Table 57 Equation parameters and their descriptions
| Equation Parameters |
| F(t)week (j) |
weekly variation in flow for headwater dam j |
| G |
flow scaling factor in kcfs |
| an, bn |
Fourier coefficients |
| t |
day of the year |
 |
offset for day of week alignment |
| Fday |
daily variation in flow in kcfs at headwater dam |
| r |
deterministic rate of change of flow per unit of flow (the range is confined such that 0 < r < 1) |
 |
intensity on the random variations in flow |
| w(t) |
Gaussian white noise process describing the temporal aspects of the flow variation |
| FD(r) |
flow output at dam immediately below reach r |
| FL(r) |
new flow loss at reach r, as adjusted for mass imbalance |
| FM(r) |
flow maximum at reach r |
| FM(i) |
flow maximum at reach i |
| FR(j) |
flow at regulation point j |
| n |
number of upstream regulated points |
| p |
number of reaches between dam r and all regulation point |
| Floss (i) |
modulated flow loss at downstream dam i |
| i |
the standard deviation of the difference in flows (kcfs) at dam i and i +1 as computed by daily observed flows at all dams over the years 1979-1981 |
| FTU(r) |
total unregulated flow input to dam r |
| p |
number of regulated flows in region |
| FD(r) |
flow output at dam r |
| FR(j) |
flow output at regulation point j |
| Ki |
flow coefficient at unregulated headwater i |
| q |
number of adjacent unregulated headwaters in region |
| FU max (i) |
maximum flow at unregulated headwater i or j |
| FD(r) |
flow output at dam immediately below reach r |
| FL(r) |
new flow loss at reach r, as adjusted for mass imbalance |
| FM(r) |
flow maximum at reach r or i |
| FR(j) |
flow at regulation point j |
| FU (i) |
flow at unregulated headwater i |
| m |
number of unregulated headwaters above r (m = 3 in Fig. 17) |
| n |
number of regulated points adjacent to nearest upstream regulation point (n = 2 in Fig. 17) |
| p |
number of reaches between dam r and all upstream regulation points (p = 9 in Fig. 17) |
| Fi (t) |
flow at regulation point i at reservoir time increment t |
| FL(i) |
flow loss at reach i |
| Fj (t) |
flow at regulation point j immediately upstream at reservoir time increment t |
| F(t)i |
modulated flow at dam i |
| F(t)arch(i) |
archive flow at dam i |
| F(t)day(j) |
daily modulated flow in regulated headwater j |
| F(t)week(j) |
weekly modulated flow in regulated headwater j |
| Floss(i) |
loss modulated flow in river segment upstream of dam i |
| Fmin(i) |
minimum allowable flow at dam i |
| J |
number of regulated headwaters upstream of dam i |
| I |
number of dams upstream of dam i, including dam i |
| t |
julian day (t = 1 to 365) |
| Yt |
estimated daily flow |
| m |
mean annual flow computed over a 10 year period |
| p |
fraction of mean annual flow for the scenario |
| et |
stochastic error term |
| Ft |
Fourier term |
| ak, bk |
Fourier coefficients estimated for each river |
 |
2 /365 |
| rt |
randomly generated variable from a normal distribution centered on 0 with variance appropriate for dry and wet years as described above |
| e0 |
0 |
| dV |
change in reservoir volume in acre-ft |
| dt |
time increment, typically 1 day |
| FU |
unregulated natural flow into the reservoir in kcfs |
| FR |
regulated flow out of the reservoir, which is controlled by the user under volume constraints in kcfs |
| V(i) |
reservoir volume time step i with units of acre-ft |
 t |
one day increment |
| FU |
unregulated flows in kcfs |
| FR |
regulated flows in kcfs |
| c |
1983.5, which is a conversion factor |
| FR |
outflow from reservoir according to the constraints |
| FU |
unregulated inflow to reservoir |
| Vrequest |
requested volume from reservoir |
| Frequest |
requested outflow from reservoir |
| V(i) |
reservoir volume in reservoir time step i |
| Vmax |
maximum reservoir volume |
| Vmin |
minimum reservoir volume |
|
2/365 |
| k |
value between 0 and 4 |
| Hu |
full pool depth at the upstream end of the segment |
| Hd |
full pool depth at the downstream end of the segment |
| L |
pool length at full pool |
| x |
pool length at lowered pool |
| E |
pool elevation drop below full pool elevation |
| W |
pool width averaged over reach length at full pool |
 |
average slope of the pool side |
| F |
flow through the pool in kcfs |
| Ufree |
velocity of free flowing river |
| V(E) |
pool volume (ft3) as a function of elevation drop E in feet |
| F |
flow in 1000 cubic feet per second (kcfs) |
| L |
segment length in miles |
| x |
pool length defined by eq (27) and with units of feet |
| Ufree |
velocity of water in the free stream (kfs) |
| T |
residence time in this calculation is in kilo seconds (ks) |
| Hu |
full pool depth at the upstream end of the segment |
| U |
average river velocity in ft/s |
| Ufree |
the velocity of a free flowing stream in ft/s |
| F |
flow in kcfs |
| E |
elevation drop (positive downward) in ft |
| Hu |
depth of the upper end of the segment in ft |
| V1 and V2 |
volume elements defined by eq (31) and (32) |
| V(0) |
pool volume at full pool |
| Fi(t) |
flow from headwater i through the river segment in question on day t |
| i(t) |
temperature from headwater i on day t |
| (t) |
temperature for selected river segment on day t |
| X |
position of a fish down the axis of the river |
| dX /dt |
velocity of fish in migration |
| r |
average velocity of fish in the segment; this is a combination of water movement and fish behavior |
|
spread parameter setting variability in the fish velocity |
| W(t) |
Gaussian white noise process to represent variation in velocity |
 |
cumulative distribution of the standard normal distribution |
| L |
segment length |
| r |
average migration velocity through the segment |
| r(t) |
migration rate (miles/day) |
| t |
julian date |
| b's |
regression coefficients, described above |
| a1 , a2 |
slope parameters |
| TSEASN |
seasonal inflection point (in julian days) |
| TRLS |
release date (in julian days) |
| r(t) |
determined from eq (48) |
| V(i) |
variance factor that varies between releases only |
| S |
measure of smolt density in the river segment and can be taken as the total number in the segment |
 |
mortality rate from all causes |
| Mp |
mortality rate from predation with units of time-1 |
| Mtdg |
mortality rate from total dissolved gas supersaturation with units of time-1 |
| S |
number of smolts leaving reservoir per day (smolts reservoir -1) |
|
combined mortality rate as used in eq (54) |
| S0 (tj | ti) |
potential number of fish that enter the segment on day ti and survive to leave the segment on day tj |
| S (tj | ti) |
actual number of fish that enter the segment on day ti and leave on day tj |
| t |
reservoir computational time increment |
| N (ti) |
number of fish that enter the river segment on day ti |
| P (tj | ti) |
probability that a fish entering on day ti survives to exit on day tj (defined by eq (46)) |
| T |
temperature (°C) |
| Pij |
the predator density in the ith zone (forebay, tailrace, or reach) for the jth project |
| ai |
the predator activity coefficient in the ith reservoir zone |
| f(T) |
the temperature response equation |
| CMAX |
the maximum consumption rate |
 T |
a slope parameter |
| TINF |
the inflection point of the curve |
| H |
forebay (tailrace) depth at full pool |
| h |
forebay (tailrace) depth at a lowered pool |
| P |
predator density at full pool for the forebay (tailrace) |
| Gs |
percent TDG above 100% as measured at the surface |
| Gc |
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 G-1 day-1 determining the initial rate of increase of mortality per %-increase in TDG |
| b |
species-specific gas mortality rate coefficient with units of G-1 day-1, determining the change in mortality rate at Gc |
| 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 |
| 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 |
| Mtdg(L) |
TDG mortality rate as a function of fish length |
| L |
fish length in mm |
| a |
0.000472 mm-1, length coefficient for TDG mortality rate (regression of all data from the 112% shallow tank experiments conducted by Dawley et al. (1976)) |
| L |
length of fish in environment |
| Le |
length of fish in TDG mortality experiments |
| Gright , Gleft |
percent TDG in the flow entering the reach on the respective sides |
| Sfr |
percent of river in the right-bank flow |
| Gmix |
flow weighted average of the TDG values in each flow |
| Gdif |
difference between the original concentrations of the two flows |
| E |
percent TDG in water at equilibrium, 100% saturation or 0% supersaturation |
| q |
diffusion rate constant in units of (mile)-1, a model parameter set for each reach |
| v |
average water velocity through the river segment |
| x |
distance downstream |
| t |
average water travel-time |
| x |
distance downstream and , where L is the pool length (miles) |
| c1 |
Gmix - E |
| c2 |
Gdif . (1-Sfr) for the right-bank flow |
| c2 |
- Gdif . Sfr for the left-bank flow (see eq (103) and eq (104)) |
| q |
reservoir mixing coefficient in (miles)-1 |
| E |
equilibrium value (0% supersaturation) |
| z |
fish depth |
| m |
a slope parameter |
| gc |
critical gas supersaturation at the surface where GBD mortality rate changes more rapidly towards more lethal levels |
| n |
indexes the julian day |
| i |
indexes the side of the river and hence the level of TDG on that side of the river; 1 indexes the right-bank and 2 indexes the left-bank |
| a |
species-specific gas mortality rate coefficient with units of G-1 day-1 determining the initial rate of increase of mortality per %-increase in TDG |
| b |
species-specific gas mortality rate coefficient with units of G-1 day-1, determining the change in mortality rate above Gc |
| S |
cumulative survival |
| Mtdg |
TDG mortality rate at a specific level of supersaturation |
| t |
exposure time |
| Gs |
TDG at the surface |
| gcorrection |
TDG experienced by the fish |
| z |
fish depth |
| m |
a slope parameter |
| S(t) |
reservoir survival after t days of migration |
| A |
14.07 |
| B |
0.1822 |
| S(t) |
reservoir survival after t days of migration |
| A |
6.73 e-06 (TURB1); 8.623 e-04 (TURB4); 8.87 e-06 (TURB5) |
| B |
3.16 (TURB1); 1.43 (TURB4); 3.02 (TURB5) |
| G |
percent total dissolved gas saturation above equilibrium (100%) |
| Qs |
total amount of spill in kcfs |
| m, b |
empirically fit slope and intercept parameters |
| G |
percent total dissolved gas saturation above equilibrium (100%) |
| Qs |
total amount of spill in kcfs |
| qs |
amount of spill through an individual spillbay |
| a,b,c |
empirically fit model parameters |
| G |
percent total dissolved gas saturation above equilibrium (100%) |
| Qs |
total amount of spill in kcfs |
| a, b and k |
coefficients specific to each dam derived from TDG rating curves provided by the Bolyvong Tanovan of the U.S. Army Corps of Engineers |
| G |
percent total dissolved gas saturation above equilibrium (100%) |
| Qs |
total amount of spill in kcfs |
| a, b and h |
coefficients specific to each dam and can be derived from TDG rating curves available from the U.S. Army Corps of Engineers |
| Q |
total flow in kcfs |
| Qs |
spillway flow in kcfs |
| Gsb |
TDG concentration exiting the stilling basin in mg/l |
| Gfb |
TDG concentration in the forebay in mg/l |
| Geq |
TDG equilibrium concentration as a function of temperature (°C) at one atmosphere of pressure (mgl-1 atm-1) |
| L |
length of the stilling basin in feet |
| P0 |
barometric pressure in atmospheres (assume P0 is 1) |
|
density of water (0.0295 atm/ft) |
| 0 |
specific gravity of the roller at the base of the spill |
| W |
spillway width |
| D |
water depth at the end of the stilling basin |
| Y0 |
thickness of the spill at the stilling basin entrance, where |
| H |
hydraulic head expressing the forebay elevation minus the elevation of the spilling basin floor (H is in ft and gravity constant g is 32 ft s-2) |
|
differential pressure factor defined |
| Ke |
bubble entrainment coefficient with units of ft s-1atm-1/3 and is defined |
| T |
temperature (°C) |
| K20 |
temperature compensated entrainment coefficient |
| E |
energy loss rate expressed as total headloss divided by residence time of water in the stilling basin |
| P |
forebay percent saturation |
| a, b, and c |
dam dependent empirical coefficients |
| Gspill |
percent TDG in the spill side flow exiting the tailrace |
| Gphouse |
percent TDG in the powerhouse side flow exiting the tailrace |
| Sfr |
percent of river in the spill side flow |
| Gmix |
flow weighted average of two gas levels |
| Gdif |
difference between the original concentrations of the two flows |
| Gspill |
percent TDG in the spill side flow exiting the tailrace |
| Gphouse |
percent TDG in the powerhouse side flow exiting the tailrace |
| Gforebay |
percent TDG in the forebay |
| Qs |
total amount of spill flow |
| Gright , Gleft |
percent TDG in the flow entering the reach on the respective sides |
| Sfr |
percent of river in the right-bank flow |
| Gmix |
flow weighted average of the TDG values in each flow |
| Gdif |
difference between the original concentrations of the two flows at the head of the reach |
| E |
percent TDG in water at equilibrium, 100% saturation or 0% supersaturation |
| q |
diffusion rate constant in units of (mile)-1, a model parameter set for each reach |
| k |
dissipation rate constant in units of (day)-1, a model parameter calculated for each reach based on the river depth, velocity and a diffusion constant (see eq (107)) |
| x |
longitudinal distance, where x is in miles |
| v |
river velocity, in miles per day |
| Q |
total flow through the dam in kcfs |
| Qs |
spill flow in kcfs |
| G |
tailrace TDG supersaturation (in percent) |
| Gfb |
forebay TDG supersaturation (in percent) |
| Gsf |
spill water TDG in percent saturation as defined by an empirical or mechanistic saturation equation |
| Qi |
flow in kcfs in segment i |
| Gi |
TDG in percent supersaturation in segment i of the confluences |
|
flux of TDG across the air/water interface |
| G |
TDG supersaturation concentration in the segment |
| Geq |
TDG equilibrium concentration |
| A |
surface area of the segment |
| Kd |
transfer coefficient defined |
| Dm |
molecular diffusion coefficient of TDG |
| U |
hydraulic stream velocity |
| D |
depth of the segment |
| Dm |
order of 2 x 10-5 cm2s-1 (Richards 1965) |
| U |
order of 3 cm/s (20 miles/day), note this changes on a daily basis and for each reach in the model |
| D |
order of 900 cm, note this changes on a reach specific basis and is dependent on reservoir elevation |
| the constant 700.75 gives the coefficient k in unit of day-1 |
| Geq |
TDG equilibrium concentration |
| G(0) |
tailrace concentration defined by eq (109) |
| k |
dissipation coefficient defined by eq (115) |
| t |
time in a river segment |
| Volume |
pool volume at a specific elevation |
| W |
average pool width at full pool |
| L |
length of pool |
 t |
instantaneous probability of passage |
| p |
proportion of time step during day |
| (1-p) |
proportion of time step during night |
| Vt |
upstream river velocity in mi/day |
| SPt |
proportion of river spilled |
| Dt |
julian date |
's and  's |
parameters that vary by dam and species |
| Y |
fraction of total fish passed in spill |
| X |
fraction of water spilled |
| a and b |
regression coefficients |
| e |
error term (var) selected from random distribution |
| fge |
fish guidance efficiency |
| z |
median depth of fish in the forebay at a distance from the dam where fish are susceptible to being drawn into the intake |
| D |
screen depth relative to full pool forebay elevation |
| Dc |
FGE calibration parameter |
| E |
amount the pool is lowered below full pool elevation |
| t |
fish age since the onset of smoltification |
| t0 |
onset of change in FGE relative to the onset of smoltification, set in the Release window |
| t |
increment of time over which FGE changes |
| z0 |
initial mean fish depth (at age t equals 0) in the forebay |
| z1 |
final mean fish depth (at age t equals t0 + t) in the forebay |
| fge0 |
FGE at onset of smoltification |
| E(t) |
elevation drop |
| D |
fraction of fish that pass dam during spill hours |
| Fsp |
fraction of daily flow that passes in spill |
| SE |
fraction of fish that pass in spill relative to the fraction of flow passing in spill |
| FGE |
fraction of fish passing into turbine intake that are bypassed |
| x |
deterministic part of the random parameter fixed for each species and dam |
| x' |
stochastic part of the parameter taken from a broken-stick distribution (see Section II.7.1 Stochastic Parameter Probability Density) over each dam time slice |
| y |
spill efficiency |
| x |
percent flow |
| a and b |
deterministic parameters |
| e |
stochastic parameter selected from a normal distribution |
| Ntu |
number of fish passing in a time increment (6 hours) |
| Nfo |
number of fish in forebay ready to pass in the increment |
| p |
probability of passing during the increment (1 - P1 from eq (119)) |
| mfo |
mortality in forebay (see Section II.4.2 Predation Mortality) |
| mtu |
mortality in turbine passage |
| fge |
fish guidance efficiency for a day or night period |
| Y |
proportion of fish passage in spill defined by spill efficiency equation (see eq (120)) |
| mby |
mortality in the bypass |
| mtr |
mortality in the transport |
| msp |
mortality in the spill passage |
| T/C |
ratio of survival of transport fish to control fish from transport experiments |
| Vn |
in river survival of spring chinook smolts |
| a, b, Vt |
a = 5.8259, b = 5.3533, Vt = 0.98 |
| T/C |
ratio of survival of transport fish to control fish from transport experiments |
| Vn |
in river survival of spring chinook smolts |
| a |
0.3281 (TURB1); 0.3330 (TURB4); 0.3292 (TURB5) |
| b |
0.1936 (TURB1); 0.1596 (TURB4); 0.1868 (TURB5) |
| x |
unit uniform random deviate range 0 < x < 1 |
| yl |
lower limit of the distribution range |
| ym |
distribution of the median value |
| yu |
upper limit of the distribution range |
| r(t) |
determined from eq (48) |
| V(i) |
variance factor which is different for each release i |
| G |
percent supersaturation above 100% |
| Qs |
spillway flow volume in kcfs |
| a, b and h |
coefficients specific to each dam, derived from TDG rating curves provided by the U.S. Army Corps of Engineers |
| K20 |
entrainment coefficient |
| E |
energy loss rate |
| P |
forebay percent saturation |
| a, b, and c |
coefficients calculated from multiple linear regression of data in Table 51 |
| T |
water temperature in the forebay in °C |
| Qs |
spill in kcfs |
| W |
spillway width (gates x width/gate) |
| L |
stilling basin length in feet |
| T |
water temperature in the river segment |
| r(t) |
migration rate (miles/day) |
| t |
julian date |
| b0, b1, bFLOW |
migration rate regression coefficients |
| Vf |
average river velocity during the average migration period |
| 1, a2 |
slope parameters |
| TSEASN |
inflection point of flow dependent term (julian day) |
| TRLS |
release date (julian day) |
