Numerical Modelling of High-frequency Internal Waves Generated
by River Discharge in Coastal Ocean
I. K. Marchevsky
1,2 a
, A. A. Osadchiev
3 b
and A. Y. Popov
1 c
1
Bauman Moscow State Technical University, 5, 2-ya Baumanskaya Str., Moscow, Russia
2
Ivannikov Institute for System Programming of the RAS, 25, Alexander Solzhenitsyn st., Moscow, Russia
3
Shirshov Institute of Oceanology, Russian Academy of Sciences, 36 Nahimovskiy Prospekt, Moscow, Russia
Keywords: River Plume, Internal Waves, Coastal Ocean, Small River, Numerical Modelling, LagrangianEulerian
Method.
Abstract: A method for numerical simulation of internal waves generation by discharges of small and rapid rivers
flowing into a coastal sea is proposed. The method is based on PFEM-2 (Particle Finite Element Method, 2
nd
version) and utilizes particles to simulate convection as well as transfer salinity. Main simplifying
assumptions and the mathematical model are presented for this problem. The numerical scheme is split into
predictor (particles motion) and corrector (finite element method solution) steps. The resulting method is
expected to be efficient in terms of mesh fineness and length of simulation time.
1 INTRODUCTION
Satellite imagery detects internal waves with short
wavelength (<100 m) that are generated in the areas
adjacent to estuaries of small rivers and propagate
offshore within river plumes (Fig. 1). This process is
regularly observed in mountainous coastal areas, in
particular, the Ring of Fire (the Pacific coasts of
Mexico, Peru, and Chile; Taiwan, New Guinea, New
Zealand), in Western Balkans, Western Caucasus.
A mechanism of generation of these internal
waves by discharges of small and rapid rivers
inflowing to coastal sea was recently described by
Osadchiev (2018). Friction between river runoff at
high velocity and the subjacent sea of one order of
magnitude lower velocity causes abrupt deceleration
of a freshened flow and increase of its depth, i.e., a
hydraulic jump is formed. Transition from
supercritical to subcritical flow conditions effectively
transforms kinetic energy of river flow to potential
energy and induces generation of high-frequency (65
90 s) internal waves. These internal waves
propagate off a river mouth at a stratified layer
between a buoyant river plume and subjacent ambient
sea with phase speed equal to 0.45 0.65 m/s and
a
https://orcid.org/0000-0003-4899-4828
b
https://orcid.org/0000-0002-6659-0934
c
https://orcid.org/0000-0002-2744-4889
dissipate within the plume or at its lateral border.
These internal waves increase turbulence and mixing
at this layer and, therefore, influence structure and
dynamics of the river plume.
The process of generation of internal waves by
river discharge described above was reported and
analyzed for small river plumes located off the
northeastern coast of the Black Sea (Osadchiev,
2018). It was shown that river runoff forms a
hydraulic jump and generates internal waves under
certain conditions defined by properties of a river
flow, ambient sea water, and a local topography. In
particular, a river current has to be fast enough to
form a supercritical freshened flow in vicinity of a
river mouth. On the other hand, kinetic energy of a
freshened flow has to be low enough to be inhibited
by friction with ambient sea along a strongly stratified
bottom boundary of a river plume. This condition is
satisfied if river discharge rate is low, i.e., river is
rapid but small.
Despite a certain progress in study of high-
frequency internal waves referred above, many
aspects of their generation, propagation, and
dissipation remain unaddressed. In particular,
emerging of internal waves with a certain period is
384
Marchevsky, I., Osadchiev, A. and Popov, A.
Numerical Modelling of High-frequency Internal Waves Generated by River Discharge in Coastal Ocean.
DOI: 10.5220/0007840203840387
In Proceedings of the 5th International Conference on Geographical Information Systems Theory, Applications and Management (GISTAM 2019), pages 384-387
ISBN: 978-989-758-371-1
Copyright
c
2019 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
Figure 1: Sentinel-2 ocean color composite from 28 December 2017 indicating internal waves generated by small rivers along
the northeastern coast of New Guinea.
presumed to be caused by oscillation of a quasi-
stationary hydraulic jump. However, the exact
background physical mechanism is largely unknown
and requires a detailed study. Thorough description
of this mechanism will reveal dependence of
parameters of internal waves (frequency, phase
speed, wavelength) on external conditions (river
inflow velocity, river plume height, salinity and
vertical stratification, salinity of ambient sea, etc.).
These dependences are crucial for evaluation of
wave-induced turbulence and mixing intensity in
bottom and lateral frontal zones of a river plume that
govern its structure and mixing dynamics (Osadchiev
and Korshenko, 2017; Osadchiev and Sedakov,
2019). In this study we apply a novel Eulerian-
Lagrangian approach to reveal these issues. Based on
results of numerical experiments we present new
insights into the processes describe above.
2 GOVERNING EQUATIONS
For numerical simulation of such phenomenon the
following simplifying assumptions can be taken into
account (Milne-Thomson, 2011):
1) the water is incompressible, so the velocity field
u
is divergence-free:
0u
;
(1)
2) the temperature of the water is considered to be
constant, its density
depends only on the
salinity
S
(in per mille), we confine the
approximate dependency to linear term
0
S

(2)
3) where
0
is the density of fresh water (at
0S
),
0.65
kg/m
3
for salt water;
4) the Boussinesq-type approximation is considered
in order to take into account the buoyancy-driven
flow which arises due to the density difference
caused by non-uniform salinity distribution:
0
1
()
u
u u p u g S
t

(3)
where
p
is the pressure field,
is the kinematic
viscosity coefficient, which we assume to be constant
(
6
10
m
2
/s for the water);
g
is the gravity
acceleration;
5) the salinity distribution in general case is
described by the convection-diffusion equation
S
u S D S
t
(4)
where
D
is the diffusivity coefficient; for saline
solution it weakly depends on concentration and can
Numerical Modelling of High-frequency Internal Waves Generated by River Discharge in Coastal Ocean
385
be considered equal to
9
1.1 10D

m
2
/s. The latter
means that the Schmidt number
Sc
D
,
(5)
which defines the ratio of momentum diffusivity
(kinematic viscosity) and mass diffusivity, has the
order of
3
10
, so the salinity diffusion process can be
neglected. As the result, we have the equation for
salinity
0
dS S
uS
dt t
,
(6)
which means that salinity is being transferred along
the velocity streamlines as a passive impurity.
3 COMPUTATIONAL DOMAIN
AND BOUNDARY CONDITIONS
Preliminary numerical simulation can be provided for
2D and 3D axisymmetric (a semicircular sector is
considered) cases. It seems reasonable to consider a
rectangular computational domain shown in the
Fig. 2.
Figure 2: Computational domain.
The boundary conditions on the computation domain
boundaries are the following:
Inlet: given values of the velocity and salinity which
refer to the fresh water from the river.
Wall: zero velocity and zero gradient of salinity and
pressure.
Outlet: mixed boundary condition: а) zero gradient
of velocity and salinity if normal component of
velocity is directed outside the flow domain, b) given
value of the total pressure and given value of salinity,
which refers to the salt water in the sea, when normal
component of velocity is directed inside the flow
domain.
Surface: it is assumed that the surface is flat which
means that the surface waves are not simulated;
normal (vertical) velocity is zero, pressure is equal to
a fixed value which refers to atmospheric pressure,
shear stress is equal to zero. Surface waves can still
be simulated in a more precise simulation, that would
require description of the water surface position and
the Boussinesq boundary conditions consideration on
the free surface. However, even in this case the
surface tension effect should be neglected because the
Weber number
2
We
ul
has the order of
34
10 ...10
4 NUMERICAL METHOD
The described problem can be solved numerically by
using, for example, finite volume method,
implemented in a number of computational codes.
However, the flow in the considered case is
convection-dominated, it means that the convective
term in the NavierStokes equation is prevalent over
all other terms: pressure gradient and viscous
diffusion. Moreover, the salinity distribution
evolution is considered as pure convective transport,
as it was mentioned above. It means that the precision
of approximation of the convective terms is much
more crucial than of the pressure and diffusive ones.
The computational domain can be rather large (in
comparison to the size of the river outlet), and
physical time in simulation also should be long
enough to obtain a quasi-steady regime of flow.
Such requirements make it reasonable to use
hybrid LagrangianEulerian methods for numerical
simulation. PFEM-2 (particle finite element method,
the 2-nd version; Idelsohn, Onate and Del Pin, 2004,
as well as Idelsohn et al., 2013) seems to be the most
suitable among them. Its main features are the
following:
1) Particles, which move along the streamlines of
the velocity field, are introduced in order to
approximate the convective term in the Navier
Stokes equation. These particles are ‘immaterial’, i.e.
they do not transfer the mass and other intrinsic
properties except of marker of the phase (for
multiphase flows). In the considered problem, it
seems to be convenient to consider salinity as the
other property transferred by the particles.
2) Fixed mesh, which can be rather coarse, is
introduced in the flow domain for finite-element
approximation of the other terms pressure gradient,
viscous term and buoyancy term.
inlet
surface
wall
wall
axis position for
axisymmetric case
outlet
ONM-CozD 2019 - Special Session on Observations and Numerical Modeling of the Coastal Ocean Zone Dynamics
386
3) Interpolation technique is used in order to
transmit velocities and salinity from the particles to
mesh nodes and vice versa.
So each time step is split into two stages. The first
one is predictor, which consists of the particles
motion phase. Note that the step size can be rather
high (CFL number is allowed to be more than 1),
while for particles motion simulation it is normally
split into sub-steps in order to provide CFL not more
than 0.10…0.15. At the same time, the velocity field
is considered to be known at the mesh nodes from the
previous time step, so its finite element reconstruction
and Euler (or RungeKutta) explicit method for
particles motion integration make this operation not
time-consuming and easy to parallelize.
After the particles motion sub-steps have been
performed, the velocity and salinity values, having
been transferred with the particles, should be
projected onto mesh nodes. Now the convection
(predictor) stage is finished.
The correction stage consists of solution of the
equation
0
1u
p u g S
t

,
(7)
where the initial value is known from the prediction
step. This equation no longer contains the convection
term, which is highly sensitive to numerical
approximation. Implicit finite element scheme is
used, both monolithic and coupled strategies
(including fractional step approach) can be used for
calculation of new values of
u
and
p
. The velocity
field
u
should satisfy the incompressibility
equation (1) while this property is broken for the
velocity field after ‘predictor’ stage.
5 CONCLUSIONS
The proposed method is expected to be an efficient
means of simulation of a mechanism of generation of
internal waves by discharges of small and rapid rivers
inflowing to coastal sea. The heavy use of particles
makes computation possible on rather coarse meshes
as well as long periods of simulation time, which is
required to reach a quasi-steady state. Current
implementation has been provided for 2D and
axisymmetric problems but can be further developed
for solution of fully 3D simulation cases.
ACKNOWLEDGEMENTS
This research was funded by the Ministry of Science
and Education of Russia, theme 0149-2019-0003
(collecting and processing of satellite data) and the
Russian Science Foundation, research project 18-17-
00156 (study of river plumes).
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high-frequency internal waves in coastal ocean.
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Osadchiev, A. A., Sedakov, R. O., 2019. Spreading
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coast of the Black Sea observed by Landsat 8 and
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