Content uploaded by Sergei Badulin
Author content
All content in this area was uploaded by Sergei Badulin on Nov 24, 2015
Content may be subject to copyright.
J.
Fluid
Mech.
(1993),
vol.
251,
pp.
21-53
Copyright
0
1993 Cambridge University
Press
21
On the irreversibility
of
internal-wave dynamics
due
to
wave trapping
by
mean
flow
inhomogeneities. Part
1.
Local analysis
By SERGE1 I. BADULIN
AND
VICTOR I. SHRIRA
P.
P.
Shirshov Institute
of
Oceanology, Russian Academy
of
Sciences, Krasikova
23,
Moscow, 117218, Russia
(Received
1
December 1991 and in revised
form
15 September 1992)
The propagation of guided internal waves on non-uniform large-scale flows of
arbitrary geometry is studied within the framework of linear inviscid theory in the
WKB-approximation. Our study is based on a set
of
Hamiltonian ray equations, with
the Hamiltonian being determined from the Tayloraoldstein boundary-value
problem for a stratified shear flow. Attention
is
focused on the fundamental fact that
the generic smooth non-uniformities of the large-scale flow result in specific
singularities of the Hamiltonian. Interpreting wave packets as particles with momenta
equal to their wave vectors moving in a certain force field, one can consider these
singularities as infinitely deep potential holes acting quite similarly to the ‘black holes’
of astrophysics. It is shown that the particles fall for infinitely long time, each into its
own ‘black hole’. In terms
of
a particular wave packet this falling implies infinite
growth with time of the wavenumber and the amplitude, as well as wave motion
focusing at a certain depth. For internal-wave-field dynamics this provides a robust
mechanism
of
a very specific conservative and moreover Hamiltonian irreversibility.
This phenomenon was previously studied for the simplest model of the flow non-
uniformity, parallel shear flow (Badulin, Shrira
&
Tsimring 1985), where the term
‘trapping’
for
it was introduced and the basic features were established. In the present
paper we study the case of arbitrary flow geometry. Our main conclusion is that
although the wave dynamics in the general case is incomparably more complicated, the
phenomenon persists and retains its most fundamental features. Qualitatively new
features appear as well, namely, the possibility of three-dimensional wave focusing and
of ‘non-dispersive’ focusing. In terms
of
the particle analogy, the latter means that a
certain group
of
particles fall into the same hole.
These results indicate a robust tendency
of
the wave field towards an irreversible
transformation into small spatial scales, due to the presence of large-scale flows and
towards considerable wave energy concentration in narrow spatial zones.
1.
Introduction
The problem of describing interactions between internal waves and large-scale
oceanic motions, which is crucial in understanding oceanic internal-wave dynamics
and energetics is still very far from finally being resolved (e.g. Miropolsky 1981
;
Olbers
1983
;
Levine 1983). The present state of internal-wave theory is characterized by the
search for the strongest mechanism of interaction, which can lead
to
considerable
energy exchange between waves and currents, internal-wave energy transfer into
distant spectral bands, wave breaking, etc.
22
S.
T.
Badulin and
V.
I.
Shrira
One such mechanism
-
the trapping of internal waves which propagate in a guide
(horizontally varying due to the non-uniformity of stratification and currents)
-
was
found by Badulin, Shrira and Tsirnring (Badulin, Tsimring
&
Shrira 1983; Badulin,
Shrira, Tsimring 1985, hereafter referred to as BSTl and BST2). The essence of this
mechanism is as follows. In the fra.mework
of
linear inviscid theory there are regions
of
‘
non-transparency
’
for comparatively short (typically with wavelengths less than
lo3 m) guided internal waves propagating on non-uniform currents. A wave packet
approaches the boundaries of these: regions of non-transparency asymptotically, i.e. it
tends to stop in a frame of reference moving with the current and the intrinsic wave
frequency tends
to
the maximum frequency of linear internal waves (for example, the
maximum value over depth
of
the Brunt-Vaisala frequency in the model
of
shearless
basic flow). All this enables us to speak about
wave trapping
by the current and to refer
to the boundary
of
non-transparency (for a given wave) as the
wave trapping layer.
(In
fact we have a
line
of
trapping, bul: we preserve the ‘historic’ term.) Hereafter we use
the term ‘trapping’ in this sense only. It is also often used for waves propagating in a
certain wave guide
-
we shall refer to these waves as ‘guided’ or ‘localized’.
The effectiveness of the wave trapping mechanism and the very fact of its existence
were demonstrated in BSTl and RST2 for the simplest type
of
non-uniformity (the
ambient fields of current velocity
U(x,z)
and stratification
N(z)
were considered to
depend on one of the horizontal spatial coordinates only). We list below some of the
results derived within this model.
1
t was shown that when a wave packet approaches
the trapping layer, wavenumber and amplitude grow with time while wave motion tends
to concentrate at a certain depth. Within the bounds of the linear inviscid theory the
growth
of
wavenumber and amplitude is unbounded.
Firstly, this picture
of
wave trapping was derived in the WKB-approximation of
linear theory. Exact solutions of the model problem (Erokhin
&
Sagdeev 1985a,
b)
confirm the main conclusions
of
the approximate theory, including the validity of the
WKB-approximation. Some experimental evidence of internal-wave trapping in
a
strong horizontally inhomogeneous current (Lomonosov current) has been reported
recently (Badulin, Vasilenko
&
Golenko 1990). It has been found that under certain
conditions the trapping definitely takes place and produces strong effects like wave
breaking or, at least, stimulates small-scale turbulence generation at the depth of
maximal stability, i.e. at the maximum of the Brunt-Vaisala frequency. We note that
the range of wave and current parameters where the phenomenon has been observed
in its simplest form is not
so
typical of ocean conditions as to imply that the trapping
in this form is a widespread phenomenon. We believe that another formulation
of
this
problem is
of
primary interest: whether the trapping takes place as a general tendency
of internal-wave-field evolution and how this tendency reveals itself.
The asymptotic character of the approach of the packet towards the trapping layer
in this context means that the internal-wave dynamics becomes principally irreversible
even in the inviscid approximatiorl of linear theory (i.e. within the framework of the
theory not taking into account, in particular, wave-wave interactions, wave breaking
processes or the existence of critical layers).
Let us specify the place
of
our problem within the vast corpus of recent works, where
the term ‘irreversibility
’
has been used in the context of internal wave dynamics.
The first group of works worth mentioning here, initiated by McComas
&
Bretherton (1975), starts with the kinetic equation for the internal-wave field or certain
analogues of this equation. Irreversibility in these problems comes from certain
ensemble averaging of wave characteristics as in the kinetic theory of gases. This
irreversibility is connected with the conservative mechanisms and provides energy
The irreversibility
of
internal-wave dynamics. Part
I
23
exchange among motions of different scales. For the internal waves these mechanisms
are the resonant wave-wave (McComas
&
Miiller 1981
;
McComas
&
Bretherton 1977)
or wave-current (Watson 1985) interactions. In the latter case, the current is
represented by a large-scale wave motion assumed to be stochastic.
Very close to the first group lies the one where both the deterministic and stochastic
fields are under consideration (Raevsky 1983
;
Bunimovich
&
Zhmur 1986). The typical
problem statement is as follows: a wave packet (the deterministic part of the wave field)
propagates through
a
stochastic environment. In this case, ‘irreversibility
’
reflects the
scattering of the deterministic component.
‘Pure deterministic irreversible’ problems can be considered as those dealing with
the elementary interactions in an appropriate wave field, that is, as ‘bricks’ of the
problems mentioned above. Commonly, consideration
of
these problems involves such
physical mechanisms as strong nonlinearity (e.g. wave breaking) or wave instability. In
particular, the inhomogeneity
of
ocean currents causes transformation of internal
waves in the small-amplitude approximation, and then significant nonlinear and
viscous effects become apparent, which in their turn provide ‘classical
’
irreversibility.
Within the ‘pure deterministic irreversible’ class of problems there is a special group
concerned with asymptotic-with-time effects, the most well-known of which is the
critical layer phenomenon (e.g. Booker
&
Bretherton 1967). The number of review
works devoted to different aspects of wave dynamics near critical layers is too large to
be listed. (For the basic references see the book by LeBlond
&
Mysak 1979.) In view
of recent results, the critical layer should not be considered to be of prime importance
for the internal wave field energy balance; earlier overestimations of its role was a
consequence
of
the problem idealization (Borovikov 1988). Other types of wave
asymptotic behaviour due to singularities of different natures, including the ‘trapping
singularity that we are interested in, have been studied by Basovich
&
Tsimring (1984)
and Olbers (1981). In the latter, the relation between critical layer and trapping layer
phenomena was demonstrated fairly well. The mathematical difference lies in the
statement of the physical problem. Trapping layers and critical layers can be viewed as
the two limits of the same problem and their physical models have their own range of
applicability. However, in both these important works the approximation of constant
Brunt-Vaisala frequency was essential and inhomogeneity of the mean hydrophysic
fields was, in fact, one-dimensional.
However, the physical picture of trapping differs significantly in one-, two- and
three-dimensional problems. The increase of dimensions changes the mathematical
nature of the problem significantly and leads to quite different physical consequences.
In particular, the concentration of wave energy occurs in a plane in the one-
dimensional problem, in a line in the two-dimensional problem and at one point,
according to one
of
possible scenarios, in three-dimension. Thus, it is fundamental to
an understanding of the phenomenon
of
trapping to analyse the situations where both
horizontal and vertical inhomogeneities are important together.
In this paper we pose the question concerning the structural stability of the effect of
trapping and the tendency towards irreversible transformation
of
internal waves to the
small scales in mean hydrophysic fields of arbitrary geometry. We have some grounds
to suppose that other effects which will not
be
considered here
do
not fundamentally
change the results of our investigation. For an asymptotical regime
of
transformation,
wave-dynamical irreversibility occurs regardless of the way nonlinearity or viscous
effects are specified. The phenomenon of trapping is a high-frequency one (relating
to
the internal-wave frequency range). That makes the effect, unlike in the critical layer
case, structurally stable and therefore important. We recall that as the wave frequency
24
S.
T.
Badulin
and
V.
I.
Shrira
near the critical layer tends to zero, the wave dynamics becomes strongly influenced by
low-frequency variations of the mean current, inertial waves
(see
e.g. LeBlond
&
Mysak 1979; Broutman 1986; Broutman
&
Grimshaw 1988), Earth rotation, etc. An
essential feature of wave trapping is that both the vertical and horizontal wave scales
tend to zero, while in a critical layer only the vertical scale diminishes, and in problems
of
wave instabilities the spatial scales, as a rule, remain the same or vary slowly.
While studying general properties of internal-wave dynamics on inhomogeneous
currents
a
number of questions. arise in connection with the phenomenon of trapping.
The first is whether the tendency of the field of guided internal waves to irreversible
transformation into the small-scale band holds when the waves propagate on
horizontally and vertically inhomogeneous currents of arbitrary geometry. This
question can be paraphrased in terms
of
questions regarding internal-wave trapping in
an arbitrary inhomogeneous ambient current field. In particular, questions about the
existence (non-existence)
of
trapping and relations between the universality of its basic
features established in BSTl, 13ST2 and the modification of these features. This work
presents some answers to these questions and is organized as follows.
In $2 we give the problem formulation. We start with the known set of Hamiltonian
ray equations which describes the propagation of small-amplitude internal waves in a
horizontally non-uniform ocean in the WKB-approximation (e.g. Voronovich 1976;
Miropolsky 198 1). The characteristics
of
the vertical inhomogeneity (vertical profiles
of mean stratification and large-scale current) are incorporated into the Taylor-
Goldstein boundary-value problem. Its solution yields the local dispersion relation, i.e.
the Hamiltonian
of
the system.
Thus, within the accepted WKB-approximation the problem
of
a complete
description
of
the wave-packet dynamics in the non-uniform
flow
field reduces to
solving a set of four Hamiltonian equations. This system describes the motion
of
a
particle-wave packet (with momentum
k)
in a certain force field. We stress that the
structure of this field is determined not only by the dependence of the mean
flow
on
horizontal spatial coordinates, but also by kinematic characteristics (wave vector) of
the internal-wave packet itsell.. The internal-wave dispersion law leads (in terms of the
particle-wave analogy) to the occurrence of special singularities of the force field
potential (‘potential holes
’).
Some particles ‘fall’ into these holes for aninfinitely long
time (each ‘type’ of particle falls into its own holes), while the particle momentum also
grows infinitely. Thus, the trapping layer acts as
a
specific perfect absorber or, using
the astrophysical analogy, as
a
‘a
black
hole’.
Hence, the problem of describing the
phenomenon of trapping can be formulated as the problem of particle dynamics in the
hole neighbourhood. We recall that the infinitely long falling of particles into these
holes is interpreted as the waves undergoing an irreversible transformation to small
scales. Further we shall study only this specific conservative and, moreover,
Hamiltonian irreversibility.
The discussion of the methods and approximations in $2, however, does not touch
upon the important question:;, which weakly depend on the geometry
of
the basic flow.
Namely, in justifying the validity of the WKB-approximation, the calculation
of
dissipative and nonlinear effects was not given. These questions have been discussed in
detail in BST2 for a plane mean
flow.
In particular, the validity of the WKB-
approximation for a proper description of wave trapping was demonstrated (BSTl
,
BST2). Later this fact was also confirmed by analysing the exact solution for the
corresponding ‘reference’ equation of a linearized problem in the vicinity of the
singular point (Erokhin
&
Sagdeev 1985a,
b).
In these works the importance of
viscosity and packet bandwidth for the final stage of the packet evolution was shown
The irreversibility
of
internal-wave dynamics. Part
1
25
as well. A quantitative description of these factors can be easily provided for arbitrary
flow geometry in a similar way. Nevertheless, we confine ourselves to the paradigm of
the ideal fluid and the quasi-monochromatic wave, aiming to understand and
demonstrate the basic qualitative features of internal-wave dynamics in an inhomo-
geneous flow field
per se.
We try to achieve this goal by studying a chain of
comparatively simple models. The models are chosen
so
that, on the one hand, they can
be analysed in detail, and on the other hand, they are proper ‘bricks’ for the synthesis
of
a consistent view.
The present paper, which is the first part of the work, is based on an approximation
which we refer to as the
‘local analysis approximation’.
Its essence is an expansion of
the Hamiltonian (or straightforwardly the flow field) in power series in the horizontal
coordinates. The analysis based on the leading terms of this expansion can be justified
for a certain range of spatic+temporal scales and in this sense is ‘local’. The scales of
validity are not small owing to the small horizontal gradients of the flow and to the
small group velocity of short internal waves. The questions concerning the quantitative
evaluation of the validity range and the ‘non-local’ effects, in particular the global
behaviour of trajectories, are the subjects of the second part of this work (Badulin
&
Shrira
1993).
We note that the local analysis approximation simplifies greatly the
mathematics
of
the problem.
Section
3
(the main results
of
which were briefly reported in Badulin
&
Shrira
1985)
is concerned with the model where the basic current is considered to be vertically
uniform while the basic density stratification
is
assumed to
be
horizontally
homogeneous. The main types of internal-wave dynamics are distinguished and
quantitatively described. The presence of the flow’s vertical shear drastically
complicates the packet dynamics in the case of two-dimensional
flow
non-uniformity
.
Still, in
$4
we obtain a quantitative description for a number of important models and
acquire a qualitative understanding of the general situation. The tendency to
irreversibility holds for internal-wave conservative dynamics under rather general
conditions.
Section
5
presents a brief discussion of the results and directions of further
investigation.
2.
Formulation
of
the
problem
2.1.
Geometric optics approximation
;
basic equations
We will study internal-wave dynamics on a steady ambient stratified large-scale shear
flow with given
U(U(z,
x),
V(z,
x),
0)
and stratification
N(z,
x);
here
x
is the horizontal
and
z
is the vertical coordinate respectively, and
N(z,
x)
is the Brunt-Vaisala frequency.
The horizontal scale
L
of the flow variability greatly exceeds the typical vertical scale
d.
We shall consider small-amplitude internal waves with wavelength
h
much smaller
than the characteristic horizontal scale
L
(A
4
L).
Under these scale relations we can
naturally use the ray optics approach (WKB-approximation) for describing the wave
propagation (Miropolsky
1981
;
Voronovich
1976).
Then the wave is supposed to
be
locally plane with local wave vector
k
and absolute (i.e. measured in a laboratory frame
of reference) frequency
o.
At each point on the horizontal plane
x
the dependence of
o
on
k,
as well as the vertical mode structure, are found by solving the Taylor-Goldstein
boundary-value problem
(o-k-U)2a:zw+[N2JkJ2-(w-k- U)a:z(w-k.U)-lk12(w-k.U)2]w
=
0,
(2.1)
with the standard boundary condition at the sea surface and the bottom. We shall take
2
PLY
251
26
S.
T.
Hadulin and
V.
I.
Shrira
homogeneous (zero) boundary conditions. The choice of boundary conditions does not
influence our subsequent analysis and results. Here
w(z,
x)
is the vertical wave-velocity
component which depends parametrically on the horizontal coordinate
x.
The ray equations
Xi
=
aklw;
k,
=
-aa,,w,
(2.2)
give us the wave packet coordinates
x
and wave vector
k
as functions of time
t.
The
evolution of the wave vertical velocity
w
is governed by the equation of conservation
of wave action
I
[!+V'(aklwI)
=
0,
(2.3
a)
where
+w2[g(w-k.
U)-3+3~-k-
U)z(~-k*
U)-z
~k~-']~z+o~.
(2.3b)
Here
w
is not a normalized eigenfunction as is usually supposed in linear homogeneous
problems, but it is determined by initial conditions at a point
xo.
We shall use the term
'internal-wave amplitude
'
and notation
A
for the maximum-over-depth value of the
wave vertical velocity
w.
The last term on the right-hand side of
(2.3b)
describes the
'direct' influence of the surface boundary conditions. Further, we shall neglect this
influence by imposing rigid-lid homogeneous conditions at the free surface and the
bottom, or the condition of decaying
w
at infinite
IzI.
Thus within the accepted approximations the problem is reduced to that of solving
:
(i) the local Taylor-Goldstein boundary-value problem (2.1) (this gives us the
(ii) the set of Hamiltonian ray equations (2.2) (this describes the wave kinematics);
(iii) the transport equation (2.3~) which governs the full spatial dependence w(z,
x)
and hence the amplitude
A.
It should be noted that the three-dimensional problem is now (in the WKB-
approximation) split into two independent parts
:
deriving the local (at each point
(x,
y))
wave vertical structure via (2.1) and then describing the evolution of the wave
parameters (determined by
w(k,
x)
and
I(k,
x))
in two spatial dimensions. However,
even such
'
two-dimensionalization
'
of the three-dimensional problem does not give
any hope for its complete analysis in the case of an arbitrary
flow,
as not only can the
dispersion law
w(k,x)
be rather capricious, but the very problem of deriving it from
(2.1) is not generally solvable. The proper choice of a number of somewhat simplified
models illuminating the main features of the phenomenon under study and its detailed
subsequent analysis seems to
be
the most natural way forward in this situation. We
start by recalling some results derived within the simplest model in BSTl and BST2.
2.2.
The phenomenon
of
trapping
in
the simplest
flow
geometry
Hamiltonian
o(k,
x)
and the vertical structure
of
w(z,
x));
Consider the simplest model of horizontally non-uniform stratified mean
flow
N
=
"z);
u=
(V(y),O,O).
Then, splitting into the horizontal and vertical problems becomes almost complete. In
the Doppler relation
w(dk,x)-k*U=
Q(k)
(2.4)
(w
is the frequency in a laboratory system,
Q
is the intrinsic wave frequency in the
reference frame moving with the ambient flow)
Q(k)
can be found from boundary-
The irreversibility
of
internal- wave dynamics. Part
I
27
value problem (2.1) with
U
=
0,
that is
Q(k)
is
a
function with well-known properties
and is independent
of
x.
Let us consider
internal-wave
kinematics near the point
yt,,
specified as follows
where
N,,,
is the maximum-over-depth value of
N(z)
and
k,
is the x-wave-vector
component. As
a
packet approaches
ytr,
G!
tends
to
N,,
and, according to the known
asymptotics of
SZ(k),
the wavenumber tends to infinity and the y-component of the
group velocity decreases
so
rapidly that it takes an infinite time for the packet to reach
y
=
ytr, i.e. the packet approaches this point asymptotically. This is easily seen from the
expression for time
~~(7~
is the time for the packet with initial wave vector
k(k,,
1,)
to
acquire the value
k(k,,l)),
which is calculated straightforwardly if we put
U,
to be
constant
:
According to the known properties of the boundary-value problem, the wavenumber
growth causes a transformation of the vertical mode structure: as
lkl
grows, the wave
motion tends to concentrate at the depth
z,
corresponding
to
the maximum value of
the Briint-Vaisala frequency
N,,,
(see figure 1). We refer to this transformation as
vertical focusing
and
z,
as
the depth
of
localization.
We also use the term the
trapping
layer
for the vertical plane y
=
ytr,
where the intrinsic group velocity of the packet
asymptotically tends to zero. The slowing down of the packet near the trapping layer
and zero reflection leads to
spatietemporal focusing.
All the energy of a mono-
chromatic wave is focused in the immediate vicinity of
ytr
at the depth
z,.
These two
factors (spatio-temporal and vertical focusing) cause the infinite growth
of
wave
amplitude near the trapping layer. We stress that this infinite growth is not an artifact
of our use of the WKB-approximations, which remains valid here up
to
the singularity
(BSTl
;
BST2). These results were also confirmed later within the framework of exact
linear theory (Erokhin
&
Sagdeev
1985a,
b).
2.3.
Internal waves
on
a flow with arbitrary non-uniformities
:
preliminary remarks
The simplicity of the analysis in the previous subsection has mainly been based on the
possibility of the easy natural separation of the two wave transformation mechanisms
mentioned above, namely, the mechanisms connected with the evolution of the wave
kinematic characteristics ('horizontal
'
factors of transformation) and the factors of the
mode structure transformation ('vertical
'
factors). Commonly, these two classes of
mechanism are intricately interrelated and, evidently, an analysis similar to that of the
previous subsection is impossible. However, the concepts introduced on this basis (the
trapping layer, vertical and spatietemporal focusing, the depth
of
localization) prove
to be very useful tools
for
the general case as well.
Some features
of
internal-wave dynamics in the general case can be more easily
understood via the following trick. Let
us
introduce the effective Brunt-Vaisala
frequency
N,,,
(2.6)
N2(z)(u-
k*
U(z0,
x))~
(o-k.
U(Z',X))~(~*
U(Z,
x))~~
(w-
k.
U(Z,
x))
lk12
+
Xff
=
(w
-
k
-
U(z,
x))2
The idea of this trick is an attempt to reduce the problem formally to the case of a
vertically uniform current. Let us take a certain arbitrary depth
zo
and define the wave
frequency
a,,,
in the reference frame moving with the flow at this depth as follows:
(2.7)
52,ff
=
w
-
k
-
U(z0,
x).
2-2
28
S.
T.
Badulin and
V.
I.
Shrira
FIGURE
1.
The trajectories
of
a packet trapped by the flow inhomogeneity
of
the current
U
=
U,(O)yi
and the vertical focusing of a wave trapped by inhomogeneity
of
the current.
(a)
The current
profile.
(b)
The trajectories of four packets of the same (modulus) initial (at
y
=
0)
frequency and
wave vector
k
projections
(k,f).
The solid line corresponds to one of the two identical cases
(antisymmetric curves) when
k-a,(U)l,-,
is negative. The unlimited monotonic increase of
I
(while
k
remains constant) and decrease
of
C,,
(the y-component
of
the group velocity
C,)
is demonstrated
by showing
k
and
C
at three points (numbered
1,2,3)
of the trajectory. Near the trapping layer
C,,
tends to zero; the difference
(C,,-
U(y
=
ytr))
also tends to zero. Two of the dashed trajectories
correspond to packets with positive initial values of
ka,(U).
Before the same process
of
trapping
begins these packets should first pass over the simple reflection point (where
C,,
and
1
change signs).
(c)
The profiles of
N(z)
(solid lines) and of the first-mode eigenfunction
w(z)
(dot-dashed lines)
corresponding
to
the points
1,2,3
of the trapped packet trajectory are plotted. The dashed lines mark
the value of the intrinsic packet frequency
R(k).
The irreversibility
of
internal-wave dynamics. Part
1
29
In contrast to the Doppler relation (2.4), here
Qef,
explicitly depends not only on
k
but
also
on the horizontal coordinates. Thus, the question of the possibility of the existence
of the effect of trapping can be formulated as
a
question about the way the effective
frequency
Q,,,
approaches or does not approach
N,,,.
Fortunately, to answer these
questions it is enough to know only the short-wave asymptote of
Q,,,
and
N,,,.
The
general tool for further study will be the concept of
the effective depth
of
localization
z,(k)
defined as the depth where
N,,,
reaches a maximum for the given
k.
2.4.
Local analysis
of
internal-wave dynamics
Short internal waves are localized near the depth of localization
z,(N(z,)
=
N,,,,
,).
We shall derive the short-wave asymptotics
of
the boundary value problem (2.1)
following a slightly generalized procedure of BSTl and BST2. Expanding the
effective Brunt-Vaisala frequency
Net,
near
z,
in powers of
(Z-Z~),
we get the
boundary-value problem for the Weber equation
with boundary conditions at infinity
w,,
+
{n(zm(k,
X,
w),
k,
X,
w)
+
nZz(z,(k,
X,
w),
k,
X,
w)
[gz-z,)']}
w
=
0,
w+o;
IzI+ao.
(2.8)
Here
The eigenfunctions
w,,(z)
of (2.8) corresponding to a discrete spectrum are Hermite
polynomials (Abramovitz
&
Stegun 1964)
The dispersion relation
w(k,
x)
is found from the equation
w(z, k,
x)
=
exp
{
-
3
-
~l7,,)~
(z
-
Z,,,)~}
Ha{
(-~Z7,,)~
(z
-
z,)}.
(2.10)
(2.1 1)
n(z,(k,
X,
w),
k,
X,
0)
[
-
Z7,,(zm(k,
X,
w),
k,
X,
w)]-i
=
(2n+ 1)/2/2.
Using (2.10) and (2.3
b)
we straightforwardly get the expression for wave action
Z
(for
details see BST2)
Z
x
r+
N2w2(w
-
k.
U)-3
dz
x
N2(z,) AZd(z,)
(w
-
k.
U(Z,))-~ @(n).
(2.12)
Here we suppose
V,,
to be small and hence the second term on the right-hand side of
(2.3b) may be neglected. The third and the fourth terms in (2.3b) are zero when
boundary conditions at infinity are used. In (2.12)
d(z,)
=
[
-fI7,,(zm(k,
x,
w),
k,
x,
w)]-'(2n
+
l)-'
is the characteristic vertical scale
of
mode
wn(z),
while
@(n)
is
a universal function
of
mode number
n.
It should be stressed that all the main parameters
of
short internal waves are thus
expressed via (2.9H2.12) through the local characteristics of a large-scale flow only.
Hence the mechanisms of the short internal-wave transformation to be studied are
immediately related to the local spatial structure of the ambient flow (contrary to the
case of waves of larger scales), rather than to the integral characteristics of the flow.
Throughout the paper we shall exploit the first-order expansion in the horizontal
coordinates of the basic flow
I
N(z,
x)
=
N,(z)
+
N,
x
+
N,
y,
U(Z,
x)
=
U&Z)
+
V,
x
+
V,
y,
(2.13)
30
S.
T.
Badulin and
V.
I.
Shrira
or the Hamiltonian
o(k,
x)
expansion directly. We refer to this first-order expansion as
the local analysis approximation. Generally, this approximation has a finite temporal
interval of validity. The quantitative evaluation of the validity range and 'non-local'
effects in wave dynamics will be given in the second part
of
this work. Here we only
note that some 'local' scenarios of trapping can be found within this approximation
for which validity of an expansion of the type (2.13) for arbitrary times can be easily
confirmed a posteriori.
3.
Wave dynamics in
a
vertically homogeneous velocity field
3.1. Internal-wave kinematics
It has already been mentioned @2.3) that, in the case of vertically homogeneous flow,
the problem
of
finding the dependence
Q(k,x)
reduces to the analysis of the well-
known boundary-value problein for internal waves in a horizontally homogeneous
stationary ocean. This allows us to make no preliminary assumptions about the
dependence
N(z).
But for the field U(x) we shall use the local (in the vicinity of some
point
x,
=
0)
representation (2.13)
u=
u,+u,,x+u,y,
v=
v,+V,x+v,y. (3.1)
Rotation of the coordinate system through the angle
q5
=
2
tan-' [2Uz(
U,
+
V,)-']
reduces (3.1)
to
the form
Here
U,
and
V,
differ from their values in (3.1); however, we shall use the same
notation. For the
flow
(3.2) a streamline function can be introduced:
(3.3)
The streamlines
Y
=
const are the second-order curves (conic sections) and their
Y
=
+U,(
y
+
u,/
V,)'
-
;
V,(X
+
v,/
V,)?
type is determined by the sign of
A2:
A2
=
V.
U,.
(3.4)
(In terms of the original variables
A2
=
U,
5-
Vi).
The remarkable feature of the velocity field approximation considered here is that
the system of ray equations splits into two sets of equations that could be solved
successively and thus becomes completely integrable.
As
far as we know, Jones (1969)
was the first to notice and exploit this fact. We also mention that approximations
of
the type (3.2) reveal some remarkable properties which are relevant to nonlinear waves
as well and have become the subject of intense studies (Craik 1989; Craik
&
Criminale
1986).
For the wave vector components we have
k
=
-
V.1;
I
=
-U,k. (3.5)
We stress that the type of wave evolution (the form of solution of (3.5)) is determined
by the sign of
A2
only, i.e. by the flow geometry exclusively:
(3.6)
c)
=
-(
1 k,
-
V,
1,I.A
)exp(At)+-(
1
k,
+
V,
lop
)exp(-At).
2
I-
U,
k0ld4 2 l+U,k,/A
Consider the behaviour of solutions (3.6) and streamlines
!P
=
const determined by
the sign of
A'.
The irreversibility
of
internal- wave dynamics. Part
1
31
(a)
A’
=
0
(‘parabolic’ point). The velocity is parallel to one of the horizontal
coordinate axes. The streamlines are straight. We have considered this case briefly in
$2.2 and it has been analysed earlier in detail in BSTl and BST2 and Erokhin
&
Sagdeev (1985). See also figure
1.
(b)
d2
>
0
(‘hyperbolic’ point, see figure 2a). The streamlines (3.3) are hyperbolas
and curves
on
the plane
(k,l)
given by (3.6) are also hyperbolas. The asymptotes of
hyperbolas
Y
=
const and (3.6) are pairwise orthogonal. It is easy to understand the
behaviour of wave-packet trajectories in this case.
We first make a qualitative analysis. Since in the model under consideration the
internal-wave intrinsic frequency cannot exceed the maximal value of the Brunt-
Vaisala frequency, the Doppler shift (scalar product
(k.
U))
cannot increase
infinitely as
Ikl
grows infinitely. Therefore, the angle
x
=
cos-’
[(k.
U)/J(k.
U)l]
must
tend to
in.
Thus, the internal-wave-packet motion in phase space at large times is
constituted by the motion of vector
k
along the appropriate branch of a hyperbola on
the plane
(k,
l)
which is accompanied by ‘gluing’
of
the wave-packet trajectory in the
plane (x,y) to a certain streamline. This ‘gluing’ is caused by peculiarities of the
internal-wave dispersion law: namely, the limited value of internal wave intrinsic
frequency and, hence, the rapid decrease of intrinsic group velocity with the growth of
wavenumber.
As
will be shown below, the behaviour of the wave-packet trajectories is qualitatively
different for waves of other types with unlimited intrinsic frequencies (surface waves,
for
example). In this case the trajectories are not glued to the streamlines of the mean
flow. The possibility of a concentration of trajectories in the vicinity of
a
certain curve
at large times could lead to infinite growth of wave amplitude.
(c)
A2
<
0
(‘elliptic’ point, figure 2b). The streamlines are ellipses. The wave vector
on
the plane
(k,l)
also moves along an elliptic curve oriented at a right angle to the
ellipse
Y
=
const on the coordinate plane. In this case there is no tendency to infinite
growth of the wavenumber but still
a
significant wave transformation is possible.
Indeed, the ratio of the maximum
to
the minimum wavenumber is determined by the
ratio
of
velocity shears
R
=
I
V,/U,)i.
The large increase of the wavenumber for large
values of
R,
as well as in case (b), provides conditions for the manifestation of
mechanisms of dissipation and nonlinear interaction which are not considered here.
This will lead to a ‘practically irreversible
’
type of wave-packet evolution.
3.2. Internal-wave dynamics
Qualitative speculations on the behaviour of the wave-packet trajectories and the
possibility
of
significant growth of wave amplitude can be supported by the exact
solutions of the system (2.2H2.3).
It
is
convenient to consider the problem in terms of new variables
c,,,
=
Rx+y;
52
=
#/R+l).
(3.7)
Indices 1,2 correspond to the upper and lower signs respectively. Then the solutions
to the system of ray equations take the form
K1,
2
=
K:,
2
exp
(T
(3.8~)
f,2
=
$2exp(+A0+exp(fA0 exp(TAOaQ/aK,,,dt. (3.86)
When
dz
>
0
the axis of the new coordinate frame coincides with the asymptotes of
hyperbolic streamlines while
K~,
K~
represent components
of
the ‘new’ wave vector in
this system. Thus, the basis for choosing the new variables becomes clear: we consider
c
32
S.
T.
lbdulin and
V.
I.
Shrira
IY
1)
=
const
--
X
1.'
k
A'
=
0
A'>O
r
In
i/r,
FIGURE
2.
Wave-vector evolution depending
on
the mean current geometry in the
'
totally-local' flow
field.
(a)
Hyperbolic streamlines
:
infinite growth of wavenumber independent
of
the initial wave
parameters (initial wavenumbers and coordinates) and wave type (wave dispersion relation) is
depicted in the right-hand sketch. The
internal-wave-packet
trajectory tends to the streamline
(depicted on the left-hand sketch) since the wave
group
velocity rapidly decreases. While the short-
wave packet propagates along the streamline, the wave vector goes along the hyperbola branch which
is perpendicular
to
this streamline.
(b)
'Elliptical streamlines
:
the wave-vector trajectory is an ellipse
(right) and the wave evolution
is
reversible in k-space (within the framework
of
linear, nonviscous
theory) irrespective of the wave type and wave parameters.
(c)
Evolution of internal-wave amplitude
(right) and wavenumber (left), depending on the mean current geometry (sign of
A2): Ap
c
0
is the
region of reversibility where wave vector and wave amplitude are finite;
d2
>
0
is the irreversibility
region where wave vector and amplitude grow infinitely with time.
The irreversibility
of
internal- wave dynamics. Part
1
33
the motion of a wave packet with reference to the asymptotes of hyperbolic streamlines
in order to describe properly the effect of the ‘gluing’
of
trajectories to these
streamlines.
According to
(3.8a)
the wavenumber increases exponentially as
t
+
00.
The wave
front tends to become parallel to asymptotes of hyperbolic streamlines. We note that
this type of wave-vector behaviour does not depend on the nature (in this context, on
the dispersion relation) of waves propagating upon the shear flow given by
(3.1).
On
the contrary, the form of the dependence
c,,
2(~)
results from the specific features of the
wave dispersion law, or, to be more precise, it is described by the behaviour
of
the
integral term on the right-hand side of
(3.8b),
when
t+
00.
The first term on the right-
hand side
of
(3.8b)
describes the wave-packet motion together with the mean
flow.
Evidently, the behaviour of the second term responds to the intrinsic packet motion
relative to the mean flow and allows one to distinguish between two principal possible
kinematic regimes for the waves depending
on
whether the intrinsic group velocity
aQ/aK
decreases rapidly enough when
lkl+00
or not. If not, the term under
consideration increases infinitely
as,
for example, in the case of surface gravity waves.
The behaviour of trajectories of internal waves is quite different: the term in
(3.8
6)
describing the intrinsic packet motion has a finite limit when
t+
co
(lkl+
00).
The
packet covers
a
finite distance relative to the water mass in which it was originally
placed. Short-wave transformation
of
internal waves in the problem concerned turned
out to
be
local in the coordinate frame that moves with the flow. Thus we can refer to
the ‘gluing’ of trajectories
of
internal waves to the mean flow streamlines.
It is natural to investigate the relations between internal-wave dynamics and the
specific features of wave kinematics mentioned above. From
(2.3a)
we obtain the law
of conservation of wave action
Z
for a stationary case in the form
where
6
is a parameter of a family of trajectories. Usually
E
is chosen as
xo(yo).
Neglecting the intrinsic packet motion relative to the mean flow we obtain
Z
=
const.
The increase of packet dimensions along
1,
is compensated by its decrease along
c2.
Conservation
of
the packet volume on the (x,y)-plane can be associated with non-
divergence of the mean flow.
Thus,
the variation of
I
is due to the intrinsic packet motion relative to the mean
flow. From
(2.9)
in our case we obtain
Here
(3.10)
(indices
1
,
2
correspond to the upper and lower signs, respectively), and the parameter
is taken in such a way that the equation
wwc
-X(W
+
XwE(C
-XPN
=
1
is satisfied. In these terms the equation of caustics
Z,/Z
=
0
can be rewritten in the form
(c
-X(N
(G
-.m)
=
9
(3.11~)
or in terms of the current packet coordinates
cl,z(t),
(ClO)
-fm
(f(0
-fe(f))
=
Ml
Y
(3.1
1
b)
34
S.
T.
Badulin and
V.
I.
Shrira
-
-
-
-__
-
-
/
L
F
I-?’
(a)
c
FIGURE
3.
Internal-wave-packet
trajectories in the case
of
the irreversible evolution
(A’
>
0).
‘Unnatural
’
caustics are formed when time increases infinitely. Trajectories tend to the caustic line
asymptotically. Trajectories which started from a hyperbola (solid line) ‘glue’ themselves
asymptotically to the same hyperbola (with the same focal parameter) transferred parallelly (dashed
hyperbola). The location
of
the ‘initial’ hyperbola determines the parallel shift
of
the caustic
line
(mean current streamline) as
5,
=h(co):
5,
=f2(co).
wheref,,
=
l/d
852/8q, and the constant
M
is determined by the initial parameters
of the wave packet.
Using the system
(3.1
1
a,
b) the: behaviour of the caustic surfaces can easily be
analysed. Let the wave vector
KO
=
(K!,
$)
be
fixed and let us assign
a
certain value to
t.
Thereby, as is seen from
(3.1
1
b), we wl1 determine the centres of hyperbolas
(3.1
1
a,
b). Let the wave packet be located at the point
c*
=
(c,c).
Having specified the
constant
M
as
we prescribe
a
certain spatial cross-section of the ray tube. Rays with the initial wave
vector
KO,
originating from this cross-section, will be focused at moment
t
on the
hyperbola determined by
(3.1
16). Varying the constant
M,
we can represent the whole
wave packet in the form of the superposition of its cross-sections by hyperbolas
(3.11
a)
at the initial moment, as well as by superposition of caustic surfaces at an arbitrary
subsequent moment
t.
Plotting the curves in the way Jhown in figure
3
is adequate for the case of an
ordinary caustic and provides a qualitative explanation of the wave amplitude
limitation in the vicinity
of
an ordinary caustic: spatial ‘smearing’ of the initial
conditions (or finite packet spectrum width) results in the spreading of the caustic
surface.
In the present problem the formation of caustics of quite another type occurs when
t
+
00
owing to the aforementioned peculiarity
of
the internal-wave dispersion relation.
For internal waves, as can easily be seen from
(3.1
1
a,
b),
such caustics exist and are
associated with the mechanism of ‘ghing’ of trajectories to the streamlines. In this case
for caustic surfaces we have
(3.12)
The irreversibility
of
internal-wave dynamics. Part
I
35
The family of hyperbolas (3.12) is one-parametric, contrary
to
the multi-parametric
family (3.1
1
a,
b)
in the case of ordinary caustics. This means that the explanation of
the mechanism of wave-packet amplitude limitation for ordinary caustics given above
cannot be applied in the literal sense to the case of caustics formed when
t
+
00.
Making use of (3.3), (3.12) it is not difficult to obtain the asymptotic (when
t
-+
m)
laws of growth of the wave amplitude
A:
A
-
exp(:At). (3.13)
In the case of closed streamlines (the ‘elliptic’ case) the dynamics of internal-wave
packets can be considered in a manner quite similar to the case of
A’>
0.
The
fundamental difference between them lies in the fact that only ordinary caustics are
possible when
A’
<
0
and the finite value
oft
corresponds to them through the system
of equations (3.11
a,
b).
The qualitative behaviour of the wave amplitude and
wavenumber is depicted on figure 2(c).
Note that while the evolution of the packet wave vector is
of
periodic character, the
trajectory on the coordinate plane is aperiodic and has the form of an intricate spiral.
When
t
--f
m
the packet tends to infinity in x-space. Thus, the local analysis of the
elliptic case remains valid only for a limited time interval.
3.3.
Degenerate case
of
internal-wave dynamics in
a
vertically homogeneous
velocity jield
Within the framework of the model considered here we studied the specific features of
internal-wave dynamics. First, we have to pay attention to the structural stability of the
effect of ‘gluing’. This structural stability, of course, can disappear in less idealized
models. However, the opposite happens
to
be
true: the structurally unstable regime of
motion in the model presented here could correspond to a robust physical mechanisms
of wave transformation.
+
0,
when
A’
>
0.
These initial conditions determine the motion along the separatrix
of
the saddle
generated by the streamlines of the mean flow. In this case the wavenumber also grows
infinitely and the internal-wave packet asymptotically approaches the saddle point
located at the coordinate origin (see figure 4 and its caption). This phenomenon proved
to be similar to that of ‘trapping’. Note, however, that in the case considered here the
asymptotic approach of the packet to the coordinate origin can occur not only for
internal waves, but also for surface gravity waves (see (3.8
b))
(the group velocity
should decrease with the wavenumber growth).
A
remarkable new feature should be pointed out: all harmonics asymptotically
approach the saddle point, i.e. all spectral components
of
wave packets focus at one
point, and, therefore, dispersive spreading is not able
to
weaken possible strong
dynamic effects which result from the packet trapping. We shall refer to such a
phenomenon as
non-dispersive focusing.
The asymptote for internal-wave amplitude
similar to (3.13) can easily
be
found
:
A
-
exp (:At). (3.14)
If we give up our assumption that the basic flow has zero divergence, then the
structurally unstable situation in the model represented here becomes of true physical
interest. In the real ocean there often exist weak vertical motions that lead to
divergence of the horizontal velocity field. The simplest model
of
such flow is the
longitudinally non-uniform flow
U
=
U,xi,
that corresponds exactly to the case just
considered of motion along the separatnx.
Consider trajectories with initial conditions
K!
=
0,
K:
9
0,
6
=
0,
36
S.
T.
Budulin and
V.
I.
Shrira
k
FIGURE
4.
The ‘longitudinal’ trapping
of
waves on the mean current. The wave with initial intrinsic
frequency
D
=
w
(when
U
=
0)
propagates against the mean horizontally inhomogeneous current,
varying its kinematic parameters according to the formula:
w
=
Q+kU
=
const. Points
A,
B
give its
parameters at the intersection
of
functions
D
=
Q(k)
and
w
-kU
(the Doppler-shifted frequency).
Before the junction point
of
these two solutions the wave parameters are determined by the point
A.
Then point B gives the wave evolution. The sign
of
the group velocity changes and the wave travels
to
the point
U
=
0.
As
wavenumber grows, infinitely,
group
velocity tends to zero, and
it
takes an
infinitely long time
for
the packet to reach the point where
U
=
0.
3.4.
Conclusions
In discussing the results of this section we would like to dwell upon some points which
are important for subsequent analysis.
Applying the local analysis approximation to internal-wave dynamics we have
distinguished and described two basic types of wave-packet evolution on horizontally
non-homogeneous flows
:
(i) irreversible inviscid transformation of internal waves into the region of small
horizontal and vertical scales (models
a,
b
of
$3.1
-
A2
>
0);
(ii) reversible transformation, i.e. with quasi-periodic time variation of packet
kinematic parameters
(A2
<
0).
A specific type
of
evolution in our models
is
determined by the mean flow field
structure only (namely, by the sign of
A2).
What is the physical meaning of this
relation? Using the geometric optics approximation accepted here it is natural to
consider the internal-wave propagation as the motion of packet particles in a certain
force field. The motion of such particles is determined by the force-field structure and
particle inertia. Depending on the relationship between these factors, i.e. the properties
of the particles and the force field, effective acceleration of the particle (growth of
wavenumber momentum) may or may not take place. The specific features of an
internal-wave-packet particle is that its
‘
acceleration
’
(trapping), as was found in this
section, takes place under rather general conditions and we stress that it does not
depend on particle characteristics.
We emphasize one more important feature of internal-wave particles. At large values
of the wavenumber, internal waves have
so
little ‘inertia’ that the ‘gluing’ of packet
trajectories
to
the streamlines
of
the mean
flow
becomes possible. The ‘gluing’, as far
as the internal waves are concerned, is particularly interesting in connection with the
possible manifestation
of
strong dynamic effects, investigation of which is beyond the
The irreversibility
of
internal-wave dynamics. Part
I
37
scope
of
this paper. We stress also that the 'gluing' is an important qualitative concept
that will be used for the further analysis of more complicated situations.
We also note that the exponential temporal growth of the amplitude
of
the trapped
waves does not mean instability of these disturbances of the ambient flow in the
commonly accepted sense
of
the term (i.e. as an unlimited growth of the energy of
disturbances). In our case the energy of the packet continues to be finite during the
course of evolution, the unlimited growth of the amplitude being caused mainly by
wave energy transfer into another spectral range.
4.
Internal waves in three-dimensionally non-uniform shear
flows
4.1.
Formulation
of
the problem
To understand when and how trapping occurs in three-dimensional flows in generic
situations one should analyse the properties
of
a dynamical system generated by the
four ordinary equations
(2.2)
with an implicitly given Hamiltonian (via
(2.1)).
The
complete answer requires knowledge of
all
the trajectories for given arbitrary flow.
This task lies far beyond the capabilities of modern theory of dynamical systems. Even
if it were possible, the subsequent problem of separating the different mechanisms
contributing to this picture would also be a formidable problem. The accepted
approximations ('local analysis
'
and short-wave approximations) which are, in fact,
expansions in 1kJ-l and
x,
although they simplify the problem greatly, still do not
provide a general solution in generic situations.
We confine ourselves to considering
a
model flow with the form
u
=
yF,(z),
v
=
xqz),
N
=
N(z)
(4.1)
(&,F,,
N
are arbitrary functions of depth
z).
(The model
(3.1)
of
$3
is a particular case
of the model
(4.
l).)
This flow satisfies the Euler equations for a non-rotating ideal fluid
and allows a wide class of flow variations within the model. The first-priority questions
are
:
to identify the basic physical mechanisms governing internal-wave propagation in
flow
(4.
I),
and to clarify the role
of
the vertical structure of the current specified by the
functions
4, 4
and the stratification profile given by
N(z).
Consider a short-wave equation of the internal-wave dispersion relation (Borovikov
&
Levchenko
1987).
(4-2)
w
=
min
A
max
"(2)
+
k.
U(z)]
+
O(lk1-p)
where
p
is
a
positive value which will be specified below for particular models.
As
it was
pointed out in
$2.3
short-wave evolution is determined predominantly by the local
hydrodynamic field structure at
a
certain depth
z,,
which corresponds, according to
(4.2),
to the extremum
of
the expression in square brackets. Depending on the type of
stratification, three different situations occur:
z,
is determined mainly by the density
stratification
N(z);
z,
is determined by the current profile
U(z);
z,
depends on both
factors.
We note also that
z,
depends upon
U(z)
only in the combination
(k-
U).
Thus
z,
also depends on the wave parameters, via
(k.
U).
In the previous section we dealt with
an example of a situation with
z,
being fixed, (i.e.
z,
did not depend upon the wave
parameters). Besides the case of
$3,
where
z,
corresponds to the maximum of
N(z),
from
(4.2)
one can easily see other examples of this kind, e.g.
zm.
corresponding to the
ocean surface or the bottom. To describe wave kinematics in the cases with
z,
independent of
k,
one can repeat all the results of
$3,
inserting
U(zm)
into all
expressions in place of
U.
38
S.
T.
Badulin and
V.
I.
Shrira
Here we shall concentrate our attention on another class of situations, namely those
where
z,
strongly depends on
k
and the wave dynamics is therefore much more
complicated. Distinguishing the influence of stratification and of shear upon
z,
as
different mechanisms governing the wave-guide parameters, one should bear in mind
that the prevalence
of
one or oth.er mechanism is determined either by the specifics of
the vertical structure of the ,given flow, or by the wavenumber range under
consideration. By varying the flow structure and the wave scales within the bounds of
the model (4.1) we shall elucidate the basic features of guided internal-wave dynamics
in shear flows. 4.2. The general properties
of
the model (4.1)
The main advantage of the model flow (4.1) is the additional symmetries in the ray
equations (2.2) in the short-wave limit, which greatly simplifies the study. In particular,
the depth
of
localization
z,
(given by (4.2)) appears to be a function of the two new
independent variables
a
(a
=
ky) and b
(b
=
lx)
only. Thus, the short-wave expansion
of
the Hamiltonian takes the form
w
=
w,(a,
b)
+
o(
1).
(4.3~)
In the general case,
z,
and
wo
are functions of four arguments
:
x, y, k,
1.
This symmetry
also yields an additional first integral of motion
S
S
=
kx+ly.
(4.3
b)
It should be stressed that the specific form
of
S
does not depend on the functions
4,
F,
and
N
(within the model (4.1)). This remarkable fact enables us to find general
properties of the wave dynamics within this model.
Taking
S
and one
of
the variables
a
or
b
as new canonical momenta we can perform
a canonical transformation. In Appendix A we give four basic types
of
this
transformation. In terms of these new variables new canonical momenta
Q,
(here the
indice is the number
of
the type of transformation in (A
1))
are cyclic. One can also
exploit this fact and decrease the: number
of
conjugated variables down to
a
single pair.
Still, we prefer to use the variables of the type (A l), which do not depend on the
specifics of
4,
F,
or
N.
In terms of these new variables the set
of
ray equations in the short-wave limit takes
the form (see Appendix A)
si
=
0,
Q,
=
hopsi,
(4.4~)
k,
=
-aw0/aQ,,
R,
=
aw,/aR,.
(4.4
b)
We have neglected here the intrinsic wave motion, but the wave kinematics are not
trivial here because of the dependence
of
the depth of localization
z,
on the wave
vector. Thus in the short-wave limit we have reduced the fourth-order system to a
second-order one. Moreover, the zeroth-order Hamiltonian
wo(a,
b)
is the first integral
of
the system (4.4) with parametric dependence on
S,
and our model can be analysed
in detail. There are two qualitatively different regimes in the cases of closed/unclosed
curves
wo
=
const on the (a,b)-plane. Closed curves
wo
=
const correspond to time-
periodic motion (in terms of
a
and b, and hence, in terms of
Q,
R,
defined by
(A
1)).
The general solution of the ray equation can easily be found in terms of the original
variables, using the new variables (A 1) as follows:
k
=
k,exp(-rA,dt),
rO
x
=
x,exp([d,dt), (4.5
a)
(4.56)
The
irreversibility
of
internal-
wave
dynamics.
Part
I
39
where
A,
=
&J,(E,,
R,,
S,)/aSi.
For the periodic motion in
(a,
b)
there is a periodicity of
the integrals in the exponents in
(4.5a)
and
(4.5b):
l+T
A,
dt
=
A,
dt
=
A,
dt
=
[+T
A,
dt.
(4-6)
Here t is an arbitrary moment of time and
T
is the period of the motion in the
(a,
b)-
plane. Note that solutions
(4.5a,
b)
look like solutions
(3.6)
for shearless flow but
integrals over period
T
in
(4.6)
are evidently not zero and hence strictly periodic
motions in the model under consideration are not structurally stable.
One can draw some qualitative conclusions using the notion that flow streamlines in
each horizontal cross-section are the second-order curves, together with the result of
the previous section. We recall that the canonical variables
Qi
are tangents
of
the angles
between the wave (or radius) vector and one of the coordinate axes. Thus periodic
variation of
a
and
b
leads to a periodic variation of
Qi
as well. This means that a certain
fixed range of angles is passed by a packet in the same time interval.
Let the packet motion
be
determined by the flow at two fixed depths z* and z**.
Within the angular intervals
(0,
q5)
and
(x, x
+
q5)
the packet moves at the
z*
depth and
within the angular intervals
($,
x),
(x
+
$,
2n)
at the z** depth. Let the flow streamlines
at z* and
z**
be, for example, ellipses with different parameters. The packet evolves in
accordance with
(4.9,
where instead of
U
and
N,
their values at z* (or
z**)
are taken.
In contrast to the case of the previous section, the wavenumber does not evolve
periodically, but grows (or decreases) with every cycle. The wavenumber growth
(decrease) with time averaged over the cycles is exponential. The packet trajectories are
composed of the flow streamlines at z* and
z**
(see figure
5a).
One can easily show that
helixes of these trajectories will twist (untwist) exponentially as well. These conclusions
also hold qualitatively when the next-order terms in the Hamiltonian are taken into
account. Owing to the intrinsic packet motion, drift that is linear with time adds to the
zeroth-order motion along the streamlines.
The physical interpretation of the relations
(4.6)
given above can also
be
applied to
cases with more than two depths of localization z, and where the flow streamlines are
not necessarily closed. Figure
5(b)
and
6
illustrate this case.
Thus the principal difference between the simplest model of previous section and the
model under consideration lies in the effect of variation of the depth of localization
z,.
Before turning to the study of concrete models we would like to
fix
some general
properties of the wave dynamics.
First, we stress again the principal role of the z,(k)-dependence in the packet
evolution. In shearless flows the type of wave evolution is determined for all the packets
exclusively by the sign of the universal value
A2,
which is prescribed by the mean flow’s
horizontal gradients and does not depend on the stratification’s vertical structure
N(z).
In shear flows an analogue to
A’
also occurs, namely
A:rf.
This is the same combination
of the flow’s horizontal gradients, but taken at different levels z,. In its turn
z,
is a
function of k and of the vertical structure
U
and
N.
Thus, the type of packet evolution
in the shear flows is determined not only by the flow’s horizontal gradients but by the
flow’s vertical structure and by the initial wave vector and position of the packet.
Second, we note that periodic regimes in shearless flows turn out to be structurally
unstable, when the presence of the shear
is
taken into account.
1’
S.
T.
Badulin
and
V.
I.
Shrira
1’
-c
k
I
/
--c
X
/
FIGURE
5.
Short internal-wave kinematics in the two-depths-of-localization model. Streamlines
(dashed-lines) are
(a)
elliptical over the two depths
or
(b)
elliptical at depth
I1
and hyperbolic at the
depth
I.
Short internal waves propagate dong the streamline of the first depth in the sectors
I
and
along the streamline of the second depth in the sectors
I1
(its trajectory is the solid line).
For
the wave
packet to come back to the same streamline it started from, a certain relation between the parameters
of
these ellipses
(or
ellipse and hyperbola) of regions
I
and
I1
should
be
strictly satisfied. This does
not
occur
in the general situation when
the
wave trajectories (in the x-plane
or
in the &-plane) are not
closed and the wave-packet parameters grow (decrease) parametricallly.
For
the model discussed,
sectors
I
and
I1
are fixed
for
the different wave vectors. Thus, the type
of
wave evolution (reversible
or
irreversible) is determined by the current geometry only. In the irreversible case when wavenumber
grows infinitely, the wave packet tends to the point
x
=
0,
y
=
0.
Then the local approximation
is
adequate for this model.
4.3.
Internal waves in the ocean with prevailing vertical non-uniformity
of
density
stratijcation
In the previous subsection we found out some general properties of the model
(4.1)
and
showed how to interpret them on the basis
of
concepts derived in
$3
for shearless flows.
We proceed now to study concrete models to distinguish characteristic regimes
of
short-wave transformation in certain ‘basic’ types of
flow
vertical structure.
We start with the problem which seems
to
be a natural continuation and extension
The irreversibility
of
internal-wave dynamics. Part
I
41
X
FIGURE
6.
Examples
of
short internal-wave trajectories (in the
x-
and &-planes) described by the
model
(4.1).
The trajectories may
(u)
fill sectors in the
x-
and &-planes,
or
(b)
propagate throughout
the whole angle range. Since the localization depth depends on the wave parameters (wave-vector and
wave-packet coordinates) these sectors and the type
of
wave evolution are not solely determined by
the mean-current field geometry. The trajectories in this model twist
or
untwist exponentially.
of the model
of
shearless flow (3.1). According to (4.2) the wave guide in the short-wave
limit is determined by the velocity shear structure.
On
the other hand, the vertical non-
uniformity of stratification usually dominates in the real ocean. That is why the
question about 'transient' regimes (from
one
dominating factor to another) is
of
interest.
4.3.1.
Wave kinematics
in
the model with constant vertical shear
Consider
a
particular case of (4.1)
:
u
=
Ay(1
+cL(z-ZJ),
v
=
Bx(1
+/?(z-ZJ),
N2
=
Nt(1
-+-y"*).
(4.7)
The constants
a,
/?
and
y
in
(4.7)
prescribe inverse vertical scales of the
flow
variability.
In
this case
a
wave guide exists owing to the presence of the maximum
N(z).
The
presence of mean shear only shifts its position. That is why we refer to the flow
(4.7)
as an example of
a
model with the role of density stratification prevailing.
42
S.
T.
Badulin and
V.
I.
Shrira
It is easy to obtain all the kinematic parameters in the short-wave approximation
using the results of the previous discussion
(see
Appendix
B).
The expression for
z,
demonstrates the main qualitative features of all shear flows mentioned above
:
dependence of the depth of localization on wave parameters. In the short-wave limit
The sign
of
zg
is determined by the initial wave parameters, namely by the sign of the
first integral of motion
S.
The leading term of the Hamiltonian
wo
depends on
S
in
a
similar way. This is also true for the next-order term of the Hamiltonian,
wl.
Thus we
can illustrate the properties of the system
(4.1)
generated by parametric dependence of
wo
on
S,
using the model
(4.7)
as the simplest example.
In the short-wave limit the problem of describing the wave kinematics reduces to the
case studied in detail in
g3.1
and
3.2.
The type of wave evolution is determined by the
flow
geometry at the corresponding depth. The value
prescribes the type of wave evolution in the same way as
A2
in the case of shearless flow.
The existence of the two different wave guides can generate some interesting
qualitative effects. Consider
flows
where
A:+
df
=
A2B2[(
1
--/3~,)~
-
2$y-?]
[(
1
-
O~Z~)'
-
2a2y-*]
<
0.
(4.10)
Then, wave evolution in one wave guide will be reversible, while in the other one it will
be irreversible. For an oceanic seasonal thermocline, typical vertical scales of current
variability often exceed those of stratification, i.e. in terms of the scales
a,
/3
and
y,
(a2+p2)
4
y2.
Under this condition the inequality
(4.10)
reduces
to
a realistic
assumption that one of the current velocity components is small at the pycnocline.
Then the wave propagating in one direction (with
S
positive) will be trapped, while the
wave with negative
S
will not. This can cause noticeable anisotropy of the wave field.
4.3.2.
Internal-wave dynamics
in
stratijied flows with constant vertical shear
Wave dynamics within the model
(4.7)
can be treated in a very similar way
to
the case
of
$3.2.
Non-trivial features
of
the dynamics are mainly caused by vertical
redistribution of the wave energy. Here we shall discuss qualitative features of the
internal-wave dynamics. Some formulae are given in Appendix
C.
The presence of the vertical shear of the mean current changes short-wave
asymptotes of the dispersion relation and eigenfunctions principally.
So,
these
asymptotes do not tend
to
those of
$3.2
when
a+O,
/3+0.
This
is an additional
illustration
of
the principal importance of taking the presence of even weak shear into
account for wave dynamics in the short-wave limit.
Infinite growth
of
the wave amplitude is caused both by
a
vanishing of the
Brunt-Vaisala frequency at the depth of localization and by
a
faster decrease of the
wave's vertical scale than in the shearless flows in
$3.2
(wave vertical focusing). The
simplest asymptotic procedure gives an expression like
(3.1
1
a,
b)
for caustics
:
2;
=
&
2/2/y. (4.8)
4*
=
M1
-/3(z2+2/2/Y)l[1-a(z,+2/2/Y)l
(4.9)
(4.1
1)
Here
f,
and
f,
are functions
($2)
of the initial wave parameters and mean flow
characteristics only, quite similar to
f,
and
f2
in
$3.2.
The expressions for
el,
e2
as
functions
of
a,
/3,
y
are too cumbersome to write here. It seems that only ordinary
caustics (at finite values of
t)
ciin exist in this case. As
t
--f
00
the wave-packet 'volume'
grows linearly in the
(x,
y)-plane and this causes a function-amplitude decrease.
The irreversibility
of
internal-wave dynamics. Part
1
/
/
/
/
1
43
FIGURE
7.
The refraction mechanism
of
internal waves which are trapped by the three-dimensional
current. While the depth of the internal-wave localization depends on the wave parameters (its wave
vector and frequency) different wave-packet spectral components are ‘glued’ to the mean current at
the different depths. Only dispersion
of
the wave energy, but not its focusing, may occur in this case.
The volume of the wave packet (phase volume in coordinate space) is shown by solid lines. This
volume is determined by the streamlines (dashed)
to
which the wave-packet’s harmonics are ‘glued’
and the dispersive diffusion
of
the packet.
Figure
7
gives a sketch of this effect. Each spectral component has its own limiting
streamline, at its own depth of localization.
As
the streamlines at the two neighbouring
depths diverge infinitely in our model, different harmonics ‘glued’ onto their own
streamlines diverge too. This can be considered as a shortcoming of the model
(4.7).
The ray family in figure
7
is shown in the case when the depth of localization is a
function of the wave parameters. Dotted lines are streamlines at different depths and
the bold line depicts the packet trajectory in the
(x,
y)-plane and the change of depth
of
its localization. Note that in the model under consideration the dependence of the
depth of localization
d(z,)
is in the second term
w1
of the Hamiltonian (see Appendix
B).
Hence the variation of d(z,)
is
a relatively weak effect
of
the same order as the
intrinsic motion
of
the wave packet.
Some
of
the properties of wave dynamics derived here are artifacts
of
the model.
Actually, the processes of interest are localized at the periphery of the pycnocline,
where
ZV+
0.
In typical ocean conditions with strong stratification the wave-amplitude
growth due to
z,
pushing out to the periphery of the pycnocline is obviously limited.
Thus, the expressions given in Appendix
C
have a clear physical meaning of
intermediate asymptotics and, therefore, we can use them for a qualitative description
only. The asymptotics
(C
4)
show the direction for further development of our models.
We conclude that vertical focusing
of
wave motion at a particular depth is the
principal determinant of the internal-wave dynamics in the case
of
arbitrary flow.
Other mechanisms, as we have seen, can dominate at the intermediate stage of
trapping. Thus we come to the necessity to consider models in which wave propagation
depends on the vertical inhomogeneity of the mean shear.
44
S.
T. Badulin and
V.
I.
Shrira
4.4. Internal waves in the ocean with dominant vertical mean flow inhomogeneity
The key result of the previous $4.3 is the principal influence of the variation
of
the
depth
of
localization on the dynamics of short internal waves. This influence may be
weak when a vertical inhomogeneity
of
density stratification prevails and it vanishes
when intrinsic motion
of
wave packet is neglected. The artifacts
of
the model discussed
above make it necessary to consider advanced models. With more realistic features of
the waveguide (for example, buoyancy frequency does not tend to zero) these models
can show new physical effect of fundamental importance
:
the variation of the depth of
localization occurs in the short-wave limit when the intrinsic motion of
a
wave packet
is neglected. Further, we shall discuss one
of
the models of this type:
U
=
Ay(1 +$X~(Z-Z~)~),
V
=
Bx(1 +$‘(z-z~)~),
N
=
No
=
const. (4.12)
Here constants
A,
B,
(al,
IpI
have a similar sense as in (4.7)
(a2,
p2
can be negative). For
the depth of localization we get
aZAaz, +p2Bbz2.
a2Aa
+
$Bb
’
z,
=
(4.13)
z,
is the key parameter for our problem. We shall concentrate our further discussion
on this characteristic. Expressions for the other characteristics are given in Appendix
D.
4.4.1. The behaviour
of
the depth
of
localization
The specific feature
of
the model (4.12)’ unlike the model (4.7), is that the depth of
localization’s dependence on the wave parameters appears in the short-wave limit. In
our model d(z,) depends on the two variables
a
and b only. The curves w(a, b)
=
const
which determine the variation of the depth
of
localization are the second-order curves
and, hence, the system can easily be analysed. We shall confine our analysis to
qualitative illustrations.
The type of the curves w(a, b)
=
const depends on the sign of
Aib
=:
a’p
-
:(a2
+
p2
+
h2)
exclusively (i.e. on the vertical inhomogeneity parameters only). From the evident
inequality
(kx-
1~)~
2
0
in the (a, b)-plane we have
S2
2
4ab. (4.14)
The inequality (4.14) gives us ‘transparency zones
’
for a harmonic with fixed
S.
A
wave
packet moves in the
(a,
6)-plane dong the segments of the curve w,(a, b)
=
const
bounded by the points
of
intersections of this curve with the hyperbola
S2
=
4ab. The
possible trajectories in the (a,b)-plane are shown in figure
8.
We distinguish two
different regimes depending on the behaviour of z,.
The finite region of motion in the (a, b)-plane corresponds
to
the periodic evolution
of
z,.
When
AEb
c
0
(figure 8a, b) the motion is ‘finite’ regardless of the initial wave
parameters
(S,
a,, b,,). The case
of
figure
8
(c)
is a special one as a wave can move along
two different segments of the curve, depending on
a,
and b, (but not
S).
We distinguish
this case because there are
no
similar transformation regimes in the model (4.7), where
transformation
is
determined by parametric dependence on
S
only.
We note that
z,
tends to infinity at the point
a
=
0,
b
=
0.
This fact may be
considered
as
a shortcoming of the model (4.7) and interpreted as follows. In the model
The irreversibility
of
internal- wave dynamics.
Part
I
45
FIGURE
8. Short internal-wave trajectories in the model
(4.1)
in the (a,b)-plane depending on the
mean-current field geometry (values
of
constant
A,
B,
N,
a,fl
and wave-packet parameters (value
of
S).
The hatched region is the non-transparency region where
S3
<
4ab
(i.e.
S2
<
4k010x,y0).
When
o(a,
b)
=
const is an ellipse
(a,
b,
c)
a(t)
and
b(t)
vary periodically with time (this does not mean that
x(t)
or
k(t)
are periodic functions). When
w(a,
b)
=
const
is
a hyperbola, the situation when
la1
and
lbl
grow infinitely is also possible
(see
e).
It is easy to show that intersections
of
the curves
w(a,
b)
=
const and
S2
=
4ab
give the simple reflection point where the reflection lasts
a
finite interval
of
time. Points where
@(a,
6)
=
const is tangent
to
S2
=
4ub
are reached asymptotically
in
an infinite
time,
V).
(4.12)
a particular ‘degenerate’ direction exists where there is no extremum of the
current velocity which determines the short-wave propagation. It can also occur in the
more general situation
(4.1)
as well and more than
a
single direction
of
‘degeneracy’
may exist. In these cases wave propagation is determined by the density field guide or
by density and current values at the ocean surface or bottom. In our model we have
not taken into account these bounds on
z,.
It does not appear to be of great
importance as it can be shown that the wave achieves these critical points within a finite
time. Moreover, trajectories exist which do not intersect the point
a
=
0,
b
=
0
(figure
8
c).
When
A$
>
0
the evolution depends significantly on the initial wave parameters.
Besides the case of figure
8(e)
in which
o(a,
b)
=
const and
S2
=
4ab
have
no
common
points, the situation exists when both finite and infinite regimes are possible (figure
8.3
depending on the initial values of
a,,
b,.
We stress that all the criteria which determine whether a regime of transformation
occurs are given by inequalities in terms
of
the initial wave parameters (by the number
of
points of intersections
of
w,(a,
6)
=
const,
S2
=
4ab).
This means that effects related
to the type
of
trajectory in the
(a,
b)-plane are structurally stable. In other words, the
neighbour harmonics
of
the wave packet will evolve in a similar way. It should be
46
S.
T.
Badulin
and
V.
I.
Shrira
stressed in connection with the case presented in figure
8
(f),
that it takes infinite time
for the packet
to
achieve the point
of
tangency of
S2
=
4ab and
w,,
=
const. One should
consider the situation of figure
80’)
as structurally unstable.
4.4.2. Internal-wave kinematics in ihe model
of
dominant mean shear vertical
inhomogeneity
The qualitative analysis of the behaviour
of
the depth of localization
z,
presented
above can easily be combined with the general properties of the model (4.1) which were
considered in $4.2. This synthesis allows one to analyse the internal-wave kinematics
in original variables and to preview the possible types
of
wave evolution. The case of
the monotonic variation of
a
and
b
and, hence, relatively simple variation of
z,
is of
less interest as it does not differ principally from the case of the model of g3.2.
The most interesting dynamic regime occurs when
z,
is changing periodically with
time. The terms
A,
are changing periodically as well. But the mean (over period) value
Ji‘+*didf does not equal zero in general case. Hence there are no structurally stable
periodic motions in original variables for the model under consideration.
So,
this
model demonstrates fairly well one of the properties
of
the internal-wave dynamics in
the general model
(4.1).
Figure
7
shows sketches of the different possible types of internal-wave trajectory for
the general model (4.1). The same picture holds for the model (4.12). The wave
trajectories are more intricate than in $3. The reasons for reversibility of the motion in
terms of changes in the angle variables are evident
:
the current at the different depths
has different directions and the wave ‘feels’ it through changes in
z,.
Two principally different types
of
wave evolution can be distinguished depending on
the sign of
A,dt
<
0
(i= 1,2,3,4).
I:w
In the first one the wave trajectory twists in the (k,I)-plane and untwists in the
(x,
y)-plane (perhaps in limited range of angles). When our local analysis breaks down
the WKB-approximation will become invalid as well while
(kl+
0.
In the second case all our approximations remain valid while
JkJ
+
co
and
1x1
-to.
The wave packet ‘falls’ into the point
x
=
0.
It is easy to take into consideration
intrinsic wave motion and to show the validity of local analysis in this case as well. We
shall not dwell upon this here. Whik there is a range (in
k,
x)-space)
of
the wave initial
conditions, where this ‘falling’ into the centre of eddy (4.1) takes place, we can say that
a certain share of the internal-wave-field energy will be localized in the immediate
vicinity of the eddy axis when
t
is sufficiently large.
Evidently, both horizontal and vertical inhomogeneities are taking part in the
complication (relating to the case of $3) of trajectories. Thus, there is no sense in
interrelations of elliptic trajectories in the model
of
$3 and spiral trajectories
of
the
model under discussion, and new features of the short-wave evolution discussed above
prompt us to introduce a new concept of ‘spectral-spatial focusing’ of an internal wave.
4.5.
On
the spectral-spatial focusing
of
internal waves due to trapping
We distinguish these two concepts (spectral and spatial focusing) on the basis of
concepts that were introduced in
$3.
When
z,
=
const, different spectral components
are generally trapped at different locations owing to the dispersive effects taking place.
However, different trajectories with a fixed wave vector are focused along some curves,
that is spatial focusing takes place.
The irreversibility
of
internal-wave dynamics. Part
1
47
To understand the spectral focusing mechanism we shall return to the analysis of
$3.3.
We have noted that the effect of spatial focusing of different harmonics at the
same point is not the consequence of the ‘degenerate’ model only. We have obtained
the same phenomenon in a non-solenoidal current velocity field in the two-layer-of-
localization model of
$4.2.
This note allows
us
to sketch the following model for this
mechanism.
While internal waves in the short-wave
limit
feel an ‘effective’ velocity field
depending on
z,
only, we can simplify our problem by reducing it to this two-
dimensional velocity field. What are the specific features of
this
field? Let
us
calculate
its derivatives, taking into account that
z,
depends on
x:
For the divergence in the model
(4.12)
we have
(4.15)
By integrating
(4.15)
over a range of angles we can obtain the total
flux
for the waves
of
fixed intrinsic frequency from this angle range:
*
sin
cp
cos
cp
(sin
cp
+
cos
cp)
(a2Acoscp+$Bsincp)3
v:.
U,,,
=
n-’A4B2a
(4.16)
We shall specify
cpl,cpz
below.
Taking
cpl
=
0,
cp2
=
2n
we obtain an expression containing a singularity. The
principal value of
(4.16)
does not equal zero in the general case. To avoid this
integration and to make our model more realistic we shall confine the depth range and,
thus, the integration range. Assume the fluid surface to be at
z
=
0
and the bottom at
z
=
H.
The depth
z,,
consequently, varies from
0
to
H.
When
z,
=
0
or
z,
=
H
and
the velocity field is solenoidal, and then the divergence of the effective velocity field is
zero in a certain range of angles. But the integral divergence
(4.16),
generally, does not
equal zero. Its sign may be different and different types of wave evolution can occur:
divergence or convergence
of
the wave trajectories such as in
g4.2
and
4.4
takes place.
We note that the divergence integral
(4.16)
does not depend on
lkl
as we take the
short-wave limit and allow vertical non-uniformity of the velocity to dominate
absolutely. By taking into account density stratification it is easy to obtain the
dependence of the divergence on
lkl
even in the short-wave approximation. Then it can
be shown that the efficiency of spectral and space focusing increases as
Ikl
increases.
As we have seen, a good understanding of these phenomena and, sometimes, some
evaluations of its efficiency can
be
achieved by analysis of the two-layer-of-localization
model. In such a model the divergence of the effective velocity field depends on the
velocity discontinuity across the rays dividing the regions of different depths of
localization.
4.6.
Discussion
All
the
models considered above are particular cases of a more general model
(4.1).
The
analysis of these particular cases was necessary in order to introduce some new
concept, which are helpful for understanding the general problem.
We have established the dominating role of vertical inhomogeneity of the current in
the kinematics and dynamics of internal waves. We have introduced and advanced the
48
S.
T.
Badulin and
V.
I.
Shrira
arctan
Ra
FIGURE
9.
The analogy between the wave packet evolution (within the model
(4.1))
and the classical
particle motion in a potential field along a funnel-like surface. There are two first integrals
of
this
motion
:
the energy of the particle (the sum of kinetic and potential energies) which is similar to the
wave frequency
o,
and the kinetic impulse projection which is like the integral
S
for an internal wave
packet. The particle which falls into thi.s ‘funnel’
is
accelerated .and tends infinitely to the point
x
=
0
like the internal-wave quasi-particle.
new concept of ‘the depth of localization’
z,,
depending on both the hydrophysical
mean fields and the wave kinematical parameters (wave vectors). The depth of
localization is a good physical concept which illustrates the transformation of
a
real
three-dimensional velocity field into some family of two-dimensional ‘effective velocity
fields’. Each element of this family
is
determined by the initial wave parameters
because of the dependence of
z,
on
k.
Thus, the concept enables us to reformulate (to
some extent) our original three-dimensional problem in terms of well-known problems
of two-dimensional hydrodynamics. The specific features of this new problem lie in the
dependence of the type of effective velocity field on the spatial variables, and in the
parametric dependence of this field on the initial wave parameters. Even when the real
velocity field is smooth and solenoidal, the effective current appears non-solenoidal
and singular. The effective velocity singularitik act similarly
to
the point sources and
sinks in hydrodynamics.
We now elaborate on the above-mentioned analogy with the motion of classical
particles in
a
potential field
of
a
special type (figure
9).
Consider a particle moving on
a funnel-like surface in a common gravitational field. The vertical component
of
its
kinetic momentum is conserved (an analogue of our first integral
S).
The particle has
a radial component of velocity and its angle varies periodically with time (an analogue
The irreversibility
of
internal-wave dynamics. Part
1
49
of angle variable
RJ.
The depth of the particle (or its momentum
Qt)
grows infinitely
and the particle falls infinitely along the spiral-like trajectory. Variables
a
and
b
are
kinetic momentum components, which vary periodically with time under gravitational
force (an analogue of gyroscope precession).
This analogy gives us a new view on the internal-wave dynamics problem. The
existence of
‘
funnels
’
for internal-wave-packet particles means specific irreversibility in
a conservative system. We have got ‘black holes’, well-known in astrophysics, in a
classical
‘
Internal-Wave Universe’. The singular surface of an infinite funnel
corresponds to the singular effective velocity field in our problem. We stress that these
singularities or ‘holes’ occur in generic situations with smooth ambient density and
velocity fields and therefore the ‘effective’ mean flows should have a multi-hole-like
structure.
5.
Discussion and conclusions
A part
of
our programme, whose final goal is to find out the role of large-scale
inhomogeneities of hydrophysical ocean fields in guided internal-wave dynamics has
been realized in this paper. We have identified and investigated some generic
mechanisms governing the evolution
of
internal waves in an inhomogeneous ocean,
and some general tendencies of internal-wave evolution. This has been done using
comparatively simple models.
A set of models which corresponds to the different types of vertical structure on the
mean flows has been considered. It has been shown that different scenarios can occur
depending on the type of inhomogeneity.
The main conclusion
of
our investigations is that
:
irreversible dynamical evolution
of internal waves into small-scale horizontal and vertical ranges inevitably takes place
under some very general assumptions about the structure of the ocean’s hydrophysical
fields and over
a
wide range of internal-wave parameters.
Moreover, the most intense internal-wave transformation by currents in many cases
appears to occur mainly in horizontally and vertically strongly localized regions
(because of wave packet ‘gluing’ to the mean current and the non-dispersive focusing
of internal waves as mentioned above). This conclusion has many physical
implications; for example, it permits us not only to simplify the mathematics
of
the
problem (by exploiting the analogy with particle motion in a simple potential field), but
as
the problem reveals a conceptual resemblance with well-known problems of physics,
these might help one in determining strategic directions of future investigation.
The main question for oceanic internal-wave investigators still remains the one
formulated by Briscoe
(1975):
‘where does the internal wave energy come from, where
does it go, and what happens to it along the way?’ In this paper we have attempted to
answer part of this question
:
what
is
the role of the internal-wave evolution tendencies
discovered here in the internal-wave degeneration processes in the ocean? This
question has an essential resemblance with the well-known astrophysical problem
of
‘black holes’ as the ‘substance devourer’.
Another analogy is the hydrodynamical one, which has been introduced on the basis
of
the model
(4.1).
The singularity
of
the effective velocity field connected with the
phenomenon
of
trapping is similar to a well-known problem of hydrodynamics
-
the
problem of point sources and sinks. Hence, a straightforward idea for further
investigation is to construct these sources and sinks by transforming generic current
and density fields, which internal wave kinematics depends on, into an effective current
field
V,,,.
As the specific features of this transformation are investigated, the
50
S.
T.
Badulin
and
V.
I.
Shrira
hydrodynamical analogue should help us to develop our analysis for more general
problems.
Another natural direction for further investigation is the extension to a higher order
of
approximation of the analysis of the ray equations
(2.2),
to consider the intrinsic
motion of the internal wave packet.
As
wave-packet 'gluing' to the mean flow takes
place it allows one to change to the! transformed curvilinear coordinate system moving
with the mean flow. This promises to be useful both for the analysis of local situations
discussed above and for developing the 'non-local' analysis of trapping.
This is an outline of the most promising, from our point
of
view, avenues
of
further
investigation
of
the effect of trapping within the same paradigm.
Our investigation has provided sufficient grounds to suppose the existence
of
a
certain strong tendency for internal-wave-field evolution into the small-scale range.
Now we certainly cannot specify quantitatively the importance of this tendency in
internal-wave dynamics in the ocean.
To
answer this question we must come out
of
the
framework of our models and, first
of
all, take nonlinearity into account. Work in this
direction is in progress now.
The authors are grateful to
G.
Watson for his valuable comments on the first draft
of the work. The hospitality of Royal Netherlands Meteorological Institute at De Bilt
and l'lnstitut Mecanique Statistiqiie de la Turbulence at Marseille (Luminy), where
part of this work was done by
ow:
of the authors
(V.
I.
Shrira), is appreciated.
Appendix A
canonical momenta
:
There are four basic types of canonical transformations with
S
and
a
or
b
as
(A
1)
S,
=
kx
+
ly,
Q,
=
-In (klp),
El
=
a
==
ky,
R,
=
-Ilk;
S,
=
kx
+
ly,
Q,
=
-In
(l/p),
1
E
2
=
b
==
lx,
R,
=-k/l;
s3
=
kx
+
6,
Q3
=
In
(YIP),
1
E3=a==
ky, R3=x/y;
S,
=
kx+ly,
Q,
=
In(xp),
1
E
4
=
b
==
Ix,
R4
=
y/x.
J
Here
p
is
a
constant chosen to make
yp,
klp,
xp,
l/p
dimensionless,
S,,
E,
(i
=
1,2,
3,4)
are new canonical momenta,
Q,,
R,
are canonically conjugated coordinates.
Appendix
B
be found from
(4.2):
The explicit expression for the depth
of
localization
z,
in the model
(4.7)
can easily
2(aAa+pBb)
y(w
-
Aa
-
Bb
+
a,
Aaz
+
p,
Bbz)
.
(B
1)
z,
=
___
The dispersion relation is obtained from
(2.10)
in the form
The irreversibility
of
internal- wave dynamics. Part
1
51
where G
=
w-Aa-Bb+aAaz,+/3Bbz2, g
=
1/2(aAa+bBb)/y.
powers of
lkl:
The explicit expression for
w
can be presented in the form of an expansion in inverse
w
=
@,(a, b)
+
q(k,
x,
1,
y)
+
o(lkl-i),
wo
=
Aa+ Bb-aAaz,-BBbz,f 1/2(aAa+BBb)/y,
(2n
+
1);
N2yo(aAa
+
BBb)
4
w1
=
&
2
[
1/2(k2+f)
I-
Using
(B
2)
we get the short-wave asymptotes
of
(B
1):
Appendix
C
velocity
A
(amplitude) can be written in the
form
According
to
definition
(2.12)
the maximal-over-depth value of the wave vertical
A2
-
[w-k.
U(z,)13/[N2(~,)d(~,)1,
(C 1)
where
[w-k-
U(z,)I3
x
G3(l -g2/G2)3,
N2(z,)
x
Nt(1
-gZ/G2), (C
2)
Quite similar to
(3.6)
we get for the case
A:
>
0
the following asymptotes:
k-exp(A,t); I-exp(A,t); x-exp(A,t); y-exp(d,t).
Then the parameters in (C
2),
(C
3)
can also be expressed in terms of
t:
[w
-
k.
U(z,)I3
-
const,
N2(z,)
-
exp
(-
24,
t),
d(z,)
-
exp
(-
A,
t),
A2
-
exp
(361)
@(x,
Y)/a(E,
or1.
(C
4)
In particular, the equality (C
1)
can be rewritten
A2
=
const
N(z,)
-
exp
(A,
t)
const
(C
5)
Appendix
D
Consider the model
U
=
Ay(
1
+
!#(z
-
zJ2),
V
=
Bx(
1
+
L#(z
-
z$),
N
=
No
=
const.
(D
1)
have
a
similar sense as in
(4.7)
(a2,
p2
can be negative). For the
(D
2)
(D
3)
Constants
A,
B,
lal,
horizon of localization we get
z,
=
(a2Aazl +$Bbzz)/(a2Aa
+
$Bb).
N;-G:+~,
G,
=
((2n+ 1)/1/2)(2~;g,+g;
G,);G~
While dispersion relation
is
52
where
S.
T.
.Badulin
and
V.
I.
Shrira
g,
=
(a2Aa
+
p2Bb)
(k2
+
P)-'?
G,
=
w
-
Aa
-
Bb
-
ABubh2(a2Aa+
P2Bb)-',
From
(D
3)
we obtain
w
=
wo(a,
b)
+
w,i:k,
x)
+
o(lkl-;),
w,,
=
No
+
Ac!
+
Bb
+
ABubh2(a2Aa
+
TBb)-',
(D
4)
We emphasize that
w1
tends to zero as
Ikl-4,
while in the previous case
w1
-
1kl-i.
Short-
wave asymptotic
(D
4)
corresponds to the structurally stable case (Borovikov
&
Levchenko
1987)
in contrast to the intermediate asymptotics
of
the model
(4.7).
w,
=
f
(n
+
i)
Ni(a2Aa
+
$Bb)t
(k2
+
P)-!
REFERENCES
ABRAMOVITZ,
M.
&
STEGUN,
I.
A.
(eds) 1964
Handbook
of
Mathematical Functions.
US Govt.
Printing
Office.
BADULIN,
S.
I.
&
SHRIRA, V. I. 1985 The trapping of internal waves by horizontally inhomogeneous
currents of arbitrary geometry.
Izv.
Akad. Nauk
SSSR,
Fiz. Atmos. Okeana
21,
982-992.
BADULIN,
S.
I.
&
SHRIRA,
V.
I.
1993
On
the irreversibility of inertial-wave dynamics due to wave
trapping by mean flow inhomogeneities. Part 2.
BADULIN,
S.
I.,
SHRIRA,
V.
I.
&
TSIMRING, L. SH. 1985 The trapping and vertical focusing of internal
waves in
a
pycnocline due to the horizontal inhomogeneities of density and currents.
J.
Fluid
Mech.
158,
199-218 (referred to herein as BST2).
BADULIN,
S.
I., TSIMRING,
L.
SH.
&
SHRIRA, V. I, 1983 On the trapping and vertical focusing of
internal waves in a pycnocline due
to
the horizontal inhomogeneities of stratification and
current.
Dokl. Acad. Nauk SSSR
273,
459-463 (referred to herein as BSTl).
BADULIN,
S.
I., VASILENKO, V. M.
&
GOLENKO,
N. N. 1990 Transformation of internal waves in the
equatorial Lomonosov current.
Izv.
Akad. Nauk
SSSR,
Fiz.
Atmos.
Okeana
26,
279-289.
BASOVICH, A. YA.
&
TSIMRING,
L.
SH. 1984 Internal waves in horizontally inhomogeneous flows.
J.
Fluid Mech.
142,
233-249.
BRISCOE,
M.
G. 1975 Introduction to collection of papers on oceanic internal waves.
J.
Geophys.
Res.
80,
289-290.
BOOKER, J. R.
&
BREMERTON,
F.
P.
1967 The critical layer for internal gravity waves in
a
shear flow.
J.
Fluid Mech.
27,
513-539.
BOROVIKOV,
V.
A.
1988 Critical layer formation in a stratified fluid with mean currents.
Reprint
N309. Instut Problem Mehaniki
AN
SSSR, Moscow 58p (in Russian).
BOROVIKOV, V. A.
&
LEVCHENKO, E.
S.
1987 The Green's functions
of
the internal wave equation
in the layer of stratified fluid with average shear flows.
Morsk. Gidrofiz.
J.
1,
24-32.
BROUTMAN,
D.
1986 On internal wave caustics.
J.
Phys. Oceanogr.
16,
1625-1635.
BROUTMAN,
D.
&
GRIMSHAW, R. 1988 The energetics
of
the interaction between short small-
amplitude internal waves.
J.
Ffuid M'ech.
196,
93-106.
BUNIMOVICH, L.
A.
&
ZHMUR,
V. V. 1986 Internal waves scattering in a horizontally inhomogeneous
ocean.
Dokl. Acad. Nauk
SSSR
286,
197-200.
CRAIK,
A.
D.
1989 The stability of unbounded two- and three-dimensional flows subject to body
forces: some exact solutions.
J.
Fluid Mech.
198,
275-292.
CRAIK, A.
D.
&
CRIMINALE,
W.
0.
1986 Iholution of wave-like disturbances in shear flows: A class
of exact solutions of the Navier-Stokes equations.
Proc.
R.
SOC.
Lond.
A
406,
13-26.
EROKHIN, N.
S.
&
SAGDEEV, R.
Z.
1985a On the theory of anomalous focusing of internal waves in
a two-dimensional nonuniform fluid. Part
1.
A stationary problem.
Morsk. Gidrofiz.
J.
2,
15-27.
EROKHIN,
N.
S.
&
SAGDEEV,
R.
Z.
1985b On the theory of anomalous focusing of internal waves in
a horizontally inhomogeneous fluid. Part 2. Precise solution of the two-dimensional problem
with regard to viscosity nonstationanty.
Morsk. Gidrofiz.
J.
4,
3-10.
JONES,
W.
L.
1969 Ray tracing for internal gravity waves.
J.
Geophys.
Res.
74,
2028-2033.
The irreversibility
of
internal- wave dynamics. Part
I
53
LEBLOND,
P.
H.
&
MYSAK, L.
A.
1979
Waves in the Ocean.
Elsevier.
LEW,
M.
D.
1983 Internal waves in the ocean:
a
review.
Rev. Geophys. Space Phys.
21,1206-1216.
MCCOMAS, C.
H.
&
BRETHERTON,
F.
D.
1977 Resonant interactions
of
the oceanic internal wave
MCCOMAS,
C.
H.
&
MULLER,
P.
1981 The dynamic balance
of
internal waves.
J.
Phys. Oceanogr.
11,
MIROPOLSKY,
Yu.
Z.
1981
Dynamics
of
Internal Gravity Waves in the Ocean.
Leningrad:
OLBERS,
D.
1981 The propagation
of
internal waves in a geostrophic current.
J.
Phys. Oceanogr.
11,
OLBERS,
D.
J.
1983 Models
of
the oceanic internal wave field.
Rev. Geophys. Space Phys.
21,
RAEVSKY,
M.
A.
1983 On the propagation
of
gravity waves in randomly inhomogeneous currents.
VORONOVICH,
A.
G. 1976 The propagation
of
surface and internal gravity waves in the geometric
WATSON,
K.
M. 1985 Interaction between internal waves and mesoscale flow.
J.
Phys.
Oceanogr.
15,
field.
J.
Phys. Oceanogr.
7,
836845.
970-986.
Gidrometeoizdat.
1224-1 230.
1
567- 1606.
Izv. Acad. Nauk SSSR. Fiz. Atmos. Okeana
19, 639-645.
optics approach.
Izv. Akad. Nauk SSSR,
Fiz.
Atmos. Okeana
12, 850-857.
12961311.
