Revisit Lorenz 1982 using ECMWF and NCEP EPS

Ting-Chi Wu

MPO674 Predictability Final Project05/03/2012

Motivation and background• Revisit Lorenz’s error growth model on global 1°x1° ensemble

forecasts from ECMWF and NCEP.• JJA of 2008 (100 consecutive days) are chosen to study the

error growth and predictability of summer season in NH, Tropics and SH because so far most of the work were done in winter season.

• Further more, apply Lanczos filter (Fourier-based filter) to study the error growth and predictability of systems with different scales.

• Apply Dalcher and Kalnay (1987) modified error growth model. The most recent work from Buizza (2011) and many previous work have proven it to be a useful tool to investigate forecast error growth.

Methodology (I)• Error growth and predictability studies:• Lorenz (1982) proposed that the nonlinear term in the equation

governing the growth of E were quadratic, and assumed that

• Where E is r.m.s. error of forecast verified against forecast/analysis; a measures the growth rate of small errors; E∞ is asymptotic value at where errors saturate.

• One can fit a quadratic curve in least-square sense to the discritized version of (1) and find the coefficient α and β.

• Then , doubling time is

• In Lorenz (1982) DJF 1980/1981, E∞ is about 110 m, and the doubling time for small errors is 2.4 days.

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dEdt

= αE − βE 2 = aE(1 −EE∞

) (1)

€

a = α ,E∞ =αβ

€

ln2α

ECMWF NCEP

NH

Tropics

SH

RMSE v.s. forecast time (k)

• Smaller initial error and final error in ECMWF• Contours of k-j > 3 are closer in ECMWF

• Globally, ECMWF has larger error growth rate of small error which leads to a smaller doubling time.

• However, the asymptotic value of error is larger in NCEP. • Crosses and dots are closer in ECMWF.

a = 0.52; E∞ = 78.07; Tdouble=.1.33

a = 0.50; E∞ = 82.90Tdouble=1.39

€

dEdt

= αE − βE 2 = aE(1−EE∞

) discretize ⏐ → ⏐ ⏐ ⏐ ΔEΔt

= α E − βE2

€

E€

ΔE

€

E€

ΔE

North Hemisphere Tropics South Hemisphere

Geopt H. ECMWF NCEP

a E∞ Tdouble a E∞ Tdouble

NH 0.56 70.67 1.24 0.50 76.04 1.39

Tropics 0.27 13.60 2.57 0.30 16.59 2.31

SH 0.50 102.39 1.39 0.50 107.95 1.39

Least-square fit of discretized error growth model (1)

• In terms of 500mb geopotential height:• Tropics has the slowest growth rate (longer doubling time ~ 2.3-2.5

days), also smaller asymptotic error value. NH and SH has similar value of growth rates, but SH generally has larger asymptotic value than NH.

• ECMWF v.s. NCEP:o ECMWF and NCEP behave similar in SH in terms of error growth

rate, but ECMWF has smaller asymptotic value in general.o A generally larger value of asymptotic error in NCEP

Least-square fit of discretized error growth model (1)U-wind ECMWF NCEP

a E∞ Tdouble a E∞ Tdouble

NH 0.47 7.42 1.47 0.46 7.57 1.51

Tropics 0.30 4.64 2.31 0.38 4.46 1.82

SH 0.42 9.30 1.65 0.45 9.49 1.54

V-wind ECMWF NCEP

a E∞ Tdouble a E∞ Tdouble

NH 0.46 7.65 1.51 0.48 7.63 1.44

Tropics 0.34 3.48 2.04 0.46 3.35 1.51

SH 0.44 9.74 1.58 0.48 9.90 1.44

• In terms of 500mb wind fields, Tropics is where ECMWF and NCEP differ the most. Especially in terms of error growth rate. ECMWF generally has slower growth rate than NCEP.

Cuf-off wavelengths (I)

7 10 12

Geopt H. ECMWF NCEP

a E∞ Tdouble a E∞ Tdouble

Cut-off = 7 0.50 70.84 1.39 0.49 75.55 1.41

Cut-off = 10 0.51 71.38 1.36 0.50 76.23 1.39

Cut-off = 12 0.51 67.28 1.36 0.50 72.07 1.39

Only includes wavelength above the cut-off value ( e.g. 7 ≈ 700 km)

By including smaller scale features, error growth rate does not change much, but asymptotic value. The asymptotic value is about 5 m more in NCEP.

Methodology (II)• Leith (1978, 1982) argued an “external” source of error

growth, the deficiencies of model, must be included. • Dalcher and Kalnay (1987) and Simmons et al (1995)

extended Leith and Lorenz (1982)’s work:

• S simulates the effect of model error deficiencies. E is the error variance in DK87. Then, the normalized solution is

• DK suggested using predictability time limits (τ), defined as the forecast times when the forecast error equals a fraction η of E∞, is a better indicator than doubling time.

€

dEdt

= αE − βE 2 + γ = (aE + S)(1−EE∞

) (2)

€

η(t) =E(t)E∞

=1 −1+ S aE∞

1+C1eC2t

where C1 =E(0) + S /aE∞ − E(0)

; C2 = a+ S /E∞

Error growth model fitting

25%

50%

71% (1/√2)

95%

€

E(t)E∞

Cuf-off wavelengths (II)Geopt H. ECMWF

a E∞ S S/E∞

Cut-off = 7 0.38 74.13 2.58 0.035

Cut-off = 10 0.39 74.46 2.46 0.033

Cut-off = 12 0.41 69.89 1.98 0.028

meaning

28.57 days

30.30 days

35.71 days

Geopt H. NCEP

a E∞ S S/E∞

Cut-off = 7 0.31 80.06 4.75 0.059

Cut-off = 10 0.32 80.68 4.72 0.059

Cut-off = 12 0.33 76.20 4.09 0.054

meaning

16.95 days

16.95 days

18.52 days

It indicates that from model errors alone, the rmsd would reach E∞ in about ? days.

Summary• Two error growth models are examined with 500 mb geopt.

height (u and v) from ECMWF/NCEP ensemble forecasts.• NCEP has larger asymptotic value than ECMWF.• Tropics is where ECMWF disagrees most with NCEP in terms

of error growth rate. Based on Lorenz’s model.• When smaller scales are included, asymptotic value is larger. • If one is only looking at model error, NCEP needs less days to

achieve its asymptotic value than ECMWF. Based on DK87 model.

• Apply Dalcher and Kalnay (1987) model on other variables and regions, and compare with deterministic forecasts.

• How to include ensemble spread?

Future Work