3rd Japanese DA workshop, Jan 18th, 2013, ISM, Tachikawa

Pathways of the North PacificIntermediate Water identified
through the tangent linear andadjoint models of an OGCM
Presenter: Yosuke Fujii (JMA/MRI)Coauthor: Toshiya Nakano*1, Norihisa Usui*2, Satoshi
Matsumoto*1, Hiroyuki Tsujino*2, and Masafumi Kamachi*2*1JMA, *2JMA/MRI

Fujii et al. 2013, JGR, accepted1Outline
Part I: Theoretical Introduction
1 Derivation of an adjoint model2 Backward tracing using an adjoint model
3 Joint likelihood and Probability Density (PD)Part II: Practical Application to NPIW.
4 Purpose of tracing NPIW
5 Results (Pathway of NPIW from the subarctic region
to the south of Japan)
6 Discussions (Is the pathway realistic?)
7 Concluding Remarks(Benefits from adjoint models)Part I: Theoretical Introduction
1 Derivation of an adjoint model3

Derivation of adjoint model (I)Consider a state vector, x, containing n variables (all variables have 3D).
e.g. x  ( x1,t , x2,t , x3,t , x4,t )  (T , S , u, v)x M (x)
The forecast model of x:tConsider a cost function, J [x( )]  J [ x1 ( ), x2 ( ),, xn ( )](For simplicity, we assume that J directly depends on the state vector attime  alone.)
The 1st-order variation of J is
 J

J
J
J
x1 ( ) x2 ( )  
xn ( ) dV , x( )  λ ( ), x( )V x ( )x2 ( )xn ( )x( ) 1J   

Definition of the inner product:y , x    y1 x1  y2 x2   yn xn dVVForward variables: x(t), Adjoint variables: λ(t) J  J
J
J 

 
,
,,
x(t)  x1(t) x2 (t)xn (t) Derivation of adjoint model (II)If x is specified at a certain time t, the cost function J is also determined.
Therefore J 
This leads toJJ, x( ) , x(t )  λ (t ), x(t )  const.
x(t )
x( )λ (t  t ), x(t  t )  λ (t ), x(t )λ (t  t ), x(t  t )  λ (t  t ), x(t )tλ (t ), x(t )  λ (t  t ), x(t )tx(t  t )  x(t )λ (t )  λ (t  t )
λ (t  t ),, x(t )
t
t
λ
x

λ (t ),, x(t )t ( t )Derivation of adjoint model (III)The forecast model leads to (x  x)
 M (x  x)  M (x)  Mx
tx
 Mx (Tangent Linear model)
t

x
λ
*
λ,
, x
 λ , Mx  M λ , x t
 (t )

Use the definition ofadjoint operatorsλ M *λ (Adjoint model) (t )
Backward time evolution of adjoint variables λ(t)  J x(t) can be calculatedby the adjoint model.
Therefore,

In a simple 4DVAR system,J  [x( )  x obs ( )]T R 1[x( )  x obs ( )] / 2

λ( )  J x( )  R1[x( )  xobs( )]Information of errors or data misfit

Thus, the adjoint model propagates the information of errors or data misfit
backward in time.

Conservation of the forward-adjoint inner productIn the derivation, we use
J 

J
J
, x( ) , x(t )  λ (t ), x(t )  const.
x(t )
x( )

This means that
The inner product of forward and adjoint variables are conserved.
Evolved by tangent linear model

Evolved by adjoint model・This fact holds for any set of tangent linear and adjoint models.
・If the cost function is defined properly, the conservation property has aphysical meaning.
・In this study, we use this property for the tracing of an ocean water mass.

2 Backward tracing using an adjoint
model

8

Forward and Backward Conditions
Origin Area

Forward Condition
The fluid particle is observed in theorigin area at the initial time.

Destination Area
Backward Condition
The fluid particle is observed in thedestination area at the final time.

Forward Tracing with an advection-diffusion model・Linear advection-diffusion model for a incompressible fluidS u  S    κS 
t・Now, we try to trace fluid particles thatsatisfy the forward condition.
If we set S=1/V in the origin area
S=0 out of the areaat the initial time.
V: Volume of the origin area

If we set S=1
S=0

in the origin areaout of the area

Probability density (PD) of theparticles that satisfy the forward
condition (PF(x)).
Fraction of the particles thatsatisfy the forward condition.
(i.e., Forward likelihood, LF(x))

Tangent linear model is written in the same form.
PF(x) and LF(x) can be considered as a forward variable.

Derivation of the adjoint operator
 S    u  S    κS 
MS   t 

λ, M S  M * λ, S u  S    κS dV  [uS ]dydz   [uS ]dzdx   [uS ]dxdy     (u)SdV  (  u)SdV   (  u)SdV  S (u   )dVVx x  y 
y  vz z  vv

   κS dV     κS dV   (κ  )  S dV   S  (κ  )dV
V

V

T

T

V

V

 , M S   [u  S    κS ]dVV  [Su    S  κ T  ]dV  M *  , SV
 M *   u       κ T  (t )



v

Backward Likelihood
DefineJ   DS ( )dV  D, S ( )v

D = 1 in the destination area
D = 0 out of the area
In a small area
(Time tc)

+sV

×

J   (t ), S (t )In a Destination area
(Time )+dJ=+sV

Deviates by +sS(xc,tc)
The adjoint variable (x,t) can be considered as the fraction of the fluid
particles that satisfy the backward condition.
We refer to this fraction as the backward likelihood, LB(x).

Backward Probability Density
Consider the case where S(tc)= ( is constant in space).

S(t)= for any t.

J   ( ), S ( )  D,   VD
In a Destination area
(Time )In a small area
(Time t)V
Volume: V

Total Number: VD

PD of particles that satisfy the backward condition is represented by
PB ( x, t )   ( x, t )V ( VD  V )   ( x, t ) / VD

We refer to this PD as the backward Probability Density.

3 Joint likelihood and Probability
Density (PD)14

Joint Likelihood
Origin area
at initial time

Where a particle will go depends only
on the current position of the particle
and does not depend on where theparticle came from.

The forward and backward
likelihoods are independent.

Destination Area
at final time

LB(x,t)
LF(x,t)Position x
Time t

LJ(x,t) = LF(x,t) LB(x,t)

LJ(x,t) : Joint Likelihood (i.e., the fraction of particles that satisfy boththe forward and backward condition)
The joint likelihood is expressed by the product of forward and adjoint
variables.

Joint Provability Density (PD)Bayes Theorem leads toHere,p( x | F )  PF ( x) ,PJ ( x)  p( x | F , B) p ( x | F ) p ( B | x, F )p( B | F )p ( B | x, F )  p ( B | x)  LB ( x)andp( B | F )   p( x | F ) p( B | x, F )dV   PF ( x) LB ( x)dV  PF ( x), LB ( x)  N J
VV

NJ: Number of particles that satisfy both forward and backward conditions.
A constantForward-adjoint Inner product is conserved.
Finally,

PJ ( x)  PF ( x) LB ( x) N J

Joint PD is also written as a product of forward and adjoint variables.

Time Evolution of the joint likelihood and PD
Forward Model:S u  S    κS t
 u      κ T 
Adjoint Model: (t )



Joint Probability Density

Consistent with theconservation property offorward-adjoint inner product.



Gauss’s Theorem leads to that Integration
over the model domain is conserved.

( S )  u  ( Sλ)    κS   S  κ T λ     Su  κS  Sκ T λ
t









Advection term: same as in the forward model
Diffusion term: Near the initial time → The forward diffusion is dominant.
Near the final time → The adjoint diffusion is dominant.

Part II: Practical Application to NPIW
4 Purpose of tracing NPIW

18

What is NPIW?

On 26.8 surface (Hydrobase)

NPIW:
North Pacific Intermediate Water
・NPIW distributes around 26.8
surface in the North Pacific
137E Clim. (Obs.)
subtropical region.

?

165E Clim. (Obs.)

・NPIW is generated in
the mixed water region
and the Kuroshio
Extension.
・Low salinity in NPIW originates from the near surface low salinity water inthe subarctic region (Okhotsk and Bering Seas?).
・The pathway of NPIW that travels from MWR to the western part of thesubtropical gyre is still controversial (e.g. Kouketsu et al. 2007, 2008).

Freshening Trend in NPIW (from Nakano et al. 2007)
Salinity on 26.8σθ

137E Line

Freshening Trend in vertical sections

P2 Line・A decadal-scale freshening trend isfound in the upper part of NPIW in the
137E section.
・Kouketsu et al. (2007) also indicateda similar freshening trend of NPIW.
・Nakano et al. (2007) speculated
existence of the shortcut and long-way
routes of NPIW.

Purpose and Method
Purpose: Identify the shortcut route of NPIW.
Method: Trace water masses that migrate fromthe low-salinity (<34.0) near-surface layer in the subarctic region (origin) to
the NPIW layer in the sea south of Japan (destination).
Current and diffusion coefficient fields: Free-simulation of an ocean model(MRI.COM, Tsujino et al. 2006) integrated using NCEP-R1 forcing.
Forward model: TL model of the original
ocean model with setting density
perturbation to 0 at every time step.
(Salinity perturbation is employed as apassive tracer.)
Adjoint model: The exact transpose
operator of the TL model
Period of migration (in main analysis):Jan. 1997 ー Dec. 2006 (10-year period)

Background Field (model simulation)
Model

Observation

・The first and second ridges of KE and its southward sift between 155-160E
is well reproduced although a spurious ridge exists at 155E.
・There are two low salinity tongues at 20N and 30N in the simulation, these
are combined into one in the observation.

5 Results

23Comparison of Forward, backward, joint likelihood
At Jan. 1st, 2002, on the 26.8σθ surface
ForwardBackward

Joint

・Forward: Particles cross the Kuroshio to the south mainly in 155E-170E.
・Adjoint： Although particles are mainly returned to the east in thesubtropical gyre, a part of the crosses the Kuroshio and reaches the mixed
water region and the Okhotsk Sea.
・Joint： Particles cross the Kuroshio around 160E to the north and reaches
the mixed water region and the Okhotsk Sea.

31Jan1997

01Jan2002

01Jan1998

01Jan2004

01Jan1999

01Jan2005

01Jan2000

01Jan2006

01Jan2001

01Oct2006

Maximum joint likelihood in the migration period
Vertically integrated value with SSH

On 26.8σθ with acceleration potential

・This figure roughly visualizes the pathways of NPIW.
・The main pathway is advected by Oyashio → 1st ridge of KE→ advected by KE → Shatsky Rise → Subtropical gyre → Destination
・The pathway north of the Kuroshio is more remarkable in verticallyintegrated value (because of the effect of the vertical diffusion).

Distribution on different density surfaces
26.0σθ

26.3σθ

26.5σθ

26.8σθ

27.0σθ

27.2σθ

Intrusion to the Kuroshio Extension（26.8σθ）
10-14Feb1998

02-06Mar1998

Salinity

Joint Likelihood

21-25JAN1998

・Particles that intrude into the Kuroshio and are advected to the east
corresponds to the low salinity water.

Ejection from the Kuroshio around Shatsky Rise (1)27Nov-01Dec2004

07-11Dec2004

17-11Dec2004

27-31Dec2004

06-10Jan2005

16-20Jan2005

Particles deviate from the Kuroshio around the meander over
the Shatsky Rise!?

Why do particles are ejected at the meander?
Stretch/shrink of vortex
tubes and ACP (27Dec)

Joint likelihood and ACP(solid: 27Dec, dashed 01Jan)

Water mass in the western side of the meander gets anticyclonicvorticity due to the shrinking of the vortex tube.
Enhance the deviation from the Kuroshio current
Promote the migration of the meander to the eastEjection from the Kuroshio around Shatsky Rise (2)15-19Jul1997

25-29Oct1997

04-08Aug1997

14-18Aug1997

24-28Aug1997

03-07Sep1997

・Some particles enter the subtropical gyre with eddies separated from themeander of the Kuroshio Extension.

Probability Transport
U 

tf

ti

zb

y2

0

y1

 

up( x, y, z , t )dydzdt

Velocity (m/sec)

Joint PD (% / m3)

Transport（％）：
Fraction of particles thatpasses the vertical section
between y1-y2 in the period ofti - tf .
Fraction of particles thatcrosses 37.5N is almost 100%.
The effect of diffusion canbe ignored.

y2
0

y1

zb

U

u(x,y,z,t)p(x,y,z,t)

Integration from 140E (annual)Integration from 140E (winter)

Integration from 140E (summer)

Transport for 1 degree interval

Map of the Transport
Transport Vector (integrated over
the whole migration period)

100% - Meridional transport
integrated from east (%)
Contours can be regarded as stream lines in
the area without shading.

・Okhotsk or Bering Seas → Oyashio→ Kuroshio Extension → Subtropical
Gyre → Destination%

・Particles mainly enter the subtropicalgyre over the Shatsky Rise and thewest of the Emperor Seamount Chain.

6 Discussions

34

Dependency on the migration period
2002-2006 （5-year period）

1987-2006（20-year period）

・Pathways is not severely changed with the migration period.
・The longer the period is, the more the pathways become diffusive.
→ The longer periods allow various pathways.

Effect of the Shatsky Rise

Mean stretch/shrink of vortex on 26.8σθ
(with ACP and 4000m bottom-depth)

Mean axis of KE
and its STD

STD of the gradientof the latitude

STD of the latitudeof KE axis
STD of the gradientof the latitude

Standard Deviations (STDs) of
the latitude of KE axis and its
gradient is relatively large east of
155E.
Anticyclonic vorticity tends to begenerated at the upstream sideof the Shatsky Rise.

On the long-way route・We hope the shortcut route is identified
in this study.
・The long-way route is not identified
because it passes through theoutside of the ocean model.
・But the simulation result shows some hints on the existence of the longway route.
Salinity on 26.8σθ

Forward likelihood on 26.8σθ
？
？

7 Concluding Remarks

38Benefits from Adjoint Models
An adjoint model is not just the tool for data assimilation, but a
propagator of a sensitivity (gradient of any cost function) backwardfrom output to input data.



According to the definition of the cost function, the sensitivity (adjoint
variable) can represent, information of data misfit (errors), thefraction of particle that reach an certain area at the final time, or the
potential impact of the input data on the target physical phenomenon
at the final time.
Evaluation of the impact of input data, parameters, observation etc.



The connection from the output to the input data represented by theadjoint model is based on the physics in the original forward model(not on the statistical information like with ensemble methods).

Conservation of the forward-adjoint inner-product may also have an
essential physical mean.

Adjoint models (at least partly) hold this property even if the originalmodels have nonlinearity.