NPGNonlinear Processes in GeophysicsNPGNonlin. Processes Geophys.1607-7946Copernicus PublicationsGöttingen, Germany10.5194/npg-24-695-2017Brief communication: Multiscaled solitary wavesDerzhoOleg G.Institute of Thermophysics, Russian Academy of Sciences, Novosibirsk, RussiaOleg Derzho (oderzho@mun.ca)23November201724469570020February20176March201722October201723October2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://npg.copernicus.org/articles/24/695/2017/npg-24-695-2017.htmlThe full text article is available as a PDF file from https://npg.copernicus.org/articles/24/695/2017/npg-24-695-2017.pdf
It is analytically shown how competing nonlinearities yield multiscaled
structures for internal solitary waves in stratified shallow fluids. These
solitary waves only exist for large amplitudes beyond the limit of
applicability of the Korteweg–de Vries (KdV) equation or its usual
extensions. The multiscaling phenomenon exists or does not exist for almost
identical density profiles. The trapped core inside the wave prevents the
appearance of such multiple scales within the core area. The structural
stability of waves of large amplitudes is briefly discussed. Waves of large
amplitudes displaying quadratic, cubic and higher-order nonlinear terms have
stable and unstable branches. Multiscaled waves without a vortex core are
shown to be structurally unstable. It is anticipated that multiscaling
phenomena will exist for solitary waves in various physical contexts.
Introduction
The typical horizontal scale (or scales) is a major characteristic of a plane
disturbance propagating in a nonuniform medium. Usually, in an ideal
density-stratified shallow fluid, a wave of small albeit finite amplitude has
one typical scale resulting from the (local) balance between nonlinearity and
dispersion like in the realm of the Korteweg–de Vries (KdV) equation
. Solitary waves of permanent forms for which capillary
dispersion is on the same order as the gravitational one may have oscillatory
outskirts as predicted by . When viscosity is taken into
account, transient effects leading to various length scales are discussed for
the KdV-type equation with cubic nonlinearity, for example by
. In the present note it is shown that, for the
gravitational dispersion, ignoring all other previously mentioned effects,
solitary waves with multiple scales are possible. These solutions exist only
for disturbances of finite amplitude exceeding the range of applicability of
the extended KdV model, which incorporates both quadratic and cubic
nonlinearities. Higher nonlinearity in the existing small-amplitude KdV or
mKdV models leads to the correction of the wave length scale without
generation of multiscaling. For the appearance of multiscaling, the various
competitive nonlinearities should be on the same order, and that order needs
to be higher than the cubic one, as analytically discussed below. This effect
was initially noticed by in a Russian journal, but the
result was not widely disseminated. Recently presented a
numerical procedure that provides fast calculations for gravitational waves
between rigid lids. This model is able to work with fine-density
stratifications. reported two-humped and usual one-humped
solitary internal wave solutions for nearly identical density profiles in a
two-pycnocline density stratification. have numerically
shown that in some stratifications with two pycnoclines three conjugate flow
solutions leading to two-humped solitary waves were present.
theoretically considered continuous stratification in order to characterize
the role of the vertical structure of the fluid density in the context of
waves close to the limiting amplitude. To the best of the author's knowledge,
neither specific nonlinearity in terms of power series of wave amplitudes
necessary to reveal a two-humped structure nor regions of density profiles
with a single pycnocline at which such structures exist have been examined in
the literature. derived a KdV-like equation with quadratic and
quartic nonlinear terms for interfacial transient waves for the specific
three-layer geometry. Assumption of a small albeit finite wave amplitude was
essential to balance nonlinearity and dispersion in that study. Table-top
limiting solutions were reported, and they were stable within the accuracy of
their numerical scheme. In the current paper, an asymptotic model presented
in earlier papers by the author addresses multiscaling phenomena for internal
solitary waves under free surfaces in the framework of the
Dubreil–Jacotin–Long (DJL) equation . Special attention is
given to the case of complicated nonlinearity involving both quadratic, cubic
and quartic nonlinear terms for the case of continuous stratification with a
single pycnocline. Solitary waves of permanent form, their existence and
structural stability are discussed. It is worth noting that the family of
solutions is richer than two-humped structures. It is expected that such
multiscaled solitary waves will exist in other physical systems where
complicated competitive nonlinearities are balanced by dispersion.
Model for internal waves
Let us consider the two-dimensional steady motion of an ideal
density-stratified fluid in a framework of a reference moving with the phase
speed of wave c. The approach is asymptotic, being based on the DJL
equation for waves without a priori limitation on amplitude. This approach
has started from the pioneering work by . Let us consider the
stratification in the form
ρ0(z)=ρ00(1-σ(z+δf(z))),δ≪1,σ≪1,f∼1,
where σ denotes the Boussinesq parameter. In it was
shown that for this case the dimensionless (primed) streamfunction
ψ′=-ψ/cH of a solitary disturbance obeys the equation
ψzz+μ2ψxx+λ(ψ-z)-σ2(ψz2-1-2ψλ(ψ-z))+δλ(ψ-z)fψ(ψ)=o(σ,δ,μ2),
where μ is the aspect ratio H/L and λ=σgHc2.
In Eq. (2) z denotes the vertical axis, taken as positive upwards, and x
corresponds to the horizontal axis; z and x are scaled with H and L,
the given vertical and horizontal scales, respectively. Expecting no
confusion, we have, for simplicity, dropped the primes in Eq. (2). Let us
locate the bottom and the surface at the dimensionless heights z=-0.5 and
z=0.5+η(x), respectively, where η(x) denotes surface displacement.
The boundary conditions at the bottom and surface are
ψx=0atz=-0.5,σ(ψxψzψzz-ψz2ψzx)+λψx=o(σ)atz=0.5+η(x),ψx=-ηxψz.
The solution of Eqs. (2)–(5) is sought in the form
ψ=ψ(0)+μ2ψ(1)+…,λ=λ(0)+μ2λ(1)+…,η=η(0)+μ2η(1)+…,
where zeroth-order variables are of order unity. Below we shall provide a
solution for the first mode, which is most frequently observed in nature. The
analysis for the higher modes is similar. In the zeroth order,
ψ(0)=z+A(x)cos(πz),λ(0)=π2,η(0)=0,
where the amplitude function A(x) is to be determined at a higher order.
For the solution to the first-order equation to exist, the solvability
condition (Fredholm alternative) demands that
Axxx+λ(1)Ax-σμ2(2Ax-8πAAx+2π2A2Ax),+2δμ2Qx(A)=0Q(A)=A∫-0.50.5cos2(πz)fψ(ψ=ψ(0))dz.
In order to (locally) balance nonlinearity and dispersion, we have to require
max(σ/μ2,δ/μ2)∼1, thus determining L.
suggested considering the nonlinear terms as a power series
in the Boussinesq parameter instead of the small-amplitude parameter.
somewhat extended this idea to account for a more general
undisturbed flow state. After straightforward integrations, Eqs. (8)–(9) can
be reduced to
Ax2+λ(1)A2+2δμ2∫0AQ(A′)dA′+A28πA3-2-π2A23=0.
The Weierstrass approximation theorem states that every continuous function
defined on a closed interval can be uniformly approximated as closely as
desired by a polynomial function. A recent account of the topic is reviewed
in . Thus, the integral below can be represented with the help
of some Nth-order polynomial according to the Weierstrass approximation
theorem. In the current study only a polynomial formula for stratification is
considered; thus, it directly leads to nonlinearities in the polynomial form.
∫0AQ(A′)dA′=A2PN(A)
For the wave of amplitude A0, Eq. (10) yields
Ax2A2=(A0-A)Φ(A,A0),Φ(A,A0)=2δμ2PN(A0)-PN(A)A0-A+σμ2π238π-A-A0,λ(1)=σσμ2-8πA3+2+π2A23-2δμ2PN(A0).
Equations (12)–(14) determine completely both
the profile and phase velocity of a solitary wave with amplitude A0.
Multiscaling
The function f in the form of an Mth-order polynomial generates PN
with the index N=M-1. The power index of Φ is thus max(1,M-2). The
condition for Eq. (8) to possess a multiscaled solution reduces to the
condition that Φ(A,A0) must be sign-defined with several extrema
within [0,A0]. Thus it must have more than two imaginary roots on that
interval. It determines that M≥4; i.e., for a stratification in the form
of a cubic polynomial or if the wave amplitude is small enough to neglect
A4 and higher-order nonlinearities, multiscaled solitary waves do not
exist because f has no imaginary roots for this case. This is why classical
KdV or mKdV can not provide multiscaled solitary waves over a flat bottom.
Let us consider wave structures for the density stratification in the form
ρ0(z)=ρ(1-σz+0.5σ2z2+ασ2z4),
which produces quadratic, cubic and quartic terms in Eq. (8). Thus Eq. (12)
for this case of stratification becomes
Φ(A,A0)=σμ2-8π313+2α-160α9π2+π23A+A0+128απ275A2+A02+AA0.
Two-humped solitary waves for the stratification given by Eq. (17) exist in
the domain shown in Fig. 1.
Existence domain for the two-humped solitary wave.
The two-humped solitary wave with amplitude A0=0.1885 for the
particular stratification profile Eq. (16) with α=-1.39 and
σ=0.01 is shown in Figs. 2 and 3.
Amplitude function and surface displacement for the
two-humped solitary wave; α=-1.39.
Streamlines for the two-humped solitary wave: α=-1.39; A0=0.1885.
Indeed, the maximum derivative on x in the dimensionless coordinates is of
order unity. However, the wave has a pronounced two-scale structure with
typical length scales, which are much larger than the length L used to
scale the derivative. A solitary wave with three typical length scales (a
three-humped one) is shown in Fig. 4.
Amplitude function for the three-humped solitary wave.
For this case the stratification profile is
ρ0(z)=ρ(1-σz+σ2(1.206z2-4.37z3-3.435z4-33.407z6)),
which produces in Eq. (8) nonlinear terms up to A6. Generally, one can
expect at most M/2 different scales for a stratification in the form of a
polynomial with an even power index M, and (M-1)/2 otherwise.
Further, we wish to examine the structure of solitary waves of a permanent
form for the stratification given by Eq. (16). We only consider the case
α=-1.39 and focus on the waves of a permanent form under a free
surface, their domain of existence, limiting forms and structural stability.
Other values of α lead to more extensive consideration with a number
of particular cases. Such a study is beyond the scope of the present paper.
First, for α=-1.39 there exist only permanent waves with positive
amplitudes. Wave phase velocity is defined by the following expression:
c(1)(A0)=c-c(A0=0)μ2=4A03π2α+13-160α9π2-A026+64αA0375π.
Phase velocity versus wave amplitude. Solid line: c(1)(A0,
dashed lines 0<A0=A1 and A0=A2). Dotted line:
A0=A*=1/π, the critical amplitude above which the model does not
work as a vortex core arises inside the wave. A2=0.1311;
A1=0.1793.
Figure 5 shows that the phase velocity is an increasing function for
0<A0<A2 and A0>A1. For A1<A0<A2 the phase velocity
decreases with amplitude and there are no steady solitary wave solutions.
When 0<A0<A2 solitary waves widen as amplitude increases with a
table-top limiting shape with a local maximum for the wave velocity as shown
in Figs. 5 and 6.
Profiles of stable solitary waves are shown by solid lines. Dashed
lines correspond to A2=0.1311 and A1=0.1793. The limiting amplitude
is reached when A0=A2.
Such waves are structurally stable according to as both the
wave energy E=∫-∞∞A2dx and the wave
velocity increase as amplitude increases.
For A0>A1 wave profiles are shown in Fig. 7. Waves change from the
table-top solution to solitary waves with a single scale via multiscaled
structures.
Profiles of unstable solitary waves are shown by solid lines. Dashed
lines: A2=0.1311; A1=0.1793. The lower limiting wave amplitude is
A0=A1.
For the particular stratification considered here, waves are structurally
unstable since wave energy decreases as shown in Fig. 8, but
the wave velocity increases with the increase in wave amplitude. An
interesting observation is that these waves of sufficiently large amplitude
could be stable as the energy is eventually increased as shown in Fig. 8. For
the stratification considered here, it does not matter because the solution
with the vortex core appears at a lower amplitude when energy is still the
decreasing function of amplitude. However, it leads to an interesting
phenomenon – waves with vortex cores could stabilize the wave. The idea is
that the vortex core leads to widening of a wave and
consequently to the increase in its energy; thus, the structural stability
criterion will be satisfied. For the considered particular stratification,
waves with vortex cores are initially unstable as an increase in energy due
to the vortex core and associated widening does not compensate for the
decrease in energy in the wave outside the vortex core. Nonetheless, above
some amplitudes, waves become structurally stable. When wave amplitude
further increases, the permanent wave of the limiting amplitude becomes
infinitely wide, as shown by .
Energy versus wave amplitude. The dotted line corresponds to
A0=A*.
The theory described above is valid for wave amplitudes below
A*, a certain amplitude at which a vortex core started to appear inside
the wave. For the nearly linear density profile A*=1/π,
have shown that
Bx2∼R(A*)(1-B)-8ν151-B5/2,
where ν is the supercriticality parameter defined such that B varies
from zero to one as wave amplitude does from A* to the maximum value
allowed to be predicted there. R(A*) depends on the stratification
profile and is fixed. It is straightforward to notice that B(x) is
monotonic and that therefore multiscaling in the vortex core area does not
exist when A>A*.
Multiscaling effects similar to those discussed above could be observed in
various physical media. reported that solitary Rossby waves
in channels obey the same KdV-type equation with complicated nonlinearity due
to the mean shear variations. A Coriolis force for Rossby waves plays the
same role as gravitational force for the internal gravity waves. The results
of multiscaling for Rossby waves with and without a trapped core will be
reported elsewhere.
Conclusions
For the particular case of a nonlinear dispersive medium such as a
density-stratified fluid, we have addressed multiscaled solitary waves which
are predicted when there exists competition of several different types of
nonlinearity. The mechanism leading to these solutions differs from the
mechanism of multiscaling due to the competition of different types of
dispersion or effects due to the dissipation. We have shown that the length
used to scale the x-derivative does not simply coincide with the typical
length scale of the wave, as for KdV. Moreover, multiscaled (multi-humped)
disturbances exist for sufficiently large amplitudes; at the least, terms in
the fourth order of wave amplitude should be accounted for. The multiscaling
(multi-humped) phenomenon exists or does not exist for almost identical
density profiles; the two-pycnocline case studied earlier is not necessary
for the existence of multiscaling. The continuous stratification given by
Eq. (16) was studied in more detail. The structure of permanent solitary
waves and how multiscaling appeared were presented. Structural stability was
examined using the criterion proposed by . It was shown that
both stable and unstable solutions of the KdV-type equation with quadratic,
cubic and quartic nonlinearities are available. Multiscaled waves without a
trapped core belong to the unstable solutions. A trapped core inside the wave
prevents the appearance of such multiple scales within the core area.
However, the trapped core could stabilize the multiscaled solution in the
sense of structural stability. The case when the trapped core and
multiscaling are combined together is beyond the scope of the present study
and will be presented elsewhere. It is noted that multiscaling phenomena
exist for solitary waves in various physical contexts, for example, for
Rossby waves on a shear flow or inertial waves in swirling
flows .
No data sets were used in this article.
The author declares that he has no conflict of
interest. Edited by: Roger
Grimshaw Reviewed by: Tatiana Talipova and one anonymous
referee
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