Boundary conditions control in ORCA2 · 2021. 1. 2. · Boundary conditions control in ORCA2 Eugene...

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Boundary conditions control in ORCA2Eugene Kazantsev

To cite this version:Eugene Kazantsev. Boundary conditions control in ORCA2. 4ème Colloque National surl’Assimilation des données (2012), Dec 2012, Nice, France. �hal-00764588�

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Boundary conditions control in ORCA2

Eugene Kazantsev

INRIA, Moise

December, 17, 2012

Eugene Kazantsev Boundary conditions control for ORCA2 page 1 of 21

The model (Dynamics)

ORCA2 Configuration : 182 × 149 × 31 nodes in curvilinear (x, y) coordinates with zlevels.

∂u

∂t= v(ω + f) −

∂(u2 + v2)/2

∂x− w

∂u

∂z︸︷︷︸

Advection

−

∂Ahuξ

∂x+

∂Ahuω

∂y︸︷︷︸

Hor.Dissipation

+

+∂

∂zA

zu

∂u

∂z︸︷︷︸

Vert.Dissipation

+ g

∫ z

0

∂ρ(x, y, ζ)

∂xdζ

︸︷︷︸

Pressure gradient

+ g∂(η + Tcφ)

∂x︸︷︷︸

SSH

∂v

∂t= −u(ω + f) −

∂(u2 + v2)/2

∂y− w

∂v

∂z−

∂Ahuξ

∂y−

∂Ahuω

∂x+

+∂

∂zA

zu

∂v

∂z+ g

∫ z

0

∂ρ

∂ydz +

∂(η + Tcφ)

∂y

ξ =∂u

∂x+

∂v

∂y, ω =

∂u

∂y−

∂v

∂xDivergence, Vorticity

w =

∫ z

H

ξ(x, y, ζ)dζ; w(x, y,H) = 0 Vertical velocity

φ =∂η

∂tGrav.Waves filter A

hu = const


The model (Thermodynamics)

∂T

∂t= −∂uT

∂x− ∂vT

∂y− ∂wT

∂z︸︷︷︸

Advection

+AhT

(∂2T

∂x2+

∂2T

∂y2

)

+

︸︷︷︸

Hor.diffusion

+∂

∂zAz

T

∂T

∂z︸︷︷︸

Vert.diffusion

+Solar Radiation + Geothermal Heating + BBL + Surface

∂S

∂t= −∂uS

∂x− ∂vS

∂y− ∂wS

∂z+Ah

T

(∂2S

∂x2+

∂2S

∂y2

)

+∂

∂zAz

S

∂S

∂z+ BBL + Surface

AzT = Turbulent closure: ∼ max(Az

0, Ck lk√e) (1)

∂e

∂t= Az

v

[(∂u

∂z

)2

+

(∂v

∂z

)2]

−AzT N2 +

∂

∂z

[

Azu

∂e

∂z

]

− cǫe3/2

lǫ

ρ = ρ(T, S), N2 = N2(T, S), R = R(T, S)


Space discretization

∂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

b

T1/2b

T3/2b

T5/2b

T7/2b

TK−1/2b

TK−3/2b

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 (2)

· · ·

hz1/2︷︸︸︷

hz3/2︷︸︸︷

hz5/2︷︸︸︷

hzK−1/2︷︸︸︷

hzK−3/2︷︸︸︷

b

u1/2

︸︷︷︸

hz1

b

u3/2

︸︷︷︸

hz2

b

u5/2

︸︷︷︸

hz3

b

u7/2b

uK−1/2

︸︷︷︸

hzK−1

b

uK−3/2

︸︷︷︸

hzK−2

b

uK−5/2


Time Discretization

un+1 − un−1

2dt= γ(un+1 − 2un + un−1)

︸︷︷︸

Asselin filter

−(

SxSyvn

)

Sy(ωn + f)−

− DxSx(un)2 + Sy(vn)2

2− Sz

(

SxwnDzu

n

)

+

+ DxAhuξ

n−1 +DyAhuω

n−1

︸︷︷︸

Explicit Euler

+DzzAzuu

n+1

︸︷︷︸

Impl.Euler

+

+ g

∫ z

0DxSzρ

n(x, y, ζ)dζ + gDx(ηn+1 + Tcφ

n+1)︸︷︷︸

Implicit

Tn+1 − Tn−1

2dt= γ(un+1 − 2un + un−1)

︸︷︷︸

Asselin filter

−Dx(unSxT

n)−Dy(vnSyT

n)−

− Dz(wnSzT

n) +AhT

(

DxxTn−1 +DyyT

n−1

)

︸︷︷︸

Explicit Euler

+

+ Dzz(AzT )Tn+1

︸︷︷︸

Implicit

+Solar Radiation + Geothermal Heating + BBL


Adjoint

The models solution depend on initial conditions and a number of parameters:

∂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.


Data

January, 1-20 2006.

Number of observation per time stepduring the 1 – 20 Jan.2006

Probability density function for thedifference (observation-background).


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,Test time — 20 days.


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 13 of 21

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 17 of 21

Velocity in the Gulf stream

North Atlantic on the Jan.,30,2006

Original model, depth 65 m Optimal BCz model, depth 65 m

Original model, depth 160 m Optimal BCz model, depth 160 mEugene Kazantsev Boundary conditions control for ORCA2 page 18 of 21

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.


Conclusion

Boundary Conditions influence is important

The cost function decreases more under BCz control than under IC control inthe assimilation window

The models forecast is closer to observations with optimal BCz than withoptimal IC

Stream jets are refined under BCz control

Tapenade allows 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.

But, it requires more memoryResolution TimeStep Model Size AM Size Traj. size

per day per 10 daysORCA 2 181× 149× 31 15 650M 1.6G 660MORCA05 722× 511× 46 40 13G 33G 35G


Conclusion

BUT

As well as for any adjoint parameter estimation

The control may violate the model physics;

The physical meaning of α is difficult to understand;

The set of α is not unique;

The problem of identifiability is not addressed yet;

The problem of stability is not even posed.

Consequently:

It is not a parameter estimation study, but

a way to compensate model errors

showing the most influent parameter (vertical BC for ORCA-2, lateral BC forShallow-Water).


En vous remerciant pour votre attention

j’apprecierai beaucoup vos commentaires et critiques

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


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