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Sw(x + 2πm) |x| ≥ π, m = integer
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

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. 87
( 136 .)



>>