In succession to our paper dedicated to Ed Spiegel, we proceed to estab-

lish a proportionality relation between the solar-cycle variation of the sky-

brightness and that of the global warming. The increase of the optical depth

appearing in the sky brightness may cause the solar-cycle global warming of

a few degrees from the minimum to the maximum.

We wish to dedicate this paper to Douglas, in celebration of his 60th birthday

anniversary.

28.1 Introduction

Solar magnetism not only controls the solar activity but also in¬‚uences sig-

ni¬cantly the structure of the convection zone (Gough, 2001). On the other

hand, the in¬‚uence of solar activity on terrestrial meteorology such as found

in tree rings, etc., has long been the subject of discussion (Eddy, 1976)

but without ¬nding the de¬nitive causal relation explaining the physics in-

volved. Recently, however, Sakurai (2002) analysed data of the sky back-

ground brightness observed with the Norikura coronagraph over 47 years

(1951-1997) and found a clear 11.8-year periodicity as well as the marked

annual variation, both exceeding the 95 per cent con¬dence level.

The annual variation is apparently meteorological, e.g., the famous Chi-

nese yellow soil particles (rising up to 100 thousand feet high! “ old Chinese

sayings). The solar-cycle variation is also considered to be caused by in-

creased aerosol formation (Sakurai, 2002); but if the solar activity changes

the chemistry in the upper atmosphere; the observed time lag of 2 to 4 years

of the sky-brightness variation relative to sunspot maximum is somewhat

enigmatic.

In our preceding paper (dedicated to Ed Spiegel; Unno et al., 2002), we

have proposed a model of the di¬usion wave propagation of aerosol seed

411

Unno & Shibahashi

412

particles to explain the solar-cycle variation of the sky brightness and its

phase delay.

The sky brightness is about 75 in units of 10’6 I on the average, where I

means a spectral intensity of the solar disk center at around 5300 ˚. Random

A

monthly ¬‚uctuations of about 30 to 40 in the same units are superposed on

the annual and the solar-cycle components; the latter are some 15 % and

10 % in amplitude, respectively, in units of the average sky brightness.

The variations in sky brightness imply an optical-depth variation which

would a¬ect the global warming through the greenhouse e¬ect. The present

study attempts to coordinate the sky brightness and the greenhouse e¬ect by

solving the radiative transfer and to estimate the solar-cycle global warming

from the sky-brightness variation.

28.2 Radiative transfer in the earth atmosphere

The equation of radiative transfer in the terrestrial atmosphere is given by

dIν

= (κν + σν )Iν ’ κν Bν ’ σν Jν ’ (κν + σν )S direct ,

µ (28.1)

ν

dt

where isotropic scattering is assumed for simplicity. Here Iν , Jν and Bν are

the (monochromatic) speci¬c intensity, mean speci¬c intensity and Planck

function respectively; µ is the direction cosine of the radiation, κν and σν

denote the absorption and scattering coe¬cients, dt ≡ ’ρdz (t is measured

inward), and the last term denotes the contribution from the intensity of

the direct solar radiation.

This equation describes both the thermal radiation ¬eld dominated by

the infrared radiation and the scattered radiation ¬eld dominated by the

visible solar radiation. Integrating equation (28.1) over both the infrared

and visible frequency ranges, we obtain

dI κ σ κV —

S0 e’„V

=I’ B’ J’

µ (28.2)

d„ κ+σ κ+σ κ+σ

and

dIV σV

S e’„V /µ ,

= IV ’

µ (28.3)

d„V κV + σV

respectively, where

d„ = (κ + σ)dt,

—

d„V = κV dt,

d„V = (κV + σV )dt,

Solar-cycle global warming and sky brightness variation 413

2

1 R

σT 4 ,

S0 =

16π a

and

2

µ R

σT 4 = 4µ S0 .

S=

4π a

Here σ denotes the Stefan-Boltzmann constant, T (=5780 K) the e¬ective

temperature of the sun, and R /a (=2.3 light-sec/500 light-sec = 4.61 —

10’3 ) is the solar radius in AU; 4πS0 is the solar energy ¬‚ux per unit area

averaged over the entire earth surface, while 4πS is the solar constant

(1.37 kW/m2 ); µ is the cosine of the angle of the sun from the zenith.

28.3 Radiative equilibrium model

For simplicity, we discuss here the radiative model to calculate the green-

house e¬ect in the earth atmosphere.

Equation (28.2) averaged over the whole solid angle results in

dH κ κV —

S0 e’„V ,

(J ’ B) ’

=

d„ κ+σ κ+σ

1 1

1 1

where J ≡ Idµ and H ≡ µIdµ. Assuming radiative equilibrium,

2 2

’1 ’1

so

∞

κν (Jν ’ Bν )dν = 0 ,

0

and grey absorption (κν = κ), we have J = B and H = S0 e’„V . As will be

discussed in a subsequent section, the radiative model seems to be not so

bad, perhaps because of the large heat capacity of the ground and oceans.

Multiplying equation (28.2) by µ and averaging over the entire solid angle,

—

we obtain dJ/d„ = 3H = 3S0 e’„V by using the Eddington approximation

1

1 1

K≡ µ2 Idµ = J.

2 3

’1

Integrating this equation with respect to „ , and using the boundary condi-

tion J(0) = 2H(0) at „ = 0, we obtain

—

J = B = π ’1 σT 4 = 2S0 + 3[(κ + σ)/κV ]S0 (1 ’ e’„V ),

—

S0 (3„ + 2) for small „V .

Hence,

2

3 2 R

4

T4.

T= „+ (28.4)

16 3 a

Unno & Shibahashi

414

Thus the temperature in the uppermost atmosphere T (0) is given by

1/2