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EVIDENCE OF PLANETARY SCALE, FREE ROSSBY WAVES IN THE NCEP/NCAR REANALYSIS

This study examines the presence of free Rossby waves in the NCEP/NCAR reanalysis data and compares them with theoretical predictions. It explores the theoretical background, searches for Rossby waves in the data set, and analyzes the observed Rossby-gravity waves. The study also investigates the mixed Rossby-gravity waves in the filtered reanalysis data.

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EVIDENCE OF PLANETARY SCALE, FREE ROSSBY WAVES IN THE NCEP/NCAR REANALYSIS

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  1. EVIDENCE OF PLANETARY SCALE, FREE ROSSBY WAVES IN THE NCEP/NCAR REANALYSIS By Roland Madden NCAR, Emeritus

  2. OUTLINE THEORY LEADING TO PREDICTED STRUCTURES AND PERIODS • SEARCH IN NCEP/NCAR REANALYSIS DATA FOR PREDICTED STRUCTURES AND PERIODS - no time filtering • A CLOSER LOOK AT THE OBSERVED ROSSBY-GRAVITY WAVES (l-s = 0) - time filtering

  3. 1) OUTLINE OF THE THEORY

  4. NONDIVERGENT ROSSBY WAVES ON A PLANE (NO BACKGROUND WIND) PHASE SPEED = -ßL²/4π² Rossby et al., 1939: Relations between variations in the intensity of the zonal circulation of the atmosphere and the displacements of the semipermanent centers of action. J. Mar. Res., 2, 38-55.

  5. NONDIVERGENT ROSSBY WAVES ON A SPHERE (NO WINDS) LOCAL PERIOD DAYS s = zonal wave number l = meridional index Haurwitz, B., 1940: The motion of atmospheric disturbances on the spherical earth. J. Mar. Res., 3, 254-267.

  6. Horizontal structures of the stream function of nondivergent Rossby waves on a sphere are Associated Legendre Functions, P s l where: s = zonal wave number l = meridional index

  7. STREAM FUNCTIONS

  8. PERIODS FOR NONDIVERGENTROSSBY WAVES ON A SPHERE (NO WIND) l(l+1) 2s FOR s = 1

  9. DIVERGENT ROSSBY WAVES ON A SPHERE LaPlace Tidal Equation Haurwitz, 1937: Gerl. Beit. Zur Geophys. Hough, 1898: Phil. Trans. Roy. Soc., A. Margules, 1893: Sitz.-Ber. Kaiserl. Akad Wissensch. Wein, IIA. Divergence slows waves

  10. Zonal Wave 1 s (m here) = 1 Westward gravity waves Rossby waves

  11. PERIODS IN DAYS Divergent Rossby Waves on a Sphere – DJF winds (Kasahara, 1980: JAS) s\l-s 0 1 2 3 4 1 1.20 4.85 9.91 18.39 28.08 2 1.71 3.84 7.27 14.23 21.47 3 2.30 4.28 7.40 13.65 xxx 4 2.90 5.21 8.20 13.55 xxx s = zonal wave number l = meridional index

  12. THEORETICAL HORIZONTAL STRUCTURES

  13. 2) SEARCH FOR ROSSBY WAVES IN THE NCEP/NCAR DATA SET (no filtering in time except that seasonal variations are removed)

  14. PREVIOUS OBSERVATIONS Kuboto and Iida, 1954: Pap. Meteorol. Geophys. Deland, 1964: Tellus Eliasen and Machenhauer, 1965; 1969: Tellus And many more

  15. METHOD • 40 YEARS OF TWICE DAILY 300hPa STREAMFUNCTION • LARGE-SCALE ONLY: s = 1, 2, 3, 4; l –s = 0,1, 2, 3, 4, 5 • EMPIRICAL ORTHOGONAL FUNCTIONS (EOFs) OF • EACH ZONAL WAVE NUMBER, s • EOFs COMPARED WITH THEORETICALLY • PREDICTED LATITUDINAL STRUCTURES • CROSS-SPECTRA BETWEEN PRINCIPAL • COMPONENTS OF PAIRS OF EOFs COMPARED • WITH THEORETICALLY PREDICTED PERIODS, • AND CHECKED FOR WESTWARD PROPAGATION

  16. DATA ARE 29220 OBSERVATIONS OF LARGE-SCALESPHERICAL HARMONICS FOR EACH ZONAL WAVE NUMBER, s = 1, 2, 3, 4

  17. COMPUTE A 12 x 12 COVARIANCE MATRIX DETERMINE ITS 12 EIGENVECTORS (EOFs) (“leading” EOFs explain most variance) DETERMINE ITS 12 EIGENVALUES (eigenvalues give amount of variance explained) PROJECT DATA ONTO EOFs GIVING 29220-VALUE TIME SERIES OF PRINCIPAL COMPONENTS

  18. EOFs FOR s = 1

  19. No Zero Crossings Looks Like MRGW

  20. CROSS SPECTRA PCs of EOF pairs for s = 1

  21. EOFs FOR s = 2

  22. CROSS SPECTRA PCs of EOF pairs for s = 2

  23. EOFs FOR s = 3

  24. CROSS SPECTRA PCs of EOF pairs for s = 3

  25. EOFs FOR s = 4

  26. CROSS SPECTRA PCs of EOF pairs for s = 4

  27. EVIDENCE FOR ROSSBY WAVES s\l-s 0 1 2 3 4 1 yes yes yes yes no 2 yes yes yes yes no 3 yes yes yes no xxx 4 no no no no xxx s = zonal wave number l = meridional index

  28. 3) A CLOSER LOOK AT THE MIXED ROSSBY-GRAVITY WAVEIN FILTERED (IN TIME) REANALYSIS DATA

  29. For s=4 s=3 s=2 s=1 And For l–s = 0, MRGW Filter Frequencies from Kasahara (1980) for MRGW

  30. NOW BASED ON FILTERED DATA: *COMPUTE A 12 x 12 COVARIANCE MATRIX *DETERMINE ITS 12 EIGENVECTORS (EOFs) (“leading” EOFs explain most of filtered variance) *DETERMINE ITS 12 EIGENVALUES (eigenvalues give amount of filtered var. explaned) *PROJECT FILTERED DATA ONTO EOFs GIVING 29220-VALUE TIME SERIES OF PRINCIPAL COMPONENTS

  31. 1.2day

  32. 1.7day

  33. 2.3day

  34. 2.9day

  35. 2.9day

  36. EVIDENCE FOR ROSSBY WAVES s\l-s 0 1 2 3 4 1 yes yes yes yes yes 2 yes yes yes yes yes 3 yes yes yes no xxx 4 yes no no no xxx s = zonal wave number l = meridional index

  37. SUMMARY THERE IS CONSIDERABLE EVIDENCE FOR THE EXISTENCE OF THE THEORETICALLY PREDICTED, LARGEST-SCALE, FREE ROSSBY WAVES IN THE REANALYSIS (AND IN EARLIER STUDIES THAT WERE BASED ON OTHER DATA)

  38. WHAT I WANT TO DO RELATE RESULTS HERE TO OTHER OBS. DETERMINE SEASONAL VARIATIONS LEARN HOW WAVES MAYAFFECT NUMERICAL WEATHER FORECASTS LEARN HOW THE WAVES ARE FORCED

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