Bathtub Model Equations & Options 

Table of Contents
  Symbol Definitions
Output Variable Descriptions
  Conservative Substance Balance
  Phosphorus Sedimentation Models
  Nitrogen Sedimentation Models
  Chlorophyll-a Models
  Secchi Depth Models
  Longitudinal Dispersion Models
  Application of Phosphorus Calibration Factors
  Application of Nitrogen Calibration Factors
  Application of Nutrient Availability Factors
  Calculation of Nutrient Mass Balance Tables
  Equations for Other Trophic Response Variables

 

Conservative Substance Balance 
Option Description
0 Do Not Compute (Set Predicted = Observed) [default]
1 Compute Mass Balances

 

Phosphorus Sedimentation Models       (see discussion)

Unit P Net Sedimentation Rate (mg/m3-year) = CP A1 PA2 

Solution for Mixed Segment: 

     Second-Order Models (A2 = 2): 

              P = [-1 + (1  + 4 K A1 Pi T)0.5 ] / (2 K A1 T)

     First-Order Models (A2 = 1): 

              P = Pi / (1 + K A1 T)
 
Option Model Description A1 A2
0 Do Not Compute (Set Predicted = Observed) [default]  -  -
1 Second-Order, Available P       [default]
Inflow Avail P = 0.33 Pi + 1.93 Pio
See options for specification of available P

0.17 Qs/(Qs + 13.3)
Qs = Max(Z/T,4)

2
2 Second-Order Decay Rate Function
Fot = Tributary Ortho P / Total P Load
Requires specification of inflow total & ortho P loads		 0.056 Fot-1Qs/(Qs + 13.3)
Qs = Max(Z/T,4)		 2
3 Second-Order 0.10 2
4 Canfield & Bachman (1981), Reservoirs

0.114 (Wp/V)0.589

 1
5* Vollenweider (1976), Northern Lakes 

T-0.5

 1
6* Simple First-Order

1

 1
7* First-Order Settling 1/Z 1
8* Canfield & Bachman (1981), Natural Lakes

0.162 (Wp/V)0.458
9* Canfield & Bachman (1981), Reservoirs + Lakes

0.129 (Wp/V)0.549

 1

For purposes of computing effective rate coefficients (A1), Qs, Wp, Fot,  T, and V are evaluated separately for each segment group based upon external loadings and segment hydraulics.

* These models are not calibrated to CE reservoir data.  They are likely to require calibration by the user to site-specific data.

Nitrogen Sedimentation Models     (see discussion)

Unit N Net Sedimentation Rate (mg/m3-year) = CN B1 NB2 

Solution for Mixed Segment: 

     Second-Order Models (B2 = 2): 

              N = [-1 + ( 1  + 4 K B1 Ni T )0.5 ] / ( 2 K B1 T )

     First-Order Models (B2 = 1): 

              N = Ni / (1 + K B1 T)
 
Option Model Description B1 B2
0 Do Not Compute (Set Predicted = Observed) [default]  -  -
1 Second-Order, Available N [default]
Inflow Avail N = 0.59 Ni + 0.79 Nin
See options for specification of available N

0.0045 Qs/(Qs + 7.2)
Qs = Max(Z/T,4)

2
2 Second-Order Decay Rate Function
Fin = Tributary Inorganic N/Total N Load
Requires specification of inflow total & inorganic N loads		 0.0035 Fin-1Qs/(Qs + 17.3)
Qs = Max(Z/T,4)		 2
3 Second-Order

0.00315

 2
4* Bachman (1980), Volumetric Load

0.0159 (Wn/V)0.59

 1
5* Bachman (1980), Flushing Rate

0.693 T-0.55

 1
6* Simple First-Order

1

 1
7* First-Order Settling 1/Z 1

Nitrogen Model 1 differs slightly from that developed in Walker (1985).  The coefficients have been adjusted so that predictions will be unbiased if inflow inorganic nitrogen data are not available (inflow available N = inflow total N). These adjustments have negligible influence on model error statistics< /A > .

For purposes of computing effective rate coefficients (B1), Qs, Wn, Fin, T, and V are evaluated separately for each segment group based upon external loadings and segment hydraulics.

* These models are not calibrated to CE reservoir data. They are likely to require calibration by the user to site-specific data.

 

Application of Nutrient Availability Factors    (see discussion)
Option Description Equations
0 Do Not Apply Availability Factors [default] Select if Ortho P & Inorganic N loadings are not specified

       Inflow Available P  =  Pi
       Inflow Available N =  Ni
 
1 Apply to P & N Sedimentation Model 1 Only When P Model 1 or N Model 1 is selected, calculate nutrient balances based upon available nutrient loads:

     Inflow Available P = 0.22 Pi + 1.93 Pio
     Inflow Available N = 0.59 Ni + 0.79 Nin

Require specification of ortho P & inorganic N loadings if P or N sedimentation model 1 is selected.
Coefficients used to compute availability factors can be edited on the 'Edit Model Coefficients' screen.

With other P or N models, nutrient balances using Total P and Total N loads (same as Option 0)
2 Apply to All P & N Sedimentation Models
Except Model 2		 Same equations as Option 1
Requires Specification of Ortho P & Inorganic N Loadings

Account for differences between dissolved & particulate nutrient forms with respect to sedimentation rates and/or bioavailability. 
Although consideration of these factors reduced prediction error in the CE model development data set, the default option (0) ignores them because  ortho P and inorganic N loadings are not typically measured.

 

Chlorophyll-a Models   (see discussion)

 Applicability Constraints
Option Description / Limiting Factors Equations a (N-150)/P Ninorg/Portho Fs
0 Do Not Compute Predicted = Observed        
1 P, N, Light, Flushing

Xpn  =  [ P-2 + ((N-150)/12)-2 ]-0.5

Bx  =  Xpn1.33 / 4.31

G  =  Zmix (0.14 + 0.0039 Fs)

B  =  K Bx /[ (1 + b Bx G) (1 + Ga) ]

        
2 P, Light, Flushing [default]

Bp = P1.37/4.88                                                

G = Zmix (0.19 + 0.0042 Fs)

B = K Bp / [(1 + b Bp G) (1  + Ga)]

   >12 >7  
3 P, N, Low Turbidity B = K 0.2 Xpn1.25    <0.9     <25
4 P, Linear B = K 0.28 P                    <0.9 >12 >7 <25
5 P, Exponential, Jones & Bachman (1976) B = K 0.081 P1.46 <.4 >12 >7 <25
6 P, Carlson TSI (1977), Lakes B = K 0.087 P1.45 <0.4 >12 >7 <25

Options 1 & 2 require estimates of non-algal turbidity for each model segment.  These are entered with observed water quality data on the 'Edit Segments' screen.  If non-algal turbidity is not specified, it is estimated from observed Secchi depth and chlorophyll-a.   If the latter are not specified, an error message is generated.

 

Secchi Depth Models     (See discussion) Applicability Constraints
Option Description Equations (N-150)/P Ninorg/Portho
0 Do Not Compute Predicted = Observed    
1 Secchi vs. Chl a and Turbidity [default] S = K / (a + b B)    
2 Secchi vs. Composite Nutrient          S = K 16.2 Xpn-0.79    
3 Secchi vs. Total P, CE Reservoirs S = K 17.8 P -0.76 >12 >7
4* Carlson TSI (1977) , Lakes S = K 48 / P >12 >7

 

Longitudinal Dispersion Models     (see discussion)
Option Description Equations
0 Do Not Compute E = 0
1 Fischer et al. (1979) Dispersion Equation
as adapted by Walker (1985)    [default]		 Width                      W = As/L
Cross-Section           Ac = W Z
Velocity                   U = Q/Ac
Dispersion                D = KD 100 W2 Z-0.84 Maximum (U,1)
Numeric Dispersion    Dn = U L/2
Exchange                 E = MAX( D - Dn , 0 ) Ac/L
2 Fixed Dispersion Rate Same as Model 1, except with fixed D = 1,000 km2/year
D = 1000 KD
3 Input Exchange Rates Directly E = KD
4 Fischer Equation, Not Adjusted for Numeric Dispersion E = D Ac/L                (D as defined in Option 1)
5 Constant Dispersion, Not Adjusted for Numeric Dispersion E = 1,000 KD Ac / L

Estimate Exchange Flows (E) between Adjacent Segment Pairs.
For all options, E = 0. always for segments discharging out of network     (outflow segment number = 0).

 

Error Analysis  (see discussion)
Option Description
0 Do Not Perform (Output CV's = 0)
1 Consider Model Error & Input Error  [default]
2 Consider Model Error Only (reflect inherent model error only)
3 Consider Input Error Only (reflect uncertainty in user-specified inputs only)

 

Application of Phosphorus Calibration Factors   (see discussion)
Option Description
0 Apply Calibration Factors to Predicted Sedimentation Rates  [default]
1 Apply Calibration Factors to Predicted Concentrations

 

Application of Nitrogen Calibration Factors   (see discussion)
Option Description
0 Apply Calibration Factors to Predicted Sedimentation Rates  [default]
1 Apply Calibration Factors to Predicted Concentrations

 

Calculation of Mass Balance Tables   (see discussion)
Option Description
0 Use Observed Segment Concentrations to Calculate Outflow and Storage Terms
Mass balance tables are based entirely on observed inflow, outflow, & segment concentrations specified by the user (independent of predicted reservoir or outflow concentrations).  If observed outflow or segment concentrations are missing, predicted values are used.
1  Use Predicted Segment Concentrations to Calculate Outflow and Storage Terms [default]

Influences output from the 'List Balances ' procedures.  Does not influence predicted concentrations.

 

Equations for Other Trophic Response & Morphometric Variables   (see discussion  & output variable descriptions) 

Variable

 Equations

Organic Nitrogen

 Norg =  K ( 157 + 22.8 B + 75.3 a )

Total P - Ortho P

 P - Portho = K Maximum [ -4.1 + 1.78 B + 23.7 a , 1 ]

Hypolimnetic Oxygen Depletion Rate (Near-Dam)

 HODv = K 240 Bm.5 / Zh   (for Zh > 2 m)

Metalimnetic Oxygen Depletion Rate (Near-Dam)

 MODv = K 0.4 HODv Zh 0.38
Principal Components
With Chl-a, Secchi, Nutrient, & Organic N Data		 PC-1 = 0.554 log(B) + 0.359 log(Norg) + 0.583 log(Xpn) - 0.474 log(S)  
PC-2 = 0.689 log(B) + 0.162 log(Norg) - 0.205 log(Xpn) + 0.676 log (S) 
Principal Components
With Chl a & Secchi Data Only		 PC-1 = 1.47 + 0.949 log(B) - 0.932 log(S)
PC-2 = 0.13 + 0.673 log(B) + 0.779 log(S)
Trophic State Indices (Carlson 1977) TSIp  =  4.15 + 14.42 ln(P)
TSIc  =  30.6 + 9.84 ln(B)
TSIs  =  60.0 - 14.41 ln(S)

Algal Bloom Frequencies (Walker 1984)
Percent of time during growing season that Chl-a exceeds bloom criteria of 10, 20, 30, 40, 50, or 60 ppb.

Calculated from Mean Chl-a (B) assuming that temporal variations in chl-a
are represented by a log-normal frequency distribution with a 
coefficient of variation = 0.62 (user-adjustable via the Model Coefficients< /A > screen)
Non-Algal Turbidity a = 1/S - b B  (minimum value = 0.08 1/m)
Applied to observed S and B values in each segment if non-algal turbidity values are not input directly (=0) on the Edit Segments screen.  The parameter b (default = 0.025) is entered on the Model Coefficients screen.
Mean Depth of Mixed Layer

log (Zmix) =       -0.06 + 1.36 log (Z) - 0.47 [log (Z)]2        (R2 = 0.93, SE2 = 0.0026)
     Constraint:     Zmix <= Z 
Used to estimate Zmix value for each segment if not input directly