# Stability and accuracy of Runge-Kutta based split-explicit time-stepping algorithms for free-surface ocean models

Florian Lemarié, Nicolas Ducousso, Laurent Debreu, Gurvan Madec

► To cite this version:

Florian Lemarié, Nicolas Ducousso, Laurent Debreu, Gurvan Madec. Stability and accuracy of Runge-Kutta based split-explicit time-stepping algorithms for free-surface ocean models. EGU 2022 - General Assembly on European Geosciences Union, May 2022, Vienna, Austria. pp.1-6. hal-03769267

HAL Id: hal-03769267

https://inria.hal.science/hal-03769267v1

Submitted on 5 Sep 2022

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# Stability and accuracy of Runge-Kutta based split-explicit time-stepping algorithms for free-surface ocean models

Nicolas Ducousso<fnref n="1" />,<fnref n="3" />, Florian Lemarié<fnref n="1" />, Laurent Debreu<fnref n="1" /> and Gurvan Madec<fnref n="2" />,<fnref n="1" />

<footnote n="1">¹ Inria, Grenoble, France</footnote>

<footnote n="2">² LOCEAN, Paris, France</footnote>

<footnote n="3">³ SHOM, Brest, France</footnote>
## Advantages of RK methods:

* 2-time-level schemes (no computational mode)

* Good stability properties for advection and diffusion

* Abundant literature (e.g. RK-IMEX, RK-SSP, DIRK, etc)

* Easy to implement (convex combination of Euler steps)

## Main questions:

* Many variants of RK schemes

* RK with mode-splitting ?

## Existing work:

* Many RK-based atmospheric models (e.g. WRF, Cosmo, Homme-NH, etc)

* « *Pseudo-second order Runge-Kutta time stepping scheme* » in HIM/MOM6 [Hallberg, 1997]
# Runge-Kutta methods in the oceanic context

<figure><img src="image_2.png" /></figure>

## Many variants of RK schemes

* Low-storage and « order=stage » method for advection [Wicker & Skamarock, 2002]

$$ \phi^{n+1/3} = \phi^n + \frac{\Delta t}{3} \mathcal{F}(\phi^n) $$

$$ \partial_t \phi = \mathcal{F}(\phi, t) \rightarrow \phi^{n+1/2} = \phi^n + \frac{\Delta t}{2} \mathcal{F}(\phi^{n+1/3}) $$

$$ \phi^{n+1} = \phi^n + \Delta t \mathcal{F}(\phi^{n+1/2}) $$

* Introduce Shuman averaging for internal gravity waves (« order=stage-1 »)

<figure><img src="image_3.png" /></figure>

## Additional difficulty inherent to oceanic models

* 2D barotropic / 3D baroclinic mode splitting to improve computational efficiency (fast and slow dynamics are split into separate subproblems)

* Separation of slow and fast modes is non-orthogonal (depth-independent barotropic mode assumption)

$$ \partial_t \bar{u} = -g \partial_x \left( \eta + \frac{1}{\rho_0 g H} \int_{-H}^{0} p_h dz \right) $$

$$ \partial_t \eta = -H \partial_x \bar{u} $$

<figure><img src="image_4.png" /></figure>

Some form of filtering is required to stabilize the splitting
# Robust mode-splitting method based on RK3

**Peculiarity of oceanic models:** 2D barotropic / 3D baroclinic mode splitting to improve computational efficiency

<figure><img src="image_5.png" /><figcaption>Gravity waves integration (without dissipation in 2D mode)</figcaption></figure>

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

**Single-first approach:**

* Only one integration of the 2D within the RK3 time-step

* Split-explicit approach with a dissipative time-stepping for the barotropic mode

* Linear stability analysis can be conducted following Demange et al. (2019)
# Robust mode-splitting method based on RK3

## Peculiarity of oceanic models: 2D barotropic / 3D baroclinic mode splitting to improve computational efficiency

$$ \bar{u}^{n+1} = \bar{u}^n - g\Delta t_0 \partial_x \eta^n $$

$$ \eta^{n+1} = \eta^n - H \Delta t_0 \partial_x ((1+\theta)\bar{u}^{n+1} - \theta \bar{u}^n) $$

<figure><img src="image_7.png" /><figcaption>(b) Baroclinic mode amplification</figcaption></figure>

<figure><img src="image_8.png" /></figure>

* Linear stability analysis provides the minimal dissipation to stabilize the splitting

* Stability constrained by barotropic mode and not by baroclinic ones
**Summary:**

► Design of an RK3 based time-stepping algorithm for split-explicit free-surface ocean models

► The single-first strategy appears to be more robust than other alternatives

► Thought for a quasi-Eulerian vertical coordinate

► A non-linear z\* 2D x-z model has been developed to check results from stability analysis

**Ongoing work:**

► Implementation in NEMO (see next talk)

► Accuracy analysis and consequences of mode-splitting

♦ Ducousso, N., Lemarié, F., Debreu, L., Madec, G.: *Stability and accuracy of Runge-Kutta-based time-stepping algorithms for split-explicit free-surface ocean models*. In preparation for J. Adv. Model. Earth Syst.