1 First results and methodological approach to parameter perturbations in GEM-LAM simulations PART I Leo Separovic, Ramon de Elia and Rene Laprise

2 MOTIVATION Sub-grid parameterization schemes are source of “parametric uncertainty”: - well-known processes that can be exactly represented (e.g. radiation transfer) but need to be approximated so that they do not take excessive computational time; - less-well understood processes (e.g. turbulent energy transfer) that are situation dependent; parameters rely on mixture of theoretical understanding and empirical fitting; - measurable parameters - measurement error, - non-measurable parameters uncertainty associated with representativity. Tuning can eliminate only reducible component of the model error. Parametric uncertainty can be (at least theoretically) quantified by perturbing parameters and measuring the impact on model output.

3 CONTENTS Detection of the model response to perturbations of parameters in a large domain - noise: brief analysis of internal variability - signal: sensitivity of seasonal climate to selected perturbations of a trigger parameter of KF convection - statistical significance (signal-to-noise ratio) - trade-off between statistical significance and computational cost Detections of model response in a small domain - effects of reduction of domain size on magnitude of the signal and noise Future work - intermediate domain size, next parameter, multiple parameter perturbations

4 EXPERIMENTAL CONFIGURATION GEM-LAM 140x140 DX=0.5 deg (max 55.5 km at JREF=65), NLEV=52

5 EXPERIMENTAL CONFIGURATION Five start dates:November GMT End date:November GMT 4 seasons DEC01-NOV30 Time step:30 min Nesting data:ERA 40 PTOPO:npex=4npey=4 Estimated time:12hrs/year Output frequency: once per 6 hours

6 Physics package Version: RPN-CMC4.5 RADIA:CCCMARAD SCHMSOL:ISBA GWDRAG:GWD86 LONGMEL:BOUJO FLUVERT:CLEF SHLCVT:CONRES, KTRSNT_MG CONVEC:KFC KFCPCP:CONSPCPN STCOND:CONSUN Stomate:.false. Typsol:.true. Snowmelt:.false.

7 Trigger vertical velocity in KFC scheme The KFCTRIG values that are deemed to be appropriate at the limits of the resolution interval in which the KFC scheme is to be used (B. Dugas, 2005): KFCTRIG (170 km) = 0.01 KFCTRIG (10 km) = 0.17 It is assumed that:KFCTRIG (RES) * RES = 1.7 = C (#) KFCTRIG is a function of the grid-tile area: KFCTRIG = KFCTRIG0 * RES0 / sqrt (DXDY) At the nominal resolution of 50km (#) gives KFCTRIG=0.034 (REFERENCE) We performed 2 perturbations (ONE PER TIME): KFCTRIG1=0.020 and KFCTRIG2=0.048 These values would be deemed appropriate at resolution of 85km and 35km.

8 THREE ENSEMBLES WKLCL =0.034(REFERENCE)5 members WKLCL=0.029(-) single5 members WKLCL=0.048(+) single5 members

9 Internal variability in the reference 5-member ensemble TA-ESTD-PCP

10 Internal variability in the reference 5-member ensemble TA-ESTD-PCP normalized

11 ESTD-TA-PCP

12 (ESTD-TA-PCP)/(EA-TA-PCP)

13 ESTD-TA-Tscn

14 Detection of the model response to parameter perturbations follows the Student’s distribution with (n R +n P -2) degrees of freedom. Null hypothesis: The two means are computed from two samples drawn from a unique distribution. Signal: difference between the ensemble averages of - reference ensemble X R : n R =5 members - perturbed-parameter ensemble X P : n P = members Error: sample STD of the ensemble averages:  E 2 (X R )/n R and  E 2 (X P )/n P. If the true variances of X R and X P are equal then the quantity

15 PCP KFCTRIG=0.020 (-)

16 PCP – level of rejection

17 Signal (TTscn)

18 TTscn: rejection level

19 Trade-off between number of parameter perturbations and significance We need to find a trade-off between P and t Let’s relate the two ensemble sizes: then Significance t Internal variability σ Computational resources $ N o of parameter perturbations P signal and i)One should invest in n R because of its low cost but not more than b=5 (diminishing returns) ii)n p =1 & b>>0 minimizes the cost but also minimizes the signal-to- noise ratio