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					Limnol. Oceanogr., 38(7), 1993, 1403-1419
0 1993, by the American   Society of Limnology   and Oceanography,   Inc.



A spectrally averaged model of light penetration and photosynthesis
Thomas R. Anderson
James Rennell Centre for Ocean Circulation, Natural Environment Research Council, Gamma House, Chilworth
Research Centre, Chilworth, Southampton, U.K. SO1 7NS

         Abstract
            A model was developed which predicts the daily photosynthesis of a vertical pigment profile divided
         into a number of homogeneous layers. A spectral model (irradiance divided into a large number of
         wavebands) was used to derive simple empirical equations for calculating spectrally averaged values of
         two parameters- the vertical light attenuation coefficient and the chlorophyll-specific absorption of algae-
         for each layer as a function of its pigment content and position in the water column. The empirical
         equations are not dependent on the layer depths chosen, i.e. the same equations can be used for any given
         set of depths. The spectrally averaged parameters can be used with analytic integrals to give a compu-
         tationally rapid and accurate result. The model is therefore ideally suited for general circulation models.


   When choosing models of light penetration                                   production in the water column by as much
and photosynthesis for a given application, one                                as 50% or more (e.g. Sathyendranath et al. 1989;
should consider the need for both accuracy and                                 Platt and Sathyendranath 1991). The spectral
computational efficiency of predictions. Some-                                 properties of PAR must be considered to
times both are required, particularly when em-                                 achieve accuracy. Comprehensive models have
bedding ecosystem models in general circula-                                   been developed that divide PAR into a large
tion models (GCMs) of the world’s oceans.                                      number of wavebands and explicitly include
The penetration of photosynthetically       active                             the wavelength-dependent      properties of vari-
radiation (PAR) into seawater depends on the                                   ous parameters which affect light attenuation
nature of the light field and the absorbing prop-                              and photosynthesis (Morel 1988, 199 1; Sath-
erties of water and substances present in it.                                  yendranath and Platt 1988, 1989). Computa-
Photosynthesis depends on the distribution of                                  tional speed can be achieved with analytic so-
algal pigments and their photosynthetic         re-                            lutions for integrating photosynthesis       over
sponse to PAR. The complicated nature of                                       depth and time. Currently available analytic
these processes means that achieving a balance                                 integrals, however, require PAR to be modeled
between accuracy (achieved by introducing de-                                  as a single entity and not split into wavebands.
tail) and computational      efficiency (achieved                                  If the water column is divided into one or
through simplicity) is not easy.                                               more homogeneous layers, a spectrally aver-
    Both light attenuation and photosynthesis                                  aged parameterization    can be achieved by cal-
are affected by the spectral properties of the                                 culating an average value for each spectrally
light field. For example, red wavelengths are                                  dependent parameter in each layer. Platt and
rapidly absorbed in the upper 10 m of the water                                Sathyendranath (199 1) used a spectral model
column, whereas blue wavelengths penetrate                                     (i.e. PAR divided into wavebands) to derive
much deeper (Jerlov 1968). Furthermore, light                                  the necessary average values and then used
at different wavelengths is absorbed with dif-                                 these to calculate daily water-column      photo-
ferent efficiencies by algae. The use of non-                                  synthesis. They showed that one- or two-lay-
 spectral models of light attenuation and pho-                                 ered models could be used with a correction
tosynthesis can overestimate daily primary                                     factor (which they suggested was small and
                                                                               stable) to yield satisfactory results, although
                                                                               their analyses were limited to water columns
                                                                               with uniformly distributed biomass. They also
Acknowledgments                                                                showed that greater accuracy was achieved
  This research was supported by the Natural Environ-                          when several layers were modeled instead of
ment Research Council and a grant from the IJ.S. DOE                           only one or two, because of the variation in
(DE-FG02-90ER6 1053).
  I thank M. J. R. Fasham, A. Morel, and S. Sathyen-                           the spectral composition of PAR with depth.
dranath for comments on the manuscript, and D. Ruiz-                           However, changes in the vertical distribution
Pino for providing inspiration.                                                of pigment or in the spectral properties of ir-
                                                                            1403
1404                                               Anderson

                                                           concentration and its position in the water col-
              Derived parameter to calculate light ab-
                                                           umn. Daily photosynthesis can then be cal-
                 sorption by algae: Eq. 13, dimension-     culated by using the spectrally averaged values
                 less                                      in conjunction with analytic integrals. In ad-
              Spectrally averaged a# for layer L           dition, the model is applicable to non-uniform
              Calculated a# just below ocean surface       pigment profiles. It can be applied to any pro-
                for pigment concn G
              Chl-specific absorption of algae, m2 (mg     file where the ocean is represented by one or
                 Chl) - ’                                  more layers, each with a homogeneous chlo-
              Derived parameter: Eq. 9                     rophyll content. Its application is, however,
              Chl-specific absorption of algae in wave-    restricted to “case 1” waters, which are those
                band j, dimensionless                      in which ‘phytoplankton     and their covarying
              Photosynthetic efficiency in white light,
                 mg C (mg Chl) ’ h-’ (PEinst m-*           products play the predominant role in deter-
                s ‘)-I                                     mining optical properties, and the contribu-
              Maximum photosynthetic efficiency, mg        tion due to other substances is small (Morel
                C (mg Chl)- i h ’ (PEinst m-2 s-l)-’       and Prieur 1977). Most oceanic waters are
              Chl biomass, mg m -3
              Parameter for light attenuation due to       case I.
                algal pigments, waveband j, m2 mg-’            A user-friendly listing of the FORTRAN
              Daylength, h                                 code for the model is available on request.
              Quantum yield, mg C PEinst ’
              Pigment biomass, mg m 3                      Background
              Parameter for attenuation due to algal
                pigments, wavcband j                          Perhaps the most frequently used equation
              Downwelling PAR just below ocean sur-        for the attenuation of PAR in ecosystem mod-
                face and at depth z, PEinst m-2 s-’        els is (e.g. Steele and Frost 1977; Hofmann and
              Scalar PAR, PEinst m-2 s ’                   Ambler 1988; Fasham et al. 1990)
              Spectrally averaged k,,, for layer L, m- ’
              Vertical attenuation coefficient, m -’                      I, = I,exp( - kparz).        (1)
              Diffuse attenuation coefficient for water
                of waveband j, m ’                         I, is total PAR at depth z, I0 is the PAR just
              Daily depth-integrated photosynthesis,       beneath the ocean surface, and k,,, is the ver-
                mgCm-2d        ’                           tical attenuation coefficient. (A list of notation
              Assimilation number, mg C (mg Chl)-’
                h ’                                        is provided.) The value of k,,, is influenced by
              Total irradiance penetrating ocean sur-      the different components of the optical system
                face, Einst m 2 d- ’                       (water itself, phytoplankton,  other substances).
              Depth of base of layer i, m                  A single exponential may be a poor approxi-
              Fraction downwclling PAR just below
                ocean surface and at depth z in wave-
                                                           mation to the true irradiance distribution       be-
                band j                                     cause red wavelengths are rapidly absorbed in
              Fraction scalar PAR just below ocean         the uppermost 10 m of the ocean, whereas blue
                surface in waveband j                      wavelengths are absorbed more slowly. An im-
              Polynomial coefficients for calculating      proved prediction can be made with a double
                daily photosynthesis
                                                           exponential (Paulson and Simpson 1977; Carr
                                                            1986):
radiance impinging on the ocean surface cause               I, = Io[R exp( - kaz) + (1 - R)exp( - kbz)].
the values of spectrally averaged parameters
to change. To determine the new values in a                                                           (2)
multilayered system, Platt and Sathyendran-                k, and kb are attenuation coefficients, and the
ath had to rerun their spectral model.                     constant R is determined empirically for var-
   The model described here extends Platt and              ious water types. A similar arctangent approx-
Sathyendranath’s approach. The novel aspect                imation has been developed by Zaneveld and
of the work is that simple empirical equations             Spinrad ( 19 SO). Only rarely have such for-
are derived for calculating spectrally averaged            mulations for irradiance been used in phyto-
values for two parameters-the         vertical light       plankton models (e.g. Fasham et al. 1983; Tay-
attenuation coefficient and the chlorophyll-               lor et al. 199 1). Different models of light
specific absorption of algae-for any given lay-            attenuation were compared by Simpson and
er solely as a function of the layer’s chlorophyll         Dickey (198 1), who concluded that the most
                                         Spectrally averaged model                                   1405

accurate method was to divide PAR into a                   Cullen (1990) has recast traditional     P-I
large number of spectral wavebands, each of             curves such as Eq. 4 in terms of more physi-
which is then independently attenuated. A good          ologically based parameters, allowing spectral
example is the model of Morel (1988), where             influences on photosynthesis to be addressed.
the photosynthetic   spectrum is split into 61          The maximum photosynthetic         rate, PB,, is
wavebands. For a uniform pigment profile,               generally considered to be independent        of
Morel’s model is                                        wavelength (Pickett and Myers 1966). Param-
                                                        eter aB can be recast as the following product
                                                        (Platt and Jassby 1976; Platt 1986):
    I, = IO 5 ti,jexp[--(k,j         + ~~Gu)z].   (3)
             j=l                                                        a”@> = 4,a,(O                (6)
G is pigment biomass (chlorophyll         + pheo-       a”(X) describes the spectral dependency of aB,
phytin), kw,j is the diffuse attenuation coeffi-        $A is the quantum yield, and a,(X) is the chlo-
cient for water in waveband j, xj and rj are            rophyll absorption cross-section. The assump-
parameters that describe light attenuation by           tion of a spectrally invariant quantum yield is
phytoplankton     pigments, and the wavelength          reasonable when defined on a per photon basis,
corresponding to band j (X, nm) is 395 + 5j.            except at the extreme blue end of the PAR
The coefficients o,,~ determine the fraction of         spectrum (Lewis et al. 1985). The absorption
total downwelling PAR just below the ocean              spectrum, and hence the photosynthetic action
surface in each of 6 1 wavebands and vary with          spectrum, is markedly wavelength-dependent
time of year, location, and atmospheric con-            (e.g. Morel 199 1). Parameter a,(X) can be use-
ditions.                                                fully normalized to the maximum value of a,(X)
    The relationship   between photosynthesis,          observed in the PAR spectrum (e.g. Morel
irradiance, and chlorophyll is commonly de-              199 1), after which the spectral dependence of
scribed with photosynthesis-irradiance        (P-I)     aB is simply
curves. If light extinction is described by a
                                                                                  a*(h)
                                                                      a”(X) = aBmax                    (7)
single light extinction coefficient (Eq. 1), then
various P-I curves can be depth integrated (e.g.        where anmax the maximum photosynthetic ef-
                                                                      is
Platt et al. 1977). Recently, Platt et al. (1990)       ficiency and a*@) the dimensionless chloro-
derived a particularly useful algorithm that in-        phyll absorption cross-section.
tegrates photosynthesis with respect to both                The light attenuation model developed here
depth and time, assuming a sinusoidal pattern           treats irradiance as a downwelling flux. When
of irradiance over a day. The P-I curve on              using a*(A) values, however, irradiance must
which this integral is based is (Platt et al. 1980)     be specified as a 2calar flux (Morel 199 1). Sca-
                                                        lar irradiance, I(X), can be calculated from
        PB = P”,[l    - exp( - ~r~l/P~,].    (4)        downwelling irradiance as (Morel 199 1)
PB is biomass-normalized    photosynthesis, B is
chlorophyll biomass, I is PAR, PB, is assim-                           f(A) = I($$.
ilation number, and aB is the initial slope of
the P-I curve. Moreover, Platt et al. (1990)            k&X) is the downwelling attenuation coeffi-
closely approximated      their integral solution       cient (which can be calculated from Eq. 3), and
with a simple fifth-order polynomial, so depth-         a(A) is (Morel, 199 1)
integrated daily photosynthesis Pd is
                                                                      +
                                                        a(X) = [a,+,(X) 0.06AChl(X)G0.65][l + 0.2v(X)]
        Pd = BDPB,,J(?rk,&i-&(I*)x.                                                                (9)
                                                  (5)
                               x=1                      with
I* is defined as ctBI,JPB,, D is daylength, Is is               y(X) = exp[-0.014(X     - 440)].      (10)
noon irradiance just below the ocean surface,           Prieur and Sathyendranath        (198 1) tabulate
and Q, are polynomial coefficients. Currently,          values for parameters a,(A) and AC&).
analytic depth integrals of photosynthesis with             In the model developed here, all photosyn-
formulations for light attenuation more com-            thetic parameters require irradiance to be spec-
plex than Eq. 1 are not available.                      ified as a scalar flux. If PAR is assumed to
1406                                                         Anderson

                                                                  Note that a”(X) is influenced     by G because of
        pigment                                  1.o
                                                                  its influence on k,,,(X).           -
        (mg m-3 1                                10.0
                                                                  Model descriplion
                     profile number                                   The modeling approach adopted here in-
                       3     45              6 depth cm>          volves deriving simple equations for calculat-
                                                  -               ing average values for the spectrally dependent
                                                 10               parameters k,,, and a# for any given layer in a
                                                                  layered vertical pigment profile. This param-
                                                23
                                                40                eterization then permits daily photosynthesis
                                                                  to be rapidly calculated with analytic integrals
                                                61                such as Eq. 5.
                                                                      To derive equations for these parameters, it
                                                                  is necessary to determine the true spectral dis-
                                                                  tribution of PAR as a function of depth and
                                                                  chlorophyll    distribution. The spectral model
                                                123               of Morel (1988) was run to provide the best
                                                                  estimate of this. For a water column that is
                                                                  divided into a number of homogeneous ver-
                                                                  tical layers, this model is


                                               200                                     ‘i - (kw,j+ XjGi’J)
                                                                                       i=l
  Fig. 1. Six pigment profiles used to test the model (based
on the GCM ofsarmiento et al. 1993).
                                                                                               ’ (Uj- Ui-1)
                                                                                                          1
encompass wavelengths between 400 and 700                                    x ew[-(kw,j     + x~G,‘J)(z-    ur-,>I
nm (e.g. Morel 1991), Eq. 4 can be recast as

                                                                                              r > 1, u. = 0.     (14)
    PB = PB,                   -cxB,,,/PBm
                                                                  z is in the rth layer, and Gi and Ui denote the
                                                                  pigment biomass and depth of the base of layer

                                                     )1
                               700
                       X              a*(X)i(A) dX       .        i. Equation 3 can simply be applied for r = 1.
                           s x=400                   -             Using this model ignores upward irradiance
                                                         (11)     due to backscattering. This irradiance is typ-
   Light attenuation is, however, modeled as a                    ically 0.3-3% of downward irradiance (Paul-
downwelling flux. Equation 11 was converted                       son and Simpson 1977) and therefore can
for this purpose as follows:                                      probably be safely neglected. Values must be
                                                                  assigned to parameters as,j (the fraction of IO
                                                                  in each spectral band) before Eq. 14 can be
    PB = PB,                    -cY~,,,/P~,~                      used. An unchanging spectral distribution just
                                                                  below the ocean surface was assumed (i.e. pa-
                               700                                rameters w,y,jare constant), thereby allowing a
                                      a”(X)I(A) dX       .
                                                         (12)
                       X                                          general empirical model to be derived. This
                           s x=400                                distribution   was obtained by running Gregg
                                                                  and Carder’s (1990) atmospheric model for
a”(X) is a new parameter:                                         maritime atmospheres.
                                                                     The synthesis of the model involved three
                a”(X) = a*(X) -
                                     ‘k,AV                        main stages: definition of the spectral distri-
                                      a@) .                       bution of irradiance just below the ocean sur-
                                     Spectrally averaged model                                         1407

face, derivation of equations for calculating an       Table 1. Values of atmospheric paramctcrs.
average k,,, for a layer as a function of its
                                                     Prccipitable water, 2 cm
position in the water column and its pigment         Ozone, 0.35 cm
content, and the derivation of equations for         Visibility, 23 km
calculating an average a” for a layer as for k,,,.   Windspccd (instantaneous), 4 m s ’
The success of the new model in predicting           Mean windspced (over 24-h period), 4 m s-’
daily photosynthesis was assessed for the six        Relative humidity, 80%
                                                     Air mass type, 1
hypothetical pigment profiles shown in Fig. 1.       Atmospheric pressure, 1,000 mb
The layer depths of these profiles are based on
the North Atlantic GCM used by Sarmiento
et al. (1993). The wide range of pigment dis-
tributions encompassed by these profiles pro-         180, the fractions of quantum irradiance at
vides a good test of the model.                      different wavelengths for zenith angles of 0”
  Spectral distribution below ocean surface-         and 80”, as predicted by the model, are shown
Running Morel’s ( 198 8) light extinction model      in Fig. 2. The two distributions are remarkably
(Eq. 3) requires that 61 values be assigned to       similar. Because the spectral distribution      of
parameters o,,~, which define the spectral dis-      irradiance was predicted to be relatively in-
tribution (quantum units) of downwelling ir-         sensitive to zenith angle, a midrange zenith
radiance just below the ocean surface. It was        angle of 45” was set to generate the spectral
assumed that this spectral distribution      does    distribution    just below the ocean surface in
not change over time, in order to derive a gen-      bandwidths of 1 nm, using Gregg and Carder’s
eral empirical model. Baker and Frouin (1987)        (1990) model. Parameters o,,~ were then cal-
showed that the spectral distribution      imme-     culated by averaging across 5-nm wavebands.
diately above the ocean surface varied little for    Parameters o,, 1 and a,.61 were weighted by 0.5
a range of atmospheric and geometrical (sun          because they are on the edge of the 400-700-
geometry) conditions. Solar zenith angle has,        nm spectrum (which actually divides into sixty
however, been thought to significantly affect         5-nm intervals); the resulting values are listed
transport across the air-sea interface (e.g. Mo-     in Table 2.
rel 199 1). The spectral solar irradiance model          Derivation of k,,,- Having generated the
of Gregg and Carder (1990) was implemented            spectral distribution just below the ocean sur-
 to generate a spectral distribution just below       face, my next step was to determine how ir-
the ocean surface. Values of atmospheric pa-          radiance is attenuated as a function of depth
rameters were set mostly to those used by Mo-        and the vertical pigment profile. If the water
rel (199 1) and are listed in Table 1. With these    column is divided into a number of layers,
inputs and a day number (from 1 January) of          each with a separate value of spectrally aver-




     1      0.0053      13      0.0157      25       0.0164       37       0.0186       49          0.0190
     2      0.0103      14      0.0156      26       0.0172       38       0.0186       50          0.0189
     3      0.0108      15      0.0158      27       0.0178       39       0.0169       51          0.0183
     4      0.0113      16      0.0163      28       0.0177       40       0.0178       52          0.0173
     5      0.0014      17      0.0169      29       0.0174       41       0.0181       53          0.0182
     6      0.0014      18      0.0156      30       0.0180       42       Cl.0187      54          0.0194
     7      0.0107      19      0.0161      31       0.0180       43       0.0187       55          0.0191
     8      0.0126      20      0.0169      32       0.0181       44       0.0186       56          0.0189
     9      0.0127      21      0.0164      33       0.0179       45       0.0191       57          0.0188
    10      0.0140      22      0.0170      34       0.0182       46       0.0188       58          0.0180
    11      0.0154      23      0.0171      35       0.0180       47       0.0187       59          0.0164
    12      0.0154      24      0.0163      36       0.0185       48       0.0190       60          0.0170
                                                                                        61          0.0089
1408                                                               Anderson




                                                                              E
                                                                              s     -50

  ‘2
                                           -0O”
   3                                       .......... . 80”
  ai   0.001        1        I        I            I        I                jmIoo~l                              .   , , . . , .~
           400             500                  600               700        a     0.01                   0.1                    I 0
                                 h   b-0                                     e              PAR (as fraction of surface)
  Fig. 2. Predicted spectral distribution of PAR just be-                    Fig. 3. Percentagedifference from the full spectral model
low the ocean surface for zenith angles of 0” and 80”.                    of the irradiance profiles generated with the spectrally av-
                                                                          eraged model, with zero pigment, for n = 1, 2, 3, 4.

aged kpar, then PAR at depth z is
                                                                          gression package (SAS 1990: procedure NLIN)
                  r-l
                                                                          was used for this purpose. Resulting parameter
       L          z [-ki(Ui - Ui- I)]                                     values are shown in Table 3, and the relative
                   i=l
                                                                          errors (the difference between resulting irra-
                 x exP[-k(z          - ur- 111                            diance profiles and those generated from Mo-
                                                                          rel’s spectral model) are shown in Fig. 3. A
                             r > 1, u. = 0. (15)
                                                                          vast improvement in accuracy was made when
z is the rth layer and ki is the k,,, value for the                       two layers were used rather than only a single
ith layer. For r = 1, Eq. 1 can be applied instead                        layer, with further gains for three and four lay-
ofEq. 15.                                                                 ers. A value of n = 3 was chosen as providing
    Consider the case where pigment is zero, and                          an acceptable level of accuracy balanced against
so parameters ki are influenced solely by the                             the need to minimize the number of layers (in
attenuation due to water itself. If the water                             order to minimize overall amounts of com-
column (O-200 m) is divided into n layers, then                           putation). For convenience, the model was re-
the question arises: how large should n be to                             parameterized for n = 3, but setting ul = 5.0
accurately simulate light attenuation with Eq.                            m and u2 = 23.0 m; the resulting ki values are
 15? The true irradiance distribution for a zero                          also listed in Table 3. The predictive error for
pigment profile was estimated by running Mo-                              these settings was only very slightly greater
rel’s spectral model (Eq. 14). Then, least-                               than that shown for n = 3 in Fig. 3.
squares best-fit values for parameters in Eq.                                 Next, the effect of the vertical pigment pro-
 15 (ki and Ui) were found for n = 1, 2, 3, 4 by                          file on light attenuation must be added to the
fitting against the best estimate of the true dis-                        model. This addition is problematical, in that
tribution for depths between 0 and 180.8 m                                the rate of light extinction at any depth z de-
(the predicted 1% light level for zero pigment)                           pends not only on the pigment concentration
with data points every 0.1 m. A nonlinear re-                             at that depth, but also to some extent on the

  Table 3. Fitted k, (m I) and u, (m) values for n = 1,2,3,4 and for n = 3 with U, and uZ set at 5.0 and 23.0 m
rcspcctively.
       n            k,                     k,                      k,                k4          UI            U2            %
       1         0.06029                                           -                 -           -             -             -
       2         0.11833             0.02589                       -                 -          7.64           -             -
       3         0.13553             0.04339                    0.02236              -          4.60         22.28           -
       4         0.14629             0.06736                    0.03186           0.02093       3.08         10.15         35.84
       3         0.13117             0.04 177                   0.02233              -          5.00         23.00
                                         Spectrally averaged model                                           1409




            0    2       4      6     8    10                       0     2      4      6     8    10
         pigment in thin layer at 10 m (mg m-3)                  pigment in thin layer at 50 m (mg mD3)
  Fig. 4. Effect of pigment at depth z (in a very thin layer), and above depth z, on -dlogl/dz
                     - -                                                                         at depth z for two
depths, predicted with the full spectral model.


pigment profile above it (which influences the             obtain ki values for each of the three layers for
spectral properties of the light field). The rel-          different pigment concentrations. A range of
ative importance of these two factors, for                 O-15 mg m-3 was considered, with a square
depths of 10 and 50 m, is shown in Fig. 4,                 root transformation     to ensure a bias toward
which was generated by applying Eq. 14. Here,              smaller values which are more frequently en-
 -dlogZ/dz (k,,,) is plotted for various pigment           countered in nature. Pigment was assumed to
concentrations in a very thin (0.2 pm) layer at            be uniformly distributed in the water column.
depths of 10 (Fig. 4a) and 50 m (Fig. 4b) and              Parameters kl, k2, and k3 were determined with
also as a function of the pigment concentration            least-squares fits (the same procedure as for
(uniform distribution)     from the surface to the         zero pigment described above); the depth of
thin layer. At both depths, - dlog1/dz was much            the bottom layer used for fitting was either
more strongly influenced by the pigment at                  180.8 m (the 1% light level for zero pigment)
depth z rather than the pigment profile above              or the depth at which the 0.00 1% light level
it, except when there was a very large concen-             was reached if this was < 180.8 m. The re-                 ’
tration at depth z with a low pigment above.               sulting fitted ki values are plotted against pig-
This finding provides the foundation for a ma-             ment concentration for each of the three layers
jor simplifying assumption of the model: that              (Fig. 5). The much higher values for the top
 k,,, at a given depth can be treated as inde-             layer are caused by the rapid attenuation of
pendent of pigment above that depth and is                 red light near the surface. Fifth-order     poly-
then a function of depth itself and pigment at             nomials were fitted for each layer relating ki to
that depth. It then follows that an average k,,,           the square root of pigment concentration in
 can. be derived for any given layer, ki, which            that layer. With uI = 5.0 m and u2 = 23.0 m,
 is influenced only by the position and chlo-              k, , k2, and k3 are then
 rophyll content of the layer and not by the
 pigment profile above.                                                  ki = bo,i + bl,ic + b2,iC2
    Having derived parameters u1 = 5.0 m, u2                                                  +
                                                                               + b3,iC3+ bd,iC4 bs,iC5 (16)
 = 23.0 m, these were then fixed. Admittedly,              where c = G1j2, mg m-3. Values for the coef-
 the optimum depths for these partitions will               ficients are listed in Table 4.
 change for different pigment profiles, but the                A three-layer model of light attenuation is
 need for a generalized model forces them to               now complete: Eq. 15 and 16. It can be applied
 remain constant. If we accept the three-layer             to any vertical profile with a given number of
 model as derived above, it is now possible to             layers, each with a homogeneous pigment con-
1410                                                    Anderson




  k Par 0.3
  (m-9
         0.2




               0            1       4          9            16
                            pigment (mg m-3)                                      PAR (as fraction of surface)
  Fig. 5. Fitted k,, values for 12= 3 for pigment con-              Fig. 6. Percentagedifference from the full spectral model
centrations between 0 and 15 mg m-3. Lines are fifth-order       of irradiance predictions of the spectrally averaged model
polynomial fits (Table 4).                                       for the six pigment profiles of Fig. 1.

tent. It is necessary to subdivide those layers              w,,~ is the fraction of total downwelling     irra-
in the profile which encompass depths 5.0 and                diance in waveband j at depth z (calculated
23.0 m unless model layers exactly match these               from Eq. 14), and the remaining parameters
depths. Consider the profiles shown in Fig. 1.               correspond to those in Eq. 13 for waveband j.
The uppermost layer of O-10 m must be sub-                   The a*, spectrum was that of Morel (199 1).
divided into two layers of O-5 and 5-l 0 m. By               Profiles of a#=for the six vertical pigment pro-
pure coincidence, the depth of the second layer              files of Fig. 1 are shown in Fig. 7. It is clear
is 23 m, and so no subdivision of layers be-                 that a#=depends not only on the pigment con-
neath the top layer is required. The three main              centration at depth z, but also the pigment in
layers dictated by u1 and u2 can be subdivided               the water column above (e.g. profiles 2 and 3
further, in which case Eq. 16 is then applied                both have a pigment concentration of 1.O mg
separately to each sublayer. The percentage er-              m-3 at 23 m, but have very different values for
ror incurred by the new model when predicting                a#=at that depth). The rate of change of a# with
light attenuation for the six profiles shown in              z, daVdz, depends on pigment concentration
Fig. 1, as compared to that predicted with Mo-               at depth z, on depth z itself, and on the pigment
 rel’s spectral model, is shown in Fig. 6. This              profile above. It is also apparent that there are
 error was consistently small (< 10%) down to                sudden changes in a#= which are caused by
 the 10% light level and < 20% down to the 1%                changes in pigment concentration.        For uni-
 light level. Errors do increase if even smaller             form pigment profiles, the relative impor-
 light levels are investigated, but such increases           tances of the pigment at depth z and the profile
 are relatively unimportant      when calculating            above are shown for two depths, 10 and 50 m,
 overall depth-integrated photosynthesis.                    in Fig. 8 (generated as for Fig. 4). At 10 m,
    Derivation of a# - A spectrally averaged val-             pigment at depth z generally had a much larger
 ue of a# can be calculated at any depth based                effect on da#ldz than pigment above (Fig.
 on the spectral distribution     of light and the            8a), although the effect is less simple at 50 m
 pigment concentration (which affects k,,,) at                (Fig. 8b). In the model it is assumed that daV
 that depth:                                                  dz is independent of the pigment profile above
                                                              depth z.
                         61 m,,ja*jkpar,j                        Equations 14 and 17 were used to calculate
                    a% = 2       a.     -               (17)
                            j=l         J                     da#/dz for pigment concentrations between 0
  Table 4. Polynomial coefficients relating k, to square root of pigment in layer i: Eq. 16.
     i                b,,
                       ,             b,,              b,,               b 3,                b 48                 b 5.1

     1             0.13096        0.030969         0.042644         -0.013738          0.00246 17          -0.00018059
     2             0.04 1025      0.0362 11        0.062297         -0.030098          0.0062597           -0.0005 1944
     3             0.021517       0.050 150        0.058900         -0.040539          0.0087586           -0.00049476
                                             Spectrally averaged model                                                 1411

                                                                Table 5. Coefficients g,-g,, for predicting da#ldz as a
                                                              function of pigment and depth: Eq. 18.
        1.0
                                                              g, =   0.048014                     g, = 0.0027974
        0.8                                                   g, =   0.00023779                   g, = 0.00085217
  a#
                                                              g3 =   -0.023074                    g* = -0.0000039804
        0.6                                                   g, =   0.0031095                    g, = 0.0012398
                                                              g, =   -0.0090545                   g,, = -0.00061991
        0.4

        0.2                                                   variations with depth (which cause the quo-
                                                              tient k,,,la to change) are simulated by apply-
            o!
                 0   20    40       60      80       100      ing an offset to a# when crossing between mod-
                           depth (m)                          el layers. The equivalent offset at the ocean
  Fig. 7. Depth dependence of a# for the six pigment          surface was calculated for the pigment con-
profiles of Fig. 1 calculated with the full spectral model.   centrations of the two layers, and it was used.
                                                              In reality this offset is depth-dependent,    but
and 15 mg m-3 (square-root bias) assuming                     the approach adopted here is a useful generic
uniform pigment profiles and depths between                   approximation.
0 and 200 m (log bias). Results are shown in                     An average value of a# for any given model
Fig. 9a. A quadratic equation was fitted to these             layer L, a#L, can now be calculated by inte-
data to describe da#/dz as a function of the                  grating Eq. 18 with respect to z through the
square root of pigment concentration (c = G1j2)               layer and dividing by two to give a value for
and the natural logarithm of depth [v = log(z                 the layer midpoint (term 3 in Eq. 19), adding
+ l), m]:                                                     it to a# at the base of the layer above (terms 1
                                                              and 2), and then adding the offset (terms 4 and
 da#/dz(c,v) = g, + g,c + g,v + g,cv
                                                              5).
               + g,c2 + g,v2 + g,c3                                                 1 UI.-1
               + g,v3 + ggc2v + g10cv2.      W-9                   a#,, = asL-, + -          daVdz(c, v) dz
                                                                                   2 s UI.-2
g,-g,, are the fitted coefficients; their values
are listed in Table 5. Predicted values of da#/                                 1         UL

                                                                                               daVdz( c, v) dz
dz for the same range of pigment concentra-                                +    2   s   UL-,
tions and depths are shown in Fig. 9b; com-
parison with Fig. 9a shows a reasonable agree-                             + CGL - a#.,, ,--,
ment.
   Sudden changes in a#= caused by pigment                                             G,
                                                                                    a#3, = a%+         a% = a#.,.,     (19)
       a)




                0    2       4      6     8    10                       0     2      4      6     8    10
             pigment in thin layer at 10 m (mg m-3)                  pigment in thin layer at 50 m (mg m-3)
                                            Fig. 8. As Fig. 4, but on da#/dz.
 1412                                                  Anderson

         a>                                                          b>




                                  10              100 200                                         10                                100 200
                                  depth (ml                                                       depth (ml
  Fig. 9. Dependence of da#ldz on depth and pigment concentration predicted with the full spectral model (a) and
according to the fitted function (b) (Eq. 18).


where a#S, is a# at the ocean
            GI                            surface calcu-
lated for pigment concentration            GL. The first                  = i (z + l)(g, +             g2c      +         &C2        +       &C3)
integral term in Eq. 19 is only           applied for L
 > 1. Equation 17 was used to             generate a”,,                      L+fi -iz+ U(g3+                        g4c         +    g9c2)

values (Fig. 10). A polynomial            equation was                        +    “6 {z   +   1 lk,     +     ia&)
fitted to these data:
                                                                                                   UI.
     a#,, = h, + h,c + h2c2 + h3c3 + h,C4(20)
                                                                              +f,-iz+          ugs
where c is G lj2. Values for the fitted coefficients                                             1UI.I
are given in Table 6.                                                                                                                               (21)
  The integral terms in Eq. 19 solve as
        UI.
                                                              where
                da#ldz(c, v) dz                               f,{z+       l}=(z+        l)log(z+             l)-(z+
                                                                                                              (22)                   l),
    s UI.-I
                                                              fi{z    + l} = (z + l)log2(z + 1) - 2fi {z + 1},(23)



              o.7i
              0.6

  #
 a s,G
              0.5




              0.3
                    0     1         4         9        16
                         pigment, G (mg m-3)
   Fig. 10. Predicted a#,y,, as a function of pigment con-                                     depth (m)
centration calculated with full spectral model. Fitted line
is the polynomial fit (Eq. 20).                                            Fig. 11. As Fig. 6, but of predicted a#.
                                              Spectrally averaged model                                                              1413

  Table 6. Coefficients ho-h, for fitting a#s,,;as a function   and for simplicity         a value of G = 0 was as-
of surface pigment: Eq. 20.                                     sumed, giving
h, = 0.36796                    h, = -0.065276                                 ~Bnlax = 2.602 aBw.          (26)
h, = 0.17537                    h, = 0.013528
                                h, = -0.0011108                    The best estimate of daily water-column
                                                                photosynthesis, against which the predictions
                                                                of the spectrally averaged model were com-
and                                                             pared, was obtained by executing the “full
                                                                spectral model”: Gregg and Carder’s (1990)
f,{z + 1) = (z + l)log3(z + 1) - 3f2{z + 1}.(24)                atmospheric model to provide the total irra-
Equations 19-24 can now be used to calculate                    diance and its spectral distribution just below
the value of a# for any given layer, a#L. The                   the ocean surface, Morel’s (198 8) spectral
errors in a#=incurred when compared with the                    model (Eq. 14) to attenuate light, and Platt et
full spectral model for the six vertical profiles               al.‘s (1980) P-I curve (Eq. 4) with spectrally
of Fig. 1 are shown in Fig. 11 (bear in mind                    dependent parameters (Eq. 12) to calculate
that a#=is constant within any one layer in the                 photosynthesis. The result, Pf (mg C me2 d-l),
new model). Profile 3 showed significant pos-                   is then
itive errors near the surface and large negative
errors below 23 m. Predictions were generally
within f 10% for the other profiles, except those
with deep chlorophyll maxima which had er-
rors as large as -20%.
Calculating photosynthesis with the model
    Having derived simple equations for cal-
culating spectrally averaged values for param-
eters k,,, and a# for each model layer, we can                  Iz,tj is downwelling PAR at depth z, time t, and
now test their ability to accurately calculate                  in waveband j and a#,j is a# at depth z in wave-
daily water-column photosynthesis. Equations                    band j (derived from Eq. 13). The integrations
4 and 5 were used to model photosynthesis.                      were done numerically, with a time step of 5
The two parameters which define the P-I curve                   min and a depth interval of 10 cm.
described by Eq. 4 are PB, and aR. Nonspectral                      The spectrally averaged model can be used
measures of aR are commonly made in white                       to calculate daily photosynthesis, integrating
light, with irradiance measured in scalar units.                analytically with respect to depth, or both depth
If white light is assumed to have the same                      and time. In the former case, daily photosyn-
spectral distribution as irradiance just beneath                thesis, Pk, (the subscript refers to spectrally
the ocean surface, then the photosynthetic ef-                  averaged k,,, and a#), is (the basic integral is
ficiency in white light, aRW,is                                 provided by Platt et al. 1990)

                                                        (25)

~Blnax is the maximum photosynthetic efficien-
                                                                              X
CY, which occurs at X = 435 nm, and ij~,j is the                                  s   0
                                                                                          D 2       (-1)x
                                                                                                             x.x!
                                                                                                                      + 1
                                                                                            x=1
fraction of surface scalar irradiance in wave-
band j, which can be calculated from param-                                   x   kLax            a#LIL,t,          l lP”,)x
eters w,g,jand Eq. 8. Consequently, from a given
value of ayBW, value for cyBmnX be calcu-
                a                   can                                       -   h%ax          a%,L2               /P”,)“]    dt.    (28)
lated. However, this calculation depends on                     I L.&l is the total PAR entering the layer at time
pigment concentration, because of its impact                    t, and IL,r,2is the light leaving the layer (both
on kpw For an aBWof 1, calculated values of                     calculated from Eq. 15). Total PAR just below
~Bnlaxrange from 2.602 for G = 0 to 2.371 for                   the ocean surface was calculated by running
G = 10. Sensitivity to G is thus relatively low,                the atmospheric model of Gregg and Carder.
1414                                                                                     Anderson
  Table 7. Summary of daily integrations.
  Estimate                                      Spectral   distribution                                k “A                                a*

    Pf                              Gregg and Carder model                                     Morel model                        Morel model
    pfi                             Paramctcrs w,~(fixed)                                      Morel model                        Morel model
    Pk,,,                           Parameters w,,, (fixed)                                    Spectrally averaged                Spectrally averaged
    P kod                           Parameters w,~(fixed                                       Spectrally averaged                Spectrally averaged
    pk                              Paramctcrs w,~,,(fixed)                                    Spectrally averaged                Morel model
    PO                              Parameters w,, (fixed)                                     Morel model                        Spectrally averaged


   If irradiance through the day is assumed to                                                   IZ,l is calculated by means of Eq. 15, with Gregg
vary sinusoidally, then water-column      photo-                                                 and Carder’s model to generate total irradiance
synthesis can be analytically integrated with                                                    just below the ocean surface. The same equa-
respect to time as well as depth with the ap-                                                    tion is used for Pkad,with a sinusoidal variation
proximation of Platt et al. (1990) (Eq. 5). Daily                                                 of irradiance about the noon value. The nu-
photosynthesis as. predicted by the spectrally                                                    meric solutions for Pk, and P,&dcalculated from
averaged model, Pka& is then                                                                      Eq. 30 consistently differed from the equiva-
                                                                                                  lent analytic solutions (Eq. 28 and 29) by 1.5%
Pkad      =     2 BLDPn,l(akL)                                                                    or less (usually much less).
                L=l                                                                                  The difference Pka - Ps arises from three
                                                                                                 sources: the assumption of a set spectral dis-
                                                                                                 tribution just below the ocean surface and the
                                                                                                 differences associated with the parameteriza-
                                                                                                 tion of k,,, and a#. Thus,
                                                                . (29)
                                          - ((XBm,,a”LZL,,,,lPBm)Xl
I L,4, and IL, 4,2are the irradiances entering and
     1                                                                                                   “p,    pf= (DSp+ Dk + D,)/lOO.                 (31)
leaving layer L at noon.
   A problem arises when comparing Ps, Pka,
and Pkad:the methods for integrating over depth                                                  DSp,Dk, and D, are the % differences from the
and time (i.e. numeric or analytic) differ. To                                                   spectral model (normalized to Pi) associated
avoid this problem when comparing the dif-                                                        with the spectral assumption and the param-
ferent daily estimates, I obtained all the results                                                eterization of k,,, and a#, respectively, nor-
displayed here by applying numeric integra-                                                       malized to the best estimate. The magnitude
tions for both depth and time. The daily pho-                                                     of each was estimated as follows.
tosynthesis can then be predicted from the                                                           D,,: The full spectral model was run (as for
spectrally averaged model as                                                                      Pf), except that the set spectral distribution just
                                                                                                  below the ocean surface (parameters w,,j) was
                                                                                                  imposed. If the resulting daily photosynthesis
ha=&,                                                                                             is PJw,  then D,, is
          L=l
                 D UL
          X
                SS    I=0        z=u,.-    [
                                               P%[l         -    exP(-~Hn,axa#LL,l
                                                                                                                     DS,,= 1.00
                                                                                                                                  Pfw -   Pf
                                                                                                                                                .       (32)
                                                                          + PBn,)] dz dt.                                            Pf
                                                                                        (30)
                                                                                                     DIC and Da: It is possible to run the full spec-
                                                                                                  tral model with the set spectral distribution (as
  Table 8. Latitudes and day numbers (from 1 January)                                             for PfJ, except that the empirical equations of
used to test the model.                                                                           the spectrally averaged model for kT, and a#,_
   Scc-                                                          Sce-
                                                                                                  can be imposed to determine either k,,, or a#
  nario                                        Day              nario         Lat (“)   Day       (both at once would give Pka). The two result-
                                                                                                  ing figures, Pk (spectrally averaged k,,, but not
    :                       70
                             0                 172
                                               310                e
                                                                  f            50       139
                                                                                        355       a#) and Pa, give an estimate of the relative
                                                                                                  impact of the parameterization      of k,,, and a#
    :                       60                 172
                                               264                E            80
                                                                               70       240
                                                                                        210       on the difference Pka - Pp Dk and Da are then
                                                                   Spectrally averaged model                                                                1415


                                                layer 1              lzsl layer 3                 0         layers 5,6,7,8

                                   l!zzi            layer 2          @j
                                                                     El     layer      4

             2       1.2

              2 1.0
               %I
              ‘3 0.8
               E
               g     0.6
              \
                g    0.4
              %
              3      0.2
                Ei
                Q    0.0
                                    abc~abc~abc~abc~abdabc
                                      1       2        3        4                                                      5                      6
                                                    profile number
  Fig. .12. Comparison between the full spectral model and the spectrally averaged model of predicted daily photo-
synthesis for scenario e (50”, day number 355) for the six pigment profiles of Fig. 1: a-P!; b--P,,; C-P,,,.



   Dk= 100
                     pk    -
                                   Pk
                                    - PjL
                                   pfi     +   pa    -   pfu
                                                                                      The assumption of a sinusoidal irradiance pat-
                                                                                      tern through the day, used to calculate Pk&,
                                                                           ,- a.      introduces an additional source of error, Dd
                                                                           (33)       (normalized to Ps, % units):

and
                                                                                                                           P       -    pka
                                                                                                            Dd=       100 ICad
                                                                                                                                   Pr             ’
   Da       = 100
                                   pa- pfi                                            A summary of the different methods for cal-
                     Pk    - Pfw+ Pa - PJu                                   .        culating daily photosynthesis  is given in
                                                                           (34)       Table 7.


  Table 9. Best estimates of daily water-column photosynthesis (O-200 m), Pr, for the six vertical profiles (Fig. 1) and
each scenario (Table 8). Units are mg C m-2 d- *.
                                                                                        Profile
   Scenario                    I                               2                  3                     4                      5                        6

                                                         410.33            1,269.67                   345.56               219.38                     156.32
        E                   2.76
                           99.83                          12.42               39.56                    11.53                 4.38                       2.82
        :                  63.22
                          130.81                         270.07              852.78                   238.99               121.82                      82.24
                                                         543.26            1,689.39                   464.79               277.37                     194.53
        e                  25.26                         111.53              359.48                   101.29                44.03                      28.90
        f                 120.01                         495.39            1,535.99                   420.02      ’        259.88                     183.94
                                                         492.57            1,545.ll                   430.65               233.47                     159.92
        i                 116.64
                           64.60                         284.14              913.48                   257.38               113.86                      74.95
1416                                                Anderson




                -40
                      1 O      a
                                                                                              0
                                                                                              d




  Fig. 13. Predicted errors associated with the spectrally averaged model for the six pigment profiles of Fig. 1 and
eight scenarios of Table 8 divided into their subcomponents: D,, Dk, and D,.


Results                                                    ber 35 5) was selected because it represents a
   The model was tested for the six pigment                midlatitude    site and was also investigated in
profiles shown in Fig. 1 and for a wide range              detail by Platt and Sathyendranath (1991) in
of latitudes and day numbers (scenarios a-h),              their layered model of water-column           photo-
corresponding to those examined by Platt et                synthesis. Predictions of the spectrally aver-
al. (1990), and listed in Table 8. Settings for            aged model are compared to those of the full
photosynthetic   parameters are based on me-               spectral model in Fig. 12. Values of P,are tab-
dian values for midlatitude phytoplankton     de-          ulated in Table 9. Daily water-column           pho-
scribed by Harrison and Platt ( 1986); they are            tosynthesis, Pka, was successfully predicted to
PB = 2.39 mg C (mg Chl)-’ h-l, and CY~,,,       =          within - 10% of the best estimate, except for
O.o”l mg C (mg Chl)-’ h-l (PEinst m-2 s-‘)-~.              those profiles with deep chlorophyll maxima,
It was assumed that chlorophyll biomass, B,                which showed slightly larger errors. Using the
was 80% of total pigment biomass, G (e.g. Trees            daily approximation     (sine irradiance assump-
et al. 1986).                                              tion), Pk&, changed the error in predictions by
   A single scenario was chosen for a detailed              - 10%. Not only were the daily totals predicted
analysis: to investigate the suitability of each           with reasonable accuracy, but the vertical dis-
of the two approximations    for predicting daily          tribution   of the photosynthesis       within the
production in each model layer for each of the             model layers was also predicted with success.
six pigment profiles. Scenario e (50°, day num-                Next, Pka was calculated for all the scenarios
                                             Spectrally averaged model                                            1417

                                                                                                 averaged over the
   Table 10. Predicted errors associated with assuming a sinusoidal irradiance pattern, Ddand Ddc,
six profiles for each scenario.
      Scenario             D,,-~95%   C.I.                                                      Dd< f95%   C.I.

          :               -0.11 kO.03
                            9.12kO.25                 0.4564
                                                     46.4 1              51.49
                                                                          0.4558                - 1.25k0.64
                                                                                                -0.02+0.01


          i                11.22kO.19
                           17.18k0.66                55.16
                                                     20.98               23.79
                                                                         66.98                  - 1.34kO.73
                                                                                                -2.72t- 1.39
          e                10.83f0.34                 6.63                7.47                  -0.BO-tO.45
           f               12.34k1.46                53.57               61.76                  - 1.99+: 1.01
          hp               26.45kO.59
                           28.6 120.77               17.41
                                                     42.39               55.94
                                                                         23.22                  -1.92k1.09
                                                                                                -3.02+ 1.61




 (a-h). For each scenario and profile, the dif-              found by Platt et al. (1990), the sinusoidal as-
 ference Pk, - P,was split into its components:              sumption overestimates the radiation entering
 Dsp, Dk, and Da. When observing these, one                  the water column per day. This error can be
 must remember that summing positive and                     corrected for by multiplying   &ad by the quo-
 negative values will cause an overall reduced               tient Qcc/QS, where Qcc is the total irradiance
 difference. Results are shown in Fig. 13. Dis-              penetrating the ocean surface per day (Einst
 regarding profiles 5 and 6, Pka was within - 7%             m-2 d-l) ca1 1 d using Gregg and Carder’s
                                                                           CU ate
 (and usually much less) of Pr for all scenarios             model, and Q, is that calculated from a sinu-
 except b. Moreover, this success was due to all             soidal function, with Gregg and Carder’s mod-
three components of the difference (Dsp, Dk,                 el to calculate noon irradiance. The new ad-
and Da) being small, and not to large values                 ditional error caused by assuming a sinusoidal
of these components of opposite sign having                  irradiance, DdC,is then
a compensating effect. The difference due to
the parameterization of a#, Da, ranged between                       DdC= 100 PkadQG,/Qs         -   pku
                                                                                                            .     (36)
 10 and 20% for profiles 5 and 6 but was offset                                            Pf
to some extent by Dk, which was typically 3-
4% in these profiles, and of opposite sign to                Predicted values of DdC,averaged for each sce-
Da. The difference associated with the spectral              nario, are also shown in Table 10. Assuming
distribution, Dsp, was very marked in the case               a sinusoidal irradiance now causes only a very
of scenario b. This scenario is a high-latitude              small negative error, because the higher irra-
site (70”) with a very short day (5.2 h). The                diances resulting from the sinusoidal assump-
resulting zenith angles for radiation ranged en-             tion were being used less efficiently for pho-
tirely between 85 and 90”. Although not shown                tosynthesis. Platt et al. (1990) showed that a
in Fig. 2, zenith angles much greater than 80”               sine-square approximation     was more suitable
cause a significant increase in the proportion               for high latitudes (the analytical solution for
of irradiance in the blue wavelengths, with less             such a case was also presented).
in the red. Because red light is rapidly absorbed
by water at the top of the water column, this
means that the spectral distribution defined by              Concluding remark
parameters o,,~ underestimates light available                  A model is presented here that is an empir-
to phytoplankton      for scenario b and, hence,             ical approximation      of Morel’s (1988, 199 1)
also daily photosynthesis.                                   spectral model of light attenuation and ab-
    Finally, the additional error caused by ex-              sorption by phytoplankton         (with Gregg and
ploiting the daily approximation        (i.e. by as-         Carder’s 1990 atmospheric model to generate
suming a sinusoidal pattern of irradiance dur-               light at the ocean surface), used in conjunction
ing the day) was assessed (I&). Results for the              with Platt et al.‘s (1980) P-I curve (Eq. 4). The
six profiles were averaged for each scenario                 new model is designed to predict daily pho-
and are shown in Table 10. Assuming a si-                    tosynthesis for a vertical pigment profile di-
nusoidal irradiance pattern introduced a vari-               vided into a number of homogeneous layers
able, and sometimes large (high-latitude        sites        and is computationally     efficient.
with long days), positive error. As was also                    The accuracy of the new model was assessed
1418                                             Anderson

by comparison with the output of the full spec-             Nevertheless, the spectrally averaged model
tral model. It was tested for several vertical           is a computationally    efficient tool for predict-
pigment profiles and scenarios (latitudes and            ing water-column photosynthesis, which takes
day numbers) and predicted daily water-col-              account of the spectral influences on light at-
umn photosynthesis to within 15%, and usu-               tenuation and photosynthesis and how these
ally to within 5%, of the spectral model (except         are influenced by the vertical pigment profile.
for profile b).                                          It is particularly well suited for use in GCMs,
    It should be remembered that the new model           where accuracy and computational          speed of
has only successfully approximated            another    predictions are at a premium.
model. It is necessary to speculate on the suit-
ability of incorporating the new model in glob-
al-scale GCMs, i.e. on the general applicability         References
of the full spectral model as it was imple-
                                                         ATLAS, D., AND T. T. BANNISTER. 1980. Dependence of
mented to generate the approximation.           When         mean spectral extinction coefficient of phytoplankton
running the spectral model, the phytoplankton                on depth, water color, and species. Limnol. Oceanogr.
parameters x, y, and a* were assigned fixed                  25: 157-159.
values for each of the 6 1 spectral wavebands.           BAKER, K. S., AND R. FROUIN. 1987. Relation between
In the real ocean, phytoplankton        communities          photosynthetically available radiation and total in-
                                                             solation at the ocean surface under clear skies. Lim-
are composed of different species with different             nol. Oceanogr. 32: 1370-1377.
characteristics     and may be physiologically           BRICAUD, A., A.-L. B~DHOMME, AND A. MOREL. 1988.
adapting to their particular environment. Al-                Optical properties of diverse phytoplanktonic species:
gal absorption spectra differ significantly        be-       Experimental results and theoretical interpretation. J.
                                                             Plankton Res. 10: 851-873.
tween different phytoplankton         groups accord-     CARR, M. R. 1986. Modelling the attenuation of broad
ing to the kinds and amounts of accessory                    band light down a water column. Statistician 35: 325-
pigments accompanying chlorophyll a and pig-                 333.
ment packaging effects (Atlas and Bannister              CLEVELAND, J.S.,M.J. PERRY, D.A. KIEFER,AND M.C.
 1980; Bricaud et al. 1988). Photosynthetic light            TALBOT. 1989. Maximal quantum yield of photo-
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