Extracting envelopes of non-zonally propagating Rossby wave packets. Submitted to Monthly Weather Review Aleksey V. Zimin 1 Istvan Szunyogh 2 Brian R. Hunt 3 Edward Ott 4 March 15, 2005 1 Corresponding author: Aleksey Zimin, Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742; alekseyz@ipst.umd.edu. 2 Institute for Physical Science and Technology and Department of Meteorology, University of Maryland, College Park, Maryland 20742 3 Institute for Physical Science and Technology and Department of Mathematics, University of Maryland, College Park, Maryland 20742 4 Institute for Research in Electronics and Applied Physics, Department of Physics and Department of Electrical and Computer Engineering, University of Maryland, College Park, Maryland 20742
Abstract In this note we propose an improved technique to extract envelopes of wave packets of synoptic scale Rossby waves. In the currently used techniques the wave packets are assumed to be propagating along latitude circles. In our new technique, we take into account the possibility that the waves may propagate in a direction that is not purely zonal. We demonstrate the advantages of the new technique both on analytical and real-world examples.
1 Introduction Techniques to extract envelopes of Rossby wave packets are useful tools for tracking the propagation of the influence of localized changes in the initial conditions of numerical weather prediction models (e.g.persson 2000; Szunyogh et al. 2002). In Zimin et al. (2003) we proposed such a technique, that has later been used at the European Centre for Medium Range Weather Forecasts (ECMWF) to track the origin of unusually large localized errors in early medium range forecasts (2-5 days), and to track the propagation of targeted weather observations (Frederico Grazzini, personal communication). While the technique has performed well on most occasions, there has also been a small number of cases with less satisfactory results. In these cases, the technique indicated multiple wave packets at nearby latitudes, although all other information suggested a single wave packet (Frederico Grazzini, personal communication). An example of such a behavior is shown in Figure 1, where the technique of Zimin et al. (2003) is applied to the dropsonde signal from a targeted observation mission. The dropsonde signal is defined as the difference between the NCEP global model forecast that was initiated by assimilating all targeted and standard (non-targeted) observations and the NCEP global model forecast that was initiated by assimilating only the standard observations. While the effects of dropsonde observations on the forecast of the 300hPa geopotential height have a wave-like structure (Figures 1 (a) and (b)), suggesting the existence of a single wave packet, the technique of Zimin et al. (2003) indicates two distinct wave packets at nearby latitude bands (Figures 1 (c) and (d)). We conjecture that this problem is due to the assumption of Zimin et al. (2003) that the wave packets are zonal structures [we note that other techniques, such as the complex demodulation of Chang and Yu (1999), Chang (2000) are also based on the same assumption]. In reality, the large scale flow guiding the propagation of the wave packets in the upper troposphere is not always zonal, and wave packet envelopes may, therefore, not be aligned along latitude circles. Thus, identifying the wave packets by analyzing the data independently for the different latitudes can lead to spurious results. Our new method is based on the assumptions that (i) the atmospheric flow can be decomposed into a large scale basic flow and a synoptic scale Rossby wave component and (ii) the Rossby waves tend to propagate along the direction of the basic flow. Phenomenology indicates that while these assumptions are not exactly true, they are reasonable approximations. We also show that the technique based on these assumptions can markedly improve upon the alternative, zonal assumption. We construct an illustrative analytical example to demonstrate how an analysis restricted to data along a latitude circle [as in Zimin et al (2003), Chang and Yu (1999), Chang (2000)] can lead to a distorted result. In this example we simulate a case in which the angle between the direction of the large scale flow and the zonal direction is 45 degrees. In particular, we assume a wave packet with an envelope modulating a spatially sinusoidal wave, where the envelope is extended along a line at 45 degrees to the supposed zonal direction (taken as horizontal in Figure 2) and narrow in the direction perpendicular to the assumed flow direction. Referring to Figure 2, if we call the vertical coordinate y and the horizontal (zonal) coordinate x placing the origin at the center of the figure, the wave packet has the following 1
functional form: φ(x, y) = A(x, y) sin(k(x + y) + γ) = exp { α(x + y) 2 β(x y) 2} sin(k(x + y) + γ), where α = 1/350, β = 1/40, k = 7(2π/144) and γ = 1. All variables are specified on the 73 144 grid. As shown in Figure 2, the technique of Zimin et al. (2003) distorts the shape of the wave packet envelope A(x, y), indicating two zonally elongated centers of the wave packet. On the other hand, as shown in Figure 2(d), the envelope recovered with the technique to be described in Section 2 is essentially identical to the envelope of the original signal (Figure 2(b)). 2 Our New Technique We start the description of our new technique by briefly reviewing the technique described in Zimin et al. (2003). The method of Zimin et al. (2003) recovers the packet envelope from a scalar atmospheric field φ(s) given as a function of the position s along a latitude circle. A Fourier transform is applied to φ(s) to obtain ˆφ(k). The field transformed into wavenumber space, ˆφ(k) is then multiplied by a filter function f(k): f(k) = 1, 0 < k min k k max f(k) = 0, otherwise. This filter removes all wavenumber components except for a band between positive wavenumbers k min and k max. Then the wave packet envelope along the latitude circle A(s) is computed as the magnitude of the inverse Fourier transform of f(k)ˆφ(k) by two. (Note that, since the filter accepts only positive k components, the inverse transform of f(k)ˆφ(k) is complex.) In the new method we apply the filter algorithm of Zimin et al. to the atmospheric variable φ(s) along a streamline instead of along a latitude circle. Suppose we have an atmospheric variable φ(x, y), and the background flow has latitudinal and longitudinal components u(x, y) and v(x, y). We assume that all variables are specified on a rectangular grid {x j, y j }. The algorithm is as follows: Step 1. For each grid point (x 0, y 0 ) we find a piecewise-linear approximation of a streamline defined by (u, v) in the neighborhood of (x 0, y 0 ). We calculate the angle between the piecewise linear streamline and the x direction by: ( ) θ(x, y) = sin 1 v(x, y). v(x, y)2 + u(x, y) 2 The next coordinates along the streamline (x ±1, y ±1 ) are found by: x ±1 = x 0 ± δ cos(θ), y ±1 = y 0 ± δ sin(θ). 2
The parameter δ sets the step size. We proceed to find the coordinates of the approximate local streamline up to distance ±δn away from the (x 0, y 0 ). The distance δn should be comparable to the size of the longest wave packet we intend to identify. Step 2. We now have points with coordinates {x i, y i }, i = N... + N. We interpolate the atmospheric variable of interest φ onto each such point (x i, y i ). We then localize φ(x i, y i ) using a Gaussian filter function centered at i = 0: ) φ(x i, y i ) = φ(x i, y i ) exp ( i2. (2N) 2 Step 3. We taking s = iδ, with φ(s) given by φ(x i, y i ) for s in, we input this new definition of φ(s) into the algorithm described in Zimin et al., (2003) to obtain the amplitude A(x i, y i ). We estimate the wave packet envelope at (x 0, y 0 ) by the amplitude at the center point A(x 0, y 0 ). We repeat the above three steps for every grid point. The C++/MATLAB Mex code for the technique is available from the authors. Figure 1 (e) and (f) demonstrate that the new technique yields a single wave packet for the data discussed in the introduction. Acknowledgments This research was inspired by Frederico Grazzini s thought provoking comments on the performance of the technique described in Zimin et al. 2003. This research was supported by a NOAA THORPEX grant, the Office of Naval Research (Physics), National Science Foundation (Grants 0104087 and 0098632), Army Research Office (Grant DAAD190210452), a James S. McDonnell 21st Century Research Award and a NASA AIRS grant. 3
3 References Chang, E. K. M. and D. B. Yu, 1999: Characteristics of wave packets in the upper troposphere. Part I: Northern Hemisphere Winter. J. Atmos. Sci., 56, 1708-1728. Chang, E. K. M., 2000: Wave packets and life cycles of troughs in the upper troposphere: Examples from the Southern Hemisphere Summer Season of 1984/85. Mon. Wea. Rev., 126, 25-50. Persson, A., 2000: Synoptic-dynamic diagnosis of medium range weather forecast systems. Proceedings of the Seminars on Diagnosis of models and data assimilation systems. 6-10 September 1999, ECMWF, Reading, U.K., 123-137. Szunyogh, I., Z. Toth, A. V. Zimin, S. J. Majumdar, and A. Persson: On the propagation of the effect of targeted observations: The 2000 Winter Storm Reconnaissance Program. Mon. Wea. Rev., 130, 1144-1165. Zimin, Aleksey V., Szunyogh, Istvan, Patil, DJ, Hunt, Brian R., Ott, Edward, 2002: Extracting Envelopes of Rossby Wave Packets. Mon. Wea. Rev., 131, (5), 10111017. 4
Figure 1: Propagation of the difference between a forecast that uses the information from the dropsondes at initial time and a forecast that does not use this information. Shown is the difference (color contours) between the two forecasts at 36hr and 48hr forecast lead times for the geopotential height at the 300hPa level. The packet envelopes identified by the method of Zimin et al. (2003) are shown in panels (c) and (d), while the envelopes recovered by the method introduced here are shown in panels (e) and (f). 5
Figure 2: The wave packet φ(x, y) (a), and the packet envelope A(x, y) (b), defined by Equation 1. The wave packet amplitude is shown in color, and the background flow is shown by the straight lines. The packet envelopes recovered by the technique of Zimin et al. (2003) (c) and by the technique described in this paper (d) are also shown. 6