Evolution equations for strongly nonlinear internal waves
Lev A. Ostrovsky
Zel Technologies/NOAA Environmental Technology Laboratory, Boulder, Colorado 80305
John Grue
Mechanics Division, Department of Mathematics, University of Oslo, Oslo, Norway
͑Received 25 February 2003; accepted 2 July 2003; published 3 September 2003͒
This paper is concerned with shallow-water equations for strongly nonlinear internal waves in a
two-layer fluid, and comparison of their solitary solutions with the results of fully nonlinear
computations and with experimental data. This comparison is necessary due to a contradictory
nature of these equations which combine strong nonlinearity and weak dispersion. First, the
Lagrangian ͑Whitham’s͒ method for dispersive shallow-water waves is applied to derivation of
equations equivalent to the Choi–Camassa ͑CC͒ equations. Then, using the Riemann invariants for
strongly nonlinear, nondispersive waves, we obtain unidirectional, evolution equations with
nonlinear dispersive terms. The latter are first derived from the CC equations and then introduced
semiphenomenologically as quasistationary generalizations of weakly nonlinear Korteweg–de Vries
and Benjamin–Ono models. Solitary solutions for these equations are obtained and verified against
fully nonlinear computations. Comparisons are also made with available observational data for
extremely strong solitons in coastal zones with well expressed pycnoclines. © 2003 American
Institute of Physics. ͓DOI: 10.1063/1.1604133͔
I. INTRODUCTION
Internal solitary waves are ubiquitous phenomena in
many areas of the ocean. ͑Below we call them internal soli-
tons, notwithstanding the formal definitions of the notion of
soliton in literature. It is our opinion, shared by a number of
other researchers that at the present stage of soliton science,
all solitary waves which, like particles, retain their structural
wholeness upon interaction, even if with some radiation as in
most nonintegrable equations, upon their interaction, should
be called solitons.͒ Many observations of internal solitons
are made in coastal zones where internal waves are excited
by tides and have the form of ‘‘solibores,’’ undulating groups
consisting of pulse-like displacements of isopycnes, close to
a series of solitary waves. General properties of these pro-
cesses have been described in detail; see, e.g., the review
papers by Ostrovsky and Stepanyants
1
and Grimshaw
2
and
references therein.
Many theoretical models of internal solitons are based
on the weakly nonlinear Korteweg–de Vries ͑KdV͒ equation
and its modifications that include cylindrical divergence of
the wave or small losses,
3
effect of Coriolis force,
4
and other
perturbation factors. In some cases these theories agree well
with the observations, such as for the solitary wave groups
observed in the Sulu Sea.
3
At the same time, the main re-
striction of the KdV model, namely, weakness of nonlinear-
ity ͑which for internal waves means smallness of an isopycne
displacement, A
i
, in comparison with its initial depth, h
i
), is
certainly violated in a number of observations, especially for
‘‘solibores’’ in coastal zones. For example, for the internal
waves observed in the Celtic Sea,
5
the ratio A
i
/h
i
could ex-
ceed 2. In the COPE experiment, performed on the north-
western shelf of USA, solitary wave groups were observed
for which this ratio reached even larger values, up to 4 and
5.
6–8
For these observations, weakly nonlinear models such
as KdV are completely inapplicable. The number of obser-
vations of strong internal solitons has increased recently, as
evidenced by the recent Internal Solitary Wave Workshop.
9
One way to address these data would be the direct nu-
merical computation of strong solitons from the hydrody-
namic equations ͑the references are given below͒. However,
of significant theoretical and practical interest is to develop
simplified approaches resulting in evolution equations for
strongly nonlinear waves which generalize the known
Korteweg–de Vries ͑KdV͒ and Benjamin–Ono ͑BO͒ equa-
tions and tend to them in the weakly nonlinear limit. This
approach would make both analytical and numerical treat-
ment of these nonlinear processes significantly easier which
is especially important for nonstationary motions. Besides, it
would make the qualitative features of the process much
clearer.
Some model equations for strong solitons were sug-
gested, in a rather phenomenological way, by one of the
authors
6,10
and have been successfully used to approximate
the COPE observational data. A more consistent approach is
associated with conservation laws for long waves, following,
in particular, from an expansion of a Lagrangian or a Hamil-
tonian in a parameter of wave slope or its analogs. Such an
approach, for surface waves of an arbitrary amplitude, was
first used by Whitham
11
who obtained a set of shallow-water
equations including a small dispersive term. Unfortunately,
in the aforementioned paper, the author has immediately re-
duced the system to the weakly nonlinear limit so that the
subsequent studies have apparently missed this result which
generalizes the one-dimensional Boussinesq equations to the
case of strong nonlinearity. Later on, the description of
PHYSICS OF FLUIDS VOLUME 15, NUMBER 10 OCTOBER 2003
29341070-6631/2003/15(10)/2934/15/$20.00 © 2003 American Institute of Physics