Generation of undular bores and solitary wave trains in fully nonlinear shallow water theory

Generation of undular bores and solitary wave trains in fully nonlinear shallow water theory Gennady El 1, Roger Grimshaw 1 and Noel Smyth 2 1 Loughborough University, UK, 2 University of Edinburgh, UK SIAM Conference on Nonlinear Waves and Coherent Structures Rome, July 21

Undular bore Undular bore: the wave-train transition between two different basic states, each characterized by a constant depth and horisontal velocity (dispersive resolution of a large-scale nonlinear wave-breaking). Internal structure: modulated nonlinear periodic wave transforming into a solitary wave at the leading edge and into a vanishing amplitude wave-packet at the trailing edge. s 2 s 1 u t 0 u Generally the speeds of the leading (soliton) and trailing (harmonic) edges s 1,2 are different, s 1 s 2, and, as a result, the classical jump conditions for the depth and horisontal velocity are not applicable. Here we consider only conservative undular bores (no dissipation)!

η 3 2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 300 350 400 450 500 550 600 650 700 750 800 Two model problems involving formation of undular bores Dispersive Riemann problem 2,u2 Rarefaction wave 0,u0 s Undular bore s 1,u1 Given the initial jumps in depth η and horisontal velocity u: (η 2, u 2 ) (η 1, u 1 ) one needs to find the intermediate state (η 0, u 0 ), the undular bore edge speeds s +, s and the amplitude of the lead soliton a +. Generation of a soliton train (out of decaying initial profile). x t 0 t t b 0.8 x Given the initial distributions η(x), u(x) one needs to find the amplitude distribution function f(a) for a solitary wave train.

Two analytic approaches to the undular bore description Semi-classical asymptotics of the IST solutions: V.I. Karpman (1967) Phys. Lett. A, 25, 708. P.D. Lax and C.D. Levermore (1983) Comm. Pure Appl. Math. 36, 253, 571, 809. Applicable to integrable models like KdV, Camassa-Holm, Kaup-Boussinesq... Whitham modulation theory: G.B. Whitham (1965) Proc. Roy. Soc. London, A283, 238. A.V. Gurevich and L.P. Pitaevskii (1974) Sov. Phys. JETP 38, 291. Applicable to both integrable and non-integrable models provided the periodic travelling wave solution exists and sufficient number of conservation laws is available.

KdV undular bores: asymptotic results. u t + uu x + u xxx = 0 Decay of an initial step = u u + (Gurevich & Pitaevskii 1974) Speeds of the undular bore edges: s =, s + = 2 3. Lead soliton amplitude: a + = 2 Generation of a soliton train from a large-scale positive hump u(x, 0) = u 0 (εx) > 0, ε 1 u 0 0 as x. Soliton-amplitude distribution function (Karpman 1967) f(a) = 1 1 dx 8πε 6 u0(x) a/2. Tallest soliton amplitude a max = 2 max u 0 Total number of solitons N = 1 + 6πε u0(x) dx Note that all the above formulae could be obtained via both the IST and the Whitham modulation theory.

From the KdV modulations to the modulations in fully nonlinear shallow-water theory There is a strong numerical evidence that qualitative features of formation and evolution of undular bores in integrable systems are true for the systems that are not completely integrable but are structurally similar to integrable ones. 1.4 1.35 1.3 η 1.25 1.2 1.15 1.1 1.05 1-100 -50 0 50 100 150 200 250 300 350 x Numerical solution for an undular bore in the fully nonlinear shallow-water equations: qualitative similarity to the corresponding KdV solution, but, at the same time, noticeable quantitative differences.

From the KdV modulations to the modulations in fully nonlinear shallow-water theory The main obstacle to obtaining the full modulation solution for a non-integrable system is the absence of the Riemann invariant form, i.e. non-diagonalizability. However, if one accepts some plausible assumptions about the structure of the undular bore modulation solution, then it becomes possible to bypass this obstacle and to derive main physical parameters of the undular bore: the transition conditions across the bore, the speeds of the edges and the amplitude of the lead solitary wave. G.A. El (2005) Resolution of a shock in hyperbolic systems modified by weak dispersion, Chaos, 15, 037103.

Fully nonlinear 1D shallow-water equations Consistent long-wave asymptotic expansion (no amplitude restrictions!) of the 1D Euler equations leads to the system (Su and Gardner 1969) η t + (ηu) x = 0, (1) u t + uu x + η x = 1 [ ] 1 η 3 η3 (u xt + uu xx (u x ) 2 ). Here η is the total depth and u is the layer-mean horizontal velocity. SG system (1) coincides with the 1D reduction of the Green-Naghdi (GN) equations (Green and Naghdi 1976) obtained within the "directed fluid sheet" model (no long-wave asymptotic expansions, instead the condition of linear dependence on z of the vertical velocity is imposed.) Same equations, different models. x

SG (GN) equations: applications, modifications and generalisations S.L. Gavrilyuk (1994) Large amplitude oscillations and their "thermodynamics" for continua with memory", Eur. J. Mech., B/Fluids 13, 753. S.L. Gavrilyuk, and V.M. Teshukov (2001) Generalized vorticity for bubbly liquid and dispersive shallow water equations, Continuum Mech. Thermodyn. 13, 365. P.J. Dellar (2003) Dispersive shallow water magnetohydrodynamics, Physics of Plasmas 10, 581. W. Choi and R. Camassa (1999) Fully nonlinear internal waves in a two-fluid system J. Fluid Mech. 396, 1. Universal model for fully nonlinear weakly dispersive waves.

SG (GN) equations: basic properties Dispersionless limit: ideal shallow-water equations η t + (ηu) x = 0, u t + uu x + η x = 0. Linear dispersion relation for modulated harmonic waves propagating on the slowly varying hydrodynamic background η, ū ( η 1/2 ) ω = ω 0 (k, η, ū) = k ū + (1 + η 2 k 2 /3) 1/2 Speed-amplitude relationship for a solitary wave c s = u + η + a Also: Four conservation laws, P 1 = [η 1]dx, P 2 = [u + 1 6 η2 u xx ]dx, P 3 = [ηu]dx, P 4 = [ 1 2 (η 1)2 + 1 2 ηu2 + 1 6 η3 u 2 x ]dx. Periodic travelling wave solution

Modulation description of undular bores: Gurevich-Pitaevskii problem t x (t) linear wave ( a 0) solitary wave ( k 0) shallow-water equations for,u Whitham equations for, u, k, a x (t),u,u shallow-water equations for,u x x x x The undular bore is modeled by the modulated periodic solution of the SG equations. The slowly varying parameters of the travelling wave solution (for instance, η, u, k, a) satisfy the Whitham equations t P j( η, ū, k, a) + x Q j( η, ū, k, a) = 0, j = 1, 2, 3. k t + (ω(η, u, k, a)) x = 0. Matching conditions at the free boundaries x ± (t): x = x (t) : a = 0, η = η, u = u, x = x + (t) : k = 0, η = η +, u = u +. where η ±, u ± is the external solution of the dispersionless ±

Dam-break problem t = 0 : η = 0 > 1, u = 0 for x < 0; η = 1, u = 0 for x > 0 1.9 1.8 1.7 1.6 1.5 η 1.4 1.3 1.2 1.1 1-300 -200-100 0 100 200 300 x Figure: Numerical solution of the dam-break problem for η. Initial total depth jump 0 = 1.8, t = 150 Transition conditions across the undular bore?

Simple undular bore transition conditions The transition across the undular bore forming in the dispersive Riemann problem is asymptotically as t characterised by a zero jump for one of the Riemann invariants of the dispersionless limit equations (as in the simple wave of compression!) (GE, Chaos 2005). Therefore, for the SG equations, the undular bore propagating to the right is characterised by the transition condition 1 2 u η = 1 2 u+ η + Note that this does not coincide with the classical shock jump condition for the shallow-water equations. 1.8 1.6 1.4 1.2 1 u- 0.8 0.6 0.4 0.2 0 1 1.5 2 2.5 3 η Figure: Riemann invariant (dashed line) and the shock-wave (dotted line) transition curves. u + = 0, η + = 1

The edges of an undular bore We define the leading x + (t) and the trailing x (t) edges of the undular bore by the kinematic conditions: The trailing edge (a = 0) speed is equal to the linear group velocity of the trailing wave packet dx dt = s = ω 0 k (η, u, k ), Here k is the wavenumber at the trailing edge. The leading edge (k = 0) speed is equal to the lead soliton velocity dx + = s + = c s (η +, u +, a + ). dt Here a + is the lead soliton amplitude and c s (η, u, a) is the soliton speed. To find s ± we need to express k, a + in terms of the hydrodynamic jumps (η η + )/η + and (u u + )

The edges of a simple undular bore The values k and a + are found from three sets of constraints imposed by the the structure of the undular bore: 1 The simple undular bore transition conditions across the bore must be satisfied 2 The edges of the undular bore must be the characteristics of the modulation system (actually, multiple characteristics) so one cannot specify all three values k, η, u independently at the trailing edge x (t) (as well as a, η, u at the leading edge x + (t) ). 3 The linear wave packet at the trailing edge and the lead solitary wave are not independent, but rather are constrained by the condition of being parts of the same undular bore.

Linear dispersion equation for modulations The key ingredient in the analysis of the modulations in the asymptotic limits as a 0 and k 0 is the linear dispersion relation of a small-amplitude wave propagating against the slowly varying background η(x, t), u(x, t): a = 0 : ( η 1/2 ) ω = ω 0 ( η, ū, k) = k ū +. (1) (1 + η 2 k 2 /3) 1/2 We incorporate the simple wave relation into (1) to obtain Ω 0 (k, η) = ω 0 (η, u(η), k) = 2k(η 1/2 1) + u(η) = 2(η 1/2 1) (2) kη 1/2 (1 + η 2. (3) k 2 /3) 1/2 Note that (2) is consistent with the simple undular bore transition condition, in which w.l.o.g. we set η + = 1, u + = 0.

Determination of of the trailing edge wavenumber k Exact reduction of the full modulation system: η t + V (η) η x = 0, a = 0, u = 2(η 1/2 1), k t + Ω 0(k, η) = 0. x Here V (η) = u(η) + η 1/2 = 3η 1/2 2 is the characteristic velocity of the right-propagating shallow-water simple wave. Then the dependence k = k(η) along the linear group velocity characteristic dx/dt = Ω 0 (k, η)/ k is given by the ODE dk dη = Ω 0 / η V (η) Ω 0 / k. The solution of (*) yields the local constraint on the admissible combinations of k and η at the trailing edge, i.e k = k(η ). The constant of integration is determined from the global consistency condition k(η + ) = 0 (if η = η + there is no undular bore so one must have k = k + = 0). ( )

Leading (soliton) edge analysis The analysis of the soliton reduction as k 0 is more delicate but the result is completely analogous to that for the trailing edge. The key observation is that the soliton velocity can be calculated as ω c s = lim k 0 k = ω 0 k where ω 0 ( k, η, u) is the linear dispersion relation for the conjugate SG system obtained by the change of independent variables t i t, x i x i.e. ω 0 = iω 0 (η, u, i k). Here k is the soliton wavenumber (inverse half-width) which is related to the SG soliton amplitude a s by a s = 1 k 2 η 2 3 1 ( k 2 η 2 )/3

Leading (soliton) edge analysis Asymptotic analysis of the modulation system for k 1 leads to the ODE for k(η) along the soliton characteristic dx/dt = Ω 0 / k: d k dη = Ω 0 / η V (η) Ω 0 / k. ( ) Here, for the SG system kη 1/2 Ω 0 (η, k) = iω 0 (η, i k) = 2 k(η 1/2 1) (1 η 2 k2 /3). 1/2 The initial condition for (*) is found from the global consistency condition for an undular bore k(η ) = 0, which is derived from the condition that the amplitude of the lead soliton must turn zero if one takes η + = η.

Edges of the SG undular bore Integrating the ODEs we obtain k and k + = k(η + ) and then find the speeds of the undular bore edges as s = Ω 0 k (η, k ), s + = Ω 0 (η +, k + ) k + As a result we get for s ± ( ), where = η /η + : β ( 4 β 3 ) 21/10 ( ) 2/5 ( 1 + β 2 + s 1/3 = 0, where β = 2). 2 and ( ) 21/10 ( ) 2/5 3 2 s + 4 s + 1 + s + = 0.

Edges of the SG undular bore For small jumps δ = 1 1 we have the expansions s + = 1 + δ 5 12 δ2 + O(δ 3 ), s = 1 3 2 δ + 23 8 δ2 + O(δ 3 ), which agree to first order with the KdV results (Gurevich and Pitaevskii 1974) 2 1.8 1.6 s+, s- 1.4 1.2 1 0.8 0.6 1 1.2 1.4 1.6 1.8 2 Figure: The leading s + (upper curve) and the trailing s (lower curve) edge speeds vs the depth ratio across the simple undular bore. Dashed line: modulation solution; Diamonds: values of s +, s extracted from the full numerical solution; Dotted line: the KdV modulation solution.

Lead solitary wave amplitude a + For a + ( ) we get ( ) 21/10 ( ) 2/5 (a + + 1) 3 1/4 4 2 a + + 1 1 + = 0. ( ) a + + 1 For small jumps δ = 1 1 we have a + = 2δ + δ 2 /6 + O(δ 3 ) which agrees to first order with the well-known KdV expression a + KdV = 2( 1) (Gurevich and Pitaevskii 1974). 2.5 2 a 1.5 1 0.5 0 1 1.2 1.4 1.6 1.8 2 Figure: The lead solitary wave amplitude vs depth ratio. Solid line: numerical solution; dashed line: SG modulation solution (*). Growing discrepancy for 1.5

η Formation of finite-amplitude rear wave front The origin of the discrepancy: linear degeneracy of the modulation characteristic field at the trailing edge. This starts for = cr 1.43 where cr is the minimum of the curve s ( ). When > cr a rapidly (exponentially) varying finite-amplitude rear wave front forms, the effect absent in the KdV theory but present in the more sophisticated Camassa-Holm model (Marchant and Smyth 2006). 1.4 1.35 1.3 1.25 η 1.2 1.15 8 7 6 5 4 1.1 3 1.05 2 1-100 -50 0 50 100 150 200 250 300 350 x 1-100 -50 0 50 100 150 200 250 300 350 400 x Figure: Subcritical = 1.2 < cr (left) and supercritical, = 3.5 > cr (right) SG undular bores. In both cases the initial velocity jump satisfies the simple wave conditions. Notice the rarefaction wave forming in the supercritical case. Remarkably, the supercritical undular bore satisfies the classical shock jump condition!

Formation of finite-amplitude rear wave front The idea: in the linear modulation theory we have k t + ω 0 (k)k x = 0 ( ) If for some k = k one has ω 0(k ) = 0 (i.e. k the linear degeneracy point for (*) ) then the wave packet cannot propagate with a speed lower (greater) than ω 0 (k ); as a result, a wavefront forms beyond which the wave amplitude decays rapidly to zero. General condition of linear degeneracy of the j-th characteristic field λ j of the quasilinear system of conservation laws (Lax) r j gradλ j = 0, ( ) where r j is the right eigenvector corresponding to the eigenvalue (characteristic speed) λ j. Then for the SG system the restriction of condition (**) to the trailing edge of the undular bore yields the critical value cr 1.43 corresponding to the occurrence of the wavefront exactly at the trailing edge.

Generation of solitary wave trains: initial data We consider the SG system with the decaying initial depth profile η(x, 0) = η 0 (x) > 1 : η 0 (x) 1 as x ( ) and the velocity profile connected with (*) by the simple wave relationship u(x, 0) = u 0 (x) = 2( η 0 (x) 1), so that u 0 (x) 0 as x. Let max[η 0 (x)] = 1 + A and let w be the typical width of η 0 (x). We shall assume that A 1/2 w 1. Also, for convenience, we assume that η 0 (x) = 1 for x > 0, which will guarantee that solitary waves propagate on the undisturbed background η = 1.

η 3 2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 300 350 400 450 500 550 600 650 700 750 800 x Generation of solitary wave trains: : conserved quantities Numerically: the SG equations with the initial conditions, η 0 (x) = 1 + A sech 2 [x/w], u 0 (x) = 2( η 0 (x) 1). ( ) t 0 t t b 0.8 Integrals: J 1 = [η 1]dx, J 2 = [u + 1 6 η2 u xx ]dx, J 3 = [ηu]dx, J 4 = [ 1 2 (η 1)2 + 1 2 ηu2 + 1 6 η3 u 2 x]dx. Comparison of the values of J 1, J 2, J 3, J 4 computed for two initial profiles having the form (*) with A = 0.4, w = 13 and A = 2, w = 10 with their values calculated for the solitary wavetrain at large t shows that the relative change, due to radiation, is O(10 2 ).

Total number of solitons The wave conservation law: k t + ω x = 0. For the decaying initial profile u 0 0, η 0 (x) 1 as x one has ω 0 as x so (*) implies conservation of the total number of waves N = 1 + kdx = constant. ( ) 2π Since the decaying initial disturbance eventually decomposes into a soliton train, formula (**) can be used to compute the total number of solitons as t. Need to know k(x, t) on the whole x-axis at some t = t 0. ( )

Total number of solitons The wave conservation law: k t + ω x = 0. For the decaying initial profile u 0 0, η 0 (x) 1 as x one has ω 0 as x so (*) implies conservation of the total number of waves N = 1 + kdx = constant. ( ) 2π Since the decaying initial disturbance eventually decomposes into a soliton train, formula (**) can be used to compute the total number of solitons as t. Need to know k(x, t) on the whole x-axis at some t = t 0. t ( ) x k( x, t) not available x k( x, t) not defined k( x, t) not defined x

Extending the definition of k(x, t) to the whole x-axis. x > x + : For the modulation solution describing undular bore we have k(x +, t) = 0 so we define k = 0 for x > x +. x < x : Since the trailing edge x (t) is the charactersitic of the modulation system we have from the previous analysis : k(x, t) = k (η(x, t)), where k (η) is the modulation characteristic integral for a = 0. For x < x the function η(x, t) satisfies the simple-wave equation η t + V (η)η x = 0, so we set k(x, t) = k (η(x, t)) for x < x, which is consistent with the matching condition at x = x (t). t x k( x, t) not available x k( x, t) k ( ( x, t)) k( x, t) 0 So at t = 0 we have k(x, 0) = k (η 0 (x)) x R. x

Total number of solitons As a result, we obtain the formula for the total number of solitons generated in the SG system out of the large-scale decaying initial condition η(x, 0) = η 0 (x), u(x, 0) = u 0 (x) = 2( η 0 (x) 1) : N = 1 + 2π where α 0 (x) is found from the equation 3(1 α 2 0 (x)) dx, η 0 (x)α 0 (x) η 0 (x) = 1 ( ) 21/10 ( ) 2/5 4 α0 1 + α0. α0 3 2 For small (η 0 (x) 1) 1 we obtain to leading order 3 1 + N η0 (x) 1dx, 2 π which argees with the KdV result obtained by the IST.

Total number of solitons η 0 (x) = 1 + A sech 2 [x/w], u 0 (x) = 2( η 0 (x) 1). ( ) 15 15 14 14 13 13 12 12 N 11 N 11 10 10 9 9 8 8 7 6 7 8 9 10 11 12 13 A 1/2 w 7 6 7 8 9 10 11 12 13 A 1/2 w Figure: Total number of solitary waves forming due to the decay of the initial disturbance (*). The initial parameters are w = 10 and A varying (left); A = 0.4 and w varying (right). Solid line is the SG modulation formula; symbols are the numerical solution; the dashed line is the KdV formula. The accuracy of the modulation solution is O(A 1/2 w) 1

Amplitude distribution function Number of solitons having their amplitudes in the interval [a, a max ] N(a) = 1 x2 3(1 α 2 0 (x, λ)) dx, 2π x 1 η 0 (x)α 0 (x, λ) where x 1,2 (λ) are the zeros of (1 α 0 (x, λ)) and α 0 (x, λ) is found from η 0 (x) = 1 ( ) 21/10 ( ) 2/5 4 α0 1 + α0. λ α0 3 2 The relationship between a and λ has the form ( ) 21/10 ( ) 2/5 λ = (1 + a) 1/4 3 4 2 1 + a 1 +. 1 + a In particular, a max corresponds to λ = max[η 0 (x)]

Amplitude distribution function Expanding N(a) for a 1 we obtain to leading order the integrated Karpman formula for the KdV equation: 3 1 x2 N(a) (η0 (x) 1) a/2 dx ( ) 2 π x 1 10 18 9 16 N(a) 8 7 6 5 4 3 2 N(a) 14 12 10 8 6 4 1 2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 a 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 a Figure: Number N(a) of solitary waves with amplitudes in the interval [a, a max] generated from the initial conditions with A = 0.4, w = 13 (left) and A = 2, w = 10 (right). Solid line is the SG modulation formula; symbols are the numerical solution; dashed line is the integrated Karpman formula (*).

Solitary wave train: amplitude profile From the speed-amplitude relationship for the SG solitons c s = 1 + a we get that for t 1 the amplitude distribution in the SG solitary wave train is a SG = (x/t) 2 1 ( ) while for the KdV soliton train we have the triangle distribution a KdV = 2(x/t 1) ( ) 4.5 4 3.5 3 2.5 a 2 1.5 1 0.5 0 1400 1600 1800 2000 2200 2400 x 2600 2800 3000 3200 3400 Figure: Amplitude profile of the SG solitary wavetrain at t = 1500. The parameters of the initial profile are A = 2 and w = 10. Solid line is the formula (*); symbols are the numerical solution; dotted line is the KdV soliton train amplitude profile (**)

References G.A. El, R.H.J. Grimshaw, and N.F. Smyth (2006) Unsteady undular bores in fully nonlinear shallow-water theory, Phys. Fluids, 18, Art No 027104, 17 pp. G.A. El, R.H.J. Grimshaw, and N.F. Smyth (2008) Asymptotic description of solitary wave trains in fully nonlinear shallow-water theory, Physica D in press.