Modelling oceanic internal waves to enhance marine and naval predictions and practices
Submitting InstitutionLoughborough University
Unit of AssessmentMathematical Sciences
Summary Impact TypeEnvironmental
Research Subject Area(s)
Earth Sciences: Oceanography
Engineering: Maritime Engineering, Interdisciplinary Engineering
Summary of the impact
Large-amplitude horizontally propagating internal solitary waves commonly
occur in the interior of the ocean. This case study presents evidence to
demonstrate the impact of research conducted by Professor Grimshaw at
Loughborough University on the development and utilisation of Korteweg- de
Vries (KdV) models of these waves, which has formed the paradigm for the
theoretical modelling and practical prediction of these waves.
These waves are highly significant for sediment transport, continental
shelf biology and interior ocean mixing, while their associated currents
cause strong forces on marine platforms, underwater pipelines and
submersibles, and the strong distortion of the density field has a severe
impact on acoustic signalling.
The theory developed at Loughborough University has had substantial
impact on the strategies developed by marine and naval engineers and
scientists in dealing with these issues.
Large-amplitude internal solitary waves are usually generated by the
interaction of the barotropic tide with the shelf break, topographic sill
or other prominent bottom features. This leads to the formation of an
internal tide, which then deforms and evolves into a train of very
large-amplitude internal waves, with associated large pycnocline
displacements and strong currents.
Research on internal solitary waves in the coastal ocean is a very active
area, conducted worldwide by physical oceanographers and marine engineers.
Intensive observations have been obtained in several locations by
ship-based experiments, supplemented by global satellite observations.
Theoretical modelling has been based either on numerical simulations, or
on the development of analytical models. It is in this latter area where
the research at Loughborough University (LU) has made a major
contribution, to the extent that the theoretical models developed by
oceanographers worldwide now form the basic paradigms for interpreting and
predicting the dynamics of internal solitary waves.
The special focus on large amplitude internal waves, which are common and
a robust feature of the coastal ocean, and also in the atmospheric
boundary layer, began with the arrival of Professor Grimshaw to
Loughborough in 2000, and the subsequent establishment of his research
group consisting of Dr. El (appointed 2005) and Dr. Khusnutdinova
(appointed 2003), together with several post-doctoral fellows and
post-graduate students, and has continued through to 2013. The group has
collaborated internationally, notably with Pelinovsky (Institute of
Applied Physics, Russia) and Helfrich (Woods Hole Oceanographic Institute,
Research has focussed on four main areas: generation mechanisms, the
structure of internal wave trains, the effect of bottom topography, the
role of the earth's rotation. One of the most effective generation
mechanisms is transcritical flow over topography where the basic model is
the forced KdV equation, [3.1, 3.2]. The basic feature of the
forced KdV equation is the appearance of undular bores, which are
generated both upstream and downstream of the topography, linked by a
quasi-steady structure over the topography. In [3.1] it is
demonstrated, essentially for the first time, that the width and polarity
of the obstacle are important parameters. Then in [3.2] the forced
KdV model is extended to finite-amplitude, initially in the surface wave
context. One of the significant current issues is the observation that in
the generation region, often several internal wave modes are observed. In
[3.3] this topic is addressed in a novel theoretical model, which
is currently being implemented on a global scale, by Dr. Stephen Griffiths
at Leeds University and collaborators.
The standard KdV model has only quadratic nonlinearity, whereas observed
internal solitary waves often have very large amplitudes. Hence it has
become common to include a cubic nonlinear term, leading to an extended
KdV (Gardner) equation, which is now the standard model [3.4] and
can be used for arbitrary density stratification and background current
fields. One of the main thrusts of the Loughborough University research
has been the investigation of how internal solitary waves deform, and
possibly even disintegrate, as they propagate over variable topography.
This involves a detailed study of the properties of the
variable-coefficient extended KdV equation, and these have been applied to
actual oceanic locations, to demonstrate the wide variety of outcomes [3.4].
In [3.5] this approach was extended to a study of how undular
bores deform over a slope which focussed on how tsunamis deform over the
Recently, it has been shown that the earth's background rotation has a
marked effect on the long- time development of internal solitary waves,
and significant new results are described in [3.6]. Importantly
and surprisingly, it transpires that the long-time outcome is the
formation of nonlinear wave packets, and this has profound implications
for inter alia the interpretation of satellite observations.
References to the research
3.1. Grimshaw R.H.J., Zhang D-H. and Chow K.W., (2007),
Generation of solitary waves by transcritical flow over a step, Journal
of Fluid Mechanics, 587, 235-254, DOI: 10.1017/S0022112007007355
3.2. El G.A., Grimshaw R.H.J. and Smyth N.F., (2009), Transcritical
shallow-water flow past topography: finite-amplitude theory, Journal
of Fluid Mechanics, 640, 187-214, DOI:
3.3. Griffiths S.D. and Grimshaw R.H.J. (2007), Internal tide
generation at the continental shelf modeled using a modal decomposition:
two-dimensional results, Journal of Physical Oceanography, 37,
428-451, DOI: 10.1175/JPO3068.1
3.4. Grimshaw, R., Pelinovsky, E., Talipova, T. and Kurkina, A. (2010)
Internal solitary waves: propagation, deformation and disintegration,
Nonlinear Processes in Geophysics, 17, 633-649. http://www.nonlin-processes-geophys.net/17/633/2010/npg-17-633-2010.pdf
3.5. El G.A., Grimshaw R.H.J. and Kamchatnov A.M., (2007),
Evolution of solitary waves and undular bores in shallow-water flows over
a gradual slope with bottom friction, Journal of Fluid Mechanics,
585, 213-244, DOI: 10.1017/S0022112007006817
3.6. Grimshaw, R. and Helfrich, K.R. (2012). The effect of rotation on
internal solitary waves, IMA Journal of Applied Mathematics, 77,
326-339, DOI: 10.1093/imamat/hxs024
Grants to Professor Grimshaw:
EPSRC GR/N63642/01, Dynamics of finite-amplitude internal and
inertial waves, Grimshaw R.H.J., 01/ 2001-01/2004, £128,521.
EPSRC EP/D003342/1,The effect of friction on undular bores,
Grimshaw R.H.J., 01/2005-08/2005, £4,200.
EPSRC EP/C530586/1, Nonlinear internal gravity wave beams,
EPSRC EP/I007180/1, Interaction of oceanic vortices with steep
topography, 09/2010-11/2010, £9,794.
EPSRC, Interaction of vortices with topography, 2009, £15,700.
Office of Naval Research, Generation of the Internal Tide,
Office of Naval Research, Internal Solitary Waves, 2000-2,
Royal Society, Large amplitude waves with trapped cores, 2011,
Royal Society, Nonlinear waves in coastal seas, 2008, £8,200.
Royal Society, Solitary waves propagating over rough topography,
Royal Society, Analytical theory of frictional undular bores,
Leverhulme Visiting Professor (Georgi Sutyrin), 2003, £34,000.
Leverhulme Visiting Professor (Efim Pelinovsky), 2008, £67,500.
This research has originality, rigour and significance, indicated by its
publication in premier refereed journals, external funding of a total of
£463,605 since 2002, and the external collaborations with two world class
institutes, viz. Woods Hole Oceanographic Institute, USA and Institute of
Applied Physics, Russia.
Details of the impact
The original and rigorous research at Loughborough University by
Professor Grimshaw and colleagues since 2000 has resulted in a new
modelling paradigm for non-linear oceanic waves. This research has had
impact not only for fundamental science, but also for the marine engineers
and ship operators worldwide who are concerned with the effect of these
waves on offshore structures and submersibles. This has given tools to
experimentalists and numerical modellers to find theoretical
interpretations of their observations and results.
The research means that the KdV model, with its various extensions and
modifications, is now routinely used as the basic paradigm for the dynamical
understanding of the nonlinear internal waves commonly found in the coastal
oceans. The large number of keynote presentations by Grimshaw, for example [5.1-5.3]
provide evidence of the widespread dissemination of this research to end
users and the interest the work has attracted.
This major change extends to end-users such as marine and oil industry
engineers, and to the navy scientists concerned with submersibles and
acoustic signalling [5.4]. Examples of applications of our
research comes from funded experimental work from the Office of Naval
Research (ONR). Professor Louis Goodman, now at the School for Marine
Science at the University of Massachusetts has carried out considerable
work for the ONR using the theories of Grimshaw, [5.5]. Oil
companies, such as BP, Exxon Mobil, StatOil, have also benefitted from
this work as working offshore needs the capacity to predict the occurrence
of these large waves, and to assess the impact of waves on structures and
pipe lines. Submersibles routinely communicate using sound signals, which
are hugely distorted by internal waves. For this reason the US Navy
through ONR has conducted several major ocean experiments, utilising
theory first developed at Loughborough University.
The benefits to these users include the better prediction of the
occurrence of large waves leading to an improved assessment of the impact
of the waves on under-sea structures and pipelines; and improved
communication for submersibles whose communication is largely disrupted.
Sources to corroborate the impact
The following sources of corroboration can be made available at request:
5.1. Invitation to be a speaker at Pacific Institute for the
Mathematical Sciences (PIMS) Workshop "Waves in the Atmosphere and Ocean",
Vancouver, 2008: http://www.math.wisc.edu/~milewski/CRG-SFU/Home.html
5.2. Invitation to be a principal lecturer at Geophysical Fluid
Dynamics (GFD) Summer School at Woods Hole Oceanographic Institution, on
Nonlinear Waves, 2009: http://www.whoi.edu/fileserver.do?id=52147&pt=2&p=19387
5.3. Invitation to be a plenary lecturer at the Third DNVA-RSE
Norway-Scotland Waves Symposium, Oslo, 2013:http://www.mn.uio.no/math/english/research/groups/fluid-mechanics/events/3rd-norway-scotland-waves-symposium.html
5.4. Email from Woods Hole Oceanographic Institution, USA
5.5. The Office of Naval Research in the USA uses Grimshaw's work: