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 X k that falls off like k -5/3 for large k Solve system of n 2 nd order ODEs in Z k 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
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 t k Defined as time at which Z k passes Y k = 0.5 (X k +X k+1 ) = energy in k th 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 – bE 2 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 Project 05/03/2012
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)
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; T double =.1.33 a = 0.52; E ∞ = 78.07; T double =.1.33 a = 0.50; E ∞ = 82.90 T double =1.39 a = 0.50; E ∞ = 82.90 T double =1.39
Ensemble-based error metrics associated with tropical cyclogenesis: Basin-wide perspective Komaromi, W. A. and S. J. Majumdar Monthly 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
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 Growth Lorenz 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”