1574

JOURNAL OF PHYSICAL OCEANOGRAPHY

VOLUME 39

Scattering of Internal Waves with Frequency Change over Rough Topography
TOMOHIRO NAKAMURA
Pan-Okhotsk Research Center, Institute of Low Temperature Science, Hokkaido University, Sapporo, JapanTOSHIYUKI AWAJI
Department of Geophysics, Graduate School of Science, Kyoto University, Kyoto, Japan(Manuscript received 8 March 2007, in final form 3 November 2008)
ABSTRACTThe frequency change in internal gravity waves upon scattering from a rough topography is investigatedanalytically. For this, sets of appropriate and tractable governing equations for various parameter regimes arederived using the method of multiple scales under the assumption that the amplitude of the bottom topography is small. A solution is shown for a simple case in which an incident internal wave is approximately
linear and monochromatic. The solution has the following features: the intrinsic frequencies of the scatteredwaves are given as the sum and difference of the incident-wave frequency and the Doppler shift (or lee-wave
frequency). This Doppler shift causes the change in the frequency. Hence, the assumption of frequency
conservation is not valid if the Doppler shift is significant, that is, when the horizontal scale of the bottomroughness (or the length scale in the plane of the slope) is on the order of or much less than that of theincident-wave flow excursion. This condition can be satisfied in a realistic parameter range.
The occurrence of such a frequency change has the following implications: first, it affects the estimate of theboundary mixing induced by the scattering because the energy redistribution in the vertical wavenumber
space on scattering differs from that estimated using the assumption of frequency conservation. This effecthappens because for a given horizontal wavenumber, the change in the frequency alters the vertical wavenumber of the scattered waves through the dispersion relation. Furthermore, if the incident waves are not
monochromatic, even the leading-order scattered waves cannot be obtained by the superposition of thesolutions for all the Fourier components of the incident waves because of the difference in the Doppler shift.
Second, the effects of the background flow associated with the incident and primary reflected waves are
significant when the frequency change occurs such that the background flow can create a critical level and/or
advect scattered waves. The former causes mixing and background-flow acceleration, and the latter is favorable for the amplification of the scattered waves through superposition. Third, the resulting energy redistribution in frequency space could modify the spectrum shape of the oceanic internal waves, which isconsidered to affect both interior and boundary mixing.

1. Introduction
The reflection and scattering of oceanic internal
gravity waves off bottom topography can redistributeinternal wave energy to higher vertical wavenumbers,leading to enhanced mixing near the bottom boundary.
Such internal wave-induced boundary mixing has been
estimated to contribute significantly to the basinwide
cross-isopycnal mixing (e.g., Eriksen 1985; Müller andCorresponding author address: Tomohiro Nakamura, Instituteof Low Temperature Science, Hokkaido University, Sapporo
060-0819, Japan.
E-mail: nakamura@lowtem.hokudai.ac.jp
DOI: 10.1175/2008JPO3795.1
Ó 2009 American Meteorological Society

Xu 1992). The total amount and the distribution ofcross-isopycnal mixing strongly affect the global thermohaline circulation (e.g., Munk 1966; Bryan 1987;
Munk and Wunsch 1998; Hasumi and Suginohara 1999);
therefore, a better understanding of the reflection andscattering process is important not only for its theoretical interest but also for a better understanding andmodeling of the thermohaline circulation.
Previous theories have revealed an interesting varietyof reflected waves corresponding to various shapes ofthe bottom topography. Theories on the reflections overa plane slope showed that the wavenumber normal
to the slope changes upon reflection based on the conservation of both the frequency and the wavenumber

JULY 2009

NAKAMURA AND AWAJI

parallel to the slope. Such a change in the wavenumber
results in intense mixing near a critical slope, where thebottom slope and the ray path of the reflected waves areparallel (e.g., Phillips 1977; Eriksen 1982, 1985).
Nonlinear effects, which become important in such asituation, were investigated further (e.g., Thorpe 1997).
Theories on the reflections over a nonplanar but‘‘smooth’’ topography in comparison with the wavelength of an incident plane wave have shown the occurrence of split reflection (Baines 1971b), different
responses over convex and concave topographies (e.g.,
Gilbert and Garrett 1989), and the effects of finite topographic height and/or depth on the change in thevertical wavenumber (Müller and Liu 2000a,b). Numerical models were invoked to investigate the effectsof three-dimensionality (Johnston and Merrifield 2003)
and nonlinearity (Legg and Adcroft 2003) for this kindof topography.
When the bottom topography is rough compared to
the wavelength of an incident wave (e.g., when a small
sinusoidal variation with a wavenumber kb is superimposed on a plane), Baines (1971a) found that the reflected waves comprise three components: a ‘‘primary’’reflected wave and two types of ‘‘scattered’’ waves. The
primary reflected wave is the same as that reflected froma plane. On the other hand, the scattered waves havedifferent wavenumbers in the direction parallel to theplane. Their wavenumbers are determined as the sum
and difference of the wavenumbers of the incident wave
and the topographic wavenumbers (i.e., ki 6 kb, where
ki denotes the wavenumber of the incident wave). Miedand Dugan (1976) relaxed the condition of the smallamplitude bottom roughness in the case with no meanslope by expanding the scattered wave solution withrespect to the slope of the bottom roughness. The horizontal wavenumber of the scattered waves is then given
as the sum and difference of the wavenumbers of anincident wave and integer multiples of those of the topography (i.e., ki 6 nkb for the nth order). Thorpe(2001) extended the theory to consider rough topographies with a mean slope, and Legg (2004) incorporated
the Coriolis effect.
These theories were applied to the empirical spectral
representations of the bottom topography and internal waves in order to estimate the redistribution of
the incident-wave energy flux in wavenumber space(Rubenstein 1988; Müller and Xu 1992). In particular,
Müller and Xu (1992) reformulated scattered wave solutions in a more revealing manner and estimated thatscattering might be equally or more efficient than reflection in causing mixing near the bottom.
All of these previous theories have assumed that thefrequency is conserved on reflection. In fact, absolute

1575

frequency (i.e., frequency in a reference frame fixed tothe bottom) is conserved on reflection. Nevertheless,when a background flow is present, intrinsic frequency(frequency in a frame moving with the backgroundflow) should be used; otherwise, the Doppler shift must
be taken into account to determine the wave ray slope
or wavenumbers of scattered waves. Previous theories
on scattering neglected the effect of advection andhence implicitly assumed that intrinsic frequency isequal to absolute frequency and is hence conserved.
However, the validity of this assumption is questionable in a certain parameter range. For example, consider the generation of internal waves by a barotropic
tidal flow over rough topography. The generated internal waves are expressed as the superposition of thehigher harmonics (Bell 1975a; Balmforth et al. 2002;
Khatiwala 2003), and their intrinsic frequencies areapproximately given by the sum and difference of thetidal frequency and the Doppler shift (or lee-wave frequency) (Nakamura et al. 2000; Nakamura and Awaji
2001). In a similar manner, intrinsic frequency may bechanged by scattering over rough topography.
St. Laurent and Garrett (2002) utilized without derivation Bell’s (1975b) theory to estimate the energy flux
scattered from the mode-1 internal tide at the MidAtlantic Ridge by using mode-1 internal tide currents inplace of a sinusoidally oscillating barotropic current.
Through the use of Bell’s formula, the estimate shouldbe affected by the effect of frequency change. Nevertheless, because their main focus was not on scattering,neither the validation of the use of Bell’s formula forthat case nor the condition, mechanism, or effects offrequency change was investigated.
In this paper, we shall analytically demonstrate thatthe frequency conservation for internal wave reflection
is violated in certain parameter regimes. The conditionand effects of the violation of the frequency conservation are then investigated.
After a heuristic derivation of governing equations, ascattered wave solution with advection effect is presented in section 2 for a simple case in which both theincident wave and the bottom topography are monochromatic and in which the Boussinesq approximation
is used with a uniform buoyancy frequency. The occurrence of frequency change and its relation to higher
harmonics are shown in section 3. The condition forfrequency change and its extent in various parameter
ranges are investigated in section 4. A more rigorous
derivation of the leading-order equations is given in
section 5, which validates the governing equations usedin section 2 and further reveals that the appropriate
governing equations differ for various parameter regimes (or equivalently, shows which terms are negligible

1576

JOURNAL OF PHYSICAL OCEANOGRAPHY

to what order). Possible effects and implications arethen discussed in section 6. Finally, the results are
summarized in section 7.

2. Scattered wave solution with advection by
primary waves
a. Governing equations
Consider internal wave reflection over small-amplitude,rough topography. Then the primary wave field consistsof the incident wave and its reflection on a plane horizontal boundary (i.e., the primary reflected wave). For asinusoidal incident wave, these waves form an exact
solution of the nonlinear governing equations with
Boussinesq approximation for uniform stratification.
The equations for the scattered waves are then obtainedby linearizing about this solution. The result will be theusual linearized internal wave equations, with extra advection terms such as ui1r  =fs 1 us  =fi1r, where thesubscripts i 1 r and s denote the sum of the incident
and primary reflected waves and scattered waves, respectively, and f denotes momentum or density. In theusual linear theory such advection terms are neglected,
and the scattered wave field satisfies the usual homogeneous linear equations forced by the bottom boundary
condition.
We consider here the effect of the advection by the
primary wave field, ui1r  =fs. This is done by assuming
momentarily that it is possible to neglect the spatial
derivative of the primary wave field in the scatteredwave equations when the primary wave field is slowly
varying relative to the topography and hence to thescattered wave field.
Then the two-dimensional equations for the scatteredwaves in an inviscid, rotating, and uniformly stratified
Boussinesq fluid become
Dus
1 ›ps
 f ys 5 
r0 ›xsDt
Dy s
1 fus 5 0,
Dt
Dws
›p
5  s  r9s ,
Dt
›zs
Dr9s
›r
5
ws .
›zs
Dt
›us
›ws
1
5 0,
›xs
›zs

(2.1)

D
›
›
›
5 1 ul
1 wl
.
Dt
›t
›xs
›zs

(2.2)

where

VOLUME 39

The corresponding bottom boundary condition is
ws (z 5 0) 5 ul

›h
.
›xs

(2.3)

Here, the standard notations are used. The subscripts
l and s denote large- and small-scale variables associatedwith the primary and scattered wave fields, respectively.
The density is separated into the reference value, background stratification, and perturbation as
rtotal 5 r0 1 r(z) 1 r9(x, t).

(2.4)

The validity of the neglect of the nonlinear terms likeus  =fi1r will be examined later in section 5. Here we note
beforehand that the above governing equations are valid
in a parameter regime where ksUl/vl (5 ksUi1r /vi) ; 1,
vs ; vl, and klUl/vl  1 (k, v, and U are the horizontal
wavenumber, intrinsic frequency, and the amplitude ofhorizontal velocity, respectively). Because of the lastcondition, klUl/vl  1, the primary waves can be approximated to be linear in the leading order, even when
the advection by the primary waves is taken into consideration. In other parameter regimes, the appropriate
governing equations take somewhat different forms.
This is especially so in the regime where the incident
wave is nonlinear in the leading order.
The above governing equations can be solved using
the Fourier transform, as shown in the rest of this section, because the spatial derivative of the large-scalewave field (i.e., the primary wave field) is assumed to be
negligible.

b. General solution
Governing equations (2.1) can be reduced to a single
equation for ws as
D2
Dt(=2s ws ) 1 N 2
2

2
›2 ws
2 › ws
1
f
50
›x2s
›z2s

(2.5)

because the large-scale wave field does not depend on
the small-scale coordinates (xs,zs) and thus D/Dt and=s 5 (›/›xs,›/›zs) commute.
For the same reason, (2.5) can be solved using theFourier transform, and thus ws may be written as thesum of Fourier components:ðð^ s (ks , t) eiks xs dks dms ,(2.6)
ws 5 As (xl )w
where ks 5 (ks,ms). The equation for each component isthen given as2›^ s (ks , t) 5 0,
^ s (ks , t) 1 V2 (ks )w
1 iks ul 1 ims wl w
›t
(2.7)

JULY 2009

1577

NAKAMURA AND AWAJI

where
N 2 k2 1 f 2 m2

V(k) 5!1/2

k2 1 m2.

(2.8)

To solve (2.7), we introduce a frame [Xs 5 (Xs,Zs)]
moving with ul; that is,
Xs 5 xs ðtHere, subscripts i and b denote the incident wave andthe bottom topography, respectively. The incident wave
satisfies the linear internal-wave Eqs. (5.34) in section 5c.
For convenience, we define a notation u, which works
in a manner similar to Einstein’s summation rule but
represents a summation for the cases of u 5 1 and u 5 21,
such thateiukx 5 eikx 1 eikx ,ul [xl (t9), t9]dt9,

(2.9)

0

which is an extension of the moving frame introduced^ s is describedby Bell (1975a). In the moving frame Xs, w
as


ðt
^
^m
(k
,
t)
5
w
(k
,
t)
exp
ik

u
[x
(t9),
t9]dt9(2.10)
w
s
s
s
s
l
l
s
0

› iukx
5 iukeiukx 5 ikeikx 1 i(k)eikx .
e›x
The wavenumbers or frequencies with the u notation
are defined as positive or zero. When there are u notations with different subscripts, each notation is independently summed such thatei(uk kxuv vt) 5 ei(kxuv vt) 1 ei(kxuv vt)
5 ei(kxvt) 1ei(kx1vt) 1ei(kxvt) 1ei(kx1vt)

and (2.7) becomes
^m
›2 w
s
^m
1 V2 (ks )w
s 5 0.
›t2

5 4 cos(kx) cos(vt).
(2.11)

This can be solved using a Fourier transform with respect to time,
ð‘
^m
w
5
F(v) eivt dv,(2.12)s

The use of this notation allows us to write the relation
between the sign of the wavenumber and the frequency
explicitly and neatly.
Using this u notation, the above topography and incident wave are written as‘h5yielding the dispersion relationv2 5 V2 (ks ).

(2.13)

Because v and ks are specified from the bottom boundary condition described below, ms is determined tosatisfy the dispersion relation along with the radiation
condition.

c. Topography and large-scale waves

1h 5 H cos(kb xs ) 5 H(eikb xs 1 eikb xs ),2(kb . 0),(2.14)

ui 5 Ui (ki )cos (ki xl 1 mi zl  vi t)
1
5 [Ui (ki )ei(ki xl 1mi zl vi t) 1 Ui (ki )ei(ki xl 1mi zl vi t) ].
2(2.15)(2.16)

1
ui 5 Ui (ui ki )eiui (ki xl 1mi zl vi t) .
2

(2.17)

Here, Ui 5 (Ui,Vi,Wi) comprises amplitudes of the velocity components, which can be written in terms of Ui(which is set to a constant value) from the polarization
relation as
Vi 5

To determine the properties of scattered waves fromthe bottom boundary condition, we next specify thebottom topography and large-scale waves in a very
simple fashion to provide basic insight into more complicated situations.
We consider bottom topography and an incident wavethat are both monochromatic as

1 iub kb xs
He
,
2

f
Ui,
iui vi

Wi 5 

ki
Ui,
mi(2.18)and vi is the incident-wave frequency given from thedispersion relation asvi 5 V(ki ).

(2.19)The primary reflected wave, whose governing equations
are the same as those of the incident wave, is determined from the bottom boundary condition, wl 5 0 at
z 5 0, and from the radiation condition as1
ur 5 Ui (ui ki )eiui (ki xl mi zl vi t) ,2

(2.20)

where subscript r denotes the primary reflected wave.
The horizontal velocity (along the x direction) of theselarge-scale waves at z 5 0 is thus given by

1578

JOURNAL OF PHYSICAL OCEANOGRAPHY

ul 5 ui 1 ur 5 U i eiui (ki xl vi t) [5 2U i cos(ki xl  vi t)].
(2.21)

d. Bottom boundary condition and the solution
Given the bottom topography and the large-scalewave field, a specific solution is determined from thebottom boundary condition.
The substitution of (2.6), (2.16), and (2.21) into thebottom boundary condition (2.3) yieldsðð^ s (ks , t)eiks xs dks dmsw
As (xl , 0)
5 Uie

iui (ki xl vi t) 1

2

iub kb xs

iub kb He.

(2.22)The bottom boundary condition is then described in themoving frame for consistency with the general solutionasðð
iks X s^m
As (xl , 0)w
dks dms ,s (ks , t)eÐt
1
iu k X 1iu k
u j dt9
5 iub kb HU i eiui (ki xl vi t) e b b s b b 0 l z50 ,2(2.23)
which implies1
As (xl , 0) 5 iub kb HU i eiui ki xl ,2Ðtiui vi t1iub kb ul jz50 dt9
0^mw.
s (ks , t) 5 d(ks  ub kb )e(2.24)(2.25)

Applying this bottom boundary condition to (2.12) andapplying the inverse Fourier transform yields
ð1 ‘ ife dt,(2.26)
F(v) 52p ‘ðtf 5 ui vi t 1 ub kb ul jz50 dt9 1 vt.
(2.27)
0

The solution in the fixed frame is thus obtained asðð^ s eiks xs dks dmswððÐtiks  ul [xl (t9),t9]dt9 iks xs
0
^m
5 As
e
dks dmsw
s e

ws 5 A s

1
5 iub kb HU i eiui ki xl
2
Ð

Ðt
t
i(u ku (x (t9),t9)dt9sign(v)ms wl (xl (t9), t9)dt903e b b 0 l lð‘
3F(v)ei[ub kb xs sign(v)ms zs vt] dv,
‘

VOLUME 39

positive in order to explicitly represent the radiation
condition.1
In the moving frame, the solution takes a more revealing form:1
iui ki xlwm
s 5 iub kb HU i e
2 ð‘
3F(v)ei[ub kb Xs sign(v)ms Zs vt] dv.

(2.29)

‘

3. Change in intrinsic frequency
a. Frequency of scattered waves
The frequency of major components of the scatteredwaves is determined from the function F(v), which sets thecomplex amplitude of the scattered waves and is specified
in (2.26). According to the stationary phase method (e.g.,
Lighthill 1978), the major contributions to the integral(2.26) comes from the intrinsic frequency v that satisfies›f
5 ub kb ul jz50  ui vi 1 v 5 0.
›t

(3.1)

The above condition is satisfied twice during one cycleof the incident wave for one value of intrinsic frequency(or once for ul|z50 5 62Ui), and the integral is estimatedby the sum of these main contributions.
Conversely, this means that the main components ofscattered waves generated at a time t0 have the intrinsic
frequency ofv 5 6(kb ul jz5 0, t5t0 6vi ) ,(3.2)

where ul|z50 is given by (2.21).2 After the generation,
the scattered waves with the above intrinsic frequency
propagate through the primary wave field, and wm
s is

1
It should be noted that the wavenumber in the fixed frame is,strictly speaking, different from that in the frame moving with thebackground flow that has nonzero convergence/divergence in thex or z direction. The extent of the difference depends on the ratioof wavenumbers of incident and scattered waves (i.e., dk 5 ki/ks
and dm 5 mi/ms in the x and z directions, respectively, as discussed
in section 5). The order of the associated error is the same as that
arising from the neglect of the spatial variation of an incident wave
in the derivation of the leading-order equations for the scatteredwaves and is small when dk and dm  l.
2
Equation (3.2) is obtained because the exponential function in
the integral in (2.29) becomes, after the substitution of (3.1) into
(2.29),

ei[ub kb X s sign(v)ms Zs (ub kb ul jz50 1ui vi )t]
5 ei[kb X s sign(v)ms Zs (kb ul jz50 1vi )t](2.28)

5 ei[kb X s sign(v)ms Zs (kb ul jz50 vi )t]

where the vertical wavenumber ms, whose absolute value
is determined from the dispersion relation, is defined as

5 ei[kb X s sign(v)ms Zs (kb ul jz50 1vi )t]
5 ei[kb X s sign(v)ms Zs (kb ul jz50 2vi )t].

JULY 2009

1579

NAKAMURA AND AWAJI

obtained by the sum of these main components generated during the time duration under consideration.
Equation (3.2) indicates that the intrinsic frequency ofthe scattered waves is given as the sum or the differenceof the incident-wave frequency and the Doppler-shift
frequency (i.e., lee-wave frequency); that is,(intrinsic freq.) 5 (Doppler shift)
6 (incident wave freq.).
The mechanism of frequency change is essentially thesame as that in the case of internal wave generation by atidal flow (Nakamura et al. 2000). In fact, the bottom
boundary condition is almost identical to the case considered in Bell (1975a) and Nakamura et al. (2000).

b. Relation to the higher harmonics
It should be noted that the above expression yields
the same wave field described using the higher harmonics like Bell’s (1975a) solution within the approximation of the stationary phase method.
To bring out this point, let us focus on the advection
effect in the kinematic bottom boundary condition, assuming that the total time derivative is approximated as
D ››’ 1 ui1r (t)
Dt ›t
›x

at z 5 0

(3.3)

›h
›x

at z 5 0.

(3.4)

These correspond to (2.2) and (2.3), respectively, but xs
is here written as x and the spatial variation of ui1r is
ignored for simplicity. Despite this simplification, theresults are essentially the same both because kl affects
only the slowly varying amplitude of scattered waves inthis parameter range as shown in (2.29) and because of
the boundary condition wl 5 0 at z 5 0.
The essence of the frequency change may be summarized as follows: Consider a monochromatic, plane
incident wave that yields
ui1r (z 5 0) 5 U i1r cos vi t.
Introduce the Fourier transformð‘b5()()eikx dx

0

(3.8)
^ m (z 5 0) with respect to time
The Fourier transform of w
yieldsð‘m^ s (z 5 0) 5
F(z 5 0, v)eivt dv,
(3.9)
w
‘

with inverseð1 ‘ m^ (z 5 0)eivt dtw
2p ‘ sððtikh^ U i1r ‘
exp i 6vi t 1 k ui1r (z 5 0, t)dt 1 vt dt.
5
2p 4 ‘0F(z 5 0, v)5(3.10)
According to the stationary phase method, the main contributions to the integral on the right-hand side comesfrom the intrinsic frequency v that satisfiesðt
›
6vi t 1 k ui1r (z 5 0, t)dt 1 vt
05
›t
,
0
(3.11)
5 6vi 1 kui1r (z 5 0, t) 1 v,
and thus the main component of internal waves generated at a time t0 has the intrinsic frequency ofand the kinematic boundary condition is
ws 5 ui1r (t)Then the bottom boundary condition becomes
 ðt

m
^
^ s (z 5 0) 5 ikui1r (z 5 0, t)h exp ik ui1r (z 5 0, t)dt .
w

(3.5)

(3.6)

v 5 kui1r (z 5 0, t 5 t0 ) 6 vi ,(3.12)as shown in the previous section.
On the other hand, Bell’s solution method directly
expresses the integral on the right-hand side of (3.9)
using a series expansion asð‘F(z 5 0, v)eivt dv
2‘
 ðt

5 ikui1r (z 5 0, t)h^ exp ik ui1r (z 5 0, t)dt ,^ m (z 5 0) 5w

0(3.13) ðt
›^
5 h exp ik ui1r (z 5 0, t)dt ,
›t
0
5 h^

‘


›
kUi1r
,
einvi t J n›t n5‘vi

å

(3.14)(3.15)

‘and the transformation to a reference frame moving withthe background flow ðtb m 5 ()
b exp ik ui1r (z 5 0, t)dt .
()
(3.7)
0

so that the total wave field is given by the sum of higher
harmonics.
The main difference between Bell’s method andthe stationary phase method is the way of estimating
the integral in (3.13) [and (3.9)]. A minor difference is the

1580

JOURNAL OF PHYSICAL OCEANOGRAPHY

transformation from (3.13) to (3.14), which is absent in
the stationary phase method and can be omitted in
Bell’s solution method.
Thus, within the approximation of the stationary phasemethod, internal wave components generated at time
t have intrinsic frequencies of 2kui1r(z 5 0, t) 6 vi,
whereas the total scattered-wave field, which is composed of the superposition of such wave components
and is subjected to the advection by the backgroundflow, is described with the sum of the higher harmonics.
The relation between the above results of the stationary
phase method and Bell’s solution may be analogous,
although it is not quite precise, to the relation between anarrow internal-wave beam described by integration
along wave rays and that expressed as the sum of multiple vertical modes. [Although the approximation usedin the stationary phase method could have a significant
error in the vicinity of kUi1r 5 vi, the results of ournumerical experiments (in preparation) suggest that theprediction of the stationary phase method is robust in aqualitative sense even in the case kUi1r 5 vi.]
Note also that the higher harmonics in Bell’s solution
contain not only the effect of the change in intrinsic frequency but also the effect of the oscillation of theadvecting flow. To reveal this fact, consider the situationin which an internal wave of the tidal frequency v0 is being
advected by a barotropic tidal flow U(t) 5 U0cos(v0t).
The vertical flow associated with the internal wave at a
certain point in the moving frame may be written as
wm 5 

k
U i cos (v0 t),m(3.16)

where Ui is the complex amplitude of the horizontal
flow of the internal wave with k and m the horizontal
and vertical wavenumbers, respectively. The verticalflow can be rewritten as
wm 5

i Ui
ikU 0 cos(v0 t)
m U0(3.17)and thus is described in the frame fixed to the bottom asÐtik
U (t )dt
0w 5 wm eÐi U i › ik t U (t)dt0e5
(3.18)
m U 0 ›t
5

‘


i Ui
kU 0
.
inv0 einv0 t J n
m U 0 n5‘v0

å(3.19)

This shows that an internal wave that is advected by asinusoidal tidal flow and has intrinsic frequency of thetidal frequency is expressed as the sum of the higher
harmonics in the fixed frame. Similarly, an internal
wave with the tidal frequency in the fixed frame be-

VOLUME 39

comes the sum of the higher harmonics in the frame
moving with the sinusoidal tidal flow.
In this sense, the physics of frequency change ismasked in the expansion into a series of higher harmonics, and therefore the following discussion is basedon the results of the stationary phase method.

4. Condition and regimes
a. Condition for frequency change
Because the change in intrinsic frequency arises fromthe Doppler shift as indicated by (3.2), the extent of thefrequency change can be estimated from
2kb U i
,
vi

(4.1)

where 2Ui is the amplitude of the horizontal flow that isparallel to the topographic wavenumber vector and isassociated with the incident and primary reflected waves
at the bottom. The frequency change is significant when
this ratio is on the order of, or much greater than, unity.
It is noteworthy that this condition for frequencychange can be satisfied in a realistic parameter range.
For example, consider an incident wave of a semidiurnal
tidal frequency with a flow amplitude of 5 cm s21; in thatcase, the topographic wavelength 2p/kb required is approximately 4 km when 2kbUi/vi 5 1 (and hence vs ; vi)
or approximately 400 m when 2kbUi/vi 5 10 (and hencevs  vi). These estimates suggest that the occurrence of
scattering with frequency change could be a common
phenomenon in the ocean because the bottom roughnessof such scales is widely distributed in the oceans.
More precisely, the horizontal wavenumber of scattered waves is the sum and difference of the horizontal
wavenumbers of the incident wave and the topography(i.e., khs 5 khi 1 kb ; Baines 1971a), although the horizontal wavenumber of the scattered waves is approximated as that of the topography in the above [i.e.,ks ’ (ubkb 5) 6 kb] through the assumption that theincident wave is slowly varying relative to the scatteredwaves (i.e., jkhi j/jkb j ’ 0). Then the more precise measureof the frequency change may be given by
jkhs  uhi1r j
,
vi

(4.2)

where the superscript h the horizontal component.

b. Parameter regimes of frequency change
The ratio (4.2) implies the presence of the following
three regimes:(i) jkhs  uhi1r j/vi  1: The scattered wave frequency ismuch greater than the incident-wave frequency.

JULY 2009

NAKAMURA AND AWAJI(ii) khs  uhi1r /vi ; 1: The scattered wave frequency ison the order of, or much smaller than, the incidentwave
frequency.
(iii) khs  uhi1r /vi  1: The scattered wave frequency isalmost equal to the incident-wave frequency.
The regimes (i)–(iii) correspond to the regimes in which
quasi-steady lee waves, mixed tidal-lee (MTL) waves, andinternal tides are generated by a barotropic tidal flow,
respectively (Nakamura et al. 2000). Frequency conservation is not valid in the first two regimes, but it provides agood approximation in the last regime, which has been
considered in the previous studies.
The parameter regimes can be also viewed in terms ofhorizontal scale. Frequency change occurs when the
topography is sufficiently rough because the ratio (4.2)
can be approximated as that of the flow excursion at thebottom (juhi1r j/vi ) to the topographic length scale(jkb j1 ). This roughness of the topography is reflectedby the difference in scale between the horizontal
wavenumbers of incident and scattered waves. Denoting these wavenumbers as Ki and Ks (5jkhi 1 kb j), respectively, we can scale the ratio (4.2) as 
j(khi 1 kb )  uhi1r j
Ks U i1r
Ks
Ki U i1r
,
;
5
vi
vi
Ki
vi

(4.3)

where U i1r 5 juhi1r j. Here, the ratio in the last bracket,
D5

K i U i1r
,
vi

(4.4)

is the wave Froude number (or temporal Rossby number) of the incident and primary reflected waves.
Using D and Ki/Ks, the extent of frequency change on
scattering can be estimated as follows (also shown in
Fig. 1):
1) KsUi1r/vi ; 1 when Ki/Ks  1 and D  1
Significant frequency change occurs (the MTL wave
regime) when the horizontal wavenumber of the topography is much higher than that of the incident
wave, even if the incident and primary reflected waves
are approximately linear.
2) KsUi1r/vi  1 when Ki/Ks  d9  1 and D  1(where d9 is a small parameter)
The scattered wave frequency is much higher than
the incident-wave frequency (the lee-wave regime)
when the horizontal wavenumber of the topography
is higher than that of the incident wave by more than
two orders, if the incident and primary reflected waves
are approximately linear.
3) KsUi1r/vi ; 1 when Ki/Ks ; 1 and D ; 1
When the incident and primary reflected waves arehighly nonlinear, frequency change is significant (the1581

MTL wave regime), even if the topographic wavenumber is on the order of the incident-wave wavenumber.
4) KsUi1r/vi  1 when Ki/Ks  1 and D ; 1
When the incident and primary reflected waves arehighly nonlinear, a much lower topographic wavenumber is sufficient to make the scattered wave
frequency much higher than the incident-wave frequency (the lee-wave regime).
5) KsUi1r/vi  1 when Ki/Ks ; 1 and D  1Frequency change hardly occurs (the internal tide
regime) when the horizontal wavenumber of the
topography is on the order of that of the incident
wave, if the incident and primary reflected waves areapproximately linear. In this regime, the linear
scattering considered in the previous studies occurs.
6) KsUi1r/vi  1 when Ki/Ks  1
Reflection rather than scattering occurs.
Hence, the scattered waves have much higher horizontal wavenumbers than the incident wave when thefrequency change occurs, except for case 3. In cases 1, 2,
and 4, the corresponding topographic wavenumber Kb
is almost equal to the horizontal wavenumber of thescattered waves (Ks) because the topography is very
rough compared to the wavelength of the incident wave.
Note that the wave Froude number D should not begreater than unity because this would be unphysical—theincident wave would break. This parameter range is excluded from the above list.

5. Derivation of leading-order equations for
scattered waves in various regimes
The discussion has hitherto been based on the governing Eqs. (2.1) and the boundary condition (2.3), assuming that the spatial derivative of the primary wave
field is negligible. However, as shown in the previous
section, there are several parameter regimes in which
frequency change occurs, and the difference between
spatial or temporal scales of the primary and scatteredwaves varies among these regimes. Also, the consideration of the effect of the advection by the primary waves
implicitly assumes that the incident wave has finite
amplitude. It is therefore required to know which terms
can be neglected in what regime and the resulting error.
To discuss these issues, we present a more rigorous
derivation that leads to different governing equations
for different regimes, as will be shown in section 5c.

a. Scaling 1: Separation of the scattered wave field
To derive the appropriate governing equations, wefirst isolate the scattered wave field from the total wave

1582

JOURNAL OF PHYSICAL OCEANOGRAPHY

VOLUME 39

FIG. 1. Diagram of the parameter regimes of KsUi1r/vi, which is a measure of the change in scattered wave frequency, as functions of thelength scale of topography with respect to that of incident wave (Ki/Kb) and the wave Froude number of an incident wave (D 5 KiUi/vi).
Here, K, U, and v are the horizontal wavenumber, current amplitude, and frequency, respectively. The subscripts s, i, i 1 r, and b denotescattered, incident, incident and primary reflected waves, and bottom topography, respectively. Also shown are scales of the bottom
topography (K 1b ), current excursion of incident and primary reflected waves (Ui1r/vi), and incident-wave wavelength (li). Significant
frequency change occurs in the shaded regimes. Note that Kb ; Ks when Ki/Kb ; 1 or  1.

JULY 2009

1583

NAKAMURA AND AWAJI

field in this subsection in order to derive equations for
scattered waves through a scaling argument in subsequent subsections.

1) GOVERNING EQUATIONS OF THE TOTAL FIELD
As in the previous sections, we consider the reflectionof a monochromatic internal wave over small-amplitude
bottom topography. Although the discussion here islimited to the topography with no mean slope, it canbe easily extended to the case with a mean slope. The
equations for an inviscid, rotating, and uniformly stratified Boussinesq fluid are›u$p r9g1 (u  $)u 1 f z 3 u 5 
z,
›t
r0
r0
$  u 5 0,
›r9›r
1 u  =r9 1 w 5 0,›t›z3) MULTIPLE SCALE METHOD(5.1)

where standard notations are used and z is a unit vector
in the vertical direction. The density is separated into areference value, background stratification, and perturbation as in (2.4), and the hydrostatic balance for a
resting fluid is subtracted from the momentum equation. The kinematic boundary condition at the bottom is
w 5 uh  $h h 5 0

at z 5 h(xh ).

the results of the nonlinearity of incident and primaryreflected waves are excluded, whereas nonlinear interactions involving scattered waves are all included in the
scattered wave field. This enables a clear separationof scattered waves from incident and primary reflected
waves of higher orders (or higher harmonics) and thus
simplifies the treatment of incident waves. Nevertheless,it should be noted that the definition (5.3) can be unimportant for the case D  1 in which nonlinear terms
are neglected in the leading order and the spatial scaleof incident and scattered waves differs greatly, as shownin Fig. 1.

(5.2)

The governing equations retain the advection terms toconsider the effects of the background flow on scatteredwaves; thus, they cannot be solved in their original form.
To transform these equations into a tractable yet relevant
form, we define the scattered wave field in a manner inwhich it is convenient to apply the multiple scale method.

2) DEFINITION OF SCATTERED WAVE FIELD
As noted in section 2, the total reflected wave fieldcomprises the primary reflected waves on the order of e0and scattered waves on higher orders (e.g., Müller andXu 1992), where e is a nondimensional scale of topographic height; its precise definition in this study is givenin (5.24) or appendix B. Also, the sum of the incident
and primary reflected waves is identical to the solution
for reflection on a plane.
Based on these facts, we define the scattered wave
field as the difference between the solution for reflection on a rough topography and that on a plane; that is,

The discussion in section 4 indicates that when thefrequency change occurs, the horizontal wavenumber ofthe scattered waves is generally much higher than thatof the incident and primary reflected waves because D
should not be greater than unity. We thus utilize themethod of multiple scales by separating the wave field
into large- and small-scale fields. The large-scale field
is scaled by the incident-wave frequency and wavenumbers (which are the same as those of the primaryreflected waves), whereas the small-scale filed is scaledby those of the scattered waves, as given by
›
›
›
5 vs
1 vl ,
›t
›~ts
›~tl
›
›
›
1 kl
,
5 ks
›x
›~
xs
›~
xl
›
›
›
5 ms
1 ml
,
›z
›~
zs
›~
zl

(5.4)

where the large and small scales are denoted by thesubscripts l and s, respectively, and where ; denotes
nondimensional variables.
Then, dependent variables are separated into thosefor the large and small scales and are scaled as~ l ) 1 (U s u~s , V s ~ys , W s w~s)(u, y, w) 5 (U l u~l , V l ~yl , W l w
p 5 Pl p~l 1 Ps p~s ,r9 5 Rl r~l9 1 Rs r~s9 .

(5.5)

Note that the large-scale variables are functions of only
xl, zl, and tl, such as(scattered) def (total)  (reflection on a plane). (5.3)

ul 5 U l u~l (xl , zl , tl ),us 5 U s u~s (xs , zs , ts ; xl , zl , tl ).

This roughly corresponds to an expansion of the wave
field with respect to e. This definition isolates the scattered wave field and thus it is helpful for the scaling of
the governing equations in section 5b. In this definition,

This is because nonlinear interactions do not occur
between a monochromatic incident wave and the primary reflected wave over a horizontal plane. In the caseof sloping topography or nonmonochromatic incident

(5.6)

1584

JOURNAL OF PHYSICAL OCEANOGRAPHY

waves such as tidal beams, nonlinear interactions cangenerate higher harmonics (e.g., Thorpe 1997; Lamb2004). Nevertheless, these higher harmonics can beneglected when D  1 as long as a resonant interaction
does not occur; without resonance, the higher harmonics
are of higher orders of magnitude with respect to D. If
needed, the inclusion of such effects is possible owing to
the definition (5.3).
For simplicity, we consider the bottom topography witha single small length scale of 1/ks, namely,~ s ).
h 5 H h(x(5.7)

We do so because the frequency change is insignificant
when the topography varies only on a large scale [i.e.,h 5 h(xl)], except for case 3 in section 4b. Further, thescattering off topography with various length scales canbe obtained to the leading order by the superposition of
the waves scattered off each topographic Fourier component for small-amplitude topography.

4) RATIOS OF THE SCALES
The multiple scale method yields three ratios of scales:
dv 5

vl
,
vs

dk 5

kl
ml
, dm 5
.
ks
ms

(5.8)

Each ratio must be much smaller or much larger thanunity for the multiple scale method to be applicable tothe corresponding direction (or time). We thus examine
the scales of these ratios.
To scale the ratios, we utilize the dispersion relationand the condition for frequency change. The dispersion
relation of linear internal waves can be approximated as
v ’ Nk/m if f  v  N, where N is the buoyancy
frequency. The above three ratios are therefore related
to each other by
dv ;

dk
dm

(5.9)

when the condition f  v  N is satisfied for both vsand vl. (The cases v ; f and v ; N are discussed inappendix A.)
In section 4, we found that the condition for frequencychange requires
ks U l 1 or ;1.
vl(5.10)The nondimensional parameter ksUl/vl corresponds to
the ratio (4.2) and thus the former regime (ksUl/vl  1)
corresponds to the lee-wave regimes 2 and 4 in section
4b, whereas the latter (ksUl/vl ; 1) corresponds to the

VOLUME 39

MTL wave regimes 1 and 3. This parameter can be rearranged as 
ks U lks
kl U lD
5(5.11)
; ,
dk
vl
kl
vl
where D (5 klUl/vl), which is redefined here and corresponds to (4.4), is now the wave Froude number (ortemporal Rossby number) of the large-scale waves andis thus the measure of their nonlinearity. (Note that D
takes, in general, various values from infinitely smallvalues for infinitesimal waves to around unity for highly
nonlinear waves, and it cannot be specified unless theincoming wave is specified.)
When ksUl/vl  1, it follows that vs  vl. This isbecause in this parameter regime, the scattered wave
frequency is almost equal to the lee-wave frequency,
which can be scaled as vs ; ksUl; thus,
dv 5



vl
ks U l 1
;
.
vs
vl

(5.12)

Note that the last term is much smaller than unity inthis regime. The combination of (5.9), (5.11), and (5.12)
yieldsdv ; d,dk ; dD, dm ; D.

(5.13)

Here, we define a small parameter d as d 5 dv. The
parameter d will be used later to represent the ‘‘small
parameters’’ arising from the multiple scale method(i.e., dv, dk, and dm) by a single parameter.
When ksUl/vl ; 1, two cases are possible: vs ; vl andvs  vl. In the former case (vs ; vl), vs is still on theorder of ksUl and thus (5.12) holds in a similar manner.
Therefore, the use of (5.9), (5.11), and (5.12) yields
dv ; 1, dk ; dm ; d ; D,

(5.14)

where we define d 5 dk because dv (5 vl/vs) ; 1.
The latter case (vs  vl) arises when the Doppler
shift is almost equal to the incident-wave frequency andthus their difference, which is equal to the scatteredwave frequency under consideration, becomes muchsmaller than the incident-wave frequency. In this case,(5.12) does not hold becausevs  vl ; ks U l .

(5.15)

Using (5.15) instead of (5.12) and defining d 5 d1
v , weobtain the following from (5.9) and (5.11):
dv ; d1 , dk ; D,

dm ; dD.

(5.16)

The above scales of the three ratios are summarized inTable 1. It is apparent that the multiple scale method is

JULY 2009

1585

NAKAMURA AND AWAJI

TABLE 1. Scales of dv, dk, and dm. The ratio df is defined as fl/fs where subscripts l and s indicate large and small scales. In the regimes
vs  vl and vs  vl, the relation between d and D is situation dependent. Once the incident wave and bottom topography are given, we
may write dn ; D, for D  1, where n is a natural number when d $ D or the inverse of a natural number when d # D.
RegimeIntrinsic frequency
Scales of ratios in terms of d, D

when D  1 (D ; dn)

when D ; 1

ksUl/vl  1
vs  vl
dv (1) 5 d
dk ; dD
dm ; Ddv (1) 5 d
dk ; dn11  1
dm ; dn  1
dv (1) 5 d ; dk
dm ; 1applicable in most cases. In the linear case D  1, it isapplicable to all the directions and time, except for thecase vs ; vl, in which it is inapplicable to time. Even in
the nonlinear case D ; 1, the multiple scale method isapplicable to the t and x directions in the lee-wave regime (vs  vl) and to the t and z directions in the casevs  vl.
It would be useful if these scales of dv,, dk, and dm
were written in terms of d alone so that the governing
equations for the higher-order scattered waves could be
systematically obtained if desired. To do so, D should be
written in terms of d, when D  1 (it is unnecessary whenD ; 1 because the order of magnitude of D is alreadygiven).
In the regime vs ; vl, (5.14) indicates that d ; D. In
the other two regimes, there is no general relation between d and D because D depends on the properties ofan incident wave, whereas d is determined by both the
topographic wavenumber and the flow excursion of theincident wave (2Ui/vi, where 2Ui 5 Ui1r). Nevertheless,once the incident wave and topography are given, themagnitudes of both D and d become known values.
Then, this specific value of D may be written using thespecific value of d as, for example,
D ; dn .

(5.17)

Here, n is a natural number (if d $ D) or the inverse ofa natural number (if d # D). The value of n should bedetermined according to the situation under consideration. The results of the substitution are shown in Table 1.
On the other hand, the results become significantly simpler when D ; 1, as also shown in Table 1.

b. Scaling 2: Nondimensional equations
By separating the scattered wave field from the total
and using the scales described in section 5a(3), we nondimensionalize the equations of the large- and smallscale fields. The multiple scale method is formally applied to t, x, and z in order to derive nondimensional

ksUl/vl ; 1
vs ; vl
dv ; 1
dk 5 d ; D
dm ; ddv ; 1
dk 5 d  1
dm ; d  1
dv ; dk ; dm ; 1

vs  vld1
v ( 1) 5 d
dk ; Ddm ; dDd1v ( 1) 5 d
dk ; dn  1
dm ; dn11  1
d1
v ( 1) 5 d ; dm
dk ; 1equations that are useful for various parameter regimes.
The applicability of the multiple scale method will betaken into consideration after the nondimensional equations are obtained.

1) LARGE SCALE
From definition (5.3), the large-scale wave field satisfies the original governing equations even without
scattered waves. Also, the large-scale wave field varies
only on the large scale, as noted in (5.6).
The continuity equation thus becomes(kl U l )

~l
›~
ul
›w
1 (ml W l )
50
›~
xl
›~
zl

(5.18)

kl U l ; ml W l .

(5.19)and hence

The nondimensional momentum equations are obtained similarly. We divide the equations by the scale ofeach local temporal change term (instead of the Coriolis
term used in the derivation of quasigeostrophic equations, since the motion being considered comprises propagating internal waves). Using (5.19), it follows that
!


›~
ul
›~
ul
›~
ul
f2
›~
p
~l
~yl 5  l ,
1 D u~l
1w
›~
xl
›~
zl
›~
xl
›~tl
v2l


›~yl
›~yl
›~yl
~l
1 u~l 5 0,
1 D u~l
1w
›~
xl
›~
zl
›~tl
!


~l
~l
~l
k2l ›w
›w
›w
›~
p
~l
5  l ~1 D u~l
1w
rl9 .
2
~
›~
x
›~
z
›~
zl
›tl
ml
l
l
(5.20)In the derivation, we performed the scalings Pl ; r0vlUl/kl,
Rl ; mlPl/g, and Vl ; ( f/vl)Ul. The first implies thatthe horizontal pressure gradient term is on the orderof the local temporal change term, since both terms
are fundamental to internal wave dynamics. The second

1586

JOURNAL OF PHYSICAL OCEANOGRAPHY

implies that the gravity acceleration term is on the orderof the vertical pressure gradient term (i.e., the hydrostatic balance is significant). The last one is for convenience and is valid when vl ; f. In the other case, theCoriolis term is automatically neglected using this form
(because the case vl , f is not considered).
Similarly, the density equation becomes!

k2l N 2
›~
rl9
›~
rl9
›~
rl9
~l
~l ,
5
1 D u~l1w(5.21)
w
›~
xl
›~
zl
›~tl
m2l v2lwhere N 5 [(g/r0 )(dr/dz)]1/2 . Note that the coefficientof the right-hand side is O(1), as obtained from the hydrostatic dispersion relation (v ’ Nk/m).
The bottom boundary condition for the large-scale~ l 5 0 at z~ 5 0, since these waves
waves is given as w
comprise the incident and reflected waves on a horizontal plane.

2) SMALL SCALE
Nondimensional equations of the small-scale waves
are obtained from the use of the multiple scale method(5.4) to (5.6) and the subtraction of the correspondinglarge-scale equation.

(i) Continuity equation

when scaled by ksUs after the subtraction of the largescale continuity Eq. (5.18). Because both dk and dm aremuch less than unity as long as D  1 (Table 1), thethird term in (5.22) must be O(1); hence,(5.23)

When D ; 1, dk or dm (or both) is O(1) and thus themultiple scale method is not applicable along the corresponding direction. Even in such a case, (5.23) is valid
as the scaling for leading-order scattered waves whose
wavenumber is scaled as ks ; kl or ms ; ml (or both).

(ii) Bottom boundary condition
The assumption of small-amplitude topography comprises two conditions: (i) the height of the topography issmaller than the vertical scale of the waves under consideration and (ii) the slope of the topography is smaller
than the wave slope. These conditions are satisfied, as
argued in appendix B, whene 5 Hms  1 (and e & d when dv  1),

where e is the nondimensional topographic height scale
used in this study. Under this condition, the bottom boundary condition (5.2) can be expanded around z 5 0 as
1 2 ›uh    5 0 at z 5 0,hw  $h  (huh )  $h 2›z(5.25)
following Müller and Xu (1992).
Noting that h 5 h(xs) and wl 5 0 at z 5 0, the nondimensional bottom boundary condition, which is scaledby Ws, becomes"   ~
ks U l HUs›h~ s (z 5 0) 5
u~l 1
u~s
w
Ws
›~
xs
Ul
 

  
U s ~ ›~us›~ul
U s ›~ush
1 dk h~
1
1
Ul
›~
xs
›~
xl
U l ›~xl
1 (O(e)).

(5.26)

The condition that the scale of the leading-order terms
on both sides be the same gives 1
ks U l H
Us
1;
; Hms
,
(5.27)
Ws
Ul
using (5.23). It follows that

A nondimensional continuity equation for the small
scale is obtained as



~s
~s
›~
us
›~
us
ms W s ›w›w
5 0, (5.22)
1 dk
1
1 dm
›~
xs
›~
xl
ks U s›~zs›~zl

ks U s ; ms W s .

VOLUME 39(5.24)

e;

Us
.
Ul

(5.28)

This relation should be emphasized. The bottom boundary condition relates the topographic height scale to theratio of the amplitudes of the incident and scatteredwaves (or the large and small scales). The relation iscommon to the case of linear steady lee waves (cf.
Baines 1995) and suggests that Us/Ul  1 for smallamplitude topography, which satisfies (5.24).
The nondimensional boundary condition can be rearranged using (5.27) as


›h~
›~
ul
~ s (z 5 0) 5 u~l
1 dk h~
w
›~
xs
›~
xl
(


›h~ ›~us ~›~us~
h 1 dk h
1
1 e u~s
›~
xs ›~
xs
›~
xl
"

#)
2~
›h~ ›~
ul
1
s
2 › u
~
~
1 dk h1 dm h
›~
xs ›~
2
zl
›~
xs ›~
zs
1 [O(e2 )],(5.29)

where the terms of the next order in e are also written.
It is interesting that the order of each term depends
not only on e but also on dk and dm. This suggests thatthe relative magnitude of these three nondimensionalparameters must be specified to obtain scattered waves

JULY 2009

1587

NAKAMURA AND AWAJI

of various orders. For example, when dk ; e (&dm), theeffect of the spatial variation of an incident wave
~ ul /›~
xl ) is excluded from the leading order and is in(h›~cluded in the second-order scattered waves, which also
include the scattering of the leading-order scattered~ This point has not been
~ xs 1 ›~us /›~
xs h).
waves (~us ›h/›~
considered in the previous studies.

(iii) Momentum and density equations
The nondimensional momentum equations for thesmall scale, which are scaled by the corresponding local
temporal change terms, are rearranged using (5.19), (5.23),
and (5.28) to give
!



D~
us
ks U l
›~
ul
›~
ul
f2
~s
~ys
1 dm w
dk u~s

1
Dt
vs
›~
xl
›~
zl
v2s


›~
ps
›~
p
5
1 dk s ,
›~
xs
›~
xl



D~ys
ks U l dk ›~yldm ›~yl~s
1 u~s 5 0, (5.30)
u~s
w
1
1
Dt
vs
dv ›~
xl
dv ›~
zl
"
#


~s
~l
~l
Dw
ks U l d2k ›w›w~s
1 dk w
u~s
1
Dt
vs
dm ›~
xl
›~
zl
!


m2s
›~
ps
›~
p
1 dm s 1 r~9s ,
5 2
›~
zs
›~
zl
ks
where
D
›
›
1 dv
5
~
Dt ›ts
›~t

l


ks U l
›
›
1
us )
1 dk
(~
ul 1 e~
›~
xs
›~
xl
vs
 

 

dk
dm
›
›
~l 1 e~s. (5.31)
1
1 dm
w
w
›~
zs
›~
zl
dm
dk
Similar to the derivation of the large-scale equations,
we used the scalings Ps ; r0vsUs/ks, Rs ; msPs/g, andVs ; (f/vs)Us.
Similarly, the density equation is nondimensionalized
as
#

"
dv d2m ›~
D~
rs9
ks Ul
›~
rl9
rl9
~s
(dv dm )~
1
us
1
w
Dt
vs
›~
xl
dk
›~
zl
!
k2 N 2~s.
5  s2 2 w
ms vs(5.32)

c. Scaling 3: Leading-order equations
Using the above nondimensional equations and parameters, which are derived without any assumptions
about D or d except that D , 1, this section shows theleading-order equations in D for the large scale, presents
the method to obtain equations of various orders in d forthe small scale, and shows the leading-order equations
for various regimes.

1) LARGE SCALE
To allow a linear approximation for the large-scale
waves, we consider the case D  1 and expand the largescale variables with respect to D, as given by
(0 )

(1)

ul 5 ul 1 Dul 1    .

(5.33)The substitution of (5.33) into (5.20) and (5.21) and therearrangement with respect to D yields large-scale equations of various orders. The leading-order equations(i.e., the zeroth order in D) are
!
(0 )
(0)
›~
ul
›~
pl
f 2 (0 )
~

5

,
y
›~
xl
›~tl
v2l l
(0 )

›~yl
(0 )
1 u~l 5 0,›~tl
! (0 )(0)~l›~plk2l ›w(0 )5
 r~9l ,
›~
zl
›~tl
m2l
!
(0 )
›~
rl9
k2l N 2~ (l 0) ,5w›~tl
m2l v2l(5.34)(0 )~ (l 0)
›~
ul
›w
1
5 0.
›~
xl
›~
zl

These are the governing equations of the primary waves
used in section 2. Higher-order equations, which arerequired to consider the higher-order scattered waves,
can be obtained similarly, although these are omitted
for conciseness.
Note that if we consider monochromatic incident
waves, sinusoidal solutions are possible even if D ; 1 for
the Boussinesq fluid of a constant buoyancy frequency.
Even in this case, the equations for the small-scale field
can be obtained by the method described in the next
section. In other words, taking into consideration theeffects of advection by the primary waves is possible
even for approximately linear primary waves in certain
parameter ranges. This enables us to extend the results
obtained in this study to a case with a mean slope, a casefor polychromatic incident waves, and so on.

2) SMALL SCALES
Small-scale equations and the bottom boundary condition in the various regimes and of various orders ind can be systematically obtained using the following
procedure: (i) represent the parameters dv, dk, and dm interms of d for each regime, as indicated in Table 1; (ii) if

1588

JOURNAL OF PHYSICAL OCEANOGRAPHY

VOLUME 39

TABLE 2. Leading-order equations. The scattered wave equations to the zeroth order in both e and d (or D for the case ks Ul /vl  1).
The equations in the regime ks U l /vl ; 1 and vs  vl are omitted. Note that Ul 5 Ui1r 5 2Ui for a monochromatic incident wave. The
definitions of e, d, and D are provided in (5.24) and in section 5a(4).
D1vs  vl (i.e., ksUl/vl  1), d ; D

vs ; vl (i.e., ksUl/vl ; 1)

vs / vl (i.e., ksUl/vl  1)

D~
u(s0)
›~
p(0)
5 s ,
Dt
›~
xs

!
D~
u(s0)
›~
p(0)
f 2 (0)~y 5  s ,
2
Dt
›~
xs
v2s s

!
›~
u(s0)
›~
p(0)
f 2 (0)~y 5  s
2
›t
›~
x
v2s s

~ (s0)Dwm2
5  2s
Dt
ks

!"
#
›~
p(s0)
(0)
1 r~9s›~zs!
9(0)k2 N 2D~rs~ (s0)
5  s2 2 w
Dt
ms vs

D~ys
1 u~(s0) 5 0
Dt!"#~ (s0)
Dw
m2s ›~
p(s0)
(0)
5 2
1 r~9s
Dt
›~
zs
ks
!
D~
rs9(0)
k2 N 2~ (s0)
5  s2 2 w
Dt
ms vs

›~ys
1 u~(s0) 5 0›t!"#~ (s0)›w
m2s ›~
p(s0)
(0)
5 2
1 r~9s
›t
›~
zs
ks
!
›~
rs9(0)
k2 N 2~ (s0)
5  s2 2 w
›t
ms vs

~ (s0)
›~
u(s0)
›w
1
50
›~
xs
›~
zs

~ (s0)
›~
u(s0)
›w
1
50
›~
xs
›~
zs

~ (s0)
›~
u(s0)
›w
50
1
›~
xs
›~
z

where

where

where

D
›
(0) ›
5
1 u~l
Dt
›~
xs
›~ts
~
(0) ›h(0)~ s 5 u~lat z 5 0
w
›~
xs

D
›
›
(0) ›
~ (l 0)
5
1 u~l
1w
Dt
›~
xs
›~
zs
›~t
~
(0) ›h(0)~ s 5 u~lw
at z 5 0›~xs~ (s0) 5 u~(l 0)
w

(0)

(0)

(0)

›~
u
›h~
1 h~ l›~xs›~x

at z 5 0

D;1
vs  vl (i.e., ksUl /vl  1)

vs ; vl (i.e., ksUl/vl ; 1)

vs / vl (i.e., ksUl/vl  1)

!

(0)

›~
u
D~
u(s0)
›~
p(0)
~ (s0) l 5  s
1w
Dt
›~
z
›~
xs

D~
u(s0)
›~
p(0)
f 2 (0)(0)~y 5  s
1 As (~ul ) 
Dt
›~
x
v2s s
(0)

~ (s0)Dwm2
5  2s
Dt
ks

!"
#
›~
p(s0)
(0)
1 r~9s
›~
z
(0)

!

›~
r9
D~
r9s(0)
k2 N 2~ (s0) l~ (s0)
1w
5  s2 2 w
Dt
›~
z
ms vs
~ (s0)
›~
u(s0)
›w
50
1
›~
xs
›~
z
where
D
›
(0) ›
5
1 u~l
Dt
›~
xs
›~ts
~
›
h
(
0
)
~ (s0) 5 u~lw
at z 5 0
›~
xs

D~ys
(0)
1 As (~yl ) 1 u~(s0) 5 0
Dt!"#~ (s0)
Dw
m2s ›~
p(s0)
(0)
(0)
~
1 As (wl ) 5  2
1 r~s9
Dt
›~
z
ks
!
D~
r9s(0)
k2 N 2(0)~ (s0)
1 As (~
r9l ) 5  s2 2 w
Dt
ms vs~ (s0)
›~
u(s0)
›w
1
50
›~
x
›~
z
›f
›f
~ (s0)
1w
where As (f) 5 u~(s0)
›~
x
›~
z
D
›
(0) ›
(0) ›
~l
5
1w
1 u~l
Dt
›~
x
›~
z
›~t
(
0
)
~
›~
u
›
h
(
0
)
~ (s0) 5 u~l
1 h~ lat z 5 0
w
›~
x
›~
x

dv, dk, or dm is O(1), quit applying the multiple scale
method along the corresponding direction (e.g., if dv ; 1,then substitute dv 5 1 and set ›/›~ts 1 dv ›/›~tl back tovs ›/›~t ); (iii) relate e and d in a manner similar to relating d and D by assuming that em ; d; (iv) expand thevariables with respect to d in a manner similar to (5.33);
and (v) substitute the above parameters and variables
into the nondimensional equations and boundary con-

No scattered waves(Reflection)

dition (5.22) and (5.29)–(5.32) and rearrange them withrespect to d, noting that ksUl/vs ; 1 when dv & 1 while
ksUl/vs ; dv when dv  1. Then we obtain equations ofvarious orders for various regimes.
The leading-order equations in both d and e for thevarious regimes are shown in Table 2. The equations
and/or the bottom boundary condition are different
for different parameter regimes. The difference in the

JULY 2009

NAKAMURA AND AWAJI

governing equations implies that the dispersion relationof scattered waves differs and hence affects the determination of the vertical wavenumber of scattered waves,the propagation after the scattering, and so on.
The governing equations and the bottom boundary
condition used in section 2 correspond to those in theregime where vs ; vl and D  1 (and hence ksUl /vl ; 1
and ki/ks  1). Note that the equations of this regime
include the leading-order equations of the other two
regimes (ksUl/vl  1 and  1) in the corresponding
limits when dn ; D  1, where n $ 1 for ksUl/vl  1.
Accordingly, the examination of the intermediate
regime ksUl/vl ; 1 will reveal the basic concept of
scattered waves in these three regimes. In contrast, theneglect of the advection of the primary wave field assumed in section 2 is no longer appropriate for highly
nonlinear waves for which D ; 1. Other features of thegoverning equations and the bottom boundary condition are described as follows:
When vs ; vl and D  1 (and hence ksUl /vl ; 1 andki/ks  1): The equations in this intermediate regime
vs ; vl for an approximately linear incident wave (D  1)
to the zeroth order in both d and e are those shown insection 2. The equations are quasi-linear and thus havenearly sinusoidal solutions.
When vs  vl and D  1 (i.e., ksUl/vl  1 andki/ks  1): The leading-order equations in this regime
are almost the same as those when vs ; vl and D  1, if
we consider the case that d ; D. (The order of e is notvery important as long as em ; d  1). The differences(0)
are twofold. First, ~ys 5 0 because f 2 /v2s is smaller than
dv (5 d). Second, the vertical advection term disappears
in the leading order; that is,D›(0 ) ›
5
1 u~l
.
~
Dt
›~
xs
›t s

(5.35)

When vs  vl and D  1 (and hence ksUl/vl ; 1 andki/ks  1): The small-scale equations in this regime arecomplicated. Because vs  vl, the incident and primaryreflected waves have a small time scale whereas thescattered wave field has a large time scale. Further, theleading-order equations become~ (l 0)w
ð0Þ

›f(s0)
50›~
zs(5.36)

when f ; vs, where fs is the scattered wave variable ofthe zeroth order, and the equations for the next-orderscattered waves involve the higher-order terms. The
derivation of appropriate equations needs a careful
treatment and is omitted here.
When vs / vl and D  1 (i.e., ksUl/vl  1 andki/ks ; 1): This is the regime that previous studies have1589

considered. Indeed, we regain the usual linear scatteredwave equations from the nondimensionalized Eqs. (5.22)
and (5.29)–(5.32), noting that ks U l /vs ; Ddv d1
k ; D.
When D ; 1: Even for a highly nonlinear incident
wave (D ; 1), the governing equations are tractable in
the regime vs  vl. The equations of the leading order(d0) are the same as those for D  1 and vs  vl, except
for two differences. One difference is that the multiple
scale method is not applicable to the z direction. The
other is that both the momentum equation for the x
direction and the density equation now include the advection of the large-scale variables by the scattered(0 )(0)~ (s0) ›~~ (s0) ›~ul /›~z and w
r9l /›~z, respectively) atwave (i.e., w
the leading order.
As for the other two regimes, the leading-order equations become nonlinear in the regime ksUl/vl ; 1 (andvs ; vl) and scattering hardly occurs in the regime
ksUl/vl  1 (i.e., vs / vl).

6. Effects and implications
Some possible effects and implications of frequencychange on scattering are discussed in this section. The
discussion, however, is limited to showing that these arelikely to occur and leaves quantitative investigations for
future works.

a. Effects on the change in vertical wavenumber
The change in the vertical wavenumber upon scattering is considered to be important for causing boundary mixing. Thus, the effects of frequency change on thechange in the vertical wavenumber upon scattering are
discussed first.
Consider the ratio of the vertical wavenumber ofscattered waves (ms) to that estimated assuming frequency conservation (ms0). It is given by
!1/2
!


2 1/2
2
2
2
 ms 
v

f
N

v
s
i


,
m  5 N 2  v2s0v2s  f 2
i

(6.1)

since the vertical wavenumber is determined from thedispersion relation (2.8) and the horizontal wavenumbers are the same. To consider the basic change according to the parameter regimes, let us approximate thedispersion relation as v ’ Nk/m. Then, the above ratio
becomes1    ms   vi  ks ul ’  ’m  v   v 6 1 ,
s0
s
i

(6.2)

where ul 5 ui 1 ur [given in (2.21)]. This states thatthe change in the vertical wavenumber depends on the

1590

JOURNAL OF PHYSICAL OCEANOGRAPHY

VOLUME 39

TABLE 3. Change in vertical wavenumber due to frequency change when D  1; ms6 and ms0 are the vertical wavenumbers of thescattered waves and those calculated by assuming frequency conservation, respectively. Subscripts s and i represent the scattered andincident waves, respectively, and subscripts 1 and 2 denote the sum and difference frequency waves, respectively.
RegimeIntrinsic frequency
Vertical wavenumber

ksUi1r/vl  1
vs  vi
ms6  ms0

ksUi1r/vl ; 1
vs ; vi
ms1 , ms0

nondimensional parameter ksUi1r/vi, which is a measure of frequency change (4.1) because Ui1r 5 Ul 5 2Ui
in section 2. This dependence is summarized in Table 3.
When ksUi1r /vi  1 (i.e., the lee-wave regime), thevertical wavenumber of the scattered waves is much lower
than that estimated assuming frequency conservation.
This is less favorable for causing boundary mixing.
When ksUi1r/vi ; 1 (i.e., the MTL wave regime) andvs ; vi, the values of ms and ms0 are different but areon the same order. The vertical wavenumber of the sum
waves (|ksul/vi| 1 1) becomes lower (i.e., ms , ms0),
whereas that of the difference waves (|ksul/vi| 2 1) becomes higher. The former is less favorable for boundary mixing; the latter is more favorable. In particular,when vs  vi, the vertical wavenumber becomes muchhigher (i.e., ms  ms0), which is favorable for boundary mixing.
In this manner, the occurrence of frequency change
affects the energy redistribution in the vertical wavenumber space. Accordingly, the estimates of boundary
mixing due to internal wave scattering based on theenergy redistribution in the vertical wavenumber can beaffected by the frequency change.

ms2 . ms0

When viewed in the frame fixed to the ground, thehorizontal phase velocity nearly vanishes (i.e., crp 1
ul|z50 ’ 0) at the time of the generation in the lee-wave
regime. Accordingly, the scattered waves are trappedhorizontally at the generation site and could effectively grow through superposition. The scattered-wave
propagation has many similarities to that of the tidally
generated internal waves presented in Nakamura andAwaji (2001).

c. Critical levels
As well as the advection of scattered waves, the flow
associated with the primary wave field will cause therefraction of scattered waves as the waves propagate
away from the generation site. One interesting implication of this is the presence of critical levels, whereU 2 cp 5 6f/k and where the wave energy converges
and can cause mixing and background-flow acceleration(cf. LeBlond and Mysak 1978). In the present case, cp isthe horizontal phase velocity of the scattered waves observed in the frame fixed to the ground and is given as
cp 5 6b. Horizontal phase velocity
The horizontal phase velocity of the scattered waves
relative to the moving fluid becomes [from (3.2)]vvi
5 ul jz50, t5t0 6
ub kb
kb
"

 #
ul jz50, t5t0
2kb U i 1
5 2U i 
6
.
2U i
vi

crp 5

vi
kb

The ratio 2kbUi/vi, which corresponds to (4.1), characterizes the scattered-wave propagation. Because the magnitude of the nondimensional advecting flow, |ul/2Ui|,does not exceed unity, the propagation of scatteredwaves is dominated or deeply affected by the advection
due to the primary wave field when 2kbUi/vi  1 (thelee-wave regime) or when 2kbUi/vi ; 1 (the MTL wave
regime), respectively. In contrast, when the frequencychange is negligible (i.e., 2kbUi/vi  1), the advection is
not effective.

(6.4)

at the time of the generation. The background horizontal flow velocity U is associated with the large-scale
waves ul and thus its maximum absolute value becomes
2Ui. Using these values, a critical level can be present if
2U i  vi /kb $ f /kb .

(6.3)

vs  vi
ms2  ms0

(6.5)

At the same time, the condition for frequency change
states from (4.1) that the primary wave field has the
maximum flow speed of 2Ui * vi/kb. Hence, in mostcases, there are one or more critical levels within one
vertical wavelength of the large-scale waves (2p/mi). In
such cases, a large portion of the scattered waves could
encounter a critical level somewhere in the course of
their upward propagation.
In particular, scattered waves would encounter acritical level when they propagate upward so fast thatwith the initial group velocity, the waves would travel
over the distance 2p/mi before the temporal variation of
the large-scale wave field can become significant; that is,

JULY 2009

NAKAMURA AND AWAJI (2p/vi )(2p/m )/c
i

gz



  1.
(6.6)

Here, cgz is the initial vertical group velocity of thescattered waves and can be approximated as 2vs/ms
when N 2  v2s  f 2 . Using the scattered wave frequency (3.2), (6.6) becomes(2p/vi )mi 1’(ks ul 6 vi )2(2p/mi )/cgz vi Nks
2
ki ks ul’
6 1  1,
ks vi
2

(6.7)

 v2i  f 2

is also used andwhere the condition N
where ks ’ kb, in the present parameter range. The
above situation can take place in the lee-wave regime
where ksul/vi  1, if kb is sufficiently high that
D2  dk(6.8)

is satisfied. In this situation, the scattered waves willencounter a critical level within one vertical incident
wavelength from the bottom and a large portion of the
waves will be absorbed there, causing both mixing andthe acceleration of the large-scale wave flow.
This suggests the possibility that the scattered wavesof the lee-wave regime could break at a level well above
the bottom when generated by an incident wave of a low
vertical mode or could break near the bottom whengenerated by an incident wave of a high vertical mode.

d. Nonmonochromatic incident waves
Thus far, we have focused on the case in which theincident wave was monochromatic. In realistic situations, however, incident waves are usually polychromatic and the flow often has a steady component. A
solution for nonmonochromatic incident waves can beobtained in a similar fashion to section 2, as long as D  1
so that the incident waves can be linearly superposed.
(A solution for the topography considered above can beeasily obtained but is omitted for conciseness.)
In such a case, the Doppler shift (2kbul) is caused by
the total large-scale flow associated with all of the incident waves, their primary reflected waves, and otherflow components. However, if a scattered-wave solution
is obtained separately for each incident-wave Fouriercomponent, the Doppler shift term does not take thetotal flow into account, but only the flow associated with
only one incident-wave component and its primary reflected wave. Because of this difference in the Doppler
shift, the scattered wave frequency and hence the vertical wavenumber are not correctly obtained from asolution for each incident-wave component.

1591

The above argument has an important implication
that, in the parameter regimes where the frequencychange occurs, the scattered waves associated withnonmonochromatic incident waves can no longer be
represented by the superposition of the scattered wave
solutions obtained separately for each incident-wave
Fourier component. This fact will make it difficult toanalytically estimate the energy made available formixing due to scattering.

7. Summary
The scattering of internal gravity waves off a roughtopography redistributes the incoming energy flux in thewavenumber space, which often causes enhanced crossisopycnal mixing near the bottom (boundary mixing).
This process is considered to contribute significantly to
the globally averaged mixing and could affect the global
thermohaline circulation (e.g., Müller and Xu 1992). In
previous studies on such internal wave scattering and/orreflection, it was assumed that the frequency is conserved. However, the present study has shown that thefrequency changes during reflection from a rough topography in a certain parameter range, which includes
the realistic range.
To demonstrate this fact analytically, we have derived sets of appropriate and tractable governing equations for scattered waves by a scaling argument, assumingthat the amplitude of the topography is small. The derivation procedure has two points: the use of the multiple
scale method and the definition of a scattered wave field.
The multiple scale method is useful because in mostcases, the scattered waves and the incident (and primaryreflected) waves have very different scales in time and/or
space when change in frequency occurs, as argued in
section 4. The ratios of the scales of the scattered andincident waves, in turn, can be determined using thedispersion relation and the condition for the change inthe frequency; they are summarized in Table 1.
Also, for a clear separation of the scattered waves andincident and primary reflected waves, we define thescattered wave field as the difference between the total
wave field and the wave field created by the reflection
on a plane. This definition roughly corresponds to the
expansion of the wave field with the nondimensional topographic height scale e. The definition requires that theincident and primary reflected wave field exactly satisfy
the governing equations, and thus it simplifies the treatment of incident waves even in the nonlinear regime.
The combination of these two points allows us tosystematically derive the leading-order equations forvarious parameter regimes; that is, D  or ; 1, and/or
KbUi1r/vi , ;, or  1, where D is the wave Froude

1592

JOURNAL OF PHYSICAL OCEANOGRAPHY

number for the incident waves and Kb, Ui1r, and vi are
the topographic wavenumber (which is on the order ofthe scattered wave wavenumber), the flow amplitudeof the sum of incident and primary reflected waves,and the incident-wave frequency, respectively. Althougha heuristic argument such as the one described in section 2 is helpful for understanding the basic idea, thelengthy but more rigorous derivation shown in section 5,
which provides the order of each term, is useful for thefurther development of the theory. For example, themethod shown here also enables us to derive higher-orderequations (although the equations are not shown for conciseness). In addition, the extension to a three-dimensionalcase or to the case with a mean slope is straightforward
although the equations shown here are for a vertically
two-dimensional case with no mean slope.
A solution in the parameter regime dv ; 1 and D  1(i.e., KbUi1r /vi ; 1 and Ki/Kb  1, where Ki is thehorizontal wavenumber of an incident wave) is shown
for a simple case in which both the incident wave andthe topographic roughness are monochromatic. Thissolution is archetypal because it includes those in theother two regimes of KbUi1r/vi  and  1 in the corresponding limits, when D  1.
The solution indicates that the intrinsic frequencies ofthe scattered waves are the sum and difference of theincident-wave frequency and the Doppler shift (or thefrequency of quasi-steady lee waves). This Doppler shift
causes the change in the frequency. Hence, the assumption of frequency conservation is not valid if the Doppler
shift is significant. This condition is met in the case thatthe horizontal scale of the bottom roughness (or thelength scale on the plane of the slope) is on the order of, ormuch smaller than, that of the incident-wave flow excursion, since the horizontal wavenumber of the scattered waves is mainly determined from that of the bottomroughness in this parameter regime. Simple estimates ofthis condition in section 4a suggest that the frequencychange could be a common phenomenon in the ocean.
The underlying physics of the frequency change is thesame as that in the case of the internal wave generation
by a barotropic tidal flow over a topographic feature,
which was shown in earlier investigations by Nakamura
et al. (2000) and Nakamura and Awaji (2001); nevertheless, the generalization to internal wave scattering hasvarious implications, such as those summarized below.
First, the frequency change in the scattering affects
the energy redistribution in the vertical wavenumber
space. This will in turn influence the strength of theinternal wave-induced boundary mixing. In addition,
the resulting energy redistribution in the frequency
space could affect the shape of the internal wave spectra, at least locally. Second, when the frequency change

VOLUME 39

occurs, the scattered waves are affected significantly by
the background flow associated with the incident andprimary reflected waves. This results in the advectionof the scattered waves so that the waves are arrested
horizontally near the generation site and superposed to
grow. Further, the background flow causes the refraction of the scattered waves. In most cases, critical levels
appear and can cause both mixing and flow acceleration.
In particular, the scattered waves in the lee-wave regime(i.e., KbUi1r/vi  1) would encounter a critical level,
which can take place well above the bottom for lowvertical-mode incident waves and near the bottom forhigh-mode incident waves. Finally, the scattered waves
associated with nonmonochromatic incident waves
cannot be obtained by summing the solutions for each
incident-wave Fourier component even at the leading
order if the frequency changes. This is because theDoppler shift depends on the total flow at the bottom.
These results suggest that an estimate of the energy
redistribution under the frequency change would be oneof the necessary steps toward estimating the boundary
mixing induced by the scattering and in understanding
the maintenance of the spectrum shape of oceanic internal waves, which affects both interior and boundary mixing. For this goal, important issues still remain.
For example, the effects of large-amplitude topography,
wave nonlinearity, polychromatic incident waves, andnonconstant N are out of scope of this study. In particular, the consideration of large-amplitude topography isimportant because the ratio of the amplitude of thescattered and incident waves is on the order of the nondimensional topographic-height scale e 5 Hms and hence
is much less than unity for a small-amplitude topography,
as the scaling argument in section 5b(2) suggested. These
effects should be included for the application to the realistic situation and are left for future works.
Acknowledgments. We sincerely appreciate the helpful comments of three anonymous reviewers. This study
was supported by Category 7 of the MEXT RR2002
Project for Sustainable Coexistence of Humans, Natureand the Earth and by a Grant-in-Aid for the 21st Century COE Program (Kyoto University, G3). T.N. waspartly supported by Grant-in-Aid for Scientific Research.
This study was done when T.N. was with the Departmentof Geophysics, Kyoto University.

APPENDIX A
Applicability around v ~ N or v ~ f
For scaling the ratios dv , dk, and dm in section 5a(4),we used the approximate dispersion relation (v ’ Nk/m)

JULY 2009

1593

NAKAMURA AND AWAJI

that results in relation (5.9), dv ; dk/dm . This approximation of the dispersion relation becomes invalid asv approaches N or f. Here, we consider the frequency
range in which dv is on the order of dk/dm so that relation(5.9) is valid, and thus the scalings in section 5 based on
this relation are appropriate.
When v ; N, the dispersion relation (2.13) can beusually approximated as
v2 5

N 2 k2(A.1)

2

k 1 m2

because N  f in most cases. Hence, if k2 1 m2 and m2
are on the same order, v is on the order of Nk/m andhence (5.9) is valid. This condition is met if k2 & m2, that is,
v2 &

N2
.
2

(A.2)

N 2 k2
1 f 2.
m2(A.3)

Accordingly, (5.9) is valid when N2k2/m2 * f 2. Thiscondition can be rewritten using (A.3) as
f2
v * .
2
2

(A.4)

This condition may be relaxed to some extent, if boththe incident frequency and the scattered frequency areon the order of f (i.e., vl ; vs ; f ). The ratio dk /dm now
becomesd2k

v2l  f 2dmv2s  f 2

52.

The nondimensional scale of topographic height (e)used in section 5 is defined here based on the assumption of a small-amplitude topography, which consists of
two conditions: (i) The height of the topography is muchsmaller than the vertical scale of the waves under consideration and (ii) the slope of the topography is smaller
than the wave ray slope.
Condition (i) is required to expand the bottom boundary condition at z 5 0 and is satisfied if
Hml  1 andHms  1,kb H

ml
, 1 and
kl

kb H

ms
, 1,
ks

(B.3)

where kb is the wavenumber of the corresponding topographic Fourier component. When vs  vl or vs ; vl,
these inequalities are satisfied if
e2 5kb
Hml , 1,kl

(B.4)

since |ms/ks| / |ml/kl| ; vl/vs & 1 in these regimes using
the approximate dispersion relation, v ’ N|k/m|. Conversely, when vs  vl, (B.3) requireskb
Hms ; Hms , 1,
ks

(B.5)

where the scaling kb ; ks is used (see section 4b).
The scales e1 and e2 are related as follows:
(A.6)

dk
e2 for vs  vl ,dmdk
; e2 ; e2 for vs ; vl ,dm
; e2 for vs  vl .

e1 ;are satisfied, dv and dk/dm are on the same order. The
inequalities can be approximated aspﬃﬃﬃﬃﬃ
10
and vs .
f,
3

(B.2)

because dm(5 ml/ms) is much smaller than or on theorder of unity, as shown in Table 1.
Condition (ii) states that there is no critical or supercritical slope. This condition is met if

e2 5

2

v2  f
1 2, 10d2v
dv , l10v2s  f 2

(B.1)

where H is the topographic height scale introduced in
section 5a(3). These two inequalities are satisfied if(A.5)

Thus, when the inequalities

pﬃﬃﬃﬃﬃ
10
fvl .
3

Nondimensional Topographic Height Scale ee1 5 Hms  1,

Similarly, when v ; f, the hydrostatic approximation
is usually valid, so that
v2 5

APPENDIX B(A.7)

assumingp
vﬃﬃﬃﬃﬃs ; vl, which yields 10vs  vl and 10vl  vs.
Because 10/3 ’ 1.054, the intrinsic frequency should
be greater than f by at least 5% for the scaling in section
5 to be applicable.

(B.6)

Either e1 or e2 is available for the expansion of the bottom boundary condition and/or the reflected wave field.
Here, we choose e1 as the nondimensional topographic
height scale e because the definition of e2 is not common
to the three-parameter regimes. From (B.6) and (B.5),
conditions (i) and (ii) can be written in terms of e as

1594

JOURNAL OF PHYSICAL OCEANOGRAPHYdk(;d)  1 for vs  vl ,dm
 1 for vs ; vl or vs  vl .

e(5 e1 ) &(B.7)

Note that the relation between e and d is specified by theabove scaling only in the casevs  vl :REFERENCES
Baines, P. G., 1971a: The reflexion of internal/inertial waves frombumpy surfaces. J. Fluid Mech., 46, 273–291.
——, 1971b: The reflexion of internal/inertial waves from bumpy
surfaces. Part 2. Split reflexion and diffraction. J. Fluid Mech.,49, 113–131.
——, 1995: Topographic Effects in Stratified Flow. Cambridge
University Press, 498 pp.
Balmforth, N. J., G. R. Ierley, and W. R. Young, 2002: Tidal
conversion by subcritical topography. J. Phys. Oceanogr., 32,
2900–2914.
Bell, T. H., 1975a: Lee waves in stratified flows with simple harmonic time dependence. J. Fluid Mech., 67, 705–722.
——, 1975b: Topographically generated internal waves in the open
ocean. J. Geophys. Res., 80, 320–327.
Bryan, F., 1987: Parameter sensitivity of primitive equation
ocean general circulation models. J. Phys. Oceanogr., 17,
970–985.
Eriksen, C. C., 1982: Observation of internal wave reflection fromsloping bottoms. J. Geophys. Res., 87, 525–538.
——, 1985: Implications of ocean bottom reflection for internal wave spectra and mixing. J. Phys. Oceanogr., 15,1145–1156.
Gilbert, D., and C. Garrett, 1989: Implications for ocean mixingof internal wave scattering off irregular topography. J. Phys.
Oceanogr., 19, 1716–1729.
Hasumi, H., and N. Suginohara, 1999: Effects of locally enhanced
vertical diffusivity over rough bathymetry on the world ocean
circulation. J. Geophys. Res., 104, 23 367–23 374.
Johnston, T. M. S., and M. A. Merrifield, 2003: Internal tide scattering at seamounts, ridges, and islands. J. Geophys. Res., 108,
3180, doi:10.1029/2002JC001528.
Khatiwala, S., 2003: Generation of internal tides in an ocean offinite depth: Analytical and numerical calculations. Deep-Sea
Res., 50, 3–21.

VOLUME 39

Lamb, K. G., 2004: Nonlinear interaction among internal wave
beams generated by tidal flow over supercritical topography.
Geophys. Res. Lett., 31, L09313, doi:10.1029/2003GL019393.
LeBlond, P. H., and L. A. Mysak, 1978: Waves in the Ocean.
Elsevier Scientific, 602 pp.
Legg, S., 2004: Internal tides generated on a corrugated continental slope. Part I: Cross-slope barotropic forcing. J. Phys.
Oceanogr., 34, 156–173.
——, and A. Adcroft, 2003: Internal wave breaking at concave andconvex continental slopes. J. Phys. Oceanogr., 33, 2224–2246.
Lighthill, M. J., 1978: Waves in Fluids. Cambridge University Press,
504 pp.
Mied, R. P., and J. P. Dugan, 1976: Internal wave reflexion froma sinusoidally corrugated surface. J. Fluid Mech., 76,
259–272.
Müller, P., and N. Xu, 1992: Scattering of oceanic internal gravity
waves off random bottom topography. J. Phys. Oceanogr., 22,474–488.
——, and X. Liu, 2000a: Scattering of internal waves at finite topography in two dimensions. Part I: Theory and case studies.
J. Phys. Oceanogr., 30, 532–549.
——, and ——, 2000b: Scattering of internal waves at finite topography in two dimensions. Part II: Spectral calculations
and boundary mixing. J. Phys. Oceanogr., 30, 550–563.
Munk, W. H., 1966: Abyssal recipes. Deep-Sea Res., 13, 707–730.
——, and C. Wunsch, 1998: Abyssal recipes. II: Energetics of tidal
and wind mixing. Deep Sea Res. I, 45, 1977–2010.
Nakamura, T., and T. Awaji, 2001: A growth mechanism for topographic internal waves generated by an oscillatory flow.
J. Phys. Oceanogr., 31, 2511–2524.
——, ——, T. Hatayama, K. Akitomo, T. Takizawa, T. Kono,
Y. Kawasaki, and M. Fukasawa, 2000: The generation oflarge-amplitude unsteady lee waves by subinertial K1 tidal
flow: A possible vertical mixing mechanism in the Kuril
Straits. J. Phys. Oceanogr., 30, 1601–1621.
Phillips, O. M., 1977: The Dynamics of the Upper Ocean. Cambridge University Press, 336 pp.
Rubenstein, D., 1988: Scattering of inertial waves by rough bathymetry. J. Phys. Oceanogr., 18, 5–18.
St. Laurent, L., and C. Garrett, 2002: The role of internal tides
in mixing the deep ocean. J. Phys. Oceanogr., 32,
2882–2899.
Thorpe, S. A., 1997: On the interactions of internal waves reflecting from slopes. J. Phys. Oceanogr., 27, 2072–2078.
——, 2001: Internal wave reflection and scatter from sloping roughtopography. J. Phys. Oceanogr., 31, 537–553.