An Adjoint Sensitivity Analysis of An Adjoint Sensitivity Analysis of the Southern California Current the Southern California Current

Circulation and EcosystemCirculation and Ecosystem

Andy Moore, Andy Moore,

Emanuele DiLorenzo, Hernan Arango, Emanuele DiLorenzo, Hernan Arango,

Craig Lewis, Zack Powell, Craig Lewis, Zack Powell,

Arthur Miller, Bruce CornuelleArthur Miller, Bruce Cornuelle

OutlineOutline

MotivationMotivation Model configuration and circulationModel configuration and circulation Sensitivity and the adjointSensitivity and the adjoint Indices of interestIndices of interest Examples of sensitivitiesExamples of sensitivities Seasonal variationsSeasonal variations SummarySummary

MotivationMotivation The California Current System is controlled The California Current System is controlled

by a number of different regimes (i.e. by a number of different regimes (i.e. upwelling, instability, topographic control).upwelling, instability, topographic control).

Sensitivity analysis can help to unravel this Sensitivity analysis can help to unravel this complex system.complex system.

Test hypotheses about other potentially Test hypotheses about other potentially important processes (i.e. stochastic forcing).important processes (i.e. stochastic forcing).

Sensitivity analysis is also an important Sensitivity analysis is also an important precursor for data assimilation and precursor for data assimilation and predictability analysis.predictability analysis.

The ROMS SCB DomainThe ROMS SCB Domain

7-20km resolution; forced by NCEP climatological winds and surface fluxes.ROMS has been used before in the CCS and validated by others (Marchesiello et al, 2003; Powell et al, 2005).

Outer domain:20km res, 20 levels.

Inner domain:7-20km res, 20 levels.Derives boundaryconditions from the outer domain.

A 4-Component Nitrogen-Based Trophic A 4-Component Nitrogen-Based Trophic ModelModel

NDissolvedNitrogen(Nitrate)

DParticulateNitrogen(Detritus)

PPhototrophic

Phytoplankton

ZHerbivorous

(macro)Zooplankton

ConstantSWR

N uptake byphotosynthetic

growth of P(Michaelis-Menten)

Excretion andmetabolism

Linear Mortalityof Z at constant rate

Linear Mortalityof P at constant rate

Sinking5 m day-1

Grazing onP by Z

(saturating)Remineralizationof D at constant

rate

A variant of theNPZD model ofPowell et al. (2005)

Seasonal CirculationSeasonal Circulation

April

October

Mesoscale Eddy VariabilityMesoscale Eddy Variability

ROMS

AVHRR

Ecosystem CirculationEcosystem Circulation

Surface P April Average

Adjoint Approach to SensitivityAdjoint Approach to Sensitivity We must first define “sensitivity.”We must first define “sensitivity.” Consider the model state vector:Consider the model state vector: Consider a function or index, , defined in terms of space Consider a function or index, , defined in terms of space

and/or time integrals of .and/or time integrals of . Small changes in will lead to changes in Small changes in will lead to changes in

where:where:

We will define sensitivity as etcWe will define sensitivity as etc..

( , , , , , , , , )Tu v T S N P Z D

J

d J

J J J J JdJ du dv dT dS d

u v T S

J J J JdN dP dZ dD

N P Z D

, , ,J J J

u v T

dJ

Sensitivity AnalysisSensitivity Analysis

Consider a function Consider a function

ClearlyClearly

ButBut

So So

( )J G Φ

TJ G Φ Φ

( ) ( , ) ( )f i f it t t t Φ R Φ

( , )Tf iJ t t G Φ R Φ

Validity of the TL AssumptionValidity of the TL Assumption

TL assumption valid for ~30 days for perts TL assumption valid for ~30 days for perts that grow to an amplitude of:that grow to an amplitude of:

|SST|~0.5-1.0C|SST|~0.5-1.0C |v|~0.2 m/s|v|~0.2 m/s

These are lower bounds!These are lower bounds!

~ 0.08m

Seasonal Circulation Index RegionsSeasonal Circulation Index Regions

JSST

JKE

J90

IndicesIndices

21 CSST

A T

J SST SST dtdATA

0.15 For ,1

2 21 C CKE

A T Z

J u u v v dtdATAZ

For , 0.005

1BI f z N v

1BI BI

A T Z

J dtdATAZ

710

“Eady Index”

An index ofbaroclinic instability

1

IndicesIndices 2

90

1N C

X Z T

J N N dtdzdxTXZ

For ,

For ,

1 9.6

2

90

1P C

X Z T

J P P dtdzdxTXZ

0.3 1

What Physical Processes are likely to What Physical Processes are likely to Influence J?Influence J?

Advection

Q, P-E+R

Advection

InstabilityLong Rossby Waves

Short Rossby Waves

CoastallyTrappedWaves &

Tides

Turbulence/wave breaking

Note: All processes indicatedcan be significantly influencedby stochastic forcing.

A 4-Component Nitrogen-Based Trophic A 4-Component Nitrogen-Based Trophic ModelModel

NDissolvedNitrogen(Nitrate)

DParticulateNitrogen(Detritus)

PPhototrophic

Phytoplankton

ZHerbivorousZooplankton

ConstantSWR

N uptake byphotosynthetic

growth of P“Sloppy feeding”

and excretion

Linear Mortalityof Z at constant rate

Linear Mortalityof P at constant rate

Sinking5 m day-1

Grazing onP by Z

Remineralizationof D at constant

rate

A variant of theNPZD model ofPowell et al. (2005)

Physical Circulation Physical Circulation SensitivitiesSensitivities

The Signature of Advection in The Signature of Advection in

Day 5 Day 10 Day 15

Day 20 Day 25 Day 30

SSTJ Q

Seasonal Variations in Sensitivities ISeasonal Variations in Sensitivities IThe change in over the target arearequired to yield one change in

for .

SSTJ1

The change in Q over the target arearequired to yield one change in

for .

SSTJ

The change in v over the target arearequired to yield one change in

for .

SSTJ1

1

Low sensitivity

Low sensitivity

Low sensitivity

High sensitivity

High sensitivity

High sensitivity

0.035

0.01

32

20

2.3

0.3

Seasonal Variations in Sensitivities IISeasonal Variations in Sensitivities II

The change in over the target arearequired to yield one change in

for .

The change in over the target arearequired to yield one change in

for .

KEJ1

BIJ

1

Low sensitivity

Low sensitivity

High sensitivity

High sensitivity

0.0045

0.002

0.003

0.0003

Seasonal Variations in Sensitivities Seasonal Variations in Sensitivities IIIIII

The change in Q over the target arearequired to yield one change in

for .

The change in v over the target arearequired to yield one change in

for .

BI

J

BIJ

1

1

Low sensitivity

Low sensitivity

High sensitivity

High sensitivity

220

15

1

< 0.01

Interdependencies: Sensitivity of KE Interdependencies: Sensitivity of KE to Baroclinic Instabilityto Baroclinic Instability

Change in required to yield a one change in when varying only v for . Recall that

BI KEJ 1 710BI

Low sensitivity

High sensitivity

Log

scal

e

910

1310

Summary for Physical CirculationSummary for Physical Circulation

SST anomaly in coastal upwelling regions SST anomaly in coastal upwelling regions equally sensitive to variations in and Q, equally sensitive to variations in and Q, with v a close second.with v a close second.

Highest (Lowest) sensitivity in Fall (Spring)Highest (Lowest) sensitivity in Fall (Spring) KE anomaly most sensitive to variations in KE anomaly most sensitive to variations in

and baroclinicity.and baroclinicity. Highest (Lowest) sensitivity Summer/Fall Highest (Lowest) sensitivity Summer/Fall

(Winter/Spring).(Winter/Spring).

Biological Circulation Biological Circulation SensitivitiesSensitivities

Adjoint Sensitivity for Ecosystem ModelAdjoint Sensitivity for Ecosystem Model

Oct: on day 13 3 day 30NO NO Oct: on day 13 day 30

NO P

3 day 30NO Z Oct: on day 1 Oct: on day 13 day 30

NO D

Mar: on day 13 day 30NO P

Jul: on day 13 day 30

NO P

Seasonal Variations in Sensitivities ISeasonal Variations in Sensitivities I

Change in required to yield a one change in for . Note the log-scale!

90PJ 1

Low sensitivity

High sensitivity

Log

scal

e

110

510

Seasonal Variations in Sensitivity IISeasonal Variations in Sensitivity IIThe change in N over the target area required to yield one change in

for .

The change in P over the target area required to yield one change in

for .

The change in Z over the target area required to yield one change in

for .

90PJ

90PJ

90PJ

1

1

1

Low sensitivity

Low sensitivity

Low sensitivity

High sensitivity

High sensitivity

High sensitivity

Summary of Biological CirculationSummary of Biological Circulation

For all NPZD-based indices, variations in are For all NPZD-based indices, variations in are found to be important.found to be important.

Variations in NPZD equally important (internal Variations in NPZD equally important (internal interactions important).interactions important).

NPZD concentrations strongly influenced by the NPZD concentrations strongly influenced by the physical environment.physical environment.

Highest (Lowest) sensitivities in Spring/Summer Highest (Lowest) sensitivities in Spring/Summer (Fall/Winter).(Fall/Winter).

Extraordinary sensitivities during some Spring Extraordinary sensitivities during some Spring periods suggestive of linear instability (i.e. we are periods suggestive of linear instability (i.e. we are perhaps the TL assumption a little too far!).perhaps the TL assumption a little too far!).

Other Ongoing ApplicationsOther Ongoing Applications

Intra-Americas SeaIntra-Americas Sea Monterey BayMonterey Bay

Intra-Americas SeaIntra-Americas Sea

Seasonal Sensitivity Dependence, JSeasonal Sensitivity Dependence, J22

1010-2-2JJ22 SST (K)SST (K) V (m sV (m s-1-1)) (N m(N m-2-2)) (m) (m) Q (W mQ (W m-2-2))

JanJan 10.810.8 2.1 (1.1)2.1 (1.1) 0.0180.018 8.88.8 148148

FebFeb 10.510.5 1.7 (0.8)1.7 (0.8) 0.0180.018 8.18.1 158158

MarMar 10.710.7 1.3 (0.6)1.3 (0.6) 0.0160.016 7.87.8 149149

AprApr 11.811.8 1.5 (0.8)1.5 (0.8) 0.0190.019 7.57.5 158158

MayMay 9.79.7 1.4 (0.7)1.4 (0.7) 0.0160.016 8.48.4 137137

JunJun 8.88.8 1.7 (0.9)1.7 (0.9) 0.0130.013 7.47.4 124124

JulJul 9.59.5 1.5 (0.7)1.5 (0.7) 0.0150.015 9.39.3 141141

AugAug 10.710.7 1.6 (0.7)1.6 (0.7) 0.0170.017 8.78.7 149149

SepSep 10.010.0 1.8 (1.0)1.8 (1.0) 0.0140.014 8.18.1 137137

OctOct 8.58.5 1.7 (1.1)1.7 (1.1) 0.0130.013 8.28.2 124124

NovNov 8.68.6 1.5 (0.9)1.5 (0.9) 0.0140.014 8.68.6 128128

DecDec 9.29.2 1.2 (0.6)1.2 (0.6) 0.0150.015 8.88.8 128128

MeanMean 9.99.9 1.6 (0.8)1.6 (0.8) 0.0160.016 8.38.3 140140

Basic State MnBasic State Mn 1010 ~1~1 ~0.1~0.1 ~0.2~0.2 ~100~100

RankRank 22 22 11 33 22

Rank based on percentage of basic state mean

Seasonal Sensitivity Dependence, JSeasonal Sensitivity Dependence, J4 4 (N)(N)

Rank based on percentage of basic state mean

NONO3 3 (mmol Nm(mmol Nm-3-3)) P P (mmol Nm(mmol Nm-3-3)) Z Z (mmol Nm(mmol Nm-3-3)) D D (mmol Nm(mmol Nm-3-3)) (N m(N m-2-2)) V (m sV (m s-1-1))

JanJan 0.0350.035 0.0720.072 0.0410.041 0.0640.064 7.5X107.5X10-4-4 0.0780.078

FebFeb 0.0340.034 0.0610.061 0.0410.041 0.0620.062 6.2X106.2X10-4-4 0.0700.070

MarMar 0.0390.039 0.0900.090 0.0460.046 0.0730.073 10.0X1010.0X10-4-4 0.0830.083

AprApr 0.0370.037 0.0070.007 0.0200.020 0.0670.067 0.8X100.8X10-4-4 0.0040.004

MayMay 0.0360.036 0.0970.097 0.0420.042 0.0660.066 9.8X109.8X10-4-4 0.0930.093

JunJun 0.0480.048 0.1120.112 0.0560.056 0.0930.093 12.1X1012.1X10-4-4 0.1120.112

JulJul 0.0400.040 0.0960.096 0.0490.049 0.0770.077 11.9X1011.9X10-4-4 0.0730.073

AugAug 0.0320.032 0.0680.068 0.0360.036 0.0540.054 6.6X106.6X10-4-4 0.0560.056

SepSep 0.0300.030 0.0760.076 0.0330.033 0.0520.052 4.8X104.8X10-4-4 0.0470.047

OctOct 0.0320.032 0.0720.072 0.0370.037 0.0590.059 6.1X106.1X10-4-4 0.0730.073

NovNov 0.0340.034 0.0820.082 0.0400.040 0.0610.061 6.7X106.7X10-4-4 0.0820.082

DecDec 0.0370.037 0.0750.075 0.0430.043 0.0680.068 7.1X107.1X10-4-4 0.0810.081

MeanMean 0.0360.036 0.0760.076 0.0400.040 0.0660.066 7.5X107.5X10-4-4 0.0710.071

Basic StateBasic State ~10~10 ~10~10 ~10~10 ~1~1 ~0.1~0.1 ~1~1

RankRank 11 11 11 22 11 22

Seasonal Sensitivity Dependence, JSeasonal Sensitivity Dependence, J4 4 (P)(P)

Rank based on percentage of basic state mean

NONO3 3 (mmol Nm(mmol Nm-3-3)) P P (mmol Nm(mmol Nm-3-3)) Z Z (mmol Nm(mmol Nm-3-3)) D D (mmol Nm(mmol Nm-3-3)) (N m(N m-2-2)) V (m sV (m s-1-1))

JanJan 3.13.1 2.82.8 3.33.3 10.710.7 6.8X106.8X10-3-3 0.450.45

FebFeb 8.48.4 6.56.5 10.710.7 17.117.1 7.9X107.9X10-3-3 0.360.36

MarMar 3.13.1 3.33.3 3.23.2 4.74.7 1.1X101.1X10-3-3 0.060.06

AprApr 1.41.4 0.090.09 2.62.6 4.74.7 0.8X100.8X10-3-3 0.0050.005

MayMay 3.03.0 3.43.4 5.55.5 5.85.8 1.1X101.1X10-3-3 0.110.11

JunJun 4.54.5 4.54.5 5.95.9 9.29.2 3.9X103.9X10-3-3 0.310.31

JulJul 2.82.8 2.72.7 4.04.0 5.25.2 2.4X102.4X10-3-3 0.230.23

AugAug 8.58.5 1.31.3 1.31.3 1.51.5 0.6X100.6X10-3-3 0.050.05

SepSep 7.17.1 7.37.3 7.97.9 11.611.6 2.3X102.3X10-3-3 0.080.08

OctOct 10.510.5 7.67.6 15.915.9 29.329.3 3.3X103.3X10-3-3 0.080.08

NovNov 2.52.5 2.72.7 3.13.1 4.84.8 6.8X106.8X10-3-3 0.110.11

DecDec 12.512.5 12.012.0 15.515.5 21.021.0 9.4X109.4X10-3-3 0.580.58

MeanMean 5.05.0 4.54.5 6.46.4 10.510.5 6.8X106.8X10-3-3 0.200.20

Basic StateBasic State ~10~10 ~10~10 ~10~10 ~1~1 ~0.1~0.1 ~1~1

RankRank 11 11 11 33 22 11

Seasonal Sensitivity Dependence, JSeasonal Sensitivity Dependence, J4 4 (Z)(Z)

Rank based on percentage of basic state mean

NONO3 3 (mmol Nm(mmol Nm-3-3)) P P (mmol Nm(mmol Nm-3-3)) Z Z (mmol Nm(mmol Nm-3-3)) D D (mmol Nm(mmol Nm-3-3)) (N m(N m-2-2)) V (m sV (m s-1-1))

JanJan 14.214.2 11.711.7 4.74.7 3232 0.030.03 1.81.8

FebFeb 12.612.6 12.612.6 7.87.8 2525 0.040.04 1.11.1

MarMar 9.09.0 7.17.1 6.46.4 1616 0.020.02 0.30.3

AprApr 0.40.4 0.250.25 0.80.8 22 0.0030.003 0.020.02

MayMay 10.910.9 6.96.9 4.14.1 3434 0.030.03 0.60.6

JunJun 26.526.5 16.816.8 9.39.3 7777 0.030.03 1.41.4

JulJul 7.57.5 7.57.5 5.45.4 1717 0.0080.008 0.20.2

AugAug 12.612.6 10.110.1 5.25.2 2525 0.010.01 0.20.2

SepSep 9.99.9 6.56.5 5.45.4 1616 0.0050.005 0.090.09

OctOct 25.425.4 15.115.1 9.29.2 6565 0.0040.004 0.10.1

NovNov 8.58.5 7.17.1 3.73.7 2222 0.020.02 0.40.4

DecDec 26.726.7 17.117.1 10.710.7 6565 0.070.07 1.81.8

MeanMean 13.713.7 9.99.9 5.65.6 3333 0.020.02 0.70.7

Basic StateBasic State ~10~10 ~10~10 ~10~10 ~1~1 ~0.1~0.1 ~1~1

RankRank 22 22 22 33 11 22

Seasonal Sensitivity Dependence, JSeasonal Sensitivity Dependence, J4 4 (D)(D)

Rank based on percentage of basic state mean

NONO3 3 (mmol Nm(mmol Nm-3-3)) P P (mmol Nm(mmol Nm-3-3)) Z Z (mmol Nm(mmol Nm-3-3)) D D (mmol Nm(mmol Nm-3-3)) (N m(N m-2-2)) V (m sV (m s-1-1))

JanJan 2525 1515 2020 88 0.030.03 1.21.2

FebFeb 2424 1717 2424 1111 0.060.06 1.81.8

MarMar 1717 1212 1717 88 0.010.01 0.70.7

AprApr 22 0.10.1 0.40.4 44 0.0020.002 0.090.09

MayMay 2424 1313 1818 66 0.010.01 0.90.9

JunJun 4646 2525 4444 1111 0.040.04 2.12.1

JulJul 1111 99 1212 55 0.010.01 0.30.3

AugAug 1212 66 1010 55 0.0070.007 0.20.2

SepSep 2121 1313 2020 77 0.0060.006 0.40.4

OctOct 5353 2222 3535 77 0.0090.009 0.30.3

NovNov 1515 1010 1414 55 0.030.03 0.70.7

DecDec 5959 3333 5353 1111 0.090.09 3.43.4

MeanMean 2626 1515 2222 77 0.020.02 1.01.0

Basic StateBasic State ~10~10 ~10~10 ~10~10 ~1~1 ~0.1~0.1 ~1~1

RankRank 33 22 33 44 11 22

The Adjoint OperatorThe Adjoint Operator

ConsiderConsider

Perturbations in given by: Perturbations in given by:

Sensitivity given by:Sensitivity given by:

is the adjoint of ROMS.is the adjoint of ROMS. The adjoint provides the Green’s functions The adjoint provides the Green’s functions

for -functions at all points in space-time.for -functions at all points in space-time.

21

A T

J SST dAdt

T T T1 2 2 2 (0) ( )

A T T T

J SST SSTdAdt dt t dt H R

1J

T2 ( )T

Jt dt

R

T ( )tR

Validity of Tangent Linear AssumptionValidity of Tangent Linear Assumption

TLM and NLM perturbedby first 10 energy SVs.(|SST|~0.5-1C; ~6cmat day 30)

Summary for CalCOFI Line90 Indices, JSummary for CalCOFI Line90 Indices, J44

Most thru least sensitive: N, P, Z, DMost thru least sensitive: N, P, Z, D N: (1) N,P,Z, wind; (2) D,VN: (1) N,P,Z, wind; (2) D,V P: (1) N,P,Z,V; (2) wind; (3) DP: (1) N,P,Z,V; (2) wind; (3) D Z: (1) wind; (2) N,P,Z,V; (3) DZ: (1) wind; (2) N,P,Z,V; (3) D D: (1) wind; (2) P,V; (3) N,Z; (4) DD: (1) wind; (2) P,V; (3) N,Z; (4) D N,P,Z,D: Extraordinary sensitivity in AprilN,P,Z,D: Extraordinary sensitivity in April N,P,Z,D: Lowest sensitivity typically during N,P,Z,D: Lowest sensitivity typically during

fall and winter. fall and winter.

( ), ( ), ( ), ( ), ( ), ( ), ( ), ( )v t T t S t t N t P t Z t D t

J J

u u

A Reminder…A Reminder…