432
J O U R N A L OF P H Y S I C A L O C E A N O G R A P H Y
Volume 43
A Note on the Role of Mean Flows in Doppler-Shifted Frequencies
Theo G er k em a, Leo R. M. M a a s, a n d H ans v a n H ar en
N IO Z Royal Netherlands Institute fo r Sea Research, Texel, Netherlands
(Manuscript received 21 May 2012, in final form 28 September 2012)
ABSTRACT
The purpose of this paper is to resolve a confusion that may arise from two quite distinct definitions of
“Doppler shifts": both are used in the oceanographic literature but they are sometimes conflated. One refers
to the difference in frequencies measured by two observers, one at a fixed position and one moving with the
mean flow—here referred to as “quasi-Doppler shifts." The other definition is the one used in physics, where
the frequency measured by an observer is compared to that of the source. In the latter sense, Doppler shifts
occur only if the source and observer move with respect to each other; a steady mean flow alone cannot create
a Doppler shift. This paper rehashes the classical theory to straighten out some misconceptions. It is also
discussed how wave dispersion affects the classical relations and their application.
1. Introduction
A m uch studied situ atio n in physical o ceanograp h y is
lin ear w ave p ro p ag a tio n in th e p resence o f a m ean flow.
R ig h t at th e outset, it is useful to identify th e th ree key
players in this problem : th e w ave source, th e observer,
an d th e m e an flow. T he source and o b se rv e r m ay m ove
relative to each o th e r as well as relative to the m ean flow.
In th e m id n in ete en th century, C hristian D o p p le r id e n ­
tified th e c irc u m sta n c e s u n d e r w hich th e fre q u e n c y
m e asu red by an o b serv er will be d ifferen t from th e
frequency at w hich th e source em its its w aves. In phys­
ics, this difference is called th e D o p p ler shift.
In th e o ceanographic literatu re , how ever, th e te rm is
o ften used in an o th e r sense, fo r exam ple, in B re th e rto n
an d G a rre tt (1968), O lbers (1981), K unze (1985), an d
B ü h ler (2009). T h eir startin g p o in t lies in th e eq uatio n s
o f m otion, w here a m ean flow U gives a com binatio n of
term s
F o r sinusoidal w aves sin(wf — kx), w ith frequency co an d
w aven u m b er k, this becom es co — Uk. It is th e n com m on
to define co1 = co — Uk, th e so-called “ intrinsic fre ­
q u ency.“ L ighthill (1978) com m ents o n th e n o m en cla­
tu re o f this expression:
Corresponding author address: Theo Gerkema, Royal NIOZ,
P.O. Box 59,1790 AB Den Burg, Texel, Netherlands.
E-mail: [email protected]
DOI: 10.1175/JPO-D-12-090.1
© 2013 American Meteorological Society
"In the context of moving sources, equation [co' = co —
Uk\ has long been called the D oppler relationship, and it
is natural to call it by this nam e also for waves propa­
gating through non-uniformly moving fluids” (p. 326, our
italics).
T hus, it has b eco m e com m o n to sp eak of U k as a
“ D o p p le r sh ift,“ re fe rrin g to th e d ifferen ce b etw e en
th e freq u en cy co m e asu red by an o b serv er a t rest an d th e
freq u en cy co’ m e asu red by an o b se rv e r m oving w ith the
m ean flow. So, h ere th e te rm is used to indicate th e dif­
ference in frequency betw een two observers, in contrast
to n o rm al parlan ce in physics, w here it refers to th e dif­
ference in frequency b etw een source and observer.
F o r th e sake o f clarity, w e shall re fe r to th e D o p p le r
shift in th e fo rm er, “ B re th e rto n /G a rre tt“ sense as quasiD o p p ler shift, reserving th e te rm D o p p ler shift fo r its
n o rm al m ean in g in physics. T h e d istin ctio n is im p o rtan t,
fo r an o b serv er m ay ex p erien ce a q u asi-D o p p ler shift,
w hile having no D o p p le r shift— an d vice versa. A m ean
flow can b e said to im ply a q u asi-D o p p ler shift, b u t a
m e an flow ca n n o t cre ate a D o p p le r shift.
Is this m erely a m a tte r of sem antics? In th e afo re ­
m e n tio n e d referen ces, sources w ere left o u t o f th e
eq u atio n ; th e w aves are sim ply assum ed to b e th ere. T he
fixed o b serv er m easu res freq u en cy co, w hile th e o b serv er
m oving w ith th e flow m easu res co1, an d th ese frequencies
can be co m p ared (q u asi-D o p p ler shift), b u t no co m ­
pariso n w ith th e forcing freq u en cy can b e m ad e because
n o n e is involved. So far, no confusion is possible. B ut, as
so o n as o n e applies th ese th e o ries to a specific o c e a n o ­
graphic situ atio n , sources inevitably com e in to play— for
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G E R K E M A ET AL.
exam ple, th e forcing o f n ea r-in e rtial w aves by th e w ind
o r o f in te rn al tides by tid al flow ov er topography. In ­
deed, in o b servatio ns it is m o re com m on to lo o k into th e
rela tio n b etw een th e m e asu red frequency and the fo rc­
ing frequency th a n it is to com p are a frequency from
a m o o red in stru m e n t w ith sim ultaneous m easurem en ts
fro m a d rifte r. It is, th e re fo re , cru cial fo r th e c o rre c t
in te rp re ta tio n of o ceanographic d a ta to u n d ersta n d
precisely u n d e r w h at circum stances a D o p p le r shift m ay,
o r m ay not, occur. H ow ever, th e distinct m eanings of
q u asi-D o p p le r shifts and D o p p le r shifts are som etim es
conflated. It is, fo r exam ple, n o t uncom m on to see th e
n o tio n o f a q u asi-D o p p ler shift being m o rp h ed into th e
(in correct) id e a th a t a m ean flow w ould give rise to a
D o p p ler-sh ifted frequency at a m ooring. Som e instances
are discussed in section 4.
T he p u rpose of this p a p e r is tw ofold: to bring into
clear focus th e distinction b etw een q u asi-D o p p le r shifts
an d D o p p le r shifts, and to provide a concise overview of
th e role o f m ean flows in D o p p le r shifts. R e g ard in g th e
la tte r point, this p a p e r has partly th e c h a racter o f a r e ­
view; m uch o f it w as already know n by th e late n in e ­
te e n th ce n tu ry , alth o u g h th e se insig h ts se em to h av e
som ew hat slipped into oblivion. A n additio n al elem en t
is the inclusion of dispersion. T he classical theory typi­
cally considers dispersionless w aves (sound, light w aves),
b u t in th e oceanographic context, dispersion is usually
im portant.
W e sta rt by reh ashing th e basic th e o ry o f D o p p le r
shifts (section 2) follow ing its historical origins. This
section also serves to stress th e im p o rtan ce of specifying
an d distinguishing th e m ovem ents o f th e source, m e­
dium , an d observer. W e th e n exam ine an analytical ex ­
am ple in w hich n o t only th e classical D o p p le r rela tio n
fo r th e shift in frequency is derived, b u t also expressions
fo r am plitude an d w avelength (section 3). U n til here,
w aves are assum ed to be dispersionless. In section 4, it is
show n th a t th e effects o f dispersion can be s tr a ig h tf o r ­
w ard ly a c c o m m o d a te d in th e classical r e la tio n fo r
D o p p le r shifts. In section 5 w e discuss som e o b se rv a­
tions on in te rn al w aves from th e lite ra tu re an d revisit
th e ir in te rp re ta tio n w ith reg ard to D o p p le r shifts. C o n ­
clusions follow in section 6 ; h ere w e also discuss, briefly,
th e effect o f n o n stead y m ean flows but, in preced in g
sections, steadiness is assum ed.
2. Recapitulation of the Doppler theory
O f th e several pap ers in w hich D o p p le r laid o u t th e
effect n am ed afte r him , th re e are o f p articu la r in te rest
here. T hey w ere published in 1842, 1846, and 1847 an d
la te r re p rin te d in D o p p le r (1907). T his b o o k is m o re
useful th a n th e original p ap ers th an k s to extensive notes
by H . A . L o ren tz, w hich co n tain n u m ero u s co rrectio n s
an d clarifications.
In his th e o ry o f 1842, D o p p le r distinguished th e tw o
cases o f a m oving source and a m oving observer. T he
m ed iu m is assum ed to b e at rest. W aves are em itte d
from th e source at p erio d Ts; th ey p ro p ag a te aw ay at
p hase sp eed c. A n im p o rta n t restriction, h ere an d in th e
follow ing section, is th a t w aves are assum ed to b e dis­
persionless. In o th e r w ords, th e p ro p e r phase sp eed c is
a co n stan t an d is in d e p en d e n t o f th e w ave period; it
d ep en d s solely o n th e p ro p ertie s o f th e m ed iu m (e.g., on
te m p e ra tu re in th e case o f sou n d w aves).
T h e first case is o n e in w hich th e o b serv er is at rest,
w hile th e source m oves. L et its sp eed b e vs, d irected
to w ard th e observer. E ac h w ave crest has to brid g e th e
d istance b etw een th e p o sitio n o f em ittan ce an d th e p o ­
sition o f th e observer, b u t this d istan ce decreases by an
am o u n t o f vsTs fo r every n ex t w ave crest becau se o f th e
m o v em en t of th e source. T h e p erio d b etw een th e arrival
o f successive crests is th e re fo re n o t Ts, b u t th e D o p p ler
shifted Ti = Ts — vsT Jc = Ts( l — v j c ) . T h e w avelength
being shifted as Ai = (c — vs)Ts, th e rela tio n c = Ai/T) is
satisfied from th e view point of th e observer.
T h e second case is on e in w hich th e source is at rest
w hile th e o b serv er m oves. Suppose th e o b serv er m oves
aw ay from th e source at sp eed v 0. A t a m o m en t w hen
a w ave crest passes th e observer, th e follow ing crest is
still at a d istance A = cT s, th e w avelength. A p p ro ach in g
th e o b serv er at a relativ e sp eed of c — v 0, th a t follow ing
crest will arrive afte r a tim e in terv al T2 = Al(c — v 0) =
77(1 — v j c ) . This is now th e D o p p le r shifted period.
T h ere is no change in w avelength in this case.
T h e first case w as so o n p u t to th e te st by th e physicist
(and la te r m eteorologist) Buys B allot. In 1845, on a newly
built steam train n ea r U trech t, h e h ad m usicians play
tones o n a tru m p et, w ith o th e r m usicians standing along
th e railw ay to register th e pitch of th e tones before and
afte r th e passage of th e train. T h e change in p itch was
fo u n d to b e in good ag reem en t w ith D o p p le r's form ula.
This was in itself a triu m p h an t confirm ation of his p rin ­
ciple, b u t D o p p ler's chief in terest lay in th e application to
th e color of th e stars, w hich proved to b e less straig h t­
forw ard (for m o re o n this history, see T o m an 1984).
In his 1846 p ap er, D o p p le r co n sid ered th e situ atio n in
w hich th e source an d o b serv er are b o th in m ovem ent.
T hus, com bining th e previous tw o cases, th e observ ed
p erio d T becom es
T=Ts
C — V
U
C ~ Vo
(2)
F o r a p ro p e r ap p licatio n of this form ula, positive values
sh o u ld be assigned to vs an d v 0 fo r m o v em en t to th e
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J O U R N A L OF P H Y S I C A L O C E A N O G R A P H Y
V o l u m e 43
th e o b serv ed freq u en cy m ust b e th e sam e as th e fre ­
quency of em ittance. This rem ark ab ly sim ple argum ent,
given by L o re n tz , h an g s o n th e sta tio n a rity o f th e se t­
tin g an d rem a in s v alid if th e m e a n flow is sp atially
no n u n ifo rm .
By th e late n in e te e n th century, this resu lt h a d already
fo u n d its w ay in to th e tex tb o o k s, fo r exam ple, L o rd
R ay leig h (1896, p .154):
o\ er
or
Fig. 1. Illustration from Doppler's 1847 paper (Doppler 1907),
showing the effect of a medium M flowing to the left (as indicated
by the arrows). The source is at rest at Q, as are the observers at A
and B. The mean flow produces a leftward translation of the cir­
cular wave pattern; at this instant, previously generated wave crests
have their centers at O, O1, O11, etc. Here the mean flow is assumed
to be lower than the proper phase speed of the waves (otherwise no
waves would move to the right at all).
right; negative values, fo r m o v em en t to th e left. H o w ­
ever, w e will ta k e c always positive. If th e o b serv er is to
th e left o f th e source, w e are co n cern ed w ith w aves
m oving to th e left and, accordingly, should replace c
by —c in th e form ula.
E x pression (2) has an obvious b u t im p o rta n t co ro l­
lary: if th e source an d o b serv er m ove at th e sam e sp eed
an d in th e sam e d irectio n (i.e., vs = v 0), no D o p p le r shift
in frequency occurs. B u t then, in a fram e o f referen ce
th a t m oves to g e th e r w ith th e source an d observer, th e
m e d iu m will be p erceived to be m oving. O n e thus arrives
im m ediately at th e conclusion th a t a m oving m edium
creates no D o p p le r shift in frequency.
T his brings us to th e case co n sid ered by D o p p le r in
his 1847 p ap e r, in w hich he studied th e effect of a m ov­
ing m edium o n w aves in e th e r (i.e., light w aves), air, or
w ater. T he source an d o b serv er are h ere assum ed to be
at rest. F o r an o b serv er lo cated u p stream (A in Fig. 1),
th e w aves trav el at a low er speed and th e w avelength is
sh o rten ed , w hile fo r an o b se rv e r lo cated d ow nstream
(B ) th e sp eed and w avelength are enlarged. B ut, w hat
h ap p en s to th e frequency? F o r an o b serv er at any fixed
position, successive w ave crests p ro p ag a te u n d e r id e n ­
tical circum stances from source to o b serv er an d hence
need equal tim es to bridge that distance; as a consequence,
"If the source and the observer move with the same v e­
locity there is no alteration of frequency, w hether the
m edium be in m otion, or n o t.”
T h e only w ay th e o b serv ed freq u en cy can b e h igher
(o r low er) is w h en th e rela tio n b etw e en source an d o b ­
serv er is nonstatio n ary : if th ey m ove to w ard (o r aw ay
from ) each o th er. In th a t case, successive w aves do n o t
p ro p ag a te u n d e r id en tical circum stances, fo r th e dis­
tan ce b etw een source an d observer, w hich th ey have to
bridge, changes.
T h e d ifferen t situations, discussed h ere, are elegantly
ca p tu red in a single fo rm u la d eriv ed by B a te m an (1917):
A gain, w e ad o p t th e co n v en tio n w ith reg ard to th e sign
o f c as in ( 2 ).
B a te m an 's E q. (3) is a n a tu ra l g en eralizatio n o f (2) in
th a t th e p ro p e r p hase sp eed c in th e la tte r is b eing re ­
p laced by th e effective p hase sp eed c + U. E q u a tio n (3)
confirm s th a t a m e an flow U does n o t c re ate a D o p p le r
shift; a m ean flow m erely m odifies an alread y existing
D o p p le r shift if th e source and o b serv er are in relativ e
m o v em en t (i.e., vs v0). T his fo rm u la will b e red eriv ed
in th e n e x t se c tio n , w h e re w e e x p lo re an a n a ly tic a l
exam ple.
3. An analytical example
W e will exam ine th e p ro p ag atio n of lin ear long in te r­
facial w aves in th e p resen ce o f a stead y an d u niform
b ack g ro u n d flow U. T his sim ple exam ple allow s us to
p in p o in t, by m eans o f an explicit solution, th e distinct
ro le s o f th e m o v e m e n t o f th e so u rce , m e d iu m a n d
o b se rv e r.
a. B a sic eq ua tio n s
W e assum e w ave p ro p ag a tio n in th e h o rizo n tal x d i­
rec tio n in a tw o-layer system ; fo r th e o th e r h o rizo n tal
d irection: S/Sy = 0. W ith in each lay er th e h o rizo n tal
c u rren t velocity asso ciated w ith th e (long) w aves is in ­
d e p e n d e n t of th e vertical; th e se v elo cities a re called iq
an d » 2 , fo r u p p e r a n d lo w er layer, resp ectiv ely . T h en
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G E R K E M A ET AL.
u = u2 — ui, th e change in h o riz o n ta l velocity across
th e in te rfa ce , satisfies th e m o m e n tu m e q u a tio n
(4)
111 + Uux = ~ S*V X,
w ith g* den o tin g red u c ed gravity; 17 is th e interfacial
displacem ent. W e ignore th e C oriolis force here, b u t will
discuss its dispersive effect in section 4.
F ro m th e continuity eq u a tio n and th e b o u n d ary co n ­
d itions (rigid lid at surface, flat b o tto m ), o n e obtains
*x+c(t—r)
dr
v ( ^ ) = Tc
di € F T(r,€ ).
(9)
x —c (t—t )
F ro m ( 6 ) w e o b tain , after tran sfo rm atio n to th e m oving
fram e o f referen ce an d tak in g th e deriv ativ e to t ,
F t {t , Û = (os eo s(wsr ) G '( f + ( U -
vs) t )
+ { U ~ vs) sin(Wsr ) G " ( f + ( U - vs) r ) .
T he in n er integral o f (9) now yields, after som e rew riting,
r)t + U r)x + hjpUx = F(t, x ) .
(5)
T iere h * is th e red u ced w ate r d ep th (h * = h i h 2l(hi + h 2),
w ith h¡ th e u n p e rtu rb e d thickness o f layer i).
In (5), an ad hoc source is in tro d u ced th ro u g h th e
forcing te rm F. F o r la te r use, w e specify th e source as
a w av em ak er oscillating at frequency cos an d m oving at
sp eed vs:
F { t,x ) = sin(w j ) G ' ( x — V t).
X+CU TW
(sin (WsT ) [ G ( l j - G ( X +)]}
T= f
x -a t-t )
0T
+ csin (w sT ){ G '(JW ) + G ' ( ! + )},
(10)
w ith X ± = x — ( ± c ) t + ( ± c + U — vs) t . T aking th e in ­
te g ral Jgdr---, th e first te rm o n th e rhs of ( 1 0 ) vanishes
entirely, so (9) becom es
(6)
F o r positive (negative) vs, th e source m oves to th e right
(left). H ere, G ' is th e derivative, w ith resp ect to its a r­
gum ent, of an arb itra ry function G. This w ay o f w riting
tu rn s o u t to be convenient in w h at follows.
W ,x ) = 2
dTsm(ü)sT ) { G '( X _ ) + G ' ( ! + )}.
T ran sfo rm in g back to th e original fram e o f referen ce
(x = x — Ut), w e g et th e final result:
b. So lu tio n
T o facilitate solving (4) and (5) we m ake a transform a­
tion to a coordinate system moving along w ith the flow:
x = x — Ut;
an d sim ilarly fo r
rule
17
u ( t,x ) = u ( t,x + Ut)
and F. H en ce, by applying th e chain
u t = üt — Uü-\
x’
an d sim ilarly fo r
(5) becom e
17.
(7)
uX = ü -X
V(t,x)=-
dTsm(o)sT ) { G \ X _ ) + G '( X + )},
(11)
w ith X ± = x — ( ± c + U)t + ( ± c + U — vs) t .
T h e rem ain in g in teg ral in (11) can b e solved an aly ti­
cally if w e choose a (m oving) p o in t source, d escrib ed by
a d e lta distribution: G ' = 8. F ro m th e g en eral p ro p erties
o f d e lta d istrib u tio n s (see, e.g., M essiah 1961), o n e has
5(JT± ) = S (( ± c + U -
v
){ t -
t
%)) = -
8{ t — T *)
± c + U —u j ’
T ran sfo rm ed to this system , (4) an d
w ith
= [ (± c + U )t — x ]/(± c + Í7 —vs). T h e d eltad istrib u tio n 8 { X ± ) yields a c o n trib u tio n if 0 < t J < 7,
a n d is z e ro o th erw ise. T hus, fro m (11) w e can w rite th e
so lu tio n as
iit = - g * f 7 r]t + h*ii~ = F ( t ,x ) .
T h ey can be co m bined into o n e eq u a tio n fo r
17:
V = V(8 )
w ith c = (g*/i*)1/2, th e phase speed o f w ave p ro p ag a tio n
in th e absence o f a back g ro u n d flow. L ike in th e p revious
section, w e define c to be positive (unlike U and vs). F o r
initial conditions f), tj, = 0 , startin g from a system at rest,
th e solution o f ( 8 ) reads (see, e.g., Z a u d e re r 1989, §4.5):
(12)
+ i?4
w ith
sin(wsT *)
V±{ t , x) =
21
± c + Í7 —u
if
0
< T * < 7,
an d j]± = 0 for all o th e r t %. In th e absence of m ean flows,
th e term s
an d 17+ in ( 1 2 ) describe w aves propagating
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J O U R N A L OF P H Y S I C A L O C E A N O G R A P H Y
o b serv er 1
observer 2
o
m ean flow
U
V o l u m e 43
a) U=0, V =0
1
0
-1
>
b) U=0.5C, V = 0
F ig . 2. Sketch of a situation used to illustrate the implications of
(13). A source, initially located halfway between two fixed ob­
servers, may move to the right. The presence of a rightward mean
flow is also considered.
1
0
■1
C) U=C, V =0
1
0
■1
to th e left and right, respectively. U sing th e definition of
t %, w e can w rite r¡+ m ore sim ply as
d) U=1 5c, V =0
1
0
V+(t,x) =
21 ± c + U —
smw
( ± c + U)t — x
± c + U — v„
■1
(13)
fo r all x satisfying e ith e r ( ± c + U)t < x < vst o r vst < x <
( ± c + U)t; for all o th e r x, r)± = 0 .
c. R esu lts
W e first consider th e case o f tw o fixed observers, as
illu strated in Fig. 2. O ne is lo cated to th e left of th e initial
lo cation of th e source at a distance o f tw o w avelengths;
th e o th e r to th e right o f th e source, also at tw o w ave­
lengths. (W avelength is h ere defined in th e absence of
a m ean flow an d w ith a source at rest.) T he signal r e ­
ceived by th e observers, 17, is calculated from (12), (13).
In Fig. 3 th ey are p lo tte d in gray an d black, respectively.
W ave forcing starts at t = 0. F igure 3a show s th e re fe r­
ence case, w hich has no m e an flow and no m o v em en t of
th e source; as expected, b o th observers sta rt to receive
th e sam e signal after tw o forcing periods.
In Figs. 3b,c,d, w e add a backg ro u n d m ean flow U, b ut
th e source still d oes n o t m ove. C learly, th e observ ed
w ave p erio d is unch an g ed in each o f th ese cases; this
confirm s th e classical result, discussed in th e previous
section, th a t a background flow does n o t create a D o p p ler
shift in frequency. In Fig. 3b, th e effective phase speed
becom es 3c/2 for rightw ard propagating waves, leading
to an earlier arrival at the observer's position and a re ­
duction in am plitude (black line). F o r leftw ard p ro p a­
gating waves, the background flow reduces the effective
phase speed to c/ 2 , leading to a la ter arrival and an en ­
han ced am plitude (gray line). In Figs. 3c,d the flow b e­
com es to o stro n g fo r th e w aves to p ro p a g a te leftw ard ,
an d th e o b se rv e r o n th e left d etec ts no w aves at all. In
Fig. 3d, they are even drifted to the right so th a t the o b ­
server on the right starts to receive them after four w ave
periods.
In Figs. 3e,f w e have no backg ro u n d flow; instead, th e
source m oves to th e right. N ow w e see, as expected ,
a D o p p le r shift in frequency. In Fig. 3e, w e see th e
source passing th e o b serv er o n th e right at t = 4; at th a t
e) U=0, V =0.5c
1
0
-1
f) U=0, V =0.9c
1
0
■1
g ) U=0.5c, V =0.5c
1
0
■1
0
1
2
3
4
5
6
time {forcing periods)
7
8
9
10
Fig. 3. Solutions of interfacial displacement 17, defined by (12)
and (13) for various U and vs, showing the received signal for an
observer two wavelengths to the left of the (initial) position of the
source (in gray), and another observer two wavelengths to the right
(in black).
in stan t, th e w ave freq u en cy d ro p s sharply. F ro m th e n
on, b o th observers d etect th e sam e signal in term s of
am plitude and (D oppler-shifted) w ave period. T he nearcritical case is show n in Fig. 3f, w here w e see a “ b o o m "
w hen th e source passes th e o b serv er o n th e rig h t (the
p ea k is cu t off in th e figure, its am p litu d e is, in fact,
nearly fo u r tim es la rg e r th a n show n).
T h e co m b in ed effect of a m oving source an d b ack ­
g ro u n d flow in show n in Fig. 3g; th e D o p p le r shift d u e to
th e m oving source is now m odified by th e b ack g ro u n d
flow (cf. Fig. 3e).
d. D isc u ssio n
W e can readily ad ju st (13) to th e case in w hich th e
o b serv er m oves as well, by th e su b stitu tio n x = x 0 + v0t.
T h e arg u m en t o f th e sine in (13) th e n becom es
± c + U —v o
-c o t-± c + U —u
± c + U — v„
(14)
This expression n eatly sum s up all th e possibilities. T he
first te rm in (14) gives th e D o p p le r shift in frequency:
February
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437
G E R K E M A ET AL.
± c + U — vs s '
': Ë
Fig. 4. Solution (13) after 10 forcing periods with the source at
rest atx = 0 and in the presence of a background flow U = c/2. This
case corresponds to the case of Fig. 3b. On the horizontal axis, x has
been scaled by the wavelength A = cT, being the unperturbed
wavelength in the absence of a background flow.
± c + U —v o
± c + U — tu
co' = co — U k
(q u asi-D o p p ler s h ift),
(17)
(15)
T h e m ovem ent o f th e o b serv er (v0) im plies a D o p p le r
shift in frequency, unless it m oves at th e sam e velocity
as th e source (v0 = vs). A m ean flow U m odifies th e
D o p p le r shift if th e source and o b serv er have a relativ e
m ovem ent, b u t ca n n o t by itself cre ate a D o p p le r shift.
T h e expression (15) confirm s exactly th e classical resu lt
(3) by B a te m an (1917).
F ro m th e second te rm in (14) w e see th a t th e w ave­
n u m b e r is m odified by a m o v em en t of th e source an d by
a m ean flow:
k =■
± c + U — v„
N otice th a t both are affected by th e m o v em en t o f th e
source and, w h en ev er source an d o b serv er are in re la ­
tive m o v em en t, by th e m ean flow as well. W a v en u m b er
k, given by (16), is also affected by th e m e an flow a n d th e
m o v em en t o f th e source. B u t to g eth er, th ey form th e
sim ple and fam iliar rela tio n
(16)
This, too, is fully in ag reem en t w ith th e classical results
(see section 2). A n exam ple is show n in Fig. 4, illus­
tratin g the effect o f a m ean flow o n d ow nstream an d
u p stream p ro p ag atin g waves.
T he analysis o f S ection 3b yields, as an ex tra result, th e
am plitude facto r l / |± c + U — vs\ in (13), dem o n stratin g
how th e am plitude is influenced by the m ean flow (see
also Fig. 4) an d by th e m ovem ent o f th e source.
e. Q u a s i- D o p p le r shifts
R e tu rn in g now to th e ¿/z/asz-Doppler shift (as w e call
it), w hich is th e concept o ften u sed in th e oceanograp h ic
literatu re , as discussed in th e introduction. In th e quasiD o p p le r shift, U k is th e difference b etw e en tw o o b ­
served frequencies, one by a fixed o b serv er an d one by
an o b serv er m oving along w ith th e m ean flow. W e can
now easily verify this relation. A ccording to (15), a fixed
o b serv er (v 0 = 0 ) m easures frequency
±c+ U
± c + U — v„
A n o b se rv e r m oving along w ith th e m ean flow (v 0 = U),
m eanw hile, m easures frequency
w hich is valid w h atev er th e value of vs. P recisely because
o f this inv arian ce fo r vs, how ever, th e rela tio n hides th e
fact th a t co1, co, an d k are, each individually, affected by
th e m o v em en t o f th e source. T o unveil this d ep en d en ce,
o n e has to exam ine th e D o p p le r shift (sensu stricto ), as
above.
4. Dispersive waves
So far, w e have co n sid ered dispersionless waves; th e
p hase sp eed w as given by a co n stan t c, w hich is in d e ­
p e n d e n t o f th e w ave p erio d . It d ep en d s only o n p ro p ­
erties o f th e m edium ; fo r exam ple, in th e previous
section, it w as given by c = ( g j i f ) 1'2.
If w e co n sid er a m oving source (speed vs to w ard an
o b serv er at rest) em ittin g dispersive w aves at forcing
p erio d Ts, w e can again use th e reaso n in g o f th e th ird
p ara g ra p h o f section 2 , b u t w ith on e im p o rta n t m odifi­
cation: th e em itte d w aves now trav el at som e (as yet
un k n o w n ) sp eed C. M o reo v er, a distinction has to be
m ad e b etw een th e w aves traveling ah ead an d b eh in d
th e source, fo r th e ir p h ase speeds will b e different.
In all o th e r respects, th e e a rlie r reaso n in g is valid r e ­
gardless o f w h eth e r th e w aves are dispersive o r not: each
w ave crest has to brid g e th e d istance b etw een th e posi­
tio n of em ittan ce an d th e position o f th e observer, b u t
this d istance d ecreases by an am o u n t o f vsTs fo r every
n ex t w ave crest because o f th e m o v em en t o f th e source.
T h e p erio d b etw een th e arrival o f successive crests is
th e re fo re n o t Ts, b u t th e D o p p le r shifted
C — V,
T = T -
C
(18)
This eq u atio n , how ever, is n o t im m ed iately useful b e ­
cause it involves an u n k n o w n T as w ell as an un k n o w n C,
w hich itself d ep en d s o n th e D o p p ler-sh ifted T. A dis­
p ersio n rela tio n is n e e d e d to close th e problem .
A s an exam ple, w e will in tro d u ce disp ersio n d u e to
th e C oriolis force (p a ra m e te r ƒ ), know n from P o in care
w aves. W e also include a stead y u n ifo rm b ack g ro u n d
flow U:
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J O U R N A L OF P H Y S I C A L O C E A N O G R A P H Y
* = U k + ( f + c.2 k7,2)\ 1/2
(19)
W ith co = I ttI T and th e definition of phase speed C = colk,
w e can solve C from (18) an d (19):
_ -(V /
C\ =
V o l u m e 43
term s o f a d o u b le in teg ral involving th e B essel func­
tio n J 0 (Z a u d e re r 1989). C hoosing again a m oving p o in t
so u rce , o n e o f th e se in te g ra ls ca n b e so lv ed . A fte r a
tran sfo rm atio n to th e original co o rd in ates, th e final ex ­
pression reads
- Ucoj) ± <oa[< ?tâ - f 2) + { U ~ v f f 2]1'2
.,2
_
z2
=
(20)
T hus, th e tw o ro o ts o f th e p hase sp eed C are expressed
in term s o f know n p aram eters. W ith this, th e D opp lershifted p erio d T can now be calcu lated from (18).
E q u a tio n (20) d em o n strate s th a t th e phase speed of
dispersive w aves is n o t only affected by a m ean flow b ut
also by th e m ovem ent o f th e source.
F inally, a m ovem ent o f th e o b serv er does n o t in any
w ay change th e characteristics o f th e w ave itself (in
p articular, it leaves C unaffected). So, th e reasonin g of
section 2 dealing w ith a m oving o b se rv e r can be applied
regardless o f dispersion. T he upsh o t is a gen eralizatio n
o f B a te m an 's form ula:
c ± ~ vs
T = TSCZ - v „
o
A c t -~ TZ
A r -- Tv -- U t+ ( U - v )t]
x An L
" \, I c2(r
X ©{c2(f — t
)2
— [x — Ut + ( U — us) r] 2} .
(23)
H e re © is th e H eav isid e step fu n ctio n (eq u al to o n e for
positive arg u m en t, zero elsew here). T h e in teg ral in (23)
is n o t am en ab le to fu rth e r analytical tre a tm e n t, b u t it
can b e easily solved num erically. F o r various values of
vs, U, an d ƒ w e have ch eck ed w h e th e r th e w ave p erio d
fo u n d num erically is th e sam e as th e on e o b ta in ed from
th e p ro ce d u re (18) an d (20). In all cases, th ey w ere fo u n d
to b e in ag re em e n t to w ith in n u m erical accuracy, c o n ­
firm ing th e co rrectn ess of th e above pro ced u re.
(21)
5. Observations and their interpretation
H e re C± is again given by (20), follow ing fro m (18) [not
from (21)!] and th e dispersion rela tio n (19). T he m ean
flow e n ters (21) via C but, as before, does n o t by itself
lead to any D o p p le r shift.
N o tic e th a t fo r ƒ = 0, (20) w ould le a d b ac k to C± =
± c + U, th e effective p hase sp e ed in th e dispersion less
case, red u cin g (21) to (3).
A n altern ativ e ap p ro ach to exam ining th e effects of
dispersion w ould be to red o the analysis o f section 3 w ith
dispersion included. T o som e extent, this is feasible, b ut
th e p ro b lem can no longer be fully solved analytically,
alth ough a closed-form solution can be o btained.
F o r exam ple, if w e ad d C oriolis term s to (4) an d (5),
we g et as a starting point, w ith an additio n al eq u a tio n for
th e transverse velocity co m p o n en t v:
111 + Uux ~ f v = ~ g*V x
v ( + U vx + fit = 0
r)t + U r)x + hipUx = F(t, x ) .
A fte r a tran sfo rm atio n to a system m oving w ith th e
m ean flow, this set can be red u c ed to one eq u atio n , for
exam ple, fo r u:
+ / 2» = - g * F A t , x ) ,
d r sinwT
—
(22)
a forced K le in -G o rd o n equation. A s befo re, c is th e
co n stan t c = (g*/¡*)1/2. E q u a tio n (22) can be solved in
T h e conflation of q u asi-D o p p le r shifts an d D o p p le r
shifts has som etim es featu red in interpretations of m oored
o b serv atio n s o n in te rn al w aves (e.g., F ran k ig n o u l 1970;
W h ite 1972; O rv ik an d M o rk 1995). T h e conflation
com es ab o u t by using th e rela tio n fo r q u asi-D o p p ler
shifts (U k) w hile exam ining th e co n n ectio n b etw een a
forcing freq u en cy an d an o b serv ed frequency, o n w hich
th e q u asi-D o p p le r rela tio n has, in fact, no bearing. In ­
stead , th e p e rtin e n t th eo ry is th a t o f D o p p le r shifts.
In a revisit of th ese in te rp re tatio n s, th e first q u estio n
th a t arises is w h eth e r th e re really w as a “ sh ift" in th e
first place. In this resp ect it is of g reat significance th a t
th ese studies focused o n n ea r-in e rtial w aves ra th e r th a n
o n in te rn al tides. In te rn a l tid es are fo rced at precise an d
w ell-know n freq u en cies (sem idiurnal lu n a r M 2 etc.). In
co n trast, th e forcing o f n ea r-in e rtial w aves is m o re
blurred. First, because th e m echanism prim arily resp o n ­
sible for th e ir g en eratio n , g eo strophic ad ju stm en t, g e n ­
e r a te s a ra n g e o f n e a r-in e rtia l fre q u e n c ie s. S eco n d ,
becau se th e n o tio n o f “ in e rtia l" changes, by definition,
w ith latitu d e. So, if on e finds a sp ectral p ea k at ƒ + S
(local C oriolis p a r a m e te r /) , does this signify a D o p p le r
shift o r does it sim ply resu lt from g en e ratio n at a h igher
la titu d e w h ere th e local ƒ w as larger? T h ese u n ­
certain ties give ro o m fo r seeing “ D o p p le r shifts" w hen,
in fact, n o n e is there.
F o r in te rn al tides, o n th e o th e r han d , th e co n trad ictio n
b etw e en th e ex p ected “ D o p p le r sh ifts" (on th e basis of
February
2013
G E R K E M A ET AL.
439
th e q u asi-D o p p ler relatio n !) an d th e absence th e re o f in
th e m o o red o b serv ations w as m o re p aten t. T o find a w ay
o u t o f this co ntradiction, one h ad to reso rt to ra th e r
im plausible explanations: F ran k ig n o u l (1970) puts fo r­
w ard th e id e a th a t the w aves com e from all directio n s so
th a t th e D o p p le r effects cancel in th e m ean; W hite
(1972) assum es th a t the spectral tidal p eak is entirely
barotropic, in w h ich case th e p re su m e d D o p p le r sh ift
w o u ld be negligible.
T hese p roblem s d isa p p ea r at once w hen one realizes
th a t m e a n flows c a n n o t c r e a te a D o p p le r sh ift. Y et,
th e y affec t th e o b s e rv a tio n s in o th e r w ays, as w e w ill
no w discuss.
a. N ear-inertial w aves
F o r n ea r-in e rtial w aves w e can use th e dispersio n r e ­
latio n (19). (This e xpression is b ased o n th e “ T rad itio n al
A p p ro x im atio n ,“ w hich m ay n o t always b e valid, b u t for
sim plicitly it is assum ed to hold h ere.) T he dispersion
rela tio n w ith an d w ith o u t a m e an flow is show n in Fig. 5.
T h e m ean flow brings ab o u t tw o changes. F irst, fo r a
given frequency th e w av en u m b er is m odified. Second,
th e low er b o u n d o f th e range o f allow able frequencies
decreases. T he frequency can now a ttain values low er
th a n ƒ. C o n trary to w h at w as suggested by Z h a i e t al.
(2005), how ever, this does n o t im ply a D o p p le r shift.
T h ere is no difference b etw een th e e m itte d and received
frequencies fo r sources and observers th a t do n o t m ove
relativ e to each o th e r, see (21). F igure 5 sim ply m eans
th a t w aves forced at frequencies slightly below ƒ can now
freely p ro p ag a te aw ay, w hereas they could n o t if no
m ean flow w ere present.
A ll this is n o t to say th a t D o p p le r shifts can n o t occur
fo r n ea r-in e rtial waves. A fte r all, w e m ay be d ealing w ith
a m oving source if th e w aves are due to th e passage of
an atm ospheric depression. In a m o o red instru m en t, th e
observed frequency will th e n be different from th e forc­
ing frequency but, since th e forcing involves a range of
n ea r-in e rtial frequencies, th e actual D o p p le r shift m ay
b e difficult to estab lish . A n o th e r c o m p lic a tio n arises
from th e fact th a t o n e n eeds to ta k e into account th e
angle b etw een th e trajecto ry o f th e source and th e line
co n n ectin g th e source a n d observ er. T his asp ect goes
b ey o n d th e scope o f this p a p e r b u t has b e e n d e a lt w ith
in th e lite ra tu re (e.g., Y o u n g 1934).
b. I n tern a l tides
F o r in te rn al tides, g e n e rate d o v er to pography , th e
source is n o t m oving. If th e w aves are d e te c te d at a fixed
m ooring, th e re will be no D o p p le r shift in frequency,
irrespective o f th e p resence o f backg ro u n d flows.
A lth o u g h g en e ratio n ov er to p o g rap h y is th e typical
an d m ost im p o rta n t m echanism fo r in te rn al tides, an
Fig. 5. The dispersion relation (19) plotted for two different
cases: without mean flow (dashed), and with mean flow (solid).
Notice that in the latter case, u> attains values lower than/.
altern ativ e m echanism can b e at w o rk in som e configu­
ratio n s, n am ely w h en th ey are p ro d u ce d dynam ically by
a h o rizo n tal d isp lacem en t o f fro n ts (O u an d M aas 1988).
A s fro n ts h av e sloping isopycnals, h o rizo n tal tid al m o ­
tio n pushes isopycnals periodically aw ay from th eir
original position. T h e situ atio n is co m p licated since th e
d ensity g rad ien t associated w ith th e fro n t is norm ally in
g eo strophic eq u ilib riu m w ith th e C oriolis force acting on
a sh e ared m ean flow p arallel to th e front. B ut, th e tide
displaces this g eo strophic flow laterally as well, such th a t
o v er a flat b o tto m th e g eo strophic eq u ilib riu m is actually
re ta in e d . T o g e n e ra te in te rn a l tid e s th is b a la n c e of
forces n eed s to b e b ro k en , e ith e r by b o tto m slopes (O u
an d M aas 1988) o r by instability o f th e g eo strophic front.
T h e com plexity o f th e tid al conversion w ith in a sh eared
fro n ta l system m akes it h ard to q u an tify th e p articu lar
sp eed vs a t w hich th e source m oves. B u t o n e can th in k of,
fo r exam ple, m esoscale v ariatio n s th a t le ad to a m o v e­
m en t o f th e in tersectio n b etw een th e fro n t (w hich itself
oscillates w ith th e b aro tro p ic tid al flow) an d a to p o ­
graphic featu re. T h e lo catio n of th a t in tersectio n form s
th e source, w hich is thus in m o vem ent. T h e crucial p o in t
in th e p rese n t co n tex t is th a t such a m o v em en t of th e
source w ould cre ate a D o p p le r shift fo r fixed o b servers
(a m ooring).
T his in tricate m echanism deserves fu rth e r study, b u t
th e lite ra tu re alread y provides an exam ple, from th e
Straits of F lo rid a, w h ere such a m echanism m ay b e at
w ork. Soloviev e t al. (2003) show th a t in sum m er, w hen
fronts are strongest, a large an d am plified baroclinie
spectral p ea k occurs n ea r periods of ab o u t 1 0 h, w hile the
com m on sem idiurnal p eak (still p resen t in th e b arotropic
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J O U R N A L OF P H Y S I C A L O C E A N O G R A P H Y
surface tide) is absent. It is n o t im m ediately clear th a t
this p ea k actually rep rese n ts an in te rn al tid e since evi­
d en ce th a t o n e is dealing w ith o n e is norm ally b ased on
its d o m in a n t frequency. P erh ap s fo r this reason, w hile
acknow ledging th e possibility th a t th ey m ay have m ea­
sured a D oppler-shifted internal tide, Soloviev et al. prefer
to in te rp re t this p ea k as show ing th e n ea r-reso n a n t r e ­
sponse of a local seiche, transverse to th e ch an n el b e ­
tw een F lo rid a and the B aham as. A lternatively, th ey
suggest a shift o f th e “ effective" in e rtial frequency aw ay
from th e p la n eta ry inertial frequency by local subinertial
vorticity o f ab o u t 4/.
In th e w in ter season th e baroclinie sp ectral peak,
w hile w eaker, does a p p e a r at th e sem idiurnal frequency.
T hus, it is plausible th a t also in sum m er th e sem idiurnal
in te rn al tid e m ust be th e re , but, app aren tly , “ blue shif­
te d ." T his suggestion is rein fo rced by a seasonal blue
shift in th e diu rn al tid e (Soloviev e t al. 2003). In view of
th e previous discussion, ap p a ren tly th e n a tu re o f th e
tid al conversion changes from w in ter to sum m er, an d w e
n o te th a t instabilities o f th e F lo rid a C u rren t an d nonu n iform b athym etry, re q u ire d to b rea k th e geostrophic
balance, are in d e ed p rese n t (D avis e t al. 2008).
L et us now m ake a crude estim ate o f th e req u ired
source speed, vs, b ased o n th e o b served D o p p ler-sh ifted
in te rn al tidal frequency. F o r this, it is im p o rta n t to r e ­
ite ra te th a t, although m ean flows ca n n o t cre a te D o p p ler
shifts, th ey will m odify an existing one; see (3) and (15).
F o r an average F lo rid a C u rren t speed U ^ 2 m s~ x, a
typical first m o d e (o r interfacial) long w ave phase sp eed
c = 0 ( 0 .2 ) m s _1, and a fixed o b serv er (v 0 = 0 ), th e
o b served 2 -h p erio d shift of th e sem idiurnal and diu rn al
tid es suggest vs « Ul5 = 0.4 m s~ x. Interestingly, (13)
also predicts th a t, sim ultaneously w ith th e blue shift, th e
am plitude of th e in te rn al tid e g en e rate d by a m oving
source (as co m p ared to th a t fo r a source at rest, vs = 0 ),
is b o o ste d by a fac to r o f 5A, n o t unlike th e sum m er am ­
plification o f th e shifted in te rn al tid e found on th e
so u th east F lo rid a Shelf (Soloviev et al. 2003).
T hus, the observations offer som e su p p o rt fo r th e
sp e cu la tio n th a t w e are h e re d ea lin g w ith a D o p p le r
shifted in te rn al tide, due to a m oving source. H ow ever,
we em phasize th a t this case should nevertheless be co n ­
sidered exceptional for, as m entioned above, internal tide
g e n e ra tio n o cc u rs p re d o m in a n tly o v e r to p o g ra p h y , in
w hich case th e source d oes n o t m ove an d th e re will b e no
D o p p le r shifts fo r m o o red observers.
6. Conclusions
T he m ain p u rp o se o f this p a p e r is to p o in t o u t th e
im p o rtan ce o f distinguishing q u asi-D o p p le r shifts (i.e.,
th e difference in frequency b etw een tw o observers: on e
V o l u m e 43
fixed an d on e m oving w ith th e m e an flow) fro m D o p p le r
shifts in th e n o rm al physical sense, w h ere o n e com pares
th e freq u en cy o f an o b serv er w ith th a t o f a source.
R egarding th e latter, th e principal points can be sum ­
m arized as follows.
(i) A D o p p le r shift in freq u en cy can o ccur only if th e re
is a re la tiv e m o v e m e n t o f so u rc e an d o b se rv e r
(sections 2 an d 3).
(ii) T h e p resen ce o f a m ean flow does n o t change this
fact (sections 2 an d 3).
(iii) T h e w avelength and am p litu d e are affected by
a m ean flow as w ell as by a m o v em en t o f th e source
(section 3).
(iv) N o m eaningful in te rp re ta tio n o f o b serv atio n al d a ta
in term s of D o p p le r shifts can b e m ad e unless th e
m o v em en t o f th e o b serv er (e.g., fixed as in m o o red
in stru m en tatio n o r m oving as in a drifting platform )
as w ell as th e m ovem ent of th e source has b een as­
certain ed (Section 5).
(v) F o r dispersive w aves, a straig h tfo rw ard g en e raliz a­
tio n o f th e classical fo rm u la fo r D o p p le r shifts can
be d eriv ed (section 4). T h e afo rem e n tio n e d points
all rem a in valid in th a t case.
R eg ard in g q u asi-D o p p ler shifts an d its cen tral re ­
latio n (17), w hich plays a key ro le in B re th e rto n an d
G a rre tt (1968) an d follow ing studies, it should b e n o te d
th a t th e m o v em en t of a source does n o t fe a tu re in this
relatio n ; it is in v arian t fo r th a t m o v em en t (section 3e).
B ut, this sh o u ld n o t b e m isin te rp rete d as im plying th a t
th e m o v em en t o f a source does n o t influence th e in ­
d ividual term s co, co', an d k. T h e inv arian ce ju st m eans
th a t (17) can n o t tell us anything ab o u t th a t influence; for
this, it is necessary to co n sid er D o p p le r shifts in th e strict
sense o f th e w ord.
In this p a p e r w e have restric ted ourselves to th e re l­
atively straig h tfo rw ard cases in w hich all velocities are
p arallel an d in w hich th e m e an flow is spatially uniform .
T h e m o re g en eral cases o f arb itra ry angles b etw e en th e
v arious velocities an d o f sp atial v ariatio n s in th e m ean
flow w ere analyzed by B atem an (1931) an d Y oung (1934).
In p articu lar, th e m ean flow at th e lo catio n o f th e source
m ay b e different from th a t at th e location o f th e observer.
T h e u p sh o t is th a t m ean flows still create no D o p p ler
shifts. T his is actually already clear from th e “ stationarity
arg u m en t" discussed in section 2 .
In reality, m ean flows vary n o t only spatially b u t also
in tim e. T his aspect, how ever, brings in a new elem en t
q u ite distinct from D o p p le r shifts, n am ely n o n lin e ar in ­
teractio n , th e g en e ratio n o f h ig h er harm onics, etc. Such
in teractio n s do n o t bring ab o u t a shift in th e sp ectral
peak , b u t ra th e r th ey c re ate new ad d itio n al p eak s at th e
sum an d d ifferen ce freq u en cies o f m ean flow an d w ave
February
2013
G E R K E M A ET AL.
(see, e.g., B ell 1976). If th e flow oscillates slowly co m ­
p are d to th e w ave o r if th e stratification varies slowly,
th ese p eaks will be very close to th e m ain peak, effec­
tively m aking it b ro a d e r (depending o n th e spectral
resolution; in th e tim e dom ain, this is m anifested as interm ittency). T o add to th e already existing confusion,
this effect is o fte n ca lle d “ D o p p le r sm e a rin g ,“ ev en
th o u g h it has n o th in g to do w ith D o p p le r shifts or, in ­
deed, w ith anything C hristian D o p p le r ev er did.
A cknow ledgm ents. W e th a n k D irk O lbers an d an
anonym ous re fe re e for th e ir critical com m ents.
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