The mechanism of hypolimnion warming induced by

LIMNOLOGY and

OCEANOGRAPHY

Limnol. Oceanogr. 00, 2015, 1–15 C 2015 Association for the Sciences of Limnology and Oceanography V

doi: 10.1002/lno.10109

The mechanism of hypolimnion warming induced by internal waves Aminadav Nishri, Alon Rimmer,* Yury Lechinsky Israel Oceanographic & Limnological Research, Yigal Allon Kinneret Limnological Laboratory, Migdal, 14950 Israel

Abstract The measured temperature of the hypolimnion of Lake Kinneret, Israel, reveals an average rise of 0.6 6 0.38C between April and December. Three mechanisms are suggested as the cause of this rise: (i) direct radiation; (ii) an entire-lake vertical diffused heat transfer; and (iii) a mechanism of “indirect warming,” which is investigated here for the first time. The indirect warming mechanism prevails in the sublittoral zones, which are affected by internal seiche activity, including Poincare and Kelvin waves with cycle periods of 12 h and 24 h, respectively. During part of the seiche’s time span, the warm epilimnetic water comes into contact with the underlying bottom sediments and heats them. During the rest of the seich’s cycle, part of the sublittoral bed sediments become overlain by cold hypolimnetic water and cause the previously heated sediments to emit most of their heat, slightly warming the hypolimnion. The daily cycle of this indirect warming mechanism is superimposed on the seasonal cycle of the sublittoral bed sediments, which are cold in the winter and warm in the summer, in accordance with the seasonal pattern of the epilimnion’s temperature. Empirical evidence that suggests that indirect warming actually takes place in response to welldocumented seiche water motions is presented in this article. For example, according to the proposed analysis of sediment temperature changes, in the month of June a daily heat flux of 7.88 W m22 (0.68 MJ m22 d21) was estimated. It was found that during the first 5 months of stratification this mechanism contributes 41% of the thermal energy needed for the 0.68C temperature increase. This mechanism is expected to occur at any thermally stratified lake that is significantly affected by internal seiche activity.

evident from the findings of Rimmer et al. (2005), who showed that during the stratified period, Cl concentration in the hypolimnion remains stable while in the epilimnion it rises considerably due to an inflow of on-shore brackish springs and evaporation. Long term weekly temperature profiles, monitored during the entire year between 1969 and 2014, reveal the systematic seasonal temperature changes that take place during the stratification period. While the temperature of the epilimnion varies between a typical winter mixed lake temperature of 158C (Fig. 1) and a late summer temperature of 308C, the hypolimnion shows only minor seasonal changes, with an average rise of just 0.68C 6 0.38C (Fig. 1) between April and December. The hypolimnetic warm up rate is usually higher during the first months of stratification (April–July), during which the internal seiche activity is intensive (Imberger and Marti 2014). It is followed by a slower warming rate during August–November, when the thermocline deepens, the Mediterranean Sea breeze changes its daily pattern, and the internal seiche activity weakens (Antenucci et al. 2000; Antenucci and Imberger 2003). A slightly higher average rate of hypolimnion heating was observed during December, resulting from the full mixis (a mix with slightly warmer epilimnion water) that sometimes occurs before the end of

Thermally stratified lakes are characterized by a relatively hot epilimnion layer on top of a colder hypolimnion layer. A steep temperature gradient between the layers may cause restricted mixing and a transfer of heat between them. In the monomictic Lake Kinneret, Israel, stratification occurs between the end of March and December. During most of the stratified period the stability of the water column is much larger than the wind energy available to disturb it (Imberger and Marti 2014), and, therefore, it can facilitate the development of two thermally distinct layers, separated by a temperature transition metalimnion, which includes the thermocline. Consequently, various solutes that originate from the mineralization of organic matter settling from above (i.e., NANH4, PASRP, CACO2, and HCO2 3 ) accumulate in the hypolimnion (Imberger 1998; Nishri et al. 2000; Rimmer et al. 2006). This restricted transport from the nutrient rich hypolimnion to the depleted epilimnion must have ecological implications for the biological activity in the photic layer. Restricted mixing between the layers was also

Additional Supporting Information may be found in the online version of this article

*Correspondence: [email protected]

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Mechanism of hypolimnion warming

gradient of roughly 158C per every 6 m of depth, and is, therefore, expected to induce heat diffusion downward. The larger difficulty in evaluating this heat flux results from our insufficient knowledge of the spatial and temporal vertical diffusion coefficient relevant to the metalimnion. However, it should be noted that temperature change measurements (Fig. 1) indicated that the temporal pattern of hypolimnion warming (faster during April–July and slower during August– November) does not follow the respective pattern of the temperature gradient (low during April and high during August) across the metalimnion. The two mechanisms may explain the hypolimnion warming pattern (Fig. 1) during April (direct radiation) and between August and November (diffusion through the metalimnion), but they can offer only limited justification for the additional temperature rise that occurs during May–June– July. Therefore, it is suggested that a third mechanism that causes hypolimnion warming may exist. In the present study, we introduce this new “indirect warming” mechanism, which we believe is responsible for a significant flux of heat to the hypolimnion (relative to the other two mechanisms), especially during May–July. The proposed mechanism is strongly associated with internal seiching, which is temporarily deflected by the Coriolis force, giving rise to waves with propagating phases like Kel€ est and Lorke 2003). Both types of vin or Poincare waves (Wu Coriolis-affected waves were observed (Antenucci et al. 2000; Saggio and Imberger 2001; Antenucci and Imberger 2003) and modeled (Hodges et al. 2000) in Lake Kinneret. The latter’s comprehensive study revealed the existence of a Kelvin wave with a period of about 24 h, as well as 3 different modes of Poincare waves with durations of about 12 h. Lake Kinneret’s size and oval shape, as well as its strong diurnal forcing of the Mediterranean Sea breeze (Shilo et al. 2007; Rimmer et al. 2009), favor the occurrences of basin-scale Kelvin waves (Bennett 1973; Ou and Bennet 1979), particularly in areas closer to the lake boundaries. An illustrative description of these waves using three dimensional model results (ELCOM) can be found in Hodges et al. (2000), Gomez–Giraldo et al. (2006), and Marti and Imberger (2008). An additional example of lake wide seiching, more specific to the proposed mechanism, illustrates the spatial and temporal distribution of the depth in which the water temperature is 238C, from 21 June 2001 10:00 h to 22 June 2001 08:00 h (Fig. 2). Based on simultaneous temperature measurements in six monitoring stations, TF, TG, TK, TC, TA, and TB (See map in Fig. 3a), and on the linear interpolation within the polygon formed by containing these stations, the illustrated depth isotherms of the 238C plane demonstrate the counterclockwise propagation of the Kelvin wave during a 24 h period. At several time points (e.g., at 10:00 h and 18:00 h), stations located 10 km apart reveal a depth difference of > 5 m in the 238C plane. Taking into account that the node of the seiches is located at the center of the lake’s

Fig. 1. (a) The average temperature measured throughout the years at a depth of 35–39 m in Station TA (Fig. 3). Shaded time intervals are periods of full mixing, while stratification periods are not shaded. (b) The average temperature change (8C) and the standard deviation with respect to the temperature at the beginning of the stratification period (the 31st of March for each season).

December. The occasional temperature drop observed between April and November (Fig. 1) is not systematic, but a natural fluctuation produced by various causes. Hypolimnion warming in lakes is often considered to be the result of two mechanisms: (1) radiation heat that penetrates directly into the hypolimnion (Bachmann and Goldman 1965; Jassby and Powel 1975) and (2) Heat diffusion across the thermocline (Cowgill 1967). Regarding the first mechanism, in Lake Kinneret the constant of the attenuation of light at the lake surface I0 , given by the equation IðzÞ5I0 expð2Kd zÞ, typically decreases with depth z by Kd 5 0.5 m21 (Yacobi 2006), and, therefore, at a depth of z 5 10 m, radiation IðzÞ diminishes to less than 1% of its value at the water surface. Thus, its contribution to hypolimnion warming (which usually occurs deeper than 15 m) in the lake’s pelagic zone is considered negligible during the entire stratification period. However, in the lake’s sublittoral areas this mechanism should be taken into account. The vertical motions of the thermocline due to internal seiches may periodically bring parts of the hypolimnion closer to the lake’s surface, where light may penetrate directly into this layer and cause some additional heating which would not be plausible without the internal wave activity. This mechanism of light intensity reaching the deep layers of the lake was studied (Rimmer et al. 2008) in the context of the photosynthetic activity of Chlorobium phaeobacteroides in the metalimnion. Since the mechanism of direct radiation is not the main objective of this study, it will not be fully analyzed here. Nevertheless, it is suggested that hypolimnion warming as a result of direct radiation might be considered feasible only during limited hours in a day and only in a limited surface area during April–May, when the thermocline’s depth is relatively shallow. Regarding the second mechanism, the intermediate layer (the Metalimnion) is characterized by a steep temperature 2

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Fig. 2. The spatial and temporal distribution for the 23 8C depth from 21 June 2001 10:00 h to 22 June 2001 08:00 h. The figures use Israeli local coordinates, and the locations of the monitoring stations [TA, TB, TF, TG, TK, TC] in the lake (top left figure at 10:00 h) are shown in Fig. 3a. proposed by Dutton and Bryson (1959) as they analyzed heat fluxes in Lake Mendota, Wisconsin, U.S.A. The authors concluded that the internal seiche that imposes heat flow through the bottom sediments may account for 40% of the heat transfer to the hypolimnion. However, their approach was purely theoretical, and lacked specific measurements and/or empirical evidence. Specifically for Lake Kinneret, Ostrovsky et al. (1996) qualitatively discussed the effect of bottom temperature “memory” and water mixing in what they called “the washing zone”—the bottom area affected by the periodic seiche-related replacement between the epilimnion and hypolimnion. They showed that just after the hypolimnion water had been replaced by epilimnetic water, the temperature near the bottom surface was 28C lower than the temperature of the overlying water, and “thus, the bottom apparently ‘remembers’ the temperature of the water that covered it earlier for a certain time” (Ostrovsky et al. 1996).

pelagic zone, while their largest amplitudes are at the bound€ est and Lorke 2003), a depth difference of 10 m for ary (Wu the 238C plane is expected for sediments located on opposite sides of the lake. The newly proposed “indirect warming” mechanism of the hypolimnion suggests that during seiche activity the thermocline surface tilts with respect to the horizontal (Fig 3a–c). For part of the seiche’s duration, the warm water of the epilimnion comes into contact with the underlying lake bottom sediments, which extend over the area affected by the tilting of the thermocline. During this time period heat is transferred downward into the lake sediments, where it is temporarily stored. During the rest of the seiche’s duration, the warm water above the sediments is replaced by hypolimnetic cold water, and the previously heated sediment bed emits most of the heat upward, into the colder hypolimnetic water (Fig. 3d,e), in accordance with the temperature gradient. This type of indirect warming mechanism was 3

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Fig. 3. (a) The bathymetric map of Lake Kinneret, including seven monitoring stations: TA, TB, TF, TG, TH, TK, TC. The schematic shades illustrate the location of the thermocline plane and the heat exchange area at certain points in time (see text). (b and c) Internal heat waves (seiches) measured in opposite sides of the lake (stations TF and TC) during June 22–24. (d and e) A schematic side view of the thermocline plane that tilts with respect to the horizontal. The attributed heat exchange areas and the direction of the heat transfer in the sediments are shown.

stratification period. The second is to test how significant the indirect heat transfer can be, compared to other possible heat fluxes, such as direct radiation that heats the hypolimnion and diffusion from the epilimnion to the hypolimnion across the thermocline.

Before the proposed “indirect warming” mechanism is elaborated the term “boundary mixing” should be briefly discussed. Marti and Imberger (2008) wrote that the boundary mixing layer (bbl) may be defined as the turbulent layer adjacent to the lake bed in which many interrelated physical processes and chemical and biological transformations actively take place. They indicated that suspended particles and nutrients are transported directly from the bbl into the metalimnion. This term is too general to be considered as a mechanism that transfers heat from the epilimnion into the hypolimnion. From the following discussion, it will be apparent that the proposed mechanism may be part of the general concept of “boundary mixing.” This study has two objectives. The first is to formulate and quantify the physical exchange of heat between the epilimnion and hypolimnion via the sediments at the bottom of the lake, and to provide empirical evidence that the proposed mechanism of hypolimnion warming actually takes place in a lake that has well documented seiches during the

Methodology Study area Lake Kinneret is a warm monomictic freshwater lake located in the northern part of the Afro-Syrian Rift valley. The lake surface area is 168 km2 (22 km long by 12 km wide) and its total volume is 4 3 109 m3 (Fig. 3a). Average and maximum depths are 24 m and 42 m, respectively. The Jordan River is the major inflow, while water pumped into the National Water Carrier (NWC) constitutes the main outflow. Since water input and exploitation varied throughout the years and within each season, during the past 45 yr the lake level fluctuated between 208.9 m and 214.87 m below 4

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T5T1 1DT;

mean sea level. According to a detailed analysis (Rimmer et al. 2011), the lake is thermally stratified from the end of March (the 81st day of year 6 21 days) until December/January, with a typical 286 6 26 days stratification period. During the summer months (May–August) the lake’s surface is strongly forced by a daily westerly Mediterranean sea breeze that occurs between 10:00 h and 20:00 h (e.g., Serruya 1975; Antenucci et al. 2000; Shilo et al. 2007; Rimmer et al. 2009; Imberger and Marti 2014). The thermocline depth is initially (in April) 9–13 m; it nearly stabilizes during the summer at a depth of 15–20 m, and begins to decline gradually in October until full mixis occurs (late December–mid January; Rimmer et al. 2005).

@T 50 @z

a5

(3)

(4)

Subject to the boundary condition: T1 ðt; 0Þ5T0 1TA sin½xðt1kÞ

(5)

This is the daily temperature variation given by a cyclic seasonal analysis, where T0 is the average seasonal temperature at the SWI (8C); TA is the seasonal temperature amplitude (8C); x52p=365:25 (d21) is the annual angular frequency; and k is the phase shift (d). Second, the equation for the short term (half day or diel) cyclic process is: @DTðt; zÞ @DT 2 ðt; zÞ 5a @t @z2

(6)

Subject to the boundary condition: DTðt; 0Þ5Tðt; 0Þ2T1 ðt; 0Þ

(7)

Here, Tðt; 0Þ is the measured temperature at the SWI, and T1 ðt; 0Þ is the optimal evaluation of Eq. 5, after taking into account the measured temperature at z 5 0, the optimal value of a that is suitable also for the solution of Eq. 6, and the seasonal propagation of the temperature into the sediments. The separation of the process into two components enables the estimation of the contribution of each of them to the process of heat transport across the SWI. Superposition of the solution for Eq. 4, subject to the boundary condition in Eq. 5, and of the solution for Eq. 6, subject to the boundary condition in Eq. 7, results in complete Tðt; zÞ variations. The vertical heat flux in the sediments, qT (J m22 d21), is the term which determines the exchange of heat transfer between the sediments and the water body. It is associated with Eq. 2 as follows:

(1)

k qcp

as z ! 1

@T1 ðt; zÞ @T1 2 ðt; zÞ 5a @t @z2

The cyclic temperature changes within the vertical profile of the sediments can be described by the classic heat transport equation: @Tðt; zÞ @T 2 ðt; zÞ ; 5a @t @z2

z50;

Here a is the diffusion coefficient (m2 d21), a product of the thermal conductivity (k; J m21 d21), the specific heat (cp; J kg21 K21), and the bulk density (q; kg m23) of the sediments’ material. For practical purpose and simplicity, it is suggested that the sediment medium is uniform so that k, cp, q, and a are constants. Theoretically, using the property of linearity for the heat transport equation, a superposition solution can be applied by dividing the problem into these two components. First, the seasonal temperature changes of the sediments can be evaluated by solving the equation for the annual cyclic temperature:

Theory Heat transport governing equations In this part of the analysis the sediments’ temperature changes throughout time were evaluated to provide empirical evidence that the proposed mechanism actually exists. Since the typical time scale of the periodic temperature changes that occur within the sediments due to seiches is part of a day, and the typical time scale of the seasonal changes is 365 d (a year), the unit of days (d) was selected to represent the time scale for the analysis. It is assumed that the sediments’ temperature, Tðt; zÞ in 8C, changes as a function of time t (d) and depth z (m). The z is a vertical coordinate directed downward where z 5 0 at the sediments-water interface (SWI). These changes are forced by the heat of the water above the sediment layer. Two simultaneous temperature change processes were hypothesized for the top 0.3 m of the sediment layer during the stratification. The first process is T1 ðt; zÞ, a continuous, long term (annual) sediment temperature change. The T1 ðt; zÞ process follows the seasonal trend of the epilimnion’s temperature. During the stratification period, this process is driven by the exposure of sediment areas to the slowly changing surrounding temperature, and differs in time for sediments in various depths of the lake floor, due to the slow deepening of the thermocline. The second process,DTðt; zÞ, the 12–24 h cyclic temperature change, is directly associated with the nature of the internal waves (Kelvin and Poincare waves; Antenucci et al 2000; Hodges et al. 2000). The DTðt; zÞ procedure is relative to the long term trendT1 ðt; zÞ, so that it is defined by: DTðt; zÞ5Tðt; zÞ2T1 ðt; zÞ

at

qT ðt; 0Þ52kð@T=@zÞjz50

(2)

(8)

Here, the flux at the SWI at any timestep is a product of the calculated temperature gradient at the sediments’ surface

Subject to the measured boundary conditions: 5

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the hypolimnion’s volume, and the three heat fluxes are described in this section. Following Rimmer et al. (2005), the change in the hypolimnion’s volume VH due to the deepening of the thermocline is:

and the thermal conductivity k. The heat flux which contributes to the hypolimnion’s warming occurs only during the time periods when the heat flux’s direction is from the sediments to the water, that is, qT < 0 (Fig. 3d,e).

dVH 52Qm dt

Hypolimnion warming equations This part of the analysis tests how significant the mechanism of sediments is as a heat exchange area (the “indirect warming”), compared to the actual measured warming of the hypolimnion during the stratification. A simple model of hypolimnion warming is applied on a monthly time interval during the stratification period (March–December), when the epilimnion and the hypolimnion are differentiated by a well-defined thermocline. Before this test is explained, clarification is needed regarding the reasons the monthly time interval was selected, and reservations must be stated regarding the limitations of accuracy in the following process. It is evident from Fig. 1 that the average monthly temperature change of the hypolimnion is in the order of 0.18C, with standard deviations larger than this change. Testing the daily temperature change (and, thus, remaining in the time scale of the heat transport equations in the sediments from the previous section) necessitates that the measured temperature in the hypolimnion will be accurate to the 0.0018C order of magnitude. This accuracy is obviously not feasible when averaging the measured temperature of the entire volume of the hypolimnion. Furthermore, unlike the previous section, which is a physical analysis of measurements at a point, the following section discusses the subject of entire-lake heat balance. It relies on rough estimations for upscaling the results of the previous section. The required results are, therefore, used only to test the possible magnitude of the indirect warming mechanism at the whole-lake scale. It is assumed that the monthly hypolymnion warming is a result of three different potential heat flux components: The first flux is qR , the radiation heat that penetrates directly into the hypolimnion. Typically in Lake Kinneret (see Introduction) the contribution of this component to hypolimnion warming is considered negligible. However, during the beginning of the stratification period in April, when the hypolimnion is relatively shallow (10 m below the lake surface), the effect of seiches on the elevation of the upper hypolimnion can be taken into account (Rimmer et al. 2008), and it is possible that direct radiation is still available for a short time. The second flux is qD , the heat diffusion that occurs throughout the entire area of the thermocline (Cowgill 1967; Jassby and Powel 1975). The third component is the new proposed indirect warming flux qT , the negative part of the heat flux (Eq. 8), which originates from direct contact between the cold hypolimnion and the relatively hot sediments. The monthly heat storage change equation, the temperature of

(9)

whereas Qm 5vth 3AHth is the vertical mixing component, which is always directed upward - from the hypolimnion to the epilimnion-since the thermocline is continuously deepening. It is expressed as the volume of water (m3) that is affected by the rate of the thermocline’s deepening (vth in m 3 d21), multiplied by the lake area (AHth ; m2) at the depth of the thermocline, at a given point in time t. Assuming that potential and kinetic energy changes are negligible compared to changes in thermal energy (Imboden € est 1995), the hypolimnetic heat storage change in and Wu time ðdGH =dtÞ can be coupled to the hypolimnetic water mass change (Eq. 9), and taking into account the three fluxes above (radiation, diffusion, indirect): dGH 52QH 1ðqR AR Þ1ðqT Ased Þ1ðqD AHth Þ dt GH ðtÞ5VH TH cPw qw QH ðtÞ5Qm TH cPw qw qD ðtÞ5Dm

(10)

½TE ðtÞ2TH ðtÞ Wm ðtÞ

Here GH ðtÞ and QH ðtÞ represent the hypolimnetic heat storage (in J) and the vertical heat flux from the hypolimnion to the epilimnion (in J d21), respectively. It is assumed that the main heat transport between the two layers is controlled by the vertical mixing component Qm ðtÞ. However, despite the scale of this component (300 million m3 month21) (Rimmer et al. 2005), it does not contribute heat to the hypolimnion due to its upward direction, leaving its temperature unchanged. Other variables in Eq. 10 are the residual radiation heat qR , multiplied by an unknown area AR ; the indirect warming flux qT , multiplied by an unknown area Ased ; and the diffusion flux qD , from the epilimnion’s temperatures TE ðtÞ to the hypolimnion’s temperatures TH ðtÞ, through the metalimnion of Wm width. The parameters in Eq. 10 are the heat capacity cPw , the density of water qw , and the diffusion coefficient, Dm . Combining Eqs. 9 and 10 results in: dTH cPw qw VH ðtÞ (11) 5ðqR AR Þ1ðqT Ased Þ1ðqD AHth Þ dt It means that the temperature change of the hypolimnion is determined by the contributed heat from either 1. The residual radiation; 2. The sediments; or 3. The epilimnion. A discrete numerical procedure to calculate the additional heat 6

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Fig. 4. (a) The volume V in m3, and the area A in m2, as a function of the lake’s absolute level H. Example values of thermocline depth and the seiches’ amplitude (Hth 6 S) are indicated, with their conversion to the volume of the hypolimnion V(Hth), and the potential heat exchange area AF. (b) The Bathymetry map of Lake Kinneret, with the example values from (a) and AF marked with a shaded area.

entiation between the three fluxes ½qD AHth 1qT Ased 1qR AR : diffusion from the epilimnion, heat exchange from the sediments, and residual radiation, respectively. First, the scale of the diffused heat flux from the epilmnion qD AHth can be approximated based on the definition of qD (Eq. 10), with the temperatures TE ðtÞ and TH ðtÞ and the metalimnion’s width Wm, estimated according to the stratification analysis (Cook and Rimmer 2010; Rimmer et al. 2011; Supporting Information A). The diffusion area AHth is the surface area of the thermocline at each time-step, and the diffusion coefficient Dm is subject to physically based estimations (see the Results section). Second, to estimate whether the proposed mechanism of heat exchange through the SWI is a significant mechanism, it was suggested to quantify it based on the following assumptions: 1. The internal wave is of a S m amplitude centered on the absolute level of the thermocline Hth (m), and the entire sediments-water heat exchange interface AF is enclosed between the upper ðHth 1SÞ and lower ðHth 2SÞ internal wave levels. Based on the measured scale of the internal waves, S was considered on a scale of several meters, and taking into account the lake bathymetry, it was suggested that AF falls into logical and physically accepted values of 30–60 km2 (see Fig. 4); 2. Due to the 12 h or 24 h cyclic movement of the thermocline plane, the actual heat contributing area Ased t is only a portion b of AF , where sediment water is warmer than the hypolimnetic water; 3. Point measurements and evaluation of qT (Eq. 8) are typical to the entire contributing area Ased (m2). These three assumptions are summarized with the following equations:

flux to the hypolimnion on a time step Dt is therefore: ½qD AHth 1qT Ased 1qR AR Dt5cPw qw VH t11 ½TH t11 2TH t

(12)

The left-hand side of Eq. 12 includes the three daily heat fluxes into the hypolimnion between time t and time t 1 1, and Dt is the number of days in a month, so that the contributing areas and fluxes present equivalent values between these two points in time, and are, therefore, marked with qA . While this expression includes several unknowns, the right-hand side of Eq. 12 can be calculated based on a predetermined (monthly) time interval. The hypolimnion volume at any time step VH t (m3) can be calculated using: VH t 5VðHth Þjt Hth 5ðHL 2Zth Þjt

;

(13)

whereas Hth is the absolute thermocline level above sea level (a.s.l.), calculated based on the measured lake water level HL (a.s.l.) and the thermocline depth (Zth (m), Cook and Rimmer 2010, Rimmer et al. 2011; Supporting Information A). The volume VðHÞ in m3 and the area AðHÞ in m2 are the hypsographic curves of Lake Kinneret, evaluated as a function of the absolute elevation H in m (Fig. 4a) (Mekorot 2003; Supporting Information B). If VH t can be determined at every monthly time step then the sum of the three heat fluxes that enter the hypolimnion can be calculated (right-hand side of Eq. 12), and the unknowns at the left-hand side can be approximated, as explained in the following section. Evaluating the three heat fluxes that enter the hypolimnion With the right-hand side of Eq. 12 calculated on a monthly basis, the remaining challenge is to find the differ-

AF 5AðHth 1SÞ2AðHth 2SÞ 7

(14)

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Ased 5bAF X ðqT Ased Þ5 qT DtAsed

(15)

using a STD-12 Plus (Applied Microsystems); and currently, since 2013, using a STD minos X (AML Oceanographic). Measurements were conducted every 1 cm with a standard deviation of 6 0.0058C, and averaged for every 1 m. At least 2035 weekly measured temperature profiles from the Yigal Allon Kinneret Limnological Laboratory (KLL) database were analyzed. For each profile, the location of the thermocline was calculated using a simple empirical temperature-depth function (Cook and Rimmer 2010; Rimmer et al. 2011; Supporting Information A), which systematically defines the depth of the thermocline, the width of the metalimnion, and the mean temperatures in the epilimnion and hypolimnion. (4) The lake level, measured on a daily basis by the Israeli Hydrological Service.

(16)

month

The order of magnitude of the components in Eqs. 14-16 is known, while the value of b is determined from optimization, compared to the measured heat flux. This coefficient may assume very small monthly values (e.g., 0 < b < 0.01), and, thus, it indicates that the entire mechanism is negligible, or it may assume very large values (b > > 1), indicating that the entire proposed mechanism is probably wrong. However, it can assume values which are physically accepted (0.1 < b < 1), thus indicating that the proposed mechanism is significant. Finally, the residual heat flux is attributed to the contribution of direct radiation ðqR AR Þ. The quantity of this flux is fairly unknown because the contributing area AR (m2) cannot be determined. The only logic that we can follow here is that the contribution of this flux is relevant only for the first 1–2 months of the stratification period, since after these months the hypolimnion is too deep to receive direct radiation. A summary of the symbols which were used for variables and constants in Eqs. 1–16 is presented in Table 1, including the units and the equation index. Constants which were found suitable for Lake Kinneret during this study and previous studies are given in the “Value” column.

Results and discussion Heat flux from the sediments Sediments and water temperature The variations in the measurements of the sediment temperature during April–August 2014 are presented in Fig. 5b (dataset #1; only measurements at z 5 0 and z 5 0.3 m). Detailed measurements, taken over 3 d in all depths below the SWI, are presented in Fig. 5c. The average seasonal temperature trend for April–August 2014 is given based on the analytical solution of Eq. 4, subject to the boundary condition in Eq. 5 (Supporting Information D):

Monitoring data The proposed mechanism was tested using four types of measured datasets:

T1 ðt; zÞ5T0 1TA exp½2zðx=2aÞ1=2 sin½xðt1kÞ2zðx=2aÞ1=2 T0 522:65o C;

(1) Specific measurements of temperatures above 0.40, 0.30, 0.20, 0.10, 0.05, and 0.03 m, at 0 m, and 0.03, 0.05, 0.08, 0.10, 0.20, 0.30 m below the SWI. Measurements were conducted in station TH, which is located 850 m off the western-northern lake shore (Fig. 3a), where the depth of the water profile is 13 m. The measurements were taken between April and October of 2014 using temperature sensors (‘Hobo pendant ua-002-32’ with an accuracy of 6 0.478C and a resolution of 0.108C) attached to a central pole (Fig. 5a), with the measurements taken at 10 min intervals. Additionally, measurements of the temperature 1.5 m below the SWI, were taken only between August 21 and October 4, to verify the seasonal temperature cycle Eq. 5. (2) Continuous water temperature profiles (18 thermistors at a water depth of 0.20, 0.50, 0.75, 1.00, 1.25, 1.50, and 2.00, and 11 additional thermistors, 1 m apart, at a depth of 3 m to 13 m) measured at the TH monitoring station with a thermistor chain (PME Temperature Sensor with measurement accuracy of 6 0.0108C and resolution of 0.00058C) at 10 min intervals (Supporting Information C). (3) Weekly temperature profiles measured at the TA monitoring station, from 1969 to 1986, using an underwater thermometer (Whitney-Montedoro); from 1987 to 2013,

TA 56:23o C;

a50:036 m2 d21 ;

x52p=365:25 d21 ;

(17)

k52162:2 d

The seasonal temperature changes from 17.08C to 28.08C are sinusoidal (t 5 1 at the 1st of January), as expected from the boundary conditions (Eq. 5). The results are in line with the theoretical aspects of the warmup mechanism described in Eqs. 1-7. If the long term solution of the sedimental warmup process would have been applied on its own (Eq. 17, see explanation about the value of a in the next section), the average temperature difference between the upper and the lower thermistors (z 5 0 m and z 5 0.3, m respectively) would have been 2 0.68C at the beginning of April (when the sediments are still warmer than the water), 0.08C at the beginning of May, and 1.28C at mid-August (the sediments are colder than the water). Moreover, according to the measurements taken in August, the temperature difference between the 0 m and the 21.5 m thermistors (which is not affected by the daily cycle) is 58C. Thus, according to the seasonal analysis alone, starting at the beginning of May, the direction of the heat flux would have been into the sediments, reaching 300,000 J m2 d21 in August. 8

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Table 1. List of the symbols used for variables and constants in Eqs. 1-16. General constants and those which were found suitable for Lake Kinneret are given in the “Value” column (See also Supporting Information C) Symbol

Variable/constant description

Value

Units 2

Equations

A(-) AR

Surface area at a certain depth Contributing area to heating with radiation.

m m2

14 10, 11, 12

Ased

Contributing area to warming from sediments.

m2

10, 11, 12

AHth Dm

Contributing area to warming from diffusion. Diffusion coefficient of the metalimnion

m2 21 21 21 Jm d K

10, 11, 12 10

51,644*

HL

The absolute level of the lake’s surface

m

13

GH Hth

Hypolimnetic heat storage The absolute level of the thermocline

joule m

10 14

L

The contour length of the absolute level H

QH

Vertical heat flux: hypo to epi.

Qm

Vertical water volume transfer: hypo to epi.

S TH

km

Supporting

J month21

Information 10

m3 month21

9

The internal wave amplitude Temp. of the hypolimnion

m 8C

14 10, 11, 12

TE

Temp. of the epilimnion

8C

10

T T1

Temp. of the sediments Temp. of the sediments due to seasonal changes

8C 8C

1, 2, 3 1, 3, 4, 5

8C

1

8C 8C

5 5

DT

Temp of the sediments from daily seiches

T0 TA

Temp. parameter in the cyclic seasonal profile Temp. parameter in the cyclic seasonal profile

VH

The hypolimnion volume

m3

9, 10, 11, 12, 13

V(-) Zth

The volume of the lake from a certain depth downward The thermocline’s depth

m3 m

13 13

Z

The vertical coordinate of lake depth.

m

Supporting

22.65† 6.23†

cp

The specific heat of the sediments

1586†,‡

J kg21 K21

Information 2, 4, 6, 8

cpw

The specific heat capacity of water

4186.8

J kg21 K21

10, 11, 12

J m21 d21 J m22 d21

2, text 10, 11, 12

J m22 d21

8, 10, 11, 12

J m22 d21 various units

10, 11, 12

†,‡

k qR

The thermal conductivity of the sediments The Heat flux to the hypolimnion from radiation

82,800

qT

Vertical heat flux in the sediments

qD t

Diffused heat flux from the epilimnion Time (general)

vth

The thermocline deepening velocity

m d21

1

z a

The vertical coordinate of sedimental depth Thermal diffusivity of the sediments

0.036†

m m2 d21

1-8 2, 4, 6

aW

Thermal diffusivity of water

0.0124

m2 d21

text

b k

The fraction of AF which contributes heat flux The phase shift of the seasonal sediment temp.

2162.17†

(-) day

15 5

q

Sedimental density

1450‡

kg m3

2, 4, 6, 8

qw x

Water density The angular frequency of the seasonal sediment temp.

1000 2p=365:25

kg m3 rad day21

10, 11, 12 5

*Based on molecular diffusion of water: Blumm and Lindemann (2003/2007). This study ‡ Tibor et al. (2014) †

file (dataset #2), is presented in Fig. 6a. The temperature measurements, taken between 0.4 and 20.3 m with respect to the SWI (dataset #1), are shown in Fig. 6b. The 0–0.3 measurements of the sediments (Fig. 6b) are identical to the measurements described in Fig. 5b,c. Both figures demonstrate how

However, after taking into account the seiching phenomena, the measured data are different. A contour map of the temperature measurements at station TH, taken between the 20 June 2014 and the 23 June 2014 with a time interval of 10 min, using an 18 thermistors chain in the 13 m water pro9

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Fig. 6. (a) A contour map of the time and depth of the temperature, taken from 18 thermistors distributed along the 13 m water profile at station TB, during 20 June 2014–23 June 2014, with a time interval of 10 min. (b) Simultaneous temperature measurements taken from 13 thermistors: The water section—from 0.4 m to 0 m (sediment–water interface [SWI]); The sediments section—from 0 m to 0.3 m below the SWI. Fig. 5. (a) The water—sediment interface thermistor device. (b) Variations of the temperature measurements in two thermistors, from the SWI downward (0 m and 0.30 m below the interface) during April 2014– August 2014, and of the annual sinusoidal change T1. (c) The details of the temperature measurements in the sediments over 3 d.

of the change in lake sediment heat storage, which can be obtained by integrating the temperature profiles of the lake sediment (Fang and Stefan 1998; Kirillin et al. 2009). qT 5qcp

the 24 h cycle seiches bring cold hypolimnetic water to the SWI (the most obvious seiche effect is shown in Fig. 6a, between the afternoon hours and midnight of 21 June 2014), which causes a negative temperature gradient in the sediments and induces heat flux from the sediments to the water. An obvious daily (or sometimes half daily, see Supporting Information C) heat diffusion mechanism in the sediments (Eq. 6) can be observed in both Figs. 5, 6. It is evident from two typical properties of the diffusion mechanism that: (i) the time of heating and cooling waves becomes delayed as the measured temperature is deeper in the sediments; (ii) the amplitude of the wave is reduced with depth. Both properties appear as slanted contours from the SWI down and right, at the sediment part of Fig. 6b.

X D z50:3 Tz Dt z50

(18)

While the volumetric heat capacity of water, qw cpw at 258C, is 4.1813 3 106 J m23 K21 and the thermal diffusivity of water, aw at 258C, is 0.143 3 1026 m2 s21 (0.0124 m2 d21; Blumm and Lindemann 2003/2007), according to Fang and Stefan (1998) the values of the thermal diffusivity and heat capacity of lake sediments depend on the sediment’s composition and range from sand to very organic materials. They reported thermal diffusivity of lake sediments, a, ranging from 0.01 to 0.11 m2 d21, and sedimental heat capacity, qcp , ranging from 1.4 3 106 to 3.8 3 106 J m23 K2l. For their own models they used a 5 0.035 m2 3 d21 and 6 23 2l qcp 5 2.3 3 10 J m K . To identify an optimal a value and its standard deviation for the present study, the full solution of Eq. 2, T(t, z), for the 0 to 0.3 depths (dataset #1) must be calculated. It includes the analytical solution of T1(t, z) (Eq. 17 and Supporting Information D) and the numerical solution of DT(t, z) (Supporting Information E). The modeled T(t, z) was compared to the measured temperatures, at six time intervals of 6 d each, between April and August of 2014. At each time interval the minimal rootmean-squared error (RMSE) between the measured and the modeled temperatures was calculated for a range of a values (0.01 to 0.09 m2 d21), and the selected a was the one with the lowest RMSE. The optimal thermal diffusivity and the standard deviation that was reached in the present study, a 5 0.036 6 0.015 m2 d21, is the average of the 6 a values and their standard deviation. These values were reached using RMSE ffi 0.258C (Supporting Information F). A

Constants, parameters, and heat flux at the sediments In this section, we evaluate the constants and parameters to quantify the heat transport in the sediments. A heat flux distribution throughout a sediment-water interface, qT , is difficult to measure directly since the heat fluxes change with time and depth. However, it can be estimated according to two methods, using sediment temperature profiles. The first method is to estimate the optimal thermal diffusivity, a, by searching the minimal root-mean-squared error (RMSE) between the measured and the calculated temperatures. This procedure, often referred to as the solution of the “invers problem” of Eq. 2, was described in detail by Adams et al. (1976) as they estimated the thermal conductivity of lake sediments based on field observations in Lake Waiau, Hawaii. The second method is to calculate the rate 10

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Fig. 7. (a) A contour map of the solution applied for Eqs. 1-3 within the 0.3 m sediments, with the optimal diffusion parameter a 5 0.036 m2 d21. (b) The contour map of the vertical heat flux qT ðt; zÞ5kð@DT =@zÞ (W m22; Eq. 8 in minus sign) for the 0.3 m sediments. The parameters were density q 5 1450 kg m23 and the heat capacity of the saturated lake sediments cp 5 2.30 E 1 06 J kg21 K21 and k 5 a/(q 3 cp) J m21 d21. (c) The vertical heat flux at the sediment surface (W m22; Eq. 8 in minus sign) and the change in the integrated sediment temperature profiles (Eq. 18).

@[email protected] K m21. Finally, the calculated heat flux at the surface during these 10 min (600 s) is qT 582,800 3 33.03/ 86400 5 31.65 W m22. This is the left-most value presented in Fig. 7c. It is assumed that the positive heat flux in Fig. 7c is the instantaneous thermal energy added to the hypolimnion. The daily cumulative amount of this flux was calculated for the entire period, from 11 April 2014 to 14 August 2014. The summation was performed as follows: Let DðqT Þ (units of J) be the daily contribution of the sediments to the hypolimnion. Starting from DðqT Þj0 50 at the hour 00:00 of each day:

contour map of the full solution applied to Eqs. 1-7, for the 0–0.3 m sediments is presented in Fig. 7a. This is the modeled contour map of the measured temperature in the sediment section of Fig. 6b. Based on the optimal diffusion parameter, the vertical heat flux qT ðt; zÞ5kð@DT=@zÞ could be evaluated within the 0.3 m sediments (Fig. 7b), and specifically qT ðt; 0Þ could be evaluated at the SWI, as presented in Fig. 7c. Note that in both Fig. 7b,c qT is directed opposite to Eq. 8, to give positive value to the thermal flux from the sediment to the hypolimnion. Additionally, the change of the integrated sediment temperature profiles (Eq. 18) is presented in Fig. 7c. Both calculated time series in Fig. 7c are independent: the first stems from the calibrated solution, while the second is calculated directly from the measured data, and, therefore, has larger fluctuations. Nevertheless, the values of qT ðt; 0Þ are basically the same (Kirillin et al. 2009). Here is an example of a detailed calculation of the heat flux at the SWI, using estimated sediments parameters from Lake Kinneret (Supporting Information G): first, assuming the typical porosity of 70% in the upper 1.0 m of sediments, with the specific density of the sedimental material of 2500 kg m23 (Tibor et al. 2014), the bulk density of the saturated sediments is q 5 1450 kg m23 and cp 5 1586 J kg21 K21, so that k5 aqcp 582; 800 J m21 d21 K21. Second, the calculated temperature difference between 0.03 m and 0 in the sediments, on 20 June 2014, during the 10 min between 00:00 h and 00:10 h, was 0.9918C (compared to 1.047 6 0.078C based on direct measurements, which are practically similar), and, therefore,

if qT jt11 > 0 ðt11 is the next 10 minÞ; DðqT Þjt11 5DðqT Þjt 1qT jt11 3600 else D ðqT Þjt11 5DðqT Þjt Here qT is in W m22, and 600 is the number of seconds in each 10 min time interval. The daily cumulative thermal flux resulted in a trend line shaped like a hyperbola (Fig. 8) that crosses the zero line at two time spots: (1) At the onset of stratification at the beginning of April, when the temperature difference between the hypolimnion and epilimnion is small; and (2) At the middle of August, when the thermocline is at a depth of 17 m, and the effect of the internal seiches ceases to influence the area of the measurement point, which is located at a depth of 13 m. At the peak of the hyperbola, mid-June, a daily average contribution of 726,000 J m22 d21 is reached (8.4 W m22, lower than the 33.44 W m22 in the example above, since 11

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sedimental heat flux in two characteristics: it is derived from the temperature difference between the lake and the air above it (similar to the flux between the sediments and the water above it); and it is also a flux that is positive during part of the day (the air warmer than the water at noon), and negative during the other part (the water warmer than the air at night).

the result of the integration in time includes values much smaller than the values used in this example). For comparison, the aforementioned calculated sediment heat flux to the hypolimnion in June 2014 is more than double the net sensible heat flux (334,000 J m22 d21) added to the lake surface during June (Fig. 4 in Rimmer et al. 2009). The sensible heat flux is a typical variable that represents the heat transfer between the lake and its environment. It is similar to the

Change in the heat storage of the hypolimnion The second objective is to estimate the order of magnitude of the indirect warming and how significant it is to the heating of the hypolimnion. Prior to this estimation, it is important to note that the entire analysis was carried out in a single location, and, therefore, the following results are only a first approximation of the scale of this flux compared to other fluxes. Nevertheless, it appears to be a significant component of thermal energy, and will be analyzed further based on a comparison to the thermal energy required to heat the entire hypolimnion by 0.68C. The 2035 weekly measured temperature profiles taken at the TA monitoring station (dataset #3) were grouped into monthly averages and standard deviations (180 samples each month). Over the years, the lake’s level fluctuated significantly, so that the analysis of the average hypolimnion volume and surface area may differ significantly between one year and another. Nevertheless, it is assumed that this rough estimation is sufficient to meet the requirement of

Fig. 8. The measured heat flux (J m22 d21) contributed from the sediments to the hypolimnion. The typical trend line has the shape of a hyperbola that crosses the zero line at two time spots: 1. At the onset of the stratification period in the beginning of April; 2. Mid-August, when the thermocline is at a depth of 17 m, and the effect of the internal seiches ceases to influence the measurement point at a depth of 13 m.

Fig. 9. A Lake-wide analysis of variables associated with Eq. 12. Hth ðtÞ: The absolute thermocline level (m a.s.l.) and its associated standard deviations;

AHth ðtÞ: the thermocline surface (km2); VHth ðtÞ: The hypolimnion volume (Mm3); LHth ðtÞ: Contour length (the perimeter of the thermocline plane, in km); GHth ðtÞ: hypolimnion heat storage (MJ); DGHth ðtÞ=Dt: The heat storage change (MJ month21) in the hypolimnion (circles) compared to the cumulative monthly heat inflow through diffusion (light gray), the indirect warming mechanisms (sediments, gray), and the direct radiation (dark gray).

12

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Table 2. Lake-wide heat fluxes into the hypolimnion, based on monthly averages month

Epi. Temp. 8C

Hypo. Temp. 8C

Meta. width m

Gradient DT/DX 21

8C m

Diffusion Flux 22

Jm

d

21

Therm. area 2

Diffusion Flux 21

km

J month

13

Sediment flux

Radiation flux

Summary

21

21

21

J month

14

J month

15

J month

15

RHS Eq. 12 J month21

4 5

19.55 23.15

15.03 15.25

11.51 11.15

0.39 0.71

20280.0 36628.2

139.96 138.45

8.51 3 10 1.52 3 1014

1.59 3 10 7.46 3 1014

1.5 3 10 2.0 3 1014

1.74 3 10 1.10 3 1015

1.75 3 1015 1.18 3 1015

6

26.21

15.26

10.05

1.09

56292.7

130.66

2.21 3 1014

8.89 3 1014

0

1.11 3 1015

6.20 3 1014

14

14

14

7 8

27.84 28.74

15.34 15.45

8.03 6.86

1.56 1.94

80406.6 100161.1

121.08 114.87

2.92 3 10 3.45 3 1014

6.53 3 10 1.03 3 1014

0 0

9.45 3 10 4.48 3 1014

8.85 3 1014 2.56 3 1014

9

28.19

15.46

6.98

1.82

94193.4

109.56

3.10 3 1014

0

0

3.10 3 1014

1.58 3 1014

14

0 0

14

2.89 3 10 2.34 3 1014

2.08 3 1014 -5.1 3 1013

8.92E113

3.37 3 1014

15

5.35 3 1015

10 11

26.32 22.63

15.62 15.50

5.79 4.14

1.85 1.72

95478.3 88854.8

100.91 87.91

2.89 3 10 2.34 3 1014

0 0

12

18.59

15.58

3.29

0.92

47290.0

62.90

8.92 3 1013

0

15

0 15

2.55 3 10 40.70%

15

Sum %

2.02 3 10 32.18%

1.70 3 10 27.12%

6.27 3 10 100%

delineating the order of magnitude of the sedimental heat transport. The results of the analysis are shown in Fig. 9. The absolute thermocline level for each month, Hth ðtÞ, and its associated standard deviations are the independent value. The other monthly values of the thermocline surface (km2), the hypolimnion volume (Mm3), the contour length (the perimeter of the thermocline plane in km, which enables an assessment of the size of the sediments’ contributing area), and the hypolimnion heat storage (MJ) are calculated according to Eqs. 9-16. Change in the heat storage (MJ month21) of the hypolimnion is calculated first from the right-hand side of Eq. 12. For example, the average temperature change in July is ½TH 7 2TH 6 ffi0.138oC (Fig. 1a), where 7 and 6 stand for July and June, respectively, and the average hypolimnion volume is VH 7 ffi1,533 106 m3. When multiplied by qw 5 1000 kg m23 and cpw 5 4186.8 J kg21 K21 the total heat flux into the hypolimnion in July is cPw qw VH 7 ½TH 7 2TH 6 5 8.85 3 1014 J (the “RHS Eq. 12” column in Table 2). This is the sum of the three heat fluxes on the left-hand side of Eq. 12. The average diffusion flux from the epilimnion during July is calculated as follows: The temperature difference between the epilimnion and hypolimnion is TE7 2TH7 527.84 2 15.34 5 12.58C and the average metalimnion thickness is Wm 5 8.03 m. Assuming Dm 5 51,644 J m21 d21 K21 is in the order of magnitude of molecular diffusion, the result is qD7 ffi 80; 406 J m22 d21. Taking into account 30 d in a month and the fact that the area of the thermocline plane in July is AHth7 5121.08 km2, the total average heat flux is qD AHth 52.92 3 1014 joule (the “Diffusion Flux” column in Table 2). This is only 33% of the total amount of heat that is added to the hypolimnion during an average July. Since the hypolimnion depth in July is too deep to be heated by direct radiation, the remaining option is to assume that 67% of the heat originates from the sediments. The monthly sum of the measured heat flux during July 2014 (Fig. 8) is 1.634 3 1013 J km2. To complete the missing

5.93 3 1014 joule in July, a total of Ased7 5 (5.93 3 1014/ 1.634 3 1013) 5 36.29 km2 is needed. This number is indeed in agreement with the expected contributing area, since the perimeter of the thermocline plan in July is 44.4 km (Supporting Information H), and the calculated AF for Hth 5228 m b.s.l. and S 5 5 m is 47.4 km2 and, therefore, bffi0.76. These are estimations which suggest that the proposed mechanism has a significant contribution to the heating of the hypolimnion in July. The monthly Ased t time-series, which is the optimal sediment contributing area, was calculated as 43, 42, 41, 40, and 39 km2 in April, May, June, July, and August, respectively. Although these estimations are rough and are subject to large errors, they fall into a logical and physically accepted range of a sedimental contribution area. A summary of the proposed monthly heat contribution to the hypolimnion is given in Table 2. Summary and conclusions It is proposed that an indirect warming mechanism, which involves the observed internal seiches with cycle period of 12 h and 24 h, accounts for at least part of the heat transfer into the hypolimnion of Lake Kinneret. This mechanism is expected to occur at any thermally stratified lake which is significantly affected by internal seiching. The entire analysis performed in this study is based on detailed temperature measurements performed in a single vertical column where the depth of the lake’s floor is 13 m. The initial hypothesis (Table 2) is that 41% of the heat is supplied by the heat flux that comes from the lake’s sediments, as a result of internal wave activity that occurs during the seasonal warmup of the hypolimnion. The remaining 59% is divided between a 32% contribution by way of diffusion from the epiliminion and a 27% contribution by way of direct radiation, mainly during April. A complete and accurate quantification of these processes must include 13

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T. Berman, and A. Nishri [eds.], Lake Kinneret—ecology and management. Springer, Heidelberg. € est. 1995. Mixing mechanisms in Imboden, D. M., and A. Wu lakes, p. 83–138. In A. Lerman, D. M. Imboden, and J. R. Gat [eds.], Physics and chemistry of lakes. Springer Verlag. Jassby, A., and T. Powell. 1975. Vertical patterns of eddy diffusion during stratification in Castle Lake, California. Limnol. Oceanogr. 20: 530–543. doi:10.4319/ lo.1975.20.4.0530 Kirillin, G., C. Engelhardt, and S. Golosov. 2009. Transient convection in upper lake sediments produced by internal seiching. Geophys. Res. Lett. 36: L18601. doi:10.1029/ 2009GL040064 Marti, C. L., and J. Imberger. 2008. Exchange between littoral and pelagic waters in a stratified lake due to wind induced motions: Lake Kinneret, Israel. Hydrobiologia 603: 25–51. doi:10.1007/s10750-007-9243-6 Mekorot water resources and water supply departments. 2003. Kinneret Lexicon. Mekorot report. Tel Aviv. (in Hebrew). Nishri, A., J. Imberger, W. Eckert, I. Ostrovosky, and J. Geifman. 2000. The physical regime and the respective biogeochemical processes in lower water mass of Lake Kinneret. Limnol. Oceanogr. 45: 972–981. doi:10.4319/ lo.2000.45.4.0972 Ostrovsky, I., Y. Yacobi, P. Walline, and I. Kalikhman. 1996. Seiche-induced mixing: Its impact on lake productivity. Limnol. Oceanogr. 41: 323–332. doi:10.4319/ lo.1996.41.2.0323 Ou, H.W., and J. R. Bennett. 1979. A theory of the mean flow driven by long internal waves in a rotating basin, with application to Lake Kinneret. J. Phys. Oceanogr. 9: 1112–1125. doi:10.1175/1520-0485(1979)009 < 1112: ATOTMF>2.0.CO;2 Rimmer A., Y. Aota, M. Kumagai, and W. Eckert. 2005. Chemical stratification in thermally stratified lakes: A chloride mass balance model. Limnol. Oceanogr. 50: 147– 157. doi:10.4319/lo.2005.50.1.0147 Rimmer A., W. Eckert. A. Nishri, and Y. Agnon. 2006. Evaluating hypolimnetic diffusion parameters in thermally stratified lakes. Limnol. Oceanogr. 51: 1906–1914. doi: 10.4319/lo.2006.51.4.1906 Rimmer A., G. Gal, T. Opher, Y. Lechinsky, and Y. Z. Yacobi. 2011. Mechanisms of long-term variations of the thermal structure in a warm lake. Limnol. Oceanogr. 56: 974–988. doi:10.4319/lo.2011.56.3.0974 Rimmer A., I. Ostrovsky, and Y. Z. Yacobi. 2008. Light availability for Chlorobium phaeobacteroides and its biomass production in a stratified lake. J. Plankt. Res. 30: 765–776. doi:10.1093/plankt/fbn037 Rimmer A., R. Samuels, and Y. Lechinsky. 2009. A comprehensive study across methods and time scales to estimate latent and heat fluxes: The case of Lake Kinneret, Israel. J. Hydrol. 379: 181–192. doi:10.1016/j.jhydrol.2009.10.007

measurements that cover more parts of the basin morphomety. Nevertheless, the existence of the mechanism was shown and proven by the measurements and the physical analysis. Better quantification may be achieved through a larger operation of measurements.

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Acknowledgments We acknowledge the technical and monitoring field staff, and the data base managers from the Kinneret Limnological Laboratory (Israel). The authors thank Prof. Yehuda Agnon from the Technion IIT for his illuminating comments and advice regarding the mathematical analysis of the measurements. We are also thankful to the two anonymous reviewers who helped to improve the manuscript significantly. Submitted 7 December 2014 Revised 25 February 2015 Accepted 20 April 2015 Associate editor: Dr.John Melack

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