## 1. Introduction

The transmission of solar energy in the upper ocean, and thus ocean radiant heating normalized to the surface irradiance, is regulated by variations in chlorophyll biomass, solar zenith angle, and cloud amount (Katsaros et al. 1985; Gordon 1989; Lewis 1987; Morel and Antoine 1994; Ohlmann et al. 2000). On cool skin and warm layer depth scales (*z* < ∼3 m), solar zenith angle and clouds can be of primary importance in regulating solar transmission. On the sort of depth scales resolved by present-day climate models (near 10 m), solar transmission is primarily controlled by changes in chlorophyll biomass and covarying materials (hereafter chlorophyll). Lewis et al. (1990) indicate that observed variations in chlorophyll concentration in the equatorial Pacific can result in *O*(10 W m^{−2}) changes in heat trapped within an upper ocean layer. In situ data from the western equatorial Pacific show a chlorophyll biomass bloom to be responsible for a near 20 W m^{−2} change in the solar flux divergence over a 10-m layer (Ohlmann et al. 1998).

A variety of solar transmission parameterizations are available for modeling ocean radiant heating (e.g., Denman 1973; Paulson and Simpson 1977; Simpson and Dickey 1981; Zanaveld and Spinrad 1980; Woods et al. 1984; Morel 1988; Morel and Antoine 1994; Ohlmann and Siegel 2000). Mathematically, the parameterizations express transmission as a sum of exponential terms. In the older transmission parameterizations, coefficients and exponents are given as a function of Jerlov water type, a discrete integer index (I, IA, IB, II, or III) developed in the 1960s as a proxy for chlorophyll concentration (Jerlov 1976). Jerlov water type–based parameterizations are computationally simple, but fail to accurately represent chlorophyll concentration, a continuous quantity. Recent advances in measurement techniques allow for near–real time global observations of chlorophyll concentration [Sea-viewing Wide Field-of-view Sensor (SeaWiFS), Ocean Color and Temperature Scanner (OCTS), Moderate Resolution Imaging Spectroradiometer (MODIS)] rendering Jerlov numbers obsolete. More recent transmission parameterizations rely on the physical factors that force variations in solar transmission, most notably chlorophyll concentration (e.g., Morel 1988; Morel and Antoine 1994; Ohlmann and Siegel 2000). The newer chlorophyll-based parameterizations generally come with increased computational costs to resolve depth scales not accounted for by present-day climate models.

One-dimensional mixed layer modeling studies have long illustrated the importance of solar transmission on sea surface temperature (SST) and mixed layer depth evolution (Denman 1973; Charlock 1982; Simpson and Dickey 1981; Dickey and Simpson 1983; Martin 1985; Ohlmann et al. 1998). Solar transmission parameterizations have typically been developed for implementation in mixed layer models. Modern transmission parameterizations have fine vertical resolution and can be computationally intensive. Three-dimensional ocean general circulation models (OGCMs) do not generally include state-of-the-art transmission parameterizations. Rather, several OGCMs uniformly distribute all the available solar energy in the topmost model layer/level, or use antiquated Jerlov water type–based parameterizations. Increased model accuracy is not believed to be worth the computational overhead associated with modern chlorophyll-based solar transmission parameterizations.

The role of solar transmission variations on ocean climate evolution has only recently been addressed (Schneider et al. 1996). Schneider and Zhu (1998) investigate effects of considering radiant heating beyond the top model level (15 m) in the equatorial Pacific. Mixed layer depth increases by 15 m, SST increases by as much as 1°C, and the annual mean temperature near 50 m increases by as much as 5°C when solar energy is allowed to heat beyond the top level. The deeper mixed layer causes a decrease in the sensitivity of SST to upwelling, leading to lighter easterly winds and a reduction in zonal currents along the equator that exceeds 4 cm s^{−1}. The Schneider and Zhu (1998) modeled SST fields are much closer to observed values when solar heating is allowed beyond the top level.

Subsequent climate modeling studies addressing solar transmission variations do so by considering radiant heating beneath the top model layer. Nakamoto et al. (2001) compare results of ocean climate model runs for the equatorial Pacific using a Jerlov water type parameterization (Paulson and Simpson 1977) and a chlorophyll-dependent transmission parameterization (Morel and Antoine 1994) forced with mean Coastal Zone Color Scanner (CZCS) chlorophyll values. Mixed layer depths decrease by more than 20 m and SST values decrease by up to 2°C when the chlorophyll-dependent transmission parameterization is incorporated. Murtugudde et al. (2002) compare ocean climate model simulations for the Pacific, Atlantic, and Indian Oceans using a constant ad hoc solar transmission parameterization and a chlorophyll-based parameterization (Morel 1988) forced with mean CZCS data in effort to “recommend the simplest possible formulation for penetrative radiation [solar transmission] for the coupled ENSO prediction models without adding computational penalties.” The Murtugudde et al. (2002) study concludes that a simple single exponential formulation for solar transmission as a function of annual mean chlorophyll values will greatly improve climate model simulations and add minimal computational overhead.

The purpose of this work is to present a simple, chlorophyll-dependent, solar transmission parameterization for use in OGCMs. The parameterization is based upon empirical fits to radiative transfer model results. The parameterization takes advantage of near-real-time chlorophyll concentration values from remote sensing (SeaWiFS, OCTS, MODIS) to give transmission that is much more accurate than with Jerlov formulations, but still computationally inexpensive. The parameterization allows solar transmission to vary with chlorophyll biomass concentration as is the case in nature. A brief summary of ocean radiant heating and solar transmission is given in the next section (section 2). The atmosphere–ocean radiative transfer model system used to generate in-water solar flux profiles for the range of chlorophyll values observed in open ocean waters is briefly described in section 3. The solar transmission parameterization for use in climate models is presented in section 4. Neglect of second-order processes (sun angle and clouds) is discussed in section 5. Parameterized transmission values are validated against in situ data from the eastern and western equatorial Pacific in section 6. Conclusions are given in section 7.

## 2. Ocean radiant heating

*z,*or ocean radiant heating rate (RHR), iswhere

*E*

_{n}(0

^{−}) is the total (spectrally integrated) net flux of solar radiation just beneath the sea surface,

*E*

_{n}(

*z*) is the total net solar flux at the base of the layer (depth

*z*),

*ρ*is the density of seawater,

*c*

_{p}is the specific heat of seawater, and

*z*is positive beneath the sea surface. Values of

*E*

_{n}(0

^{−}) are parameterized from the total incident solar flux just above the sea surface [

*E*

_{d}(0

^{+})] by accounting for sea surface albedo (

*α*), defined as the fraction of the incident irradiance just above the sea surface that is reflected back to the atmosphere and does not contribute to ocean radiant heating (e.g., Payne 1972; Katsaros et al. 1985). Thus,

*E*

_{n}(0

^{−}) =

*E*

_{d}(0

^{+})(1 −

*α*). Values of

*E*

_{n}(

*z*) are parameterized with a solar transmission function [Tr(

*z*)] that can be defined in either of two ways. First, transmission can be defined after accounting for surface albedo as the fraction of the net irradiance just beneath the sea surface that exists at depth, or(e.g., Morel and Antoine 1994). Alternatively, transmission can implicitly include effects of sea surface albedo and be defined as the fraction of the surface incident irradiance that exists at depth

*z,*or(e.g., Ohlmann and Siegel 2000).

The Tr^{−}(*z*) formulation [Eq. (2a)] is the proper definition for solar transmission parameterizations used in coupled climate model systems. An explicit albedo calculation is necessary to conserve solar energy and to accurately determine *E*_{n}(0^{−}). The primary drawback of this formulation lies in parameterization validation. Parameterized values of Tr^{−}(*z*) are difficult to validate because *E*_{n}(0^{−}) is not an easily measurable quantity. In addition, values of transmission relative to *E*_{n}(0^{−}) can be less intuitive because *E*_{n}(0^{−}) is not a well-recognized quantity compared with *E*_{d}(0^{+}).

Solar transmission is discussed here relative to the incident surface flux [*E*_{d}(0^{+}); Eq. (2b)] for ease of understanding. Parameter formulation is ultimately given relative to the net solar flux just beneath the sea surface [Tr^{−}(*z*); Eq. (2a); Table 1a], and relative to the incident surface flux [Tr^{+}(*z*); Eq. (2b); Table 1b]. Parameterization error is expected to be similar or slightly less for the subsurface transmission formulation [Eq. (2a)] as parameter fits for both Tr^{+}(*z*) and Tr^{−}(*z*) are based on the same set of modeled in-water solar flux profiles, and the subsurface formulation is independent of any error introduced by the albedo calculation in the atmosphere–ocean radiative transfer model.

*λ,*defined as

*E*

_{n}(

*z,*

*λ*), can thus be computed from the spectral surface irradiance

*E*

_{d}(0

^{+},

*λ*) aswhere

*K*

_{d}(

*λ,*

*w*) and

*K*

_{d}(

*λ,*chl) represent spectral diffuse attenuation coefficients due to pure water and chlorophyll, respectively, and

*α*defines the surface albedo. The total net solar flux at depth is generally parameterized as a sum of exponential terms where the

*A*

_{i}and

*B*

_{i}parameters loosely represent the fraction of the surface irradiance within a spectral region and the corresponding attenuation value for energy in the spectral region, respectively, orThe

*A*

_{i}and

*B*

_{i}parameters take on an increased physical meaning as the number of exponential terms increase (e.g., Lewis et al. 1990; Ohlmann et al. 1996). In one and two exponential term models, the

*A*

_{i}and

*B*

_{i}values are little more than empirically fit parameters.

## 3. Radiative transfer model

A radiative transfer model system is used to create a set of simulated in-water solar flux profiles for a predefined set of chlorophyll concentration, solar zenith angle, and cloud amount combinations. In-water solar transmission profiles are then computed for each of the modeled flux profiles (hereafter modeled profiles). Parameters for the double exponential in-water transmission parameterization developed here are determined by fitting curves to each of the modeled profiles, giving double exponential parameters as a function of chlorophyll concentration. Variations in solar zenith angle and cloud amount are included in the set of simulations to quantify errors associated with neglect of these dependent variables in the parameterization.

The HYDROLIGHT in-water radiative transfer model provides an exact solution to the radiance transfer equation for a plane parallel marine environment using invariant imbedding techniques (Mobley 1989, 1994; Mobley et al. 1993). The version of HYDROLIGHT used here resolves the entire solar spectrum, from 250 to 2500 nm, and is thus appropriate for radiant heating applications (Ohlmann et al. 2000; the standard HYDROLIGHT model resolves only the visible wavebands, 350 to 700 nm, sufficient for bio-optical studies). The modeled profiles are computed with 10-nm spectral resolution from 250 to 700 nm, and 25-nm spectral resolution from 700 to 900 nm. Wavelengths beyond 900 nm are ignored as *e*-folding depths for these near-infrared wavebands are much less than 1 m (Hale and Querry 1973; Smith and Baker 1981; Pope and Fry 1997) and this study is interested in solar transmission beyond the top 2 m (more than 99.9% of the solar energy beyond 1 m lies within the 250–900-nm spectral range for clear ocean waters). In-water solar flux values are computed every 2 m from the surface to 50 m, and every 5 m from 50 to 80 m. Solar flux values beyond 80 m are sufficiently small that even extreme variations in transmission result in negligible changes in terms of absolute energy values. The spectral resolution and depth values used in the HYDROLIGHT runs are chosen arbitrarily to maintain model accuracy while keeping computation times reasonable.

The surface incident spectral radiance distribution, sea surface geometry, and upper ocean optical properties are required as input to HYDROLIGHT. Spectral radiance distributions as a function of depth are then computed. Integration over wavelength and solid angle gives net irradiance as a function of depth [*E*_{n}(*z*)]. Optical properties required by HYDROLIGHT are pure water scattering, pure water absorption, particulate scattering, and particulate absorption. Pure water values are from Hale and Querry (1973) and Pope and Fry (1997) studies. The particulate values, functions of chlorophyll concentration, are from Gordon and Morel (1983) and Morel (1991). A Petzold (1972) volume scattering function is used. Sea surface geometry is computed from the Cox and Munk (1954) wave slope statistics (cf. Preisendorfer and Mobley 1986). A more detailed discussion of inputs to the full spectral version of HYDROLIGHT is given in Ohlmann et al. (2000).

The surface incident radiance distribution used as input to the HYDROLIGHT in-water model comes from the Santa Barbara Discrete Ordinate (DISORT) Atmospheric Radiative Transfer Model (SBDART; Ricchiazzi et al. 1998) run with the spectral resolution indicated earlier. SBDART uses a discrete ordinate method (Stamnes et al. 1988) to solve the radiative transfer equation in the atmosphere. Atmospheric optical properties are from a Low Resolution Transmittance (LOWTRAN) molecular absorption model (Pierluissi and Maragoudakis 1986), Rayleigh and Mie scattering codes, and an aerosol model. Clouds are quantified through a cloud index defined as one minus the ratio of downwelling irradiance to downwelling clear-sky irradiance (Siegel et al. 1999; Ohlmann et al. 2000). The cloud index corresponds to the cloud forced reduction in surface insolation relative to the clear-sky value. Surface incident radiance distributions are computed for solar zenith angels (*θ*) of 0, 15, 30, 45, 60, and 75°, and cloud indices (ci) of 0 (clear sky), 0.2 (a 20% reduction in surface insolation by clouds), 0.4, 0.6, and 0.9.

HYDROLIGHT generated in-water solar flux profiles are computed for all *θ* and ci combinations with ocean chlorophyll values of 0.03, 0.3, and 3.00 mg m^{−3}. Simulations for 19 additional chlorophyll values ranging from 0.01 to 3.00 mg m^{−3} (0.01 and 0.02, every 0.05 mg m^{−3} from 0.05 to 0.50 mg m^{−3}, every 0.10 mg m^{−3} from 0.50 to 1.00 mg m^{−3}, and every 0.50 mg m^{−3} from 1.00 to 3.00 mg m^{−3}) are run with *θ* = 15° and ci = 0.4 only, as detailed later. Modeled transmission profiles are computed by applying Eq. (2) to the modeled in-water solar flux profiles (Fig. 1). Modeled transmission extends from ∼0.05 to 0.31 at 10 m, the depth of a top level or layer within a typical OGCM, due to varying chlorophyll concentration. Such a 0.26 change in transmission is equivalent to a 52 W m^{−2} change in absolute solar flux (assuming a 200 W m^{−2} incident solar flux, typical of equatorial regions). The parameterization being developed here is based on the transmission profiles shown in Fig. 1, and resolves the illustrated variability.

## 4. Solar transmission parameterization for climate models

*A*

_{i}and

*B*

_{i}values are chlorophyll-dependent parameters determined empirically from curve fits to the radiative transfer model generated profiles. Equation (5) follows from substituting the two-term form of Eq. (4) into Eq. (2) (e.g., Paulson and Simpson 1977). Two exponential terms are necessary and sufficient for accurately parameterizing solar fluxes beyond 2 m. Only the first exponential term need be evaluated for quantifying solar transmission at depths greater than 8 m (the second term becomes negligible). Chlorophyll concentration, necessary for getting at the

*A*

_{i}and

*B*

_{i}parameters in practice, is available globally from remotely sensed ocean color data.

Identifying the influence of *θ* and ci on solar transmission for specific chlorophyll cases is the first step in parameterization development. Modeled transmission profiles for low (0.03 mg m^{−3}), medium (0.30 mg m^{−3}), and high (3.00 mg m^{−3}) chlorophyll concentration cases and all combinations of *θ* and ci values considered are illustrated in Fig. 2. Transmission [relative to *E*_{d}(0^{+})] variations due to *θ* and ci, for each of the low-, medium-, and high-chlorophyll cases, are less than 0.107 beneath 2 m, and less than 0.075 below 10 m. The *θ* and ci induced variations in transmission are significantly less than variations due to chlorophyll concentration changes over the range of values observed in oligotrophic, open ocean, waters.

The three sets of modeled profiles shown in Fig. 2 are used to determine three mean profiles for variations about *θ* and ci. Root-mean-square (rms) error between the mean profile for each chlorophyll concentration set and each of the individual profiles within the set are then computed to determine the *θ* and ci values that give transmission closest to the mean profile for each of the three chlorophyll concentration cases. Profiles computed with *θ* = 15° and ci = 0.4 minimize the error with the mean profile for the low and medium chlorophyll concentration cases. Transmission computed with *θ* = 30° and ci = 0.4 minimizes error with the mean profile for the high-chlorophyll case; however, rms error for the *θ* = 15° and ci = 0.4 profile is essentially the same. The rms transmission errors between mean and modeled profiles with *θ* = 15°, ci = 0.4 are less than 5 × 10^{−4} for each of the three chlorophyll cases. Extension of this analysis for additional chlorophyll concentration cases requires additional computing time for radiative transfer calculations, and is unlikely to give significantly different results.

The required *A*_{i} and *B*_{i} in-water solar transmission parameters are determined from profiles modeled with *θ* = 15° and ci = 0.4 for varying chlorophyll concentrations (0.01–3.0 mg m^{−3}; Fig. 1). This is done by first fitting single exponential curves to the modeled profiles over the 8–80-m depth range using a least squares method. Above 8 m, energy within the quickly attenuating ultraviolet (UV) and red wavebands is substantial enough to impact the goodness of log-linear fits for low-chlorophyll cases. Solar transmission values beneath 80 m are sufficiently small (<0.05 in the clearest waters) that even large relative errors result in only small absolute flux errors [*O*(1 W m^{−2})]. Transmission values from the single exponential fits are then subtracted from their respective modeled profiles at depths less than 8 m. A second exponential least squares fit is then made to the transmission differences over the 2–8-m depth range. The exponential fits ultimately give the *A*_{i} and *B*_{i} parameters in Eq. (5) as a function of chlorophyll concentration. Parameters are computed for solar transmission defined in terms of both the net irradiance just beneath the sea surface (Table 1a), and the incident surface irradiance (Table 1b).

The method used for computing double exponential fits gives a parameterization that allows solar transmission at depths of 8 m and beyond to be accurately determined by evaluating only the first exponential term. Both exponential terms must be evaluated to resolve the solar transmission at depths from 8 to 2 m. The parameterization is not valid for depths shallower than 2 m. Since parameters are from curve fits beneath 2 m, the sum *A*_{1} + *A*_{2} is not expected to equal 1 [Eq. (2a)] or 1 − *α* [Eq. (2b)]. The parameterization keeps computational overhead to a minimum when used in models with present-day (∼10 m) vertical resolution and remains optimal for future use as long as top model layer (or level) depths are greater than 2 m. The double exponential fits to each of the modeled transmission profiles result in coefficient of variation (*r*^{2}) values that are always greater than 0.998, absolute errors that are always less than 8.0 × 10^{−3}, and rms errors that are always less than 4.5 × 10^{−3}. The *r*^{2} values decrease slightly, and the greatest error increases more than 50%, when the first exponential is fit up to 6 m.

Values of the double exponential fit parameters as a function of chlorophyll concentration for solar transmission normalized to the incident surface irradiance are illustrated in Fig. 3. The *A*_{1} parameter shows a logarithmic increase with chlorophyll for low-to-moderate chlorophyll values, and is fairly constant for large chlorophyll values (>∼1.5 mg m^{−3}). The *A*_{2} parameter shows the reverse, a logarithmic decrease with chlorophyll for low-to-moderate chlorophyll values, and fairly constant values for large chlorophyll concentrations (>∼1.5 mg m^{−3}). The *B*_{1} parameter increases with chlorophyll concentration as a square root function. The *B*_{2} parameter increases logarithmically with chlorophyll concentration to near 1.5 mg m^{−3}, and shows a near-linear increase with chlorophyll beyond this.

*A*

_{1},

*A*

_{2}, and

*B*

_{2}parameters. A square root function is fit to the

*B*

_{1}parameter. The nonlinear fits are performed in a least squares sense using a gradient-expansion algorithm (Bevington 1969). Equations for the

*A*

_{i}and

*B*

_{i}parameters as a function of chlorophyll concentration (mg m

^{−3}) that minimize the errors between parameterized and modeled profiles are(Fig. 3). These equations are from fits over the 0.01–2.5 mg m

^{−3}chlorophyll range for the

*A*

_{1},

*B*

_{1}, and

*B*

_{2}parameters, and over the entire 0.01–3.0 mg m

^{−3}chlorophyll range for the

*A*

_{2}parameter. The 0.01 and 0.02 mg m

^{−3}chlorophyll values are likely smaller than the minimum values observed in nature, but are included for curve-fitting purposes so that the more realistic chlorophyll minimum, 0.03 mg m

^{−3}is not an end-member. The curves shown in Fig. 3 are not exactly reproducible with Eqs. (6a)–(6d) due to round-off error. The exact formulations can be obtained by fitting curves of the given forms to the Table 1b (or 1a) data.

Transmission errors introduced by curve fitting alone [differences between modeled and parameterized profiles relative to *E*_{d}(0^{+})] for the 0.03–3.0 mg m^{−3} chlorophyll values are everywhere less than 0.015 (Fig. 4). Total rms error for each profile is always less than 0.003. A transmission error of 0.01 corresponds to an absolute solar flux error of ∼2 W m^{−2}, based on a characteristic equatorial climatological surface irradiance value of 200 W m^{−2}. Curve-fitting errors between modeled and parameterized transmission relative to *E*_{n}(0^{−}) are similar.

## 5. Sun angle and cloud influences

*F*(

*θ*)

*G*(ci)] so that

*z,*

*θ,*

*z,*

*F*

*θ*

*G*

*z,*chl) is from Eq. (5), and

*F*and

*G*are simple functions of

*θ*and ci, respectively. The mean factor formulation is selected to account for changes in transmission due to variations in

*θ*(largely through surface albedo) that follow variations in clouds (diffuse light). The mean factor provides a computationally simple way of considering

*θ*and ci dependencies in a scaling value applied to solar transmission parameterized in terms of chlorophyll concentration only.

Mean factors are determined with radiative transfer model generated transmission profiles for the case of *θ* = 15°, ci = 0.4 [Tr(*z,* chl) in Eq. (7)] and the varying *θ* and ci cases for low (0.03), medium (0.30), and high (3.00 mg m^{−3}) chlorophyll values. The mean factor as a function of *θ* and *z,* for varying cloud indices, is illustrated in Fig. 5. The mean factor as a function of ci and *z,* for varying *θ* cases is illustrated in Fig. 6. Mean factors are shown for the low-chlorophyll case only. Values for the medium- and high-chlorophyll cases are within a few percent of those for the low-chlorophyll case, and display the same general pattern.

*θ*and ci are computed with a depth-weighted least squares scheme to determine

*F*and

*G*functions that can be combined for mean factor estimates. The weighting scheme, whereby weights decrease exponentially with depth, mimics the exponential decay of solar energy. Mean factors change by less than ∼10% over the range of

*θ*values for all but the clearest sky cases (ci = 0, 0.2; Fig. 5), and over the range of ci values for all but the highest

*θ*cases (

*θ*= 60, 75°; Fig. 6). For the clear-sky case, the mean factor ranges from ∼1.1 for the low-sun-angle case, to less than 0.8 for the high-sun-angle case. For a high solar zenith angle (75°), the mean factor ranges from ∼0.8 for a clear sky, to ∼1.1 for a very cloudy sky. The modified transmission parameterization [Eqs. (7) and (8)] allows for the large (>∼10%) changes in transmission forced by

*θ*and ci variations that occur mostly for the clear-sky case to be resolved. The mean factor is thus modeled as a linear function of

*θ*for clear-sky periods (Fig. 5; ci = 0) givingwhere ci = 0.1 is an arbitrarily determined “clear-sky” threshold (e.g., Ohlmann and Siegel 2000).

Errors between transmission from the radiative transfer model results and determined with this modified parameterization are shown in Figs. 7a–c. Error profiles for the simple chlorophyll-only scheme are shown in Figs. 7d–f, for comparison. For the chl = 0.03 mg m^{−3} case, the modified parameterization serves to reduce the transmission error during low cloud, large solar zenith angle cases by nearly 35% (Fig. 7a versus 7d). Total rms error decreases from 0.013 for the chlorophyll-based parameterization to 0.009 for the modified parameterization. The chl = 0.30 mg m^{−3} case shows a similar decrease in rms error (0.007–0.004) for the low cloud, high solar zenith angle cases (Fig. 7b versus 7e). The addition of solar zenith angle and cloud dependencies have little impact for the chl = 3.00 mg m^{−3} case as absolute solar flux values are small (Fig. 7e versus 7f). The large biomass concentration gives such rapid solar attenuation that errors are differences in very small transmission values. It is reiterated that the *θ* and ci enhancement to the chlorophyll-dependent parameterization is presented to estimate errors associated with neglect of second-order processes, and to provide a way to account for the errors when they are large. Sun angle and cloud effects are likely of little importance in regulating solar transmission on scales considered by climate model studies.

## 6. Model validation

The parameterization is validated using in situ solar flux, and in situ and remotely sensed chlorophyll concentration data, from the eastern and western equatorial Pacific. Data from the eastern Pacific have been collected aboard the R/V *Ron Brown* from 12 September to 4 October 2001 as part of the Eastern Pacific Investigation of Climate Processes in the Coupled Ocean–Atmosphere System (EPIC) project. In-water solar flux values are measured with a Satlantic SeaWiFS Profiling Multichannel Radiometer (SPMR) that records downwelling spectral irradiance in 11 narrow (nominally 10-nm bandwidth) wavebands from the UV (325 nm) to the red (683 nm), and upwelling radiance in the same spectral bands. Calibration values are traceable to National Institute of Standards and Technology (NIST) standards. The SPMR returns 6-Hz data during each free-fall descent to 100-m depth (descent rate = ∼1 m s^{−1}). Surface incident irradiance is sampled every 10 s (averaged to 1 min) with a pair of Eppley PSP pyranometers mounted atop a tall scaffold located on the ship's bow. Chlorophyll *a* concentration is measured at 1, 20, 40, 60, 80, 100, 120, and 150 m with water samples and Turner fluorometry.

The EPIC cruise track is comprised of 19 days on station at 10°N, 95°W, followed by a southward transect to 1.0°S, along 95°W. Roughly 8 SPMR casts were performed each day during daylight hours while at 10°N, 95°W. Water samples were collected once daily near local noon. Pyranometer data were recorded continuously with a data-logging problem during high rain rates. In-water measurement activities were suspended on 23 September due to inclement weather. During the southward transect, the ship stopped at every 0.5° of latitude for water and SPMR (during daylight stations) sampling. A total of 143 in-water solar flux profiles were collected on station (10°N, 95°W) and 8 were recorded during the southward transect.

Solar transmission is computed with the EPIC data by averaging in-water spectral irradiance data for each cast over 1-m-depth bins, integrating depth-averaged spectral irradiance measurements over the 325–683-nm range, and normalizing by the average of the two pyranometer values closest in time to the beginning of the optics cast. Transmission profiles are only computed if a pyranometer record exists within 1 min of the start of an optics cast. Resolving the 325–683-nm spectral range gives accurate transmission values for depths beneath ∼10 m, as solar energy beyond these wavebands is almost completely attenuated within the top few meters of the water column. The complete set of solar transmission profiles collected at 10°N, 95°W is shown in Fig. 8. A mean transmission profile for EPIC at 10°N, 95°W is determined by fitting a single exponential profile to the 1-m depth-averaged transmission values, over the 10–40-m depth range [Fig. 8; Tr = 0.525 exp(−*z* × 0.076)]. Chlorophyll concentration is fairly homogenous and absolute flux values are significant above ∼40 m. The mean in situ profile is representative of the spatial (depth) and temporal scales that the transmission parameterization developed here is designed to resolve.

Parameterized transmission profiles are determined using the first exponential term in Eq. (5), with *A*_{1} and *B*_{1} parameters from Eqs. (6a) and (6c), respectively, and in situ and remotely sensed (SeaWiFS) chlorophyll values. The mean in situ chlorophyll value for 10°N, 95°W (0.28 mg m^{−3}) is computed by averaging values from the surface to 40 m over the 19-day sampling period. The mean remotely sensed chlorophyll value (0.26 mg m^{−3}) is determined by averaging the September 2001, SeaWiFS, level 3, 9-km value for the pixel that includes 10°N, 95°W (0.34 mg m^{−3}) and the four adjoining pixels. Transmission profiles parameterized with in situ and remotely sensed chlorophyll concentrations are Tr = 0.484 exp(−*z* × 0.071) and Tr = 0.482 exp(−*z* × 0.069), respectively (Fig. 8). The parameterized profiles are in excellent agreement with the mean in situ profile. The largest difference between parameterized transmission (considering both in situ and remotely sensed chlorophyll concentration) and the mean measured profile is <0.01.

The region encompassing 10°N, 95°W is characterized as Jerlov type II water (Jerlov 1976). The transmission profile parameterized with the Paulson and Simpson (1977) single exponential for Jerlov type II water gives transmission differences from the mean measured profile that exceed 0.13 at 10 m and 0.06 at 20 m (Fig. 8). Transmission differences correspond to absolute solar flux discrepancies of more than 26 and 12 W m^{−2} at 10 and 20 m, respectively (for a climatological incident surface irradiance of 200 W m^{−2}). For comparison, the largest difference between measured mean transmission and transmission from the chlorophyll-based parameterization is less than 2 W m^{−2}. A 15 W m^{−2} discrepancy over a 10-m layer corresponds to nearly a 1.0 C month^{−1} temperature change for the layer (excluding feedbacks).

The western equatorial Pacific is characterized by low chlorophyll concentrations and large solar transmission values when compared with the eastern equatorial Pacific. Solar transmission and chlorophyll concentration data from the western Pacific have been collected aboard the R/V *John Vickers* from 21 December 1992 through 19 January 1993, as part of the Tropical Ocean Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE). In-water solar fluxes, incident surface irradiance, and chlorophyll concentration are measured in a similar manner to that described earlier, using a Biospherical Instruments MER-2040 radiometer package. The MER-2040 measures downwelling and upwelling spectral irradiance in 13 wavebands between 340 and 683 nm, and is traceable to NIST standards. In-water solar flux values are determined from net irradiance, the difference between downwelling and upwelling irradiance. More than 700 optical profiles were made during daylight as part of TOGA COARE. The profiles were all recorded within 5 km of 2.08°S, 156.25°E. A thorough description of the TOGA COARE optical data is provided in Siegel et al. (1995) and Ohlmann et al. (1998).

Daily averaged solar transmission profiles computed from the complete 1-m binned TOGA COARE dataset are illustrated in Fig. 9. The data show two transmission regimes caused by a large chlorophyll bloom near the middle of the sampling period (e.g., Siegel et al. 1995; Ohlmann et al. 1998). The high-transmission period (pre–chlorophyll bloom) is characterized by a deep chlorophyll maximum between ∼70 and 90 m, and a corresponding change in slope of the transmission curves near 70 m. Transmission curves are straighter during the low-transmission period (post–chlorophyll bloom). A mean observed transmission profile is determined by fitting a single exponential curve to the 1-m daily average values over the 10–40-m depth range for the entire 30-day sampling period [Fig. 9; Tr = 0.411 exp(−*z* × 0.047)]. This mean TOGA COARE observed profile is extremely close to the mean profile reported by Ohlmann et al. (1998), fit over an expanded depth range.

A parameterized transmission profile is computed using the single exponential form of the model described here with the mean in situ chlorophyll value for the top 40 m during the TOGA COARE optical sampling period (0.13 mg m^{−3}). The parameterized transmission profile [Tr = 0.459 exp(−*z* × 0.058)] underestimates the mean measured profile at almost all depths (Fig. 9). The largest transmission underestimate is 0.015 at 31 m; the rms error between the two profiles over the 10–80-m depth range is 0.009. The largest transmission difference corresponds to only a 3 W m^{−2} discrepancy, assuming a 200 W m^{−2} climatological surface flux. Such a discrepancy is well within the limits of both measurement and radiative transfer model error.

The region encompassing 2°S, 156°E is characterized as Jerlov type 1B water (Jerlov 1976). Transmission from Paulson and Simpson's (1977) single exponential parameterization for Jerlov type 1B water underestimates mean observed TOGA COARE values by more than 0.05 between 10 and 25 m (Fig. 9). The 0.07 transmission difference at 10 m corresponds to a 14 W m^{−2} error, much larger than the 3 W m^{−2} discrepancy that occurs with the chlorophyll-based parameterization presented here.

Addressing the two TOGA COARE transmission regimes separately allows for model validation over a slightly expanded range of chlorophyll values. Mean observed chlorophyll concentrations for the pre- and post-bloom periods are 0.09 and 0.16 mg m^{−3}, respectively. Mean transmission profiles from observations are Tr = 0.369 exp(−*z* × 0.037) and Tr = 0.428 exp(−*z* × 0.053). The parameterized profile for the pre-bloom period underestimates the observed mean profile at all depths beneath 15 m. The largest difference for the pre-bloom period is 0.022 at 38 m. The rms error between the two profiles for the pre-bloom period over the 10–80-m depth range is 0.015. For comparison, the maximum difference between mean observed transmission and transmission for Jerlov type 1B water for the pre-bloom period is 0.074 (Jerlov transmission underestimates at all depths), the rms error is 0.045. For the post-bloom period, parameterized transmission underestimates observed mean values at all depths. The largest difference is 0.016 at 26 m; the rms error is 0.009. The maximum difference between mean observed and Jerlov transmission for the post-bloom period is 0.068; the rms error is 0.025. In both cases, the Jerlov water type–based transmission parameterization gives significantly larger errors than the chlorophyll-based parameterization when compared with in situ transmission.

For further validation, parameterized transmission profiles are compared to individual profiles measured at various latitudes along the 95°W transect made during EPIC (Fig. 10). The individual transect profiles contain diurnal signals that are unresolved in the transmission parameterization. However, the set of transect profiles spans a broad range of depth-averaged chlorophyll values (0.14–0.73 mg m^{−3}), and is a validation alternative in the absence of additional data. Transmission from the chlorophyll-based parameterization is in good agreement with measured transmission values. For the 8°N comparison (chl = 0.49 mg m^{−3}), the largest difference between measured and parameterized transmission values is 0.015 at 18 m. The rms error over the 10–80-m depth range is <0.01. The measured transmission profile at 8°N changes slope dramatically near 45 m due to a deep chlorophyll maximum between ∼40 and 60 m. The transmission difference beneath 40 m is <3 × 10^{−3} even though the parameterized profile fails to resolve the deep chlorophyll maximum. The small transmission difference is due to the small transmission values (<∼0.01) that exist beneath 40 m.

Comparisons for the low-to-moderate chlorophyll cases at 5, 4.5, and 2°N (chl = 0.17, 0.14, and 0.27 mg m^{−3}) give similar results to comparisons for the mean EPIC and mean TOGA COARE cases. Individual profile comparisons show transmission differences that are everywhere <0.045, with rms errors ranging from 0.008 (4.5°N) to 0.015 (5°N). No obvious bias is evident with parameterized transmission values mostly underestimating observed values in one case (5°N), and mostly overestimating in the other two cases (4.5° and 2°N). Comparisons with in situ transmission for the largest chlorophyll concentration case (0.73 mg m^{−3}; at the equator) show the smallest differences of all “transect” cases. The rms errors are <0.006 for both individual profile comparisons.

Differences between the measured and parameterized profiles discussed earlier are summarized in Fig. 11. The largest difference is 0.045 at 10 m, corresponding to a 9 W m^{−2} discrepancy (assuming a 200 W m^{−2} climatological surface flux). The rms error for the individual measured-parameterized difference profiles range from 0.004 to 0.015. For comparison, the largest difference between measured and Jerlov water type–based transmission profiles exceeds 0.16, corresponding to an absolute solar flux discrepancy of more than 32 W m^{−2}. In water, solar flux errors of order 10 W m^{−2} can have a marked impact on upper ocean evolution, subsequent heat exchanged with the atmosphere, and atmospheric evolution, if integrated over significant periods of time. The rms errors from comparisons between measured and Jerlov water type–based parameterized transmission profiles are a factor of 2 to 10 greater than when using the chlorophyll-based transmission parameterization presented here.

## 7. Conclusions

An improved solar transmission parameterization for use in climate studies is presented. The parameterization is based on empirical fits to in-water solar flux profiles generated with an atmosphere–ocean radiative transfer model system. The double exponential model is as computationally simple as possible, while still adequately resolving the characteristic depth scales in climate system models. The two-term model accurately resolves solar transmission beyond ∼2 m. Only the term with the largest *e*-folding depth need be evaluated for accurate determination of solar transmission beneath 8 m. Model parameters depend solely on upper ocean chlorophyll concentration, the physical quantity primarily responsible for regulating transmission on climatological scales. Remotely sensed chlorophyll concentration data are presently available and should continue to be collected well into the future. Model parameters are given for solar transmission computed relative to the incident surface irradiance, and relative to the net irradiance just beneath the surface. The latter quantity is computed from the surface incident value and sea surface albedo.

Proper chlorophyll concentration scales for driving the solar transmission parameterization in climate simulations are not yet known. Chlorophyll concentration is not generally a prognostic quantity in climate models, making climatologies necessary. Selecting a single climatological chlorophyll concentration value to represent the entire ocean is the crudest approach. Spatially varying annual, seasonal, or monthly climatologies are likely more reasonable options. Simulations that address the sensitivity of upper ocean evolution to chlorophyll representation must be performed to determine the impact of various scales of solar transmission variability. Chlorophyll concentration may eventually become a prognostic term in climate models. The possibility of forecasting chlorophyll concentration (to first order) using simple ocean physics is beginning to be explored by Ohlmann and colleagues.

Comparisons with in situ solar transmission profiles indicate that rms errors decrease by between 50% and 90% when the chlorophyll-based parameterization is used, compared to a common Jerlov water type–based parameterization. Maximum differences between solar transmission from the double exponential chlorophyll-based parameterization and in situ transmission are less than 8 W m^{−2}. Differences between solar transmission from a commonly used Jerlov water type–based parameterization (Paulson and Simpson 1977) and in situ transmission and can exceed 25 W m^{−2} (both cases assume a 200 W m^{−2} surface flux). The double exponential, chlorophyll-based solar transmission parameterization is easily incorporated into existing ocean models. The parameterization enables solar transmission to be discussed in terms of chlorophyll concentration, the measurable quantity on which it depends, rather than Jerlov water type or some attenuation parameter.

The need for an improved solar transmission parameterization specific to climate-scale applications became apparent at the 2001 Surfside Climate Workshop, held at Scripps Institution of Oceanography. Bill Large and Peter Gent provided information regarding the workings and requirements of climate models that helped shape the parameterization. Curt Mobley allowed the use of the HYDROLIGHT model. Catherine Gautier supplied the SBDART model. Chris Fairall, Jeff Hare, and Frank Bradley kindly shared pyranometer data recorded during EPIC. Discussions with Dave Siegel were helpful, as always. Support is from the National Science Foundation (OCE-0002902).

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Table 1a. The *A*_{1}, *B*_{1}, *A*_{2}, and *B*_{2} values for solar transmission normalized to the net irradiance just beneath the sea surface [see Eqs. (2a) and (5)]

Table 1b. The *A*_{1}, *B*_{1}, *A*_{2}, and *B*_{2} values for solar transmission normalized to the total incident irradiance just above the sea surface [see Eqs. (2b) and (5)]