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Semi-inertial velocity variations along the Gulf Stream axis
Deep-Sea Research, 1965, Vol. 12, pp. 893 to 897. Pergamon Press Ltd. Printed in Great Britain.
Semi-inertial velocity, variations along the Gulf Stream axis CHESTER W . NEWTON National Center for Atmospheric Research, Boulder, Colorado
(Received 15 June 1965)
Abstract--It is suggested that velocity variations along the Gulf Stream core may be characterized by a Lagrangian periodicity o f a full p e n d u l u m day, on the average, as in the atmospheric jet stream. This is consistent with the condition that particle velocities vary at twice the rate o f change o f the geostrophic velocity.
INTRODUCTION
WEBSTER (1963) and others have demonstrated the presence of water motions having an inertial component of a half pendulum day. This is indicative of a more-or-less uniform swaying of the total water mass, with no accommodation of the pressure field to the water oscillations. In a narrow current system such as the Gulf Stream, which is on the average in geostrophic balance, it is reasonable to expect that a partial accommodation should take place, by local rearrangements of the mass distribution in its baroclinic environment. In this note, a class of inertial motion is discussed, which is apparently related to the dimensions of high-velocity streaks in the Gulf Stream. Ross~v (1951) has stressed the viewpoint that the behaviours of atmospheric and oceanic currents, whose structures are basically similar, are governed by the same physical mechanisms. Thus it is appropriate to draw upon the results of analyses of the atmospheric system, in which the pressure and velocity fields are more readily observed, to suggest analogies with the oceanic system.
PARTIAL-INERTIAL
OSCILLATIONS
Figure 1 shows an analysis of Gulf-Stream eddy " E d g a r " (FuGLISTER and WORTHINGTON, 1951), to which the field of isotachs has been added*. Along the axis of the Stream, alternating regions of faster and slower current speeds are evident, similar to the variations observed along the atmospheric jet stream. It will be supposed that the geopotential field in the neighbourhood of such velocity streaks is similar to that typical of the atmosphere (Fig. 2). In a perturbation of this kind superimposed on a nearly straight basic current, it is generally observed that where the actual current speeds reach maximum values, the geostrophic current speed is also greatest. *The GEK surface current measurements were kindly made available by Mr. L. V. Worthington from the files of the Woods Hole Oceanographic Institution. In drawing the isotachs, GEK-observed speeds have been "corrected" by a k-factor of 1.53, the average value for the whole of Operation Cabot, determined by yon ARX (1951). Use of a constant k-factor gives an overestimate of current speed in regions of anticyclonic curvature, and. an underestimate in regions of cyclonic curvature. 893
894
CHESTER
W.
NEWTON
6':0
G~Oo
!*q ~
ii
Fig. 1. Gulf-Stream eddy "Edgar." Only 15 ~and 20"~isotherms (mean temperature in upper 200-M layer) areshown; for detailed analysis see FUGUSTERand WORTFtnqGTON(1951). Velocity symbols show uncorrected surface current measurements by geomagnetic.electrokinetograph; a pennant represents 100 cm/sec, and a full barb 20 cm/sec. Solid symbols refer to measurements made on 17 June; dashed symbols on 16 or 18 June. [sotachs (full lines) after NEWTON(1961), reanalyzed by multiplying GEK velocities by 1.53. Particles in the jet core overtake the slower-moving speed maximum, passing through it from the upstream toward the downstream side. In doing so, they accelerate then decelerate, necessitating a flow first toward lower then toward higher geopotentiat contours. In frictionless flow.
where V, Vg and Va denote the actual, geostrophic, and ageostrophic velocities, K ix an upward-directed unit vector and f is the Coriolis parameter. A meander of the current axis in the manner shown in Fig. 2 requires that the current be supergeostrophic where the speed is greatest, and subgeostrophic where it is least. Thus following a particle, the actual speed generally changes in the same sense as the geostrophic speed, but at a greater rate. Although velocity oscillations of various types might be imagined, the simplest hypothesis would be a circular variation, as indicated at the bottom of Fig, 2 and in Fig. 3, according to the relationship V,j ..... n ~ ' ;
0 • n
1.
I2~
From equation (2), the variable part of the velocity is V' := (1 - - n ) -1 Va. According to equation (1), V' rotates with time in a clockwise sense. Its angular velocity is, from Fig. 3, f2 fVa/V' ::: (I n)f The period of the oscillation is 2~r/.Q, or
Semi-inertial velocity variations along the Gulf Stream axis
--..___
V mm
. _ . - -"~- - - ' ~ -/
~
=0
7Legend: .....,kyg , ~
_
~ I ~~ J ~"
!
t =a-To
1 ,-gTa
/~/g+,~o - - 7 ]-=->~
895
V m,~...---
!
i
t:1¼T a
t: Ta
-~'2>
Fig. 2. Schematic geopotential contours (solid lines) and isotachs (dashed) in neighbourhood of a velocity perturbation superimposed on a straight basic current. Heavy dashed line is axis of current; velocity variations of a particle in it are indicated in lower part. After NEwtoN (1959).
TC l
r
2~
Ti l --)7
in)J'-
(I
(3)
where Ti is the inertial period, 1 day/sin 4'.
/
/ ¥
j J
,)'\ /
,
, /V':(I-n)-
% '/~x
n~>''
go
t
,, ftodOgr aph
/
. of (Va*,V'g)
/
Fig. 3. Velocity variation of a particle in the current core, drawn for the case V' See text.
A
PREFERRED
SEMI-INERTIAL
2V,,.
PERIOD
In a previous investigation applying the above principles to the jet stream (NEWTON, 1959), it was predicated (without any logical basis for the choice) that n in equations (2) and (3) might have a value of-~. This prescribes an oscillation having the properties -- 29g
= 29a:
Ta
= 2T~.
(4)
T~ corresponds (Fig. 2) to thc time required for a fluid particle, in the core of the
896
CHESTER W. NEWTON
current, to pass through the velocity maximum, from the minimum speed point ot~ the upstream side to the one on the downstream side. The path described is a cycloid, having wavelength and amplitude Lj ---=21;: (17 -- cj);
Aj --~ 4 V a i l
(.5)
where f" is the mean speed in the core of the current, and cj is the speed with which the isotach pattern progresses downstream. Tests against jet-stream isotach analyses indicated that on the average, equation (5) is nearly satisfied*. In Fig. 1, there are five maximum-speed streaks in the distance of 1500 km between A and B, giving an average value ofL~, = 300 kin. In the core of the Gulf Stream, t7 was about 225 em/sec. F e a t u r e " C " a p p e a r e d to progress about 55 km downstream between 12-13 June and 20-21 June, at a rate of 8 cm/sec. This value may not be typical, but it suggests that, as in the jet stream, the velocity streaks move much more slowly than the central current speed. Using the above values, the mean period (/7 .... cj)/Lj is found to be 1.60 days, compared with 2Ti = 1-57 days. Considering uncertainties in analysis, and the fact that only one synoptic case is available for which the velocity field is mapped in such detail, this close agreement with equations (4) and (5) may be considered fortuitous. Nevertheless, it is suggested that semi-inertial oscillations are present in the Gull Stream, as well as in the atmospheric j e t stream, with average periods approximating a full pendulum day in the core o f the current. F r o m Fig. 3 with the condition specified by equation (4), Va ~ (Vm~× - V"min). In Fig. 1, 17max~ 272 cm/sec and 17mi. ~ 176 cm/sec, so that f~, ~ 24 cm/sec. The amplitude of the meander due solely to the velocity perturbation would then be. from equation (5), about 10 kin. According to the above values, the maximum geostrophic current at Vm~Xor ~'],~,. points would be about 10 per cent under- or over-estimate of the true current speed, if the velocity perturbation were superimposed on a straight basic current as in Fig. 2. From equation (1) it can be shown that, neglecting variations of curvature with depth, (~ V/~z)/(3Vg/~z) = 1/(1 - - 2 V a / V ) , so that in the bends of the current a 21 per cent error in the vertical shear would result?. No speculation is offered, as to the origin of the velocity perturbations. It is not suggested that they are directly related to the meanders described by FUGLISTE~ and WORTHINGTON(1951) and others; rather they are superimposed on these. The approximately daily local speed variations and the " shingled " structure observed in the Florida Current portion of the Gulf-Stream System (YON ARX et al., 1955) also evidently have no direct connection. Recognizing the hazards in drawing conclusions from a single synoptic case, the above can only be taken as a suggestion that the dimensions of velocity streaks in the deep-ocean portion of the Gulf Stream, *There was a great deal of variation in individual cases, partly attributable to uncertainties in analysis. In 2/3 of the cases, the period of the oscillation was between 1.5 and 2.5 T~; the average periods in different latitude belts varied in accord with the latitudinal variation of Tz. In middle and higher latitudes, perturbations of the kind in Fig. 2 are almost always superimposed on a curved basic current, as is the case in Fig. t. tThis is small compared with Uae error in the computed decay with depth, in the upstream anticyclonic bend in Fig. 1. With the conservative assumption that the GEK k-factor is 1, the maximum current speed would be about 2.2 m/sec, Va = 0"65 m/sec, and 3V/bz would be 2"4 times 3 Vg/3z. In the subtropical jet stream, ratios of this size are common, in layers of restricted depth. The discrepancy may be enhanced; if the variation of trajectory curvature with height is considered (NEwror~ and P~gssorq, 1962).
Semi-inertial velocity variations along the Gulf Stream axis
897
like those in the jet stream, are b r o a d l y related to a d o u b l e inertial p e r i o d in the core o f the current. REFERENCES FUGLISTER F. C. and WORTHINGTON L. V. (1951) Some results of a multiple ship survey of the Gulf Stream. Tellus 3, 1-14. NEWTON C. W. (1959) Axial velocity streaks in the jet stream : Ageostrophic '" inertial " oscillations. J. Metearal. 16, 638-645. NEWTON C. W. (1961) Estimates of vertical motions and meridional heat exchange in Gulf Stream eddies, and a comparison with atmospheric disturbances. J. geaphys. Res. 66, 853-8?0. NEWTON C. W. and PERSSON A. V. (1962) Structural characteristics of the subtropical jet stream and certain lower-stratospheric wind systems. Tellus 14, 221-241. RossBv C. G. (1951) A comparison of current patterns in the atmosphere and in the ocean basins. Union G~od. et G~ophys. Intern., 9th Assembly, Assoc. de M~;t~or., Brussels, 9-31. YON ARX W. S. (1951) Some measurements of the surface velocities in the Gulf Stream. Tech. Rep., Reference 51-96, Woods Hole Oeeanogr. Inst., 22 pp. (unpublished manuscript). YON ARX W. S., BUMPUS D. F. and RICHARDSON W. S. (1955) On the fine structure of the Gulf Stream front. Deep-Sea Res. 3, 46-65. WEBSTER F. (1963) A premiminary analysis of some Richardson current meter records. Deep-Sea Res. 10, 389-396.
Semi-inertial velocity, variations along the Gulf Stream axis CHESTER W . NEWTON National Center for Atmospheric Research, Boulder, Colorado
(Received 15 June 1965)
Abstract--It is suggested that velocity variations along the Gulf Stream core may be characterized by a Lagrangian periodicity o f a full p e n d u l u m day, on the average, as in the atmospheric jet stream. This is consistent with the condition that particle velocities vary at twice the rate o f change o f the geostrophic velocity.
INTRODUCTION
WEBSTER (1963) and others have demonstrated the presence of water motions having an inertial component of a half pendulum day. This is indicative of a more-or-less uniform swaying of the total water mass, with no accommodation of the pressure field to the water oscillations. In a narrow current system such as the Gulf Stream, which is on the average in geostrophic balance, it is reasonable to expect that a partial accommodation should take place, by local rearrangements of the mass distribution in its baroclinic environment. In this note, a class of inertial motion is discussed, which is apparently related to the dimensions of high-velocity streaks in the Gulf Stream. Ross~v (1951) has stressed the viewpoint that the behaviours of atmospheric and oceanic currents, whose structures are basically similar, are governed by the same physical mechanisms. Thus it is appropriate to draw upon the results of analyses of the atmospheric system, in which the pressure and velocity fields are more readily observed, to suggest analogies with the oceanic system.
PARTIAL-INERTIAL
OSCILLATIONS
Figure 1 shows an analysis of Gulf-Stream eddy " E d g a r " (FuGLISTER and WORTHINGTON, 1951), to which the field of isotachs has been added*. Along the axis of the Stream, alternating regions of faster and slower current speeds are evident, similar to the variations observed along the atmospheric jet stream. It will be supposed that the geopotential field in the neighbourhood of such velocity streaks is similar to that typical of the atmosphere (Fig. 2). In a perturbation of this kind superimposed on a nearly straight basic current, it is generally observed that where the actual current speeds reach maximum values, the geostrophic current speed is also greatest. *The GEK surface current measurements were kindly made available by Mr. L. V. Worthington from the files of the Woods Hole Oceanographic Institution. In drawing the isotachs, GEK-observed speeds have been "corrected" by a k-factor of 1.53, the average value for the whole of Operation Cabot, determined by yon ARX (1951). Use of a constant k-factor gives an overestimate of current speed in regions of anticyclonic curvature, and. an underestimate in regions of cyclonic curvature. 893
894
CHESTER
W.
NEWTON
6':0
G~Oo
!*q ~
ii
Fig. 1. Gulf-Stream eddy "Edgar." Only 15 ~and 20"~isotherms (mean temperature in upper 200-M layer) areshown; for detailed analysis see FUGUSTERand WORTFtnqGTON(1951). Velocity symbols show uncorrected surface current measurements by geomagnetic.electrokinetograph; a pennant represents 100 cm/sec, and a full barb 20 cm/sec. Solid symbols refer to measurements made on 17 June; dashed symbols on 16 or 18 June. [sotachs (full lines) after NEWTON(1961), reanalyzed by multiplying GEK velocities by 1.53. Particles in the jet core overtake the slower-moving speed maximum, passing through it from the upstream toward the downstream side. In doing so, they accelerate then decelerate, necessitating a flow first toward lower then toward higher geopotentiat contours. In frictionless flow.
where V, Vg and Va denote the actual, geostrophic, and ageostrophic velocities, K ix an upward-directed unit vector and f is the Coriolis parameter. A meander of the current axis in the manner shown in Fig. 2 requires that the current be supergeostrophic where the speed is greatest, and subgeostrophic where it is least. Thus following a particle, the actual speed generally changes in the same sense as the geostrophic speed, but at a greater rate. Although velocity oscillations of various types might be imagined, the simplest hypothesis would be a circular variation, as indicated at the bottom of Fig, 2 and in Fig. 3, according to the relationship V,j ..... n ~ ' ;
0 • n
1.
I2~
From equation (2), the variable part of the velocity is V' := (1 - - n ) -1 Va. According to equation (1), V' rotates with time in a clockwise sense. Its angular velocity is, from Fig. 3, f2 fVa/V' ::: (I n)f The period of the oscillation is 2~r/.Q, or
Semi-inertial velocity variations along the Gulf Stream axis
--..___
V mm
. _ . - -"~- - - ' ~ -/
~
=0
7Legend: .....,kyg , ~
_
~ I ~~ J ~"
!
t =a-To
1 ,-gTa
/~/g+,~o - - 7 ]-=->~
895
V m,~...---
!
i
t:1¼T a
t: Ta
-~'2>
Fig. 2. Schematic geopotential contours (solid lines) and isotachs (dashed) in neighbourhood of a velocity perturbation superimposed on a straight basic current. Heavy dashed line is axis of current; velocity variations of a particle in it are indicated in lower part. After NEwtoN (1959).
TC l
r
2~
Ti l --)7
in)J'-
(I
(3)
where Ti is the inertial period, 1 day/sin 4'.
/
/ ¥
j J
,)'\ /
,
, /V':(I-n)-
% '/~x
n~>''
go
t
,, ftodOgr aph
/
. of (Va*,V'g)
/
Fig. 3. Velocity variation of a particle in the current core, drawn for the case V' See text.
A
PREFERRED
SEMI-INERTIAL
2V,,.
PERIOD
In a previous investigation applying the above principles to the jet stream (NEWTON, 1959), it was predicated (without any logical basis for the choice) that n in equations (2) and (3) might have a value of-~. This prescribes an oscillation having the properties -- 29g
= 29a:
Ta
= 2T~.
(4)
T~ corresponds (Fig. 2) to thc time required for a fluid particle, in the core of the
896
CHESTER W. NEWTON
current, to pass through the velocity maximum, from the minimum speed point ot~ the upstream side to the one on the downstream side. The path described is a cycloid, having wavelength and amplitude Lj ---=21;: (17 -- cj);
Aj --~ 4 V a i l
(.5)
where f" is the mean speed in the core of the current, and cj is the speed with which the isotach pattern progresses downstream. Tests against jet-stream isotach analyses indicated that on the average, equation (5) is nearly satisfied*. In Fig. 1, there are five maximum-speed streaks in the distance of 1500 km between A and B, giving an average value ofL~, = 300 kin. In the core of the Gulf Stream, t7 was about 225 em/sec. F e a t u r e " C " a p p e a r e d to progress about 55 km downstream between 12-13 June and 20-21 June, at a rate of 8 cm/sec. This value may not be typical, but it suggests that, as in the jet stream, the velocity streaks move much more slowly than the central current speed. Using the above values, the mean period (/7 .... cj)/Lj is found to be 1.60 days, compared with 2Ti = 1-57 days. Considering uncertainties in analysis, and the fact that only one synoptic case is available for which the velocity field is mapped in such detail, this close agreement with equations (4) and (5) may be considered fortuitous. Nevertheless, it is suggested that semi-inertial oscillations are present in the Gull Stream, as well as in the atmospheric j e t stream, with average periods approximating a full pendulum day in the core o f the current. F r o m Fig. 3 with the condition specified by equation (4), Va ~ (Vm~× - V"min). In Fig. 1, 17max~ 272 cm/sec and 17mi. ~ 176 cm/sec, so that f~, ~ 24 cm/sec. The amplitude of the meander due solely to the velocity perturbation would then be. from equation (5), about 10 kin. According to the above values, the maximum geostrophic current at Vm~Xor ~'],~,. points would be about 10 per cent under- or over-estimate of the true current speed, if the velocity perturbation were superimposed on a straight basic current as in Fig. 2. From equation (1) it can be shown that, neglecting variations of curvature with depth, (~ V/~z)/(3Vg/~z) = 1/(1 - - 2 V a / V ) , so that in the bends of the current a 21 per cent error in the vertical shear would result?. No speculation is offered, as to the origin of the velocity perturbations. It is not suggested that they are directly related to the meanders described by FUGLISTE~ and WORTHINGTON(1951) and others; rather they are superimposed on these. The approximately daily local speed variations and the " shingled " structure observed in the Florida Current portion of the Gulf-Stream System (YON ARX et al., 1955) also evidently have no direct connection. Recognizing the hazards in drawing conclusions from a single synoptic case, the above can only be taken as a suggestion that the dimensions of velocity streaks in the deep-ocean portion of the Gulf Stream, *There was a great deal of variation in individual cases, partly attributable to uncertainties in analysis. In 2/3 of the cases, the period of the oscillation was between 1.5 and 2.5 T~; the average periods in different latitude belts varied in accord with the latitudinal variation of Tz. In middle and higher latitudes, perturbations of the kind in Fig. 2 are almost always superimposed on a curved basic current, as is the case in Fig. t. tThis is small compared with Uae error in the computed decay with depth, in the upstream anticyclonic bend in Fig. 1. With the conservative assumption that the GEK k-factor is 1, the maximum current speed would be about 2.2 m/sec, Va = 0"65 m/sec, and 3V/bz would be 2"4 times 3 Vg/3z. In the subtropical jet stream, ratios of this size are common, in layers of restricted depth. The discrepancy may be enhanced; if the variation of trajectory curvature with height is considered (NEwror~ and P~gssorq, 1962).
Semi-inertial velocity variations along the Gulf Stream axis
897
like those in the jet stream, are b r o a d l y related to a d o u b l e inertial p e r i o d in the core o f the current. REFERENCES FUGLISTER F. C. and WORTHINGTON L. V. (1951) Some results of a multiple ship survey of the Gulf Stream. Tellus 3, 1-14. NEWTON C. W. (1959) Axial velocity streaks in the jet stream : Ageostrophic '" inertial " oscillations. J. Metearal. 16, 638-645. NEWTON C. W. (1961) Estimates of vertical motions and meridional heat exchange in Gulf Stream eddies, and a comparison with atmospheric disturbances. J. geaphys. Res. 66, 853-8?0. NEWTON C. W. and PERSSON A. V. (1962) Structural characteristics of the subtropical jet stream and certain lower-stratospheric wind systems. Tellus 14, 221-241. RossBv C. G. (1951) A comparison of current patterns in the atmosphere and in the ocean basins. Union G~od. et G~ophys. Intern., 9th Assembly, Assoc. de M~;t~or., Brussels, 9-31. YON ARX W. S. (1951) Some measurements of the surface velocities in the Gulf Stream. Tech. Rep., Reference 51-96, Woods Hole Oeeanogr. Inst., 22 pp. (unpublished manuscript). YON ARX W. S., BUMPUS D. F. and RICHARDSON W. S. (1955) On the fine structure of the Gulf Stream front. Deep-Sea Res. 3, 46-65. WEBSTER F. (1963) A premiminary analysis of some Richardson current meter records. Deep-Sea Res. 10, 389-396.