## 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.

**y**

*t*

_{2}

**L**

*t*

_{1}

*t*

_{2}

**y**

*t*

_{1}

**y**is the perturbation and

**L**

**L**

**L**

**L**

**L**

**L**

**L**

*t*

_{1}=

*t*

_{present}and

*t*

_{2}→ ∞ (referred to as forward SVs) at initial time. The backward Lyapunov vectors are the SVs calculated for

*t*

_{1}→ −∞ and

*t*

_{2}=

*t*

_{present}(referred to as backward SVs) at final time. Initial-time forward SVs should lose their dependence on

*t*

_{2}at sufficiently long optimization times as they converge to the forward LVs. Likewise, final-time backward SVs should lose their dependence on

*t*

_{1}at sufficiently long optimization times as they converge to the backward LVs. Legras and Vautard (1995) also point out that for backward LVs, the final-time leading (fastest growing) LV becomes independent of metric. Likewise, for forward LVs, the initial-time trailing (fastest decaying) LV becomes independent of metric.

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 *t*_{1} = 0, *t*_{2} = 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**

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**^{1/2}**LK**^{−1/2}, where **K****x**, **x**^{T}**K****x** gives the measure of perturbation growth, or metric. For the kinetic energy metric, **K****x**^{T}**K****x** gives the kinetic energy of the system, and the SVs of **K**^{1/2}**LK**^{−1/2} are referred to as the KE SVs. Likewise, for the enstrophy (EN) SVs, **K****x**^{T}**K****x** gives the enstrophy of the system. Because the state vector is streamfunction, **K**

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.

*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.

### a. Horizontal spatial patterns

Figure 3 shows the 500-mb streamfunction associated with the leading (fastest growing) initial-time KE SVs for different optimization times. The SVs shown here are from one of the three “forward” sets calculated using weak dissipation. For forward SVs, the start time of the trajectory remains fixed while the end time varies (Fig. 2). Note that the sign of the SV is arbitrary. As optimization time is increased, the SVs become less spatially localized. 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, although Molteni and Palmer (1993) also find that 8-day SVs based on a steady-state background flow are less localized than the 2-day SVs based on the same flow. There is a strong resemblance between the 40-day, 20-day, 10-day, and to a lesser extent, 5-day SVs, indicating that the leading initial-time forward SVs start to converge toward the leading forward LV at optimization times as short as 5 days. The SVs for the 1-day or 3-day optimization times do not resemble the SVs for the longer optimization times. The initial-time backward SVs (not shown) also become less spatially localized, but, because they verify at different times, do not converge to a single pattern.

The 500-mb streamfunction associated with the leading (fastest growing) final-time KE SVs for different optimization times is shown in Fig. 4. These are backward SVs calculated using weak dissipation. Backward SVs share a common final time but have different initial times (Fig. 2). As with the initial-time forward SVs, the 10-day through 40-day final-time backward SVs are very similar and share some characteristics with the 5-day SVs. However, the 1-day and 3-day SVs are considerably different from the SVs for longer optimization times. The final-time forward SVs (not shown) do not converge toward a single pattern, because they verify at different times. For both the initial-time forward SVs and final-time backward SVs, strong convergence toward the leading LV appears to start for optimization times between 5 and 10 days for the cases considered above. However, there is large case-to-case variability in the convergence rates, as will be shown in section 4.

### b. Vertical cross sections

Vertical cross sections are presented to provide information on the vertical structures of the SVs and their spatial extent at all three levels. Vertical cross sections at a latitude circle at 35°N for the initial-time forward KE SVs for weak dissipation are shown in Fig. 5. The 1-day and 3-day SVs have little in common with the 10-day, 20-day, or 40-day SVs. The 1-day and 3-day SVs are primarily restricted to a region between 90°E and 180°, while the longer optimization-time SVs have significant amplitude over a much larger longitudinal range. The SVs have a distinct westward tilt with height, discussed in detail in previous studies (Molteni and Palmer 1993; Buizza and Palmer 1995), for optimization times as long as 5 days. For optimization times of l0 days or longer, the vertical structure becomes more ambiguous, with the anomalies at 500 and 800 mb often appearing out of phase. For all optimization times, the maximum amplitude of the SVs is found at 500 mb. Kinetic energy wavenumber spectra examined in the next section show this to be true for the global fields as well.

At final time, the vertical cross sections of the leading final-time backward SVs (Fig. 6) also indicate convergence toward the backward LV for optimization times of 5 days and greater. Consistent with many previous studies of SVs optimized for a few days, the strong westward tilt with height apparent at initial time is weaker at final time. For longer optimization times, there appears to be no significant tilt with height. For all optimization times, the maximum energy appears at 200 mb. Kinetic energy spectra shown in the next section confirm this to be true for the global fields as well.

### c. KE as a function of total wavenumber

Figure 7 shows kinetic energy as a function of the total wavenumber at initial and final time for the weak-dissipation KE SVs. Each curve represents the six case average (three backward and three forward cases) and is also averaged over the first five leading singular vectors. The SVs at both initial and final time are normalized to have unit kinetic energy. At initial time, the maximum energy is at 500 mb in wavenumbers 8 and higher. As optimization time increases, there is a systematic increase in energy in the smallest wavenumbers (largest spatial scales) at initial time. At final time, a strong spectral peak is found at the 200-mb level between wavenumbers 7 and 12. This peak decrease as optimization time increases, and there is also a systematic increase in energy at the smallest spatial scales (largest wavenumbers). Similar trends are seen for the medium-dissipation SVs (not shown).

Figure 8 illustrates the case-to case variability in the kinetic energy spectra. Each thin solid curve represents one of the six cases for the weak-dissipation KE SVs for the 1-day and 30-day optimization times. The six-case mean is given by the thick dashed curve. There is considerable case-to-case variability for the 1-day SVs. However, the differences between the 1-day and 30-day SVs at the smallest wavenumbers at initial time and the largest wavenumbers at final time lie outside the range of this variability.

## 4. Convergence between SVs

### a. Convergence to the LV

As indicated in Figs. 3–6, the initial-time forward SVs and the final-time backward SVs appear to converge toward a single pattern for optimization times of 5 days and longer. As described by (2), the sum of the squared KE inner product between the first 40-day KE SVs (our proxy for the LVs) and a subset of the leading KE SVs for shorter optimization times is used to quantify this convergence. This value is the percent variance of the LV (measured in terms of KE) that can be explained using a subset of the first SVs for the shorter optimization times. The LVs are both a function of dissipation and trajectory. For Figs. 9–14 results for the medium (weak) SVs are indicated by solid (dashed) lines.

The percent variance of the first forward KE LV that is explained by the first two initial-time forward KE SVs is shown in Fig. 9a. In general this percentage increases with optimization time. However, the case-to-case variability is very large, indicating a large sensitivity to trajectory. In four of the six cases, the first two initial-time forward 30-day SVs explain over 93% of the forward LV and in the two other cases explain 78% of the LV. This implies that in four of the six cases, the 40-day SV is probably a good approximation of the LV. However, in the other two cases, convergence to the true LV has most probably not occurred by 40 days. For the 5-day optimization time, the first two forward SVs explain between 5% and 60% of the variance of the forward LV. The initial-time backward SVs (not shown) do not converge toward the LV, as they verify at different times. There does not appear to be a systematic difference between convergence rates for the weak- and medium-dissipation cases.

As the number of SVs is increased (Figs. 9b,c), the sensitivity to trajectory decreases for the longer optimization times, although for the 1–3-day SVs, this sensitivity is still very large. Larger ensembles describe considerably more of the variance of the forward LV than do the first two SVs. A 10-member ensemble of the initial-time forward SVs (Fig. 9b) explains 40%–80% of the forward LV when using 5-day SVs, greater than 90% of the LV when using 20-day SVs, and greater than 97% of the LV when using 30-day SVs. When using a 30-member ensemble (Fig. 9c), in five of the six cases, initial-time forward 2-day SVs describe between 30% and 50% of the variance of the forward LV. This percentage increases to between 64% and 88% for the 5-day SVs. The results for the SN and EN SVs are similar, except that the first 30 EN SVs explain significantly more of the EN LV variance (between 45% and 75% for the 1-day optimization time).

Results for the final-time backward SVs are shown in Fig. 10. As with the initial-time forward SVs, in four of the six cases the first two 30-day SVs explain more than 93% of the LV; however, in the remaining two cases, the percent is much lower, indicating convergence to the LV has not occurred on these timescales (Fig. 10a). [When the first five SVs are included (not shown) then the percentages increase to over 87% for all six cases.] The convergence rates are very sensitive to the trajectory. The impact of the dissipation on the convergence rates appears to be of secondary importance. For the 5-day optimization time, the first two final-time backward SVs explain between 20% and 60% of the backward LV (Fig. 10a). The percentages shown here are significantly smaller than those found in Szunyogh et al. (1997). They found that in their southern wintertime integration, the first two 5-day SVs explained 93% of the KE variance of the leading LV. This difference may be attributable to the higher horizontal resolution in the model used in this study, or the stronger zonal asymmetry present in a northern wintertime integration. The fastest growing SVs and LVs found in Szunyogh et al. (1997) are found in the Southern Hemisphere and are not spatially localized.

When considering an ensemble of the first 10 final-time backward SVs (Fig. 10b), then 45% to 89% of the backward LV variance can be explained using the 5-day SVs. If considering the first 30 SVs (Fig. 10c), then 47% and 75% of the backward LV variance can be explained using the 2-day SVs, and between 70% and 93% using the 5-day SVs (Fig. 10c). This indicates that, for this model, a significant percent of the LV may be captured within the subspace spanned by a typical ensemble of SVs optimized for short time intervals. Should these results hold for a more complex model, then this is consistent with the idea put forth by Toth and Kalnay (1993) that the perturbations in a breeding cycle can be described by linear combinations of the leading SVs, given that bred modes and LVs are similar. Results for the EN and SN SVs give similar convergence rates.

### b. Convergence between different metrics

As pointed out by Legras and Vautard (1995) and others, the final-time leading (fastest growing) LV is independent of metric. [For a discussion of metrics and ensemble prediction, see Palmer et al. (1998).] To see at what optimization time this insensitivity to metric occurs, the percent variance of the leading SN SV described by the two leading KE SVs for each optimization time is shown in Fig. 11. The percent variance of the leading EN SV described by the two leading KE SVs for each optimization time is shown in Fig. 12.

Figure 11a shows the percent variance of the first SN SV explained by the first two KE SVs at initial time, which should be dependent on the metric. The solid (dashed) lines denote results for the six medium (weak) dissipation cases. In all but one case, the first two KE SVs explain between 50% and 80% of the first SN SV for short optimization times, and between 55% and 70% for longer optimization times, with no discernable difference due to dissipation strength. As expected, the leading KE and SN SVs are much more similar to each other at final time (Fig. 11b) than at initial time (Fig. 11a). At final time, all SVs show at least a 90% similarity at optimization times of 10 days or greater. In five of the six cases, the first two 40-day KE SVs explain at least 99% of the first 40-day SN SV.

The EN and KE SVs are much less similar than the SN and KE SVs. The percent variance of the leading EN SV explained by the first two KE SVs at initial time (Fig. 12a) is almost always less than 20%. At final time (Fig. 12b), the similarity between the leading EN SV and the first two KE SVs increases rapidly as the optimization time increases. In five of the six cases, the first two 40-day KE SVs explain at least 95% of the first 40-day SN SV at final time, despite the fact that for short optimization times, the final-time KE and EN SVs are very dissimilar. The results are consistent with the results of Palmer et al. (1998), who found that, for short optimization times, the KE and SN SVs are more alike than the KE and EN SVs, and that the SVs for different metrics are more alike at final time than at initial time. These results do show the expected independence of metric for the leading final-time SVs, although the rate at which this independence is reached exhibits considerable sensitivity to trajectory and metric.

## 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.

## Acknowledgments

The quasigeostrophic model, written by Franco Molteni, was provided to the first author while visiting ECMWF, and the help and guidance of Tim Palmer and Jan Barkmeijer are especially appreciated. Support of the sponsor, Office of Naval Research, Project Element 0601153N, is gratefully acknowledged. The Department of Defense High-Performance Computing Program provided computing support.

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Schematic diagram depicting the optimization intervals along the nonlinear trajectory for the forward (dark solid) and backward (light dash) SVs. The optimization interval for the 40-day SVs is the same for both the forward and backward sets (dark dash). The *x*-axis units are days.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

Schematic diagram depicting the optimization intervals along the nonlinear trajectory for the forward (dark solid) and backward (light dash) SVs. The optimization interval for the 40-day SVs is the same for both the forward and backward sets (dark dash). The *x*-axis units are days.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

Schematic diagram depicting the optimization intervals along the nonlinear trajectory for the forward (dark solid) and backward (light dash) SVs. The optimization interval for the 40-day SVs is the same for both the forward and backward sets (dark dash). The *x*-axis units are days.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The 500-mb streamfunction associated with the first (fastest growing) forward KE SVs at initial time for (a) 1-day, (b) 3-day, (c) 5-day, (d) 10-day, (e) 20-day, and (f) 40-day optimization times for weak dissipation. Contour intervals at ±5, ±15, ±25, ±35 × 10^{6} m^{2} s^{−1}, with values greater than ±15 × 10^{6} shaded.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The 500-mb streamfunction associated with the first (fastest growing) forward KE SVs at initial time for (a) 1-day, (b) 3-day, (c) 5-day, (d) 10-day, (e) 20-day, and (f) 40-day optimization times for weak dissipation. Contour intervals at ±5, ±15, ±25, ±35 × 10^{6} m^{2} s^{−1}, with values greater than ±15 × 10^{6} shaded.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The 500-mb streamfunction associated with the first (fastest growing) forward KE SVs at initial time for (a) 1-day, (b) 3-day, (c) 5-day, (d) 10-day, (e) 20-day, and (f) 40-day optimization times for weak dissipation. Contour intervals at ±5, ±15, ±25, ±35 × 10^{6} m^{2} s^{−1}, with values greater than ±15 × 10^{6} shaded.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The 500-mb streamfunction associated with the first (fastest growing) backward KE SVs at final time for weak dissipation. Optimization times, contouring, and shading as in Fig. 3.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The 500-mb streamfunction associated with the first (fastest growing) backward KE SVs at final time for weak dissipation. Optimization times, contouring, and shading as in Fig. 3.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The 500-mb streamfunction associated with the first (fastest growing) backward KE SVs at final time for weak dissipation. Optimization times, contouring, and shading as in Fig. 3.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The vertical cross section of streamfunction at 35°N for the first (fastest growing) forward KE SVs at initial time for (a) 1-day, (b) 3-day, (c) 5-day, (d) 10-day, (e) 20-day, and (f) 40-day optimization times for weak dissipation. Contour intervals at ±4, ±8, ±12, ±24, ±32 × 10^{6} m^{2} s^{−1}, with values greater than ±12 × 10^{6} shaded. The *x*-axis is longitude and the *y*-axis is pressure.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The vertical cross section of streamfunction at 35°N for the first (fastest growing) forward KE SVs at initial time for (a) 1-day, (b) 3-day, (c) 5-day, (d) 10-day, (e) 20-day, and (f) 40-day optimization times for weak dissipation. Contour intervals at ±4, ±8, ±12, ±24, ±32 × 10^{6} m^{2} s^{−1}, with values greater than ±12 × 10^{6} shaded. The *x*-axis is longitude and the *y*-axis is pressure.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The vertical cross section of streamfunction at 35°N for the first (fastest growing) forward KE SVs at initial time for (a) 1-day, (b) 3-day, (c) 5-day, (d) 10-day, (e) 20-day, and (f) 40-day optimization times for weak dissipation. Contour intervals at ±4, ±8, ±12, ±24, ±32 × 10^{6} m^{2} s^{−1}, with values greater than ±12 × 10^{6} shaded. The *x*-axis is longitude and the *y*-axis is pressure.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The vertical cross section of streamfunction at 35°N for the first (fastest growing) backward KE SVs at final time for weak dissipation. Optimization times, shading, and contouring as in Fig. 5.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The vertical cross section of streamfunction at 35°N for the first (fastest growing) backward KE SVs at final time for weak dissipation. Optimization times, shading, and contouring as in Fig. 5.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The vertical cross section of streamfunction at 35°N for the first (fastest growing) backward KE SVs at final time for weak dissipation. Optimization times, shading, and contouring as in Fig. 5.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The kinetic energy as a function of total wavenumber averaged for the first five KE SVs averaged for the six cases for weak dissipation. The KE at initial time at (a) 200 mb, (b) 500 mb, and (c) 800 mb, and at final time at (d) 200 mb, (e) 500 mb, and (f) 800 mb. Each curve represents a different optimization time, given in key in first panel.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The kinetic energy as a function of total wavenumber averaged for the first five KE SVs averaged for the six cases for weak dissipation. The KE at initial time at (a) 200 mb, (b) 500 mb, and (c) 800 mb, and at final time at (d) 200 mb, (e) 500 mb, and (f) 800 mb. Each curve represents a different optimization time, given in key in first panel.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The kinetic energy as a function of total wavenumber averaged for the first five KE SVs averaged for the six cases for weak dissipation. The KE at initial time at (a) 200 mb, (b) 500 mb, and (c) 800 mb, and at final time at (d) 200 mb, (e) 500 mb, and (f) 800 mb. Each curve represents a different optimization time, given in key in first panel.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The kinetic energy as a function of total wavenumber averaged for the first five KE SVs for weak dissipation. The thin lines represent the individual cases and the thick dashed line represents the six-case average. The KE at initial time at 500 mb for the (a) 1-day SVs and (b) 30-day SVs. The KE at final time at 500 mb for the (c) 1-day SVs and (d) 30-day SVs.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The kinetic energy as a function of total wavenumber averaged for the first five KE SVs for weak dissipation. The thin lines represent the individual cases and the thick dashed line represents the six-case average. The KE at initial time at 500 mb for the (a) 1-day SVs and (b) 30-day SVs. The KE at final time at 500 mb for the (c) 1-day SVs and (d) 30-day SVs.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The kinetic energy as a function of total wavenumber averaged for the first five KE SVs for weak dissipation. The thin lines represent the individual cases and the thick dashed line represents the six-case average. The KE at initial time at 500 mb for the (a) 1-day SVs and (b) 30-day SVs. The KE at final time at 500 mb for the (c) 1-day SVs and (d) 30-day SVs.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The percent variance of the first (fastest growing) initial-time 40-day KE SV explained by the (a) first two, (b) first 10, and (c) first 30 initial-time KE SVs for all optimization times for the three weak- and three medium-dissipation forward cases. The solid (dashed) lines are for medium (weak) dissipation.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The percent variance of the first (fastest growing) initial-time 40-day KE SV explained by the (a) first two, (b) first 10, and (c) first 30 initial-time KE SVs for all optimization times for the three weak- and three medium-dissipation forward cases. The solid (dashed) lines are for medium (weak) dissipation.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The percent variance of the first (fastest growing) initial-time 40-day KE SV explained by the (a) first two, (b) first 10, and (c) first 30 initial-time KE SVs for all optimization times for the three weak- and three medium-dissipation forward cases. The solid (dashed) lines are for medium (weak) dissipation.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The percent variance of the first (fastest growing) 40-day final-time KE SV explained by the (a) first two, (b) first 10, and (c) first 30 final-time KE SVs for all optimization times for the three weak- and three medium-dissipation backward cases. The solid (dashed) lines are for medium (weak) dissipation.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The percent variance of the first (fastest growing) 40-day final-time KE SV explained by the (a) first two, (b) first 10, and (c) first 30 final-time KE SVs for all optimization times for the three weak- and three medium-dissipation backward cases. The solid (dashed) lines are for medium (weak) dissipation.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The percent variance of the first (fastest growing) 40-day final-time KE SV explained by the (a) first two, (b) first 10, and (c) first 30 final-time KE SVs for all optimization times for the three weak- and three medium-dissipation backward cases. The solid (dashed) lines are for medium (weak) dissipation.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The percent variance of the first (fastest growing) *n*-day SN SV explained by the first two *n*-day KE SVs at (a) initial time and (b) final time for all optimization times for all six weak- and six medium-dissipation cases. The solid (dashed) lines are for medium (weak) dissipation.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The percent variance of the first (fastest growing) *n*-day SN SV explained by the first two *n*-day KE SVs at (a) initial time and (b) final time for all optimization times for all six weak- and six medium-dissipation cases. The solid (dashed) lines are for medium (weak) dissipation.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The percent variance of the first (fastest growing) *n*-day SN SV explained by the first two *n*-day KE SVs at (a) initial time and (b) final time for all optimization times for all six weak- and six medium-dissipation cases. The solid (dashed) lines are for medium (weak) dissipation.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The percent variance of the first (fastest growing) *n*-day EN SV explained by the first two *n*-day KE SVs at (a) initial time and (b) final time for all optimization times for all six weak- and six medium-dissipation cases. The solid (dashed) lines are for medium (weak) dissipation.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The percent variance of the first (fastest growing) *n*-day EN SV explained by the first two *n*-day KE SVs at (a) initial time and (b) final time for all optimization times for all six weak- and six medium-dissipation cases. The solid (dashed) lines are for medium (weak) dissipation.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The percent variance of the first (fastest growing) *n*-day EN SV explained by the first two *n*-day KE SVs at (a) initial time and (b) final time for all optimization times for all six weak- and six medium-dissipation cases. The solid (dashed) lines are for medium (weak) dissipation.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The growth rates [ln(singular value)/optimization time] of the first (fastest growing) (a) KE SVs, and the first (fastest growing) (b) EN SVs, for all six weak- and medium-dissipation cases. The solid (dashed) lines are for medium (weak) dissipation. Units are day^{−1}.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The growth rates [ln(singular value)/optimization time] of the first (fastest growing) (a) KE SVs, and the first (fastest growing) (b) EN SVs, for all six weak- and medium-dissipation cases. The solid (dashed) lines are for medium (weak) dissipation. Units are day^{−1}.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The growth rates [ln(singular value)/optimization time] of the first (fastest growing) (a) KE SVs, and the first (fastest growing) (b) EN SVs, for all six weak- and medium-dissipation cases. The solid (dashed) lines are for medium (weak) dissipation. Units are day^{−1}.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The percent of total perturbation variance accounted for by the first (a) 1, (b) 10, and (c) 30 KE SVs as a function of optimization time assuming that the initial perturbation is isotropic with respect to all the initial-time SVs. The solid (dashed) lines indicate the medium (weak) dissipation averages. The circles (triangles) indicate the medium (weak) dissipation individual cases.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The percent of total perturbation variance accounted for by the first (a) 1, (b) 10, and (c) 30 KE SVs as a function of optimization time assuming that the initial perturbation is isotropic with respect to all the initial-time SVs. The solid (dashed) lines indicate the medium (weak) dissipation averages. The circles (triangles) indicate the medium (weak) dissipation individual cases.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The percent of total perturbation variance accounted for by the first (a) 1, (b) 10, and (c) 30 KE SVs as a function of optimization time assuming that the initial perturbation is isotropic with respect to all the initial-time SVs. The solid (dashed) lines indicate the medium (weak) dissipation averages. The circles (triangles) indicate the medium (weak) dissipation individual cases.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The percent of the total perturbation variance accounted for by the first *n* KE and EN SVs, where *n* = 1, 30. Different curves represent different optimization times as given in key. EN (KE) SVs are denoted by the thick (thin) curves.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The percent of the total perturbation variance accounted for by the first *n* KE and EN SVs, where *n* = 1, 30. Different curves represent different optimization times as given in key. EN (KE) SVs are denoted by the thick (thin) curves.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2

The percent of the total perturbation variance accounted for by the first *n* KE and EN SVs, where *n* = 1, 30. Different curves represent different optimization times as given in key. EN (KE) SVs are denoted by the thick (thin) curves.

Citation: Monthly Weather Review 127, 10; 10.1175/1520-0493(1999)127<2309:COSVTL>2.0.CO;2