NASA/TM-1999-104606, Vol. 16

Technical Report Series on

Global Modeling and Data Assimilation

Max J. Suarez, Editor

Goddard Space Flight Center, Greenbelt, Maryland

Volume 16

Filtering Techniques on a Stretched Grid

General Circulation Model

Lawrence Takacs

General Sciences Corporation, Laurel, Maryland

William Sawyer

University of Maryland, College Park, Maryland

Max J. Suarez

Goddard Space Flight Center, Greenbelt, Maryland

Michael S. Fox-Rabinowitz

University of Maryland, College Park, Maryland

National Aeronautics and

Space Administration

Goddard Space Flight Center

Greenbelt, Maryland 20771

November 1999
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NASA Center for AeroSpace Information

7121 Standard Drive

Hanover, MD 21076-1320

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Available from:

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Price Code: A10
Abstract

This report documents the techniques used to filter quantities on a stretched grid general circulation model. Standard high-latitude filtering techniques (e.g. using an FFT

to decompose and filter unstable harmonics at selected latitudes) applied on a stretched

grid are shown to produce significant distortions of the prognostic state when used

to control instabilities near the pole. A new filtering technique is developed which

accurately accounts for the non-uniform grid by computing the eigenvectors and eigenfrequencies associated with the stretching. A filter function, constructed to selectively

damp those modes whose associated eigenfrequencies exceed some critical value, is used

to construct a set of grid-spaced weights which are shown to effectively filter without

distortion. Both offline and GCM experiments are shown using the new filtering technique. Finally, a brief examination is also made on the impact of applying the Shapiro

filter on the stretched grid.

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Contents

List of Figures vii

1 Introduction

2 Stretched Grid GEOS GCM

3 Uniform Grid High-Latitude Filter 2

4 Stretched Grid High-Latitude Filter

4.1 Convolution Filter ................................

4.2 GEOS GCM Results ...............................

4.3 Uniform Grid Interpretation ...........................

5

8

9

9

5 Global Shapiro Filter 10

6 Conclusion 12

Appendix 14

References 47

V
List of Figures

=

=

4

5

6

7

8

9

10

11

12

13

14

15

16

Longitude and Latitude Grid Increments using stretched grid ......... 18

Stretched grid structure used for GCM experiments .............. 19

Initial temperature and Day 5 temperature change (top panel), and 0.2 mb

temperature change and zonal wind field (bottom panel) using the standard

FFT filter on the stretched grid ......................... 20

Filter function and grid-space weighting coefficients at 85N for uniform grid. 21

Filter function and grid-space weighting coemcients at 60N for uniform grid. 22

Initial conditions for gravity-wave experiment on a uniform grid ....... 23

Timeseries of single harmonics on a uniform grid using a linearly stable

timestep and no filtering ............................. 24

Timeseries of single harmonics on a uniform grid using an unstable timestep

with FFT filtering ................................. 25

Timeseries at longitude 0 of single harmonics on a uniform grid comparing

filtered (thick) and non-filtered (thin) runs ................... 26

A comparison of the uniform and stretched-grid zonal increments ...... 27

Initial conditions for gravity-wave experiment on a stretched grid ....... 28

Timeseries of single harmonics on a stretched grid using a linearly stable

timestep and no filtering ............................. 29

Timeseries of single harmonics on a stretched grid using an unstable timestep

with FFT filtering ................................. 30

Timeseries of single harmonics on a stretched grid using the FFT filter squared. 31

Single harmonics at time = 3 hrs using the standard FFT filter and FFT

filter squared on the stretched grid ........................ 32

Wavenumber 3 as seen by Uniform Grid, Stretched Grid, and Stretched Grid

interpretted as Uniform Grid ........................... 33

vi
17

18

19

20

21

22

23

24

25

26

27

28

29

Spectraldecompositionof singleharmonicwavesasinterprettedby FFT on

the StretchedGrid.................................

Filter functionandgrid-spaceweightingcoefficientsusingstandardFFT on

a stretchedgrid..................................

Eigenvaluesassociatedwith the uniformandstretchedgrid..........

Eigenvectorsassociatedwith theuniformgrid.................

Eigenvectorsassociatedwith the stretchedgrid.................

Filter function and grid-spaceweightingcoefficientsusingthe convolution

filter ona stretchedgrid..............................

Timeseriesof singleharmonicsonastretchedgrid usinganunstabletimestep

with convolutionfiltering.............................

Initial temperatureandDay5 temperaturechange(top panel),and0.2mb

temperaturechangeandzonalwindfield (bottompanel)usingthe convolution filter onthe stretchedgrid..........................

Day5sea-levelpressureand300-mbheightfieldsforthestandardFFT filter

andthe convolutionfilter.............................

Grid-spaceweightingcoefficientsusingtheempiricalmethod,andcompared

with theeigenvaluemethod............................

Timeseriesof singleharmonicson a stretchedgrid with weightsobtained

usingthe empiricalmethod............................

Singleharmonicsat time = 30 hrs comparing the eigenvalue and empirial

methods ......................................

Filter function and grid space weights for an 8th-order Shapiro filter on the

uniform and stretched grid ............................

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35

36

37

38

39

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1 Introduction

Version 2 of the Aries/GEOS dynamical core (Suarez and Takacs,1995) was developed at

the Goddard Space Flight Center for use in data assimilation and climate prediction applications. In this paper we present a generalization of this dynamical core that allows for a

non-uniform latitude/longitude (AOj, AAi) grid. This capability allows use of the model for

regional data assimilation with the Goddard Earth Observing System (GEOS) Data Assimilation System (DAS) (see Rood, 1996) and for regional prediction studies using the NASA

Seasonal to Interannual Prediction Project's (NSIPP) coupled prediction system. The use

of this dynamical core in conjunction with a non-uniform grid has been demonstrated by

Fox-Rabinovitz et al. (1997) for the geld-Suarez (1994) dynamics benchmark. That work

demonstrated the usefulness of achieving high resolution results over selected regions with

much greater efficiency than employing high-resolution uniformly over the sphere.

The Aries/GEOS dynamical core relies on two filtering strategies to ensure stability and

efficiency. The first, high-latitude filtering, is done to control linear instability due to the

converging meridions near tile pole. This filter selectively damps the tendencies of fastest

modes. Thus, on a uniform grid, the filter acts near the poles on the smallest scales. The

second technique is the application of the Shapiro (1970) filter. This filter is applied globally

to damp small-scale dispersive waves and to prevent computational nonlinear instability

(Phillips, 1959) to occur. This report will examine the properties of these two filtering

techniques on the stretched grid, and develop a stretched grid convolution filter applicable

to controlling linear instability in high latitudes.

2 Stretched Grid GEOS GCM

Following the work of Fox-Rabinovitz et al. (1997), the stretched grid dynamical core

was interfaced with the uniform grid physics package of the GEOS General Circulation

Model (GCM). While the ultimate goal will be to perform the physics on the stretched grid

itself, this more conservative approach was used to avoid unforseen problems within the

parameterizations in regions of high resolution. In practice, a copy of the dynamic state

variables (u, v, O, q, Ps) are transformed (via cubic interpolation) to the uniform grid to force

tile physics packages. The tendencies of the dynamic state variables due to the physics are

then transformed to the dynamics stretched grid to act as external diabatic forcing in the

time integration scheme. As a result, the dynamics state is always updated and preserved

on the stretched grid.

The generation of the stretched grid used for this study employs a cosine mapping function

to smoothly create grid increments which vary with wavenumber 1 in the longitudinal

direction, and increase to a maximum at the farthest pole in the latitudinal direction.

These longitude and latitude grid increments are shown in Fig. 1. For this study, the
uniformgrid resolutionis definedby a 2.5°x2.0° Ion-lat grid (144x91 points). These points

are re-distributed in longitude and latitude so as to produce uniform 1.0°xl.0 ° resolution

over the eastern United States (_ 28N-48N, I00W-70W). Figure 2 shows the resulting 2-

dimensional grid structure over the globe. Note that in the coarsest region (the South-East

quadrant) an effective resolution of .-- 5°x4 ° is used.

Starting from an arbitrary, uniform grid January simulation, initial conditions were transformed and balanced on the stretched grid to begin stretched grid forecasts. The top panel

of Fig. 3 shows the initial zonal mean temperature distribution and the day 5 temperature

change from the intitial condition. During this 5-day forecast, a stratospheric warming

event has taken place (between day 4 and 5) which is associated with strong cross-polar

flow. The bottom panel shows the temperature change and day 5 zonal wind at 0.2 mb

from latitudes 50N to 90N (longitnde 0 at bottom). We see that near the poles in the region

of highest longitudinal resolution, computational small scale features have developed. This

occurs even though the timestep chosen for the integration was adequately reduced to reflect

the increased resolution in the area of interest. It was n0ted in Fox-Rabinovitz et al. (1997)

that, even for the simple-physics Held-Suarez test, excessive polar noise developed when

_ became large (>_ 16). In those cases, ad hoc strengthening total stretching factors (A_m,, J

of the polar filters was required. It is clear from these results that, while the standard polar

filtering technique is effective in preventing linear instability in many cases, it is not robust

enough for use oll the stretched grid for general applications.

3 Uniform Grid High-Latitude Filter

To understand why the standard high-latitude filtering technique is not adequate when using

the stretched grid, we must first review the technique employed on the uniform grid. As

pointed out in Suarez and Takacs (1995), polar Fourier filters in the Aries/GEOS dynamical

core are applied to the tendencies of all prognostic variables. This is done to avoid linear

computational instability due to the convergence of the meridians near the poles. The

filter acts poleward of a critical latitude ¢c (nominally 45°), and its strength is gradually

increased toward the pole by increasing the number of affected zonal wavenumbers and the

amount by which they are damped.

Consider the linearized, one-dimensional shallow water equations with no mean flow given

by:

Ou _ g oh (1)

Ot a cos ¢ 0h '

Oh H Ou

_ (e)

Ot a cos ¢ 0h "
Discretizingin spaceon astaggeredC-grid,wemaywrite

0u_+1⁄2 g (hi_- hi) Ot - acos¢ \ A_ ' (3)

Ohi g ( ui+1⁄2 - ui-1⁄2"_

Ot -- acos¢ k AA ] " (4)

Here, AA = 2_ and IM is the zonal dimension. Assuming wave solutions of the form:

ui+ 1⁄2(t) ---- _et(k(i+1⁄2 )A)'-_'t) , (5)

hi(t) = ]_e_(kiA_-Èt) , (6)

where i denotes the grid-space location in longitude and t = v/L_, it can be shown that the

frequency associated with each harmonic component k is given by

u=+ 2v H sin(k ). (7)

a cos CA

We see that the frequency increases as a function of wavenumber and as we approach the

pole.

Using the leapfrog time scheme, linear stability requires that

uAt __ 1. (8)

This forces restrictions on the timestep governed by

1 acos CAA 1

at < 2 sin(k )" (9)

To eliminate the requirement that the timestep goes to zero as we approach the pole, a

wavenumber-dependent filter, Fk, is applied to the time tendencies of the prognostic fields

resulting in a modified frequency function given by

2v_H sin (k-_) . (10) u=4-Fk acosCAA

By requiring that the frequency be no larger than UMAX (e.g., the unfiltered value associated

with the shortest wavelength at the critical latitude ¢c), the functional form of Fk may be

obtained:

Fk=min 1,_,cos¢_ sin P'x
Here, n is an arbitrary factor used by Suarez and Takacs (1995) to increase the strength of

the filter. While not needed for these linear studies, it has proven beneficial to use n = 2 for

the non-linear uniform grid GCM. Unless otherwise stated, results shown here were made

using n = 1. It is important to remember that applying the filter in this manner simply

slows down each harmonic component by an amount which produces linear stability. More

precisely, the computational propogation speed of each zonal harmonic is adjusted to keep

its transport to no more than 1 A), per timestep. The filter has no impact on the magnitude

nor energy of the harmonic.

Figures 4 and 5 show the filter function at latitudes 85N and 60N, respectively. For these

results, the zonal dimension was set to IM=144 (corresponding to a 2.5 ° resolution). Near

the pole (85N) the filter strength is quite strong with only wavenumbers 0-5 remaining

untouched, ttowever, away from the pole (60N) the filter is much weaker with only moderate

damping even at the smallest scales. Also shown in Figs. 4 and 5 are the equivalent

grid-space weighting coefficients at various longitudes obtained through convolution of the

spectral filter functions. Since the grid is uniform, the weighting stencil is identical for each

longitude location. For the strong spectral filter function near the pole, the convolution

weights are quite broad in longitude and have a peak amplitude which is relatively small

(0.3). Away from the pole where the filter strength is weak, the convolution weights resemble

that of a Delta function whose amplitude is near 1.0 at the grid-point location and almost

0.0 elsewhere. The breadth of the convolution weights are thus intimately connected to the

physical scale of the waves being filtered.

To illustrate the effect of filtering, a set of initial conditions consisting of the single harmonics

1, 3, 5, and 7 (see Fig. 6) is used. To benchmark this test, the integration is first run

without filters by using a sufficiently small timestep which is linearly stable, Fig. 7. These

single harmonics simply oscillate in time with no translational motion. The test is then

repeated using a timestep comparable to that used for GCM simulations. Without filtering,

instability occurs within a few iterations (not shown). Figure 8 shows the results from the

uniform grid using the standard FFT polar filter described above. The filtering associated

with latitude 85N was used for these runs. With tendency filtering, the solutions look quite

similar to the non-filtered case with no discernable instabilities. Figure 9 compares the

filtered and non-filtered solutions at longitude 0 °, with additional wavenumbers 9 and 12

also shown. As the wavenumber increases, the frequency is increasingly slowed compared

with the non-filtered run. Also, the frequencies for wavenumbers 7, 9, and 12 are identical

for the filtered run since they are forced to be no greater than the maximum allowed value.

However, the magnitude of the oscillation remains untouched, with no distortion of the

wave form.

Z

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i

u

4
Theexperimentisthenrepeatedusingthestretchedgrid. Theuniform2.5° grid isstretched

to 1° betweenlongitudes75Eand 105E,Fig. 10,with initial conditionsshownin Fig. 11.

Figure 12showsthe resultsfor the stretchedgrid usinga sufficientlysmall timestepto

ensurestability without filtering. Thestretchedgrid solutionisverysimilarto theuniformgrid solution,creatingnegligabledistortiondueto the stretching.Theexperimentis again

repeatedusinga largertimestepcomparableto that usedfor GCM simulationsand thus

requiringtendencyfiltering, Fig. 13. It is clearthat the useof the standardFFT filter

hascreated significant distortions of the pure waveforms and has affected the solution at

all longitudes. In addition, significant small-scale features have been created even though

linear stability has been achieved. Arbitrarily increasing the strength of the filter to n = 2

creates even further distortions, Fig. 14. A snapshot of the solution at 3 hours is shown in

Fig. 15 clearly demonstrating the inability of the standard FFT filter to adequately control

noise and prevent distortions on the stretched grid.

4 Stretched Grid High-Latitude Filter

Since the standard filtering technique is done in spectral space, it is clear that the functions

upon which stability is based (the trigonometric zonal harmonics) must have the same

relevance on the stretched grid for the filtering technique to be valid. Figure 16 depicts the

single harmonic wa_enumber 3 on the uniform grid (top panel) and on the stretched grid

(center panel). Here, the term "wavenumber 3" is used to describe a field whose structure

is repeated three times in the longitudinal direction. Using this definition we see that for

each zonal harmonic k there is an associated geophysical scale whose wavelength Lk is given

by:

27ra cos ¢ (12) Lk-- k

This physically-based wavelength is independent of any finite-resolution grid used to construct the field. Assuming this physically-based wave propogates uniformly in the zonal

direction, the filter strength required to slow it down must clearly be a function of the

local zonal grid increment just as it is a function of latitude (i.e., the required strength to

restrict transport to no more than 1 AI per timestep would need to be greater in an area

of high resolution than in an area of coarse resolution). However, as shown above for the

standard filtering technique, there is no longitudinal dependence on the filter strength Fk.

Moreover, since the FFT implicitly assumes a uniform grid distribution, the FFT interprets

this wavenumber 3 example as a field which is made up of both longer and shorter wave

components (Fig. 16, bottom panel). The varying strengths of the filter as defined by (11)

for these wavenumbers would be misapplied if the intent was to filter the geophysical scale

associated with wavenumber 3.
The top panelof Figure 17showsthe FFT harmonicdecompositionof the wavenumber3

examplefor both the uniformgrid andthestretchedgrid. Interestingly,the wavenumber3

harmonicis almostcompletelymissingfrom the stretchedgrid interpretation.Insteadwe

seeaspreadof harmonicsto eithersideof the actualinput wavenumber.Thebottom panel

of Fig. 17repeatsthis analysisfor eachwavenumberfom 0to IM/2. Eachsingleharmonic

wavecomponentis projectedonto the strethedgrid, and thenusedasinput to the FFT

analysis.As before,eachcomponentis interpretedashavingboth longer waves and shorter

waves, with little energy in the actual wavenumber used as input.

Figure 18 examines the convolution weights obtained from the standard FFT filter but

applied on the stretched grid. The top panel, as before, shows the spectral filter function

independent of longitude. The bottom panel of Fig. 18 shows the convolution weights

plotted on the stretched grid. We see that the peak amplitude of the weights is constant

in longitude. However, in the fine-resolution region the convolution weights have narrowed

in longitude while in the coarse-resolution region they have broadened. This implies that

the effect of the standard filtering technique on the stretched grid will be to filter less in

the fine-resolution region and filter more in the coarse-resolution region, exactly opposite to

that required from previous considerations.

It has become clear that to accurately filter on a stretched grid, we must first determine

the stretched grid basis functions upon which the model solutions may be projected. We

again start by writing the discretized linear shallow water equations for the stretched grid:

Ot a cos ¢ _, AAi+ '

Ohi Ot _- acos¢ H ( u- l - u ,-_ ' ),-r_"

Here, AAi+ 1⁄2 is the distance in longitude between mass points, and

A_,+1⁄2 + AA i

AAi = -_-

2

(13)

(14)

(15)

Using (13) and (14), the wave equation is formed given by:

02hi gH hi+l - hi hi - hi-1

Otz (a cos ¢) 2 AAiAA/+1⁄2 AAiAAi_ 1⁄2

Assuming a solution of the form:

hi(t) = ]_,ie-'t ,

(16)

(17)
wefind

-v _(a cos CA_) 2

gH

/

+ ) '

(18)

where A,1 2,_ Note that for the uniform grid (AAi AA) the weights used in (18) simply

become 1,-2, 1. Defining:

AA 2

ri = AAi+_AAi_ , (19)

5 2

AA 2

r_- AA/AAi+! ' (20)

AA 2

r_-= AAiAAi_! ' (21)

2

we may write (18) in matrix form as:

-2rl r_

r_ -2r2 r]

r 3 ". -.

''. --2FIM_I FI't'M. 1

r+ M riM --2rim

rl

fl = ,Xfi, (22)

where A is the eigenvalue associated with the eigenfrequency v:

A=- u_(ac°s_A_)2

gH

(23)

From (22) and (23), we may compute the eigenvalues (or eigenfrequencies) and corresponding eigenvectors associated with the stretched grid (see Appendix for detail). Figure 19

shows the eigenvalues computed by solving (22) for both the uniform grid and the stretched

grid, in addition to the analytic uniform grid solution associated with (7). Here we have

ordered the eigenvalues by magnitude. For the uniform grid, the analytic and numerical

solutions are indistinguishable. For the stretched grid, we see a significant increase in the

magnitude of the eigenvalues associated with the highest mode index (smallest scales).

7
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:=

The eigenvectors associated with a sampling of modes are shown in Fig. 20 for the uniform

grid and Fig. 21 for the stretched grid. The uniform grid solutions are simply discretized

sines and cosines, analogous to the analytic solution. The eigenvectors for the stretched grid

are similar for the low-index (slow) modes. However, as the mode index (or eigenfrequency)

increases the structure of the eigenvector is confined to smaller and smaller regions. For the

highest eigenfrequency, the eigenvector is totally concentrated in the finest resolution area.

By ordering the eigenvalues and associated eigenvectors in this manner, a filter may be

constructed to selectively damp those modes whose associated eigenfrequencies are faster

than some critical value. Using (23) we see that

v=4- IAI1⁄2gv_/ (24)

a cos CAA

As in the uniform grid case, the maximum frequency allowed will be defined as the unfiltered

frequency associated with the shortest wave at tile critical latitude ¢_:

= 2gv (25)

a cos ¢cAAmi n

Requiring that the filtered frequency be no faster than the maximum frequency allowed,

F v < VMaX , (26)

the filter function for the stretched grid becomes:

[ co ¢] f = mill 1, [Al1⁄2AAmi n cos¢_j

(27)

4.1 Convolution Filter

To construct the stretched grid convolution filter, consider for example the height field

defined by a vector array h constructed from the set of modes found in Section 4:

h = MA, (28)

where M are the stretched grid modes or eigenvectors and A are the mode projection

amplitudes. The amplitudes may be explicitly obtained by

A = M :i h. (29)

A new set of amplitudes are now created by multiplying (29) by a diagonal matrix F defined

by the convolution filter function (27):

A F=FA=FM -I h. (30)
The reconstructedfilteredfield,hF, is thereforegivenby:

hr =_ M A r = M F M -1 h . (31)

Weights in Grid Space

Tile weights obtained in grid space are analogous to the weights obtained through convolution of tile standard FFT for the uniform grid. Through convolution we may now examine

in spectral space the response of these weights as a function of longitude. The top panel

of Fig. 22 shows the spectral response of the stretched grid convolution filter for various

longitude locations, while the bottom panel shows the grid space weighting coefficients. We

see that the amplitude of the coefficients is now a function of longitude while the breadth of

tile stencil is constant. This is exactly opposite to that obtained by the use of the standard

FFT filter applied to the stretched grid, with the impact being that the strongest, filtering

now takes place in the region of highest resolution. The results from using the stretched grid

convolution filter are shown in Fig. 23. The stretched grid convolution filler successfully

prevents linear instability with no distortion of the waveforms nor generation of additional

small scale features.

4.2 GEOS GCM Results

tlaving seen the effectiveness of using the stretched grid convolution filter in simple off-line,

gravity wave test cases, we now proceed to analyze its performance within the GEOS GCM.

For this test, the forecast experiment described in Section 2 is repeated by replacing the

standard FFT polar filtering technique with that of the stretched grid convolution filter.

Figure 24 once again shows tile initial zonal mean temperature distribution and the day

5 temperature change, as well as the 0.2 mb day 5 temperature change and zonal wind

fields. Compared with Fig. 3, the stratospheric warming event is again simulated but with

no evidence of small-scale noise near the polar, high-resolution region. It should be noted

that tile noise generated with the standard filtering technique was primarily evident at

high stratospheric altitudes where the wind field is particularly strong. At lower levels the

standard filter and the stretched grid convolution filter produce very similar results. Figure

25 shows the sea-level pressure and 300-mb height fields at day 5 for the two simulations.

The two solutions are virtually identical with no evidence of noise in either case.

4.3 Uniform Grid Interpretation

While the preceeding sections derived exactly the proper filter function for the linearized

shallow water equations on a stretched grid, it is possible to approximate this solution

through a uniform grid interpretation. Recall from Section 3 that the filter function derived

9
for the uniformgrid for n = 1 was given by

Fk =min [1, (cos \cos¢_

(32)

for k=0, ]-_-. This may be rewritten as:

[ /iac°_seA)_i F/,k, = min 1, \a cos ¢cA$mi n

(33)

for

k' = 0,--Ni (34)

2 '

Ni -- 2rr (35)

A,_i

We 2_ (32) and (33) are identical. However, for the see that for the uniform grid, A,ki = i-if,

stretched grid we define Ni at each longitude location as the number of points a uniform

grid would have assuming a uniform resolution of AAi. This will produce a uniform grid

filter function which, through convolution, may be represented by the grid-space weighting coefficients. The stretched grid weighting stencil may then be constructed by simply

interpolating the uniform grid stencil to the stretched grid locations.

The top panel of Fig. 26 shows the grid-space weighting coefficients obtained using this

empirical method, while the bottom panel shows the differences compared with the eigenvalue method. The empirical method produces coefficients very similar to the eigenvalue

method, with the resulting simulation shown in Fig. 27. This implies that the stretched

grid convolution filter developed in Section 4.1 acts to treat each local A)_i as if it were

global, and constructs a filter function based on a uniform grid of that resolution. Interestingly the minor differences which are obtained using this empirical method are ultimately

pathological giving rise to instabilities as shown in Fig. 28 after 30 hours of simulation.

However, as will be seen in the next section, this technique can be useful to analyze the

impact of arbitrary uniform grid filter response functions applied to a stretched grid.

5 Global Shapiro Filter

As previously noted, the uniform grid GEOS GCM also employs the Shapiro (1970) filter

to globally damp small-scale dispersive waves. This filter also prevents computational nonlinear instability (Phillips, 1959) to occur. The Shapiro filter is applied as a tendency to

the winds, potential temperature, and tracers (including specific humidity). Thus, only a

fraction of the full Shapiro filter is incorporated at each time step. This is done to reduce

dynamical imbalances and diabatic responses caused by the filter.

10
TheShapirofilter tendencyfor a quantityq is defined by

Oq) qF _ q

_- SF T '

(36)

Here q and qF are the unfiltered and filtered quantities, and T is an adjustable timescale. For

the uniform grid GEOS GCM, r is set to 1.5 hours which effectively removes the smallest

two-grid interval wave in approximately 6 hours. The filter is applied separately in the

longitudinal and latitudinal direction using

where

q ,j F = [1 - (r;)2 n] [1- qi,j (37)

1

F (qi,j) - (qi+l,j - + qi_l,j),

1

F_(qi,j) - _(qi,j+l - 2qi,j + qi,j-1) ,

(38)

and n is the Shapiro filter order/2. At 2.5°x2 ° resolution the GEOS GCM uses an 8th-order

(n=4) filter. Lower-order filtering corresponds to stronger damping.

We may examine the Shapiro filter response function in one dimension by assuming a wave

solution defined by

qi = qk e_kiA'\ • (39)

Doing so we find

= 1 - sin2_(k_) . (40)

The left-hand column of Fig. 29 shows the frequency response for the 8th-order filter on a

2.5 ° uniform resolution grid as well as the grid space weights at selected longitudes obtained

through convolution. We see that the grid space averaging is very local (only 9 grid points)

and produces a sharp delineation between the small scales which are heavily filtered and

the longest scales which are virtually untouched. The center column of Fig. 29 shows the

response of the standard technique described above but applied to the stretched grid. In

this case the basic local stencil is simply used without concern for the non-uniform mesh.

Due to the success of the uniform grid interpretation in approximating the convolution

weights obtained by the eigenvalue method for the stretched grid high-latitude filter, it is

useful to employ this technique to estimate the proper stretched grid convolution weights

derived from the uniform grid Shapiro filter frequency response. The right-hand panel of

Fig. 29 shows the results from assuming that each local A,ki was globally uniform. The

convolution weights associated with the Shapiro filter response for each implied uniform

resolution were then interpolated to the actual stretched grid locations. The frequency

response associated with these empirically derived coefficients was then plotted for selected

11
longitudelocations.Dueto theverylocalnatureoftheShapirofilter andthefilter response's

non-dependencyon critical wavelengths,very little impactis seenas a resultof the nonuniformgrid. This issignificantlydifferentthantheimpactseenfor thehigh-latitudefilter

case,Fig. 22,wherelongitudinallocalitygreatlyinfluencedfilter strength.Asaresultof this

analysisin additionto empiricalevidencefrom numerousstretchedgrid GCM simulations

and the greaterefficiencyof the standardalgorithm,no obviousadvantageis realizedby

usingthe (empirical)eigenvaluemethodfor theglobalShapirofilter.

4

-1

6 Conclusion

In this report the filtering characteristics of a stretched grid GCM have been reviewed. It

was shown that using the sta_ndard FFT polar filtering technique, as performed by FoxRabinovitz et el. (1997) for the stretched grid Held-Suarez dynamics benchmark, computational small-scale noise is generated near the poles in tile region of high longitudinal

resolution. While this noise is associated with high wind speeds ill the stratospheric domain, Fox-Rabinovitz et el. found similar problems when total stretching factors were large

(> 16). For those cases, ad hoc strengthening of the filter coefficients were used to control

stability.

A review of the standard high-latitude filtering procedure was made and an examination of

the physical scales associated with the filter were analyzed on both the uniform grid and

the stretched grid. It was shown that the implicit assumption of a uniform grid within

the FFT algorithm results in a misrepresentation of the zonal harmonic composition of the

input field. Standard filter strengths which are derived from uniform grid considerations

and stability analyses are thus misapplied to the stretched grid harmonic decomposition,

resulting in weaker filtering within the high resolution region and stronger filtering in the

coarse resolution region. This is opposite to the requirement of stronger filtering in areas

of high resolution.

A new filtering technique has been developed which accurately accounts for the non-uniform

grid by computing the eigenvectors and eigenfrequencies associated with the stretching. |t

is shown that the convolution of the required damping function within mode space yields

grid-spaced weights which can be efficiently used to perform the filtering without distortion.

Offline tests showed that the stretched grid convolution filter correctly filtered more in the

high resolution region and less in the coarse resolution region. Online GCM experiments

further showed that computational small-scale noise was no longer generated in the region of

highest longitudinal resolution near tile poles. Away from high latitudes, no adverse effects

from using the stretched grid convolution filter were seen. In the limit that the stretched

grid becomes uniform, this technique reduces to the standard uniform grid filter.

In addition to the complete eigenvalue/eigenvector solution, an empirical method was dei

12
velopedwhichtreatseachlocal A)_i as if it were global. The convolution weights associated

with the filter frequency response for the implied uniform resolution are then interpolated

to the actual stretched grid locations. This empirical estimation of the convolution weights

for a stretched grid was used to analyze the impact of applying the Shapiro filter on the

stretched grid. It was shown that due to the Shapiro filter's very local grid stencil (9 points

for the 8th-order filter), the standard application of the Shaprio filter produced very small

differences compared with the (empirical) eigenvector method. Due to the greater efficiency of the standard algorithm, no obvious advantage is realized by using the (empirical)

eigenvector method for global filtering.

Acknowledgements

This work was supported by the Global Atmospheric Modeling and Analysis Program,

NASA/Office of Earth Science.

13
N

Appendix

Linear Algebra Aspects

of the Stretched Grid Convolution Filter

The stretched grid convolution filter requires the eigenvalues and eigenvectors of a nonsymmetric matrix (which we define here as R) from (22). The matrix R is a tridiagonal

matrix with the addition of the two corner elements h+M and r 1. It has several interesting

properties. From (19), (20), and (21) it is clear that all row sums are zero. That is,

Vi : -2ri + r_ + r[ = O, or in matrix form,

Rc = 0 (41)

where c is a vector of constant elements c = (c, c,..., ¢)T. In other words R is singular.

From the same definitions (19), (20), and (21) it is easily shown the product of the ratios

is unity:

r i

1-I ri+ = 1 (42)

i=1 ri

since every AAi, AAi+1⁄2 and AAi_ 1⁄2 appears in both the numerator and the denominator.

If (19), (20), and (21) are all multiplied through by AA_, or equivalently, (22) is multiplied

on the left by

A __

AA2

AA_M

(43)

a generalized eigenvalue problem containing a symmetric left-hand side It = AR side can

be obtained:

I_h = ,kAle. (44)

It would therefore be straightforward to use a generalized eigen-solver for symmetric matrices to solve the eigenvalue problem. But the problem can be further simplified. Since

14
both1_andA arerealand symmetric,it isclearthat all A are real. This means that R is

symmetrizable, that is, a similarity transform can be applied to R to create a symmetric

matrix with the same eigenvalues:

ZRZ-I=s=s T " (45)

In order to find Z by construction, consider the diagonal matrix

Z

(_IM

(46)

Clearly, Z -1 is also a diagonal matrix with _ on the diagonal. 1 If the similarity transform

is applied using this Z, the result is a matrix with the same structure as R:

S=ZRZ-I=

_2rl _rl + _M15 _,,-

_r_- -2r2 _r+_32

r_" "". "'.

• 8IM--1 _4

"" --2rIM--i _'IM--1

51Mr+ 515M--_--r_ --2riM

_1 IM

(47)

This matrix is symmetric if, and only if, the elements r+ and r_- are non-zero, and the

following conditions are satisfied:

_i+12 w r_.._/+6i2 , (48)

ri+l

r +

The conditions (48) wrap around (i.e., 512 = :-u2t_5IM) and can only be satisfied if

r I

r 1 r 2 •.. riM

-- 1. (49)

From (42) we already know this to be the case for the matrix R. One degree of freedom is left

to set, for example, 51 = 1. Thus, with a simple similarity transform, a symmetric matrix

lit is assumed that the _, are non-zero.

15
canbeobtainedwhichis moreefficientto factorize2 than the original R. It also improves

the algorithm by indicating that the eigenvalues are necessarily real -- a non-symmetric

eigensolver might return very small imaginary components due to round-off errors. Finally,

the eigen-decomposition of a symmetric matrix yields orthogonal eigenveetors Q,

!

!

|

i

!

|

l

i

|

QT S Q = diag(Al,;_2,...,A_M) (50)

which can be a considerable advantage.

Equation (41) indicates that R has a zero eigenvalue, i.e., At = 0. Although most eigensolvers can handle such a case, it is conceivable that floating point errors can result. In

order to further stabilize the algorithm, it may be worthwhile to remove the zero eigenvalue

by the following technique. The eigenvector to the zero eigenvalue of R is clearly c, a vector

of constants, as shown in (41). The equation

Rc=Z -1 S Zc=0 (51)

indicates that y = Zc is the eigenvector of S corresponding to its zero eigenvalue. Assuming

a normalized y it is possible to create an orthonormal basis,

where Y can be constructed iteratively from Gram-Schmidt orthogonalization.

second similarity transformation can be applied on S using Y:

Thus a

yT S Y = ,_, [S] [y Y] (53)

= [ ,yTSy yTSy "_-T YTS_r S _r I (54)

= [00 ] 0 s (,55)

Note, for example, that yTSY = yTsT'_[ = (Sy)T_ r = 0 T. This similarity transform ensures that S is also symmetric and contains all the eigenvalues of S except the zero eigenvalue

corresponding to eigenvector y. The symmetric matrix S has the eigen-decomposition,

2For example, with the LAPACK routine dsyev.

w

I

i

16
QT s c) = aiag(_=,..._,M) • (.50)

From (53) and _(56)_it is straightforward to show that, the eigenvectors of the non-zero 3

eigenvalues are YQ. Thus the full set of eigenvectors of S is:

It is now possible to specify the eigen-decomposition of R in terms of all the components

of tile fully stabilized eigen-solver:

M-1RM = diag(0,)_2,...AxM)

= QTSQ

= [ yT

L ] Z-'RZ[ Y (6o)

Thus, with the eigenvectors Q of S;, the normalized eigenvector y, the set of constructed

orthonormal vectors Y, and the diagonal matrix Z, it is possible to recreate the eigenvectors

M of R:

M=Z[ y _rQ ] (61)

which are required in the stretched grid convolution filter.

3The proof is omitted here that, all other eigenvalues of S are non-zero.

17
ml

.4

Longitude Grid Increments

5.5

5.0

4.5

4.0

3.5-

03

(]3

3.0

ET_ 2.5 <33

C]

2.0

1.5

1.0

0.5

0.0 8o _2'ow 66w 6 66E 12'oE

Latitude Grid Increments

03

(13

O

E7_

(33

r-_

ii1

315]

3.0

2.5

2.0

1.5"

1.0

0.5

0.09os 6_s 36s Eb _6N 66N

Figure I: Longitude and Latitude Grid Increments using stretched grid.

18
la.i

Figure 2: Stretched grid structure used for GCM experiments.

19

%
q

_a

.:Y.

GEOS-2 GCM Standard FFT Filter

Initial Temperature (cint: 5 K)

60S 30S EO 30N 60N

Day 5 Temperature Change

0.02

0.04 -

0.1-

0,2

q

0,4

2

3

20

30

200 I_ 300

1I_

90N 90S 6(3S 3()S

(cint: 2 K)

Day 5 Temperture Change (cint: 5 K) Day 5 Zonal U-Wind (cint: 15 m/s)

Figure 3: Initial temperature and Day 5 temperature change (top panel), and 0.2 mb

temperature change and zonal wind field (bottom panel) using the standard FFT filter on

the stretched grid.

2O
1.1

1.0

0.9-

0.8-

0.7-

0.6 _

0.5-

0.4-

0.3-

0.2

0.1

0.0

o

Uniform-Grid FFT Filter Lot: 85N

Spectral Filter Function

1'0 1_5 2'0 2'5 3'0 3'5 4'0 4'5 5'0 5_5 6_0 6_5 7_0

Wavenumber

0.35

Grid Convolution Weights

0.30

0.25

0.20

0.15 )

0.10 !

0.05

o.oo

-0.05

-180 - 1'20 - (SO 6 6'0 t1⁄20

Longitude

Figure 4: Filter function and grid-space weighting coefficients at 85N for uniform grid.

21
1.1

Uniform-Grid FFT Filter Lot: 60N

Spectral FilterFunction

1.0

0.9

0.8

0.7-

0.6

0.5-

0.4

0.3-

0.2

0.1

0.0, I'0 (8 2'0 2'8 3'0 3'5 4'0 4's s'o 5's 6'0 6'5 7'0

Wavenumber

Grid Convolution Weights

0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.00- _3:1_

-180 -1"20 - 60 6 6'0 11⁄20

Longitude

Figure 5: Filter function and grid-space weighting coefficients a,t,60N for uniform grid.

22
8 .... [ 7 ; 7 ; _ _

m

0

.4..a,

¢-

"0

(.9

I

E

0

¢.-

e",

E

c

o

._m

if31

¢1

g

Figure 6: Initial conditions for gravity-wave experiment on a uniform grid.

23
==

Figure 7: Timeseries of single harmonics on a uniform grid using a linearly stable timestep

_nd no filtering.

24
,._ ,lJ

v

,._#f,

,_,---,/!

v

o

,.6

Figure 8: Timeseries of single harmonics on a uniform grid using an unstable timestep with

FFT filtering.

25
Filtered (Thick)

Wevenumber 1

vs. Non-Filtered (Thin)

Uniform Grid

Wavenumber 3

Time (hrs)

Wavenumber 5

1

/

/

Time (hrs)

Wovenumber 9

.

\J

i

Time (hrs)

Wavenumber 7

1t

t

Time (hrs)

Wavenumber 12

t

Time (hrs) Time (hrs)

Figure 9: Timeseries a,t longitude 0 of single ha,rmonics on a uniform grid comparing filtered

(thk-k) and non filtered (thh,) runs.

26
rE

Q)

$-

O

¢-

-O

°_

$-

(.9

tO

$-

O9

E

k.

O

41_

.m

¢-

saaJBa(]

°

I.u

o

.o

o

o

g

0_

0

Figure 10: A comparison of the uniform and stretched-grid zonal increments.

27
z

r-,-

Figure 11: Initial conditions for graxity-wave experiment, on a stretched grid.

i 28
®

Figure 12: Timeseries of single harmonics on a stretched grid using a linearly stable timestep

and no filtering.

29
Z

m

0

-I

}

E

'u

'V

@

iE

Figure 13: Timeseries of single harmonics on a stretched grid using an unstable timestep

with FFT filtering.

30

==
Figure 14: Timeseries of single harmonics on a stretched grid using the FFT filter squared.

31
FFT Filter Wavenumber 1

FEE Filter Wovenumber 3

FEE Filter Squored Wovenumber I

FFT Filter Squored Wovenumber 3

FFT Filter Wovenumber 5 FFT Filter Squored Wovenumber 5

FFT Filter Wovenumber 7 FFT Filter $quored Wovenumber 7

Figure 15: Single harmonics at time = 3 hrs using the standard FFT filter and FFT filter

squared on the stretched grid.

=

32
1.2

Uniform Grid

1.0

0.8

0.6

0.4

0.2-

0.0

-0.2 _

-0.4-

-0.6-

-0.8

-1.0

-1.2

I80 120W 6(]W 6 6(_E 12'0E

1.2"

Stretched Grid

1,0"

0.8.

0.6-

0,4-

0.2

0.0

-0.2

-0.4

-0.6

-0.8

-1.0

-1.2

180 120W 60W 0 60E 12'0E

1.2

Stretched Grid Seen by FFT

1.0-

0.8-

0.6

0.4

0.2

0,0

-0.2

-0.4

-0.6

-0.8

-1.0-

-1.2

I80 12bw 66w 6 60E 12'0E

Figure 16: Wavenumber 3 as seen by Uniform Grid, Stretched Grid, and Stretched Grid

interpret.ted as Uniform Grid.

33
Wove 3 Uniform Grid Wave 3 Stretched Grid

1.1

I 1.0

0.91

0.8-

0.7

0.6-

0.5"

04

03

0.2

0.1

O0

-0,1

0 1 2 3 4 5 6 7 8 9 10 1t t2 0 t 2 3 4 5 6 7 8 9 10 tl 12

Wovenumber Wavenumber

Stretched-Grid (Light) vs Uniform-Grid (Dark)

m

B

=EE

=

FFT Response

Figure 17: Spectral decomposition of single harmonic waves as interpretted by FFT on the

Stretched Grid.

34
1.1

1.0

0.9

0.8

0.7.

0.6-

0.5-

0.4-

0.3

0.2

0.1

0.0

Stretched-Grid FFT Filter Lat: 85N

Spectral Filter Function

1'0 1'5 2'0 2'5 3'0 3'5 4'0 4'5 5'0 5'5 6'0 6'5 7'0

Wavenumber

Grid Convolution Weights

Longitude

Figure 18: Filter function and grid-space weighting coefficients using standard FFT on a

stretched grid.

35
1.00

Eigenvalues (Uniform Grid)

0.50

0.00

-0.50

- 1.00

-1.50

-2.00-

-2.50

-3.00

-3.50

-4.00

-4.50

-5.00 1'0 2'0 3'0 4'0 5'0 6'0 7'0 8'0 9'0 160 _{0 1_0 150 14.0

Mode Index

Eigenvalues (Stretched Grid)

5.00

0.00

-3.00

-6.00-

-9.00-

-12.00-

-15.00-

-18.00 -

-21.00

-24.00

-27.OO

1--_02'0 #0 4'0 s'o 6'0 7'0 8'0 9'0 160 li0 1_0 1._0t_0

Mode Index

Figure 19: Eigenvalues associated with the uniform and stretched grid.

36
Idode IndeN 1

Mode Index 20

Ik_de In,dex 50

Mode Index 5

Mode Index 25

Mode Index 60

Mode Tndex t0

i

J

Mode I_x 30

Mode Index 70

Uode Index 15

Mode Index 35

Mode h_x 80

Uo_ I_ex gO Mode ii'_4x 110 Index I_

Mode Index 130 Mode Index 13._. Mode Index T_.O

Fig_]re 20: Eigenvectors associated with the uniform grid.

37
lndol 1

fndcx Z0

Index

Uode tndel 90

Uode In_x 130

Figure 21:

Hode )_ 100

Mode lndexl_ Mode Index 140

Uo_ I_x 15

Eigenvectors associated with the stretched grid.

q_r

Index

llode Index 8_

Uodo Index 120

Mode Index 144

38
1.1

1.0

0.9

0.8-

0.7

0.6

0.5

0.4

0'5 t

0.2

0.1

0.0

0

Stretched GHd Convolution Filter Lot: 85N

Spectral Filter Function

I'0 (5 2'0 2'5 3'0 3'5 4'0 4'5 5'0 5'5 6'0 6'5 7'0

Wovenumber

Grid Convolution Weights

0.90

0.80

0.70

0.60

0.50-

0.40

0.30

0.20

0.10

0.00-

- 80

ZJj LZ__j

-1'2o -_o 6 6'0 1_o

Longitude

)

Figure 22: Filter function and grid-space weighting coefficients using the convolution filter

on a stretched grid.

39
!

I

Figure 23: Timeseries of single harmonics on a stretched grid using an unstable timestep

with convoiut:ion filtering.

4O
GEOS-2 GCM Convolution Filter

Initial Temperature (cint: 5 K)

0.02 -

0.04 -

0.1

0.2

0.4

1

2

3

20

30

200-

300 --__

90S 60S 30S EQ 3ON 60N

0.02 Day 5 Temperature Change (cint: 2 K)

0.04 4

0.1

02

0,4

1.

2-

8-

20-

30

200 -

300 -

90N 90S 60S .30S EO 3ON 60N 90N

Day 5 Temperture Change (cint: 5 K) Day 5 Zonal U-Wind (cint: 15 m/s)

%

Figure 24: Initial temperature and Day 5 temperature change (top panel), and 0.2 mb

temperature change and zonal wind field (bottom panel) using the convolution filter on the

stretched grid.

41
J

"x j

tl

,i

)r

J

o

f_

tn_

:2::

t

Figure 25: Day 5 sea-level pressure and 300-mb height, fields for the standard FFT filter

and the convolution filter.

42
Grid Convolution Weights (Empirical Method)

0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.00

-180

I

-1'2o -_o 6 6'o i_o

Longitude

0.0040

0.0035

O.OO30

0,0025

0.0020-

0.0015-

0.0010

0.0005

0.0000

-O.0005

-0.0010

-0.0015]

-180

Empiricol Method - Eigenvalue Method

0

-1'2o -_o 6 6'o 1_o

Longitude

Figure 26: Grid-space weighting coefficients using the empirical method, and compared

with the eigenvalue method.

43
o o

o

4

Figure 27: Timeseries of sfi_gle harmonics on a stretched grid with weights obtained using

the empirica,| method.

44
Eigenvalue Method Wavenumber 1 Empirical Method Wavenumber 1

Eigenvalue Method Wavenumber 3

_,_c_  ̧ p _11 I _1

Eigenvalue Method Wavenumber 5

VVVVV

Eigenvalue Method Wavenumber 7

t'

;. : .5 ̧ [ ,/,; , , ,,

Empirical Method Wavenumber 3

]

Empirical Method Wavenumber 5

a/ j ;

I

I

f

!

Empirical Method Wavenumber 7

Figure 28: Single harmonics at time = 30 hrs comparing the eigenvalue and empirial methO8S.

45
=_

"r_"_ I _ I_ J_i

}

_I / I_E _ _ °

Figure 29: Filter function and grid space weights for an 8th-order Shapiro filter on the

uniform and stretched grid.

46
References

Fox-Rabinovitz, M., G.L. Stenchikov, M.J. Suarez, and L.L. Takacs, 1997: A FiniteDifference GCM Dynamical Core with a a.rlable-Resolutlon Stretched Grid. Mon. 7- •

Wea. Rev., 125, 2943-2968.

Gates, W. Lawrence, 1992: AMIP: The Atmospheric Model Intercomparison Project. Bull.

Am Met. Soc., 73, 1962-1970

Golub, G. and C. \.ran Loan, 1989: Matrix Computations. Johns Hopkins University Press, Baltimore, MD.

Held., I.M., and M.J. Suarez, 1994: A benchmark calculation for the dry dynamical cores of

atmospheric general circulation models. Bull. Amer. Meteor. Soc., 75, 1825-1830.

Molod, Andrea, H.M. Helfand, and L.L. Takacs, 1996: The Climatology of Parameterized

Physical Processes in the GEOS-1 GCM and their Impact on the GEOS-1 Data

Assimilation System. J. Climate, 9, 764-785.

Phillips, N.A., 1959: An example of non-linear computational instability. The Atmosphere

and the Sea in Motion, Rossby Memorial Volume, Rockefeller Institute Press, .501- 504.

Rood, R., 1996: Algorithm Theoretical Basis Document for Goddard Earth Observing

System Data Assimilation System (GEOS DAS) With a Focus on Version 2. Data

Assimilation Office, NASA, Goddard Space Flight Center, Greenbelt, MD.

Schubert, S.D., J. Pfaendtner and R. Rood, 1993: An assimilated data set for Earth Science

applications, Bull. Am Met. Soc., 74, 2331-2342

Schubert, S.D., C.K. Park, C.Y. Wu, W. Higgins, Y. Kondratyeva, A. Molod, L. Takacs, M.

Seablom, and R. Rood, 1995: A Multiyear Assimilation with the GEOS-1 System:

Overview and Results, NASA Technical Memorandum 104606 Volume 6, NASA,

Goddard Space Flight Center, Greenbelt, hiD.

Schubert, S.D., and R. Rood, 1995: Proceedings of the Workshop on the GEOS-1 Five-3%ar

Assimilation, NASA Technical Memorandum 104606 Volume 7, NASA, Goddard

Space Flight. Center, Greenbelt, MD.

Shapiro, R., 1970: Smoothing, filtering and boundary effects. Rev. Geophys. Space Phys., 8, 359-387.

Suarez, M. J., and L. L. Takacs, 1995: Documentation of the Aries/GEOS Dynamical Core

Version 2, NASA Technical Memorandum 104606 Volume 5, NASA, Goddard Space

Flight Center, Greenbelt, MD.

Takacs, L. L. , A. Molod, and T. Wang, 1994: Documentation of tile Goddard Earth

Observing System (GEOS) General Circulation Model-Version 1. NASA Techni47
calMemorandum104606Volume1, GoddardSpaceFlight Center,Greenbelt,MD

20771.

Takacs,L.L., andM.J. Suarez,1996:Dynamicalaspectsof climatesimulationsUsingthe

GEOSGeneralCirculationModel,NASA TechnicalMemorandum104606Volume

10,NASA,GoddardSpaceFlight Center, Greenbelt, MD 20771.

J

48

.i

i

i

i

|
Previous Volumes in This Series

Volume 1

September 1994

Documentation of the Goddard Earth Observing System

(GEOS) general circulation model - Version 1

L.L. Takacs, A. Molod, and T. Wang

Volume 2

October 1994

Direct solution of the implicit formulation of fourth order

horizontal diffusion for gridpoint models on the sphere

Y. Li, S. Moorthi, and J.R. Bates

Volume 3

December 1994

An efficient thermal infrared radiation parameterization for

use in general circulation models

M.-D. Chou and M.J. Suarez

Volume 4

January 1995

Documentation of the Goddard Earth Observing System

(GEOS) Data Assimilation System - Version 1

James Pfaendtner, Stephen Bloom, David Lamich,

Michael Seablom, Meta Sienkiewicz, James Stobie,

and Arlindo da Silva

Volume 5

April 1995

Documentation of the Aries-GEOS dynamical core: Version 2

Max J. Suarez and Lawrence L. Takacs

Volume 6

April 1995

A Multiyear Assimilation with the GEOS-1 System:

Overview and Results

Siegfried Schubert, Chung-Kyu Park, Chung-Yu Wu,

Wayne Higgins, Yelena Kondratyeva, Andrea Molod,

Lawrence Takacs, Michael Seablom, and Richard

Rood

Volume 7'

September 1995

Proceedings of the Workshop on the GEOS-1 Five-Year

Assimilation

Siegfried D. Schubert and Richard B. Rood

49
zz

=

m

===

i

Volume 8

March 1996

Volume 9

March 1996

Volume 10

April 1996

Volume 11

May 1997

Volume 12

August 1997

Volume 13

December 1997

Volume 14

August 1998

Volume 15

june 1999

Documentation of the Tangent Linear Model and Its Adjoint

of the Adiabatic Version of the NASA GEOS-1 C-Grid GCM:

Version 5.2

Weiyu Yang and I. Michael Navon

Energy and Water Balance Calculations in the Mosaic LSM

Randal D. Koster and Max J. Suarez

Dynamical Aspects of Climate Simulations Using the GEOS

General Circulation Model

Lawrence L. Takacs and Max J. Suarez

Documentation of the Tangent Linear and its Adjoint Models

of the Relaxed Arakawa-Schubert Moisture Parameterization

Package of the NASA GEOS-1 GCM (Version 5.2)

Weiyu Yang I. Michael Navon, and Ricardo Todling

Comparison of Satellite Global Rainfall Algorithms

Alfred T.C. Chang: and Long S. Chiu

Interannual Variability and Potential Predictability in Reanalysis Products

Wie Ming and Siegfried D. Schubert

A Comp_ison of GEOS Assimilated Data with FIFE

Observations

Michael G. Bosilovich and Siegfried D. Schubert

A Solar Radiation Parameterization for Atmospheric Studies

Ming-Dah Chou and Max J. Suarez

5O

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1. AGENCY USE ONLY (Leave b/ank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

November 1999 Technical Memorandum

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Technical Report Series on Global Modeling and Data Assimilation

Volume 16--Filtering Techniques on a Stretched Grid General Circulation

Model

6. AUTHOR(S)

Lawrence L. Takacs, William Sawyer, Max J. Suarez, and

Michael S. Fox-Rabinowitz

7. PERFORMING ORGANIZATION NAME(Si AND ADDRESS (ES)

Climate and Radiation Branch

Goddard Space Flight Center

Greenbelt, Maryland 20771

9. SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS (ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

Code 910

91-92910167-26

8. PEFORMING ORGANIZATION

REPORT NUMBER

2000-00443-0

10. SPONSORING / MONITORING

AGENCY REPORT NUMBER

TM- 1999-104606, Vol. 16

11. SUPPLEMENTARY NOTES

L. Takacs: General Sciences Corporation, Laurel, Maryland; W. Sawyer and M. Fox-Rabinowitz: University of

Maryland, College Park, Maryland

12a. DISTRIBUTION I AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE

Unclassified-Unlimited

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13. ABSTRACT (Maximum 200 words)

This report documents the techniques used to filter quantities on a stretched grid general circulation model.

Standard high-latitude filtering techniques (e.g., using an FPT to decompose and filter unstable harmonics at

selected latitudes) applied on a stretched grid are shown to produce significant distortions of the prognostic state

when used to control instabilities near the pole. A new filtering technique is developed which accurately accounts for the non-uniform grid by computing the eigenvectors and eigenfrequencies associated with the

stretching. A filter function, constructed to selectively damp those modes whose associated eigenfrequencies

exceed some critical value, is used to construct a set of grid-spaced weights which are shown to effectively filter

without distortion. Both offiine and GeM experiments are shown using the new filtering technique. Finally, a

brief examination is also made on the impact of applying the Shapiro filter on the stretched grid.

14. SUBJECT TERMS "

Numerical Methods, Atmospheric Models, Regional Models, Filtering Techniques,

Finite-Difference Methods

17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION

OF REPORT OF THIS PAGE

Unclassified Unclassified

NSN 7540-01-280-5500

19. SECURITY CLASSIFICATION

OF ABSTRACT

Unclassified

15. NUMBER OF PAGES

58

16. PRICE CODE

20. LIMITATION OF ABSTRAC I

UL

Standard Form 298 (Rev. 2-89)

Prescribed by ANSI Std z3g.18

298-102

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