In a stratified water column, the nitracline is a layer where the nitrate concentration increases below the nutrient-depleted upper layer, exhibiting a strong vertical gradient in the euphotic zone. The subsurface chlorophyll maximum layer (SCML) forms near the bottom of the euphotic zone, acting as a trap to diminish the upward nutrient supply. Depth and steepness of the nitracline are important measurable parameters related to the vertical transport of nitrate into the euphotic zone. The correlation between the SCML and the nitracline has been widely reported in the literature, but the analytic solution for the relationship between them is not well established. By incorporating a piecewise function for the approximate Gaussian vertical profile of chlorophyll, we derive analytical solutions of a specified nutrient–phytoplankton model. The model is well suited to explain basic dependencies between a nitracline and an SCML. The analytical solution shows that the nitracline depth is deeper than the depth of the SCML, shoaling with an increase in the light attenuation coefficient and with a decrease in surface light intensity. The inverse proportional relationship between the light level at the nitracline depth and the maximum rate of new primary production is derived. Analytic solutions also show that a thinner SCML corresponds to a steeper nitracline. The nitracline steepness is positively related to the light attenuation coefficient but independent of surface light intensity. The derived equations of the nitracline in relation to the SCML provide further insight into the important role of the nitracline in marine pelagic ecosystems.

Nitrogen availability, especially the nitrate upward supply to the euphotic zone where light intensity is sufficient to support net photosynthesis, limits the primary productivity in a stratified water column (Falkowski et al., 1998). Specifically, the nitrate supply from below and the light attenuated from above with the depth collaboratively affect the growth of phytoplankton and lead to the subsurface chlorophyll maximum (SCM) (Riley et al., 1949; Steele and Yentsch, 1960; Herbland and Voituriez, 1979; Cullen, 1982). The SCM layer (SCML) has attracted much attention since Riley et al. (1949) because the layer contributes significantly to new primary production (NPP) in stratified waters (Probyn et al., 1995; Ross and Sharples, 2007; Fernand et al., 2013). The synergistic physical and biological interaction leads to a strong vertical nitrate gradient, conventionally referred to as the nitracline (Eppley et al., 1978; Herbland and Voituriez, 1979; Cullen and Eppley, 1981). Depth and steepness of the nitracline are important measurable parameters in regulating the supply of nitrate to the euphotic zone and hence affecting NPP (Lewis et al., 1986; Bahamón et al., 2003; Aksnes et al., 2007; Cermeno et al., 2008; Omand and Mahadevan, 2015).

The nitracline depth physically depends on the degree of water-column stratification and the magnitude of momentum transfer associated with wind stress (Denman and Gargett, 1983; Laanemets et al., 2004). It also depends on momentum transfer from below (Lipschultz et al., 2002) and, in some cases, vertical advection such as upwelling (Laanemets et al., 2004). However, in a relatively stable environment, the SCML may restrict the diffusive flux of nitrates to the euphotic zone and continually erode the nitracline supposing that sufficient light is available (Probyn et al., 1995). The SCML thereby acts as an effective nutrient trap, regulating the nitracline depth (Banse, 1982; Beckmann and Hense, 2007; Klausmeier and Litchman, 2001; Probyn et al., 1995). However, variation of the nitracline steepness, which is critical to determine the nitrate supply, was poorly understood due to a lack of high vertical resolution data; e.g., both bottle data and Argo data tend to have low vertical resolution sampling. Some studies showed that nitrogen flux is dependent more on the nitracline steepness than on the density gradients regulating turbulent diffusion (Bahamón and Cruzado, 2003; Bahamón et al., 2003; Lavigne et al., 2015). Thus, these measurable features of the nitracline and their correlation with the SCML may provide insightful information for mechanisms of the productivity in pelagic ecosystem, and the analytic solutions for these parameters may fill the knowledge gap.

Although a close relationship between the nitracline and the SCML is always observed, the quantitative nature of the nitracline in relation to the SCML formation has not been studied. The system of phytoplankton and the limiting nutrient on the vertical axis has often been utilized to study the depth, intensity and persistence of the SCML. Major theoretical results include photoacclimation (increase in chlorophyll per cell) (Steele, 1964; Fennel and Boss, 2003), bistability (Yoshiyama and Nakajima, 2002; Ryabov et al., 2010), oscillating SCM (Huisman et al., 2006), hysteresis conditions (Kiefer and Kremer, 1981; Navarro and Ruiz, 2013) and the ESS (evolutionary stable strategy) depth obtained by game-theory approach (Klausmeier and Litchman, 2001; Mellard et al., 2011). Recent mathematical studies solved the persistence and uniqueness of the steady-state solution (Du and Hsu, 2010; Hsu and Yuan, 2010; Du and Mei, 2011) and gave rigorous proofs for the abovementioned ESS depth and the game-theory approach (Du and Hsu, 2008a, b). Additionally, several modeling studies have been conducted to quantitatively assess the importance of different physical–biological processes leading to the SCML (Jamart et al., 1977, 1979; Varela et al., 1994; Klausmeier and Litchman, 2001; Hodges and Rudnick, 2004; Beckmann and Hense, 2007).

Among the studies using the nutrient–phytoplankton model, Klausmeier and Litchman (2001) first analytically derived the vertical nutrient distribution with the development of the SCML. In this model, the concentration of the limiting nutrient was found to be low and constant above the SCML and linearly increasing with depth below this layer in a poorly mixed water column. Building on that model, Mellard et al. (2011) added stratification and surface nutrient input, which can make phytoplankton grow in both the surface mixed layer and deep layer (SCML) simultaneously. Fennel and Boss (2003) determined that the sum of nutrients and phytoplankton at steady state will increase monotonically below the surface mixed layer until it equals the fixed nutrient concentration. By incorporating a generalized Gaussian function for a vertical chlorophyll profile into the nutrient–phytoplankton dynamic equation, Gong et al. (2015) found that the steady-state nitrate concentration increased from the upper community compensation depth to the SCML depth. None of the studies, however, focused on the quantitative nature of the nitracline in relation to the SCML in the stratified waters.

In this paper, we modified the nutrient–phytoplankton model by Gong et al. (2015) to study the roles of SCM in reshaping the nitracline. Two additional terms – atmospheric input, which promotes the growth of phytoplankton in the surface mixed layer, and phytoplankton self-shading, which regulates the light penetration – were introduced into the previous model. Accordingly, a piecewise function comprising a constant value within the surface mixed layer and a Gaussian function below this layer was used as a fit to the steady-state vertical chlorophyll profiles simulated by the nutrient–phytoplankton model. By incorporating the piecewise function into the nutrient–phytoplankton model, we derived the analytic solutions for the properties of the nitracline and the SCML in steady state, and the relationship between them was examined in response to light availability, surface nutrient input and vertical diffusivity.

We consider the following equations for phytoplankton and nutrient dynamics
in stratified waters (Eqs. 1–2), where light and nitrogen are two limiting
factors for phytoplankton growth (Fig. 1). The change in phytoplankton at
depth

Schematic picture of vertical profiles of nitrate gradient and
chlorophyll (Chl

Nitrogen (in units of mmol N m

Light intensity

The sinking velocity of phytoplankton

List of symbols and their values used in models at SEATS station in northern SCS.

Superscripts refer to the references that provide the source for
the parameter value, and the citations are as follows:

To describe the water-column stratification, we assume that the vertical
eddy diffusivity

The zero-flux boundary condition for the phytoplankton at the surface is
used. At the bottom boundary of the model domain (

Examples of the vertical profiles of chlorophyll (black solid line). (Red dash–dotted lines represent the parts of Gaussian fitting curves, not the actual chlorophyll.)

In many stratified water columns, the vertical distribution of chlorophyll concentration is homogeneous within the surface mixed layer and appears as a Gaussian below this layer (Fig. 2a), which is typical in open oceans (Uitz et al., 2006), shelf seas (Sharples et al., 2001), stratified estuary (Lund-Hansen, 2011) and Arctic waters (Martin et al., 2012). The nonuniform vertical profile of chlorophyll within an SCML was first modeled by a generalized Gaussian function (Lewis et al., 1983), which has subsequently been widely used with small modifications. For example, Platt et al. (1988) superimposed a constant background on the generalized Gaussian and fitted it to field data on the vertical distribution of chlorophyll from coastal, upwelling, open oceans and Arctic waters. Afterward, some studies introduced a parameter to represent the slope of the Gaussian curve (Matsumura and Shiomoto, 1993; Mu Oz Anderson et al., 2015). In particular, to account for the observed characteristic that surface values always exceed the bottom ones (Fig. 2a), the generalized Gaussian functional form has been modified with a superimposition of a background linearly or exponentially decreasing with depth (Uitz et al., 2006; Mignot et al., 2011; Ardyna et al., 2013).

For simplicity, to analytically study the role of the SCML in shaping the nitracline, we therefore propose a piecewise function comprising a constant value in the surface mixed layer and below a general Gaussian function (Eq. 6) to approximate the vertical profile of chlorophyll concentration in Fig. 2a.

The piecewise function approximation (Eq. 6) was evaluated and justified
through numerical simulation of the nutrient–phytoplankton system (Eqs. 1–2), which is solved with a semi-implicit time stepping scheme. The
vertical resolution is uniform (2 m), extending down to 200 m. We assumed a
small uniformly distributed concentration of phytoplankton (

Figure 3 shows the numerically simulated equilibrium distributions of nitrogen, light and chlorophyll. In addition, the simulated vertical profile of chlorophyll is fitted well by the piecewise function of chlorophyll using the least square method (Fig. 3). Many numerical solutions of the nutrient–phytoplankton system have reproduced the vertical chlorophyll profile with the SCML (Fennel and Boss, 2003; Huisman et al., 2006; Ryabov et al., 2010). Thus, analogous to the study by Klausmeier and Litchman (2001), we incorporate the piecewise function (Eq. 6) into the nutrient–phytoplankton system (Eqs. 1–2) at steady state to examine the roles of the SCML in reshaping the nitracline. We note that the useful delta function approximation in Klausmeier and Litchman (2001) was verified by both simulation and rigorous mathematics (Du and Hsu, 2008a, b). As presented above, the assumption of the piecewise function approximation is physically practical.

The vertical distributions of nitrate often exhibit a strong gradient in depth (the nitracline), but the features of the nitracline (depth, steepness) are variable in euphotic zones due to the combined effect of physical and biological processes.

Steady-state vertical distributions of chlorophyll, nitrate and
light determined by numerical solutions of Eqs. (1) and (2). Horizontal red line indicates the
depth of the surface mixed layer. Black dash line represents the fitting
curve of vertical Chl

Many studies define the nitracline depth as the location where the maximum
vertical gradient in nitrate concentrations occurs (Eppley et al., 1979;
Bahamón et al., 2003; Wong et al., 2007; Beckmann and Hense, 2007;
Martin et al., 2010). To measure the defined depth, a high vertical
resolution of nitrate concentrations is needed, and this has been a big technical
challenge for a long time. Thus, some definitions were also
proposed to make the depth measurable. For example, one definition is the
depth where the nitrate concentration reaches a prescribed concentration,
e.g., 0.05, 0.1, 1.0, or 12 mmol N m

With the development of nearly continuous nitrate profile measurement using
the in situ ultraviolet spectrophotometer (ISUS) optical nitrate sensor
(Johnson and Coletti, 2002; Johnson et al., 2010), the detection of the
maximum nitrate gradient could be more accurate than before. In this study,
we adopt the location of the maximum nitrate gradient (

Below the surface mixed layer, the steady-state version of Eq. (2) reduces
to

In this study the nitracline steepness is defined as the nitrate gradient at
the nitracline depth (

The nitrate profiles were obtained from the ISUS measurement at the SEATS station during the CHOICE-C (Carbon Cycling in China Seas – budget, controls and ocean acidification) 2012 summer cruise. Nine casts were conducted during 6–7 August 2012. The raw ISUS nitrate data, which employed temperature compensation, were first calibrated by the AutoAnalyzer 3 (AA3) and then smoothed to remove noise. The sampling frequency was set to 5 Hz, and the raw data were thus smoothed with a 25-point moving average in the surface mixed layer, a 5-point moving average in the SCML and a 15-point moving average below the SCML. The data were then interpolated by a cubic spline function. The corresponding temperature was obtained from conductivity–temperature–depth (CTD) measurements. Overall, nine sets of profiles are available to examine our analytical solutions.

At steady state, multiplying Eq. (1) by

To illustrate the relationship between the nitracline steepness and the
SCML, by integrating Eq. (8) from depth

Incorporating Eq. (11) into Eq. (13) leads to

By substituting the general Gaussian function for chlorophyll below the
surface mixed layer (Eq. 6) into Eq. (1), we obtain the steady-state net
growth rate of phytoplankton below the surface mixed layer:

Clearly,

Furthermore, the insertion of Eq. (3) into Eq. (17) yields another expression of
the nitracline depth:

Importantly, Eq. (18) predicts that the nitracline depth has no relation with subsurface diffusivity. Aksnes et al. (2007) also proposed a similar result, i.e., that a shoaling nitracline per se cannot be taken as an unequivocal sign of increased upwelling, as well as eddy diffusion. However, this does not mean that fluid dynamics are unimportant in shaping the vertical distribution of nitrate.

Equation (18) also indicates that both a higher recycling efficiency (

In steady state, integrating Eq. (2) from the nitracline depth

Similar to methods used by Gong et al. (2015), the piecewise function for the vertical chlorophyll profile (Eq. 6) was incorporated into the nutrient–phytoplankton model (Eqs. 1–2) at steady state to derive the three SCM characteristics (SCML thickness, its depth and intensity).

For

The expression of the SCML intensity is different from the results presented
in Gong et al. (2015). Integrating Eq. (7) from the surface of the water to the
bottom of the surface mixed layer (

Adding Eqs. (25) and (26) yields

Because recycling processes are assumed to not immediately convert dead
phytoplankton back into dissolved nutrients below the surface mixed layer,
i.e.,

Our results (Eqs. 28–29) also show that enhanced subsurface diffusivity
(

We now examine how the steady-state nitracline, in relation to the SCML, depends on light availability, especially the light level at the nitracline depth.

Substituting Eq. (17) to Eq. (28) and rearranging, we have

The nitracline depth deepens with increasing surface light intensity but
with decreasing light attenuation coefficients (

The predicted effect of surface light intensity and light attenuation coefficient on the nitracline depth (Eq. 18) implies that the nitracline depth in stratified waters may have seasonal variations. In the North Pacific Subtropical Gyre, Letelier et al. (2004) found that the nitracline depth differences between winter and summer disappeared when nitrate concentrations were plotted against light level in the water column. Aksnes et al. (2007) found that the seasonal pattern of nitracline depth was governed by seasonality in the light attenuation coefficient, rather than in surface light intensity. In particular, the inverse proportional relationship between the nitracline depth and light attenuation coefficient (Eq. 18) has also been derived from a steady-state model by Aksnes et al. (2007), which is consistent with observations in the coastal upwelling region off Southern California (Aksnes et al., 2007). Teira et al. (2005) found a significant positive correlation between the nitracline depth and the depth of 1 % surface light intensity (the proportion of reciprocal light attenuation coefficient) in the eastern North Atlantic Subtropical Gyre. Bahamón et al. (2003) showed that the nitracline depth remained relatively constant around 1 % surface light intensity depth in the western Sargasso Sea.

The nitracline steepens with a higher light attenuation coefficient
(

The inverse effects of the light attenuation coefficient on the nitracline steepness and its depth imply that the nitracline becomes steeper as it shoals. Aksnes et al. (2007) found this consistent pattern in the upwelling area off the coast of southern California.

Current evidence and modeling analyses suggest that climate warming will increase ocean stratification and hence reduce nutrient exchange between the ocean interior and the upper mixed layer (Cermeno et al., 2008; Chavez et al., 2011). Therefore, nutrient input directly to the euphotic layer due to atmospheric deposition may become a relatively more important nutrient supply mechanism to the euphotic layer (Mackey et al., 2010; Okin et al., 2011; Mellard et al., 2011). However, few model studies (e.g., Mellard et al., 2011) have explored the influence of external surface nutrient supply on vertical phytoplankton distribution.

Observations show that an interzone exists between the transition of the
surface mixed layer and the deep layer, where the nutrient gradient equals
nearly 0

Accordingly, we treat the vertical phytoplankton distribution as a piecewise function, comprised a linear function in the surface mixed layer and a Gaussian function below, which is more realistic than the general Gaussian function. The assumption of the piecewise function for phytoplankton is also consistent with the assumption of piecewise vertical diffusivity. For simplicity, we assume that the transition layer between the surface mixed layer and the deep one is infinitely thin and that the chlorophyll is continuous within the transition layer. By assuming that the SCML depth is significantly deeper than the base of the surface mixed layer, we obtain the steady-state solutions for the SCML depth and thickness, similar to the solutions using the general Gaussian function. However, the intensity of the SCML is affected by a surface nutrient supply with an associated positive increase in phytoplankton concentration.

Observations and numerical simulations showed that the SCML played a role as a nutrient trap in some regions, restricting the diffusive flux of nitrates to the surface mixed layer (Anderson, 1969; Klausmeier and Litchman, 2001; Navarro and Ruiz, 2013).

From Eq. (10), we know

According to the definition of the depth

Because the SCML acts as a nutrient barrier, it is easy to understand that
the rate of NPP in the SCML (

From the monotonicity of the quadratic function of depth

The negative gradient of nitrate below the surface mixed layer
(

The model in this study integrates a number of physical, chemical and biological processes that act together to determine the vertical distribution of phytoplankton and nitrate, under the assumption that the system is strictly vertical and in steady state. A few processes such as oxygen status, photoacclimation, luxury uptake of nutrients, phytoplankton motility, concentration-dependent herbivory and depth-dependent herbivory are not included, although they can affect the vertical distribution of phytoplankton and nitrate. Detritus, dissolved organic matter and zooplankton are not included explicitly, and all loss processes, except for sinking, are set to be linearly proportional to phytoplankton. The sinking velocity of phytoplankton is assumed to be independent of density gradients. Further, the vertical transport of nutrients is only by eddy diffusion in our model; in reality, nutrients can be supplied by many processes (turbulence, internal waves, storms, slant-wise and vertical convection), especially by upwelling (Katsumi and Hitomi, 2003; Aksnes et al., 2007).

Vertical nitrate gradient; ISUS nitrate and temperature at SEATS station (2012, cast 36) (horizontal line indicates the depth of the surface mixed layer, horizontal dashed line indicates the depth of the nitracline and the vertical dash–dotted line represents zero nitrate gradient).

In this study, the sinking velocity of phytoplankton is set to be independent of nitrate concentration. A vertically varying sinking velocity has been observed as a physiological response to variations in light or nutrient levels (Steele and Yentsch, 1960; Bienfang and Harrison, 1984; Richardson and Cullen, 1995). The sinking velocity reduces with decreased light level and with increased nutrient concentration, and the resulting divergence in sinking velocity can be large enough to affect the location of the phytoplankton particle maximum. However, numerical results given by Fennel and Boss (2003) showed analytically that the divergence of the sinking rate contributes to the location of the SBM layer in a significant way only when the divergence in sinking rates occurred above the compensation depth in stable, oligotrophic environments. They also found that in stable, oligotrophic environments with a predominance of small cells, the biomass maximum is located at the depth where growth and losses are equal, leaving little influence by sinking divergence.

It is worth pointing out that, in extreme oligotrophic regions, the SCML is
very deep and attributable mostly to photoacclimation of chlorophyll content
rather than to a peak in biomass (Steele, 1964; Fennel and Boss, 2003;
Cullen, 2015). The process of photoacclimation is also important for the
nutrient–phytoplankton system (with stratified conditions) we focused on. To
explore the influence of photoacclimation on the nitracline, we parameterized
Chl : C using the mathematical description by Cloern et al. (1995), i.e.,
Chl : C

The piecewise equation (Eq. 6) can be used to mimic a large variety of
vertical chlorophyll profiles from coastal, upwelling, open oceans and high-latitude waters (Fig. 2). For example, for

Choosing the values of model parameters representative of the system in the northern South China Sea (given in Table 1), we can retrieve the nitracline depth and steepness, the optimal depth, and the three SCM characteristics. To make calculation easy, we neglect the term of self-shading by phytoplankton in the calculation because a higher self-shading parameter has the same effect as an increasing light attenuation coefficient by water. The calculated and observed values of these parameters are listed in Table 2. All these parameters calculated are in a reasonable range, although there are some discrepancies compared with observations. In fact, this is not surprising, considering that we assume a single phytoplankton group and neglect the microbial loop and the dynamics of the dissolved organic matter and detritus pools.

We stress that the analytical solutions of the nitracline are valid only for
estimates of

Estimated results and observed values at SEATS station.

Superscripts refer to the references that
provide the source for the parameter value, and the citations are as
follows:

We have presented a theoretical framework to investigate the interaction of phytoplankton and nutrient in a stratified water column. A piecewise function for chlorophyll profiles comprising a linear function in the surface mixed layer and a Gaussian function below is assumed in the nutrient–phytoplankton model at steady state. A number of important findings are obtained under the conditions of the model equations imposed.

In steady state, the nitracline is confined to between two depths where the gross growth rate equals the recycling efficiency of dead phytoplankton, indicating that within the nitracline, nitrate consumption by phytoplankton has to be replenished by the upward flux of nitrate. This layer thereby is the major contributor to new primary production.

The nitracline depth is located below the SCML depth; both depths deepen with either increased surface light intensity or a decreased light attenuation coefficient. The nitracline depth does not depend on the value of the subsurface diffusivity. The nitracline is steeper with a thinner SCML. The nitracline steepness is positively influenced by the light attenuation coefficient, yet it responds insignificantly to surface light intensity.

Our analytical solutions show that phytoplankton in the SCML acts as an efficient nutrient trap, filtering out the upward nitrate supply. The light level at the nitracline depth has a positive relation with the depth-integrated chlorophyll concentration in the whole water column and with the maximum rate of NPP, acting as the indicator of integrated NPP. The NPP is constructed from the model equations that rely on Blackman's law of limiting factor for the growth rate.

The data used in this study are available upon request to the corresponding author.

The authors declare that they have no conflict of interest.

The authors thank the State Key Laboratory of Marine Environmental Science, Xiamen University, for providing ISUS nitrate and CTD data; we especially acknowledge C. J. Du. We are very grateful to the associate editor (Jack Middelburg) and two reviewers (A. W. Omta, and another, anonymous reviewer) for their constructive and helpful suggestions. We also would like to thank Xiaohuan Liu, Yang Yu and Xiaokun Ding for valuable advice and programming assistance. This work is funded in part by the National Key Basic Research Program of China 2014CB953700 and the National Nature Science Foundation of China (41406010, 41210008 and 91328202). Edited by: J. Middelburg Reviewed by: A. W. Omta and one anonymous referee