Numerical and Physical Delicacies in Coupled Ocean...

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)

http://www-ljk.imag.fr/membres/Florian.Lemarie/

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

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






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.

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






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






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






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)

Numerical and Physical Delicacies in Coupled Ocean...

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