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Submitted on 13 Dec 2012
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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�
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
Eugene Kazantsev Boundary conditions control for ORCA2 page 2 of 21
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)
Eugene Kazantsev Boundary conditions control for ORCA2 page 3 of 21
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
Eugene Kazantsev Boundary conditions control for ORCA2 page 4 of 21
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 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
Eugene Kazantsev Boundary conditions control for ORCA2 page 4 of 21
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) = 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
Eugene Kazantsev Boundary conditions control for ORCA2 page 4 of 21
Space discretization
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
Eugene Kazantsev Boundary conditions control for ORCA2 page 4 of 21
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
Eugene Kazantsev Boundary conditions control for ORCA2 page 5 of 21
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.
Eugene Kazantsev Boundary conditions control for ORCA2 page 6 of 21
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.
Eugene Kazantsev Boundary conditions control for ORCA2 page 6 of 21
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.
Eugene Kazantsev Boundary conditions control for ORCA2 page 7 of 21
Data
January, 1-20 2006.
Number of observation per time stepduring the 1 – 20 Jan.2006
Probability density function for thedifference (observation-background).
Eugene Kazantsev Boundary conditions control for ORCA2 page 8 of 21
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.
Eugene Kazantsev Boundary conditions control for ORCA2 page 9 of 21
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.
Eugene Kazantsev Boundary conditions control for ORCA2 page 10 of 21
Optimal IC and Optimal BCz
SSH, North Atlantic, January,1-30 2006.
Optimal IC Optimal BCz
Eugene Kazantsev Boundary conditions control for ORCA2 page 11 of 21
Optimal IC and Optimal BCz
SSH, North Pacific, January,1-30 2006.
Optimal IC Optimal BCz
Eugene Kazantsev Boundary conditions control for ORCA2 page 12 of 21
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
Eugene Kazantsev Boundary conditions control for ORCA2 page 13 of 21
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
Eugene Kazantsev Boundary conditions control for ORCA2 page 14 of 21
Vertical velocity
North Atlantic, January, 30, 2006, y − z section
Original model Optimal BCz
Eugene Kazantsev Boundary conditions control for ORCA2 page 14 of 21
Vertical velocity
North Atlantic, January, 30, 2006, x− z section
Original model Optimal BCz
Eugene Kazantsev Boundary conditions control for ORCA2 page 14 of 21
Tourbillon
Levels z = 28 and z = 29
Velocity u Velocity v Velocity w
Velocity u Velocity v Velocity w
Eugene Kazantsev Boundary conditions control for ORCA2 page 15 of 21
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
Eugene Kazantsev Boundary conditions control for ORCA2 page 16 of 21
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.
Eugene Kazantsev Boundary conditions control for ORCA2 page 19 of 21
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
Eugene Kazantsev Boundary conditions control for ORCA2 page 20 of 21
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).
Eugene Kazantsev Boundary conditions control for ORCA2 page 20 of 21
En vous remerciant pour votre attention
j’apprecierai beaucoup vos commentaires et critiques
http://www-ljk.imag.fr/membres/Kazantsev/orca2/index.html
Eugene Kazantsev Boundary conditions control for ORCA2 page 21 of 21
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