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Review of PAR parameterizations in ocean ecosystem models
Estuarine, Coastal and Shelf Science xxx (2014) 1e6
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Review of PAR parameterizations in ocean ecosystem models Do-Seong Byun a, *, Xiao Hua Wang b, Deirdre E. Hart c, Marco Zavatarelli d, e a
Ocean Forecasting Team, Korea Hydrographic and Oceanographic Administration, Busan 606-806, Republic of Korea School of Physical Environmental and Mathematical Sciences, University of New South Wales at Australian Defence Force Academy, Canberra, ACT 2600, Australia c Department of Geography, University of Canterbury, Private Bag 4800, Christchurch, New Zealand d di Bologna, Dipartimento di Fisica e Astronomia, Viale Berti-Pichat 6/2, 40127 Bologna, Italy Alma Mater Studiorum Universita e di Bologna, Centro Interdipartimentale per la Ricerca sulle Scienze Ambientali, Via S. Alberto 163, 48100 Ravenna, Italy Alma Mater Studiorum Universita b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 January 2014 Received in revised form 8 May 2014 Accepted 10 May 2014 Available online xxx
Commonly-used empirical equations for calculating downward ‘photosynthetically available radiation’ or PAR were reviewed in order to identify a more theoretically-sound parameterization for application to ocean biogeochemical models. Three different forms of broadband PAR parameterization are currently employed in biogeochemical models, each of them originating from the downward irradiance formulations normally applied to ocean circulation models, which produce poor attenuation estimates for PAR. Two of the PAR formulations, a single-exponential function and a double-exponential function, are parameterized by multiplying surface irradiance by a coefficient determining the portion of underwater PAR. The third formulation uses the second term of the double-exponential function. After elucidating the theoretical problems of modeling PAR using these parameterizations, we suggest an improved, Rmodified double-exponential PAR formulation, including Paulson and Simpson's (1977) parameter values. We also newly estimate PAR penetration via least-squares fitting of values digitized from Jerlov's (1976) observations in different oceanic water types, and compare this PAR-observation derived parameterization with our new, theoretical, R-modified parameterization. Finally, we discuss a universal limitation inherent in current theoretical approaches to PAR parameterization. © 2014 Elsevier Ltd. All rights reserved.
Keywords: light parameterization PAR solar radiation ecosystem model
1. Introduction Around 90% of all marine life lives in the euphotic zone. The biological and physical processes that characterize this ocean surface layer, such as primary production and thermal dynamics, are fundamentally controlled by the penetration of sunlight through the water column. Modeling these processes, which sustain life across two thirds of our planet, requires an accurate and computationally-viable parameterization of the vertical distribution of underwater irradiance (Anderson, 1993; Liu et al., 2002; Kara et al., 2005). To date, a simple light parameterization has been almost exclusively used in ocean general circulation models (OGCM) and biogeochemical models, implemented as standalone or coupled models. More accurate, spectrally-dependent bio-optical parameterization methods are typically not employed in such models because they are computationally expensive and due to the difficulties in parameterizing sunlight penetration through seawater
* Corresponding author. E-mail addresses: [email protected], [email protected] (D.-S. Byun).
with suspended phytoplankton, detritus and optically-active dissolved organic matter (e.g. Gallegos and Correl, 1990; Anderson, 1993). In this study, we explore the characteristics of the simple downward irradiance parameterizations commonly used in OGCM and biogeochemical models in order to understand how to develop a more accurate, but still efficient, light parameterization for biogeochemical models. We examine three different types of photosynthetically active radiation (PAR) parameterizations, derived originally from the downward irradiance parameterizations used in physical models, before proposing a slightly-modified and theoretically-improved light parameterization for computing phytoplankton production. In addition, we explain a key limitation of using downward irradiance-derived PAR parameterizations compared to observed PAR behavior. 2. Methods: parameterizations of downward irradiance and PAR penetration Biogeochemical models and OGCM are interested in the penetration of sunlight underwater for fundamentally different reasons.
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In OGCM, the parameterization of total downward irradiance (300e2500 nm) is used to study ocean heat flux while in biogeochemical modeling PAR (350e700 nm) is parameterized in order to simulate primary production by phytoplankton. In both types of model, however, the vertical profile of downward irradiance (i.e. total or PAR) is generally calculated using a simple LamberteBeer's Law formula, with a constant (depth-averaged) attenuation coefficient k (m1) and an assumption of optically homogeneous waters (e.g. Denman, 1973; Alexander and Kim, 1976; Chen et al., 1999) such that:
et al., 2003; Byun and Cho, 2006; Byun et al., 2007). Such PAR apportioning is usually defined by inserting a PAR to total irradiance ratio (3 PAR) into Eq. (1) as follows:
IðzÞ ¼ I0 expð kzÞ;
IPAR ðzÞ ¼ I0 3 PAR ½R expð k1 zÞ þ ð1 RÞexpð k2 zÞ
(1)
where I(z) is the downward irradiance at depth z, and I0 is the seasurface irradiance at z ¼ 0. It is recognized that using a singleexponential function in physical models yields a poor approximation of irradiance in the upper few meters (Paulson and Simpson, 1977; Zaneveld and Spinrad, 1980), leading to the overestimation of vertical light penetration (Simpson and Dickey, 1981). This is because longer wave lengths are disproportionately absorbed in near-surface waters. Jerlov (1976, Table XXVIII) proposed an optical classification of seawater types with respect to water clarity or turbidity, based largely on the amount of particulate and dissolved matter. He defined 5 oceanic and 5 coastal water types, ordered from 1 to 5 in terms of increasing turbidity. According to this study, the infrared (IR > 700 nm) portion of the total downward solar radiation spectrum (300e2500 nm) is rapidly-absorbed within the first vertical meter of the oceans. Meanwhile PAR (350e700 nm) is more successful in penetrating below the surface, but undergoes a shift from the blue-green spectrum (400e500 nm) in open ocean waters to the green spectrum (500e550 nm) in coastal waters, according to increasing water column turbidity. Jerlov's (1976) observations of selective downward irradiance absorption and penetration prompted other authors to develop improved heat flux parameterizations. A double-exponential parameterization (Paulson and Simpson, 1977) and an arctangent parameterization (Zaneveld and Spinrad, 1980) of the downward solar radiation profile in oceanic waters were recommended as more accurate approximations of the irradiance field for OGCM modeling. Simpson and Dickey (1981) revealed that these two parameterizations of the downward irradiance flux yield essentially equivalent results. The simpler, Paulson and Simpson (1977) double-exponential formulation was thereafter widely adopted in OGCMs such as the Princeton Ocean Model (POM) and Regional Ocean Modeling System (ROMS), using different empirical attenuation coefficients for Jerlov's different seawater types (e.g. Martin, 1997; Mellor, 2003; Vichi et al., 2003; Marchesiello et al., 2003). According to this parameterization, the vertical irradiance profile through optically homogenous waters is given by:
IðzÞ ¼ I0 ½R expð k1 zÞ þ ð1 RÞexpð k2 zÞ;
(2)
where R is the empirical apportioning constant, k1 and k2 are the empirical vertical attenuation coefficients (m1), and these three empirical coefficients are adjusted to fit the observed downward irradiance distribution (see Appendix A for more details). Although widely adopted in OGCM, the above simple formulation for the underwater attenuation of total irradiance (300e2500 nm) is of little use in biogeochemical models, where accurate irradiance estimates for the PAR (350e700 nm) spectrum specifically are crucial for modeling photosynthetic processes. It is typically assumed that incoming (sea surface) PAR comprises between 42 and 50% of the total incoming solar radiation (e.g. Parsons et al., 1984; Baretta-Bekker et al., 1997; Jacovides et al., 2003; Vichi
IPAR ðzÞ ¼ I0 3 PAR expð kzÞ;
(3)
A double-exponential parameterization has sometimes been used for the estimation of downward PAR in biogeochemical models. For example, introducing the concept of the conversion factor from Eq. (3) into Eq. (2), Fasham et al. (1983) parameterized downward PAR as:
(4)
According to this method, the term (1 R) accounts for the PAR fraction belonging the blue-green wavelengths and R accounts for the remaining PAR fraction. Note that Fasham et al. (1983) also considered the effect of phytoplankton shelf-shading in order accurately to reproduce the development of the spring phytoplankton bloom in the Celtic Sea. Consideration of this effect is, however, beyond the scope of the present study. Further, Kara et al. (2005), and Hamme and Emerson (2006), using the double-exponential parameterization in dynamic-mixedlayer models, pointed out that the second term on the right-hand side of Eq. (2) is related to the vertical PAR distribution, which is expressed as:
IPAR ðzÞ ¼ I0 ð1 RÞexpð k2 zÞ:
(5)
Eq. (5) is similar to Eq. (3) but it is derived from the doubleexponential irradiance parameterization of Eq. (2). 3 PAR and (1 R) appear identical but in fact they differ as 3 PAR (as stated above) is the ratio of PAR to total solar radiation just above sea surface whereas both the R and k2 terms of Eq. (5) are determined via a least-squares fit of downward irradiance observations from the subsurface water column, where only a portion of PAR exists. In the following section we investigate the vertical light penetration characteristics exhibited by each PAR parameterization mentioned above and suggest a new R-modified PAR formulation.
3. Results and discussion 3.1. Evaluation of existing and new downward-PAR parameterizations The double-exponential parameterization shown in Eq. (2) was derived from the spectrally-and depth-dependent attenuation of total solar radiation (300e2500 nm) in sea-surface waters, as explained in the previous section. It has already been noted that this double-exponential formulation, used in OGCM, is not useful for calculating PAR (350e700 nm), since PAR accounts for only about 42e50% of the total downward irradiance, and narrow bandwidths within the PAR spectrum are disproportionately able to penetrate the sea surface waters depending on particulate conditions. In contrast, the second exponential term in Eq. (2) could be used to estimate PAR without alteration since it explains only a portion of the visible irradiance, based on the fact that only a narrow band of PAR penetrates below the surface layer in oceanic waters. In Eq. (4) the conversion factor for PAR (3 PAR) is multiplied by a portion of the visible irradiance in the second term on the righthand side, leading to the underestimation of downward PAR compared to when 3 PAR is not included in calculations, particularly for the subsurface layers. Additionally Eq. (5), which corresponds to the second exponential function of Eq. (2), typically underestimates PAR in the surface water column compared to observations. Based
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Table 1 Summary of PAR parameterizations and parameter values used in comparative tests. Experiments and equations
PAR parameterization (IPAR(z)¼)
Parameter values Type I
Type III
Mellor (1998) Fasham et al. (1983)
Exp. 1 [Eq. (3)] Exp. 2 [Eq. (4)]
I03 PARek z I0 3 PAR ½R ek1 z þ ð1 RÞek2 z
k ¼ 0.037 R ¼ 0.58 k1 ¼ 2.875 k2 ¼ 0.043
k ¼ 0.127 R ¼ 0.78 k1 ¼ 0.714 k2 ¼ 0.127
Hamme and Emerson (2006)
Exp. 3 [Eq. (5)]
I0 ð1 RÞek2 z
R ¼ 0.49 k2 ¼ 0.043
R ¼ 0.49 k2 ¼ 0.127
This study
Exp. 4 [Eq. (8)]
I0 ½R0 ek1 z þ ð1 RÞek2 z
R ¼ 0.58 k1 ¼ 2.875 k2 ¼ 0.043
R ¼ 0.78 k1 ¼ 0.714 k2 ¼ 0.127
Reference
Note that parameter values for Experiments 2, 3 and 4 for oceanic water types I and III were derived from calculations by Paulson and Simpson (1977), while those of Exp. 1 were obtained from Mellor (1998). R is the empirical apportioning constant while R0 ¼ (R þ 3 PAR 1), with 3 PAR ¼ 0.49. The units of k, k1 and k2 are m1.
on the characteristics of the above irradiance penetration formulations, we arrived at the following, R-modified double-exponential function for calculating PAR in biogeochemical models, with no change in the attenuation coefficients (k1, k2) used in physical models. The conceptual idea for the new R-modified doubleexponential PAR function is to use the second exponential function of Eq. (2), which accounts for a portion of the visible irradiance, without considering the conversion factor for PAR (3 PAR) to downward irradiance. In addition, the calculated amount of PAR (1 R) should be extracted from the total incoming PAR (I03 PAR) just above the sea-surface, which is estimated at about 49% (i.e. 3 PAR ¼ 0.49) of the total incoming irradiance, in order to calculate the amount of PAR in sea-surface waters (I0R0 ). This may be expressed as:
I0 R0 ¼ I0 3 PAR I0 ð1 RÞ:
(6)
Accordingly, the PAR apportioning constant in the surface water (R0 ) is written as:
R0 ¼ 3 PAR ð1 RÞ ¼ R þ 3 PAR 1:
(7)
By inserting Eq. (7) into the R parameter in the first exponential on the right-hand side of Eq. (2), the downward PAR with constant attenuation coefficients from the incoming solar radiation may be expressed as:
IPAR ðzÞ ¼ I0 ½R0 expð k1 zÞ þ ð1 RÞexpð k2 zÞ:
(8)
Fig. 1. Comparison of underwater PAR attenuation profiles for Jerlov's (1976) oceanic water types I and III: PAR observations digitized from Jerlov's Fig. 130 versus results calculated using the four different parameterizations listed in Table 1. IPAR0 is the sea-surface PAR at z ¼ 0.
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As shown in Fig. 1a, for oceanic water type I the results from our modified parameterization (Exp. 4) approximated the irradiance observations more closely than those from the other parameterizations, particularly where the attenuation rate was above 90% (i.e. IPAR/IPAR0 > 0.1). The RMS (Root Mean Square) errors between observations and Exp. 1 to Exp. 4 were 0.059, 0.074, 0.051 and 0.028. In contrast, for oceanic water type III our parameterization (Exp. 4, Eq. (8)) underestimated PAR penetration slightly (Fig. 1b), whereas Exp. 1 (Eq. (3)) and Exp. 3 (Eq. (5)) yielded slight overestimates. As expected, Exp. 2 (Eq. (4)) significantly underestimated PAR penetration. The RMS errors between observations and Exp. 1 to Exp. 4 were 0.053, 0.0891, 0.053 and 0.046, respectively. 3.3. Limitation of the R-modified parameterization These results indicate that our R-modified parameterization is very simple and effective for certain ocean water conditions but still has a limitation. In order to investigate the source of this limitation, we examined the attenuation characteristics of total downward irradiance (I) and PAR (IPAR) through oceanic water types I, II and III. These data were derived from Jerlov's (1976) Table (XXVIII) for total downward irradiance and were digitized from his Fig. 130 (p. 197) for PAR. Fig. 2 shows that I and IPAR have different attenuation behaviors. In oceanic water types I, II and III the total downward irradiance (I-S, II-S, III-S) is rapidly attenuated within 1 m of the ocean surface (as explained earlier), leading to the doubleexponential parameterization. In contrast, the vertical attenuation of PAR (I-P, II-P, III-P) tends to be relatively slow, with its rate changing little between the ocean surface and subsurface. This may be the reason why Eq. (3) (Exp. 1) produces relatively good results (Fig. 1) and has been commonly used in calculating phytoplankton production. Specifically, for oceanic water type I, the vertical attenuation behaviors between I-P and I-S are very similar. However, the I-P behavior is different from the II-P and III-P behaviors in the surface waters. These results reveal that the assumption used in Eq. (8), our new parameterization, is reasonable for oceanic water type I but not for oceanic water types II and III. Furthermore, we observe that the vertical attenuation gradient of PAR below the subsurface is similar to that of total solar radiation, since their two profiles share the same slope, as illustrated in Fig. 2. We have shown that our R-modified formulation produces a more accurate depth-dependent PAR profile for oceanic water type I, as demonstrated via comparisons with Jerlov's observationderived PAR values. Here, we compare the double-exponential PAR parameter values derived from our approach with those from the PAR observation-derived formulation in order to quantitatively identify attenuation behavior characteristics and similarities. We applied the same approach as Paulson and Simpson (1977) for computing the downward-irradiance double-exponential parameter values. That is, the PAR double-exponential parameter values (Rp and the attenuation coefficients kp1 and kp2) were calculated for oceanic water types I, II and III using PAR observation data
Fig. 2. Comparison of underwater attenuation profiles for oceanic water types I, II and III: total irradiance values (I-S, II-S, III-S) were plotted from Jerlov (1976, Table XXVIII) while the PAR irradiance values (I-P, II-P, III-P) were digitized from Jerlov (1976, Fig. 130).
Note that Eq. (8) is parameterized on the assumption that the attenuation behavior of the downward PAR is similar to that of the downward solar radiation. This assumption is discussed in Section 4. Meanwhile, in the next section the above-mentioned PAR parameterizations are tested and compared with PAR observations. 3.2. Comparison of observed and calculated PAR We compared downward PAR values calculated from the three existing, and one new, parameterization with Jerlov's (1976) PAR observations for oceanic water types I and III. The values of R, k1 and k2 for oceanic waters I and III used in our calculations were derived from the study of Paulson and Simpson (1977). Table 1 lists each parameterization of broadband photosynthetic irradiance and the associated parameter values used in experiments.
Table 2 Parameter values determined by least-squares fitting for each of the three sets of oceanic PAR values digitized from Jerlov (1976, Fig. 130) and used in our R-modified formulation.
Oceanic waters
IPAR ðzÞ ¼ I0 ½R0 expðk1 zÞ þ ð1 RÞexpðk2 zÞ
IPAR ðzÞ ¼ IPAR0 ½Rp expðkp1 zÞ þ ð1 Rp Þexpðkp2 zÞ
Water type
I II III
Rp
kp1 (m
0.19 0.32 0.35
0.796 0.594 0.464
1
)
kp2 (m 0.041 0.077 0.130
1
)
R
R0
k1 (m1)
k2 (m1)
0.58 0.77 0.78
0.07 0.26 0.27
2.875 0.666 0.714
0.043 0.071 0.127
IPAR0 is the PAR just above the sea-surface; R is the empirical apportioning constant for downward solar radiation; and R0 ¼ (R þ 3 PAR1), with 3 PAR ¼ 0.49. The parameter values of R, k1 and k2 were calculated by Paulson and Simpson (1977).
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Acknowledgments The authors would like to thank two anonymous reviewers for their constructive and helpful comments on the manuscript. Appendix A. Parameterizing light penetration through ocean surface waters Using the observed vertical irradiance values, Paulson and Simpson (1977) parameterized each oceanic water type as the sum of two exponential functions through the least squares fitting method. They based this method on the fact that only a portion of irradiance tends to penetrate deeply into ocean waters because of their strong absorption of the infrared spectrum. The calculation procedure is as follows. Irradiance values observed in the subsurface (below 6 m depth) can be expressed as a linear function: Fig. 3. PAR attenuation below the surface for ocean water types I, II and III from Jerlov (1976, Fig. 130).
digitized from Jerlov's (1976) Fig. 130 (see Table 2 and Fig. 3). Note that Rp, kp1 and kp2 are derived from PAR whereas R, k1 and k2 in Section 3 are derived from total irradiance. As expected, the attenuation coefficient values for PAR below the surface (kp2) in ocean water types I, II and III were very similar to those for total irradiance (k2) calculated by Paulson and Simpson (1977) using Jerlov's (1976) observations. In contrast, the attenuation coefficient values for PAR penetration in the surface waters (kp1) were smaller than those for total irradiance (k1), especially for oceanic water type I (Table 2). This result supports our physical understanding that PAR is attenuated more slowly in the surface waters compared to the rest of the irradiance spectrum (i.e. total irradiance e PAR). In addition, the Rp values (0.19e0.35) calculated directly from the downward PAR were slightly greater than our total-irradiance-derived R0 values (0.07e0.27) (Table 2). The larger Rp(or R0 ) and/or kp1(or k1) values produced higher attenuation rates in the downward PAR (or solar radiation) calculations, generally leading to the underestimation of PAR penetration in surface waters compared to observations. As illustrated in Table 2, for oceanic water type I, even though our formulation's R0 value (0.07) is 0.12 smaller than the Rp value (0.19), its k1 value (2.875 m1) is considerably larger than the kp1 value (0.796 m1), resulting in better agreement with the observed PAR attenuation (Fig. 1a). For oceanic water type III, our formulation's R0 value (0.27) is slightly smaller than the Rp value (0.32), and its k1 value (0.714 m1) is greater than the kp1 value (0.464 m1), showing that our formulation underestimates the downward PAR (Fig. 1b) in such turbid ocean conditions.
z ¼ b lnðI=I0 Þ þ b0 ;
(A1)
where the coefficients b and b0 can be found from least squares fitting. Rearranging Eq. (A1) after taking the natural exponent gives:
IðzÞ ¼ ð1 RÞexpðz=bÞ; I0
(A2)
where R ¼ 1 expðb0 =bÞ: The irradiance estimated from Eq. (A2) should be subtracted from the surface observation values to conduct the second fitting for the surface observation values, resulting in:
Iup ðzÞ ¼
IðzÞ ð1 RÞexpðz=bÞ: I0
(A3)
Applying the same procedure as used above for the data sets of Iup (z) calculated from Eq. (A3), each Iup(z) in the upper layers is given as:
Iup ðzÞ ¼ R expðz=aÞ:
(A4)
Substituting the resulting Iup(z) into Eq. (A3), the irradiance attenuation in the entire water column can be expressed as (Paulson and Simpson, 1977):
IðzÞ ¼ R expðz=aÞ þ ð1 RÞexpðz=bÞ: I0
(A5)
References 4. Conclusions We have produced an irradiance derived, R-modified doubleexponential formulation for calculating the downward profile of PAR below the sea surface. This new method produces improved estimates under certain oceanic conditions, without the need to change attenuation coefficients. In order to replicate Jerlov's PAR observations, we assumed temporally- and vertically-constant PAR attenuation coefficients, a condition that approximates that found in low turbidity offshore ocean waters but which is seldom encountered in the coastal ocean. In the latter areas, attenuation due to wave, tide and nearshore-circulation suspended sediments and self-shading by phytoplankton can create dynamic and complicated attenuation coefficient profiles. Tests against field data, which incorporate temporal and vertical variability in PAR attenuation, are needed as the next step in improving downward PAR formulations for biogeochemical models.
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Review of PAR parameterizations in ocean ecosystem models Do-Seong Byun a, *, Xiao Hua Wang b, Deirdre E. Hart c, Marco Zavatarelli d, e a
Ocean Forecasting Team, Korea Hydrographic and Oceanographic Administration, Busan 606-806, Republic of Korea School of Physical Environmental and Mathematical Sciences, University of New South Wales at Australian Defence Force Academy, Canberra, ACT 2600, Australia c Department of Geography, University of Canterbury, Private Bag 4800, Christchurch, New Zealand d di Bologna, Dipartimento di Fisica e Astronomia, Viale Berti-Pichat 6/2, 40127 Bologna, Italy Alma Mater Studiorum Universita e di Bologna, Centro Interdipartimentale per la Ricerca sulle Scienze Ambientali, Via S. Alberto 163, 48100 Ravenna, Italy Alma Mater Studiorum Universita b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 January 2014 Received in revised form 8 May 2014 Accepted 10 May 2014 Available online xxx
Commonly-used empirical equations for calculating downward ‘photosynthetically available radiation’ or PAR were reviewed in order to identify a more theoretically-sound parameterization for application to ocean biogeochemical models. Three different forms of broadband PAR parameterization are currently employed in biogeochemical models, each of them originating from the downward irradiance formulations normally applied to ocean circulation models, which produce poor attenuation estimates for PAR. Two of the PAR formulations, a single-exponential function and a double-exponential function, are parameterized by multiplying surface irradiance by a coefficient determining the portion of underwater PAR. The third formulation uses the second term of the double-exponential function. After elucidating the theoretical problems of modeling PAR using these parameterizations, we suggest an improved, Rmodified double-exponential PAR formulation, including Paulson and Simpson's (1977) parameter values. We also newly estimate PAR penetration via least-squares fitting of values digitized from Jerlov's (1976) observations in different oceanic water types, and compare this PAR-observation derived parameterization with our new, theoretical, R-modified parameterization. Finally, we discuss a universal limitation inherent in current theoretical approaches to PAR parameterization. © 2014 Elsevier Ltd. All rights reserved.
Keywords: light parameterization PAR solar radiation ecosystem model
1. Introduction Around 90% of all marine life lives in the euphotic zone. The biological and physical processes that characterize this ocean surface layer, such as primary production and thermal dynamics, are fundamentally controlled by the penetration of sunlight through the water column. Modeling these processes, which sustain life across two thirds of our planet, requires an accurate and computationally-viable parameterization of the vertical distribution of underwater irradiance (Anderson, 1993; Liu et al., 2002; Kara et al., 2005). To date, a simple light parameterization has been almost exclusively used in ocean general circulation models (OGCM) and biogeochemical models, implemented as standalone or coupled models. More accurate, spectrally-dependent bio-optical parameterization methods are typically not employed in such models because they are computationally expensive and due to the difficulties in parameterizing sunlight penetration through seawater
* Corresponding author. E-mail addresses: [email protected], [email protected] (D.-S. Byun).
with suspended phytoplankton, detritus and optically-active dissolved organic matter (e.g. Gallegos and Correl, 1990; Anderson, 1993). In this study, we explore the characteristics of the simple downward irradiance parameterizations commonly used in OGCM and biogeochemical models in order to understand how to develop a more accurate, but still efficient, light parameterization for biogeochemical models. We examine three different types of photosynthetically active radiation (PAR) parameterizations, derived originally from the downward irradiance parameterizations used in physical models, before proposing a slightly-modified and theoretically-improved light parameterization for computing phytoplankton production. In addition, we explain a key limitation of using downward irradiance-derived PAR parameterizations compared to observed PAR behavior. 2. Methods: parameterizations of downward irradiance and PAR penetration Biogeochemical models and OGCM are interested in the penetration of sunlight underwater for fundamentally different reasons.
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In OGCM, the parameterization of total downward irradiance (300e2500 nm) is used to study ocean heat flux while in biogeochemical modeling PAR (350e700 nm) is parameterized in order to simulate primary production by phytoplankton. In both types of model, however, the vertical profile of downward irradiance (i.e. total or PAR) is generally calculated using a simple LamberteBeer's Law formula, with a constant (depth-averaged) attenuation coefficient k (m1) and an assumption of optically homogeneous waters (e.g. Denman, 1973; Alexander and Kim, 1976; Chen et al., 1999) such that:
et al., 2003; Byun and Cho, 2006; Byun et al., 2007). Such PAR apportioning is usually defined by inserting a PAR to total irradiance ratio (3 PAR) into Eq. (1) as follows:
IðzÞ ¼ I0 expð kzÞ;
IPAR ðzÞ ¼ I0 3 PAR ½R expð k1 zÞ þ ð1 RÞexpð k2 zÞ
(1)
where I(z) is the downward irradiance at depth z, and I0 is the seasurface irradiance at z ¼ 0. It is recognized that using a singleexponential function in physical models yields a poor approximation of irradiance in the upper few meters (Paulson and Simpson, 1977; Zaneveld and Spinrad, 1980), leading to the overestimation of vertical light penetration (Simpson and Dickey, 1981). This is because longer wave lengths are disproportionately absorbed in near-surface waters. Jerlov (1976, Table XXVIII) proposed an optical classification of seawater types with respect to water clarity or turbidity, based largely on the amount of particulate and dissolved matter. He defined 5 oceanic and 5 coastal water types, ordered from 1 to 5 in terms of increasing turbidity. According to this study, the infrared (IR > 700 nm) portion of the total downward solar radiation spectrum (300e2500 nm) is rapidly-absorbed within the first vertical meter of the oceans. Meanwhile PAR (350e700 nm) is more successful in penetrating below the surface, but undergoes a shift from the blue-green spectrum (400e500 nm) in open ocean waters to the green spectrum (500e550 nm) in coastal waters, according to increasing water column turbidity. Jerlov's (1976) observations of selective downward irradiance absorption and penetration prompted other authors to develop improved heat flux parameterizations. A double-exponential parameterization (Paulson and Simpson, 1977) and an arctangent parameterization (Zaneveld and Spinrad, 1980) of the downward solar radiation profile in oceanic waters were recommended as more accurate approximations of the irradiance field for OGCM modeling. Simpson and Dickey (1981) revealed that these two parameterizations of the downward irradiance flux yield essentially equivalent results. The simpler, Paulson and Simpson (1977) double-exponential formulation was thereafter widely adopted in OGCMs such as the Princeton Ocean Model (POM) and Regional Ocean Modeling System (ROMS), using different empirical attenuation coefficients for Jerlov's different seawater types (e.g. Martin, 1997; Mellor, 2003; Vichi et al., 2003; Marchesiello et al., 2003). According to this parameterization, the vertical irradiance profile through optically homogenous waters is given by:
IðzÞ ¼ I0 ½R expð k1 zÞ þ ð1 RÞexpð k2 zÞ;
(2)
where R is the empirical apportioning constant, k1 and k2 are the empirical vertical attenuation coefficients (m1), and these three empirical coefficients are adjusted to fit the observed downward irradiance distribution (see Appendix A for more details). Although widely adopted in OGCM, the above simple formulation for the underwater attenuation of total irradiance (300e2500 nm) is of little use in biogeochemical models, where accurate irradiance estimates for the PAR (350e700 nm) spectrum specifically are crucial for modeling photosynthetic processes. It is typically assumed that incoming (sea surface) PAR comprises between 42 and 50% of the total incoming solar radiation (e.g. Parsons et al., 1984; Baretta-Bekker et al., 1997; Jacovides et al., 2003; Vichi
IPAR ðzÞ ¼ I0 3 PAR expð kzÞ;
(3)
A double-exponential parameterization has sometimes been used for the estimation of downward PAR in biogeochemical models. For example, introducing the concept of the conversion factor from Eq. (3) into Eq. (2), Fasham et al. (1983) parameterized downward PAR as:
(4)
According to this method, the term (1 R) accounts for the PAR fraction belonging the blue-green wavelengths and R accounts for the remaining PAR fraction. Note that Fasham et al. (1983) also considered the effect of phytoplankton shelf-shading in order accurately to reproduce the development of the spring phytoplankton bloom in the Celtic Sea. Consideration of this effect is, however, beyond the scope of the present study. Further, Kara et al. (2005), and Hamme and Emerson (2006), using the double-exponential parameterization in dynamic-mixedlayer models, pointed out that the second term on the right-hand side of Eq. (2) is related to the vertical PAR distribution, which is expressed as:
IPAR ðzÞ ¼ I0 ð1 RÞexpð k2 zÞ:
(5)
Eq. (5) is similar to Eq. (3) but it is derived from the doubleexponential irradiance parameterization of Eq. (2). 3 PAR and (1 R) appear identical but in fact they differ as 3 PAR (as stated above) is the ratio of PAR to total solar radiation just above sea surface whereas both the R and k2 terms of Eq. (5) are determined via a least-squares fit of downward irradiance observations from the subsurface water column, where only a portion of PAR exists. In the following section we investigate the vertical light penetration characteristics exhibited by each PAR parameterization mentioned above and suggest a new R-modified PAR formulation.
3. Results and discussion 3.1. Evaluation of existing and new downward-PAR parameterizations The double-exponential parameterization shown in Eq. (2) was derived from the spectrally-and depth-dependent attenuation of total solar radiation (300e2500 nm) in sea-surface waters, as explained in the previous section. It has already been noted that this double-exponential formulation, used in OGCM, is not useful for calculating PAR (350e700 nm), since PAR accounts for only about 42e50% of the total downward irradiance, and narrow bandwidths within the PAR spectrum are disproportionately able to penetrate the sea surface waters depending on particulate conditions. In contrast, the second exponential term in Eq. (2) could be used to estimate PAR without alteration since it explains only a portion of the visible irradiance, based on the fact that only a narrow band of PAR penetrates below the surface layer in oceanic waters. In Eq. (4) the conversion factor for PAR (3 PAR) is multiplied by a portion of the visible irradiance in the second term on the righthand side, leading to the underestimation of downward PAR compared to when 3 PAR is not included in calculations, particularly for the subsurface layers. Additionally Eq. (5), which corresponds to the second exponential function of Eq. (2), typically underestimates PAR in the surface water column compared to observations. Based
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Table 1 Summary of PAR parameterizations and parameter values used in comparative tests. Experiments and equations
PAR parameterization (IPAR(z)¼)
Parameter values Type I
Type III
Mellor (1998) Fasham et al. (1983)
Exp. 1 [Eq. (3)] Exp. 2 [Eq. (4)]
I03 PARek z I0 3 PAR ½R ek1 z þ ð1 RÞek2 z
k ¼ 0.037 R ¼ 0.58 k1 ¼ 2.875 k2 ¼ 0.043
k ¼ 0.127 R ¼ 0.78 k1 ¼ 0.714 k2 ¼ 0.127
Hamme and Emerson (2006)
Exp. 3 [Eq. (5)]
I0 ð1 RÞek2 z
R ¼ 0.49 k2 ¼ 0.043
R ¼ 0.49 k2 ¼ 0.127
This study
Exp. 4 [Eq. (8)]
I0 ½R0 ek1 z þ ð1 RÞek2 z
R ¼ 0.58 k1 ¼ 2.875 k2 ¼ 0.043
R ¼ 0.78 k1 ¼ 0.714 k2 ¼ 0.127
Reference
Note that parameter values for Experiments 2, 3 and 4 for oceanic water types I and III were derived from calculations by Paulson and Simpson (1977), while those of Exp. 1 were obtained from Mellor (1998). R is the empirical apportioning constant while R0 ¼ (R þ 3 PAR 1), with 3 PAR ¼ 0.49. The units of k, k1 and k2 are m1.
on the characteristics of the above irradiance penetration formulations, we arrived at the following, R-modified double-exponential function for calculating PAR in biogeochemical models, with no change in the attenuation coefficients (k1, k2) used in physical models. The conceptual idea for the new R-modified doubleexponential PAR function is to use the second exponential function of Eq. (2), which accounts for a portion of the visible irradiance, without considering the conversion factor for PAR (3 PAR) to downward irradiance. In addition, the calculated amount of PAR (1 R) should be extracted from the total incoming PAR (I03 PAR) just above the sea-surface, which is estimated at about 49% (i.e. 3 PAR ¼ 0.49) of the total incoming irradiance, in order to calculate the amount of PAR in sea-surface waters (I0R0 ). This may be expressed as:
I0 R0 ¼ I0 3 PAR I0 ð1 RÞ:
(6)
Accordingly, the PAR apportioning constant in the surface water (R0 ) is written as:
R0 ¼ 3 PAR ð1 RÞ ¼ R þ 3 PAR 1:
(7)
By inserting Eq. (7) into the R parameter in the first exponential on the right-hand side of Eq. (2), the downward PAR with constant attenuation coefficients from the incoming solar radiation may be expressed as:
IPAR ðzÞ ¼ I0 ½R0 expð k1 zÞ þ ð1 RÞexpð k2 zÞ:
(8)
Fig. 1. Comparison of underwater PAR attenuation profiles for Jerlov's (1976) oceanic water types I and III: PAR observations digitized from Jerlov's Fig. 130 versus results calculated using the four different parameterizations listed in Table 1. IPAR0 is the sea-surface PAR at z ¼ 0.
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As shown in Fig. 1a, for oceanic water type I the results from our modified parameterization (Exp. 4) approximated the irradiance observations more closely than those from the other parameterizations, particularly where the attenuation rate was above 90% (i.e. IPAR/IPAR0 > 0.1). The RMS (Root Mean Square) errors between observations and Exp. 1 to Exp. 4 were 0.059, 0.074, 0.051 and 0.028. In contrast, for oceanic water type III our parameterization (Exp. 4, Eq. (8)) underestimated PAR penetration slightly (Fig. 1b), whereas Exp. 1 (Eq. (3)) and Exp. 3 (Eq. (5)) yielded slight overestimates. As expected, Exp. 2 (Eq. (4)) significantly underestimated PAR penetration. The RMS errors between observations and Exp. 1 to Exp. 4 were 0.053, 0.0891, 0.053 and 0.046, respectively. 3.3. Limitation of the R-modified parameterization These results indicate that our R-modified parameterization is very simple and effective for certain ocean water conditions but still has a limitation. In order to investigate the source of this limitation, we examined the attenuation characteristics of total downward irradiance (I) and PAR (IPAR) through oceanic water types I, II and III. These data were derived from Jerlov's (1976) Table (XXVIII) for total downward irradiance and were digitized from his Fig. 130 (p. 197) for PAR. Fig. 2 shows that I and IPAR have different attenuation behaviors. In oceanic water types I, II and III the total downward irradiance (I-S, II-S, III-S) is rapidly attenuated within 1 m of the ocean surface (as explained earlier), leading to the doubleexponential parameterization. In contrast, the vertical attenuation of PAR (I-P, II-P, III-P) tends to be relatively slow, with its rate changing little between the ocean surface and subsurface. This may be the reason why Eq. (3) (Exp. 1) produces relatively good results (Fig. 1) and has been commonly used in calculating phytoplankton production. Specifically, for oceanic water type I, the vertical attenuation behaviors between I-P and I-S are very similar. However, the I-P behavior is different from the II-P and III-P behaviors in the surface waters. These results reveal that the assumption used in Eq. (8), our new parameterization, is reasonable for oceanic water type I but not for oceanic water types II and III. Furthermore, we observe that the vertical attenuation gradient of PAR below the subsurface is similar to that of total solar radiation, since their two profiles share the same slope, as illustrated in Fig. 2. We have shown that our R-modified formulation produces a more accurate depth-dependent PAR profile for oceanic water type I, as demonstrated via comparisons with Jerlov's observationderived PAR values. Here, we compare the double-exponential PAR parameter values derived from our approach with those from the PAR observation-derived formulation in order to quantitatively identify attenuation behavior characteristics and similarities. We applied the same approach as Paulson and Simpson (1977) for computing the downward-irradiance double-exponential parameter values. That is, the PAR double-exponential parameter values (Rp and the attenuation coefficients kp1 and kp2) were calculated for oceanic water types I, II and III using PAR observation data
Fig. 2. Comparison of underwater attenuation profiles for oceanic water types I, II and III: total irradiance values (I-S, II-S, III-S) were plotted from Jerlov (1976, Table XXVIII) while the PAR irradiance values (I-P, II-P, III-P) were digitized from Jerlov (1976, Fig. 130).
Note that Eq. (8) is parameterized on the assumption that the attenuation behavior of the downward PAR is similar to that of the downward solar radiation. This assumption is discussed in Section 4. Meanwhile, in the next section the above-mentioned PAR parameterizations are tested and compared with PAR observations. 3.2. Comparison of observed and calculated PAR We compared downward PAR values calculated from the three existing, and one new, parameterization with Jerlov's (1976) PAR observations for oceanic water types I and III. The values of R, k1 and k2 for oceanic waters I and III used in our calculations were derived from the study of Paulson and Simpson (1977). Table 1 lists each parameterization of broadband photosynthetic irradiance and the associated parameter values used in experiments.
Table 2 Parameter values determined by least-squares fitting for each of the three sets of oceanic PAR values digitized from Jerlov (1976, Fig. 130) and used in our R-modified formulation.
Oceanic waters
IPAR ðzÞ ¼ I0 ½R0 expðk1 zÞ þ ð1 RÞexpðk2 zÞ
IPAR ðzÞ ¼ IPAR0 ½Rp expðkp1 zÞ þ ð1 Rp Þexpðkp2 zÞ
Water type
I II III
Rp
kp1 (m
0.19 0.32 0.35
0.796 0.594 0.464
1
)
kp2 (m 0.041 0.077 0.130
1
)
R
R0
k1 (m1)
k2 (m1)
0.58 0.77 0.78
0.07 0.26 0.27
2.875 0.666 0.714
0.043 0.071 0.127
IPAR0 is the PAR just above the sea-surface; R is the empirical apportioning constant for downward solar radiation; and R0 ¼ (R þ 3 PAR1), with 3 PAR ¼ 0.49. The parameter values of R, k1 and k2 were calculated by Paulson and Simpson (1977).
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Acknowledgments The authors would like to thank two anonymous reviewers for their constructive and helpful comments on the manuscript. Appendix A. Parameterizing light penetration through ocean surface waters Using the observed vertical irradiance values, Paulson and Simpson (1977) parameterized each oceanic water type as the sum of two exponential functions through the least squares fitting method. They based this method on the fact that only a portion of irradiance tends to penetrate deeply into ocean waters because of their strong absorption of the infrared spectrum. The calculation procedure is as follows. Irradiance values observed in the subsurface (below 6 m depth) can be expressed as a linear function: Fig. 3. PAR attenuation below the surface for ocean water types I, II and III from Jerlov (1976, Fig. 130).
digitized from Jerlov's (1976) Fig. 130 (see Table 2 and Fig. 3). Note that Rp, kp1 and kp2 are derived from PAR whereas R, k1 and k2 in Section 3 are derived from total irradiance. As expected, the attenuation coefficient values for PAR below the surface (kp2) in ocean water types I, II and III were very similar to those for total irradiance (k2) calculated by Paulson and Simpson (1977) using Jerlov's (1976) observations. In contrast, the attenuation coefficient values for PAR penetration in the surface waters (kp1) were smaller than those for total irradiance (k1), especially for oceanic water type I (Table 2). This result supports our physical understanding that PAR is attenuated more slowly in the surface waters compared to the rest of the irradiance spectrum (i.e. total irradiance e PAR). In addition, the Rp values (0.19e0.35) calculated directly from the downward PAR were slightly greater than our total-irradiance-derived R0 values (0.07e0.27) (Table 2). The larger Rp(or R0 ) and/or kp1(or k1) values produced higher attenuation rates in the downward PAR (or solar radiation) calculations, generally leading to the underestimation of PAR penetration in surface waters compared to observations. As illustrated in Table 2, for oceanic water type I, even though our formulation's R0 value (0.07) is 0.12 smaller than the Rp value (0.19), its k1 value (2.875 m1) is considerably larger than the kp1 value (0.796 m1), resulting in better agreement with the observed PAR attenuation (Fig. 1a). For oceanic water type III, our formulation's R0 value (0.27) is slightly smaller than the Rp value (0.32), and its k1 value (0.714 m1) is greater than the kp1 value (0.464 m1), showing that our formulation underestimates the downward PAR (Fig. 1b) in such turbid ocean conditions.
z ¼ b lnðI=I0 Þ þ b0 ;
(A1)
where the coefficients b and b0 can be found from least squares fitting. Rearranging Eq. (A1) after taking the natural exponent gives:
IðzÞ ¼ ð1 RÞexpðz=bÞ; I0
(A2)
where R ¼ 1 expðb0 =bÞ: The irradiance estimated from Eq. (A2) should be subtracted from the surface observation values to conduct the second fitting for the surface observation values, resulting in:
Iup ðzÞ ¼
IðzÞ ð1 RÞexpðz=bÞ: I0
(A3)
Applying the same procedure as used above for the data sets of Iup (z) calculated from Eq. (A3), each Iup(z) in the upper layers is given as:
Iup ðzÞ ¼ R expðz=aÞ:
(A4)
Substituting the resulting Iup(z) into Eq. (A3), the irradiance attenuation in the entire water column can be expressed as (Paulson and Simpson, 1977):
IðzÞ ¼ R expðz=aÞ þ ð1 RÞexpðz=bÞ: I0
(A5)
References 4. Conclusions We have produced an irradiance derived, R-modified doubleexponential formulation for calculating the downward profile of PAR below the sea surface. This new method produces improved estimates under certain oceanic conditions, without the need to change attenuation coefficients. In order to replicate Jerlov's PAR observations, we assumed temporally- and vertically-constant PAR attenuation coefficients, a condition that approximates that found in low turbidity offshore ocean waters but which is seldom encountered in the coastal ocean. In the latter areas, attenuation due to wave, tide and nearshore-circulation suspended sediments and self-shading by phytoplankton can create dynamic and complicated attenuation coefficient profiles. Tests against field data, which incorporate temporal and vertical variability in PAR attenuation, are needed as the next step in improving downward PAR formulations for biogeochemical models.
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