Vertical Coordinate Issues:Sigma vs Eta or Eta-like

Fedor Mesinger1, Dušan Jović2,Sin Chan Chou3,

Jorge L. Gomes3, and Josiane F. Bustamante3

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 WorkshopMay 10-12, 2006, Ubatuba, Brazil

Vertical coordinate: an unsettled subject !

Principles ?

• Pressure-gradient force (PGF), “hydrostatic consistency”;

• Accuracy (formal: Taylor series): No help!

• Change in layer thickness in horizontal !?

• . . .

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?

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

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 pressure-gradient force (PGF);

• . . . . .

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 pressure-gradient 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”)

Why eta (motivation) ?

What is the sigma PGF problem?In hydrostatic systems:

−∇ pφ → −∇σ φ − RT ∇ ln pS

The way we calculate things, in models,

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 !

φ =φS − Rd TvpS

p

∫ d ln p

In nonhydrostatic models: similar

The best type of sigma scheme:

will depend on , 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)

Example, continuous case:PGF should depend on,

and only on,variables from the ground

up to the p=const surface:

Tj+1/2,k+1

pS

pS

vj,k

T j-1/2 ,k

T j+1/2 ,k

T j-1/2 ,k-1

T j+1/2 ,k+1

•••

p = const

φ

φ

φ φ

φ

φ

σ = const

•••

Thus, the (“step-topography”) eta:

In early tests eta/ sigma, and in those somewhat later in NCEP’s

full-physics “Eta Model“, eta did extremely well:

Sigma Eta

André RobertMemorial Volume:

Quite a few more !

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?

Some of the relatively recent Eta results that suggest

the eta is a good choice

Eta domain and topographyused for NCEP Reg. Reanalysis:

(Domain same as that of the NCEP’s operational Eta)

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)

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?

For an answer, I have looked into:

• precip scores, 24 accumulations, 00-48 h vs 36 to 84 h,May 2001-April 2002;

• rms fist to raobs as a function of time;

00-24, 12-36, 24-48 h

Eta

Avn

Eq. threat scores

36-60, 48-72, 60-84 h

Relative QPF skill, Eta vs GFS, stayed about the same !

Eq. threat scores

RMS fits to raobs:upper tropospheric winds presumably ~ the best indicator

of the largest scales (jet stream !)

“Warm Season”

Eta GFS

Note: done on diff. resolution grids !

“Cold Season”

Eta

GFS

The Eta in relative terms improves a little with time !

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

Reason ?

One can argue that a major contributor to the Eta strength at extended forecast times is

the eta coordinate

How?

• Relative to the GFS the Eta does best in winter(when jet stream is at its southernmost latitudes);

• Considerable benefit from its large domain

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

Domains:

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 !!

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 !?

“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 threeclosed isobars (analyzed at 4 mb intervals). A “closed isobar” is here onethat 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 closedisobars, counting from the inside;

iii) has to be located east of the Continental Divide, over land or inlandwaters (e.g., Great Lakes, James Bay); and

iv) must be stamped on “four-pane” 60-h forecast plots of both the Eta andthe Avn.

In the second verification,

Same, except that at least two closed isobars are required

Done manually

(NCEP HPC analyses used for verification,

hand-edited, at 12 h intervals, not available electronically)

Table 1. Forecast position errors, at 60 h, of "major lows”, east of the Rockies and over land or inland waters, Dec. 2000 - Feb. 2001

——————————————————————————————————————–

Valid at HPC depth Cl. isb. Ctr. Avn error Eta error

12z 7 Dec. 1002 mb 3 SD 875 km 425 km

00z 12 Dec. 997 mb 4 In 125 km 275 km

12z 12 Dec. 988 mb 7 NY 325 km 150 km

12z 17 Dec. 1001 mb 4 Sk 100 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. 1001 mb 3 Co 100 km 25 km

02z 18 Dec. 1010 mb 2 Mo 650 km 500 km

12z 19 Dec. 1006 mb 3 Ab 425 km 175 km

00z 20 Dec. 997 mb 5 Sk 250 km 350 km

12z 20 Dec. 1002 mb 2 ND 175 km 175 km

12z 21 Dec. 1008 mb 3 Mi 100 km 175 km

00z 22 Dec. 1007 mb 3 Mi 100 km 50 km

12z 22 Dec. 1011 mb 2 On 125 km 375 km

12z 24 Dec. 1015 mb 3 On 325 km 150 km

etc.

Summary

Winter #1 (2000-2001): 41 cases, 18 events;Average errors: Avn 319 km, Eta 259 kmMedian errors: Avn 275 km, Eta 275 km# of wins: Eta 25, Avn 15, 1 tie

Winter #2 (2001-2002): 38 cases, 16 events;Average errors: Avn 330 km, Eta 324 kmMedian errors: Avn 262.5 km, Eta 250 km# of wins: Eta 19, Avn 17, 2 ties

Eta somewhat more accurate both winters, in spite of thisbeing at 2.5 days lead time, plenty in winter for the

western boundary error to make it into the contiguous U.S.!

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

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

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

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)

Any relevant results since ?

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

Suggested explanation

of the downstorm windstorm problem :

pS

pS

pS

T3

T5 T6

T1

T9

T2

v

v

v

vv v

v

v

vv

Flow attempting to move from box 1 to 5 is forced to enter

box 2 first.

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.

Refined (sloping steps) eta (Mesinger and Jović) :

Discretization accounting for slopes. Approach:

Define slopes at v points, based on four surrounding hpoints. 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” ?

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

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 boxdown by one layer thickness, linearly, to the centers of T2, T3 and T4

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 ofthe four: No slope

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.

Example of slopes with an actual model topography:

Sloping steps Eta: runs operationally at least at one place

Other possibilities. Other models using or having anoption 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)

Precip:Three-model precipitation scores,

on NMM ConUS domains ("East" ,…, "West"),available since Sep. 2002

• 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. 2002 T254 (55 km) resolution, sigma

Eta

“West”

“East”

The first 12 months of three model scores:

East

ETS (Equitable Threat Score) Bias

Is the GFS loosing (winning) because of its bias difference?

What can one do ?

BIAS NORMALIZED

PRECIPITATION SCORES

Fedor Mesinger1 and Keith Brill2

1NCEP/EMC and UCAR, Camp Springs, MD

2NCEP/HPC, Camp Springs, MD

J12.6

17th Prob. Stat. Atmos. Sci.; 20th WAF/16th NWP (Seattle AMS, Jan. ‘04)

Method 1, assumption:

can be solved;

dH

dF= a (O − H ), a = const,

GFSEta

NMM

Bias adjusted eq. threats

East

Eta

NMM

GFS

(Five very heavy el Niño precip events)

West

An example of precip at one of these events:

(8 Nov. 2002,

red contours:3 in/24 h)

An extraordinarychallenge to do well in QPF sense !

“The last 12 months”: Feb. 2004 - Jan. 2005

(includes high impact California precip,winter 2004-2005)

East

Eta

GFS

NMM

West

EtaGFS

NMM

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 ?

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

East

ETS

Bias

West

ETS

Bias

A fortuitous by-product of quasi-horizontal coordinates:

Model grid-boxes have approximately the samemass 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

Numerous tests failed to demonstrate any benefitfrom 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 (?)

Eta vs the RSM:2 years of scores, 1996-1997, at ~50 km resolution:

Note: Eta is using 12 h “old” Avn LBCs, RSM is using current Avn LBCs

EtaRSM

Why ?

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

However: Arakawa-style dynamics features (many in the Eta) are enforced on boxes; inconsistency less of a role !

With conservations achieved by communicationbetween neighboring grid boxes:

approximately same masses of grid boxes in horizontal seems desirable/

may have helped the Eta do well !?

Manuscript by Gary Russell, of NASA GISS (NY):

• Original version:

“Allowing the number of layers in a column to varyimproves 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 varyimproves the vertical mass fluxes, the flow around and through mountains, and the precipitation distribution, especially so in the Andes”

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

Model Domain and

Topography (m)

LLJ Case, Initial condition: 1200 UTC 15 Jan. 2003Left: sloping steps eta fcst; Right: sloping steps - traditional discr.

Zonda Case: Synop Observations

20050517 12Z 20050517 18Z

20050517 00Z 20050518 06Z

Zonda Case: Synop Observations

Infrared Water vapor

20050517 18Z

Zonda Case: Satellites Images

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

Sloping steps etaEta

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

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;

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;

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;

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

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 ! ! !