersive (kuxxxx ) terms follow behind the nonlinear terms. η+ is a known function of the coordinate and the time, which defines the amplitude, and the motion of initial ripples. Coefficients in (4) can depend on t and x. Versions of Eq. (4) can also describe nonlinear waves propagating in bubbly liquids, gas and elastic bodies [2,4]. While the general solution of (4) is unknown, we treat here an interesting special solution that can be readily obtained. Let us consider the traveling wave: u = J (r).
(5)
In this case, we have directly from (4) that 2 c0 − a02 − βJ − β1 (J )2 − β2 (J )3 − β3 (J )4 J = µc0 J + kJ + 0.5gηx+ .
(6)
If µ = k = β2 = β3 = 0, r∗ = 1.5ββ1−1 , c02 − a02 = −2−2/3R ∗ β1 and c02 − a02 = βJ + β1 (J )2 we have (J )3 + r∗ (J )2 + 3R ∗ J /22/3 + 1.5gβ1−1η+ + C = 0. (7) ∗ Here R is a trans-resonant parameter [1–4] and C is a constant of integration. Let us introduce a new function F : F = J + r∗ /3. If C = 0 and R = R ∗ − 22/3r∗2 /9, G(r) = 2r∗3 /27 − r∗ R ∗ /22/3 + 1.5gβ1−1η+ , then Eq. (7) yields (F )3 + 3R/22/3F + G = 0,
(8)
where G = G(r). It is necessary to distinguish four cases: (1) Let R = 0, then Eq. (8) is satisfied if F = (−G)1/3.
(9) F
is unique, single(2) Let R > 0, then function valued and continuous F = −2D sinh 13 arcsin h |R|−1.5 G , (10) where D = (sign sin ωt)(|R|2−2/3 )0.5 . (3) Let R < 0 and 0.25(R 3 + G2 ) 0. In this case there is no continuous single-valued solution and a solution with discontinuities was constructed [2,4]. However, nature often manifests multi-valued solutions (for example, breaking waves and turbulence).
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Indeed, nonlinear systems often exhibit two and more dynamic equilibrium states for the same values of parameters. Some states may be chaotic, while others are periodic. We will construct here regular multi-valued solutions with the help of the following smooth singlevalued solutions: Fi = −2D cos 13 arccos |R|−1.5 G + 2iπ/3 , (11) where i = 0, 1, 2. (4) If R < 0 and 0.25(R 3 + G2 ) > 0, we have one real and two complex solutions: F = −2D cosh 13 arccos h |R|−1.5 G , (12) 1 −1.5 G F± = D cosh 3 arccos h |R| √ 1 −1.5 ± 3 D sinh 3 arcsin h |R| (13) G . Thus, we have obtained the set of solutions (9)–(13) which describe waves generated within and near the trans-resonant band.
where η = hJ , R = 25/3 β1−1 (a02 − c02 ), Φ(c0 t + x) = −1.5gh3 β1−1 η+ and h is the thickness of the layer. If h → 0 then a difference a02 − c02 is small and depends on dispersion, nonlinearity and vertical acceleration. We assume that the trans-resonant parameter R can vary near the layer edge from a positive to a negative value. Thus, the problem is reduced to the solution of a cubic algebraic equation (15). Using the analytical solutions of this equation (see Section 2) we studied the nonlinear trans-resonant evolution of harmonic waves near and at the edge of the layer. It is assumed that the breaker is formed where the amplitude/thickness ratio approximately equals 0.88 [7]. The temporal and spatial evolution of the harmonic wave near the layer edge is shown in Fig. 1 for L = 100 m, c0 = (gh0 )0.5 , R = h/ h0 + 2(Ah0 / h) sec h[0.5Lπ −1 (0.88h − 2A)] cos[0.25(c0t + x)], h0 = 1 m, h varying from 1 m to 0, Φ(c0 t + x) = Ah2 sin 0.25(c0t + x1 ) and A = 0.05 m.
3. The trans-resonant evolution of the edge waves The solutions (9)–(13) have not taken into account the dispersive effect. This effect can be important for sufficiently thick layers. Let us consider a wave u1 = cos ωc0−1 (c0 t + x1 ) propagating in a thick layer. In this case, Eq. (4) yields c02 = a02 − kω2 c0−2 .
(14)
In (14) the most important terms for the analysis were preserved. It follows from (14) that due to the dispersion, waves of different wavelengths travel at different phase velocities c0 . According to (14) the dispersion can change a sign within the resonant band and every new Fourier component generated by nonlinear modification of the initial wave shape has a different velocity. This effect prevents the formation of breaking waves or vortices. If h → 0 then the dispersive effect is reduced [7], the velocity of the wave c0 → a0 (14), and resonance occurs. Consider this resonance and the trans-resonant evolution of waves traveling along the layer slope using the results of Section 2. Eq. (7) is rewritten so that [4] η3 − 0.75hη2 + 1.5h2 Rη/22/3 + Φ(c0 t + x) = 0, (15)
Fig. 1. Trans-resonant evolution and the breaking (curling) of the edge waves. The trans-resonant parameter R is strongly varied near the edge. An example of this variation is given for t = 0 (curve R).
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various systems. In particular, they qualitatively simulate a generation of drops and bubble in water and granular waves. Closed loops in Fig. 2 may be also considered as vortices. The generation of the vortices by the nonlinear waves begin to study during the last time [1,8,9].
4. Stability and trans-resonant evolution of waves We shall treat Eq. (6) assuming that β = β2 = µ = ηx+ = 0. (1) Consider the stability of a disturbance Jd passing the resonant band. In this case, (6) yields 3 3RJd /22/3 + (Jd )3 + β3 β1−1 (Jd )5 + 3kβ1−1Jd = 0. 5 (16)
Fig. 2. Trans-resonant evolution of the edge waves and the generation of breakers, bubbles and drops. Curves R show the variation of the trans-resonant parameter R.
Coastal wave evolution is a function of a bottom slope. Due to the slope the smooth wave transforms into a breaking wave (Fig. 1). The breaker overturns at t ≈ 2.5 s. As a result, an air cavity is formed within the wave. The maximum amplification of the initially harmonic wave is approximately 5. The wave do not produce a large splash and is reminiscent of a collapsing breaker on the water surface [7, Fig. 8.5]. The collapsing-type breaker can be transformed into a plunging-type breaker when the slope is changed. We assume L = 300 m, h0 = 5 m, h varying from 5 m to 0. Results of the calculations are presented in Fig. 2. It is seen from Fig. 2 that nonlinear transresonant wave evolution and the overturning process can produce air cavities and splashes near the coastal line. As a result, drops, bubbles and vortices can be generated due to the overturning [7, Fig. 8.4]. It is possible to give different interpretations of the results presented in Figs. 1 and 2 [1–4]. The similar trans-resonant wave processes may be generated in
Eq. (16) takes into account dispersion and nonlinearity. A solution of (16) is sought in the form: Jd = A cos ωc0−1 r. We substitute Jd into (16) to find the approximate equation for ω: ω2 = B, where B = (R/22/3 + 14 A2 + 18 β3 β1−1 A4 )β1 k −1 c02 . If B < 0, then Jd is amplified according to expression Jd = √ A cosh |B| c0−1 r. It is seen that the amplification of Jd depends on competition between the trans-resonant parameter R and the amplitude A. If A is small, then the disturbance Jd can be amplified. If the amplitude is large enough, so that A2 + 0.5β3β1−1 A4 > −4/22/3 (we assume that |R| 1 within the trans-resonant band) and β1 k −1 > 0, then the disturbance Jd is not amplified. (2) Let us consider the trans-resonant wave evolution. We assume J = J∗ + Jd
(17)
in (6), where J∗ = J − Jd . At the resonance J∗ Jd and Eq. (6) yields kJ∗ + µc0 J∗ + β1 (J∗ )2 + β3 (J∗ )4 + β1 R/22/3 J∗ + β1 R/22/3Jd = 0.
(18)
Thus, the trans-resonant evolution is determined by effects of dispersion, dissipation and nonlinearity. If the nonlinear effect is very small and Jd = A sin ωc0−1 (r + π/2ωc0−1 ), (18) yields the pendulum type linear equa-
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tion y + 2γ ω∗ y + ω∗2 y = p0 sin ωc0−1 r + π/2ωc0−1 ,
(19)
where y = J∗ , ω∗2 = k −1 β1 R/22/3 , γ = 0.5µc0 × k −1 ω∗−1 , p0 = Ak −1 β1 R/22/3 and C = 0. The wave evolution is determined by parameters R, µ, k and the following solution y = J∗
√ √ 2 2 = e−γ ω∗ r C1 eiω∗ 1−γ r + C2 e−iω∗ 1−γ r 2 −0.5 + p0 ω∗−2 1 − Ω 2 + (2γ Ω)2 × cos ωc0−1 r − arctan 2γ Ω/ 1 − Ω 2 , (20) √ where Ω = ωc0−1 ω∗−1 and i = −1. The first term of (20) is determined by the factors exp r[−0.5µc0k −1 ± i(k −1 β1−1 R/22/3 − 0.25µ2k −2 c02 )0.5 ], where coefficients µ and k vary in the trans-resonant band [1–4]. Therefore, the influence of R, µ and k on the factors and the wave stability is very complex. For simplicity we assume that µ and k are positive constants. In this case, the waves are stable if R > 2−4/3 β1 k −1 µ2 c02 . If R < 2−4/3 β1 k −1 µ2 c02 , then the waves are unstable. It follows from this linear analysis that the waves are unstable within the resonant band, if R < 0. (3) Consider nonlinear effects on the trans-resonant wave evolution. We assume that k = µ = 0 and β1 (J∗ )2 + β3 (J∗ )4 + β1 R/22/3 = 0. In this case, (18) is transformed into the form −0.11(J∗ )5 + (J∗ )3 + 3R/22/3J∗ + 3R/22/3 Jd = 0.
(21)
We assumed in (21) that 3β3 β1−1 /5 = −0.11 and C = 0. Fig. 3 presents solutions of (21) for the case R = −0.2 + 0.002Y (r) + tanh[Y (r)/50], where Y is changed according to the linear law from 30 to −120. Waves in Fig. 3(a) and (c) are calculated for Jd = A cos 0.5r, where A = 0.3 (Fig. 3(a)) or A = 0.7 (Fig. 3(c)). Waves in Fig. 3(b) is calculated for Jd = 0.3(1 + 0.75 cos 0.5r) cos 0.5r. It is seen that within the resonant band the continuous solution bifurcates into closed loops (vortices) [1]. They can generate Karman’s so-called “vortex street” [10,11]. Growth of the amplitude A from 0.3 to 0.7 (Fig. 3(c)) strongly complicates the trans-resonant evolution of the wave. As a result, shock-like waves, breakers,
Fig. 3. Trans-resonant evolution of the disturbance and the generation of mushroom-like waves, breakers and a cluster of the vortices described by the fifth-order algebraic equation. The vortices reduce and disappear when R reduces.
and mushroom-like structures can be generated. The harmonic wave can transform into the saw-like curve (Fig. 3(c)). The calculations showed that very weak long waves are not amplified in the resonant band. Moderately small waves can be amplified (Fig. 3(a), (b)). Strong disturbances can lose stability within the band (Fig. 3(c)). As a result, they are not amplified.
5. Generation of vortices within the trans-resonant band Vortices can be generated because of instability of waves in the trans-resonant band. Consider the influence of a cubic nonlinearity on the formation of vortices. In this case Eq. (21) yields (J∗ )3 + 3R/22/3J∗ + 3R/22/3 Jd = 0.
(22)
The analytical solutions of this equation are similar to (9)–(13). Function J = J∗ + Jd is shown in Fig. 4. First we assume that Jd = A cos 0.5r, A = 0.3, R = 0.2 − αr (or R = −(αr)2 ) and 0 < αr < 0.7. The
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The singularities occur when J∗ = −R/22/3 , J∗ = − −R/22/3 .
(24) (25)
We expand the equation for small displacements around (24) and (25). In particular, let J∗ = g ∓ −2−2/3R, (26) r = r0 + z, where r0 is some value of r in the neighbourhood of which we want to study the equation. Substituting in (23), we have g g ± 2−4/3(−R)−0.5 R = ±2−4/3(−R)0.5 A × C∗ + ωa0−1 z cos ωa0−1 r0
± 22/3 b1 r0 cos ωa0−1 r0 (−R)0.5 g ,
(27)
where b1 = −0.5ωβ1a0−3 and C∗ = sin ωa0−1 r0 + 2−2/3 b1 r0 R cos ωa0−1 r0 . In most instances the constant C∗ is not zero. Consider points where C∗ vanishes [15]. The points are determined by the following equation Fig. 4. Trans-resonant evolution of the disturbances into mushroom-like waves and mushroom vortices described by Eq. (22). The straight lines are the singular lines, the focal points are shown by black spots. The nonlinear effect begin to form the vortices when r ≈ 23 (R ≈ 0).
evolution of smooth waves into mushroom-like waves and mushroom vortices is shown in Fig. 4(a) (see also Fig. 3). This phenomenon was simulated with the help of numerical methods and observed on liquid surfaces [12], and in different fluid systems [11,13, 14]. On the other hand, within the band there are R where waves are excited having various amplitudes and signs. According to Fig. 4(a) the system exhibits a hysteresis effect. In other words the system does not behave the same when R is increased as it does when R is deceased. We may expect that the vortices are connected with the multi-valued solutions and depend on the behaviour of the singularities of (18). Assume in (18) that Jd = A cos ωc0−1 r and k = µ = β3 = 0. As a result, we have 2 (J∗ ) + R/22/3 J∗ + AR/22/3 sin ωc0−1 r = 0. (23)
tan ωa0−1 r0 + 2−2/3 b1 r0 R(r0 ) = 0.
(28)
The points situate in curves (24) and (25) and are shown in Fig. 4(b)–(d) as black spots. The leading terms of (27) near those points yield a bilinear equation g = ±az + (b ∓ E)g /g, (29) where a = 2−4/3ωa0−1 A(−R)0.5 cos ωa0−1 r0 , b = −2−2/3 b1 r0 AR cos ωa0−1 r0 , E≈2
−4/3
(−R)
−0.5
and
R.
The solution is determined by eigenvalues λ, 2 λ− 1,2 = 0.5 (b − E) ± (b − E) + 4a , 2 λ+ 1,2 = 0.5 (b + E) ± (b + E) − 4a .
(30) (31)
We emphasise that eigenvalues λ− 1,2 (30) correspond to + the singular line (24), and λ1,2 (31) correspond to the singular line (25). The behaviour and the stability of solution curves in J –r plane depend on λ∓ 1,2 and may
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be studied following [16]. For simplicity we assume that b1 ≈ 0, b = 0 and R 0. Then (28) yields that r0 ≈ Nπa0 ω−1 (N = 1, 2, 3, . . .). According to (30) and (31) unstable nodes and saddles are formed if both roots (b − E)2 ± 4a real. When the roots become imaginary (E reduces) the unstable nodes can be transformed into the spirals (focuses). This evolution is determined by R and A [4]. For example, if E = 0 (exact resonance, R = 0) and √ = ±2i |a| (vortex) and N = 1 we have a < 0, λ− 1,2 √ = ±2 |a| (saddle). The vortex is located on the λ+ 1,2 singular line (24), and the saddle is located on (25). Let E (or R) reduce from zero. Then the vortex transforms into the unstable spiral. A similar transformation takes place if N = 2 and a > 0. If N is odd, then the spirals are located on (24) and the saddles on (25). If N is even, then the spirals are located on (25) and the saddles are located on (24). The generation and the evolution of the unstable spirals are qualitatively shown in Fig. 4(c), (d). It is seen that Fig. 4(c), (d) resembles results of experiments [10,11,17] better than Fig. 4(a), (b). Indeed, profiles in Fig. 4(c), (d) qualitatively describe the generation and the evolution of waves and vortices in various cases: on a jet surface, behind of a obstacle in the fluid stream, in a mixing layer and in boundary-layer flow. It is interesting to ask oneself how vortices do in fact arise. It is obviously due to nonlinearity. The viscous effect may be connected with R, which can take into account the linear boundary friction of the layer [4]. According the theory the vortex generation is connected with the resonance phenomena. Perhaps, small perturbations are amplified due to the resonance. Additionally, we emphasise that Fig. 4(c) simulates well the vortex formation in an air mixing layer which is shown in pages 160–161 in [17]. Fig. 4(c) simulate well a Karman vortex street in water flow behind the circular cylinder ([11], see, also, Fig. 7.10 from [10]). The next application involves two horizontal dimensions. We assume that r = c0 t −x −y. In this case, (21) yields −0.11(J∗ )5
+ (J∗ )3 + 3R/22/3J∗ + sin ωc0−1 (c0 t − x − y) = 0,
(32)
where J∗ = dJ /dr. Let the variation of R within the resonator is determined by the soliton-like function: R = 0.5 − 2 sec h{0.01[(25 − x)2 + y 2 ]}. The
Fig. 5. Trans-resonant evolution of initially harmonic waves into mushroom-like waves and vortices in x–y plane.
evolution of incident harmonic waves propagating in the negative y-direction is shown in Fig. 5. We assumed that ωc0−1 = 1 and t = 0. Far from the resonator centre (x = 25, y = 0) the waveform is slightly changed and is reminiscent of the incident harmonic wave (y = 11). Then near the line x = 25 the waves curl and mushroom-like waves are generated (y = 9.9 and 9.75). As a result of the instability the mushroomlike waves break, nucleating the closed loops. We may consider these loops as vortices. It is possible to give different interpretation of Figs. 3–5. In particular, they qualitatively resemble the formation shock waves and vortex pairs in the Bose– Einstein condensate resonator [18].
6. Conclusion The complex evolution of waves may be within the trans-resonant band. The harmonic waves transform into the shock-like waves when R → 0. Then the singularities of wave fronts are generated. These singularities can evolve into drop-like, bubble-like,
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jet-like and vortex-like structures. This evolution is a result of the strongly nonlinear processes. Today, many scientists consider nonlinear science as the most important frontier for gaining a fundamental understanding of nature. Close to their critical point, greatly different systems exhibit strongly similar nonlinear dynamics. Similar nonlinear waves can be generated in various fields ranging from fluids, plasmas and the Bose–Einstein condensate to solidstate, chemical, biological, astrophysical and geological systems. In particular, the transition from the smooth waves to vortices have been observed in many natural and artificial (for example, the Bose–Einstein condensate) systems [1,18]. The generation of wave turbulence may be connected with this transition. Following [1] we have shown that this transition is determined by the strongly nonlinear trans-resonant process. This process is described by ordinary differential equation (6) or algebraic equation (8). Since equations (1), (6) and (8) simulate the properties of many perturbed wave equations in the trans-resonant band, we think that similar evolution can take place in different physical system ranging from microresonators to the early Universe [1].
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Sh.U. Galiev, Chaos 11 (2001) 686. Sh.U. Galiev, J. Phys. A: Math. Gen. 32 (1999) 6963. Sh.U. Galiev, Phys. Lett. A 266 (2000) 41. Sh.U. Galiev, Geophys. J. Int., in press. Sh.U. Galiev, T.Sh. Galiev, Phys. Lett. A 246 (1998) 299. M. Remoissenet, Waves Called Solitons, Springer, 1996. L. Debnath, Nonlinear Water Waves, Academic Press, 1994. D.C. Dunn, N.R. McDonald, E.R. Johnson, J. Fluid Mech. 448 (2001) 335. O. Bühler, T.E. Jacobson, J. Fluid Mech. 449 (2001) 313. T.E. Faber, Fluid Dynamics for Physics, Cambridge Univ. Press, Cambridge, 1995. C.H.K. Williamson, Annu. Rev. Fluid Mech. 28 (1996) 477. L.D. Cloutman, M.F. Wehner, Phys. Fluid A 4 (1992) 1821. A.S. Sabry, J.T.C. Liu, J. Fluid Mech. 231 (1991) 615. P. Scandura, G. Vittori, P. Blondeaux, J. Fluid Mech. 412 (2000) 355. J. Jimenez, J. Fluid Mech. 59 (1973) 23. W.J. Cunningham, Introduction to Nonlinear Analysis, McGraw–Hill, 1958. C. Suplee, Physics in the 20th Century, Abrams, New York, 1999. Z. Dutton, M. Budde, C. Slowe, L.V. Hau, Science 293 (2001) 663.
Trans-resonant evolution of wave singularities and vortices Sh.U. Galiev Department of Mechanical Engineering, The University of Auckland, Private Bag 92019, Auckland 1, New Zealand Received 12 June 2000; received in revised form 9 March 2003; accepted 13 March 2003 Communicated by A.R. Bishop
Abstract Nonlinear resonant wave phenomena are treated. It is assumed that near and at the resonance first-order linear terms in perturbed wave equations annihilate each other. As a result, the perturbed wave equations reduce to basic highly nonlinear ordinary differential equation or the basic algebraic equation for traveling waves. These equations determine the evolution of smooth waves into shock waves. Then the jump curls and eventually breaks, nucleating drops, bubbles or vortices. 2003 Elsevier Science B.V. All rights reserved. PACS: 03.40.Kf; 43.25; 91.30.Mv Keywords: Wave universality; Wave singularity; Nonlinearity; Stability
1. Introduction Wave nonlinearity is an important element of nature. This nonlinearity is often focused near and at critical points where greatly different physical systems exhibit a strong similarity. These points were called ‘resonance’ [1]. In the resonant band the nonlinear effects can increase strongly and the first-order linear effects drop to zero. In particular, harmonic waves can be transformed into anomalous waves and wave structures (breakers, mushroom-like waves, jets and vortices). The traditional methods of nonlinear physics fail at the resonance because of the singularity. Therefore a development of the theory is required to describe nonlinear trans-resonant wave phenomena. This development was presented in [1–4], where the trans-resonant wave processes in different nonlinE-mail address: [email protected] (Sh.U. Galiev).
ear, dispersive–dissipative systems were considered. It is assumed that near and at the resonance linear terms in perturbed wave equations can annihilate each other (ut t ≈ a02 ∇ 2 u). As a result, the perturbed wave equations reduce to a following basic highly nonlinear ordinary differential equation for traveling waves [1,4] kJ + µ∗ J + D2 + D1 J + D3 (J )2 + D5 (J )3 + D6 (J )4 J + D∗ (J )2 + s2 J |J | = X ,
(1)
where J is the one-sided traveling wave and J = dJ /dr, X = dX/dr. For one-dimensional waves r = c0 t + x. If k = µ∗ = D∗ = s2 = 0, then (1) yields the basic resonant algebraic equation D2 J + 0.5D1 (J )2 + 0.333D3(J )3 + 0.25D5(J )4 + 0.2D6 (J )5 = X + C.
0375-9601/03/$ – see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0375-9601(03)00492-4
(2)
Sh.U. Galiev / Physics Letters A 311 (2003) 192–199
Coefficient D2 in (1) and (2) changes from negative to positive value within the resonant band. It was emphasized in [1,4] that the model equations (1) and (2) describe the trans-resonant evolution of waves in different systems ranging from microresonators to the early Universe. Indeed, waves in these systems are described by similar perturbed wave equations [1]. Although the above equations can have quite different intrinsic characteristics, coefficients and different solutions they also have a class of similar resonant solutions. Some of them were constructed in [1–4]. These solutions demonstrate the analogies between surface waves, nonlinear and atom optics, field theories and acoustics of the early Universe. The solutions describe forced, parametric-excited and free waves, and wave patterns, which have both classical and quantummechanical features [1]. Here we study the following particular case of Eq. (1) kJ + µ∗ J + D2 + D1 J + D3 (J )2 + D5 (J )3 + D6 (J )4 J = X . (3) Eq. (3) determines the evolution of smooth waves into shock waves and mushroom-like waves. According to (3) the shock-jump curls as a result of the instability and eventually breaks, nucleating drops, bubbles and vortices. Here high nonlinear effects are considered. Transresonant evolution of quadratic nonlinear waves was treated in [4,5].
2. Governing equations and solutions for traveling resonant wave phenomena As an example of the trans-resonant evolution we consider here one-dimensional waves in strings, lattices and surface layers. The following equation is valid for these waves [1,4,6] ut t − a02 uxx = βux uxx + β1 u2x uxx + β2 u3x uxx + β3 u4x uxx + µut xx + kuxxxx + 0.5gηx+ ,
(4)
where u is the longitudinal displacement, a0 is the sound velocity, t and x are time and coordinate, respectively. Here and below letter subscripts denote
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differentiation with respect to the corresponding value. In (4) there are quadratic, cubic, fourth and fifth order terms with regard to u. The dissipative (µut xx ) and dispersive (kuxxxx ) terms follow behind the nonlinear terms. η+ is a known function of the coordinate and the time, which defines the amplitude, and the motion of initial ripples. Coefficients in (4) can depend on t and x. Versions of Eq. (4) can also describe nonlinear waves propagating in bubbly liquids, gas and elastic bodies [2,4]. While the general solution of (4) is unknown, we treat here an interesting special solution that can be readily obtained. Let us consider the traveling wave: u = J (r).
(5)
In this case, we have directly from (4) that 2 c0 − a02 − βJ − β1 (J )2 − β2 (J )3 − β3 (J )4 J = µc0 J + kJ + 0.5gηx+ .
(6)
If µ = k = β2 = β3 = 0, r∗ = 1.5ββ1−1 , c02 − a02 = −2−2/3R ∗ β1 and c02 − a02 = βJ + β1 (J )2 we have (J )3 + r∗ (J )2 + 3R ∗ J /22/3 + 1.5gβ1−1η+ + C = 0. (7) ∗ Here R is a trans-resonant parameter [1–4] and C is a constant of integration. Let us introduce a new function F : F = J + r∗ /3. If C = 0 and R = R ∗ − 22/3r∗2 /9, G(r) = 2r∗3 /27 − r∗ R ∗ /22/3 + 1.5gβ1−1η+ , then Eq. (7) yields (F )3 + 3R/22/3F + G = 0,
(8)
where G = G(r). It is necessary to distinguish four cases: (1) Let R = 0, then Eq. (8) is satisfied if F = (−G)1/3.
(9) F
is unique, single(2) Let R > 0, then function valued and continuous F = −2D sinh 13 arcsin h |R|−1.5 G , (10) where D = (sign sin ωt)(|R|2−2/3 )0.5 . (3) Let R < 0 and 0.25(R 3 + G2 ) 0. In this case there is no continuous single-valued solution and a solution with discontinuities was constructed [2,4]. However, nature often manifests multi-valued solutions (for example, breaking waves and turbulence).
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Indeed, nonlinear systems often exhibit two and more dynamic equilibrium states for the same values of parameters. Some states may be chaotic, while others are periodic. We will construct here regular multi-valued solutions with the help of the following smooth singlevalued solutions: Fi = −2D cos 13 arccos |R|−1.5 G + 2iπ/3 , (11) where i = 0, 1, 2. (4) If R < 0 and 0.25(R 3 + G2 ) > 0, we have one real and two complex solutions: F = −2D cosh 13 arccos h |R|−1.5 G , (12) 1 −1.5 G F± = D cosh 3 arccos h |R| √ 1 −1.5 ± 3 D sinh 3 arcsin h |R| (13) G . Thus, we have obtained the set of solutions (9)–(13) which describe waves generated within and near the trans-resonant band.
where η = hJ , R = 25/3 β1−1 (a02 − c02 ), Φ(c0 t + x) = −1.5gh3 β1−1 η+ and h is the thickness of the layer. If h → 0 then a difference a02 − c02 is small and depends on dispersion, nonlinearity and vertical acceleration. We assume that the trans-resonant parameter R can vary near the layer edge from a positive to a negative value. Thus, the problem is reduced to the solution of a cubic algebraic equation (15). Using the analytical solutions of this equation (see Section 2) we studied the nonlinear trans-resonant evolution of harmonic waves near and at the edge of the layer. It is assumed that the breaker is formed where the amplitude/thickness ratio approximately equals 0.88 [7]. The temporal and spatial evolution of the harmonic wave near the layer edge is shown in Fig. 1 for L = 100 m, c0 = (gh0 )0.5 , R = h/ h0 + 2(Ah0 / h) sec h[0.5Lπ −1 (0.88h − 2A)] cos[0.25(c0t + x)], h0 = 1 m, h varying from 1 m to 0, Φ(c0 t + x) = Ah2 sin 0.25(c0t + x1 ) and A = 0.05 m.
3. The trans-resonant evolution of the edge waves The solutions (9)–(13) have not taken into account the dispersive effect. This effect can be important for sufficiently thick layers. Let us consider a wave u1 = cos ωc0−1 (c0 t + x1 ) propagating in a thick layer. In this case, Eq. (4) yields c02 = a02 − kω2 c0−2 .
(14)
In (14) the most important terms for the analysis were preserved. It follows from (14) that due to the dispersion, waves of different wavelengths travel at different phase velocities c0 . According to (14) the dispersion can change a sign within the resonant band and every new Fourier component generated by nonlinear modification of the initial wave shape has a different velocity. This effect prevents the formation of breaking waves or vortices. If h → 0 then the dispersive effect is reduced [7], the velocity of the wave c0 → a0 (14), and resonance occurs. Consider this resonance and the trans-resonant evolution of waves traveling along the layer slope using the results of Section 2. Eq. (7) is rewritten so that [4] η3 − 0.75hη2 + 1.5h2 Rη/22/3 + Φ(c0 t + x) = 0, (15)
Fig. 1. Trans-resonant evolution and the breaking (curling) of the edge waves. The trans-resonant parameter R is strongly varied near the edge. An example of this variation is given for t = 0 (curve R).
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various systems. In particular, they qualitatively simulate a generation of drops and bubble in water and granular waves. Closed loops in Fig. 2 may be also considered as vortices. The generation of the vortices by the nonlinear waves begin to study during the last time [1,8,9].
4. Stability and trans-resonant evolution of waves We shall treat Eq. (6) assuming that β = β2 = µ = ηx+ = 0. (1) Consider the stability of a disturbance Jd passing the resonant band. In this case, (6) yields 3 3RJd /22/3 + (Jd )3 + β3 β1−1 (Jd )5 + 3kβ1−1Jd = 0. 5 (16)
Fig. 2. Trans-resonant evolution of the edge waves and the generation of breakers, bubbles and drops. Curves R show the variation of the trans-resonant parameter R.
Coastal wave evolution is a function of a bottom slope. Due to the slope the smooth wave transforms into a breaking wave (Fig. 1). The breaker overturns at t ≈ 2.5 s. As a result, an air cavity is formed within the wave. The maximum amplification of the initially harmonic wave is approximately 5. The wave do not produce a large splash and is reminiscent of a collapsing breaker on the water surface [7, Fig. 8.5]. The collapsing-type breaker can be transformed into a plunging-type breaker when the slope is changed. We assume L = 300 m, h0 = 5 m, h varying from 5 m to 0. Results of the calculations are presented in Fig. 2. It is seen from Fig. 2 that nonlinear transresonant wave evolution and the overturning process can produce air cavities and splashes near the coastal line. As a result, drops, bubbles and vortices can be generated due to the overturning [7, Fig. 8.4]. It is possible to give different interpretations of the results presented in Figs. 1 and 2 [1–4]. The similar trans-resonant wave processes may be generated in
Eq. (16) takes into account dispersion and nonlinearity. A solution of (16) is sought in the form: Jd = A cos ωc0−1 r. We substitute Jd into (16) to find the approximate equation for ω: ω2 = B, where B = (R/22/3 + 14 A2 + 18 β3 β1−1 A4 )β1 k −1 c02 . If B < 0, then Jd is amplified according to expression Jd = √ A cosh |B| c0−1 r. It is seen that the amplification of Jd depends on competition between the trans-resonant parameter R and the amplitude A. If A is small, then the disturbance Jd can be amplified. If the amplitude is large enough, so that A2 + 0.5β3β1−1 A4 > −4/22/3 (we assume that |R| 1 within the trans-resonant band) and β1 k −1 > 0, then the disturbance Jd is not amplified. (2) Let us consider the trans-resonant wave evolution. We assume J = J∗ + Jd
(17)
in (6), where J∗ = J − Jd . At the resonance J∗ Jd and Eq. (6) yields kJ∗ + µc0 J∗ + β1 (J∗ )2 + β3 (J∗ )4 + β1 R/22/3 J∗ + β1 R/22/3Jd = 0.
(18)
Thus, the trans-resonant evolution is determined by effects of dispersion, dissipation and nonlinearity. If the nonlinear effect is very small and Jd = A sin ωc0−1 (r + π/2ωc0−1 ), (18) yields the pendulum type linear equa-
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tion y + 2γ ω∗ y + ω∗2 y = p0 sin ωc0−1 r + π/2ωc0−1 ,
(19)
where y = J∗ , ω∗2 = k −1 β1 R/22/3 , γ = 0.5µc0 × k −1 ω∗−1 , p0 = Ak −1 β1 R/22/3 and C = 0. The wave evolution is determined by parameters R, µ, k and the following solution y = J∗
√ √ 2 2 = e−γ ω∗ r C1 eiω∗ 1−γ r + C2 e−iω∗ 1−γ r 2 −0.5 + p0 ω∗−2 1 − Ω 2 + (2γ Ω)2 × cos ωc0−1 r − arctan 2γ Ω/ 1 − Ω 2 , (20) √ where Ω = ωc0−1 ω∗−1 and i = −1. The first term of (20) is determined by the factors exp r[−0.5µc0k −1 ± i(k −1 β1−1 R/22/3 − 0.25µ2k −2 c02 )0.5 ], where coefficients µ and k vary in the trans-resonant band [1–4]. Therefore, the influence of R, µ and k on the factors and the wave stability is very complex. For simplicity we assume that µ and k are positive constants. In this case, the waves are stable if R > 2−4/3 β1 k −1 µ2 c02 . If R < 2−4/3 β1 k −1 µ2 c02 , then the waves are unstable. It follows from this linear analysis that the waves are unstable within the resonant band, if R < 0. (3) Consider nonlinear effects on the trans-resonant wave evolution. We assume that k = µ = 0 and β1 (J∗ )2 + β3 (J∗ )4 + β1 R/22/3 = 0. In this case, (18) is transformed into the form −0.11(J∗ )5 + (J∗ )3 + 3R/22/3J∗ + 3R/22/3 Jd = 0.
(21)
We assumed in (21) that 3β3 β1−1 /5 = −0.11 and C = 0. Fig. 3 presents solutions of (21) for the case R = −0.2 + 0.002Y (r) + tanh[Y (r)/50], where Y is changed according to the linear law from 30 to −120. Waves in Fig. 3(a) and (c) are calculated for Jd = A cos 0.5r, where A = 0.3 (Fig. 3(a)) or A = 0.7 (Fig. 3(c)). Waves in Fig. 3(b) is calculated for Jd = 0.3(1 + 0.75 cos 0.5r) cos 0.5r. It is seen that within the resonant band the continuous solution bifurcates into closed loops (vortices) [1]. They can generate Karman’s so-called “vortex street” [10,11]. Growth of the amplitude A from 0.3 to 0.7 (Fig. 3(c)) strongly complicates the trans-resonant evolution of the wave. As a result, shock-like waves, breakers,
Fig. 3. Trans-resonant evolution of the disturbance and the generation of mushroom-like waves, breakers and a cluster of the vortices described by the fifth-order algebraic equation. The vortices reduce and disappear when R reduces.
and mushroom-like structures can be generated. The harmonic wave can transform into the saw-like curve (Fig. 3(c)). The calculations showed that very weak long waves are not amplified in the resonant band. Moderately small waves can be amplified (Fig. 3(a), (b)). Strong disturbances can lose stability within the band (Fig. 3(c)). As a result, they are not amplified.
5. Generation of vortices within the trans-resonant band Vortices can be generated because of instability of waves in the trans-resonant band. Consider the influence of a cubic nonlinearity on the formation of vortices. In this case Eq. (21) yields (J∗ )3 + 3R/22/3J∗ + 3R/22/3 Jd = 0.
(22)
The analytical solutions of this equation are similar to (9)–(13). Function J = J∗ + Jd is shown in Fig. 4. First we assume that Jd = A cos 0.5r, A = 0.3, R = 0.2 − αr (or R = −(αr)2 ) and 0 < αr < 0.7. The
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The singularities occur when J∗ = −R/22/3 , J∗ = − −R/22/3 .
(24) (25)
We expand the equation for small displacements around (24) and (25). In particular, let J∗ = g ∓ −2−2/3R, (26) r = r0 + z, where r0 is some value of r in the neighbourhood of which we want to study the equation. Substituting in (23), we have g g ± 2−4/3(−R)−0.5 R = ±2−4/3(−R)0.5 A × C∗ + ωa0−1 z cos ωa0−1 r0
± 22/3 b1 r0 cos ωa0−1 r0 (−R)0.5 g ,
(27)
where b1 = −0.5ωβ1a0−3 and C∗ = sin ωa0−1 r0 + 2−2/3 b1 r0 R cos ωa0−1 r0 . In most instances the constant C∗ is not zero. Consider points where C∗ vanishes [15]. The points are determined by the following equation Fig. 4. Trans-resonant evolution of the disturbances into mushroom-like waves and mushroom vortices described by Eq. (22). The straight lines are the singular lines, the focal points are shown by black spots. The nonlinear effect begin to form the vortices when r ≈ 23 (R ≈ 0).
evolution of smooth waves into mushroom-like waves and mushroom vortices is shown in Fig. 4(a) (see also Fig. 3). This phenomenon was simulated with the help of numerical methods and observed on liquid surfaces [12], and in different fluid systems [11,13, 14]. On the other hand, within the band there are R where waves are excited having various amplitudes and signs. According to Fig. 4(a) the system exhibits a hysteresis effect. In other words the system does not behave the same when R is increased as it does when R is deceased. We may expect that the vortices are connected with the multi-valued solutions and depend on the behaviour of the singularities of (18). Assume in (18) that Jd = A cos ωc0−1 r and k = µ = β3 = 0. As a result, we have 2 (J∗ ) + R/22/3 J∗ + AR/22/3 sin ωc0−1 r = 0. (23)
tan ωa0−1 r0 + 2−2/3 b1 r0 R(r0 ) = 0.
(28)
The points situate in curves (24) and (25) and are shown in Fig. 4(b)–(d) as black spots. The leading terms of (27) near those points yield a bilinear equation g = ±az + (b ∓ E)g /g, (29) where a = 2−4/3ωa0−1 A(−R)0.5 cos ωa0−1 r0 , b = −2−2/3 b1 r0 AR cos ωa0−1 r0 , E≈2
−4/3
(−R)
−0.5
and
R.
The solution is determined by eigenvalues λ, 2 λ− 1,2 = 0.5 (b − E) ± (b − E) + 4a , 2 λ+ 1,2 = 0.5 (b + E) ± (b + E) − 4a .
(30) (31)
We emphasise that eigenvalues λ− 1,2 (30) correspond to + the singular line (24), and λ1,2 (31) correspond to the singular line (25). The behaviour and the stability of solution curves in J –r plane depend on λ∓ 1,2 and may
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be studied following [16]. For simplicity we assume that b1 ≈ 0, b = 0 and R 0. Then (28) yields that r0 ≈ Nπa0 ω−1 (N = 1, 2, 3, . . .). According to (30) and (31) unstable nodes and saddles are formed if both roots (b − E)2 ± 4a real. When the roots become imaginary (E reduces) the unstable nodes can be transformed into the spirals (focuses). This evolution is determined by R and A [4]. For example, if E = 0 (exact resonance, R = 0) and √ = ±2i |a| (vortex) and N = 1 we have a < 0, λ− 1,2 √ = ±2 |a| (saddle). The vortex is located on the λ+ 1,2 singular line (24), and the saddle is located on (25). Let E (or R) reduce from zero. Then the vortex transforms into the unstable spiral. A similar transformation takes place if N = 2 and a > 0. If N is odd, then the spirals are located on (24) and the saddles on (25). If N is even, then the spirals are located on (25) and the saddles are located on (24). The generation and the evolution of the unstable spirals are qualitatively shown in Fig. 4(c), (d). It is seen that Fig. 4(c), (d) resembles results of experiments [10,11,17] better than Fig. 4(a), (b). Indeed, profiles in Fig. 4(c), (d) qualitatively describe the generation and the evolution of waves and vortices in various cases: on a jet surface, behind of a obstacle in the fluid stream, in a mixing layer and in boundary-layer flow. It is interesting to ask oneself how vortices do in fact arise. It is obviously due to nonlinearity. The viscous effect may be connected with R, which can take into account the linear boundary friction of the layer [4]. According the theory the vortex generation is connected with the resonance phenomena. Perhaps, small perturbations are amplified due to the resonance. Additionally, we emphasise that Fig. 4(c) simulates well the vortex formation in an air mixing layer which is shown in pages 160–161 in [17]. Fig. 4(c) simulate well a Karman vortex street in water flow behind the circular cylinder ([11], see, also, Fig. 7.10 from [10]). The next application involves two horizontal dimensions. We assume that r = c0 t −x −y. In this case, (21) yields −0.11(J∗ )5
+ (J∗ )3 + 3R/22/3J∗ + sin ωc0−1 (c0 t − x − y) = 0,
(32)
where J∗ = dJ /dr. Let the variation of R within the resonator is determined by the soliton-like function: R = 0.5 − 2 sec h{0.01[(25 − x)2 + y 2 ]}. The
Fig. 5. Trans-resonant evolution of initially harmonic waves into mushroom-like waves and vortices in x–y plane.
evolution of incident harmonic waves propagating in the negative y-direction is shown in Fig. 5. We assumed that ωc0−1 = 1 and t = 0. Far from the resonator centre (x = 25, y = 0) the waveform is slightly changed and is reminiscent of the incident harmonic wave (y = 11). Then near the line x = 25 the waves curl and mushroom-like waves are generated (y = 9.9 and 9.75). As a result of the instability the mushroomlike waves break, nucleating the closed loops. We may consider these loops as vortices. It is possible to give different interpretation of Figs. 3–5. In particular, they qualitatively resemble the formation shock waves and vortex pairs in the Bose– Einstein condensate resonator [18].
6. Conclusion The complex evolution of waves may be within the trans-resonant band. The harmonic waves transform into the shock-like waves when R → 0. Then the singularities of wave fronts are generated. These singularities can evolve into drop-like, bubble-like,
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jet-like and vortex-like structures. This evolution is a result of the strongly nonlinear processes. Today, many scientists consider nonlinear science as the most important frontier for gaining a fundamental understanding of nature. Close to their critical point, greatly different systems exhibit strongly similar nonlinear dynamics. Similar nonlinear waves can be generated in various fields ranging from fluids, plasmas and the Bose–Einstein condensate to solidstate, chemical, biological, astrophysical and geological systems. In particular, the transition from the smooth waves to vortices have been observed in many natural and artificial (for example, the Bose–Einstein condensate) systems [1,18]. The generation of wave turbulence may be connected with this transition. Following [1] we have shown that this transition is determined by the strongly nonlinear trans-resonant process. This process is described by ordinary differential equation (6) or algebraic equation (8). Since equations (1), (6) and (8) simulate the properties of many perturbed wave equations in the trans-resonant band, we think that similar evolution can take place in different physical system ranging from microresonators to the early Universe [1].
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