MPO 674 Lecture 5

1/27/15

This morning in Boston

Lorenz (1969)

• Hypothesis: “… certain formally deterministic fluid systems possessing many scales of motion may be observationally indistinguishable from indeterministic systems, in that they possess an intrinsic finite range of predictability which cannot be lengthened by reducing the error of observation to any value greater than zero.”

Method

• Take vorticity equation• Streamfunction fields and error fields• Linearize equation• Energy E, Error energy G• Convert into spectral space, produce ODE for

spectral components of error• X(K) = energy per unit wave number x wave

number• Z(K) = error energy per unit wave number x wave

number

Method

• Choose Xk that falls off like k-5/3 for large k

• Solve system of n 2nd order ODEs in Zk

• Start with initially tiny, small-scale perturbation• Compute error-energy spectra at selected times• Equal areas = equal amounts of energy• Area under a thin curve = total error energy G

at a selected time.• Area under heavy curve = E

Lorenz (1969)

K-¼

K¼ Z(K) K¼ X(K)

Fig. 2

• Error energy G doubles quickly when confined to smaller scales

• By 3 days, G is half of E, and growth is much less rapid.

Range of predictability tk

• Defined as time at which Zk passes Yk = 0.5 (Xk+Xk+1) = energy in kth resolution interval

Experiment B

• Confined initial error to largest scale of motion– Errors in larger scales do not amplify rapidly– Errors in smaller scales quickly induced, as if they

had been present initially– Predictability ranges for all scales of motion

comparable to Experiment A

Experiment C

• How much predictability can one gain by reducing the initial error by half?– C1 – C8 decreasingly smaller initial errors– C1: synoptic-scale systems >1 day predictability– C1: planetary-scale systems >1 week predictability– C1: smaller-scale systems (<40 m) <1 sec pred.– C1: range of predictability doubles as wave length

doubles

Predictability fails to double as initial error is halved?

Predictability fails to double as wavelength is doubled?

Experiment C

• How much predictability can one gain by reducing the initial error by half?– C2: range of predictability almost twice that of C1– Ultimately, for each scale there is a point where• Cutting the initial error in half fails to double the range

of predictability• Doubling the wave length fails to double the range

– Spread of errors from smaller to larger scales becomes appreciable

– C8 hardly distinguishable from C7

Conclusions

• System has an intrinsic finite range of predictability

• At any particular range, there is a definite limit beyond which the expected accuracy of a prediction cannot be increased by reducing the uncertainty of the initial state further

• Linearizing tends to overestimate growth rate• Do real fluid systems possess a similar lack of

predictability?

Lorenz (1982)

• Prior to Lorenz (1969), it may have appeared that the range of acceptable weather forecasts could be extended by 3 days simply by reducing ‘observational’ errors to half their present size.

• But small-scale features were ignored.• Lorenz (1969): even if larger scales were observed

perfectly, the uncertainties in the smaller scales would induce errors in the larger scales, comparable to the larger-scale initial errors. These errors would grow.

Lorenz (1982)

• Therefore, the accuracy of short-range (say 1-day) forecasts is bounded

• A doubling time places an upper bound on the extent to which prediction a few days or weeks is possible

• Lower bounds to predictability have not been examined.

• What is the lower bound? A forecast.• Use ECMWF model to determine upper and

lower bounds to atmospheric predictability

Method

• Took 1-10 day ECMWF forecasts, prepared daily

• RMS differences in 500 hPa Z, in spectral space• Compared the difference between j- and k-day

forecasts (thin lines) against the difference between the corresponding analyses and k-j day forecasts (thick lines)

• k-j = 1, 2, 3, …

Lorenz (1982)Rate of increase of forecast error with increasing time range

Interpretation

• Thick line: rate at which the ‘truth’ and the model diverge with time (i.e. rate of increase of forecast error)

• Thin lines: upward slope as k increases is the rate at which two solutions of the same system of equations diverge

• Excess slope of thick line = maximum amount by which model can be improved

• If model is improved, the thick line will drop, and the thin line will move upward.

• Larger errors amplify less rapidly than smaller errors• [Note: mean analysis not same as mean forecast]

Simple analog for error growth

• dE/dt = aE – bE2

• a = growth rate of small errors• b > 0 curtails error growth

• Fig. 2: increase in RMS error versus average RMS error

• In other words, dE/dt versus E

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.

(1)

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

North Hemisphere Tropics South Hemisphere

Ensemble-based error metrics associated with tropical cyclogenesis: Basin-wide perspective

Komaromi, W. A. and S. J. MajumdarMonthly Weather Review, 2014

850-200 hPa wind shear (m/s)June

July

August

September

October

November

700 hPa relative humidity (%)June

July

August

September

October

November

850-700 hPa circulation (s-1)June

July

August

September

October

November

2txtxt af

Verifying analysis of x at time t

Forecast of x for forecast time t

How quickly do errors in forecasts of our metrics grow in the ECMWF ensemble?

Is there a geographical preference to where errors grow the fastest?

Mean square error

Error GrowthLorenz 1982 – Z500

Circulation errorsShear errors

Moisture errors

Errors in moisture and circulation beginning to asymptote by d10, while errors in shear continue growing rapidly, suggesting greater predictability

“conceptual model”