Nonequilibrium Systems in Natural Water Chemistry

2 Time to Chemical Steady-States in Lakes and Ocean ABRAHAM LERMAN

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Canada Centre for Inland Waters, Burlington, Ontario

In water and sediments, is controlled (diffusion,

sedimentation)

species.

the time to chemical

magnitude transport When

is weak,

of

distances,

advection

diffusion

steady-states

transport

mechanisms

and reaction

(water

controls

the

solute

persal and, hence, the time to steady-state.

Models of

states include

species in two-

salt between radium-226 conservative

brine

and three-layer

layers

in the oceanic species in

transport

in the

Dead

water column,

rates

flow, rate

sient and stationary chemical

Tn

the

advection),

of chemical

A

by

of

tran-

conservative

lakes, transport Sea,

of dis-

of

oxygen

and

and reacting

and

sediment.

n a t u r a l systems of l a r g e d i m e n s i o n s — b o d i e s

of w a t e r ,

sediments,

a t m o s p h e r e — m a n y c h e m i c a l processes are c o n t r o l l e d b y t h e t r a n s p o r t

of r e a c t i n g species t h r o u g h the system.

T h e d i s t r i b u t i o n of

chemical

species i n n a t u r a l systems is o n l y too often not h o m o g e n e o u s ; c o n c e n t r a t i o n gradients a n d m o r e o r less a b r u p t changes i n a b u n d a n c e f r o m p a r t of a n e n v i r o n m e n t t o a n o t h e r are c o m m o n p l a c e .

one

I n general, the

n o n h o m o g e n e o u s d i s t r i b u t i o n s of c h e m i c a l species are a c o m b i n a t i o n of (i)

the g e o m e t r y

of t h e e n v i r o n m e n t :

its shape

a n d l o c a t i o n of

the

"sources" a n d " s i n k s " of t h e c h e m i c a l species; (it)

physics: mechanisms

of transport of m a t t e r t h r o u g h t h e s y s t e m ; a n d (Hi)

c h e m i s t r y : the n a t u r e

a n d rates of the c h e m i c a l reactions i n w h i c h the species enter. K n o w l e d g e of these three facets of a n a t u r a l system is i n d i s p e n s a b l e w h e n w e n e e d to u n d e r s t a n d its present c h e m i c a l state a n d also to p r e d i c t q u a n t i t a t i v e l y the changes i n the c h e m i c a l state a n d t h e i r d u r a t i o n , as w o u l d o c c u r w h e n the present characteristics of the system c h a n g e . I n o r d e r to v i s u a l i z e t h e significance of t h e geometric, p h y s i c a l , a n d c h e m i c a l factors, one m i g h t c o n s i d e r a system c o n s i s t i n g of a s e d i m e n t 30 In Nonequilibrium Systems in Natural Water Chemistry; Hem, J.; Advances in Chemistry; American Chemical Society: Washington, DC, 1971.

2.

LERMAN

Time to Chemical

Steady-States

31

a n d a w a t e r c o l u m n a b o v e i t . T h e g e o m e t r i c factors i n t h i s case are t h e l o c a t i o n of the sources

of t h e c h e m i c a l s p e c i e s — f o r

e x a m p l e , at t h e

s e d i m e n t - w a t e r i n t e r f a c e , w i t h i n the sediment, or d i s t r i b u t e d t h r o u g h o u t the w a t e r c o l u m n — a n d its sinks, s u c h as r e m o v a l b y a c h e m i c a l or b i o c h e m i c a l r e a c t i o n o c c u r r i n g t h r o u g h o u t t h e system, r e m o v a l i n outflow, or e v a p o r a t i o n . T h e r e l e v a n t c h e m i c a l aspects of s u c h a system are the c o n c e n t r a t i o n o r rate of p r o d u c t i o n of the c h e m i c a l species at the source a n d t h e n a t u r e a n d rates of t h e reactions i n v o l v i n g the species.

Biological production

a n d c o n s u m p t i o n of a d i s s o l v e d substance c a n i n c e r t a i n cases b e t r e a t e d Downloaded by TUFTS UNIV on December 11, 2015 | http://pubs.acs.org Publication Date: June 1, 1971 | doi: 10.1021/ba-1971-0106.ch002

( I ) as i f it w e r e a c h e m i c a l r e a c t i o n of a s i m p l e order. T h e transport m e c h a n i s m s i n c l u d e d i s p e r s a l b y d i f f u s i o n a l processes, w a t e r flow ( a d v e c t i o n ) , s e t t l i n g of b i o l o g i c a l or d e t r i t a l p a r t i c l e s t h r o u g h the w a t e r c o l u m n , a n d a c c u m u l a t i o n of s e d i m e n t o n the floor. I n v i e w of the p r i m a r y significance of d i f f u s i o n i n the transport of d i s s o l v e d m a t t e r i n a w a t e r c o l u m n , this m e c h a n i s m a n d its b e a r i n g o n a n u m b e r

of

c h e m i c a l processes w i l l b e discussed i n d e t a i l i n this p a p e r . D i f f e r e n t d i f f u s i o n a l processes a n d the m a g n i t u d e of the c h a r a c t e r i s tic d i f f u s i o n coefficients

are i d e n t i f i e d i n F i g u r e 1.

W i t h reference

to

v e r t i c a l m i g r a t i o n of c h e m i c a l species t h r o u g h w a t e r - f i l l e d sediments a n d w a t e r c o l u m n of lakes a n d ocean, the r e l e v a n t d i f f u s i o n a l processes are the m o l e c u l a r a n d e d d y d i f f u s i v i t y , respectively. T h e difference of several orders of m a g n i t u d e b e t w e e n t h e m o l e c u l a r a n d e d d y d i f f u s i o n coefficients reflects the m u c h m o r e r a p i d d i s p e r s a l b y t u r b u l e n t eddies i n n a t u r a l b o d i e s of w a t e r . T h e m u c h h i g h e r values of t h e e d d y diffusivities i n s u r face waters are o w i n g to the greater effect of the w i n d - g e n e r a t e d t u r b u lence, as c o m p a r e d w i t h the d e e p e r parts of the b a s i n . T h e values of the diffusion coefficients w i t h i n a p a r t i c u l a r t y p e of e n v i r o n m e n t

(such

as

porous m e d i a or t h e r m o c l i n e layers ) m a y v a r y b y several orders of m a g n i t u d e , a n d there is some o v e r l a p b e t w e e n different e n v i r o n m e n t s ( F i g u r e 1).

T h e large v a r i a t i o n i n the values of the d i f f u s i o n coefficients r e p o r t e d

i n the l i t e r a t u r e for different c h e m i c a l species i n different e n v i r o n m e n t s a n d the laboriousness of t h e i r d e t e r m i n a t i o n i n n a t u r a l e n v i r o n m e n t s m a k e it difficult i n m a n y cases to o b t a i n accurate estimates of the t i m e r e q u i r e d for a c e r t a i n c h e m i c a l process to go to c o m p l e t i o n .

H o w e v e r , w h e n the

diffusivities are not w e l l k n o w n , it is s t i l l possible i n some systems to choose " r e a s o n a b l e " l o w e r a n d u p p e r l i m i t s of the d i f f u s i o n

coefficients

a n d t h e r e b y to b r a c k e t the m o d e l i n short a n d l o n g t i m e estimates. T h e effects of the m a g n i t u d e of e d d y d i f f u s i v i t y o n the t r a n s p o r t of a d i s s o l v e d species i n a stratified b o d y of w a t e r are discussed i n some s i m p l i f i e d l a k e m o d e l s a n d a n e x a m p l e f r o m a r e a l l a k e i n t h e next t w o sections.

In Nonequilibrium Systems in Natural Water Chemistry; Hem, J.; Advances in Chemistry; American Chemical Society: Washington, DC, 1971.

32

NONEQUILIBRIUM

SYSTEMS IN N A T U R A L

WATERS


-EDDY DIFFUSION : HORIZONTAL , SURFACE WATERS

- EDDY DIFFUSION : VERTICAL, THERMOCLINE AND DEEPER REGIONS IN LAKES AND OCEAN.

HEAT

IN H,0

MOLECULAR DFFUSION : SALTS AND GASES IN H 0 2

id h 6

r

I PROTEINS IN H,0 THERMAL DIFFUSION :

Figure 1. Diffusion various environments;

-IONIC SOLUTES IN POROUS MEDIA (SEDIMENTS, SOILS)

coefficients characteristic of sources of data: Ref. 2, 3, 4, 5,

6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16

Effect of Eddy Diffusivity

on Transport in a Stratified Water Column

A n i d e a l i z e d p i c t u r e o f a stratified b o d y o f w a t e r is a w e l l - m i x e d l a y e r at the surface, a l a y e r w i t h a m o r e or less p r o n o u n c e d d e n s i t y g r a d i ent ( p y c n o c l i n e ) b e l o w i t , a n d a w e l l - m i x e d l a y e r b e l o w the p y c n o c l i n e . I n m a n y fresh w a t e r lakes, the d e n s i t y stratification is t h e r m a l i n o r i g i n , a n d t h e concentrations

o f major d i s s o l v e d solids a r e t h e same i n t h e

l i g h t e r a n d denser layer.

A difference

i n concentrations

between t w o

layers m i g h t arise, f o r e x a m p l e , w h e n a large i n f l u x of w a r m e r raises the l a k e l e v e l a p p r e c i a b l y .

water

A certain amount of m i x i n g is likely

to o c c u r i n the i n i t i a l stages of flooding, w i t h the result t h a t a c h e m i c a l species d i s t r i b u t e d h o m o g e n e o u s l y i n the o r i g i n a l l a k e retains its h o m o geneous d i s t r i b u t i o n i n t h e d e e p e r l a y e r , b u t a c o n c e n t r a t i o n

gradient

comes i n t o b e i n g i n the m i x e d layer. S u c h cases o f flooding o f a saline l a y e r b y a l a y e r o f l i g h t e r w a t e r h a v e b e e n discussed for some A n t a r c t i c ,


2.

Time

LERMAN

to Chemical

33

Steady-States

A r c t i c , a n d P a c i f i c C o a s t lakes (17, 18, 19). T h i s is s h o w n d i a g r a m m a t i c a l l y i n the inset o f F i g u r e 2 ; the i n i t i a l c o n c e n t r a t i o n i n the u p p e r l a y e r (C °) 2

is n i l . A n o t h e r p o s s i b l e a p p l i c a t i o n o f t h e m o d e l is w h e n a d i s

s o l v e d species has b e e n i n t r o d u c e d i n t o o n e o f t h e m i x e d l a y e r s .

Dif

f u s i o n t h r o u g h the p y c n o c l i n e s u b s e q u e n t l y establishes a c o n c e n t r a t i o n g r a d i e n t , a n d t h e m a t e r i a l r e a c h i n g the o t h e r m i x e d l a y e r i s u n i f o r m l y d i s p e r s e d w i t h i n i t . T h e m o d e l m a y a p p l y f r o m the e a r l y stages o f s u c h a process, after some m a t e r i a l has crossed t h e m i d d l e l a y e r , p r o v i d e d t h e c o n c e n t r a t i o n g r a d i e n t is a p p r o x i m a t e l y l i n e a r . W h e n a t h r e e - l a y e r sys t e m r e m a i n s c l o s e d a n d t h e d i m e n s i o n s o f the w a t e r layers d o n o t c h a n g e ,


a c o n s e r v a t i v e c h e m i c a l species i n o n e o f t h e m i x e d layers w o u l d r e d i s t r i b u t e itself b e t w e e n

the t w o layers b e c a u s e o f t h e d i f f u s i o n a l

flux

d o w n the c o n c e n t r a t i o n g r a d i e n t f r o m o n e m i x e d l a y e r i n t o t h e other. F o r a case o f t r a n s p o r t f r o m the l o w e r i n t o t h e u p p e r m i x e d l a y e r , c h a n g e i n t h e c o n c e n t r a t i o n i n t h e u p p e r l a y e r ( C ) as a f u n c t i o n o f t i m e 2

^UMIT

C /C, 2

WHEN

e

"

h

2

0.5

0.3

0.1

-

1/ / I u

—

-

•0

jC -C (t-*' 2

(6)

0

Ah

2

, W

e-« « >]

Ah

2

C

[1 "

hi + Ah/2 hi(h + Ah/2)

-

( 7 )

2


C o n s t a n t k m a y b e e v a l u a t e d as f o l l o w s . W h e n the flux ( F ) t h r o u g h the p y c n o c l i n e is e d d y d i f f u s i o n a l i n n a t u r e a n d the c o n c e n t r a t i o n g r a d i e n t is l i n e a r , t h e n

= *Ît = S

F

( C I

-

C 2 )

(8)

w h e r e Κ is the e d d y d i f f u s i o n coefficient i n the p y c n o c l i n e l a y e r ( c m sec

-1

2

·

) a n d AC is the difference b e t w e e n the concentrations at t h e p y c n o

c l i n e b o u n d a r i e s . F o r the case w h e n e d d y d i f f u s i v i t y i n the p y c n o c l i n e is constant, c o m p a r i n g E q u a t i o n s 8 a n d 1 gives k = K/Ah

(9)

A s i m i l a r d e r i v a t i o n has b e e n g i v e n i n R e f . 20. F o r the m o d e l s h o w n i n F i g u r e 2, the thickness of t h e Ah =

pycnocline

10 m , a n d e d d y d i f f u s i o n coefficient i n the p y c n o c l i n e w a s g i v e n the

values of Κ =

5 Χ 10 , 1 χ 3

10" , a n d 5 Χ 10" c m 2

2

2

· sec . 1

From Equa

t i o n 9, the values of k are 1.58, 3.16, a n d 15.8 m · y r " . W h e n t h e i n i t i a l 1

c o n c e n t r a t i o n i n the u p p e r l a y e r C ° = 2

0, E q u a t i o n 6 c a n b e w r i t t e n i n

the f o l l o w i n g f o r m .

T h e c o n c e n t r a t i o n - t i m e curves s h o w n i n F i g u r e 2 w e r e c a l c u l a t e d u s i n g R e l a t i o n s h i p 10 w i t h the values of the layers' thickness h = x

Ah =

h = 2

25 m ,

10 m , a n d the values of k d e r i v e d as e x p l a i n e d a b o v e . C o n s t a n t k defined i n E q u a t i o n 2 is i d e n t i c a l w i t h the c o n c e p t of

e n t r a i n m e n t v e l o c i t y (U ) e

w h i c h has b e e n s t u d i e d b y T u r n e r (21)

in

experiments o n the t r a n s p o r t of salt a n d heat across the i n t e r f a c e of a density-stratified t w o - l a y e r w a t e r c o l u m n . T h e d e f i n i t i o n of k i n t h i s sec t i o n also a p p l i e s to a t w o - l a y e r m o d e l w i t h a s t a t i o n a r y i n t e r f a c e ; i n a t w o - l a y e r w a t e r c o l u m n , Ah =

0 in Equations 3-7.

T h e n , h o w e v e r , the


2.

Time to Chemical

LERMAN

Steady-States

r e l a t i o n s h i p b e t w e e n k a n d t h e e d d y d i f f u s i o n coefficient Κ i n E q u a t i o n 9 becomes i n v a l i d . Calculation of Concentration—Time Curves for a T w o - L a y e r Model (Figure 3 ) .

I n a t w o - l a y e r system, w h e n the d i f f u s i o n coefficients i n the

t w o layers are e q u a l , the c o n c e n t r a t i o n of a d i s s o l v e d substance o r i g i n a l l y c o n f i n e d to one l a y e r is g i v e n b y the f o l l o w i n g r e l a t i o n s h i p

(22,

p. 15) =


C

where C

0

Σ

VC 2

0

{erf

h

+

2

" ^ *

+

erf

h

'

(H)

+ A

is the i n i t i a l c o n c e n t r a t i o n i n one l a y e r ( 0
ft), a n d ρ is a p a r a m e t e r d e p e n d e n t o n t h e densities, specific heats, a n d d i f f u s i o n coefficients of t h e t w o layers.

I n aqueous

solutions that are n o t h i g h l y c o n c e n t r a t e d b r i n e s , t h e

p r o d u c t pc is close to 1 a n d varies o n l y s l i g h t l y w i t h c o n c e n t r a t i o n . T h u s , Downloaded by TUFTS UNIV on December 11, 2015 | http://pubs.acs.org Publication Date: June 1, 1971 | doi: 10.1021/ba-1971-0106.ch002

E q u a t i o n 13 m a y b e s i m p l i f i e d to ρ = [1 -

V(*V*.)]/[1

+

(14)

ViKJKt))

T h e flux across t h e p l a n e ζ — ft m a y b e d e r i v e d f r o m E q u a t i o n 12.

X

2-

7

h

= y* ° |/f I - * - 0:C = C at* = 0

(56)

C = C at ζ = h

(57)

2

z

T h e s o l u t i o n of E q u a t i o n 54 w i t h t h e i n i t i a l a n d b o u n d a r y c o n d i t i o n s of E q u a t i o n s 5 5 - 5 7 is

C = C_

exp ( < 2 » 1

+

.,

0

+ Vo ( C ~ C*) exp (C7*/2tf) 2

χ erfc

+

X

j/(T^>] -


2.

Time to Chemical

LEHMAN

l/(S

+

x

>]-"p('

C)


H (C,

S{

EXP

( -[(2η+1)Λ

KS

+

X

)]

+EXP

Steady-States

[(2n + 2)h -

e x p (U(z -

)/S-

2

('

+

z]

|/Jg +

h)/2K)

X

X

(2n + ^)xerfc;. 2ν/(Λ)

[(2n + 1) Λ -

ζ)i l / — + M

ζ

X

(58) The

term C

i = = 0

is t h e i n i t i a l steady-state c o n c e n t r a t i o n g i v e n b y

R e l a t i o n s h i p 55. I n E q u a t i o n 58, t h e t i m e - d e p e n d e n t terms b e t w e e n t h e braces c o n t a i n t h e d e c a y constant λ. T h e r e f o r e , the rate o f c h a n g e i n R a - 2 2 6 c o n c e n t r a t i o n at a n y d e p t h (dC/dt)

d e p e n d s o n t h e d e c a y rate constant. T h u s ,

i n the case of a first-order r e a c t i o n ( r a d i o a c t i v e d e c a y ) , t h e rate of c h a n g e i n c o n c e n t r a t i o n d e p e n d s o n t h e r e a c t i o n rate constant, w h e r e a s i t h a s b e e n s h o w n i n t h e p r e c e d i n g section that f o r a z e r o - o r d e r r e a c t i o n ( o x y g e n c o n s u m p t i o n ) , t h e rate o f c h a n g e i n c o n c e n t r a t i o n (dC/dt)

is i n d e

p e n d e n t of its rate constant. A n e w steady-state c o n c e n t r a t i o n ( C

i = = Q 0

)

w i l l be attained w h e n t

tends t o infinity i n t h e t i m e - d e p e n d e n t terms of E q u a t i o n 58.


66

NONEQUILIBRIUM SYSTEMS I N N A T U R A L WATERS

C

i

- ~

-

β χ ρ ( ^ ) - β χ ρ ( ^ )

exp (βιΛ) T h e constants R i a n d R

2

'

( f i

—

^ exp

(# A)

e

exp

x

p

(

β

ι

2

)

+

(59)

(«*)

2

w e r e defined u n d e r E q u a t i o n 52.

T h e t i m e i t takes the R a - 2 2 6 c o n c e n t r a t i o n t o b u i l d u p t o a steadystate m a y b e c o n s i d e r e d , as before, the t i m e w h e n the c o n c e n t r a t i o n has a t t a i n e d the 9 5 %

v a l u e o f the difference

between

theo l d and

new

steady-state c o n c e n t r a t i o n s : Downloaded by TUFTS UNIV on December 11, 2015 | http://pubs.acs.org Publication Date: June 1, 1971 | doi: 10.1021/ba-1971-0106.ch002

C -

Figure

C =o = 0.95 ( C t

£

e

-

C_ ) 0

(60)

15. Time to steady-state of Ra-226 concentration at ζ = 1 km. Notation as in Figure 13. C from Equation 58.


2.

Time to Chemical

LERMAN

67

Steady-States

T h e values o f t i m e w h i c h satisfy E q u a t i o n 6 0 w e r e c a l c u l a t e d f o r different values of t h e e d d y diffusion coefficient

( r a n g e 0.2-2.0 c m

2

·

s e c ' ) a n d a d v e c t i v e v e l o c i t y ( r a n g e f r o m + 2 . 4 X 1 0 ' t o —2.4 Χ 1 0 " 1

5

c m · sec" ) a n d p l o t t e d i n t h e t-K-U 1

5

g r a p h i n F i g u r e 15. T h e curves

i n F i g u r e 15 are s y m m e t r i c a l a b o u t [7 = 0, i n d i c a t i n g t h a t t h e d i r e c t i o n of flow ( u p o r d o w n ) has n o effect o n the t i m e i t takes t o r e a c h a steadystate c o n c e n t r a t i o n .

W h e n t h e t u r b u l e n c e i n t h e w a t e r c o l u m n is r e l a

t i v e l y h i g h (K i n t h e v i c i n i t y of 2.0 c m · s e c " ) , t h e a d v e c t i o n has l i t t l e 2

1

effect o n t h e t i m e t o steady-state; t h e t i m e values a r e i n t h e range 3 0 0 400 years. W h e n t u r b u l e n c e is l o w ( l o w values of K ) , t h e n a d v e c t i o n Downloaded by TUFTS UNIV on December 11, 2015 | http://pubs.acs.org Publication Date: June 1, 1971 | doi: 10.1021/ba-1971-0106.ch002

d o m i n a t e s t h e p i c t u r e . T h e t i m e t o steady-state decreases f r o m a p p r o x i m a t e l y 2000 years w h e n U is near 0 t o 300 years w h e n the absolute v a l u e of U is h i g h . T h e r e are i n d i c a t i o n s t h a t release of R a - 2 2 6 b y d e c o m p o s i n g

organic

m a t t e r is a m e c h a n i s m of some significance i n m a i n t a i n i n g t h e Ra-226 concentrations i n o c e a n w a t e r ( 4 2 , 4 3 ) . I f this a d d i t i o n a l s u p p l y of R a - 2 2 6 i s expressed as a constant p r o d u c t i o n rate Ç ( g r a m s · l i t e r ' · 1

y r " ) , t h e n a steady-state c o n c e n t r a t i o n - d e p t h profile m a y b e o b t a i n e d 1

f r o m the d i f f e r e n t i a l e q u a t i o n U f - \ C dz

dz

2

(61)

+ Q = 0

the s o l u t i o n of w h i c h f o r constant b o u n d a r y concentrations ( C at ζ = 0 0

a n d C i at ζ = h) is r

t„

_ Q , Ci ~ Co exp (RJi) - [1 - e x p (R h)]Q/X ^ , ι\ ix e x p \ti\Z) -+· λ e x p (Rih) — e x p (/? *) 2

— r- i

/

D

/

(

p

2

C . e x p (R>h) - C

1

[1 -

+

exp ( ^ ) 1 Q / X

exp (Rih) — exp (J? A) 2

T h e constants Ri a n d R w e r e d e f i n e d u n d e r E q u a t i o n 52. 2

F o r n e w concentrations at the b o u n d a r i e s of the w a t e r c o l u m n , C

2

at

ζ = 0 a n d C at ζ = h, transient concentrations m a y b e e v a l u a t e d f r o m 3

the f o l l o w i n g e q u a t i o n C =

+ Λ (C — Co) e x p (Uz/2K) λ

2

X [summation terms from e q u a t i o n 58]

+ Λ (Cz — Ci) exp [U(z — h)/2K] X [ s u m m a t i o n terms f r o m E q u a t i o n 58] λ

(63) w h e r e the i n i t i a l d i s t r i b u t i o n C

i=

0

i s g i v e n b y E q u a t i o n 62. A t the n e w

steady-state, the concentrations are g i v e n b y E q u a t i o n 62 w i t h C r e p l a c 3

ing C i and C replacing C . 2

0


68

NONEQUILIBRIUM SYSTEMS IN N A T U R A L WATERS

E s t i m a t e s of the p r o d u c t i o n rate of R a - 2 2 6 at i n t e r m e d i a t e d e p t h s i n the P a c i f i c are i n the r a n g e 1 Χ 1 0 " - 5 0 Χ 1 0 18

1 8

grams · l i t e r " · y r " 1

T w o steady-state profiles c o m p u t e d f r o m E q u a t i o n 62 u s i n g Q = 10"

18

grams · l i t e r " · y r ' 1

1

(43).

1

21

are s h o w n i n F i g u r e 14 ( d a s h e d c u r v e s ) .

X

The

differences b e t w e e n the concentrations s h o w n b y the s o l i d curves 1 a n d 2 i n F i g u r e 14 ( n o p r o d u c t i o n , Q =

and the corresponding

dashed

curves are 1 0 % or less. T h e s m a l l difference b e t w e e n the t w o

0)

models

shows that the s a m p l i n g a n d a n a l y t i c a l a c c u r a c y m u s t be h i g h i f the differences

i n the R a - 2 2 6 p r o d u c t i o n i n the w a t e r c o l u m n are to

be


i n f e r r e d f r o m observations.

Figure

16.

Diagrammatic

Ra-226 profiles in oceanic

sediment

Profile at t = 0: steady-state, Equation 64, diffusion coefficient of Ra-226 in sedi ment Κ = 1 X 10~ cm · sec' , sedimentation rate U = 9.5 X 10~ cm · sec' (3 mm11000 yr; (5)j. Concentration scale normalized to the value of initial Ra-226 at the sediment-water interface C = 1. New concentration at the interface C i = 1.6Co. Profiles at 100 and 1000 years after the change in boundary concentration computed from Equation 70. New steady-state (t = oo ) from Equation 71. Con stant Th-230 concentration at the interface taken as C T ° = 100C . 9

2

1

12

1

c

o

R A - 2 2 6 IN SEDIMENT.

M i g r a t i o n of R a - 2 2 6 i n the s e d i m e n t c o l u m n i n

d e e p ocean has b e e n i n f e r r e d f r o m the d i s e q u i l i b r i u m of R a - 2 2 6 a n d its p a r e n t i o n i u m ( T h - 2 3 0 ) d e t e c t e d i n a n u m b e r of sediment cores

(5).

W h e r e a s T h - 2 3 0 t a k e n u p b y the s e d i m e n t p a r t i c l e s f r o m t h e sea w a t e r


2.

LERMAN

Time to Chemical

69

Steady-States

shows n o t e n d e n c y to m i g r a t e n o r r e d i s t r i b u t e itself i n the s e d i m e n t 44),

(5,

R a - 2 2 6 m i g r a t e s , a n d this process results i n the flux of R a - 2 2 6 f r o m

the s e d i m e n t

into water column.

A

generalized

concentration-depth

profile of R a - 2 2 6 i n the s e d i m e n t , a d o p t e d f r o m the I n d i a n O c e a n d a t a , is s h o w n i n F i g u r e 16, l a b e l l e d t =

0. T h e c o n c e n t r a t i o n p r o f i l e , c o n s i d

e r e d s t a t i o n a r y , is m a i n t a i n e d b y a b a l a n c e b e t w e e n the s u p p l y of R a - 2 2 6 f r o m t h e d e c a y of the p a r e n t T h - 2 3 0 ( w h i c h is b e i n g a d d e d to the s e d i m e n t at a constant rate a n d constant c o n c e n t r a t i o n )

a n d t h e d e c a y of

R a - 2 2 6 a n d its m i g r a t i o n t h r o u g h the s e d i m e n t c o l u m n . F r o m t h e s t e a d y state profile of R a - 2 2 6 i n the I n d i a n O c e a n , the d i f f u s i o n coefficient


R a - 2 2 6 i n the s e d i m e n t has b e e n e s t i m a t e d as 1 Χ

10" c m 9

· sec'

2

for (5).

1

T h i s v a l u e is a p p r o x i m a t e l y t h r e e orders of m a g n i t u d e l o w e r t h a n t h e values of the d i f f u s i o n coefficients of i o n i c solutes i n a q u e o u s solutions, a n d i t is also m u c h too l o w to b e a c c o u n t e d for b y t h e t o r t u o s i t y of t h e p o r e space i n the sediment.

C h e m i c a l i n t e r a c t i o n of R a - 2 2 6 w i t h

the

s e d i m e n t m a y b e the reason for the l o w v a l u e o f the d i f f u s i o n coefficient obtained

(5).

A stationary c o n c e n t r a t i o n - d e p t h

profile of a c h e m i c a l

species i n the s e d i m e n t m a y b e p e r t u r b e d b y a n y c o m b i n a t i o n of s u c h factors as a c h a n g e i n the rate of d e p o s i t i o n , c h a n g e i n the rate of s u p p l y , a n d c h a n g e i n the c o n c e n t r a t i o n at the s e d i m e n t - w a t e r i n t e r f a c e r e s u l t i n g f r o m e x t e r n a l causes. S u c h changes, d i s t u r b i n g the e x i s t i n g c h e m i c a l steady-state, w o u l d cause the c o n c e n t r a t i o n of the species to v a r y as a f u n c t i o n of t i m e u n t i l a n e w steady-state has b e e n e s t a b l i s h e d . T h e t i m e r e q u i r e d to a t t a i n a n e w steady-state for R a - 2 2 6 i n the s e d i m e n t w i l l b e

e v a l u a t e d for

a

s i m p l e , b u t h y p o t h e t i c a l , case of the R a - 2 2 6 c o n c e n t r a t i o n at the s e d i m e n t - w a t e r i n t e r f a c e i n c r e a s i n g b y a f a c t o r of

1.6.

S u c h a n increase

w o u l d k e e p the R a - 2 2 6 / T h - 2 3 0 a t o m r a t i o at t h e s e d i m e n t - w a t e r i n t e r face at the v a l u e of 1.6/100, w h i c h is s t i l l b e l o w the e q u i l i b r i u m v a l u e of a p p r o x i m a t e l y 2 / 1 0 0 ; the p r e s e n t - d a y r a t i o is n e a r 1 / 1 0 0 . T h e present (t

=

0)

R a - 2 2 6 profile i n F i g u r e 16 is g i v e n b y t h e

f o l l o w i n g r e l a t i o n s h i p d e s c r i b i n g a steady-state d i s t r i b u t i o n of a d e c a y i n g species C = A exp [-zkr/U]

+

(Co

A

=

exp

A)

[(à

-

Ϋ&+ i>]

(64)

x

w h e r e A is a constant.

\

T

"KTC T°

λ* -

IT

-

KkJ/U*

is the d e c a y constant o f T h - 2 3 0 , C ° T

the s e d i m e n t - w a t e r i n t e r f a c e , λ

Β

(65)

is T h - 2 3 0 c o n c e n t r a t i o n a t

is t h e d e c a y constant of R a - 2 2 6 , C


0

is

70


the R a - 2 2 6 c o n c e n t r a t i o n at the s e d i m e n t - w a t e r i n t e r f a c e , 17 is the rate of s e d i m e n t a t i o n , a n d Κ is the d i f f u s i o n coefficient

of

Ra-226 i n the

sediment. A r e l a t i o n s h i p for a non-steady-state

c o n c e n t r a t i o n of R a - 2 2 6

may

be obtained b y solving the following differential equation

|£

=

exp

XTC ° T

(-ZXT/U)

+

Κ

^

-

U ^

1C

-

(66)

R

w h e r e the t e r m λ Ο °

exp ( —X z/U)

(grams · c m "

i n the s e d i m e n t o w i n g to d e c a y of T h - 2 3 0 .

τ

is the rate of p r o d u c t i o n of R a - 2 2 6

T

τ

· yr" )

3

1

The


c o n c e n t r a t i o n s at the i n t e r f a c e , rate of s e d i m e n t a t i o n , a n d d i f f u s i o n co efficient are c o n s i d e r e d constant.

E q u a t i o n 66 is to b e s o l v e d w i t h t h e

following conditions. Initial conditions : at / = 0 : C

=

C o

^

A

exp

[-zk /U]

+

T

-

(C

0

A)

exp

-

| / ^

+

^ ]

(67) B o u n d a r y c o n d i t i o n s : a t t > 0 : C = Ci a t ζ = 0 C = 0 at ζ =

(68)

oo

(69)

T h e s o l u t i o n of E q u a t i o n 66 is

c - c_. + y

2

(c, -

e.)

exp

[(^

-

| / ^

+

^ ]

χ

-P[*|/S +Ï ] X « * [ ^ + ^ ( S ^ > Î

(70)

W h e n a n e w steady-state has b e e n a t t a i n e d , t h e R a - 2 2 6 c o n c e n t r a t i o n as a f u n c t i o n of d e p t h b e c o m e s

C _„ (

= A exp [-zkr/V]

+

(Ci -

A) exp [(^

-

|/^-

2

+

^ ]

w h i c h is a n a l o g o u s to R e l a t i o n s h i p 64 f o r t h e i n i t i a l steady-state.

(71) The

c o n c e n t r a t i o n - d e p t h profile for a n e w steady-state, w i t h t h e R a - 2 2 6 c o n c e n t r a t i o n at t h e s e d i m e n t - w a t e r i n t e r f a c e t a k e n as C i = i n F i g u r e 16 i n t h e c u r v e l a b e l l e d t =

1 . 6 C , is s h o w n 0

oo. T w o curves for transient c o n -


2.

LERMAN

Time

to Chemical

71

Steady-States

centrations at 100 a n d 1000 years after the c h a n g e i n the b o u n d a r y c o n c e n t r a t i o n , c o m p u t e d u s i n g E q u a t i o n 70, are also i n F i g u r e 16. A f t e r a t i m e as short as 1000 years, the c o n c e n t r a t i o n - d e p t h profile is a l r e a d y v e r y close to the n e w steady-state profile; the differences i n c o n c e n t r a t i o n b e t w e e n the t w o curves are 7 % a n d less. I n the i n i t i a l profile, the R a - 2 2 6 c o n c e n t r a t i o n increases f r o m the s e d i m e n t - w a t e r interface d o w n .

Such a

c o n c e n t r a t i o n g r a d i e n t is a p r e r e q u i s i t e c o n d i t i o n for m a i n t a i n i n g d i f f u s i o n a l flux of R a - 2 2 6 f r o m the s e d i m e n t i n t o t h e o v e r l y i n g w a t e r . the n e w

steady-state, h o w e v e r ,

the c o n c e n t r a t i o n

decreases f r o m

At the

interface d o w n , w h i c h indicates that there w o u l d be n o R a - 2 2 6 flux out


of the sediment.

T h e n e w steady-state profile w o u l d b e a t t a i n e d i n a p

p r o x i m a t e l y 3000 years; c o n c e n t r a t i o n c u r v e for t = of the steady-state

concentration.

3000 is w i t h i n

T h e t i m e is o b v i o u s l y

v i e w e d i n p e r s p e c t i v e of the h i s t o r y of o c e a n i c sediments.

short

1%

when

It m a y

be

verified f r o m E q u a t i o n 70 for transient concentrations that, i n g e n e r a l , r a p i d rates of s e d i m e n t a t i o n ( l a r g e U) s e d i m e n t ( l a r g e K) state.

or h i g h diffusivities w i t h i n the

w o u l d result i n a m o r e r a p i d a t t a i n m e n t of a steady-

T h e h a l f - l i f e of the c h e m i c a l species (1620

y r i n the case of

R a - 2 2 6 ) has r e l a t i v e l y little effect o n the l e n g t h of t i m e i t takes to estab l i s h a n e w steady-state. n u c l i d e (λ;? =

E v e n w h e n the m i g r a t i n g species is a stable

0 i n E q u a t i o n s 64, 70, a n d 7 1 ) , it w o u l d t a k e less t h a n

10,000 years for its c o n c e n t r a t i o n to c o m e to w i t h i n 5 % of the steady-state v a l u e i n the u p p e r 1 0 - 2 0 c m of the sediment. T h e g e n e r a l i t y of the a r g u ments m a y be stressed b y p o i n t i n g out t h a t the t i m e to

steady-state

d e p e n d s o n h o w fast the t i m e - d e p e n d e n t terms (those b e t w e e n the braces i n E q u a t i o n 70) t e n d to t h e i r l i m i t i n g values of 2 a n d 0 as ί tends to infinity.

T h e s e terms d e p e n d o n U, K, a n d

b u t not o n the c h e m i c a l

nature of the p a r e n t species (i.e., n e i t h e r o n its c o n c e n t r a t i o n C ° T

d e c a y constant

n o r its

λ ). τ

T h e shortness of t i m e for t r a n s i t i o n f r o m one stationary c o n c e n t r a t i o n profile to another demonstrates that e v e n i n the s l o w l y d e p o s i t e d

deep

o c e a n i c sediments it m i g h t be difficult to detect near the s e d i m e n t - w a t e r interface a n y changes ( i f s u c h o c c u r r e d ) i n the past c h e m i c a l h i s t o r y of the ocean. Appendix T h e second-order p a r t i a l d i f f e r e n t i a l equations g i v e n i n the text of the p a p e r c o n t a i n t i m e derivatives of c o n c e n t r a t i o n

(dC/dt)

a n d terms

c o n t a i n i n g dC/dz a n d C. T h e solutions of these equations, unless r e f e r r e d to a l i t e r a t u r e source, w e r e o b t a i n e d b y the m e t h o d of L a p l a c e transfor m a t i o n w i t h the a i d of s t a n d a r d tables of L a p l a c e transforms.

Good

w o r k i n g s u m m a r i e s of the L a p l a c e t r a n s f o r m a t i o n m e t h o d as a p p l i e d to


72


s o l u t i o n of p r o b l e m s i n heat flow a n d diffusion are i n References 22 a n d 32.

T a b l e s of L a p l a c e transforms i n References 22, 32, a n d 45 are g i v e n

i n the f o r m w h i c h is p a r t i c u l a r l y c o n v e n i e n t for s o l v i n g equations w i t h constant coefficients, of the t y p e u s e d i n this p a p e r . T h e functions erf ( e r r o r f u n c t i o n )

a n d erfc

(error function

com

p l e m e n t ) a p p e a r i n m a n y of the solutions g i v e n i n the p a p e r . T h e s e f u n c tions a p p e a r

i n the process of i n t e g r a t i o n of terms c o n t a i n i n g e'

y2

(y is

some f u n c t i o n of K, U, λ, t, a n d ζ ) a n d i n the process of i n v e r t i n g ( w i t h the a i d of the tables) the t r a n s f o r m e d c o n c e n t r a t i o n v a r i a b l e C b a c k to the o r i g i n a l c o n c e n t r a t i o n C , to be g i v e n i n the s o l u t i o n as a f u n c t i o n of


ζ a n d t ( a n d the c o n s t a n t s ) .

D i s c u s s i o n a n d m a t h e m a t i c a l definitions

of the error f u n c t i o n are g i v e n i n m a n y texts a n d , a m o n g those l i s t e d i n the references of this p a p e r , i n the H a n d b o o k of M a t h e m a t i c a l F u n c t i o n s (46),

C a r s l a w a n d Jaeger ( 3 2 ) , a n d C r a n k (22).

T h e error f u n c t i o n is

defined as

erf χ =

X

f

(72)

e~y dy 2

w h e r e y is i n t e g r a t i o n v a r i a b l e , a n d it m a y be a f u n c t i o n of x. T h e error f u n c t i o n c o m p l e m e n t is defined as

erfc χ =

4= f Vr,J

~ V

e

(^

y2d

73

x

E r f χ a n d erfc χ are i n t e r r e l a t e d , erfc χ =

1 — erf χ

(74)

F o r negative a r g u m e n t , erf ( — ζ ) = erfc ( — x) =

— erf χ

(75)

1 + erf χ

= 2 — erfc χ (76) I n m a n y of the solutions g i v e n i n the p a p e r , the l i m i t i n g values of c o n c e n t r a t i o n C w h e n either ζ or t a p p r o a c h 0 or infinity m a y be v e r i f i e d b y s u b s t i t u t i o n of the a p p r o p r i a t e values of the functions erf a n d erfc. T h e values of these functions w h e n the a r g u m e n t is zero, p l u s - or m i n u s i n f i n i t y are: erf (0) = 0

erfc (0) = 1

erf ( « ) =

erfc ( « ) =

erf (-co)

1 =

-

1

erfc ( - « )

0 =

2


2.

LERMAN

Time

to Chemical

73

Steady-States

E r f c χ is a r a p i d l y d e c r e a s i n g f u n c t i o n of x.

W h e n χ increases i n

definitely, erfc χ tends to zero faster t h a n e? tends to infinity. O w i n g to 2

this, the p r o d u c t e ~ erfc χ a n d , c o n s e q u e n t l y , the p r o d u c t e enc χ t e n d x

x

to z e r o as χ increases i n d e f i n i t e l y . P r o d u c t s of exponentials a n d e r r o r f u n c t i o n s a p p e a r i n some of the solutions d i s c u s s e d i n the p a p e r . T a b l e s of erf χ are a v a i l a b l e for values of χ b e t w e e n 0 a n d 2.00, i n steps of 0.01 (46).

E r f x, erfc x, a n d several r e l a t e d functions have b e e n

t a b u l a t e d for values of χ b e t w e e n 0 a n d 3.0, i n steps of 0.05 a n d 0.1 32).

(22,

References to o l d e r tables are i n C a r s l a w a n d Jaeger (32, p. 4 8 2 ) . I n this p a p e r , the values of erf χ a n d erfc χ w e r e c o m p u t e d f r o m a n


a p p r o x i m a t i o n for erf χ g i v e n i n R e f . 47 for 0 $ζ χ < 3. erfc χ = where α

1/(1

+ αχ + ax λ

2

2

+ α χ + α α + α^ ) 3

3

4

χ

= 0.14112821

α

4

=

α

2

= 0.08864027

α

5

=

α

3

= 0.02743349

F o r values of χ S* 3, the f o l l o w i n g series w a s u s e d f

=

e~

J

/ l

1

.

3

15

.

105

4

-

5

(77)

8

0.00039446 0.00328975

(32). 945

10365\

(

.

R e l a t i o n s h i p s 77 a n d 78 are easily p r o g r a m m a b l e for use i n a d i g i t a l computer.

I n c o m b i n a t i o n w i t h R e l a t i o n s h i p s 75 a n d 76, they

allow

c o m p u t a t i o n of erfc χ b e t w e e n the l i m i t s m i n u s - a n d p l u s - i n f i n i t y . O t h e r forms of series e x p a n s i o n , r a t i o n a l a p p r o x i m a t i o n s , a n d m e t h ods of i n t e r p o l a t i o n f r o m tables of erf χ a n d erfc χ are g i v e n i n the H a n d b o o k of M a t h e m a t i c a l T a b l e s (46, p p . 2 9 7 - 9 , 3 0 4 ) .

Acknowledgment T h e m a t e r i a l i n the p a r t of this p a p e r d e a l i n g w i t h the D e a d Sea a n d L a k e T i b e r i a s was p r e p a r e d i n 1969 at the Isotope D e p a r t m e n t , W e i z m a n n Institute of Science, R e h o v o t , Israel. A t that t i m e , I

benefited

f r o m c o n s t r u c t i v e discussions w i t h J o e l R . G a t a n d A a r o n N i r of

the

W e i z m a n n I n s t i t u t e , a n d I t h a n k D r . G a t for c o m m u n i c a t i o n of u n p u b l i s h e d d a t a o n R a - 2 2 6 i n the D e a d Sea. F o r c r i t i c a l r e a d i n g a n d d i s c u s s i o n of the p a p e r I a m i n d e b t e d to J . S t e w a r t T u r n e r ( C a m b r i d g e , U . K . , a n d W o o d s H o l e , M a s s . ) , H a r m o n C r a i g ( L a J o l l a , C a l i f . ) , a n d the e d i t o r i a l reviewers, E d w a r d D . G o l d b e r g ( L a Jolla, Calif. ) a n d D e r e k W . Spencer (Woods Hole, Mass.).


74

NONEQUILIBRIUM SYSTEMS IN N A T U R A L

WATERS


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2.

LERMAN

Time

to Chemical

Steady-States

75

(24) Lovering, T. S., "Heat Conduction in Dissimilar Rocks and the Use of Thermal Models," Bull. Geol. Soc. Am. (1936) 47, 87-100. (25) Eckart, C., "Hydrodynamics of Oceans and Atmospheres," p. 57-71, Pergamon, New York, 1960. (26) Kato, H., Phillips, Ο. M., "On the Penetration of Turbulent Layer into Stratified Fluid," J. Fluid Mech. (1969) 37, 643-55. (27) Turner, J. S., "A Note on Wind Mixing at the Seasonal Thermocline," Deep-Sea Res. (1969) Suppl. toVol.16, 297-300. (28) Neev, D., Emery, K. O., "The Dead Sea, Depositional Processes and En vironments of Evaporites," Bull. Israel Geol. Surv. (1967) 41, 1-147. (29) Neumann, J., "Tentative Energy and Water Balances for the Dead Sea," Bull. Res. Council Israel (1958) 7G, 137-63. (30) McEwen, G. F., "A Mathematical Theory of the Vertical Distribution of Temperature and Salinity in Water Under the Action of Radiation, Conduction, Evaporation, and Mixing Due to the Resulting Convec tion," Bull. Scripps Inst. Oceanog. (1929) Tech. Ser. 2, 197-306. (31) Gat, J. R., Gilboa, G., Isotope Dept., Weizmann Institute of Science, Rehovot, Israel, "Radium-226 in the Dead Sea," personal communica tion, 1969. (32) Carslaw, H. S., Jaeger, J. C., "Conduction of Heat in Solids," 2nd ed., 50-496, Oxford University Press, Oxford, U.K., 1959. (33) Redfield, A. C., Ketchum, Β. H., Richards, F. Α., The Influence of Or ganisms on the Composition of Sea-Water, "The Sea," M. N. Hill, Ed., Vol. 2, p. 26, Interscience, New York, 1963. (34) Bowen, V. T., Noshkin, V. E., Volchok, H. L., Sugihara, T. T., "Strontium-90: Concentrations in Surface Waters of the Atlantic Ocean," Science (1969) 164, 825-7. (35) Broecker, W. S., "Radioisotopes and the Rate of Mixing Across the Main Thermoclines of the Ocean," J. Geophys. Res. (1966) 71, 5827-36. (36) Münnich, K.-O., Roether, W., "Transfer of Bomb C-14 and Tritium from the Atmosphere to the Ocean. Internal Mixing of the Ocean on the Basis of Tritium and C-14 Profiles," Symp. Radioactive Dating and Methods of Low Level Counting, p. 93, International Atomic Energy Agency, Vienna, 1967. (37) Rooth, C., Ostlund, H. G., "Tracing the Oceanic Tritium Transient," Tech. Rept., Univ. of Miami, Rosenstiel School of Marine and Atmospheric Sciences, 1970, 1-27. (38) Sverdrup, H. U., Johnson, M. W., Fleming, R. H., "The Oceans," p. 161, Prentice-Hall, Englewood Cliffs, N. J., 1942. (39) Wyrtki, K., "The Oxygen Minima in Relation to Ocean Circulation," Deep-Sea Res. (1962) 9, 11-28. (40) Lerman, Α., "Sea Water—Geochemical Balance," R. W. Fairbridge, Ed., "Encyclopedia of Earth Sciences," 4A, Van Nostrand Reinhold, New York, 1971 (in press). (41) Koczy, F. F., "Natural Radium as a Tracer in the Ocean," Proc. 2nd U.N. Intern. Conf. Peaceful Uses of Atomic Energy (1958) 18, 351-7. (42) Craig, H., "Dissolved Gases, Deuterium, Oxygen, and Carbon Isotopes in the Ocean," Bat-Sheva Seminar on Marine Geochemistry, Intern. Sum mer School, Weizmann Institute of Science, Rehovot, Israel, June 1969 (unpublished). (43) Craig, H., Scripps Institution of Oceanography, La Jolla, Calif., "Rates of the Ra-226 Production in the Pacific," personal communication, 1971. (44) Bernat, M., Goldberg, E. D., "Thorium Isotopes in the Marine Environ ment," Earth Planet. Sci. Lett. (1969) 5, 308-12. (45) Nixon, F. E., "Handbook of Laplace Transformations," 2nd ed., Prentice -Hall, Englewood Cliffs, N. J., 1965, 260 pp.


76


(46) Abramowitz, M., Stegun, I. Α., Eds., "Handbook of Mathematical Func tions with Formulas, Graphs and Mathematical Tables," p. 279-329, National Bureau of Standards, Washington, D. C., 1966. (47) Hastings, C., "Approximations for Digital Computers," p. 186, Princeton University Press, Princeton, N. J., 1955 (also cited in Ref. 46, p. 299). 27,

1970.


RECEIVED May


Nonequilibrium Systems in Natural Water Chemistry

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