J oumalofG'laciology, Vot. 44,
.Vo.
143, 1998
Diffusion of isotopes in the annual layers of ice sheets
J. F. NYE
H. H. TI'ills
P/~J!sics
L aborat07). Ulliversi{J! if Bristol, 7jndall Avenue, Bristol BS8 ] TL Englalld
ABSTRACT.
18
Th e a nnu al laye rin g in the Greenl a nd a nd Anta rcti c ic c sheets re\'ca led
by th e 8 0 reco rd b eeom es less di stinc t with depth because o f self-difTusio n. H owever, th e
ca lculated diITu sio n rates are too slow to expla in th e ob e rvations. It is suggested th at th e
presence of \'ein s of I iquid wa ter in e l-eases the eITective difTusion consta nt by a fac tor or
a bo ut 20.
1. OUTLINE OF THE PROBLEM AND THE
SUGGESTED SOLUTION
Th e ox yge n isotope ra tios used to identify the annu al laye rs
of th e Anta rctic a nd Greenl a nd ice sh ee ts gradua lly lose
th e i r initi a l contras t by self-difTusio n through th e solid ice.
H owever, recent wo rk Uo hnsen a nd o the rs, 1997) on ice fro m
th e GRIP co re, Green la nd, concludes th a t the usua l processes of solid-sta te difTusion, including g ra in-boundar y diffu sio n, a rc LOO slow to p roduce th e obser ved loss of contras t.
Th e diITusion con ta nt in the - 32°C co ld H olocene ice need s
to b e a bo ut 10 times g reater th an th e m eas ured single-c r ys ta l
d iffu sion consta nt a t th e sa me te mpera ture (R am se ie r,
1967a, b ) to expla in the d ata. Thi s no te suggests a new process, no t previo usly recog nised, th a t will e nha nce the ra te.
It depe nds o n the prese nce within th e ice of veins o f unfro ze n wa ter at the three-grain junc tions (;-.lye a nd Fra n k,
19 73; M a der, 1992a, b; N ye, 1992). Th e ve ins fo rm a co ntinuou s ne t work th a t re nde rs the ice slig htl y p e rmeabl e to wate r.
Th ey d o not fr eeze wh e n the tempera ture is lowe red beca use
o f two physica l eITec ts: onc is lower ing o f the freez ing p o i nt
by the c ur vature o f the ice- water int e rfaces a t the tripl e
junc ti o ns (the ice is co m 'ex LOwa rd s the wa ter); the oth e r is
th e lowe rin g of th e fr eez ing point by so luble impuriti es in
th e ve in wa ter. As th e te mperature is reduced th e \'Cins te nd
to fr eeze, but this in c rea ses the cur vature', a nd so lowe rs th e
fr eez in g point sti ll furth er; a lso, because the ice latti ce
rej ec ts impuriti es, th ey rema in in th e liqu id , wh ich beco m es
m o re conce ntrated. The res ult is th a t a ne w equilibrium is
ro und a t th e lower temperature, with th e veins na rrowe r
but still open.
To sec th e eflect o r th e \'eins on the diITusion of iso to pe
r a ti os, consider two laye rs with iso tope ra ti os 8 = 80 + 6.
a nd D = 80 - 6. se p a rated by a di sta nce >- / 2, where >- is th e
sp ac in g of the a nnu a l laye rs. The sta nda rd diffusion process
is dr ive n by the g ra d ient 46./ >- . H o weve r, suppose we co nce ptu a ll y suspend thi s process a nd co nn ect the laye rs by
wa te r ve ins. Isoto pe diffusion within th e ve ins will be so fas t
co mpa red with so li d-state diffusion th a t D within th em wi ll
reac h the mea n \'a lue 80 virtua ll y insta nta neously (with in
ho urs or days ). Thu s, th e ra te-co ntrollin g process is now la tera l difTusio n within the laye rs towa rd s a nd away from th e
\Tin s, and the g ra die nts dri\'ing thi s process a re ±6. /T1,
whe re T1 is half the spacing of the veins, which is approx-
im ate ly th e g ra in-size b. 1£"1' 1 < Aj.t th a t is b < >- /2 . diffusion by thi s process co uld o ut pace the sta nd a rd process. On
dim ensio n a l g round s o ne wo uld ex pect th e ra te fo r the la tera l process to be abo ut (>- /2 b)2 times th e ra te fo r the co nve nti o na l ve rtica l diffusio n process. In o th er word s, th e
veins mi g ht short-circ uit th e usua l \'e rtica l difTusion a nd
the ra te o f a pproach to uniro rmit y wo uld the n b e co ntrolled
by how c lose a point is to a vein, rath er th a n by th e laye r
spacing. Th a t is the sugges tio n.
2. MORE DETAILED MODEL
To exa min e the geome tr y m ore ca refu ll y, with both processes ta kin g pl ace at o nce, co nsider a lo ng ye rti ca l cylinder
of'i ce o f o uter radius 7'1 a nd inner radiu s TO . Th e inn er cylindri ca l ho le represe nts a ve in , a nd o n its surface 15 is ma inta ined co nsta nt at 150 . Th e ra dius of th e o ute r cylindri cal
bound a r y is ta ken as "'t = b/ 2, where b is th e g ra in-size. Diffusion o f 8 ta kes pl ace with diffusion co nsta nt D in th e \'ertical z directi o n, a nd a lso ra di all y w ith th e bo unda r y
conditi o n 88/81' = 0 o n r = 1"1, acco rding to th e equ ati o n
2
2
?
8
18
( ) D.
-88 = D'V-15
= D ( -+--+2
8t
Eh·
T
A)'
Wr iting 8(1". z. t) = F(r)Z(z)T(t), a nd
sepa ra tes t he va ri ables to g ive
T' = - DkT.
(1)
8 z2
substituting,
- 1,; / z.
(2)
- k}F(T),
(3)
Z" =
a nd
F"(r)
+ ~F'(r) =
r'
where th e k's a re consta nts sati sfying I,;
solutio n ca n be written
8 - 80
=
A(sink:Z)F(r)e- "Df
= 1,; ,2 + k/
Thus, a
(A = con s t a n t)
(4)
with F(r) obeying Equ a ti o n (3) with bo und a r y co nditi ons
F(ro) = 0, F'(rl) = O. With th e substituti o ns k,.1' = R .
k,·T'o = R o. k,.1'1 = RI. Equa ti on (3) beco m es th e Besse l
equa ti o n
F" (R) + ~F' ( R) + F (R ) = O.
(5)
with bo unda r y conditi o ns F (Ro) = O. F' (R I ) = O.
The fac to r sin k:z in Equ a ti on (4) represe nt s th e a nnua l
laye rin g. Thus wc rega rd k z as prescribed, but k,. (th e lowes t
+67
JournaL qfCLacioLog))
frequency th at fits the boundary conditions) remains to be
fo und, a lthough it mu st be o r order 1/1'\. Th e questi on will
be the relati\·e co ntributions of k: a nd kr to the decay
co nsta nt k.
Th e genera l solutio n of Equation (5) is
F( R) = Jo(R)
+ BYo(R),
(B = constan t)
(6)
the ot her arbitrary co nstant being absorbed into A.
The boundary co nditions th en imply
Jo(Ro)
+ BYo(Ro) =
0
(7)
a nd
(8)
which is equi vale nt to
mic onl y. For a gralll-size of 6 mm instead of 3.5 mm, as
ta ken above, the factor would be 8.7.
Beca use RI is always roug hl y 0.5 in the range o rinterest,
a useful approximation is to take k,. ;:::: l l b, which gives the
simpl e formu la.f;:::: 1 + (A2/47r 2 b2 ).
Samples taken from the interi ors of the indiv idual ice
grains ought to give the bes t contras t.
S.]. J ohnsen (personal communicati on) remarks that
the excess diffusion onl y acts in relatively clean glacier ice
like the H o lo cene ice; samples from the dusty, last glacial,
ice show no evidence of it. H e suggests that the veins may
be blocked by dust particl es in th e lalter case. Of course,
a ny blockage would have to ac t on the dilTusio n process
along the veins, not just on the bulk motion of the water.
(9)
a re given. At - 30°C the expected vein size is
abo ut TO = I {ml (M a d er, 1992 b). With the grain-size in
H olocene ice obsen ·ed to be abo ut 3- 4 mm (Thorsteinsson
a nd oth ers, 1997), we ta ke 1'1 = 3.5/2 = 1.8 mm .
Although TO and 1'] are given, Ra and RI a rc not,
because kr is unknown. Ra is given in terms of RI b y
Ra = (ro l rl)RI. Th en, elimin ating B between Equations
(7) a nd (9) gives the fo llowing implicit equation for RI
TO
a nd
1'1
Jo[(roITt} Rd _ JI( Rd = 0
Yo [(To l rl) R I ] ~(Rd
'
(10)
which is solved by itera tion to give RI =0.544. It follows that
k, = RJ I TI = 302 m- I
Thi s is to be compa red with kz . For ice 10000 yea rs old
the laye r spacing is A = 0.1 m, which g ives kz = 27r1 A =
63 m I. Thus, k,., from the la teral diffusion process, makes a
larger co ntribution to the rate constant k = k r 2 + kz 2 than
does k z, from conventi o nal vertical diffusion . k,. increases k
by a factor .f = (k,2 + k/) I k} = 24; that is, the diffusion
co nsta nt will appear to be 24 times greater than normal.
The result depends a lmost entirely on the relative values of
1'1 a nd A, the dependence on th e vein size TO being logarith-
ACKNOWLEDGEMENT
I am grateful to S.]. Johnsen for kindl y supplying me with
essential data on thi s probl em.
REFERENCES
J ohnsen, S. J. Cl /ld 14 otil m . 1997. The .) 18 0 record along the Greenland Ice
Core Project deep ice core and the problem of possible Ee m ian ciimalic
instability. J Geo/JI!),s. Res., 102 (C I2), 26,397- 26,410.
l\lader, H. M . 1992a. Observations o f the water-vein system in pol ycr),stallin e ice. J Glacio!., 38(130), 333- 347.
l\[ader, H. NI. 1992 b. The thermal beh ",·iour of the water-,·e in system in
polycrystal lin e ice. J Glacio!. , 38 (130), 359- 37+.
l'\ye, J. F 1992. Water veins and lenses in pol ycrvsta lli ne ice. [ /I Maeno. :'>.
and T. H ondoh, eds. Procerdillgs if the i n/emaliollal Symposium 01/ the PI!),sics
alld Chemisl1y if lee, Sa/J/)ol"O, ]a/Jall. Sapporo, Hokkaido Universit y Press,
200-205.
Nye, J. F. and F. C. Frank. 1973. Hydrology of the illle rg ranu lar veins in a
temperate g lacier. IIl/ematiollal Association ojScielltijic fiJ'drology Publica/ion
95 (Symposium at Cambridge 1969 - H),drology rifGlacim ), 157- 161.
Ramsei er, R. 0. 1967a. Self-di ffusion in ice monocrystals. CRREL Res. Re/).232.
Ramseier, R. O. 1967b. Self-diffusio n o f tritium in natural and synthetic ice
monocrysta ls. J ,-lfJPl. PI!),s., 38 (6), 2553- 2556.
Thorslcinsson, Th. , J Kipfstuhl and H . Mill er. 1997. "Iex tures and fabrics in
the G R IP ice co re.]. Geopl!),s. Res., 102 (C I2), 26,583- 26,599.
MS received 24 October 1997 and accepted in revisedform 5 J anua7J 1998*
Publication of thi s paper was delayed due to its loss by the
pos ta l se rvices.
468