response of the biological pump to perturbations in the ...€¦ · [sarmiento et al., 2004;...

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])

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

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

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


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


mmol Si / m3

10 30 50 70 90

Latitude


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


ηP(SO→SO), base case

mmol P / m3

0.4 0.2 0 0.2 0.4

Depth

(km

)

Latitude

δηP(SO→SO), TROx0


5

4

3

2

1

0

mmol P / m3

0.4 0.2 0 0.2 0.4

Latitude

δηP(SO→SO), TROx100


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


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


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


E (P)

E (Si) Sireg

, base case

mmol Si / m3

8 4 0 4 8

De

pth

(km

)

Latitude

δSireg

, TROx0bio

bio


5

4

3

2

1

0

mmol Si / m3

8 4 0 4 8

Latitude

δSireg

, TROx100


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


+

00



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


+

00



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


+

00



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)

response of the biological pump to perturbations in the ...€¦ · [sarmiento et al., 2004;...

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