Vertical Coordinate Issues: Sigma vs Eta or Eta-like

Transcription

1 Vertical Coordinate Issues: Sigma vs Eta or Eta-like Fedor Mesinger 1, Dušan Jović 2, Sin Chan Chou 3, Jorge L. Gomes 3, and Josiane F. Bustamante 3 1) Earth System Science Interdisciplinary Center (ESSIC), University of Maryland, College Park, MD, USA 2) NCEP Environmental Modeling Center (EMC), Camp Springs, MD, USA 3) Center for Weather Prediction and Climate Studies (CPTEC) National Institute for Space Research (INPE) Cachoeira Paulista, SP RAMS/BRAMS/OLAM Users Workshop May 10-12, 2006, Ubatuba, Brazil

2 Vertical coordinate: an unsettled subject! Principles? Pressure-gradient force (PGF), hydrostatic consistency ; Accuracy (formal: Taylor series): No help! Change in layer thickness in horizontal!?...

3 Vertical coordinate: an unsettled subject! Principles? Pressure-gradient force (PGF), hydrostatic consistency ; Accuracy (formal: Taylor series): No help! Change in layer thickness in horizontal!?... As we increase resolution, does the choice of principles matter?

4 Eta: Akio Arakawa: Design schemes so as to emulate as much as possible physically important features of the continuous system! Understand/ solve issues by looking at schemes for the minimal set of terms that describe the problem Help is not expected from increased formal accuracy, nor just from increased resolution The religion of the Eta Model

5 What is being done? (in the Eta) gravity-wave terms scheme on the B/E grid that enables propagation of a height point perturbation to its nearest-neighbor height points; horizontal advection scheme that conserves energy and C-grid enstrophy, on the B/E grid, in space differencing (Janjić 1984); conservation of energy in transformations between the kinetic and potential energy, in space differencing; the eta vertical coordinate, ensuring hydrostatically consistent calculation of the pressure gradient ( second ) term of the pressuregradient force (PGF);.....

6 What is being done? (in the Eta) gravity-wave terms scheme on the B/E grid that enables propagation of a height point perturbation to its nearest-neighbor height points; horizontal advection scheme that conserves energy and C-grid enstrophy, on the B/E grid, in space differencing (Janjić 1984); conservation of energy in transformations between the kinetic and potential energy, in space differencing; the eta vertical coordinate, ensuring hydrostatically consistent calculation of the pressure gradient ( second ) term of the pressuregradient force (PGF);..... split-explicit (economical) time differencing for gravity-inertia waves/ neutral with time steps twice leapfrog; fairly well posed LBCs (no Davies boundary relaxation )

7 Why eta (motivation)?

8 What is the sigma PGF problem? In hydrostatic systems: p φ σ φ RT ln p S The way we calculate things, in models, φ =φ S R d p T v d ln p p S Thus: PGF depends only on variables from the ground up to the considered p=const surface! We could do the same integration from the top; but: we measure the surface pressure, thus, calculation from the top not an option! In nonhydrostatic models: similar

9 Example, continuous case: PGF should depend on, and only on, variables from the ground up to the p=const surface: φ T j-1/2,k-1 φ T j-1/2,k φ v j,k φ T j+1/2,k φ T j+1/2,k+1 φ p = const σ = const p S The best type of sigma scheme: will depend on Tj+1/2,k+1, which it should not; will not depend on Tj-1/2,k-1, which it should. The problem aggravates with resolution! (If the steepness does) p S

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11 Thus, the ( step-topography ) eta:

12 In early tests eta/ sigma, and in those somewhat later in NCEP s full-physics Eta Model, eta did extremely well:

13 Sigma Eta

14 André Robert Memorial Volume: Quite a few more!

15 However, a 10-km Eta in 1997 did a poor job on a case of the so-called Wasatch downslope windstorm, while a sigma system MM5 did well; also: Gallus, Klemp (MWR, 2000) Eta: bad press ever since: ill suited for high resolution prediction models Schär et al., Mon. Wea. Rev., 2002; Janjic, Meteor. Atmos. Phys., 2003; Steppeler et al., Meteor. Atmos. Phys., 2003; Mass et al., Bull. Amer. Meteor. Soc., 2003; Zängl, Mon. Wea. Rev., 2003; more?? Thus, should we go back to sigma?

16 Some of the relatively recent Eta results that suggest the eta is a good choice

17 Eta domain and topography sed for NCEP Reg. Reanalysis: (Domain same as that of the NCEP s operational Eta)

18 Eta vs the Avn/GFS: (1) The Eta is driven for some years now with GFS forecasts initialized 6 h ago. The LB jet stream gets into the CONUS area in ~ two days, even less; In addition: (2) there is the mathematical LB error, e.g., the contamination at the lateral boundaries limits the operational usefulness of the LAM beyond some forecast time range (Laprise et al., MWR 2000, emphasis FM)

19 Eta vs the Avn/GFS: (1) The Eta is driven for some years now with GFS forecasts initialized 6 h ago. The LB jet stream gets into the CONUS area in ~ two days, even less; In addition: (2) there is the mathematical LB error, e.g., the contamination at the lateral boundaries limits the operational usefulness of the LAM beyond some forecast time range (Laprise et al., MWR 2000, emphasis FM) Thus, can one detect the impact of the advection of the LB error?

20 For an answer, I have looked into: precip scores, 24 accumulations, h vs 36 to 84 h, May 2001-April 2002; rms fist to raobs as a function of time;

21 Eq. threat scores Avn Eta 00-24, 12-36, h

22 Eq. threat scores 36-60, 48-72, h Relative QPF skill, Eta vs GFS, stayed about the same!

23 RMS fits to raobs: upper tropospheric winds presumably ~ the best indicator of the largest scales (jet stream!)

24 Note: done on diff. resolution grids! Warm Season Eta GFS

25 Cold Season Eta GFS

26 The Eta in relative terms improves a little with time!

27 No relative loss of skill, Eta vs GFS at extended forecast times, identified! Reason?

28 How? One can argue that a major contributor to the Eta strength at extended forecast times is the eta coordinate Relative to the GFS the Eta does best in winter (when jet stream is at its southernmost latitudes); Considerable benefit from its large domain

29 The Early vs the Meso Eta Early : 48 km, 12 h old Avn LBCs, Meso : 29 km, current Avn LBCs; Domains:

30 2 years of scores: Scores of the the Early and the Meso Eta about the same! The benefit of the large domain compensates the combined benefit of more accurate LBCs and higher resolution!!

31 The only way a model can benefit from a large domain is if it is doing well the largest scales it can accommodate!? (jet stream!) But if I am correct re improvement of the largest scales, how could one verify this? Position forecast errors of major lows at 60-h time!?

32 Major lows : On consecutive HPC analyses, at 12 h intervals, in the first verification, i) the analyzed center has to be the deepest inside at least three closed isobars (analyzed at 4 mb intervals). A closed isobar is here one that has all of the isobars inside of it, if any, appear only once; ii) must not have an L analyzed between the 1st and the 2nd of its closed isobars, counting from the inside; iii) has to be located east of the Continental Divide, over land or inland waters (e.g., Great Lakes, James Bay); and iv) must be stamped on four-pane 60-h forecast plots of both the Eta and the Avn. In the second verification, Same, except that at least two closed isobars are required

33 Done manually (NCEP HPC analyses used for verification, hand-edited, at 12 h intervals, not available electronically)

34 Table 1. Forecast position errors, at 60 h, of "major lows, east of the Rockies and over land or inland waters, Dec Feb Valid at HPC depth Cl. isb. Ctr. Avn error Eta error 12z 7 Dec mb 3 SD 875 km 00z 12 Dec. 997 mb 4 In 125 km 12z 12 Dec. 988 mb 7 NY 325 km 12z 17 Dec mb 4 Sk 100 km 425 km 275 km 150 km 75 km 12z 17 Dec. 990 mb 7 On 175 km 425 km 00z 18 Dec. 984 mb 7 Qc 450 km 575 km 12z 18 Dec. 963 mb 11 Qc 75 km 100 km 00z 18 Dec mb 3 Co 100 km 25 km 02z 18 Dec mb 2 Mo 650 km 500 km 12z 19 Dec mb 3 Ab 425 km 175 km 00z 20 Dec. 997 mb 5 Sk 250 km 350 km 12z 20 Dec mb 2 ND 175 km 175 km etc. 12z 21 Dec mb 3 Mi 100 km 00z 22 Dec mb 3 Mi 100 km 12z 22 Dec mb 2 On 125 km 12z 24 Dec mb 3 On 325 km 175 km 50 km 375 km 150 km

35 Summary Winter #1 ( ): 41 cases, 18 events; Average errors: Avn 319 km, Eta 259 km Median errors: Avn 275 km, Eta 275 km # of wins: Eta 25, Avn 15, 1 tie Winter #2 ( ): 38 cases, 16 events; Average errors: Avn 330 km, Eta 324 km Median errors: Avn km, Eta 250 km # of wins: Eta 19, Avn 17, 2 ties Eta somewhat more accurate both winters, in spite of this being at 2.5 days lead time, plenty in winter for the western boundary error to make it into the contiguous U.S.!

36 Eta 3D-Var vs Eta GFS interpolated IC an 8 months parallel, wind rms at 48 h

37 There is also one experimental result, eta/ sigma, done in 2001:

38 Eta (left), 22 km, switched to use sigma (center), 48 h position error of a major low increased from 215 to 315 km ~ Just as in earlier experiments at lower resolution

39 However: the downslope windstorm problem; also: Statements made claiming that sigma is better than eta in placing precip over topography; (Expected impact, as NOAA-wide announced in 2002, of the NCEP NMM s switch from eta to sigma)

40 Any relevant results since?

41 The eta downslope windstorm problem : Flow separation on the lee side (à la Gallus and Klemp 2000)

42 Suggested explanation of the downstorm windstorm problem : v T1 v T2 v T3 v Flow attempting to move from box 1 to 5 is forced to enter box 2 first. p S v T5 T 6 p S v v T 9 p S v v Missing: slantwise flow directly from box 1 into 5! As a result: some of the air which should have moved slantwise from box 1 directly into 5 gets deflected horizontally into box 3.

43 Refined (sloping steps) eta (Mesinger and Jović) : Discretization accounting for slopes. Approach: Define slopes at v points, based on four surrounding h points. Slopes discrete, going down one layer thickness from one to the other neighboring h point. Slantwise transports calculated as appropriate. Discretized version of Adcroft et al. shaved cells?

44 The sloping steps, vertical grid The central v box exchanges momentum, on its right side, with v boxes of two layers:

45 Horizontal treatment, 3D Example #1: topography of box 1 is higher than those of 2, 3, and 4; Slope 1 Inside the central v box, topography descends from the center of T1 box down by one layer thickness, linearly, to the centers of T2, T3 and T4

46 Example #2: topographies of boxes 1 and 2 are the same, and higher than those of 3, and 4; Slope 2 Topography descends from the centers of T1 and T2 down by one layer thickness, linearly, to the centers of T3 and T4 Etc.: Slopes 3, 4,, 8 If two opposite, or if three topography boxes are the highest of the four: No slope

47 Slantwise advection of mass, momentum, and temperature, and ωα : Velocity at the ground immediately behind the mountain increased from between 1 and 2, to between 4 and 5 m/s. lee-slope separation as in Gallus and Klemp ~ removed. Zig-zag features in isentropes at the upslope side removed.

48 Example of slopes with an actual model topography:

49 Sloping steps Eta: runs operationally at least at one place Other possibilities. Other models using or having an option of using quasi-horizontal coordinates: Univ. of Wisconsin (G. Tripoli); RAMS (R. Walko); DWD Lokal Modell (LM); MIT, Marshall et al. (MWR 2004); NASA GISS (NY), G. Russell, MWR (submitted)

50 Precip: Three-model precipitation scores, on NMM ConUS domains ("East",, "West"), available since Sep Operational Eta: 12 km, driven by 6 h old GFS forecasts (a considerable handicap compared to GFS of the same initial time); NMM: 8 km, sigma, driven by the Eta; GFS (Global Forecasting System) as of the end of Oct T254 (55 km) resolution, sigma

51 East West Eta

52 The first 12 months of three model scores: East ETS (Equitable Threat Score) Bias Is the GFS loosing (winning) because of its bias difference?

53 What can one do?

54 J th Prob. Stat. Atmos. Sci.; 20th WAF/16th NWP (Seattle AMS, Jan. 04) BIAS NORMALIZED PRECIPITATION SCORES Fedor Mesinger 1 and Keith Brill 2 1 NCEP/EMC and UCAR, Camp Springs, MD 2 NCEP/HPC, Camp Springs, MD

55 Method 1, assumption: dh df = a (O H ), a = const, can be solved;

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57 Bias adjusted eq. threats Eta NMM GFS East

58 (Five very heavy el Niño precip events) West Eta GFS NMM

59 An example of precip at one of these events: (8 Nov. 2002, red contours: 3 in/24 h) An extraordinary challenge to do well in QPF sense!

60 The last 12 months : Feb Jan (includes high impact California precip, winter )

61 GFS East NMM Eta

62 West NMM GFS Eta

63 Thus, NCEP s QPFs: strong indication that (even) the step-topography eta works better than sigma!? Should we not trust these QPFs? If so, why?

64 Two systems at present being tested at NCEP: NMM (NCEP WRF), using a new GSI data assimilation system; Operational Eta, using the Eta 3D-Var

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66 East ETS Bias

67 West ETS Bias

68 A fortuitous by-product of quasi-horizontal coordinates: Model grid-boxes have approximately the same mass in horizontal; Vertical sides of model grid boxes in horizontal are about the same! Why this might be favorable? A flux-type model such as the Eta becomes approximately a finite-volume model

69 Numerous tests failed to demonstrate any benefit from high formal (Taylor-series) accuracy: In very large sample tests the 2nd order Eta did significantly better than 4th order accurate NGM; Infinite order, in horizontal, NCEP s Regional Spectral Model (RSM, e.g., Juang et al. BAMS 1997) Cullen et al. (Robert Mem. Vol. 1997): Gary Russell (MWR 2006, submitted); unpublished: RAMS (?)

70 Eta vs the RSM: 2 years of scores, , at ~50 km resolution: RSM Eta Note: Eta is using 12 h old Avn LBCs, RSM is using current Avn LBCs

71 Why?

72 Hypothesis: We are facing an inconsistency in our doing physics (boxes!) and dynamics (points)? Physics forcing of boxes produces box-to-box noise. This works against the way we traditionally do dynamics (Taylor series expansion), assuming smoothness, and values valid at points

73 However: Arakawa-style dynamics features (many in the Eta) are enforced on boxes; inconsistency less of a role! With conservations achieved by communication between neighboring grid boxes: approximately same masses of grid boxes in horizontal seems desirable/ may have helped the Eta do well!?

74 Manuscript by Gary Russell, of NASA GISS (NY): Original version: Allowing the number of layers in a column to vary improves the vertical mass fluxes, the flow around and through mountains, and the precipitation distribution spectacularly Revised version: Allowing the number of layers in a column to vary improves the vertical mass fluxes, the flow around and through mountains, and the precipitation distribution, especially so in the Andes

75 Recent effort at CPTEC: Evaluate the impact of this sloping steps eta discretization in two major wind related phenomena near the Andes: South American low-level jet (LLJ); Downslope zonda wind in the lee of the Andes

76 Model Domain and Topography (m)

77 LLJ Case, Initial condition: 1200 UTC 15 Jan Left: sloping steps eta fcst; Right: sloping steps - traditional discr.

78 Zonda Case: Synop Observations

79 Zonda Case: Synop Observations Z Z Z Z

80 Zonda Case: Satellites Images Infrared Water vapor Z

81 Zonda Case, Initial condition: 1200 UTC 15 May 2005 Left: traditional eta fcst; Right: sloping steps - traditional discr.

82 Zonda Case, Initial condition: 1200 UTC 15 May 2005 Relative humidity (%), Equiv. potential temp. (deg) Eta Sloping steps eta

83 Conclusions, 1 LLJ case: Seems well done: associated with pronounced downward motion along the lee slopes; Downward motion considerably stronger in the lee of the main barrier due to sloping steps (up to ~20%), appears welcome;

84 Conclusions, 1 LLJ case: Seems well done: associated with pronounced downward motion along the lee slopes; Downward motion considerably stronger in the lee of the main barrier due to sloping steps (up to ~20%), appears welcome; Zonda case: Seems to have been done quite satisfactory with standard step-topography discr. (Seluchi et al. WF 2003); Downward motion still stronger all along the lee of the main barrier with sloping steps (up to ~20%); appears welcome;

85 Conclusions, 2 appears welcome : Based on two Eta real data cases of downslope windstorms (Wasatch, Santa Ana), Gallus-Klemp experiment, and its explanation shown here; Efforts at verification against actual data planned;

86 Conclusions, 2 appears welcome : Based on two Eta real data cases of downslope windstorms (Wasatch, Santa Ana), Gallus-Klemp experiment, and its explanation shown here; Efforts at verification against actual data planned; Should be recalled: Downstorm windstorms rare; The operational Eta QPF performance over the U.S. West remains consistently better than that of NCEP s operational sigma system models

87 Concluding comments: Finite volume: not removing the sigma PGF problem; Dynamical core issues matter!! Very little work, in US, a few percent? Thorpex: ~ zero? Communication between communities working on/with different models/ dynamical cores!!!

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