CALIFORNIA STATE UNIVERSITY, NORTHRIDGE
A !JftATHEr-1ATICAL r-TODEL FOR KELP GROVvTH
;r
· A thesis submitted in partial satisfaction of the
lrequirements for the degree of Master of Science in
;
Biology
by
Bruce Nicholas Anderson
May, 1973
..
··-·-
...
-··-·
.......
--.~·-······--·
---~--·-
-----~--
...... __________ _
._
The thesis of Bruce
California State University, Northridge
Nay, 1973
I
___,_j
il
ACKNO\r.LEDGEMENTS
I wish to express my,rapprecl.ation to Dr. vlheeler
North for providing me with copies of the Kelp Investigation Program reports and for his review,of the model
during its development.
I want to thank r.1r. Robert de
Violini of the Point Mugu Geophysical Division for
solarimeter data and Mr. Frank Hovore for doing the
illustrations.
Also I wish to express my gratitude to Dr. Ross
Pohlo and Dr. John Swanson for their review of the
manuscript.
I especially want to thank my research
advisor, Dr. Richard Swade who has shown an extraordinary
interest in my progress and encouraged me accordingly.
My thanks go to my mother, Mra. Paula Anderson,
for helping \vith .preliminary> drafts.
I
Most of all, I want to
I family: to my wife, Linda,
r
I
express appreciation to my
for her help, encouragement
and devotion, and to my children, Mark, Angela, Roger
I
I
and Cindy for their understanding and endurance.
!
l'
#
;
l-·-·--- -·- ··---~···---~--- _,. __ -· --- ----.--·~--- --~-~~ --~--- ..
iii".
---·--·-------,I
TABLE OF CONTENTS
ACKNOV/LEDGEMENTS •
~
• • • • • • •· •· • • •· •. • • •
iii
LIST OF FIGURES
• • • • •· • • • • • • • • • • • •
v
LIST OF STI'IBOLS
• • • • • • • • • • • •
• • • • •
vi
.ABSTRACT • • • • • • • • • • • • • • • •
BIOLOGY OF KELP • • • • • • • • • • • •
HISTORY OF KELP BED INVESTIGATION • • •
MATHEfJIATICAL MODEL CONSTRUCTION. • • • •
Introduction to Model Development •
Derivation of Relationships • • • •
• • • • •
viii
• • • • •
1
• • • • •
8
• • • • •
• • • • •
• • • • •
11
11
12
RESULTS • • • • • • • • • • • • • • • • • • • • •
Optimization of Ne\'/ Growth • • • • • • • • •
Optimization of Canopy Regrowth • • • • • • •
Turbidity Influence • • • • • • • • • • • • •
Discussion. • • • • • • • • • • • • • • • • •
35
35
•
LITERATURE CITED •
.. . •
• • • • • •
Appendix 1 • • •
Appendix 2
Appendix 3
AP~ENDICES
• • • • • • • •
.....
...
•
•
•
•
•
•
0
•
•
•
•
•
•
•
•
0
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
0
•
•
•
•
•
•
•
~
••
•
•
•
••
•
•
0
•
•
~
•
•·
•
•·
•
36
36
41
45
47
47
51
52
I
!
,---
------------
1
LIST OF FIGURES
Page
Figure
..
Kelp beds' • • •
• • • • • • • • • • •· • •
Macrocystis pyrifera • • •· • ,
•
1·
15>
4.
Daylength • • •
• •
• • • • • • • • • •
Radiation, annual
• • • • • • • • • • •
5.
Radiation, daily
18
6.
Sunrise • • • • • • •
Declination •· • • • •
Reflected light •• • •
Photosynthetic rate
1.
2~
3.
. .. .. .. .. ..
..
0
.. ..
• • •·
.. • . •
.
• • • • •
5)
16
..
• • • • • • • • • • •
19
• • • • • • •
21
• • • • • • • • • • •
23
• • • • • • • • • • •
27
29
11 •
Water temperature • • • • • • • • • • • • •
Growth rate simulation •
• • • • • • • •
32'
12 •.
New growth
371
7.
s •.
9.
10.
13.
14 •.
..
.. •·
• • •
0
..
• • • • • • • • • • • • •
Growth rate envelope • • • • • • • • • •
Canopy regrowth •· • • •
• • • • • • • •
.
0
115:_-.. Turbidity • • • • •
.
0
0
v
0
• • • • • • • • • • •
38
39
40
LIST. OF STI-ffiOLS
DESCRIPTION
ABSO
absorbancy
APHOR
average photosynthetic rate
ASR
angle of sun,
ASD
angle of sun, degrees
ATEMP·
average water temperature
CAUL
canopy length
CTRANS
% light transmitted
C1, C2
particle concentration
CC1, CC2, ••• CC4
coefficients in photosynthetic rate pmly,nomial
DA
declination (angle), radians
DAD
declination (angle), degrees
DAYS
duration of simulation run
DFRL
rate of change of frond
length
DINC
daily time increment
DL
day length
DTF'
depth, surface to a point
on frond
FRL
frond length
HA
hour angle, radians
HAD
hour angle, degrees
ID
through canopy
,
illumination at depth
I
!
~adians
~-~··- ~~-~----------~--~--- ---~~--~----------------~-------
·- - ---· ------····
-··
vi
LISTOF SYNBOLS (cont.)
! SYMBOL
DESCRIPTION
II
illumination, incident
upon surface
IO
illumination, after reeflected light
IUC
illumination under canopy
LA
latitude (angle)
PHOR
photosynthetic rate
P1, P2, ••• P6
coefficients in reflected
light polynomial
REF
percent reflected light
RLMD
radiation, instantaneous
throughout day
RLMY.
SD
radiation, noon, throughout
the year
stipe density, stipes/m2
SDT
summation of delta time
SRIS
sunrise
TEHP
water temperature:
viii_
\
------,
I
ABSTRACT
I
A MATHEMATICAL MODEL FOR KELP GROWTH
I
by
Bruce Nicholas Anderson
I
Master of Science in Biology
!
May, 1973
A deterministic model for growth of species of giant
kelp, Macrocystis, of the southern California coast was
developed and analyzed using FORTRAN.
Factors modifying
only light and temperature are required to simulate empirically derived growth curves.
The model was used to
determine optimum time of the year for new growth which
was found to be early June, and optimum time of the year
for regeneration of harvested canopy, found to be early
July~
Critical values for turbidity which will prevent
new growth for any given kelp bed depth were determined.
At twenty meters depth, reduction of transmitted light
to 82% per meter caused by introduction of particles,
e.g. sewage effluent, into coastal waters reduces growth
time by only
76~
20%,
but when transmitted lig4t is below
per meter growth time is zero.
Possibilities for
further development of the model by the quantification
of additional variables are discussed.
--····-··-~·---"- ~-·
viii
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-···- ····------!
I
l
BIOLOGY OF KELP
Due to their ecological and economic importance, the ;
}
species of giant kelp, Macrocystis pyrifera and M. angusti-.
I
folia, which occur along the southern California coast
and islands have been studied for many years.
\vi th the
advent of World War I, the United States Government
sought to stimulate commercial harvesting
fu~d
of kelp as a source of potash for munitions.
processing
Immediately
following the war years harvesting all but ceased.
A
resurgence of commercial interest began in the nineteen
thirties with annual harvests of 10,000 wet tons recorded.
Harvesting has increased steadily with recent harvests of
140,000 tons per year being·made by three companies under
regulation by the California State Department of Fish
and Game (Figure 1).
The external structure of a Macrocystis plant is
shown in Figure 2.
The fronds (stipes, pneumatocysts
and blades taken collectively) extend from the holdfasts
to the surface.
When fully mature the surface
canopy~;,.
buoyed by the pneumatocysts located at the base of each
blade, may extend along the surface for a distance
approximately equal to the water depth.
The life span
of individual fronds is approximately 6 months, and fronds
are constantly-replaced by new growth.
There is consider-
able sloughing; stipe breakage occurs from the activity
1;
•')")
e.~._
----~-l
Figure 1:
Numerical designation of southern
California kelp beds.
Harvesting
is regulated by the California
State Department of Fish and Game
by assignment of beds to companies
(Maples, 1966 and North, 1971).
.
I
I
'---·--····· . ---·-·~-. ·------··-·---- . -.. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . --------·-·--..···-· . . . . . . . . . . . . . . . . ___j
.
. . · · · -· ·- ·-·- -· · ·-·-·-· - - - - l
Conception
I
,~ 33
36
.
3-z-~~~­
~~ 37 38 ·3s
~"-""="'""
17
16 . 15
. . ""
•Los Angeles·
35
~
40
~
41
0
~-
50
~
45"~"~
5
·~
eSan....--·D~:.:_.~
·-·
·····-------J
\.)oJ
,,
4
---------------~
I
I
!
Figure 2_:
Diagra~atic
representation of an
adult giant kelp, l-1acroc:t_stis
pyrifera.
A.
B.
c.
D.
E.
F.
G.
H.
Holdfast:.
Primary stipe
Remains of old stipes
Sporebearing reproductive blades
Young frond
Senile frond
Main stipe bundle
Growing tip of a mature frond
(Neushul, 1957)
5
6
~~f
1
grazing organisms, wave and storm damage, and senea-
cence.
I
Photosynthesis in kelp, unlike most
I plants, is conducted in all parts of the
1
I
denning and Sargent, 1957).
I the
1
terrestrial
plant (Clen-
At a given position along;;
frond, the stipes, pneumatocysts and blades have
about the same photosynthetic capacity per unit area •
.; The stipe and pneumatocyst portions of the fronds Rhotosynthesize at a rate sufficient to compensate for res. piration losses and can sustain the plant even without
"leaves" (blades).
An equally important propert:y.repre-
senting decentralized growth characteristics of the
plant is its grasslike "intercalary growth".
Blade and
stipe elongation occurs throughout the plant (Clendenning and Sargent,: 1957).
Reproductioll: occurs by a sexual process in Macrocystis
~fera
in Macrocystis
and by vegetative and sexual processes
an~1stifolia
(Neushul, 1971a).
Vegeta~
tively reproduced fronds arise from the holdfasts of
established plants and are nourished by the translocation of photosynthate downwards from fronds extending
toward and along the water surface.
This is an important
characteristic, as the light reaching the holdfast beneath thick canopies is often insufficient to support
photosynth~sis
at a level which will support growth (i.e.
compensate for respiration losses).
Sexual reproduction
7
consists of the release of millions of zoospores from
rsporophyll fronds
I: clusters
I
th:-~
located near the base of the stipe
(Figure 2) •.
Sexual reproduction is not sea-
!1
l
1
I sonallyfrestricted, but substratum conditions, submarine
I illumination, and grazers, influence gametophyte su:r1
1i vival.
i
The la..rge beds of Macrocystis_represent an important
1
i
component in a very/ complex biological community..
Ri:bbon- ,
like "forests" of Nacrocystis extend for miles along the
shore in water ranging from eight to about thirty five
meters in depth (Neushul, 1957 and Clarke and Neushul,
1967).
These stands, in conjunction ·~..lith other algae
and the Vhytoplankt·on, represent the producer level for
the community consumers.
Macrocystis_further influences
the diversity of the community by providing cover and
a substratum for the growth and activity of other
ganisms.
or~
The standing forests of kelp may contribute
"breakwater" effects which physically influence the
shoreline.. Kelp and associated organisms which break up
and wash ashore provide food for some intertidal species.
Thus the presence or absence of kelp affects the faunal
diversity and distribution, not only in the sublittoral
kelp region, but in the intertidal region as well (North,
197n) •.
HISTORY OF KELP BED INVESTIGATION
Intensive study of the ecoiogy of Macr.ocystis began
in 1957 \vith the work of North et al (1964).
Although
the life cycle and developmental morphology were well
kno\vn at that time, the systematics of the genus was
confused, and the ecology was essentially unlalown..
Con-
tinuing reduction of once extensive kelp beds (Leighton
_ et al, 1966) prompted the California State Department of
Fish and Game to fund a five-year Kelp Investigation
Program.
The investigation concentrated on the practi-
cability of kelp bed culturing, analysis of kelp bed
distribution, and determination of the natural causes
contributing to kelp bed deterioration.
Additional
funding v-ras soon provided by the State Water Quality-,
Control Board
fo~
investigation of pollution, and by
the National Science Foundation for investigation of
_food chain intermediates.
These programs collectively_-
became known as the Institute of Marine Resources Kelp
Program, \>Jhich lasted from 1956 to 1963.
led to two major pubiications (North et
North and Hubbs,l968).
These studies
~
1964 and
Funding was made available for
continued study under the Kelp Habitat Improvement Project by the Kelco: Company of San Diego, later joined by,
governmeni7al supporters.
A final report of the Institute·
of Marine Resources Kelp Program was published in 1971
8
9
r----1 (North,
1971).
More recently, a conglomerate of
local~
I
governments has sponsored continuing investigation of
the ecology of southern California Coastal waters.
One of the most significant findings of these
studies, particularly with respect to the present study,
has been the assessment of the influence of sewage on
kelp gro\vth.
The volume of materials from sewage pipes
extending into the ocean (outfalls) has increased tremendously·in recent years.
Currently.municipal and in-
dustrial waste waters amounting to 1.65 x 109 cubic
meters annually enter the coastal waters of southern
California (Isaacs, 1972).
Of the aspects of sewage
examined .in the Kelp Investigation Program (toxicity,
grazing, sedimentation, phytoplankton, pathogens, and
turbidity) North et
to be most
~
(1964) found turbidity and grazing
deleterious~
With increasing use of the coastal environment for
commercial, recreational, and municipal purposes has
come the need for a better understanding of its complexities.
Investigation was made difficult by the very
fundamental problem of simply getting into the
ment.
envi~on­
However, with the aid of self-contained under-
water breathing apparatus (SCUBA) this problem has been
largely circumvented.
I
A second device, the high speed
digital cdmputer, has enabled development of systems
!L. analysis
models for investigation of ecological systems.
I
10
North (1968) has constructed mathematical models
with up to seven variables for investigation of photosynthetic aspects of Macrocystis.
However, the effort
presented herewith represents the first attempt to quantify many more variables into a system simulation model
and to employ computers in its solution •.
I
· · -· . ... . . .. .------·· .· · · · . . . · -· · . .. . . . . .. .-·· · · ··-·. ............. -·-····-·J
MATHEMATICAL MODEL CONSTRUCTION
Introduction to l.Jlodel Development
The growth of giant kelp requires analysis of a
complexity of variables: solar radiation,
water· depth, and temperature.
photoperiod,~
Light attenuation through
the water column is, in turn, influenced by numerous natural and man-made variables: canopy shading, bed density,
and concentration of light absorbing or scattering particles.
These particles may consist of plankton, suspended
particles attributable to bottom surge, and sewage effluent from outfalls located along the coast.
consists .of quantification of each variable.
Modeling
Simpli-
fications must be made in a:ny effort to simulate a "realworld" phenomenon.
This does not seriouSly diminish the
usefulness of models.
"Contrary to the feeling of many
skeptics when it comes to modeling complex nature,.
information about only a relatively small number of
variables is often a sufficient basis for effective
models because 'key factors' ••• often dominate or control
a large percentage of the action. 11
(
Odum,,.
197~,
P. 7,).
In deriving mathematical expressions for the kelp
growth system, the equations were set up such that up
to a two year simulation run on the computer is possible.
A simulation "run" consists of determining frond elongation during each time increment, i.e •. the elongation
11
122
~puted for each 0.01
I former :frond length.
I contributing to
Ii re-evaluated.
day (14.4 minutea) is added to
the~
With each such iteration, variables
that elongation are re-calculated or
This procedure is an application of the
Ii Euler iterative integration method (Kowal, 1971). Dev.ei!
j opment of the model, then, becomes a matter of expressing
the system variables as a function of time.
Derivation of Relationships
The following variables are time independent; in
essence they are constants.
They are entered as initial
conditions for simulation runs.
Day of year at which simulation is to begin.
Number of days to b8 simulated.
Effect of canopy shading.
Kelp bed density.
Water absorbancy (turbidity).
Water depth.
Latitude.
Coefficient relating photosynthetic rate to
frond elongation rate.
The time dependent variables are light energy and
water temperature.
Of these two ,1 the amount of light
available for photosynthesis is much more important and
its derivation more complex.
The amount of light varies
with the time of year, daylength or photoperiod (DL),
~
and the maximum light available at local noon (RLMY).
I
1
I
1
1.1
13
Assuming a sinewave approximation
o~
light intensity
from~
dawn to dusk, with RLMY as a maximum, the instantaneous
I
I
radiation at any time of the day (RLMD) can be calculated !
at a point just above the water surface.
But a water
surface will reflect much light striking it at an angle,.
so reflected light, albedo (REF) must be calculated.
Albedo changes throughout the day with the sun's angle
above the horizon (ASD) which, in turn, changes seasonally
. with the sun's declination (DA), and throughout the day
with the sun's hour angle (HA).
A third angle influencing
, ASD is latitude (LA) treated as a constant.
Radiation (RLMD) is converted to illumination (II).
The incident light at the sea's surface (II) is diminished by the amount
re~leeted,
leaving the illumination
(IO) that actually enters the water.
The light which
, penetrates to depth is subject to attenuation by canopy
'
shading and water turbidity.
The illumination under the
c·anopy (IUC) is reduced to light at depth (ID) by absorb,..
ing particles distributed within the water column..
mathematical
~ormulation
of these values follows.
constants and variahles are in FORTRAN symbolism.
The
The
A
tabulation for ease in referencing is given in the List
of Symbols.
Solar Radiation.
Values for the daily total
irra~
. diance (langleys per day) in conjunction with daylength
I)were
t
L--··-· ~-~ . --
used to obtain the noon radiation value in langleys
14
per minute.
l
Daylength is given by the formula
DL = 0.50695 + 0.09235 sin (2n' ~~;- + t.558)Tf)
I
derived as explained in Appendix 1 and shown in Figure 3.
!
i
I
The noon values for radiation in langleys per minute, :
as a function of day of the year are shown in Figure 4.
The data points are the actual values, derived in the
manner described in Appendix 2, and the curve represents
sinewave and linear fitting of these points.
Within
the FORTRAN program, it is a straightforward matter to
direct the computer to leave the sinewave equation for
RLMY given by
RLMY
=
.980 + .370 sin (2Tf ~~; + 1:.558,tf)
and substitute a linear expression when the appropriate
point in time has been reached, i.e. 91 days.
It should
be noted that the use of actual data truces into account
the appropriate fog and cloud attenuation.
The cloudihesa;
of Jnne and the clearness of July, typical of the Los
Angeles coastal area, are evident in Figure 4.
Points on the curve of daily noon values serve as
the maximum in an expression for the instantaneous: value
M
of radiation from sunrise to sunset.
This expression is:
---..-
0.62
t-
-------·-------------.
FIGURE 3
Daylength
.
~
.....
0.58
[f)
~
rd
tr-f
0
Oct 54
til
,.q
..p
rd
(1)
~
rd
s:l
Oo50
.E
...
,.q
+='
till
s:l
0.46
~
~
R
0.42
0
60
120
180
240
300
360
Day of the yea:r, measured from Jan 1i
~
~
-l.
I
FIGURE 4
•
~
Radiation, annual
·r-1
~
~
I
..
(1,)
0
§
·r-i
td
ru
1.2
):..j
~
·r-i
~
0
0
~
..
1.0
:>,
~
oo:a·.
0 •.6
0
60
120
180
240
300
360
Day of the year measured from Jan 1;
...&.
0\
1.7
RLMD = RLMY
DINC;
=
.
.« DINC-SRIS
s~nu
where:·
DL
summation of time increment in
o..ot
day.·uni ts (and is rezeroed each day,
at midnight) •.
SRIS
= sunrise; hundredths of days past
midnight
DL
= daylength, hundredths of days
That the daily increase and decrease in solar radiation could be closely approximated by the use of a sinewave was determined from analysis of solar radiation
charts of Point Mugu Geophysics Division, whose solarimeter was located a few feet above sea level.
The
actual data and the sinewave approximation are shown
in Figure 5.
Curve A is for l-1arch 30, 1970 at
Mugu, a
day.
the
clear:~
Point'~
The sinewave approximation illustrates
built~in
compensation for average cloud and fog
attenuation.
Sunrise (:Figure 6) is approximated by
SRIS = 0.24305 + 0.04725 cos (2tr ~~§ t .058 1!)
#
An approximate conversion for radiation to illumina-
-·---~·-··· ···---·~-----··-"-·"
~·--···--.
.
FIGURE 5
Radiation, dailY./
1. 2.
--·~·~····-
·····-·· ---.....
Hou;iy:
·1
~h~g-e-~i-n___
radiation intensity
I
March, 30
j
.
Curve A]· maximum
Curve B: sinewave simulation
1,
Curve C: minimum *
*
1 oO
Shape of curves
A & C based upon
data from Pt.
Mugul
•
o.a
.~
s
'>;
H
..
0.6
~
0
·~
.~
0.4
·~
.rd
ro
;P:::
0.2
0
~
I
0.60
Oo50
0.40
0.30
Time of day, in hundredths of days (24 hrs.
-
Oo-70
= 1o00)
---------- --····· ---
- _______. ______ ---------------------------....1
..,),
CJ
19
~
0
\.0
t<'\
•
§
1-;)
a
0
~
ct--1
0
0
t<'\
rd
(I)
~
~
ro
ro
(I)
s
0
--=!N
..
~
(I)
:;:..,
(I)
\.0
(I)
~
p
·ri
c!J
H
rr..
~
ro
0
~
'r"
co
s:l
ct--1
0
ro
~
~
(f)
A
0
N
'r"
0
\.0
0
N
t<\
0
•
co
N
•
0
.qN
0
•
0
C\1
0
•
·..
20
f tion
(~eifsnyder
and Lull, 1965;) is given by
= 72,000 RLMD
II
wherEr;
II, incident illumination, is in units of
Lux ( Lumen/m2 ),
Albedo.
The angle of the sun with the horizon deter-
mines the amount of incident light which will be
reflected~
The angle, AS, is given by the expression
where
Sin
<1>
= sin LA sin DA
+ cos LA cos DA cos HA
(Gates, 1962) where
LA= latitude (34°, a constant for the model)
DA
= declination
HA
= hour
Declination.
"DA
= .410
angle
Declination is given by
sin (2 !( ~~~ - •.438 'It' ).
see Figure T
- -·-·--
.....
--·--···----·-··----·-·---~-------,
FIGURE 7
(I)
(!)
I
Declination
+30
Q)
~
M
(!)'
.. +20
rO
8
(I)
(!)
,.q
+>
+10
4-1
0
p
0
·r-i
~
>=:
·r-i
rl
0
()
(!)
R
-10
-20
-30
60
0
120
180
240
300
Day of the year, measured from Jan 1:
~·· ~- ,
....-~""--·~····"··-~·· .··- ----------~···---
.---··. ·-·-·-· -"··· -· ·--· ···--.
360
I
······-·-··-··------·--------'
i\)
~
22
----~our Angle.
The hour angle is the angle
between~
the sun's meridian and the meridian of the observer.
noon the angle is zero
and~
symmetricalon each side
of noon tm.,ard sunrise and sunset.
HA
= -DINC
HA
=
(
tr
50
At
) + 1!
It is given by
midnight to noon
noon
0
HA = + DINC (
rr
50
) + 1(
noon to midnight
Again, it should be noted that in the actual program,
computation of this variable (as well as all psxameters
pertaining to sunlight) is oypassed in the hours between
sunset and sunrise.
With the sun's angle lmown, the reflected light
(REF) of all wavelengths, albedo, is obtained by the
following polynomial expression
REF
= P1
+
+
P2•AS + P3•AS 2 + P4•AS3
P-5·~·AS 4 + P6•AS 5
For P values, see Appendix 3 •
.
This empirical relationship represents a curve fit of
the data shown in Figure 8 which illustrates the high
I
'
23
-----------------·--------FIGURE 8
Reflected light
100
Curve A: Tiata from
North, 19_68
Curve B: Polynomial
simulation
80
A
~40
___________
0
60
40
ASD, angle of sun with horizon, degraeS>
20
....,.
80
~
24
r-percentage of reflected
li~t
at low angles, and
low percentage at high angles.
th~
The amount of illumina-
tion actually entering the sea's surface, IO, is then,
IO = IL - II ?o~
Canopy Shading.
, LUX
Kelp beds which have developed a
canopy of surface fronds will present an immediate
str.iction
to the transmission of light.
re~
North (1968)
states that thick canopies suitably developed for harvesting typically transmit low percentages of light •.
Obviously there is a mathematical relationship between
canopy cover and transmitted light.
The relationship
probably could be quantified in terms of canopy length
and stipe density, both of which are growth (time) de-p~ndent
variables.
In the absence of information as
to the exact nature of this relationship, I have chosen
to specify canopy transmitted light (CTRANS) as an
independent variable.
Illumination which penetrates
the canopy is given by
Iuc - =- IO CTRANS
100
where
CTRANS = light transmitted through the canopy,.
Il
I
percent
1I
25
IUC
= illumination
under canopy, LUX
j
Turbidity.
l late
matter from various sources which are incorporated
1
The dispersing medium contains particu-
!
as exponents in the Beers-Lambert law as follows:
ID
= IUC
ID
= illumination
e- (C1 C2 ••• Cn) DTF
Cl C2 · ••• Cn
where
at depth, LUX
= absorption
coefficients based
upon concentration of absorbing
particles of various origins
DTF = DEPTH, m
Coastal oceanographic waters are subject to light
reflecting particles from several causes.
Among these
are natural background (composed of planktonic organisms
and suspended detrital material) and particulate matter
resulting from the introduction of sewage from numerous
outfalls located along the coast.
North(l964) introduced
these as exponents in the foregoing expression in terms
of absorbancy, defined as the percent of_ light reduc·tion
per meter of transit through the water column.
ground
absorbancy.~
The
back~
for coastal waters is eight percent •.
26
!Hence, increase in turbidity from any source can be ex1 pressed
l have
in terms of higher percentage absorbancy.
l
I
used this approach and "collected" all turbidity
!I resulting from the concentration of particulate matter
I
I into_: the
expression above (in the exponent, Ct).
The
i stipes extending from the holdfasts to the surface, while
!
not homogeneously distributed, can be treated as light
absorbing "particles 11 in the Beers-Lambert law (North
and Hubbs, 1968).
This factor is introduced in the ex-
ponent, c2.
Photosynthesis.
The relative rate of photosynthesis
as a function o.f light intensity, PHOR, was obtained by
curve
fi~ting
data ·.for mature kelp .fronds (Clendenning
and Sargent, 1957).
CC2•ID + CC3•ID 2 + CC4•ID 3
PHOR
= CC1
PHOR
= relative
+
where
rate of photosynthesis (in
terms of rate at any wavelength to
rate at red wavelength, 6700 !)
For CC values, see Appendix 3.
This relationship (Figure 9) illustrates the approximately linear increase in photosynthetic rate with
increasing radiant energy, a "knee" marking the beginning
;
- . - _.,. -- -·-·--··---···-
~---·-· ---,-------~ ··----~-- -~ ~~"~~..l
·-··-··-·-··
-~,-·-~·---
FIGURE 9
Photosynthettic rate
14:
.
····---------------.
t
I
l
II
.. 12
I
Ul
·r-1
[/)
Q)
I
10
l
~
..p
~
[/)
0
..p
Curv.e A: data from
Clendenning and
Sargent
Curve Bi polynomiaL.
simulationl
8
0
~
ct-t
0
6
a)
itl
~
<1>
4
t>
•r-1
~
rl
Q)
2
~
~
0
p::·
P-1
0
2
4
-·
-·-~.-·--~~--·--·"~' "~
6
... -·· ·-·-
10
16
Illumination, LUX x 10-3
8
·-~·-·
._ ...... _.,,, ·- -· -
12
14:
18
20
~··-·
rg
28
~~t~ation
I the
intensity and finally a flat portion of
curve for which no increase in photosynthetic rate
I
I is realized.
North and Hubbs (1968) state that these
I data,
for mature blades, can be applied to the entire
I plant
'llli th
!
I'
introduction of only very minor error •.
The rate of photosynthesis to be applied in com-
! puting
frond elongation, APHOR, was obtained by aver-
'
.f
i aging values taken at half meter intervals over the total
!
· frond length •.
Temperature.
An approximation of water temperature
at times throughout the year (Figure 10) was obtained
by curve-fitting to data pertinent to a kelp bed of
twenty meters depth (North 1964).
A sinewave gives a
satisfactory approximation for any depth..
To obtain
values at any depth it was necessaryto incorporate depth
dependent variabl.es:
TEMP = ADD + MUL sin ( 2 1T ~~§ + PHIN 'IT)
where
ADD, MUL, and PHIN are functions of the depth
and determine which of a family of
curves (illustrated
"ltli
th three ex-
amples in Figure 10.) is appropriate •.
For expressions in terms of ADD, MUL, and
PHIN, see Appendix 3.
FIGURE 10
\'later temperature
• .Actual data, surface
Actual data, 20 m
'
19
• •
18
•
•
17
Surface
16
•
:o
'0
i
! ..
!IU
15
Depth
:~
:~-
~ 8m
•
;~
ro
~
<D
~
Q)
14
:8
13
•
12
Depth
= 20m
~
11
fit
10
0
~
240
180
120
60
Days of the year, from Jan 1
300
360
30
r-As with photosynthetic rate, water temperature was
averaged at each half meter increment of frond length.
I The average values
! water
l
for photosynthetic rate, APHOR, and
temperature, ATEMP, were used in c·emputing the
I
!
rate of frond elongation for each time increment, SDT.
The average thus applied, represents a compromise between the use of only one value (of PHOR and TEI1P) for
each iteration and a more accurate (but computer time
consuming) integration over the frond length in much
smaller increments.
Frond Elongation.
The frond rate of grm'lth v:as de-
termined by application of a modification of the logistia;
growth equation:
DFR1
= FRL
FRL
= frond
K
= 40
(R1 R2
R1 R2 FRL)
K
where·
length
meters, a value for maximum frond
length·for kelp growing in water of
20meter depth.
A logistic gro\ITth equation was chosen on the basis
of its similarity to empirically derived curves for frond
growth obtained by North (1971) in which he found that
~
gro\1th over extended time periods ( 100 days} exhibited
S-shaped curves characteristic of growth in many organisms.
I
I
I
r=---
The incorporation of a variable rather than a constant
growth rate, R1 R2, modifies the slope of the pure logistic growth curv.e· (Figures 11 and 13).
The rates R1 and R2:are determined by photosynthesis
and water temperature respectively.
R1
= RR1
APHOR
where:
APHOR
= average
photosynthetic rate described
above
RR1
= CANL
(x/20) + x
In this expression, x is a constant . coefficient relating
photosynthetic rate to
~rond
elongation rate.
It was
determined (Figure 11) by trial and error to fit
empiri~
cally derived frond grmvth data pertinent to water depth
of twenty meters (North, 1971).
The fit divergence be-
ginning at approximately 55 days (Figure. 11) was intentional, for North stated (personal communication) that
the breakover point of the curve was abnormally sharp
and the data should not be considered typical beyond the
55th day •. Excessive water temperature (above 20°)
caused blade deterioration and measurements were stopped
I
for fear of gathering atypical data.
Attention is called
! to the use of a log scale (Figure 11) which unrealisti-
1
j
FIGURE 11
Growth rate simulation
50
40
B
30
..,. ..,.
20
/
//
---
A
/
/
/
I
/
.....
Ol
~
<I>
I
I
10
I
.p
<I>
a
.
,.q
.p
~
<1>
rf
rO
~
0
5
Curve A: data from
North, 1971
Curve B: modified
logistic
simulation
4
~
~
3
2
1
0
20
40
60
80
100
Days of the year, measured from May 9
33
call~'" magnifies
curve disparity between ([; and 55
day~~~
In actuality the disparity is not significant, amounting
to no more than one meter frond length.
The beneficial aspects of the canopy, i.e. production of photosynthate sufficient for translocation
dov.mward., derive from its favored position with respect
to light.
An exact quantification of canopy influence
in terms of' all related variables has not been determined.
(Neushul, 1971b) •.
I incorporated it by assuming the
linear relationship
RR1 = CANL (x/20) + x
CANL
= canopy
'.vhere
length
W~en the canopy length is zero (e.g •. the growing frond
has not reached the surface) RR 1 = x.
As canopy deve±opa,
RR1 is increased to 2RR1 maximum, at fUll canopy length
of twenty meters.
The influence of temperature, rate R27 was determined on the basis of experimental data (North,.l971)
in which it was found that the effect of water temperature upon kelp growth as it increases from 10 to l8PC
is beneficial.
Above 20° however, whole beds have been
known to disappear, probably due to a bacterial induced
"black rot" (Chapman, 1970) .:
34
r
I
R2 =
0.09 ATEMP - .125
This is a linear relationship which increases
temperature increases.
18~6°0
R~
as
Water temperature never exceeds
in the model simulation,, thus the influence of the
higher, detrimental, temperatures are not encountered •.
,----·RESULTS
The developed model may be used to investigate
1
I various
aspects of kelp growth.
Program input and output
' format requires only mihor changes to do this.
Several
examples of such applications are presented below;
Optimization of New Growth
By designating a set of initial conditions, and by
starting the simulated plant growth at various times of
the year, the projected length of time required for
growth to maximum length can be determined.
A comparison
of these projections will then reveal the optimum starting time .for maximum gro-v1th rate.
by the follO\ving example.
This is illustrated
Consider an initial frond
length of one meter at the base of an existing kelp bed
with a canopy which is transmitting
flected light.
?~b
of the unre-
Bed density is defined as 6 stipes/m~,
and water absorbancy . is 10%.
These conditions would simu- ·
late the growth of a new frond in an established bed.
Time is then "started" and the model simulation is run
until the one meter frond has reached forty meters, the
equivalent of mature length.
The results may then be
expressed as the average frond elongation rate in meters
per day required to grow to forty meters.
Simulation in
•
this manner was repe.ated with starting times at thirtY;'
day intervals throughout the year to derive the data
35
36
shown in Figure 12.
rate is
ma~imal
These results show that
gro~th ~
for fronds which begin growth in early
June (day 160) and minimal .for fronds beginning in early
October (day 300).
Growth curves for·these extremes,
sho"m in Figure 13, represent an "envelope·" of all possibilities.
Optimization of Canopy Regrowth
In a similar manner the time required to regenerate
a canopy after its removal byy harvesting can be determined.
California state law penmits kelp harvesting to
a depth of four feet.
Thus, virtually all canopy and
(subject to the influence of currents, waves and tidal
conditions) probably the first two meters of the vertical
stipes are removed. In simulation, the initial value for
frond length is set at 18 meters.
The light transmitted
through the canopy (which is now non-existant) is set at
10o%.
With this, the adjustment of input parameters is
complete.
Regrowth of canopy is then determined for
different starting times of the year as in the previous
example.
The results (Figure 14) indicate that optimum
harvest time occurs in early JulY.•
Turbidity Influence
According to Clendenning and Sargent
(19~7~
1;60 LUX
or more light must be present for growth to occur.
This
amount may not reach the ocean depths of ten to thirty
meters when turbidity is high.
In Figure 15 three curv.es
I
II
FIGURE 13
GrO\'lth rate envelope
I
I
so
1
40
30
20
ro:
Curve A: growth
at day
Curve B: growth
at day
5
4
{tj
~
starts
160
starts
300
3
(1)
..p
(1)
a
..
,.q
2
..p
~
(1)
,r-i
•re
~
:o
f..f
1%1
1
0
#
20
60
40
80
100
Days of growth
~. . .
~----~~-·· ~-~--~----·~-----·--«·--~H·-·-···--·-
• -••
·-~~•·••<
>
120
140
··-····---¥-'
-~-·'
........
- --
..
,.----··--~---~····----·---·-·-- ~-
---~-----------
I
FIGURE 14
Canopy regro\'li;h
I
.. 1.00
.90
~
<d
'S-
o80
...
,.q
.p
;::
0
~
QO
.70
Q)
~
p.,
PI
0
@
.60
0
0
60
180
120
240
300
360
Starting day of the year (at 18 m•. frond length)
measured from Jan 1:
·
--~-·------~-- .,_____________ _j
~
1..0
;r~··~---'--·---··-
. -·-··. -·- ·--- ..
"~·-·
-
···-··--·-----·- -- --·---·---
··-------·-·-··------~--
·- --------·-··-·"'
,...
_-··-·-···-----·
FIGu;RE 1i5'
Turbidity
><
·stoo
0
\.0
T-"
AI
~
so
0
.,;
..p
ro
\'later depth, m
~
.,;
8 60
irl
rl
·r-1
(!)
rl
~ 40
rl
• •r-1
ro
~
• >, 20
rl
·~
g
§
·~
. ..p
. s:::
;Q)
.()
.~
(!)
P-i
56
44
60
40
-64
68
36
32
Percent; light
Percent light
72
80
76
28
24
20
transmitted/m of depth
absorbed/m of depth
84
16
88
12.
92
.
96·
a~
I
--l
..r-:0
,·
41
are shown, each curve representing different
oce~
de.pths:.!
Turbidity in terms of percent light transmission per
meter of water column is plotted on the abscissa.
sorbed light
Ab-
corresponds to one minus the transmitted
light and this is also shown on the abcissa.
The ordi-
nate represents the percent of the daylight hours out
of a year when light is sufficient for growth.
As
explained above, growth of fronds can occur with low
light intensities because photosynthate is passed down
to them from fronds reaching the surface.
But new
plants developing from zygotes deposited on the bottom
are inhibited without at least 160 LUX.
Figure
15~·
shows,
for example, that at twenty meters, light transmission
of 82 percent per meter, even if it persisted throughout
the year, probably would not seriously curtail growth
because light intensity of 160 LUX is still available
78 percent of the time.
However, for light transmission
below 82 percent·, the curve drops off sharply, and at ..
76 percent, growth is prohibited.
The net effect of
water turbidity is to restrict kelp growth to shallow;
water, a prediction of the model supported by observation
in the field (North, 1971.) •.
Discussion
The predictions of the model appear credible.
two prime•contributing factors to growth
The
which the model
evaluates are radiant energy and water temperature.
The
42
:P-hotoperiod is greatest during the sUmm.e:v·solstice which
coincides closely with the predicted growth rate optimum.
Ligb.tintensity is also maximum since the sun's angle is
greatest.
Although bottom water temperature is minimal
in June (so that R2 is minimum and the least benefit is
derived from this factor) fronds reach the surface, and
hence more favorable light and temperature regimes,. in
a shorter time.
The rate of growth exhibited by the fastest growing
fronds appears reasonable.
Sargent and Lantrip (1952:)
state that whole fronds can grow as much as 45 centimeters
per day, the most rapid plant growth known.
Caution must
be applied in the interpretation of results in such absolute terms however, due to the manner in which the photosynthetic rate to frond elongation rate relationship was
e:·stabl.ished.
The trial and error selection of a coeffi-
cient to fit empirical frond growth data could be replaced by a more exacting method.
This would entail
development of the model to include the actual penetration of individual wavelengths of light.
chromatic light was used.
Only polyr
In conjunction, the action
and absorption spectrum of Macrocystis must be incorporated.
These improvements would take into account the
fact that photosynthetic rate is not constant along the
frond (Clendenning,., 19611).
The light attenuation factors considered in the
43
model were stipe density and absorbency,;.
seasonally
v~iable
There are oth-::-l
and non-uniformly dispersed factors.
Plankton blooms of summer can severely diminish light.
Strong bottom surge accompanies the long period swells
I of winter months.
As a consequence sediments are agi-
ltated into suspension causing extreme murkiness near the
j
I
!
ocean floor.
It will not be a difficult matter to quan-
.~1
I tify and include these as variable exponents in the
'
:! Beers-Lambert equation.
Turbidity can significantly reduce growing time
1 (Figure 15).
\'Thile vegetatively produced fronds (charac-
1
! teristic
j
!
of Macrocystis angustifolia) might survive from
translocated nutrients, light attenuation at the ab-
I sorbancy
levels illustrated would prohibit the growth of
i gametophytes.
This factor alone could explain the diffi-
cu,lties experienc-ed in attempting to successfully reintroduce kelp by sowing laboratory
gro~rn
gametophytes,
for values of water absorbancy as high as 40 percent
(light transmission as low as 60 percent) are common
along the California coast (North and Hubbs, 1968).
Further analysis should be made after incorporating
seasonally variable absorbancy capability into the model.
The growth rate enhancement due to canopy should be
examined by alternative expressions to the linear one
used.
Possibilities discussed with North (personal
communication) suggest parabolic curves concave up and
I
)
44
concave down would represent extremes for initial investigation.
This illustrates another utilitarian value of
:simulation models, i.e. hypothetical relationships can
I
I be
tested at will.
I
The influence of canopy removal on the growth of new .
Ifronds
I also
!
I
1
from the holdfasts· of Macroc:ystis angustifoli'! is
a factor to be considered.
The harvesting process
represents an interesting trade-off to the species, for
light penetration to submarine fronds is enhanced but
nutrient translocation is sacrificed.
The model will
require some additional sophistication to examine this
phenomenon.
Fin~lly,
frond elongation for very small frond
elements (much smaller than the half meter steps used)
j
\AJ"ould improve accuracy.
I integration
1
l
l
j
The Euler method of iterative
has become wasteful of computer time in
this regard, for up to ten minutes are required for
single runs.
I used.
Another, more efficient method must be.
Possiblyc- the Runge-Kutta method (Kowal, 1971)
I will provide this improvement in a second generation
I
I model •.
I:
!
"
·----···-·-----------------------------------,
I
LITERATURE CITED
<lhapman,
V.~ J. 1970..
&i: Co., LTD~'
Sea\'leeds and Their Uses.
Methuen
Clarke, \1/illia.m. D. and 1<1ichael Neushul. 1967. Subtidal
ecology of the southern California coast,, pp.
~9-4Z~
In Theodore A. Olson and Fredrick
J. Burgess ( ed.), Pollution and marine ecology •.
John Wiley and Sons.
Clendenning, Kenneth A. 1961. Photosynthesis and growth
in 11Iacrocystis 12.,yrifera. Proc. IV Int. Seaweed
~.
Trondheim, Norway.
: Clendenning,, Kenneth A. and Harston C. Sargent. 1957.
Physiology and biochemistry of giant kelp.
Ann. RPT. Kelp Inv. Prog., Univ. Calif. Inst •.
Marine Resources, IMR Ref. 57-4:~ 25-36.
:Gates, David M. 1962. Energy exchange in the biosphere.
Harper and Row, Inc. New York.
Isaacs, John D. 1972~ The ecology of the southern Cali~
fornia bight: implications for water quality
management. Vol. 1. Southern Calif. Coastal
Water Res. Proj., El Segundo •.
Kowal, N. E. 1971. A rationale for modeling dynamic
ecological systems, pp. 123-194. In B. c.
Patten (ed.), Systems analysis and simulation
in ecology·:;.:..V:ol. t •. Academic Press. Inc.,~
New York and London •.
Lei'ghton, D. L., L. G. Jonesand W. J. North. 1966.
Ecological relationships between giant kelp
and sea urchins in southern California. Proc.
V. International Seaweed Symp., pp. 141-153.'
Pergamon Press, Oxford.
Maples, R. L. 1966. The southern California kelp industry. A study in the utilization of a natural
resource. M. A. Thesis. San Fernando Valley
State College, Geography •.
Neushul, r1ichael. 1957. Growth and reproduction. Ann.
·Rpt. Kelp Inv. Prog., Univ. Calif. Inst.
Marine Resources, IMR Ref. 57-4: 37r?6~
45 ..
46
-------'------~
------- ·~ 197la. The species of Macrocystis with
particular reference to those of North and
South America, pp. 211-222. In W. J. North
(ed.), T-he biology of giant kelp beds (Macrocystis) in California. Beihefte zur Nova
Hedwigi.a. Heft 32. J. Cramer, Germany .•
!
l
• 197lb. Submarine illumination in Macrocystis beds, pp. 241-254. In W. J. North
( ed.), The biology of giant kelp beds (JY1ac:r.9-=
cystis) in California. Beihefte zur Nova
Hedwigia. Heft 32. J. Cramer, Germany.
I
i.
North, Wheeler J. 1968. 11easurements of bottom light
intensities. Kelp Habitat Improvement Project~
Calif. Inst. Tech. KHIP Ann. Rpt. 1967-68:
85-98 •
• 1971. Growth of individual fronds of the
mature giant kelp, I>1acrocystis, pp. 123-168.
In W. J. North (ed.), The biology of giant
kelp beds (Macrocystis) in California.
.
Beihefte zur Nova Hedwigia. Heft 32. J. Cramer,:
Germany.
North, W. J. and Carl L. Hubbs. 1968. Utilization of
kelp bed resources in southern California.
Calif. Fish and Game, Fish Bull. 139~
North,, W. J., K. A. Clendenning, L. G. Jones, J. B.
Lackey, D. L. Leighton,. N. Neushul, M. c.
Sargent, and H. L. Scotten. 1964. An investigation of the effects of discharged
wastes on kelp. Calif. State \Vater Quality
Control Board Pub. 26: l-124 .•
Odum, E. P. 1971. Fundamentals of Ecology.
Saunders Co. Philadelphia.
Vl. B.
Reifsp.y:¢ler W. E. and H. W. Lull. 1965). Radient ene:cgyt
in relation to forests. Tech. Bull. 1344.
u. s. Dept. Agriculture. For. Serv.
Sargent, Marston C. and Lester Vl. Lantrip. 1952. Photosynthesis, growth and translocation in giant
kelp. Amer. Jour. Bot. 39: 99-107•
•
APPENDIX 1
In many instances, the phenomena pertinent to this
investigation were found to vary_:rin a cyclic manner and
adequate simulation was achieved by the use of a sinewave.
The seasonal change in daylength is a case in point
and will be used to illustrate the general sinewave function and its application:
Y
= V1
+ V
2 sin (
e+
1/1 )
1::
~~--~-~------(}=~~----------~
The computer makes computations in radians, hence
all angles are converted to radians through the relationship: 2 1r radians
the variable,
(J
= 365
e
=
360°.
The x axis is in terms of
One cycle (period)corresponds to
d.ay~.
47
4-8
At 3+0 latitude daylength varies approximately
between a low of 10 hours and a high of 14 hours.
For
l
simplicity, assume Y1 = 10 hours, Y2 = 14 hours. v1
and v 2 then are determined as follows: the nominal daylength, 12 hours, is assigned to V1 •
The difference
. between nominal and maximum (or nominal and minimum) •
which equals 2 hours, is assigned to
v2 •
The desired
: period for one complete oscillation, in this case 365
i
days, is assig;ned to 8 •
i
Sine 8 varies between + 1 .0,
at the maximum amplitude, and -1.0 at the minimum.
Hence,
;'
• V2 will be added to or subtracted from V 1 in accordance
with the instantaneous value of the sine of 8 •
The .sivewave is placed at the proper starting
point with respect to the origin with the plase angle
term,
~
•
When
~
= o,
the sinewave begins at the origin
as shO\V'n:
By the use of phase angle, any desired point on the sine-wave can be placed at the origin.
In the daylength ex-
ample, with no phase angle, the origin of the sinewave
would correspond to the spring equinox, 81 days after
.·,'1;
~~-
•· .•
~
49
"' =
~t 5 <2~ -rr )
=
;:~;~
n
== •.442.
rr
Hence, the sinewave can be moved to the right by
a phase angle of
~.442
If or to the left by + 1.55R IT
•
(i.e. 1. 558 1l + •.442.. rr =2 1T ,, one period).
The completed simulation formula for daylength as a
function of fraction of the year appears as follows:
DL = 0 •.50695 + 0 •.09235 sin (2.11 ~~§ + 1 •.558, tr )
0~.50695
= nominal· daylength = 12.17 hours
0 •.09235 = difference between nominal and
maximum (or minimum) d:aylength
=
2.22 hours
2-11" ~~§ = the day (along the abscissa) for
which DL is to be calculated
Note that v
and v 2 are in units of 0.01 days (not hours)
1
corresponding to the exact daylength, maximum andminimum,
for our latitude of 14 hours 22 minutes and 9 hours
58 minutes respectively (U.
s.
Coast,and Geodetic Survey,
50
1973 )..
The variable, SDT, equals the sUmmation of delta
time, i.e. the accumulation of successive 0.01 day inc·re:...
ments.
•
APPENDIX 2?
L.os;Angeles coastal data between the years 1955·
and 1969 (United States Weather Bureau publication
I Climatological
Data National Summary) vrere tabulated.
These data were given in terms of total daily irradiation RT, langleys per day.
The noon value, RLMY, in
terms of langleys per minute was desired (.for use in a
subseguent expression for instantaneous value at any
i
~ time of day).
1
I
I
A sinewave approximating the daily
build-up to and decline from the noon maximum, was
assumed, relating these variables as follows:
i
RT
=
/DL 14.4 (RLMY sin 'IT D~ )
dx
where
x=.01
RT
= total
DL.
= daylength (Figure
daily radiation
(langley~/day)
3)
A computer program was written for solution of
this equation and solved by trial and error to obtain
RLMY.
... I
RLMY values thus derived were plotted in Figure 4.•
51
APPENDIX 3
The program logic flm'i chart and a list of the
I program
follow.
I either symbols
The logic compartments are labeled with
or numbers.
Symbols, keyed to the program
i list, indicate points of logical decision.
1
1
respond to the controlling statements.
Numbers cor-
The algebraic
·:
I! signs
indicate relative position with respect to state-
: ment numbers at left margin.
The flow chart depicts
'
j only the logical steps of major importance; there are
I
i
numerous minor ones which can be determined from a de-·
Itailed review of the program,
I
I
!
I
I
52
53
Jl'L0¥1 CHART OF PROGR.AJ.i LOG !C
1Set initial conditions, constants
and -wariables DAYS, CTR.A.NS, SD,
ABSO
I
G
yes
Final. day of calculations: reached
• lsToPj
no
C$
First calculation for the day?
Inn
I
yes
4-to 12+
Calculate RLMY, SRIS, DL, DA
~
_yes
Time is before sunrise or after
· sunset?
llO;
14 to 19
Calculate HA, RLIID ' II, ASD, REF,
IO, IUC
.
20+to 29+
r·1oving along frond in half meter
increments, calculate ID, APHOR,
ATEI'. IP
ll
29+
Calculate R1, R2, DFRL, FRL
32
..
~Print
results
~·
54
c
MODELKELPMODELKELPMODELKELPMODELKELPMODELKELP
REAL IOoiDoiUCtl!tMULtLA
ASINIY)
= ATAN(Y/SORTilo-Y*Y))
DAYS = 700o •
DT = oOl
CTRANS : lOOo
II
·f
i
1
Il
2
3
I
I
e
SO = 6a
ABSO = tOa
POI = 10. o
OPTM = 20a
LA = o594
CCl
-Oo29fl022
..__
CC2 = Io574879E-03
CC3 : -lo241379E-07
CC4 = 3~350447E-12
Pl = 69e4561021
P2 = -6 0 98350423
P3 = Oo304002
P4 = -6o403187E-03
P5 = 6o381565E-05
P6
-2.408547E-07
YR. = 365o
PI
3ol'•l6
JMAX = IFIXIDAYS/DTi
READ 160,34) JtTI
IF IJoEOo40000) GO TO 33
FRL "' 18o
CANL = Oo
X = o035
RRl = X
WRITE .16lt35)
J2 = -1
J2 = J2+1
IF IJ2oFO~l00) GO TO 2
IF IJ.EOoJMAX~ GO TO l
J = J+1
D = FLOATIJ21
= FLOATIJ)
SDT = FJ*DT
IF I J2 .GT .o I GO TO 13
IF IJ.GT.9126oAND.J.LTol4450) GO TO 4
IF IJ.GEol4450.~NDoJoLTol67311 GO TO 5
IF IJ.GEol673l.AND.JoLTol97731 GO TO 6
IF IJoGE.l9773.ANDoJoLTo243361 GO TO 7
IF IJoGTo45630oANDoJoLTo509541 GO TO 8
IF 1JoGE~50954oAND.JoLTo532351 GO TO 9
IF CJ.GEo53235oANDoJoLTo562771 GO TO 10
IF IJoGEo56277oANDoJoLTo608401 GO TO 11
RLMY = o98+1o370*SINI2o*Pl*SDT/YR+lo558*PI)l
GO TO 12
RLMY = lo05/53e231*SDT+o973
GO TO 12
RLMY = -la03/22o81l*SDT+lo28
GO TO 12
RLMY = lol5/30o42)*SDT+o235
GO TO 12
RLMY = -loll/45~63)*SDT+lo686
GO TO 12
RLMY = c.05/53.23)*CSDT-YRI+o973
GO TO 12
RLMY =#-Ce03/22a8l)*ISOT-YR)+le28
FJ
I
•
I
.,I.
I
4
5
6
7
8
9
I
I
L__ _
55.
---'"---GO TO 12 ~
10 RLMY = C.l5/30o42)*1SOT-YR)+o235
GO TO 12
11 RLMY = -l.ll/45e63l*ISDT-YR)+lo686
12 SRIS = o24305+1.04725*COSI2t*PI*SDT/YR+o058*PI))·OL = o50695+1o09235*SINI2o*PI*SDT/YR+lo558*PI))
OA = o410*SINI2.*PI*SDT/YR-e219*2o*PI)
.
DAD = l80o/PI*DA
13 DINC = D/100•
IF IOINC.LT.SRIS) GO TO 18
IF IDINCoGT.DL+SRISl GO TO 18
IF IJ2oLTo501 GO TO 14
IF !J2oEOa5Q) GO TO 15
IF IJ2oGTo50) GO TO 16
14 HA = -IPI/50ol*D+PI
GO TO 17
15 HA = Ot .
GO TO 17
16 HA = PI/50a*D-PI
17 RLMD = RLMY*SINIPI*IIDINC-SRIS)/DL~)
HAD = 180o/PI*HA
GO TO 19
18 RLMD = O·e
19 II = RLMD*6700o*l0o76
IF I!IoEOoOel GO TO 31
TSIN = SINILAl*SINIDAl+COSILA)*COSIOA)*COS(HAl
ASR = ASINITS!Nl
ASD = 180t/PI*ASR
IF IASDoLE.O.) GO TO 31
REF = Pl+P2*ASD+P3*ASD**2+P4*ASD**3+P5*ASD**4+P6*ASD**5
IO = II*Il.-o01*REF)
IUC = IO*ICTRANS/lOOol
Cl = lo0133*ABS0)-.025
C2 = II.044/9oi*SD!+lo
SPHOR = Oo
STENP ::. Oo
NI = 0
...
co
o.
20 CO
CO+oS
NI = NI+l
IF ICOoGEoFRL) GO TO 29
IF ICOoGEo20ol GO TO 24
DTF = 20.-CO
ID = JUC/IEXPCCl*C2*DTFJ)
IF IIO.GEol6150.) GO TO 21
IF IIDoLTo300.) GO TO 22
PHOR = CCl+CC2*ID+CC3*ID**2+CC4*ID**3
GO TO 23
21 PHOR = 6t80
GO TO 23
22 PHOR = O.
23 ADD= <C-4.2/20.)*DTFl+16o4
MUL = CC-o2/20ol*DTF)+2a2
PHIN = (C-.83/20o)*OTFl+l•42
TEMP= ADD+IMUL*SINI2o*PI*SDT/YR+PHIN*PI))
GO TO 28
24 IF (!OaGEtl6150•l GO TO 25
IF IIO.LT.300.) GO TO 26
PHOR = CC1+CC2*IO+CC3*I0**2+CC4*I0**3
*·
i
L~,--w;o--,
GO TO 27
25 PHOR = 6.80
GO TO 27
26 PHOR = o.
27 TEMP = 16.4+C2.2*SINI2o*PI*SDT/YR+le42*Pl))
28 SPHOR = SPHOR+PHOR
STEMP = STEMP+TEMP
GO TO 20
29 APHOR = SPHOR/FLOA T CNI)
ATEMP = STEMP/FLOATCNil
Rl = RRl*APHOR
R2 = 1.09-li-ATEMP)-.125
DFRL = Rl*R2*FRL*C140o-FRL)/40 0 )
FRL = FRL+IDT*DFRLI
CANL = FRL-DPTM
IF ICANL.GToOol GO TO 30
CANL = Oo
30 RRl = IX/20ol*CANL+X
31 IF IJ.EQ.SOl GO TO 32
IF ISDT.LToPO!*TI+o5l GO TO 3
32 WRITE 16lt36l SDTtSRIStDLtiitDADoHADtASDtREFt!OtCTRANStiUCtlDt
$APHORtFRLtDTFtRRltRltR2
IF IJoEOoSOl GO TO 3
IF CABSIFRL-40.loLToo0ll GO TO 1
TI = TI+le
GO TO 3
.c
33 STOP
34 FORMAT
35 FORMAT(
$
I
I
I
!0
II5o2XtF3~0l
DAY
cTRANS
t
DL
II
ID
APHOR
DAD
FRL
HAD
DTF
ASD
RRl
REF
Rl
$R2')
36 FORMAT C2XoF6o2t2Xo2CF4.2t2XItF6oOt2Xt4CF5olt2X)tF6oOt2XtF4o0t2Xt2
$IF6.0o2X!tF4o2t2XtF4olt2XtF5olt312XtF5o3ll
END
. f
I
I
I
. SRIS
!UC
#