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Wind-forced continental shelf waves from a geographical origin
Continental Shelf Research, Vol 3, No 3, pp 215 to 232, 1984 Printed m Great Britain
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0278-4343/84 $3 00 + 0 00 1984 Pergamon Press Lid
Wind-forced continental shelf waves from a geographical origin JASON H. MIDDLETON* and ANDREW CUNNINGHAM* (Received 23 June 1983; in revisedform 17 January 1984, accepted 31 January 1984) Abstrsct--The response of a barotropm coastal ocean on a step-shaped continental shelf to a travel-
ing sinusoldal wind stress forcing is predicted theoretically usmg a frictional force proportional to the alongshore current velocity. Tlus theory is compared to a small set of observations from the northeast coast of Australia where a sudden widening of the continental shelf provides a geographical origin. The comparison is accomplished by means of frequency response functions relating alongshore wind stress with alongshore velocity. Amplitudes of the response functions are predicted to increase with alongshore distance equatorward and also to decrease with frequency at any location. These predictions are verified by the measurements Predicted phase lags are generally less than about 30 ° , with observations agreeing with theory to within about 20 ° . In general, the measurements provide reasonable evidence to support the theory of wind-forced continental shelf waves from a geographical origin
1_ INTRODUCTION THE ROLE o f wind stress as the principal driving mechanism for those topographically t r a p p e d waves o f sub-inertial frequency known as continental shelf waves was first formalized b y ADAMS and BUCHWALD (1969). They showed that the alongshore c o m p o n e n t o f the wind stress was responsible for the generation of the sea level fluctuations observed first by HAMON (1962, 1966) on the East Australian coast. GILL and SCHUMANN (1974) argued that a b o u n d a r y layer approximation could be m a d e for the equations governing the response if, as is usually the case, the wind stress forcing (and the response) have length scales greatly exceeding the continental shelf width. Their solution showed that the response can be expressed as a sum o f modes, each satisfying a first order wave equation in the alongshore direction. To allow for frictional attenuation, they suggested a form for the friction factor, and discussed how friction might modify the solutions. M o r e recently, BRINK and ALLEN (1978) introduced friction formally into the b a r o t r o p i c equations o f motion by means o f a perturbation expansion and showed that the alongshore c o m p o n e n t of current was brought m o r e nearly into phase with the driving force than for the frictionless case where the alongshore current lags the alongshore c o m p o n e n t o f the wind stress by lt/2. Observations o f wind-forced shelf waves now abound in the hterature and have been comprehensively reviewed by LEBLOND and MYSAK (1978), MYSAK (1980a,b), and ALLEN (1980), but to date there appear to have been no observations of the wind generation of shelf waves from a geographical origin. THOMSON and CRAWFORD (1982) have, however, presented
* Faculty of Science, University of New South Wales, Kensington, NSW 2033, Australia. 215
216
J . H . MIDDLETON and A. CUNNINGHAM
theoretical and observational arguments in support of tidally driven shelf waves from a geographical origin. In this paper we present the theory and observations of wind-forced continental shelf waves opposed by a frictional force linear in alongshore velocity, and trapped on a stepshaped continental shelf. The present solution differs from that of Brink and Allen as a consequence of the inability, in this case, to expand the wind stress in terms of the across-shelf modes in a situation where only one mode exists. As a result, the present model is not limited to small values of friction. The model is verified by comparison with a limited data set from the Great Barrier Reef region of the continental shelf on the northeast coast of Australia where the sudden widening of the continental shelf from about 70 km south of the Capricorn Channel (23°S) to about 180 km north of the Capricorn Channel provides a geographical origin for wind-forced shelf waves. While THOMSONand MIDDLETON 0 9 8 3 ) have outlined the salient features for identification of wind-forced frictional continental shelf waves in general, the flat shelf simplifies the algebra substantially, obviating the need for those rotary techniques (HSIEH, 1982; MIDDLETONet al., 1982) which are useful in continental slope regions. Here we simply use the frequency response function pertaining to the alongshore current (output) and the alongshore wind stress (input). The theoretical derivation is described in Section 2 and is followed in Section 3 with a general description of the field experiment and the observations. The direct comparison of theory with experiment is described in Section 4 with the aid of frequency response functions, while Section 5 concludes with a general discussion. 2. G O V E R N I N G E Q U A T I O N S
We consider the non-divergent equations applicable to wind-forced shelf waves along a continental shelf oriented parallel to the y axis with the coast lying at x = 0. Variations in bottom topography are assumed to be independent of alongshore station y, giving the depth h = h(x). For waves having frequency ¢0 small compared to the coriolis parameter f, wavelengths sufficiently long that they are much larger than the scale width of the shelf L, and negligible surface divergence ~ = f 2 L 2 / g h ¢g 1; then the inviscid, linearized, vertically averaged shallow water equations are at;
fv = g -~x
(l)
av at; ~w -~ +fu =-g-~ + p----~
(2)
a
a (hu) + ~ (hv) = O. ~y ~x
(3)
Here u and v are the depth-averaged components of velocity in the x and y directions, respectively, t; is the surface elevation adjusted for variations in atmospheric pressure, ~'w and
Wind-forced continental shelf waves
217
x~ are the alongshore components of wind stress and bottom stress, and p is the density of the water. Neglect of the across-shelf components of both wind and bottom stress follows from the assumptions outlined above, and is justified by GILL and SCHUMANN(1974), LEBLOND and MVSA~: (1978), and BRINK and ALLE~ (1978). By virtue of the non-divergence expressed by (3) a stream function ¥ can be defined such that u-
1 by
h by
v-
1
by
h
bx
(4)
and eliminating the surface elevation from (1) and (2) yields 0 bxbt
-~x
+ h 2 dx
by - bx ([~w - ~ ]/ph).
(5)
This is the basic Gill and Schumann formulation with the inclusion of bottom fraction. The object is to solve for ¥ for a given forcing function x~,, with appropriately parameterized bottom friction ~ and subject to the following boundary conditions ¥ = 0at
by bx
x = 0
(6)
= 0 for x > L.
(7)
These imply that there is no flow through the coast, and that the alongshore velocity v becomes appropriately small in the region off the shelf. In addition, we shall be concerned here with the implementation of boundary conditions at the edge of the continental shelf. Since the continental shelf in the region of interest is relatively flat, and the slope remarkably steep, we shall consider a simple fiat shelf geometry with constant depth h on the shelf and constant depth h 0 in the ocean. We seek solutions for the stream function in regions on the shelf and in the ocean, subject to (6) and (7), and then match solutions appropriately at the shelf break. The bottom friction on the shelf is parameterized by
pr by =On,=
h
bx '
(8)
where r is a friction parameter with dimensions of velocity which has been observed to take values from about r = 10 -4 to 10-3 m s -~ (W1NAN'r and BEARDLSEV, 1979) depending upon the region. The across-shelf structure may be obtained from (5) by assuming separation of variables such that V(x, y, t ) = F(x)~JO,, t).
(9)
If there is no across-shelf variation in wind stress across a fiat shelf (h = constant) then substitution of (9) into (5) shows the across-shelf structure to be governed by ~ F / b x 2 = O.
218
J . H . MIDDLETON and A. CUNNINGHAM
Implementing the boundary condition (6), and incorporating the first constant of integration into # gives the across-shelf structure F(x) = x
0 < x < L.
(10)
If we further assume the surface elevation ~ to be given on the shelf by ~(x, y, t ) = Z(x)t[~.V, t)
(11)
then substituting (11) into (1), and using (10) gives Z(x)=Z(L)---~(L--x).
(12)
A direct substitution of (9) to (12) in (2) gives an expression for the alongshore structure
4g.v, O; a~
a~
T ; + tghZ(L) -
r
~'w
+ -g ÷ :
and writing (13) gives the forced, frictional wave equation 1 i)~ ~# r -- -- +-+-- ~=
c ~t
ay
~w
ch
(14)
pc
The form of (14) was suggested by Gill and Schumann, and found by Brink and Allen for general across-shelf depth variation. The theory of Brink and Allen cannot be applied here because of the mathematical difficulties associated with the expansion of the wind stress in terms of an infinite series of across-shelf eigenfunctions, in a case where only one across-shelf eigenfunction exists. We can solve (14) to find d~V, t) for any ~,0', t) but require Z(L) for any full solution. This may be obtained by considering flow in the deep ocean, and then by matching elevation and fluxes at the shelf break. Assuming geostrophy and non-divergence in the ocean gives f u = --g ~ y ,
at fv = g ~.ff ,
au av --ax + -~y = 0 .
(15)'
If we further assume separation of variables in the deep ocean such that 1
~= Zo(x)~.v, t)
and
a~b
- Fo(x) ~ y (y, t), u = -- -ho
(16)
where ~b(y, t) is the same function as for the shelf, then the first of (15) gives Zo(x) = Fo(X)
f gho
(17)
Wind-forced continental shelf waves
219
Matching volume fluxes at the shelf break x = L gives or
F(L) = F o ( L ) ,
L = F0(L),
(18)
while matching elevations at the shelf break gives Z ( L ) = Z o (L),
f
or
Z ( L ) = F0 (L) - gho
(19)
Substituting (18) into (19) allows us to determine the speed c from (13) as c = -fL(l
-- h/ho).
(20)
The general solution for ~ may now be found by integrating (14) along a characteristic s = t - - y / c (s =constant) as outlined by Gill and Schumann. Then (14) reduces to
dO dy
+ ¥~ = --,
(21)
W
where ¥ r(ch) -I is the friction factor and the sub- and superscripts ofxYw have been dropped for convenience. Employing a change of variable from y and t to ~ and s, such that y = ~, t = s + ~ c gives =
~ d
1 (O¢'r;)d~ = - ~ f ~ eV~xff,, s + ~/c)d~.
(22)
As pointed out by Gill and Schumann, solutions calculated from a wind stress operating from (a) initial time t = 0, have ~ = ~-- ct and (b) geographical origin y = 0, have ~o = 0. The solution for ~ from (22), in conjunction with (10) and (4) may be used to determine the forced response from any imposed wind forcing x(y, t) on a flat shelf, provided only that the scale of the forcing greatly exceeds the shelf width. In the following we will be interested in seeking solutions from a geographical origin (the mouth of the Capricorn Channel) and choose the traveling forcing function. = x0 cos(k0y - tot).
(23)
The general solution of (22), assuming no energy input at y = 0 is 0=%
os koy -- tot + ~ -- 5
--
-vy eos
e
- -c-
tot + ~- -- 8
,
(24)
where
~o
"Co =
(25)
and tan 5 -
¥ (to/c - k o )
(26)
220
J . H . MIDDLETON 811d A. CUNNINGHAM
The solution (24) thus consists of forced modes having phase speed ¢0/ko (traveling with the forcing) and free modes with phase speed c which are damped exponentially by friction from the origin y = 0. When observing the response (24), Fourier techniques will view the two components as a single component of the form ~ / ~ = G cos(0 - cot + ~t/2 - 5),
(27)
where k o ] y ) + e-2~'y
G2 = 1 - 2e-Wcos ( [ ~ -
(28)
and sin k0y -- e-~sin
coy c
tan 0 =
(29) cos k0y - e-~'Ycos
coy c
Thus the alongshore wind stress f o r c i n g t = t0 c o s alongshore current v given by
(koy - •t)
p r o d u c e s a response in the
v = Vo c o s ( O - o)t + n / 2 - 6),
where the ratio of amplitudes is given by
Vo/to =
G p c h [ y 2 + (ko - (o/c) 21 ~
(30)
and the phase lag u of response behind the forcing is given by u =
0 -
koy
+ lt/2 -
6.
(31)
At sufficiently small distances from the origin that yy ~ 1 the free wave component is little affected by friction and exp (--yy) = 1 - yy. If, in addition, the wavenumbers of the free and forced modes are sufficiently small that (o)/c - k o ) y ,~ 1 then G simplifies to G = (co/c - ko)y. In the low-frequency limit the amplitude ratio (30) shows a linear dependence with distance y from the origin, i.e.,
Vo/%
Y =
plF /k~ +
(32)
F h21½ "
Finally, we note that it is the difference between the forced and free wave amplitudes in (24) which allows this linear increase in response function with distance from the origin. The forced wave amplitude, by itself, is constant and equal to the value given by (30), but with G - 1, while the free wave amplitude decreases slowly from the origin. 3
THE FIELD EXPERIMENT
In this section we discuss observations made on the continental shelf region of the northeast coast of Australia, where the sudden widening of the continental shelf from about
Wind-forced continental shell" waves
221
70 to 180 km at 23°S comprises a geographical origin for wind-forced continental shelf waves.
From April 1980 to February 1981, three current meter moorings were deployed and maintained at the mtd-shelf position on a region of flat shelf in the south-central Great Barrier Reef region from the C S I R O R.V. Sprightly (Fig. 1). Each mooring had one Aaoderaa R C M 4 current meter with sampling interval set at 30 mln located I0 m from the bottom m total depths of 56 m (Hook mooring), 54 m (Bushy mooring), and 76 m (Bell mooring). Hydrographic measurements taken by Sprightly at the ttmes the meters were deployed and servmed showed the water column to be well mixed throughout the year, and we assume the observed currents to be barotropic. As a result of the damping o f longer period swell by the reef, and also because the depth of the meters in all cases exceeded 40 m, aliasing of the current speed data due to wave surge ts thought to be negligible (PEARSON et al., 1981). In add,tion, an Aanderaa W L R 5 water level recorder was deployed at Penrith Island, and wind velocity data from Pine Island obtained from the Bureau of Meteorology. Current meter time series were calibrated dtrecttonally to correct for the magnettc non-linearity inherent m A a n d e r a a current meters and adjusted for the local magnetic variation to give directions relative to true North. Sea levels were modified to account for fluctuations in atmospheric mean sea-level pressure.
ll-6
20
ll-8
150
152
IS& I
20
-
%,"
%e~BUSHY
22 __
~t
~ BROAD
-
SOU~
"-..
BELLe ~
ReSFs ~x
~
~
;;
Z2
C,aJ:qCORN ROCKHA
2~
GLADSTONE
_
......
26 --
) ~*E~N
Apprommnte
BUNDABERO
edge of r H f s
*
Current
•
W ~ e r level recorder
meter
•
Anm~omet¢~
I
1
I
I
1&6
l&g
iS0
152
,
(I 15&
Fig. I The region of study showing the locations of mstruments and the sudden wtdenmg of the continental shell"at 23°S comprising the geographical origin of wind-forced continental shelf waves.
222
J.H.
MIDDLETONand A. CUNNINGHAM
Current and sea-level data were low-pass filtered using a fast Fourier transform filter
(WALTERS and HESTON, 1982) with 10% of the remaining Fourier coefficients cosine tapered about the cut off frequency (half power at 1 cycle per 40 h) to reduce leakage in time. All transforms used prime factors <9"/as a base (BRENNER, 1967) and were individually tadored to the length of each series. The filtered data was then decimated at daily intervals. The filtered data and the daily wind are shown in Fig. 2. Except for the Penrith Island sea levels, all other (vector) quantities are shown as vector stick diagrams which represent speed and direction as a function of time, with the oceanographic convention (specifying the direction lo which the wind blows) for the wind velocity. Average currents flow to the southeast at each of the mooring locations in contrast to the prevailing wind which blows to the northwest. The mean currents are principally oriented in the along-channel direction, which =s also basically parallel to the coast. There is some seasonal variaUon in these mean currents with stronger flow being observed in the period after August. Also noteworthy are current fluctuations at periods of" 7 to 20 days wh,ch appear highly correlated from mooring to mooring. These fluctuations have amphtudes in excess of 15 cm s-j and are at times strong enough to reverse the direction of flow. There is also variabdity in the wind data (Pine) and the sea-level data (Penrith) of about the same period as the current fluctuations. Time series were detrended by a linear least-squares method and tapered prior to transforming for spectral calculations. The spectral estimates were adjusted to account for the loss of energy due to tapering, and band averaged over 5 adjacent esUmates giving 10 d.E Prior to calculation of cartesian spectral estimates, preferred directions of orientation for low-frequency motions were found by plotting the outer rotary autocoherence (ellipse
u 5~
t~
/~
_[~
. . . .
/'~
PENRITH
BELL ~10 I
I M
i J
i J
I A
I S
I 0
i N
I D
I J
1980
Ftg. 2. The time series of low-passed current velocity (Hook, Bushy, Bell), wind velocity (Pine), and sea level (Penrith) recorded during 1980 The scale 10 is appropriate for current velocity in cm s -1 and wind velocity m m s - L
223
Wind-forced continental shelf waves
10
HOOK
W W
Z
'"05 m 1" Q L~
"r Q.
95%
0
i
I
i
I
0
i
I
i
I
01
i
I
I
-'n
02
03
FREOUENCY (CPD) Fig. 3. Outer rotary autocoherence squared (solid hne) and phase (dashed line) for the Hook current velocity time series. For low frequencies the observed phase ~ = 70 ° and the orientation of the major axis of the current ellipse is a = - x z ~ = - - 3 5 ° in the complex plane Low-frequency current elhpses are therefore oriented along 125 to 305°T.
PERIOD 100 21 gllll
I
I
( DAYSI
10
~1111||
o
I
I
100
]111111 I
!
""'
'
10
I" ......
1'""'
' '
/-', PENRITH
2sg
0 PINE
1300)
0
~. 2 o
HOOK ..... _
1305l
~
z
. 5 ~
~0
b
BUSHY
(325)
m
o ~: 0
Ul
,,/'.
BELL
13601 •
w:EO
~
V
. . . . .
.....
U I
0"01
..
O|
I i ill
i
I illilll
i
001
C
lilllll[
i
lllli
01
FREQUENCY (CPD) Fig 4. Alongshore (V), across-shelf(U), anticlockwise (A), and clockwise (C) spectral estimates in energy preserv.ng form for the current vector time series at Hook, Bushy, and Bell, wind stress time series at Pine, and sea level at Pennth. Note that the energy scales are the same for both the U, V and C, A spectra for each time series, hut differ for each individual time series The orientations (degrees true) of the alongshore components are shown in brackets for each vector time senes.
224
J.H. MIDDLETON ~
A. CUNNINOHAM
stability) and phase, the phase being minus twice the orientation o f the ellipse m a j o r axis. By this means, principal axes for the low-frequency current ellipses were found to be oriented in directions o f 3 6 0 ° T (Bell), 3 2 5 ° T (Bushy), 3 0 5 ° T (Hook), and 3 0 0 ° T (Pine). The plot for the H o o k d a t a is shown in Fig. 3. In energy preserving form (whereby equal areas under a spectral plot represent equal
energies), cartesian and rotary spectra for all time series are shown in Fig. 4. For the cartcsian currcnt spectra the V(U) components arc aligncd with the major (minor) axes of thc low-frequency current clhpscs as outlined abovc. Thc cartesian spectra for wind and current vectors show the majority of cncrgy m the alongshore (V) componcnts for periods of 5 to about 30 days. In cach case thcrc appears to bc more energy at Bushy than at Bell, and more cncrgy at Hook than at Bushy, with an approximate linear incrcasc in current amplitudc north from thc Capricorn Channcl. The rotary spectral levels indicate thc cncrgy to bc almost cqually partitioned into anticlockwise (A) and clockwisc (C) components for Hook and Bushy, indlcatlng thc long period velocity fluctuations to be almost rcctilincar,although the vclocity vectors at Bell rotate in a clockwise scnsc in the horizontal planc. Thc shapes of thc wind strcss spectra and the velocity spectra arc similar as might bc cxpcctcd for wind-driven flows. To gain an idea of the rclationshlpbetween the wind stress vector time series and thc current velocity vector timc series, thc inner rotary cohcrcncc squared and phase between the wind stress at Pine and the current velocity at H o o k were calculated (Fig. 5). The vcctors arc generally coherent for periods longer than about 4 days, 10 PINE
-
HOOK
~m~O..5 =* o /
%/ 0
I
I
I
~0 -It'
-03
I
I
I
i
I
i
I
I
I
I
I
i
I
l
i
i
f
,
i
-0-2
*
- 0-1
0 FREQUENCY
01
0-2
03
[CPD|
Fig. 5. Rotary coherence squared and phase between wind stress vector (Pine) and current vector (Hook) time series. The intercept and gradient of a line of best fit for inner rotary phase at lowfrequency provide estimates ofthe difference in orientation of major axes and time lags, respectLvely, between the vector time series. Solid (dashed) lines represent tuner (outer) coherence and phase.
Wind-forced continental shelf waves 10
225
BELL- HOOK
I,u
~os
/ /
o¢=J
\
/
S
.........
/7-
- ~'7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t
I
i
I
|
I
i
I
i
I
i
I
|
I
i
It
I
"-,
I I "'
I
Z o.
! ,! i
- a- 0-3I
'
-0'2
-01
o FREQUENCY
I
ol
t
i
02
i
03
(CPDI
Fig. 6. Rotary coherence squared and phase between current vector series at Bell and at Hook. The increase in inner phase (solid line) wsth frequency indicates phase propagation toward the equator and a linear least-squares fit to the inner rotary phase as a function o r frequency (for those frequencies with significant coherence) gives an intercept o f 9 ° and a gradient or 154 ° per cpd. These features imply (a) that the major axis of the wind stress elhpse is oriented 9 ° anticlockwise from the wind stress vectors at low frequencies, and (b) that the currents lag the wind stress by about 0.43 days. Thus the response to the wind stress at Hook is predominantly alongshore and is the same direction as the wind stress, with a small phase lag. To gain a general idea of the phase propagation characteristics of the current fluctuations, the rotary coherence squared and phase between the Hook current velocity series and the Bell current velocity series were calculated and these are plotted in Fig. 6. Phase propagation is equatorward as shown by the general increase in inner rotary phase as the frequency increases. Simslar calculations with the other combinations of velocity series also show an e q u a t o r w a r d phase propagation. Lastly, for a direct comparison o f experiment with theory we require a knowledge of the wind forcing as a function of distance. R o t a r y spectra calculated between wind stress vectors at Pine and 120 days of wind stress data from Heron Island (285 km to the southeast of Pine Island) give phase lags consistent with a horizontal wavelength o f 9200 km at periods o f 8 and 20 d a y s with the principal axis oriented basically alongshore. This represents a forcing wavenumber of about k 0 : 6.8 x 10 -7 m -[ independent o f frequency. 4 COMPARISON OF OBSERVATIONS WITH THEORY In this section we directly relate the observations described m the preveous section with the theory presented in Section 2.
226
J.H. MIDDLRTONand A. CUNmlqOUAM
Theoretically, topographically trapped waves of low frequency will result in the velocity vector on the continental shelf tracing out a highly eccentric ellipse in the anticlockwise dmrection in the horizontal plane, with major axis oriented approximately in the local alongshore direction. Cartesian and rotary spectral levels from the current vectors of the Hook and Bushy t~me series are consistent with these features, although the spectral levels at Bell are inconsistent at periods longer than about 8 days where an antlclockwise direction of rotation is observed. This, however, is not statistically significant to w~thin 95% confidence. Techniques of identification of forced continental shelf waves have been fully discussed by THOMSON and MIDDLETON (1983) and here we elect to use the cartesian system rather than rotary quantities by calculating the frequency response of the alongshore velocity component to the alongshore wind stress component. In particular, if X(~) and Y(m) represent the Fourier coefficients of the alongshore components of wind stress and current vector time series at radian frequency 0~, then the frequency response function H(o~) is defined by Y(o~) = n ( ~ ) X ( m )
= In(to)le-~')X(o~).
For a system in which there exists incoherent noise distributed across the current spectrum, the coherence C~y(m) specifies the fraction of energy in the current which is coherent with the wind such that =
'
where the overbars indicate either band or ensemble averaging in frequency space and the * denotes the complex conjugate. It follows that the magnitude of the frequency response is given by IH((.)I
=
Y*(o))Y((o)/X*(o))X(o))]÷
while the phase a(e), which represents the phase lag of the current time series behind the wind stress time series at frequency m, is given by [" lmag {X*(~)Y(o~)}] II(a)) = --tan -I L
Real ~
J"
Calculations of the magnitude of the frequency response function IH((~)[ and the phase a(m) from the data by means of the above expressions may then be directly compared with the amplitude ratio (30) and the phase lag (31) predicted by theory. Before a direct comparison may be made, appropriate values must be selected for the theoretical parameters contributing to the amplitude ratio (30) and the phase lag (31). In the previous section we determined the wavenumber of the forcing to be ko = 6.8 x 1 0 -7 m -~ , and a suitable value for the density is p = 1027 kg m -3 . It remains to find appropriate values for the mean depth h, the free wave phase speed c, and the friction coefficient r. These parameters determine y and 5 so the response can be predicted as a function of downstream distance y at any frequency ~. Reference to the bathymetry (Fig. 1) shows the origin of motions (y = 0) to be in the mouth of the Capricorn Channel, some 200 km southeast of the Bell mooring location. To examine the dependence of the solution on these quantities, a set of curves drawn from (30) and valid for ~ ~ 0 is shown in Fig. 7 on which is also plotted the observed response at
227
Wind-forced continental shelf waves
2
CC
BELL
BUSHY
I
1
I
HOOK
.[.
h =35m
~
I
o
lOO
I
I
I
200
I 300
I
I L00
I
I
1
500
y[KM) Fig. 7.
Amphtude ratios representing the amplitude of the theoretical frequency responsefunction
m the low-frequency limLt (o~ --, 0) for selected values of the friction coefficient r and mean depth h (sohd and dashed lines) as a function of distance from the origin, the Capricorn Channel (CC) The crosses indicate the observed values
low frequency (the crosses). The observed low-frequency response for Bell and Bushy is approximately linear with distance alongshelf from the mouth of the Capricorn Channel (CC), while the response at Hook is larger, possibly as a result of the narrowing of the continental shelf in the region of the Whitsunday Islands due west of the Hook mooring. Plots of the low-frequency amplitude ratio response function with distance are shown for two depths (h = 35 and 45 m) and two values of the friction parameter (r = 10 ~ m s -k and 5 x 10-4 m s -I ). The free wave phase speed here is given by (20) and choosing L = 180 km and h/ho = 35/500 gives a value of c = 8.75 m s - ' . These curves show that increasing the friction reduces the magnitude of response and that increasing the depth also reduces the magnitude of the response. In general the rate of increase of amplitude near the origin depends on the value of h, while at larger distances the effect of variation caused by the friction coefficient r is more pronounced as the free waves generated at the origin become increasingly damped. The present theory takes no account of the existence of the Great Barner Reef on the outer shelf. MIDDLETON(1983) has shown that for this section of the continental shelf the existence of the Reef increases the phase speed of higher frequency free shelf waves to about 14.6 m s - ' at 0.25 cpd, while shelf waves with frequencies lower than about 0.05 cpd remain unaffected. This variation of phase speed with frequency corresponds to Middleton's dimensionless reef opacity parameter ~ = 7, this being a value supported by observation. Since the phase speed c appears in the denominator of (30), an increase in phase speed with frequency will result in a decrease in response amplitude as frequency increases, a feature observed in all the present data as discussed below. MIDDLETON'S (1983) unforced wave theory cannot be readily incorporated into the forced theory presented here, and so we have elected here to use simply the phase speed c(co) determined from the earlier work and plotted here for reference in Fig. 8.
228
J.H. MIDDLETON 8yld A. CUNNINOI-IAM
1/-
I 0
I
I
0-1
I 0-2
FREQUENCY {CPD]
Fig. 8. Phase speed plotted as a functton of frequency for free shelf waves propagating along the shelf in the region of observation as predicted and observed by MIDDLETON(1983). The presence of the Great Barrier Reef results in an increase of phase speed with frequency from the low-frequency limit of 8.75 m s-I as shown here.
As a function of frequency, the response function and phase for each of the moorings are shown in Fig. 9 for the selected values of r = 2.5 × 10 -4 m s -j , h = 35 m, and c as given in Fig. 8, these values providing the best all around fit. Although the value for h may appear low when compared to the mooring depths the presence of shallower water toward the coast and the existence of the spatially intermittent reefs extending to the surface on the outer shelf (where the depth between reefs is about 50 m) renders h -- 35 m a plausible average value. From the data, calculations were made of the amplitude and phase of the frequency response function. Band averaging over 10 adjacent estimates in the frequency domain gave 20 d.f. in the spectral estimates used to evaluate [H(m)[ and a(c0). Since most of the energy in the currents exists at frequencies lower than about 0.25 cpd, we have limited calculations to this range. For the Bell data the amplitude response provides an excellent fit for all periods longer than 5 days, while the observed phases lag more than predicted by theory. In general the predicted and observed phases agree to within about 20 ° and the predicted trend of an increase in phase lag with frequency is observed. For the Bushy data the phase lag results also show a general increase of phase lag with frequency as predicted by the data, but the amplitude results are less convincing. In conjunction with the higher frequency response data, an average value of the two lower frequency data points (each from the two separate segments of current data) shows the theory to underpredict by about 10%. The observed phases at Hook show good general agreement at both very long periods (50 days) and at very short periods (44 to 5 days), while at intermediate periods the observed currents lag the predicted currents by 10 to 30 °. The amplitude response at Hook is, however, clearly anomalous. The shape of the response is in general agreement although the amplitudes are underpredicted (solid lines) by a factor of about 50%. Figure 1 shows that the protruding coastline and islands due west of the Hook mooring effectively narrow the continental shelf here. Wind-forced continental shelf waves propagating toward this region will
z 0 a.. m i,i
u.I
-r 13..
,,,
_J
co
o
0
o.5
-20
0
20
~0
0
BELL
I (CPD)
0'2
u.l
z 0 a.
w
v.
0 5 -
15
-20
0
20
/.0
BUSHY
FREQUENCY
(CPD)
I 0 2
I 0.1
I
F,g. 9 (a) The frequency response function amphtude and phase for alongshore components of current velocity (Bell) and wind stress (Pine)Iplotted as a function of Jfrequency as predicted by the theoretical model (sol/d lines) and as calculated from the data (individual points). (b) As for (a) but for the alongshore velocity component from the Bushy time series.
FREQUENCY
0'1
(o)
•
w
--
LLI ¢n ~C "1" n.
_J
o
15
w
o
(b)
=
t,J
<
go
¢t
=.
O O
t=L
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?-
o z
z
c~
(P)
L
HIlHN3d
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o
m
7O
X
,-I
S'O ~
::0 m (.n 7o 0 Z
AON3nO3H~ L,0
(ODD) ;'0
%
I
%.
I
%.
_i'.
(o)
-~.u~ = m . LIIUU~d gLllmoJj suopem:n U I=a.,7-u~s =ql JOJ lnq (u) JOj sV (p) 'Ix~Igql m, I~ssn~s!p se ~1ooH :J.e IOpIA~ J]~)qs J~IMO.LmU ~ l q l JOJ lunooou o1 polsn.l'pu gPmildum pal~!paJCl aq3 sluoso.ld~J aU!l poqsep aL~ 'sau~s OLUI1 ]1OOH aql UJOJJ lUauodmoo ,(l!OOlaA aJoqsSuole aql ~oj ]nq (e) ~oJ sV 06 '~tzl
L'O
((]dO) XON3nO3~ld E'O
I
t
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SL
--OZ
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~"
m~ (~
0
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:l>
U~ m f--
R
2
Wind-forced continental shelf waves
231
have their energy concentrated into this narrower shelf width, contributing to the higher currents observed at the Hook mooring compared with those predicted. An estimate of the increase in velocity due to the narrowing shelf may be obtained through energy considerations. From Rockhampton to Mackay the shelf width remains approximately constant at L = 180 km; however, at Hook the effective width of the shelf is only about L = 90 kin. For non-divergent waves the total energy of flow in any cross-section of shelf is proportional to v2Lh, and with the cross-sectional width L effectively halved at Hook, the alongshore velocity v will be increased by a factor 2½ over that for a constant w~dth shelf. Theoretical amplitudes of the response function, adjusted to this factor, are shown m Fig. 9c as a function of frequency by a dashed line. The agreement is much improved and the prediction generally within about 10% of the observations except at the higher frequencies where there is, in any case, little energy. Finally, the amplitude response of the sea level at Penrith for the wind stress at Pine ~s shown m Fig. 9d. As for the alongshore current response diagram for Bushy, the observed sea-level response is underpredicted, possibly as a result of the narrowing of the shelf equatorward of Broad Sound, while the phase lag lies within about 20 ° of that predicted, and is substantially smaller than the value of 90 ° predicted by the frictionless theory. 5
DISCUSSION
In this paper we have presented theory and measurements for low-frequency wind-forced continental shelf waves from a geographical origin, this being the first such evidence of this type of motion to appear in the literature to the best of our knowledge. The geographic origin exists as a consequence of the sudden widening of the continental shelf on the northeast coast of Australia from about 70 km south of 23°S to about 180 km north of 23°S. The wider shelf provides a greater surface area of shallow water on which the wind stress may act, wind stress forcing on the deeper ocean surface being ineffective in terms of shelf-wave generation as a result of the stratification in the deep ocean. The problem is, however, complicated by the presence of a section of the Great Barrier Reef system on the outer shelf (this being known to modify the phase speeds and structure of free wave modes), and the narrowing of the contlnental shelf at 20"S. Theory and experiment have been compared by means of frequency response functions relating alongshore current (output) to alongshore wind stress (input). Response amplitudes calculated from the data recorded at the two mooring s~tes nearest the Capricorn Channel (the origin of motion) show reasonable support for the theory, while observed phase lags are generally within 20* of those predicted. In particular amphtudes and phase lags are observed to decrease and increase, respectively, with frequency. Data from the Hook location ~s, at first s~ght, less convincing because the theoretical amplitudes underestimate the observed amplitudes by a factor of about 50%. However, the shelf in the region of the Hook mooring is narrower than that further south and this would be expected to concentrate the shelf wave energy as it propagates equatorward leading to observed currents exceeding those predicted. Adjustment of the theoretical response to allow for the narrower shelf width provides a much improved agreement with theory. For an ideal experiment location, the matching of theory with experiment would serve to determine a value for the frictional parameter r, and in this case the 'best' fit appears to occur for a value of r - - 2 . 5 x 10 -4 m s-~, with a selected mean depth of 35 m. However, in thin case, the flow m the region of observation (the development region for the flow) is insensltwe
232
J.H. ]~[IDDLETON and A. CUNNINGHAM
t o t h e v a l u e o f r. I n a d d i t i o n , w e h a v e elected t o u s e a q u a d r a t i c w i n d s t r e s s f r i c t i o n c o e f f i c i e n t h a v i n g a v a l u e e q u a l to 1.6 a n d 2.5 × 1 0 -3 f o r w i n d s p e e d s < 7 a n d > 1 0 m s-~ , r e s p e c t i v e l y , w i t h a l i n e a r t r a n s i t i o n for s p e e d s b e t w e e n 7 a n d 10 m s -~ (AMoROCHO a n d DEVRIES, 1980), a n d the o b s e r v e d r e s p o n s e is l i n e a r l y d e p e n d e n t o n t h e selected v a l u e s . I n v i e w o f t h e a b o v e a r g u m e n t s a n d s e v e r a l o t h e r f a c t o r s i n c l u d i n g (a) t h e p r e s e n c e o f t h e G r e a t B a r r i e r R e e f o n the o u t e r s e c t i o n s o f t h e c o n t i n e n t a l shelf, (b) u n k n o w n c o n d i t i o n s e x i s t i n g in t h e C o r a l S e a in t e r m s o f v a r i a b l e a l o n g s h o r e p r e s s u r e g r a d i e n t s , a n d (c) t h e r a t e o f p r o p a g a t i o n o f e n e r g y into the C a p r i c o r n C h a n n e l f r o m f u r t h e r s o u t h , t h e m e a s u r e m e n t s p r o v i d e r e a s o n a b l e e v i d e n c e for t h e e x i s t e n c e o f w i n d - f o r c e d c o n t i n e n t a l w a v e s f r o m a g e o g r a p h i c a l origin.
Acknowledgements--Th~s work was supported by the Australmn Marine Sciences and Technologies Grarlts Scheme under Grant 80/23. We wish to thank Gregory Nippard for his mooring work and Alison Dorr for her data analys~s. In addition, we thank the CSIRO Division of Oceanography for their kindness in allowing us to deploy our instruments from R V Sprightly. REFERENCES ADAMS J. K. and V. T. I]UCHWALD(1969) The generation of continental shelf waves. Journal of FlmdMechamcs, 35, 815-826. ALLEN J. S. (1980) Models of wind driven currents on the continental shell Annual Reviews of Fluid Mechanics, 12, 389-433. AMOROCHO J. and J.J. DEVRIES (1980) A new evaluataon of the wind stress coefficient over water surfaces. Journal of Geophyswal Research, gs, 433-442. BgEN~R N. M. 0967) Three Fortran programs that perform the Cooley-Tukey Fourier transform. Techmcal Note 1967-2, Lincoln Laboratory, M.I T., 29 pp. BRINK K. H. and J. S. ALLEN (1978) On the effect of bottom fncUon on barotropic motion over the continental shelf. Journal of Physical Oceanography, 8, 919-922. GILL A. E. and E. H. SCHUMANN (1974) The generation of long shelf waves by the wind. JournalofPhyslcal Oceanography, 4, 83-90. HAMON B. V. (1962) The spectrums of mean sea level at Sydney, Coifs Harbour and Lord Howe lsland. Journal of Geophysical Research, 67, 5147-5155. HAMON B. V. (1966) Contanental shelf waves and the effects of atmospheric pressure and wind stress on sea level. Journal of Geophysical Research, 71, 2883-2893. HSlEH W. W. 0982) On the detection of continental shelf waves. Journal of Physical Oceanography, 12, 414-427. LEBLOND P. H. and L.A. MYSAK (1978) Waves in the ocean. Elsevier, New York, 602 pp. MIDDLETON J H. (1983) Low-frequency trapped waves on a wide reef-fnnged continental shelf. Journal of Physical Oceanography, i$, 1371-1382. MIDDLETON J. H., T. D. FOSTER and A. FOLDVlK (1982) Low-frequency currents and continental shelf waves in the southern Weddell Sea. Journal of Physical Oceanography, 12, 618-634. MVSAK L.A. (1980a) Recent advances in shelf wave dynamics. Rewews of Geophysics and Space Phystcs, 18, 211-241 MYSAK L.A. (1980b) Topographically trapped waves. Annual Reviews of Fired Mechanics, 12, 45-76. PEARSON C. A., J. D. SCHUMACHER and R. D. MUENCH (1981) Effects of wave mduced mooring noise on udal and low frequency current observations. Deep-Sea Research, 215, 1223-1229. THOMSON R. E. and W. R. CRAWFORD (1982) The generataon of dmrnal period shelf waves by tidal currents. Journal of Physical Oceanography, 12, 635-643. THOMSON R.E. and J.H. MIDDLETON (1983) On wavenumber estimates for forced continental shelf waves. Journal of Physical Oceanography (submitted). WALTERS R. A. and C. HESTON 0982) Removing t~dal period vanations from time series data using low-pass digital filters. Journal of Physical Oceanography, 12, 112-115. WINANT C. D. and R.C. BEARDSLEY (1979) A comparison of some shallow wind-driven currents. Journal of Physical Oceanography, 9, 218-220.
~
0278-4343/84 $3 00 + 0 00 1984 Pergamon Press Lid
Wind-forced continental shelf waves from a geographical origin JASON H. MIDDLETON* and ANDREW CUNNINGHAM* (Received 23 June 1983; in revisedform 17 January 1984, accepted 31 January 1984) Abstrsct--The response of a barotropm coastal ocean on a step-shaped continental shelf to a travel-
ing sinusoldal wind stress forcing is predicted theoretically usmg a frictional force proportional to the alongshore current velocity. Tlus theory is compared to a small set of observations from the northeast coast of Australia where a sudden widening of the continental shelf provides a geographical origin. The comparison is accomplished by means of frequency response functions relating alongshore wind stress with alongshore velocity. Amplitudes of the response functions are predicted to increase with alongshore distance equatorward and also to decrease with frequency at any location. These predictions are verified by the measurements Predicted phase lags are generally less than about 30 ° , with observations agreeing with theory to within about 20 ° . In general, the measurements provide reasonable evidence to support the theory of wind-forced continental shelf waves from a geographical origin
1_ INTRODUCTION THE ROLE o f wind stress as the principal driving mechanism for those topographically t r a p p e d waves o f sub-inertial frequency known as continental shelf waves was first formalized b y ADAMS and BUCHWALD (1969). They showed that the alongshore c o m p o n e n t o f the wind stress was responsible for the generation of the sea level fluctuations observed first by HAMON (1962, 1966) on the East Australian coast. GILL and SCHUMANN (1974) argued that a b o u n d a r y layer approximation could be m a d e for the equations governing the response if, as is usually the case, the wind stress forcing (and the response) have length scales greatly exceeding the continental shelf width. Their solution showed that the response can be expressed as a sum o f modes, each satisfying a first order wave equation in the alongshore direction. To allow for frictional attenuation, they suggested a form for the friction factor, and discussed how friction might modify the solutions. M o r e recently, BRINK and ALLEN (1978) introduced friction formally into the b a r o t r o p i c equations o f motion by means o f a perturbation expansion and showed that the alongshore c o m p o n e n t of current was brought m o r e nearly into phase with the driving force than for the frictionless case where the alongshore current lags the alongshore c o m p o n e n t o f the wind stress by lt/2. Observations o f wind-forced shelf waves now abound in the hterature and have been comprehensively reviewed by LEBLOND and MYSAK (1978), MYSAK (1980a,b), and ALLEN (1980), but to date there appear to have been no observations of the wind generation of shelf waves from a geographical origin. THOMSON and CRAWFORD (1982) have, however, presented
* Faculty of Science, University of New South Wales, Kensington, NSW 2033, Australia. 215
216
J . H . MIDDLETON and A. CUNNINGHAM
theoretical and observational arguments in support of tidally driven shelf waves from a geographical origin. In this paper we present the theory and observations of wind-forced continental shelf waves opposed by a frictional force linear in alongshore velocity, and trapped on a stepshaped continental shelf. The present solution differs from that of Brink and Allen as a consequence of the inability, in this case, to expand the wind stress in terms of the across-shelf modes in a situation where only one mode exists. As a result, the present model is not limited to small values of friction. The model is verified by comparison with a limited data set from the Great Barrier Reef region of the continental shelf on the northeast coast of Australia where the sudden widening of the continental shelf from about 70 km south of the Capricorn Channel (23°S) to about 180 km north of the Capricorn Channel provides a geographical origin for wind-forced shelf waves. While THOMSONand MIDDLETON 0 9 8 3 ) have outlined the salient features for identification of wind-forced frictional continental shelf waves in general, the flat shelf simplifies the algebra substantially, obviating the need for those rotary techniques (HSIEH, 1982; MIDDLETONet al., 1982) which are useful in continental slope regions. Here we simply use the frequency response function pertaining to the alongshore current (output) and the alongshore wind stress (input). The theoretical derivation is described in Section 2 and is followed in Section 3 with a general description of the field experiment and the observations. The direct comparison of theory with experiment is described in Section 4 with the aid of frequency response functions, while Section 5 concludes with a general discussion. 2. G O V E R N I N G E Q U A T I O N S
We consider the non-divergent equations applicable to wind-forced shelf waves along a continental shelf oriented parallel to the y axis with the coast lying at x = 0. Variations in bottom topography are assumed to be independent of alongshore station y, giving the depth h = h(x). For waves having frequency ¢0 small compared to the coriolis parameter f, wavelengths sufficiently long that they are much larger than the scale width of the shelf L, and negligible surface divergence ~ = f 2 L 2 / g h ¢g 1; then the inviscid, linearized, vertically averaged shallow water equations are at;
fv = g -~x
(l)
av at; ~w -~ +fu =-g-~ + p----~
(2)
a
a (hu) + ~ (hv) = O. ~y ~x
(3)
Here u and v are the depth-averaged components of velocity in the x and y directions, respectively, t; is the surface elevation adjusted for variations in atmospheric pressure, ~'w and
Wind-forced continental shelf waves
217
x~ are the alongshore components of wind stress and bottom stress, and p is the density of the water. Neglect of the across-shelf components of both wind and bottom stress follows from the assumptions outlined above, and is justified by GILL and SCHUMANN(1974), LEBLOND and MVSA~: (1978), and BRINK and ALLE~ (1978). By virtue of the non-divergence expressed by (3) a stream function ¥ can be defined such that u-
1 by
h by
v-
1
by
h
bx
(4)
and eliminating the surface elevation from (1) and (2) yields 0 bxbt
-~x
+ h 2 dx
by - bx ([~w - ~ ]/ph).
(5)
This is the basic Gill and Schumann formulation with the inclusion of bottom fraction. The object is to solve for ¥ for a given forcing function x~,, with appropriately parameterized bottom friction ~ and subject to the following boundary conditions ¥ = 0at
by bx
x = 0
(6)
= 0 for x > L.
(7)
These imply that there is no flow through the coast, and that the alongshore velocity v becomes appropriately small in the region off the shelf. In addition, we shall be concerned here with the implementation of boundary conditions at the edge of the continental shelf. Since the continental shelf in the region of interest is relatively flat, and the slope remarkably steep, we shall consider a simple fiat shelf geometry with constant depth h on the shelf and constant depth h 0 in the ocean. We seek solutions for the stream function in regions on the shelf and in the ocean, subject to (6) and (7), and then match solutions appropriately at the shelf break. The bottom friction on the shelf is parameterized by
pr by =On,=
h
bx '
(8)
where r is a friction parameter with dimensions of velocity which has been observed to take values from about r = 10 -4 to 10-3 m s -~ (W1NAN'r and BEARDLSEV, 1979) depending upon the region. The across-shelf structure may be obtained from (5) by assuming separation of variables such that V(x, y, t ) = F(x)~JO,, t).
(9)
If there is no across-shelf variation in wind stress across a fiat shelf (h = constant) then substitution of (9) into (5) shows the across-shelf structure to be governed by ~ F / b x 2 = O.
218
J . H . MIDDLETON and A. CUNNINGHAM
Implementing the boundary condition (6), and incorporating the first constant of integration into # gives the across-shelf structure F(x) = x
0 < x < L.
(10)
If we further assume the surface elevation ~ to be given on the shelf by ~(x, y, t ) = Z(x)t[~.V, t)
(11)
then substituting (11) into (1), and using (10) gives Z(x)=Z(L)---~(L--x).
(12)
A direct substitution of (9) to (12) in (2) gives an expression for the alongshore structure
4g.v, O; a~
a~
T ; + tghZ(L) -
r
~'w
+ -g ÷ :
and writing (13) gives the forced, frictional wave equation 1 i)~ ~# r -- -- +-+-- ~=
c ~t
ay
~w
ch
(14)
pc
The form of (14) was suggested by Gill and Schumann, and found by Brink and Allen for general across-shelf depth variation. The theory of Brink and Allen cannot be applied here because of the mathematical difficulties associated with the expansion of the wind stress in terms of an infinite series of across-shelf eigenfunctions, in a case where only one across-shelf eigenfunction exists. We can solve (14) to find d~V, t) for any ~,0', t) but require Z(L) for any full solution. This may be obtained by considering flow in the deep ocean, and then by matching elevation and fluxes at the shelf break. Assuming geostrophy and non-divergence in the ocean gives f u = --g ~ y ,
at fv = g ~.ff ,
au av --ax + -~y = 0 .
(15)'
If we further assume separation of variables in the deep ocean such that 1
~= Zo(x)~.v, t)
and
a~b
- Fo(x) ~ y (y, t), u = -- -ho
(16)
where ~b(y, t) is the same function as for the shelf, then the first of (15) gives Zo(x) = Fo(X)
f gho
(17)
Wind-forced continental shelf waves
219
Matching volume fluxes at the shelf break x = L gives or
F(L) = F o ( L ) ,
L = F0(L),
(18)
while matching elevations at the shelf break gives Z ( L ) = Z o (L),
f
or
Z ( L ) = F0 (L) - gho
(19)
Substituting (18) into (19) allows us to determine the speed c from (13) as c = -fL(l
-- h/ho).
(20)
The general solution for ~ may now be found by integrating (14) along a characteristic s = t - - y / c (s =constant) as outlined by Gill and Schumann. Then (14) reduces to
dO dy
+ ¥~ = --,
(21)
W
where ¥ r(ch) -I is the friction factor and the sub- and superscripts ofxYw have been dropped for convenience. Employing a change of variable from y and t to ~ and s, such that y = ~, t = s + ~ c gives =
~ d
1 (O¢'r;)d~ = - ~ f ~ eV~xff,, s + ~/c)d~.
(22)
As pointed out by Gill and Schumann, solutions calculated from a wind stress operating from (a) initial time t = 0, have ~ = ~-- ct and (b) geographical origin y = 0, have ~o = 0. The solution for ~ from (22), in conjunction with (10) and (4) may be used to determine the forced response from any imposed wind forcing x(y, t) on a flat shelf, provided only that the scale of the forcing greatly exceeds the shelf width. In the following we will be interested in seeking solutions from a geographical origin (the mouth of the Capricorn Channel) and choose the traveling forcing function. = x0 cos(k0y - tot).
(23)
The general solution of (22), assuming no energy input at y = 0 is 0=%
os koy -- tot + ~ -- 5
--
-vy eos
e
- -c-
tot + ~- -- 8
,
(24)
where
~o
"Co =
(25)
and tan 5 -
¥ (to/c - k o )
(26)
220
J . H . MIDDLETON 811d A. CUNNINGHAM
The solution (24) thus consists of forced modes having phase speed ¢0/ko (traveling with the forcing) and free modes with phase speed c which are damped exponentially by friction from the origin y = 0. When observing the response (24), Fourier techniques will view the two components as a single component of the form ~ / ~ = G cos(0 - cot + ~t/2 - 5),
(27)
where k o ] y ) + e-2~'y
G2 = 1 - 2e-Wcos ( [ ~ -
(28)
and sin k0y -- e-~sin
coy c
tan 0 =
(29) cos k0y - e-~'Ycos
coy c
Thus the alongshore wind stress f o r c i n g t = t0 c o s alongshore current v given by
(koy - •t)
p r o d u c e s a response in the
v = Vo c o s ( O - o)t + n / 2 - 6),
where the ratio of amplitudes is given by
Vo/to =
G p c h [ y 2 + (ko - (o/c) 21 ~
(30)
and the phase lag u of response behind the forcing is given by u =
0 -
koy
+ lt/2 -
6.
(31)
At sufficiently small distances from the origin that yy ~ 1 the free wave component is little affected by friction and exp (--yy) = 1 - yy. If, in addition, the wavenumbers of the free and forced modes are sufficiently small that (o)/c - k o ) y ,~ 1 then G simplifies to G = (co/c - ko)y. In the low-frequency limit the amplitude ratio (30) shows a linear dependence with distance y from the origin, i.e.,
Vo/%
Y =
plF /k~ +
(32)
F h21½ "
Finally, we note that it is the difference between the forced and free wave amplitudes in (24) which allows this linear increase in response function with distance from the origin. The forced wave amplitude, by itself, is constant and equal to the value given by (30), but with G - 1, while the free wave amplitude decreases slowly from the origin. 3
THE FIELD EXPERIMENT
In this section we discuss observations made on the continental shelf region of the northeast coast of Australia, where the sudden widening of the continental shelf from about
Wind-forced continental shell" waves
221
70 to 180 km at 23°S comprises a geographical origin for wind-forced continental shelf waves.
From April 1980 to February 1981, three current meter moorings were deployed and maintained at the mtd-shelf position on a region of flat shelf in the south-central Great Barrier Reef region from the C S I R O R.V. Sprightly (Fig. 1). Each mooring had one Aaoderaa R C M 4 current meter with sampling interval set at 30 mln located I0 m from the bottom m total depths of 56 m (Hook mooring), 54 m (Bushy mooring), and 76 m (Bell mooring). Hydrographic measurements taken by Sprightly at the ttmes the meters were deployed and servmed showed the water column to be well mixed throughout the year, and we assume the observed currents to be barotropic. As a result of the damping o f longer period swell by the reef, and also because the depth of the meters in all cases exceeded 40 m, aliasing of the current speed data due to wave surge ts thought to be negligible (PEARSON et al., 1981). In add,tion, an Aanderaa W L R 5 water level recorder was deployed at Penrith Island, and wind velocity data from Pine Island obtained from the Bureau of Meteorology. Current meter time series were calibrated dtrecttonally to correct for the magnettc non-linearity inherent m A a n d e r a a current meters and adjusted for the local magnetic variation to give directions relative to true North. Sea levels were modified to account for fluctuations in atmospheric mean sea-level pressure.
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222
J.H.
MIDDLETONand A. CUNNINGHAM
Current and sea-level data were low-pass filtered using a fast Fourier transform filter
(WALTERS and HESTON, 1982) with 10% of the remaining Fourier coefficients cosine tapered about the cut off frequency (half power at 1 cycle per 40 h) to reduce leakage in time. All transforms used prime factors <9"/as a base (BRENNER, 1967) and were individually tadored to the length of each series. The filtered data was then decimated at daily intervals. The filtered data and the daily wind are shown in Fig. 2. Except for the Penrith Island sea levels, all other (vector) quantities are shown as vector stick diagrams which represent speed and direction as a function of time, with the oceanographic convention (specifying the direction lo which the wind blows) for the wind velocity. Average currents flow to the southeast at each of the mooring locations in contrast to the prevailing wind which blows to the northwest. The mean currents are principally oriented in the along-channel direction, which =s also basically parallel to the coast. There is some seasonal variaUon in these mean currents with stronger flow being observed in the period after August. Also noteworthy are current fluctuations at periods of" 7 to 20 days wh,ch appear highly correlated from mooring to mooring. These fluctuations have amphtudes in excess of 15 cm s-j and are at times strong enough to reverse the direction of flow. There is also variabdity in the wind data (Pine) and the sea-level data (Penrith) of about the same period as the current fluctuations. Time series were detrended by a linear least-squares method and tapered prior to transforming for spectral calculations. The spectral estimates were adjusted to account for the loss of energy due to tapering, and band averaged over 5 adjacent esUmates giving 10 d.E Prior to calculation of cartesian spectral estimates, preferred directions of orientation for low-frequency motions were found by plotting the outer rotary autocoherence (ellipse
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223
Wind-forced continental shelf waves
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224
J.H. MIDDLETON ~
A. CUNNINOHAM
stability) and phase, the phase being minus twice the orientation o f the ellipse m a j o r axis. By this means, principal axes for the low-frequency current ellipses were found to be oriented in directions o f 3 6 0 ° T (Bell), 3 2 5 ° T (Bushy), 3 0 5 ° T (Hook), and 3 0 0 ° T (Pine). The plot for the H o o k d a t a is shown in Fig. 3. In energy preserving form (whereby equal areas under a spectral plot represent equal
energies), cartesian and rotary spectra for all time series are shown in Fig. 4. For the cartcsian currcnt spectra the V(U) components arc aligncd with the major (minor) axes of thc low-frequency current clhpscs as outlined abovc. Thc cartesian spectra for wind and current vectors show the majority of cncrgy m the alongshore (V) componcnts for periods of 5 to about 30 days. In cach case thcrc appears to bc more energy at Bushy than at Bell, and more cncrgy at Hook than at Bushy, with an approximate linear incrcasc in current amplitudc north from thc Capricorn Channcl. The rotary spectral levels indicate thc cncrgy to bc almost cqually partitioned into anticlockwise (A) and clockwisc (C) components for Hook and Bushy, indlcatlng thc long period velocity fluctuations to be almost rcctilincar,although the vclocity vectors at Bell rotate in a clockwise scnsc in the horizontal planc. Thc shapes of thc wind strcss spectra and the velocity spectra arc similar as might bc cxpcctcd for wind-driven flows. To gain an idea of the rclationshlpbetween the wind stress vector time series and thc current velocity vector timc series, thc inner rotary cohcrcncc squared and phase between the wind stress at Pine and the current velocity at H o o k were calculated (Fig. 5). The vcctors arc generally coherent for periods longer than about 4 days, 10 PINE
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Wind-forced continental shelf waves 10
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Fig. 6. Rotary coherence squared and phase between current vector series at Bell and at Hook. The increase in inner phase (solid line) wsth frequency indicates phase propagation toward the equator and a linear least-squares fit to the inner rotary phase as a function o r frequency (for those frequencies with significant coherence) gives an intercept o f 9 ° and a gradient or 154 ° per cpd. These features imply (a) that the major axis of the wind stress elhpse is oriented 9 ° anticlockwise from the wind stress vectors at low frequencies, and (b) that the currents lag the wind stress by about 0.43 days. Thus the response to the wind stress at Hook is predominantly alongshore and is the same direction as the wind stress, with a small phase lag. To gain a general idea of the phase propagation characteristics of the current fluctuations, the rotary coherence squared and phase between the Hook current velocity series and the Bell current velocity series were calculated and these are plotted in Fig. 6. Phase propagation is equatorward as shown by the general increase in inner rotary phase as the frequency increases. Simslar calculations with the other combinations of velocity series also show an e q u a t o r w a r d phase propagation. Lastly, for a direct comparison o f experiment with theory we require a knowledge of the wind forcing as a function of distance. R o t a r y spectra calculated between wind stress vectors at Pine and 120 days of wind stress data from Heron Island (285 km to the southeast of Pine Island) give phase lags consistent with a horizontal wavelength o f 9200 km at periods o f 8 and 20 d a y s with the principal axis oriented basically alongshore. This represents a forcing wavenumber of about k 0 : 6.8 x 10 -7 m -[ independent o f frequency. 4 COMPARISON OF OBSERVATIONS WITH THEORY In this section we directly relate the observations described m the preveous section with the theory presented in Section 2.
226
J.H. MIDDLRTONand A. CUNmlqOUAM
Theoretically, topographically trapped waves of low frequency will result in the velocity vector on the continental shelf tracing out a highly eccentric ellipse in the anticlockwise dmrection in the horizontal plane, with major axis oriented approximately in the local alongshore direction. Cartesian and rotary spectral levels from the current vectors of the Hook and Bushy t~me series are consistent with these features, although the spectral levels at Bell are inconsistent at periods longer than about 8 days where an antlclockwise direction of rotation is observed. This, however, is not statistically significant to w~thin 95% confidence. Techniques of identification of forced continental shelf waves have been fully discussed by THOMSON and MIDDLETON (1983) and here we elect to use the cartesian system rather than rotary quantities by calculating the frequency response of the alongshore velocity component to the alongshore wind stress component. In particular, if X(~) and Y(m) represent the Fourier coefficients of the alongshore components of wind stress and current vector time series at radian frequency 0~, then the frequency response function H(o~) is defined by Y(o~) = n ( ~ ) X ( m )
= In(to)le-~')X(o~).
For a system in which there exists incoherent noise distributed across the current spectrum, the coherence C~y(m) specifies the fraction of energy in the current which is coherent with the wind such that =
'
where the overbars indicate either band or ensemble averaging in frequency space and the * denotes the complex conjugate. It follows that the magnitude of the frequency response is given by IH((.)I
=
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while the phase a(e), which represents the phase lag of the current time series behind the wind stress time series at frequency m, is given by [" lmag {X*(~)Y(o~)}] II(a)) = --tan -I L
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Calculations of the magnitude of the frequency response function IH((~)[ and the phase a(m) from the data by means of the above expressions may then be directly compared with the amplitude ratio (30) and the phase lag (31) predicted by theory. Before a direct comparison may be made, appropriate values must be selected for the theoretical parameters contributing to the amplitude ratio (30) and the phase lag (31). In the previous section we determined the wavenumber of the forcing to be ko = 6.8 x 1 0 -7 m -~ , and a suitable value for the density is p = 1027 kg m -3 . It remains to find appropriate values for the mean depth h, the free wave phase speed c, and the friction coefficient r. These parameters determine y and 5 so the response can be predicted as a function of downstream distance y at any frequency ~. Reference to the bathymetry (Fig. 1) shows the origin of motions (y = 0) to be in the mouth of the Capricorn Channel, some 200 km southeast of the Bell mooring location. To examine the dependence of the solution on these quantities, a set of curves drawn from (30) and valid for ~ ~ 0 is shown in Fig. 7 on which is also plotted the observed response at
227
Wind-forced continental shelf waves
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low frequency (the crosses). The observed low-frequency response for Bell and Bushy is approximately linear with distance alongshelf from the mouth of the Capricorn Channel (CC), while the response at Hook is larger, possibly as a result of the narrowing of the continental shelf in the region of the Whitsunday Islands due west of the Hook mooring. Plots of the low-frequency amplitude ratio response function with distance are shown for two depths (h = 35 and 45 m) and two values of the friction parameter (r = 10 ~ m s -k and 5 x 10-4 m s -I ). The free wave phase speed here is given by (20) and choosing L = 180 km and h/ho = 35/500 gives a value of c = 8.75 m s - ' . These curves show that increasing the friction reduces the magnitude of response and that increasing the depth also reduces the magnitude of the response. In general the rate of increase of amplitude near the origin depends on the value of h, while at larger distances the effect of variation caused by the friction coefficient r is more pronounced as the free waves generated at the origin become increasingly damped. The present theory takes no account of the existence of the Great Barner Reef on the outer shelf. MIDDLETON(1983) has shown that for this section of the continental shelf the existence of the Reef increases the phase speed of higher frequency free shelf waves to about 14.6 m s - ' at 0.25 cpd, while shelf waves with frequencies lower than about 0.05 cpd remain unaffected. This variation of phase speed with frequency corresponds to Middleton's dimensionless reef opacity parameter ~ = 7, this being a value supported by observation. Since the phase speed c appears in the denominator of (30), an increase in phase speed with frequency will result in a decrease in response amplitude as frequency increases, a feature observed in all the present data as discussed below. MIDDLETON'S (1983) unforced wave theory cannot be readily incorporated into the forced theory presented here, and so we have elected here to use simply the phase speed c(co) determined from the earlier work and plotted here for reference in Fig. 8.
228
J.H. MIDDLETON 8yld A. CUNNINOI-IAM
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As a function of frequency, the response function and phase for each of the moorings are shown in Fig. 9 for the selected values of r = 2.5 × 10 -4 m s -j , h = 35 m, and c as given in Fig. 8, these values providing the best all around fit. Although the value for h may appear low when compared to the mooring depths the presence of shallower water toward the coast and the existence of the spatially intermittent reefs extending to the surface on the outer shelf (where the depth between reefs is about 50 m) renders h -- 35 m a plausible average value. From the data, calculations were made of the amplitude and phase of the frequency response function. Band averaging over 10 adjacent estimates in the frequency domain gave 20 d.f. in the spectral estimates used to evaluate [H(m)[ and a(c0). Since most of the energy in the currents exists at frequencies lower than about 0.25 cpd, we have limited calculations to this range. For the Bell data the amplitude response provides an excellent fit for all periods longer than 5 days, while the observed phases lag more than predicted by theory. In general the predicted and observed phases agree to within about 20 ° and the predicted trend of an increase in phase lag with frequency is observed. For the Bushy data the phase lag results also show a general increase of phase lag with frequency as predicted by the data, but the amplitude results are less convincing. In conjunction with the higher frequency response data, an average value of the two lower frequency data points (each from the two separate segments of current data) shows the theory to underpredict by about 10%. The observed phases at Hook show good general agreement at both very long periods (50 days) and at very short periods (44 to 5 days), while at intermediate periods the observed currents lag the predicted currents by 10 to 30 °. The amplitude response at Hook is, however, clearly anomalous. The shape of the response is in general agreement although the amplitudes are underpredicted (solid lines) by a factor of about 50%. Figure 1 shows that the protruding coastline and islands due west of the Hook mooring effectively narrow the continental shelf here. Wind-forced continental shelf waves propagating toward this region will
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Wind-forced continental shelf waves
231
have their energy concentrated into this narrower shelf width, contributing to the higher currents observed at the Hook mooring compared with those predicted. An estimate of the increase in velocity due to the narrowing shelf may be obtained through energy considerations. From Rockhampton to Mackay the shelf width remains approximately constant at L = 180 km; however, at Hook the effective width of the shelf is only about L = 90 kin. For non-divergent waves the total energy of flow in any cross-section of shelf is proportional to v2Lh, and with the cross-sectional width L effectively halved at Hook, the alongshore velocity v will be increased by a factor 2½ over that for a constant w~dth shelf. Theoretical amplitudes of the response function, adjusted to this factor, are shown m Fig. 9c as a function of frequency by a dashed line. The agreement is much improved and the prediction generally within about 10% of the observations except at the higher frequencies where there is, in any case, little energy. Finally, the amplitude response of the sea level at Penrith for the wind stress at Pine ~s shown m Fig. 9d. As for the alongshore current response diagram for Bushy, the observed sea-level response is underpredicted, possibly as a result of the narrowing of the shelf equatorward of Broad Sound, while the phase lag lies within about 20 ° of that predicted, and is substantially smaller than the value of 90 ° predicted by the frictionless theory. 5
DISCUSSION
In this paper we have presented theory and measurements for low-frequency wind-forced continental shelf waves from a geographical origin, this being the first such evidence of this type of motion to appear in the literature to the best of our knowledge. The geographic origin exists as a consequence of the sudden widening of the continental shelf on the northeast coast of Australia from about 70 km south of 23°S to about 180 km north of 23°S. The wider shelf provides a greater surface area of shallow water on which the wind stress may act, wind stress forcing on the deeper ocean surface being ineffective in terms of shelf-wave generation as a result of the stratification in the deep ocean. The problem is, however, complicated by the presence of a section of the Great Barrier Reef system on the outer shelf (this being known to modify the phase speeds and structure of free wave modes), and the narrowing of the contlnental shelf at 20"S. Theory and experiment have been compared by means of frequency response functions relating alongshore current (output) to alongshore wind stress (input). Response amplitudes calculated from the data recorded at the two mooring s~tes nearest the Capricorn Channel (the origin of motion) show reasonable support for the theory, while observed phase lags are generally within 20* of those predicted. In particular amphtudes and phase lags are observed to decrease and increase, respectively, with frequency. Data from the Hook location ~s, at first s~ght, less convincing because the theoretical amplitudes underestimate the observed amplitudes by a factor of about 50%. However, the shelf in the region of the Hook mooring is narrower than that further south and this would be expected to concentrate the shelf wave energy as it propagates equatorward leading to observed currents exceeding those predicted. Adjustment of the theoretical response to allow for the narrower shelf width provides a much improved agreement with theory. For an ideal experiment location, the matching of theory with experiment would serve to determine a value for the frictional parameter r, and in this case the 'best' fit appears to occur for a value of r - - 2 . 5 x 10 -4 m s-~, with a selected mean depth of 35 m. However, in thin case, the flow m the region of observation (the development region for the flow) is insensltwe
232
J.H. ]~[IDDLETON and A. CUNNINGHAM
t o t h e v a l u e o f r. I n a d d i t i o n , w e h a v e elected t o u s e a q u a d r a t i c w i n d s t r e s s f r i c t i o n c o e f f i c i e n t h a v i n g a v a l u e e q u a l to 1.6 a n d 2.5 × 1 0 -3 f o r w i n d s p e e d s < 7 a n d > 1 0 m s-~ , r e s p e c t i v e l y , w i t h a l i n e a r t r a n s i t i o n for s p e e d s b e t w e e n 7 a n d 10 m s -~ (AMoROCHO a n d DEVRIES, 1980), a n d the o b s e r v e d r e s p o n s e is l i n e a r l y d e p e n d e n t o n t h e selected v a l u e s . I n v i e w o f t h e a b o v e a r g u m e n t s a n d s e v e r a l o t h e r f a c t o r s i n c l u d i n g (a) t h e p r e s e n c e o f t h e G r e a t B a r r i e r R e e f o n the o u t e r s e c t i o n s o f t h e c o n t i n e n t a l shelf, (b) u n k n o w n c o n d i t i o n s e x i s t i n g in t h e C o r a l S e a in t e r m s o f v a r i a b l e a l o n g s h o r e p r e s s u r e g r a d i e n t s , a n d (c) t h e r a t e o f p r o p a g a t i o n o f e n e r g y into the C a p r i c o r n C h a n n e l f r o m f u r t h e r s o u t h , t h e m e a s u r e m e n t s p r o v i d e r e a s o n a b l e e v i d e n c e for t h e e x i s t e n c e o f w i n d - f o r c e d c o n t i n e n t a l w a v e s f r o m a g e o g r a p h i c a l origin.
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