Dilts (1985) Modern reinvention of Schuster™s scheme

Brown (1985) Fast spherical harmonic transform

Orszag (1986) Fast transform for any basis satisfying

3-term recurrence relation

Elowitz&Hill Compares fast transforms of Dilts (1985),Brown (1985)

&Duvall (1989) but conclusion is false

Alpert&Rokhlin(1991) Fast Legendre-to-Chebyshev transform;

does not generalize to associated Legendre

Boyd (1992c) Showed FMM could be applied to non-Chebyshev basis

Dutt&Rokhlin (1993,1995) FMM nonequispaced FFT

Jakob-Chien&Alpert(1997) Grid-to-spherical-harmonics-to-grid

Yarvin&Rokhlin(1998) projective ¬lter for longitude/latitude

time-dependent double Fourier series

Foster&Worley(1997) Comparison of parallel spherical harmonic transforms

Healy&Rockmore& Fast spherical harmonic transform;

Kostelec&Moore(1999) freeware at www.cs.dartmouth.edu/ geelong/sphere

Swarztrauber&Spotz(2000) “Weighted Orthogonal Complement” algorithm reduces

Spotz&Swarztrauber(2000) storage by O(N ); faster than alternatives

because algorithm mostly stays in the on-chip cache

18.13 Equiareal Resolution and the Addition Theorem

De¬nition 40 (Equiareal Resolution) A numerical algorithm which has the property that its

numerical characteristics are invariant to a rotation of the north pole of the coordinate system so

that features of a given size are resolved equally well or badly regardless of whether they are located

at the poles, equator, or anywhere in between.

A so-called “triangular truncation” of a spherical harmonic basis has this property be-

cause of the following theorem and its corollaries.

Theorem 36 (ADDITION THEOREM:) Let (» , θ ) denote longitude and latitude as measured

relative to a set of coordinate axes rotated with respect to the original axes. Let (», θ) denote the

angles measured relative to the original, unrotated coordinate system. Then

n

(n)

m m

(18.45)

Yn (» ,θ)= Dmm (R) Yn (», θ)

m =’n

(n)

where the coef¬cients Dmm are functions of the rotation angle.

COROLLARY 1: The spherical harmonics of degree n form a (2n+1)-dimensional representation of

the continuous rotation group.

COROLLARY 2: A triangular truncation of spherical harmonics, that is, keeping only those har-

monics such that

n¤N all m (18.46)

gives equal resolution to equal areas on the globe, regardless of where those areas are located.

PROOF: Eq. (18.45) is a classical result discussed in most quantum mechanics texts such as

Merzbacher (1970).

18.14. SPHERICAL HARMONICS: LIMITED-AREA MODELS 409

The ¬rst corollary is merely a way of restating the theorem in the jargon of group theory.

m

When we rotate the pattern that is a particular spherical harmonic Yn , we create a new

function of latitude and longitude. Like all such functions, this may be expanded in a

spherical harmonics series, but there is no obvious reason why these series should not

contain an in¬nite number of terms. The theorem shows, however, that the series contains

at most (2n+1) terms, and all the non-zero harmonics have the same degree n as the function

that is rotated. Thus, the spherical harmonics of degree n form a closed subset under rotation

through arbitrary angles ” and this is what is required to form a “representation of the

rotation group”.

The collection of spherical harmonics that are retained in a “triangular truncation”

therefore are closed under rotation, too. This implies the property of equiareal resolution.

In the words of Orszag (1974), “The basic mathematical reason is that, under arbitrary ro-

tations, expansions truncated at harmonics of degree N remain truncated at degree N so

that the resolution of such series must be uniform over the sphere.”

18.14 Variable Resolution Spherical Harmonics Models

“Limited-area” weather forecasting models offer the advantages of very high resolution

over a small portion of the earth™s surface ” higher than would be affordable if this resolu-

tion were extended over the entire globe. Most national weather services run limited-area

models targeted at the country that pays for them. Limited-area models are also used to

track tropical hurricances, which have such small scales that it is dif¬cult for uniform res-

olution global models to track them accurately.

One obvious strategy is to employ a dense but uniform grid over a small portion of the

globe and specify in¬‚ow-out¬‚ow conditions at the sides, usually from climate data or from

interpolation of a global model. Such non-global limited-area codes are in wide use, but

there are dif¬culties. One is that specifying lateral boundary conditions turns out be very

hard; rather elaborate blending procedures are necessary to avoid corruption of the high-

resolution data within the domain by the low-resolution data at the boundaries. Moreover,

as models incorporate more and more physics and chemistry, it has become increasingly

painful to maintain two completely separate models, one global and one limited-area, us-

ing different numerical schemes, tuning parameters and so on.

An alternative is to use the global model as the limited-area model, too. This can be

done by a smooth change of coordinates that maps the surface of the sphere into itself. In

physical space, the transformed grid has a high density of points over the region of interest,

but decreases to lower and lower density as one moves away from the target region. No

arti¬cial sidewall boundary conditions are needed because there are no sidewalls. A single

model can be used for both global and regional forecasting by switching on or off a few

metric factors in the evaluation of derivatives.

Schmidt(1977, 1982) proposed a conformal sphere-to-sphere mapping which he tested

successfully in a simple code. With re¬nements, this has been adopted by M´ t´ o-France

ee

for its primary operational weather forecasting code (“Arpege”) as described by Courtier

& Geleyn(1988), Courtier et al.(1991), and D´ qu´ &Piedelievre(1995). Hardiker(1997) has

ee

shown that such mappings are equally effective for tracking hurricanes.

These variable-resolution models have been suf¬ciently successful that the desire to

combine global and regional models into a single code is not a major threat to continued use

of spherical harmonics for weather forecasting. Rather, the big Thing-That-Goes-Bump-in-

the-Night is the switch to massively parallel machines, which may or may not be happy

doing PMMTs for very large N .

CHAPTER 18. SPHERICAL & CYLINDRICAL GEOMETRY

410

Table 18.6: Variable Resolution Spherical Harmonic Models

References Comments

Schmidt(1977,1982) Conformal mapping to give high local resolution

to track cyclones

Courtier&Geleyn(1988) Variable resolution (mapped) weather prediction

Courtier et al.(1991) Arpege project at M´ t´ o-France:

ee

global weather model with high resolution in Europe

D´ qu´ &Piedelievre(1995)

ee Experience with Arpege variable resolution weather model

Hardiker(1997) Conformal mapping for variable resolution

18.15 Spherical Harmonics and Physics

The spherical harmonics are the eigenfunctions of the two-dimensional Laplacian operator

Yn = ’n(n + 1) Yn

2 m m

(18.47)

where

‚2 ‚2

‚ 1

≡ 2 + cot(θ)

2

(18.48)

+

sin2 (θ) ‚»2

‚θ ‚θ

Because of this, the spherical harmonics are fundamental solutions to many problems in

physics.

In geophysics, for example, Haurwitz (1940) showed that the streamfunction for lin-

ear, barotropic Rossby waves was proportional to a spherical harmonic, that is to say, the

spherical harmonics are the quasi-geostrophic normal modes of the earth™s atmosphere.

Similarly, Longuet-Higgins (1968) has shown that for gravity waves in the barotropic limit,

the velocity potential is proportional to a spherical harmonic. In both cases, elementary

identities show that the velocities and other quantities may be written as pairs of spherical

harmonics.

Barrett (1958) observed that since the (purely westward) phase velocity is

2„¦

cphase = ’ [Rossby-Haurwitz waves] (18.49)

n (n + 1)

one may synthesize a uniformly propagating disturbance from an arbitrary sum of spheri-

cal harmonics of the same degree n. In particular, an “eccentric” spherical harmonic, that is,

a spherical harmonic rotated so that its pole at some arbitrary latitude θp , is a steadily prop-

agating Rossby wave because of the addition theorem of Sec. 18.13. In more recent times,

such rotated spherical harmonics have been the basis for constructing nonlinear modons

in spherical geometry as explained by Tribbia(1984b).

One could multiply these examples with dozens from other ¬elds (Morse and Fesh-

bach, 1953). This close and intimate connection between the spherical harmonics and the

physics, as well as their good numerical properties, have helped to make spherical har-

monics popular.

18.16 Asymptotic Approximations I: Polar-Cap and Bessel

Functions

At high latitudes,

cos(θ) ≈ 1 sin(θ) ≈ θ (18.50)

& θ 1

18.16. ASYMPTOTIC APPROXIMATIONS, I 411

and the horizontal Laplacian operator becomes

‚2 1 ‚2

1‚

≈

2

(18.51)

+ +2 2

‚θ2 θ ‚θ θ ‚»

which is identical in form with the Laplacian in plane polar coordinates if we identify θ

with radius r and » with the polar angle. For a given zonal wavenumber m, the second

»-derivative in (18.51) becomes multiplication by (’m2 ) and the eigenequation 2 Yn =

m

’n(n + 1) becomes Bessel™s equation:

d2 m2

1d

+ k’ 2

2

(18.52)

+ Jm (kθ) = 0

dθ2 θ dθ θ

where

k≡ (18.53)

n (n + 1)

A more heuristic (but equally correct) way of justifying this planar, polar coordinate

approximation is to simply look down on a globe from above one of the poles. As one

moves closer and closer to the poles, the sphere ¬‚attens into a plane, and the meridians

form a network of radial lines.

Since this “polar-cap” approximation becomes exact near the pole, the Bessel functions

must be consistent with the known behavior of the spherical harmonics at the pole. We

m

note that Jm (r) has an m-th order zero at r = 0 ” mimicing the m-th order zero of Yn at

m

the pole. If we expand Yn as a power series in θ, we ¬nd that the expansion contains only

every other power of θ; even powers of θ when m is even and odd powers of θ (because of

the “parity factor”, sin(θ)) when m is odd. Similarly, the expansion of Jm (r) is in alternating

powers of r; since the expansion begins with rm , the powers are the odd powers of r when

m is odd and the even powers of r when m is even.

The Bessel function (for m > 0) rises monotonically to a turning point at r ≈ m and

then oscillates for larger values of its argument. Fig. 18.9 illustrates three typical Bessel

functions. The turning colatitude is

m

θt ≈ [“turning colatitude”] (18.54)

n (n + 1)

When m is roughly equal to n (recall that n cannot be smaller than m), then the predicted

turning latitude is θt ≈ 1, that is, only about 30 degrees from the equator. The polar cap

approximation is not accurate that close to the equator, so what the Bessel approximation

tells us in this case is simply that the spherical harmonic has most of its amplitude at low

latitudes (where we shall use a different approximation given in the next section). Near the

m

pole, the Bessel approximation is still valid ” but the behavior of both Jm (kθ) and Yn is

dominated by the m-th order zero at the pole, so the polar cap approximation does not tell

us anything we did not already know.

When n m, however, the predicted turning point is close to the poles and the Bessel

approximation is more useful. There is some region equatorward of θt where the Bessel

approximation is legitimate, and we can use the asymptotic approximation

1

4 (2m+1) π

4

1 1

[n(n+1)] θ ∼ cos [n(n+1)] 2 θ ’ (18.55)

Jm θ θt

2

2 n(n+1) θ 2

π 4

Eq. (18.55) shows explicitly that a spherical harmonic oscillates equatorward of its turning

latitude. It also shows that the amplitude of the oscillation decreases in the direction of

√

the equator (as 1/ θ) ” the maximum of the harmonic is just equatorward of the turning

latitude.

The “polar-plane” approximation, which invented by B. Haurwitz, has been systemat-

ically developed for geophysical applications (Bridger and Stevens, 1980).

CHAPTER 18. SPHERICAL & CYLINDRICAL GEOMETRY

412

Figure 18.9: Three Bessel functions, illustrating the behavior of spherical harmonics near

the pole, r = 0. Note that J10 (r), which is representative of high degree harmonics, has a

“turning latitude” at about r = 10: the function is exponentially decaying towards the pole

for small r, and oscillatory for larger r. (r ≡ n(n + 1) θ increases toward the equator.)

18.17 Asymptotic Approximations, II: High Zonal Wavenum-

ber & Hermite Functions

Abramowitz & Stegun (1965) give the asymptotic approximation

√

1

Pm+n [cos(θ)] ∼ q exp(’ mφ2 ) Hn ( m φ)

m

(18.56)

2