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High levels of total dissolved gas in the river lead to the development of gas bubble disease (GBD) in smolts, as well as other aquatic life. This condition involves the formation of bubbles in the fish's organs, tissues, and vascular system. GBD is also suspected of compromising the fish's vitality by increasing its susceptibility to predators, bacteria and disease (Dissolved Gas Abatement Interim Letter Report,1994). Because of the varied symptoms and effects of total dissolved gas, GBD will be considered an independent force of mortality.
There is uncertainty as to the significance of GBD-induced mortality at low levels of supersaturation (<110%) but it is clear in all studies that as the amount of supersaturation increases (> 110%) the rate of mortality increases significantly. The transition between low levels of generally sublethal effects to the higher level lethal condition involves a shift in the bubble-related mechanisms that lead to death. Specifically, at levels of supersaturation below the threshold fish are more susceptible to death related to infection and stress while above the threshold fish experience death from large intravascular bubbles (White et al. 1991).
Theory
In CRiSP, the level of total dissolved gas is represented by percent nitrogen saturation. Nitrogen is generated by spill at the dams and then dissipated as the water moves downstream. In the model, the effects of both lethal and sublethal levels of nitrogen are considered as well as the changes in the effective nitrogen concentration resulting from depth and distance downstream.
The relationship between migration factors and gas bubble disease is illustrated in Fig. 36. Nitrogen supersaturation can be defined with any of three submodels selected from the Nsat equation window opened from the DAM button menu.
Mortality rate equation
To incorporate both the lethal and sublethal effects of gas bubble disease the model uses a piecewise-linear function that expresses the rate of mortality Mn as a function of Ns, the level of nitrogen above supersaturation (see figure below). This piecewise-linear characteristic is accomplished by using the Heaviside function H() which switches from 0 to 1 as its argument changes from negative to positive. This allows the model to assume a moderate linear increase in mortality (slope a) at low levels of nitrogen supersaturation. When the lethal threshold of saturation Nc is reached, the Heaviside function turns on and the mortality curve increases linearly but now at a higher rate (slope a + b). Using the work of Dawley et al. (1976) the empirical mortality rate equation is
(100)
where
- Ns = percent nitrogen saturation above 100% as measured at the surface.
- Nc = threshold above 100% at which the gas bubble disease mortality rate is observed to change more rapidly towards more lethal levels.
- a = species-specific gas mortality rate coefficient with units of N-1 day-1 determining the initial rate of increase of mortality per %-increase in nitrogen saturation.
- b = species-specific gas mortality rate coefficient with units of N-1 day-1, determining the change in mortality rate at Nc.
- H() = Heaviside function, also known as the unit step function; equal to zero when its argument is negative, and equal to one when its argument is positive.
Eq(100) is illustrated in Fig. 37
Depth dependent critical values
Fidler and Miller (1994) demonstrated that the critical nitrogen supersaturation concentration (Nc) is depth dependent, with Nc increasing as depth increases. They observed a decrease in effective nitrogen supersaturation of about 10% per meter below the surface. In other words, fish at lower depths are less susceptible to nitrogen supersaturation. Based on this work, CRiSP utilizes a linear relationship to relate Nc to fish depth:
(101)
where
- z is fish depth,
- mc is a slope parameter,
- nc is the critical nitrogen supersaturation concentration at the surface.
Thus to obtain an effective Nc for a stock, eq(101) is multiplied by fish density as a function of depth, and then this term is integrated over the reservoir depth. Calibration for juvenile salmon converted into the model units are nc = 10.9, the surface critical level with units of %saturation of nitrogen above 100%, and mc = 2.96, the rate of decrease of Nc with units of %saturation of nitrogen above 100%.ft-1.
Downstream dissipation
As fish move downstream in a reservoir their mortality rate due to nitrogen supersaturation decreases because dissolved gas levels are highest at the upstream end and dissipate as the water moves downstream. The saturation level can be expressed as
(102)
where
- N0 = is percent supersaturation above 100% at the upstream end of a river segment, which may be a tailrace
- k = dissipation constant defined by eq (129).
To express the supersaturation in spatial coordinates the time coordinate is transformed to distance downstream by
(103)
where
- Uave = average water velocity through the river segment
- t = time
Transforming time to downstream distance using eq (103) and defining a new parameter
(104)
the surface supersaturation above 100% is
(105)
where
- x = distance downstream and
, where L is the pool length (miles).
The rate of mortality as a function of fish depth and distance downstream can be expressed by substituting eq (101) and eq (105) into eq (100) to give
(106)
Isopleths of constant mortality rate through the reservoir can be generated by setting eq (106) to a constant and solving for z as a function of x. The resulting isopleths of constant mortality are illustrated in Fig. 38.
Vertical distribution
A population of fish from a given species will spread out vertically. A number of distribution functions have been hypothesized (Zabel, 1994). For simplicity, CRiSP.1 uses an isosceles triangular distribution given by:
(107)
where
- zD = depth of the reservoir
- zb = maximum depth of fish distribution
- zm = mode of fish distribution
- m0 = slope of distribution function above mode
- m1 = slope of distribution function below mode
The fish depth distribution is illustrated in Fig. 39.
The work of Zabel (1994) shows that fish of a given species tend to seek specific depths that are correlated to level of illumination.
Integrate for average rate through pool.
The average mortality rate for a fish while it is in a pool is given by the equation:
(108)
where
= the mortality rate due to gas bubble disease averaged throughout the length and depth of the pool.
Calibration
The calibration was done in a three step process. First, cumulative mortality curves were fit with the CRiSP nitrogen mortality equation to generate estimates of the mortality rate. Second, these mortality rates for different nitrogen levels were plotted and fit with the CRiSP rate equation expressing rate vs. supersaturation. Third, mortality rates were corrected for fish length where the correction was developed using the first two steps and experiments conducted by Dawley et al. (1976).
Mortality Rates
The first step in the calibration was to fit cumulative mortality curves. The nitrogen mortality equation is given by eq (58) setting the predator mortality to zero. The equation to fit is then
(109)
where
- S = cumulative survival
- Mn = nitrogen mortality rate at a specific nitrogen supersaturation.
The results of fitting the cumulative mortality for fall chinook and steelhead are illustrated in Fig. 40 and Fig. 41 below. The estimated mortality rates from the regressions are given in Table 30. For steelhead (Fig. 40) the model fit is close for all levels of nitrogen and for all exposure times. For fall chinook (Fig. 41), with low levels of supersaturation, mortality rate is overestimated for short duration exposures and the rate is underestimated for the longer exposures. For higher levels the fit is better.
Size Difference
Dawley et al. (1976) demonstrated that large fish have higher levels of mortality. In a shallow tank using fall chinook of different sizes exposed to 112% and 115% supersaturation they determined cumulative mortality curves were significantly different (Fig. 10 in Dawley et al. 1976). These data can be used to infer the effect of fish length on nitrogen mortality in reservoirs since the study also demonstrated that shallow tank mortality curves had the same pattern as deep tank mortalities with higher nitrogen supersaturation levels. The studies indicated that mortality curves in shallow tanks at 112% saturation were equivalent to mortality curves in a deep tank with 122% supersaturation.
The resulting mortality-length relationship can be used to extrapolate experimental results using one fish length to field conditions where the fish were larger. The first step was to determine an empirical relationship relating nitrogen supersaturation mortality to fish length. With this relationship the results of fall chinook studies in the Dawley experiments are extrapolated to fall and spring chinook in the Lower Granite reservoir using different average fish lengths for each stock (Table 31). The study also determined the mortality rate for steelhead. This value is used to extrapolate to steelhead in the Lower Granite reservoir.
Size-mortality relationship
To determine the relationship between fish size and nitrogen supersaturation mortality the mortality rate is first estimated by fitting eq(109) to cumulative mortality vs. exposure time for different sized fall chinook and steelhead (Fig. 42). The estimated rates are given in Table 31.
The resulting mortality rates plotted against fish length are illustrated in (Fig. 43). The graph combines fall chinook ranging from 40 to 67 mm and steelhead 180 mm in length. The line in the figure is a linear fit with a least squares regression constrained to pass through zero.
Although no mechanism has been developed justifying a linear relationship, qualitatively the ability of a fish to establish gas equilibrium with its environment should be related to its volume to surface area ratio, which is proportional to fish length. Thus on physical principles of gas exchange a length relationship should be involved with nitrogen supersaturation mortality. The regression in Fig. 43 is
(110)
where
- Mn(L) = nitrogen mortality rate as a function of fish length
- L = fish length in mm
- a = 0.0466 mm-1, length coefficient for nitrogen mortality rate with an r2 of 0.95.
From eq(110) the nitrogen mortality rate can be corrected for fish length using
(111)
where
- L = length of fish in environment
- Le = length of fish in nitrogen mortality experiments
Length-corrected nitrogen mortality rates are given in Table 32.
Relating Mortality Rate to Nitrogen
To relate the length-adjusted mortality rates to nitrogen supersaturation eq (100) is applied where
- N = percent nitrogen saturation above 100%
- N crit = critical supersaturation level
- a = a species specific gas mortality rate (low slope)
- b = a species specific gas mortality rate (high slope).
It was fit one segment at a time using the Splus lsfit() function. A critical value was selected and lsfit() was used to fit mortalities at supersaturations between 0% and Ncrit, constraining the line through 0% mortality at 0% supersaturation. The remaining points were fit to a line constrained to meet the first at N = Ncrit. The resulting coefficients for the three species, and 
2 values for the fit are given in Table 33.
The fits of the rate curves are illustrated in Fig. 44
Fish Depth
The gas bubble disease rate depends on fish depth which is characterized by a mode depth and bottom depth. Fish depths vary continuously over day and night, fish age, and position in the river. For the current model a representative depth is required for each species. These were selected after reviewing the data on fish vertical distributions. The literature and essential elements are given in Table 34.
The modal depth is taken to represent most typical fish depth (Table 35).
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Columbia River Salmon Passage Model CRiSP.1.5 Theory, Calibration & Validation Manual
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