# A few comments on the continuous and discrete formulation of the ocean-atmosphere coupling problem

Florian Lemarié, Eric Blayo, Valentina Schüller, Simon Clement, Sophie Thery, Charles Pelletier

## ► To cite this version:

Florian Lemarié, Eric Blayo, Valentina Schüller, Simon Clement, Sophie Thery, et al.. A few comments on the continuous and discrete formulation of the ocean-atmosphere coupling problem. Rencontres annuelles 2024 du GDR "Défis théoriques pour les sciences du climat" - Interfaces dans le système climatique, May 2024, Grenoble, France. hal-05485783

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https://inria.hal.science/hal-05485783v1

Submitted on 30 Jan 2026

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LABORATOIRE
JEAN KUNTZMANN
MATHEMATIQUES APPLIQUÉES - INFORMATIQUE

# A few comments on the continuous and discrete formulation of the ocean-atmosphere coupling problem

F. Lemarié¹, E. Blayo¹, V. Schüller²,¹, S. Clément¹, S. Thery³,¹, C. Pelletier⁴,¹

¹ Inria (Airsea Team), Univ. Grenoble-Alpes, LJK, France

² Centre for mathematical sciences, Lund University, Lund, Sweden

³ Institut für Mathematik, Augsburg University, Augsburg, Germany

⁴ ECMWF, Earth System Predictability Section, Bonn, Germany
Context

<figure><img src="image_4.png" /></figure>
Air-sea interactions

<figure><img src="image_5.png" /></figure>

Articulation of several "*components*"

* resolved fluid dynamics

* unresolved fluid dynamical aspects (i.e., those represented by physical parameterizations)

* nonfluid dynamical aspects such as radiation and microphysical processes

These *components* are developed mostly independently of each other and then coupled together in some manner
# COCOA: Comprehensive Coupling approach for the Ocean and the Atmosphere

Jan. 2017 - Sept. 2021

Partners : LJK (Grenoble), LSCE (Paris-Saclay), LMD (Paris-Sorbonne), LOPS (Brest), CNRM/CERFACS (Toulouse)

<figure><img src="image_6.png" /></figure>

**Objective** : revisiting the representation of ocean-atmosphere interactions in climate models

* Parameterization of boundary layers and air-sea fluxes

* Improvement of the algorithmic formulation of the OA coupling

* Evaluation of the impact of these developments using coupled SCMs and GCMs
Content

1. Formulation of the OA coupling problem

2. Coupling schemes in OA context

3. Analysis of a coupled Ekman boundary layer problem

4. Numerical experiments
1

Formulation of the OA coupling problem
# Coupling problem in the single-column framework

## Geophysical approximations :

$$ (\partial_z \bar{\nu}, \partial_z \bar{\Theta}) \gg (\partial_{x,y} \bar{\nu}, \partial_{x,y} \bar{\Theta}) \quad \text{(horizontal homogeneity)} $$

$$ \partial_z (\overline{w'v'}, \overline{w'\Theta'}) \gg \overline{w} \partial_z (\bar{\nu}, \bar{\Theta}) $$

atmosphere = moist mixture of dry gas with water vapor

Ocean

$$ \partial_t \bar{\nu}_{\text{oce}} = -f \mathbf{e}_z \times (\bar{\nu}_{\text{oce}} - \mathbf{v}_{g,o}) - \partial_z (\overline{w'v'})_{\text{oce}} $$

$$ \partial_t \bar{\Theta}_{\text{oce}} = -\partial_z (\overline{w'\Theta'})_{\text{oce}} - \frac{\partial_z Q_s}{\rho_0 c_{p,o}} $$

$$ \partial_t \bar{S}_{\text{oce}} = -\partial_z (\overline{w'S'})_{\text{oce}} $$

$$ \rho_o = \rho_{\text{eos}}(\bar{\Theta}_{\text{oce}}, \bar{S}_{\text{oce}}, p_0(z)) $$

Atmosphere

$$ \partial_t \bar{\nu}_{\text{atm}} = -f \mathbf{e}_z \times (\bar{\nu}_{\text{atm}} - \mathbf{v}_{g,a}) - \frac{1}{\rho_a} \partial_z (\rho_a (\overline{w'v'})_{\text{atm}}) $$

$$ \partial_t \bar{\Theta}_{\text{atm}} = -\frac{1}{\rho_a c_{p,a}} \partial_z (\rho_a c_{p,a} (\overline{w'\Theta'})_{\text{atm}} + \mathcal{F}_{\text{SW}} + \mathcal{F}_{\text{LW}}) $$

$$ \partial_t \bar{q}_{\text{atm}} = \frac{1}{\rho_a L^{(l)}} \partial_z (\rho_a L^{(l)} (\overline{w'q'})_{\text{atm}}) $$

$$ \rho_a = \rho(\bar{\Theta}_{\text{atm}}, p_{\text{atm}}) $$
Coupling problem in the single-column framework

Ocean

$$
\begin{align*}
\partial_t \bar{\mathbf{v}}_{\text{oce}} &= -f \mathbf{e}_z \times (\bar{\mathbf{v}}_{\text{oce}} - \mathbf{v}_{g,o}) - \partial_z (\mathbf{w}' \mathbf{v}')_{\text{oce}} & \partial_t \bar{\mathbf{v}}_{\text{atm}} &= -f \mathbf{e}_z \times (\bar{\mathbf{v}}_{\text{atm}} - \mathbf{v}_{g,a}) - \frac{1}{\rho_a} \partial_z (\rho_a (\mathbf{w}' \mathbf{v}')_{\text{atm}}) \\
\partial_t \bar{\mathbf{\Theta}}_{\text{oce}} &= -\partial_z (\mathbf{w}' \mathbf{\Theta}')_{\text{oce}} - \frac{\partial_z Q_s}{\rho_0 c_{p,o}} & \partial_t \bar{\mathbf{\Theta}}_{\text{atm}} &= -\frac{1}{\rho_a c_{p,a}} \partial_z (\rho_a c_{p,a} (\mathbf{w}' \mathbf{\Theta}')_{\text{atm}} + \mathcal{F}_{\text{SW}} + \mathcal{F}_{\text{LW}}) \\
\partial_t \bar{\mathbf{S}}_{\text{oce}} &= -\partial_z (\mathbf{w}' \mathbf{S}')_{\text{oce}} & \partial_t \bar{\mathbf{q}}_{\text{atm}} &= \frac{1}{\rho_a L^{(l)}} \partial_z (\rho_a L^{(l)} (\mathbf{w}' \mathbf{q}')_{\text{atm}}) \\
\rho_o &= \rho_{\text{eos}} (\bar{\mathbf{\Theta}}_{\text{oce}}, \bar{\mathbf{S}}_{\text{oce}}, p_0(z)) & \rho_a &= \rho (\bar{\mathbf{\Theta}}_{\text{atm}}, p_{\text{atm}})
\end{align*}
$$

Atmosphere

• Internal energy interface condition

$$
\rho_0 c_{p,o} (\overline{\mathbf{w}' \Theta}')_{\text{oce}} = \underbrace{\rho_a c_{p,a} (\overline{\mathbf{w}' \Theta}')_{\text{atm}}}_{\mathcal{F}_T:\text{sensible heat flux}} + \underbrace{\mathcal{F}_{\text{SW}} + \mathcal{F}_{\text{LW}}}_{\text{net radiation flux}} + \underbrace{\rho_a L^{(l)} (\overline{\mathbf{w}' q}')_{\text{atm}}}_{\mathcal{F}_L:\text{latent heat flux}}
$$

• Continuity of stresses (under the wind-wave equilibrium assumption)

$$
\rho_0 (\overline{\mathbf{w}' \mathbf{v}'}_{\text{oce}}) = \rho_a (\overline{\mathbf{w}' \mathbf{V}'}_{\text{atm}}) = \tau^a, \quad p_{\text{oce}}|_{z=\eta} = p_{\text{atm}}
$$

<figure><img src="image_7.png" /></figure>
Notion of "boundary layer" in OA context

Lemone et al. (2019) :

* Atmospheric turbulent boundary layer: "part of the troposphere that is directly influenced by the presence of the earth's surface, and responds to surface boundary conditions with a timescale of about an hour or less"

* In the **surface boundary layer** the velocity passes from $O(0)$ to $O(1)$ over a short distance

<figure><img src="image_8.png" /></figure>
Coupling problem in the single-column framework

• In turbulent boundary layers (BLs) :

$$ (w'X')^{\text{param}} \equiv -\nu^t \partial_z \overline{X} + \text{coherent structures} $$

$\nu^t$ given by an appropriate closure($k, k-\epsilon, \text{K-profile}, \dots$)

• In Surface Boundary Layer (SBL) :

$$ (w'X')^{\text{param}} \equiv u_* X_* $$

<figure><img src="image_9.png" /></figure>
# Coupling problem in the single-column framework

## In Surface Boundary Layer (SBL) :

$$ (w'X')^{\text{param}} = u_{*}X_{*} $$

## Semi-empirical Monin-Obukhov similarity theory (MOST)

$$ \bar{X}_{\text{atm}}(z) = \bar{X}_{\text{oce}}(zX) - \frac{X_{*}}{\kappa} \left[ \ln\left(\frac{z}{z_X(u_{*}, X_{*})}\right) - \psi_X \left(\frac{z}{L_{\text{Ob}}(u_{*}, X_{*})}\right) + \psi_X \left(\frac{z_X(u_{*}, X_{*})}{L_{\text{Ob}}(u_{*}, X_{*})}\right) \right] $$

* Generalization of the classical law of the wall (Von Kármán, 1930) to stratified conditions

* Usually implemented via “bulk algorithms”

* Internal time-scale $\Delta t_{\text{blk}}$ to keep uncertainty on turbulent flux estimates at a "reasonable level"

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>
**OA coupling problem** = imbrication of governing equations of Fluid Dynamics with nonlinear Thermodynamics, turbulent BLs parameterizations leading to nonlinear interface conditions, ...

From a mathematical point of view :

* Well-posedness of the systems of equations "with turbulence" ?

- Bernardi et al. (2002), Thery (2024) : coupling of 2 stationary turbulent fluids (no stratification)

- Deleersnijder et al. (2008), Chacón Rebollo et al. (2014) : oceanic single column with BLs closure

- Pironneau & Mohammadi (1994), Chacón Rebollo & Lewandowski (2014), Jiang et al. (2021) : mathematical foundation of eddy viscosity models based on TKE (no stratification)

- Work of S.Thery continued as part of **PEPR Maths-Vives** (Master's thesis of P. Dufour)

* Mutual benefits are to be expected for modelers

- Mathematical analysis is all the more tractable when parametrization schemes are based on first principles under clearly identified approximations and hypotheses

- Mathematical analysis could exhibit conditions of compatibility for the imbrication of turbulence parameterizations

⇒ What would be a representative continuous problem for the formulation of realistic coupled models ?
# Articulation of SBL with resolved flow

## Usual practice in FSI community

<figure><img src="image_12.png" /></figure>

## OA context

<figure><img src="image_13.png" /></figure>

## Treatment of boundary cells in practical implementations

<figure><img src="image_14.png" /></figure>

## Resulting inconsistencies :

* The implementation of MOST in coupled models ignores the ocean vertical discretization (inherently *one-sided*)

* Dual treatment of the boundary cells

* ⇒ not possible to properly formulate the continuous problem the coupled models discretize

* ⇒ problem should theoretically be formulated with 3 subdomains
Concept of *two-sided* bulk closure

Specific parameterizations for near surface ocean-specific processes are often developed in a way that is disconnected from the bulk closure → Concept of *two-sided* bulk closure

<figure><img src="image_15.png" /></figure>

* Framework within which incorporating new parameterizations can be done, without altering the global consistency

* Occurrences with significant differences in turbulent air-sea flux determination (up to 20%)
Concept of two-sided bulk closure

Specific parameterizations for near surface ocean-specific processes are often developed in a way that is disconnected from the bulk closure → Concept of two-sided bulk closure

<figure><img src="image_16.png" /></figure>

* Framework within which incorporating new parameterizations can be done, without altering the global consistency

* Occurrences with significant differences in turbulent air-sea flux determination (up to 20%)
Dual treatment of the boundary cells

▷ The extent of the SBL $\delta_{SL}$ is imposed by the discretization
⇒ SBL is both treated by a standard numerical scheme and in a parameterized form

▷ Quantities at the first model level are interpreted in a finite-difference sense

Concept of **log-layer mismatch** in LES community : the simulated vertical profiles do not match the logarithmic relation predicted by MOST.

Some work to improve the numerical treatment of the SBL

* Nishizawa & Kitamura (2018): computation of surface fluxes using Volume-Averaged quantities

* Kawai & Larsson (2012): use other levels than the first one to evaluate the wall law (neutral case)
  → take an elevated level $z_{sl}$ at which the flow is well resolved
  → improved grid convergence

* Maronga et al. (2020): going beyond the Instantaneous Logarithm (IL) method
  → combines local and spatially averaged quantities for the MOST prediction
# Development of a dedicated FV discretization

* Consider subgrid reconstructions consistent with MOST in SBL

* The choice of $\delta_{SL}$ is free

<figure><img src="image_17.png" /></figure>

⇒ nearing a discretization consistent with the continuous coupled problem formulated earlier

Clément S., F. Lemarié, E. Blayo, 2023: *Towards a Finite Volume Discretization of the Atmospheric Surface Layer Consistent with Physical Theory*, in: Finite Volumes for Complex Applications X, Springer

Clément, S., 2022: *Numerical analysis for a combined space-time discretization of air-sea exchanges and their parameterizations*. Ph.D. thesis, Université Grenoble Alpes (2022), (tel-04066324)
# Development of a dedicated FV discretization

* Consider subgrid reconstructions consistent with MOST in SBL

* The choice of $\delta_{SL}$ is free

## Atmosphere-only (unstable case)

<figure><img src="image_18.png" /></figure>

## Coupled case

<figure><img src="image_19.png" /></figure>

Clément S., F. Lemarié, E. Blayo, 2023: *Towards a Finite Volume Discretization of the Atmospheric Surface Layer Consistent with Physical Theory*, in: Finite Volumes for Complex Applications X, Springer

Clément, S., 2022: *Numerical analysis for a combined space-time discretization of air-sea exchanges and their parameterizations*. Ph.D. thesis, Université Grenoble Alpes (2022), (tel-04066324)
# Partial conclusion

## Comments on the formulation of the OA coupling problem :

* Encourage the mathematical analysis of problems incorporating physical parameterizations

→ better understanding of the interactions between parametrizations developed independently and in an uncoupled framework

→ put into perspective the assumptions and approximations underlying parametrizations

* Reconcile and improve thermodynamics formulations and conservation principles between model components (e.g. Olbers et al., 2012; Mayer et al., 2017; Lauritzen et al., 2022)

* Numerical representation of flow near the interface could be more consistent

→ The approach used to nest bulk formulation in coupled models leads to ambiguities:

1. Dual treatment of the boundary cells

2. Impossibility to properly formulate the continuous problem that coupled models discretize

→ Two-sided Bulk concept: explicitly parameterizing the oceanic surface layer (+ wavy boundary layer)

ANR proposal AWARE ( A consistent oceAn-Wave-AtmospheRE coupling: from the processes to climate models), M.-N. Bouin (CNRM)
2

Coupling schemes in OA context
Generalized OA Coupling Problem

$$
\left\{
\begin{aligned}
\mathcal{L}_{\text{atm}} \mathbf{U}^a &= f_{\text{atm}}, && \text{in } \Omega_{\text{atm}} \times [0, T] \\
\mathcal{B}_{\text{atm}} \mathbf{U}^a &= g_{\text{atm}}, && \text{in } \partial \Omega_{\text{atm}} \times [0, T] \\
\mathbf{U}^a|_{t=0} &= \mathbf{U}_0^a, && \text{in } \Omega_{\text{atm}} \\
C_{\text{atm}} \mathbf{U}^a &= \mathbf{F}_{\text{oa}} (\mathbf{U}^a, \mathbf{U}^o) && \text{in } [0, T]
\end{aligned}
\right.
$$

$$
\left\{
\begin{aligned}
\mathcal{L}_{\text{oce}} \mathbf{U}^o &= f_{\text{oce}}, && \text{in } \Omega_{\text{oce}} \times [0, T] \\
\mathcal{B}_{\text{oce}} \mathbf{U}^o &= g_{\text{oce}}, && \text{in } \partial \Omega_{\text{oce}} \times [0, T] \\
\mathbf{U}^o|_{t=0} &= \mathbf{U}_0^o, && \text{in } \Omega_{\text{oce}} \\
C_{\text{oce}} \mathbf{U}^o &= C_{\text{atm}} \mathbf{U}^a && \text{in } [0, T]
\end{aligned}
\right.
$$

<figure><img src="image_20.png" /></figure>
Temporal discretization of the coupling problem for $t \in [0, T]$

• Model components communicate at the end of a *coupling window* [$t_n, t_{n+1}$]

• Model components use dynamical time steps smaller than the coupling window size

• Averaged in time quantities are exchanged (conservation issues + physical considerations)

→ Decoupling of dynamical and coupling time steps

Interface conditions are the result of processes in turbulent boundary layers

⇒ Semi-empirical Monin-Obukhov similarity theory is the basis to derive turbulent quantities from the mean variables available from the models

<figure><img src="image_21.png" /></figure>

Large (2006) "Surface fluxes for practitioners of global ocean data assimilation"

▷ For hourly fluxes on a spatial scale of 10 km, there is at least a factor of 2 uncertainty due to transfer coefficient variability on these scales

▷ Long-term averaging is required before the uncertainty in boundary fluxes is minimized
Parallel Algorithm

<figure><img src="image_22.png" /></figure>
Atmosphere-first Algorithm

<figure><img src="image_23.png" /></figure>
Parallelized Atmosphere first Algorithm

<figure><img src="image_24.png" /></figure>
... but does the coupling lag matter ?

⇒ alternative coupling algorithm : **Schwarz Waveform Relaxation (SWR)**

<figure><img src="image_25.png" /></figure>

Black box approach: implementation is primarily based on the coupler, not the components
Partial conclusion

* A numerical coupling method can be referred to as

1. **Tightly coupled**: state variables across different models are synchronized at all times

2. **Loosely coupled**: state variables are shifted by one time-step or a sequence of time-steps

→ Current methods used in CMs can be classified as loose coupling methods
Partial conclusion

* A numerical coupling method can be referred to as

1. **Tightly coupled**: state variables across different models are synchronized at all times

2. **Loosely coupled**: state variables are shifted by one time-step or a sequence of time-steps

→ Current methods used in CMs can be classified as loose coupling methods

* Keyes et al. (2013): "Using an approach that ignores strong couplings between components gives a false sense of completion"

→ Set up a tight coupling method (e.g. SWR) to evaluate loose coupling methods

→ Tight coupling between components requires smoothness
## Partial conclusion

* A numerical coupling method can be referred to as

1. **Tightly coupled**: state variables across different models are synchronized at all times

2. **Loosely coupled**: state variables are shifted by one time-step or a sequence of time-steps

⇒ Current methods used in CMs can be classified as loose coupling methods

* Keyes et al. (2013): "Using an approach that ignores strong couplings between components gives a false sense of completion"

⇒ Set up a tight coupling method (e.g. SWR) to evaluate loose coupling methods

⇒ Tight coupling between components requires smoothness

* What level of analysis is tractable to study the convergence of the iterative process ?

* Impact in realistic CMs ?
3

Analysis of a coupled Ekman boundary layer problem
## Coupled Ekman boundary layer problem

$$ \rho_a \partial_t \bar{v}_{\text{atm}} = -\rho_a f \mathbf{k} \times \bar{v}_{\text{atm}} - \partial_z (\rho_a \overline{w'v'_{\text{atm}}}) $$

$$ \rho_0 \partial_t \bar{v}_{\text{oce}} = -\rho_0 f \mathbf{k} \times \bar{v}_{\text{oce}} - \partial_z (\rho_0 \overline{w'v'_{\text{oce}}}) $$

$$ \text{Interface condition : } \rho_0 \overline{w'v'_{\text{oce}}}|_{z=0} = \rho_a \overline{w'v'_{\text{atm}}}|_{z=0} $$

Common closure for turbulent fluxes at $z = 0$:

$$ \begin{aligned} \overline{w'v'_{\text{atm}}}|_{z=0} &= C_D \| \bar{v}_{\text{atm}}|_{z=z_1} - \bar{v}_{\text{oce}}|_{z=z_1} \| (\bar{v}_{\text{atm}}|_{z=z_1} - \bar{v}_{\text{oce}}|_{z=z_1}) \\ &= \mathcal{F}_{\text{sl}} (\bar{v}_{\text{atm}}|_{z=z_1} - \bar{v}_{\text{oce}}|_{z=z_1}) \end{aligned} $$

Some specific issues for OA coupling :

* Current implementations: discretizations and $\mathcal{F}_{\text{sl}}$ are linked → convergence analysis needs to be at the semi-discrete in space level

* Nonlinear transmission condition $\mathcal{F}_{\text{sl}}$

<figure><img src="image_26.png" /></figure>
Nonlinear and linearized systems: convergence properties

<figure><img src="image_27.png" /></figure>

* Development of tools for the semi-discrete convergence analysis of Schwarz methods (linear case)

* Well posedness (close to a given equilibrium) using the inverse function theorem (cf. Chacon Rebollo et al., 2014)

* Convergence analysis of a linearized problem

* Convergence analysis with linearized condition is representative of the nonlinear problem

⇒ Opens up the possibility of optimizing convergence using a relaxation parameter

$$ \overline{w'v'}_{\text{atm}}|_{z=0}^k = C_D \| \overline{v}_{\text{atm}}|_{z=z_1}^{k-1} - \overline{v}_{\text{oce}}|_{z=z_{-1}}^{k-1} \| (\theta \overline{v}_{\text{atm}}|_{z=z_1}^{k-1} + (1-\theta) \overline{v}_{\text{atm}}|_{z=z_1}^k - \overline{v}_{\text{oce}}|_{z=z_{-1}}^{k-1}) $$

For more details :

* S. Clement, F. Lemarié, E. Blayo: *Semi-discrete analysis of a simplified air-sea coupling problem with nonlinear coupling conditions*, In: R. Bank et al. (eds), Domain Decomposition in Science and Engineering XXVII, Springer-Verlag, 2023

* S. Clement, F. Lemarié, E. Blayo: *Discrete analysis of Schwarz waveform relaxation for a diffusion reaction problem with discontinuous coefficients*, SMAI J. Comput. Math., 2022
4

Numerical experiments
Application of Schwarz iterations in realistic CMs :

* **IPSL-CM**: full version (Atm. 3.75° × 1.875° + 39 vertical levels; Oce. 2° × 2° + 31 vertical levels)
  → mostly focused on the impact on sea surface temperature

* **CNRM-CM**: single-column version (CINDY/DYNAMO case)

* **EC-Earth**: single-column version (PAPA buoy)
# Resynchronization of the diurnal cycle

24h simulations with the 1-D version of CNRM-CM (S. Valcke and A. Voldoire)

Sea surface temperature evolution at $z = -0.5$ m

<figure><img src="image_28.png" /></figure>

Parallel algorithm

<figure><img src="image_29.png" /></figure>

SWR

⇒ SWR more robust to changes in coupling window size
SST "errors"

<figure><img src="image_30.png" /><figcaption>Monthly average of the ensemble mean SST difference between SWR vs parallel coupling</figcaption></figure>

<figure><img src="image_31.png" /><figcaption>2-year simulations with 50 members</figcaption></figure>

* O. Marti, S. Nguyen, P. Braconnot, S. Valcke, F. Lemarié, E. Blayo, 2021: *A Schwarz iterative method to evaluate ocean-atmosphere coupling schemes: implementation and diagnostics in IPSL-CM6-SW-VLR*, Geosci. Model Dev.
Comparison of Coupling Schemes

Results from EC-Earth SCM (312 48h-forecasts) – MSc thesis V. Schüller

<figure><img src="image_32.png" /></figure>

Weighted error $e/e_{max}$ ($e = |SST - SST^{SWR}|$)
Control experiment at PAPA station (50°N, 145°W)

Sea surface temperature

<figure><img src="image_33.png" /></figure>

→ phase shift for parallel &
sea-first

Atmospheric temperature + humidity (at the lowest model level)

<figure><img src="image_34.png" /></figure>

→ no obvious phase shift,
larger differences between
algorithms
Control experiment at PAPA station (50°N, 145°W)

<figure><img src="image_35.png" /></figure>

DC : dry convective,

DS : dry stable,

Cu : cumulus topped boundary layer,

Sc : stratocumulus topped boundary layer

The mass flux scheme in OpenIFS reacts particularly sensitively to changes in the coupling algorithm
Results from EC-Earth SCM (312 48h-forecasts)

<figure><img src="image_36.png" /></figure>

SWR runtime convergence Criteria :

$$|c_n^{k+1} - c_n^k| \le \text{TOL} \cdot |c_n^{k+1}|$$

$$\|c^{k+1} - c^k\|_{\infty} \le \text{TOL} \cdot \mathcal{A}(c^{k+1})$$

with

$$\mathcal{A}(c^{k+1}) = \max_{n \in 0,N} c_n^{k+1} - \min_{n \in 0,N} c_n^{k+1}$$

c = all variables which are part of the OASIS *namcouple*
Connecting model stability and model uncertainty

* Simplified equation sets

* Numerical experiments with uniformly distributed noise of 10% in $C_D$ and $C_H$

* 3 different coupling methods

1. TWN ↔ Schwarz

2. TWP-GA ↔ Parallel Algorithm (local-in-time)

3. OWP-GA ↔ Parallel Algorithm (local-in-time, no currents)

<figure><img src="image_37.png" /><figcaption>**Fig. 14** Expectation of AST (solid line) and adding or subtracting one standard deviation (dotted lines)</figcaption></figure>

<figure><img src="image_38.png" /><figcaption>**Fig. 12** Expectation of AST using $\Delta t = 1/50$ (left) and $\Delta t = 1/1,000$ (right) for $t \in [0, 1]$</figcaption></figure>

AST = Average Surface Temperature

J. Connors, B. Ganis: *Stability of algorithms for a two domain natural convection problem and observed model uncertainty*, Comput. Geosci., 2011
In a more realistic setting

* **Numerical codes:** WRF/ROMS coupling

* **Testcase:** simulation of tropical cyclone Erica (New Caledonia, 03/2003)

$$ \Delta x_{\text{atm}} = \Delta x_{\text{oce}} = 25 \text{ km}, \Delta t_{\text{atm}} = 180 \text{ s}, \Delta t_{\text{oce}} = 1800 \text{ s} $$

* **Ensemble approach:** perturbation of initial conditions and coupling frequency (3h vs 6h + linear reconstruction of surface fluxes)

→ Two ensembles: atmosphere-first vs Schwarz method

<figure><img src="image_39.png" /><figcaption>F. Lemarié, P. Marchesiello, L. Debreu, E. Blayo: *Sensitivity of Ocean-Atmosphere Coupled Models to the Coupling Method: Example of Tropical Cyclone Erica*, Research Report RR-8651, 2014</figcaption></figure>

<figure><img src="image_40.png" /></figure>

<figure><img src="image_41.png" /></figure>

<table><thead><tr><th></th><th>atm-first</th><th>Schwarz</th></tr></thead><tbody><tr><td>Trajectory</td><td>125 km</td><td>92 km</td></tr><tr><td>Intensity</td><td>6.2 m s<sup>-1</sup></td><td>4.1 m s<sup>-1</sup></td></tr></tbody></table>
## Theoretical aspects:

- Consider mass flux term in the convergence analysis

- Analysis of the atmosphere/sea-ice coupling (In IPSL-CM the iterative coupling was slow over sea ice + no convergence in EC-Earth)

- Higher-order time interpolations & FV discretization of SBL

- Mathematical analysis of problems with turbulence (inclusion of stratification and/or time derivatives)

## Practical aspects:

- Confirm numerical results with a larger set of locations, time spans, and atmospheric forcing data and/or an other AOSCM (e.g. CNRM-CM)

- Cost of the method: use ML to learn an extrapolation in time (LEFE ESM2, O. Marti) or reduced models

- Review the features that couplers should be able to support (e.g. suitability of preCICE coupling library (TUM & University of Stuttgart) for OA coupling ?)

<figure><img src="image_42.png" /><figcaption>Figure 6: 10 WR simulations, MOSAiC test case. Standard AOSCM runs correspond to the first iteration (in brackets)</figcaption></figure>

WR iterations, MOSAiC test case. Standard AOSCM runs correspond to the first iterate (courtesy of V. Schüller, Lund Univ.)