1. Introduction

The appropriate method for determining the initial perturbations for ensemble forecasting has been the subject of much lively debate. Legras and Vautard (1995) explain that the bred-growing modes (BGMs) employed at the National Centers for Environmental Prediction (NCEP; Toth and Kalnay 1993) and the singular vectors (SVs) employed at the European Centre for Medium-Range Weather Forcasts (ECMWF; Molteni et al. 1996) are related to backward and forward Lyapunov vectors (LVs), respectively. The LVs are defined for infinitely long optimization times. However, because these vectors are defined in terms of the tangent linear approximation, it is of interest to see how quickly these vectors asymptotically approach their limiting (t → ∞) values. Specifically, we would like to determine if the convergence to these limiting values occurs over time intervals for which the tangent linear approximation is valid, given current analysis error magnitudes. Szunyogh et al. (1997) found evidence that singular vectors converge toward Lyapunov vectors on 2-day to 5-day timescales based on a southern winter integration of a T10 L18 model. In this study, we investigate how quickly singular vectors converge toward Lyapunov vectors using a model with higher horizontal resolution and a northern winter integration, which contains more zonal asymmetry.

As stated previously, LVs may be related to two types of perturbations used for ensemble forecasting. The BGMs, used in the ensemble forecasting system at NCEP, have similarities to the leading backward LVs, as pointed out by Legras and Vautard (1995) and Szunyogh et al. (1997), who refer to them as local Lyapunov vectors (aside from the fact that BGMs are calculated using a nonlinear model and are somehow rescaled to reflect spatial differences in analysis uncertainties). The usefulness of BGMs for ensemble prediction has been examined by Houtekamer and Derome (1994, 1995) using the same quasigeostrophic (QG) model employed in this study. The LVs for this QG model, their relationship to initial error growth, and their sensitivity to dissipation strength and metric have been examined by Vannitsem and Nicolis (1997). They found that the mean error evolution of scale-independent, small-amplitude perturbations are closely related to the spectral distribution of the Lyapunov vectors.

On the other hand, the initial-time forward SVs for t₁ = 0, t₂ = 48 h, used in the ensemble forecasting system at ECMWF (Buizza 1997), may show some resemblance to forward LVs, depending upon how quickly the SVs asymptotically approach their limiting structures as optimization time increases. The structures of SVs with optimization times as long as 8 days were examined in Molteni and Palmer (1993) using simple barotropic and baroclinic models with steady-state background flows. They found that the final-time 8-day SVs share some characteristics with the leading normal modes, indicating that by 8 days, convergence to a limiting structure is starting to occur. There is also some evidence that SVs converge toward LVs over timescales for which the tangent linear approximation may still be valid given initial perturbations the size of current analysis errors. Toth and Kalnay (1993) found that the growth rates of the BGMs converge to a rate of about 1.5 day⁻¹ after 3 or 4 days in the NCEP ensemble prediction system. Szunyogh et al. (1997) have compared singular (optimal) and Lyapunov vectors in a southern wintertime integration of a T10L18 dry version of the NCEP Medium Range Forecast Model (MRF). They found that the growth rates of initially random perturbations converge to a similar rate (about 0.83 day⁻¹) within 3–5 days of integration. They also found that the first two kinetic energy (KE) SVs optimized for 5 days explain at least 90% of the variance of the backward (local) LV by day two of their evolution.

The specific purpose of this study is to look at how quickly the structures of SVs of increasingly long optimization times, based on a temporally varying background flow, converge to a single structure represented by the Lyapunov vector. The percent variance of the leading LV that is explained by an ensemble of short-optimization-time SVs is also examined. This should provide some information on the similarities between LV-based and SV-based ensembles. The dominance of the leading final-time singular vectors based on an initially random perturbation as a function of optimization time is also investigated. This is done to ascertain the applicability of LVs to behavior of finite size perturbations in a nonlinear model. In this study, SVs for increasingly long optimization times are calculated using a T21 L3 QG model. The model used here has significantly coarser vertical resolution than the one used in Szunyogh et al. (1997), but offers higher horizontal resolution and more zonal asymmetry. This may lead to a slower convergence of the SVs, at least for their horizontal structure, because greater asymmetry allows the SVs to be more spatially localized. Because of computational constraints, the longest optimization time considered in this study is 40 days, and the SVs for this optimization time are used as an approximation of the LVs. The validity of this approximation will be examined.

Previous studies have found the structures and amplification rates of SVs and LVs to be sensitive to dissipation. Buizza (1998) using a primitive equation model, found that increasing the strength of the horizontal diffusion decreases the amplification factors and has a significant impact on the structures of the SVs. For the T21 spectral truncation, the energy at small scales decreases as the strength of the horizontal diffusion is increased. Vannitsem and Nicolis (1997) found that the Lyapunov exponents and vectors are also strongly dependent on the magnitude of the horizontal dissipation. In this study, the sensitivity of the results to dissipation strength is likewise examined.

2. Approach

The model used in this study is the QG potential vorticity (PV) model described in Marshall and Molteni (1993). The model is run with a T21 truncation and has three levels corresponding to 800, 500, and 200 mb. The SVs for short optimization times for this model, and their usefulness for ensemble prediction and forecast and analysis correction, have been examined in detail in several previous studies (e.g., Molteni and Palmer 1993; Mureau et al. 1993; Buizza and Palmer 1995; Reynolds and Palmer 1998). The model forcing is composed of specified source terms of PV that are spatially varying but temporally constant and correspond to a northern winter climatology. The model has three types of dissipative forcing. For the control case, referred to as weak dissipation, the dissipative terms are Newtonian relaxation of temperature between levels with a relaxation coefficient of 25 day⁻¹, an 800-mb linear drag term that varies with topography from 3 to 1.5 day⁻¹, and horizontal scale-selective (∇⁸) dissipation such that spherical harmonics of the time-varying potential vorticity with a total wavenumber of 21 are damped on a 2-day timescale. For “medium” dissipation, all the dissipation timescales are reduced by 25%. The time mean and standard deviation of the 500-mb height fields for weak and medium dissipation are shown in Fig. 1. The maximum in the standard deviation in the 500-mb geopotential height in the Northern Hemisphere is 180 m for the weak-dissipation case (similar to observed) and 140 m for the medium-dissipation case. A “strong” dissipation configuration, where all the dissipation timescales were reduced by 50%, was also examined. The maximum in the standard deviation in the 500-mb geopotential height in that case was less than 30 m (not shown). Since the temporal variability in the strong-dissipation case was so small that the trajectory was approximately steady state, only results from the weak- and medium-dissipation cases will be presented. The model, with 1449 degrees of freedom, is well suited for this type of study, since it is complex enough to capture baroclinic synoptic-scale processes important in forecast error growth, as well as being small enough so that L and the corresponding full set of SVs can be calculated for relatively long optimization times.

SVs have been calculated based on intervals of the last 80 days of a 180-day integration of the QG model. SVs have been calculated for 1-, 2-, 3-, 5-, 10-, 15-, 20-, 30-, and 40-day optimization times for a “backward” set (ending at the same time), and a “forward” set (starting at the same time). The intervals of the trajectory over which the SVs are calculated are illustrated in Fig. 2. The initial-time forward SVs are expected to converge for sufficiently long optimization times because they verify at the same time. Likewise, the final-time backward SVs are expected to converge. In contrast, the final-time forward SVs and initial-time backward SVs all verify at different times, and therefore neither set should converge toward a single pattern, given their trajectory dependence. Note that the 40-day SVs serve as both forward and backward SVs. To examine case-to-case variability, these experiments are repeated over different portions of the trajectory two more times, resulting in three forward sets and three backward sets of SVs. These SVs have been calculated for two different dissipation strengths (weak and medium), and three metrics (kinetic energy, enstrophy, and squared streamfunction). The SVs are actually calculated for K^1/2LK^−1/2, where K is a diagonal matrix such that for a state vector x, x^TKx gives the measure of perturbation growth, or metric. For the kinetic energy metric, K is defined such that x^TKx gives the kinetic energy of the system, and the SVs of K^1/2LK^−1/2 are referred to as the KE SVs. Likewise, for the enstrophy (EN) SVs, K is defined such that x^TKx gives the enstrophy of the system. Because the state vector is streamfunction, K is the identity matrix for the squared streamfunction (SN) SVs.

In this study, the 40-day SVs are taken as a proxy for the LVs. The initial-time 40-day SVs will approximate the forward LVs, and the final-time 40-day SVs will approximate the backward LVs. The validity of this approximation will be apparent from the convergence of the SVs at increasing optimization times. Although an optimization time longer than 40 days is desirable, computational expense limits the length of the maximum optimization time. This may result in an overestimation of how quickly the SVs converge to the true Lyapunov vectors, which are presumably less similar to the short-optimization-time SVs than are the 40-day SVs. Therefore these results should provide an upper bound on the convergence rates.

As an objective measure of convergence, the sum of the squared inner product between a subset of the SVs and the proxy for the LV will be used. For the leading KE LV, this would be

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where 〈 ; 〉 denotes a KE inner product. This sum represents the percent variance of the LV explained by the subset of the SVs, measured in terms of kinetic energy. All of the SVs and LVs at both initial and final times have been normalized to have kinetic energy equal to 1.0. Since the KE SVs form an orthonormal set, this sum (2) will be equal to 1.0 for m = 1449. When using SN or EN SVs, a squared streamfunction or enstrophy inner product is used instead of the KE inner product. Subsets of size m = 2, 10, and 30 are considered here. The subset of size two is chosen because the first and second SVs are often in quadrature with each other (i.e., they have similar spatial structures that are approximately 90° out of phase). Larger subsets are also included in order to examine how much of the leading LV would be explained by an ensemble of the leading SVs. In addition to examining the convergence of the SVs toward the LVs, the dependence of the SVs on the metric is also investigated.

3. Description of the forward and backward SVs

Before calculating the convergence of the SVs toward the LVs, it is instructive to examine the horizontal and vertical characteristics of the SVs as a function of optimization time.

4. Convergence between SVs

5. Singular values

The natural logs of the leading singular values divided by optimization time (the inverse of the e-folding time), corresponding to the fastest growing KE and EN SVs, are shown in Fig. 13. These growth rates should asymptotically approach that of the leading LV as optimization time is increased. Note that the growth rate decreases as optimization time increases and as dissipation is increased. Buizza (1998) also shows a decrease in the leading singular values as horizontal diffusion is increased, and Vannitsem and Nicolis (1997) show the leading Lyapunov exponents are likewise sensitive to diffusion. The growth rates exhibit the most case-to-case variability for short optimization times for weak dissipation, indicating a larger sensitivity to the interval of the trajectory over which they are optimized. Although the growth rates appear to be asymptoting, they are still decreasing between the 30-day and 40-day cases, indicating that the leading Lyapunov exponent has not yet been attained. Note that for small optimization times the growth rates for the KE SVs (Fig. 13a) are significantly smaller than the growth rates for the EN SVs (Fig. 13b). However, for the long optimization times, the growth rates are very similar, averaging 0.30 and 0.20 day⁻¹ for the KE weak and medium dissipation SVs, respectively, and 0.31 and 0.22 day⁻¹ for the EN weak and medium dissipation SVs, respectively.

The growth rates corresponding to the weak dissipation leading 40-day KE singular values range from 0.29 to 0.31 day⁻¹. These values are larger than the corresponding average value of 0.23 day⁻¹ found by Vannitsem and Nicolis (1997), but within the expected variability found by them. On the other hand, this growth rate is considerably smaller than the 0.83 day⁻¹ (amplification factor of 2.3) found as the Lyapunov exponent by Szunyogh et al. (1997) for their truncated dry version of the MRF. These results appear to indicate that 40 days is not a sufficiently long optimization time to give a very accurate approximation of the leading Lyapunov exponent. Szunyogh et al. (1997) had found that most of the “super-Lyapunov” growth associated with SVs for short optimization times was associated with geostrophic adjustment. The growth rates for the leading 1-day SVs in this study are comparable to those in Szunyogh et al. (1997). In fact, the difference between the 1-day optimal growth rates and the estimated Lyapunov growth rate is considerably larger in our study, indicating that large super-Lyapunov growth is possible even in the absence of geostrophic adjustment.

If one makes the assumption that initial condition errors are approximately isotropic with respect to the initial-time SVs (i.e., they have a white noise spectrum in terms of the SVs), then the tangent linear growth of the variance of that error over the optimization time can be computed by taking the mean square of the singular values (Lorenz 1965; Farrell 1990). Lorenz (1965) points out that the dominance of the leading singular value should increase as optimization time increases. We have computed the ratio between the leading squared-KE singular values and the sum of all the squared-KE singular values (Fig. 14). This ratio is the percent of the total error variance represented by the leading 1, 10, and 30 KE SVs. This should indicate the time span over which random errors may be dominated by the leading SVs (which should be the leading LVs for sufficiently long optimization times). In this figure, the solid (dashed) lines give the average for the medium (weak) dissipation cases. Circles (triangles) give the individual medium (weak) case results. It is clear that as optimization time is increased, the leading vectors become more dominant. The first 40-day SV accounts for, on average, 53% of the forecast error variance in the weak-dissipation case, and 67% of the forecast error variance in the medium-dissipation case (Fig. 14a). Houtekamer and Derome (1994), using the same model, found that the first empirical orthogonal function explained 48% of the variance in different bred perturbations after 20 days of breeding, which is consistent with the average percentages for the 20-day SVs shown here.

The trajectory dependence of these percentages is very large, particularly at longer optimization times. (The range of values for the 40-day optimization time is between 33% and 90%.) This sensitivity to trajectory at large optimization times becomes much smaller as larger sets of the leading singular values are considered. From these results it is clear that the leading singular vector alone does not account for a majority of the forecast error variance over linear timescales, although small ensembles may. For the 5-day SVs, the leading singular vector accounts for between 5% and 21% of the forecast error variance (Fig. 14a). An ensemble of the first 10 5-day SVs (Fig. 14b) would account for between 36% and 48% of the forecast variance, while an ensemble of the first 30 5-day SVs (Fig. 14c) would account for between 60% and 70%. The difference between the weak and medium dissipation averages in Fig. 14a is not significant given the wide spread of the individual cases.

These results are dependent on the metric. Figure 15 shows the average percent forecast variance explained by the first n final-time SVs, where n = 1, 30. The different curves are for different optimization times, and the thin (thick) curves are for the KE (EN) metric. Note that for short optimization times, the leading EN SVs are far more dominant than the KE SVs (for a 1-day optimization time, the EN SVs explain twice as much percent variance as the KE SVs). However, this metric dependence diminishes as optimization time increases. This is consistent with the fact that the leading EN singular values are much larger than the leading KE singular values for short optimization times, but similar for longer optimization times (Fig. 14). Also note that a small ensemble of short-optimization-time SVs can account for a significant portion of the total error at final time. The first 30 3-day KE SVs or first 30 2-day EN SVs account for almost 50% of the total error variance. Likewise, the first eight 5-day EN SVs, and the first 13 5-day KE SVs account for half of the total error variance.

6. Summary and conclusions

The results presented in this study are meant to give an estimate of how quickly SVs converge to LVs in a simple yet fairly realistic atmospheric model with significant zonal asymmetry. The results must be interpreted within light of the simplifications and limitations of this study. First, only three sets of forward and three sets of backward SVs have been calculated for both medium and weak dissipation. The similarities between the convergence rates for the final-time backward and initial-time forward SVs exhibited in Figs. 9 and 10 indicate that results for the 10- and 30-member ensembles should be fairly robust to sampling error. However, the significant scatter in the results when including only two SVs indicates that these results can be expected to vary largely from one case to another. [See Vannitsem and Nicolis (1997) for diagnoses of variability in the Lyapunov exponents.]

The 40-day SVs are taken as a proxy for the LVs. The large percent of the 40-day SV variance explained by the first two 30-day SVs in four of the six cases implies that the 40-day SVs are most likely a good approximation to the LVs in these cases (Figs. 9 and 10). However, it is also likely that the 40-day SVs are more similar to the SVs for shorter optimization times than are the true LVs. This would mean that the true convergence rates have been overestimated here. It also appears that the leading singular values have not fully converged onto the leading Lyapunov exponent by 40 days (Fig. 13).

The estimation of the dominance of the leading SV in the forecast error is based on the assumption that the initial perturbation is approximately isotropic with respect to the initial SVs. Analysis differences have been shown to be approximately isotropic with respect to 2-day SVs for this model (Reynolds and Palmer 1998). The projection of true analysis errors onto SVs is unknown, of course, but there is some indirect evidence that the analysis errors are isotropic at least with respect to the first 30 or so leading SVs used in the ensemble prediction system at ECMWF (Gelaro et al. 1998). Should analysis errors have a preferential projection onto the initial-time leading SVs, then the dominance of the leading SV in the forecast error has been underestimated here. Also, the application of a local projection operator [i.e., constraining the final-time SVs to optimize perturbation growth within a specified region (Barkmeijer 1992; Buizza 1994)] can increase the dominance of the leading SVs significantly (Reynolds and Palmer 1998).

With these caveats in mind, the results of the study are summarized. The results here are in general agreement with the definition of LVs. Initial-time forward SVs converge toward a forward LV and final-time backward SVs converge toward a backward LV. The leading final-time KE, SN, and KE and EN SVs, exhibit the expected independence of metric for long optimization times.

In general, the 1-day and 3-day SVs are more spatially localized than the SVs for longer optimization times. This may be due to the fact that using a very long optimization time has the effect of averaging over moving, local regions of instability. For the leading SVs, the maximum in kinetic energy is found at the 500-mb level at initial time and at the 200-mb level at final time for all optimization times considered here. As optimization time is increased, there is more energy at the largest spatial scales at initial time and more energy at the smallest spatial scales at final time (Fig. 7).

Strong convergence of the first two SVs toward an LV occurs at optimization times of five days or longer, although there is considerable case-to-case variability. This would indicate that, for this model, in most cases, convergence of the SVs to the LV does not occur over timescales short enough for the tangent linear approximation to be valid (Figs. 9 and 10). We conjecture that, for a more realistic atmospheric model, with far more degrees of freedom, this convergence would occur even more slowly, not more rapidly. The percent variance of the leading LV described by the first 30 2-day SVs is significant (47%–74% at final time). These results suggest that, for this model, a significant portion of the leading LV may be contained within the subspace of an ensemble composed of the first 30 SVs optimized over a short time interval. These percentages would probably be smaller if based on more accurate LVs and a more complex model. However, if the overestimation of these percentages is small, then these results are consistent with the idea that the leading LV can be described by a linear combination of the leading SVs (Toth and Kalnay 1993). The rate of convergence of the final-time backward SVs to the backward LV and the initial-time forward SVs to the forward LV displays large case-to-case variability but is not a monotonic function of the dissipation strength. This indicates considerable sensitivity to the trajectory, though not necessarily to the dissipation strength itself.

The percent variance of the tangent-linear evolved perturbation that is explained by the leading final-time SV increases as optimization time increases. However, given an initial-time perturbation that is isotropic with respect to the initial-time SVs, the leading final-time 5-day SV accounts for only 5%–22% of the total perturbation variance. This indicates that while the leading final-time SV will represent a significant part of the global evolved perturbation variance, it will not account for a majority of that variance on short timescales. However, a small ensemble of 10–30 SVs may account for a majority of the perturbation variance over linear timescales, depending on the trajectory and the metric.