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2511

A Growth Mechanism for Topographic Internal Waves Generated by an
Oscillatory Flow
TOMOHIRO NAKAMURA AND TOSHIYUKI AWAJI
Department of Geophysics, Graduate School of Science, Kyoto University, Kyoto, Japan
8 December 1999 and 26 February 2001
ABSTRACTOn the basis of ray tracing of individual waves generated at various phases of the tidal flow, an amplification
mechanism is presented for a new class of topographically generated internal waves identified by Nakamura etal., which develop across a broad latitude range and can exist even above the critical latitude where the tidal
frequency equals the inertial frequency. The results show that unsteady lee waves are always amplified whenthe maximum frequency is much smaller than the buoyancy frequency because their phase speeds (amplitudes)
are equal (proportional) to the tidal flow speed at their time of generation. Fast mixed tidal–lee waves are also
effectively amplified, when the rotation effect is significant. Accordingly, amplification of unsteady lee and fast
mixed tidal–lee waves can occur even if the requirements of previous theories (e.g., the critical slope and critical
Froude number conditions) are not satisfied. Since the result here covers the generation and amplification
processes of topographic internal waves across a broader parameter range than earlier theories, it should contributeto a better understanding of boundary mixing processes, especially in high-latitude regions.

1. Introduction
The generation of large-amplitude internal waves by
tidal currents is a subject of increasing research interest
because recent studies reveal that vertical mixing effects
caused by the breaking of tidally generated internal
waves at open-ocean boundaries—the so-called boundary mixing problem (e.g., Lukas et al. 1996)—can significantly affect large-scale processes in the open ocean.
The interactions between waters of the Okhotsk Sea andthose of the North Pacific Ocean via the Kuril Straits,
for example, can be considered to be a typical boundary
mixing problem. Several studies (Talley 1991; Kono andKawasaki 1997) show that sill-related vertical mixing
in the Kuril Straits is an important factor in the production of the North Pacific Intermediate Water and suggest that interactions between swift tidal currents andsill topography at the Kurils is a likely cause for theintense vertical mixing. In order to clarify this mechanism, Nakamura et al. (2000, hereafter NA) recently
performed numerical simulations of tidally generated
internal waves over a representative sill in the Kuril
Straits (;478N; Fig. 1), using a two-dimensional nonhydrostatic f -plane model to distinguish the physics ofCorresponding author address: Dr. Tomohiro Nakamura, Dept. ofGeophysics, Graduate School of Science, Kyoto University, Kyoto
606-8502, Japan.
E-mail: nakamura@kugi.kyoto-u.ac.jp

q 2001 American Meteorological Society

the waves generated for both subinertial K1 and superinertial M 2 cases. Their results showed a number ofinteresting phenomena that are difficult to explain on
the basis of present internal wave theories.
The formation of large-amplitude internal waves bya tidal flow is normally considered to involve the following three stages. First, tide–topography interactions
generate internal waves at the tidal frequency (internal
tides), and the waves are considerably amplified if theslope of their characteristics is equal to that of the topography (the ‘‘critical slope’’ condition: Wunsch 1969;
Baines 1982; Craig 1987; Sherwin 1988; Sherwin andTaylor 1990; reviews by Wunsch 1975; Huthnance1989). The second stage is the amplification of internal
tides at their generation regions. This process is thought
to be most important and was investigated by Hibiya(1986, hereafter HI), using the modal wave approximation. He showed that when a barotropic tidal flow is
critical [i.e., the internal Froude number, defined as theratio of the barotropic flow speed U(t) to phase velocityof the nth mode c n , F n 5 U(t)/c n , is unity], internal tides
propagating upstream are trapped within the generation
region and amplified to form large-amplitude internal
waves. When the tidal flow falls below the critical
Froude number, the waves begin to propagate upstream
away from the generation region. The third stage represents the nonlinear evolution of the large-amplitudeinternal waves as they propagate away from the gen-

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FIG. 1. The characteristics of the M 2 internal tides in the model of
NA. The slope of the right-hand side of the sill is supercritical.

eration region. Investigations using the KdV equation
or its alternatives showed that when the wave amplitude
is sufficiently large, internal tides grow and disintegrate
into large-amplitude solitary wave trains (e.g., Lee andBeardsley 1974; Mazé 1987; Willmott and Edwards
1987; Smyth and Holloway 1988; Gerkema and Zimmerman 1995). Generally, this process follows the second stage, after the amplified waves have propagated acertain distance from the generation region (Liu et al.
1985).
In the superinertial M 2 case of NA, short internal
waves generated around the sill break were trapped andamplified within the generation region. As shown in Fig.
1, the right-hand slope in such a situation was significantly steeper than that of the characteristics of the M 2internal tides, given by [(s 2f 2 f 2 )/(N 2 2 s 2f )]1/2 whereN, f , and s f are the buoyancy frequency, the Coriolis
parameter, and the tidal frequency, respectively. This
blocks leftward propagation of the internal tides generated on the right-hand slope. Thus, the growth of theshort wave is not consistent with the critical slope theory. Assuming the conventional modal wave theory near
the sill top for convenience, we estimated the contribution rates and Froude numbers for the lowest seven

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modes at the right-hand break (see appendix A). Table
1 shows that the major vertical modes were the lowest
three modes, but the maximum values of F1 , F 2 , andF 3 were 0.33, 0.51, and 0.87, respectively. This suggeststhat the wave trapping and amplification in the M 2 caseof NA is out of the range of the HI model.
Observations by Kuroda and Mitsudera (1995) captured similar phenomena in the shelfbreak region of theEast China Sea, in which the first mode wave wastrapped and amplified, eventually leading to a hydrauliclike jump in the isopycnal surfaces when F1 5 0.71.
They interpreted the wave trapping with a steady hydraulic control theory, taking into consideration the effect of the decrease in wave speed due to advection atlarge amplitudes (Grimshaw and Smyth 1986). However, their theoretical model does not effectively explain
the wave trapping in the NA case in which the Froudenumber is considerably less than unity (for example, F1; 0.3). In addition, the isopycnal displacement of theshort wave in NA is about 20 m, which in a depth of
550 m is too small to cause the wave speed decreases
reported in Kuroda and Mitsudera (1995).
The other important result presented in NA was that
the K1 flow over the sill generates large-amplitude internal waves that propagate away from the sill, eventhough the K1 tide is subinertial at the Kuril Straits (seeFigs. 6 and 7 of NA). This posed the question as to how
these waves come about since subinertial internal tides
are topographically trapped waves.
From theoretical considerations, NA revealed the nature of internal waves generated by a tidal flow using
the new nondimensional parameter kU 0 /s f , where k is
the horizontal wavenumber and U 0 is the tidal flow amplitude. As is summarized in Table 1, in the case of kU 0 /s f k 1, the generated waves are characterized as unsteady lee waves with intrinsic frequencies of 2kU(t),and amplitudes depending on U(t)h x (h x is the slope ofthe bottom topography). Thus, all wave properties except for | k | are defined at the generation time. In contrast, when kU 0 /s f K 1, internal tides are generated,
whose frequency is almost constant and is identical to
the tidal frequency in the limit kU 0 /s f → 0. Internal

TABLE 1. The modal decomposition result of vertical velocity for the M 2 case after 1/8, 1/4, 3/8, and 1/2 period. In order to focus on the
short internal wave over the right-hand sill break, velocity data at the break are analyzed here. C.T.: contribution rate, R.M.: relative magnitude,
and Fr: Froude number.
1/8 tidal period

2/8 tidal period

3/8 tidal period

4/8 tidal period

Mode
no.

C.R.
(%)

R.M.
(1021)

Fr

C.R.
(%)

R.M.
(1021)

Fr

C.R.
(%)

R.M.
(1021)

Fr

C.R.
(%)

R.M.
(1021)

Fr

1
2
3
4
5
6
7
sum

48.76
40.94
6.27
2.84
1.15
0.66
0.47
101.08

21.55
2.01
20.71
0.52
20.29
0.25
20.20

0.23
0.37
0.61
0.80
0.99
1.19
1.39

21.72
54.85
12.48
7.00
2.42
1.33
0.81
100.60

21.69
3.81
21.63
1.33
20.69
0.58
20.43

0.33
0.51
0.87
1.13
1.40
1.69
1.97

8.33
29.73
29.92
21.25
6.12
2.63
1.28
99.26

0.75
2.01
21.81
1.65
20.78
0.58
20.38

0.23
0.37
0.61
0.80
0.99
1.19
1.39

21.88
69.76
0.01
4.51
3.33
1.12
0.33
100.92

1.38
23.50
0.03
0.87
20.66
0.43
20.22

0.00
0.00
0.00
0.00
0.00
0.00
0.00

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2513

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tides propagate in both directions, as previous studies
indicated. In the intermediate range of kU 0 /s f ; 1,
waves with properties intermediate between those of lee
waves and internal tides (and are named ‘‘mixed tidallee waves’’ by NA, hereafter MTL waves) are generated,with intrinsic frequencies of 2kU(t) 6 s f . Thus, theMTL wave splits into two components propagating withdifferent phase velocities.
The nature of the unsteady lee and MTL waves not
only allows us to understand the generation mechanismof free-propagating internal waves by a subinertial flow
(as described above) but can also account for their trapping at sills, which leads to significant wave amplification. The intrinsic horizontal phase speeds of the internal tides (6s f /k) are constant. However, those of theunsteady lee waves and MTL waves [2U(t) and 2U(t)6 s f /k, respectively] vary with the phase of the tidal
flow at their generation time. This affects the way inwhich the individual wave components of these waves
superpose. For example, the phase speeds of unsteadylee waves are equal to the basic flow speed at theirgeneration time for all current speeds (as long as f ,| kU 6 s f | , N ), suggesting a tendency to be trappedand superposed in the generation region. A similar propagation feature was found in numerical experiments onan oscillatory flow over the shelf slope with small
length-scale ripples by Thorpe (1996), who suggested
that the propagation time of internal motions from thegeneration region to the shelf break depends on a parameter kU(t)/s f , though his theory (Thorpe 1992, 1996)
assumed a steady flow condition. However, such effects
have not been investigated in any depth in previous
theories, though it is clear that the dynamics of lee
waves, and in particular their transient propagation processes, should be incorporated into any representative
account of wave growth.
Lott and Teitelbaum (1993a,b) studied atmosphericlee waves using a similar approach to NA. Though theydescribed many interesting features of unsteady lee
waves, the effect of the time variation of forcing on
wave excitation was neglected and, hence, the shortwave growth in an oscillatory flow regime lies outside
the scope of their work. Also, Bell (1975) considereda situation similar to NA, and Bannon and Zehnder(1985) added a steady flow component to Bell’s model.
However, since internal waves in these studies are treated as an infinite series of modes in the temporal domain,
the characteristic features of unsteady lee waves and
MTL waves were not clearly described.
In this study, we investigate the growth mechanismof topographic internal waves generated by an oscillatory flow, focusing specifically on the unsteady lee
waves and MTL waves. In doing so, we use a ray tracing
approach for individual wave growth because a modal
decomposition is not strictly valid for the finite-amplitude sill case of NA due to complicated energy conversion between different modes. In addition, application of the modal decomposition is less suitable for tran-sient waves (see Fig. 7 of NA) that are generated aroundthe sill top and have not been reflected at the surface(a variety of vertical modes are required for identification and characterization of the aspects of wave
growth, Rattray et al. 1969). This is so, in a sense, forunsteady lee and MTL waves since their frequencies
vary with the phase of a tidal flow. To gain an understanding of the wave growth mechanism, ray tracing
might therefore be a simpler and more relevant approachthan a modal expansion.
The outline of this paper is as follows. In section 2
a new amplification mechanism for tidally generated
internal waves is proposed, and in section 3 ray tracingof individual waves is performed to demonstrate theeffectiveness of our amplification mechanism. The conclusions are summarized and discussed in section 4.
2. The growth mechanism of topographic internalwaves
In this section, we examine the growth mechanismof a compound wave formed by the superposition ofindividual waves, taking into consideration the transient
propagation process of individual wave components.
a. Wave growth in each wave regime
First, we investigate the propagation and associatedsuperposition of wave components in the horizontal direction. To highlight the difference due to wave types,
we begin with the convenient case in which the groupspeed | c g | is nearly equal to the phase speed | c p | inboth horizontal and vertical directions:

1 21
2
s
f
. |c | 11 2 211 2 2 . |c |.
Ns|c g | 5 |c p | 1 2ps 2i 1 2 f 2 /s 2i
N 2 1 2 f 2 /N 2
2
i

2

2

2
i

p

(1)

The above case corresponds to assuming individual
waves with frequencies of f K s i K N (other cases
will be discussed later). To define the fundamental
mechanism of wave growth clearly, the following simplifications are made, which can be easily relaxed: The
basic flow and stratification are uniform and, as modal
waves still have not been formed, the wavenumbers canbe determined from the horizontal scale of forcing, theintrinsic frequency s i , and the dispersion relation. In
this situation, forcing at t 5 t 0 generates two individual
waves with horizontal phase velocities of

1 U 6 kU 2 ,c6px (t0 ) 5 U0 2

U(t0 )
0

sf

(2)

0

where the upper (lower) signs correspond to each other.
Thus, the growth of compound waves formed by thesewave components depends on the parameter kU 0 /s f , as

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this determines the shapes of ray paths and hence theirsuperposition.
The positions, x6 (t), of the two wave components
mentioned above arex6 (t) 5 c6gx (t0 )(t 2 t0 ) 1

EtU(t ) dt ,

(3)

t0

where the generation point is set to be x 5 0. We consider a basic flow of U(t) 5 U 0 sin(s f t), which begins
to accelerate rightward from U 5 0 at t 5 0, as in NA.
Figure 2 shows the loci of individual wave components
during one tidal cycle for various kU 0 /s f values, inwhich the tidal excursion U 0 /s f is kept constant. Wave
components of phase velocity c 2px (c1px ) are shown in Fig.
2-column 1 (Fig. 2, column 2), and their amplitudes are
represented by line thickness.
1) UNSTEADY LEE WAVE REGIME
When kU 0 /s f k 1, both horizontal phase velocities
c6px(t 0 ) are almost equal and in the opposite direction to
the basic flow velocity at their generation time [i.e.,
2U(t 0 )]. Accordingly, Figs. 2(1a) and 2(1b) are almost
identical and have the following features: Wave components generated during the acceleration stage of rightward flow (0 ; 1/4 period) move downstream from thegeneration point until the basic flow decelerates tospeeds that match those at their generation time. Thisis because the phase speed of an individual wave component remains equal to the flow speed at the generation
time, while the basic flow continues to accelerate andso becomes faster than the speed of the wave component. On the other hand, wave components generated
during deceleration of the rightward flow (1/4 ; 1/2
period) move upstream relative to the ground and accelerate since the basic flow slows down after their generation. After the flow turns to the left, all of these wave
components move leftward since the direction of thebasic flow becomes the same as that of the phase velocities of wave components generated during the rightward flow. The propagation of wave components generated in the latter half period (1/2 ; 1 period) is thesame as in the former half period but with the directions
reversed.
The small time variation in flow speed around thetime of maximum flow results in the formation of a
distinct wave front. Now a condition persists underwhich the phase speeds of the individual wave components are almost equal to the basic flow speed. Accordingly, individual waves generated around this time

VOLUME 31

are effectively trapped so that they superpose within thegeneration region. In addition, because the amplitudes
are also proportional to the basic flow speed, these wave
components have the largest amplitudes and hence thecompound wave shows an apparent amplitude dispersion. Thus, individual waves generated around the timeof maximum flow form a leading wave front that propagates with maximum phase speed and little dispersion
in the upstream direction at the generation time andexports most of the wave energy generated in the half
period. The other individual waves form a highly dispersive tail in which waves of low amplitude graduallydisappear. Similar growth and propagation processes for
short internal waves generated on the downstream sideof a sill have been reported in observations (Farmer andSmith 1980).
In the K1 case of NA, the growth process to the largeamplitude wave at the right-hand break can be understood by the above mechanism because here kU 0 /s f is
5.4, indicating that this wave is roughly in the unsteadylee wave regime. This can be also interpreted in termsof a Froude number for individual waves, F i , definedas the ratio of the basic flow speed U(t) to the phasespeed of each wave component. The Fi values are almost
unity for individual unsteady lee waves when they are
generated. As the basic flow accelerates (decelerates)and becomes supercritical (subcritical) where F i . 1(,1) after generation, the individual waves move downstream (upstream) relative to the seabed. At maximum
flow speed, the condition F i . 1 is still maintained, andindividual wave components generated at this time aretrapped and superposed on each other at the generation
point.
Although the application of modal wave theory to atransient wave field might not be suitable, the above
description becomes qualitatively similar to the interpretation by conventional modal wave theories (e.g.,HI), if the definition of the Froude number is extended
to individual wave components (F i ). This enables us toeasily understand the trapping process of tidally generated transient waves at sills. It should be noted, however, that transient wave phase speeds are equal to thebasic flow speeds at their generation time regardless ofthe phase speeds of modal waves.
2) INTERNAL TIDE REGIME
Because the horizontal phase velocities of wave components characterized as internal tides are always almost
equal to 6s f /k, when kU 0 /s f K 1, the maximum valueof the Froude number for internal tides F i (5kU 0 /s f )→FIG. 2. The loci of individual wave components in x–t space during one tidal cycle, when the parameter kU 0 /s f is 10 (a), 2 (b), 1 (c), 0.5(d), and 0.1 (e). Those of phase velocity c1px and c 2px are shown in the left (1) and right (2) columns. Waves are generated at x 5 0, the basic
flow is directed rightward during 0 # period , 0.5 and leftward during 0.5 # period , 1, and the amplitudes of individual waves are
represented by line thickness (for the case of symmetric forcing). The horizontal scale is dedimensionalized using the tidal excursion U 0 /sf .

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VOLUME 31

TABLE 2. Properties of tidally generated internal waves.

Regime

kU0
k1
sf

kU0
;1
sf

kU0
K1
sf

Wave type
Intrinsic frequency
Relative phase velocity

Unsteady lee waves
2kU(t)
2U(t)Mixed tidal–lee waves
2kU(t) 6 sf
2U(t) 6 sf /kInternal tides
6s f
6sf /kis much less than unity. Hence the effect of advection,
which slightly slows down (speeds up) wave components propagating upstream (downstream), is too small
to enable these components to superpose (Fig. 2e), andthus wave amplification never occurs. Therefore, largeamplitude waves are not formed except in the vicinityof the critical slope, as is indicated by previous studies.
3) MIXED TIDAL–LEE WAVE REGIME
When kU 0 /s f ; 1, the propagation of wave components with phase velocities of c1px and c 2px are quite
different, making their loci asymmetric. Since the magnitudes of lee-wave [2U(t)] and internal tide (6s f /k)constituents are comparable in this range, the difference
between horizontal phase velocities c1px and c 2px is significant, as with internal tides, while these phase velocities vary in time and the advection effect is significant, as with unsteady lee waves. For this reason, we
separate MTL wave components on the basis of whether
the phase velocity of their internal-tide constituent is in
the same direction as the lee wave constituent, that is,c px 5 c pxf 5 2U 1 sgn(2U )

sf
,
k

(4)

sf
.
k

(5)

or in the opposite direction, that is,s
c px 5 c px
5 2U 1 sgn(1U )We name the waves with phase velocities c fpx and c spx‘‘fast’’ and ‘‘slow’’ MTL waves, respectively. As is apparent from the definition, fast MTL waves are fasterthan both lee-wave and internal-tide constituents,
whereas slow MTL waves propagating upstream (downstream) are slower than unsteady lee waves (internal
tides).
In Fig. 2, the components of fast (slow) MTL waves
generated in the former half period of rightward flowcorrespond to those with phase speeds c 2px (c1px), andthose in the latter half period to components with c1px
(c 2px). The advection effect together with the significant
differences in phase velocity between individual MTL
waves make wave superposition possible in the following way, based on consideration of the Froude number.
At the generation time, F i , 1 for individual fast MTL
waves and F i . 1 for individual slow MTL waves propagating upstream. As the basic flow speeds up (slows
down), the F i values for the fast (slow) MTL wave
components before (after) the maximum flow increase(decrease) toward unity. Thus, a superposition of fast
(slow) MTL wave components generated in the acceleration (deceleration) stage before (after) the maximum
flow in effect takes place.
It seems useful to seek the values of kU 0 /s f , for which
the propagation of MTL waves becomes similar to thatof internal tides or unsteady lee waves. A situation similar to internal tides could be created by a condition
under which slow (fast) MTL waves always move inthe downstream (upstream) direction at their generation
times. This condition requires that the signs of the absolute phase velocities [c6px (t 0 ) 1 U(t)] are always sameas those of the intrinsic phase velocities [c6px (t 0 )]. It isthen straightforward to derive that this is satisfied whenkU 0 /s f , 12 (kU 0 /s f , 1) for the slow (fast) MTL waves,as confirmed by Figs. 2c and 2d. For a situation similar
to unsteady lee waves, the propagation process shown
below is probably best regarded as being similar to thatof unsteady lee waves. Around the maximum flow, bothslow and fast MTL wave components with maximum
amplitude form wave fronts, which subsequently movein the upstream direction after the basic flow decelerates
to speeds comparable to the wave speed (Figs. 2a,b).
Such a situation occurs if both fast and slow MTL waves
generated by the maximum flow reach their greatest
phase speed. This condition always holds for fast MTL
waves. For the slow MTL waves, the condition is realized when kU 0 /s f . 2.
It is noteworthy that slow MTL wave components
generated around the maximum flow have very small
phase speeds (almost zero when kU 0 /s f 5 1) and large
amplitudes, indicating that these components stayaround the generation region for a relatively long timeand simply move back and forth under the influence of
advection. This implies that these MTL waves can besuperposed and amplified over many periods and thereby can affect the mean state considerably. Since thebehavior of such waves may depend on the approximation used in the relation between fixed and moving
frames, particularly when kU(t)/s f 5 1, a more detailed
analysis may be required to clarify this effect.
The above features of MTL waves enable us to understand the reason why the short wave at the sill breakin the M 2 case of NA is amplified, since kU 0 /s f is 2.5,
indicating that this wave is in the MTL wave regime.
Moreover, our theoretical result also enables us to explain one unexplained numerical result of Lott and Teitelbaum (1993b). For the case « 5 4.8 in their study,
which roughly corresponds to kU 0 /s f ; 4, waves are

AUGUST 2001

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NOTES AND CORRESPONDENCE

FIG. 3. Ray paths of individual wave components in x 2 z space for a half tidal cycle, when the parameter kU 0 /s f is 10 (a), 1 (b), and
0.1 (c).

also present on the upstream side at the maximum flow
time. This cannot be explained by unsteady lee waves
but can be explained on the basis of MTL waves, because individual fast MTL waves are faster than thebasic flow at the generation time. An asymmetric wave
pattern shown in the case of « 5 0.4 (i.e., kU 0 /s f ;0.5) can also be explained by the difference in phasespeeds between individual fast and slow MTL waves,and especially by our result that the phase speeds of thelatter decrease with increase in the basic flow speed.
b. Propagation in the vertical direction
Next, we investigate propagation in the z direction.
The vertical positions of wave components z6 at timet under consideration are
z6 (t) 5 c6gz (t 0 )(t 2 t 0 ) . | c6pz (t 0 ) | (t 2 t 0 ).

(6)

Using the same approximations for Eq. (1), the phasespeed c6pz can be written in terms of horizontal phasespeeds as|c6pz (t0 )| .

1

k 6
k
s
(c px (t0 )) 2 .
U0 7 f
N
N
k

2,
2

(7)

and hence the propagation in z–t space shows basicallythe same tendency as that in x–t space, when advection
is absent.
Figure 3 summarizes the features of topographic internal wave propagation in terms of ray paths in x–z
space for a half period. Unsteady lee wave components
are highly dispersed around the locus of the wave component generated by the maximum flow, forming paths
into a leaflike shape (Fig. 3a). They then continue todisperse and propagate upward at an oblique angle to2518

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FIG. 4. The ratio of horizontal group velocity to phase velocities as a function of N/sf and kU/sf , when f/sf is 0,0.1, 0.5, 1, and 2. Left- and right-hand columns correspond to fast and slow mixed tidal lee waves, respectively.
Black regions indicate that wave components are not freely-propagating waves (subinertial), and shaded regions
indicate that the ratio is less than 0.8.

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AUGUST 2001

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2519

FIG. 5. The ray paths of (a) fast and (b) slow waves generated at the right-hand sill break in
the K1 case until half period. Calculations are performed using the numerical model results. The
numbers indicate the time (hour) when the individual wave packets are launched. The superposedis the distribution of vertical velocity after half period. The contour interval is 0.5 cm s 21 . Values
in nonshaded areas and the thicker shaded areas are positive and negative, respectively.

the vertical. This is basically the same as the result of
Lott and Teitelbaum (1993b). In contrast, internal tidecomponents are distributed in the well known V shape,with relatively little dispersion (Fig. 3c). When kU 0 /s f
; 1, both ray paths of fast and slow MTL wave components also form leaf shapes, and adding the two forms
gives an asymmetrical V shape (Fig. 3b). Such a transition from internal tides to unsteady lee waves enhances

the concentration of wave energy near the generation
point. These features are also supported by the resultsof additional numerical experiments using cosine-shapeisolated sills with various scales (not shown).
Superposition of wave components generated in thesame half period occurs most effectively in the horizontal direction for the unsteady lee wave case and in
the vertical direction for the MTL wave case. When

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downward reflection at the sea surface occurs, superposition in the vertical direction could readily takeplace, thus enhancing the growth of these waves.
c. Influence of the difference between phase velocity
and group velocity
As is clear from Eq. (1), the group speed is smaller
than the phase speed in all wave regimes. The difference
depends on the ratios of wave frequency to buoyancy
frequency s i /N and wave frequency to the inertial frequency s i / f . Figure 4 shows the ratio of horizontal
group speed to phase speed as a function of kU/s f and
N for various f .
In the unsteady lee wave regime, the ratio c gx /c px decreases with increasing kU/N (i.e., with increasing frequency or decreasing stratification). This is marked in
the very high frequency region, and hence waves forthat region are effectively advected leeward since thebasic flow speed is large and the group speed is significantly smaller than the phase speed, with the latter
equal to the basic flow speed, as past studies haveshown. The lower panels show that the decrease in theratio c gx /c px with increasing f /s f is significant for waves
close to internal tides. These waves are finally trappedat the topography when the basic flow is subinertial.
Slow MTL waves are also effectively trapped whenkU(t)/s f . 1 since their frequencies become very small
in this range. Note that unsteady lee waves and fast
MTL waves can propagate freely even when the flow
is subinertial, although the rotation effect is not negligible. In addition, the reduction in group speed of fast
MTL waves caused by rotation may lead to enhanced
superposition of their individual wave components in
the generation region since the wave speeds decrease
toward the basic flow speed.
A series of our experiments suggests that the assumption of | c gx | . | c px | is most valid when the ratio
c gx /c px is greater than around 0.9. Otherwise the effectof the difference between the group and phase speeds
should be added to the superposition processes of individual wave components discussed so far.
The ratios (smax /N, smax / f , c gx /c px ) are (0.08, 0.31,0.90) of the values presented in NA for the M 2 case,and (0.10, 0.24, 0.94) and (0.07, 0.34, 0.89) of the NA
values for the fast and slow MTL waves, respectively,
for the K1 case. Hence, the approximation used in theprevious sections will, in general, be valid, except forslow MTL waves in the present M 2 case, which aresubinertial (here the maximum intrinsic frequency,
2kU 0 1 sf , is 0.7 3 10 24 s 21 ). In this case the waves
are trapped by the sill.
3. Ray tracing in the calculated field of NA
To demonstrate the effectiveness of our new amplification mechanism, we perform ray tracing of wave
components evolving to large-amplitude compound

VOLUME 31

waves in the K1 and M 2 cases (see appendix B for the
method). The basic states are determined from the calculated velocity and density fields of NA in order totake account of the effect of both spatial variations inflow and stratification on wave propagation. In particular, the strong stratification near the surface and thesignificant basic flow variation due to the presence ofthe large-amplitude sill are expected to influence thewave propagation considerably.
The ray paths traced up to 1/2 period for the K1 andM 2 cases are shown in Figs. 5 and 6a, respectively.
These figures show that the paths of MTL waves andunsteady lee waves eventually form leaf-shaped distributions with well-defined leading edges. The ray paths
correspond to the positions of the wave packets (i.e.,the center of wave energy), which agree well with thelocations of strong vertical velocity for both cases. After
1/2 period, wave energy in the surface layer is muchlarger than in lower layers due to stronger stratification
there. Accordingly, using ray tracing techniques, we cansuccessfully explain the trapping and amplification of
topographic internal waves in the K1 and M 2 cases, a
process that is difficult to explain using previous theories. Thus, our theoretical model is considered to beeffective in describing wave growth processes within
the limits of nonlinearity, duration, and the general applicability of the WKB approximation inherent in thesecases.
For comparison, Figs. 6b and 6c show ray tracing
results for initial wave frequencies set to those of thelee-wave (b) and internal-tide (c) constituents for theM 2 case. Clearly, the MTL wave case shown in Fig. 6acan best explain the position of the wave observed over
the right-hand break in the model result. This also indicates the importance of the concept of MTL waves.
It should be noted that, when the amplitude of thebottom topography is finite, flow separation often takes
place, so the particle flow paths are not strictly guided
by the bottom topography. This corresponds to the present K1 and M 2 cases for which the horizontal wavenumber of the generated waves cannot be simply determined from the curvature of the topography. In fact,
the main streamline may be controlled by various parameters. For example, the shape of topography, thebackground flow speed, the buoyancy frequency, andthe Coriolis parameter can play a role. Our understanding of the factors determining the relationship between
the shape of the main streamline and such basic parameters remains partial so that here we have determined
the horizontal wavelength from the calculated verticalvelocity distribution for convenience. Though our estimate shows that the horizontal scale derived from thecurvature of the sill break topography (;5 km) is roughly comparable to that of the waves modeled here, thebest means of determining horizontal wavelength still
remains to be clarified. Nevertheless, the fact that arough agreement between wave and sill scales exists
suggests that the theoretical considerations made above

AUGUST 2001
NOTES AND CORRESPONDENCE

FIG. 6. As in Fig. 5 but for (a) the ray paths of fast MTL waves in the M 2 case.
The results of additional experiments where initial frequencies are set to thoseof (b) lee waves and (c) internal tides are also shown. Contour interval of thevertical velocity after half period is 0.1 cm s 21 .

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VOLUME 31

TABLE 3. Horizontal group velocities of topographic internal waves.

f50

Wave regime

kU0
k1
sf

kU0
;1
sf

kU0
K1
sf

parameter range

unsteady lee waves

mixed tidal–lee waves

internal tides

N2 2 (kU)2
U(k2 1 m2)

k[N2 2 (kU 6 sf)2](kU 6 sf)(k2 1 m2)

k[N2 2 s2f ]sf (k2 1 m2)

2U(t)

2U(t) 6 sf /ksi /N ; 1
si /N K 1

6sf /kf , sfAlmost the same as above

3(1 2 f 2/s2f )f $ sf3(1 2 f 2/s2i )Topographically trapped waves

on the basis of a linear approximation may help us tounderstand the basic features of the wave growth in thepresent numerical experiments.
The finite sill height also affects lee wave responses
by producing variations in the basic flow as follows 1)
A basic flow increases with decreasing depth so thatwave trapping occurs more effectively on the downstream side (i.e., for waves propagating toward shallower water), as seen in Fig. 7 of NA. This fact can beeasily confirmed by ray tracing with a spatially varying
basic flow (not shown); 2) the generation regions ofinternal tides, which may work as a basic current forlee waves, are spatially broad; and 3) an overflow induces an intensified bottom current, which may enhance
the generation of unsteady lee waves.
4. Summary and discussion
Table 3 shows an overview of the amplification mechanism discussed so far in terms of the intrinsic horizontal
group speeds of topographic internal waves for eachregime.
Unsteady lee waves, which are always amplified,
grow most effectively in all three wave types when the
maximum frequency is sufficiently small relative to thebuoyancy frequency. Fast MTL waves are also effectively amplified as unsteady lee waves, when the reduction of the group speed due to rotation is so significant as appeared in the M 2 case. Thus, when the conditions f K kU 0 K N or

f ; s f , kU 0 K N(8)is satisfied, individual wave components generated
around the maximum flow time are effectively trappedand superposed. Therefore, the topographic internal
waves readily reach large amplitudes if the duration and/or magnitude of the forcing are sufficiently large. These
effects can be estimated by [(NU 0 /s f )h x ] 2 , which indicates the increase of potential energy associated withthe vertical excursion of isopycnal surfaces caused by
the basic tidal flow (e.g., Baines 1982).
It should be noted that such trapping and amplification can occur even if the critical Froude number or thecritical slope conditions required by previous theories
are not satisfied. The reason for this is that these earlier

conditions for wave growth cover only a part of theparameter range of topographic internal waves. For thecritical slope theory, the corresponding range is kU 0 /s f
K 1 so that the generated waves are all identical tointernal tides, and advection effects in both generation
and amplification are neglected. The HI model assumed
that 1) only internal tides are generated, 2) these waves
form modal waves immediately after the generation, and3) the group speed is almost equal to the eigenvalue ofthe corresponding mode and hence the phase speed(though implicitly). Thus, the HI model corresponds to
the range of kU 0 /s f K 1, f 2 K s 2f , and s 2i K N 2 .
Accordingly, it is appropriate for long internal tides at
low latitudes and/or when the typical horizontal scales
are at most the internal radius of deformation. Also, if
the HI model is extended to incorporate the effect of
the change in frequency, it could be useful to describe
the situation of almost flat bottom topography in shallow
water where waves form modal structures immediately
after the excitation.
Many observations show the generation of high-frequency waves over topography by an oscillatory flow,
particularly in the context of continental shelfbreak features (e.g., Chereskin 1983; Sandstrom and Elliott 1984;
Holloway 1985, 1987; Park 1986; Loder et al. 1992;
New and Pingree 1992; Liu et al. 1998). Some of the
waves could be excited as unsteady lee and/or MTL
waves, which may be amplified by the mechanism discussed above. For example, Holt and Thorpe (1997)
suggested that around 25% of the high-frequency internal wave packets observed in the Celtic Sea haveorientations consistent with generation by an alongslope
flow over rough topography on the slope. Considering
the dominance of tidal currents there and the smallnessof the horizontal scale of the topography, these packets
are expected to be excited as unsteady lee waves. Another example is the observation by Kuroda and Mitsudera (1995) in the East China Sea, which was mentioned in the introduction. The growth of the observed
waves could be explained on the basis of the new concepts introduced in this paper, though Kuroda and Mitsudera (1995) considered the growth to result from nonlinearities due to finite wave amplitudes.
Care should be taken, however, when identifying unsteady lee waves with observations at a fixed point, such

AUGUST 2001

2523

NOTES AND CORRESPONDENCE

as a mooring, since discrimination of unsteady lee waves
from internal tides is difficult around the generation
region because the former are trapped until the flow
direction changes. This can be easily understood by considering the Doppler relation. The intrinsic frequencies
of topographic internal waves generated at x 5 x 0 andt 5 t 0 are 2kU(x 0 , t 0 ) 6 s f , but the frequencies at a
fixed point x are observed as 1kU(x, t) 2 kU(x 0 , t 0 ) 6
sf .
In this study, we have revealed the topographic internal wave growth by focusing only on the relation to
the important parameter, kU 0 /s f . In doing so, because
ray tracing strictly applies only when the waves are in
slowly varying ambient conditions, our investigation israther qualitative. Also, some physical factors are missing in our model. We have neglected the nonlinear effects caused by large wave amplitudes, though thismight be acceptable because our primary interest is in
how topographic internal waves are amplified to suchlarge amplitudes. However, after the waves have grownsufficiently, we should incorporate nonlinear wave effects such as breaking of lee waves at the generation
region. In addition, wave generation becomes more
complicated in three dimensions due to the more complex flow and the changes in buoyancy frequency thatdevelop around obstacles. Background flows, such as
wind-driven and thermohaline circulations, modify thebasic flow and thus may affect the generation and evolution of the waves. These problems are left for futurework.
Acknowledgments. We wish to acknowledge Prof. J.
P. Matthews at Kyoto University for his critical reading.
We also thank anonymous reviewers for their invaluablecomments. Numerical calculations were done on the FACOM M1800 and VP2600 at the Data Processing Centerof Kyoto University.
APPENDIX A

APPENDIX B
Method of Ray Tracing in the Calculated Fieldof NA
If we neglect the effect of currents induced by waves
themselves, the propagation of wave packets in a gradually varying basic state can be described by the following equations (Lighthill 1978):dx i]s
5 Ui 1
,
dt
]k i

(B1)

dk i
]U
]s
5 2k j j 2
,
dt
]x i
]x i

(B2)

where U i (U1,2 5 U, W) and k i (k1,2 5 k, m) denote thebasic flow speed and wavenumber in the x i (x1,2 5 x, z)
direction, respectively, and d/dt is the temporal change
along the ray. Equation (B1) gives the velocity of a wave
packet along its ray path, which consists of the basic
flow vector responsible for advecting the packet and thegroup velocity of the wave. Equation (B2) describes therefraction of a wave packet due to the spatially varying
basic flow and stratification. To complete the set of equations we use the linear dispersion relation,s 2 5 (N 2 K 2 1 f 2 m 2 )(k 2 1 m 2 ) 21 .
The ray tracing equations then becomedx
k N2 2 s2
5U1,
dt
s m2 1 k2(B4)dz
m s2 2 f 2
5W2,
dt
s m2 1 k2

(B5)

(B6)
1
2
dm
]U
]W
1
k
]N
5 2 1k1m 22. (B7)dt]z]z
2s m 1 k ]z
dk
]U
]W
1
k2
]N 2
52 k
1m
2
,
dt
]x
]x
2s m 2 1 k 2 ]x22Method for Estimating the Contribution Rate andRelative Magnitude
In this study, the ‘‘contribution rate’’ of the mth mode(m 5 1, 2, · · · ) is defined, for convenience, as

1W E N R dz2@1E N w dz2 3 100(%),
2
m

2

2
m

2

2

where R m and W m are the eigenfunction and its coefficient for the mth mode (i.e., w 5 S m W m R m ). Theoretically, the sum of the contribution rates for all m should
be exactly 100%, so the excess from this value gives
the error magnitude (here approximately 2%) in calculating the decomposition. Also, the ‘‘relative magnitude’’ of the mth mode is defined here as the valueof the coefficient corresponding to the eigenfunction
when normalized to a maximum value of unity.

(B3)

2

2

Here we set W 5 0 in order to avoid the influence of
currents induced by the traced wave itself. Numericalintegration of these ordinary differential equations witha fourth-order Runge–Kutta method enables us to trace
individual wave components generated at various times.
For the initial wave structure, we determine the horizontal wavelength as 8 km and 6 km for the K1 and M 2case, respectively, from the vertical velocity distribution. The wave frequency is taken as 2kU(t) 6 s f , andwe require the vertical wavenumber to satisfy the dispersion relation. Then, the effect of temporal variationsin the basic state is incorporated through the basic statevariables (U, N) and the initial wave frequency. The
initial position of wave packets is set to be the righthand sill break where the large-amplitude waves are
generated. When wave packets reach the surface, they
are perfectly reflected.

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