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RESIDU, a computer code for the evaluation of the mean residence time of water in monomiCtic lakes
J. HARTMAN
LEGAL NOTICE
This document was prepared under the sponsorship of the Commission of the European Communities.
Neither the Commission of the European Communities, its contractors nor any person acting on their behalf:
make any warranty or representation, express or implied, with respect to the accuracy, completeness, or usefulness of the information contained in this document, or that the use of any information, apparatus, method or process disclosed in this document may not infringe privately owned rights; or
assume any liability with respect to the use of, or for damages resulting from the use of any information, apparatus, method or process disclosed in this document.
This report is on sale at the addresses given on page 3 of cover.
Published by the Commission of the European Communities Directorate General" Scientific and technical information and information management" 29, rue Aldringen LUXEMBOURG (Grand-Duchy)
The present document contains knowledge resulting from the execution of the research programme of the European Economic Community .
COMMISSION OF THE EUROPEAN COMMUNITIES
environment and quality of life
1975
RESIDU, a computer code for the evaluation of the mean residence time of water in monomicti·c lakes
J. HARTMAN
Joint Nuclear Research Centre
lspra Establishment -Italy
EUR 5361 e
Table of contents
Abstract
§ 1. Introduction
§ 2. The code
2.1 General Remarks
2.2 Flowchart
2.3 Input description
2.4 Listing of the code
2.5 Some examples
References
2
4
7
7
8
9
11
15
21
3
Abstract
In this paper a method is presented to evaluate the mean residence time
of water in a monomictic lake.
A certain knowledge, concerning the hydrologic cycle and the time evo
lution of the thermocline of the lake to be treated, is assumed.
A computer code has been established to serve the purpose and some ex
amples have been calculatedo
4
§ 1. Introduction
In the general case of a monomictic lake, i.e. a lake which has a water
turnover, whether partial or total, only once a year, the evaluation of
the mean residence time of the water in the lake proceeds as follows
(ref. 1,2).
We define an annual hydrologic cycle of the lake with a partial water
turnover, which we will refer to as a partial cycle.
We further have a pluriannual cycle, composed of N partial cycles (oc
casionally N might be equal to zero) and one cycle with a complete water
turnover, which will be called a total cycle. Every cycle is divided into
intervals of constant epilimnion volume. The subdivision into intervals
is based upon a statistical knowledge of the time evolution of the thermo
cline in the lake during the year. The last interval of every cycle re
presents the condition of greatest homogeneity (occasionally total homo
geneity) of the lake, while the f~rst interval of every cycle is that of
transition between the turnover and the stratification period.
The calculation gives, at the end of a partial cycle, a dilution coeffi-
cient for the original water in the epilimnion called ~ so that after ep' N identical partial cycles the dilution coefficient for the original water
in the epilimnion becomes ~N • ep
The total cycle, appearing after N partial cycles, gives the dilution co
efficient 0( for the whole lake.
If the pluriannual cycle is composed of N+l = T annual cycles, the mean
residence time '1:' of the water in the lake is obtained by means of the
following relationships.
+ (I N) 0( N 2. N +' T + - - -- -·-"' 2.
· f .. l N } 'T = ( I - o<.) I l I + 3 0<. ... ~ 0\ 1" - -- -+ ( 2 N T t) 0( T - - - -
<::.0
- ( i - o0 T L (l N + ') ~ N 2. N::o
- (I -o() T { f_ 'l N 0{, N + f. o<.. N J 2 N~o N=c
= (1-cX}- 2. L_ N 0\ ·+ --· T {· ~ N \ }
'1 N:o 1-(X.
where
oo N oo ~ 0(~ NNN.-1 L__No( = L ~ N.:O N:o
we therefore obtain
T ·= (' -ex.) .T { l 0<.. . - ' . + I } 2 (I _ !X)2 i _ o<..
_ ·1 { 2. <X. + 1 } 2 t-o<.
_ T 1o<. ·+ \ - oc _T -- ·-.l. l - 0(_ 2.
6
The calculation of c<ep (or 0( ) is based upon the general assumption
that the mixing of inflowing waters with the original lake water is
limited to the region above the thermocline, or epilimnion.
The volume of the original water in the epilimnion at the end of
interval i is determined by the equation
, .n. (t) =
J.
1 where ~ is the probability that a water volume element, present
somewhere in the lake at the beginning of time interval i, flows
out of the lake during the same interval.
We obtain .n.. = n
i oi t
exp ( - - ) Bi
where !loi is the volume of the original water in the epilimnion
at the beginning of the interval i.
If we define a retention coefficient
vi ep
vi + qi a tJ.. ep
where vi = volume of the epilimnion in interval i, A t1. = length ep
of interval i, and qi = mean rate of inflow water in interval i,
we then have 1
Bi"-
Substituting (3) into equation (2) gives
n. ~
=A .l..L. 0~
exp ( -
t
vi + q. A ti ep ~
t )
Although v!P is constant during interval i, the rate of inflow
water usually varies. Therefore the interval i is subdivided into
m subintervals, each with a constant rate of inflow water qik
(k=:l, ••••• m) and equation (4) becomes
A. = .n.. J. 0~
m
exp ( - L k=l
) i
v + q 'k /itk ep ~
(1)
(2)
(3)
(4)
7
§ 2. The code
t 2.1 General Remarks
The program RESIDU is a PL/1 code composed of a main program and one sub
routine called FUN. This subroutine calculates o( , the dilution coefep
ficient for the original water in the epilimnion at the end of an annual
cycle.
To facilitate the reading of the code, the following description of the
main variables appearing in the program might be useful.
NPAR
NTO"r
NIP
NIT
V(i)
Vl(i)
TRANS(j)
DAYS(i)
Q(i)
ALFA
TAU
number of partial cycles
number of total cycles (=1)
number of intervals in a partial cycle
number of intervals in the total cycle
epilinmion volume (m3) of interval i in
epilimnion volume (m3) of interval i in
the partial cycle
the total cycle
average inflow-outflow 3 rate (m /day) per month j
total number of days in interval i
total volume of transit (inflow-outflow) water (m3
) in interval i
0( = dilution coefficient for the original water at the end Of ep 1 1 an annua eye e
"t = mean residence time
§ 2.2 FLOWCHART
8
READ INPUT DATA AND PRINT
NO
PREPARE DATA FOR PARTIAL CYCLE AND CALL FUN
FUN
CALCULATE DILUTION COEFFICIENT AT THE END OF ALL PARTIAL CYCLES
PREPARE DATA FOR TOTAL CYCLE AND CALL FUN
FUN
DETERMINE MEAN RESIDENCE l TIME AND PRINT
§ 2.3
Card 1
Card 2
Card 3
9
Input description
TEXT = title (col. 1-80)
NPAR = number of partial cycles. NIP = number of intervals in the partial cycle. NIT = number of intervals in the total cycle.
3 TRANS(j) (j=l,l2) = average inflow-outflow rate (m /day)
for each month; 12 values are required.
If NPAR = 0 cards 4, 5 and 6 are omitted.
Card 4
Card 5
Card 6
V(i) (i=l,NIP) = epilimnion volume (m3
) of interval i in the partial cycle.
N = number of different subintervals of which interval i is composed.
NDAYS(k) = number of days of subinterval k. NMONTH(k) = identification of the month of
subinterval k. Identification code to use:
JAN = january FEB = february MAR = march APR = april MAY = may JUN = june JUL = july AUG = august SEP = september OCT = october NOV = november DEC = december
The values NDAYS(k) and NMONTH(k) define the sub:·, nterval k (k=l ,N).
Cards 5 and 6 are repeated NIP times.
Card 7
Card 8
Card 9
V(i) (i=l.~IT) = epilimnion volume (m3
) of interval~ in the total cycle.
as card 5.
as card 6.
10
Cards 8 and 9 are repeated NIT times.
Further, two general PL/1 rules have to be observed:
1. All input v~lues are separated by blanks and/or commas.
2. All alphanumerical values (as TEXT and NMONTH) are enclosed with
quotes.
12
I I I RrSID" I I ((At Ctllf\TFS THE MEl\1\J RfSIOF.:~CE TIME OF wATER TN MONOMICTIC LAKES)! T I
NPAR "-JTtJT NIP NIT V ( I) Vl (I )
N l I~· BE R C F P tH< T I At C VC L E S ~~liMBE-R fF TCTA' CYCI ES (=1) ~I.H-13FR nF INTEP,VALS P~ A PARTIAL CYCI. E Nt!Mt3ER CF INTERVALS I'J THE TOTAl CYCL!: fPit.IMNI!lN 'vO'-.UME (M3) UF INTERVAl. I 1"4 THE PARTIAl CYClE fPitlf'NION \lllli'~E (M3) DF JNTFRVAL I I~J THE TOTA' CYCLE
T R Al\j$ ( J) O.~YS("!) {.,) ( I )
AVERAGF. INfl.OW·-OlfTFl OW RATE (M3/DAY) PER M'lNTH J T D TAt N t If~ BE R OF D A Y S I N I NT E R VAL I TOTtd VOt. l!Mf OF TRANSIT ( INFLOw-OIJTFtClW) W-\TER (M3) IN
.O.tFA
Tl\f.l .
INTER VAL I Dit IJT IC:~J COEFFICIENT FnR THE ORIGI~ Al WATER AT THE FNO OF
AN A~\JtiAt CYCLE
~-~----------------------------~-~-~-~-~-------------~--~~--~---------
I!.A •• PRflC flPTIONS (MAl~),• f'l(f TFXT CHJ.,R(80) VAR,CtJ(20),. !1Cl \/(20) ,Q( 20) ,DAYSf20) ,Vl(20), Q.l (20) ,DAYS! (20),. DCI RHO(?OJ,~H01(2Q),. DC1 NOAYS(*)CT!, NMONTH(*)CHAR(9) CTf ,o DC t T P. A "J S { J 2 ) , c DC t '>1 0 ~ S C H A R ( 4 8 )
I N I T ( ' J A f\ $ FE B $ f'J A R $APR $MAY $ Jl.fl\j $ .J II l. $ A I J G $ S r.: P $ !J C T $ N Q V $ DE C $ 1 ) , •
r] ~! F R f.:. 0 Q S N l\ P Pl.! T P ~ G F D .6. T t\, o
PllT PA(,E tDI T (I INPliT C/\ROS' '-----------' J C SKIP ( 4) , A, SKIP, A), o PUT SKIP ( 5 J, •
G~T f.lST ( TFXT,NPAR,NIP 1 NIT) COPY,. GtT t I ST ( CTRA~JS fi) On l =1 Til 12)) COPY, • NTflT=J to
IF ~.!pf\R=O THFN CCTC HR • GET ((V{I) D(l l=J TO "JfP)) CflPY,c nc1 1=1 TO N!P,e GET (N) COPY,. Alt0CATE NDAYS(N),f\MCNTH(N),. r, F T t ( ~ !1 A Y S ( ~~ ) , N" C NTH C M ) n 0 i\11 = 1 T U N ) ) C OP Y, o
13
PAYS(J_)=~:t,. G(I)=O,. Sl~M=n,. 0 fl .J = 1 T -, N , ., K = ( I N 0 f '< ( M CN S , S II 8 S T R ( N f'i 0 NTH ( ,J ) , 1 , 3 ) ) -t- 3 ) I 4, o IF K=O THJ":N f1 1J,. PIJT ·::oJT ( 1 MISTAK~ TN INP!tT OATA 1 ,NMONTH(J))
P=TPANS(Kl*~CAYS(J),e S ll ~1 = S 1. 1 ~ + P I ( V ( I ) + F ) , o O~Y~(J)=O~YS(l)+~DAVS(J),. t.J(J l=Q{!)-t-P,. rNDto F: HO ( J. ) = :; X P ( - S! 1 ~ ) , • ENO,o
( .A , X ( 3 ) , A ) , c G nT r, EO J , • F ~J 0, •
BR.coGf:T l_IST f(VJ(!) :JC ·r=J TO NtT)) CflPY,. 0 :J 1 ~ J TC N I T,. G E T ( ~J) C 0 P V , o ~ ~ t t c c ~ T i: f\J o A v s < ~ 1 ) , t\ \1 n "·' liH c N ) , • GET ((~~AYS(~},N~rNTH(~1 DO ~=1 TO~)) COPY,. :JAYSl(!)=lJ,o {Jl(!)=O,o S~P..,=O,o f""'G .J=l Tll N, c K = ( J N ') E X ( M ON S , S II P. S T K ( N t-' fl NTH ( J ) , l , 3 ) ) + 3 ) I 4, • P = T R 1\ N S ( K ) * N [L\ Y S ( ,I ) , o Sl.IM=St 1M+P/ (V~ (I) +P), o DAYSl (I )=DAYSlti )+NDAVS(J),. Q l ( J ) =IJ l { I ) + P, o
):;ND,o RHi1J (J)=fXP(-S!tMt,. c ~-'n '• p•_tT PtGE F['lT (TFXT,SI1PSTR{(80)'= 1 ,J,I. E\JGTH(TF.XT)))
(SI<IP(4) ,J',,SKIP, _6) ,. PPT St<IP(4) EDITf 1 AVERAGE li\JF10W-f11JTFifJW RATF.(M3/0.t\YJ P'.:R MONTH',
·---------------------------------------------· t (A, SKIP),. P!.IT ~DJT ((SIIRSTR(ti.ONS,1-+{l-})*4,3J,TRI\\JS{!) DO 1=1 P112))
(SKrP,~,'l(3) ,~(15,S,7) ),o PIIT SK1P(3) fDIT ( 'l\JTERVt\1 •, 1 0AVS' ,•Vol o fPI1 IM~If1N(M.3)' ,•vnt. TRANSIT W/lTFR(M3) •, . 1--------•,•----•,•-------------------•,•----------------------•)
( Cr::t ( 2 R), .~, Cf.ll ( 40), A, C'JL (C)~), A, COt ( 7 5), A),.
I~ ~PAR=O lHF~ GCTC (C,o P\IT ::DIT {'PARTJAt CYCtE 1 ) (SK!PC?),Cnt.(l~),A),c
~ 1 •• F ,..., R ',1 ,~ T ( C 1 t ( 2 53 ) , F ( 5 ) , C Cif. ( 4 ·J ) , F ( It ) , C 'J l. ( 5 ·~ ) , [ ( 13 , 3 , 5 ) , Cl L ( 7 5 ) , £:(14,3,5)),0
fVl r=1 T:J "rr,. P!IT EOlT(J,DAYSCI),V(I)Ji..l(I)) {R(Fl)),.r:ND,.
CC.oPIIT ~OIT ('TnTA• CYCtt-• (SKIP(2),COI(llJ,~),. OJ I=J TO NITto
14
!'!IT !:: [)IT ( 1 t 0 A Y S 1 ( I·) , V l ( I ) , Q 1 ( I ) ) ( R ( F 1 ) ) , • f. \J 0, •
P''T ~niT C tf'-Jt'f"'3[F CF P!1RTJ .61 CYCl fS =• ,~'JPAR, '"JI.l\1')F.R 1F TtJTAI CYClES =1 ,"-JTflT) ( S K I 0 ( 4 ) , C C1. ( !. 5 ) , .h, F ( 3) , SKI P, C Ot ( 1 7) , :\ , F ( ~ ) ) , •
IF ~.JPAk=O THEN GCTO FF,o PitT ;:ryr ( 1 Nt 1MBFP C~ JNTEPVf\t.S Jr\ PA~TTAl CYCtf = 1 ,N!P)
(([!t (3).tA,F{3))1• FFoo 0 t1T [;~HT ( 1 Nti~HFt< OF INTtRVALS IN TOTAL CYC' E =1 ,N!T)
(cr. 1. ( ':) ) , A, F ( 3 ) ) , • P!.l T SKI P ( 3) , •
T =!'J P t\ P +NT fl i, c wr::P=~ to J F ~'> J P A R = l) T H F N D r , • A 1 F A= 1 , • "-1 T 0 T = 0, • G J T :J D C, • E "~ n , • WEP=Fl'N(N! P,V,RHC,WFP, fJM,O), c PliT ~r!T{'fd FL\ AFTER 1 YEAR =•,w~p,• PERCFNT 1 )
(X(l ),~,F(9,3,2),1\),. ~ 1 F I\ = WE P * * t\ P t R , c
DD •• at Ff\=FitN(~.IJT,Vl,RH[1 ,At f('.,C~~,NTOT),. P"T EOIT( 1 A1FA tFTEP. 1 ,T, 1 YFAR', 1 S 1 , 1 •,• = 1 ,AtF~,• PEqCE"tT')
(SKIP,X(~),A 1 F(4),X(l),A,A(T GT l),t\(T If J),A,F(9,3,2),A),. T AII=T /?.* (1 +f. t ~A) I ( l-.Al. FA), •
P''T FUIT(•~EI'N PFSIDFtJCr: TIME =1 ,T.~!f,• Yf:ARS•) ( SK J P {? ) , X (?) , A, r ( 9, J), A.), •
EOJ •• f~\JD .. AA,.
F I I N o " P iHl ( t r--! , 'I , P H rJ , W ·~ P , C ~ , N T C T ) , c
DCt VC*l,RHI]{*) 7 (~(*),.
' 0 0 ~ = ~. T ;J N t e If 1=1 THEN GO TC PH,. IF 1 NE 2 THEN GC Tr CCto WN=wt?Pto Wf:P=~)M(l )/\1( J) ,.
!lBoo IJr-.1rJ=V (!) *wFP tG GC TrJ r.t:, o
c c. G. I f v ( i ) ~. F v ( 1. ) i HE~·! G c T {1 f) f) ' 0
wEP=1,. I F ~.J T D T = 1. ~ ·~l r I N F N T ._, ~=: N W !'-.: P = W N , •
DO • o 0 f'J.fJ = n tJ. { I -1. ) + ( V ( I ) - V ( I- 1 ) ) *WE P, o
E E •• U ~q I ) = F. H lJ ( I ) * lJ ~ r , • CNO, o R !~ T PR N ( c ~ ( 1\ ) I v ( ~! ) ) '0 ENO t-=UN,.
Lake
Lake
15
§ 2.5 Some examples
The code has been applied to the Lake Maggiore (Italy), to the
Lake Lugano and the Bay of Agno (Switzerland).
Lake Maggiore has a pluriannual cycle composed of 5 equal years
with a partial turnover, followed by 1 year with a total
turnover, while Lake Lugano has a pluriannual cycle of 9 equal
years with a partial turnover and then 1 year with a total
turnover.
The Bay of Agno is characterized by an annual cycle with a total
turnover.
Table I presents some significant data concerning the different
examples, together with the values for the mean residence time.
Table I
number of number of 0( t2p rx intervals intervals at the end at the end mean
number of in in of the of the residence partial partial total lstpartial total time cycles cycle cycle cycle cycle (years)
Maggiore 5 7 8 81,.582 % 67,.245 % 15,318
Lugano 9 7 8 85,.438 % 51,273 % 15,523
Bay of Agno 0 - 7 - 64.248 % 2.297
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22
References
1) R. Piontelli e V. Tonolli - Il tempo di residenza delle acque lacustri
in relazione ai fenomeni di arricchimento in sostanze immesse, con par
ticolare riguardo al Lago Maggiore. Mem. Ist. Ital. Idrobiol., 17, 247-
266 (1964).
2) G. Rossi, V. Ardente, F. Beonio -Brocchieri, E. Diana - Il tempo medio
di residenza delle acque nel Lago di Lugano, da pubblicare (1974).