Vertical mixing is an important physical process in a small, shallow, stratified lake during the summer and it affects aquatic bio-chemical processes both before and after mixing (Jones et al., 2008; Tsai et al., 2008, 2011). Vertical mixing is normally caused by wind-driven circulation (Kullenberg, 1976), cooling-related convection (Read et al., 2012), and inflow from heavy rainfall inputs (Laborde et al., 2010). In general, temperate lakes experience destratification because of cooling and the strong wind-induced mixing that occurs during winter in a seasonal cycle of mixing (Sundaram and Rehm, 1973). However, owing to severe rainstorms, stratified temperate lakes may also get mixed during summer (Porter et al., 2005). This study focused on mixing that is strongly affected by severe rainstorms in a small, thermally stratified, shallow lake. The mixing is similar to that shown in the simple sketch in Fig. 1a, which represents a typical temporal vertical profile of water temperatures during rainstorm-induced mixing. This transient thermal structure consists of a homogenous surface layer mixed and maintained by wind-generated turbulence (hereafter referred to as surface mixed layer, SML), a thermocline depth maintained by the net heat balance at the surface, and a bottom mixed layer maintained by bottom-shear turbulence (Fischer et al., 1979). In this vertical mixing process, cooling and wind-induced turbulence are the dominant driving factors (MacIntyre and Melack, 1982; MacIntyre, 1993; MacIntyre et al., 2002). The definition of SML is sketched in Fig. 1b.
In general, a small and shallow lake has a small volume for stored water; therefore, heavy rainfall events can bring an amount of inflow much larger than the lake volume (Kimura et al., 2012b). This situation suggests that heavy rainfall-induced inflow may play an important role in lake mixing (Fig. 1c). We hypothesised that in addition to cooling and wind-induced turbulence (Fig. 1b), an inflow effect, caused by changes in the discharge volume and temperature, is also a major factor influencing vertical mixing, although it is generally minor (Fig. 1c). Using heat content as an effective indicator (Wetzel and Likens, 2000), might be helpful for assessing the effect of inflow. Moreover, comparing the contributions of these different physical factors to vertical mixing requires a quantitative analysis using the same physical variables, for example, energy and force. Some studies use the surface energy budget to compare the effects of forcing on lake mixing (MacIntyre et al., 2002). However, to the best of our knowledge, no factor analysis has been performed to determine the relative effect of physical factors on vertical mixing.
The purpose of this study was to determine which physical process among the major driving forces related to meteorological (MET) forcing (i.e., wind, heating/cooling, inflow) is the primary cause of mixing related to heavy rainfall events in a small, shallow, stratified lake. We focused on both vertical mixing throughout the entire water column and the SML. The former describes complete mixing from the lake surface to the bottom, caused mainly by wind, heating/cooling, and inflow. The latter describes mixing within the surface layer directly affected by wind forcing and heating/cooling (Imberger, 1985; Ishikawa and Tanaka, 1993), which typically extends from the lake surface to the upper thermocline (MacIntyre et al., 2002). For entire vertical mixing, we analysed the relationships between the forces using a mixing indicator of thermal stability and an indicator that can measure the effect of inflow on vertical mixing. In the SML, a comparison between the wind-induced friction velocity and the heating/cooling-related convection velocity was conducted. In these analyses, we assumed that implementing the seasonal trend frame work could be performed for short-term events like rainstorm-induced mixing. In addition, we performed a quantitative statistical analysis using principal component analysis (PCA) to determine which forces played significant roles in lake mixing and to describe a correlation between physical and chemical factors.
Instrumented buoy and field site
Yuan-Yang Lake (YYL) is a shallow (4.5 m maximum depth and approximately 1.7 m mean depth) and small (3.6×104 m2) lake situated 1670 m above mean sea level in north-central Taiwan (24°35′ N, 121°24′ E). The lake is partly surrounded by wetlands (2.2×104 m2) (Fig. 2). It has only one outlet, one dominant inlet and minor inlets from the northwest to the northeast. The steep catchment and part of the headwater area of the Tahan River (3.7×106 m2) are dominated by cypress forest. The lake is subjected to three to seven typhoons each year in summer and autumn, which generally accounts for more than 40% of the annual precipitation. Wind speed over the lake is relatively weak (normally 0-1.0 m s–1). The dominant wind directions are from the east and southwest due to the V-shaped valley oriented east-west (http://conservation.forest.gov.tw/). Seasonal thermal stratification starts developing from late spring on, is stable with a >10°C gap during summer, and then weakens due to mixing processes. Additionally, several cycles of destratification and stratification occur every year owing to typhoons. During a severe rainstorm, significant runoff occurs that partly includes ground water (hereafter referred to as inflow in this study). The inflow contains less-turbid water because the lake is surrounded primarily by pristine dense forest and is partly connected to wetland.
Wireless auto-data collection systems (Porter et al., 2005) have been used since spring 2004 in YYL. Measurements from spring 2004 to autumn 2010 were obtained with an instrumented buoy placed in the western, deepest portion of the lake (Fig. 2). The YYL buoy was equipped with an anemometer (model 03001, R.M. Young, Traverse City, MI, USA) 2.0 m above the water surface and a thermistor chain (Templine, Apprise Technologies, Duluth, MN, USA) below the water surface with 10 sensors spaced at intervals of 0.25 to 0.5 m, to a depth of 3.5 m. Additionally, we placed a dissolved oxygen (DO) sonde (600-XLM, YSI Inc., Yellow Springs, OH, USA) at 0.25 m, which was fitted with an oxygen-temperature electrode (model 6562, YSI). The water temperature profile between 3.0 m and the lake bottom (normally 4.1 to 4.5 m) was assumed to be relatively constant as evidenced by point-based data that exhibited only 1.5°C difference between 3.2 and 4.1 m when the temperature at the surface was 23.1°C and 12.1°C at the bottom (July, 2009). The temperature difference between the surface and the bottom normally ranged from 9 to 13°C (during summer at the buoy) according to the oxygen-temperature electrode. An observatory station on the eastern lakeshore recorded water level (PS 9800(1), Instrumentation Northwest Inc., Kent, WA, USA). A land-based MET station approximately one km from YYL monitored air temperatures (HMP50-L, Vaisala, Vantaa, Finland), the relative humidity (same sensor as for the air temperature), air pressures 1.7 m above the ground level (090D, Met One Instruments, Inc., Grants Pass, OR, USA), shortwave radiation at 4.0 m above the ground level (Model PSP, Eppley Laboratory, Inc., Newport, RI, USA), and precipitation (rain gauge, Takeda Co., Osaka, Japan). These datasets were mostly unaffected by the ground or water surface where they were measured, except for humidity. Humidity in the surrounding environment was similar to that over the lake as it is typically high everywhere during typhoon events. At the buoy, water level and MET data were measured every 10 min. The field data from the instrumented buoys are accessible from the Global Lake Ecological Observatory Network website at http://www.gleon.org. The dataset data used in this study is an extension of a previous study (Kimura et al., 2014).
Field data analyses
Because of heavy-rainfall events caused by typhoons during summer to early autumn, vertical mixing often occurs in the thermally stratified YYL in Taiwan. We define this mixing as heavy-rainfall-induced mixing. This vertical mixing may be caused primarily by three MET-forcing factors: wind, heating/cooling and inflow. The quantitative aspects of these factors can be measured as wind shear stress (τ), net heat balance (Hnet) and increase of the water level (hGAP), which is defined by the difference, or gap, between the minimum and maximum water levels during mixing. hGAP was employed as an inflow-related variable. The variables τ and Hnet were computed using theoretical and empirical equations, given by
To measure the strength of stratification and to compare the different factors, a quantitative indicator (Schmidt stability St) was introduced,
Taking into account the energy balance in the water column determined by the net surface heat flux, wind-induced turbulence and inflow temperature; is crucial for understanding the internal response to the forcing factors: wind, heating/cooling, and inflow. We introduced net heat content (Qt) (J m–2) to compare the energy balance between the three factors, given by
Modified principal component analysis
In the entire water body, each factor can be partially categorised by their physical features, such as wind-induced turbulence and the heating/cooling process. We used the PCA, a linear transformation technique that ranks a set of variables, in order to produce components (data grouped by similar characteristics) in descending order from higher to lower score components (Hotelling, 1933). We followed the method described by Lin et al. (2005) (Supplementary Material II). We assumed that these components corresponded to some physical features in the lake. To indicate the relative strength of stratification, we introduced a ratio of the difference in St, defined as [St(pre) – St (mixing)]/St(pre) where ‘pre’ is a pre-mixing period, hereafter referred to as the St ratio. This St ratio is used as an indicator to show vertical mixing intensity. The pre-mixing period was defined as the one day before the mixing started because these days were assumed to be a non-typhoon event. Variables during the pre-mixing period were averaged over each day. Four variables (St ratio, τ, Hnet and hGAP) were used in our PCA procedures. The detailed procedure of the PCA is described in Supplementary Material II.
During data processing, missing values were not interpolated nor extrapolated from the available data. Consequently, the incomplete heavy-rainfall-induced vertical mixing observed data for analysis was excluded from Tab. 1.
Field data was collected at the YYL buoy from spring 2004 to autumn 2010 (except for 2007 due to insufficient complete datasets for analysis). Fig. 3a shows the seasonal variation of hourly averaged water temperatures and thermocline heights (zt = zm – zE) from the surface. The hourly averaged variations in MET variables (shortwave radiation, air temperature, wind, and precipitation) corresponding to the measurement period of water temperature are shown in Fig. 3 b,c. Mixing often occurred in YYL in summer and early autumn (June to October) when large amounts of precipitation and high wind speeds (caused mostly by typhoons) were observed. In the winter, mixing also occurred at times when there were colder air temperatures and reduced shortwave radiation. The temporal variation of the water column temperature indicates that typhoons caused complete or partial mixing (Fig. 3a, Tab. 1).
Major physical process in vertical mixing
Using thermal profiles and the lake bathymetry information, the strength of heavy-rainfall-induced vertical mixing in the water column can be represented by the St ratio. Using St ratios, Fig. 4 shows the relationships during heavy rainfall events between the strength of vertical mixing and three forcing factors: wind shear stress, net heat balance at the water surface and water level gap between peak and pre-mixing levels, which represents the effect from inflow/runoff inputs to the lake. Note that the water level gap was considered a forcing factor in this study although it is an indirect forcing factor through inflow/runoff. These key factors were averaged over each mixing period induced by a heavy rainfall event. The duration of the mixing period was defined as the period from the beginning of mixing to the end of mixing (Fig. 1a).
We assumed that vertical mixing is stronger when the St ratio is higher and that an extreme weather event occurs when the air pressure gap (the difference in air pressure between pre-mixing and mixing periods) is higher. In Fig. 4, the Pearson product–moment correlations (R) between St ratio and each MET forcing factor are 0.51 for τ, 0.02 for Hnet and 0.71 for water level gap (P<0.05 for all variables). The relationship between τ and the St ratio shows that larger τ(> 10–2N m–2) caused stronger vertical mixing (St ratio >0.4) (Fig. 4a). The relationship between Hnet and the St ratio in Fig. 4b indicates that Hnet was not an important factor in the promotion of vertical mixing. Large St ratios (>0.5) were strongly correlated with large water level gaps (>1.0 m) (Fig. 4c). However, even for small water level gaps (<1.0 m), some large St ratios were observed because weakly stratified waters caused by cool weather were already formed prior to the mixing event.
Surface layer mixing
In general, cooling and wind-driven turbulence cause deeper mixing; well below the surface layer. The inflow effect from discharge volume and temperature on surface mixing cannot be quantified, owing to the fact that inflow-related discharge does not spread out spontaneously and homogenously over the lake surface. Nevertheless, we quantitatively examined only the contribution that the penetrative convection (w*) and the surface friction velocity () in the lake surface layer reaching down to the upper thermocline make to vertical mixing throughout the entire water column. We assumed that three variables (w*, and thermocline depth) represent the degree of cooling, wind-driven turbulence and deepening of mixing respectively. The quantitative comparison between w* and shows that for lower (<0.004 m s–1), w* is dominant. On the other hand, was slightly higher than w* for some events with a higher air pressure gap (Fig. 5a). Fig. 5b shows that the ratio for some events was higher than the threshold (0.75) indicating that for relatively large air pressure gaps, the turbulent energy from contributed more to the SML. In addition, a positive slope between and thermocline height from the water surface was strongly marked after the threshold. This implies that larger winds accompanying larger air pressure gaps promoted vertical mixing. However, the contribution from inflow to the SML is unclear because inflow potentially intrudes into the subsurface (e.g., thermocline) or lake bottom, depending on the water density of the inflow. Therefore, to evaluate the inflow effect on YYL vertical mixing throughout the entire water column, we had to introduce the heat content as a total heat-storage indicator. The heat content accounts for the variation in water levels, caused by inflow to the lake.
Examples of typhoon-induced mixing events
We assessed the inflow effect on heat content using two typhoon-induced mixing events with a similar wind shear stress: stronger vertical mixing during Typhoon Fungwong and weaker vertical mixing during Typhoon Rananim although we assumed that indirect effects of heat flux from the surface on heat content were weak (Fig. 4b). Fig. 6 shows MET forcing (wind, dry-air temperature and rainfall), water temperatures, water level and heat content with/without the inflow effect during the sampled events. The inflow effect was estimated indirectly from the change in lake volume by the increase in the water level. The effect of ‘no inflow’ was assumed to correspond to no change in the lake volume. Water temperatures were weakly stratified before Typhoon Rananim (Fig. 6a). Because of this, the difference between heat content with and without an inflow effect was smaller (~67 MJ m–2 h–1) (Fig. 6 b,c). In contrast, the strong vertical mixing and high inflow in the stratified YYL during Typhoon Fungwong resulted in a larger difference (~115 MJ m–2 h–1) between inflow and no inflow heat content (Fig. 6 d,e,f). The changes of heat content suggest that the inflow effect becomes significant for vertical mixing accompanied by a large increase in water level and that the heat content is a good indicator for describing the strength of the inflow effect on vertical mixing.
Heat content indicator of vertical mixing
We used the Burgean heat budget Bh (eq. 7) as an indicator to evaluate inflow effects, partial effects of cooling and indirect effects of wind-induced turbulence on vertical mixing. The inflow effect can be a dominant factor in heat content because of the large change in lake volume and the difference between the temperature of inflow and the lake surface. The increase in lake volume during mixing periods ranged from ~10 to 92% with a mean of 41% and a standard deviation of 23%. The mean difference in temperature for all mixing events was ~2.5°C with 1.3°C standard deviation, computed by an empirical regression between inflow and air temperatures if inflow temperature is uniformly spread from the inlet (Kimura et al., 2014). The mean and standard deviation value for inflow temperatures can be ~10% and ~5% of the entire lake temperature during typhoons. Note that, in reality, inflow could affect lake temperatures on a local scale. This trend typically has an annual cycle extending from the end of complete mixing to the beginning of the next complete mixing event and through the stratification period. We assumed that this annual cycle corresponds to a period of heavy-rainfall-induced mixing (Fig. 1a). Bh with log10 scale for heavy-rainfall-induced mixing events is moderately correlated with the St ratio with an R value of 0.60 (P<0.05) (Fig. 7a). This implies that the effect of inflow on complete vertical mixing is moderate in the YYL.
However, it is necessary to verify the implementation of Bh to heavy-rainfall-induced mixing. Seasonal trends of heat content have been investigated in numerous studies. For the YYL study, a short-term Bh was computed for each vertical mixing. Fig. 7b shows that the Bh values for heavy-rainfall-induced mixings were less than half the magnitude (approximately 1 to 100 MJ m–2) of the empirical relationship between Bh and Zm (Supplementary Material I). Bh values are different from the empirical relationship because of the different dynamics between polymictic and meromictic lakes, and the difference in lake size and depth. Fig. 7b shows that the Bh values gradually increased as maximum depth increased. The slope of the linear regression for higher air pressure gaps (>5.2 m for maximum depth) was similar to that (~0.45) of the empirical relationship.
Our goal in this section is to determine the major factor governing vertical mixing using a quantitative comparison of results in the previous section. The data analysis revealed three implications. In Fig. 4, the strength of the vertical mixing (St ratio) had a moderate relationship with wind shear stress (τ) and water level gap (hGAP), and a weak relationship with net heat balance (Hnet) based on the interpretation of the R s (Evans, 1996). The moderate relationships of τ and hGAP were supported by other studies. For instance, wind forcing is a dominant factor for vertical mixing in a thermally stratified, shallow lake (Tuan et al., 2009). The relationship between wind speed and precipitation was positively strong for rainstorm events that caused vertical mixing in a small, shallow lake (Kimura et al., 2012a).
If focusing only on the SLM, mixing events accompanying large air pressure gaps were likely dominated by wind-turbulence mixing in the lake surface where the effects of wind and heating/cooling reached (Fig. 5). This implication supports the moderate relationship between τ and St ratio (Fig. 4a) when the SML in each event was approximately equivalent to the depth from surface to bottom owing to surface layer mixing being deeper except for one event. In contrast, w* is dominant for low u*w (Fig. 5a), which suggests that the SML extended to greater depths because of cooling (MacIntyre et al., 2002). The result from extreme weather events accompanying strong wind is different from those of other studies that show the change of dominant forcing, Hnet or τ for diurnal variation (MacIntyre et al., 2002), and the dependency on lake surface size for seasonal variation (Read et al., 2012).
The impact of inflow on entire vertical mixing was also very significant, as revealed by the heat content indicator for two larger, extreme weather events (Fig. 6) as well as 20 other mixing events (Fig. 7). These inflow effects on vertical mixing are described over the lake under the assumption that horizontally homogenous temperatures exist in a small lake, to which typhoon-induced inflow/runoff waters move from any near-shore area. The large magnitude difference of heat content between off- and on-effects of inflows (~100 MJm–2 h–1 in Fig. 6f) likely corresponded to the seasonal difference of heat content in a small, shallow template water-body (Rodríguez-Rodríguez and Moreno-Ostos, 2006). The impact of typhoon-induced inflow was large although the inflow effect is usually minor in physical processes in large lakes (Michalsky and Lemmin, 1995). For understanding localized mixing for stratified water bodies, numerous spot measurements are necessary to reveal more precise characteristics of inflow-induced vertical mixing in specific areas with density plume phenomena (Laborde et al., 2010; Hogg et al., 2013). The overall suggestion is that the inflow had an important effect on vertical mixing although wind shear stress was also influential.
As a result, the above implications only describe a qualitative trend where statistical methods can determine which forcing factor is dominant in vertical mixing. For a more quantitative interpretation of these results focusing on common components among the observed data, we employed the PCA, modified by partial correlation in the pre-processing. The PCA was performed using the variables of Schmidt stability, wind shear stress, net heat balance, and water level gap (i.e., St ratio, τ, Hnet and hGAP). The St ratio was included as an index variable of the mixing in order to compare with the other variables. The matrix data of variance and covariance (Tab. 2) between two variables in each combination (e.g. τ and Hnet) were used for the PCA computation. However, the P-values for the covariance between Hnet and hGAP or St were high because of their weak relationships, which were likely caused by double counting of the wind effect even in Hnet. Their relationships partly controlled by the third variable are likely spurious. To reduce these P-values, eliminating the spurious relationships was performed using partial correlation. The improved matrix data are shown in Tab. 3.
The first and second principal components (PC1 and PC2) are listed in Tab. 4. They accounted for 90% of the total variance and might explain the nature of the four variables. The eigenvectors (sets of weighted coefficients) for each principal component (PC) are also shown in Tab. 4. The large magnitudes of eigenvector elements in each PC potentially indicate that the element dominantly contributes to the PC (Liu et al., 2011). The values of loadings (i.e., eigenvector elements for τ, hGAP and St ratio) for the PC1 (~54% of the variance) were relatively high, almost equivalent to each other, and their signs were the same. We assume that the PC1 represents the relatively strong effect of vertical mixing owing to the large value of St ratio. Therefore, the implication is that the strength of the vertical mixing was significantly affected by both wind and inflow to the lake. They change in the same direction to increase or decrease the degree of stratification. In the quantitative comparison, the influence of inflow on mixing was slightly larger (~10%) than that of wind turbulence. Another assumption is that the PC2 represents a relatively weak effect of vertical mixing owing to the low value of the St ratio. Thus, the factor related to cooling/heating (Hnet) was dominant, but it was weakly related to the intensity of mixing. Therefore, the intensity of the YYL mixing dynamics was caused primarily by wind turbulence and inflow intrusion, and the inflow effect was slightly stronger than that of wind shear stress. These conclusions are supported by our analytical results in the Results section.
As an extended discussion, to determine the effects of physical processes on water quality during vertical mixing, we examined the relationships between DO concentrations near the surface (i.e., the depth of 0.25 m) and the dominant physical factors (i.e., wind and inflow). The difference in DO concentration between pre-mixing and mixing periods, normalised to the concentration during pre-mixing, was defined as [DO(pre) – (DO(mixing)]/DO(pre) (hereafter DO conc. ratio). However, because the number of DO concentration measurements available during typhoons was limited, the relationships are shown only for a few DO values (Fig. 8). The values of the DO conc. ratio were negative when DO concentration increased during strong mixing events. The DO conc. ratio shows a moderate negative correlation to the St ratio (Fig. 8a). For mixing events, the weak, negative correlation between the DO conc. ratio and St ratio could result from the balance among several mechanisms, such as the aeration of water by wind-induced turbulence (Melack, 1984), the cooling process in the surface water due to increasing oxygen solubility (Connell and Miller, 1984), the intrusion of inflow water with high DO concentration and the entrainment of anoxic hypolimnetic water to the surface layer (Tsai et al., 2008, 2011). The relation between the DO conc. ratio and the water level gap is more strongly correlated than that between the DO conc. ratio and wind shear stress (Fig. 8 b,c). These quantitative relationships imply that DO concentration in YYL was mixed more as a result of inflow, even though DO concentration is typically affected by wind-induced mixing and water temperature (Herb and Stefan, 2004). Thus, a small, shallow lake might be more strongly affected by inflow than by other MET factors. However, it is still necessary to understand how strong mixing increased the DO concentration, and this is planned for future work.
Using field data from an instrumented buoy in a small, shallow, stratified lake (YYL), we studied the characteristics of mixing induced by heavy rainfall events. We examined which MET forcing factor among three physical variables (wind, heating/cooling, and inflow) was dominant in the mixing process using: an indicator of thermal stability (St ratio) for mixing intensity, an indicator of heat storage (Bh) for inflow over the entire lake volume, and the energy balance between friction velocity and penetrative convection at the lake surface. We revealed that:
the strength of the vertical mixing is moderately related to wind shear stress;
the effect of wind-induced turbulence on surface layer mixing is dominant during large wind-dominant rainfall events; and
the effect of inflow on vertical mixing is significant for higher air pressure gaps as shown by the heat content indicator of large extreme-weather events.
In addition, further data analysis was performed quantitatively using PCA modified with partial correlation. The result showed that the PC1 consisted predominantly of wind and inflow inputs to the lake, with a relatively high value of vertical mixing intensity, and that the net heat balance was less correlated with the intensity of vertical mixing in the PC2. The relationships between DO concentration and physical factors also suggest that inflow may affect vertical mixing.