Mixing Dynamics in Lakes Across Climatic Zones

S MacIntyre and J M Melack, University of California, Santa Barbara, CA, USA

© 2009 Elsevier Inc. All rights reserved.

Introduction

Interactions between climatic and geographic conditions lead to variations and gradients in hydrological, physical, chemical, and biological conditions in lakes. The annual cycle of solar irradiance with seasonal amplitude increasing at higher latitudes causes well-known gradients in photoperiod and temperature. Continental interiors differ from coastal regions that tend to be moister and to experience more moderate temperatures. Hence, because of the larger proportion of the Earth's land in the northern hemisphere, oceanic conditions tend to have a greater influence in the southern hemisphere. Globally, annual average precipitation is greater than evaporation in low and middle latitudes, but evaporation generally exceeds precipitation in the subtropics. However, hydrologi-cally closed basins, where evaporation exceeds local precipitation, occur on all continents.

During the twentieth century, studies of lakes throughout the world have produced a wealth of information about individual lakes as well as regional syntheses. Based on these results, latitudinal correlations with limnological characteristics have revealed several patterns. Based on limited variations in solar irradiance in the tropics, maximum temperatures in the mixed layer of lakes are similar across the tropics but tend to decline through the temperate latitudes, and most of the geographical differences in near bottom temperatures are explained by latitude and elevation. Dissolved oxygen concentrations tend to be lower in deep waters of tropical lakes than higher latitude lakes. Though considerable monthly variation in photosynthetic rates of phytoplankton occurs in tropical lakes and corresponds to seasonality in rainfall, river inflows and vertical mixing, no latitudinal trend in primary productivity exists among tropical lakes. When extended to temperate and arctic lakes, maximum variability about doubles and a significant correlation with latitude emerges. In contrast to fish, species richness of phytoplankton, zooplankton, and zoobenthos does not tend to be higher in tropical lakes versus temperate lakes.

Recent investigations of physical processes in lakes have advanced our understanding and permit mechanistic explanations of climatic differences in stratification and mixing. Because these processes are fundamental to biological and chemical conditions, we focus our examination of lakes across climate zones on these processes.

Patterns in Stratification and Mixing Background

Differences among lakes in stratification and mixing dynamics derive from momentum and energy exchanges with the atmosphere and inflows modulated by climatic conditions and basin morphometry. Schemes to classify lakes based on temporal differences in vertical mixing compiled in G.E. Hutchinson's Treatise on Limnology and modified by W.M. Lewis Jr. led to an empirical classification system. Eight types of mixing (amictic, cold monomictic, dimictic, warm monomictic, and four types of polymixis) were distributed in relation to water depth and latitude, adjusted for elevation. Meromictic lakes remain stratified for long periods of time due to processes that lead to accumulation of solutes at depth.

For lakes deep enough to seasonally stratify, the major mixing types defined by Lewis depend upon seasonal weather patterns. Cooling in autumn or during a monsoon combined with the shear due to wind forcing erodes seasonal stratification. Polymictic lakes are shallow and tend to mix frequently due to the combination of cooling and wind at night. Cold monomictic and polymictic lakes are found at high latitudes where water temperatures do not exceed 4 °C; these lakes mix in summer when ice free. Warm monomictic and polymictic lakes occur where water temperatures exceed 4 °C for some part of the year. Warm monomictic lakes mix in winter. Warm polymictic lakes are found over a wide range of latitudes including the Arctic.

As understanding of mixing processes in lakes has improved with applications of new instrumentation and numerical models, underlying mechanisms can be incorporated into analyses of the frequency, intensity, and depth of mixing. Detailed descriptions of the physical processes are provided in the Physical Limnology Section of the Encyclopedia. Computation of surface energy budgets using meteorological data and time series measurements of temperature is particularly important for understanding controls on mixed layer dynamics. With these, as will be illustrated later, we are beginning to develop a predictive understanding of how mixed layer dynamics vary in large and small lakes from the tropics to the poles and the implications for aquatic organisms. Dimensionless indices provide insights into the mechanisms inducing mixing in the metalimnion and hypolimnion.

Understanding how these vary with latitude will allow predictions of the connectivity between the upper and lower levels in the water column with implications for aquatic ecosystem function.

Via surface energy budgets, we learn how much heat is lost or gained via latent heat exchange (LE, evaporation), sensible heat exchange (SE, conduction), and long- and short-wave radiation and, with measurements of light attenuation, we can predict whether temperatures will be stably stratified to the surface, whether a mixed layer will form, and how deep it will be (Figure 1).

The heat fluxes at a lake's surface (Figure 1) are computed as Qtot = LWin + LWout + LE + SE where Qtot is the surface energy flux and LWin and LWout are incoming and outgoing long-wave radiation, respectively. LWout depends on surface water temperature in degrees Kelvin to the fourth power; LWin depends on cloud cover; LE depends on the vapor pressure gradient between the lake and overlying water, temperature, and wind speed; and SE depends upon the air-water temperature difference and wind speed. The effective heat flux into the actively mixing zone is computed as the sum of the surface energy fluxes plus the net short wave radiation into the actively mixing layer. When net solar radiation exceeds the heat losses at the air-water interface, the water column will stratify. When they are less,

Figure 1 Terms of a surface energy budget and examples of resulting temperature profiles. The heat losses or gains which occur at the surface of a lake include latent heat flux (LE) and sensible heat exchange (SE). Outgoing long wave radiation (LWo) and incoming long wave radiation (LWi) are added to the other terms to compute the surface energy fluxes (SEF). Incoming short wave radiation (SWi) induces heating to a depth dependent on the diffuse attenuation coefficient. A portion of this heat is lost (SWo) before entering the lake due to reflection with the amount lost dependent upon the albedo. The sum of the SEF and net shortwave is called the effective heat flux (EHF). Depending upon this latter term, plus mixing induced by wind, temperature profiles (T) will either be stratified to the surface (left profile) or a mixed layer will form with a thermocline below (right profile). Depending upon previous heating and cooling, multiple thermoclines may form.

as occurs at night, during cloudy periods, during fall cooling, and often during monsoon periods, the upper mixed layer will lose heat and deepen. The deepening is induced by thermals due to the cooling at the lake's surface. The turbulence produced by the heat loss can be quantified via the turbulent velocity scale for heat loss. This term, w*, is computed from the buoyancy flux (Jbo) due to the effective heat loss (H) and the depth of the actively mixing layer (h). That is, /bo = agH/pcf where a is the thermal coefficient of expansion, g is gravity, p is density, and Cp is the heat capacity, and w* = (/b0h)1/3; w* only exists when a lake is losing heat. Via comparisons of both the magnitude of this term across latitudes and the length of time it is positive during the day, we are positioned to understand the role of heat loss in mixed layer dynamics for lakes in different climate zones.

Wind-induced mixing is often assumed to dominate mixed layer deepening. However, whether this assumption is true can only be determined by comparing the turbulent velocity scale from wind with that from heat loss. The momentum flux from the wind depends upon the shear stress t at the air-water interface and is generally computed based on the wind speed squared times a drag coefficient. That is, t = pau*2 = pwuw*2 = CdpaU2 where u* is friction velocity in the atmosphere, u*w is water friction velocity, pa and pw are density of air and water respectively, and Cd is a drag coefficient. The shear stress is equivalent on both sides of the air-water interface. The turbulent velocity scale from wind can be characterized by the water friction velocity u*w.

Wind forcing and heat loss can both generate turbulence in the upper water column. By computing the turbulent velocity scales for both, we learn whether heat loss contributes to the mixing and whether wind or heat loss is dominating the mixing in the upper waters (Figure 2). The turbulent kinetic energy flux (TKF) into the mixed layer depends upon the cube of both these terms. The magnitude of this term thus allows comparison of surface forcing during periods of calm and during storms. By computation of these different terms, we are poised for comparisons which enable us to understand the factors leading to temporal changes in mixed layer dynamics in one lake as well as between lake differences.

Wind forcing not only energizes currents and turbulence in the upper mixed layer, it also leads to upwelling and downwelling of the thermocline in stratified lakes and the ensuing production of internal waves and turbulence (Figure 2). By computing Wed-derburn (W) or Lake numbers (LN), we learn whether wind forcing at the surface is sufficient to tilt the thermocline (Figure 3) W = g Aph2/(pu*2L), where g

Interactions leading to mixing

Interactions leading to mixing

Figure 2 Schematic of processes leading to turbulence production in the upper mixed layer (UML), metalimnion, and hypolimnion of lakes. Turbulence is quantified in a number of ways, the most common being the rate of dissipation of turbulent kinetic energy (e), the coefficient of eddy diffusivity (Kz), and the turbulent velocity (u) and length scales (/) where u is computed from u*w and w* or from the relation e = u3// when e and / have been measured directly.

Figure 2 Schematic of processes leading to turbulence production in the upper mixed layer (UML), metalimnion, and hypolimnion of lakes. Turbulence is quantified in a number of ways, the most common being the rate of dissipation of turbulent kinetic energy (e), the coefficient of eddy diffusivity (Kz), and the turbulent velocity (u) and length scales (/) where u is computed from u*w and w* or from the relation e = u3// when e and / have been measured directly.

Figure 3 The degree of tilting of the thermocline depends on the wind speed and stratification as well as the depth of the mixed layer (h) and length of the lake (L) and can be computed from the Wedderburn and related Lake numbers. Four hypothetical cases with the same wind forcing and density stratification show the greater degree of upwelling of the thermocline for long basins relative to ones with shorter fetch (a vs b and c vs d) and the greater degree of upwelling for shallow relative to deep mixed layers (a vs c and b vs d). For each case, a possible Wedderburn number is indicated, but values will shift with the assumed values of h and L. The lower the value of the Wedderburn or Lake number, particularly as they drop below 10, the greater the degree of turbulence in the metalimnion and the hypolimnion below.

Figure 3 The degree of tilting of the thermocline depends on the wind speed and stratification as well as the depth of the mixed layer (h) and length of the lake (L) and can be computed from the Wedderburn and related Lake numbers. Four hypothetical cases with the same wind forcing and density stratification show the greater degree of upwelling of the thermocline for long basins relative to ones with shorter fetch (a vs b and c vs d) and the greater degree of upwelling for shallow relative to deep mixed layers (a vs c and b vs d). For each case, a possible Wedderburn number is indicated, but values will shift with the assumed values of h and L. The lower the value of the Wedderburn or Lake number, particularly as they drop below 10, the greater the degree of turbulence in the metalimnion and the hypolimnion below.

is gravity, p is density, Ap is density difference across the thermocline, h is depth of the mixed layer, L is length of the lake, and u*w is the water friction velocity derived from the shear stress and which can be approximated as 0.001 U where U is mean wind speed. W is essentially the ratio of the buoyancy forces resisting mixing divided by the inertial forces that induce mixing times the aspect ratio of the lake.

Thus, for the same density difference across the ther-mocline and the same wind forcing, lakes with a shallower mixed layer or longer fetch will have lower values of W (Figure 3). The Lake number is an integral form of the Wedderburn number. If these numbers are low (^1), a lake mixes rapidly by wind forcing. If near 1, the thermocline upwells to the surface, if between and 10, partial upwelling occurs, and if >10, no tilting occurs. If the ther-mocline tilts, increased shear occurs at the base of the mixed layer and can contribute to its deepening. Similarly, increased shear can occur within the meta-limnion and hypolimnion leading to enhanced mixing and fluxes of biologically important solutes. In fact, the coefficient of eddy diffusivity (Kz), an index of turbulence used to compute fluxes in stratified flows, depends upon the Lake number. When LN > 10, Kz is ~10"7m2 s"1, equivalent to the molecular conductivity of heat. As it decreases below 10, values of Kz increase by up to three orders of magnitude. Comparison of these two dimensionless indices in lakes across latitude allows a predictive understanding of the influence of wind in generating fluxes below the surface mixed layer.

Comparative Mixing Dynamics

Developing a predictive understanding of mixing processes in lakes at different latitudes requires consideration of the magnitude and phase of the forcing factors outlined in Figure 1 and the resulting changes in terms of the surface energy budget, turbulent velocity scales, and dimensionless indices (Figure 2). As mentioned earlier, the large scale differences in forcing across latitudes are well known, and heat budgets based on monthly averages for a diversity of lakes have been published and led to the seasonal differentiation of lake mixing dynamics developed by Hutchinson and Lewis. What is not well known is the magnitude of parameters controlling and describing mixing dynamics on shorter time scales during the stratified period. To that end, we compare critical parameters based on a suite of lakes for which high frequency meteorological data are available (Tables 1 and 2).

The lakes for which we present data range in size from 1 ha to 670 km2 and are found near the equator (Pilkington Bay, Uganda; Lake Calado, Brazil), in the Arctic (Toolik Lake, Alaska), and at a range of elevations in the temperate region. Three of the temperate lakes are only a few hectares in size and are dimictic; the three larger lakes, due to their larger size or salt content (Mono Lake, CA), are warm monomictic. Attenuation coefficients are variable, ranging from 0.09 in clear Lake Tahoe to 5 in highly strained Trout Bog. While mixed layer depths are anticipated to be shallower for higher attenuation, and deeper for longer fetches, the mixed layer depths noted here do not fully follow those patterns. In fact, the deeper mixed layer depths are found for the larger lakes in the temperate zone and in Pilkington Bay, and the smaller mixed layer depths occur in the sheltered dimictic lakes and in L. Calado, Brazil. Typically, the mixed layer depth in Toolik Lake is similar to the other small lakes, but when a cold front passes, it may deepen to a depth similar to those typically found in summer in larger lakes. Similar mixed layer depths occur in a nearby lake which is 10 times smaller than Toolik Lake. Thus because of differences in the surface energy budget and resulting stratification for Toolik Lake, it can have attributes similar to larger temperate lakes. Analysis of similar anomalies for other lakes will provide insights into how climate structures mixing dynamics.

Comparative Mixing Dynamics: The Upper Mixed Layer

By comparing the surface energy budgets and turbulence parameters, we can begin to understand the factors controlling the differences in the hydrodynamics

Table 1 Characteristics of lakes with high quality meteorological data in different climate zones arranged from small to large from the Arctic to the tropics

Lake

Latitude, longitude

Surface area (km2)

Zmax (Zmean) (m)

kd (m1)

Depth UML (m)

Elevation (m)

Toolik Lake

68° 38' N, 149° 38' W

1.5

25 (7.1)

0.5-0.9

0-10

760

Emerald Lake

36° 35' N, 118° 40' W

0.027

10(6)

0.2-0.3

0-3

2800

Trout Bog

46°3' N, 89°41' W

0.011

7.9 (5.6)

2.5-5

0-1.5

494

Lawrence

42° 44' N, 85° 35' W

0.05

12.5, 5.9

0.4

0-4

270

Lake

Mono Lake

38° N, 119° W

160

46 (18)

0.3

0-10

1944

Lake Tahoe

39° 1' N, 20° 1' W

490

501 (301)

0.09

0-22

1899

(near-shore)

Lake Biwa

35°11.5' N,

670

103.8 (45.5 N basin);

0.3-1.9

0-18

86

135°58.8' E)

(3.5 m S basin)

Lake Calado

3° 15' S, 60°34' W

2-8

1-12 m

1.5

0-3

30

Pilkington Bay

(00° 17' N, 33° 20' E

40

16 (4.5)

1-1.7

0 to bottom

1240

Maximum depth, Zmax; mean depth, Zmean; diffuse attenuation coefficient for photosynthetically available radiation (kd); and depth of the upper mixed layer (UML) during stratification.

Maximum depth, Zmax; mean depth, Zmean; diffuse attenuation coefficient for photosynthetically available radiation (kd); and depth of the upper mixed layer (UML) during stratification.

Table 2 Comparison across latitudes of dominant terms in surface energy budgets, Wedderburn or Lake numbers, and coefficients of eddy diffusivity in the metalimnion when W or LN are low using data from lakes in Table 1

Tropical lakes

Temperate (small, sheltered)

Temperate (moderate to large)

Arctic

LE (U = 4 ms"1)

250

180-300

150

50

LE (U = 8 ms"1)

400

NA

400

<200

Max LWnet

~100

150

200

130

Largest term of SEF

LE

LWnet

LE

LE

Relative size of u*w and w*

u*w <= w*

u*w <= w*

u*w >=w*

u*w > w*

Maximum of uw* and w*

1, 1.3

0.75, 0.75

1.5, 1.2

1.5, 1

Low values of W or LN

1-3(Calado)

100

1

0.1-5

in seasonal thermocline

Metalimnetic Kz

10~6

Molecular rates

10~5-10-4

10~6-10~5

Units of LE and LWnet are Wm 2, of u*w and w* are m s 1, and of Kz are m2s 1. Positive values of LE, SE and LWnet represent heat losses.

Units of LE and LWnet are Wm 2, of u*w and w* are m s 1, and of Kz are m2s 1. Positive values of LE, SE and LWnet represent heat losses.

of lakes. Cumulatively, the surface forcing and resulting surface energy budgets, in concert with the attenuation coefficient for light, determine lake temperatures. They contribute to the temperature difference across the thermocline, which is also set by air temperatures during the coldest time of year, either winter or the monsoon in tropical areas. The temperature difference is generally smallest for tropical lakes and largest for temperate lakes. Surface water temperatures in tropical lakes can be near 30 °C; they are between 20 and 30 °C in large temperate lakes; near 30 °C in small, low altitude temperate lakes and cooler as elevation or latitude increases. In arctic lakes in Alaska during summer, surface water temperatures range from 11 to 20 °C. These temperatures, in turn, lead to variations of different processes in the surface energy budget. For example, surface water temperatures determine the magnitude of evaporation and long wave outgoing radiation as both are higher at warmer temperatures. For many lakes, evaporation is the largest term in a surface energy budget, and thus the magnitude of evaporation is influential in determining how much the mixed layer will deepen due to heat loss (Table 2). For example, for low to moderate wind forcing, latent heat fluxes in tropical and temperate lakes in summer are 3-5 times larger than in arctic lakes. In addition, the rate of change of density with temperature is larger at warmer temperatures, thus cooling more quickly erodes stratification. Thus, given that sensible heat exchanges do not vary widely across latitudes once stratification has set up, and net long wave radiation rarely exceeds 150 Wm"2, the larger latent heat fluxes in warm water lakes contribute considerably to mixing the upper water column.

Wind is another major forcing factor. For the lakes in Table 1, wind forcing is somewhat dependent on lake size. For all the larger lakes, wind speeds in excess of 8 ms"1 occur frequently. However, for the small, sheltered lakes, winds rarely exceed 5 ms"1. Exceptions occur for L. Calado, where wind speeds are generally less than 6 ms"1 and for Toolik Lake, where winds exceed 5 ms"1 on a near daily basis and are frequently above 8 ms"1. In consequence, maximum values of u*w are higher for the larger lakes and for exposed small lakes such as Toolik, indicating that the greater importance of wind forcing for turbulence production in the upper mixed layer in these lakes than in smaller lakes.

Based on the latitudinal differences in the magnitude of evaporation and the size, and in part latitudinal related differences in wind speed, patterns begin to emerge in the role of heat loss versus wind mixing in setting mixed layer depth in lakes in diverse locations. In general, the magnitude of u*w is lesser than w* in tropical lakes. In fact, in Pilkington Bay, w* is sometimes 6 times higher than u*w. In contrast, in arctic lakes, u*w is generally larger than w*. The relative magnitude also depends on the size of the lakes. u*w exceeds w* in the large temperate lakes; the converse is true for the small lakes. That is, heat loss contributes more toward turbulence production and mixed layer deepening in tropical lakes and in small temperate lakes than in lakes with cooler surface water temperatures. Because the magnitude of evaporation depends on wind speed, w* is lowest in the small sheltered temperate lakes; the combination of low w* and low u*w leads to the shallow mixed layer depths in such lakes.

The diel variations in w* are a major cause of the diurnal variations in mixed layer depth. That is, since w* goes to zero on sunny days, the upper water column may thermally stratify even if it is windy. It is through this mechanism that diurnal thermoclines form. With the higher evaporation rates in warm water bodies, the onset of cooling begins earlier in the day such that the combination of wind mixing and heat loss can lead to more rapid deepening and deeper mixed layers. For example, as a result of typical heat losses from sensible and latent heat exchanges and net long wave radiation of 200-300 Wm-2 at Pilkington Bay, the mixed layer is likely to deepen from about 0.5 to 6 m from late afternoon until early morning. For the arctic lake, similar heat losses occur with cold fronts with sustained cold temperatures and moderate winds.

Cloud cover or its absence can influence mixed layer dynamics in different ways at different latitudes. Cloud cover reduces net short- and long-wave radiation. In arctic and temperate regions, the decreased solar radiation is such that the effective heat loss drops below zero (heat loss from the lake) and the mixed layer deepens even during the day. In contrast, in tropical regions during the rainy season, warmer air temperatures, higher relative humidity, and greater cloud cover may lead to lower heat losses at the surface thus mitigating the lower insolation. Hence, stratification may be persistent even during the day. In contrast, for sheltered tropical lakes, cloud-free periods and the considerably enhanced heat loss, particularly at night, from long wave radiation may abet seasonal deepening of the mixed layer.

Comparative Mixing Dynamics in the Upper Mixed Layer: Case Studies

In the following, we compare surface forcing and resulting surface energy budgets for specific lakes in the tropics, temperate zone, and Arctic (Table 1). We first compare the more weakly stratified tropical and arctic lakes to illustrate dominant mixing processes in these environments. We then contrast surface forcing in large and small temperate lakes found in regions that are dry and in regions that are more humid.

Comparison of arctic and tropical lakes - Pilkington Bay is a shallow embayment of Lake Victoria near the equator in Uganda. Due to warm air temperatures, high irradiance and fairly turbid water, it stratifies daily with temperature increases between 1 and 3 ° C. For relatively low winds of 4 ms-1, evaporative heat loss is 250 Wm-2; for moderate winds of 8 ms-1, evaporative heat loss is 400 Wm-2. Once heat losses at the air-water interface exceed inputs from solar radiation, the mixed layer begins to deepen. For example, as a result of typical heat losses from sensible and latent heat exchanges and net long wave radiation of 200-300Wm-2, the mixed layer is likely to deepen from about 0.5 m to 6 m from late afternoon until early morning. Turbulent kinetic energy fluxes with such heat losses and light winds are (0.1-0.4) x 10-6

m3s-3. Passage of night-time squalls with winds up to 9 ms-1 can increase surface heat losses to 600Wm-2 and turbulent kinetic energy fluxes to 2.5 x 10-6m3s-3. Even if wind speeds increase, the turbulent velocity scale for cooling exceeds that for wind mixing attesting to the role of convection due to heat loss for deepening the mixed layer.

Arctic lakes, such as Toolik Lake, have evaporative heat losses during summer for winds of 4 and 8 m s-1 of 50 and 200Wm-2, respectively (Table 2). Thus, with the cold temperatures, heat loss due to evaporation is much less than in tropical lakes. Consequently, net long-wave radiation and sensible heat exchange tend to be a larger proportion of the surface energy budget. Under stationary air masses with surface energy fluxes of 100-200Wm-2, and turbulent kinetic energy fluxes less than 0.5 x 10-6m3s-3, diurnal heating induces either linear stratification to the surface or diurnal thermoclines. Nocturnal cooling erodes these features to the depth of the seasonal thermocline. However when cold fronts with over a day of sustained cold temperatures and moderate winds occur, surface energy fluxes exceed 300Wm-2, and fluxes of turbulent kinetic energy reach 2 x 10-6 m3 s-3. The mixed layer deepens to depths that are reached within the tropical embay-ment in 36 h as opposed to 12 h. Due to the longer time scale for mixing, deepening is only possible during cloudy conditions; otherwise the heating from solar radiation would restratify the upper water column. In addition, the depth of mixing is reduced relative to tropical water bodies because of the higher temperature gradient across the thermocline.

Comparison of the turbulent velocity scale for cooling to that for wind mixing illustrates the different roles of heat loss and wind in the two environments. In the arctic lake, u*w and w* are similar or u*w is up to 2 times higher. Except during cold fronts, w* drops to zero in the day and a diurnal thermocline forms. During cold fronts, w* stays high and the mixed layer deepens due to the combination of heat loss and wind mixing. In contrast, in Pilkington Bay, the values of u*w and w* are similar or w* is up to 6 times higher. During cold fronts in the arctic lake, the Lake number drops to values between 1 and 3. This decrease indicates that the thermocline tilts and shear at the base of the mixed layer contributes to mixed layer deepening for much of the period. Although Lake numbers for Pilkington Bay are nearly always less than 1, a large fraction of the turbulent kinetic energy flux into the mixed layer at night is due to cooling. Thus, thermocline tilting does not appear to be essential for mixed layer deepening.

Overall, the picture emerges that deepening of the upper mixed layer in a tropical lake occurs readily by heat loss and is relatively rapid. In contrast, for similar turbulent kinetic energy fluxes in an arctic lake, wind shear plays a larger role in mixing the upper water column and shear near the base of the mixed layer further contributes. The typical nocturnal cooling and wind leads to a shallow upper mixed layer (~2-3m) which overlays a seasonal thermocline. In contrast, in tropical lakes such as Pilkington Bay, the depth of nocturnal mixing is greater. The lake mixes to the bottom more frequently and a seasonal ther-mocline does not form or forms at deeper depths. Thus, the dominant processes causing mixing at these two latitudes differ.

Mixing during the stratified period in moderate-sized temperate lakes Mono Lake (California) is located in an arid region. Its winter climate is marked by the passage of frontal systems with high winds; in summer, diurnal winds predominate with magnitudes ~5-8ms_1. Fronts are less frequent than in winter, but can induce winds in excess of 8-10 ms-1 for periods over a day. With the absence of appreciable cloud cover in summer, heat losses due to net longwave radiation range from 100 to 200 W m-2 and are higher than in either the tropical or the arctic lake. Sensible heat exchanges are similar to those in the tropical and arctic lakes. Heat losses due to latent heat exchange are similar to those in the tropical and arctic lake when diurnal winds prevail. When winds increase to 10ms_1 and relative humidities drop to 25%, latent heat fluxes reach maxima of 500 Wm-2. The high winds which induce these large latent heat fluxes persist and latent heat fluxes can exceed 300 Wm-2 for over a day. During such conditions, u*w is only slightly larger than w*. Turbulent kinetic energy fluxes during light winds are 0.2-0.4 x 10-6m3s-3; during high winds they increase to values between 2 and 3 x 10-6m3s-3. Similar to the tropical and arctic lakes, when winds were light, nocturnal mixing through the 10 m deep upper mixed layer is driven by heat loss and the surface layer restratifies in the day. However, when frontal systems pass, the combination of heat loss and wind shear induce mixing through much of the mixed layer even during the day. The Lake Number drops to values near 1; hence, mixing is induced near the base of the mixed layer by shear. Thus, we see that mixing during the passage of fronts at Mono Lake is similar to that during the passage of fronts in arctic lakes.

During periods with light to moderate winds, the depth of mixing at night in summer in moderate-sized lakes is greater for lakes in arid regions than where humidity is high. Lake Tahoe (California/Nevada) is similar to Mono Lake and experiences low humidity and cool nights. The upper mixed layer stratifies and mixes on a daily basis, with nocturnal heat losses inducing turbulence to the top of the seasonal thermocline at 22 m. Diurnal heating restratifies the lake in the morning and dampens the turbulence induced at night. Light to moderate afternoon winds cause turbulence in only the upper 4 m. In contrast, Lake Biwa (Japan) has warm temperatures and high humidity. During periods with light winds, the upper mixed layer remains stratified even at night. For both lakes, the expectation is that as wind speeds increase, the mixing dynamics would be similar to those during fronts at Mono Lake.

Mixing during the stratified period in small temperate lakes Small, sheltered temperate lakes have received the majority of study by aquatic ecologists and experience conditions different from those in larger temperate lakes or small arctic lakes. Turbulent kinetic energy fluxes in high elevation Emerald Lake (California) and Trout Bog (northern Wisconsin) are less than 0.2 x 10-6m3s-3. With rare exceptions, wind speeds are less than 5ms-1 at Emerald Lake, and less than 2 ms-1 at Trout Bog. During spring in northern Wisconsin, cold fronts with cloudy skies and air temperatures 5-10 °C lower than surface water temperatures occur at day intervals. A similar periodicity occurs in the fall, but in summer fronts pass through every 10-15 days. Surface water temperatures are almost always slightly warmer than air temperatures; typical values of SE are 20 Wm-2. Because of the low wind speeds, latent heat fluxes do not exceed 150 Wm-2. In consequence, net longwave radiation is, on average, the largest heat loss term. Because the sum of the heat loss terms is at most 300Wm-2, and, during daylight hours net solar radiation frequently exceeds this value, diurnal thermoclines form on most days. Thus, the pattern involves formation of a shallow diurnal thermocline each day followed by nocturnal cooling. u*w is always less than 0.5 ms-1, and w* reaches 0.5ms-1 nearly every night. These low values attest to the reduced turbulence in these environments compared to larger or more exposed lakes. While the upper mixed layer mixes due to penetrative convection from heat loss at night; the reduced values of turbulent kinetic energy flux and the high density gradient at the top of the thermocline lead to a shallow mixing depth.

Cloudy periods with cool air temperatures occur infrequently at Emerald Lake in summer, but passage of fronts with air temperatures ~10 ° C colder than lake temperatures occur in autumn. During summer, relative humidity averages 50%. Similar to Trout Bog, sensible heat fluxes are low in summer although they increase to 50Wm-2 during cold fronts in autumn. Latent heat fluxes increase steadily as surface temperatures warm with maxima in mid-September. They decrease as temperatures decrease in autumn. With a few exceptions, maximum values of latent heat exchanges are less than 150 Wm-2. Due to the low wind speeds and low latent heat fluxes, net long-wave radiation dominates the surface energy budget. LWnet also varies with surface water temperatures. Consequently, surface energy fluxes increase steadily through summer. The upper water column heats diurnally and cools at night with u*w and w* similar in magnitude to values at Trout Bog. The magnitude of the heat losses at night sets the depth of nocturnal mixing and thus, with the increase in TKE due to warmer surface temperatures, the depth of the seasonal thermocline progressively deepens throughout the summer. The first cold front in autumn erodes the seasonal thermocline. Thus, the autumn cooling period is marked by diurnal heating and nocturnal mixing with successive cold fronts leading to overall decreases in water temperatures. Lake Numbers in summer at both Trout Bog and Emerald Lake always exceeded 100 and hence wind shear does not contribute to mixing at the base of the thermocline. Although Lake Numbers decrease due to the weakened stratification in fall, heat loss is the dominant source of turbulent kinetic energy due to the low winds. Hence, wind shear plays a minor role in the deepening.

Summary - For small, sheltered dimictic lakes, con-vective mixing due to heat loss sets the depth of the seasonal thermocline in summer with progressive deepening at night as surface temperatures increase with concomitant increases in evaporation and net long-wave radiation. Autumn cooling is induced by heat loss with a minor contribution from wind shear. This pattern occurs at Trout Bog where relative humidities vary from 30% to 90% and for Emerald Lake where they ranged from 15% to 85% in summer, and 5% to 90% in autumn. This dominance of convective cooling in mixed layer deepening is similar to that found in tropical lakes. For the moderate-sized to large temperate lakes, the mixed layer mixes by convection at night at low winds where relative humidities are low but not when relative humidities are higher. During periods with moderate to high winds, turbulent velocities from wind exceed those from heat loss and Lake Numbers decrease to values low enough such that mixing is induced at the base of the mixed layer by shear. Thus, for the moderate-sized lakes during the passage of fronts, wind shear plays a considerable role in energizing the mixed layer, and mixing dynamics are similar to those in the dimictic arctic lakes in summer.

Mixing Dynamics: Thermocline and Hypolimnion

When Wedderburn and Lake numbers drop below threshold values (see Background section), vertical mixing is enhanced in the thermocline and hypolim-nion by instabilities in the internal wave field. Lake numbers for Pilkington Bay rarely exceed 1. Because heat loss causes rapid deepening of the thermocline in tropical lakes and reduction of the density gradient across the thermocline, the low Lake numbers are not indicative of mixing induced by internal waves. In contrast, when Lake Calado, a dendritic lake on the Amazon floodplain, is thermally stratified, Lake numbers drop to values between 1 and 3 as winds increase diurnally. Eddy diffusivities increase to values 10 times the molecular diffusion of heat. Lake numbers during summer stratification in Emerald Lake and Trout Bog exceed 100 and indicate that the thermocline will not tilt and nonlinear internal waves will not form. Eddy diffusivities in the thermo-cline and hypolimnion of these lakes in summer are, with only a few exceptions in Emerald Lake, at molecular values. In Mono Lake, Toolik Lake, and Lake E5, a stratified lake near Toolik Lake but whose surface area is one-tenth larger, Lake numbers typically exceed 10 during periods with diurnal winds but drop to values near 1 when high winds occur in association with the passage of fronts. This greater wind forcing tilts the thermocline and induces instabilities in the internal wave field. In all three lakes, the coefficient of eddy diffusivity increases to values 10-100 times molecular in the thermocline and, in Toolik and E5, to values 10-1000 times molecular in the hypolimnion. These high values attest to the enhanced mixing associated with instabilities in the internal wave field.

In summary, depending on wind forcing relative to stratification, nonlinear waves form in the stratified waters of both small and large lakes. The instabilities are larger near the boundaries in nearly all cases examined. Thus fluxes will be larger near shore and other mechanisms will be required for transport to offshore waters. These include intrusions as well as the propagation of nonlinear waves. It is not surprising that nonlinear waves form in larger lakes with their greater fetch. Whether they are found in small lakes appears to depend upon the latitudinal gradient which sets the temperature difference across the thermocline and upon the degree with which wind forcing is reduced due to sheltering. Interestingly, the temperature gradient in small, stratified tropical lakes is small enough that Lake numbers drop to values indicative of instabilities in the metalimnion on a daily basis. In small temperate lakes, the temperature gradient is so high relative to the forcing that Lake numbers stay high. The temperature gradient in small arctic lakes is high enough that when diurnal winds predominate, Lake numbers exceed 10. However, the increase in wind speeds during the passage of fronts is sufficient to drop Lake numbers to values near 1 such that internal waves become unstable.

We thus see different patterns in lakes at the three latitudes. In tropical lakes, cooling events or events with low Lake numbers frequently occur allowing vertical exchange of solutes and particulates. In arctic lakes as small as 11 ha, exchanges occur in summer during the passage of fronts, and whether they involve full water column mixing or breaking internal waves depends on the depth of the basin and the extent of heat loss. Exchange is particularly effective during cold fronts when the mixed layer deepens and low Lake numbers occur. In moderate-sized temperate lakes in summer, exchanges occur when winds increase with the passage of fronts or major storms. In small temperate lakes during summer, the Lake and Wedderburn numbers stay high, exchange between the epilimnion and metalimnion and ultimately the hypolimnion occurs when entrainment occurs due to heat loss. The depth of this mixing depends upon the magnitude of heat loss relative to the stratification gradient. In summary, physically induced connectivity between the upper and lower water column occurs frequently in shallow, tropical lakes, occurs during the passage of fronts in stratified arctic and large temperate lakes, and rarely occurs in small, stratified temperate lakes during summer.

Implications of Differences in Mixing Dynamics

Turbulent kinetic energy fluxes which cause mixing in lakes are similar during the stratified period in tropical, temperate, and arctic lakes. The highest values occur during windy periods; for sheltered lakes the values are always low. In tropical lakes, with their warm water and consequent high evaporation rates and the small temperature differences between the upper and lower water column, mixed layer deepening can occur rapidly at night. In temperate and arctic lakes, the overall depth of mixing in summer is reduced relative to tropical lakes for the same heat loss and flux of turbulent kinetic energy due to the greater work required to entrain the more stably stratified water below. Regardless, even with light winds, heat loss due to convection causes nocturnal mixing which can redistribute solutes and particles. The depth of nocturnal mixing is less where humidity is higher.

The phasing of wind relative to heat loss also influences mixed layer deepening. In particular, the implications of higher winds depend upon whether they occur during the day or night and whether it is cloudy. If they occur on days with low cloud cover, they distribute heat downwards in the water column and diurnal thermoclines may not form. If they occur on cloudy days or at night, heat loss occurs and the mixed layer deepens. In temperate and arctic regions, the passage of fronts induces increased wind speeds. At some locations, the winds are highest near mid-day, and little mixed layer deepening occurs. In arctic lakes, winds do stay high at night during fronts and considerable deepening occurs. In some summers the arctic lakes restratify, in others the combination of winds and cold temperatures is such that the lakes mix to the bottom and do not restratify.

When wind forcing is high relative to the density gradient across the thermocline, the thermocline tilts and nonlinear waves form. The ensuing turbulence induces connectivity between the hypolimnion and epilimnion. The latitudinal differences in mixed layer depth and density gradient across the thermo-cline determine the importance of this mechanism. Instabilities in the internal wave field occur in many small arctic lakes in summer. They are less frequent in small temperate lakes due to the stronger stratification. How frequently they occur in moderate-sized lakes depends upon the frequency of frontal systems with high winds. Their importance in tropical lakes depends upon the magnitude of the wind field and resulting latent heat fluxes. When latent heat fluxes are high, mixed layer deepening may be rapid and deep enough that connectivity between the upper and lower water columns is induced by heat loss. Thus, we see that comparisons of surface energy budgets and use of dimensionless indices provides a mechanistic understanding of latitudinal differences in lake hydrodynamics. This understanding will promote improved intuition as to differences in ecosystem level function in different latitudes as well as to ecosystem responses to climate change.

See also: Currents in Stratified Water Bodies 2: Internal Waves; Density Stratification and Stability; Small-Scale Turbulence and Mixing: Energy Fluxes in Stratified Lakes.

Further Reading

Davies BR and Walmsley RD (eds.) (1985) Perspectives in Southern Hemisphere Limnology. Developments in Hydrobiology, vol. 28. Dr. W. Junk Publishers.

Hutchinson GE (1957) A Treatise on Limnology. John Wiley & Sons.

Lewis WM Jr. (1983) A revised classification of lakes based on mixing. Canadian Journal of Fisheries and Aquatic Science 40: 1779-1787.

Lewis WM Jr. (1987) Tropical limnology. Annual Review of Ecology and Systematics 18: 159-184.

Imberger J (1985) The diurnal mixed layer. Limnology and Oceanography 30: 737-770.

Melack JM (1979) Temporal variability of phytoplankton in tropical lakes. Oecologia 44: 1-7.

Melack JM (2006) Biodiversity in inland aquatic ecosystems: Natural gradients and human-caused impoverishment. In: Leybourne M and Gaynor A (eds.) Water: Histories, Cultures, Ecologies, pp. 182-190. ch. 14, University of Western Australia Press.

Maclntyre S (1997) Turbulent eddies and their implications for phytoplankton within the euphotic zone of Lake Biwa, Japan. Japanese Journal of Limnology 57: 395-410.

Maclntyre S, Romero JR, and Kling GW (2002) Spatial-temporal variability in mixed layer deepening and lateral advection in an embayment of Lake Victoria, East Africa. Limnology and Oceanography 47: 656-671.

Maclntyre S, Sickman JO, Goldthwait SA, and Kling GW (2006) Physical pathways of nutrient supply in a small, ultra-oligotrophic arctic lake during summer stratification. Limnology and Oceanography. 51: 1107-1124.

Robarts RD, Waiser M, Hadas O, Zohary T, and Maclntyre S (1998) Contrasting relaxation of phosphorus limitation due to typhoon-induced mixing in two morphologically distinct basins of Lake Biwa, Japan. Limnology and Oceanography 43: 1023-1036.

Talling JF and Lemoalle J (1998) Ecological Dynamics of Tropical Inland Waters. Cambridge University Press.

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