Variational Approach to Improve Computation of Sensible Heat Flux over Lake Superior Zuohao Cao 1, Murray D. Mackay 1, Christopher Spence 2 and Vincent Fortin 3 1 Meteorological Research Division, Environment Canada, Toronto, Ontario, Canada 2 National Hydrology Research Centre, Environment Canada, Saskatoon, Saskatchewan, Canada 3 Meteorological Research Division, Environment Canada, Dorval, Quebec, Canada

Outline Motivation and objective Methodology and data Results Conclusions

Motivation The sensible heat flux is important for characterizing the energy transfer between the atmosphere and its underlying surfaces such as the Laurentian Great Lakes. Accurate representation of the flux and the interaction in this coupled system is therefore necessary to better predict hydro-meteorological variables.

Motivation The flux computation in current numerical weather prediction models suffers from substantial inaccuracies due to limitations of Monin-Obukhov Similarity Theory (MOST)-based algorithms used in the computation, especially over heterogeneous surfaces such as lakes. The variational method can overcome these drawbacks by making full use of the observed meteorological information over the underlying surface and the information provided by MOST.

Objective In this study, the variational method is employed for the first time to improve computation of surface sensible heat fluxes over Lake Superior. The principle of this variational approach is to minimize the differences between the computed and the observed wind, temperature, and moisture so that it can adjust the computed flux toward the “true” value.

The Variational Method The cost function Optimal estimates of u * and F h

Computed u, ∆T, ∆q Flux-gradient method (e.g., Yaglom 1977)

Computed H, λE, L, Z 0 Sensible heat flux: Latent heat flux Monin-Obukhov length The roughness length z 0 (Charnock 1955)

Computation procedures A quasi-Newton method is used to find the minimum of the cost function J. The following iterative procedure is used to compute the cost function and its gradient, and to derive u * and F h : – Set up initial guesses of unknowns, e.g., u * =0.3 m s -1, and F h =0.03 K m s -1. – Calculate L. – Calculate u, ∆T and ∆q. – Calculate the cost function J and its gradients with respect to u * and with respect to F h. – Perform the quasi-Newton method to search for zeros of the gradients of J so as to minimize the cost function J; the expected u * and F h are reached at the minimum of the cost function. – After new values of u * and F h are obtained, the roughness length z 0 is updated using the Charnock’s relationship. Repeat the above 5 steps until the procedure converges.

Data Direct eddy-covariance measurements of sensible heat fluxes over the Great Lakes have become available only recently through the GLISA-funded project. As of November 2013, a total of five stations are in operation.

Stannard Rock Light, Lake Superior

Observations The dataset used in this study was collected from 2008 to 2013 at Stannard Rock Light ( o N, o W) of Lake Superior (Blanken et al. 2011; Spence et al. 2011, 2013) The measurement height is 32 m above the mean water surface. The observed variables include momentum flux, sensible heat flux, latent heat flux, wind speed and direction, temperature gradient, moisture gradient, atmospheric pressure, and incident shortwave and longwave radiation. All the data used in this study are of a time interval of half hour. The observed sensible heat flux is used for a verification purpose for both variational and flux-gradient methods.

Observed sensible heat flux (in 2008) vs computed ones (a)Variational R = 0.79 MAE = 38.0 W m -2 (b) Flux-gradient R = 0.54 MAE = 48.6 W m -2

Observed sensible heat flux vs computed ones for Julian day 277 (a)Variational (b) Flux-gradient

Observed sensible heat flux vs computed ones for Julian day 339 (a)Variational (b) Flux-gradient

Observed sensible heat flux vs computed ones for Julian day 351 (a)Variational (b) Flux-gradient

Observed sensible heat flux vs computed ones for Julian day 353 (a)Variational (b) Flux-gradient

Comparisons among the observed sensible heat flux, the variational method, and the GEM regional model

Differences between the observed flux and the calculated (the variational method and the modified GEM regional models, Deacu et al. 2012)

Conclusions The variational method yields very good agreements with the direct eddy-covariance measurements over Lake Superior. The variational approach is much more accurate than the conventional flux-gradient method. It is anticipated that in the future the variational approach can be used to improve the GEM forecasting system, for example, the variational method could be used for real-time estimate, and calibration of the flux-gradient method required in prognostic flux-coupled atmosphere-lake models.

Acknowledgements Stephane Belair, and Pierre Pellerin for their constructive suggestions and discussions.

Ice on Lake Superior (91.4% coverage) on Feb. 21, 2015