ond ‘‘inner’’ tachocline that separates a slowly rotating core from the radiative zone at a normalized radius probably in the range 0.2–0.3. We present the basic equations for r-mode oscillations in Sec- tion 2 , and we discuss the attributi on of the Rieger and relate d per iod iciti es in sola r activ ity data to r-m ode oscilla tion s in the known (outer) tachocline in Section 3 . In Section 4 , we show that certain periodicities in the Super-Kamiokande solar neutrino data ma y be att rib ute d to r- mo de os cil lat io ns in the sam e reg ion . In Se c- tion 5 , we discuss periodicities in the Mt Wilson diameter mea- surement [7] , and in Section 6 we discuss similar periodicities in nuclear decay data acquired at the Lomonosov Moscow State Uni- versity (LMSU) [8] . We dis cus s the po ssi ble ro le of neu tri no s an d the Re son ant Sp in Flavor Precession (RSFP) process in Section 7 , and we present fur- ther disc ussi on in Sect ion 8 . In App end ix A , we di scu ss the po ssi bil - ity that r-mode oscillations may be excited by a Kelvin–Helmholtz instability due to a radial gradient in angular velocity. In Appendix B , we review comm ents that have been made concerning evidence for the variability of nuclear decay rates.

2. r-Mode oscillations

We ﬁrst conside r r-mode osci llations as they occur in a uniform and unif orm ly rota ting ﬂuid sphere with siderea l rota tion fre- quency

m

R

, all frequencies being measured in cycles per year. Since r-mode oscillations are retrograde, the ‘‘absolute’’ frequency of an r-m ode oscilla tion as mea sure d in an iner tial frame is give n, to good approximation, by

m

A

ð

l

;

m

Þ ¼

m

m

R

À

2

m

m

R

l

ð

l

þ

1

Þ

;

ð

1

Þ

where

l

and

m

ar e tw o of the thr ee fa mi lia r sp he ri ca l ha rm on ic ind i- ces [5] . The allowed values of

l

and

m

are

l

¼

2

;

3

;

. . .

;

m

¼

1

;

. . .

;

l

. Sin ce this frequ ency does not dep end on the rad ial ind ex

n

, and since a thin spherical- shell wave function may be deco mpo sed into a set of spherical harmonics with different

n

values, we may infer that Eq. (1) also gives the frequency of r-mode oscillations conﬁned to a thin spherical shell. As measured by an observer on Earth, the oscillation frequency will be given by

m

E

ð

l

;

m

Þ ¼

m

ð

m

R

À

1

Þ À

2

m

m

R

l

ð

l

þ

1

Þ

:

ð

2

Þ

We now cons ider the pos sibil ity that the r-m ode oscillat ion interacts with some structure (such as a magnetic ﬂux tube) that rotates with the Sun. An arbitrary structure may be regarded as a superposition of magnetic-ﬁeld conﬁgurations with various values of the longitudinal index

m

S

. The interplay of an r-mode with the cylindrica lly symm etrical compo nent (with

m

S

= 0) will pres ent the sam e tim e- de pe nd en ce as the r-m od e its elf , a s giv en by Eq . (2) . More generally, the interplay of an r-mode oscillation with a magnetic-ﬁeld component that has a periodicity index

m

S

(which can have either sign) will lead to oscillations that, as seen from Earth, would have the frequency

m

S

ð

l

;

m

Þ ¼

m

S

ð

m

R

À

1

Þ À

m

E

ð

l

;

m

Þ

:

ð

3

Þ

The case

m

S

=

m

is particularly interesting, since it would lead to low- frequ ency oscillations with frequencies given by

m

S

ð

l

;

m

Þ ¼

2

m

m

R

l

ð

l

þ

1

Þ

:

ð

4

Þ

This is, of course, the r-mode oscillation frequency that would be measured by an observer co-moving with that region of the solar interior. It appears that, for reasons yet to be explored, in general these oscillation s seem to have a mor e prono unced inﬂuence on observational quantities than those corresponding to other values of

m

S

, perhaps simply because they have lower frequencies. How- eve r, we shall see in Sectio n 4 that, in Super -Kam iokand e solar neutrin o data,

E

-type oscillations are more signiﬁcant than

S

-type oscillations.

3. Solar activity oscillations

We now examine the possibility that the Rieger-type oscilla- tio ns, suc h as tho se id ent iﬁe d by Bai [2] , m ay be at tr ib ut ed to os ci l- lations with frequencies given by Eqs. (2) and (4) . If we assume that each of the detected periodicities may be characterized by a central frequency

m

k

and an uncertainty

D

m

k

, then the probability distribution function for the

k

0

th periodicity is given by

P

k

ð

m

Þ ¼

1

ð

2

p

Þ

1

=

2

:

D

m

k

exp

À

1 2

m

À

m

k

D

m

k

 

2

 !

:

ð

5

Þ

On the assu mp tion that the uncerta inty in the freque ncy as- signed to an oscillation, on the basis of observational data, is likely to be pr op or tio na l to the fre que nc y its elf , we ad op t the ap pr ox im a- tio n tha t the valu es of

D

m

k

are pro por tion al to

m

k

, and ado pt

D

m

k

=

m

k

/

Q

. Fig. 1 shows the sum of these curves,

F

ð

m

Þ ¼

X

k

P

k

ð

m

Þ

;

ð

6

Þ

for

Q

= 10 0. (T he re sul ts of ou r ca lcu la tio ns pr ov e no t to be se ns iti ve to the assumed value of

Q

.) We now wish to ﬁnd the sidereal rotation frequency that gives the bes t ﬁt bet we en the da ta sum ma riz ed in Eq . (6) and the r-m od e frequencies given by Eq. (4) . Focusing on oscillations correspond- i n g to

l

¼

2

;

m

¼

1,

l

¼

2

;

m

¼

2,

l

¼

3

;

m

¼

1,

l

¼

3

;

m

¼

2,

l

¼

3

;

m

¼

3,

l

¼

4

;

m

¼

1,

l

¼

4

;

m

¼

2,

l

¼

4

;

m

¼

3 , a n d

l

¼

4

;

m

¼

4, we carry out the comparison by forming the sum

H

ð

m

R

Þ ¼

F

ð

m

R

=

3

Þ þ

F

ð

2

m

R

=

3

Þ þ

F

ð

m

R

=

6

Þþ

F

ð

m

R

=

2

Þ þ

F

ð

m

R

=

10

Þ þ

F

ð

m

R

=

5

Þ þ

F

ð

3

m

R

=

10

Þ þ

F

ð

2

m

R

=

5

Þ

;

ð

7

Þ

fo r a ra ng e of va lu es of the sid er ea l ro tat io n fr eq ue nc y

m

R

. The res ult is sh ow n in Fig. 2 . Th e pr inc ip al pe ak is fo un d at

m

R

= 14 .3 0, whi ch is wit hin the ran ge 10

À

15 ye ar

À

1

that is our conven tion al sea rch band for interna l rotatio n freque ncies [9,10] . This corresponds to a per iod of 25. 54 day s, whi ch agr ees wit h the ‘‘fu nda me ntal p erio d’’ proposed by Bai [2] . (The next biggest peak is at 7.14, which is one half the frequency of the principal peak, so that it is related to the principal peak.).

0 5 1 0 1 5 0 1 2 3 4 5 6 7 8 9 10 Frequency (year

−1

)

P D F

Fig. 1.

Representation of the oscillation frequencies identiﬁed by Bai [2] .

P.A. Sturrock et al./ Astroparticle Physics 42 (2013) 62 –69