1/17
Response of the biological pump to perturbationsin the iron supply: Global teleconnections
diagnosed using an inverse model of the coupledphosphorus-silicon-iron nutrient cycles.
Benoıt Pasquier and Mark Holzer
School of Mathematics and Statistics
2/17
Motivation
Iron is a key limiting nutrient in HNLC regions [Boyd et al., 2007].
Diatom
60E 120E 180 120W 60W 090S
60S
30S
EQ
30N
60N
90N
dFe
Question: How do the global nutrient cycles respond to perturbationsin the iron supply?
High-latitude control on biological productivity (dFe not modeled)[Sarmiento et al., 2004; Primeau et al., 2013; Holzer and Primeau, 2013]Iron input perturbations using forward models
[e.g., Dutkiewicz et al., 2005; Nickelsen and Oschlies, 2015]We built a data-constrained, inverse model that couples P, Si, and Fe.
3/17
Model: Tracer Equations
T χP =∑c(SP
c − 1)Uc − γg(χP − χobsP )
T χSi = (SSi − 1)RSi:PUdia − γg(χSi − χobsSi )
T χFe =∑c(SFe
c − 1)RFe:Pc Uc + sA + sS + sH
+ (SsPOP − 1)JPOP + (SsbSi − 1)JbSi − Jdst
T = advective-diffusive transport(data-assimilated [Primeau et al., 2013])
dFe
dFe′ dFeL
Dust & Combustion
sA
sH
sS
UFe
Jdst
Jorg + JbSi
3 sources of iron:Aeolian, sA
Sedimentary, sS
Hydrothermal, sH
3 sinks:POP, JPOP
Opal, JbSi
Dust, Jdst
3/17
Model: Tracer Equations
T χP =∑c(SP
c − 1)Uc − γg(χP − χobsP )
T χSi = (SSi − 1)RSi:PUdia − γg(χSi − χobsSi )
T χFe =∑c(SFe
c − 1)RFe:Pc Uc + sA + sS + sH
+ (SsPOP − 1)JPOP + (SsbSi − 1)JbSi − Jdst
T = advective-diffusive transport(data-assimilated [Primeau et al., 2013])
dFe
dFe′ dFeL
Dust & Combustion
sA
sH
sS
UFe
Jdst
Jorg + JbSi
3 sources of iron:Aeolian, sA
Sedimentary, sS
Hydrothermal, sH
3 sinks:POP, JPOP
Opal, JbSi
Dust, Jdst
3/17
Model: Tracer Equations
T χP =∑c(SP
c − 1)Uc − γg(χP − χobsP )
T χSi = (SSi − 1)RSi:PUdia − γg(χSi − χobsSi )
T χFe =∑c(SFe
c − 1)RFe:Pc Uc + sA + sS + sH
+ (SsPOP − 1)JPOP + (SsbSi − 1)JbSi − Jdst
T = advective-diffusive transport(data-assimilated [Primeau et al., 2013])
SmallLargeDiatom
c = phytoplankton classMatsumoto et al., [2013]
dFe
dFe′ dFeL
Dust & Combustion
sA
sH
sS
UFe
Jdst
Jorg + JbSi
3 sources of iron:Aeolian, sA
Sedimentary, sS
Hydrothermal, sH
3 sinks:POP, JPOP
Opal, JbSi
Dust, Jdst
3/17
Model: Tracer Equations
T χP =∑c(SP
c − 1)Uc − γg(χP − χobsP )
T χSi = (SSi − 1)RSi:PUdia − γg(χSi − χobsSi )
T χFe =∑c(SFe
c − 1)RFe:Pc Uc + sA + sS + sH
+ (SsPOP − 1)JPOP + (SsbSi − 1)JbSi − Jdst
T = advective-diffusive transport(data-assimilated [Primeau et al., 2013])
SmallLargeDiatom
c = phytoplankton classMatsumoto et al., [2013]
dFe
dFe′ dFeL
Dust & Combustion
sA
sH
sS
UFe
Jdst
Jorg + JbSi
3 sources of iron:Aeolian, sA
Sedimentary, sS
Hydrothermal, sH
3 sinks:POP, JPOP
Opal, JbSi
Dust, Jdst
3/17
Model: Tracer Equations
T χP =∑c(SP
c − 1)Uc − γg(χP − χobsP )
T χSi = (SSi − 1)RSi:PUdia − γg(χSi − χobsSi )
T χFe =∑c(SFe
c − 1)RFe:Pc Uc + sA + sS + sH
+ (SsPOP − 1)JPOP + (SsbSi − 1)JbSi − Jdst
T = advective-diffusive transport(data-assimilated [Primeau et al., 2013])
SmallLargeDiatom
c = phytoplankton classMatsumoto et al., [2013]
dFe
dFe′ dFeL
Dust & Combustion
sA
sH
sS
UFe
Jdst
Jorg + JbSi
3 sources of iron:Aeolian, sA
Sedimentary, sS
Hydrothermal, sH
3 sinks:POP, JPOP
Opal, JbSi
Dust, Jdst
3/17
Model: Tracer Equations
T χP =∑c(SP
c − 1)Uc − γg(χP − χobsP )
T χSi = (SSi − 1)RSi:PUdia − γg(χSi − χobsSi )
T χFe =∑c(SFe
c − 1)RFe:Pc Uc + sA + sS + sH
+ (SsPOP − 1)JPOP + (SsbSi − 1)JbSi − Jdst
T = advective-diffusive transport(data-assimilated [Primeau et al., 2013])
SmallLargeDiatom
c = phytoplankton classMatsumoto et al., [2013]
dFe
dFe′ dFeL
Dust & Combustion
sA
sH
sS
UFe
Jdst
Jorg + JbSi
3 sources of iron:Aeolian, sA
Sedimentary, sS
Hydrothermal, sH
3 sinks:POP, JPOP
Opal, JbSi
Dust, Jdst
4/17
Model: Nutrient Uptake and Limitation
PO4-uptake function of temperature, light and nutrient availability
Uc = µc pc = pmaxc
τceκT
(FI FN,c
)2
(Derived from a logistic equation [Dunne et al., 2005])Nutrient limitation: product of Monod factors for each nutrient i
FN,c =∏i
χiχi + kic
≡∏i
(1−Dic)
Diatom
60E 120E 180 120W 60W 090S
60S
30S
EQ
30N
60N
90N
Large
90S
60S
30S
EQ
30N
60N
90N
Small
60E 120E 180 120W 60W 0
DPc
1 DFec
1
DSic
1
4/17
Model: Nutrient Uptake and Limitation
PO4-uptake function of temperature, light and nutrient availability
Uc = µc pc = pmaxc
τceκT
(FI FN,c
)2
(Derived from a logistic equation [Dunne et al., 2005])Nutrient limitation: product of Monod factors for each nutrient i
FN,c =∏i
χiχi + kic
≡∏i
(1−Dic)
Diatom
60E 120E 180 120W 60W 090S
60S
30S
EQ
30N
60N
90N
Large
90S
60S
30S
EQ
30N
60N
90N
Small
60E 120E 180 120W 60W 0
DPc
1 DFec
1
DSic
1
5/17
Parameter optimization: Inverse Modelling34 BGC parameters are adjusted to minimize the mismatch withobserved nutrient and phytplankton concentrations, χobs
i and pobsc :
cost =∑i
ωi
∫dV (χmod
i − χobsi )2 +
∑c
ωc
∫dV (pmod
c − pobsc )2.
Parameter optimization requires to solve the tracer equationsthousands of times!Solution: we use a Newton Solver [e.g., Kelley, 2003] to solve thediscretized nonlinear equations (∼600 000 equations and unknowns)which converges typically in ∼10 iterations.
No spin-up ⇒ fast!
f(x)
xx0x1
f(x0)
6/17
Results: Estimates of the current state of the ocean
Iron sources in literature:2 orders of magnitude range
[Tagliabue et al., 2015]
We chose a rangeof iron sources:
sA ∼ 0–15 GmolFe/yrsS ∼ 0–12 GmolFe/yrsH ∼ 0–3 GmolFe/yr
All consistent with observations:PO4: RMS ∼ 0.1 mmol/m3 (5 %)Si(OH)4: RMS ∼ 10 mmol/m3 (12 %)dFe: RMS ∼ 0.28 nM (43 %)
nM
0 0.2 0.4 0.6 0.8 1 1.2
de
pth
[km
]
6
5
4
3
2
1
0
de
pth
[km
]
6
5
4
3
2
1
0
de
pth
[km
]
6
5
4
3
2
1
0
SP 60S 30S EQ 30N 60N NP
ATL
PAC
IND
Total
0 0.5 1
nM
Aeolian
0 0.5 1
nM
Sedimentary
0 0.5 1
nM
sA
/ stot
(%)0 50 100
Hydrothermal
0 0.5 1
nM
[dFe]
7/17
Results: Carbon and Opal Export Productions are wellconstrained
0 100 200
TgC yr1
/ deg.
gC
m
2yr
1
0
20
40
60
80
100
0 5 10
Tmol Si yr1
/ deg.
mo
l S
i m
2yr
1
0
0.4
0.8
1.2
1.6
2
Longitude
La
titu
de
60E 120E 180 120W 60W 090
60
30
0
30
60
90
Longitude
La
titu
de
60E 120E 180 120W 60W 090
60
30
0
30
60
90
Cexport ~10.69 (9.9310.97) PgC /yr Zonal Integral
Siexport ~174. (168.177.) Tmol Si / yr Zonal Integral
8/17
Perturbations of the Tropical Aeolian Iron Supply
Map of aeolian iron source pattern [Luo et al., 2008]:
log
10(m
ol F
e m
2yr
1)
6.5
6
5.5
5
4.5
4
90
60
30
0
30
60
90
Latitu
de
Longitude
60E 120E 180 120W 60W 0
a) Aeolian source, sA
~ 1.9 (0.612.7) Gmol Fe yr1
Between 30S and 30N, we multiply the aeolian iron source by α, rangingfrom 0–100.
sA −→ αsA
8/17
Perturbations of the Tropical Aeolian Iron Supply
Map of aeolian iron source pattern [Luo et al., 2008]:
log
10(m
ol F
e m
2yr
1)
6.5
6
5.5
5
4.5
4
90
60
30
0
30
60
90
Latitu
de
Longitude
60E 120E 180 120W 60W 0
a) Aeolian source, sA
~ 1.9 (0.612.7) Gmol Fe yr1
Between 30S and 30N, we multiply the aeolian iron source by α, rangingfrom 0–100.
sA −→ αsA
9/17
Southern Ocean Trapping and Path Density Diagnostic
Southern Ocean (< 38 S) nutrient trapping [Holzer et al., 2014]:
Preformed nutrients are “leaked” fromthe SO via mode and intermediate waters
Regenerated nutrients are remineralizedwithin upwelling circumpolar deepwater(and trapped)
Path Density from uptake in region Ωi to uptake in region Ωf :
Concentration of nutrient i lasttaken up in SO: g↓
i (r|SO)
Fraction of nutrient to be nexttaken up in SO: f↑
i (r|SO)
Path density of nutrient “trapped”in SO→SO transit:
ηi(r|SO→ SO) = g↓i (r|SO)f↑(r|SO)
10/17
Perturbations: Increased iron input ⇒ Increased SO Trapping
NP SP
mmol P / m3
0.1 0.3 0.5 0.7 0.9
Latitude
SP 60S 30S EQ 30N 60N NP
mmol P / m3
0.1 0.3 0.5 0.7 0.9
Latitude
SP 60S 30S EQ 30N 60N NPNP SP
mmol Si / m3
10 30 50 70 90
Latitude
SP 60S 30S EQ 30N 60N NP
mmol Si / m3
10 30 50 70 90
Latitude
SP 60S 30S EQ 30N 60N NP
ηSi
(SO→SO), base case
mmol Si / m3
12 6 0 6 12
Depth
(km
)
Latitude
δηSi
(SO→SO), TROx0
SP 60S 30S EQ 30N 60N NP6
5
4
3
2
1
0
mmol Si / m3
12 6 0 6 12
Latitude
δηSi
(SO→SO), TROx100
SP 60S 30S EQ 30N 60N NP
ηP(SO→SO), base case
mmol P / m3
0.4 0.2 0 0.2 0.4
Depth
(km
)
Latitude
δηP(SO→SO), TROx0
SP 60S 30S EQ 30N 60N NP6
5
4
3
2
1
0
mmol P / m3
0.4 0.2 0 0.2 0.4
Latitude
δηP(SO→SO), TROx100
SP 60S 30S EQ 30N 60N NP
11/17
Perturbations: Export Production Response
Tg
C /
de
g.
/ yr
0
40
80
120
160
200
Tm
ol S
i / d
eg
. /
yr
0
1
2
3
4
5
Latitu
de
C-export TRO pert.
90S
60S
30S
Eq
30N
60N
90N
Latitu
de
α
Si-export TRO pert.
90S
60S
30S
Eq
30N
60N
90N
0.01 0.1 1 10 1000.01 0.1 1 10 1000
50
100
150
200
α
Tm
ol S
i/ yr
Si export TRO pert
Pg C
/ y
rC export TRO pert
0.01 0.1 1 10 1000.01 0.1 1 10 100
0
5
10
15
typicallow Sed.
typicallow Sed.
C-export zonal integral
Si-export zonal integral
C-export global integral
Si-export global integral
12/17
Perturbations: Regenerated Nutrient Response
NP SP
mmol P / m3
0.2 0.6 1 1.4 1.8
LatitudeSP 60S 30S EQ 30N 60N NP
mmol P / m3
0.2 0.6 1 1.4 1.8
LatitudeSP 60S 30S EQ 30N 60N NP
α0.01 0.1 1 10 1000
0.2
0.4
0.6
0.8
typ
Low S
mmol Si / m3
20 60 100 140 180
Latitude
reg
SP 60S 30S EQ 30N 60N NP
mmol Si / m3
10 30 50 70 90
Latitude
reg
SP 60S 30S EQ 30N 60N NPNP SP
α0.01 0.1 1 10 1000
0.2
0.4
0.6
0.8
typ
Low S
Preg
, base case
mmol P / m3
0.6 0.4 0.2 0 0.2 0.4 0.6
De
pth
(km
)
Latitude
δPreg
, TROx0
SP 60S 30S EQ 30N 60N NP6
5
4
3
2
1
0
mmol P / m3
0.6 0.4 0.2 0 0.2 0.4 0.6
Latitude
δPreg
, TROx100
SP 60S 30S EQ 30N 60N NP
E (P)
E (Si) Sireg
, base case
mmol Si / m3
8 4 0 4 8
De
pth
(km
)
Latitude
δSireg
, TROx0bio
bio
SP 60S 30S EQ 30N 60N NP6
5
4
3
2
1
0
mmol Si / m3
8 4 0 4 8
Latitude
δSireg
, TROx100
SP 60S 30S EQ 30N 60N NP
13/17
Summary and Conclusions
We have an efficient model coupling the P, Si, and Fe cycles,embedded in a data-assimilated steady circulation:
Computational efficiency allows for optimization of BGC parameters(inverse modelling) and for numerous perturbation experiments.The current sparse dFe observations are consistent with a large range ofiron source strengths.
Global response to tropical perturbations of the aeolian iron input:The initial state of the unperturbed iron cycle (e.g., low sedimentarysource) determines the sensitivity of nutrient cycles to perturbations.Tropical perturbations have a strong high-latitude influence, particularlyfor Southern Ocean productivity and nutrient trapping.Increased tropical aeolian Fe input plugs the Southern Ocean leak of thebiological pump.The Si-cycle is less sensisitive to iron perturbations than the P-cyclebecause changes of the Si:P uptake ratio compensates changes in exportproduction.
14/17
Discretized PDE
The tracer equation is ∂tx = f(x), where x represents the(3-dimensional) concentration fields of the 3 nutrients, rearranged intoa single column vector of size n ∼ 600,000
The function f combines the advective-eddy-diffusive transport
, thebiological cycling (and biogenic transport), and the iron sinks andexternal sources
f(x) =
transport (linear)
−
TT
T
p
sf
=
x
biology (nonlinear)
+∑c
bPc
bSic
bFec
sources and sinks (nonlinear)
+
00
sA + sH + sS − jsc
We solve the steady state equation f(x) = 0 using Newton’s Method,i.e. we solve 600,000 equations in 600,000 unknowns!
14/17
Discretized PDE
The tracer equation is ∂tx = f(x), where x represents the(3-dimensional) concentration fields of the 3 nutrients, rearranged intoa single column vector of size n ∼ 600,000The function f combines the advective-eddy-diffusive transport
, thebiological cycling (and biogenic transport), and the iron sinks andexternal sources
f(x) =
transport (linear)
−
TT
T
p
sf
=
x
biology (nonlinear)
+∑c
bPc
bSic
bFec
sources and sinks (nonlinear)
+
00
sA + sH + sS − jsc
We solve the steady state equation f(x) = 0 using Newton’s Method,i.e. we solve 600,000 equations in 600,000 unknowns!
14/17
Discretized PDE
The tracer equation is ∂tx = f(x), where x represents the(3-dimensional) concentration fields of the 3 nutrients, rearranged intoa single column vector of size n ∼ 600,000The function f combines the advective-eddy-diffusive transport, thebiological cycling (and biogenic transport)
, and the iron sinks andexternal sources
f(x) =
transport (linear)
−
TT
T
p
sf
=
x
biology (nonlinear)
+∑c
bPc
bSic
bFec
sources and sinks (nonlinear)
+
00
sA + sH + sS − jsc
We solve the steady state equation f(x) = 0 using Newton’s Method,i.e. we solve 600,000 equations in 600,000 unknowns!
14/17
Discretized PDE
The tracer equation is ∂tx = f(x), where x represents the(3-dimensional) concentration fields of the 3 nutrients, rearranged intoa single column vector of size n ∼ 600,000The function f combines the advective-eddy-diffusive transport, thebiological cycling (and biogenic transport), and the iron sinks andexternal sources
f(x) =
transport (linear)
−
TT
T
p
sf
=
x
biology (nonlinear)
+∑c
bPc
bSic
bFec
sources and sinks (nonlinear)
+
00
sA + sH + sS − jsc
We solve the steady state equation f(x) = 0 using Newton’s Method,i.e. we solve 600,000 equations in 600,000 unknowns!
15/17
Path densities: Definition
Regenerated nutrient = remineralized at depth that has not yetreemerged in the euphotic zoneThe path density 〈ηreg(r)〉 is the concentration of regeneratednutrients at r last taken up Ωi that is destined to reemergence in Ωf
15/17
Path densities: Definition
Regenerated nutrient = remineralized at depth that has not yetreemerged in the euphotic zoneThe path density 〈ηreg(r)〉 is the concentration of regeneratednutrients at r last taken up Ωi that is destined to reemergence in Ωf
15/17
Path densities: Definition
Regenerated nutrient = remineralized at depth that has not yetreemerged in the euphotic zoneThe path density 〈ηreg(r)〉 is the concentration of regeneratednutrients at r last taken up Ωi that is destined to reemergence in Ωf
15/17
Path densities: Definition
Regenerated nutrient = remineralized at depth that has not yetreemerged in the euphotic zoneThe path density 〈ηreg(r)〉 is the concentration of regeneratednutrients at r last taken up Ωi that is destined to reemergence in Ωf
15/17
Path densities: Definition
Regenerated nutrient = remineralized at depth that has not yetreemerged in the euphotic zoneThe path density 〈ηreg(r)〉 is the concentration of regeneratednutrients at r last taken up Ωi that is destined to reemergence in Ωf
16/17
Newton PDE solutionsteady state: ∂tx = f(x) = 0use Newton’s Method (generalized zero search)linear approximation:
f(x1) = f(x0) + Df(x0) (x1 − x0) + o (‖x1 − x0‖)
where Df is the Jacobian,a n× n sparse matrixwhere n ∼ 600,000!
To get f(x1) ∼ 0,we take
x1 = x0 −Df(x0)−1f(x0)
Kelley, 2003
f(x)
xx0x1
f(x0)
16/17
Newton PDE solutionsteady state: ∂tx = f(x) = 0use Newton’s Method (generalized zero search)linear approximation:
f(x1) = f(x0) + Df(x0) (x1 − x0) + o (‖x1 − x0‖)
where Df is the Jacobian,a n× n sparse matrixwhere n ∼ 600,000!
To get f(x1) ∼ 0,we take
x1 = x0 −Df(x0)−1f(x0)
Kelley, 2003
n
n
17/17
Path densities: Computation1. Extract nutrient’s regenerated source: sX
reg(x) where e.g., X = Si.2. Linear labelling/unlabelling equation: (∂t + T + L0)xreg = sX
reg
3. Use Green function to propagate from source on Ωi:
(∂t + T + L0)greg(t) = 0 and greg(0) = diag(sXreg)Ωi
4. Use Adjoint Green function to propagate to reemergence on Ωf :
(−∂t + T + L0)Greg(t) = 0 and Greg(0) = VL0Ωf
5. Time integral by direct inversion:
〈greg〉 = (T + L0)−1diag(sXreg)Ωi
〈Greg〉 = (T + L0)−1VL0Ωf
6. Combine into path density:
〈ηreg(r)〉 = 〈Greg(r)〉 × 〈greg(r)〉 (element-wise multiplication)