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Evidence for bottom-trapped topographic Rossby waves from single moorings
Deep-Sea
Research,1976.Vol.23, pp. 629 to 635. Pergamon Press. Printed in GreatBritain.
Evidence for bottom-trapped topographic Rossby waves from single moorings* RORY O. R. Y. THOMPSON~"and JAMES R. LUYTENt (Received 25 October 1974; in revisedform 14 August 1975; accepted 8 September 1975)
Abstract--Current-meter observations near 39°N, 70°W, on the continental rise, provide evidence that the motions with periods of I to 2 weeks are dominated by baroclinic topographic Rossby waves which decay upward from the bottom. Temperature and up-slope velocity are coherent and in quadrature at these frequencies, as predicted. The kinetic energy structure versus depth is consistent with horizontal wavelengths of 100 to 200 km. The spectra drop abruptly for periods shorter than a week, the shortest period the model says the slope and stratification around Site D can maintain. The principal axis of the velocity shifts from nearly perpendicular to the isobaths at 1-weekperiod to nearly along the isobaths at long periods, in satisfactory quantitative agreement with the model. 1. INTRODUCTION 1T HAS been suggested by various authors that the persistent low frequency variability in horizontal currents of periods of 1 to 2 weeks observed from moored current meters in the vicinity of Site D (39°10'N, 70°00'W) is related to the presence of topographic Rossby waves supported by the continental rise. THOMPSON (1971) examined the consistency of a barotropic model of the region with Site D data. Schmitz, Suarez and Rhines (personal communications) have emphasized the importance of including the baroclinic modification to the model. We wish to emphasize that some of the data are consistent with any quasigeostrophic propagating wave mode. The dispersion relation, relating the horizontal spatial and time scales, are the critical test for a particular model. We wish to show here the consistency of the data from single moorings with the topographic wave model, deferring the question of the dispersion relation to a subsequent paper (THOMPSON, in preparation). In the vicinity of Site D (39°I0'N, 70°00'W) the bottom topography is as sketched in Fig. I. 1. A reasonable, simple model for this continental rise near Site D is a plane with east-west isobaths and a constant upward slope F-----10 -2. The Brunt-Viiisiil~i frequency N, averaged over many soundings at Site D by G o r d o n Volkmann is sketched in Fig. 1.2. A reasonable, simple model
for the Brunt-Vliisiil~i frequency is N = 10 -s s -t. William Schmitz, Jr. and James Richman (personal communication) have shown that the variations in N near the surface and bottom are not likely to affect either the structure or the frequency of the bottom-trapped waves to be discussed here, so we can use the results of RHINV.S (1970, p. 284) for constant slope and N. A more detailed model would include the local variations in both the topography and the density structure. These considerations are important for a more complete understanding of the structure of the variability over the continental rise. He predicts the horizontal velocity components will be proportional to =
z
(1.1)
(z increases upwards from - - H at the ocean floor) with ~: the horizontal wavenumber, and with frequency o~ = r N (sin O) coth (~NH/f),
(1.2)
where 0 is the angle between the wavenumber vector and upslope. The angle 0 is also the angle *Contribution Number 3446 of the Woods Hole Oceanographic Institution. ~Woods Hole Oceanographic Institution, Woods Hole, Massachusetts 02543, U.S.A. 629
630
RORY O. R. Y. THOMPSONand JAMESR. LUVtEN
,:7 0 °
7i ° W
6 9°
) "I
.) ....... VN sin 0. .
200
(2.1)
m
Around Site D, N ~ 10 3 s 1, t t ,~ 2.7 kin, f ~ 10-a S- 1 , SO 'short' means wavelengths less than about 150 km. These shortwaves amount to internal Kelvin waves, whose essential physics is that of internal gravity waves for which the restoring force is provided by the stratification. The frequency, (2.1), is that of an internal gravity .n wave with displacements of the stratification at a slope F sin 0. The highest frequency occurs for 3~...... \.is/~~"'-s°°°" motions up and down the slope (0 = hi2) so the shortest period of interest is 2n/(FN) ~ 8 days. The effect of rotation is to cause upward decay of / m _ _ S - . - - - -vertical displacement of the stratification, and of the velocity field. Near the sloping bottom, an upslope velocity (V, positive) must be coupled with an upward \ ) vertical velocity, bringing denser (colder, here) rJ,llt~ ~,l water up. Similarly, higher temperature (T) near Fig. 1.1. Bottom topography in the vicinity of Site D, on the bottom will follow downslope flow, hence the continental rise south of Cape Cod. precede positive V. Thus, V and T should be in quadrature, with T leading. N IO+3SEC 2x10-3 SEC -~ 3xlO-3SEC -I The Buoy Project of the Woods Hole Oceanoi i i graphic Institution recovered a 109-day record (No. 4601) of both temperature and velocity from a bottom mooring near Site D in the autumn of 1972. This record, a low passed version of which 1000 is shown in Fig. 2.1, is the only one we know of E that is near the bottom, on a slope, and long enough to be suitable for testing the hypothesis 2000 that T and V should be coherent and in quadrature. The coherence between T and V is pre00 r*OM L / sented in Fig. 2.2. There were 18 overlapped data windows, so the 95 % significant level is as shown; Fig. 1.2. Brunl-V~iisfil/i frequency profile from Site D T and V are coherent at periods centered at 6 days averaged over many years. and greater, and are incoherent at shorter periods. between the velocity vector and the isobaths, which The phase between T and V is presented in Fig. are assumed to be oriented in the east-west 2.3; it is seen that T leads about 90° at the lowest direction. Thus U and V are the components of frequency. The agreement with theory is quite velocity in the easterly (alongslope) and northerly satisfactory. (upslope) directions. 3. VERTICAL PROFILES OF KINETIC ENERGY The longest continuous current meter records 2. TEMPERATURE AND VELOCITY in the Woods Hole data files are from 18 km For short waves, ~c is large and coth ( ~ N H / f ) south of Site D at 39°00'N, 70°00'W ('Site S') tends to 1, so (1.2) becomes ~
_~
~
- .....
t.
x
I
"~oo~
Evidence for bottom-trapped topographic Rossby waves from single moorings
1264
"-- 35
3
2
631
PERIOD (doys)
/
180"
3,1
90*
2.8
--THEORY
_
NORTH 30
l°
/,4 /",
~
t-~ r . ~ A ~ _
AI
SOUTH -30 -90* EAST 30 r
-180" WEST -50 04 SEFt 72
19
04 OCT 72
19
05 NOV 72
18
03 DEC. 72
£N Fig. 2.3. Phase between temperature and upslope velocity
from No. 4601. Fig. 2.1. Low passed records of temperature, upslope
(north-south) and along slope (east-west) velocities from instrument No. 4601 (at 2620-m depth, 100m off the bottom).
T :-:-\ ........ .4
\
5% SIGNIFICANCE LEVEL
\..
~o
°~o1~ ~ ~ ~
PERIOD (doy$]
Fig. 2.2. Coherence between temperature and upslope velocity from No. 460L
for 10 months in 1972, simultaneously at 1000 and 2500 m, the latter near the bottom at 2720 m. Each consisted of three overlapping segments. The gaussian low-pass filter used on the data passes 95 % of the power at 5 days so should have
no visible effect on the spectral estimates. A plot of the low-passed data from a segment of the record at 1000 and 2500 m is shown in Fig. 3.1. The spectra for U and V from the 2500-m record are presented in Fig. 3.2. The U-spectrum, for flow along the slope, is red because of the mean flow. However, the continental slope apparently prevents mean flow across the slope, and the V-spectrum is one of the very few current spectra that is not red. In fact, the V-spectrum very nicely exhibits most of its power between 8 days and about 20 days, just about the region in which one can expect the dynamics to be linear. The Uspectrum also exhibits little power for periods shorter than 8 days. These cut-offs fit well with the apparent lack of restoring mechanism for motions with periods less than 8 days (and longer than tidal or inertial). From (1.1), the ratio of kinetic energies at depths zt and z2 is
RORY O. R, Y. THOMPSONand JAMES R. LUYTEN
632 10
~
--<-
10 x
-10
- -10
]10
10k
.
"L,
•
,
.
O~
i
7] 0 ii
~
?iii:i:.
-I0 L
,
.!
'
....k.;
i'
Fig. 3.1. Low-passed vector time series from Site S, at 1000 m (upper)and 2500 m (lower).
5 .9.6
~
3
I
.,/..',,
? ',,v'i
I 2
/ o.O../ 6 4 32
\ ..o-.... 16
8 ~-- PER~00
5
4
{days]
Fig. 3.2. Spectrafor U and Vfrom Site S at 2500-m depth.
If the physics of topographic Rossby waves hold, and zl < % the R < 1. This does not immediately fit with the observations of WEBSTER (1969) and THOMPSON (1971), who found that the total kinetic energy at Site D decreases with depth. Conversely from (3.2), if we know the ratio of the kinetic energies of a wave at depths Zl and z2, we can infer the wave number •. If we know
the ratio of the energies at a given frequency from the spectra, (3.2) allows an (energy-weighted) estimate of the wavenumber at each frequency. In Fig. 3.3 are plotted the ratios of the spectra at 1000-m depth at Site S to those at 2500 m, for U and V separately. The coherence between the corresponding velocity components at the two levels is high, of order 0.75, as can readily be
Evidence for bottom-trapped topographic Rossby waves from single moorings with S,v the cospectrum combining the estimates estimate the wavenumber quencies, and have plotted
Sto00
$25oo
5r
T"i
633
between u and v. By of ~: and 0, we can vectors for these frethem in Fig. 3.4. They
OO
\',,
o
\.
/?-
200kin.
X,~g
400km-
~
8 -90*
7,
I
1
I
T
32
t6
8
4 "4-- PERIOD
Fig. 3.3.
r
2.3.
(days.)
Ratio of spectral estimates at 1000 to 2500 m
from Site S. seen from Fig. 3.1. We see that, while there is more energy at the upper level for low and high frequencies, the lower level indeed has more energy at the intermediate frequencies of interest. Table 1 gives R and the derived ~: for these frequencies. Table I. Ratios of the V spectra at 1000 m at Site S, to those at 2500 m, the inferred wavenumber, and the principal angles at 2500 m. Period
(days)
Rv
KNH/f
wavelength
O
32
3.77
16
.51
1.01
160 k m
-25 °
0*
Ii
.39
1.23
130
-40*
8
.19
1.78
90
-70 °
The ratio of the V spectra has been used since V seems less likely to be contaminated by non-linear phenomena and the mean flow. The angle 0 of the principal axis was derived from FOFONOFF'S(1969) formula tan
20 --
2S.v
&.-sw
,
(3.2)
Fig. 3.4. Estimates of horizontal wavenumber vectors, based on the principal axis and spectral estimates for 8-, 11-, and 16-day periods.
agree reasonably well with the wavenumber vectors computed by Thompson on the basis of phase-lags between moorings in an array around Site D. There are 12 degrees of freedom in the spectral estimates, so each of the terms in (3.2) could be in error by 50 %. Thus the wave number estimates are at best approximate. A more detailed analysis of the effects of variable N and of the confidence limits on the wavenumber estimates subsequently will be presented (Thompson). 4. P R I N C I P A L AXES VERSUS FREQUENCY Another way of looking at the statistics of current meter records is the principal axes of the band-passe d vector process. These are determined from the eigenvalues o f the correlation matrix
sS:) • The orientation of the principal axis is determined from equation (3.2). The principal axes for the 2500-m record at Site S are plotted for each frequency interval in Fig. 4.1. This seems to
RORY O. R. Y. THOMPSON and .lAMB R. LUYTEN
634
expect short waves to lose their dominance away from the bottom. Then (4. l) will become an upper bound for the magnitude of 0, since coth ~ 1. Figure 4.2 shows the principal angle computed
$25OO
PE;?/OD(dGys) 32
ll6
......
~
'
(__
\\
4,
....
i °7
\\\ II/
45 ~ ~
~
l
8
ol ~ f.,,$25oo
0
+~°
® 4
O
I\\
THEORY
.90 +
1
I
~\
\\e.,~ ~ ' I ~ u
r /"1 Fig. 4.2. Principal angle as a function of period from Site S at 2500m, compared with theoretical wavenumber orientation.
23'
Fig. 4.1. Principal axes of the band-passed current, from Site S at 2500 m, as a function of the period at the center of the band.
by equation (3.2) from the record at Site S at 2500-m depth, as well as from (4.1), using I?N -= 10-5 s-1. The agreement is amazing.
9o+[
-,-- PERIOD (days)
32 i
be a useful (and perhaps new) method for presenting the spectra and cross-spectra for a vector process, in that the orientation and amplitudes are immediately seen. In this case, it can be seen that there is much more power at low frequencies (the ellipses are bigger), the current changes, from running back and forth along the isobaths at the lowest frequencies to up and down across them at intermediate frequencies (periods down to a week), then becomes isotropic (round ellipses) at higher frequencies. The vertical structure of the kinetic energy suggested that waves of periods less than a month might be short enough for (2.1) to be a reasonable approximation. It follows that the principal angle 0 is a function solely of m (for all waves not too long), so 0 ~- sin -t (o)/FN).
(4.l)
Since short waves decay rapidly upward, we can
16 i
8 i
4 ~--
45*
o• .~./Sio00
"-45"
-90'
Fig. 4.3. Principal angle as a function of period from Site S at 1000rn, compared with theoretical wavenumber orientation.
Figure 4.3 shows the same for 1000-m depth. N o w the agreement is more as expected, with 0 less than sin -1 (o)/FN), particularly at 8 days,
Evidence for bottom-trapped topographic Rossby waves from single moorings
where the waves are shorter, so do n o t penetrate into the interior so well.
Acknowledgements--The organization, the collecting and the processing of the data and the preparation of this report involved the entire Moored Array Program, which is supported by the Office of Naval Research Contract N 00014-66-CO 241 NR 083-004. We would like to thank Dr. WILLIAMSCHMITZfor his support and encouragement in this study. REFERENCES FOFONOFF N. P. (1969) Spectral characteristics of
635
internal waves in the ocean. Deep-Sea Research, Supplement, 16, 59-71. RHINES P. (1970) Edge-, bottom-, and Rossby waves in a rotating stratified fluid. Geophysical Fluid Dynamics, 1, 273-302. THOMPSON RORY O. R. Y. (1971) Topographic Rossby waves at a site north of the Gulf Stream. Deep-Sea Research, 18, 1-19. THOMPSON RORY O. R. Y. Horizontal wave-number observations on topographic Rossby waves at Site D. In preparation. WEBSTER F. (1969) Vertical profiles of horizontal ocean currents. Deep-Sea Research, 16, 85-98.
Research,1976.Vol.23, pp. 629 to 635. Pergamon Press. Printed in GreatBritain.
Evidence for bottom-trapped topographic Rossby waves from single moorings* RORY O. R. Y. THOMPSON~"and JAMES R. LUYTENt (Received 25 October 1974; in revisedform 14 August 1975; accepted 8 September 1975)
Abstract--Current-meter observations near 39°N, 70°W, on the continental rise, provide evidence that the motions with periods of I to 2 weeks are dominated by baroclinic topographic Rossby waves which decay upward from the bottom. Temperature and up-slope velocity are coherent and in quadrature at these frequencies, as predicted. The kinetic energy structure versus depth is consistent with horizontal wavelengths of 100 to 200 km. The spectra drop abruptly for periods shorter than a week, the shortest period the model says the slope and stratification around Site D can maintain. The principal axis of the velocity shifts from nearly perpendicular to the isobaths at 1-weekperiod to nearly along the isobaths at long periods, in satisfactory quantitative agreement with the model. 1. INTRODUCTION 1T HAS been suggested by various authors that the persistent low frequency variability in horizontal currents of periods of 1 to 2 weeks observed from moored current meters in the vicinity of Site D (39°10'N, 70°00'W) is related to the presence of topographic Rossby waves supported by the continental rise. THOMPSON (1971) examined the consistency of a barotropic model of the region with Site D data. Schmitz, Suarez and Rhines (personal communications) have emphasized the importance of including the baroclinic modification to the model. We wish to emphasize that some of the data are consistent with any quasigeostrophic propagating wave mode. The dispersion relation, relating the horizontal spatial and time scales, are the critical test for a particular model. We wish to show here the consistency of the data from single moorings with the topographic wave model, deferring the question of the dispersion relation to a subsequent paper (THOMPSON, in preparation). In the vicinity of Site D (39°I0'N, 70°00'W) the bottom topography is as sketched in Fig. I. 1. A reasonable, simple model for this continental rise near Site D is a plane with east-west isobaths and a constant upward slope F-----10 -2. The Brunt-Viiisiil~i frequency N, averaged over many soundings at Site D by G o r d o n Volkmann is sketched in Fig. 1.2. A reasonable, simple model
for the Brunt-Vliisiil~i frequency is N = 10 -s s -t. William Schmitz, Jr. and James Richman (personal communication) have shown that the variations in N near the surface and bottom are not likely to affect either the structure or the frequency of the bottom-trapped waves to be discussed here, so we can use the results of RHINV.S (1970, p. 284) for constant slope and N. A more detailed model would include the local variations in both the topography and the density structure. These considerations are important for a more complete understanding of the structure of the variability over the continental rise. He predicts the horizontal velocity components will be proportional to =
z
(1.1)
(z increases upwards from - - H at the ocean floor) with ~: the horizontal wavenumber, and with frequency o~ = r N (sin O) coth (~NH/f),
(1.2)
where 0 is the angle between the wavenumber vector and upslope. The angle 0 is also the angle *Contribution Number 3446 of the Woods Hole Oceanographic Institution. ~Woods Hole Oceanographic Institution, Woods Hole, Massachusetts 02543, U.S.A. 629
630
RORY O. R. Y. THOMPSONand JAMESR. LUVtEN
,:7 0 °
7i ° W
6 9°
) "I
.) ....... VN sin 0. .
200
(2.1)
m
Around Site D, N ~ 10 3 s 1, t t ,~ 2.7 kin, f ~ 10-a S- 1 , SO 'short' means wavelengths less than about 150 km. These shortwaves amount to internal Kelvin waves, whose essential physics is that of internal gravity waves for which the restoring force is provided by the stratification. The frequency, (2.1), is that of an internal gravity .n wave with displacements of the stratification at a slope F sin 0. The highest frequency occurs for 3~...... \.is/~~"'-s°°°" motions up and down the slope (0 = hi2) so the shortest period of interest is 2n/(FN) ~ 8 days. The effect of rotation is to cause upward decay of / m _ _ S - . - - - -vertical displacement of the stratification, and of the velocity field. Near the sloping bottom, an upslope velocity (V, positive) must be coupled with an upward \ ) vertical velocity, bringing denser (colder, here) rJ,llt~ ~,l water up. Similarly, higher temperature (T) near Fig. 1.1. Bottom topography in the vicinity of Site D, on the bottom will follow downslope flow, hence the continental rise south of Cape Cod. precede positive V. Thus, V and T should be in quadrature, with T leading. N IO+3SEC 2x10-3 SEC -~ 3xlO-3SEC -I The Buoy Project of the Woods Hole Oceanoi i i graphic Institution recovered a 109-day record (No. 4601) of both temperature and velocity from a bottom mooring near Site D in the autumn of 1972. This record, a low passed version of which 1000 is shown in Fig. 2.1, is the only one we know of E that is near the bottom, on a slope, and long enough to be suitable for testing the hypothesis 2000 that T and V should be coherent and in quadrature. The coherence between T and V is pre00 r*OM L / sented in Fig. 2.2. There were 18 overlapped data windows, so the 95 % significant level is as shown; Fig. 1.2. Brunl-V~iisfil/i frequency profile from Site D T and V are coherent at periods centered at 6 days averaged over many years. and greater, and are incoherent at shorter periods. between the velocity vector and the isobaths, which The phase between T and V is presented in Fig. are assumed to be oriented in the east-west 2.3; it is seen that T leads about 90° at the lowest direction. Thus U and V are the components of frequency. The agreement with theory is quite velocity in the easterly (alongslope) and northerly satisfactory. (upslope) directions. 3. VERTICAL PROFILES OF KINETIC ENERGY The longest continuous current meter records 2. TEMPERATURE AND VELOCITY in the Woods Hole data files are from 18 km For short waves, ~c is large and coth ( ~ N H / f ) south of Site D at 39°00'N, 70°00'W ('Site S') tends to 1, so (1.2) becomes ~
_~
~
- .....
t.
x
I
"~oo~
Evidence for bottom-trapped topographic Rossby waves from single moorings
1264
"-- 35
3
2
631
PERIOD (doys)
/
180"
3,1
90*
2.8
--THEORY
_
NORTH 30
l°
/,4 /",
~
t-~ r . ~ A ~ _
AI
SOUTH -30 -90* EAST 30 r
-180" WEST -50 04 SEFt 72
19
04 OCT 72
19
05 NOV 72
18
03 DEC. 72
£N Fig. 2.3. Phase between temperature and upslope velocity
from No. 4601. Fig. 2.1. Low passed records of temperature, upslope
(north-south) and along slope (east-west) velocities from instrument No. 4601 (at 2620-m depth, 100m off the bottom).
T :-:-\ ........ .4
\
5% SIGNIFICANCE LEVEL
\..
~o
°~o1~ ~ ~ ~
PERIOD (doy$]
Fig. 2.2. Coherence between temperature and upslope velocity from No. 460L
for 10 months in 1972, simultaneously at 1000 and 2500 m, the latter near the bottom at 2720 m. Each consisted of three overlapping segments. The gaussian low-pass filter used on the data passes 95 % of the power at 5 days so should have
no visible effect on the spectral estimates. A plot of the low-passed data from a segment of the record at 1000 and 2500 m is shown in Fig. 3.1. The spectra for U and V from the 2500-m record are presented in Fig. 3.2. The U-spectrum, for flow along the slope, is red because of the mean flow. However, the continental slope apparently prevents mean flow across the slope, and the V-spectrum is one of the very few current spectra that is not red. In fact, the V-spectrum very nicely exhibits most of its power between 8 days and about 20 days, just about the region in which one can expect the dynamics to be linear. The Uspectrum also exhibits little power for periods shorter than 8 days. These cut-offs fit well with the apparent lack of restoring mechanism for motions with periods less than 8 days (and longer than tidal or inertial). From (1.1), the ratio of kinetic energies at depths zt and z2 is
RORY O. R, Y. THOMPSONand JAMES R. LUYTEN
632 10
~
--<-
10 x
-10
- -10
]10
10k
.
"L,
•
,
.
O~
i
7] 0 ii
~
?iii:i:.
-I0 L
,
.!
'
....k.;
i'
Fig. 3.1. Low-passed vector time series from Site S, at 1000 m (upper)and 2500 m (lower).
5 .9.6
~
3
I
.,/..',,
? ',,v'i
I 2
/ o.O../ 6 4 32
\ ..o-.... 16
8 ~-- PER~00
5
4
{days]
Fig. 3.2. Spectrafor U and Vfrom Site S at 2500-m depth.
If the physics of topographic Rossby waves hold, and zl < % the R < 1. This does not immediately fit with the observations of WEBSTER (1969) and THOMPSON (1971), who found that the total kinetic energy at Site D decreases with depth. Conversely from (3.2), if we know the ratio of the kinetic energies of a wave at depths Zl and z2, we can infer the wave number •. If we know
the ratio of the energies at a given frequency from the spectra, (3.2) allows an (energy-weighted) estimate of the wavenumber at each frequency. In Fig. 3.3 are plotted the ratios of the spectra at 1000-m depth at Site S to those at 2500 m, for U and V separately. The coherence between the corresponding velocity components at the two levels is high, of order 0.75, as can readily be
Evidence for bottom-trapped topographic Rossby waves from single moorings with S,v the cospectrum combining the estimates estimate the wavenumber quencies, and have plotted
Sto00
$25oo
5r
T"i
633
between u and v. By of ~: and 0, we can vectors for these frethem in Fig. 3.4. They
OO
\',,
o
\.
/?-
200kin.
X,~g
400km-
~
8 -90*
7,
I
1
I
T
32
t6
8
4 "4-- PERIOD
Fig. 3.3.
r
2.3.
(days.)
Ratio of spectral estimates at 1000 to 2500 m
from Site S. seen from Fig. 3.1. We see that, while there is more energy at the upper level for low and high frequencies, the lower level indeed has more energy at the intermediate frequencies of interest. Table 1 gives R and the derived ~: for these frequencies. Table I. Ratios of the V spectra at 1000 m at Site S, to those at 2500 m, the inferred wavenumber, and the principal angles at 2500 m. Period
(days)
Rv
KNH/f
wavelength
O
32
3.77
16
.51
1.01
160 k m
-25 °
0*
Ii
.39
1.23
130
-40*
8
.19
1.78
90
-70 °
The ratio of the V spectra has been used since V seems less likely to be contaminated by non-linear phenomena and the mean flow. The angle 0 of the principal axis was derived from FOFONOFF'S(1969) formula tan
20 --
2S.v
&.-sw
,
(3.2)
Fig. 3.4. Estimates of horizontal wavenumber vectors, based on the principal axis and spectral estimates for 8-, 11-, and 16-day periods.
agree reasonably well with the wavenumber vectors computed by Thompson on the basis of phase-lags between moorings in an array around Site D. There are 12 degrees of freedom in the spectral estimates, so each of the terms in (3.2) could be in error by 50 %. Thus the wave number estimates are at best approximate. A more detailed analysis of the effects of variable N and of the confidence limits on the wavenumber estimates subsequently will be presented (Thompson). 4. P R I N C I P A L AXES VERSUS FREQUENCY Another way of looking at the statistics of current meter records is the principal axes of the band-passe d vector process. These are determined from the eigenvalues o f the correlation matrix
sS:) • The orientation of the principal axis is determined from equation (3.2). The principal axes for the 2500-m record at Site S are plotted for each frequency interval in Fig. 4.1. This seems to
RORY O. R. Y. THOMPSON and .lAMB R. LUYTEN
634
expect short waves to lose their dominance away from the bottom. Then (4. l) will become an upper bound for the magnitude of 0, since coth ~ 1. Figure 4.2 shows the principal angle computed
$25OO
PE;?/OD(dGys) 32
ll6
......
~
'
(__
\\
4,
....
i °7
\\\ II/
45 ~ ~
~
l
8
ol ~ f.,,$25oo
0
+~°
® 4
O
I\\
THEORY
.90 +
1
I
~\
\\e.,~ ~ ' I ~ u
r /"1 Fig. 4.2. Principal angle as a function of period from Site S at 2500m, compared with theoretical wavenumber orientation.
23'
Fig. 4.1. Principal axes of the band-passed current, from Site S at 2500 m, as a function of the period at the center of the band.
by equation (3.2) from the record at Site S at 2500-m depth, as well as from (4.1), using I?N -= 10-5 s-1. The agreement is amazing.
9o+[
-,-- PERIOD (days)
32 i
be a useful (and perhaps new) method for presenting the spectra and cross-spectra for a vector process, in that the orientation and amplitudes are immediately seen. In this case, it can be seen that there is much more power at low frequencies (the ellipses are bigger), the current changes, from running back and forth along the isobaths at the lowest frequencies to up and down across them at intermediate frequencies (periods down to a week), then becomes isotropic (round ellipses) at higher frequencies. The vertical structure of the kinetic energy suggested that waves of periods less than a month might be short enough for (2.1) to be a reasonable approximation. It follows that the principal angle 0 is a function solely of m (for all waves not too long), so 0 ~- sin -t (o)/FN).
(4.l)
Since short waves decay rapidly upward, we can
16 i
8 i
4 ~--
45*
o• .~./Sio00
"-45"
-90'
Fig. 4.3. Principal angle as a function of period from Site S at 1000rn, compared with theoretical wavenumber orientation.
Figure 4.3 shows the same for 1000-m depth. N o w the agreement is more as expected, with 0 less than sin -1 (o)/FN), particularly at 8 days,
Evidence for bottom-trapped topographic Rossby waves from single moorings
where the waves are shorter, so do n o t penetrate into the interior so well.
Acknowledgements--The organization, the collecting and the processing of the data and the preparation of this report involved the entire Moored Array Program, which is supported by the Office of Naval Research Contract N 00014-66-CO 241 NR 083-004. We would like to thank Dr. WILLIAMSCHMITZfor his support and encouragement in this study. REFERENCES FOFONOFF N. P. (1969) Spectral characteristics of
635
internal waves in the ocean. Deep-Sea Research, Supplement, 16, 59-71. RHINES P. (1970) Edge-, bottom-, and Rossby waves in a rotating stratified fluid. Geophysical Fluid Dynamics, 1, 273-302. THOMPSON RORY O. R. Y. (1971) Topographic Rossby waves at a site north of the Gulf Stream. Deep-Sea Research, 18, 1-19. THOMPSON RORY O. R. Y. Horizontal wave-number observations on topographic Rossby waves at Site D. In preparation. WEBSTER F. (1969) Vertical profiles of horizontal ocean currents. Deep-Sea Research, 16, 85-98.