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### Retrospective on a 24-year Collaboration with Moyses Nussenzveig on Mie Theory and its Applications to Climate

Warren Wiscombe

NASA Goddard

Questions

- How did I, a climate person, get involved with Prof. Nussenzveig and Complex Angular Momentum Theory?
- What is this Complex Angular Momentum Theory?
- What applications in Earth science is it useful for?
- What kind of numerical machinery is needed?
- The problems we worked on?

Moyses 70th Conference

Myself

- A CalTech applied mathematician caught up in the environmental movement of the 1960s, winding up in the middle of the nascent climate revolution
- ARPA Climate Dynamics Program needed a radiation expert in 1971, so I volunteered
- Many climate problems of the 1970s had a radiative basis, and at the core of the radiation was either...

spectroscopy, or

Mie theory!

Mie theory was crucial for aerosol and cloud forcing.

• My applied math skills were perfect for Mie theory.

Moyses 70th Conference

Why are aerosols and clouds important for climate?

- Clouds exert a substantial cooling effect (15–20 W/m2) compared to 4 W/m2 for doubled CO2
- The huge differences among models reported by IPCC are due mainly to differing cloud treatments
- Small changes in mean drop size (say from 8 up to 10 microns) can have a climatic effect as big as doubled CO2
- Climate models predict too much warming since 1880, and the only way to rescue them is aerosols
- Biomass burning aerosols have lately come under intense scrutiny

Moyses 70th Conference

What Mie size parameter ranges are important for climate?

x = 2p r / l = Mie size parameter

- Clouds: x = 1 to 1000
- Aerosols: x = 1 to 20
- No good approximations were available in these size parameter ranges, except for special situations
- Mie computations were big computer hogs in the 1970s, and remain so today if you want to follow every resonance, ...
- so, people arbitrarily truncated drop size distributions to lower their Mie burden!

Moyses 70th Conference

Myself and Mie Theory

- Mie theory was analytically complete, but...
- Its numerical implementation had been a hot subject in the 1960s, when, among other things, ...
- Rayleigh’s observation about Bessel function recurrences being unstable in the upward direction was re-discovered.
- Dave’s 1969 computer program for Mie calculations quickly dominated the field.
- I studied Mie calculations starting in 1971 and made significant improvements on Dave which I made public circa 1979, and ...
- made a computer program available which supplanted Dave’s & became a new standard.

Moyses 70th Conference

Dave IBM Report, 1969

one of those rare seminal works that was never formally published

Moyses 70th Conference

Dave IBM Report – 2

Moyses 70th Conference

S1 =S cn [ an(m,x) pn (q) + bn (m,x) tn (q) ]

n=1

Mie Theoryr = sphere radius

l = wavelength

x = 2p r / l = Mie size parameter

m = mre ± i mim = refractive index

Far-field scattered electric field:

E = S(x,m,q) exp(ikR – iwt) / kR

nmax = x + 4x1/3

cn = (2n+1) / [n(n+1)]

an, bn depend on Bessel functions

pn, tn are related to Legendre polynomials

Moyses 70th Conference

Mie Series: Key Facts

- Number of terms ~ x
- Not “normal” convergence: terms more or less same size till near end, then crash to zero
- Rapid variation in angle because Legendre poly’s oscillate more and more as n increases
- Rapid variation in x because Bessel functions oscillate more and more as n increases, AND because denominators of an, bnoften approach zero

Moyses 70th Conference

Mie Series: Rapid Variation in x

Moyses 70th Conference

Practical Consequences of Extreme Variability of Mie Quantities

The rapid and extreme vacillations in Mie quantities as a function of both angle and size make them nearly impossible to interpolate.

Thus, when one wants to do any averaging, one is forced to take very small steps to resolve the vacillations, or risk the consequences.

This is a great burden, and is responsible for the bad reputation Mie calculations have for absorbing vast amounts of computer time.

Moyses 70th Conference

Integration of Mie quantities over size

Must abandon hope of convergence. As decrease integration step size, result tends to vacillate not converge. Resonances are the culprits!

For cross-sections and other integrals over angle, and for scattering angles where the Mie series does not suffer a lot of cancellation, the vacillation eventually confines itself to a band of width 5–10%.

For quantities like near-backscattering, where there is a lot of cancellation in the Mie series, the vacillation may be 100% or more.

Moyses 70th Conference

Size Averaging Damps the Ripple

But this presents the best face on the problem, since Qext is easy.

How to integrate over size for other Mie quantities was never really solved.

Moyses 70th Conference

S1 =S cn [ an(m,x) pn (q) + bn (m,x) tn (q) ]

n=1

Cancellation in the Mie Seriesnmax = x + 4x1/3

x = 2p r / l = Mie size parameter

The amount of cancellation among the terms varies widely:

• In near-forward directions, little cancellation (hence the big forward diffraction peak)

• in backward directions, terms with n < x (the “below-edge” terms) cancel almost completely, making the sum very sensitive to what happens in the “above-edge” region, n > x. Resonances occur in this region.

Moyses 70th Conference

Dave Mie program: a fragment

The door was open for an improved algorithm cleaning up many loose ends left by Dave, and written to be understood by a human not just a computer.

(Revolution in way programs were written barely touched scientists.)

Moyses 70th Conference

My 1980 Mie Algorithm Paper

Successful partly because a detailed NCAR Report was published in 1979, and the program was sent to anyone who requested it (on punched cards!).

Moyses 70th Conference

And thus we met!

I invited Khare, Moyses’ student, to NCAR in 1978 and he mentioned that his adviser was in the country and might also be able to come...

which he did!

And thus the climate person with all the skills needed to calculate anything, and a growing reputation in Mie theory computation, met the only person significantly advancing sphere scattering theory.

Mie theory was a sticking point for climate-related radiation work, so it was a natural alliance.

Moyses 70th Conference

Of course Moyses got a bit of a headstart on me...

I was just fleeing the Caltech Physics Dept. in search of something more solid than quarks... and Regge poles...

Moyses 70th Conference

Mie Series: Localization Principle

- For x= 2p r/l >> 1, each term ‘n’ corresponds to a ray with impact parameter nl/(2p).
- n=x then corresponds to an impact parameter r, i.e. rays hitting the edge of the sphere
- “Edge rays” near n=x not only participate in diffraction but cause surface waves and tunneling through the sphere (leading to resonances).
- None of these effects are easy to see in the Mie series solution. It gives no insight.
- Complex angular momentum theory turns series index ‘n’ into a complex variable with the character of angular momentum.

Moyses 70th Conference

Mie Efficiency Factors

“Efficiency factors” are just normalized in some way; e.g. for cross-sections by dividing by the projected area of the sphere.

We did extinction, absorption, and radiation pressure efficiencies.

Moyses 70th Conference

CAM Gave Best Analytic Expression for Qext

up till that time, and since...

Didn’t require any numerical integrations or Bessel functions

Moyses 70th Conference

Mie Efficiency Factors – Qabs

imag

index

contours of

log(% error)

imag

index

- • 3D plots of Qabs (absorption efficiency factor) as a function of both size parameter and imaginary index; rare in those days
- • much harder than Qext; required numerical integration, Bessel fcns
- • we distributed programs for this too
- – special functions were hard in those days!

Moyses 70th Conference

Radiation Pressure Qpr: the hardest

a key quantity for radiative transfer in Earth’s atmosphere

(the very simplest approximations, to whose development I contributed in the 1970s, have a minimum requirement of Qext, Qabs, and Qpr

Moyses 70th Conference

Numerical Machinery Required

- Many functions required for CAM work did not exist in the publicly available libraries
- esp. Bessel and Airy functions of complex argument
- So I had to develop these myself, with great care
- We also needed highly reliable quadrature methods, since we needed to understand differences from exact series solutions which were sometimes 0.01% or less
- Kronrod-Patterson method fit the bill perfectly
- I developed our own version
- Highest standards of program development used
- All programs were made publicly available

Moyses 70th Conference

Advantages of CAM Approximations

- computation time independent of size!
- good down to size parameters no one had expected: 10 for Qext, 50 for Qabs and Qpr
- averages over ripple analytically, removing need to take small steps to capture ripple
- computation superficially simple: quadrature over combinations of Bessel and Airy functions
- approximations were embraced immediately by atmospheric radiation community, although
- not improved upon
- not extended to nonspherical particles

Moyses 70th Conference

Angular Variation: Moyses gave best quantitative explanation of rainbows, ever

Moyses 70th Conference

Niagara Rainbow

Moyses 70th Conference

Moyses explained the rainbow partly in terms of complex rays

but we were already used to complex angular momentum, so how much worse could it get ...

esp. if complex rays helped us understand a problem stretching back to Newton.

Moyses 70th Conference

CAM Rainbow Approximation

Moyses 70th Conference

Angular Variation: Moyses gave the first real quantitative account of the glory

In the interests of historical accuracy, Moyses published his first paper on rainbow and glory, a 6000-page behemoth in J. Math. Phys., in 1969

Moyses 70th Conference

The Glory Explained

- Edge effects
- orbiting
- axial focusing
- cross-polarization
- surface waves
- complex rainbow rays
- geometrical resonances
- competing damping effects

Moyses 70th Conference

But the rainbow and glory theories were custom-crafted explanations for particular angular regions

and particular ranges of refractive index...

A general solution required dealing with various transition regions:

• forward diffraction

• Fock-type

• rainbow-type

• glory

• critical-angle (refr indx<1)

where asymptotic expansions of either Bessel or Legendre functions fail.

Moyses 70th Conference

It was time to face the general angular problem

in atmospheric science we were still pretty much stuck with geometric optics

Moyses 70th Conference

Diffraction as Tunneling, 1987

error in scattering amplitude

scattering amplitude

size param = 10

Note: all angles, not just a limited range any more

Moyses 70th Conference

Further improvements followed...

this seemed somewhat far from climate theory, but wherever Moyses led, I followed...anyway he told me it also applied to acoustic scattering, and who knows when that might be important?

Moyses 70th Conference

Hard Core – 2

- breakthrough: CAM uniform approximation, good at all angles
- programs again made available publicly
- (lots of unexpected activity on my ftp site ...)

Moyses 70th Conference

Hard Core – 3

CAM uniform approx.

error two orders of magnitude lower than competition

Moyses 70th Conference

Interlude: Mathematical Physics as Art

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Mathematical Physics as Art – 2

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Stories: Moyses and Scientific American

Why was he so mad at them?

He said they dumbed down his text, but it wasn’t until later I found the real reason...

Moyses 70th Conference

Scientific American:The Real Story

Moyses wanted one of these ... but Sci Am refused to include one as part of his honorarium!

Moyses 70th Conference

Mie Resonances

Moyses 70th Conference

Once a ray sneaks in at the critical angle

it can do quite a few loops before it runs out of energy...

and that’s a resonance!

Moyses 70th Conference

S1 =S cn [ an(m,x) pn (q) + bn (m,x) tn (q) ]

n=1

Mie Theory Resonances — 1an, bn = ratios of Bessel function expressions

(Bessel functions have arguments x and mx)

Resonances occur when an, bn have a maximum in x interms withn > x, justwhere an, bn are normally plummeting to zero (the “above-edge” terms)

Moyses 70th Conference

Mie Theory Resonances — 2

For x > 10 or so, Moyses has worked out a Complex Angular Momentum equation

f(z) = 0

such that

Re(zroot) = resonance location (in Mie size param. x)

Im(zroot) = resonance half-width

(error in resonance location << half-width)

Moyses 70th Conference

Qabs — Method

(1) Find resonance locations and half-widths numerically from Nussenzveig equation

(2) Use exact Mie theory to calculate value Q0 of Qabs at each resonance location

(3) For resonances which do not extend appreciably outside the x-interval of interest, add their contributions using the Lorentzian fit

(4) Add the background continuum contribution from exact Mie theory from which the terms corresponding to the resonances are removed

A difficulty: broad resonances extend outside x-interval of interest and must be included in continuum. Involves empiricism.

Moyses 70th Conference

Backscatter Gain

- 80-90% of backscatter gain, averaged over a size interval, is due to resonances
- Resonances go both up and down; not as simple as Qabs case
- Resonances also introduce a time delay
- Space lidars are going to see a gamut of resonance effects

Moyses 70th Conference

Changes needed to a standard Mie code to do resonances

- Everything double precision...resonance calculations are sensitive to exact location
- Extend number of terms in Mie series up to ~ |m| x
- Calculate Ricatti-Bessel function yn by down-recurrence, initializing with a new Lentz method
- Exclusively down-recurrence for An
- Calculate Qabs term by term, rather than ex post facto from Qext – Qsca

Moyses 70th Conference

Excursions Afield

Moyses 70th Conference

We were always taking intriguing detours...

even when they didn’t have much to do with climate...

Moyses 70th Conference

then Moyses led me into bubbles...

still not much climate connection...

but mine not to reason why..

anyway, the ocean has bubbles...

Moyses 70th Conference

CAM provided a term whose subtraction tamed the Mie bubble result...

and allowed better understanding of scattering near the critical angle

Moyses 70th Conference

CAM Quiz 101

Complex angular momentum theory (circle all that apply):

revived a flagging idea from 1950s quantum mechanics

generalizes Mie summation index to complex plane

only accounts for rays that actually hit a sphere

generalizes the Fresnel reflectivities at an interface

uses the Debye rather than the Mie series expansion

led to a new conceptual picture of diffraction, as tunneling

applies only to classical Mie scattering

led to a new art form: mathematical physics equations

Moyses 70th Conference

CAM Quiz 201 – CAM Theory...

- improves upon Fock approximations
- improves upon WKB approximations
- cannot compete with physical optics approximations
- improves upon Airy approximations for the rainbow
- explains the glory from water droplets
- disproves van de Hulst’s conjectures about surface waves & shortcuts
- presses the limits of modern numerical analysis

(no one knows that better than me...)

- is a testimony to the single-minded devotion of a single man

Moyses 70th Conference

Rainbow Physics: Nussenzveig’s Poor Predecessors

Newton had a go at the rainbow problem, with a little help from Descartes, but in spite of Airy’s later intervention (mainly in order to name a function after himself), quantitative progress was poor.

Moyses 70th Conference

1991: People still find CAM rough going...

Moyses 70th Conference

What lies ahead for Moyses,now that he has wrapped up the bubble problem?

Moyses 70th Conference

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