Appendix – Biogeochemical model equations and parameters
For each layer and grid location the time evolution of each model substance (Y) is the sum of conservative advection, diffusion and
sinking processes (ϕY) and non-conservative biogeochemical rate processes (βY):
∂Y/∂t = -ϕY + βY
(1)
where ϕY = (uΔY+ΔHKHΔY) - (∂/∂z)KZ(∂Y/∂z) + (∂wSY/∂z)
and
Δ = (∂/∂x) + (∂/∂y) + (∂/∂z) ; ΔH = (∂/∂x) + (∂/∂y)
[Here u is the velocity vector; KH and Kz are the horizontal and vertical diffusion coefficients and wS is the settling velocity.]
Table A1: Non-conservative β terms for each biogeochemical model state variable (some at Redfield 106C:16N:1P or Atkinson ratio
550C:30N:1P).
Symbol State Variable
Processes
β term
(Ratio)
Nitrogen (mg N m-3)
B1
Small phytoplankton net growth, grazing,
 B1   B1  mB B1  g1
(R)
mortality
B2
Large phytoplankton net growth, grazing,
 B 2   B 2  mB B2  g 22 / renc
(R)
mortality
B3
Microphytobenthos
growth, grazing,
bB3 = (mB3 - mB3B3 ) B3 - g2j 4 / renc
(R)
mortality
Ns
Stored N (M. pyrifera) Net uptake, fixation,
bNs = (uNsN1 + uNsN 2 ) Qmin / N f - gma Ns - dM Ns
(-)
loss
Nf
Fixed N (M. pyrifera)
Net fixation,
bNf = gma Ns - dM N f
(A)
mortality
M
Benthic Macroalgae
net growth,
 M   M  mM M
(A)
mortality
S
Seagrass (A)
net growth,
 S   S  mS S
mortality
Z1
Small zooplankton
growth, mortality
 Z 1  g1e  mZ 1 Z12
(R)
Z2
Large zooplankton
growth, mortality
 Z 2  g 2 e  mZ 2 Z 22
(R)
N1
N2
Nitrate (-)
Ammonium (-)
uptake,
nitrification(wc+sed)
, denitrification(sed)
uptake, mortality,
defecation,
remineralisation,
nitrification(wc+sed)
bN1 = - (mB1B1 + mB2 B2 + mB3B3 + mS S + mM M + mBA ) N1 / ( N1 + N2 )
-uNsN1Qmin / N f + rN N2O / ( KO +O)

 1  K OD / K OD  O rNsed O 2 N 2 / K O2  O 2

bN 2 = - (mB1B1 + mB2 B2 + mB4 B3 + mS S + mM M + mBA ) N2 / ( N1 + N2 )
-uNsN 2Qmin / N f + dM N s + (1- g ) ( mZ1Z12 + mZ 2 Z22 ) + ( g1 + g2 ) (1- e - V )
+rL ( D1 + D2 ) (1- x - J ) + rR D3 (1- J 3 )
+rD N 3 - rN N 2O / ( KO + O) - rNsed O 2 N 2 / ( KO2 + O 2 )
N3
D1
Dissolved organic N (- remineralisation
)
Pelagic labile detritus mortality, defecation,
(R)
remineralisation
D2
Benthic labile detritus
(A)
D3
Refractory detrital N
(-)
Phosphorous (mg P m-3)
P1
Dissolved inorganic P
(-)
P2
E1
E2
Dissolved organic P ()
Refractory detrital P
(-)
Unflocculated
particulate inorganic
P (-)
 N 3  rL D1  D2   3 rR D3  rD N 3
bD1 = mB ( B1 + B2 ) + mB3 B32 + g ( mZ1Z12 + mZ 2 Z 22 )
+V ( g1 + g2 ) (1- e) + dM N f - rL D1 + m BA
mortality,
remineralisation
remineralisation
 D 2  mS S  mM M  rL D2
uptake, mortality,
defecation,
remineralisation,
desorption
remineralisation
bP1 = - ((mB1B1 + mB2 B2 + mB3B3 ) RWP + (mS S + mM M ) AWP )
 D 2  rL D1  D2   rR D3


 1    mZ 1 Z12  mZ 2 Z 22 RW P  g1  g 2 1  e   RW P   BA RW P
 rL D1 RW P  D2 AW P 1       rR E1 1  3   rD P2  d 2  d 3
 P 2  rL D1 RWP  D2 AWP   3 rR E1  rD P2  d 2  d 3
remineralisation
 E1  rL D1 RW P  D2 AW P   rR E1
desorption,
flocculation,
immobilization
 E 2  d 2  rF E2
 E 2  d 2  rF E2  rI E2 sediment
E3
Flocculated
particulate inorganic
P (-)
E4
Immobilized
particulate inorganic
P (-)
Carbon (mg C m-3)
C1
Dissolved inorganic C
(-)
desorption,
flocculation,
immobilization
immobilization
uptake, mortality,
defecation,
remineralisation
 E 3  d 3  rF E2
 E 3  d 3  rI E2  rI E3 sediment
 E 3  r4 E2  rE 4 E3
bC1 = - ((mB1B1 + mB2 B2 + mB3B3 ) RWC + ( mS S + mM M ) AWC )
+ (1- g ) ( mZ1Z12 + mZ 2 Z22 ) RWC
(
+ (1- e - V ) ( g1 + ( g2 / renc ) (j 2 + j 4 )) RWC
)
 rL D1 RW C  D2 AW C 1       rR F1 1  3   rD C 2
C2
Dissolved organic C ()
F1
Refractory detrital C
(-)
Oxygen (mg O m-3)
O
Dissolved oxygen (-)
remineralisation
-mBA RWC
 C 2  rL D1 RWC  D2 AWC   3 rR F1  rD C2
remineralisation
 F1  rL D1 RW C  D2 AW C   rR F1
growth,
mortality, defecation,
remineralisation
bP1 = (( mB1B1 + mB2 B2 + mB3 B3 ) RWO + (mS S + mM M ) AWO )


 
 1    mZ 1 Z 12  mZ 2 Z 22 RW O O / K OA  O 
(
(
))
+ (1- e - V ) ( g1 + ( g2 / renc ) (j 2 + j 4 )) RWO O / ( KOA +O)
(
)
- rL ( D1RWC + D2 AWO ) (1- x - J ) + ( rR D3 (1- J 3 ) + rD N3 + mBA ) RWO O / ( KOA +O)
Mineral Suspended Solids (kg TSS m-3)
T1
Unflocculated solids
flocculation, burial
(-)
T2
Flocculated solids (-)
flocculation, burial
T 1  rF T1
 T 1  rF  rb T1
T 2  rF T1
 T 2  rF T1  rbT2
for sediment
for sediment
Table A2: Rate processes included in the state variable equations.
Symbol Process
Equation
uNB1,2,3
Maximum phytoplankton u NB    N 1 N1   N 2 N 2 
uptake of N
(mg N s-1cell-1)
uPB1,2,3
Maximum phytoplankton u PB   P1 P1
uptake of P
(mg P s-1cell-1)
uIB1,2,3
Maximum supply of light u IB  I av k a m N  r Bmax m IB
to phytoplankton portion respired
(E s-1cell-1)
uIB3
Maximum supply of light u = I m 1- e-ka 4B3dz / B dz - rm max m
IB3
botS NB
3
B
IB
to benthic algae –
Portion respired
(E s-1cell-1)
uNM
Maximum benthic
u NM   N 1 N1   N 2 N 2  / 
macroalgae uptake of N
(mg N s-1m-2)
uPM
Maximum benthic
uPM   P1P1 / 
macroalgae uptake of P
(mg P s-1m-2)
uNsN1 = (VNO3 N1 / (K NO3 + N1 ))((Qmax -Q) / (Qmax -Qmin ))
uNsN1
M. pyrifera uptake of
-1
-1
nitrate (mg N g dw d )
uNsN 2 = (VNH 4 N2 / (K NH 4 + N2 ))((Qmax -Q) / (Qmax -Qmin ))
uNsN2
M. pyrifera uptake of
Ammonium
(mg N g-1 dw d-1)
(
(
)
)
uIM
uNS
uPS
uIS

Maximum supply of light
to benthic macroalgae –
portion respired (E s-1m2)
Maximum seagrass
uptake of N (mg N s-1m-2)
Maximum seagrass
uptake of P (mg P s-1m-2)
Maximum supply of light
to seagrass – portion
respired (E s-1m-2)
Autotroph growth (d-1)


u IB  I bot 1  e  k M M  r Mmax m IM
u NS  S Smax N1  N 2  / K N
u PS  S Smax P1 / K P


u IS  I botM 1  e  k S S  r Smax m IS
   max R N RP RI found in a look-up table of solutions to 3 simultaneous equations equating
uptake and growth (to avoid explicitly modelling relative resources RN RP RI for each autotroph):
u N 1  RN    max RN RP RI mN
u P 1  RP    max RN RP RI mP
gmp
M. pyrifera growth (d-1)
lI
Light limitation for
M. pyrifera (-)
lT
Temperature limitation
for M. pyrifera (-)
Nutrient limitation for
M. pyrifera (-)
Internal quotient of N
For M. pyrifera
(mg N g-1 dw)
Height of M. pyrifera (m)
lQ
Q
h
u I 1  RI    max RN RP RI mI
gmaxlLlT lQ maximum growth multiplied by limitation due to irradiance, temperature and N
e
(e
k*h
I e-kh
-z
Is
I
-z
-e Is )
1
1+exp[-ςp (T-Tp )]
g
Q-Qmin
Q-kc
𝑄𝑚𝑖𝑛 (1 + 𝑁𝑠 𝑁𝑓−1 )
max⁡((0.00174Nf /num_fronds)1.047 , 1)
 BA
 BA   BA max min u NBA , u PBA , u CBA 
Bacterial growth (d-1)
u NBA  ( N 1  N 2 ) 2 /( K NBA  N 1  N 2 )
u PBA  P 2 /( K PBA  P ) RW P
u CBA  rLM (C 3  F2 ) / RW C
g1
g2
 diff
 rmot
 shear
rF
d2,3
g1  min ring , renc Z 1 / Z 1m B1 / B1m  where:
Small zooplankton
grazing (mg N s-1)
(cell P cell Z-1 s-1)
(cell P cell Z-1 s-1)
ring   Zmax
1 eZ1m / B1m 
renc  B1 / B1m  diff 1   rmot1   shear1 
(
ring = m
max
Z2
e (( Z2 / B2m ) + ( Z2 / B4m ))
renc = ( B2 / B2m ) (j diff 2 + j rmot 2 + j shear 2 ) + ( B4 / B4m ) (j diff 4 + j rmot 4 + j shear 4 )
Diffusive encounter rate
(m3s-1cell Z-1)
Relative motion
encounter rate (m3s-1cell
Z-1)
Fluid shear encounter
rate (m3s-1cell Z-1)
Flocculation rate (s-1)
Desorption of P (mg P
3s-1)
)
g2 = min {ring, renc } ( Z2 / Z2m ) (( B2 / B2m ) + ( B4 / B4m )) where:
Large zooplankton
grazing (mg N s-1)
(cell P cell Z-1 s-1)
(cell P cell Z-1 s-1)
m-
 diff  2 BB.T / 3 K 2  rB / rZ  rZ / rB 
rmot   rB  rZ 2  wS2 / 3U P2 / 3U Z  U Z   U P2 / 3U Z  U Z 
shear  1.3K Z / K 0.5 rB  rZ 3
rF  rF 10
for SS > 10 psu
rF  rF10 SS  6 / 4 for 6 < SS < 10 psu
d 2  pT2 pak E2 / T2 pak  P1O / K PA  O
d 3  paT2 pak E3 / T2 pak  P1O / K PA  O
Table A3: Derived model variables.
Symbol Derived Variable
X
Phytoplankton chlorophyll (mg Chl m-3)
Equation
X = ( B1 + B2 + B4 ) QXN
Kd
Attenuation coefficient (m-1)
Iav
Layer mean PAR (E m-2 s-1)
Kd = kw + k fw (35- SS ) / 35+ ka1B1 + ka2 B2 + ka4 B4
+kdt ( D1 + D2 + D3 ) + kn N3 + kt (T1 +T2 ) + km M + acs N f
I av = (QQW / AV ) ( I top - I bot ) / Kd.dz
 Kd .dz
where I bot  I tope
IbotM
PAR below macroalgae (E m-2 s-1)
IbotS
PAR below seagrass (E m-2 s-1)
IbotB4
PAR below microphytobenthos (E m-2 s1)
I botM  QQW / AV I bot e kM M
I botS  QQW / AV I botM e kS S
I botB4  QQW / AV I botS e ka 4 B4dz
Table A4: Model parameters. All parameters were taken from the study by Wild-Allen et al. (2013) on SE Tasmanian estuary, except for the M. pyrifera parameters
which were from Hadley et al. (2014).
Symbol Autotroph Parameters
rB
μmax
r
k
Radius (m)
Max growth rate (d-1)
Respired fraction of μmax (-)
Specific absorption cross section (m2mg N -1)
Large Phytoplankton
B2
1.0E-5
1.25
0.025
0.0018
mB
Mortality term (d-1)
0.14
0.14
N/A
N/A
0.0001
(d-1(mgN m-3)-1)
N/A
N/A
N/A
N/A
KN
Half saturation constant for N uptake in sediment
(mg N m-3)
KP
Half saturation constant for P uptake in sediment
(mg P m-3)
Symbol Zooplankton Parameters
rZ
Radius (m)
Small Phytoplankton
B1
2.5E-6
1.25
0.025
0.0024
Micro-phytobenthos
B4
1.0E-5
1.35
0.025
0.0018
Sea Grass
S
N/A
0.1
0.025
1.0E-5
Macroalgae
M
N/A
0.02
0.025
0.001
0.00275
0.01
15.0
N/A
15.0
N/A
Large
Zoo- Small
Zoo- Optical Parameters
Value
plankton Z2
plankton Z1
5.0E-4
12.5E-6
Background attenuation of sea 0.1
water
e
Growth efficiency (-)
0.38
0.38
CDOM attenuation coefficient of 2.2
fresh water (m-1)
μmax
Maximum growth rate at 15˚C (d-1)
0.6
3.0
Detrital
specific
coefficient (m-1)
UZ
Swimming velocity (m)
1.5E-3
2.0E-4
TSS specific attenuation
coefficient
(m-1kg-1 m-3)
30
ζ
Fraction of growth inefficiency lost to detritus (-)
0.5
0.5
Dissolved organic nitrogen
specific attenuation coefficient
(m-1mgN-1m-3)
0.0009
Mortality (quadratic) rate
(d-1(mgN m-3) -1)
γ
Fraction of mortality lost to detritus (-)
Symbol Detritus & Remineralisation Parameters
0.004
0.02
0.5
Value
0.5
Symbol
M. pyrifera Parameters
Value
rL
Pelagic labile detritus breakdown rate (d-1)
0.1
g mp
Maximum growth rate (d−1 )
0.2
rR
Refractory detritus breakdown rate (d-1)
0.0036
VNH4
Maximum uptake rate (NH4 )
(mg⁡N⁡gdw −1 d−1)
8.0
rD
Dissolved organic matter breakdown rate
(d-1)
0.00176
VNO3
Maximum uptake rate (NO3 )
(mg⁡N⁡gdw −1 d−1)
10.3
L
Fraction of labile detritus converted to DOM (-)
0.01
K NH4
Half saturation constant (NH4 )
(mg⁡N⁡⁡m−3 )
74.2
ξ
Fraction of labile detritus converted to refractory 0.19
detritus (-)
K NO3
Half saturation constant (NO3 )
(⁡mg⁡N⁡⁡m−3 )
182.0
R
Fraction of refractory detritus converted to DOM (-)
Q max
Maximum internal nitrogen
25.0
mZ
0.01
attenuation 0.0038
(mg⁡N⁡gdw −1)
rN
Maximum water column nitrification rate
(d-1)
0.1
Minimum internal nitrogen
(mg⁡N⁡gdw −1)
7.0
rNsed
Maximum sediment nitrification rate (d-1)
20.0
Kc
Half growth constant
(mg⁡N⁡gdw −1)
6.0
KO
O2 half saturation rate for nitrification
(mg O m-3)
500.0
T0
Optimal Temperature (℃)
12.0
KOD
O2 half saturation rate for denitrification
(mg O m-3)
10000.0
Tr
Range of Optimal Temperature
(⁡℃)
1.0
KOA
O2 half saturation rate for aerobic respiration
(mg O m-3)
500.0
Saturation irradiance (E⁡m−2 s−1 )
0.000134
Nitrogen Specific Shading
(m2 ⁡mg⁡N −1 )
0.0001
dm
Mortality Rate (d−1 )
0.003
Num_fronds
Number of Fronds (-)
7.0
Q min
IS
acs
Table A5: Initial concentrations of autotrophs and zooplankton
Symbol
State Variable (mg m-3) ( *g m3)
Symbol
Small phytoplankton
Value
(Water
column,
sediment)
(0.1, 0.01)
B1
State Variable
(mg m-3)
M
Value
(Water
column,
sediment)
Benthic Macroalgae (10.0, N/A)
B2
Large phytoplankton
(1, 0.01)
S
Seagrass
(10.0, N/A)
B3
Microphytobenthos
(0.01, 100.0)
Z1
Small zooplankton
(0.5, 0.001)
Ns
Stored N (M. pyrifera) *
(0.1, N/A)
Z2
(0.5, 0.001)
Nf
Fixed N (M. pyrifera) *
(0.1, N/A)
Large
zooplankton