Limnol. Oceanogr., 41(l), 1996, 157-168 0 1996, by the American Society of Limnology and Oceanography, Inc.

Vertical eddy diffusion calculated by the flux gradient method: Significance of sediment-water heat exchange

Gaboury Benoitl and Harold F. Hemond Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge 02 139

Abstract Periodically measured temperature profiles were used to calculate vertical eddy diffusivities by the heat

flux gradient method in the stratified portion of a dimictic New England lake. Loss of heat from the water column to sediments, a term commonly neglected, was explicitly calculated with a simple numerical model. This term dominated the heat budget of the stratified lake and would have caused large errors in calculated eddy difisivities had it been ignored. Heat exchange with sediments should be considered when thermal flux gradient analysis is used to calculate eddy diffusion coefficients for lakes with combined metalimnetic plus hypolimnetic thicknesses up to - 15 m. -

In the stratified portion of lakes, the distribution of chemicals is controlled by source and sink functions and by transport via turbulent mixing. For simplicity and mathematical tractability, the combined effect of these poorly understood processes is commonly combined into an eddy diffusivity, which can then be used in a diffusion equation. Lake hypolimnia are poorly mixed, and vertical concentration gradients of dissolved materials are com- monly observed in such regions. To understand the geo- chemical cycling of chemical components as reflected by their fluxes and patterns of distribution within the hy- polimnion, it is first necessary to know the magnitude of eddy diffusion as a function of depth and time.

Calculation of eddy diffusion can be carried out by monitoring the change in distribution of a stable, con- servative tracer (heat: Orlob and Selna 1970; Jassby and Powell 1975; dye: Kullenberg et al. 1974) or of a radio- active, conservative tracer with a known source function (radon: Imboden and Emerson 1978; tritium: Quay et al. 1980). Because of the relative ease of measurement and the lower cost of instrumentation, heat is more widely used than the other methods despite the need to account for multiple source and loss terms.

Heat input from radiation can be calculated based on careful measurements of insolation and water transpar- ency. Heat loss to sediments was shown to be negligible

’ Present address: Yale School of Forestry and Environmental Studies, Greeley Laboratory, 370 Prospect St., New Haven, Connecticut 065 11.

Acknowledgments Janina Benoit played a key role in all phases of data collection

at Bickford Reservoir. Tim Rozan provided assistance in run- ning the computer models with help from Mark Ducey and Anjali Mathur. The manuscript benefited from the suggestions of Peter Santschi and two anonymous reviewers.

Financial support for this work was provided by a National Science Foundation graduate research fellowship, a U.S. Geo- logical Survey water resources research grant, three National Wildlife Federation environmental conservation fellowships, the Massachusetts D.W.P.C., and a Geological Society of America graduate award.

in one seminal study (Birge et al. 1927), and since that time many others (e.g. Davison et al. 1980; Lewis 1983; Rippey 1983; Adams et al. 1987) have neglected this term, even though thermal exchange with sediments can be important to the heat budget of stratified lake waters (Likens and Johnson 1969). We report here one case in which heat exchange with sediments dominates the heat budget of the hypolimnion of a lake and argue that this situation may be common. We also present a method whereby a simple computer program can be used to es- timate the sediment heat term based on minimal water- column temperature data and minor knowledge of sedi- mentary characteristics.

Study site

The study site, Bickford Reservoir, is described in de- tail elsewhere (Benoit and Hemond 1987, 1990, 1991). The lake is oligotrophic and has an area of 60 ha, a max- imum depth of 13 m, and a mean depth of 6 m. Dimictic conditions prevail, with summer stratification lasting from May through October, and bottom waters usually becom- ing anoxic from August through October. Water residence time averages 0.23 yr but is much longer in summer, when outflow drops to near zero. Eddy diffusion calculations reported here were conducted as a key supporting com- ponent of detailed studies of radionuclide and transition metals reported elsewhere (Benoit and Hemond 1990).

Materials and methods

Field measurements-Dissolved oxygen and temper- ature profiles were measured in the field with an Orbis- phere model 27 14 temperature-dissolved oxygen meter. The manufacturer’s specifications indicate an accuracy of +0.2”C, and this was confirmed in laboratory tests with a water bath monitored with an NTIS traceable reference thermometer.

Light extinction was measured at the deepest point in the lake with a LiCor model 185 quantum-radiometer- photometer. The meter was coupled to a model LI-192s underwater quantum sensor having a cosine-corrected

157

158 Benoit and Hemond

J $czl~, F(z) Z

SEDIMENT HEATING

I

Zmax H(z) W) dz z

Fig. 1. Schematic illustration of heat flux terms in a ther- mally stratified lake.

response within 5% of ideal for all angles of incidence within 70” of normal (Roemer and Hoagland unpubl.). Manufacturer’s claimed linearity is within l%, and spec- tral response approximates an ideal quantum (not energy) response. On each date, light intensity was measured in duplicate profiles from the surface to the bottom and back, and the two profiles agreed to better than 5%.

SCUBA divers hand collected sediment samples in 3.9- cm-diameter plastic tubing which was sealed with plastic endcaps. Cores were cut immediately after return to the laboratory into sections 0.5-l .O cm thick. The weight of each section was measured and its exact length recorded in order to determine in situ density. All exposed surfaces were sliced away exposing fresh sediment, and the re- maining material from each depth interval was homog- enized before a 0.5-l .0-g subsample was taken. The sam- ple was then dried in an oven at 105°C and reweighed to determine water content and porosity. Pore-water seepage velocities were calculated with a seepage meter described by Lee (1977). In Bickford, we consistently found flow from the lake into the aquifer. To preclude the possibility that this flow might be caused by the elastic force of the collection bag squeezing water into the sediments, we ran a control consisting of a filled and disconnected bag de- ployed near a seepage meter for 2 weeks. The bag lost no volume in that time, indicating that the observed pore- water flows were not artifacts.

Theory- Eddy diffusion can be described by the eddy diffusion coefficient, K,(z), which is analogous to the mo- lecular diffusion coefficient and can be used in Fick’s laws in the same way. Eddy diffusivity can be calculated from changes in the temporal and spatial distribution of a con- servative tracer, such as heat, by the flux gradient method (Jassby and Powell 1975). According to Fick’s first law, the flux of heat is proportional to the temperature gradient and eddy diffusivity. With z positive in the downward

direction, the heat budget of a lake can be stated as

z”ax dT dt F(z) dz. (1)

T is temperature (“C), t is time (s), z is depth (cm), F(z) is lake area at depth z (cm2), Cp is heat capacity (cal g-l ‘C-l), and p is density (g cm-3).

The simple formulation of Eq. 1 assumes that eddy diffusional transport is the only way that the heat content of the stratified portion of the water column can change. In fact, at least two other processes can have a significant impact: thermal exchange with sediments and radiant heating by sunlight. These terms enter the equation as source-sink terms on the right-hand side of Eq. 1. Pre- liminary calculations showed that for Bickford, these ad- ditional terms were large and could not be ignored. Equa- tion 1 mus’t then be reformulated as

dT -p Cp K,(z) dz

zmax dT dt F(z) dz - R(z) F(z)

s Zmax

+ H(z) Z(z) dz. Z (2)

R(z) is solar radiant heat flux per area at depth z (cal cm-2 s-l), H(z) is rate of heat loss to sediments per area of sediment at depth z (cal cmm2 s-l), and (z) is wetted perimeter of the lake at depth z (cm). Each of these terms is illustrated in Fig. 1.

Eddy di:rusion can be calculated by dividing the cal- culated heat flux through a given surface by the temper- ature gradient at that surface. Rearranging Eq. 2 gives

ZlllliX

p CP dT ht F(z) dz + R(z) F(z)

S Zlllt3X - H(z) Z(z) dz

Z 1 (3)

Temperature profiles were measured at Bickford once every lo- 15 d during the study period. The stratified water column was divided into slices 1 .O or 0.5 m thick, and a polynomial describing T(t) was determined for each stratum with a least-squares method. Differentiation of these equations yielded the dT/dt terms. Likewise, each temperatu:re profile was fitted with a polynomial describ- ing T(z) in order to evaluate dT/dz. Lake area as a func- tion of depth was determined from a bathymetric map prepared earlier with a recording depth meter. Contours were drawn by a computer plotting program using a Krig- ing algorithm and second-order weighting. Areas were measured from the final plot with a mechanical planim- eter.

Radiant heating was determined at each depth for each of the 10 time intervals from calculations of the magni- tude and angle of incidence of solar radiation, information

Vertical eddy dijbsion 159

on percent cloud cover, and field measurements of the optical transmission of Bickford lake water. Determina- tion of radiant heating was accomplished by writing a short FORTRAN program, RADHEAT. Each day within a given interval was calculated separately to allow for rapid seasonal changes. Also, since the changing solar altitude throughout the day causes large changes in the optical path in the lake (and consequent light absorption), each day’s light period was broken into half-hour intervals (typically 25-30 per day). For a given half-hour interval, direct and diffuse solar radiation terms were calculated from the widely used model of the American Society of Heating, Refrigeration, and Air-conditioning Engineers (Farber and Morrison 1977, hereafter referred to as ASH- RAE). The ASHRAE model was selected over alterna- tives, such as those listed by Henderson-Sellers (1986). The improved parameters recommended by Iqbal(1983) were used, and the clearness number (CN) was taken to be equal to the value at Blue Hill, Massachusetts, given by Threlkeld and Jordan (1958). Direct solar radiation was adjusted by a factor of cos(zenith angle) to correct for oblique incidence on the horizontal lake surface. An additional correction factor was applied to eliminate all infrared and ultraviolet radiation (53% of total, Iqbal 1983) which is absorbed in the top meter of the water column. A pathlength correction factor was derived from the midinterval zenith angle and the law of sines (refrac- tion) with an average value of 1.33 for the refractive index of visible light in water. The result was introduced into Bouguer’s law, using measured extinction coefficients, to derive the radiant energy as a function of depth in the lake. The calculation was completed by summing all the half-hour intervals to give a total for the day and by combining all the days to give an average for each period between measurement of temperature profiles.

predictions of radiative flux matched measured daily val- ues to within 4% on average. This approximation is much more accurate than the formula of Black (1956), which has the added disadvantage that it cannot be used when sky cover exceeds 80%.

It is interesting to note that in winter when the sun is low in the sky, cloudy days may deliver a greater amount of solar energy to the depth of lakes than do sunny ones. Although light intensity is reduced by a factor of -3 on cloudy days, it arrives less obliquely, resulting in smaller cosine and pathlength correction terms. At the winter solstice, the minimum zenith angle at Bickford is 7 lo, corresponding to a pathlength factor of 1.42 and cosine correction of 0.33. At 10 m in Bickford, a cloudy day would deliver 2.5 times as much energy at noon as would a sunny day, and the difference would be even greater away from the noon hour.

The output from RADHEAT is a tabulation of average daily values of radiant heating at depths of 7, 8, 9, 10, 11, 11.5, 12, and 12.5 m in the water column for each of the 10 time periods between 14 June and 19 October 1985. The range in values is from 0.2 to 9.2 cal cm-2 d- l. As would b e expected, heating decreased rapidly with depth on a given date and gradually with time (after the summer solstice) at a fixed depth.

DiIhtse radiation was treated somewhat differently from direct radiation. First, a constant pathlength correction of 1.19 was used (Hutchinson 1975). On sunny days, diffuse radiation amounts to - 10% of direct radiation and is calculated exactly by the ASHRAE model. For partly sunny conditions, it was assumed that the sun was obscured for a fraction of time equal to the percent sky cover and that the radiation received when the sun was behind clouds was a third of the clear sky value:

The heat budget of the hypolimnion is influenced by thermal exchange with the sediments as well as by radiant heating and eddy diffusion. Over the course of the year, the temperature of bottom water alternates seasonally between warm and cool values, and the temperature of surficial sediments closely tracks that of the bottom water. In winter, the water cools the sediments; in summer, it warms them. Propagation of this temperature cycle deep- er into the sediments is damped by mud’s limited thermal conductivity and high thermal inertia (i.e. heat capacity), so that at a depth of a few meters, the sediments are practically isothermal throughout the year. Nevertheless, for lakes whose,zone of stratified water is thin, heat loss to sediments can be a significant thermal sink in summer.

No direct measurement of heat exchange with the sed- iments was made, but initial calculations indicated that the value was too high to be ignored in the determination of the eddy diffusion coefficient. Among investigators who have not ignored the sediment heat term, several ap- proaches have been tried.

R, = R,, + R,

= (1 -- f) x ASHRAE

+ 0.333 x f x ASHRAE. (4)

R, is total radiation on a given day, Rb is bright sun radiation for the day, R, is cloudy radiation for the day, fis the fraction of sky obscured by clouds, and ASHRAE is the ASHRAE calculated global solar radiation value.

Imboden et al. (1979) included a sediment heat term in their computer calculation of eddy diffusivity for a limnocorral in Baldeggersee, Switzerland. They found that sediment heating had a negligible effect on K, in the 12- m-diameter, 1 O-m-deep subdivision. Imboden et al. ( 1979) seem to have erroneously included only sediment pore waters in their calculations of heat diffusion, ne- glecting the role of sediment solids, which may help to explain the negligible effect they observed.

Values off were taken from the Local Climatological Priscu et al. (1986) used measurements and computer Data Summary for Worcester, Massachusetts (NCDC, modeling to derive the sediment heat term for their study Asheville, North Carolina), and a pathlength correction site, a geothermally influenced lake. In that unusual sys- of 1.19 was applied to the cloudy radiation. As a test of tem, the sediments supplied an apparently constant the assumptions, the calculation was carried out for a 1 -yr amount of heat to the water column. In Bickford, heat is period on data for Montreal taken from Iqbal(l983), and lost to sediments in summer, and our computer simula-

160 Benoit and Hemond

tions (see below) show that the magnitude of this term changes substantially with location and time.

Lehman and Naumoski (1986), using data of Birge et al. (1927) and Likens and Johnson (1969), noted that heat exchange with sediments was linearly proportional to the annual range of temperatures at the sediment-water in- terface. Lehman and Naumoski then assumed that heat flux is constant over a 6-month period. This simple ap- proach is useful as a first approximation but could lead to significant errors in a lake such as Bickford, where the flux is strongly time variable.

Stauffer and Armstrong (1984) and Stauffer (1986a,b) also considered the sediment heat term in studies of phos- phorus, silica, and trace metal dynamics of Lake Men- dota, Wisconsin, and Shagawa Lake, Minnesota. Like Lehman and Naumoski, they included variations in sed- iment heating with changing water depth, but details of their method have not been published.

Another method of estimation is to use the temperature gradient at the sediment-water interface to calculate heat flux (Nyffeler et al. 1983):

J=-KFg- . (5) Z-Zmax

J is heat flux (cal s- ‘), K is thermal conductivity of sed- iments (cal cm-l s- l “C-l), F is area of sediments (cm2), Tis temperature (“C), and z is depth in the water column (cm). This method has the disadvantage that it assumes that the temperature gradient in the upper sediments is identical to that in the adjacent water. It is probably in- accurate because of differences in heat capacity, advec- tion, and thermal conductivity between the two layers. It also depends critically on precise knowledge of the tem- perature gradient at the interface (i.e. dT/dz 1 z=z,,X). The true magnitude of this gradient may be difficult to esti- mate by extrapolation from measurements made outside the hydrodynamically unique domain near the sediment- water interface.

To avoid this difficulty, we developed a numerical model (MUDHEAT) that simulates heating of sediments by the water column. The governing partial differential equa- tion is

6T(z, t) K a2T(z, t) 61=pc,x822. (6)

The depth and time intervals used were 1 cm and 0.00 1 d. The brief time interval was necessary to achieve nu- merical stability in the model. We assumed that thermal conduction is the only method of heat transport, that the temperature of sediments at the sediment-water interface is equal to that of the overlying water column, and that the thermal conductivity could be calculated as a linear combination of the values for water and solids in the proportions (56% water, 44% solids) measured at depth in sediment profiles of this lake. The ratio was the mea- sured value at the base of sediment cores at the study location (Benoit and Hemond 1991) and is believed to be equivalent to subsurface soil material below the lake at this site, which was flooded in 1970. Water content

changes rapidly with depth in the upper 5-10 cm, but is unlikely to change significantly in the remaining l-2 m of soil underlying this area, which was sampled by the deepest sections of the cores. These conclusions are sup- ported by porosity measurements at other locations in the watershed (Genereux and Hemond 1990). A final assumption is that sediments are a semi-infinite system with a temperature at z = 00 equal to the average annual overlying .water temperature. Within the model, this con- dition was handled by truncating a time step as soon as the calculated downward heat flux fell below a negligible threshold value.

The temperature profiles in the sediments (at each wa- ter depth) at the beginning of the study period were cal- culated. As a starting point for the calculation, we as- sumed an isothermal condition. Then the sediments were allowed to interact with overlying water, the temperature of which varied according to the pattern measured in 1982-198.3 (Benoit and Hemond 1987). This pattern was similar to that observed in 1985 but covered a full year. The model calculations were run for several years until the sediment temperature profiles converged for each date.

An evaluation of the reasonableness of these assump- tions folio-ws. Assumption that thermal conduction is the only method of heat transport: the only means to trans- port heat through the sediments other than conduction is by advection of pore waters. Measurements in summer 1985 showed that seepage velocities at this location av- eraged 0.06 cm d-l (range,’ 0.03-O. 10). Approximating the depth of diffusional transport as (2Dt)“, we estimate that conduction transports heat 15 times faster than does advection to a depth of 4 m in these sediments. For a l-m depth it is 60 times faster. Assumption that thermal conductivity can be derived from values for solids and water: this approach yields a value for thermal diffusivity (thermal conductivity divided by density and heat ca- pacity) of 0.0040 cm2 s-l, similar to measured values for Lake Mendota, 0.0033 cm2 s-l, and Lake Hula, 0.004 cm2 s-l (Hutchinson 1975). The model was relatively insensitive to changes in the assumed solids : water ratio; varying the ratio alters the heat capacity and thermal conductivity in opposite ways so that the shifts tend to cancel each other. For example, nearly halving the bulk density and using a ratio of 24% detritus to 76% water caused only a 15% change in the output of the model.

As a further verification, the model was used to generate temperature profiles that were compared ’ to measure- ments on sediments at a 12-m water depth in Lake Men- dota in summer 19 19 (Birge et al. 1927). The starting condition was a measured temperature profile on 18 April. The heat capacity and density of the sediments were in- ferred from the measured thermal diffusivity. First, it was assumed that diffusivity was a linear combination of val- ues for water and rock and a solids : water ratio derived. That ratio was then applied to the heat capacity and den- sity of rock (0.2 cal g-l ‘C-l and 2.5 g cm-3) and water (1 .O cal g- I ‘C-l and 1 .O g cm-3) to get combined values of the sediment for use in the model. The results of the model are compared to measurements in Fig. 2 and show very good agreement. Integrating the heat inventory in

Vertical kddy d@usion 161

Lake Mendota sirnulat,ion

0 18 April A 13 May n 29 May

$ ‘(I 14 June 11

2 E 9

E 7

5

DEPTH (m) Fig. 2. Simulation of sediment heating in Lake Mendota in

19 19. Direct measurements of the temperature of the sediments as a function of depth indicated by closed symbols (Birge et al. 1927); open symbols are temperatures predicted with the model MUDHEAT as described in the text.

the profile on the various dates shows that the model predicts sediment heating to an average accuracy of better than 110%.

Results

Because Bickford Reservoir is dimictic, during the warm months the lake has an upper mixed layer and a lower stratified layer (Fig. 3). In deep lakes, the lower zone consists of two parts: a region where temperature de- creases rapidly with depth (metalimnion), and below that a region of nearly constant low temperature (hypolim- nion). Although there is little density stratification in the hypolimnion, water circulates very slowly because it is isolated from the direct influence of the wind. In Bickford, as can be seen in Fig. 3, the greatest depth (13 m) is at a level close to the intersection of the metalimnion and hypolimnion, so that the density gradient extends to near the very bottom. The pattern of temperature distribution displayed by Fig. 3 is like that of many other lakes except for the truncation of the bottom of the profiles. Bickford’s spring circulation takes place in April, so at the beginning of the study period in June 1985, the lake was already stratified, and the epilimnion extended from the surface to a depth of 6 m. Throughout the rest of summer, the depth of the epilimnetic-metalimnetic boundary deep- ened approximately linearly with time, reaching a depth of - 12 m just before fall overturn on - 1 November (Fig. 4). An exception to the gradual deepening was the rapid erosion of the thermocline that took place as a result of mixing caused by Hurricane Gloria on 26 September 198 5. That Gloria did not cause complete mixing of Bickford attests to the high stability imparted by the temperature gradient. Similarly, Linsley Pond was not completely

6

a

10

12

AUG OCT Fig. 3. Isopleths of temperature (“C) for Bickford Reservoir,

1985, showing typical pattern of thermal stratification. Note that the thermocline extends nearly to the bottom throughout most of this period. Ordinate is depth (m); points indicate lo- cations and times of measurements.

mixed by the much stronger hurricane of 21 September 1938 (Hutchinson 1975).

Temperature profiles in early summer show the bottom meter of the Bickford water column to be nearly isother- mal, indicating that this zone forms an abbreviated hy- polimnion. The occurrence of a hypolimnion at this depth is in agreement with data from other small temperate- zone lakes, where the base of the thermocline is between 5 and 12 m (Table 1). The upper boundary of the hy- polimnion at Bickford is at the deep limit of the range

J-UN AUG OCT Fig. 4. Depth of the epilimnetic-metalimnetic boundary as

a function of time in Bickford Reservoir, 198 5. Notice the steady deepening that took place from August through October and the failure of Hurricane Gloria (26 September) to completely de- stroy stratification.

162 Benoit and Hemond

Table 1. Depth of the top of the hypolimnion in selected small lakes.

* Depth b-0 Reference

Bickford Res., Massachusetts Linsley Pond, Connecticut Lake Mendota, Wisconsin Lake Quassapaug, Connecticut Lawrence Lake, Michigan Little Crooked Lake, Indiana Little Round Lake, Ontario Martin Lake, Indiana Crooked Lake, Indiana Lake 227, Ontario Lake 224, Ontario Crystal Lake, Wisconsin Esthwaite Water, U.K.

12 This study 8 Hutchinson 1975

10 Hutchinson 1975 12 Hutchinson 1975 5 Wetzel 1975 6 Wel.zel 197 5 6 Wei;zel 1975 9 Wei;zel 197 5

10 Wel;zel 197 5 4 Quay et al. 1980

10 Quay et al. 1980 10 Talbot and Andren 1984 10 Sholkovitz and Copland 1982

for lakes of comparable profundity. The exceptionally deep hypolimnion at Bickford may be a result of the water’s clarity, which permits radiational heating to be active to an unusually great depth.

Transmission of visible light was measured on 4, 11, and 19 September 198 5. Light extinction changed little during the period of measurement (Fig. 5). As in most lakes, there was an exponential decrease in transmitted

l 71=0.40,-l P

A 7j = 0.71 In-l /

/.-2: Sep

I

LIGHT TRANSMISSION Fig. 5. Light extinction in Bickford Reservoir, 1985. Light

transmission was nearly constant over time in the upper water column and decreased to a lower value in anoxic bottom waters, where SPM was elevated.

light as a function of depth in accordance with Bouguer’s law:

wo = exp[ - q(z)z]. (7) I, is the light intensity at depth z, I, is the intensity at the surface, z is the depth in the water column, and q is the extinction coefficient. In theory, this law applies only to monochromatic light, but in practice it seems to work reasonably well for broadband sunlight, after infrared and ultraviolet wavelengths are filtered out in the upper meter of the water column.

The value of 17 averaged 0.39 m-l in the epilimnion (n = 9, r = 0.9999). (For comparison, pure water has an extinction coefficient averaging -0.060 m-l for light be- tween 500 and 600 nm.) The value for Bickford is near the lower, end of the range found by Birge and Juday in a survey of Wisconsin lakes (cited by Hutchinson 1975, p. 393), indicating that the water in Bickford Reservoir is clearer than most. A consequence of the low extinction is that light penetrates to greater depth, heating deeper waters and causing the metalimnion to occur at a some- what greater depth than in an equivalent lake with less transparent water.

The light extinction coefficient increased markedly in the metalimnion, where suspended particulate matter (SPM) increased sharply. On 4 September 1985, q in- creased with depth from 0.40 to 0.7 1 m-l; on 11 Sep- tember, :It increased from 0.39 to 0.57 m-l; and on 19 September, it increased from 0.38 to 0.59 m-l. An equiv- alent increase in extinction coefficient due to SPM in the hypolimnion of Crystal Lake is evident in the data of Birge and Juday. In Bickford, SPM increased from < 1 .O mg liter-l in the epilimnion to >2.0 mg liter-l in the metalimnion. Such an increase in light absorption is con- sistent with theory, which states that extinction due to pure wal;er, particulate matter, and color should be ad- ditive, such that

rlt = %v -I- rlc + rip. (8) vt is the total extinction coefficient, and vw, vc, Q are those due to pure water, color, and particles. In this case, the

Vertical eddy d@usion 163

value of Q increased by an average of 0.24 m- l. Anec- dotally, the rapid decrease in light and increase in SPM was quite noticeable to SCUBA divers as they passed through the thermocline.

A least-squares polynomial was used to fit a smooth curve to the temperature vs. time data at depths of 7, 8, 9, 10, 11, 11.5, 12, and 12.5 m (Fig. 6). The resulting curve was then differentiated with respect to time to de- rive dr/dt. Likewise, temperature profiles were fit with a polynomial to get T(z), which was differentiated to get dT/dz. The value of dT/dz used in the calculation was the average of values at the given depth on the dates at either end of the experimental period. The results of the calculation are average daily values of eddy diffusivity for the eight specified water depths during the 10 time intervals from 14 June to 19 October 1985. Results were not calculated for depth-date combinations that fell in the epilimnion due to descent of the thermocline.

The results of the eddy diffusion calculations fall in the range from 0.00 1 to 0.026 cm2 s-l (Table 2). For each value, 0.0012 cm2 s-l of the calculated result is attrib- utable to molecular diffusion of heat by water. An ap- proximate error analysis was carried out in parallel with the eddy diffusivity calculations. Uncertainty as to the cross-sectional area of the basin or its perimeter length as a function of depth was assumed to contribute negli- gible error. The dT/dt and dT/dz terms (numerator and denominator in the governing equation) were estimated to be uncertain to kO.2”C over the measured time or depth interval. This value comes from the manufacturer’s specification for the temperature meter used. The inso- lation calculations were assumed to be accurate to +4% based on the average digerence between calculated and measured values for Montreal, as described earlier. Ad- ditional error may be introduced due to variation of the water-column light extinction coefficient, q, over time, and this would make the assumed uncertainties under- estimates. Nevertheless, radiant heating proved to be a

JUN A-LJG OCT Fig. 6. Temperature (“C) as a function of time at different

depths in the water column of Bickford Reservoir, 1985. The lines are polynomial regressions used in eddy diffusion calcu- lations.

minor part of the total heat budget at all depths; as a result, the uncertainty in this term usually had a negligible effect on the total uncertainty of the eddy diffusivity. The sediment heating term was taken to be uncertain to f 10% based on the average error determined in the verification run for Lake Mendota.

The total uncertainty was calculated based on these error estimates and standard propagation of error tech- niques (Bevington 1969). Uncertainties ranged from 3 to 30% of the calculated K, values (Table 2). Five of the 59 calculated values were not significantly different from the molecular thermal diffusivity of heat by water. The av- erage of all 59 values was 0.0045 +0.0040 cm2 s-l. With- out including the term for sediment heating, many of the calculated values of K, would have been substantially in error, with derived values usually lower than true values and sometimes negative (Fig. 7).

Table 2. Calculated eddy diffusion coefficients. Upper number in each pair is eddy diffusion coefficient (K,; units, cm2 s-l) and bottom number is relative uncertainty value. (Dash indicates no data.)

Depth 14 Jun- 24 Jul- 14 Aug- 25 Sep- b-0 4 Jul 4-24 Jul 14 Aug 4 Sep 4-l 1 Sep 11-19 Sep 19-25 Sep 9 Ott 9-12 Ott 12-19 Ott

7

8

9

10

11

11.5

12

12.5

0.0034 0.0029 0.0026 +4% &3% +3%

0.0013 0.0017 0.0018 +5% +4% f 4%

0.0018 0.0036 0.0034 I!I 1 1% +7% +6% 0.0028 0.0049 0.0032 +18% flO% +8% 0.0047 0.0058 0.0023 &17% &12% f 14% 0.0069 0.0064 0.0027 zk 19% + 14% I!I 16% 0.0119 0.0088 0.0046 &22% + 18% + 19% 0.026 1 0.0 172 0.009 1 k 30% 2 24% z!I 19%

-

0.0017 0.002 +3% +3%

0.0028 0.0033 +4% 14%

0.002 0.0028 &6% f5%

0.0007 0.0014 I!I 15% +9% 0.00 12 0.002 1 f 13% +16% 0.0027 0.004 1 f 14% +13% 0.0068 0.00 1 zk 19% I!I 14%

- -

-

0.004 +3%

0.0039 +4%

0.0024 +7%

0.002 *lo% 0.0023 IhI 1 1% 0.0035 AI 12%

-

-

-

0.0056 I!I 4%

0.004 +5%

0.0032 f7%

0.003 &lo% 0.004 I!z 12%

0.0063 - - + 4%

0.0045 0.0062 0.0085 _+5% +5% f5%

0.0033 0.004 1 0.0055 +8% f7% +7%

0.0035 0.0039 0.0052 +10% + 10% + 10%

164 Benoit and Hemond

Kz calculated without sediment heating

6 I I I I I

JUN AUG OCT Fig. 7. Isopleths of apparent eddy difhtsivity (cm2 s-l) in the

stratified portion of Bickford Reservoir, 1985. Values have not taken into account heat loss to sediments. Note that many values are lower than the molecular thermal diffusivity of water (0.00 12 cm2 s- I), and some are even negative. Details as in Fig. 3.

Discussion

Many investigators have measured eddy diffusivities in lakes (Table 3). Values vary considerably depending on the time and depth of measurement (e.g. de Bruijn et al. 1980) and the size of the lake (Mortimer 194 1, 1942), among other factors. It is thus difficult to compare results without detailed analysis of whole data sets. Still, it is clear from Table 3 that the eddy diffusivities the measured are near the low end of reported values. In part, this is due to the relatively small area of Bickford Reservoir, and partly it is because almost all of our measurements were made in the metalimnion, where the density gradient and resistance to mixing are greatest. When these factors are taken into consideration, eddy diffusivities in Bick- ford are comparable to values calculated for other lakes of similar size.

The relative contribution to the metalimnetic heat bud- get of the three terms- eddy diffusion, radiant heating, and cooling by sediments-varies with time and depth (Table 4). As would be expected, solar heating (normal- ized to area) drops rapidly with depth and more slowly with time. The effect of sediment heating on the heat budget of the water column has a complex pattern, in- creasing with water depth on a given date, but first de- creasing then increasing with time at a fixed depth. This pattern is a function of the rate of change of temperature at any depth, since changing water temperature drives sediment heating. Between 7 and 9 m, temperature in- creased rapidly and continuously over time, while at greater depth, it increased more slowly overall and was nearly constant until the final month of the study period (Fig. 6). The change in K, was as great as a factor of 11 over depth (during the first time-sampling interval) and as much as a factor of 3.5 over time (depth constant at 12.5 m).

Table 4 compares the relative magnitude of the three heat terms. Sediment heating was much greater than ra-

diant heating in the deeper part of the lake. At water depths of 9 m and greater, the average ratio of sediment heating to radiant heating was 10.9. Above 9 m, radiant heating was more important than sediment heat ex- change. This level (9 m) represents the depth of 3% il- lumination compared to the lake’s surface. For other small lakes, the 3% surface may thus demarcate the depth below which sediment heat exchange needs to be considered in heat budgets. Bickford Reservoir is relatively clear, and radiant heating at depth is substantial (Fig. 5, Table 4). It is significant, therefore, that the common practice of ignoring sediment heating (Robarts and Ward 197 8; Wal- ters 1980; :Liden 1983; Wodka et al. 1983) would cause larger errors when calculating eddy diffusivities than would ignoring radiant heating in the deeper parts of a lake. The effects of the two terms act in opposite directions; ignoring radiant heating causes overestimations of eddy diffusiv- ities, while neglecting sediment heating causes underes- timations. Also, radiant heating has a greater influence higher in the water column, while the importance of sed- iment heat exchange increases with depth.

The amount of heat transferred to sediments by contact with the w.ater column was substantial. Typically, heat lost to sediments was equal to or greater than that going to raise the temperature of the hypolimnion. This com- parison suggests that sediments pose no greater barrier to heat transport than does the stratified water column, which may seem surprising until the following factors are con- sidered. First, the relative volume of materials affected was about equal: the metalimnion thickness was l-5.5 m and the heated sediment thickness 3-4 m. Second, the heat capacities were very similar: water = 1.0 cal cm-3 “C- l; sedirnent = 1.08 cal cm-3 “C- l. Third, the thermal diffusivities were comparable: water = 0.0045 cm2 s-l (typical eddy diffusivity); sediment = 0.004 cm2 s-l (mo- lecular thermal diffusivity). Finally, the temperature gra-

Eddy diffusion coefficients (Kz)

6

8

10

12

JUN AUG OCT Fig. 8. Isopleths of eddy diffusivity (cm* s-l) during the

study period. Conditions are the same as in Fig. 7, except that values have been calculated including a term for heat loss to sediments. Data are uncorrected for the thermal conductivity of water, 0.00 12 cm2 s-l. Note that physically impossible values that are negative or lower than the thermal conductivity of water, which predominate in Fig. 7, are eliminated by inclusion of sediment heat exchange. Details as in Fig. 3.

Vertical eddy d$usion 165

Table 3. Previous measurements of eddy diffusivities.

Z mm

b-0

K

(cm2 s-l) , Reference

Castle Lake, California 35 0.02-1.5” Orlob and Selna 1970 Hungry Horse Res.,

California 104 0.12-1.7? Orlob and Selna 1970 Tahoe, Calif. Nev. 490 O.lO-1.2”f Orlob and Selna 1970 Fontana Res., California 130 0.17-l .2? Orlob and Selna 1970 Linsley Pond, Connecticut 14 0.002-0.006* Hutchinson 197 5 Castle Lake, California 35 0.004-0.05* Jassby and Powell 1975 Greifensee, Switzerland >31 0.05-0.2’f Imboden and Emerson 1978 L. McIlwaine, Rhodesia 27 0.07-0.5* Robarts and Ward 1978 Egg Lake, Washington 5 0.005-0.0 16” Lehman 1979 Lakes 224 & 227, ELA,

Ontario 25 0.002-0.0048* Quay et al. 1980 Esthwaite Water, U.K. 15 <o-0.05* Davison et al. 1980 Grevelingen, Netherlands 50 0.001-10.” de Bruijn et al. 1980 L. Washington, Washington 65 O-10? Walters 1980 L. Lanao, Philippines 112 0.22-l. l’f Lewis 1982 L. V alencia, Venezuela 35 0.09-0.23” Lewis 1983 Lovosundet, Sweden 14 ** Liden 1983 Baldegg, Switzerland 66 0.004-0.2” Imboden et al. 1983 Biel, Switzerland 33 0.014-0.4§(~ Nyffeler et al. 1983 Augher Lough, Northern

Ireland 14 0.005-o. 1* Rippey 1983 Onondaga Lake, New York 13 0.002-0.06” Wodka et al. 1983 Shagawa Lake, Minnesota 12.5 0.016-0.3=/’ Stauffer and Armstrong 1984 Third Sister Lake, Michigan 16.5 0.013-0.3$ Lehman and Naumoski 1986 L. Rotoiti, New Zealand 1;: I+§ Priscu et al. 1986 L. Mendota, Wisconsin 0.04-0.07-f Stuaffer 1986a,b L. Anna, Virginia 21 0.007-0.08* Adams et al. 1987 Crystal Lake, Wisconsin 21’ 0.003 11 Colman and Armstrong 1987 Devils Lake, Wisconsin 13 0.0003-0.0 1 I( Colman and Armstrong 1987 Emrick Lake, Wisconsin 24 0.016-0.061~ Colman and Armstrong 1987 Fish Lake, Wisconsin 13 0.007-0.6 11 Colman and Armstrong 1987 Green Lake, Wisconsin 67 0.009-0.0 18 II Colman and Armstrong 1987 L. Mendota, Wisconsin 24 0.0003-0.301~ Colman and Armstrong 1987 Shagawa Lake, Wisconsin 14 0.0005-0.00 1 II Colman and Armstrong 1987 Bickford Rcs.,

Massachusetts 13 0.00 l-0.0 16 This study

* Sediment heat term ignored, but probably important. $ Sediment heat term probably not important (very deep lakes). $ Eddy diffusion values calculated but results not published. 0 Sediment heat term explicitly considered. 11 Sediment heat term not relevant (radiotracer studies).

dients were similar and usually greater in the sediment near the interface. The net result was that much of the heat flux through the lower water column was taken up by sediments; little went into heating the deep hypolim- nion.

The dominant importance of the sediment term to the heat budget of the stratified portion of this particular lake is partly due to the relative thinness of the stratified zone. Still, neglecting sediment heating should also cause a sig- nificant error when thermal flux gradient analysis is used to calculate eddy diffusion coefficients for other lakes with combined metalimnetic and hypolimnetic thicknesses that are a few times those of thermally active sediments. That criterion would include almost all lakes up to 20 m deep and some deeper lakes that have profound thermoclines

due to relatively transparent water column, late stratifi- cation, high average wind shear, long fetch, or other caus- es.

Calculated K, values tended to increase with depth and decrease with time in the metalimnion and hypolimnion, at least in the period from June through August (Fig. 8). The highest values were found near the sediment-water interface at the beginning of the study period, and the lowest values were higher in the water column and later in the year. An exception to this pattern was an increase in K, beginning in mid-September in the zone corre- sponding to the rapidly eroding thermocline. The overall trend in K, was opposite that of the density gradient, dp/ dz, which decreased with depth and increased with time (Fig. 9). Many investigators have found a correlation be-

166 Benoit and Hemond

Table 4. Contribution of individual terms (cal s-l ha-l) to the metalimnetilz heat budget.

Depth 14 Jun- 24 Jul- 14 Aug- 25 h-0

Sep- 4 Jul 4-24 Jul 14 Aug 4 Sep 4-l 1 Sep 1 l-l 9 Sep 19-25 Sep 9 Ott 9-12 Ott 12-19 Ott

Total change in heat content (includes turbulent heat transport)

7 14.1 15.5 16.1 8 6.5 8.4 9.3 10.3 11.7 9 3.7 7.8 8.7 10.0 13.0 15.6

10 2.4 6.5 5.4 48 7.9 11.4 16.3 11 1.4 4.0 1.9 0.1 1.7 4.2 7.8 15.6 11.5 1.0 2.5 1.2 0.0 0.9 2.2 4.2 8.5 13.8 17.6 12 0.6 1.3 0.9 0.3 0.6 1.1 1.8 3.4 5.4 6.8 12.5 0.2 0.5 0.4 0.2 0.2 0.3 0.4 0.7 1.1 1.4

Radiant heating 7 8.0 9.2 9.2 8 5.1 5.9 5.9 3.8 2.7 9 3.3 3.8 3.8 2.4 1.7 2.8

10 2.1 2.4 2.4 1.5 1.1 1.7 1.2 11 1.4 1.6 1.5 0.7 0.5 0.8 0.6 0.5 11.5 1.1 1.2 1.2 0.5 0.4 0.6 0.4 0.4 0.5 0.4 12 0.9 1.0 0.9 0.3 0.2 0.4 0.3 0.3 0.3 0.3 12.5 0.7 0.8 0.6 0.2 0.2 0.3 0.2 0.2 0.2 0.2

Sediment heat exchange 7 0.8 0.9 0.9 8 1.0 1.1 1.2 1.2 1.2 9 2.1 2.1 2.2 2.4 2.6 2.8

10 2.7 2.5 2.3 2.3 2.5 2.7 3.0 11 3.6 3.1 2.4 2.0 2.1 2.4 2.9 4.1 11.5 4.3 3.7 2.7 2.1 2.2 2.6 3.2 4.6 6.2 7.4 12 5.7 4.8 3.5 2.7 2.8 3.3 4.0 5.8 7.9 9.4 12.5 8.4 7.1 5.2 3.9 4.1 4.8 5.9 8.5 11.5 13.8

tween eddy diffusivity and the stability of stratification as expressed by the rate of change of density with depth. To test this hypothesis, we plotted K, against the square of the Brunt-Vaisala frequency, N2 [defined as (-g/p)(dp/ dz)]. For the purpose of this calculation, the density gra- dient was assumed to depend only on temperature. Ig-

Isopleths of density gradient

6

JUN AUG OCT Fig. 9. Isopleths of density gradient (cm2 s-l) during the

study period. Eddy diffusivity generally is highest where the density gradient and stability are lowest. One exception occurs late in summer in the region where the thermocline is rapidly eroded. Details as in Fig. 3.

noring the effect of solutes in this way is reasonable for an oligotrophic lake like Bickford. During most of sum- mer, there :is a pronounced tendency for eddy diffusion to decrease with increasing water-column stability (Fig. lOA), as seems intuitively reasonable. The regression of the log-transformed variables conform to the relation K, = 1.8 x 1O-,4 (lv2)-“.45 (r = 0.67, P < 0.001). Welander’s (1968) theoretical analysis predicts that K, will be pro- portional to (Nz)-l when turbulence is caused by a shower of energy dlown from large-scale motions and to (N2)-” for turbulence generated by local shear stress. In Bickford, the exponent of N2 is close to -0.5, as it was in field studies of Quay et al. (1980) and Jassby and Powell (1975). The similarity between the Bickford summer data and those in several earlier studies is consistent with the idea that under ordinary circumstances, eddy diffusivity is controlled by stability of the water column. In late sum- mer and early fall, this relationship breaks down dra- matically, with K, correlating positively with N2 (Fig. 1 OB). This corresponds to the period when the thermocline was rapidly eroded downward. We hypothesize that the same physical processes that cause the erosion also increase eddy diffusivity. Thus, increased eddy diffusivity and con- comitant heat transport may be a precursor or means whereby destabilization and mixing of the thermocline lead to its erosion. Possible mechanisms for enhanced heat transport near the epilimnetic-metalimnetic bound- ary include heat interchange with sediments during seich-

Vertical eddy d@kion 167

31 Jun - 15 Aug

K;N2 = 4. 1O-6 -

4

0.0002 0.001 0.008

15 Aug - 31 Ott

l-----I

K;N2 = 4. IO+

. . ,- +.-. - - -.. 0.001 0.010

N 2 Fig. 10. A. Eddy diffusivity as a function of density stability

in the water column .of 13ickford Reservoir, June-August 1985. Linear regression of the data suggests a dependence of K, on Nz to the -0.45 power (P < 0.001). Like measurements from a wide range of aquatic systems, the data also cluster near a line representing K, x w = 4 x 1O-6 (see text). B. Similar data for the period from mid-August through late October. The rela- tionship evident in early summer breaks down and is replaced by a correlation between K, and I\rz with a positive slope (P < 0.001). The data still fall in the vicinity of the zone where K, x w = 4x 10-6.

ing (Dutton and Bryson 1962) or by breaking of internal waves (Quay et al. 1980).

Sarmiento et al. (1976) and Broecker and Peng (1982) have pointed out that the buoyancy flux (the product of W and KJ falls close to 4 x 10m6 for systems ranging from the deep sea to small lakes. The data for Bickford agree well with this observation, having an average value near 9 x 1 O-6. Naturally, this value cannot be a constant when K, 0~ (W)-O.“, but the data cluster in the vicinity of the line where K, = 4 x 10B6 (N2)-l (Fig. 10).

Conclusions

Heat loss to sediments can be a significant term in the heat budget of shallow (<25 m) lakes. This term must be considered to avoid significant errors when eddy dif-

fusivities are calculated by the widely used flux gradient method. The importance of heat exchange with sediments increases with proximity to the bottom. During summer, heat exchange with sediments is likely to be more im- portant than radiant heating below the surface where light intensity drops to - 3% of its value at the lake’s surface. In Bickford Reservoir, eddy diffusivity seems to be con- trolled by stability of the water column except in the vicinity of the rapidly eroding thermocline, where un- identified processes cause unexpectedly high values.

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Submitted: 21 December 1989 Accepted: 25 June 1991

Amended: 11 September 1995