The vertical structure of temperature fluctuations within an oceanic thermocline

Deep-Sea Research, 1967, Vol. 14, pp. 613 to 623. PergamonPress Ltd. Printed in Great Britain.

The vertical structure of temperature fluctuations within an oceanic thermocline R. A. WHITE* (Received 13 June 1967) A study has been made of the vertical structure of the temperature fluctuations present in an oceanic thermocline from two time-series of temperature profiles to 300 m depth, o f duration 2½ and 7 days, in the N o r t h Atlantic Ocean. The maximum variation o f temperature for both series (r.m.s. value -"- 0-5°C) occurred o n the upper boundary of the thermocline. Within the frequency ranges considered for each series (i.e. 0.2-1 c/hr and 0.05-1 c/hr) the spectral intensity of the temperature fluctuations generally tended to decrease with increase in frequency and no predominant periodic components were found. The variation of coherence between the temperature components of a given frequency at two depth levels, separated by varying distances, gave an indication of the vertical scale of the fluctuations. Throughout most o f the frequency range the fluctuations within the thermocline were significantly coherent for levels spaced by almost the whole thermocline layer. A level could be found at which the temperature fluctuations were most representative of the variation of the layer. The implications of the various spectral and coherence features on the nature of the fluctuations has been discussed.

Abstract

1.

INTRODUCTION

Tim TEMPERATUREfluctuations in a series of observations below the upper boundary of the thermocline may arise from several causes. The changes resulting from advection, seasonal heating and long period waves may introduce a trend into a short time series. Internal waves with frequencies ranging from the Brunt-V~iis~la to the inertial frequency can be expected to contribute to the fluctuations. Turbulentfluctuations, may also be present, produced by a variety of causes amongst which are current shear and breaking of internal waves. Observations of temperature fluctuations beneath the ocean surface have been made by several previous investigators. HAURWlTZ, STO~VmL and MUN~¢ (1959) analysed a long series of temperature measurements observed on the sea bottom at depths of 50 and 500 m at positions 1.5 km apart near Bermuda. The two series were not correlated and this may indicate that oscillations of the seasonal and permanent thermocline were not coupled in that area (Cox, 1962). At the same time the horizontal separation may have contributed to the lack of coherence. BAER and HAMM (1965) analysed data collected by CARSOLA, HAMM and ROQUE (1963) and found that fluctuations in water temperature at levels 20 m apart within the upper 60 m of the water column were uncorrelated (although reservations were made about the errors that may have been introduced by movement in the thermistor chain). SABININand SHULEVOV(1965) observed short period internal waves in the Norwegian Sea and found that temperatures at 25 m and 30 m depth within the discontinuity layer were highly correlated. LEE and Cox (1966) analysed a 14 day long series of temperature measurements at 5 rain intervals at 15 depths between 85 and 3803 m. They found a semi*Department of Oceanography, Liverpool University, Liverpool 3. 613

614

R.A. WHITE

diurnal periodic component with a maximum amplitude at mid-depth and also evidence of a microscale of temperature layering. VOORHISand PERKINS(1966) obtained an experimental estimate of the spatial spectrum in a horizontal plane at a depth of about 100 m of the temperature fluctuations in the near surface summer thermocline 200 km North West of Bermuda. They suggested that the temperature fluctuations were driven by turbulent fluctuations in the mean flow. This paper presents the results of an analysis of two series of temperature profiles to a depth of 300 m below the ocean surface. Both series were obtained with a BissettBerman Hytech Salinity-Temperature-Depth System, an evaluation of which has been published by HOWE and TA~ (1965). Series A consisted of profiles recorded at halfhourly intervals for 2½ days in the vicinity of ocean weather ship Juliett (52°N, 19":W) during July 1965. Series B extending over 7 days was of half-hourly profiles observed near the Canary Islands (29°N, 18°W) in January 1966. During each series the ship was at anchor and radar watch was kept on a moored buoy during series A and on Palma Island during series B. A slight drift occurred during series B due to dragging of the deep sea anchor. Values of temperature were read off each profile at selected depths between 0 and 300 m. The results described in this paper are mainly concerned with the character of the fluctuations at depths beneath the upper limit of the seasonal thermocline. A typical temperature profile from series B is shown in Fig. 1. A fine structure exists at depths below the mixed layer down to 300 m.

/

300 -

270-

240-

210-

180-

Depth (m)

150-

Y

120-

90-

60-

300_.

2O

I 19

I 18

- - ' 1

17

{ 16

--

Temperature °C

Fig. 1, A n e x a m p l e o f a t e m p e r a t u r e profil© f r o m Sefica B.

~5

615

The vertical structure o f temperature fluctuations within art oceanic thermocline

2.

ANALYSIS

OF THE D A T A

The mean temperature (73 and r.m.s, fluctuation of temperature (r.m.s. T) in series A and B are shown in Fig. 2 for the depths between the surface and 300 m. The thermocline of series A extends from roughly 30 to 100 m and below 100 m the temperature falls off more gradually with depth. The thermocline of series B extends from roughly 80 m depth to below 300 m; a well marked mixed layer exists above this thermocline. O-

O-

O-

60-

60-

60-

120-

120-

120-

DEPTH

DEPTH

(m)

(m)

DEPTH (m)

lBO-

1BO-

1B0-

240-

240-

240B

300

300

10

12

14

16

300

i

02

o.~,

&

10

oc

0C

M E A N TEMPERATURE

R.M S. VARIATION

c/hr

V)~,IS,S,L k

FREOUENCY

Fig. 2. The means, r.m.s, variations and V~iisfil/i frequencies in Series A and B.

The maximum temperature variation in both series occurs in the layer of steepest temperature gradient indicating that the fluctuations are probably caused by vertical movement of the isotherms. Both series A and B were subject to a certain amount of trend or long period variation. Series A which was shorter was more seriously affected and so was prefiltered by the procedure employed by MUNK, SNODGRASSand TUCKER (1959). This involved subtracting a weighted mean from each observation. The frequency response of this filter was such that the amplitude at a frequency of f=0.125 cycles per hour (c/hr) was almost unattenuated but at f = 0.083 c/hr a n d f = 0-042 c/hr, the amplitudes fell to 0.6 and 0-2 respectively. The accuracy of the instrumental system is described by HOWE and TAIT (1965) whereas the resolution is limited by the thickness of the analogue record trace and is about 0.02°C. This gives rise to a white noise in each auto spectrum of about 0.3 × 10 -4 °C 2 hr (see CARTWRIGHT, TUCKER a n d CATTON, 1962). 3.

SPECTRA OF THE FLUCTUATIONS

The spectra of the temperature fluctuations in the high pass filtered series A and raw data of series B were evaluated according to Tukey's method. The spectra in

616

R.A. Wui~

Figs. 2 and 4 are presented as contour diagrams in which depth (m) and frequency (c/hr) are ordinates and lines are drawn through points with equal density. The spectral units arc (°C) 9'hr x 104 and contours are drawn for 1O, 50, 100, 500, 1000, 5000, 10,000 units. A logarithmic depth scale was chosen for series A and a linear depth scale for series B for presentation purposes.

300200 -

lOO DEPTH (m) 50

• ,ooo

•

,

•

• ~

:

~

• --

~ 5

0

i

o

~

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-.

.~r-10

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0

o

o

,,

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•

~,ooo

•

~

500 ~00 50

~

20 i

10

~ 0

-~"~'"~ i

-02

0.4

T

T

T

0.6

0.8

10

f (c/hr)

Fig. 3. T h e s p e c t r u m o f t h e t e m p e r a t u r e f l u c t u a t i o n s in Series A. 500 300

. . . . . . . . . . .

? Z , o :

'

'

240

!iii

'

ii

....

180

DEPTH (m)

.

. . . .

-..<_ . . . .

~

=

120 •

•

•

•

•

o ~

~.-~oo

-.-..loo 0

L 0

0.2

0.4

0.6

0.8

f (c/hr)

Fig. 4. T h e s p e c t r u m o f t h e t e ~ n p e ~ t m ' e f l u c t u a t i o n s i n Series B.

1.0

The vertical structure of temperature fluctuations within an oceanic thermocline

617

Series A

The autocorrelation coefficients of series A did not reveal any marked periodicity. The spectrum shown in Fig. 3 was estimated at 14 depth levels between 0 and 300 m for five frequency bands centred on f----0.2r [r = 0 (1)5]. The 8 0 ~ confidence limits for the number of degrees of freedom in this case (40) are 0.73 and 1-30 of each spectral estimate. Considering the frequency range f = 0.2 to 1 c/hr at depths below 75 m and above 30 m the spectral energy tends to decrease with frequency increase whereas in the thermocline between 35 and 75 m the energy is practically constant within the constraints of the 80 ~o confidence limits. The last mentioned feature raises the problem of aliasing into the spectrum from frequencies in excess of 1 c/hr. No estimate of this had been made for this series but local values of the Brunt-V~iisiil~i frequency given by f - - [ ( g / p ) ( ~ p / b z ) ] l e and shown in Fig. 2 shows that internal waves with frequencies well in excess of 1 c/hr are possible and that aliasing could be serious particularly in the thermocline. Series B

The spectra of the temperature fluctuations at 11 levels in series B were evaluated. Each spectrum consisted of 20 frequency bands leading to 34 degrees of freedom per spectral estimate. The results are shown in Fig. 4. The main features are the maxima which occur on the boundary of the thermocline (at 105 m) and at about 240 m. At each depth the energy decreases with increase in frequency and there are no significant peaks. The aliased contribution was estimated at the 110 m level by making use of a continuous 20 hr time series of temperature at 110 m depth which had been observed prior to series B and was digitized at 5-min intervals. The spectral energy decreased rapidly for frequencies between 0.5 c/fir and 2c/hr and at f = 1 c/hr (limit of the profiling spectrum) the spectral energy was about 5 ~o of that a t f = 0.5 c/hr. The aliased contribution is probably small at all other depths between 0 and 300 m in spite of the fact that the estimate of the Brunt-Viiisiilii frequency (Fig. 2) shows that internal waves with frequencies well above 1 c/hr can be expected. The auto-correlation coefficients in series B exhibited a maximum value at a lag between 12 and 14 hr. The unfiltered spectrum with 20 frequency bands did not reveal a peak corresponding to this lag however. The possible significance of the oscillation was investigated by using a method described by LEE and Cox (1966). The series at the 120, 180, 240 m, levels were prefiltered by forming successive differences between the terms in the series at each level. This " p r e w h i t e n i n g " procedure reduces the contamination of the low frequency spectral components by a strong peak of even lower frequency energy. The spectra of the prefiltered series were then evaluated and corrected for the prefiltering procedure. The resolution in the low frequency part of the spectrum was improved by using 48 bands and 14 degrees of freedom. The spectral energy was greater in the frequency band centred on f = 0.063 c/fir than in the neighbouring bands centred on f = 0.041 c/hr and f = 0.083 c/fir, but this peak was found to be insignificant and hence there is no well-defined long period or tidal component in series B. 4. THE VERTICAL STRUCTURE OF THE TEMPERATURE VARIATIONS As a first step in determining how representative are the temperature fluctuations at a particular depth in the thermocline the cross correlation coefficient (lag 0 hr) between temperatures at pairs of levels in series A and B was evaluated. In Table 1 is

618

R . A . WmrE

Table 1.

Cross correlations R (x) o f temperatures at 40 m with other levels for Series A, and 150 m with other levels for Series B.

d

20 24 30 36 40 45 50 60 75 100 200 300

Series A

R (x)

d

0.11 0.22 0-35 0.72 1.00 0.76 0-61 0.56 0.42 0.25 0"16 0-12

90 105 120 150 180 210 240 270 300

Series B

R (x)

0.35 0.54 0.66 1.00 0-79 0.52 0.50 0.51 0.48

shown the coefficient R (x) for the 40 m temperatures and those at other levels of series A, and 150 m and other levels of series B. A significant correlation is found in the case of series A for levels between 30 and 100 m and in the case of series B for levels throughout the whole 90 to 300 range. This shows that some fluctuations at least that occur at a particular level in the thermocline occur throughout the whole discontinuity layer. The series A correlation coefficients are of approximately exponential form and this may suggest some similarity with results in atmospheric turbulence (PASQUILL, 196t). To show how the correlation is related to frequency and amplitude of the components in the series, it is necessary to evaluate the coherence. The coherence between two series (1 and 2) at a frequency f is defined as : y2(f) =

(7122 ( f ) + Q12z ( f )

ca1 (f) Cz2 (f)

where Cn, C22, C1~ and Q12 are the auto spectra, co and quadrature spectra of series l and 2. ~ (f) ---- 1 when there is a perfect linear relation between series 1 and 2 and (f) = 0 when the series are uncorrelated. In the case of a finite length record the 95 ~ confidence limits afor zero coherence are approximately 4/v (MUNK et al., 1959) where v = number of degrees o f freedom per spectral estimate. In the case of series A and B with v = 40, the 95 ~o limit of ~,2 ( f ) is about 0.1. The phase difference between series 1 and 2 is defined by :

QI~ (f) tan • ( f ) = C12 ( f ) " The 9 5 ~ limits ff (f) in radians for a true coherence have been tabulated in MUNK et al. (1959). For example, if y~ ( f ) = 0.5 and v = 40 the true phase lies within about 0-4 rads of the computed phase. Series A

The coherence between the temperature fluctuations in series A at pairs of depth levels ranging from 20 to 100 m is shown in Fig. 5 for frequency bands centred on f = 0.2, 0.4 and 1 c/hr. The ordinates are depth (meters) and lines are drawn through points with equal coherence (for ~ = 0.1, 0.3, 0-5 and 0.7). The coherence between

The vertical structure of temperature fluctuations within an oceanic thermocline

~

f = 0.05 c/hr

•

619 300

300

0.7

0,7

.0.5

0.5 •

f = 02c/hr

•

2

4

0

/

/

/

°

D

60,/" ~. 60

t o.,.~. 120

T

~"

(--~3.

180

0.1

_<~o3

240

60

300

120

18Q

DEPTH ( m )

240

300

DEPTH (m)

300

300

.3 ef -/ 0 .h4 r

240

o

°

120

LOW

60

120

180 DEPTH (m)

o

•

°

.

LOW°

•

0.1

240

300

60

120

180

240

DEPTH (m)

Fig. 5. Coherence between the temperature fluctuations at depth levels of Series

A.

the temperatures at pairs o f levels just above the thermocline boundary (i.e. levels between 30 and 40 m) exceeds 0.1 for levels spaced within 10-20 m of each other and exceeds 0.5 with levels spaced by less than 5 m. The temperature fluctuations at a level within the thermocline are representative o f fluctuations over a considerably larger vertical distance. For example, referring to Table 2 the coherencies are shown between the series A temperature fluctuations at 50 m and those at other levels between 30 and 100m and it can be seen that the coherence exceeds 0.1 over almost the whole frequency range for levels from 36 to 100 m. The coherence is in excess o f 0.5 over the portion of the layer from 40 to 45 m d o w n to roughly 70 m. Referring to Fig. 5, it appears that the vertical scale is a maximum for fluctuations centred at the 50-60 m

300

620

R.A. WrnTE

level and this implies that in this case the thermocline m o t i o n as a whole m a y be best represented by temperature changes at these levels.

Table 2. Series A. Coherence between fluctuations at 50 m and other depths. d(m)

f = 0

0.2

0"4

0.6

0.8

1-0

30 36 40 45 60 75 I00

0.03 0-23 0.58 0.68 0-67 0.39 0-17

0.04 0-16 0.50 0.67 0-68 0.44 0.15

0.05 0.11 0.36 0.64 0.66 0.47 0.09

0-06 0.13 0.31 0.42 0-62 0-38 0-06

0.06 0.11 0.26 0.57 0.66 0.29 0.06

0.05 0.12 o. 19 0.69 0-71 0-34 0.08

Series B. Coherence between fluctuations at 150 m and other depths 75 90 105 120 180 210 240 270 300

0-20 0.01 0.03 0.27 0.50 0.11 0.15 0.30 0-37

0.07 0.28 0.55 0.64 0.76 0.34 0.36 0.26 0.26

0.11 0.25 0-50 0.38 0.67 0.38 0.35 0.28 0.05

0.04 0.35 0.55 0.47 0.68 0.38 0.52 0-30 0.22

0.01 0.22 0.42 0.47 0.53 0.55 0.19 0.15 0-09

0.04 0.15 0.37 0.51 0-69 0.47 0.48 0.24 0.22

0.10 0.19 0.51 0.45 0.51 0.56 0.30 0.25 0.21

0.15 0-41 0.51 0.42 0.66 0.41 0.20 0.25 0.20

0.03 0-38 0.52 0.47 0-71 0-57 0.49 0.47 0.32

0-25 0-31 0.52 0.53 0.75 0.56 0.54 0.40 0.29

0.06 0.30 0.29 0.23 0.74 0.54 0.20 0.38 0.I 1

Series B The coherence between the temperature fluctuations at pairs o f levels between 60 and 300 m o f series B is shown in Fig. 6 for the frequency bands centred o n f = 0.05, 0.2, 0.5 and 1 c/hr. At the lowest f r e q u e n c y f ~- 0.05 c/hr, the coherence between the 105 and 240 m temperature components is less than 0. ! and reference to Table 3 shows that the coherence is relatively low for these two levels t h r o u g h o u t the frequency range covered. Thus the peaks in low frequency energy observed in the spectrum (see Fig. 4) at 105 and 240 m are uncorrelated. At depths below 120 m the coherence

Table 3. f(c/hr) v~(f)

Series B. Coherence of 105 and 240 m levels.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.02

0.18

0.18

0.21

0.10

0.29

0-11

0.21

0.23

0-30

0.02

exceeds 0.5 for levels spaced by at least 25 or 30 m. In the figures f o r f --= 0.2, 0-5 and 1 c/hr the coherence exceeds 0-1 for most pairs o f temperature series at depths of 120-300 m. The most representative level for the temperature variation in the thermocline occurs at a b o u t 150 m. The coherencies for this 150 m level with other levels is shown in Table 2 where the coherence exceeds 0.5 for most o f the frequency range covered at levels f r o m 105 m to a b o u t 2 1 0 m (which leads to a vertical scale o f a b o u t 3 times that in series A). Referring back again to Fig. 6, it can be seen that isolated high coherence peaks (9,2 = 0"6 and 0"3) occur a t f = 1 c/hr for the 180 with 270 m and 90 with 210 m levels respectively. These effects might possibly be attributed to the nodal structure o f internal waves. However, the phases between the spectral components at this frequency do not support this explanation. Table 4 shows the coherence and phase o f the t e m p e r a t u r e fluctuations at 180 m relative to those at other levels for

T h e vertical structure o f t e m p e r a t u r e fluctuations within a n oceanic t h e r m o c l i n e

180/ f • O-~ c/hr_

80~100

0,

f . 0A c/l'r

0.5 DEPTH ( m )

DEPTH ( m )

60

,

"/"

20

~

"

"

40

-

20

T

"

60 DEPTH (n9

,

1 100

80

60

.

20 20

4

. O5

*

03

~

LOW

J

I

40

60 DEPTH (m)

I 80

J

L 'O

elhr,

~ Q / ~ / "

DEPTH

40

/

07

~ ~oy

621

60

80

100

(m)

Fig. 6. C o h e r e n c e b e t w e e n t h e t e m p e r a t u r e fluctuations at d e p t h levels o f Series B.

Table 4. Series B. Coherence y2 (j)aM phase lag d~(f) of 180m temperature and other levels d (m). d(m)

f = 0.05 ¢]hr

f = 0.2 c / h r

f = 0.5 c/hr

f=

90 120 150 210 240 270 300

0"00 0-22 0"59 0.32 0.24 0"27 0"26

0.17 0.15 0"67 0.53 0.44 0.31 0.07

0"04 0"34 0-69 0-56 0"56 0-26 0"32

0-19 0"21 0.74 0"49 0"25 0"61 0.20

-0"39 0"17 0"01 0-12 0-14 0"22

~b in f a d s fl in c/hr d in m e t r e s

0-78 0.06 0"19 0"01 0.19 -- 0-14 -~b ~b

-0"07 -- 0"04 0"03 0.02 -- 0"19 - - 0"04

+ ve 180 m leads - - ve 180 m lags

lc/hr -----

0"51 0"02 0"07 0.01 0"41 0"22 0-36

. 01

• I 100

622

R.A. WrnTE

f = 0"05, 0"2, 0"5 and 1 c/hr. None of the recorded phase differences are significantly different from zero. A numerical solution to Fjeldstad's internal wave equation (applicable to low frequencies, i.e. say f < 0.2c/hr) was obtained for crt with depth distributions in series B. The method used was proposed and tested by VAPNYAR and SrIAPKINA (1964) and involved the solution of a transcendental equation to yield a single mode internal wave which can be said to be consistent with the observed distribution of ~r, with depth. At depths below 300 m
REFERENCES BAER L. and D. HAMM(1965) HOW representative is an ocean temperature. J. geophys. Res_ 70 (18), 4579-4591. BLACKMANR. B. and J. W. TUKEY(1959) Power spectra analysis. Dover, New York. CARSOLAA. Ji, D. P. HAr,iM and J. C. RoQtm (1963) Spectra of internal waves over basins and banks off S o u ~ California, Lockheed-California Co. Rept. 16795. Burbank, California. (Unpublished manuscript).

The vertical structure of temperature fluctuations within an oceanic thermocline

623

CARSOLAA. J., D. P. HAMM and J. C. RoQtm (1965) Spectra of temperature fluctuations over the continental borderland off Southern California. Deep-Sea Res., 12 (5), 685-691. CARSOLAA. J. and E. B. GALLOWAY(1962) Two short period internal wave frequency spectra. Limnol. Oceanogr., 7, 115-120. CARTWRIGHT D. •., M. J. TUCKER and D. CATTON (1962) Digital techniques for the study of sea waves, ship motion and allied processes. Trans. Soc. Instrum., Technol., 14 (1), 1-16. Cox C. S. (1962) Internal waves. The Sea, Interscience, New York, 1, 756. HAURWlTZ B., I-L STOMMELand W. H. MONK (1959) On the thermal unrest in the o c e a n - the atmosphere and sea in motion, Rossby Memorial Vol., 74-94. Howe M. R. and R. I. TAIr (1965) An evaluation of an in-situ salinity-temperature-depth measuring system. Mar. Geol., 3, 483-487. LEE W. H. S. and C. S. Cox (1966) Time variation of oceanic temperatures. J. geophys. Res., 71 (8), 2101-2111. MUNK W. H., F. E. SNODGRASSand M. J. TUCKER (1959) Spectra of low frequency waves. Bull. Scripps. Inst. Oceanogr., 7 (4), 283-362. PASQUILL F. (1961) Atmospheric diffusion. Van Nostrand, New York. SABrNIN K. D. and V. A. SntrLEeOV (1965) Short period internal waves of the Norwegian Sea. Okeznologiiz, Akad. Nauk. SSSR, 5 (2), 254-275. (Translation, Scripta Tecnica for Am. Geophys. Un.) VAPNYAR D. O. and V. F. SHAPKrNA (1964) Calculating the factors of internal tidal waves and the periodic fluctuations in water temperature associated with them. Deep-Sea Res., 11, 440--449. VooarIIS A. D. and H. T. PERrdlqS (1966) The spatial spectrum of short wave temperature fluctuations in the near-surface thermocline. Deep-Sea Res., 13, 641-654.