Field observations of fronts and high-frequency temperature fluctuations in a thermal plume

THEODORE GREEN and STEPHEN ROFFLER Marine Studies Center, University of H4sconsin-Madison, Madison, FFisconsin 53706, USA (Received 1 April 1980; revised 3 September 1980)

High-frequency temperature variations were measured at seven depths, at a point 130 m from the cooling-water outfall of a power plant. The data are studied using spectra, correlation functions and probability distributions. The sharp thermal fronts observed in the data are related to the surface thermal fronts seen using airborne infra-red scanning, and to laboratory studies of instabilities in mixing layers between uniformly flowing and quiescent fluids.

I N T R O D U C T I O N

Thermal plumes associated with power-plant cooling- water systems are of obvious ecological importance. These plumes are also a common, well-defined, full-scale example of a stratified, turbulent shear flow. For both reasons, they warrant detailed study.

A full understanding of the mechanisms involved in plume behaviour will probably only come with a large number of temperature and velocity measurements over both space and time, which are sufficiently dense to document the complete spectra of the fluctuations. The work described below represents some progress in that direction, which may guide the way to further, more comprehensive work. This paper summarizes material in a more detailed report by Green and Roffler 1.

SURFACE FRONTS

It is worthwhile to comment first on the principal motivation for our study. In a series of intensive airborne infra-red scanning studies of plumes in Lake Michigan 2, we often encountered a very regular series of (surface) thermal fronts which dominated the plume surface structure, and which spread radially outwards from the plant outfall (Figs. 1 and 2). The total temperature change across a front was usually 1-2c'C; this change occurred over a distance of a few metres or less.

These fronts seemed quite clearly to be the surface manifestations of mature Kelvin-Helmholtz instabilities, generated by the large velocity gradient at the interface between the warm, spreading plume water and the cool, fairly quiescent receiving water. The period of the fronts is about 103 sec, and they move at velocities roughly comparable to observed surface velocities in the plume. The fronts are first obvious on the thermal imagery 100- 200 m from the end of the outfall. However, they could have existed earlier, and not been measured because of saturation Of the film used in recording the scanner data. A detailed study of these fronts was presented by Scarpace and Green 3. Similar fronts in thermal plumes have also been described by Kuhlman and PrahP, and Cataldo 5.

Weaker, secondary fronts, with wavelengths and periods about a tenth of those of the primary fronts, are also observed sometimes. These were more visible on the higher resolution aerial colour photographs taken concurrently with the thermal imagery (cf. Figure 6 of Scarpace and Green3). The spatial temperature variation along several lines extending outward from the outfall is shown in Fig. 3. These values were obtained by densitizing the thermal imagery shown in Fig. 1.

In 1972. we began a program designed to measure the associated subsurface temperature and velocity fine structure of the thermal plume associated with the Point Beach Nuclear Plant, on Lake Michigan (Fig. 4). This plant has a fairly 'clean' outfall geometry, and the offshore bathymetry is relatively simple. We also had seen both primary and secondary fronts here quite often.

Representative samples of the high-frequency temperature data taken at seven depths in the plume are discussed below. The main purpose will be to document the temperature structure of a full-scale plume (and turbulent shear flow), and to compare these data, where possible, to those obtained by others in controlled laboratory conditions.

P O IN T BEACH TH ERMA L P L U M E

The Point Beach thermal plume has been studied intensively, most notably by Argonne National Laboratory, and to some extent by the University of Wisconsin. The two independent pressurized water reactors each have a peak generation capacity of 523 MW. Cooling water is withdrawn from the lake through an intake located 550 m offshore, and is returned to the lake through two surface canals 10.7 m wide, 4.2 m deep and 68 m long. Under full load, each unit produces a cooling water temperature rise of 10.7°C at a flow rate of 25 m3/sec. Thus, the outfall velocity is about 50 cm/sec. The temperature measurements were made at a point 130 m from the end of the south outfall canal. The mean water depth at the measurement point is 4 m, but fluctuates by at least 0.5 m due to the lake's annual water-level cycle, and

0309- 1708/81 "030137 0952.00 © 1981 CML Publications Adv. Water Resources, 1981, Volume 4, September 137

Temperature fluctuations in a thermal plume: T. Green and S. Roffler

0 . . . .

Figure 1. An infra-red image (8-14 I~m) of the thermal plume associated with the Point Beach Nuclear Plant at Two Creeks, Wisconsin, on the shore of Lake Michigan. Warm surface water is dark, cool surface water light. The circular cooling-water intake is 550 m from shore. The total sin?lace-water temperature di[]'erence is about IO~C

500m

Figure 2. The position of one of the fronts seen in Figure 1, as a function of time. Times are about 5 rain apart. The front moves radially outward from the outfall

wind and seiche effects. During the studies described below, only the south outfall was operating.

The plume outfall densimetric Froude number is V.(,qh. Ap/p) ~ 2 _ 3. The outfall Reynolds number is Vh/v - 2 x 106. Here, Vis the outfall velocity; h is the outfall depth: p is the density of the plume water; Ap is the density difference between the outfall water and the ambient water; v is the kinematic viscosity; g is gravity. The outfall characteristics were steady over time, as the plant was almost always operated near full load.

We have about 200 thermal images of the Point Beach plume. The plume varies widely from day to day, although the measurement point is almost always in the plume. This point is in the portion of the plume where the fronts seen in Fig. 1 are not yet fully developed, and where the plume probably has not lost contact with the lake bottom for long.

The velocity at the measurement point is about 30 cm/sec °. The plume depth here is about 3 m3; it seems to be off the bottom almost all the time. The densimetric Froude number here is about 2, and the Reynolds number about 10%

SUPPORT STRUCTURE

The instruments were mounted on a vertical pole, with an offset vertical instrument support to minimize the influence of the pole on the measurements. The first (1972-73) version of the pole was crushed by ice during the winter. The second design is shown in Fig. 5, and is described by Green and Roffler 1. This final version was eventually crushed during a fall storm; this second disaster brought the field program to an abrupt halt.

I N S T R U M E N T A T I O N

The electrical connection between the pole and the shore- based recording equipment was a dual-section cable laid along the lake bottom. Bayonet connectors mated the cable with the thermistor electronics chassis, which was housed with the recording apparatus in a trailer located just outside the power plant fence on shore, and with a junction box at the pole.

Temperature measurements were made using Fenwal Electronics GB42SMM1 Oceanographic Iso-Curve thermistors 1. These have time constants of 70 msec in quiescent water. Seven thermistors were sealed with epoxy into 16 cm standoffs on the instrument support shown in Fig. 5.

Laboratory calibration was accomplished by

! I 0"5 1 d (Km)

Figure 3. The temperature T along five lines extending radially outwards from the outfall, against distance d from the out fall. The traces have been displaced about 1.5°C in temperature, and 70 m in distance, for clarity. The middle trace is associated with a line running along the extended centreline of the outfall; the upper traces are for lines at angles of l.5 ~ and 3 ~ counterclockwise of the centreline: the lower traces are for lines at angles of l.5 ° and 3 ° clockwise of the centreline

138 Adt. V~ater Resources, 1981, Volume 4, September

Temperature.fluctuations in a thermal plume: T. Green and S. Rqflter

I \\

18

N

I I

3 0 0 ,ft

Figure 4. A schematic view of the Point Beach Nuclear Plant, showing the position of the instrument pole ( • ) and the local bathymetry (in ft)

Figure 5. A schematic view, closely to scale, of the instrument pole. A, the pole; B, the instrument support on which the thermistors were mounted at 0.5 m intervals; C, a guy wire; D, the work platform: E, a bottom joist. The flow is to the right

measuring the resistances of the thermistors when immersed in a Haake FK2 constant-temperature water bath. The indicated temperatures had a maximum deviation of 0.2°C from linearity occurring near both extrema of the operating range of 0-30~C.

High levels of distorted 60 Hz noise present at the output of the thermistor amplifiers required the construction of 3-pole active Chebychef notch filters with 100 db of rejection at 60 Hz. After passage through these filters, the data were recorded on paper stripcharts or as analogue signals on magnetic tape. The most reliable data were recorded on an Ampex SP300 AM/FM 7-track magnetic tape recorder operated in the FM mode at a tape speed of 1 and 7/'8 ips. Under these conditions the recording system has a 30 db signal-to-noise ratio, and a frequency response of 0 to 312 Hz.

DATA TAKEN

Although there was a capacitance wave gauge mounted on the pole, we were anxious to avoid complexities associated with wave-induced advection. Because of this,

data were usually only taken under very calm conditions. This allowed us to record data from a seventh thermistor (thus, not recording the wave data).

The data discussed below were selected on the basis of two conditions: the seven taped tracks had to be bounded within the recording system's dynamic range, and the lake surface had to be flat calm. The temperature data for 4 September 1974 satisfied these criteria especially nicely, being taken under what would be called a visually 'glassy' water surface. Waves of 2 cm in amplitude and 1 3 sec period were breaking and running about 10 cm up the beach. The wave gauge showed a short-term surface deviation of less than 2.6 cm. The longshore current was less than 1 cm/seC. Data were recorded for 957 sec. On the basis of perusing much other data taken on other, simikir days (mostly only recorded on strip-charts), there is no reason to believe that the data discussed below are atypical.

The raw analogue data were low-pass filtered using an active analogue filter with a 3 db cutoff set at 16 Hz, then sampled by an A/D converter at 30 Hz and stored on magnetic tape. Owing to computer core memory limitations, a 4 x 2048 array was the maximum size which could be generated. Since the entire data record would not fit into this size array with the desired sampling rate. the record was digitized in two passes, and stored in sixteen 2048-point (68.2 sec) sections, with about 8 sec overlap. The first pass processed channels 1 4 and the second, channels 4-7. Using a Univac 1110 digital computer, the digitized data were then read, converted to temperatures calibrated in °C, and stored on a magnetic tape file.

One data segment is shown in Fig. 6. The complete set of these temperature plots is found in Green and Roffler 1. The surface and 3 m depth temperatures were typically quite stable. The latter feature indicates the usual presence of ambient water (see also the discussion below of temperature distributions, and Fig. 7), the former suggests the suppression of internal-wave activity near the surface. At other depths, the observed fluctuations seemed to be a combination of turbulent fluctuations, internal waves of periods between 1 and 10 sec, and ramp-like structures with steep forward (smaller time) faces, and much gentler trailing faces. These structures were often similar at several thermistor depths, whereas the wave-like and

22

20

18

14

I I I I I I i 0 20 40 60

t,(sec)

Figure 6. Water temperature vs. time at water depths for the last of the 16 data segments. The 'surface trace' is actually 2-5 cm below the surface

Adv. Water Resources, 1981, Volume 4, September 139

Temperature fluctuations in a thermal plume." T. Green and S. Roffier

0"0

0"08

P o

• w---------v- " - ' " - I

! I J ! ! I I I

0.08

0"08

6

20

0

T.(°C)

Figure 7. Frequency histograms showin9 the temperature distribution at each level, in terms of the empirical probability P of the temperature falling in various 0.2°C intervals. The number in each panel corresponds to the water depth: 1, the surface temperature; 2, the temperature at 0.5 m, etc. to 7, for the temperature at 3 m

random oscillations were rarely coherent over more than two depths. More will be said of the structures later. They were sometimes almost obscured by the internal waves. The internal-wave activity seems to decrease with increasing depth. The temperature traces often cross in the upper layers, indicating a turbulent, convective state of motion.

The mean temperatures, standard deviations, and layer stability frequencies ( - y / p dp,/dz) 1'2 (where z is upward) are shown in Table 1. The standard deviations increase by a factor of two with increasing depth. The stability frequencies are all about 0.06 sec-1, giving minimum internal wave periods longer than those observed. This discrepancy is, of course, very likely due to wave advection by the mean current at the pole. The probability of coming within 0.1°C of the mean temperature is also shown in Table 1. Obviously, these probabilities are low; thermistors towed through this region of the plume are very unlikely to give the mean isotherm positions.

Data taken on other days, under less than ideal conditions, showed these statements to be more generally true.

TEMPERATURE DISTRIBUTIONS

Frequency histograms for each level were calculated from the entire record by quantizing the temperatures into

0.2°C intervals (Fig. 7). The distributions show some clear trends with depth. Those near the surface look fairly normal, although they are not. In fact, all the distributions fail the g 2 test for normality at any reason.able confidence limit (e.g., 80~o). Those near the bottom show a marked skew towards higher temperatures. The secondary peak on the near-bottom histograms indicates the presence of another water type at this level, which must be rather 'pure' plume water, arriving in a relatively unmixed state. Note that the secondary peak at the first level where it becomes apparent (1.5 m) is close to the surface peak, and that the lower-level secondary peaks become colder with increasing depth, suggesting some limited mixing. The two peaks are probably also present nearer the surface, but so close together that the secondary one is masked. The two separate peaks probably imply that two different kinds of mixing are occurring in the bottom portion of the plume. One is likely a 'normal' turbulent mixing process. The other process results in some almost unmixed plume water arriving at the plume bottom. This occurrence is probably linked with the coherent structures noted above, and could be due to either variations in horizontal advection, or to surface water being carried downward by the flow associated with the coherent structures. The sharp histogram peak at the 3 m depth indicates that plume water rarely reached this depth, and that the ambient water was fairly uniform in temperature.

The skewness and kurtosis of the temperature distribution are shown in Fig. 8. Both these quantities increase markedly and quite regularly with depth. The large kurtosis near the plume bottom indicates that temperature excursions from the mean are large and of short deviation. These occur when relatively unmixed plume water arrives at this level.

TEMPERATURE SPECTRA

There has been a significant amount of past work in predicting high-frequency spectra of temperature fluctuations, usually using simple force balances, in conjunction with dimensional reasoning. Thus, it may be useful to compare our results with various theoretical predictions.

The data at each depth were preconditioned by removing linear trends and hanning. Fast Fourier transforms were then calculated for each data segment, and the results averaged over all sixteen segments to give one spectrum for each depth. Those for the surface, a depth of 3 m, and the average for all seven depths are shown in Fig. 9. Only the high-frequency range has been

Table 1. Mean temperatures, standard deviations, and stability frequencies at each depth

Probability Stability of finding

Mean Standard frequency T within 0.1 °C Depth temperature deviation ( -O/P dp/dz) l/2 of the average (m) (°C) (°C) (see - 1 t value

Surface 20.7 0.6 0.061 0.16 0.5 19.8 0.6 0.049 0.12 1.0 19.2 0.7 0.059 0.12 1.5 18.3 0.9 0.057 0.11 2.0 17.4 1.1 0.071 0.08 2.5 15.9 1.2 0.070 0.06 3.0 14.3 1.I 0.25

140 Adv. Water Resources, 1981, Volume 4, September

Temperature fluctuations in a thermal plume." T. Green and S. Roffler

2 -

0

0 Figure 8.

- 10 K

• 5

0

! I

1 2 3 DAm)

Skewness (S) and kurtosis (K) of the temperature distributions vs. water depth (D)

10 -3

16 5

10 .7 I I I

o-1 1 10 f(Hz)

Figure 9. High-frequency temperature spectra at the surface (0 m), and at a depth of 3 m. The average spectrum over all depths is shown as a broken line. Straight lines with slopes of - 3 and - 5 / 3 are also shown, to allow a comparison with theory

under which the above-mentioned theories should apply are not present, and that the temperature" variance is dominated by more mechanistic processes such as internal waves and Kelvin-Helmholtz instabilities.

Longer-period phenomena may also be significant. For example, the thermal scans of Scarpace and Green suggest, in conjunction with surface velocities of about 30 cm/sec near the pole, a thermal period on the order of 103 sec at Point Beach. Cataldo 5 observed periodicities of the order of a few hundred seconds in a thermal plume on Lake Ontario. Frigo et al.~° also studied temperature and velocity oscillations at Point Beach, and found periods of about 140 and 1200 sec. The lower-frequency oscillations were though to be (perhaps) associated with the primary fronts reported by Scarpace a n d Green (although they finally hypothesize that they were somehow associated with lateral mixing and motions in the plume). The higher frequency oscillations could be associated with the secondary fronts. These secondary fronts have lengths of order 20 m, which in conjunction with a mean flow of about 30 cm/sec gives periods of order 100 sec.

All these periods are too large to be studied using standard spectral techniques, owing to the relatively short data set. However, data-adaptive spectral analysis is well suited to this situation. Maximum entropy spectral analysis (MESA) gives very high frequency resolution even at periods on the order of the length of the data set 11. In brief, one finds the spectrum from the reciprocal of the squared response of a whitening filter, There is some arbitrariness in the length of the filter; most writers seem to prefer one with a length of about 25% of the data set.

The MESA spectra for a filter length of 25°~ are shown in Fig. 10. Spectra with filter lengths of 20°,o look quite similar. There is a very pronounced peak at all depths at a period of about 103 sec, which is sharper near the surface. There is another, broader peak at most depths at about 200 sec. Thus, our data show periodicities similar to those noted before at Point Beach, suggesting their common occurrence. It is difficult to tie either of these periods to oscillation characteristics of the outfall, or of the pumping system, and more likely that they both represent aspects of the evolution of Kelvin-Helmholtz instabilities associated with high shear at the plume base.

shown; it is here that various predictions may apply. The spectra for the other depths are generally bounded

by those at the surface and 3 m. All have slopes of about - 2 . 5 o n the logarithmic scales used. There is no systematic variation of slope with depth. There is more energy nearer the surface, as should be evident from the raw data shown in Fig. 6. Here, the spectrum has a smaller slope at low frequency and a larger slope at high frequency, than at greater depths. The observed spectral slope in the high-frequency range does not agree with various theories. For example, the predicted slope for the convective subrange is - 5/3 s. The predicted slope for the buoyancy subrange is between - 1 and - 7/5 9. Internal waves with frequencies up to 1 Hz are quite obvious in the raw data. The actual frequencies are, of course, altered significantly by the (unknown) rate of convection past the thermistors. Also, the sharp fronts remarked upon earlier are a major contributor to temperature changes and would likely increase thermal variance at low wave numbers, thus increasing the spectral slope. It seems likely, then, that the homogeneous, isotropic conditions

A e--

E % o

' : fA_ 1 °"r 7" , , " y . - /

2

I I I I

3

0 0"2 0 4 f ( c p m )

10

I

0"1

I I I I

I I I I

I I I I 0 2 0 - 4

f ( c p m )

Figure 10. Maximum entropy spectra Jbr long-period temperature fluctuations at the seven water depths studied

Adv. Water Resources, 1981, Volume 4, September 141

Temperature fluctuations in a thermal plume: T. Green and S. Roffier

1 1 i ~ ~ ~ used. It should be noted that this estimate is far different than the average time between the primary temperature

s fronts seen in Fig. 1. Also, R ll does not seem to be o t decreasing steadily at the largest lags considered. The

calculated I-I should apply to the secondary fronts, but . 2 probably does not reflect the correlations imposed by the

quasiperiodic primary frontal structures. ~o ~ The cross-correlation functions R~ relating the surface

\ ~ ] c temperature to those at 0.5, 1.0 and 1.5 m depth, and the 1.5 m temperature to those at 2.0, 2.5 and 3.0 m depth, are

i shown in Fig. 13. This rather awkward reference method o was made necessary by the two-pass A/D conversion

scheme described above, in turn imposed by computer memory limitations.

o X The maxima in Rij occur at lags which increase with V depth. These lags are shown in Fig. 14. The presence of the

Figure 11. Temperature autocorrelations R at the seven thermistor depths. As in Figure 7, 1 corresponds to the surface, 2 to a depth of 0.5 m, etc.

0"5 1

RO R 0

T(sec)

0 30T (sec) 60 0"5

Figure 12. Surface-temperature autocorrelation for a large range of time lags, r

CORRELATION FUNCTIONS R 0 The autocorrelation functions R, at all depths are shown in Fig. 11. These were calculated for lags z up to 20 sec for each data segment, and then averaged over all 16 segments. In general, those nearer the surface drop off more sharply near r =0, in accord with the greater high- frequency thermal activity observed here in the raw data. The surface record is not as peaked as those at 0.5 and 1.0 m; the presence of the surface suppresses the internal- wave activity in this region.

The surface-temperature autocorrelation for lags up to one segment long is shown in Fig. 12. This estimate is, of course, less reliable than those in Fig. 11. However, it shows evidence of periodicities in the data at time scales of the order of 1 min (see also the MESA discussion above). The integral time scale for temperature fluctuations is given by

n--f R1 x(~)d~ 0

This scale gives an estimate of the length of data needed for reliable time-average statistics and, when multiplied by the mean velocity, a distance over which the temperatures are uncorrelated. For the surface temperature, FI ~ 10 sec, Here, absolute values ofR 11 were

0 10 T(sec)

Figure 13. Cross-correlations between temperatures at different depths. Curves 2, 3, 4 denote R 12, R t 3, R 14; curves 5, 6, 7 denote R45, R46, R47

3

2

E Q 1 •

0

Figure 14.

I [ 1 [ I 50 100

Tma x (sec}

The lag ~ at which Ri~ in Fig. 13 attains its maximum value, vs. water depth, D

142 Adv. Water Resources, 1981, Volume 4, September

Temperature.fluctuations in a thermal plume," T Green and S. Roffler

0 •

E •

c~2

0

2 -

0

Figure 15.

_" ¢ C

I

I 30.

I • I :

Q •

I I • 60 90

t (sec)

The times ( • ) at which fronts are encountered at various depths D, for the last six minutes of the data record. The top panel is first, the middle second, and the third last. No • at a depth means that no obvious temperature jump was seen there

1.5 m record in both digitized series, plus the assumption that the lagged maximum between the surface and, say, 2.0 m equals the sum of those between the surface and 1.5 m, and between 1.5 m and 2.0 m, were necessary to plot the data in this manner. The values of the maxima decrease with increasing depth, indicating a process which begins at the surface, with the signal propagating downward, or vice versa. These lagged maxima, which decrease in strength with increasing separation distance, are very likely associated with the thermal structures noted in the raw data. They are probably swept by the thermistor array with the mean flow, sooner at lesser depths, where the velocity is larger.

O R G A N I Z E D T H E R M A L S T R U C T U R E S

The most noticeable characteristic of the raw data is the rapid rise in temperature over several depths, occurring sooner nearer the surface. These rises or fronts are usually followed quickly be more gradual drops. One example of this behaviour is easily seen in Fig. 6, starting at about 40 sec. In each case, the maximum temperature rise is of the order of 2°C, and occurs in about 1 sec. These sharp rises are very similar to the spatial variations associated with plume surface fronts (Fig. 3). Note that in both cases, the fronts are sharp temperature rises when seen by an observer at a fixed point.

There were about 15 easily discernible structures such as this, where the sharp temperature rise occurred over at least three depths. Others may have been masked by internal waves, three-dimensional plume turbulence, and ambient thermal features. The total time it took for a structure to move past the thermistor array varied widely, in accord with the number of depths at which it occurred. About 5 sec was typical for one covering 3 depths; 30 sec was more typical of one covering 6 or 7 depths. These times are smaller than, but the same order of magnitude asl the lags for which the cross-correlations had maxima (Fig. 14). The time lags between adjacent thermistor depths were larger at greater depth. The depth at which the temperature rise was a maximum was usually near the midpoint of the depths at which the event occurred. Less than half of the structures included the surface thermistor. Thus, they would not have been evident in thermal imagery such as that shown in Fig. 1.

The times that all discernable fronts were encountered at various depths are shown as a phase diagram in Fig. 15. There is, of course, some judgment necessary in ascertaining which temperature changes are indeed fronts, and in tracing them from one depth to another. A change had to be at least 1 ~-C in less than 1 sec, at least at one depth, in order to be counted. The degree to which this criterion was followed can be judged by comparing Fig. 15 with the raw data in Fig. 6 1. Note the regularity of the fronts, over the last two-thirds of the record shown. Earlier, they are much more sparse. This change in behaviour may be associated with the point of detachment of the plume from the lake bottom. (However, no observations could be made to confirm or reject this possibility.)

Such structures were apparently first discussed in Sunyach 12, who used a vertical rake of thermistors to study the mixing properties of a two-dimensional jet. His much more controlled data contained many structures quite similar to those described above. A simple and compelling explanation involves the occurrence of wave- like motions on the shear layer between the heavy and light fluid, associated with a Kelvin-Helmholtz type instability 13. The upward motion of relatively stagnant, cold water creates an 'obstacle', about which the faster- moving warm water must flow, and behind which a wake is formed. In the wake of this structure is found a mixture of warm and cold water, probably dominated by rolling- up vortices. The upstream boundary of this obstacle is sharply defined, the downstream boundary not so.

A mature instability, and the temperature variations seen by a fixed probe as the wake and cold-water obstacle are swept by are shown in Fig. 16. This is a very simple, two-dimensional model, but is strongly supported by laboratory work. The growing waves become vortices, which tend to grow, entrain cold, less turbulent water into the upper layer, and grown even further by pairing with other vortices 14. Reviews of laboratory work in this area are given by Laufer 15 and Roshko ~6. The effects of the decaying vortices will be to further scramble the temperature signals, already heavily contaminated by

o2

COOL

5 j

Figure 16. Top panel: a simplified schematic of a rolling- up, turbulent vortex associated with a mature Kelvin- Helmhohz instability. Positions (1) and (2) indicate hypothetical temperature sensors. Middle panel: idealized temperatures at a f ixed time at the levels of (1) and (2). Bottom panel: idealized temperatures seen by sensors (1) and (2), as a function of time

Adv. Water Resources, 1981, Volume 4, September 143

Temperature f luc tuat ions in a thermal plume: T. Green and S. Roffler

internal waves. Indeed, it may be considered somewhat surprising that the thermal structures are so evident in the natural environment.

The time between structures ranges from 15 to 60 sec, and is not always well defined. On the other hand, the spacings between both the primary and secondary surface thermal fronts observed by thermal scanning, when they exist, seem quite regular. It could be that this is because the fully developed surface fronts occur later (i.e., downstream of the pole), and that selective processes such as vortex pairing determine in a fairly regular way which will now appear at the surface. Also, the small plume depth, compared to the spacing between the surface fronts, must quickly limit the vortex-pairing process. The process which results in two widely different wavelengths is not understood; we know of no previous observations of this behaviour. The pole itself was, unfortunately, probably close to the point where the plume s e p a r a t e s from the bottom. That is, the bot tom of the outfall channel is at lake bottom, so that the plume is most likely on the lake bot tom near the outfall. The point where the detachment occurs is thus likely to vary with slight changes in ambient conditions, and this variation must affect the presence and character of the fronts at the pole. The virtual absence of fronts in the first half of our data may be associated with this behaviour.

In one sense the structures described above are unsurprising; they have been seen quite often in the laboratory, and sophisticated measurements made of their properties tv. However, we are not aware of any previous field studies of such subsurface structures, at least in thermal plumes, other than that of Frigo et a lJ °, who used temperature sensors with relatively large time constants from a moving platform, and ultimately ascribed their observed fluctuations to lateral motions in the plume. The connection of these structures with the surface fronts seen using airborne infra-red scanners is almost certain, and confirms earlier suggestions concerning their origin 3 .

very deep ~8, it seems more likely to us that the bot tom was not directly responsible for the observed structures.

The subsurface structures seen in our data are quite clearly related to the fronts seen very often in the thermal imagery, although we could never mobilize our resources so as to collect both types of data concurrently. The surface fronts are very definitely not merely surface phenomena. The techniques used to inspect the subsurface data reported here all reflect properties of the structures. The histograms indicate two kinds of mixing - - one o f 'normal ' turbulent down-gradient transfer, and one where high-temperature water arrives more directly at greater depths. The maximum-entropy spectra suggest periodicities in the developing region of Kelvin- Helmholtz instabilities (the pole position) that are similar to those of both the primary and secondary fronts, interpreted from thermal imagery. Correlation analysis also suggests periodicities on the order of the secondary fronts observed in our data.

How are these flow structures different from those observed in the laboratory? The main differences other than the Reynolds number are the small depths of both the upper and lower fluids. This must limit the v o r t e x - p a i r i n g process, and could lead to the observed constant distance between surface fronts after they are clearly apparent in the imagery. This effect has not been well studied, and because of its importance in most natural situations, warrants detailed investigation, as does the relation of the primary and secondary fronts.

A C K N O W L E D G E M E N T S

This work was supported almost entirely by funds provided by the US Department of the Interior, as authorized under the Water Resources Research Act of 1964, as amended. The University of Wisconsin Sea Grant Program also provided some support. The MESA spectral calculations were performed by Mr J. Villanueva; the thermal imagery was obtained by Dr F. L. Scarpace.

C O N C L U S I O N S

The most significant contribution of this work is probably that it provides quite firm evidence of the existence of mature, two-dimensional vortical structures in a high Reynolds-number flow in the natural environment. (Note, however, that the Reynolds number is still somewhat lower than that which obtained in one experiment referred to by Roshkoa6.) The thermal imagery shows the structures to be concentric, rather than linear. However, there is little doubt that the dynamics are closely analogous. Similar structures have been seen in thermal images of laboratory experiments by Alavian t8 and Kuhlman and PrahP. It should be noted, however, that the relation to the vortical structures observed in the laboratory remains a hypothesis. There is also a possibility, e.g., that the observed structures are associated with an oscillation of the point where the plume detaches from the lake bottom. Unfortunately, we have no measurements with which to investigate such a possibility. However, in light of the large amount of evidence for ordered structures associated with instabilities of turbulent shear flows 16, and of laboratory observations of similar structures in a surface thermal plume like that at Point Beach, but where the bot tom was

R E F E R E N C E S

1 Green, T. and Roffler, S. High frequency temperature fluctuations in a power-plant thermal plume, Univ. 14'isconsin Water Resources Center Technical Completion Report WIS WRC 80-06 1980

2 Scarpace, F. L., Green, T. and Madding, R. P. Thermal plumes along the Wisconsin shore of Lake Michigan, Trans. l,~sc. Acad. Sciences 1977, 65, 86

3 Scarpace, F. L. and Green, T. Dynamic surface temperature structure of thermal plumes, Water Resourc. Res. 1973, 9, 138

4 Kuhlman, J. M. and Prahl, J. M. Observations of the Kelvin- Helmholtz instability in laboratory models and field examples of thermal plumes, J. Great Lakes Res. 1975, l, 101

5 Cataldo, J. E. Discharge plume thermal front phenomenon, Abstr. Int. Ass. Great Lakes Res. Conf. 1979

6 Wolf, P. R. and Keating, T. J. Precise water velocity measurements using photogrammetric techniques, Water Resourc. Bull. 1973, 9, 312

7 Saunders, K. L., Van Loon, L., Tome, C. and Harrison, W. Nearshore currents at Point Beach, Wisconsin, Argonne National Laboratory Report ANI2WR-76-1 1976

8 Pao, Y. H. Spectra of interval waves and turbulence in stratified fluids, Radio Science 1969, 4, 1315

9 Lin, J. T., Panchev, S. and Cermak, J. E. A modified hypothesis on turbulence spectra in the buoyancy subrange of stably stratified shear flow, Radio Science 1969, 4, 1333

10 Frigo, A. A., Zivi, S. M., King, R. F. and Levinson, E. D. Field observations of the dynamics of heated discharge jets, Proc. 17th Conf. Great Lakes Res. 1974, pp. 412-424

144 Adv. Water Resources, 1981, Volume 4, September

Temperature fluctuations in a thermal plume: T Green and S. Roffler

11 Kanasewich, E. R. Time Sequence Analysis in Geophysic.s, University of Alberta Press, 1975, 364 pp.

12 Sunyach, M. Contribution a I'etude des frontiEres d'ecoulements turbuleuts libres, Thesis, L'Universite de Lyon, 1971

13 Michalke, A. Vortex formation in a free boundary layer according to stability theory, J. Fluid Mech. 1965, 22, 371

14 Winant, C. D. and Browand, F. K. Vortex pairing: the mechanism of turbulent mixing-layer growth at moderate Reynolds number, J. Fluid Mech. 1974, 63, 237

15 Laufer, J. New trends in experimental turbulence research, A. Rev. Fluid Mech. 1975, 7, 307

16 Roshko, A. Structure of turbulent shear flows: a new look. AIAA J. 1976, 14, 1349

17 Chen, C. P. and Blackwelder, R. F. Large-scale motion in a turbulent boundary layer: a study using temperature contamination, J. Fluid Mech. 1978, 89, 1

18 Alavian, V. Investigation of thermal fronts in heated water discharges, PhD Dissertation. University of Wisconsin-Madison, 1979, 192 pp.

In te rna t iona l C o n f e r e n c e o n

SOIL DYNAMICS AND EARTHQUAKE ENGINEERING 1 3-15th July 1 982

To be held at Southampton University, England

Object ives To provide a forum for the presentation and discussion of new and advanced ideas in soil dynamics and earthquake engineering.

To encourage and enhance the role of mechanics and other disciplines as they relate to earthquake engineering by providing an opportunity for the presentation of the work of applied mathematicians, involved in solving problems closely related to the field of earthquake and geotechnical engineering.

Fields to be covered * Seismology and geology relevant to earthquake problems " Elastodynamics: wave propagation and scattering " Soil and rock dynamics * Dynamic constitutive behaviour of materials " System methodology and identification in soil mechanics * Probabilistic methods * Earthquake engineering reliability and risk analysis * Soil-structure interaction • Finite element analysis in dynamics • Case histories

Call for Papers Interested authors should send a summary of no more than 300 words to the Conference Secretary before October 1st 1 981.

Organising Commi t tee A. S. Cakmak- Princeton University A. M. AbdeI-Ghaffar- Princeton University C. Brebbia- (Conference Secretary) Southampton University

Sponsored by the In ternat ional Journal on Soil Dynamics and Earthquake Engineering, Internat ional Society for Computat iona l Me thods in Engineering ( I .S .C.M.E. ) O

Abstracts should be sent to: Dr. C. A. Brebbia, C. M.k. Centre, 125 High Street, Southampton, SO1 0AA, UK

Adv. Water Resources, 1981, Volume 4, September 145