The Scientific World Journal

Volume 2015 (2015), Article ID 729080, 9 pages

http://dx.doi.org/10.1155/2015/729080

## Time Evolution of Initial Errors in Lorenz’s 05 Chaotic Model

Department of Meteorology and Environment Protection, Faculty of Mathematics and Physics, Charles University in Prague, V Holešovičkách 2, 180 00 Prague, Czech Republic

Received 11 August 2014; Accepted 24 September 2014

Academic Editor: Ivan Zelinka

Copyright © 2015 Hynek Bednář et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Initial errors in weather prediction grow in time and, as they become larger, their growth slows down and then stops at an asymptotic value. Time of reaching this saturation point represents the limit of predictability. This paper studies the asymptotic values and time limits in a chaotic atmospheric model for five initial errors, using ensemble prediction method (model’s data) as well as error approximation by quadratic and logarithmic hypothesis and their modifications. We show that modified hypotheses approximate the model’s time limits better, but not without serious disadvantages. We demonstrate how hypotheses can be further improved to achieve better match of time limits with the model. We also show that quadratic hypothesis approximates the model’s asymptotic value best and that, after improvement, it also approximates the model’s time limits better for almost all initial errors and time lengths.

#### 1. Introduction

Forecast errors in numerical weather prediction models (NWPMs) grow in time because of the inaccuracy of the initial state, chaotic nature of the weather system itself, and the model imperfections. Due to the nonlinear terms in the governing equations, the forecast error will saturate after some time. Time of saturation or* the limit of predictability of deterministic forecast* in NWPM is defined by [1] as time when the prediction state diverges as much from the verifying state as a randomly chosen but dynamically and statistically possible state. Forecasters also use other time limits to measure the error growth.* Forecast-error doubling time * is time when initial error doubles its size. , , , and are times when the forecast error reaches 95%, 71%, 50%, and 25% of the limit of predictability. The time limit is the time when the forecast error exceeds of the saturation or asymptotic value (AV) and, by [2], it corresponds to the level of climatic variability. Lorenz [3] calculated forecast error growth of NWPM by comparing the integrations of model, starting from slightly different initial states. Present-day calculations use the approach developed by Lorenz [4], where we can obtain two types of error growth. The first is called* lower bound* and is calculated as the root mean-square error (RMSE) between forecast data of increasing lead times and analysis data valid at the same time. The second is called* upper bound* and is calculated as the root mean-square (RMS) difference between pairs of forecasts, valid at the same time but with times differing by some fixed time interval. For example, if this interval is one day, the analysis for a given day is compared with one day forecast valid for the same day, and then this one day forecast is compared with two days forecast valid for the same day and so on. This second method compares only model equations and therefore it represents growth without model error. The innovation to upper bound, that is also used, is calculated as the RMS difference between forecast and control forecast with higher resolution of the model (*perfect model framework*).

*Quadratic hypothesis* (QH) was the first attempt that was made by Lorenz [3] to quantify the error growth. QH is based on the assumption that, if the principal nonlinear terms in the atmospheric equations are quadratic, then the nonlinear terms in the equations governing the field of errors are also quadratic. Dalcher and Kalney [5] added a model error to Lorenz’s QH. A version that is used by recent researchers is the Simmons’s et al. modification [6] of [5]. The Lorenz’s QH is therefore suitable for upper bound of error growth and the Simons’s et al. modification for lower bound. Trevisan et al. [7] came out with idea that logarithmic term is more valid than quadratic and linear term in the equations governing the field of errors, but this* logarithmic hypothesis* (LH) has never been used in NWPM computations.

*Ensemble prediction systems* (EPS) are used in order to estimate forecast uncertainties. They consist of a given number of deterministic forecasts where each individual forecast starts from slightly different initial states. EPS also includes a stochastic scheme designed to simulate the random model errors due to parameterized physical processes. Recent studies of predictability and forecast error growth (e.g., [8–11]) are mostly done by models of European Centre for Medium Range Weather Forecasts (ECMWF) and the Global Ensemble Forecast System (GEFS) from the National Centers for Environmental Prediction (NCEP). They include deterministic and ensemble forecast with 1 to 70 members. Operational model of ECMWF uses 50 members plus control forecast. More detailed study [10] uses 5 members plus control forecast. The initial conditions of ensemble members are defined by linear combination of the fastest singular vectors. Horizontal resolution with spectral truncation varies from T95 to T1279 and the number of vertical levels varies from 19 to 91 (analyses use higher resolution than forecasts). The output data are interpolated to 1° latitude × 1° longitude or 2.5° latitude × 2.5° longitude resolution separately for the Northern Hemisphere (20°, 90°) and Southern Hemisphere (−90°, −20°). Forecast is usually run for 90 days at winter (DJF) or summer (JJA) season with 0 (analysis) to 10, 15 (ECMWF), or 16 days (NCEP) of* forecast length* (FL) at 6 or 12 hours intervals. The most often used variable for analyzing the forecast error is geopotential height at 500 hPa level (Z500). Others are geopotential height at 1000 hPa level (Z1000) and the 850 hPa temperature (T850). To describe the forecast error growth over the calculated forecast length, the Simmons et al.’s modification [6] of Lorenz’s QH [3] is used.

The questions that have arisen from studies of predictability and forecast error growth and that represent the key issues addressed in this work are: Is the LH [7] better approximation of initial error growth than QH [3]? Is there a possible modification of LH and QH that better approximates model data? If so, how much difference it creates in time limits that measure the forecast error growth? How precisely do the approximations describe forecast error growth over the FL (10, 15 or 16 days)? How do the approximations obtained from model values with various number of ensemble members differ from each other? Lorenz’s chaotic atmospheric model (L05II) [12] will be used. For a more comprehensive introduction to the problem of weather predictability, we refer reader to the book by Palmer and Hagedorn [13]. After this introduction, Section 2 describes the model and experimental design, Section 3 describes ensemble prediction method, Section 4 introduces quadratic and logarithmic hypotheses, and Section 5 sets experimental designs. Section 6 presents the results and their discussion and Section 7summarizes the conclusions.

#### 2. Model

Because of the limitations of NWPMs and because we want to derive the impact of initial error (perfect model framework), we use modification [13] of low-dimensional atmospheric model (L96). L96 [14] is a nonlinear model, with variables connected by governing equations: are* unspecified (i.e., unrelated to actual physical variables) scalar meteorological quantities*, is a constant representing external forcing, and is time. The index is cyclic so that and variables can be viewed as existing around a circle. Nonlinear terms of (1) simulate advection. Linear terms represent mechanical and thermal dissipation. The model quantitatively, to a certain extent, describes weather systems, but, unlike the well-known Lorenz’s model of atmospheric convection [15], it cannot be derived from any atmospheric dynamic equations. The motivation was to formulate the simplest possible set of dissipative chaotically behaving differential equations that share some properties with the “real” atmosphere. NWPMs interpolate the output data mostly to 1° latitude × 1° longitude grid. In L96, it means . Such a high resolution would create large number of waves with similar maxima “pressure highs” and minima “pressure lows”; however, to share some properties with the “real” atmosphere, we would rather have 5 to 7 main highs and lows that correspond to planetary waves (Rossby waves) and a number of smaller waves that correspond to synoptic-scale waves. Therefore, we introduce spatial continuity modification (L05II) [12] of L96. Equation (1) is rewritten to the form:where

If is even, denotes a modified summation, in which the first and last terms are to be divided by 2. If is odd, denotes an ordinary summation. Generally, is much smaller than and if is even and if is odd. For our computation, we choose , so each sector covers 1° degrees of longitude. To keep a desirable number of main pressure highs and lows, Lorenz suggested keeping ratio and therefore . Parameter is selected as a compromise between too long doubling time (smaller ) and undesirable shorter waves (larger ). We first choose arbitrary values of the variables , and, using a fourth order Runge-Kutta method with a time step or 6* hours*, we integrate forward for 14400* steps*, or 10* years*. We then use the final values, which should be free of transient effect. Figure 1 shows values of model variables with selected parameters. For this setting and by the method of numerical calculation presented in [16], the global largest Lyapunov exponent is . The definition of a chaotic system according to [3] states, that a bounded dynamical system with a positive Lyapunov exponent is chaotic. Because the value of the largest Lyapunov exponent is positive and the system under study is bounded, it is chaotic. Strictly speaking, we also need to exclude the asymptotically periodic behavior, but such a task is impossible to fulfill for the numerical simulation. The choice of parameters and* time unit* = 5* days* is made to obtain similar value of the largest Lyapunov exponent as state of the art NWPMs.