Corrigendum

OATAO is an open access repository that collects the work of Toulouse

researchers and makes it freely available over the web where possible

Any correspondence concerning this service should be sent

to the repository administrator:

[email protected]

This is an publisher’s version published in:

https://oatao.univ-toulouse.fr/

26036

To cite this version:

Pannekoucke, Olivier and Ricci, Sophie and Bathelemy, Sébastien and Menard,

Richard and Thual, Olivier

Corrigendum. (2018) Tellus A: Dynamic Meteorology

and Oceanography, 70 (1). 1-2. ISSN 1600-0870 .

Official URL:

https://doi.org/10.1080/16000870.2018.1472954

Corrigendum

Pannekoucke, O., Ricci, S., Bathelemy, S. M
enard, R., and Thual, O. (2016). Parametric Kalman filter for chemical transport models. Tellus A, 68, 31547.

https://doi.org/10.3402/tellusa.v68.31547

1. Introduction

In our previous contribution Pannekoucke et al. (2016) (P16), an error has been made in the derivation of the error diffusion tensor dynamics Eq. (20). This involves an error in the Lagrangian dynamics of the uncertainty given in the Algorithm 2, while leaving the Eulerian dyan-mics unchanged.

The corrigendum is organized as follows. The modifica-tion of the Lagrangian dynamics is presented in Secmodifica-tion 2, where a new version of the Algorithm 2 of P16 is pre-sented. The computation leading to the Eulerian dynam-ics is described in Section 3, which gives the dynamdynam-ics presented in Eq. (26) of P16.

2. Lagrangian dynamics of the diffusion tensor

The advection over a small time step dt can be viewed as equivalent of the deformation of the error field eb_{by the} transformation DðxÞ ¼ x þ uðx; tÞdt (Pannekoucke et al.,

2014). Hence, it follows that the metric tensor field gxðtÞ evolves in time as ^gxðt þ dtÞ ¼ DT_x g_D1_ðxÞðtÞD1_x , where

D1_x is the gradient of the inverse deformation D1 at x (Pannekoucke et al.,2014, see Eq. (35)). This can be for-mulated considering the diffusion tensor m, defined by mx¼1

2g1x , as:

^mb_{xðt þ dtÞ ¼ ðD}1

x Þ1mbD1_ðxÞðtÞðDTx Þ1: (1)

Eq. (1)can be simplified as follows. Since the derivative of the identiy D½D1ðxÞ ¼ x is DD1_ðxÞD1_x ¼ I, it results

thatðD1_x Þ1¼ DD1_ðxÞ. Hence, the dynamics of the

diffu-sion tensorEq. (1)writes ^mb

xðt þ dtÞ ¼ DD1ðxÞmbD1ðxÞðtÞðDD1ðxÞÞ

T_{: (2)}

ConsideringEq. (2), the Algorithm 2 of P16 now writes Algorithm 1.

Algorithm 1. Iteration process to forecast the background covariance matrix at time t¼ s from the analysis covari-ance matrix given at time t¼ 0, under local homoge-nity assumption.

Require: Fields ofma_{and V}a_{. dt ¼ s=N, t ¼ 0} for k¼ 1 : N do 1- Pure advection DðxÞ ¼ x þ uðx; tÞdt ^mb_{xðt þ dtÞ ¼ DD} 1ðxÞmb_D1_ðxÞðtÞðDD1ðxÞÞT ^mb_{xðt þ dtÞ ¼ V}b_½D1_{ðxÞ; t} 2- Pure diffusion mb xðt þ dtÞ ¼ ^mbxðt þ dtÞ þ 2jðxÞdt Vbxðt þ dtÞ ¼ ^mb_xðt þ dtÞj^m b xðt þ dtÞj 1=2 jmb xðt þ dtÞj 1=2 3- Update of the background statistics

VbxðtÞ Vbxðt þ dtÞ mb

xðtÞ mbxðt þ dtÞ t tþ dt end for

Return fields mb_{xðsÞ and V}b_xðsÞ

3. Eulerian dynamics of the diffusion tensor

This expression modifies the derivation of the Eulerian dynamics in Appendix D of P16, where this time

DD1_ðxÞ¼ Dxdtuðx;tÞþoðdtÞ¼ I þ dtruðx  dtuðx; tÞ þ oðdtÞÞ;

leading to

DD1_ðxÞ¼ I þ dtruðx; tÞ þ oðdtÞ: (3)

WithEq. (3), the computation in P16 leading to Eq. (D1) applies, and Eq. (D1) is found again: for the advection process, the dynamics of the error diffusion tensor writes:

Tellus A: 2018.# 2018 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1

Citation: Tellus A: 2018, 70, 1445379, https://doi.org/10.1080/16000870.2018.1472954

Tellus

SERIES ADYANAMIC

METEOROLOGY AND OCEANOGRAPHY

otmb_{þ urm}b_{¼ m}b_ðruÞT_{þ ðruÞm}b_{: (4)}

4. Conclusion

In this corrigendum, the dynamics of the metric tensor and of the diffusion tensor have been corrected.

Algorithm 2 in Pannekoucke et al. (2016) is now replaced by the present Algorithm 1.

This modification does not alter the numerical results pre-sented in P16.

References

Pannekoucke, O., Emili, E. and Thual, O. 2014. Modeling of local length-scale dynamics and isotropizing deformations. Q. J. R. Meteorol. Soc., 140, 1387–1398.

Pannekoucke, O., Ricci, S., Barthelemy, S., Menard, R. and Thual, O. 2016. Parametric kalman filter for chemical transport model. Tellus, 68, 31547.