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Proc Math Phys Eng Sci. 2017 January; 473(2197): 20160709.
PMCID: PMC5312131

Whitham modulation theory for the Ostrovsky equation

Abstract

This paper derives the Whitham modulation equations for the Ostrovsky equation. The equations are then used to analyse localized cnoidal wavepacket solutions of the Ostrovsky equation in the weak rotation limit. The analysis is split into two main parameter regimes: the Ostrovsky equation with normal dispersion relevant to typical oceanic parameters and the Ostrovsky equation with anomalous dispersion relevant to strongly sheared oceanic flows and other physical systems. For anomalous dispersion a new steady, symmetric cnoidal wavepacket solution is presented. The new wavepacket can be represented as a solution of the modulation equations and an analytical solution for the outer solution of the wavepacket is given. For normal dispersion the modulation equations are used to describe the unsteady finite-amplitude wavepacket solutions produced from the rotation-induced decay of a Korteweg–de Vries solitary wave. Again, an analytical solution for the outer solution can be given. The centre of the wavepacket closely approximates a train of solitary waves with the results suggesting that the unsteady wavepacket is a localized, modulated cnoidal wavetrain. The formation of wavepackets from solitary wave initial conditions is considered, contrasting the rapid formation of the packets in anomalous dispersion with the slower formation of unsteady packets under normal dispersion.

Keywords: wavepackets, Ostrovsky equation, modulation equations

1. Introduction

Oceanic internal waves are often assumed to have amplitudes small compared with the depth (weak nonlinearity) and wavelengths long compared with the depth (weak dispersion). Under these assumptions, the Korteweg–de Vries (KdV) equation

ut+uux+uxxx=0,
1.1

gives an accurate description for the interfacial displacement of the waves [14]. In KdV-type theories, the effects of background rotation are often considered negligible although observed waves can persist for several days allowing rotational effects to become important. The simplest extension of the KdV equation that takes the effects of rotation into account is the Ostrovsky (rotation-modified KdV) equation [5]

ut+uux+uxxx=γxudx,
1.2

where γ represents the relative strength of background rotation. It follows directly from (1.2) with γ≠0, that localized solutions of the Ostrovsky equation satisfy the zero-mass constraint

udx=0.
1.3

The Ostrovsky equation is derived and applies for disturbances with energy in only a finite range of wavenumbers that excludes the zero wavenumber [6]. Constraint (1.3) is equivalent to the requirement that solutions have no energy at the zero wavenumber.

The effect of adding the non-local integral term to the right-hand side of the KdV equation is to introduce ‘large-scale’ dispersion into the system. For oceanic waves, the coefficient representing the strength of rotation is usually positive, γ>0, and this parameter regime has been described as the Ostrovsky equation with normal dispersion [7]. The opposite case, when γ<0, arises in strongly sheared ocean flows and acoustic problems [7] and is described as anomalous dispersion.

In the strong-rotation limit (γ[dbl greater-than sign]1), it has been shown that the Ostrovsky equation supports localized wavepacket solutions for both cases of dispersion and several recent articles [710] have examined the dynamics of these wavepackets. In the weak-rotation limit (γ[double less-than sign]1), wavepackets have been observed for normal dispersion only. Helfrich [11], who was the first to discover these rotation-induced wavepackets, originally stated the solutions resemble modulated cnoidal wavetrain solutions of the KdV equation; however, these observations remain currently unexplained.

For the unperturbed KdV equation (1.1), it is possible to derive a set of equations, often referred to as the Whitham modulation equations [12], that describe the evolution of a modulated cnoidal wavetrain. The simplest derivation is based on the integrability of the KdV equation and thus does not extend directly to the Ostrovsky equation, which is not known to be integrable. Myint & Grimshaw [13] formulated a way of deriving the modulation equations for a KdV equation with a perturbation term,

ut+uux+uxxx=ϵV(u),
1.4

where V (u) is an arbitrary functional of u and ϵ[double less-than sign]1. Kamchatnov [14] has generalized their results but applying this work to the Ostrovsky equation is not direct because of the atypical nature of the non-local perturbation term.

This paper makes some minor changes to the method of Myint & Grimshaw [13] to derive a set of modulation equations for the Ostrovsky equation. Section 2 sketches briefly the derivation of the perturbed modulation equations which are verified against numerical integrations of the Ostrovsky equation. The derived equations are then used in §3 to describe wavepacket solutions of the Ostrovsky equation with §3a considering steady wavepacket solutions with anomalous dispersion and §3b unsteady wavepacket solutions with normal dispersion. Section 4 addresses the emergence of these wavepackets and in particular the rapid formation of packets under anomalous dispersion in contrast to their slower formation under normal dispersion.

2. Derivation of the modulation equations

Let the perturbative rotational term in the Ostrovsky equation be of order ϵ2 and take the derivative of equation (1.2), so that it is in the form

(ut+uux+uxxx)x=±ϵ2u,
2.1

where ϵ>0 and the positive and negative signs refer to normal and anomalous dispersion, respectively. If a solution of (2.1) has period 2L, is of compact support lying in |x|<L or decays sufficiently rapidly as x± then it satisfies the zero mass condition

LLudx=0,
2.2

where L is infinite in the last case.

Following Myint & Grimshaw [13], the equations governing modulated nonlinear wavetrain solutions of (2.1) can be derived straightforwardly by and seeking a solution of the form

u(x,t)=u0(θ,X,T)+ϵu1(θ,X,T)+ϵ2u2(θ,X,T)+,
2.3

where the fast variable θ and the slow space and time variables X and T are defined as

θ=ϵ1Θ(X,T),X=ϵx,T=ϵt.
2.4

The local frequency ω(X,T), local wavenumber k(X,T) and local phase velocity c(X,T) are defined by

ω=ΘT,k=ΘX,ω=kc
2.5

and are related by the consistency condition,

kT+(kc)X=0,
2.6

describing the conservation of waves.

Let the modulations be periodic with period 2L, or of compact support lying in |X|<L, or decay sufficiently rapidly as X, then an anti-differentiation operator X1 can be defined such that for any function v(X),

X1v=XLv(X)dX,
2.7

with L infinite in the last case. If, further, v satisfies (2.2), then

X1v=0at X=±L.
2.8

It is convenient to divide both sides of (1.2) by ϵ to give

(ut+uux+uxxx)X=±ϵu,
2.9

and so

ut+uux+uxxx=±ϵX1u,
2.10

for solutions vanishing as x. Then, substituting (2.3) into (2.9) gives at leading order

(kcu0θ+ku0u0θ+k3u0θθθ)X=0.
2.11

Equation (2.11) can be integrated once to give

(kcu0+12ku02+k3u0θθ)θ=C(T),
2.12

for some function C(T). Let the period in θ of the carrier waves making up the wavetrain be 2P. Then, integrating (2.12) over a period 2P in θ and using the periodicity of u0 shows that C(T) is identically zero. Hence, (2.11) can be written

cu0θ+u0u0θ+k2u0θθθ=0.
2.13

Using (2.10) allows the next term in the expansion to be written as

kcu1θ+k(u0u1)θ+k3u1θθθ+f1=0,
2.14

where

f1=u0T+u0u0X+3k2u0θθX+3kkXu0θθX1u0.
2.15

Integrating (2.14) with respect to θ over a period 2P, and using the periodicity of u1, gives

f1=0,
2.16

where the mean of any 2P-periodic function v(θ) is defined as

v=12PPPv(θ)dθ.
2.17

Multiplying (2.14) by u0, integrating with respect to θ over a period 2P, and using periodicity and the fact that u0 satisfies (2.13), gives

u0f1=0.
2.18

This averaging operator also gives a simpler form for the anti-differentiation operator. For any function v(θ,X) of the slowly varying X and rapidly varying θ, of period 2P in θ,

X1X2vdX=X1X2[v+(vv)]dX=X1X2vdX,
2.19

because the second term in the second integral vanishes in every consecutive sub-interval of length 2P in θ from the construction of left angle bracketvright angle bracket. Thus,

X1v=X1v,
2.20

which is a slowly varying function, unaffected by further averaging.

Combined with the consistency relation (2.6), equations (2.16) and (2.18) give the modulation equations for the leading-order solution u0. The remainder of the analysis is confined to this leading-order solution and because the modulation equations form a closed system the subscript zero can be dropped without ambiguity. Substituting for f1 in (2.16) and (2.18) then using (2.20) gives

uT+12u2X=±X1u
2.21a

and

12u2T+13u3X32(k2uθ2)X=±uX1u.
2.21b

These equations are precisely those of Myint & Grimshaw [13] with the operator V (u) replaced by integration with respect to X, the slow-x variable.

Equation (2.13) has the exact periodic cnoidal solution,

u=a{b+cn2[β(θθ0)]}+d,
2.22

where cn is the Jacobi elliptic function with parameter m (described in [13] as the modulus, usually used for m1/2). Choose b, so that d is the mean value of u, i.e. d=left angle bracketuright angle bracket, then

a=12mk2β2,
2.23a

b=(1m)mE(m)mK(m),
2.23b

c=d+(a3m)[2m3E(m)K(m)]
2.23c

andP=K(m)β,
2.23d

and all parameters are slowly varying functions of the form a(X,T), b(X,T), c(X,T), d(X,T), k(X,T), m(X,T) and β(X,T). The phase shift θ0(X,T) can be determined only by considering the second-order terms in the expansion (2.3), but as its value does not affect the other parameters it is ignored in the subsequent analysis.

Integrating (2.13) twice with respect to θ gives

k2uθ2=2A+2Bu+cu2u33,
2.24

where

A=(ab+d)33+c(ab+d)22a2(1m)(ab+d)6m
2.25a

and

B=(ab+d)22c(ab+d)+a2(1m)6m.
2.25b

Differentiating (2.24) with respect to θ and dividing by 2uθ gives

k2uθθ=B+cuu22,
2.26

and averaging (2.26) gives, for the cnoidal solution,

u2=2(B+cu)=2(B+cd).
2.27

Multiplying (2.26) by u and averaging gives

k2uθ2=Bd+cu2u32,
2.28

whereas averaging (2.24) directly gives

k2uθ2=2A+2Bd+cu2u33.
2.29

Then adding (2.28) and (2.29) gives

(56)u3=2A+3Bd+2cu2,
2.30

and taking 3 × (2.29)–2 (2.28) gives

5k2uθ2=6A+4Bd+cu2.
2.31

Substituting (2.22) in system (2.21) and including the consistency relation (2.6) gives the required modulation equation system

kT+(ck)X=0,
2.32a

dT+(cd+B)X=±X1d
2.32b

and(cd+B)T+[c(cd+B)A]X=±(12)[(X1d)2]X.
2.32c

using d(X1d)=(12)[(X1d)2]X. Because there are four equations (2.23ad) between the seven parameters a, b, c, d, k, m, β only three are independent and hence equations (2.32ac) are sufficient to determine the solution.

Following Whitham [12], equations (2.32ac) can be greatly simplified introducing the variables

r1=q+r,r2=r+p,r3=p+q,
2.33

where p, q, r, (rqp) given by

p=d+(am)[1E(m)K(m)],
2.34a

q=d+(am)[1mE(m)K(m)]
2.34b

andr=d(am)[E(m)K(m)],
2.34c

are the three roots of the polynomial on the right-hand side of (2.24). The modulation equations (2.32) can be written

riT+Qi(r1,r2,r3)riX=±2(X1d),fori=1,2,3,
2.35

where each ri is a Riemann invariant that propagates with characteristic velocity Qi, given by

Q1=c(a3)[K(m)(K(m)E(m))],
2.36a

Q2=c(a3)(1m)K(m)[E(m)(1m)K(m)]
2.36b

andQ3=c+(a3)(1m)K(m)[mE(m)].
2.36c

The change of variables required to derive equations (2.35) from equations (2.32) is laborious (details can be found in [13,12]).

As a function of the Riemann invariants, the solution (2.22) takes the form

u=(r1+r2r3)2+(r2r1)cn2[(r3r1)12(ξξ0)],
2.37

where

ξ=xct,c=(r1+r2+r3)6,m=(r2r1)(r3r1),
2.38

and ξ0 is a phase shift. The wavelength λ=2P/k is given by

λ=2K(m)(r3r1)/12,
2.39

and the maximum and minimum values of u are

umax=(r2+r3r1)2andumin=(r1+r3r2)2.
2.40

The wave amplitude and mean of u are given by

a=r2r1andd=(r1+r2r3)2+(r3r1)E(m)K(m).
2.41

To test the results derived above, both the Ostrovsky equation (2.1) and equations (2.35) were integrated numerically with equivalent initial conditions. To integrate both the modulation equations (2.35) and the Ostrovsky equation (2.1) a pseudo-spectral Fourier discretization on a periodic domain in x and an adaptive fourth-order Runge–Kutta time-stepping in t was used. The initial condition was chosen as a cnoidal wavetrain (2.22) with a=1+0.5exp[x2/150], d=0 and m=0.97. Figure 1 shows the results of the integration. The solutions obtained from integrating the Ostrovsky equation are shown by the solid lines and the envelope solutions obtained from integrating the modulation equations are shown by the dashed lines. Figure 1a shows the solution used for both integrations at t=0. Figure 1b shows a comparison of the integrations at t=325 when no large-scale dispersion is present. This case corresponds to the KdV equation and unperturbed Whitham modulation equations. Figure 1c shows a comparison of the integrations at t=325 for normal dispersion of magnitude ϵ=0.1 and Figure 1d shows a comparison of the integrations at t=300 for anomalous dispersion of magnitude ϵ=0.224.

Figure 1.
Numerical integrations of the Ostrovsky equation (solid lines) and the time-dependent modulation equations (dashed lines) for both normal and anomalous dispersion. (a) A cnoidal initial condition with parameters a=1+0.5exp[x2/150], ...

The solutions of the perturbed modulation equations agree qualitatively with the corresponding integrations of the Ostrovsky equation. The modulation equations, however, are only valid for sufficiently small ϵ (≈0.1, in practice). For larger ϵ, the KdV equation cnoidal wave (2.22) is no longer a close approximation to a solution of the Ostrovsky equation, and the theory becomes less accurate. This is apparent in Figure 1d, where ϵ=0.224 and the wavetrain to the left of the disturbance no longer appears cnoidal.

A striking feature of the integrations shown in figure 1 is that the modulation equations appear to correctly capture the nonlinear steepening behaviour exhibited when either normal or anomalous dispersion are present (figure 1cd). Furthermore, for the normal dispersion case in figure 1c, the three peaked structure located between x=−200 and −300 resembles that of the rotation-induced wavepackets seen in [8,9,11].

3. Application to wavepackets

(a) Anomalous dispersion

For anomalous dispersion, it is known steady localized solutions exist. In the strong-rotation limit (ϵ[dbl greater-than sign]1), these solutions originate from the point in wavenumber space where the linear phase and group velocities are equal [7], but as the strength of rotation is decreased the wavepacket structure of this family of solutions is eventually replaced by a soliton solution (figure 2a) that is described asymptotically by a KdV soliton with pedestal of order ϵ[double less-than sign]1 [15].

Figure 2.
Numerical solutions of the Ostrovsky equation with anomalous dispersion moving at constant velocity c=c0. (a) A soliton solution with ϵ=0.0324 and c0=0.213. (b) A three-peak packet solution with ϵ=0.0207 and c0=0.138. (c) A 35-peak packet ...

Section 4 describes how these single-peak solitary wave solutions arise naturally from the evolution of an initial KdV soliton in the presence of weak rotation. These are not, however, the only steadily propagating solutions of compact support. Figure 2b,c shows new steady wavepacket solutions found by a spectral Newton–Kantorovich iteration on a periodic grid.

The packet in figure 2b has the form of three adjacent KdV solitons. The packet in figure 2c resembles neither figure 2a nor figure 2b. As more peaks are added to the solution, the leading and trailing edges retain the form of solitary waves but the centre of the packet becomes more sinusoidal. Solutions with more peaks can be obtained only by decreasing the strength of the rotation, with the packet speed and rotation strength of the solutions in figure 2 related through the normalization here so that c=20ϵ/3. An approximation for the upper and lower envelopes of the multi-peak solutions was found by numerical interpolation through the peaks and troughs of the packets. It was observed that the maximum of the lower envelope occurred at the centre of the packet and its value approached zero as the number of peaks increased, a result that is derived below from the modulation theory. The small peak-number (strong rotation) limit of the solution family shown here coincides with the first member of the solution family in figure 4 of Obregon & Stepanyants [16], found, for stronger rotation than considered here, by placing two individual solitons, each similar to that in figure 2a, so that the maximum of one soliton coincides with a local minimum of the other.

Figure 4.
Solution trajectories for a steady solution, c=c0, of the modulation equations with anomalous dispersion in (r1,r3) space.The shape of the paths is independent of D which simply changes the speed, i.e. introduces a nonlinear, monotonic stretching in ...

For an anomalous wavepacket moving at constant velocity c0, the time dependence of the modulation equations (2.35) can be removed by introducing the variables ri=ri′(X′), Qi=Qi′(X′), d=d′(X′), where X′=Xc0T. Substituting the new variables into (2.35) and dropping dashes gives

dridX=2D(Qic0),wheredDdX=d.
3.1

If the wavepacket is steady, i.e. c=c0, one of the Riemann invariants, ri, can be eliminated using (2.38). Further if the packet is assumed to be symmetric, forcing D(0)=0, then integrating equation (2.32c) after transforming into the frame, X′=Xc0T, shows that

D=sgn(X)2(AA0),
3.2

where A0=A(0) and sgn(X) is the signum function. The sign of the root in (3.2) has been chosen to agree with packets like that those shown in figure 2, where the ‘mass’ is initially negative when approaching the packet from negative x. Hence, for a steady, localized, symmetric wavepacket the modulation equations are reduced to a set of two ordinary differential equations.

To test if the modulation equations have wavepacket solutions, equations (3.1) were integrated numerically, using an initial condition corresponding to the wavepacket solution shown in figure 2c. An approximation for the envelope of the solution in figure 2c was found using numerical interpolation of the peaks and troughs. Using the known velocity c0 and equations (2.40) and (2.41), approximations for r1(0) and r2(0) were then found. Rather than use the analytical expression for D given in equation (3.2), the equation dD/dX=d was used, to avoid the difficulty of integrating from a stationary point.

The results of the integration are shown in figure 3. The dashed lines in figure 3a for x<180 show the envelope obtained from integrating equations (3.1), and the solid lines show the exact wavepacket solution from figure 2c. The values for the Riemann invariants obtained from the integration are also shown in figure 3b. The numerical integration failed for |r2r3|<10−8 which corresponds to the m=1 solitary wave limit and at the point r2=r3 the solution is likely to be discontinuous. Whitham [12] found that the non-rotating equations can have discontinuities across characteristics and these discontinuities can take the form of a jump from a solitary wave to a state of no waves, exactly as seen here in front of the leading solitary wave at x≈180.

Figure 3.
(a) Comparison of the anomalous dispersion wavepacket solution of the Ostrovsky equation (solid line) in figure 2c with themodulation equations solution (dashed lines). (b) The Riemann invariants: r1 (dash-dotted line/blue online), r2 (dashed ...

In the limit m1, it can be shown r12d and r2,r3,2d+a with their characteristic velocities given by Q1d and Q2,Q3,d+a/3 [13]. If a no wave solution is required ahead of the leading solitary wave, then a=0, and r2 and r3 must jump to the r1 solution at the leading edge of the solitary wave. Therefore, only r1 is continuous across this front. Additionally, in the limit m1, it can be shown 2Ac0d22d3/3. Hence, for a localized solution d0 as |X|, the constant A0 in (3.2) must be zero because of the zero mass constraint D0 as |X|. Substituting the above into equation (3.1) gives

dddX=c0d22d3/3(dc0)forX>0,
3.3

which has the solution,

X+X0=9c06d+c0log[9c06d+9c09c06d9c0],
3.4

where X0 is a constant. The solution (3.4) is plotted for x>180 in figure 3 where the constant X0 was fixed using the solution from figure 2c. Note in figure 3b that while d is continuous across the edge of the leading solitary wave, its derivative is not. At the point r2=r3 (depicted by the vertical dotted line), d has infinite gradient.

For a steady solution with c=c0, the identity logλ/ri=1/6(Qic0) allows the modulation equations (3.1) to be written as the gradient flow

drdX=12D(r1,r2,r3)rlogλ(r1,r2,r3),
3.5

where [nabla]r=([partial differential]/[partial differential]r1,[partial differential]/[partial differential]r2,[partial differential]/[partial differential]r3) and r=r1e^1+r2e^2+r3e^3 is a point in (r1,r2,r3) space. Because (e^1+e^2+e^3)rlogλ=0 the flow in X is constrained to the plane r1+r2+r3=6c0 and so has only two degrees of freedom. For a localized solution, D=sgn(X)2A, and it can be shown that A=(3c0r1)(3c0r2)(3c0r3). The symmetry condition D(0)=0, then implies that one of the Riemann invariants equals 3c0 at X=0. For c0>0 and distinct ri(0), the restriction 0≤m≤1 means that the only possibility is r2(0)=3c0. Either side of X=0 the function D is single signed and bounded away from zero and thus affects simply the speed at which a point moves along the flow lines. Figure 4 shows the trajectories (for X>0, with arrowheads in the direction of increasing X) projected onto the plane r2=0. The flow is bounded by the lines r3=(6c0r1)/2 and r3=3c0r1 corresponding to m=1 and r2=3c0, respectively. Figure 4 shows that any trajectory starting in this sector moves to m=1 and hence the wavetrain part of the solution terminates in solitons. The condition r2(0)=3c0 also means that solutions satisfy umin=0 at X=0 as observed in the numerical solutions. The bold (red) line in figure 4 shows the trajectory of the solution in figure 3.

The comparison above shows that the modulation equations accurately describe the wavepacket solution of figure 2c. Similar direct comparison, not reproduced here, of modulation solutions with the wavepacket solution of of22b, show similar accuracy.

(b) Normal dispersion

In contrast to the anomalous case, for normal dispersion, there are no known steady wavepacket solutions. Although the wavepacket envelope propagates at a constant velocity, the carrier waves propagate at a velocity different to that of the envelope making the solution intrinsically unsteady. It has also been observed that the phase velocities are non-constant [9]. In the strong-rotation limit, the wavepacket dynamics are described asymptotically by a nonlinear Schrödinger equation bright soliton envelope solution with frequency-shifted linear carrier waves [9]. However, in the weak-rotation limit, despite being the most physically relevant parameter range, very little is known about the dynamics of the normal dispersion wavepackets.

For a normal dispersion wavepacket moving at constant velocity, s0, the time dependence of the modulation equations can be eliminated by moving into the frame Xs0T. Therefore, similar to the anomalous case, the solution in the translating frame is described by the following equations

dridX=2D(Qis0),wheredDdX=d.
3.6

Equations (3.6) are identical to the anomalous set (3.1), but with s0 replacing c0 and an opposite sign on the right-hand side representing the change in dispersion. If the packet is assumed to be symmetric, forcing D(0)=0, then integrating equation (2.32c) after transforming into the frame, Xs0T, gives

D=sgn(X)2(ϕ0ϕ),
3.7

where ϕ=2A+2(s0c)(cd+B) and ϕ0 is the value of ϕ at X=0. The sign of the root in (3.7) has been chosen to describe a solution where the ‘mass’ is initially positive when approaching the packet from negative x and hence is opposite in sign to (3.2). For a localized, symmetric wavepacket with normal dispersion, the modulation equations are therefore reduced to a set of three ordinary differential equations. There is one equation more than for the anomalous case as it is not possible to eliminate a Riemann invariant, using the expression for c in (2.38).

For steady wavepackets with anomalous dispersion, it was shown in figure 4 that all solution trajectories travel towards the limit m=1 and terminate in solitary waves. The direction of the trajectories in figure 4 is determined by sign of the dispersion. For wavepackets with normal dispersion, it is plausible to predict solution trajectories would move in the opposite direction and travel towards the limit m=0, where they terminate in infinitesimal linear waves. This hypothesis is in agreement with the wavepackets found by Whitfield & Johnson [9], where it was noted that the wavepackets had exponentially small radiating tails at their edges. Under this assumption, a solution similar to (3.4) can be found for the outer limit of the wavepacket.

Following Myint & Grimshaw [13], we consider the limit m0, where a0 with a/m fixed and approximately given by 12k2β2. In this limit, to leading order in a, it can be shown r12d12k2β2, r22d12k2β2 and r3,2d with characteristic velocities given by Q1,Q2d12k2β2 and Q3,d [13]. Additionally, in this limit, it can be shown A2d2k2β2+d3/6, B4dk2β2d2/2 and cd4k2β2. Therefore, D2=2ϕ0s0d2+2d3/3 and hence for a localized solution d0 as |X|, the constant ϕ0 in (3.7) must be zero because of the zero mass constraint D0 as |X|. Substituting these results into equation (3.6) for r3 gives

dddX=2d3/3s0d2(ds0)forX>0,
3.8

which has the solution

X+X0=6d9s02s0arctan[3s02d3s0],
3.9

where X0 is a constant. In the limit m0, the solution to leading order in a is given by u=d and hence it is expected that (3.9) would be the solution at the wavepacket edge.

In Whitfield & Johnson [9], numerical solutions were obtained for unsteady wavepackets with normal dispersion. Therefore, to test if the modulation equations have wavepacket solutions, equations (3.6) were integrated numerically using an initial condition corresponding to the wavepacket with the weakest rotation in [9]. Using the measured envelope values from [9] approximations for r1(0), r2(0) and r3(0) could be parametrized as a function of c(0). The constant ϕ0 in (3.7) was assumed to be zero for the reasons outlined in the preceding paragraph and hence imposing the symmetry condition, D(0)=0, fixed the value of s0 for a given c(0). The lack of data made it impossible to determine c(0) and therefore it was left as a parameter. As was done for anomalous dispersion, the equation dD/dX=d was used to avoid the difficulty of integrating from a stationary point.

The result of one integration is shown in figure 5. In figure 5a, the dashed lines for |x|<60 show the envelope obtained from numerically integrating equations (3.6) and the dashed lines for |x|>100 show the analytical solution (3.9) with X0=−1.7756. The dotted lines in figure 5a show the measured envelope obtained by Whitfield & Johnson [9], and the solid lines show the exact unsteady solution from [9] at its maximum value. The Riemann invariant values obtained from the integration are also shown in figure 5b. The value of c(0) in figure 5 was chosen to produce the best agreement with the measured packet envelope, however, for a wide range of different c(0), it was found that the numerical integration failed before it could reach the edge of the wavepacket. This was because Q2s0 which led to |dr2/dX| being extremely large. The reason for this behaviour is unclear. The approximate initial conditions are a possible source of error, as is the assumption of a symmetric solution, especially as it was shown conclusively in [9] that for ϵ[dbl greater-than sign]1 wavepackets had asymmetry of order 1/ϵ.

Figure 5.
(a) An unsteady, normal-dispersion wavepacket solution of the Ostrovsky equation (solid line) and its envelope (dotted lines) compared with the modulation equations solution (dashed lines). (b) The Riemann invariants: r1 (dash-dotted line/blue), r2 (dashed ...

Relaxing the localization condition by allowing ϕ0 to be non-zero allows initial conditions for the modulation equations to be chosen, so that their numerical integration can proceed indefinitely. Using the initial conditions of the wavepacket solution shown in figure 5 as a starting point, the values of s and c(0) were varied until a numerical integration was found, so that Q2 remained distinct from s0. Figure 6 gives results from one such integration, showing a periodic solution of the modulation equations and suggesting the Ostrovsky equation possesses unsteady, periodically modulated cnoidal wave solutions with the modulation steady in a frame moving at speed s.

Figure 6.
A periodic, modulated, wavetrain solution of the modulation equations with normal dispersion, at ϵ=0.0180. (a) The top, umax, and bottom, umin, envelopes from the solution. These are steady in a frame propagating with speed s=−0.0789. ...

It has been suggested [17] that rotation-induced wavepackets, like that of figure 5a, could offer an explanation as to why non-rank-ordered internal solitary wavepackets are sometimes observed instead of the expected rank-ordered wavepacket structure of solutions of the (non-rotating) KdV equation. The modulated wavetrain of figure 6a provides further evidence that rotational effects influence internal wavetrains in such a way that they form non-rank-ordered structures. It remains to be shown, however, whether wavetrains of this form arise in the Ostrovsky equation with more general initial conditions.

4. Wavepacket emergence

The variables and analysis of §3 give a context for discussing the emergence of the wavepacket solutions above from KdV soliton initial conditions under weak rotation as observed by Grimshaw & Helfrich [8], Grimshaw et al. [7] and Whitfield & Johnson [9]. In terms of the variables (2.4), equation (2.1) has the inner solitary wave solution

ui=asech2[(a12)1/2(xc(T)t)]+ϵd(T)+O(ϵ2),
4.1

where c(T)=13a+ϵd(T) gives the slowly and weakly varying speed of the solitary wave on a small, slowly varying pedestal. For a KdV soliton initial condition of amplitude a0, a=a0 and d(0)=0. As x±, uiϵd(T) and so merges smoothly onto an outer solution u=ϵuo(X,T), which satisfies, to leading order in ϵ

uTXo=±uo.
4.2

At this order the speed c can be taken to be constant with c=co=a/3>0. Moving to a frame translating to the right with speed c0 by writing ξ=Xc0T gives

uTξoc0uξξo=±uo,ξ0
4.3a

and

uo(0,T)=d(T).
4.3b

The problem is closed at this order by the mass constraint (1.3) which gives, to leading order,

a0sech2((a012)1/2θ)dθ+uodξ=0,
4.4a

i.e.[c0uξo]0=±A,
4.4b

using (4.3), where [ ]0 denotes the jump in the enclosed quantity across ξ=0 and A=43/a0. In particular, (4.3a) becomes the forced linearized reduced Ostrovsky equation

uTξoc0uξξouo=±Aδ(ξ),
4.5

where δ(ξ) denotes the Dirac delta function. Writing

uo(ξ,T)=u^(κ,T)eiκξdξ,
4.6

gives the Fourier transform of (4.5) as

u^T+iω(κ)u^=±Aiκ,
4.7

where ω(κ)=−c0κ±1/κ, with solution vanishing at T=0,

u^(κ,T)=A(eiωT1)(1c0κ2).
4.8

The solution consists of the steady solution forced by the poles at κ2=±1/c0 and a superposition of dispersive waves of frequency ω(κ) and group velocity cg(κ)=−c0[minus-or-plus sign]1/κ2. The flow evolution and final state for normal and anomalous dispersion differ.

For normal dispersion, cg is strictly negative for all κ and all waves propagate in the negative ξ direction. The poles in (4.7) and (4.8) lie on the real κ axis and thus contribute a steady downstream lee-wavetrain behind the soliton with the deviation of the solution from its final steady state decaying as T−1/2 at large T, so that as T

u{O(T1/2)ξ>0,(Ac0)sin(ξc0)+O(T1/2)ξ<0,
4.9

as in figure 7a. There is an order ϵ2 momentum flux in the negative x-direction in the lee of the soliton which exerts a drag on the soliton, leading to a decay in the soliton over the longer timescale ϵT=ϵ2t. This longer-time decay of the soliton is discussed in greater detail by Grimshaw et al. [18,7].

Figure 7.
The outer solution of (4.5) at time T=20, for speed c0=1 and amplitude A=4, shown in a frame moving with the soliton. (a) Normal dispersion. The steady lee-wave wake has been established in −20<ξ<0 with transients in ξ ...

For anomalous dispersion, the poles in (4.7) and (4.8) lie on the imaginary κ axis and thus contribute a symmetric steady contribution decaying exponentially away from a maximum at ξ=0. The group velocity vanishes when κ2=c0. Thus, transient long waves with |κ|<c01/2 propagate to ξ= in advance of the soliton, decaying as T−1/2, whereas transient short waves with |κ|>c01/2 propagate to ξ= in the lee of the soliton, decaying as T−1/2. Waves with |κ|=c01/2 remain in vicinity of the soliton. As dcg/=2/κ3 vanishes nowhere, these waves decay as T−1/3. Thus, as T,

u(A2c0)exp(|ξ|c0)+O(T1/3),
4.10

as in figure 7b. The final flow state is symmetric without waves and so suffers no wave drag. Unlike the case of normal dispersion, there is no further leading-order evolution over the longer timescale ϵT=ϵ2t, in agreement with the numerical integrations of [7]. Note that although the T asymptotic state is a symmetric zero-drag solution, the solution is asymmetric throughout the evolution to this state and so the soliton suffers a loss of momentum flux of order ϵ2, leading to a reduction in its amplitude. This higher-order effect is not captured by leading order analysis here. The departure from the analysis in [7], which leads to the prediction of a decaying soliton there, is that the far-field boundary condition on the leading order solution u(0) of [7], compatible with (4.10), is u(1)(θ)=u(1)(θ+), so u(1)=constant is an allowable solution of the adjoint equation (3.9) of [7], giving conservation of mass to leading order in γ.

5. Discussion

The Whitham modulation equations for the Ostrovsky equation, derived using the method of Myint & Grimshaw [13], have been used to find wavepacket solutions of the Ostrovsky equation in the weak-rotation limit. Considering wavepacket solutions in the modulation equations offers two advantages: first, the anti-derivative representing large-scale dispersion is simplified to an algebraic term and, second, intrinsically unsteady wavepacket solutions of the Ostrovsky equation with normal dispersion, whose phase and group velocities differ, become steady solutions.

For anomalous dispersion, a new type of cnoidal wavepacket solution was obtained and shown to be representable as a solution of the modulation equations. The modulation equations showed, in agreement with numerical integrations, that the wavepacket has minimum zero at its centre and that the wavetrain section of the packet terminates in a KdV soliton at each edge. An analytical solution for the outer section of the wavepacket where no waves exist was also found.

In contrast to the anomalous dispersion case, for normal dispersion, it is argued that any localized symmetric wavetrain solution should approach infinitesimal linear waves at its edge and more solitary-like waves at its centre. Based on this assumption an analytical solution for the outer limit of the wavepacket was found. The outer solution was compared with data from a normal dispersion wavepacket found in [9], showing good agreement. The same packet data was also compared with solutions of the modulation equations for the inner wavetrain region of the packet and agreed well over the central portion of the packet. Towards the edge of the packet, the characteristic velocity of the intermediate Riemann invariant became equal to the packet propagation speed introducing an infinite gradient in the integrations across the packet. This singularity appears to be a robust feature of the equations when applied to these packets but its physical meaning has not been determined. The good agreement of the modulation solution with the data over the majority of the wavepacket suggests that the modulation equations do have wavepacket solutions for normal dispersion, supporting the observation of Helfrich [11] that the wavepackets under normal dispersion can be regarded as modulated cnoidal waves.

Section 4 describes the emergence of these wavepackets from KdV soliton initial conditions under weak rotation. For anomalous dispersion the flow adjusts over times of order ϵ−1 to a zero-drag symmetric state that undergoes no further evolution at this order. For normal dispersion, the flow evolves over times of order ϵ−1 to an asymmetric non-zero drag form that undergoes further slow decay over times of order ϵ−2, as described by Grimshaw et al. [18].

Cnoidal wavepacket solutions are known [10] to arise from localized initial conditions in the Ostrovsky equation with normal dispersion. It remains to be shown whether the cnoidal wavepackets described above for the Ostrovsky equation with anomalous dispersion will arise from general initial conditions.

Authors' contributions

The authors are equally responsible for the contributions to the paper and both authors gave final approval for publication.

Competing interests

We declare we have no competing interests.

Funding

The funding of the research rested solely on the authors’ being employed and supported by UCL.

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