NPGNonlinear Processes in GeophysicsNPGNonlin. Processes Geophys.1607-7946Copernicus PublicationsGöttingen, Germany10.5194/npg-25-1-2018On the interaction of short linear internal waves with internal solitary wavesXuChengzhuc2xu@uwaterloo.caStastnaMarekDepartment of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, CanadaChengzhu Xu (c2xu@uwaterloo.ca)17January201825111731August20173December201730November201712September2017This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://npg.copernicus.org/articles/25/1/2018/npg-25-1-2018.htmlThe full text article is available as a PDF file from https://npg.copernicus.org/articles/25/1/2018/npg-25-1-2018.pdf
We study the interaction of small-scale internal wave packets with a large-scale internal solitary wave using
high-resolution direct numerical simulations in two dimensions. A key finding is that for wave packets whose constituent
waves are short in comparison to the solitary wave width, the interaction leads to an almost complete destruction of the
short waves. For mode-1 short waves in the packet, as the wavelength increases, a cutoff is reached, and for larger
wavelengths the waves in the packet are able to maintain their structure after the interaction. This cutoff corresponds to
the wavelength at which the phase speed of the short waves upstream of the solitary wave exceeds the maximum current
induced by the solitary wave. For mode-2 waves in the packet, however, no corresponding cutoff is found. Analysis based on
linear theory suggests that the destruction of short waves occurs primarily due to the velocity shear induced by the
solitary wave, which alters the vertical structure of the waves so that significant wave activity is found only above
(below) the deformed pycnocline for overtaking (head-on) collisions. The deformation of vertical structure is more
significant for waves with a smaller wavelength. Consequently, it is more difficult for these waves to adjust to the new
solitary-wave-induced background environment. These results suggest that through the interaction with relatively smaller
length scale waves, internal solitary waves can provide a means to decrease the power observed in the short-wave band in
the coastal ocean.
Introduction
Internal waves are commonly observed in stably stratified fluids such as the Earth's atmosphere and oceans. They exist in
a variety of environmental conditions, including those with background shear currents, and on different length and timescales.
The interaction between internal waves and other physical processes
results in energy exchange between the waves and the background environment . Based on linear wave theory,
studied internal waves in a shear background current, and found that in addition to the velocity shear across the
pycnocline, the vertical structure of the horizontal velocity profile also had a significant influence on the evolution of
internal waves. The interaction between mode-1 internal tides and mesoscale eddies was examined in . The
authors found that the interaction, essentially the bending of the paths followed by the wave energy, produced hot and
cold spots of energy flux. These took the form of beam-like patterns, and resulted in the scattering of energy from the
incident mode-1 to modes-2 and higher. The above-mentioned studies were not dependent on the presence of boundaries.
Motivated by the fact that internal waves have reflection properties that are different from classical Snell's law,
investigated the interaction between near-inertial waves and ocean fronts, and found that inertial
waves could travel with two distinct characteristics at a front, one flat and one tilted upward, implying the existence of
critical reflections from the ocean surface.
The interaction between internal waves of different length scales also occurs naturally . When the disparity in
length scales between the participating waves is large, the relatively smaller length scale wave essentially plays the
role of the disturbance to the “background flow” induced by the relatively larger length scale wave, as they interact
with each other. Previous literature has considered wave–wave interaction in a variety of contexts. For example, using
ray theory for linearized waves and the principle of wave action conservation, studied the interaction
of short high-frequency progressive internal waves and long progressive near-inertial waves, and found that there was
a net energy transfer from the inertial wave field to the short internal waves. investigated the
interaction between two mode-1 internal solitary waves (ISWs), and showed that the interaction of solitary waves did not
correspond to soliton dynamics, since energy exchange was observed and small-amplitude trailing waves of possibly higher
modes were generated. More recently, examined the interaction between mode-1 and mode-2 internal
solitary (solitary-like in cases when the mode-2 wave was breaking) waves, and demonstrated that the interaction yielded
a nearly complete disintegration of the relatively smaller mode-2 wave. In particular, the majority of kinetic energy
carried by the mode-2 wave was lost and the disturbance to the flow field after the collision no longer had a mode-2
structure. When the length scales of participating waves are similar, found that nonlinear
self-interaction might occur, which resulted in energy being transferred to superharmonic disturbances. These disturbances
were a superposition of modes such that the amplitude was largest where the change in background buoyancy frequency with
depth was largest.
In this work, we study the interaction of small-scale mode-1 internal waves initialized from linear waves with an ISW
initialized from the exact Dubreil–Jacotin–Long equation, using high-resolution direct numerical simulations in two
dimensions. Internal waves that are short in terms of wavelength compared to the fluid depth are generally less
documented in the nonlinear wave literature. In fact, the derivation of the model equations of most weakly nonlinear
theories, such as the Korteweg–de Vries equation and its variations, assumes large horizontal scales and thus filters out
short waves . Nevertheless, such waves occupy a non-negligible portion of the Garrett–Munk spectrum of
internal waves in the oceans , and it is important to understand their behaviour in order to fully describe
internal wave dynamics.
The remainder of the paper is organized as follows: theoretical descriptions of internal waves are introduced in
Sect. . The problem is formulated in Sect. . The simulation results are presented in
Sect. . A key finding is that for waves that are short in comparison to the ISW width, the interaction leads
to an almost complete destruction of the short waves; for mode-1 short waves, however, there is a cutoff determined by the
wavelength of short waves, and waves longer than this cutoff maintain their structure after
interaction. We show that this
is a key difference from mode-1–mode-2 interaction, which is examined in . The energy transfer during
the interaction is discussed in Sect. , and a summary concluding the findings of this study is given in
Sect. .
Example of (a) buoyancy frequency profile and
(b) vertical structure profiles as the wave number varies for mode-1
internal waves in a zero-background current. The amplitudes of the
eigenfunctions have been varied for clarity of visual presentation.
Internal wave theories
In the classical linear theory, the horizontal structure of internal waves is usually described by the travelling wave
ansatz exp(ik[x-cp(k)t]) where k is the horizontal wave number and is related to the wavelength λ by the
formula k=2π/λ, and cp is the phase speed. The vertical structure is described by solutions of the
eigenvalue problem often referred to as the Taylor–Goldstein (TG) equation , which is given by
ϕzz+N2(z)(cp-U)2+Uzzcp-U-k2ϕ=0,ϕ(0)=ϕ(H)=0,
where U=U(z) is the background horizontal velocity, N is the buoyancy frequency defined by
N2(z)=-dρ¯dzg,
with ρ¯ being the (dimensionless) undisturbed density profile in the background and H is the height of the
water column. If there are no critical layers (i.e. cp-U≠0 for all z), for physically relevant N(z), the TG
equation has an infinite set of discrete eigenvalues cp which decrease as k increases and as the mode number
increases. The corresponding eigenfunction ϕ(z) characterizes the vertical structure of the velocity field (e.g. the
wave-induced horizontal velocity is proportional to ϕz). It also determines the mode number of internal waves
according to the formula “one plus the number of zeros that the eigenfunction has in the interior of the water
column”. Note that the TG equation simplifies considerably when there is no background shear flow. An example of a single
pycnocline buoyancy frequency profile, together with the corresponding vertical structure functions for mode-1 waves of
particular horizontal wave numbers in the absence of a shear current, is given in Fig. .
Due to the nonlinear nature of fluid flows, purely linear waves are a mathematical idealization. For large-amplitude waves
or on timescales long enough for nonlinear effects to manifest themselves, results predicted by the linear theory do not
agree with measurements. Weakly nonlinear theory attempts to better describe internal wave dynamics by expanding flow
variables asymptotically and retaining corrections that correspond to finite amplitude (nonlinearity) and wavelength
(dispersion). The most famous weakly nonlinear model is probably the Korteweg–de Vries (KdV) equation, given by
,
Bt+clwBx+αBBx+βBxxx=0,
where B is the horizontal structure function describing the propagation and evolution of the wave in the x direction,
clw is the speed of linear long wave (i.e. waves with k=0), and the parameters α and β
measures the nonlinearity and dispersion of the wave, respectively. Analysis of the KdV equation shows that it has
solitary wave solutions of the form
B(x,t)=asech2x-Vtλ,
where a measures the wave amplitude and V is the nonlinear wave propagation speed. Internal solitary waves (ISWs) are
translating waves of permanent form, which consist of a single wave crest. They are one of the most commonly observed
types of internal waves in the field (e.g. ; also see , for a more complete
review).
While the KdV theory correctly predicts some properties of internal waves, it can only be expected to perform
well within certain asymptotic limits (e.g. the small-amplitude limit and long-wave limit). For large-amplitude waves, solutions of the
KdV equation and its variations have been shown to be different from waveforms predicted by the fully nonlinear theory
. Fully nonlinear ISWs can be computed by solving the nonlinear eigenvalue problem known as the
Dubreil–Jacotin–Long (DJL) equation, which, in a zero-background
current, takes the form ∇2η+N2(z-η)cisw2η=0,η(x,0)=η(x,H)=0,η(x,z)→0asx→±∞.
In this equation, cisw is the solitary wave propagation speed (equivalent to V in the KdV theory), and η=η(x,z) is the vertical displacement of the isopycnal relative to its far-upstream depth. The DJL equation is
equivalent to the full set of stratified Euler equations in a frame moving with the wave, where no assumptions are made
with respect to the nonlinearity of the fluid flow. Hence, its solutions are exact solitary wave solutions. For
non-constant N, the DJL equation has no analytical solutions. In this work, the DJL equation is solved numerically using
the method described in . The algorithm for solving the DJL equation is based on the variational scheme
developed in , which iteratively seeks a solution that minimizes the kinetic energy, subject to the
constraint that the scaled available potential energy is held fixed.
Graph of the model setup. Solid curves are isopycnals indicating the ISW and the linear waves in the initial field.
Problem formulationGoverning equations and numerical method
The governing equations for the present work are the incompressible Navier–Stokes equations under the rigid lid and
Boussinesq approximations, given by DuDt=-∇p-ρgk^+ν∇2u,∇⋅u=0,DρDt=κ∇2ρ,
where D/Dt is the material derivative defined by
DDt=∂∂t+u⋅∇,
and
∇=∂∂x,∂∂z.
In these equations, u=(u,w) describes the velocity field with u and w being the horizontal and vertical
velocities, respectively, ρ describes the density field, p is the pressure, g is the gravitational
acceleration, k^ is the unit vector in the vertical direction (positive upwards), ν is the kinematic viscosity,
and κ is the molecular diffusivity. As is the common practice under the Boussinesq approximation, the equations are
in dimensional form, except that the density ρ and pressure p are scaled by the reference density ρ0. For all
simulations, we fix the viscosity at ν= 10-6m2s-1 and the diffusivity at
κ= 2 × 10-7m2s-1. This gives a Schmidt number Sc=ν/κ=5. Periodic boundary conditions are used in the horizontal direction, and free-slip boundary conditions are used at the
top and bottom boundaries. We note that no-slip conditions have also been tested but the difference is insignificant,
since the majority of the velocity perturbations are found near the pycnocline. The effect of the Earth's rotation is
neglected, and hence the simulations are performed in an inertial frame of reference.
A complete description of the numerical model used in this study can be found in , where a detailed
validation and accuracy analysis through several test cases is also given. The model employs a spectral collocation
method, which yields highly accurate results at moderate grid resolutions. From a purely numerical point of view, the
spectral method requires a minimum of two points in the horizontal direction in order to completely resolve a wave
. Nevertheless, in this study we employ high resolution in order to better resolve the thin pycnocline,
with at least 10 grid points in the pycnocline and across the short waves. For spatial discretization, equally spaced
grid points are used in both horizontal and vertical directions. As appropriate for the boundary conditions, the Fourier
transform is applied in the x direction, whereas the Fourier sine or cosine transform is applied in the z direction
depending on the variable of interest. For time stepping, the model employs an adaptive third-order multistep method,
where viscous and diffusive terms are solved implicitly, and pressure is computed via operator splitting.
Model setup and parameter space
A graph showing the model setup is given in Fig. , where a right-handed Cartesian coordinate
system is considered with the origin fixed at the lower left corner of the domain. The position vector is expressed as
x=(x,z), with the x axis directed to the right along the flat bottom and the z axis pointing up towards
the surface. The two-dimensional, rectangular computational domain is on the laboratory scale and has an overall length
Lx=10m and a depth Lz=0.5m. It consists of an ISW subdomain of length Lisw=4m and a linear wave subdomain of length Llin=6m. The grid size is
Nx×Nz= 4096 × 512, which gives a horizontal grid spacing of 2.44mm and
a vertical grid spacing of 0.98mm.
We focus on flows in a quasi-two-layer stratification with a dimensionless density difference Δρ=0.01, for
which the Boussinesq approximation can be safely adopted. The background density profile, non-dimensionalized by the
reference density ρ0, is given by
ρ¯(z)=1-0.5Δρtanhz-z0d,
where z0 is the location of the pycnocline and d is the half-width of the pycnocline. The specific location of the
pycnocline does not affect the dynamics of the interaction between the ISW and the linear waves in general, except for the
case where the pycnocline is close to the surface such that the ISW could be breaking . In this work,
we set z0=0.4m in order to avoid this situation. The thickness of the pycnocline can affect the gradient
Richardson number through the buoyancy frequency profile it determines, which may have an impact on the
interaction. However, this topic is not the focus of the present work (see Sect. ). In this work, we
simply set d=0.01m for all simulations.
Solitary wave parameters. Here, the amplitude is measured by the
maximum isopycnal displacement ηmax, and the wavelength is measured
by the horizontal velocity profile along the inviscid upper boundary
according to the formula λisw=2(xR-xL), where xR
and xL satisfy the equation u(xR,Lz)=u(xL,Lz)=0.5umax.
Filled contours showing the horizontal velocity induced by the ISW,
with positive current shown in red and negative current shown in blue. The
dashed curve shows the isopycnal displacement along the pycnocline. The black
contours show the gradient Richardson number with Ri=0.25, 0.4,
0.6, and 1 from inside to outside.
The initial solitary wave is specified by interpolating a solution of the DJL equation onto the ISW subdomain. Parameters
of the particular solitary wave solution considered in this work are given in Table . Here, we compute the
Reynolds number Re based on the amplitude and maximum wave-induced current as
Re=umaxηmaxν.
While there are a variety of Reynolds number estimates available in the literature, this simple estimate is more relevant
to the length and velocity scales set by the ISW. The gradient Richardson number Ri is defined by
Ri=N2uz2,
where u is the ISW-induced horizontal current, and Rimin is the local minimum of the Richardson
number. It measures the ratio between the strength of the stratification and the shear stress in a parallel shear
flow. The horizontal velocity profile of the ISW, together with the Richardson number contours, is shown in
Fig. . The figure shows that the Richardson number has a local minimum in the high-shear region across the
pycnocline along the wave crest, and is very large outside this region. We note that while Rimin given in
Table is slightly smaller than the critical Richardson number 0.25, the Richardson number criterion
Ri<0.25 is a necessary but not sufficient condition for linear stability in a parallel shear
flow. Moreover, due to the fact that ISW-induced flow is not necessarily a parallel shear flow, the onset of shear
instability is possible only when Ri is considerably smaller than 0.25 over a region long enough for
perturbations to amplify in space . Thus, the onset of shear instability is not likely to occur.
Linear wave parameters. In the case labels, O indicates an
“overtaking” collision and H indicates a “head-on” collision, and the proceeding digits correspond to the wavelength of the linear waves.
Case labelWavelengthWave numberPhase speedGroup speedTimescaleλ (m)k (m-1)cp (cms-1)cg (cms-1)τ (s)Cases with an overtaking collision O20.231.413.441.34110O2.50.2525.133.931.65113O30.320.944.361.97117O40.415.715.062.59126O50.512.575.613.17136O60.610.476.053.69146Cases with a head-on collision H20.231.41-3.44-1.3484H2.50.2525.13-3.93-1.6582H30.320.94-4.36-1.9780H40.415.71-5.06-2.5976H50.512.57-5.61-3.1773H60.610.47-6.05-3.6970
Shaded density contours (full range of density shown, green denotes
the pycnocline centre) showing the solitary wave and the linear waves in the
case O2 (a) before, (b) during, and (c) after the
collision. Panel (d) shows the corresponding density field from the
simulation performed with the same linear wave packet but without the
solitary wave. Note the difference in x axis for each panel.
We perform a suite of simulations in which the solitary wave propagates to the right and interacts with a small-scale wave
packet initialized from linear waves. The linear waves are specified by solving the TG equation numerically using
a pseudo-spectral technique in the linear wave subdomain. In order to ensure a smooth transition across
the boundaries between the ISW subdomain and the linear wave subdomain, an envelope function is applied to the amplitude
of linear waves. The particular form of the envelope function used here is given by
env(x)=0.5tanhx-(Lisw+1)0.2-0.5tanhx-(Lx-1)0.2,
although by testing other forms we found that results are not sensitive to the exact shape of the envelope.
Same as Fig. but for O6. Vertical lines in panels (c) and (d) show the misalignment of wave crests in the two density fields.
We consider linear waves of wavelengths ranging from 0.2 to 0.6 m, whose parameters are given in
Table . For waves with a wavelength less than 0.2 m, nonlinear self-interaction
becomes evident and may affect the interaction, whereas waves with a wavelength larger than 0.6 m may no longer be
considered “short” in comparison to the ISW width. We examine two types of interaction in particular: an “overtaking
collision” means that the ISW and the linear waves propagate in the same direction, whereas a “head-on” collision means
that the two propagate in the opposite direction. The amplitude of linear waves is set to be 1 cm for all
cases. According to the linear theory, the propagation of linear internal waves is independent of their amplitude, at
least at the limit of small-amplitude waves. In fact, simulations with an amplitude of 2 cm have produced
quantitatively similar results, and thus will not be discussed in this paper. For each experiment, we measure the
time, τ, over which the solitary wave (which moves at the speed cisw) and the linear wave packet (which
moves at the speed cg) experience a full collision cycle by defining
τ=Lxcisw-cg.
At t=τ, the location of the solitary wave relative to the linear wave packet is approximately the same as it was in
the initial field. In all figures presented in this paper, reported time T is scaled by this quantity such that T=t/τ.
We would like to mention that these linear waves are in fact not purely linear during the simulations. However, by scaling
the relevant terms (i.e. Bt and BBx) in the KdV equation (Eq. ) using the amplitudes and wavelengths of
these waves, we found that the timescale at which the nonlinearity becomes important is on the order of 1000 s, at least
for waves with an amplitude of 1 cm and a wavelength larger than 0.2 m. In contrast, the timescale of
the interaction, as indicated in Table , is on the order of 100 s. Hence, for clarity of notation, the
small-scale waves will still be referred to as “linear waves”, as opposed to the “solitary wave” or the “ISW”, in
the remainder of this paper.
Simulation resultsEvolution of flow fields
An impression of the overall flow behaviour in the case O2 can be gained from Fig. . The initial density profile
is shown in Fig. a, where the disparity in both amplitude and length scale between the solitary wave and the
linear waves can be clearly seen. The linear waves have an amplitude that is approximately 10 % of the solitary wave
and a wavelength of 7.5 % of the solitary wave. Figure b shows that as the linear waves pass through the
solitary wave, they are deformed significantly such that they have lost their coherent, wave-like structure almost
entirely. Figure c shows that after the collision, the disturbance behind the solitary wave has a spatial
structure that is completely different from the initial linear waves. To demonstrate that such deformation of linear waves
does not occur naturally but is a result of the collision, we performed an additional simulation with the same linear wave
packet but without the solitary wave. The resulting density field at T=1 is shown in Fig. d.
Detailed density contours of the simulations (a) O2 and (b) H2, showing the overturning of the linear waves during the collision.
Vertical structure profiles of the linear waves with wavelengths of
(a) 0.2 m and (b) 0.6 m in the initial,
undisturbed state (solid curve) and the ISW-induced background state with an
overtaking collision (dot-dashed curve), a head-on collision (dashed
curve), and a hypothetical zero-background current (dotted curve).
The density profiles of the case O6 are shown in Fig. . The initial density profile, visible in
Fig. a, shows again the disparity between the solitary wave and the linear waves, though in this case the
wavelength of the linear waves is 3 times larger than that in the previously discussed case (or 22.5 % of the
wavelength of the solitary wave). Figure b and c show, however, that unlike in the previously
discussed case, the linear waves are able to retain their spatial structure throughout the collision. The amplitude is
also maintained, suggesting that energy loss during the collision is small. Instead, comparison to Fig. d,
which shows the corresponding density profile obtained from the simulation with linear
waves only, suggests that the primary effect of the
collision on the linear waves is a phase shift, as indicated by
the vertical lines. We will revisit the energy loss in
these cases in Sect. .
Detailed density contours showing the mode-2 wave packet (a) before and (b) after the collision with the solitary wave.
Destruction of short waves
In Fig. , we show details of the density field during the overtaking collision in case O2 (Fig. a), and
compare it with the density field during the head-on collision in case H2 (Fig. b). In both cases, the linear
waves in front of the solitary wave are unperturbed, whereas those behind the solitary waves are almost completely
destroyed. Inside the solitary wave, the deformation of linear waves in the two cases proceeds in a qualitatively
different manner. Figure a shows that for the overtaking case, overturning of the linear waves occurs above the
pycnocline centre, while Fig. b shows that for the head-on collision case, overturning occurs with and below the
pycnocline.
As suggested in Fig. , the overturning of short waves is not triggered by the shear instability since the local
Richardson number is not small enough and the high-shear region is not long enough. To understand what causes the
deformation of short waves in these cases, we performed an analysis similar to , in particular their
Fig. 9. We first extracted the background horizontal velocity and buoyancy frequency profiles at the crest of the solitary
wave. This background state consists of a pycnocline lower than that in an undisturbed situation, and a horizontal
velocity with significant shear across the deformed pycnocline. We then computed the linear wave solution with
a wavelength of 0.2 m in this background environment using the TG equation (Eq. ), and compared it with the
solution in the undisturbed background environment. The mode structure functions of these solutions are plotted in
Fig. a. This figure shows that the vertical structure of the horizontal velocity induced by linear waves is
highly dependent on the stratification and the background current, such that for both overtaking and head-on collision
cases, the structure functions at the solitary wave crest (indicated by dashed and dot-dashed curves) are completely
different from their initial, undisturbed state (indicated by the solid curve). The locations of maximum amplitude of the
structure functions are shifted downward from their undisturbed situation, in order for the linear waves to adapt to the
new solitary-wave-induced background stratification. However, there is a qualitative difference between the overtaking
and head on collision as well. Indeed, under the influence of the shear background current, the structure function in the
overtaking (head-on) collision case has its maximum value above (below) the disturbed pycnocline. This is consistent with
the observation in Fig. that inside the solitary wave, perturbations in the overtaking (head-on) collision case
have a wave-like structure above (below) the pycnocline. We also note that if there is no velocity shear in the
background, the vertical structure of linear waves of a given wavelength (e.g. λ=0.2m in this case, as
indicated by the dotted curve) depends only on the stratification, regardless of the direction they propagate in. Moreover,
the vertical structure with respect to the pycnocline centre is essentially unchanged. This suggests that the velocity
shear in the background alters the vertical structure of the short waves in a nonlinear manner and leads to the
observation that a head-on collision manifests differently from an overtaking collision.
In Fig. b, we made the same plot for linear waves with a wavelength of 0.6 m. The figure shows that the
key difference in the initial structure function is that it has a non-negligible value over a much larger vertical extent.
As a result, near the pycnocline centre, changes in the vertical structure functions are much less dramatic in both the overtaking
and head-on collision cases. Therefore, longer waves are able to adapt to the ISW-induced background environment more easily and hence are more likely
to survive the collision with the solitary wave. We also note that formally changing the amplitude of linear waves does
not change their vertical structures and thus does not affect the dynamics of the collision process, though in practice
larger amplitude waves are expected to have a different (i.e. Stokes wave) structure.
Same as Fig. but for mode-2 waves.
Comparison of mode-1 to mode-2 collisions
The above analysis suggests that as the linear waves enter into the solitary-wave-induced background state, they are
subject to a modified stratification and a velocity shear due to solitary-wave-induced current, and it is this velocity
shear across the deformed pycnocline that leads to the deformation of short waves. This process is in many ways similar to
that found in . However, a key difference is that the disintegration of mode-2 waves due to the collision
is much less dependent on their wavelength. To compare and contrast with their results, we performed an additional
simulation, with mode-2 waves of amplitude of 1 cm and wavelength of 0.6 m interacting with the same ISW
with an overtaking collision. The phase and group speeds of the mode-2 waves are cp=1.43cms-1 and cg=1.35cms-1, respectively, much smaller than their mode-1
counterparts. Figure shows that after the collision with the ISW, the mode-2 waves are almost completely
destroyed, except for some mode-1-like disturbances found near x=8m in Fig. b.
In Fig. , we plotted the vertical structure functions for mode-2 waves in the ISW-induced background
environment. The figure shows that the presence of velocity shear leads to significant changes in the vertical structures
of horizontal velocity profiles of mode-2 waves with wavelengths of both 0.2 and 0.6 m. In the latter case, the
deformed vertical structure functions show characteristics of mode-1 waves, with essentially no perturbation below (above)
the pycnocline for the overtaking (head-on) case. This is similar to Fig. 9b in , but fundamentally
different from our Fig. b, implying that mode-1–mode-2 collisions are different from mode-1–mode-1
collisions. In fact, mode-2 waves were unable to maintain their coherent structure after the collision with mode-1 waves
in all simulations in . Recent experiments (M. Carr, personal communication, 2017)
suggest that the situation is more complex when the mode-1 wave amplitude is comparable to the mode-2 wave
amplitude, though it is unclear if such a situation had relevance to situations in the ocean.
Phase speed of mode-1 linear waves in the ISW-induced background
shear current (solid curves) and a hypothetical zero-background current
(dashed curves) for (a) overtaking and (b) head-on
collisions as a function of wavelength. Dot-dashed lines indicate the maximum
(minimum) ISW-induced current.
Froude number of mode-1 linear waves in the ISW-induced background
shear current for (a) overtaking and (b) head-on
collisions as a function of wavelength.
Change of phase speed
Recall from Fig. that a secondary effect of the interaction is a phase shift of the linear waves. To explain
this observation, consider the linear long-wave speed clw in a two-layer stratification, defined by
clw=Δρgh1h2H,
where h1 is the upper layer depth, h2 is the lower layer depth, and H is the total depth. The long-wave speed sets
the limit of the phase speed of linear waves in a two-layer stratification such that cp approaches clw as
the wavelength approaches infinity. Thus clw provides a good estimate of the maximum phase speed in
a quasi-two-layer stratification. Using the long-wave speed as a guide, we note that the phase speed reaches its maximum
value when h1=h2 (i.e. when the two layers are equal in depth), provided other parameters (e.g. wavelength) remain
constant. In our simulations, since we consider an ISW of depression, the pycnocline at the wave crest is close to the
mid-depth. Hence, the linear waves will experience an increase in phase speed as they propagate through the ISW.
In addition to the stratification, the presence of background current will also modify the phase speed. In
Fig. , we explore the change of phase speed due to the presence of ISW-induced shear current for mode-1 linear
waves. For overtaking collisions shown in Fig. a, at the long-wave limit, the phase speed in the shear
background current is very close to that in a zero-background current. However, at the short-wave limit, the figure shows
that the phase speed in the shear background current approaches the maximum ISW-induced current, whereas the phase speed
in a zero-background current approaches 0 instead. This again suggests that short mode-1 waves are more likely to be
influenced by the nonlinear interaction with ISW. In particular, the critical wavelength that determines whether the phase
speed is significantly influenced by the shear current is approximately 0.5 m, where the phase speed in a zero-background
current intersects the maximum velocity of the shear current. On the other hand, for mode-2 waves (not shown),
the phase speed is altered by the shear current throughout the whole spectrum of wavelengths, since the phase speed in
a zero-background current is less dependent on the wavelength and is always much smaller than the maximum velocity of the
shear current. This is also consistent with the fact that mode-2 waves are less persistent after nonlinear interactions
with the ISW. The fact that the ISW-induced maximum current essentially sets the lower limit for the phase speed of short
waves implies that a critical layer does not exist for the ISW used in our simulations (as well as those with smaller
amplitude). While the above analysis is performed for overtaking collisions (i.e. for linear waves propagating to the
right), we also examined head-on collisions. As shown in Fig. b, at the short-wave limit, the behaviour of the
phase speed as a function of wavelength is very similar to that in the cases of an overtaking collision, except that now
the phase speed is approaching the minimum current induced by the ISW.
Power spectral density (PSD) of the initial horizontal velocity fields, computed from some of the overtaking collision cases.
We would like to note that given the nonlinear nature of the ISW, the interaction is indeed a nonlinear process, and thus
the linear theory can only provide some rough guide for the flow behaviour. To measure the nonlinearity of the fluid flows,
we shall introduce the Froude number which, in the context of internal wave dynamics, is usually defined as
Fr=Uc,
where U is the background current and c is the phase speed of the linear waves. The flow is said to be critical if
Fr=1, in which case the nonlinear effects are dominant. In our simulations, for any x the vertically
integrated U is essentially 0 since the flow is non-divergent in the simulation domain. Thus, a better estimation
of U would be the effective horizontal velocity in a reference frame moving with the ISW, which is essentially
-cisw. In this reference frame, the estimated c would be -cisw+cp where cp>0 for an
overtaking collision and cp<0 for a head-on collision. Hence, we can define the Froude number in a reference frame
moving with the ISW as
Fr=ciswcisw-cp.
Figure shows the Froude number of the linear waves in the ISW-induced background current as a function of
wavelength. The figure shows that Fr<1 for an overtaking collision and Fr>1 for a head-on
collision. In both cases, Fr approaches 1 at the short-wave limit since short waves propagate slower. This
implies that in the interaction of ISWs with short waves, the nonlinear effects become more important as the wavelength of
short waves becomes smaller.
EnergeticsDiagnostic tool: power spectral density
A function that describes a physical process can be represented either in the physical space or in the Fourier space. The
two different representations are connected through the Fourier transform. Suppose f is a function of position x in
the physical space, then the corresponding Fourier transformed variable F is a function of the horizontal wave
number k and is given by
F(k)=∫-∞∞f(x)eikxdx.
If x is bounded, then k takes discrete values
k=kn=2nπLx,n=0,1,…,∞.
Parseval's theorem states that the total power in a signal is the same whether it is computed in the physical space or in
the Fourier space . That is,
Total Power=∫0Lx|f(x)|2dx=∑n=0∞|F(kn)|2dk.
From this theorem, we can define the power spectral density (PSD) of the function f as
PSD=|F(k)|2.
The PSD is a function of the wave number k. It can be interpreted as the strength of the signal at each wave number. For
this reason, it provides a powerful tool for analyzing physical processes.
In the remainder of this section, we compute the PSD of horizontal velocity in the layer above the pycnocline and use it
to estimate the amount of wave energy being transferred during the collisions. The location of the horizontal layer chosen for
the analysis is z=0.43m (i.e. 3cm above the pycnocline), though we have also calculated the PSD at
other depths and found that results are not sensitive to the particular choice of horizontal layer. The PSD profiles of
the initial horizontal velocity fields for some of the overtaking collision cases (O2, O3, O4, and O6) are plotted in
Fig. . The figure clearly shows the wave number peaks due to the small-scale waves, which occur at
considerably larger wave numbers than those associated with the ISW spectrum (the peak near k=0). This suggests that
these small-scale waves are indeed “short” in comparison to the ISW width. For each simulation, we scale the PSD
computed at the scaled time T=1 by the maximum PSD of the considered linear waves observed in the initial field.
According to Parseval's theorem, this ratio remains the same when mapped back into the physical space. Although only the
horizontal velocity is used here, computation of the PSD of the vertical velocity yields quantitatively similar results,
as it usually decays in a way similar to that of the horizontal velocity due to the interaction. Thus, the scaled PSD of
horizontal velocity represents the relative strength of horizontal current at T=1 and hence provides an estimate of the
percentage of kinetic energy remaining after one full collision cycle.
Scaled PSD of linear waves in the simulations with (a) an overtaking collision and (b) a head-on collision at T=1.
Maximum values of the scaled PSD observed in Fig. vs. their corresponding wavelengths.
Reduction of wave energy
In Fig. we examine the energy reduction of linear waves due to collision by plotting the PSD of horizontal
velocity in the wave number domain. The figure shows that in all cases, there is a net loss of wave energy due to the
collision. It also shows that for a given solitary wave, the wavelength of the linear waves (which remains unchanged
after the collision) is the single most important factor that determines the amount of PSD (and hence wave energy)
remaining after the collision. While the longest waves may retain as much as 85 % of the kinetic energy they had
initially, the shortest waves lose almost all of their initial energy such that the peaks of the PSD can hardly be
distinguished from background noise. Among other factors, a head-on collision is slightly more efficient in destroying
the linear waves than an overtaking collision, except for the small wavelength limit. This may be explained by the fact
that during a head-on collision, the structure function (especially its peak) shifts further away from its initial state
than during an overtaking collision, as shown in Fig. , such that the new ISW-induced background environment is
more difficult for the linear waves to adjust to. In contrast, the initial amplitude of the linear waves has very little
impact on the net energy transfer due to collision, since curves produced from simulations in which the linear waves have
an amplitude of 2 cm (not shown) are almost exactly the same as their smaller amplitude counterparts shown in
Fig. a; though we did not consider large-amplitude short waves, since these will have their own complex
dynamics.
Quantitative measurement of the maximum values plotted in Fig. .
The maximum value of the scaled PSD as a function of wavelength is plotted in Fig. , along with
a quantitative measurement in terms of percentage given in Table . The figure and table show that the maximum
value of the scaled PSD increases monotonically as the wavelength increases, for both overtaking and head-on collisions. It
approaches 0 at the short-wave limit and 1 at the long-wave limit. For waves with a wavelength much longer than
0.6 m, simulation results (not shown) suggest that the maximum values of the scaled PSD at T=1 are at the level
of 90 % but are never larger than 100 %, implying that very little wave energy is being transferred from the short
waves during the collision and that no energy is transferred from the ISW to the short waves. For the longest waves, the
slight decrease in PSD is at a similar level to viscous dissipation. We note that this observation is consistent with the
result shown in Fig. , since above the critical wavelength λ=0.5m, very little energy
exchange occurs due to the interaction.
Scaled PSD of the cases (a) O6 and (b) H6 after repeated collisions.
Quantitative measurement of the peak values observed in Fig. .
For the cases O6 and H6, simulations were performed for an extended period of time in order to allow for repeated
collisions between the solitary wave and the linear wave packet. For each of these cases, four complete collision cycles
were observed, and the scaled PSD has been computed at T=1, 2, 3, and 4, as shown in Fig. . The
corresponding measurement of the scaled PSD at each peak is given in Table . The figure and table suggest
that for both cases, the scaled PSD is reduced after each subsequent collision, down to 60.20 % in the case of O6 and
33.61 % in the case of H6 at T=4. Nevertheless, they are still larger than those of the shorter waves after only one
collision, implying that the wavelength is an important factor that determines the wave energy being transferred.
Figure and Table also show that after each collision cycle, the scaled PSD
in the case of H6 is always less than that of the
case of O6, implying again that a head-on collision is more efficient in destroying the linear waves.
Time series of scaled maximum vertically integrated kinetic energy (KE).
The figure shows the difference between simulations with and without the linear
waves. Note the different scales in the y axes.
Influence of the interaction on the ISW
During the collision with the linear wave packet, the solitary wave is also affected by the linear waves that pass through
it. We note, however, that the kinetic energy carried by the linear waves is much smaller than that carried by the
solitary wave, and hence the impact of linear waves on the solitary wave is also small. Here, we define the kinetic
energy (KE) per unit mass following standard convention (which drops the reference density and hence changes the
dimensions of the quantity) by
KE=12(u2+w2).
We found that when measured in terms of vertically integrated kinetic energy at the wave crest, the linear waves are about
1 % as energetic as the solitary wave.
To analyze changes in the solitary wave and determine if they are results of the collision, we performed an additional
simulation with the same solitary wave but without the linear waves. We estimated the vertically integrated KE at the
crest of the ISW for simulations with and without linear waves, and plotted the difference as time series (i.e. as functions
of scaled time T) in Fig. over one complete collision cycle. Mathematically, this quantity is computed as
1Amax0≤x≤Lx∫0Lz(KEfull-KEisw)dz,
where A is the normalization factor defined as the maximum vertically integrated KE of the initial solitary wave. The
subscript “full” denotes variables from simulations with both solitary and linear waves, and the subscript
“isw” denotes variables from simulations with a freely propagating solitary wave. For linear waves with
a wavelength λ=0.2m shown in Fig. a, there is a net energy transfer into the solitary wave
as a result of the interaction, such that the maximum vertically integrated KE has increased by at least 1 %. We are
able to confirm that such an energy increase in the solitary wave is robust since we have also performed additional
simulations with a longer linear wave packet (not shown), and found that the maximum vertically integrated KE increases
approximately linearly with respect to the length of the wave packet. On the other hand, for waves with a wavelength
λ=0.6m shown in Fig. b, energy increase in the solitary waves after the collision is
insignificant. In all cases, the curves shown demonstrate periodicity associated with their particular wavelengths.
We have also attempted to detect the phase shift of the solitary waves from the locations of maximum vertically integrated
KE. However, we found that such a phase shift, if it exists at all, is on the order of millimetres. In other words, the
detected phase shift is on the grid scale and is subject to numerical error. For this reason, the results are not shown
here.
Conclusions
In this work we performed two-dimensional direct numerical simulations to study the interaction between a large-scale
fully nonlinear ISW and small-scale linear internal waves. We demonstrated that there was a net energy transfer from the
small-scale linear waves to the large-scale solitary waves. This contrasts the conclusion in , made for
a different type of internal wave interaction, that energy is transferred from large-scale waves to small-scale waves.
Our simulation results suggest that during the interaction, the solitary wave essentially acts as a filter through which
only long waves may pass. For waves with a smaller wavelength, the interaction leads to a reduction of their initial
energy and a destruction of their spatial structure. These processes occur in a background state set by the solitary-wave-induced
stratification and current. During the interaction, adjustment of the short waves to this new background
environment extracts their wave energy and modifies the wave structure. The fact that short waves may not survive the
interaction with a solitary wave, or more generally any localized nonlinear background environment which both deforms the
pycnocline and induces shear, implies that the observed spectrum of wavelengths of internal waves in locations with large
amplitude ISWs (such as Straits) is likely to be deficient in short waves. At the time of writing we are unaware of
measurements to support or contradict this hypothesis.
We performed analysis based on linear wave theory and showed that during the nonlinear interaction with the ISW, the
destruction of short linear waves occurs primarily due to the presence of ISW-induced velocity shear, which alters the
vertical structure of the short waves in a nonlinear manner, leading to significant wave amplitudes on only one side of
the deformed pycnocline centre. On the other hand, a shift of the location of the pycnocline plays a secondary role during the
collision, as its main effect is to alter the propagation speed of the linear waves, and shift the location of the maximum
of the vertical structure downward. However, the vertical structure is unchanged with respect to the pycnocline centre. We
also demonstrated that a critical layer is not present during the collision, regardless of the wavelength of the linear
waves, since the phase speed approaches the maximum ISW-induced current asymptotically as the wavelength approaches 0.
A clear avenue of future research is to explore the parameter space, in particular the Richardson number effect, of the
solitary wave. In the present work, we studied the ISW whose minimum Richardson number is 0.246. Although none of the
simulations show evidence of the generation of shear instability, this does not necessarily mean that the wave–wave
interaction considered in the present work is Richardson number independent. Moreover, showed that it is
not only the minimum Richardson number but also the length of the unstable region with a low Richardson number relative
to the wavelength of ISW that is jointly responsible for the generation of shear instability in an ISW. It is thus
reasonable to assume that the relative length of the region with a low Richardson number in the ISW also has an influence
on the wave–wave interaction. Future research will explore these effects in detail.
We note that our findings are in many ways similar to those in . Their study also concluded that the
direction of energy transfer during the interaction is from the small-scale weakly nonlinear wave (i.e. the mode-2 wave)
to the large-scale solitary wave (i.e. the mode-1 wave), and that such energy transfer is more efficient when a head-on,
instead of overtaking, collision is involved. The main difference is that in a mode-1–mode-1 interaction, there is a cutoff
determined by the wavelength of short waves, above which the small-scale waves maintain their structure after interaction,
whereas a mode-1–mode-2 interaction is much less dependent on the wavelength. In a mode-1–mode-1 interaction, this cutoff
corresponds to the wavelength at which the phase speed of the short waves upstream of the solitary wave exceeds the
maximum ISW-induced current. In a mode-1–mode-2 interaction, however, this cutoff does not exist since the maximum
ISW-induced current is always larger than the phase speed for any given wavelengths.
While all of the simulations discussed in this work are performed on the laboratory scale, the scaling-up of the current
experiments to the field scale is left as a topic for future work. When the field scale is considered, waves with a much
larger range of wavelengths can be expected to breakdown, including short waves affected by self-interaction
. Also, a higher Reynolds number implies that the overturning seen in Fig. may eventually
lead to significant overturns. The three-dimensionalization of the flow field should also be examined. As shown in
, two-dimensional models are unable to properly describe the physics or the consequences of the wave-breaking
process, in particular those consequences induced by the presence of a critical
layer. We also note that in two dimensions,
the only possible form of wave–wave interaction is either an overtaking collision or a head-on collision. However,
observational evidences (e.g. ; in particular, see their Figs. 2 and 8) suggest that internal waves do
not generally propagate parallel to each other but may interact at different angles. The effects of directionality of
wave propagation is another topic that can be considered in forthcoming studies.
Data is available upon request by email to the first author.
The authors declare that they have no conflict of interest.
This article is part of the special issue “Extreme internal wave events”. It is a result of the EGU, Vienna, Austria, 23–28 April 2017.
Acknowledgements
Time-dependent simulations were completed on the high-performance computer cluster Shared Hierarchical Academic Research
Computing Network (SHARCNET, www.sharcnet.ca). Chengzhu Xu was supported by an Ontario Graduate Scholarship while Marek Stastna was
supported by an NSERC Discovery Grant RGPIN-311844-37157.
Edited by: Kateryna Terletska
Reviewed by: two anonymous referees
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