numerical and physical delicacies in coupled ocean...
TRANSCRIPT
Physics dynamics coupling in geophysical models - bridging the gap, Dec. 2014
Numerical and Physical Delicacies in CoupledOcean-atmosphere Simulations
Florian Lemarie
INRIA Rhone-Alpes, EPI AIRSEA
Laboratoire Jean Kuntzmann, Grenoble
http://www-ljk.imag.fr/membres/Florian.Lemarie/
In collaboration with : E. Blayo (Univ. Grenoble), L. Debreu (INRIA), P. Marchesiello (IRD)
The INRIA AIRSEA team (previously MOISE team)
Mathematics and computing applied to oceanic and atmospheric flows
Context : Atmospheric and Oceanic Coupled Models (AOCMs)
Time0.1 yr 1 yr 10 yrs 100 yrs 1000 yrs
Climate Change PaleoclimateMonsoon
Cyclone Global ocean circulationEl-nino
Ingredients
OCEAN
ATMOSPHERE
COUPLERAIR-SEA FLUXES
COUPLINGALGORITHM
• Message-passing
• Synchronisation in time
• Interpolation/Extrapolation
• Bulk formulae
• Parameterized turbulent fluxes
• Radiative fluxes
• Asynchronous algorithms
HPCPhysics Numerics
Motivations
→ Physical issues : several semi-empirical parameterizations based on field andlaboratory experiments have been designed(Fairall et al., 2003 ; Large, 2006)
→ Computational issues : OASIS, MpCCI, ESMF, ...(Hill et al., 2004 ; Joppich and Kurschner, 2006 ; Redler et al., 2010)
→ Numerical coupling issues : .... room for innovation
It is known that coupled models exhibit strong sensitivity to model parameters(Bengtsson, 1999 ; McWilliams, 2007)
Initial conditions (Ploshay & Anderson, 2002)
coupling frequency (Terray et al., 2011 ; Masson et al., 2012)
air-sea flux formulation (Lebeaupin-Brossier et al., 2008)
⇒ How much of this sensitivity is reducible by further model improvements at a
mathematical level ?
Outline
1 Coupling problem formulation
2 Coupling algorithms : the different approaches and their stability
3 A numerical example
4 Convergence analysis
5 Conclusions and future directions
Outline
1 Coupling problem formulation
2 Coupling algorithms : the different approaches and their stability
3 A numerical example
4 Convergence analysis
5 Conclusions and future directions
Coupling problem
oce
atm
LoceUo = foce
LatmUa = fatm
BatmUa = gatm
BoceUo = goce
@extatm
@extoce
surface layer
ABL
OBL
habl
hobl
Kom(z)
Kam(z)
U = Ua(za) Uo(zo)
z
z
x, y
Turbulent fluids : X = 〈X〉+X ′,⟨w′X ′
⟩= −Km(z)
∂ 〈X〉∂z
Coupling problem
Find Uo = (uo, T o)t et Ua = (ua, T a)t that
satisfy (for a given initial condition)
LoceUo = foce in Ωoce × [0, T ]
BoceUo = goce on ∂Ωext
oce × [0, T ]
LatmUa = fatm in Ωatm × [0, T ]BatmUa = gatm on ∂Ωext
atm × [0, T ]oce
atm
LoceUo = foce
LatmUa = fatm
@extatm
@extoce
surface layer
ABL
OBL
z
x, y
Interface conditions on Γ× [0, T ] :
ρoKom∂zu
o = ρaKam∂zu
a = τ = ρaCD‖ua − uo‖(ua − uo)
ρocopKos∂zT
o = ρacapKas ∂zT
a = Qnet = Qrad + ρacapCH‖ua − uo‖(T a − T o)
Continuous problem
Is the coupling problem well-posed ?
Existence of a solution under some hypothesis (linearized primitive equations
and interface conditions, no stratification, no parameterizations)
e.g. : Lions, J.-L., R. Temam, S. Wang, 1995 : Mathematical theory for the
coupled atmosphere-ocean models, J. Math. Pures Appl., 74
Very few mathematical analysis including turbulent closures and noneof them in a coupled framework
e.g. : Deleersnijder, E., E. Hanert, H. Burchard, H.A. Dijkstra, 2008 : On the
mathematical stability of stratified flow models with local turbulence closure
schemes, Ocean Dynam., 58, 237-246
Outline
1 Coupling problem formulation
2 Coupling algorithms : the different approaches and their stability
3 A numerical example
4 Convergence analysis
5 Conclusions and future directions
Multiphysics coupling
Asynchronous coupling (based on time-mean fluxes)
oce
ti1ti2 ti
atm
ti+1
Foa(hUoii1 , Ua|i)
Foa(hUoii2 , Ua|i1)
↵i1
Foa(hUoii3 , Ua|i2)
↵i2
Foa(hUoii1 , Ua|i)
↵i
Foa(hUoii3 , Ua|i2) Foa(hUoii2 , Ua|i1)
.i.i2 .i1
ii 1i 2
Synchronous coupling (based on instantaneous fluxes)
!oce
!atm
t
t t + !to
t + N!ta t + 2N!ta
t + 2!to
Foa Foa Foa
Foa = function allowing the computation of air-sea fluxes
Main problems
Synchronous coupling (based on instantaneous fluxes)Relevance of instantaneous fluxes ?Stability with implicit schemes ?Computationally intensiveUnnatural to handle space-time non-conformities
Asynchronous coupling (based on time-mean fluxes)Retroaction of the ocean is shifted in time
Remarks on the air-sea flux computation (semi-empirical MO theory)
Quasi-stationarity, horizontal homogeneity
Constant flux layer assumption
Dimensional analysis excluding the viscous sublayer
Calibration with averaged in space/time data
⇒ air-sea fluxes must be interpreted as mean fluxes (∆tdyn 6= ∆tphys)
W. G. Large. Surface fluxes for practitioners of global ocean data assimilation. In E. Chassignet and J. Verron, editors, An
Integrated View of Oceanography : Ocean Weather Forecasting in the 21st Century. Kluwer, 2006.
Main problems
Synchronous coupling (based on instantaneous fluxes)Relevance of instantaneous fluxes ?Stability with implicit schemes ?Computationally intensiveUnnatural to handle space-time non-conformities
Asynchronous coupling (based on time-mean fluxes)Retroaction of the ocean is shifted in time
Remarks on the air-sea flux computation (semi-empirical MO theory)
Quasi-stationarity, horizontal homogeneity
Constant flux layer assumption
Dimensional analysis excluding the viscous sublayer
Calibration with averaged in space/time data
⇒ air-sea fluxes must be interpreted as mean fluxes (∆tdyn 6= ∆tphys)
W. G. Large. Surface fluxes for practitioners of global ocean data assimilation. In E. Chassignet and J. Verron, editors, An
Integrated View of Oceanography : Ocean Weather Forecasting in the 21st Century. Kluwer, 2006.
Stability analysis at a discrete level (Dirichlet-Neumann algorithm)
Parabolic equations : fully implicit time integration with explicit updating of the
interface boundary conditions (i.e. synchronous algorithm)
For z ∈ R− (i.e. k ≤ 0)
qn+1k − qnk∆t
=ν1
∆z21
(qn+1k+1 − 2qn+1
k + qn+1k−1
), k < 0
qn+10−− qn
0−
∆t=
2
∆z1
[−Fn − ν1
∆z1(qn+1
0−− qn+1−1 )
],
Fn = −ν2
∆z2(qn1 − qn0+)
For z ∈ R+ (i.e. k ≥ 0)
qn+1k − qnk∆t
=ν2
∆z22
(qn+1k+1 − 2qn+1
k + qn+1k−1
), k > 0
qn+1
0+ = qn0−
z = 0
k = 1
k = 1k = 0+
k = 0
z1
z2
| z
| z k = 2
k = 2
R R+
Cell interfaces
Cell center
Stability analysis at a discrete level (Dirichlet-Neumann algorithm)
⇒ Synchronous algorithm is only conditionally stable
→ The amplification factor A satisfies the equation
√1 +
4σ11−A−1 + rA−2
(√1 +
4σ21−A−1 − 1
)= 0, σj =
νj∆t
∆z2j, r =
∆z2∆z1
→ Asymptotic results
r 1 : A ≈ (1 + 4σ1)−1 stable discretization
r 1 : A−1 ≈ 0 unstable discretization
σj 1 : stability limit
0 < r <
√σ1σ2
⇔√ν1ν2
> 1
→ in the context of OA coupled models : r ∼ 60,
√ν1ν2∼ 1/10
Global-in-time Schwarz method (a.k.a. Schwarz waveform relaxation)
Loop over k till convergence (for a given initial condition) [0, T ] =⋃Ni=1[ti, ti+1]
LatmUak = fatm in Ωatm × [ti, ti+1]
BatmUak = gatm on ∂Ωext
atm × [ti, ti+1]
(Kam∂zu
a)k = CD‖uak − uok−1‖(uak − uok−1) on Γ× [ti, ti+1](Ka
s ∂zTa)k = CH‖uak − uok−1‖(T ak − T ok−1) on Γ× [ti, ti+1]
LoceUok = foce in Ωoce × [ti, ti+1]
BoceUok = goce on ∂Ωext
oce × [ti, ti+1]
(Kom∂zu
o)k = (ρa/ρo) (Ka
m∂zua)k on Γ× [ti, ti+1]
(Kos∂zT
o)k = (ρacap/ρ
ocop) (Kas ∂zT
a)k on Γ× [ti, ti+1]
with Uok=0 = Uo|[ti−1,ti]
Global-in-time Schwarz method (a.k.a. Schwarz waveform relaxation)
Loop over k till convergence (for a given initial condition) [0, T ] =⋃Ni=1[ti, ti+1]
LatmUak = fatm sur Ωatm × [ti, ti+1]
BatmUak = gatm sur ∂Ωext
atm × [ti, ti+1]
(Kam∂zu
a)k = CD‖uak − 〈uok−1 〉‖(uak − 〈uok−1 〉) on Γ× [ti, ti+1](Ka
s ∂zTa)k = CH‖uak − 〈uok−1 〉‖(T ak − 〈T ok−1 〉) sur Γ× [ti, ti+1]
LoceUok = foce sur Ωoce × [ti, ti+1]
BoceUok = goce sur ∂Ωext
oce × [ti, ti+1]
(Kom∂zu
o)k = (ρa/ρo)〈 (Ka
m∂zua)k 〉 sur Γ× [ti, ti+1]
(Kos∂zT
o)k = (ρacap/ρ
ocop)〈 (Kas ∂zT
a)k 〉 sur Γ× [ti, ti+1]
with Uok=0 = Uo|[ti−1,ti]
Numerical considerations
→ Asynchronous method = only one iteration of a global-in-timeSchwarz algorithm
→ Synchronous method = only one iteration of a local-in-time Schwarzalgorithm
Toy model (i.e. no errors in the flux computation)
`2-error
Schwarz synchronous Schwarz asynchronousinstantaneous fluxes instantaneous fluxes mean fluxes mean fluxes
0 0.9 10 100
⇒ even for mean fluxes, things are improved
Outline
1 Coupling problem formulation
2 Coupling algorithms : the different approaches and their stability
3 A numerical example
4 Convergence analysis
5 Conclusions and future directions
Realistic simulations
Numerical codes :
. WRF (NCAR), Compressible Euler
. ROMS (UCLA, IRD, INRIA), Primitive equations
Test-case : simulation of tropical cyclone Erica (New Caledonia, 03/2003)∆xatm = ∆xoce = 30 km, ∆tatm = 120 s, ∆toce = 2400 s
Ensemble approach : perturbation of initial conditions and couplingfrequency (3h vs 6h)
Two ensembles : asynchronous method vs Schwarz method
Iterative process
1 2 3 4 5 6 7 8Iterates k
0.0
0.2
0.4
0.6
0.8
1.0
Rk
convergence rate
ti+1 − ti = 3hti+1 − ti = 6h
0 1 2 3 4 5 6 7 8Iterates k
10−2
10−1
100
101
102
‖ek‖ 2
error `2-normti+1 − ti = 3hti+1 − ti = 6h
Rk =‖ek‖2‖ek−1‖2
=‖SST o
k − SSToM‖2
‖SST ok−1 − SST
oM‖2
, ‖ek‖2 =
√√√√nx×ny∑i=1
∣∣(SST ok
)i−(SST
oM
)i
∣∣2
Ensemble simulations
asynchronous SchwarzTrajectory 125 km 92 kmIntensity 6.2 m s−1 4.1 m s−1
0 π4
π2
3π4
π
ω[h−1]
10−4
10−3
10−2
10−1
100
Nor
mal
ized
PS
D
ti+1 − ti = 3h
ti+1 − ti = 6h
Spω‖ua
h(z = 10 m)k=M‖ − ‖uah(z = 10 m)asyn‖
150 155 160 165 170
Longitude
−24
−22
−20
−18
−16
−14
−12
Latit
ude
Schwarz method
envelopeensemble mean
150 155 160 165 170
Longitude
−24
−22
−20
−18
−16
−14
−12
Latit
ude
Asynchronous coupling
envelopeensemble mean
Outline
1 Coupling problem formulation
2 Coupling algorithms : the different approaches and their stability
3 A numerical example
4 Convergence analysis
5 Conclusions and future directions
Open questions
→ Which parameters control the convergence rate ?
→ Could we improve the convergence rate ?
→ Impact of surface/boundary layer formulation on the convergence ?
Model problem to get more insight on the convergence properties :
Lu = ∂tu− ∂z 〈u′w′〉 = ∂tu− ∂z(ν(z)∂zu) = f in [0, T ]×]− L1;L2[
We define two non-overlapping subdomains Ω1 =]− L1; 0[, Ω2 =]0, L2[.
The solution to the problem satisfies the matching conditions
u1(0, t) = u2(0, t), ν1(0)∂zu1(0, t) = ν2(0)∂zu2(0, t)
Schwarz Waveform Relaxation (alternating form)
L1uk1 = f, in Ω1 × [0, T ],
uk1(z, 0) = uo(z), z ∈ Ω1,
B1uk1(−L1, t) = g1, t ∈ [0, T ],
F1uk1(0, t) = F2u
k−12 (0, t), in Γ× [0, T ],
L2uk2 = f, in Ω2 × [0, T ],
uk2(z, 0) = uo(z), z ∈ Ω2,
B2uk2( L2, t) = g2, t ∈ [0, T ],
G2uk2(0, t) = G1u
k1(0, t), in Γ× [0, T ].
Transmission Conditions
”Naive“ approach : Dirichlet-Neumann conditions
u1(0, t) = u2(0, t), ν1(0)∂zu1(0, t) = ν2(0)∂zu2(0, t)
Optimized SWR
( ν1(0)∂z + Λ1)u1(0, t) = ( ν2(0)∂z + Λ1)u2(0, t)(−ν2(0)∂z + Λ2)u2(0, t) = (−ν1(0)∂z + Λ2)u1(0, t)
Absorbing conditions : non-local in time in physical space.
zeroth-order 2-sided approximation : λ01 ≈ Λ1, λ02 ≈ Λ2
minλ01,λ
02∈R
(max
ω∈[ωmin,ωmax]R(λ01, λ
02, ω)
)R =
∣∣∣∣ek1(0, ω)
ek−11 (0, ω)
∣∣∣∣
Problem well-posed for λ01 + λ02 > 0
Optimized Robin Transmission Conditions (Lemarie, Debreu, Blayo, 2013a)
. Case 1 : constant and discontinuous coefficients ν1 6= ν2, γ =√ν1/ν2
µ =(
ωminωmax
)1/4∼ N1/4
t , Nt time steps on a time-window [ti; ti+1]
1 2 3 4
2
4
6
8
10
12
14
16
18
3-point equioscillation
µ
2-point equioscillation(ν1 ν2) (ν1 = ν2) (ν1 ν2)
RR? 0(µ + µ−1)−
√2
(µ + µ−1) +√2
0
Optimized Robin Transmission Conditions (Lemarie, Debreu, Blayo, 2013b)
. Case 2 : spatially-variable and discontinuous coefficients ν1(z) 6= ν2(z)
→ Derivation of the analytical convergence factor ∂z (ν(x) ∂zΦn) + c2nΦn = 0 z ∈]0, L[−ν(0) ∂zΦn(0) + λΦn(0) = 0Φn(L) = 0
Impact convergence for low temporal frequenciesRobin-Robin TCs more robust than Dirichlet-Neumann TCs.
⇒ Possibility to systematically check the relevance of the constant coefficients assumption
Outline
1 Coupling problem formulation
2 Coupling algorithms : the different approaches and their stability
3 A numerical example
4 Convergence analysis
5 Conclusions and future directions
Summary
The OA coupling problem has been written in the framework of theSchwarz domain decomposition methods
→ Current coupling methods correspond toone iteration of an iterative method
Part of the variability in the response of coupled models may beattributed to numerical flaws in the coupling method
The iterative process affects mainly low temporal frequencies ⇒significant impact on coupled climate models ?
Mathematical tools have been developed to study theconvergence of iterative coupling algorithms
encouraging ... but it’s just a beginning
Perspectives and topics relevant for dynamics/physics coupling
Define ”compatibility criteria” between OBL and ABLparameterizations
. Add the formulation of parameterizations in the convergence analysis
. Define constraints on the formulation of BL schemes
The corresponding Schwarz algorithm must converge (cont.)Preserve stationary states of the continuous problem (disc.)
Development of idealized coupled testcases (mostly single-column)+ coupled LES for reference solutions (J.-L. Redelsperger)
Improve the convergence speed (limit the # of iterations)
. Nature of interface conditions
. Use of a rapid solver for the iterative process
. Impact on space-time nonconformities
Assess the impact in the context of IPCC-like climate models
. PhD thesis of Charles Pelletier (co-advis. E. Blayo, F. Lemarie and P. Braconnot)