1. TC/PR/RB Lecture 2 – Introduction to singular vectors
Roberto Buizza
European Centre for Medium-Range Weather Forecasts

2. Outline Definition of singular vectors: brief summary.
Singular vector characteristics:
geographical location;
vertical structure;
spectra.
Metrics and singular vectors.

3. 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:

4. 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:

5. 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 Al(z) defined by the trajectory. Note that the trajectory is not constant in time.

6. 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:

7. 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:

8. 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.

9. Example 1: singular vectors for 18-20 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.

10. Ex 1: singular vector 1 for 18-20 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 (0.05) deg for T at initial (final) time (the SV is normalized to have unit total energy norm at initial time).
T=0
T=+48h
500hPa
700hPa

11. Ex 1: vertical cross section of SV 1 for 18-20 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.

12. Ex 1: energy distribution of SV 1 for 18-20 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.

13. Ex 1: singular vector 4 for 18-20 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 (0.05) deg for T at initial (final) time (the SV is normalized to have unit total energy norm at initial time).
T=0
T=+48h

14. Ex 1: vertical cross section of SV 4 for 18-20 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.

15. Ex 1: energy distribution of SV 4 for 18-20 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.

16. Ex 1: average energy distribution for 18-20 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.

17. Ex 1: SVs’ and Eady index for 18-20 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 average Eady index. The contour isolines are 0.5dam for the SV’s 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.

18. 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.

19. 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.

20. 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 (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.

21. 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 (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.

22. 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 (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.

23. 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 (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.

24. 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).

25. 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).

26. 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).

27. 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.

28. 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 (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).

29. 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.

30. 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.

31. 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.

32. SVs can be used to identify unstable regions
1 Jan 97
10 Jan
30 Jan
20 Jan
The average (or a weighted average) of the geographical distribution of the singular vectors’ total energy can be used to identify unstable regions.

33. 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 E0 and E.

34. Impact of the initial and final metrics on the SVs
Singular vectors computed with four different initial time norms are compared:

35. 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.

36. 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

37. 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-TE
VO-TE

38. Impact of the initial and final metrics on the SVs
TE-TE
VO-TE
PS-TE
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.

39. Impact of the initial and final metrics on the SVs
TE-TE
VO-TE
PS-TE
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.

40. 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.

41. 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):

42. 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:

43. 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 S0 and ST) different initial and final time spectral characteristics.

44. 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.

45. 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.

46. 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,