which has the very slowly converging Fourier series

∞

(’1)j+1

2

Sw(x) ≡ ∀x (18.83)

sin(jx)

π j

j=1

Fig. 18.15 shows ¬ltering of both ¬xed order and spatially-varying order. With 100

terms, the accuracy is greater than 1/100,000 over 90% of the spatial interval. Fig. 18.16 is

a zoom plot showing a small portion of the spatial interval. It is indeed possible to recover

spectral accuracy outside of a small neighborhood of the discontinuity. The best ¬lter is of

spatially-varying order and is of high order away from the discontinuity.

18.20.6 Topographic Filtering in Meteorology

A wide variety of ¬lters have been employed to smooth topography for spherical harmon-

ics models. Because they are high order ¬lters, the exponential ¬lter of Hoskins(1980) and

Sardeshmukh and Hoskins(1984) and the spherical splines of Lindberg and Broccoli(1996)

look good, at least when the tunable order parameter is suf¬ciently large. Bouteloup(1995)

employed an optimization approximation very similar to spherical splines, but with more

elaborate smoothness penalties. In contrast, Navarra, Stern and Miyakoda™s (1994) Cesaro

¬lter is a low order ¬lter and therefore tends to smooth even the largest spatial scales too

much. However, Navarra et al. nevertheless obtained a smoothed topography that was

much more satisfactory than the truncated, un¬ltered series, which emphasizes that for

a function with a discontinuity or discontinuous slope, almost any ¬ltering is better than

none.

The spurious ocean valleys have been especially annoying to numerical modellers be-

cause darn it, one really ought to be able to represent a ¬‚at ocean surface as a ¬‚at surface.

A child with a ruler and a pencil can draw an approximation much better than the trun-

cated spherical harmonics series! This suggested to Navarra, Stern and Miyakoda(1994),

Bouteloup(1995) and Lindberg and Broccoli(1996) that it would be desirable to employ a

non-isotropic ¬lter which is strong over water, and weak over land. Lindberg and Broccoli,

for example, restricted the integral of the Laplacian to water only so that the approximation

was penalized for oscillations over water, where it should be ¬‚at, but not over land, where

the oscillations might represent the real hills and valleys.

This differential land/water ¬lter is sound strategy. However, distance from the singu-

larity ” for topography, it is distance from the coast where the slope of the atmospheric

boundary is discontinuous ” also should control the order p of the ¬lter.

CHAPTER 18. SPHERICAL & CYLINDRICAL GEOMETRY

424

Table 18.7: Filtering and Smoothing on the Sphere

References Comments

Hoskins(1980) exponential ¬lter

Sardeshmukh&Hoskins(1984) Isotropy Theorem; exponential ¬lter

Hogan&Rosmond(1991) comparison of several ¬lters;

applied Lanczos ¬lter to Navy forecast model

Navarra&Stern 2D isotropic ¬lter (Cesaro); 2D physical space

&Miyakoda(1994) ¬lter applied to oceans only

Bouteloup(1995) optimization method with multiple penalty constraints

Lindberg&Broccoli(1996) spherical spline; orography & precipitation

Gelb(1997) Gegenbauer regularization, restricted to discontinuities

Gelb&Navarra(1998) parallel to either latitude or longitude lines

Clearly, there is a need for further experimentation and creativity. Gelb (1997) and

Gelb and Navarra(1998) used an ingenious regularization procedure which replaces a high

order spherical harmonic series by a low order Gegenbauer polynomial series. Unfortu-

nately, their method requires that the discontinuities lie on the walls of a rectangle or a

union of rectangles. In its present state, the Gegenbauer method is not suf¬ciently gen-

eral to accomodate real topography. However, this strategy, which can be applied to other

spectral series, is still under rapid development and perhaps its present de¬ciencies will

be overcome. In any event, it shows that the algorithm developers have not yet run out of

ideas.

Errors in ErfcLog Filter for Sawtooth Function

0

10

-5

10

-10

p=8

10

x-Varying

Order

-15

10

0 0.2 0.4 0.6 0.8 1

x/π

Figure 18.15: Errors in ErfcLog-¬ltered, 100-sine approximations to the piecewise linear or

“sawtooth” function. Thin line: ¬xed ¬lter order p = 8. Thick line: optimal, spatially-

varying p.

18.21. RESOLUTION OF SPECTRAL MODELS 425

Errors in Spatially-Varying ErfcLog Filter: Sawtooth

0

10

-1

100 Fourier terms

10

-2

10

-3

10

-4

10

-5

10

0.9 0.92 0.94 0.96 0.98 1

x/π

Figure 18.16: Zoom plot of the errors in ErfcLog-¬ltered (one hundred term) sine series for

the sawtooth function. The discontinuity of Sw(x) is at x = π (right edge of the graph).

The thin solid curve shows the error when the order p is varied with distance from the dis-

continuity in an optimal way. The thick dashed curve is an upper bound or “envelope” for

the errors, showing (on this log-linear plot) that the error falls exponentially with distance

from the singularity at x = π.

18.21 Resolution of Spectral Models

Filtering and dealiasing raise an obvious question: How small are the features that a spec-

tral model can resolve with a given N (Laprise, 1992, and Lander and Hoskins, 1997)?

The central argument of Lander and Hoskins is that a graph of spectral coef¬cients

versus degree n can be divided into three regions. (For a Fourier series, n is wavenumber;

m

for spherical harmonics, n is the subscript of the harmonic Yn .) These regions are:

1. “Unresolved scales” < ’ ’ ’’ > “Dealiasing pad”

3

(18.84)

N <n< N

2

2. “Unbelievable scales” < ’ ’ ’’ > “Discretization-addled” wavenumbers

(18.85)

NP < n < N

3. “Believable scales”

n ¤ NP (18.86)

The “dealiasing pad” or what Lander and Hoskins call simply the “unresolved scales”

arise because the usual weather forecasting model has roughly three-halves as many points

in each of latitude and longitude as the truncation N . This allows the integrals of the

CHAPTER 18. SPHERICAL & CYLINDRICAL GEOMETRY

426

Galerkin method to be evaluated without error and is consistent with the Orszag Two-

Thirds Rule for removing aliasing error. However, these wavenumbers with n > N are not

retained in the truncation.

Lander and Hoskins™ “unbelievable scales” are those wavenumbers which are kept in

the trucation, but are untrustworthy. Unfortunately, spectral coef¬cients near the trun-

cation N are corrupted by a wide variety of errors. The schematic illustrates three such

error-mechanisms, labelled in italic fonts.

First, if a time-dependent hydrodynamics model is not strongly damped, it tends to

develop accumulate noise near the truncation limit, causing the graph of spectral coef¬-

cients to develop an upward curl, as shown by one of the dashed curves in Fig. 18.17.

Second, if the model incorporates a strong arti¬cial viscosity, then the spectral coef¬cients

may be damped to unrealistically small amplitude as shown by the downward curling

dashed curve. If damping is chosen correctly, the computed coef¬cients will curl neither

up nor down, but will be of roughly the correct order of magnitude. However, “ correct

magnitude” is as unsatisfactory to the arithmurgist as to the brain surgeon.

In addition to these sources of error, there is a third coef¬cient-addling in¬‚uence which

is always present, even for one-dimensional boundary value problems: the “discretization

error” ED de¬ned by De¬nition 9 in Chapter 2 and illustrated in Tables 3.1 and 3.2. This

error arises because the coef¬cients that we compute are always in¬‚uenced by the coef-

¬cients that are not computed. Consequently, all the calculated coef¬cients have absolute

errors which are the order of magnitude of the truncation error ET , which is the mag-

nitude of the uncomputed coef¬cients for n > N . Because the low degree coef¬cients are

large compared to the truncation error, their relative error is small and these spectral coef¬-

cients are “believable”. The coef¬cients near the truncation limit, however, are of the same

magnitude as the discretization error ED , and therefore have such large relative errors as to

be “unbelievable”.

Lander and Hoskins suggest two strategies. One is to ¬lter the wind and pressure, etc.,

that are input to the physics parameterizations to include only the “believable” scales. This

will increase cost, but reduce noise.

Their second strategy is that the physical parameterizations of chemistry, hydrologic

cycle and so on should be computed only on a coarse grid restricted to these believable

scales. (This explains the notation for the boundary of these wavenumber, NP ” “P” for

parameterizations and also “P” for physics.) Instead, it is the usual practice to compute the

chemistry and physics on the entire grid. Although the ozone concentration on such a ¬ne

grid could be in principle be expanded with harmonics up to N = (3/2)N , only coef¬cients

in the triangular truncation T N are ever calculated. Since the non-hydrodynamic calcula-

tions account for at least half the total running time, one could gain much by restricting

chemistry and physics to the “believable” scales only.

One drawback of Lander and Hoskins™ second strategy is the need to interpolate to and

from the coarse grid [where the parameterizations are evaluated] to the ¬ne grid [used

to compute nonlinear products]. However, the parameterizations are so expensive that it

seems likely that there would be signi¬cant savings even so. Furthermore, the ¬ne scales

of the ozone, water vapor and so on force the dynamics at “unbelievable” scales; the high

resolution parameterizations may actually be reducing accuracy! However, neither of their

suggestions has been tested in a real code.

One important practical issue is: What is NP , the highest believable wavenumber? Lan-

der and Hoskins discuss ¬ve choices. Their preference is for a length scale rG which is

de¬ned in terms of the “point spread function”. [discrete approximation to the Dirac delta-

function] as the distance from its global maximum to the ¬rst minimum . This is equivalent

18.21. RESOLUTION OF SPECTRAL MODELS 427

to

(3/2)

NP ≈ 0.63N ≈ (18.87)

N

2.4

This would reduce the total number of points for the non-hydrodynamic parameterizations

by a factor of 5.76 = 2.42 .

However, Lander and Hoskins hedge, and suggest that sometimes a larger NP may be

desirable for safety™s sake. The truth is that “believability” is likely to be highly problem-

dependent and model-dependent. Nevertheless, it is only by understanding that there are

three regions in the wavenumber spectrum that we can hope to make rational choices in

model architecture, ¬lters and so on.

Unbelievable

Scales

Believable Unresolved

Scales Scales

Spectral

Blocking

Discretization

Error ED

Artificial

Damping

NP N (3/2)N

Figure 18.17: Schematic of Fourier or spherical harmonics coef¬cients versus degree N .

The thick solid line shows the exact coef¬cients for n ¤ N ; the dotted line shows the exact

coef¬cients which are neglected when the series is truncated at degree N . The “believable”

coef¬cients are accurate; the length scales of the corresponding spherical harmonics de¬ne

the “believable scales”. The “unbelievable” scales are corrupted by numerical dif¬culties.

Three villians (labelled in italics) are illustrated by thick dashed curves, which show how

the computed coef¬cients are distorted by each pathology. The absolute errors for the

discretization error are roughly the same for all coef¬cients in the truncation (thin jagged

line), but the relative errors are large only for n ≈ N .

CHAPTER 18. SPHERICAL & CYLINDRICAL GEOMETRY

428

18.22 Vector Basis Functions: Vector Spherical Harmonics &

Hough Functions

When solving a system of equations for a vector of unknown ¬elds, there is no reason

why one cannot use vector basis functions. The eigenmodes of the linearized system will