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Roberto Buizza 1 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors TC/PR/RB Lecture 2 – Introduction to singular vectors Roberto Buizza European Centre for Medium-Range Weather Forecasts

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Roberto Buizza 2 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Outline Definition of singular vectors: brief summary. Singular vector characteristics: –geographical location; –vertical structure; –spectra. Metrics and singular vectors.

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Roberto Buizza 3 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Total-energy metric (and norm) definition Given two state-vectors x and y expressed in terms of vorticity, divergence D, temperature T, specific humidity q and surface pressure, the total energy metric (and the associated norms) is defined ( is the Euclidean inner product) as:

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Roberto Buizza 4 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors The adjoint operator Given any two vectors x and y, the adjoint operator L * of the linear operator L with respect to the Euclidean norm is the operator that satisfies the following property: Using the adjojnt operator L * the time-t E-norm of z can be written as:

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Roberto Buizza 5 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Singular vector definition Consider an N-dimensional system: Denote by z a small perturbation around a time-evolving trajectory z: The time evolution of the small perturbation z is described to a good degree of approximation by the linearized system A l (z) defined by the trajectory. Note that the trajectory is not constant in time.

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Roberto Buizza 6 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Singular vector definition The solution of the linearized system can be written in terms of the linear propagator L(t,0): The linear propagator is defined by the system equations and depends on the trajectory characteristics. The E-norm of the perturbation at time t is given by:

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Roberto Buizza 7 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Singular vector definition The computation of the directions of maximum growth can be stated as finding the directions in the phase-space of the system characterized by the maximum ratio between the time-t and the initial norms: The problem reduces to solving the following eigenvalue problem:

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Roberto Buizza 8 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors SVs geographical distributions and Eady index The geographical distribution of the singular vectors reflect the characteristics of the underlying basic-state flow. A measure of the baroclinic instability of the basic- state flow is given by the Eady index: which is the growth rate of the most unstable Eady mode (Hoskins & Valdes 1990). In this equation, the static stability N and the vertical wind shear can be estimated using the 300- and 1000-hPa potential temperature and wind. Results indicate that locations with maximum singular vector concentration coincide with regions with maximum Eady index.

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Roberto Buizza 9 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Example 1: singular vectors for Jan 1997 This figure shows the amplification rate (i.e. the singular value) of the leading 30 unstable singular vectors growing between 18 and 20 January The SVs were computed at the resolution T42L31 and were used to generate the EPS initial conditions.

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Roberto Buizza 10 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Ex 1: singular vector 1 for Jan 1997 This figure shows the most unstable singular vector growing between 18 and 20 Jan Left (right) panels show the SV at initial and final (i.e. +48h) time. The top panels show the SV T at model level 18 (~500hPa, shading) and the Z500 analysis; the bottom panels the SV T at model level 23 (~700hPa, shading). The contour interval is 8dam for Z, and 0.01 (0.05) deg for T at initial (final) time (the SV is normalized to have unit total energy norm at initial time). T=0T=+48h 500hPa 700hPa

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Roberto Buizza 11 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Ex 1: vertical cross section of SV 1 for Jan 1997 This figure shows, for SV 1, the vertical cross section of the T component at initial time (top, for 36N) and of the vorticity component at final time (bottom, for 44N). The two cross sections have been taken along the parallel where the SV had maximum amplitude. Note the strong initial tilt, suggesting baroclinic instability, and the final time more barotropic-type structure. Note that T is shown at initial time and vor at final time because the initial time SV has a strong potential energy part.

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Roberto Buizza 12 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Ex 1: energy distribution of SV 1 for Jan 1997 The top figure shows, for SV 1, the vertical distribution at initial time of the kinetic (red dotted, x100) and total (red solid, x100) energy, and the corresponding final time distributions (blue). The bottom figure shows the total energy spectrum at initial (red solid, x100) and at final time (blue solid). Note the upward and upscale energy transfer/growth, and the transformation from initial potential to mainly final kinetic energy.

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Roberto Buizza 13 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Ex 1: singular vector 4 for Jan 1997 This figure shows the 4th SV growing between 18 and 20 Jan Left (right) panels show the SV at initial (final, +48h) time; top (bottom) panels show the SV T at model level 18 (23) and Z500 (Z700) analysis. The contour interval is 8dam for Z, and 0.01 (0.05) deg for T at initial (final) time (the SV is normalized to have unit total energy norm at initial time). T=0T=+48h

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Roberto Buizza 14 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Ex 1: vertical cross section of SV 4 for Jan 1997 This figure shows, for SV 4, the vertical cross section of the T component at initial time (top, for 44N) and of the vorticity component at final time (bottom, for 56N). The two cross sections have been taken along the parallel where the SV had maximum amplitude. Note the strong initial tilt, suggesting baroclinic instability, and the final time more barotropic- type structure. Note that T is shown at initial time and vor at final time because the initial time SV has a strong potential energy part.

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Roberto Buizza 15 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Ex 1: energy distribution of SV 4 for Jan 1997 The top figure shows, for SV 4, the vertical distribution at initial time of the kinetic (red dotted, x100) and total (red solid, x100) energy, and the corresponding final time distributions (blue). The bottom figure shows the total energy spectrum at initial (red solid, x100) and at final time (blue solid). By contrast to SV 1, note that this SV is characterized by the superposition of two different structures, growing at different levels and with different characteristic lengths.

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Roberto Buizza 16 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Ex 1: average energy distribution for Jan 1997 The top figure shows the SV1:25 average vertical distribution at initial time of the kinetic (red dotted, x100) and total (red solid, x100) energy, and the corresponding final time distributions (blue). The bottom figure shows the SV1:25 average total energy spectrum at initial (red solid, x100) and at final time (blue solid). Note the SV typical upward and upscale energy transfer/growth, and the transformation from initial potential to mainly final kinetic energy.

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Roberto Buizza 17 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Ex 1: SVs and Eady index for Jan 1997 The top panel shows the t+24h average root-mean-square (rms) amplitude (in terms of Z500) of the first 25 singular vectors growing between 18 and 20 January The bottom panel shows the January 1997 average Eady index. The contour isolines are 0.5dam for the SVs rms amplitude and 0.5d -1 for the Eady index. Results indicate a good correspondence between areas of SV concentration and of maximum value of the Eady index.

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Roberto Buizza 18 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Ex 2: NH SVs and Eady index - JFM 1997 and 1998 The top panels show the average t+24h root-mean-square amplitude (in terms of Z500, ci=0.3dam) of the first 25 singular vectors during JFM 1997 (left) and 1998 (right) over the NH. The bottom panels show the average Eady index computed between 1000 and 300 hPa (ci=0.2d -1 ). Results indicate a good agreement between areas of large Eady index and high SV concentration.

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Roberto Buizza 19 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Ex 2: SH SVs and Eady index - JFM 1997 and 1998 The top panels show the average t+24h root-mean-square amplitude (in terms of Z500, ci=0.3dam) of the first 25 singular vectors during JFM 1997 (left) and 1998 (right) over the SH. The bottom panels show the average Eady index computed between 1000 and 300 hPa (ci=0.2d -1 ). Results indicate a good agreement between areas of large Eady index and high SV concentration.

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Roberto Buizza 20 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Ex 3: NH SVs & Eady index - JJA 1997 & 1998 The top panels show the average t+24h root-mean-square amplitude (in terms of Z500, ci=0.3dam) of the first 25 singular vectors during JJA 1997 (left) and 1998 (right) over the NH. The bottom panels show the average Eady index computed between 1000 and 300 hPa (ci=0.2d -1 ). Results confirm a good agreement between areas of large Eady index and high SV concentration.

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Roberto Buizza 21 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Ex 3: SH SVs & Eady index - JJA 1997 & 1998 The top panels show the average t+24h root-mean-square amplitude (in terms of Z500, ci=0.3dam) of the first 25 singular vectors during JJA 1997 (left) and 1998 (right) over the SH. The bottom panels show the average Eady index computed between 1000 and 300 hPa (ci=0.2d -1 ). Results confirm a good agreement between areas of large Eady index and high SV concentration.

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Roberto Buizza 22 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Ex 3: NH SVs & Eady index - JFM 2000 & 2001 The top panels show the average t+24h root-mean-square amplitude (in terms of Z500, ci=0.3dam) of the first 25 singular vectors during JFM 2000 (left) and 2001 (right) over the NH. The bottom panels show the average Eady index computed between 1000 and 300 hPa (ci=0.2d -1 ). Results confirm a good agreement between areas of large Eady index and high SV concentration.

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Roberto Buizza 23 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Ex 3: SH SVs & Eady index - JFM 2000 & 2001 The top panels show the average t+24h root-mean-square amplitude (in terms of Z500, ci=0.3dam) of the first 25 singular vectors during JFM 2000 (left) and 2001 (right) over the SH. The bottom panels show the average Eady index computed between 1000 and 300 hPa (ci=0.2d -1 ). Results confirm a good agreement between areas of large Eady index and high SV concentration.

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Roberto Buizza 24 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Impact of horizontal resolution on SVs This figure shows the time evolution of the average (10 SVs for 8 cases) total energy vertical distribution from t=0 (x30, red dotted) to t=48h (x3, red solid) every 12h for T21 (top right), T42 (bottom left) and T63 (bottom right) SVs (values have been scaled differently at each forecast step).

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Roberto Buizza 25 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Impact of horizontal resolution on SVs This figure shows the time evolution of the total energy average (10 SVs for 8 cases) spectrum from t=0 (x30, red dotted) to t=48h (x3, red solid) every 12h for T21 (top right), T42 (bottom left) and T63 (bottom right) SVs (values have been scaled differently at each forecast step).

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Roberto Buizza 26 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Seasonal variation of the NH SVs amplification rate Singular vectors location and amplification rate depend on the atmospheric flow. This figure shows the average of the amplification rate (i.e. singular values) of the SV1:3 localized in different NH regions for winter 98/99 (D98-JF99).

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Roberto Buizza 27 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Seasonal variation of the NH SVs amplification rate This figure shows the mean of the average of the amplification rate of SV1:3 for 6 different NH regions for winter 1997/1998 (top) and 1998/1999 (bottom). During winter the amplification rates are larger over the the Pacific and the Atlantic regions. Comparing the two winters, results indicate that the amplification rates were larger during winter 1998/99 especially over the Pacific and the Atlantic regions.

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Roberto Buizza 28 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Seasonal variation of the NH SVs amplification rate This figure shows the mean of the average of the amplification rate of SV1:3 for 6 different NH regions for summer 1998 (top) and 1999 (bottom). During summer the amplification rates are not so strongly depend on the geographical region. Compared to winter, summer amplification rates are smaller and less daily variable (please note that SVs during these seasons were computed with a dry tangent model).

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Roberto Buizza 29 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Seasonal variation of the NH SVs amplification rate This figure shows the seasonal average amplification rate of SV1:3 for 4 summer and 4 winter seasons.

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Roberto Buizza 30 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Seasonal variation of the NH SVs amplification rate This figure shows the seasonal variation (with respect to the seasonal average) of the amplification rate of SV1:3 for 4 summer and 4 winter seasons.

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Roberto Buizza 31 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Seasonal variation of the NH SVs amplification rate Focussing over the NH and Europe, results indicate that summer amplification rates were largest during 2000 for both areas, and that winter amplification rates were largest during during 1999/2000 for Europe but during 1998/1999 for the NH.

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Roberto Buizza 32 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors SVs can be used to identify unstable regions The average (or a weighted average) of the geographical distribution of the singular vectors total energy can be used to identify unstable regions. 1 Jan Jan 30 Jan 20 Jan

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Roberto Buizza 33 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Impact of the initial and final metrics on the SVs The computation of the leading singular vectors reduces to solving the following eigenvalue problem: It is evident from this equation that singular vectors depend on the choice of the initial and final time metrics E 0 and E.

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Roberto Buizza 34 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Impact of the initial and final metrics on the SVs Singular vectors computed with four different initial time norms are compared:

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Roberto Buizza 35 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Impact of the initial and final metrics on the SVs The total energy norm has been used at final time. Furthermore, singular vectors have been computed to maximize the final time norm inside the region A={30:80N;-30:10E}. SVs have been computed at T42L19 resolution, with a 48-h optimization time interval, with starting date 5 December 1994.

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Roberto Buizza 36 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Impact of the initial and final metrics on the SVs This figure shows the streamfunction of the dominant SV (T42L19, 48h optimization time interval) at initial time on a model level near 500hPa computed with different metrics: TE-TE (top left), KE- TE (top right), VO-TE (bottom left) and PS-TE (bottom right). VO-TE KE-TE PS-TE TE-TE

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Roberto Buizza 37 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Impact of the initial and final metrics on the SVs This figure shows the streamfunction of the dominant SV (T42L19, 48h optimization time interval) at final time on a model level near 500hPa computed with different metrics: TE-TE (top left), KE-TE (top right), VO- TE (bottom left) and PS-TE (bottom right). The contour interval is 20 times larger than at initial time. TE-TE KE-TE PS-TEVO-TE

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Roberto Buizza 38 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Impact of the initial and final metrics on the SVs Average total energy spectrum of SV1:16 (as a function of the 2D total wave-number) for SVs computed with diferent metrics:TE- TE (top), VO-TE (middle) and PS-TE (bottom). The blue lines show the spectra at initial time and the red line at final time scaled by 100 for TE-TE, 1 for VO-TE and 10 for PS-TE. TE-TE VO-TE PS-TE

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Roberto Buizza 39 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Impact of the initial and final metrics on the SVs Average vertical distribution of total energy of SV1:16 for SVs computed with diferent metrics: TE-TE (top), VO-TE (middle) and PS-TE (bottom). The blue lines show the spectra at initial time and the red line at final time scaled by 100 for TE-TE, 1 for VO-TE and 10 for PS-TE. TE-TE VO-TE PS-TE

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Roberto Buizza 40 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Impact of the initial and final metrics on the SVs These results show that there is a strong dependence of singular vectors on the choice of the metric. More precisely, the initial structure of T42L19 SVs of a primitive equation model (without moist processes) are very sensitivity to the initial-time metric. The comparison of SVs computed with different final-time metrics indicate that these SVs are less sensitive to changes in the final time norm (not shown here). The reader is referred to Palmer et al (1998) and to Barkmeijer et al ( ) for a more detailed discussion of the sensitivity of singular vectors to the choice of the metric.

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Roberto Buizza 41 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors The effect of spatial scale on SV structures To investigate the scale dependence of singular vector growth, singular vectors have been computed applying a filter at initial and/or at final time (Hartmann et al, 1995). Denote by s(n,m) the following function: where n is the total wave-number, m is the zonal wave and N identifies a region of the spectral space. Denote by Sx the multiplication of each spectral component of the vector x by the function s(n,m):

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Roberto Buizza 42 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors The effect of spatial scale on SV structures Singular vectors with chosen initial and final time spectral characteristics can be computed by applying the spectral filter S at initial and/or final time: This leads to the SV generalized eigenvector problem:

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Roberto Buizza 43 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Impact of the initial and final metrics on the SVs This figure show the total energy spectra at initial (dashed, x100 in the top 6 panels and x30 in the bottom 3 panels) and final (solid) time of T21L19 singular vectors with imposed (via S 0 and S T ) different initial and final time spectral characteristics.

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Roberto Buizza 44 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Impact of the initial and final metrics on the SVs Hartmann et al (1995) results indicate that: SVs linear growth is most rapid for the smallest scales. Disturbances (i.e. SVs) composed of sub-synoptic-scale wavenumbers may very rapidly grow to synoptic-scale waves. The final-time SV structure is insensitive to the the presence or absence of initial-time large scales. This indicates that information for the final-time synoptic-scale structure is contained in the sub-synoptic scales. SVs constrained to be of large scale at initial time grow much more slowly than unconstrained, smalelr-scale perturbations.

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Roberto Buizza 45 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Conclusions Singular vector characteristics have been discussed: –SVs are localized in areas of strong barotropic and baroclinic instability; –average SV total energy maps can be used to identify unstable regions; –SV growth is characterized by an upscale and upward energy transfer, with initial- time potential energy converted during time evolution into kinetic energy. Seasonal variability of SVs characteristics: –winter is more unstable than summer (but note that currently SVs have been computed without moist processes in the tangent model), and during winter the Pacific and the Atlantic sectors are on average more unstable than other regions. The choice of the initial time metric has a strong effect on the SVs. Small-scale disturbances grow faster than large-scale disturbances, and affect large scales at optimization time.

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Roberto Buizza 46 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors Bibliography On normal modes and baroclinic instability: –Hoskins, B. J., & Valdes, P. J., 1990: On the existence of storm tracks. J. Atmos. Sci., 46, –Orr, W, 1907: Stability and instability of the steady motions of a perfect fluid. Proc. Roy. Irish Acad., On singular vector characteristics: –Borges, M., & Hartmann, D. L., 1992: Barotropic instability and optimal perturbations of observed non-zonal flows. J. Atmos. Sci., 49, –Barkmeijer, J, Van Gijzen, J., & Bouttier, F., 1998: Singular vectors and estimates of the analysis covariance metric. Q. J. R. Meteorol. Soc., 124, –Barkmeijer, J, Buizza, R., & Palmer, T. N., 1999: 3D-Var Hessian singular vectors and their potential use in the ECMWF Ensemble Prediction System. Q. J. R. Meteorol. Soc., 125, –Buizza, R., 1998: Impact of horizontal diffusion on T21, T42 and T63 singular vectors. J. Atmos. Sci., 55, –Buizza, R., & Palmer, T. N., 1995: The singular vector structure of the atmospheric general circulation. J. Atmos. Sci., 52, –Buizza, R., Tribbia, J., Molteni, F., & Palmer, T. N., 1993: Computation of optimal unstable structures for a numerical weather prediction model. Tellus, 45A, –Farrell, B. F., 1982: The initial growth of disturbances in a baroclinic flow. J. Atmos. Sci., 39, –Farrell, B. F., 1989: Optimal excitation of baroclinic waves. J. Atmos. Sci., 46, –Hartmann, D. L., Buizza, R., & Palmer, T. N., 1995: Singular vectors: the effect of spatial scale on linear growth on disturbances. J. Atmos. Sci., 52, –Hoskins, B., Buizza, R., & Badger, J., 2000: The nature of singular vcetor growth and structure. Q. J. R. Meteorol. Soc., 126, –Molteni, F., & Palmer, T. N., 1993: Predictability and finite-time instability of the northern winter circulation. Q. J. R. Meteorol. Soc., 119, –Palmer, T. N., Gelaro, R., Barkmeijer, J., & Buizza, R., 1998: Singular vectors, metrics, and adaptive observations. J. Atmos. Sci., 55,

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