DMSTA Phosphorus Cycling Model

One CSTR at Steady-State

Unit Area Storage & Fluxes

Concs in mg/m3

Fluxes in mg/m2-yr

Storage in mg/m2

Water Column

Mass = M

Q C

Conc = C = M / Z

F_Z K₁ S C

K₂ S²

F_Z = Min ( 1, Z / Z_X )

Biomass P Storage

K₃ S

State Variables:

Water Column P Storage

mg/m2

Temporary P Storage in Biota, etc.

mg/m2

Water Column Mean Depth

Driving Variables:

P Load, Including Atmos. Deposition

mg/m2-yr

Outflow

m/yr

Parameter Values:

K₁

Maximum Uptake Rate

m3 / mg - yr

K₂

Recycle Rate

m2 / mg-yr

K₃

Burial Rate

1/yr

Steady-State Mass Balances 1 CSTR & F_Z = 1:

Storage:

K1 C = K2 S + K3

Overall:

L - Q C = K3 S

Solution for C:

C = ( K2 L + K3² ) / ( K3 K1 + Q K2 )

Solution for S:

S = ( K1 C - K3 ) / K2

For Parameter-Estimation Purposes, Model Coefficients Are Re-expressed as Follows:

Net Settling Rate at Steady State

m/yr

Fit to Concentration Time Series Data

Water Column Conc at S = 0 mg/m2

ppb

Estimated from Storage vs. Concentration Relationship

Water Column Conc at S = 1000 mg/

ppb

Estimated from Storage vs. Concentration Relationship

Steady-State Solution:

C = ( L + K C0 ) / ( K + Q )

S = 1000 ( C - C0 ) / ( C1 - C0 )

used to calibrate C0 & C1

The steady-state solution is equivalent to the familiar K/C* model with C* = C0 :

Q C

K ( C - C0 )

where,

K = K2 K1 / K2

Translations of Parameter Sets:

K3 = K ( C1 - C0 ) / 1000

K = K1 K3 / K2

K1 = K3 / C0

C0 = K3 / K1

K2 = K3 K1 / K

C1 = (1000 K2 + K3 ) / K1

Generalized Model for Non-Emergent Systems

With the SAV calibration (C0 = 12 ppb), the model predicts that storage and net removal rate will approach zero as conc approaches 12 ppb.

With the PSTA calibration (C0 = 4 ppb), net removal continues at concentrations between 4 and 12 ppb.

Given that both SAV and PSTA are actually a mixture of submersed aquatics and periphyton, it is reasonable to assume that periphyton become

increasingly dominant in non-emergent systems as water column concentrations approach zero.

The generalized calibration for non-emergents switches from the SAV to the PSTA calibrations as the average concentration decreases.

This parameter set is labeled (NEWS) for NonEmergent Wetland System.

The transition is represented by a logistic function that is keyed to the predicted storage value:

F_SAV = 1 / ( 1 + exp [ - ( S - S_M ) / S_B ] )

F_PSTA = 1-F_SAV

S_M=

transition midpoint ( 50% SAV & 50 % PSTA ) =

400

mg/m2

S_B=

transition band width =

mg/m2

Parameter Estimation Results:

Vegetation Type / Parameter Set

EMERG

PSTA

SAV

NEWS

Dataset

Boney Marsh

PSTA-8

Cell-4

WC Conc at 0 g/m2 Storage

ppb

Paired C vs. S Data

WC Conc at 1 g/m2 Storage

ppb

Paired C vs. S Data

Net Settling Rate at Steady State

m/yr

129

Fit to Conc Timeseries

Z_X

Depth Scale

Residuals

S_M

Transition Midpoint

mg/m2

#N/A

400

S_B

Transition Bandwith

mg/m2

#N/A

Uptake Rate

m3/mg-yr

0.0707

0.1071

0.0707

Storage-Dependent

= K3 / C0

Recycle Rate

m2 / mg-yr

0.0013

0.0019

0.0013

Storage-Dependent

= K3 K1 / K

Burial Rate

1/yr

0.2826

0.4284

0.2826

Storage-Dependent

= K (C1 - C0 ) / 1000

Depth Multiplier for Gross Uptake Rate:

Thought to Reflect Hydraulic Effects (increased short circuiting at shallow depths)

Effective surface area for P uptake increases with depth in SAV & emergent systems, but not for periphyton mats

F_Z = Min ( 1, Z / Z

, = 1

if Z_X = 0

Z_X = Depth Scaling Factor (cm)

= 0 for PSTA , 60 cm for SAV & Emergents

Z = Mean Depth (cm)

Z_X

Substrate

SAV/Emerg

PSTA

0.00

1.00

0.17

1.00

0.33

1.00

0.50

1.00

0.67

1.00

0.83

1.00

100

1.00

Steady-State Solutions of P Cycling Model for Each Vegetation Type

Results Shown Over Calibration Range of Each Vegetation Type, and <= 200 ppb.

Mean Depth =

Storage (mg/m2)

= ( Fz K1 C - K3 ) / K2

Storage Response Time (yrs)

= -ln (0.1) / ( K2 S + K3 )

Net P Removal (mg/m2-yr)

= K3 S

Net Settling Rate, C* = 0 (m/yr)

= K3 S / C

6/21/2003