Variational Approach to Improve Computation of Sensible Heat Flux over Lake Superior

Zuohao Cao1, Murray D. Mackay1, Christopher Spence2 and Vincent Fortin3

1Meteorological Research Division, Environment Canada, Toronto, Ontario, Canada

2National Hydrology Research Centre, Environment Canada, Saskatoon, Saskatchewan, Canada

3Meteorological 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 Fh

0*

hF

J

u

J

22

T

2

u WTTWW2

1J obs

qobsobs qquu

Computed u, ∆T, ∆q

Flux-gradient method (e.g., Yaglom 1977)

Computed H, λE, L, Z0

Sensible heat flux:

Latent heat flux

Monin-Obukhov length

The roughness length z0 (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 Fh:

– Set up initial guesses of unknowns, e.g., u*=0.3 m s-1, and Fh=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 Fh.

– 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 Fh are reached at the minimum of the cost function.

– After new values of u* and Fh are obtained, the roughness length z0 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 (47.183oN, 87.225oW) 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) VariationalR = 0.79 MAE = 38.0 W m-2

(b) Flux-gradient R = 0.54 MAE = 48.6 W m-2

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150250350450550650

-450 -350 -250 -150 -50 50 150 250 350 450 550 650

H (w

m-2

) Obs

erva

tions

H (W m-2) Variational method

R = 0.79

(a)

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H (W

m-2

) Obs

erva

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H (W m-2) Flux-gradient method

R = 0.54(b)

Observed sensible heat flux vs computed ones for Julian day 277

(a) Variational

(b) Flux-gradient

020406080

100120140160180200

0:00 2:30 5:30 8:00 10:30 13:00 15:00 17:00 19:00 21:00 23:00

Sens

ible

hea

t flux

(W m

-2)

Hours (local time)

Flux-gradientObservation Julian day 277

(b)

020406080

100120140160180200

0:00 2:30 5:30 8:00 10:30 13:00 15:00 17:00 19:00 21:00 23:00

Sens

ible

hea

t flux

(W m

-2)

Hours (local time)

VariationalObservation Julian day 277

(a)

Observed sensible heat flux vs computed ones for Julian day 339

(a) Variational

(b) Flux-gradient

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0:00 2:00 4:00 6:00 8:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00

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(W m

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Hours (local time)

VariationalObservation Julian day 339

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ible

hea

t flux

(W m

-2)

Hours (local time)

Flux-gradientObservation Julian day 339

(b)

Observed sensible heat flux vs computed ones for Julian day 351

(a) Variational

(b) Flux-gradient

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t flux

(W m

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Hours (local time)

VariationalObservation Julian day 351

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0:00 2:00 4:00 6:00 8:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00

Sens

ible

hea

t flux

(W m

-2)

Hours (local time)

Flux-gradientObservation Julian day 351

(b)

Observed sensible heat flux vs computed ones for Julian day 353

(a) Variational

(b) Flux-gradient

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0:00 2:00 4:00 6:00 8:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00

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(W m

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Hours (local time)

VariationalObservation Julian day 353

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t flux

(W m

-2)

Hours (local time)

Flux-gradientObservation Julian day 353

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)

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Calcu

lated

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ved H

(W m

-2)

UTC (= EDT + 4)

VariationalGEM-Reg3MESH4MESH3

Dec. 16, 2008

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