USGS Home
Contact USGS
Search USGS

U.S. GEOLOGICAL SURVEY
Scientific Investigations Report 2008–5076

Back to Table of Contents

Simulation Results

Hydrodynamics

The hydrodynamic model was calibrated for 2005 by adjusting parameters a and z₀, which determine the surface and bottom friction coefficient, respectively (table 3). The same values for a and z₀ were used to run the model with the three different types of wind (UW-MDL, UW-MDN, and VW). In the UW-MDL simulation, a spatially uniform wind as measured at site MDL, a raft site in the southern part of the lake, was used. In the UW-MDN simulation, a spatially uniform wind as measured at site MDN, a raft site in the northern part of the lake, was used. In the VW simulation, a spatially variable wind was used. The spatially variable wind was determined from an interpolation of six sites, including the two raft sites and four others—WMR-MET, BLB-MET, HDB-MET, and SSHR-MET—located on the shoreline (fig. 1), with the exception that the data from site SSHR-MET were not included in the interpolation until August 18 when the sensors at that site started collecting data, and the data from site MDL were not included in the interpolation between September 1 and September 8 because the sensors at that site failed. The availability of data from the various meteorological sites and the periods covered by the three simulations were discussed previously and are summarized in table 2. In all cases the model was run for 5 days or more prior to the start of any error calculations in order to allow oscillations resulting from the startup of the model to dissipate.

The gaged and simulated lake elevation at two sites, UKL at Rocky Point, near the northern end of the lake, and UKL at Klamath Falls, near the southern end, are shown in fig. 6. These two gages show that the lake surface rises at the southern end and falls at the northern end on a daily basis, in response to the diel wind fluctuation. The model reproduces the timing of the oscillations well, although the simulated amplitude is larger than observed at the gages, particularly at the Klamath Falls gage, as can be seen in detail from July 15 to July 30. The VW simulation results in simulated oscillations closer to the observed amplitude than the UW-MDL simulation (fig. 6).

Prevailing winds over UKL blow over the northern part of the lake from the west to slightly northwest and are constrained by the surrounding topography to a northwest wind over the southern two-thirds of the lake, where the wind sensor at site MDL was located (Hoilman and others, 2008, fig. 7). On a typical day, wind speed picks up in the early to midafternoon and then dies down in the late evening or early morning. Typically, early morning is the calmest time of day. The response of the currents to moderate, prevailing winds is a relatively well-defined clockwise circulation consisting of broad, shallow flow in the direction of the wind on the eastern side of the lake and passing to the east of Bare Island, and a narrow, deep flow opposing the wind through the trench along the western shoreline and passing to the west of Bare Island. Usually, a large fraction of the northward flow in the trench veers eastward both south and north of Bare Island, forming a gyre that circulates water between Bare Island and Rattlesnake Point. The rest of the northward flow continues into the northern part of the lake, completing a clockwise loop that circulates water between Rattlesnake Point and the northernmost part of UKL (fig. 8). A reversal in the wind direction that is sustained for several hours causes the currents in the trench to reverse, resulting in a stall in the circulation pattern. If the wind reversal lasts long enough (at least a day), the current loop between Bare Island and Rattlesnake Point can reverse as well (fig. 9).

All of the ADCP data collected during 2003–06 are described in Gartner and others (2007), but it is useful to describe the basic circulation pattern in the lake in order to provide context for the ADCP data used in calibrating the model. Site ADCP1 was located in the deepest part of the trench (fig. 1). The current speeds are higher in the trench than in the shallower parts of the lake; site ADCP1 was strategically located to capture some of the highest velocities in the lake (fig. 10). The direction of the currents at site ADCP1 is constrained by the bathymetry of the trench to approximately 310 degrees under prevailing wind conditions; during a wind reversal the currents change direction by 180 degrees (fig. 11). The other ADCPs also were in areas that have pronounced prevailing current directions, although not as pronounced as at site ADCP1. Site ADCP7 was located near the southern terminus of the trench at the mouth of Howard Bay. Current speeds there were the second highest of the five locations (fig. 10), and current directions were predominantly about 30 degrees east of due north, in alignment with the direction of the trench at that site. Sites ADCP3 and ADCP5 were located in shallower water in the northern third of the lake: site ADCP5 in a shallower section of the trench to the west of Eagle Point, and ADCP3 near the terminus of the trench north of Ball Bay (fig. 1). The current directions at both of these sites are not as tightly constrained by the bathymetry but still show a prevailing direction that indicates currents moving in a clockwise direction around the northern third of the lake (fig. 11). Current speeds at these sites are much lower than in the deeper sections of the trench (fig. 10). Site ADCP6 was located in the broad, shallow flow that moves over the eastern part of the lake; the current direction there was the most variable of all the sites, but it indicates flow primarily to the southeast (fig. 11). Current speeds at site ADCP6 were the lowest of those measured at the five sites (fig. 10).

Error statistics for the three simulations (UW-MDL, UW-MDN, and VW) for the midsummer overlap period between July 26 and August 31 are provided in table 4 and are presented visually for ADCP1 in fig. 12. Error statistics were calculated at each ADCP site for both the depth-averaged speed, which is calculated from all the bins in the water column, and for individual east-west and north-south velocity components at two points in the water column—the bin closest to 1 m from the bottom and the bin closest to 1 m from the surface. The mean error (ME) in the simulated currents at all of the observation sites is provided as a measure of the overall bias of the simulation at each site. Errors are calculated as observed value minus simulated value, so a positive ME indicates simulated values less than observed values. The root mean squared error (RMSE) is provided as a measure of the overall goodness-of-fit of the simulation to the observations. The mean of the half-hourly observations and simulations also is provided for comparison. During the calibration process, emphasis was placed on correctly simulating the highest velocities through the trench, and as a result the ME and RMSE at site ADCP1 are small in comparison to the depth-averaged speed and the individual velocity components at that site (table 4, row 1 and rows 6–9, and fig. 10). In addition, both the ME and RMSE at ADCP1 are smaller for the VW simulation than for either of the uniform wind simulations, particularly in comparison to the UW-MDN simulation, which used only the wind measured in the northern part of the lake.

The second-best fit appears to be at site ADCP6 (table 4, row 4), where velocities are low. Comparison of the individual velocity components table 4, rows 18–21) indicates that, while the speed at this site is captured well by the VW simulation, the direction of the simulated currents is rotated to the west of the observations (fig. 11). At site ADCP6, the VW simulation produces a better result than the UW-MDL simulation, even though site ADCP6 was coincident with the meteorological site at MDL.

The simulated mean current speeds at the remaining sites are biased low in the VW simulation, as indicated by a positive ME, particularly at sites ADCP5 and ADCP7 (table 4, rows 3 and 5) and to a lesser extent at site ADCP3 (table 4, row 2). The ME of the average current speed at sites ADCP5 and ADCP7 is less for the UW-MDL simulation than for the VW simulation (table 4, rows 3 and 5); at site ADCP5 the RMSE also is less for the UW-MDL simulation than for the VW simulation (table 4, row 3). Site ADCP5 is located in the northern part of the lake; yet, if a uniform wind forcing is used, the currents there are better simulated by using winds measured in the central part of the lake than in the northern part of the lake closer to the site. This underscores that fact that success in accurately simulating the currents anywhere in the lake is dependent on having an accurately measured wind over as much of the lake as possible, rather than an accurate wind measured nearby at a single point of interest. In the case that wind data are available at only a single point, site MDL is a better site for collecting wind to force the model because the wind data collected there probably represents a large surface area in the broad, central part of the lake where much of the momentum transfer from the wind to the water occurs; site MDN is not as good a choice because the wind data collected there represent a smaller surface area of the lake.

The ME statistics for the east-west and north-south velocity components (table 4, rows 6–25) are mixed with regard to whether the VW or the UW-MDL scenario produces the best simulation of the observations. The RMSE statistics, however, show consistent improvement with the VW simulation, with the exception of the near-bottom, north-south velocity component at site ADCP6 (table 4, row 20). Also apparent in the statistics (particularly the RMSE) for the individual components is the fact that the errors are consistently larger at the surface than at the bottom.

At site ADCP3, for example, the simulated north-south component of the surface currents has a tendency to be too large in the early afternoon, whereas the simulated east-west component has a tendency to be too small in the late afternoon (fig. 13 and table 4). The VW simulation reproduces the directional shift between the bottom and the surface better than the UW-MDL simulation (fig. 13).

At sites ADCP1, ADCP6, and ADCP7, the comparison between observed and simulated currents worsens from September 1 to September 8 (fig. 10). Because there is a data gap in the wind observations at site MDL during this time, a likely explanation is that the interpolated wind field is still dependent on observations at site MDL to provide accuracy in the wind forcing over the lake.

The model was validated using 2006 boundary conditions and forcing functions from May 20 to October 15. The values of the two calibration parameters, a and z₀, were the same as established during calibration for the 2005 season. This provided an opportunity to validate the performance of the model over nearly 5 months. The end-of-summer lake elevation in 2006 was lower than that in 2005 and revealed some inaccuracies in the representation of the bathymetry and shoreline in the model, as the simulated gages were without water after mid-September (fig. 14).

The observed and simulated current speed and direction at the two 2006 ADCP sites are shown in fig. 15. The visual comparison suggests that the underestimation of current speeds by the model simulation at site ADCP1 is greater in 2006 than in 2005, and this is confirmed by the ME statistics and the means (table 5, rows 1 and 2). In order to check whether the 2006 errors were larger because of the longer simulation period in 2006, errors also were calculated over the same period (July 26 through August 31) that was used for the error calculations in 2005. Errors over this shorter period were comparable to those over the longer simulation, which seems to suggest that errors are not growing with the length of the simulation period. Nonetheless, the 2006 errors were larger for the July 26 through August 31 period than for the same period in 2005. For example, the RMSE at site ADCP1 was 3.88 for July 26 to August 31, 2006, as compared to 3.08 for the same period in 2005. It is not obvious why the model underestimates the currents in 2006, but it indicates that the calibration obtained for the 2005 data is not necessarily optimized for multiple years.

The simulation of the currents at the second 2006 ADCP site was less successful. This site, ADCP9, was located in the northern part of the lake, farther from the trench than either site ADCP3 or ADCP5 in 2005. The mean depth-averaged speed at this site during August was 3.42 cm/s (table 5), which was less than the speed at either of the northern sites in 2005 (table 4). The RMSE of 2.49 for the July 26 to August 31 period (table 5, row 3) is comparable to the errors at the northern sites in 2005 (table 4, rows 2 and 3), but a visual examination of the data shows that the simulation does not reproduce well the fluctuations in the observations, particularly on the time scale of several days (fig. 15). Site ADCP9 was located in a place where the currents were particularly complicated but not strong. The mean direction of the currents was toward 280–290 degrees clockwise from north at the surface, rotated about 20 degrees counterclockwise from the bottom (Gartner and others, 2007), so at this site more than at the northern ADCP sites in 2005, the currents directly opposed the prevailing wind direction observed at site MDN, where the strongest winds were approximately westerly, coming from 270–280 degrees clockwise from north (Hoilman and others, 2008). Thus, errors in the wind forcing at the site, as well as inaccuracies in the characterization of surface or bottom friction, will result in more noticeable discrepancies between observations and simulations at this site.

When putting the error statistics into perspective, it is important to understand that although ADCP technology is generally reliable and accurate for measuring water-current profiles, there are some inherent limitations and assumptions associated with those measurements, including two primary assumptions. First, since the instrument determines beam velocity from Doppler (or frequency) shift in the acoustic beams from backscattered signals, it is actually measuring the motion of particles in the water, not the actual water motion, which may be different. Second, the instrument transforms the beam velocities into earth coordinates, which requires the assumption that flow is essentially homogeneous in the four beams. There are error checks to discard obviously bad measurements but the assumption of homogeneous flow may impart some error. Because the four acoustic beams are oriented 20 degrees from vertical, the validity of that assumption is a function of distance from the transducer. Further, the instrument has a limited accuracy for each acoustic measurement. That accuracy (or “single ping” standard deviation) is typically reduced by averaging multiple pings to a single ensemble average that is then recorded. The (random) standard deviation is reduced by a factor of the square root of the number of pings. In the case of the ADCP measurements in UKL, the ADCP ensemble standard deviation or accuracy was about 0.73 cm/s for the horizontal velocity.

Overall, the calibration of the hydrodynamic model for 2005 and the validation for 2006 was successful. In both years, the biggest discrepancies between the observed and simulated currents occurred near the surface of the water column and indicated difficulty with simulating the friction boundary layer, so the errors in the depth-averaged speed were larger at shallow sites where the boundary layer is a bigger proportion of the water column. The simulation of the highest velocities through the trench was accurate, however, and provides confidence that the overall circulation patterns produced are correct. The validation of the model in 2006 produced currents that were somewhat underestimated in the trench, however, which indicated that even though the currents were accurately simulated at the same location in 2005 by adjusting only two calibration parameters, the model was not necessarily optimized for multiple years.

Heat Transport

The heat transport model was calibrated by adjusting one parameter, r_SW, the amount of incoming solar radiation reflected at the water surface. Comparisons between the observed and simulated temperature at the shallow sites where there was one continuous monitor, placed 1 m from the bottom, are provided in fig. 16. The goodness-of-fit temperature statistics for the midsummer overlap period of July 26 to August 31 are provided in table 6 for all three simulations and are presented visually for a subset of the sites in fig. 17. The absolute value of the ME of the VW simulation is less than 1°C at all sites and less than the ME of the UW-MDL simulation at most sites. The RMSE of the VW simulation also is less than 1°C at most of the sites, has a maximum value of 1.12, and is less than the RMSE of the UW-MDL simulation at most sites. The ME statistics are both positive (indicating an underestimation of the temperature) and negative (indicating an overestimation of the temperature), but the negative errors are larger, so that there a slight overestimation of the overall temperature around the lake during August, which is the warmest month. Interestingly, the RMSE of the UW-MDN simulation is generally comparable to and sometimes less than the RMSE of the VW simulation, indicating that the correct simulation of the currents is not required for correct simulation of the temperature. This underscores the fact that the temperature at any point in the lake is primarily a function of the meteorological forcing at the surface rather than advection.

After September 4, there is a noticeable increase in the diel range in temperature at some of the sites, which is due in part to the use of a land-based measurement of air temperature (from WMR-MET) to calculate incoming longwave radiation and evaporative heat loss for the boundary condition after the measurement of air temperature from the raft at site MDL became unavailable. The diel range in temperature over land is greater than that over water, primarily because the daily minimum is lower (fig. 18); the use of the measurement made over the water is preferable and generates more realistic results. Rapid drops in air temperature due to weather patterns such as those occurring around June 17, September 22, and October 1 appear to produce a more extreme response in simulated water temperature than is realistic. During these times, cloud cover is likely underestimated in the model by the use of a uniform value, and the incoming longwave radiation is therefore likely underestimated as well. Thus the incorporation of a temporally variable cloud cover, or directly measured longwave radiation, would likely moderate the simulated temperature response of the lake to these weather patterns.

Another important aspect of temperature simulation, in addition to correct simulation of the daily average temperature and temperature range, is the ability to predict the difference between the upper and the lower parts of the water column at those sites where the water is deep enough to stratify, including sites MDT, EPT, and MDN (Hoilman and others, 2008). Comparisons between the observed and simulated temperature records at the deep sites where there was a near-surface and near-bottom monitor in 2005 are provided in fig. 19. The observed and simulated daily maximum and daily minimum range in temperature from top to bottom at sites MDT, EPT, and MDN are shown in fig. 20. In general the VW simulation seems to be able to correctly reproduce those periods following July 26 when there is some temperature stratification that persists through several days; these are most noticeable at site MDT, the deepest site. The UW-MDL simulation does not reproduce those periods of persistent temperature stratification prior to July 26. The model correctly reproduces the large diel variation in the near-surface layer while the near-bottom temperature remains more nearly constant during the day; this is true even at site MDN, the shallowest of the three sites, where two continuous monitors were deployed. The ME and RMSE goodness-of-fit statistics at the near-bottom monitors at the deep sites tend to be bigger than the goodness-of-fit statistics at the near-surface monitors at the same sites (table 6, rows 1–6).

The model validation run used observed 2006 meteorology between May 20 and October 15. The value of the calibration parameter, r_SW, was the same as that established during calibration for the 2005 season. The observed and simulated temperature records at the four deep sites where there was a near-surface and near-bottom continuous temperature monitor in 2006 are shown in fig. 21. The observed and simulated temperature at the eight shallow sites where there was one continuous monitor, placed 1 m from the bottom, are shown in fig. 22. In addition, in 2006 five continuous monitors were placed in nearshore locations (within 100 m of the shoreline), and the observed and simulated temperatures at these sites are shown in fig. 23. Error statistics were calculated over the entire 149-day simulation, and over July 26 to August 31 in order to compare to the results from the 2005 calibration (table 7). The ME was bigger over the longer period at some sites and smaller at others, but the overall ME was smaller over the entire simulation. The overall RMSE grew slightly from 0.75 to 0.88 from the 36-day simulation to the 149-day simulation (table 7, row 22). As in 2005, the largest errors were not at the deepest sites, where a monitor was located 1 m from the bottom and 1 m from the surface (table 7, rows 1–8; fig. 22). The top-to-bottom range in temperature at the deeper sites was generally well-simulated over the entire 149-day simulation (fig. 24).

At sites where temperature was collected in both 2005 and 2006, the RMSE was comparable over July 26 to August 31 (rows 1–10, 13,14, and 16 in table 6, and rows 1–6, 9–12, 14, 15, and 17 in table 7; fig. 17). The average of the RMSE over July 26 to August 31 at sites where temperature was collected in both 2005 and 2006 was 0.72 in 2005 and 0.75 in 2006. The ME for both the month of August and the entire simulation indicates that the model underestimated the temperatures, as there are more and larger positive errors than negative. Closer consideration shows that underestimation of temperature was more common at shallow and nearshore sites. The ME was greater than 1 at several sites (table 7)—all of these except site HDB were nearshore sites that were added in 2006 (fig. 23). The measured temperatures at the nearshore sites were generally higher than measured temperatures at the other sites (table 7). This may indicate that sediments play a role at these very shallow sites by absorbing incoming shortwave radiation that is transmitted through the entire water column and transferring some of the heat back to the water column. In the model, shortwave radiation that is not absorbed within the water column does not contribute heat to the water column. At some nearshore sites, most notably GBE and WDW, the simulated temperature shows greater diel swings than the observations (fig. 23); this may be an indication that the depth at these sites was not accurately portrayed in the model grid. At these very shallow nearshore sites, small errors in the bathymetry make a large proportional error in the depth of the water column.

Overall, the calibration of the model with 2005 temperature data and the validation with 2006 temperature data was satisfactory and adequate for the purpose of simulating the weekly to seasonal variations in temperature. The temperature is used in density calculations within the model and therefore determines the vertical mixing characteristics of the water column. The simulation of temperature captures both the daily thermal stability and those periods of time when the water column stratifies in deeper parts of the lake for several days at a time. Even though the lake is shallow, there has been discussion about the role that temporary stratification plays in water quality (Kann and Welch, 2005; Wood and others, 2006). Buoyant cyanobacteria like AFA are particularly suited to taking advantage of mild thermal stability to position colonies within the photic zone. In addition, as discussed further below, the tendency for the water column to stratify in the trench may provide a mechanism, particularly during a rapid bloom decline, for concentrating rising AFA colonies in the central part of the lake.

Back to Table of Contents