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

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Summary

Upper Klamath Lake (UKL) is a hypereutrophic lake that experiences annual blooms of the buoyant cyanobacterium Aphanizomenon flos aquae. The severe water-quality conditions associated with these annual blooms (high pH, supersaturated to undersaturated dissolved oxygen, high un-ionized ammonia) are detrimental to the survival of two species of endangered suckers in the lake. As part of an ongoing and multipronged effort to understand the ecosystem of the lake, the U.S. Geological Survey and the Bureau of Reclamation entered into a cooperative agreement to develop a three-dimensional numerical model of the hydrodynamics and heat transport in the lake.

The model is built on the UnTRIM computational core. The important features of this model include a semi-implicit finite difference solution method for the governing equations, flexibility in the use of the hydrostatic assumption, the use of a mass-conserving scheme to solve the constituent transport equation that also ensures that the solution is bounded by the initial and boundary conditions, and the use of a flux limiter in the solution of the transport equation to preserve accuracy while relaxing the stability constraint on the size of the polygons in the numerical grid. UnTRIM solves the governing equations on an unstructured orthogonal grid. The advantage of this type of grid is that it allows the size of the polygons that make up the grid to vary over the domain according to the requirements of the bathymetry and geometry, and allows for a shoreline-fitting boundary. The solution schemes used to solve the governing equations within the model provide a robust, stable, and computationally efficient platform on which to develop the UKL model.

Source and sink terms to calculate surface heat fluxes were added to the UnTRIM computational core. These include incoming shortwave radiation, atmospheric longwave radiation, reflected longwave radiation, evaporative cooling, and conduction. Two additional enhancements were made to the computational core. The first was a submodel designed to estimate a spatially variable surface wind based on mass-conserving interpolation. The second was a two-equation turbulence closure model designed to calculate spatially and temporally varying vertical diffusivities as a function of turbulent kinetic energy and its rate of dissipation.

The hydrodynamic model was calibrated for June through September 2005 by using two parameters to adjust surface and bottom shear stress. Two different spatially uniform surface winds were used to run the model from either June or July through August—one measured in the northern part of the lake and one measured in the central part of the lake. The model also was run using a spatially variable surface wind from August through September. The spatially variable surface wind was generated from the variable-wind submodel using wind data collected at two sites on the lake and four around the shoreline. Simulations resulting from all three surface winds could be compared during the midsummer overlap period. Datasets that were used to evaluate model simulations during the calibration process included surface elevation determined by 3 gages around the lake, currents measured by Acoustic Doppler Current Profilers (ADCPs) at 5 sites in the lake, and temperature measured at 14 sites by continuous monitors.

The model predicted wind-driven circulation patterns in the lake that reproduced the prevailing currents measured by ADCPs—a clockwise circulation consisting of broad, shallow flow with the wind (to the south-southeast) on the eastern side of the lake, and a narrow band of flow opposing the wind (to the north-northwest) through the deep trench along the western shoreline and passing to the west of Bare Island. The model simulations interpolate currents between the ADCP measurements sites; they reveal that under prevailing wind conditions there are two predominant modes to this circulation. The first is a smaller gyre that circulates water between Bare Island and Rattlesnake Point; the second is a larger gyre that circulates water between Rattlesnake Point and the northernmost part of UKL.

In order to evaluate the performance of the model, the mean error (ME) and root mean squared error (RMSE) of the hourly measured and simulated currents and temperature were calculated. In the process of model calibration with data collected during June to September of 2005, emphasis was placed on correctly simulating the highest velocities through the trench. The ME and RMSE of the VW simulation (calculated over 37 days between July 26 and August 31) in the depth-averaged speed at site ADCP1 (located in the middle of the trench) were small (0.50 and 3.08 centimeters per second (cm/s), respectively) in comparison to the mean of the depth-averaged speed at that site over the same period (11.23 cm/s). The simulated depth-averaged speed at two sites that were located in the northern part of the lake where velocities were lower were biased low, as indicated by a larger, positive ME of 0.80 cm/s at site ADCP3 (mean depth-averaged speed 4.09 cm/s) and 2.58 cm/s at site ADCP5 (mean depth-averaged speed 5.81 cm/s). Consideration of the velocity components near the surface and near the bottom showed that the goodness-of-fit errors were higher near the surface than near the bottom, indicating that the model may have difficulty in correctly simulating the surface boundary layer, particularly in those areas of the lake where the depth-averaged currents oppose the prevailing wind stress. The use of a spatially variable wind forcing that was interpolated between two meteorological sites on the lake and four on the shoreline generally resulted in improved error statistics. The improvement obtained by using the spatially variable surface wind was particularly evident upon closer inspection of the individual velocity components at a site in the northern part of the lake. At this site (ADCP3), the spatially variable surface wind correctly simulated the diel phase shift between the bottom and surface currents, whereas the uniform surface wind did not. In the process of comparing results among the spatially variable surface wind simulation and the two uniform surface wind simulations, it was demonstrated that, if a uniform wind must be used, the measurement should be made in the central part of the lake rather than in the northern part. In interpreting any of these results it is useful to remember that there is some inherent accuracy limitation to the ADCP measurements, and that accuracy may be a function of distance from the ADCP (in other words, near-bottom measurements may be somewhat more accurate than near-surface measurements).

The calibration of the heat transport was accomplished by adjusting one calibration parameter that specified the amount of incoming shortwave radiation reflected at the water surface. The error statistics indicated a small, high bias in the simulated temperatures over the lake, as the ME calculated between July 26 and August 31 at all the sites was more often negative than positive, although it was less than 1 degree Celsius (°C) at all sites, ranging from –0.94 to 0.73°C when a spatially variable surface wind was used. The RMSE was less than 1°C at all but one site and ranged from 0.40 to 1.12°C when a spatially variable surface wind was used. The model accurately reproduced thermal stability in the water column on a daily basis, and also correctly identified those periods when the stratification was maintained for several days.

The model validation simulations spanned 123 days between May 15 and October 15, 2006. Those simulations used a spatially variable surface wind and the same calibration parameters as established for 2005. Fewer ADCP velocity records were available to verify the model in 2006 (two sites), but one of these was site ADCP1, a site located in the deep trench where an ADCP also was deployed in 2005. This provided the opportunity to directly compare the performance of the model at this site for a comparable period, roughly the month of August, in both years. The ME and RMSE of the depth-averaged speed at this site, where the mean depth-averaged speed was 12.98 cm/s in 2006, were 2.30 and 3.88 cm/s, compared to 0.50 and 3.08 in 2005, so the model did not perform as well in 2006. The larger positive ME indicates that the velocities in the trench were underestimated; model calibration may not be optimized for multiple years. A second ADCP site in 2006 was located in an area of very low velocities (mean depth-averaged speed 3.42 cm/s) in the northern part of the lake. This site was farther from the trench and the main circulation gyre than the two sites where ADCPs were placed in the northern part of the lake in 2005; the accuracy of water currents simulated at this site was poor. The error statistics showed that most of the error was in the surface currents, as might be expected because the measured velocities at this site largely opposed the prevailing wind stress.

The results of the hydrodynamic and heat transport model for water quality in the lake indicate first, that water moving northward through the trench to the west of Bare Island is routed both to the north and to the south of the lake, and second, that the water originating from deeper in the trench is routed preferentially toward the north, while the water from nearer the surface is routed preferentially toward the south. Numerical tracer experiments were used to illustrate and quantify these observations. These tracer experiments are specific to the wind conditions, which were weak to moderate, during the dates of the simulation—from August 1 to 10, 2005. The experiments showed that 10 percent or less of the water passing through site MDL, in the southern part of the lake, originated from depths greater than 4.5 meters in the trench within the previous 5 days, whereas as much as 20 percent of the water at site MDN, in the northern part of the lake, originated from there. Similarly, the amount of water passing through site MDL that originated from depths shallower than 4.5 meters in the trench was between 90 and 100 percent, whereas only about 40 percent of the water passing through site MDN originated from there. These percentages were obtained for the specific conditions of early August 2005 and therefore may not be appropriate for other conditions. However, the more qualitative observation that surface water is circulated very effectively from the trench through the central part of the lake between Bare Island and Rattlesnake Point is robust and generally applicable, as is the observation that much less of the surface water but most of the deeper water in the trench moves instead toward the northern part of the lake. This means that during severe low dissolved oxygen events, most of the deeper water with a very low dissolved oxygen concentration moves into the prime adult sucker habitat in the northern part of the lake.

Given that dissolved oxygen is not conservative and that travel times are several days, this does not entirely explain why dissolved oxygen concentrations tend to be much lower and closer to concentrations in the trench during these events in the northern part of the lake than in the central part of the lake. Some difference in the capacity to replenish dissolved oxygen concentration between surface and deep water also is implied. Vertical velocities collected at site ADCP1 suggest that the trench is deep enough to effectively separate rising and sinking Aphanizomenon flos aquae (AFA) colonies. In conjunction with the two circulation gyres, this may provide a mechanism to concentrate healthier, rising AFA colonies in the central part of the lake, while at the same time causing the northern part of the lake to become relatively depleted of AFA colonies capable of photosynthetic oxygen production. This hypothesis could help explain why the water-quality conditions associated with rapid AFA bloom declines are more severe in the northern part of the lake, but further testing of this idea is required.

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