On the numerical stability of surface–atmosphere coupling in weather and climate models

Beljaars, Anton; Dutra, Emanuel; Balsamo, Gianpaolo; Lemarié, Florian

doi:https://doi.org/10.5194/gmd-10-977-2017

Articles | Volume 10, issue 2

https://doi.org/10.5194/gmd-10-977-2017

© Author(s) 2017. This work is distributed under
the Creative Commons Attribution 3.0 License.

https://doi.org/10.5194/gmd-10-977-2017

© Author(s) 2017. This work is distributed under
the Creative Commons Attribution 3.0 License.

Articles | Volume 10, issue 2

Development and technical paper

|

27 Feb 2017

Development and technical paper |

| 27 Feb 2017

On the numerical stability of surface–atmosphere coupling in weather and climate models

Anton Beljaars, Emanuel Dutra, Gianpaolo Balsamo, and Florian Lemarié

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Final revised paper (published on 27 Feb 2017)
Preprint (discussion started on 04 May 2016)

Interactive discussion

Status: closed

AC: Author comment | RC: Referee comment | SC: Short comment | EC: Editor comment

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- Supplement

RC1: 'A Review of "On the numerical stability of surface-atmosphere coupling in weather and climate models" by Anton Beljaars, Emanuel Dutra and Gianpaolo Balsamo', Florian LEMARIE, 14 Jun 2016
- AC1: 'Reply to reviewer 1', Anton Beljaars, 19 Aug 2016
RC2: 'Review of 'On the numerical stability of surface-atmosphere coupling in weather and climate models': excellent ideas but could be better explained', Alex West, 29 Jul 2016
- AC2: 'Reply to reviewer 2', Anton Beljaars, 19 Aug 2016

Peer-review completion

AR: Author's response | RR: Referee report | ED: Editor decision

AR by Anton Beljaars on behalf of the Authors (14 Sep 2016) Author's response Manuscript

ED: Referee Nomination & Report Request started (16 Sep 2016) by Sophie Valcke

RR by Alex West (03 Oct 2016)

RR by Jan Polcher (05 Dec 2016)

Suggestions for revision or reasons for rejection

It was a pleasure to read this paper and also a surprise to see that this issue has not yet been solved in all land-surface models. This is the typical problem of flux equation discretizations at discontinuities. I have also recently leaned that similar methods are being used for solving diffusion equations on parallel computers when the domain decomposition cuts through the diffusion.

It is not true that an implicit coupling is incompatible with a modular code structure. This has been discussed and solved since Polcher at al. 1998. Most European weather and climate model use a fully implicit coupling and have a modular structure which allows them to run their land-surface scheme off-line or the atmosphere as an aqua-planet without changing the code. Recently ORCHIDEE was coupled to WRF using OASIS while maintaining the implicit solution to the full set of vertical diffusion equations ! With some of the authors we have revisited the issue a few years ago and it is surprising that the fully implicit coupling is still not implemented at ECMWF !

I do not think that introducing an empirical equation is needed to link T1 to the heat flux at the skin temperature level. During our thesis with Frédréric Hourdin (Hourdin 1992), we proposed to extrapolate the two upper soil temperatures towards the interface and thus provide the surface energy balance equation with a flux and a heat capacity for the infinitesimal layer for which the skin temperature is computed. The benefit of this methodology for complex land surface scheme was documented during the thesis of Jan-Peter Schulz (Schulz et al. 2001).

This has recently been extended to the multi-layer snow scheme without problems (Wang et al. 2013). The referenced paper covers the numerically explicit version of the scheme but this was converted to an implicit scheme coupled to the surface energy balance equation without problems (Will be described in the ORCHIDEE documentation for CMIP6). The same extrapolation then also needs to be done at the bottom of the snow in order to link the heat diffusion in the snow pack with the diffusion in the soil while maintaining a temperature at the contact point between the two media. This allows to solve implicitly the diffusion equations from the top of the atmosphere to the bottom of the soil with clear interfaces and a flexible modular code.

We have also verified that the thermal diffusion equation is numerically stable for very thin upper layers in the soil. This needed to be done when we decided to use a common vertical discretization for the hydrology and thermodynamics (Wang et al 2016). Thus we are quite satisfied with this approach to combine the diffusion equation with an interface temperature. We would agree that the linearity assumption can be criticized on physical grounds.

Indeed, the first few millimetres away from the interface temperature are characterized by very strong gradients and changes in properties which probably break any linearity assumption. Just as a single skin temperature for a complex canopy is dubious. We are currently progressing toward a true 3D energy balance which takes into account the complexity of the vertical structure of the medium which interacts with the atmosphere (Ryder et al. 2015). Even such complex set of equations can be simply solved implicitly and with keeping a maximum of modularity in the code.

I cannot recommend the publication of this paper as it does not take into account sufficiently previous publications on this topic. The references I provide below are only the one I know well as I work everyday with that code. But I know that at Princeton and NCAR some very interesting progress has been made recently on the numerics of the surface processes but I would not know what the most pertinent reference is. Our colleagues at MPI Hamburg have also developed some interesting methods for coupling implicitly multiple surface energy balances to atmospheric columns which are relevant to the discussion proposed.

I do not think either that introducing the empirical equation 8 and its two parameters is justified when the fully implicit solution can be implemented in a modular way and the interface temperature issue solved more elegantly. The linear relation assumed in the Hourdin method is much more tolerable than the scaling relation for α and β !

Referee Report: PDF

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ED: Reconsider after major revisions (14 Dec 2016) by Sophie Valcke

AR by Anton Beljaars on behalf of the Authors (13 Jan 2017) Author's response Manuscript

ED: Publish subject to technical corrections (20 Jan 2017) by Sophie Valcke

AR by Anton Beljaars on behalf of the Authors (26 Jan 2017) Author's response Manuscript

Short summary

Coupling an atmospheric model with snow and sea ice modules presents numerical stability challenges in integrations with long time steps as commonly used for weather prediction and climate simulations. Explicit flux coupling is often applied for simplicity. In this paper a simple method is presented to stabilize the coupling without having to introduce fully implicit coupling. A formal stability analysis confirms that the method is unconditionally stable.