Boundary conditions control in ORCA2 · 2017. 1. 27. · Boundary conditions control in ORCA2...

Boundary conditions control in ORCA2

Eugene Kazantsev

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Eugene Kazantsev. Boundary conditions control in ORCA2. Journee thematique - Que peuventattendre les modelisateurs de l’assimilation ?, Feb 2013, Paris, France. 2013. <hal-00925863>

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Boundary conditions control in ORCA2

Eugene Kazantsev

INRIA, Moise

Journee thematique“Que peuvent attendre les modelisateurs de l’assimilation de donnees ?”

Paris, le 12 fevrier 2013

Eugene Kazantsev Boundary conditions control for ORCA2 page 1 of 19

Model configuration

ORCA-2 configuration of NEMO

182× 149× 31 nodes in curvilinear (x, y) coordinates;

z levels with partial steps at the bottom;

leap-frog scheme with Asselin filter;

implicit surface pressure gradient with External Gravity Waves filter;

implicit vertical diffusion with TKE Turbulent Closure Scheme;

Solar Radiation + Geothermal Heating + BBL + Surfaceevaporation/precipitation;

surface wind stress.


Space discretization on the C-grid

∂u

∂t=

(

SxSyv

)

Sy(ω + f)−DxSxu2 + Syv2

2− Sz

(

SxwDzu

)

+DxAhuξ +DyA

huω +

+ g

∫ z

0DxSzρ(x, y, ζ)dζ +Dzz(A

zuu) + gDx(η + Tcφ)

∂T

∂t= −Dx(uSxT )−Dy(vSyT )−Dz(wSzT ) +Ah

T

(

DxDxT +DyDyT

)

+

+ Dzz(AzTT ) + Solar Radiation + Geothermal Heating + BBL

ξ = Dxu+Dyv, ω = Dyu−Dxv, w =

∫ z

Hξ(x, y, ζ)dζ;w(x, y,H) = 0

Interpolations and Derivatives

(Sw)k+1/2 =wk+1 + wk

2k = 1, . . . ,K − 1

(DT )k =Tk+1/2 − Tk−1/2

hk = 1, . . . ,K − 1



∂u

∂t=

(

SxSyv

)


2− Sz

(

SxwDzu

)

+DxAhuξ +DyA

huω +

+ g

∫ z



∂T


T

(

DxDxT +DyDyT

)

+



∫ z

Hξ(x, y, ζ)dζ;w(x, y,H) = 0

Interpolations and Derivatives Modified Near the boundary

(Sw)k+1/2 =wk+1 + wk

2, k = 1, . . . ,K − 2, (Sw)1/2 = αS

0 + αS1w0 + αS

2w1

(DT )k =Tk+1/2 − Tk−1/2

h, i = 2, . . . ,K − 2, (DT )1 = αD

0 +αD1 T1/2 + αD

2 T3/2

h

· · ·w0 w1 w2 w3 wKwK−1wK−2wK−3

❜

T1/2❜

T3/2❜

T5/2❜

T7/2❜

TK−1/2❜

TK−3/2❜

TK−5/2



∂u

∂t=

(

SxSyv

)


2− Sz

(

SxwDzu

)

+DxAhuξ +DyA

huω +

+ g

∫ z



∂T


T

(

DxDxT +DyDyT

)

+



∫ z

Hξ(x, y, ζ)dζ;w(x, y,H) = α0(x, y)

Vertical velocity

wi,j,K−1 = αwb

0 − αwb

1 hzi,j,K−1/2ξi,j,K−1/2

wi,j,k−1 = wi,j,k − hzi,j,k−1/2ξi,j,k−1/2 ∀k : 2 ≤ k ≤ K − 1

wi,j,0 = wi,j,1 + αws

0 − αws

1 hzi,j,1/2ξi,j,1/2

· · ·w0

αw0

αw1 hz1/2︷︸︸︷

w1 w2 w3 wK

+αw0αw

1 hzK−1/2︷︸︸︷

wK−1wK−2wK−3



Vertical diffusion

∂

∂zA

zu

∂u

∂zis replaced by

(

Dzzu

)

i,j,1/2=

(Azu)1

hz1hz1/2

(αDzzUs

2 u3/2 − αDzzUs

1 u1/2)

(

Dzzu

)

i,j,k−1/2=

1

hzk−1/2

( (Azu)k

hzk

(uk+1/2 − uk−1/2) −(Az

u)k−1

hzk−1

(uk−1/2 − uk−3/2)

)

∀k : 2

(

Dzzu

)

i,j,K−1/2=

1

hzK−1/2

[

αDzzUb

2

(Azu)K−1

hzK−1

uK−1/2 − αDzzUb

1

( (Azu)K

hzK

+(Az

u)K−1

hzK−1

)

uK−3/

∂u

∂z

∣∣∣∣w0

= αDzzUs

0 +τx

hz1ρ0

,∂v

∂z

∣∣∣∣w0

= αDzzUs

0 +τy

hz1ρ0

,∂T

∂z

∣∣∣∣w0

=∂S

∂z

∣∣∣∣w0

= αDzzTs

0

u|bottom = v|bottom = αDzzUb

0 T |bottom = S|bottom = αDzzTb

0 (1)

· · ·

hz1/2︷︸︸︷

hz3/2︷︸︸︷

hz5/2︷︸︸︷

hzK−1/2︷︸︸︷

hzK−3/2︷︸︸︷

❜

u1/2

︸︷︷︸

hz1

❜

u3/2

︸︷︷︸

hz2

❜

u5/2

︸︷︷︸

hz3

❜

u7/2❜

uK−1/2

︸︷︷︸

hzK−1

❜

uK−3/2

︸︷︷︸

hzK−2

❜

uK−5/2


Adjoint

The models solution depend on initial and boundary conditions :

∂T

∂t= −Dx(uSxT )−Dy(vSyT )−D

(α)z (wS

(α)z T )+Ah

T

(

DxxT+DyyT

)

+Dzz(α)(AzT )T

The model x(t) = M0,t(x0, α)

We calculate the derivatives and their adjoints with respect to

x0, α

by TAPENADE 3.6 (Tropics team, INRIA) that allows us

to avoid a HUGE development/coding (a double of the classical one, at least)

to obtain immediately the derivative with respect to any parameter we want.


Adjoint

TAPENADE 3.6 (Tropics team, INRIA)with the Memory Usage Optimization:

search for push/pop

CALL PUSHREAL8ARRAY(sold, nx*ny*nz)CALL PUSHREAL8ARRAY(told, nx*ny*nz)CALL PUSHREAL8ARRAY(vold, nx*ny*nz)CALL PUSHREAL8ARRAY(uold, nx*ny*nz)CALL PUSHREAL8ARRAY(ssh, nx*ny)CALL PUSHREAL8ARRAY(s, nx*ny*nz)CALL PUSHREAL8ARRAY(t, nx*ny*nz)CALL PUSHREAL8ARRAY(v, nx*ny*nz)CALL PUSHREAL8ARRAY(u, nx*ny*nz)

replace by

call push uvts(u,v,t,s,ssh)

Procedure push/pop uvts(u,v,t,s,ssh):

does not push n− 1 step and pops appropriate values (divides the requiredmemory by 2)

does not push u, v, t, s in lower level routines

does not push values on continents (divides by 2)

pushes values in Real*4 format (divides by 2)

eventually pushes only odd timesteps and interpolate when poping (dividesby 2)

Total reduction of required memory is up to 25 times.10 hours window =⇒ 10 days window.


Data

ECMWF data issued from Jason-1 and Envisat altimetric missions andENACT/ENSEMBLES data banque.

January, 1, 2006.

Difference between observations and background during the 1st of January.


Cost function

The model: xN = M0,N (x0, α) with x = (u, v, T, S, ssh)T

Cost function J

J = ‖x0 − xbgr‖2B−1 + ‖α− αbgr‖

2B−1 +

+N∑

n=0

tn‖HM0,n(x0, α)− yn‖2R−1

Matrices: B−1 = diag(10−4),

R−1 = diag(1/σu, 1/σv , 1/σT , 1/σS , 1/σssh) where σ2u = 1

Nobs

∑(uobs − ubgr)

2

Minimization is performed by M1QN3 (JC Gilbert, C.Lemarechal)

Data Assimilation – Forecast

Assimilation window — 10 days (Jan. 1-10, 2006),Test time — 20 or 30 days (Jan. 1-31, 2006).


Distance Model-Observations

The model: x(t) = M0,t(x0, α) with x = (u, v, T, S, ssh)T

Distance: ξ(t) =t∑

n=0

‖HM0,n(x0, α)− yn‖R−1

Convergence of J and evolution of ξ

20 Cost function calls with T = 5 days and 40 calls with T = 10 days.


Optimal IC and Optimal BCz

SSH, North Atlantic, January,1-30 2006.

Optimal IC Optimal BCz


Optimal IC and Optimal BCz

SSH, North Pacific, January,1-30 2006.

Optimal IC Optimal BCz


BC for the vertical velocity

Modified formula

wi,j,K−1 = αwb

0 − αwb

2 hzi,j,K−1/2ξi,j,K−1/2

wi,j,k−1 = wi,j,k − hzi,j,k−1/2ξi,j,k−1/2 ∀k : 1 ≤ k ≤ K − 2

wi,j,0 = wi,j,1 + αws

0 − αws

2 hzi,j,1/2ξi,j,1/2


BC for the vertical velocity

α for the vertical velocity w. North Atlantic.

α0 on the surface α2 on the surface

α0 on the bottom α2 on the bottomEugene Kazantsev Boundary conditions control for ORCA2 page 10 of 19

Vertical velocity

North Atlantic, January, 30, 2006, surface

Original model Optimal BCz


Vertical velocity

North Atlantic, January, 30, 2006, y − z section



Vertical velocity

North Atlantic, January, 30, 2006, x− z section



Tourbillon

Levels z = 28 and z = 29

Velocity u Velocity v Velocity w

Velocity u Velocity v Velocity w


α0 for the operator Dzzu

∂u

∂z

∣∣∣∣w0

= αDzzUs

0 +τx

hz1ρ0,

∂v

∂z

∣∣∣∣w0

= αDzzUs

0 +τy

hz1ρ0,

u|bottom = v|bottom = αDzzUb

0

North Atlantic

Surface Bottom


Velocity components

North Atlantic, January, 30, 2006, Velocity u, y − z section

Original model Optimal BCzEugene Kazantsev Boundary conditions control for ORCA2 page 14 of 19

It is not an artefact.

North Atlantic

Modification of the SSH in the North Atlantic is strongly related to the boundaryconditions of u and v especially on the bottom.


Restrained control

Only α0 on the Bottom for u and v, only in the Vertical diffusion

∂

∂zA

zu

∂u

∂zis replaced by

(

Dzzu

)

i,j,1/2=

(Azu)1

hz1hz1/2

(u3/2−u1/2)

(

Dzzu

)

i,j,k−1/2=

1

hzk−1/2

( (Azu)k

hzk

(uk+1/2 − uk−1/2) −(Az

u)k−1

hzk−1

(uk−1/2 − uk−3/2)

)

∀k : 2

(

Dzzu

)

i,j,K−1/2=

1

hzK−1/2

[ (Azu)K−1

hzK−1

uK−1/2 −

( (Azu)K

hzK

+(Az

u)K−1

hzK−1

)

uK−3/2

]

u|bottom = αu0 v|bottom = α

v0 (2)

Control space dimension

Initial conditions: 1 707 245Full vertical boundary: 1 197 792Only bottom: 33 272


SSH in the restrained control experiment

North Atlantic


SSH in the restrained control experiment

North Pacific


Que peuvent attendre les modelisateurs de l’assimilation de donnees ?

Extending the set of control parameters we can

find a way to compensate model errors

showing the most influent parameter and the most important geographicalregions.

Automatic adjoint code generation helps us

to generate TLM/AM almost immediately,

to avoid a HUGE development/coding,

to obtain immediately the derivative with respect to any parameter we want.

http://www-ljk.imag.fr/membres/Kazantsev/orca2/index.html


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