GMDGeoscientific Model DevelopmentGMDGeosci. Model Dev.1991-9603Copernicus GmbHGöttingen, Germany10.5194/gmd-8-2231-2015EMPOWER-1.0: an Efficient Model of Planktonic ecOsystems WrittEn in RAndersonT. R.tra@noc.ac.ukGentlemanW. C.YoolA.https://orcid.org/0000-0002-9879-2776National Oceanography Centre, University of Southampton,
Waterfront Campus, European Way, Southampton SO14 3ZH, UKDepartment of Engineering Mathematics, Dalhousie
University, 5269 Morris St., Halifax, Nova Scotia, B3H 4R2,
CanadaT. R. Anderson (tra@noc.ac.uk)24July2015872231226210November20145January20156May20153July2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://gmd.copernicus.org/articles/8/2231/2015/gmd-8-2231-2015.htmlThe full text article is available as a PDF file from https://gmd.copernicus.org/articles/8/2231/2015/gmd-8-2231-2015.pdf
Modelling marine ecosystems requires insight and judgement when it comes to
deciding upon appropriate model structure, equations and parameterisation.
Many processes are relatively poorly understood and tough decisions must be
made as to how to mathematically simplify the real world. Here, we present
an efficient plankton modelling testbed, EMPOWER-1.0 (Efficient Model of Planktonic ecOsystems WrittEn in R), coded in the freely
available language R. The testbed uses simple two-layer “slab” physics
whereby a seasonally varying mixed layer which contains the planktonic
marine ecosystem is positioned above a deep layer that contains only
nutrient. As such, EMPOWER-1.0 provides a readily available and easy to use
tool for evaluating model structure, formulations and parameterisation. The
code is transparent and modular such that modifications and changes to model
formulation are easily implemented allowing users to investigate and
familiarise themselves with the inner workings of their models. It can be
used either for preliminary model testing to set the stage for further work,
e.g. coupling the ecosystem model to 1-D or 3-D physics, or for undertaking
front line research in its own right. EMPOWER-1.0 also serves as an ideal
teaching tool. In order to demonstrate the utility of EMPOWER-1.0, we
implemented a simple nutrient–phytoplankton–zooplankton–detritus (NPZD)
ecosystem model and carried out both a parameter tuning exercise and
structural sensitivity analysis. Parameter tuning was demonstrated for four
contrasting ocean sites, focusing on station BIOTRANS in the North Atlantic
(47∘ N, 20∘ W), highlighting both the utility
of undertaking a planned sensitivity analysis for this purpose, yet also the
subjectivity which nevertheless surrounds the choice of which parameters to
tune. Structural sensitivity tests were then performed comparing different
equations for calculating daily depth-integrated photosynthesis, as well as
mortality terms for both phytoplankton and zooplankton. Regarding the
calculation of daily photosynthesis, for example, results indicated that the
model was relatively insensitive to the choice of photosynthesis–irradiance
curve, but markedly sensitive to the method of calculating light attenuation
in the water column. The work highlights the utility of EMPOWER-1.0 as a
means of comprehending, diagnosing and formulating equations for the
dynamics of marine ecosystems.
Introduction
Ecosystem models are ubiquitous in marine science today; they are used to study a
range of compelling topics including ocean biogeochemistry and its response
to changing climate, end-to-end links from physics to fish and associated
trophic cascades, the impact of pollution on the formation of harmful algal
blooms, etc. (e.g. Steele, 2012; Gilbert et al., 2014; Holt et al., 2014;
Kwiatkowski et al., 2014). Models have become progressively elaborated in
recent years, a consequence of both superior computing power and an
expanding knowledge base from field studies and laboratory experiments. All
manner of models have appeared in the published literature varying in terms
of structure, equations and parameterisation. Anderson et al. (2014), for
example, commented on the “enormous” diversity seen in chosen formulations
for dissolved organic matter (DOM) in the current generation of marine
ecosystem models and asked whether reliable simulations can be expected
given this diversity. This question applies not just to modelling DOM, but
also to most processes and components considered in modern marine ecosystem
modelling (Fulton et al., 2003a; Anderson et al., 2010, 2013).
A certain amount of variability among models is to be expected because of
differing objectives among modelling studies. A distinction can, for
example, be made between models designed primarily for improving
understanding of system dynamics, as opposed to those for out-and-out
prediction (Anderson, 2010). Ultimately, however, much of the variability
seen in model structure and equations is an outcome of personal choice on
the part of the practitioner. Indeed, the art of modelling is in making
decisions regarding model structure, parameters, design of simulations,
types of output analysis, etc. The underlying root of this diversity and
seeming subjectivity is that, despite a wealth of available data, many
processes in marine ecosystems are not easy to characterise mathematically.
Modellers therefore need to consider how this uncertainty affects their
results and use it to inform how best to construct and parameterise their
models for chosen applications. Sensitivity analysis and model validation
are the obvious means to address model uncertainty, as well as model
intercomparison studies. There is however an additional problem, namely that
ocean biology is inextricably linked to physics and both incur modelling
error. An appropriate physical framework must be selected that adequately
represents mixing, advection and the seasonal changes in the depth of the
upper mixed layer. Understandably, 1- or 3-D physical frameworks
are the usual choice, given the realism thus provided. But this increased
dimensionality (or spatial resolution) comes at a price. They require
expertise and time to set up, sufficient computational resources for running
and storage of output and, last but not least, analysis of the frequently
copious output into coherent results. These constraints serve to limit the
extent to which modellers can and do carry out extensive diagnosis and
testing of their models including sensitivity analysis and validation.
In the early days of marine ecosystem modelling, it was necessary to resort
to simple empirical approaches to deal with physics given the limited power
of computers at the time. The so-called zero-dimensional “slab” models
that came to the fore were the cornerstone of their discipline in the mid
20th century. Slab models have a simple physical structure consisting of two
vertical layers. The depth of the upper (mixed) layer, which can vary
seasonally, was determined empirically from observations of vertical
profiles of temperature or density. Containing the pelagic marine ecosystem,
the upper layer was positioned above an essentially implicit (in that it is
unchanging) bottom layer that contains a (typically fixed) nutrient
concentration. Such slab models can be run quickly and straightforwardly,
enabling both a multitude of runs and ease of analysing results.
Despite the simplicity of the two-layer slab physics, these models are
sufficiently well formulated to permit realistic and insightful simulations
of marine ecosystems (e.g. Evans and Parslow, 1985; Fasham et al., 1990).
Indeed, looking back at the history of marine ecosystem modelling, it is
remarkable how simple models allowed so much progress to be made, notably by
pioneers such as Gordon Riley, John Steele and Mike Fasham (Gentleman, 2002;
Anderson and Gentleman, 2012). We admire these individuals when it came to
encapsulating the complexity of the real world with mathematical equations.
They necessarily had to think deeply about their models because they had to
build them from scratch as, in most instances, established relationships for
processes such as photosynthesis, grazing and mortality could not be
borrowed from elsewhere. A key aspect of their success, we submit, is that
they experimented extensively with their models, trying out different
formulations and parameterisations in order to see the effect on model
predictions (e.g. Anderson and Gentleman, 2012). It is this preparation
that served them so well, allowing them to set up meaningful simulations
from which they could so effectively draw conclusions and make progress in
their field of study.
The need for preparation in terms of exploring sensitivity to ecosystem
model formulations and parameterisation is no less in the modern era, indeed
it is arguably greater given our deeper knowledge of the marine biota and a
correspondingly larger multitude of mathematical formulations to choose
from. We propose that modellers can benefit from extensively “playing
with” and testing their models and that the use of simple slab physics is
an obvious choice in this regard, at least for ocean locations where the
bulk of the biological activity occurs in the surface mixed layer.
Experimentation of this kind may then be used to set the stage for the
“serious” model runs that may follow, e.g. in 1-D or 3-D, although it is
also entirely possible to undertake successful studies using only slab
physics models. In addition, because they are straightforward to understand
and do not require powerful computing resources to run, models that
incorporate simple slab physics are ideal for use in teaching future
generations of marine scientists about ecological structure and function.
Here, we present a slab a.k.a. zero-dimensional, and hence computationally
efficient, plankton ecosystem testbed, coded in the freely available R
environment, EMPOWER-1.0 : Efficient Model of Planktonic ecOsystems WrittEn
in R. Our aim is to provide EMPOWER-1.0 for general use and to demonstrate
how it can readily and easily be used both to study ecosystem dynamics at a
range of ocean sites and to assess the pros and cons of different model
choices for best representing and analysing the ecosystems in question.
EMPOWER's code is structured in a modular way to ensure maximum ease of
adjusting parameters and formulations and, indeed, the inclusion of entirely
new marine ecosystem compartments, processes and associated outputs as
required. Here, we demonstrate the use of EMPOWER-1.0 in combination with a
simple illustrative nutrient–phytoplankton–zooplankton–detritus (NPZD)
model. It should be noted, however, that EMPOWER-1.0 can be used to test and
examine the performance of simple and complex models alike. Our choice of a
simple ecosystem model is motivated by the fact that simple models are
conceptually straightforward as well as being easy to set up and analyse.
This study is structured as follows. First, a brief history of slab models
in marine science is presented to illustrate the origin and utility of these
models as research tools in marine science. The NPZD model is then described
and implemented within EMPOWER. The utility of EMPOWER as a testbed for
undertaking model parameterisation is next demonstrated by a parameter
adjustment exercise, specifically the fitting of the NPZD model to observed
seasonal cycles of chlorophyll and nutrients at each of four stations in
diverse regions of the world ocean. The sensitivity analysis is then
extended to model equations with a comparison of the performance of
different equations for calculating, first, daily depth-integrated
photosynthesis and, second, phytoplankton and zooplankton mortality.
Finally, the utility of slab models is discussed in context of ongoing
contemporary marine ecosystem modelling research.
Slab models: from pioneering studies to the present day
In this section, we provide a history of slab modelling which serves as an
introduction to how these models are constructed, as well as to demonstrate
that, despite their simplicity, the simulations these models generate can be
meaningful and realistic. Models provide the theoretical basis for our
understanding of the dynamics of marine ecosystems. One of the first
applications of theory in biological oceanography occurred around 80 years
ago when scientists were interested in the mechanisms driving the spring
phytoplankton bloom that is characteristic of many marine systems. The basic
theory as we know it today, whereby bloom initiation occurs as the water
column stratifies, was proposed in the early 1930s by Haaken H. Gran, a
Norwegian botanist (Gran, 1932; Gran and Braarud, 1935). Mathematical testing
of this proposal was essential in order to establish quantitative merit,
given the dynamic interplay between bottom-up controls on phytoplankton via
light and nutrients versus top-down control by grazing. Following on from
initial work by Fleming (1939), it was Gordon Riley, a biological
oceanographer based at the Bingham Oceanographic Laboratory in the
northeastern USA, who constructed a model of seasonal phytoplankton dynamics
for Georges Bank, a raised plateau off the coast of New England, northeast
USA (Riley, 1946), a remarkable achievement at the time (Anderson and
Gentleman, 2012). The model had a single differential equation for the rate
of change of phytoplankton biomass, expressed with terms for photosynthesis,
respiration and grazing. Using a photosynthesis–irradiance (P-I) curve based
on his own shipboard experiments, Riley developed a formula for daily
depth-averaged photosynthesis in the mixed layer that was derived from
observed seasonal irradiance at the ocean surface as calculated by
atmospheric transmission by Kimball (1928), measured light attenuation
coefficients and a nutrient limitation term. The seasonal cycle of mixed
layer depth (MLD) was imposed empirically, with calculated photosynthesis in the
euphotic zone being diminished accordingly when MLD
exceeded that of the euphotic zone (Fig. 1). Temperature was considered to
affect net primary production via regulation of respiration. Despite its
simplicity, in both biology and physics, Riley's model successfully
reproduced the spring plankton bloom at Georges Bank, highlighting the
subtle interplay between growth and grazing in controlling plankton stocks.
Forcing used by Riley (1946) in his model of George's Bank: (a)
depths of euphotic zone and mixed layer; (b) diminution in photosynthesis due
to light limitation (LV).
Although Riley's model considered depth-averaged photosynthesis over the
mixed layer, it could not be described as a slab model per se because it did
not account for fluxes of material across the pycnocline. It was John
Steele, a mathematical marine biologist from Scotland, who took the next
step by experimenting with a dynamic ecosystem embedded within multi-layer
models (e.g. Steele, 1956), arguably a coarser version of what is done
today in the more complex 1-D models. Steele's experience with this model
led him to realise that much of the net effect of vertical gradients could
be captured with just a few layers, and he further simplified the physics to
a two-layer sea in his study of the plankton in the North Sea (Steele,
1958). The resulting NPZ ecosystem was confined to the upper layer with a
lower layer that contained only nutrient, in fixed concentration. Inputs of
nutrients to the surface layer occurred due to mixing, balanced by export
via phytoplankton sinking and mixing (Fig. 2). Steele had thus constructed
the first slab model of its kind although with this, as well as his later
models including those in his seminal work The Structure of Marine Ecosystems (Steele, 1974), he used a fixed,
rather than seasonally varying, mixed layer depth. Applying the model to
study the plankton of Fladen Ground and other regions in the northern North
Sea, Steele demonstrated good agreement between the model and estimates of
production from observations. Through work such as this, Steele emphasised
that it is simplification that allows us to most easily address the
controlling factors in marine ecosystems. One of Steele's best-remembered
findings, demonstrated again using simple models, is that the form of the
zooplankton closure term has important consequences for ecosystem dynamics
and export flux (Steele and Henderson, 1992). This finding remains relevant
to modellers today and, indeed, we will examine model sensitivity to
zooplankton mortality in Sect. 4.4.
Two-layer slab physics framework (adapted from Steele, 1974).
It was Geoff Evans and John Parslow who would make the next major advance in
the development of slab models with their “model of annual plankton
cycles” (Evans and Parslow, 1985). Following Steele, they opted for an NPZ
ecosystem embedded within the same two-layer framework with the marine
ecosystem restricted to the upper layer and a fixed nutrient concentration
in the lower. Evans and Parslow provided a more complete representation of
the interaction of the marine ecosystem with its physical environment by
allowing the depth of the mixed layer to vary seasonally with direct impacts
on the model state variables. As the mixed layer deepens, nutrients are
entrained from below while phytoplankton density is diluted because their
surface layer biomass is spread over a greater depth. Conversely, as the
mixed layer shallows, the concentrations of nutrients and phytoplankton are
unchanged although losses occur on a per unit area (m-2) basis. As many
zooplankton can swim, Evans and Parslow assumed that they are able to avoid
detrainment in a similar manner to the assumptions of prior models (e.g.
Steele, 1958; Riley et al., 1949) in which case their concentration
increases as MLD decreases.
Evans and Parslow (1985) also took seasonal and daily irradiance forcing
into consideration, in combination with depth integration of a non-linear
P-I curve. As opposed to previous studies that had used observations,
variation in light at the ocean surface was calculated from standard
trigonometric/astronomical formulae (Brock, 1981), with transmission losses
in the atmosphere as 70 % of cloud cover and photosynthetically active
radiation (PAR) as three-eighths of total irradiance. Variation in light with time of
day was assumed to be triangular (Steele, 1962), permitting analytic
integration in time. A notable contribution of Evans and Parslow's (1985)
paper is the appendix which provides the equations required to construct a
model subroutine to calculate daily depth-integrated photosynthesis in a
model layer as a function of noon irradiance (PAR entering the layer from
above), day length, phytoplankton concentration, rate of light extinction
(Beer's law) and parameters for maximum photosynthesis and initial slope
that define the P-I curve.
In common with their predecessors, Evans and Parslow were interested in the
factors controlling the initiation of the spring phytoplankton bloom,
focussing on the role of vertical mixing. Bloom initiation, they concluded,
required a low rate of primary production over winter, which is to be
expected in the North Atlantic due to deep mixed layers at that time, and is
also linked to coupling between phytoplankton and grazers. The simplicity of
the slab model was key to their conclusions as articulated in their own
words: “It is worth emphasising the advantages of analysing simple models,
and simplifying models until they can be analysed”. The controls on
phytoplankton dynamics in high-nutrient low-chlorophyll (HNLC) areas such as
the subarctic Pacific has remained a topical issue ever since, in large part
because limitation by iron is also indicated (Martin et al., 1994; Coale et
al., 1996), but the role of grazing and the link between
phytoplankton–zooplankton coupling and mixed layer depth remains firmly
established as a key mechanism in these systems (Frost, 1987; Fasham, 1995;
Chai et al., 2000; Smith Jr. and Lancelot, 2004).
Perhaps the most famous slab modelling paper, published 5 years after
Evans and Parslow (1985), is the study of nitrogen cycling in the Sargasso
Sea by Fasham et al. (1990; henceforth FDM90). It is by far the most highly
cited marine ecosystem model (Arhonditsis et al., 2006, noted that it had
accumulated 405 ISI cites by November 2005; this number has increased to 758
as of May 2015). In terms of physical structure, Fasham's model used the
same basic slab construct as in Evans and Parslow (1985), with seasonally
varying mixed layer depth and irradiance forcing. The novel aspects of FDM90
were instead related to additional complexity of the ecosystem, expanding
from a simple NPZ to explicitly separate new and regenerated production by
including state variables for nitrate and ammonium (critical for calculating
the f ratio; Eppley and Peterson, 1979), as well as having a simple
microbial loop of dissolved organic nitrogen and bacteria. Sinking detritus
was also added as a state variable, facilitating the prediction of export
flux. The success of this model was due to it being the first attempt to
fully elucidate the processes involved in the recycling of nitrogen in the
euphotic zone, as well as the complimentary roles of zooplankton and
bacteria. The simplified physics of the model allowed it to be run on PCs of
that era and Fasham purportedly distributed the code on floppy disks,
allowing other researchers to run the model on their PCs.
Characteristics of published slab models.
ReferenceLocationStructureMLDIrradiancePhotosyn.Evans and Parslow (1985) Frost (1987) Fasham et al. (1990) Robinson et al. (1993) Fasham (1995) Matear (1995) Hurtt and Armstrong (1996) Popova et al. (1997) Anderson and Williams (1998) Spitz et al. (1998) Fennel et al. (2001) Natvik et al. (2001) Schartau et al. (2001) Spitz et al. (2001)Flemish Cap, subarctic Pacific subarctic Pacific Sargasso Sea Pacific upwelling subarctic Pacific, North Atlantic subarctic Pacific Sargasso Sea none (theoretical) English Channel Sargasso Sea Sargasso Sea Flemish Cap Sargasso Sea Sargasso SeaNPZ NP(Z) 2NPZDB (DOM) P2Z 2NPZDB (DOM) 2NP2ZDB (DOM) 2NPR NPZD 2NPZDB (DOM) 2NPZDB (DOM) NPZD NPZ NPZ 2NPZDB (DOM)clim. clim. clim. f(winds) clim. clim. clim. hypothet. clim. clim. clim. model 1989–1993 1989–1993astronomical data astronomical astronomical astronomical data astronomical astronomical astronomical astronomical astronomical astronomical astronomical astronomicalE&P85 numeric E&P85 numeric? E&P85 E&P85 E&P85 E&P85 A93 E&P85 E&P85 E&P85 E&P85 E&P85Hemmings et al. (2004)North AtlanticNPZclim.dataE&P85Onitsuka and Yanagi (2005)Japan SeaNPZD, 2N2P3Z (DOM)clim. data numeric Findlay et al. (2006) Mitra et al. (2007) Mitra (2009) Llebot et al. (2010) Kidston et al. (2013)None (theoretical) North Atlantic North Atlantic Mediterranean Bay Southern OceanNP 2NPZDB (DOM) 2NPZDB (DOM) 2N2PD (DOM) NPZDhypothet. clim. clim. f(R no.) modelnone astronomical astronomical astronomical modelB&P05 E&P85 E&P85 numeric E&P85
MLD: clim. (climatological from data); hypothet. (hypothetical); f(R no.)
(function of Richardson number).
Photosynthesis calculation (photosyn.): E&P85 (Evans and Parslow, 1985);
A93 (Anderson, 1993); B&P05 (Boushaba and Pascual, 2005).
The description of the marine ecosystem provided by FDM90 has largely served
as the foundation for marine ecosystem modelling ever since. With the advent
of increasing computer power, as well as increasing interest in the
spatio-temporal behaviour of plankton systems, most modelling studies are
now undertaken in 1-D or 3-D physical frameworks. Nevertheless, many slab
modelling studies have been published since FDM90 which follow the basic
design described above, or slight modifications thereof (Table 1). A range
of ecosystem models of varying complexity have been incorporated within slab
physics and applied to contrasting sites throughout the world ocean. The
basic physical construction is similar in most cases consisting of a classic
slab structure with a seasonal cycle of mixed layer depth specified from
data and seasonal irradiance from standard trigonometric equations.
Remarkably, Evans and Parslow's (1985) equations for calculating daily
depth-integrated photosynthesis have prevailed and been used in most
studies. A more sophisticated calculation method was developed by Morel (1988, 1991) and a simplified form of this (Anderson, 1993) is examined in
Sect. 4.3. The models in Table 1 have been used for a diverse range of
applications including studies of parameter optimisation (Spitz et al.,
1998; Fennel et al., 2001; Schartau et al., 2001; Hemmings et al., 2004),
parameter sensitivity analysis (Mitra, 2009; Mitra et al., 2007, 2014),
phytoplankton bloom dynamics (Findlay et al., 2006), nutrient cycling via
organic and inorganic pathways (Llebot et al., 2010), primary production in
HNLC systems (Kidston et al., 2013) and primary production and export flux
in contrasting regions (Fasham, 1995; Onitsuka and Yanagi, 2005).
Model description
We demonstrate the use of EMPOWER-1.0 using a simple NPZD ecosystem model
and forcing for four time series stations in the ocean. The code is readily
adapted to incorporate other ecosystem models, including the relatively
complex models of the modern era, and/or forcing for other ocean sites.
Slab setup and forcing
The model uses slab physics as per Evans and Parslow (1985), namely a
seasonally varying surface mixed layer that contains the ecosystem
positioned above a deep homogeneous layer containing unchanging nutrient and
no plankton (Fig. 2). We have also included temperature dependencies for the
physiological rates in the ecosystem model (see below). Our model was set up
for four stations, two in the North Atlantic (stations BIOTRANS,
47∘ N, 20∘ W and India, 60∘ N,
20∘ W) and two HNLC systems (stations Papa in the North
Pacific, 50∘ N, 145∘ W and KERFIX in the
Southern Ocean, 50∘ 40′ S 68∘ 25′ E). These
stations were chosen because of their contrasting environments, as
illustrated by the differences in forcing variables: seasonally varying
MLD, irradiance (I) and sea surface temperature (T)
(Fig. 3), as well as deep nitrate (N0; see below). Mixed layer depths
were taken from the World Ocean Atlas 2009 (WOA; Antonov et al., 2010; Locarnini et
al., 2010). In common with most previous slab modelling studies, noon (peak
daily) irradiance at the ocean surface is calculated, for a given latitude
as a function of time of year, using standard trigonometric/astronomical
equations. The effect of clouds on atmospheric transmission was calculated
using the model of Reed (1977). The equations for irradiance forcing are not
usually provided as part of published model descriptions but, for
completeness, they are listed here in Appendix A.
The bottom layer in most slab models is assumed to have a fixed
concentration of nutrient, N0. There is in reality a gradient of
nutrient with depth and this can be represented empirically in slab models
using simple functions of nutrients versus depth (Frost, 1987; Steele and
Henderson, 1993; Fasham, 1995). We adopted this approach here for stations
BIOTRANS and India, using simple linear relationships with depth (z):
N0(z)=aNz+bN.
The regression coefficients were fitted from WOA
data (Garcia et al., 2010) for subthermocline NO3 (z>100 m). Resulting values for aN and bN were 0.0174 and 3.91 for
station BIOTRANS and 0.0074 and 10.85 for station India. There were no
obvious relationships between N0 and depth for the two HNLC stations
and so mean (fixed) values of 26.1 and 14.6 mmol N m-3 were used for
N0 for KERFIX and Papa respectively.
Model forcing for stations India (60∘ N,
20∘ W), BIOTRANS (47∘ N, 20∘ W), Papa (50∘ N, 145∘ W) and KERFIX
(50∘ 40′ S, 68∘ 25′ E): (a) mixed layer depth (m), (b) noon irradiance (W m-2), (c) sea surface temperature
(∘ C).
Structure of the NPZD model.
Ecosystem model description
The NPZD ecosystem model we have implemented in EMPOWER is presented in
Fig. 4 with dissolved inorganic nitrogen (N; the sum of nitrate and
ammonium), phytoplankton (P), zooplankton (Z) and detritus (D) as state
variables. It is a simplification of the marine ecosystem inspired by that
of FDM90 with improved formulations for multiple-prey grazing, plankton
mortality, nutrient regeneration and other detrital loss terms, as well as
alterations to the parameterisation. The equations are described below;
model parameterisation is described in Sect. 4.1. The phytoplankton
equation is
dPdt=μPP-GP-mPP-mP2P2-(wmix+H′(t))PH(t),
where the terms are growth, grazing and non-grazing mortality (linear and
quadratic), physical losses due to mixing across the bottom of the mixed
layer, and dilution effects of entrainment. H(t) is mixed layer depth (m) at
time t and H′(t) denotes the rate of change of H when dH/dt is positive
(dilution). As explained above, when dH/dt is negative the change in
phytoplankton density due to detrainment of mass from the mixed layer is
exactly balanced by the increasing phytoplankton density due to decreases in
volume and therefore detrainment does not alter the concentration of
remaining biomass. Variable μP is the vertically averaged
temperature-dependent daily growth rate, defined as the product of a
temperature-dependent maximum growth rate, μPmax(T), and
non-dimensional limitation terms for nutrients and light, LN(N) and
LI(I(t,z)):
μP=μPmax(T)LN(N)LI(I(t,z)).
Note that μP is calculated on a daily basis averaging over the
time of day (t) and depth (z). Temperature and nutrients are assumed to be
uniformly distributed throughout the mixed layer, in which case μP
is
μP=μPmax(T)LN(N)24H∫024 h∫0HLI(I(t,z))dzdt.
With the assumption of balanced growth, μPmax(T) is equal to
the equivalent maximum photosynthetic rate, VPmax(T). The
temperature dependence of photosynthesis is from Eppley (1972):
VPmax(T)=VPmax(0)1.066T,
where VPmax(0) is photosynthesis at 0∘ C. Note
that this exponential relationship is equivalent to a Q10 of 1.895.
The usual way NPZD-type models characterise nutrient limitation of
phytoplankton growth rate by nutrients, LN(N), is calculated as a
Michaelis–Menten (or Monod) relationship:
LN(N)=NkN+N,
where kN is the half-saturation constant.
Photosynthesis–irradiance curves with parameter settings
VPmax= 2.5 g C (g chl)-1 h-1 and α= 0.15 g C (g chl)-1 h-1 (W m-2)-1; Smith function (Eq. 7) and
exponential function (Eq. 8).
The calculation of LI is the most mathematically complicated aspect of
characterising phytoplankton growth in this model as it takes into
consideration both seasonal and diurnal patterns of irradiance arriving at
the ocean surface (I0), attenuation of irradiance with depth and
photosynthesis as a function of light intensity. Light is assumed to vary
with depth according to Beer's law (I=I0exp(-kPARz)), where
kPAR is the attenuation coefficient, and photosynthesis calculated
using a P-I curve. The daily depth-average photosynthetic rate is calculated
over the course of the day using an assumed daily variation of light, from
which the daily average is derived. The user of EMPOWER is provided with a
choice between two P-I curves, a Smith function (Eq. 7) and an exponential
function (Eq. 8) (Fig. 5):
VP=αIVPmax(VPmax)2+α2I2,VP=VPmax(1-exp(-αI/VPmax)).
Triangular versus sinusoidal patterns of diel irradiance
illustrated for a 12 h day and noon irradiance of 200 W m-2.
Integration with depth (inner integral of Eq. 4) can be calculated
analytically for either of the two P-I curves; equations are provided in
Appendix B. The default method of handling the diurnal variation in
irradiance at the ocean surface (outer integral of Eq. 4) is to do a numeric
integration. The user may choose between assuming either a sinusoidal (Platt
et al., 1990) or triangular (Steele, 1962; Evans and Parlsow, 1985) pattern
of irradiance throughout each day, from sunrise to sunset and peaking at
noon (Fig. 6).
Analytic depth integrals require a Beer's law attenuation of light within
the water column characterised by a single attenuation coefficient,
kPAR. The simplest assumption, provided as the first of two options in
EMPOWER, is that kPAR is the sum of attenuation due to water and
phytoplankton, parameters kw and kc, respectively:
kPAR=kw+kcP.
Parameters kw and kc are often assigned values of 0.04
and 0.03 m2 (mmol N)-1 respectively (e.g. FDM90); these values
are used here.
The assumption of a single mixed layer value of kPAR is questionable
because in reality the value of kPAR varies with depth as a result of
the changing spectral properties of the irradiance field. Red light is
mostly absorbed by water in the upper few metres while blue penetrates
deepest, with relatively efficient absorption by chlorophyll at both
wavelengths. Based on a complex treatment of submarine light (Morel, 1988),
a piecewise approach to light attenuation was developed by Anderson (1993)
with different values, kPAR,i, with i=1 for depth range 0–5 m, i= 2
for depth range 5–23 m and i=3 for depths > 23 m, in each
case kPAR(i) is related to pigment (chlorophyll) concentration, C:
kPAR,i=b0,i+b1,iC1/2+b2,iC+b3,iC3/2+b4,iC2+b5,iC5/2.
This approach to light attenuation is provided as the default option for use
in EMPOWER. The values of the polynomial coefficients (b0,i-b5,i) are listed in Table 2.
Coefficients for use in the Anderson (1993) calculation of light
attenuation (Eq. 10).
The diurnal variation in light at the ocean surface over the course of a day
may be reasonably approximated by a sinusoidal function that is symmetric
about noon irradiance (Platt et al., 1990). Further simplification is possible by
use of a linear model, i.e. use of a triangular model centred at noon (e.g. Steele, 1962;
Evans and Parlsow, 1985) because this simplifies the time integration. It
should be noted here that despite Evans and Parslow's (1985) claim that
differences between the triangular and sinusoidal approximations are minimal
if the area under the curve is the same, they did not make the “equivalent
area” adjustment to their formula, nor is their statement generically true
(i.e. it depends on the peak light intensity, the attenuation of light with
depth and the non-linear P-I relationship).
In EMPOWER, the default method of handling the diurnal variation in
irradiance at the ocean surface is to do a numeric integration. Undertaking
a numerical time integral involves computational cost and two empirical
methods (Evans and Parslow, 1985; Anderson, 1993) have been published that
provide analytic calculations (i.e. pre-determined formulae) for daily
depth-integrated photosynthesis in a water column. Both are provided as
options for use in EMPOWER and have the advantage of faster run time. The
first of the two EMPOWER options is the depth-averaged light-dependent
calculation of growth of Evans and Parslow (1985) which assumes a triangular
pattern of daily irradiance, Beer's law for light attenuation (Eq. 9) and a
Smith function as the P-I curve (Eq. 7). It has been a popular choice in
previous slab modelling studies (Table 1). The second option is from
Anderson (1993), which was developed as an empirical approximation to the
spectrally resolved model of light attenuation and photosynthesis of Morel (1988) used in combination with the polynomial method of integrating daily
photosynthesis of Platt et al. (1990). It assumes a sinusoidal pattern of
irradiance through the day, a piecewise Beer's law light attenuation (Eq. 10) and an exponential function as the P-I curve (Eq. 8). Parameter α, the initial slope of the P-I curve, is also spectrally dependent. The
method of Anderson (1993) calculates the variation of α with depth
as a function of chlorophyll in the water column. Daily photosynthesis is
then calculated using a polynomial approximation. The methods for
calculating daily depth-integrated photosynthesis of Evans and Parslow (1985) and Anderson (1993) are non-trivial and, for completeness, the
equations are supplied in Appendix C.
Grazing by zooplankton is assumed to be on both phytoplankton and detritus.
This choice was made in part to illustrate how to implement ingestion on
multiple prey types, as such functions are used for more complex models
(e.g. when there are multiple phytoplankton size classes or functional types
and/or omnivory by zooplankton). Many multiple-grazing formulations,
however, comprise questionable assumptions about zooplankton feeding
behaviour (Gentleman et al., 2003). For example, the multiple-prey grazing
formula used in FDM90 is classified as an active switching response
(Gentleman et al., 2003) which can display anomalous behaviour such as
suboptimal feeding (i.e. ingestion rates decreasing when prey availability
increases). We have therefore opted to improve upon Fasham's choice by using
a different multiple-prey response, but one that is nevertheless commonplace
in the literature. Specifically, we have adopted a passive switching
response where density dependence of the prey preferences arises due to
inherent differences in the single-prey responses (see Gentleman et al.,
2003). This Sigmoidal (or Holling Type 3) response is characterised as
(Fig. 7)
GP=Imaxφ^PPkZ2+φ^PP+φ^DDZ,φ^P=φPP,φ^D=φDD,GD=Imaxφ^DDkZ2+φ^PP+φ^DDZ,
where the terms in parentheses are the zooplankton specific ingestion rates
IP and ID respectively. This formulation implies that the
single-prey responses for both phytoplankton and detritus are each sigmoidal
(Type 3). Parameter Imax is the maximum specific grazing rate, which is
the same for both phytoplankton and detritus and equates to their single-prey maximum ingestion rates. Although parameters φP and
φD are often called preferences in the literature, the actual
prey preferences associated with this response (i.e. relative amount in the
diet as compared to the environment) are density-dependent, with the
relative preference for phytoplankton to detritus determined by
prefP:D=φPPφDD=φ^Pφ^D. The φ parameters actually relate
to the half-saturation constants associated with the single-prey functional
responses. Specifically, φP=kZ2kP2,
where kP is the half-saturation value for the Type 3 single-prey
response for ingestion of phytoplankton, and φD is defined
similarly. Parameter kZ, which is often referred to as the
half-saturation value in the literature, is actually an arbitrary parameter
(i.e. this formulation is over-parameterised; see Gentleman et al., 2003)
whose value determines the assumed single-prey half-saturation constants
based on choices for the φ parameters.
Contours of the zooplankton specific ingestion rates (IP,
ID) versus densities of the two prey types (P: phytoplankton,
D: detritus) as characterised by the sigmoidal grazing response (Eqs. 11,
12) using parameters Imax= 1 d-1, kZ= 0.52 mmol N m-3, φP= 0.67 and φD= 0.33. Upper
panels illustrate assumed interference effect of one prey type over another,
e.g. for a given P, increasing D reduces IP. The lower panel
illustrates assumed optimal feeding (i.e. total ingestion, Itot, always
increases with increase in P or D) and the benefit of generalism (i.e.
increase in Itot due to consumption of P and D vs. just P).
The Sigmoidal response assumes an interference effect of alternative prey in
that as detritus increases, ingestion of phytoplankton decreases (with the
same interaction for phytoplankton and ingestion of detritus). This
interference effect is not so great as losing the benefit of generalism,
i.e. total ingestion always increases for an increase in total prey density.
The non-equal preferences reduce the interference effect for phytoplankton,
i.e. the contours in the first panel of Fig. 7 are more vertical than for
equal preferences. The corollary effect is that the increased ingestion by
consuming both phytoplankton and detritus versus just phytoplankton is
reduced as compared to when prey have equal preferences.
Regarding phytoplankton non-grazing mortality, FDM90 has the usual choice of
a linear term although non-linear approaches are also possible, e.g. the use
of a Michaelis–Menten saturating function by Fasham (1993). We opted for the
more flexible approach of using both linear and non-linear terms (Yool et
al., 2011, 2013a). The former may account for metabolic losses or natural
mortality. The use of an additional non-linear term represents
density-dependent loss processes, notably mortality due to infection by
viruses. The abundance of viruses is highly dependent on the density of
potential host cells (e.g. Weinbauer, 2004) and, as reviewed by Danovaro et
al. (2011), there is “compelling” evidence that, at least in some
instances, viruses are responsible for the demise of phytoplankton blooms
based on observations of high proportions (10–50 %) of infected cells
(e.g. Bratbak et al., 1993, 1996). A quadratic form was used for the
non-linear mortality term (e.g. Kawamiya et al., 1995; Oschlies and
Schartau, 2005) and all phytoplankton non-grazing mortality losses were
allocated to detritus.
The equation for rate of change of zooplankton density is
dZdt=(βkN(GP+GD))-(mZZ+mZ2Z2)-(wmix+H′(t))ZH(t),
where the terms are growth, mortality (linear and quadratic) and losses due
to mixing and changing MLD. Zooplankton growth can be described as the
product of gross growth efficiency (GGE) and intake, where GGEs are
typically between 0.2 and 0.3 (Straile, 1997). Gross growth efficiency is
itself the product of absorption efficiency, β (more commonly, but
incorrectly, known as assimilation efficiency; e.g. see Mayor et al., 2011)
and net production efficiency, kNZ. Splitting into these separate
parameters (Table 3) permits three-way fractionation of intake between
egestion (i.e. faecal pellet production, 1-β), growth (β⋅kNZ= GGE; first term in Eq. 13) and excretion (β(1-kNZ)).
A variety of formulations exist in ecosystem models to describe zooplankton
mortality and the appropriate functional form has been and continues to be a
hotly debated topic (Steele and Henderson, 1992; Edwards and Yool, 2000;
Mitra et al., 2014). Most common are the linear and quadratic terms, although
some authors have chosen to employ other non-linear functions (e.g. Fasham,
1993 used a Michaelis–Menten relationship). As with phytoplankton, we used
both linear and quadratic non-linear terms (Yool et al., 2011). The linear
term represents density-independent natural mortality, whereas the quadratic
term is considered to be due to predation by carnivores (whose population
tracks that of the zooplankton). The different sources of mortality result
in different fates for these terms. Loss from natural mortality is allocated
to modelled detritus, which implies a broader size class of modelled
particulates (and therefore higher sinking rates) than when just
phytoplankton death contributes to this variable.
The fate of the predation-related mortality is less obvious because the
metabolic activity of higher predators results in ingested material being
converted into dissolved nutrients as well as larger particulates (e.g.
fecal pellets and death). Moreover, the higher predators may export material
from the local region with migration. FDM90, along with a suite of follow-on
models, therefore chose to allocate predation-related zooplankton mortality
between nutrients (ammonium and DON, attributed to excretion by higher
predators) and material that is immediately exported from the system (e.g.
attributed to fast-sinking detritus generated by higher predators).
Similarly, Steele and Henderson (1992) also allocated zooplankton mortality
to export. Nevertheless, many past and recently published marine ecosystem
modelling studies allocate all of zooplankton mortality to detritus
(Oschlies and Schartau, 2005; Salihoglu et al., 2008; Hinckley et al., 2009;
Ye et al., 2012). We argue, however, that this is not necessarily realistic
given that detrital particles related to higher predators are larger and
therefore even faster-sinking than that produced by the modelled plankton.
We have therefore here adopted to follow the sage approach of the model
pioneers and assume that the predation-related mortality represented by our
quadratic term is instantly exported and thereby entirely lost from the
surface mixed layer of the model. As with phytoplankton, zooplankton are
subject to changes in concentration via mixing and changes in MLD.
The equation for the rate of change of dissolved inorganic nitrogen (DIN)
density is
dNdt=-μPP+β(1-kNZ)(GP+GD)+mDD+(wmix+H′(t))(N0-N)H.
DIN is taken up by phytoplankton (first term) and, via the food web,
regenerated with the second and third terms in Eq. (14) representing excretion by
zooplankton and remineralisation of detritus respectively. The fourth term
represents the net transport due to mixing (i.e. supply by the deep water
and loss from the surface layer). The last term represents the net effect of
volume changes, i.e. increases in DIN density due to supply of deep water
nutrients through entrainment and decreases in DIN density due to volume
increases associated with entrainment.
Finally, the detritus equation is
dDdt=mPP+mP2P2+mZZ+(1-β)(GP+GD)-GD-mDD-(wmix+H′(t)+vD)DH.
Detritus is produced by phytoplankton mortality, zooplankton natural
mortality (linear term) and as zooplankton egestion (faecal pellet
production). It is lost by zooplankton grazing and is also remineralised at
a constant rate, mD. Detritus is mixed and subject to changes via the
seasonal cycle of MLD in the same manner as phytoplankton and zooplankton
(terms six and seven), and also experiences losses due to gravitational sinking
(last term). This occurs at rate vD (m d-1) and provides direct
export of particulate organic matter to the layer below (where it is
implicitly remineralised back to DIN).
The first results Sections (4.1, 4.2) are devoted to parameterising the
model, in the first instance, for station BIOTRANS and a detailed
description of values assigned to model parameters is provided therein.
Setup in R
We have chosen to code our model in the R programming language which can be
readily downloaded for free over the Internet. Input and output files are in
ASCII text (.txt) format, avoiding the use of proprietary software. The
structure of the code is designed to be transparent, where possible using
conventional syntax common to different programming languages such as the
use of loops and block IF statements. Where possible, we have followed
what we consider to be best practice in developing the code, which includes the following.
Creation of a fixed segment of core code that handles the numerical
integration, as well as writing to output files. Being fixed, this segment
does not require alteration in the event of changes to the ecosystem model
formulation, nor indeed if an entirely new ecosystem model is implemented.
The ecosystem model formulation, i.e. the specification of the terms
in the differential equations and calculation of their rates of change, is
handled by a function (FNget_flux) that is external to the
core code.
The specification of parameter values and run characteristics (e.g.
time step, run duration, as well as flags for choices between different
formats for export to output files, choice of ocean location and for
different parameterisations of key processes) is via text files that are
read in at the onset of each simulation. Thus, there is no need to enter or
alter the model code when changing parameter values or other model settings.
When a model run finishes, the summed annual fluxes associated with
each term in the differential equations is displayed on the computer screen
along with a report as to whether mass balance is achieved for each state
variable (over the last year of simulation). Basic checking of mass balance
is useful for ensuring that the model equations are error-free.
Regimented layout for clarity with extensive commenting throughout.
The R programming language is supported by various libraries that can be
accessed via the Internet. One such library is for solving ordinary
differential equations (Soetaert et al., 2010). Using this library has the
advantage of minimising the length of the code and offers flexibility in
terms of a range of numerical methods. On the other hand, its implementation
requires that various conventions are adhered to and these can be
restrictive when it comes to producing ancillary code, e.g. the formatting
and export of output files. As such, we opted to code the numerical solution
of the ordinary differential equations (ODEs) manually within the core code of the model for the following reasons.
It offers full transparency for the interested user who wishes to see
the method of integration.
The use of manual code makes it considerably easier to export chosen
variables and fluxes to output files in desired formats and frequencies.
In our case, the user is given the choice between two integration
methods, Euler and fourth order Runge Kutta (RK4). These methods,
particularly the latter, are entirely sufficient for the numerical task at
hand and the coding of them is straightforward.
By using elementary syntax, the code can be easily altered or converted
to other programming languages.
The code is stand alone and not subject to reformulation in the event of
future changes in subroutine libraries.
Structure of the model code.
The structure of the code is shown in Fig. 8. The functions come first,
appearing prior to the core code in R. The key function call is
FNget_flux which contains the ecosystem model specification
(Sect. 3.2). The rate of change is calculated for each term in the
differential equations and allocated to a 2-D array (flux no., state
variable no.) which is then passed back to the core (permanent) code for
processing. Other functions are FNdaylcalc (calculates length of day; Eq. A7), FNnoonparcalc (noon irradiance, PAR; Eq. A5), FNLIcalcNum (undertakes
numerical (over time) calculation of daily depth-integrated photosynthesis),
FNLIcalcEP85 (calculates LI using the equations of Evans and Parslow,
1985; Appendix C1), FNaphy (calculates chlorophyll absorption, effectively
parameter α, in the water column after Anderson, 1993; Eq. C14) and
FNLIcalcA93 (calculates LI using the equations of Anderson, 1993;
Appendix C2).
Model setup comes next. Parameter values are read in from file
NPZD_parms.txt. Simulation characteristics are then read in
from file NPZDextra.txt. These include
initial values for state variables (N,P,Z,D);
run duration (years) and time step;
choice of station: BIOTRANS, India, Papa, KERFIX;
choice of photosynthesis calculation: numeric (default), Evans and
Parslow (1985) or Anderson (1993);
choice of integration method: Euler or RK4;
choice of output characteristics: none, last year only or whole
simulation, and a frequency of once per day or every time step.
Model forcing for the chosen station of interest is then assigned. Monthly
values of MLD and sea surface temperature are read in and subject to linear interpolation in
order to derive daily forcing. Other forcing variables are also set:
latitude, deep nitrate (N0; Eq. 1) and cloud fraction. At the end of
the setup section there are a few lines of code that need to be altered if
the ecosystem model is changed. These lines tell the computer how many state
variables the model has, the maximum number of flux terms associated with
any one state variable and the maximum number of auxiliary variables to be
stored for writing to output files.
An advantage of this structure is that an initial section of customisable
code is followed by a section of permanent code that does not require
adjustment in the event of changes to the equations that describe the
ecosystem model, or indeed if a completely new ecosystem model is to be
used. This code sets up a series of matrices to store fluxes and outputs and
then integrates the model equations over time. State variables are updated
and results exported to three output files: out_statevars.txt
(state variables), out_aux.txt (chosen auxiliary variables)
and out_fluxes.txt (all the terms in the differential
equations). These text files are readily imported to, for example, Microsoft
Excel.
Results are plotted graphically on the computer screen at the completion of
each simulation run. The graph plotting code is necessarily model specific
and needs to be updated by the user as required. R is a user friendly
programming language in this regard and the code provided should be
sufficient for the user to incorporate extra variables with ease.
Finally, a user guide is provided in Appendix D, outlining how to set up R,
run the code, a summary of input and output files, and guidance on
considerations when altering the ecosystem code and/or forcing.
Results
Model results are presented in four sections. First, a simulation is shown
for station BIOTRANS using parameters taken from the literature (Sect. 4.1). This station is chosen as our primary focus, inspired by the North
Atlantic Bloom Experiment in 1989 as part of JGOFS (the Joint Global Ocean
Flux Study; e.g. Ducklow and Harris, 1993; Lochte et al., 1993). It
exhibits the characteristic spring blooming of phytoplankton of temperate
latitudes, followed by relatively oligotrophic conditions over summer, and
has been the subject of previous work using slab models (Fasham and Evans,
1995). Parameter tuning is then undertaken to fit all four ocean time series
stations, BIOTRANS, India, Papa and KERFIX, to data for chlorophyll and
nitrate at each site (Sect. 4.2). Moving on from the calibration of
parameters, structural sensitivity analysis is then carried out by examining
model sensitivity to equations for the calculation of daily depth-integrated
photosynthesis (Sect. 4.3) and mortality terms for phytoplankton and
zooplankton (Sect. 4.4).
The model is compared to seasonal data for chlorophyll and nitrate within
the mixed layer, for each station. Nitrate data are climatological, from
World Ocean Atlas 2009 (Garcia et al., 2010), as is the model forcing in
terms of mixed layer depths and irradiance. Regarding chlorophyll, data are
SeaWiFS (Sea-Viewing Wide Field-of-View Sensor) 8-day averages (O'Reilly et al., 1998), for which we had access to
years 1998–2013. Averaging data across years to provide a climatological
seasonal cycle of chlorophyll is not meaningful as key features, such as the
spring phytoplankton bloom, are smoothed out because the bloom timing is
variable between years. A characteristic year was therefore chosen for each
station by firstly converting the data to log(chlorophyll), then calculating
mean log(chlorophyll) for each year and finally selecting the median year
(an odd number of years is required, so we used 1998–2012. The resulting
year selections were 2002, 1998, 2007 and 2006 for stations BIOTRANS, India,
Papa and KERFIX respectively. The entire data sets are shown with the
multiple years overlaid in Fig. 9, with data for the selected median year
highlighted.
SeaWiFS chlorophyll data (mg m-3) for each of the four
stations, years 1998–2013 overlaid, with selected median year (see text)
highlighted.
It is not our objective here to provide thorough quantitative assessment of
different model simulations in terms of objective quantification of
model–data misfit but, rather, to demonstrate the utility of EMPOWER as a
testbed for model evaluation. Different ecosystem models and associated data
sets will necessarily require different skill metrics and so a lengthy
description and use of quantitative metrics is not appropriate here. Very
often anyway, as is the case here, visual inspection of model–data misfit is
sufficient to determine the best options for model
formulation/parameterisation. If quantitative methods are required, these
are readily accessed from the literature (e.g. Lewis and Allen; 2009; Lewis
et al., 2006).
Parameter initialisation: station BIOTRANS
Adjustment of parameters is a perennial problem for modellers. Parameters
can be set from the literature, sometimes directly on the basis of
observation and experiment, but the usual starting point is to take values
from previously published modelling studies. Almost inevitably, however, the
resulting simulations will show mismatch with data and parameters are
usually selected for adjustment (tuning) to improve the agreement with data.
One option is to use objective tuning methods, such as the genetic algorithm
or adjoint method in which many or all of the model parameters are varied
simultaneously in order to try and find a best fit solution to data (e.g.
Friedrichs et al., 2007; Record et al., 2010; Ward et al., 2010; Xiao and
Friedrichs, 2014). The advantage is objectivity, but difficulties include
sloppy parameter sensitivities (parameters compensate for each other),
different values of model parameters may be similarly consistent with the
data (the problem of identifiability), exploration of a huge parameter space
may be required and local minima in misfit parameter space can make it
difficult to find the true global minimum (Slezak et al., 2010). It is
usually the case that models are underdetermined by data anyway (Ward et
al., 2010), i.e. there are insufficient data (in terms of absolute amount
and/or different types of data) to adequately constrain parameter values.
And of course, objective methods require expertise, time and computing
resources.
Modellers more often than not carry out parameter adjustment by varying
values of chosen parameters one at a time until satisfactory convergence
with data is achieved. The skill is in deciding which parameters to vary. In
principle, sensitivity analysis can be of help in this regard in that
sensitive parameters can be identified and selected for adjustment if they
can be justifiably altered (i.e. there is uncertainty regarding their
value). Here, we will demonstrate the use of EMPOWER for model calibration.
Parameter sets will be derived for the four stations, BIOTRANS and India in
the North Atlantic and the HNLC stations Papa (subarctic North Pacific) and
KERFIX (Southern Ocean). The ecosystem model we have presented uses the NPZD
structure in combination with up-to-date formulations for key processes such
as photosynthesis, grazing and mortality. As such, it has not been
previously published and so there is no readily available complete set of
parameter values to draw upon. Using our experience, we chose appropriate
parameter values from the literature and adjusted others to give a good fit
with the data for station BIOTRANS. This result is presented below along
with a discussion of how we went about achieving this parameter set. Working
from this parameter set, tuning of parameters is then undertaken to fit the
other stations to the data.
Model parameters. Fitted model solutions for stations
BIOTRANS, India, Papa and KERFIX. The initial (unfitted) parameter guesses
for BIOTRANS were as for the fitted solution, except that parameters mP
and kZ were tuned from initial settings of 0.02 d-1 and 0.86 mmol N m-3 respectively (see text and footnotes).
ParameterMeaningUnitBIOTRANSIndiaPapaKERFIXVPmax(0)αmax rate photosynthesis 0∘ C initial slope of P-I curveg C (g chl)-1 h-1 g C (g chl)-1 h-1 (W m-2)-12.5a 0.15a2.5 0.151.25b 0.075b1.25b 0.075bkNhalf-sat. constant: N uptakemmol N m-30.85c0.850.850.85mPmP2ImaxkZφPφDβZkNZmZmZ2vDmDwmixθchlphyto. mortality (linear) phyto. mortality (quadratic) zoo. max ingestion rate zoo. half-saturation for intake grazing preference: P grazing preference: D zoo. absorption efficiency zoo. net production efficiency zoo. mortality (linear) zoo. mortality (quadratic) detritus sinking rate detritus remineralisation rate cross-thermocline mixing C to chlorophyll ratiod-1 (mmol N m-3)-1 d-1 d-1 mmol N m-3 dimensionless dimensionless dimensionless dimensionless d-1 (mmol N m-3)-1 d-1 m d-1 d-1 m d-1 g g-10.015d 0.025e 1.0c 0.6g 0.67h 0.33h 0.69i 0.75j 0.02k 0.34m 6.43c 0.06c 0.13c 75n0.015 0.025 1.0 0.6 0.67 0.33 0.69 0.75 0.0l 0.34 6.43 0.06 0.13 750.015 0.025 1.25f 0.6 0.67 0.33 0.69 0.75 0.02 0.34 6.43 0.06 0.13 750.015 0.025 2.0f 0.6 0.67 0.33 0.69 0.75 0.02 0.34 6.43 0.06 0.13 75
Source: a mean of values for polar waters provided in Table 2 of Rey
(1991); b photosynthetic parameters of HNLC stations halved with respect
to BIOTRANS because of iron limitation (see text); c Fasham and Evans
(1995); d tuned for BIOTRANS; initial guess was 0.02 d-1 (Yool et
al. (2011, 2013a); e Oschlies and Schartau (2005); f tuned for HNLC
stations (see text); g tuned for BIOTRANS: initial guess was 0.86 mmol N m-3 (Fasham and Evans, 1995); h as for Fasham (1993) but adjusted
for different model structure; i Anderson (1994); j Anderson and
Hessen (1995); k Yool et al. (2011, 2013a); l tuned for station
India; m Oschlies and Schartau (2005); n Sathyendranath et al. (2009).
Station BIOTRANS was previously modelled by Fasham and Evans (1995) and we
used this publication as a starting point for the assignment of some of the
parameter values (note that we opted for the second of two optimisation
solutions in this reference). Other parameters were otherwise assigned
values from the literature where possible and/or selected as a best guess.
The resulting parameter set, along with adjusted (tuned) values (see below),
is shown in Table 3.
Photosynthetic parameters, VPmax (maximum rate) and α
(initial slope of the P-I curve) are geographically variable, in part due to
temperature (Harrison and Platt, 1986; Cullen, 1990; Platt et al., 1990;
Rey, 1991; Marañón and Holligan, 1999; Bouman et al., 2000; Huot et
al., 2013). We based parameters VPmax(0) (the maximum rate of
photosynthesis at 0∘ C) and α (initial slope of the
P-I curve) on the mean of values for polar waters provided in Table 2 of Rey (1991),
i.e. VPmax(0)=2.5 g C (g chl)-1 h-1 and
α=0.034 g C (g chl)-1 h-1 (µE m-2 s-1)-1.
Similar values were recorded more recently in the Beaufort
Sea by Huot et al. (2013). Converting units, parameter α is 0.15 g C (g chl)-1 h-1 (W m-2)-1 (1 W m-2=4.55 µE m-2 s-1, based
on the spectral distribution of white light given
in Anderson, 1993). Note that photosynthetic parameters are specified per
unit phytoplankton biomass expressed as chlorophyll, requiring unit
conversion. The Redfield C : N molar ratio of 6.625 is the obvious choice to
convert between C and N. Carbon to chlorophyll ratios are more variable and
a value of 50 g C (g chl)-1 has previously been used in modelling
studies (e.g. Fasham et al., 1990). However, C : chl ratios are known to vary
widely in response to ambient conditions. The recent study of Sathyendranath
et al. (2009) found that, in the North Atlantic, the ratio typically vary
between 50 and 100 g C (g chl)-1 and so here we use an intermediate
value of 75 g C (g chl)-1 (parameter θchl). Remaining
phytoplankton parameters are kN, 0.85 mmol N m-3 (Fasham and
Evans, 1995), mP, 0.02 d-1 (Yool et al., 2011, 2013a), and
mP2, 0.025 (mmol N m-3)-1 d-1 (Oschlies and Schartau, 2005).
Zooplankton parameters Imax and kZ were assigned directly from
Fasham and Evans (1995) with values of 1.0 d-1 and 0.86 mmol N m-3 respectively. When it comes to calculating growth, the
assimilation efficiency used by Fasham and Evans (1995) is in fact a growth
efficiency whereas our use of absorption efficiency (parameter β) is
more in keeping with contemporary zooplankton modelling (e.g. see Anderson
et al., 2013) and refers to the fraction of material absorbed across the
gut. It is multiplied by net production efficiency (parameter kNZ) to
give growth efficiency. Values of 0.69 and 0.75 were assigned to parameters
β and kNZ respectively (Anderson, 1994; Anderson and Hessen,
1995). Zooplankton ought to have a strong grazing preference for
phytoplankton and so the preference value (parameter φP) of
0.12 used by Fasham and Evans (1995) seems unreasonably low. We instead
assigned values of 0.67 and 0.33 for parameters φP and
φD, the same ratio of the equivalent preferences used in
Fasham (1993). Thus, if kZ=1 mmol N m-3, this implies that the
phytoplankton single-prey half-saturation is 1.22 mmol N m-3 and the
detritus single-prey half-saturation constant is 1.75 mmol N m-3. The
implied single-prey half-saturation constants change to 1.05 and 1.50 mmol N m-3 respectively when kZ=0.86 mmol N m-3. Mortality
parameters mZ and mZ2 were assigned values of 0.02 d-1 (Yool
et al., 2011, 2013a) and 0.34 (mmol N m-3)-1 d-1 (Oschlies and Schartau, 2005) respectively.
Detritus is composed of a range of sinking material including faecal pellets
and marine snow of between 5 and several 100 m d-1 (Wilson
et al., 2008), as well as slow-sinking material that is likely to be
remineralised in the upper water column (Riley et al., 2012). A typical
sinking rate used in ecosystem models is between 5 and 10 m d-1 (e.g,
Fasham et al., 1990; Oschlies and Garcon, 1999; Anderson and Pondaven, 2003;
Llebot et al., 2010; Kidson et al., 2013). We used a value for VD of
6.43 m d-1 (Fasham and Evans, 1995). Note also that the detritus
produced by quadratic zooplankton mortality is assumed to be very fast-sinking and is instantly exported from the upper mixed layer. The
remineralisation rate of detritus (parameter mD) was set to 0.06 d-1
(Fasham and Evans, 1995). Finally, parameter wmix was set to
0.13 m d-1 (Fasham and Evans, 1995).
Choices have to be made regarding the settings for calculating daily
depth-integrated photosynthesis. A sinusoidal pattern of daily irradiance
was set as default for this purpose, with a numeric integration over time of
day. A Smith function was chosen as the P-I curve (Eq. 7) as this permits a
straightforward analytic depth integral for photosynthesis (Appendix B).
Photosynthesis at depth can be vertically integrated analytically when light
extinction in the water column is described by Beer's law with a constant
coefficient. As default, we use the piecewise Beer's law treatment of
Anderson (1993) in which the water column is divided into three depth zones
(0–5, 5–23 and > 23 m) and a separate extinction coefficient
calculated for each as a function of chlorophyll (Eq. 10). Although this
approach is more complicated than using a single extinction coefficient, it
is easily justified a priori given the improved representation of light attenuation
and its impact on predicted primary production (Anderson, 1993). Model
sensitivity to these various assumptions regarding the calculation of light
attenuation and photosynthesis will be examined in Sect. 4.3, including an
assessment of the performance of the algorithms of Evans and Parslow (1985)
and Anderson (1993).
The model was run for 5 years, by which time it generates a repeating
annual cycle of plankton dynamics. The last year of simulation for station
BIOTRANS, with initial parameter settings as described above, is compared to
data for chlorophyll and nitrate in Fig. 10. Nitrate (model DIN) is
predicted remarkably well using these default parameter settings, whereas
the predicted seasonal cycle of chlorophyll shows a poorer match with
data. The peak of the spring bloom is more than double that observed and
post-bloom chlorophyll is also consistently elevated (by approximately 0.2 mg m-3) relative to observations (Fig. 10). Parameter adjustment is
therefore desirable in order to improve the fit with data.
Simulation for station BIOTRANS using first-guess parameters
compared to data (year 2002) for (a) chlorophyll and (b) nitrate.
Model calibration
Many modelers go about parameter adjustment on a trial-and-error basis,
making ad hoc changes to parameters and observing the outcome. A more structured
way of going about this is to undertake a systematic sensitivity analysis of
parameters and then, informed by this analysis, choose which parameters to
vary. We use EMPOWER to demonstrate this practice here. Three variables were
selected as simple measures of model mismatch with data: minimum DIN
encountered during the seasonal cycle, Nmin, which is a logical choice
because it is desirable to correctly predict DIN drawdown during the spring
period, maximum chlorophyll at the peak of the spring bloom, chlmax and
the average summer chlorophyll between days 150 and 300, chlav. Values
of these three quantities, as outputs from the run shown in Fig. 10, were
0.093 mmol N m-3 for Nmin and 2.30 and 0.58 mg chl m-3 for
chlmax and chlav respectively. Model parameters were varied
±10 % and the change in these variables quantified in terms of
normalised sensitivity:
S(p)=(W(p)-WS)/WS(p-pS)/pS,
where WS is the value of a given variable (in this case Nmin,
chlmax or chlav) for the standard parameter set with parameter
value pS, and W(p) is the value when the parameter is given value p.
Results are shown in Table 4, ordered high to low for sensitivity of
chlmax.
Model sensitivity analysis: station BIOTRANS. Variables
are chlav (average chlorophyll day 150–300), chlmax (peak bloom
chlorophyll) and Nmin (minimum nitrate during seasonal drawdown).
Parameters ranked according to sensitivity to chlmax.
The requirement for improving the model fit is to decrease chlmax and,
to a lesser extent, decrease chlav also. Looking at Table 4,
chlmax and chlav are together sensitive to zooplankton parameters,
notably kZ, Imax and βZ. In contrast, chlmax is
sensitive to phytoplankton mortality, mP, whereas chlav is not.
The initial guess for kZ of 1.0 mmol N m-3 may be somewhat high,
e.g. separate values of 0.8 and 0.3 mmol N m-3 were used for micro- and
mesozooplankton in the model of Yool et al. (2011, 2013a). Values for
kZ lower than 1.0 mmol N m-3 have also been used in other models,
e.g. values of 0.75 and 0.8 mmol N m-3 were used by Anderson and
Pondaven (2003) and Llebot et al. (2010) respectively. Mortality parameters
such as mP are poorly known and an easy choice for modellers when it
comes to parameter adjustment. We varied parameters kZ and mP and
were able to achieve a good fit to the data with kZ=0.6 mmol N m-3 and mP=0.015 d-1 (Fig. 11). The predicted
overwinter chlorophyll is somewhat too low but this is a common feature of
slab-type models. The mismatch can be improved by removing the linear
phytoplankton mortality term (i.e. setting mP=0; see Sect. 4.4 and
discussion therein). A further consideration is that phytoplankton may
adjust their C : chl ratio in winter to mitigate the effect of the low light
intensities that they experience. We consider removing this mortality term
unrealistic. It is no good getting the right result for the wrong reasons
and so chose to keep phytoplankton mortality unchanged.
Simulation for station BIOTRANS after parameter tuning (see
text): (a) chlorophyll, (b) nitrate.
The associated seasonal cycles of P,Z and D, along with primary production,
phytoplankton grazing and mortality are shown in Fig. 12. Phytoplankton
escape grazing in control in April and early May with the peak of the bloom
occurring on day 137. Zooplankton catch up a week later. Primary production
remains relatively high over summer, but tightly coupled to grazing which is
sufficient to keep phytoplankton biomass in check. Nutrient drawdown
continues after the peak of the bloom with maximum depletion occurring in
July.
Predicted state variables and fluxes for the station BIOTRANS
simulation: (a)P, Z and D and (b) phytoplankton growth, grazing and
non-grazing mortality.
It might be expected that station India is simulated accurately with the
same parameter values as those of station BIOTRANS because of their
relatively close proximity in the northern North Atlantic Ocean. In fact,
the predicted spring bloom is rather high, approximately double the maximum
in the observations for year 1998 (Fig. 13), although not outwith what is
seen in the multi-year data (Fig. 9). An improved fit is easily achieved by
setting mZ=0, i.e. removing the linear zooplankton mortality term
(Fig. 13). Other models, e.g. Fasham (1993), have similarly not included a
linear zooplankton loss term.
Simulations for station India: (a) chlorophyll, (b) nitrate. Data
are for year 1998.
The two HNLC stations can be expected to require alternative
parameterisations to the two North Atlantic stations because of their
different food web structure. In contrast to the diatom spring bloom in the
northern North Atlantic, iron-limited HNLC systems favour small
phytoplankton which are tightly coupled to microzooplankton grazers (Landry
et al., 1997, 2011), “grazer controlled phytoplankton populations in an
iron-limited ecosystem” (Price et al., 1994). Low growth rate of
phytoplankton may be expected relative to the North Atlantic because of iron
limitation. Parameters VPmax(0) and α may typically
decrease by 50 % relative to iron-replete conditions (Alderkamp et al.,
2012). For stations Papa and KERFIX, we therefore assigned VPmax(0)= 1.25 g C (g chl)-1 h-1 and α= 0.075 g C (g chl)-1 h-1 (W m-2)-1. In addition, high maximum grazing
rates may be expected because of the small size structure of the plankton
assemblage. If grazing is dominated by microzooplankton, maximum grazing
rate (parameter Imax) may be as high as 2.0 d-1 (Mongin et al.,
2006). We achieved a good fit to data with Imax=1.25 d-1 (Fig. 14). A similar exercise was carried out for station KERFIX. Using the same
parameter set as for station Papa, predicted chlorophyll was too high (by
approximately 0.05 mg m-3) during the austral summer (Fig. 15). If
Imax is further increased to 2.0 d-1, a reasonable fit to the
chlorophyll data is achieved (Fig. 15). The predicted end-of-year increase
in chlorophyll arrives a month or two too early, but this may be a
consequence of the imposed climatological cycle of mixed layer depth.
Predicted nitrate is somewhat too low (by about 4 mmol m-3) if the
BIOTRANS parameters are used but is markedly improved with the adjusted
parameters.
Simulations for station Papa before and after parameter tuning:
(a) chlorophyll, (b) nitrate. Data are for year 2007.
Sensitivity to photosynthesis algorithm
Structural sensitivity analysis is performed to assess model sensitivity to
the different assumptions for calculating daily depth-integrated
photosynthesis. The best-fit simulation for station BIOTRANS presented above
(Fig. 11) is used as the baseline for comparison, although we will comment
on sensitivity for other stations also. Default settings in the baseline
simulation were a numerical time integration (over the day), a Smith
function for the P-I curve, and a sinusoidal pattern of daily irradiance
with the piecewise application of Beer's law (Eq. 10; Anderson, 1993) for
light attenuation in the water column.
Simulations for station KERFIX before and after parameter tuning
(see text for details): (a) chlorophyll, (b) nitrate. Data are for year 2006.
Simulations for station BIOTRANS showing sensitivity to choice of
P-I curve: (a) Smith function (standard run), (b) exponential function.
Simulations for station BIOTRANS showing sensitivity to choice of
diel variation in irradiance: (a) sinusoidal (standard run), (b)
triangular.
The first sensitivity test involved changing the P-I curve from a Smith
function (Eq. 7) to an exponential function (Eq. 8). Predicted seasonal
cycles for chlorophyll and nitrate at station BIOTRANS are shown in Fig. 16.
Results changed little with respect to the baseline simulation, the only
noticeable difference being the magnitude of the spring bloom which was
about 0.2 mg m-3 greater when using the exponential P-I curve. Similar
insensitivity was seen when using the exponential P-I curve for simulating
stations India, Papa and KERFIX (results not shown). It is perhaps
unsurprising that the model shows minimal sensitivity to choice of P-I curve
as the shapes of the two curves are similar.
Reverting to the Smith function as the chosen P-I curve, model predictions
were next compared for simulations using sinusoidal versus triangular
irradiance (Fig. 17). Once again, the difference between the two simulations
is relatively minor. A larger spring bloom (approx. 0.5 mg m-3) is seen
when using the triangular assumption. Irradiance is underestimated relative
to the sinusoidal pattern (Fig. 6) leading to lower primary production over
winter, decoupling from zooplankton and a larger spring bloom. It is worth
noting that the sensitivity shown to choice of irradiance pattern is at
least as great as that for the choice of P-I curve but has generally
received much less attention in the literature.
Model sensitivity of predicted primary production to the equations
describing light attenuation in the water column was previously highlighted
by Anderson (1993), although without extending to analysis using full
ecosystem models. Model predictions for the two choices for light
attenuation (simple Beer's law, Eq. 9, versus piecewise Beer's, Eq. 10) are
shown in Fig. 18, for all four stations. Whereas chlorophyll shows little
change when switching between the two routines, predicted NO3 exhibits
markedly greater drawdown when using the simple Beer's law, especially for
station India where concentrations reached near zero by the end of June. The
difference between the simulations can be understood by comparing kPAR
as a function of phytoplankton concentration for the two algorithms (Fig. 19).
The single Beer's law of Eq. (9) predicts a modest increase in kPAR
from 0.04 m-1 at zero phytoplankton to 0.1 m-1 at P=1 mmol N m-3. The main difference with the piecewise Beer's law is the much
greater light extinction in the upper 5 m of the water column, with
kPAR of 0.13 m-1 at P=0 mmol N m-3 increasing to 0.23 m-1 at P=1 mmol N m-3. A lesser rate of light attenuation
using the simple Beer's law leads to greater penetration of light into the
water column, higher photosynthesis and greater predicted drawdown of
NO3.
Model simulations for all four stations showing sensitivity to
choice of method for calculating light attenuation in the water column: (a)
piecewise Beer's law (Eq. 10), (b) simple Beer's law (Eq. 9).
Finally, there is the option to use the routines of Evans and Parslow (1985)
and Anderson (1993) to calculate daily depth-integrated photosynthesis,
without recourse to using numerical integration over time. Evans and Parslow
used a Smith function for photosynthesis in combination with a triangular
pattern of daily irradiance. This corresponds exactly to the simulation in
Fig. 17 for triangular irradiance. Thus, running the model using the
Evans and Parslow equations (Appendix C) produces a result indistinguishable
from the numerical simulation. Matters are not so simple when using the
Anderson (1993) equations to calculate daily depth-integrated
photosynthesis. The assumptions here are an exponential P-I curve and
sinusoidal light, corresponding to the exponential P-I curve simulation in
Fig. 16. But there is the additional assumption that parameter α, in
addition to kPAR, is spectrally dependent and varies in the water
column. Thus, running the model with both light attenuation and
photosynthesis calculated as in Anderson (1993) gives rise to different
simulations for the four stations, especially India where there is no bloom
(Fig. 20). It is noticeable that, when using the method of Anderson (1993),
primary production is higher over winter, a result of elevated α,
giving rise to an earlier spring chlorophyll bloom and greater drawdown of
nitrate.
Light attenuation as predicted by Evans and Parslow
(1985; EP85) and for the three layers (0–5, 5–23, > 23 m; 1,2,3
respectively) in Anderson (1993; A93), as a function of phytoplankton
concentration.
Simulations for all four stations comparing methods for
calculating daily depth-integrated photosynthesis, standard run (numeric
integration) and the algorithm of Anderson (1993) which is an empirical
approximation of a full spectral model: (a) chlorophyll, (b) nitrate.
Simulations for all four stations showing model sensitivity to
phytoplankton mortality. Parameters mP (linear mortality) and mP2
(quadratic mortality) were set to zero in turn. (a) Chlorophyll, (b) nitrate.
Mortality terms
The model includes two mortality terms, linear and quadratic, for each of
phytoplankton and zooplankton. This approach has previously been used in
other models (e.g. Yool et al., 2011, 2013a), giving maximum flexibility.
The obvious question is whether all four terms are actually needed. As a
simple structural sensitivity analysis, we removed each of the four
mortality terms in turn and show the impact on the predicted seasonal cycles
of chlorophyll and nitrate for all four stations. The model is relatively
insensitive to the phytoplankton mortality terms although setting
mP=0 (i.e. removal of the linear term) promoted net phytoplankton
growth over winter, increasing coupling to zooplankton grazers and giving
rise to smaller phytoplankton blooms at stations BIOTRANS and India in
spring (Fig. 21). Predicted seasonality in NO3 drawdown was barely
affected by phytoplankton mortality parameters. It seems hard to justify
that loss rates should go to near zero at low population densities (the
consequence of using a quadratic term only) because all organisms have
metabolic requirements. Nearly all marine ecosystem models do, therefore,
include a linear term for density-independent phytoplankton mortality and,
for our baseline simulation (Sect. 4.2), we chose to keep this term on a
purely conceptual basis. Given deep mixing, it is surprising that
phytoplankton biomass, as seen in the data, is maintained over winter in
high-latitude waters. The reasons why this is so remain a matter of
conjecture with candidate theories including cyclic motion associated with
convective mixing (Huisman et al., 2002; Backhaus et al., 2003), and
phytoplankton motility or buoyancy to remain near the ocean surface (see
Ward and Waniek, 2007, and references therein). The slab model has
difficulty dealing with this issue but there is no evidence that this
seriously compromises results when it comes to the predicted timing and
magnitude of the spring bloom and associated ecosystem dynamics later in the
year. In contrast to the representation of linear mortality, many models do
not include a non-linear phytoplankton mortality term. Removing it only
caused minor changes to model predictions (Fig. 21) and so it may not be
necessary.
Simulations for all four stations showing model sensitivity for
zooplankton mortality. Parameters mZ (linear mortality) and mZ2
(quadratic mortality) were set to zero in turn. (a) Chlorophyll, (b) nitrate.
In contrast to the phytoplankton results, removing the linear zooplankton
mortality term had relatively little impact on model predictions, whereas
removal of the quadratic term did, for all four stations (Fig. 22). Removal
of quadratic mortality resulted in phytoplankton levels decreasing by as
much as 50 % which is unsurprising since more zooplankton means more
grazing. Perhaps less obvious is the result that removal of quadratic
closure resulted in similarly large changes in predicted post-bloom nitrate
levels. Predation-related losses, the quadratic term, were assumed to be
instantly exported and thereby lost from the surface mixed layer of the
model. Thus, when these losses are set to zero (parameter mZ2=0),
nitrate drawdown is significantly diminished because, instead of being
instantly exported, zooplankton quadratic mortality is allocated to sinking
detritus, part of which is remineralised in the mixed layer. As was noted by
Fulton et al. (2003b), quadratic closure of the upper trophic level in the
trophic web tends to be a successful way of closing the web. Overall, the
work highlights the need for careful consideration of the parameterisation
of closure in models, including the fate of material thereof.
Discussion
Marine ecosystem modelling is somewhat of a black art regarding decisions
about what state variables to include and how to mathematically represent
key processes such as photosynthesis, grazing and mortality, as well as
allocating suitable parameter values. The proliferation of complexity in
models has only served to increase the plethora of formulations and
parameterisations available to choose from. The complex ecosystem models that have
come to the fore in recent years include, for example, any number of
plankton functional types, multiple nutrients, dissolved organic matter and
bacteria, etc. (e.g. Blackford et al., 2004; Moore et al., 2004; Le
Quéré et al., 2005). Simulations are often carried out within
computationally demanding 3-D general circulation models (GCMs) and, of
course, the realism in ocean physics thus gained is to be welcomed. The
caveat is, however, that improvements in prediction can only be achieved if
the biological processes of interest can be realistically characterised
(Anderson, 2005). The key is, as described above, to undertake extensive
analysis of ecosystem model performance and we propose that the use of a
simple slab physical framework of the type used in EMPOWER is ideal in this
regard. The pioneers of the field such as Riley, Steele and Fasham employed
slab physics to test their models, trying out different formulations and
parameterisations, just to see what would happen (Anderson and Gentleman,
2012). The simplicity afforded by using a zero-dimensional slab physics
framework provides an ideal playground for familiarisation with ecosystem
models, allowing for a multiplicity of runs and ease of analysis. It is by
following this approach that the user develops an intuitive understanding of
the complex non-linear interdependencies of the model equations, a precursor
to making predictions with confidence.
Here, we have presented an efficient plankton modelling testbed,
EMPOWER-1.0, coded in the freely available language R. It provides a readily
available and easy to use tool for thoroughly evaluating ecosystem model
structure, formulations and parameterisations by coupling the ecosystem
dynamics to a simplified representation of the physical environment. EMPOWER
has several advantages in that it is fast, easy to run, its results are
straightforward to analyse and, last but by no means least, the code is
transparent and easily adapted to incorporate new formulations and
parameterisations. As such, the main purpose of EMPOWER is to provide an
ecosystem model testbed that allows users to fully familiarise themselves
with their models, allowing them to subsequently be incorporated with
greater confidence into 1-D or 3-D models, as required. It may be that some
amount of reparameterisation is required when transferring the model
ecosystem between physical codes (from slab to 1-D or 3-D), but this ought
usually to be minimal in extent and will itself be greatly informed by the
previous slab modelling work. Much better this approach, than starting out
from scratch using computationally expensive and time-consuming 1-D or 3-D
codes to undertake ecosystem model parameterisation.
Bearing in mind Steele's two-layer sea, the first slab model of its kind
(Sect. 2), it is worth noting that simple ocean box models are akin to
slab models in terms of physical structure but, whereas slab models usually
are usually set up for point locations in the ocean, box models represent
spatial areas (e.g. ocean basins or the global ocean). A mixed layer or
euphotic zone is positioned above a deep ocean layer, with mixing between
the two but usually without a seasonally changing mixed layer depth. Tyrrell (1999), for example, used a global ocean box model to study the relative
influences of nitrogen and phosphorus on oceanic primary production. Box
models were likewise used by Chuck et al. (2005) to study the ocean response
to atmospheric carbon emissions over the 21st century. Slab models,
including EMPOWER, effectively convert to simple box models if the
seasonality of mixed layer depth is switched off. Without a seasonally
varying MLD, box models have limited capacity to capture seasonal plankton
dynamics because of the role played by MLD in mediating the light and
nutrient environment experienced by phytoplankton. Our results (Figs. 18–20) demonstrate sensitivity to accurate representation of the submarine
light field (i.e. equations describing light attenuation in the water
column).
In order to demonstrate the utility of EMPOWER, we carried out both a
parameter tuning exercise and a structural sensitivity analysis, the latter
examining the equations for calculating daily depth-integrated
photosynthesis and mortality terms for both phytoplankton and zooplankton.
In the parameter tuning exercise, a simple NPZD model, broadly based on the
ecosystem model of Fasham and Evans (1995), was fitted to data (seasonal
cycles) for chlorophyll and nitrate at four stations: BIOTRANS
(47∘ N, 20∘ W), India (60∘ N,
20∘ W), Papa (50∘ N, 145∘ W)
and KERFIX (50∘ 40′ S, 68∘ 25′ E). Formal
parameter sensitivity analysis was carried out, highlighting which
parameters phytoplankton stocks and nitrate drawdown are sensitive to. The
model was successfully tuned to all four stations, the two HNLC stations
(Papa and KERFIX) requiring different parameterisations, notably a halving
of photosynthetic parameters (acting as a proxy for iron limitation)
relative to the North Atlantic sites.
Our parameterisation of the different stations highlighted the somewhat ad hoc
process that most modellers go through when assigning parameter values. Some
parameters were set directly from the results of observation and experiment.
More often than not, however, we followed the “path of least resistance”
when assigning parameters, namely to simply select values from previously
published modelling studies. Equations for processes such as photosynthesis,
grazing and mortality were likewise selected “off the shelf” from the
published literature. Previous publication does not, of course, guarantee
that equations or parameter values are necessarily best suited for a
particular modelling application. Moreover, it is all too easy for less than
ideal, even dysfunctional, formulations to become entrenched within the
discipline and used in common practice (Anderson and Mitra, 2010). Parameter
tuning is almost inevitable in order to ensure satisfactory agreement with
data and we have shown how rigorous sensitivity analysis can help in this
regard. Of course, even with a table of parameter sensitivities, there is
still a considerable subjective element to choosing which parameters to
adjust. The most sensitive parameters should be selected, but the degree of
uncertainty in parameter values is an additional consideration. It is no
good tuning a sensitive parameter if its value is already well known from
observation and experiment.
A necessary complement when ensuring that models show acceptable agreement
with data is to remember that it is important that the theories and
assumptions underlying the conceptual description of models are correct or,
at least, not incorrect (Rykiel Jr., 1996). Indeed, it is the conceptual realisation of
models that in many ways poses the greatest challenge, requiring expertise
and practice to overcome observational or experimental lacunae (Tsang,
1991). Subsequent to the parameter tuning exercise, we studied the
sensitivity of simulation results to chosen formulations for
depth-integrated photosynthesis and both phytoplankton and zooplankton
mortality. In the case of the photosynthesis calculation, some aspects
showed relatively low sensitivity, namely the choice of P-I curve and
whether to assume a triangular or sinusoidal pattern of irradiance
throughout the day. In contrast, the way in which light attenuation in the
water column is calculated showed marked sensitivity. Using a simple Beer's
law (Eq. 9) attenuation coefficient throughout the water column is clearly
oversimplified because the spectral properties of irradiance vary with
depth. Moving to a piecewise Beer's law (Eq. 10), with separate attenuation
coefficients for depth ranges 0–5, 5–23 and > 23 m (Anderson,
1993), led to more rapid light attenuation near the ocean surface.
Depth-integrated photosynthesis declined accordingly, delaying the onset of
the spring bloom and reducing its magnitude, along with drawdown of
nutrient. The difference is in part due to parameter values, rather than the
inherent difference in the equations. Additional sensitivity analysis and
parameter tuning could be used to investigate this further but in fact such
an analysis was undertaken by Anderson (1993), who showed that no amount of
parameter tuning can adequately account for the fact that attenuation will
vary with depth, and cannot be assumed to be constant, because of the
spectral properties of the irradiance field. Given the above, we conclude
that the use of the Evans and Parslow (1985) algorithm to calculate daily
depth-integrated photosynthesis, as has been the choice of many previous
studies (Table 1), is easily justified, at least for the stations we
examined, given the relative insensitivity to choice of P-I curve and choice
of triangular versus sinusoidal irradiance. Superior predictions are likely,
however, if this algorithm is used in conjunction with the piecewise
parameterisation of light attenuation (Anderson, 1993; Eq. 10) rather than
a simple Beer's law with fixed attenuation throughout the mixed layer (Eq. 9).
When it comes to biogeochemical modelling studies in GCMs, it is possible
that all manner of different methods are used to calculate light attenuation
in the water column and resulting photosynthesis. Methodologies are often
not reported in full within published texts, the assumption being that they
are in some way routine and straightforward and that, perhaps, the models
are insensitive to this choice. Consider, for example, the MEDUSA-2.0 (Model of Ecosystem Dynamics, nutrient Utilisation, Sequestration and Acidification) model
(Yool et al., 2013a), published within Geoscientific Model Development and afforded a detailed description
of equations and chosen parameter values. Despite this level of detail, the
model's calculation of light attenuation is largely overlooked and the
reader is instead summarily directed to the LOBSTER model (Levy et al.,
2001). This divides light into two wavebands, “red” and “green-blue”,
that are attenuated separately by seawater, and a Smith function (Eq. 7) is
used to calculate photosynthesis. The published description omits a number
of key details (although the model code was supplied), for instance that
there is a 50:50 division of light between the two wavebands at the ocean
surface, that the photosynthetically active fraction is 0.43 of total
irradiance, that extinction coefficients for the two wavebands are a
function of chlorophyll and that photosynthesis is calculated within each
model layer (the model uses fixed layer depths, with 13 layers in the upper
100 m) as a function of average light within the layer.
As a point of interest, we ran our model for all four stations again, this
time using the MEDUSA-2.0 method of light attenuation and a Smith function
for the P-I curve (see Appendix E for details of the parameterisation of
light attenuation). The calculation included replication of the layer structure
within the GCM in order to achieve a fair comparison. Results (not shown)
were remarkably close to the baseline, fitted simulations for each station.
In the case of station BIOTRANS, the peak of the spring phytoplankton bloom
using the MEDUSA light parameterisation was only 0.7 mg chl m-3, 0.2 mg m-3 less than that in the standard run, but otherwise predicted
seasonal cycles of chlorophyll and nitrate that were almost identical for the two
simulations. Likewise, the predicted chlorophyll and nitrate changed little
at stations India and Papa, whereas at KERFIX nitrate drawdown was slightly
greater, approximately 0.5 mmol N m-3, when using the MEDUSA light
parameterisation. The similarity between simulations using the two different
approaches to light attenuation occurs because, remarkably, calculated light
attenuation using the two red and green wavebands (MEDUSA) differs little
from that using the Anderson (1993) piecewise Beer's law. Here, in a
nutshell, is a classic example of the utility of EMPOWER. This result should
alert GCM modellers to the fact that near-identical results can be generated
for light attenuation in the water column using these two contrasting sets
of equations and a choice can be made as to which is most suitable for
implementation based on computational efficiency. From a theoretical point
of view, the result is also interesting. The equations of Anderson (1993)
are an empirical approximation of the full spectral model of Morel (1988)
which divided PAR into 61 wavebands. It would appear that this model can be
stripped down to just two wavebands, red and green, without serious
degradation in accuracy when it comes to predicting light attenuation.
We also used EMPOWER to undertake an analysis of model sensitivity to the
presence/absence of linear and non-linear mortality terms for phytoplankton
and zooplankton. Whereas the use of linear phytoplankton mortality terms is
commonplace in models (e.g. Anderson and Williams, 1998; Oschlies and
Schartau, 2005; Salihoglu et al., 2008; Llebot et al., 2010), we
investigated the performance of an additional quadratic phytoplankton
mortality term. This term is intended to represent loss processes that scale
with phytoplankton biomass that are not already accounted for in the model.
Given that both self-shading and grazing are explicitly modelled, we
considered the quadratic term to represent mortality due to viruses. Model
results were however relatively insensitive to this parameterisation,
although the potential importance of viruses in marine systems should not be
underestimated (Bratbak, 1993, 1996; Danovaro et al., 2011).
It has long been recognised that the parameterisation and functional form of
zooplankton mortality, the model closure term, can have a pronounced effect
on modelled ecosystem dynamics (e.g. Steele and Henderson, 1981, 1992, 1995;
Murray and Parslow, 1999; Edwards and Yool, 2000; Fulton et al., 2003a, b;
Neubert et al., 2004). Quadratic closure is a common choice, although other
non-linear functional forms are also in use. While it is commonly stated
that quadratic closure is dynamically stabilising, i.e. it prevents both
blooms and extinction of prey, there is a limit to this influence (Edwards
and Yool, 2000) since other processes can come into play. In our case, it is
obvious that quadratic closure had a stabilising effect on the model. Its
removal caused the bloom peak to be higher and also post-bloom phytoplankton
levels to decline to near zero.
In contrast to the community's broad recognition of the potential
sensitivity to choice of closure scheme, far less attention has been paid to
model sensitivity regarding the fate of zooplankton mortality. In reality,
there are likely various types of zooplankton mortality including grazing by
higher predators, starvation and disease. As a mathematical closure term,
one can consider the grazing loss to be partitioned between an infinite
series of higher predators (e.g. Fasham et al., 1990), with partitioning
between detritus and dissolved nutrients in both organic and inorganic form.
The fate of these losses will occur with time delays and potentially also
with spatial separation due to migration of predators. Moreover, any
detrital production by higher predators would comprise significantly larger
“particles” than those due to plankton death and would therefore be
associated with much higher sinking rates. Non-grazing mortality might lead
to production of detritus in situ. There is no consensus on best practice, despite
the fact that different approaches to partitioning of zooplankton losses
between detritus, nutrient and DOM differs markedly between models and can
have a significant effect on modelled ecosystem function (Anderson et al.,
2013). Future structural sensitivity studies should be conducted to explore
how the f ratio (the fraction of primary production fuelled by external
nutrient) and e ratio (i.e. relative export to total primary production) are
affected by the various assumptions relating to zooplankton mortality and
model closure.
Model sensitivity to choice of functional forms and parameterisation, often
manifested as surprising and unforseen emergent predictions, is classic
complexity science (Bar-Yam, 1997). Understanding emergence and the
consequences for accuracy of prediction is a key component of modelling
complex systems (Anderson, 2005). Results here, as discussed above, showed
varying sensitivities to different formulations and assumptions and
demonstrated the utility of EMPOWER in tackling this important topic. High
sensitivities have previously been documented in marine ecosystem models,
e.g. to the exact form of the zooplankton functional response (Anderson,
2010; Wollrab and Diehl, 2015) and choice of zooplankton trophic transfer
formulation (Anderson et al., 2013). Other studies have also shown
“alarming” sensitivity to apparently small changes in the specification of
biological models (e.g. Wood and Thomas, 1999; Fussmann and Blasius, 2005).
Anderson (2005) described this insidious problem, namely sensitivity of
emergent outcomes to interacting non-linear differential equations, as “all
in the interactions”. Dealing with it poses an ongoing challenge for the
modelling community.
EMPOWER-1.0 is provided as a testbed which is suitable for examining the
performance of any chosen marine ecosystem model, simple or complex. We
chose to demonstrate its use by incorporating a simple NPZD ecosystem model.
Simple marine ecosystem models are, however, all too often brushed aside in
marine science today. While our objective here is not to delve deeply into
the ongoing debate about complexity in models (e.g. Fulton et al., 2004;
Anderson, 2005; Friedrichs et al., 2007; Ward et al., 2010), we would
nevertheless like to comment on the worth of simple ecosystem models.
Complex ecosystem models are often favoured today (e.g. Blackford et
al., 2004; Moore et al., 2004; Le Quere et al., 2005) with a similar trend in
ocean physics toward large, computationally demanding models. Many
publications in recent years have involved the use of 3-D models (e.g. Le
Quéré et al., 2005; Wiggert et al., 2006; Follows et al., 2007;
Hashioka et al., 2013; Yool et al., 2013b; Vallina et al., 2014), although
1-D models are also well represented (e.g. Vallina et al., 2008; Kearney et
al., 2012; Ward et al., 2013). The caveat is that improvements in prediction
can only be achieved if the processes of interest can be adequately
parameterised (Anderson, 2005). That is a big caveat and one made harder to
achieve because it is often difficult and/or time consuming to thoroughly
test the formulations and parameterisations involved. Simple NPZD-type
models have a useful role in this regard. Albeit with tuning (but the
complex models are tuned also), our NPZD model was successfully used to
describe the seasonal cycles of phytoplankton and nutrients at four
contrasting sites in the world ocean. It was readily applied to test
different parameterisations for photosynthesis and mortality. At least in
terms of basic bulk properties, simple models produce realistic predictions
and are easy to thoroughly investigate and assess. The whole issue of model
complexity ought in any case to be question dependent (Anderson, 2010), e.g.
simple models may be useful to address questions on biogeochemical cycles
whereas more complex models may be necessary to answer more ecologically
relevant questions such as the effect of biodiversity on ecosystem function.
The use of the EMPOWER testbed allows the user to investigate and determine
whether a particular ecosystem model is sufficiently complex, or indeed too
complex, to address the question of interest.
We have described the utility of slab models as a testbed underpinning
marine ecosystem modelling research. This is however by no means their only
use. Slab models are ideal for teaching ecological modelling. They embrace
the complex interplay between primary production and the physico-chemical
environment, combined with top-down control by zooplankton. Students often
have difficulty grasping the relative significance of causal effects in
ecosystems (Grotzer and Basca, 2003), e.g. the relative roles of bottom-up
versus top-down processes in structuring food webs. A certain amount of
lecture material is of course needed, but there is no substitute for
hands-on modelling providing an interactive approach whereby students can
actively investigate ideas and interact between themselves and a teacher
(Knapp and D'Avanzo, 2010). Insight can be gained by getting students to try
simple things like switching grazing off, doubling phytoplankton growth
rates, etc. The slab modelling framework provided herein is ideal for this
purpose. The code is transparent, modular and readily adjusted to include
alternate parameterisations, it is easily set up for alternate ocean sites,
the model runs fast with graphs of results appearing on the screen on
completion, results are readily written to output files for more in depth
analysis and, by coding in R, the models can be accessed and run without
need for purchasing proprietary software.
Finally, the great advances in marine ecology that the pioneers of plankton
modelling achieved using slab models should not be forgotten. Riley, Steele
and Fasham laid the foundations of today's marine ecosystem modelling using
plankton models embedded within simple physics. Even in the modern arena,
this use of simple physics cannot be dismissed as being too simple for
practical application and there is no reason why further scientific advances
cannot be made using slab models. Models are, fundamentally, all about
simplifying reality.
Irradiance calculations
Both the Evans and Parslow (1985) and Anderson (1993) subroutines for
calculating daily photosynthesis require noon irradiance and day length as
inputs. When there are data available, these data can be used as forcing for
a model, akin to what is done for temperature. However, most typically light
data is not available and so a light submodel must be used to prescribe the
necessary forcing. A climatological approach is often used whereby these
inputs are specified using trigonometric/astronomical equations. This task
is not as straightforward as it might first appear. The basic equations are
presented in texts such as Brock (1981) and Iqbal (1983). Some adjustments
were provided by Shine (1984) and we use the equation for short-wave
irradiance at the ocean surface on a clear day published therein:
Iclear=ISCcos2(z)/Rv21.2cos(z)+e0(1.0+cos(z))/1000+0.0455.
ISC is the solar constant (e.g. 1368 W m-2: Thekaekara and
Drummond, 1971), i.e. the incoming solar radiation that would be incident
on a perpendicular plane, immediately outside the atmosphere. Iclear
also depends on solar zenith angle (z), Earth's radius vector (RV:
accounts for the eccentricity of the earth's orbit) and water vapour
pressure (e0; the partial pressure of water vapour in the atmosphere).
A typical value for e0 is 12 mb (e.g. Josey et al., 2003); the
calculation of Iclear is not sensitive to this parameter. The equation
for RV is
RV=1/(1+0.033cos(2πJ/365))1/2,
where J is day of year (Julian day; i.e. 1 = 1 January). Solar zenith
angle depends on latitude (φ), solar declination angle (δ)
and time of day (γ, where the Earth moves 15∘ per hour
and γ is difference from noon):
cos(z)=sin(φ)sin(δ)+cos(φ)cos(δ)cos(γ).
The cos(γ) term becomes irrelevant when considering noon irradiance.
Solar declination angle is given by
δ=23.45sin(2π(284+J)/365),
where h is hour angle which is the difference between the given time and
noon (where 1 h is 15∘). Note that δ is expressed in
degrees in the above equation (1 radian =180/π∘).
The flux of photosynthetically active solar radiation just below the ocean
surface at noon, Inoon, can now be calculated:
Inoon=CFACfPAR(1-ϕ)Iclear,
where fPAR is the fraction of solar radiation that is PAR (λ
between 400 and 700 nm), ϕ is ocean albedo and CFAC is the effect
of clouds on atmospheric transmission. Parameters fPAR and ϕ are
relatively invariant with typical values of 0.43 for fPAR and 0.04 for
ϕ (e.g. Fasham et al., 1990). Dealing with the effects of clouds is a
problematic issue for modellers. Simple empirical approaches have been developed,
two of the most popular being those of Reed (1977) and Smith and Dobson (1984). We have opted for the former in which CFAC is a function of
zenith angles (specified in degrees):
CFAC=1-0.62W/8+0.0019(90-z),
where W is cloud fraction in oktas. A value of W=6 was used for all four
stations.
The equation for calculating day length (DL, h) is (Brock, 1981)
DL=215arccos(-tan(φ)tan(δ)).
Analytic integrals for photosynthesis with depth
The average photosynthesis within a layer of depth H is
V¯P(H)=1H∫z=0HVP(z)dz,
where VP is photosynthesis as a function of light intensity (specified
as the P-I curve). Two P-I curves were investigated using EMPOWER, a Smith
function (Eq. 7) and an exponential function (Eq. 8). Analytic solutions to
Eq. (B1) are provided here for each of these two P-I curves. In both cases a
Beer's law attenuation with depth is assumed (parameter kPAR), i.e.
I(z)=I(0)e-kPARz, where I(0) is the irradiance entering the layer
from above.
Smith P-I curve
By performing a change of variables such that x=αI(z), the
integral above becomes
V¯P(H)=-VPmaxH∫z=0H1((VPmax)2+x2)1/2dx.
This integral is solved analytically using a trigonometric transformation
and then integration by parts, giving
V¯P(H)=VPmaxkPARHlnx0+((VPmax)2+x02)1/2xH+((VPmax)2+xH2)1/2,
where x0 is x(z=0) and xH is x(z=H).
Exponential P-I curve
In order to integrate Equation B1 using an exponential P-I curve it is first
useful to define (Platt et al., 1980)
I∗z=IzαVPmax.
The integration over depth is then (see Platt et al., 1990)
V¯P(H)=VPmaxkPARH∑n=1∞(-1)n+1n.n!((I∗0)n-(I∗H)n).
For practical purposes, we used a maximum n value of 16.
Special formulations for calculating daily photosynthesisEvans and Parslow (1985) photosynthesis calculation
Evans and Parslow (1985) provide an algorithm for calculating daily
depth-integrated photosynthesis with the assumptions of a Smith P-I curve
(Eq. 3), a triangular pattern of irradiance from sunrise to sunset and light
extinction calculated with a single Beer's law coefficient (Eq. 9). The
average daily rate of photosynthesis within the mixed layer is calculated
as
V¯P(H,τ)=2∫0τ1H∫0MVP(I,z)dzdt,
where t, measured in days, is 0 at sunrise and τ at noon and H is
layer depth. Assuming a triangular pattern of irradiance about noon,
Eq. (A3.1) can be recast as (Evans and Parslow, 1985)
V¯P(H,τ)=2VPmaxkPARH∫0τ∫β1β2tdydty(y2+t2)1/2,β1=VPmaxταInoon,β2=β1exp(kPARH).
Inoon is the photosynthetically active radiation (PAR) just below the
ocean surface at noon. This integral is solved as (Evans and Parslow, 1985)
V¯P(H,τ)=2VPmaxkPARHf(β2,τ)-f(β1,τ)-f(β2,0)+f(β1,0),f(y,t)=(y2+t2)1/2-tlnt+(y2+t2)1/2y.
Anderson (1993) photosynthesis calculation
The subroutine of Anderson (1993) was developed as an empirical
approximation to the spectrally resolved model of light attenuation and
photosynthesis of Morel (1988), used in combination with the polynomial
method of integrating daily photosynthesis of Platt et al. (1990). It is
based on an exponential P-I curve (Eq. 8) and assumes a sinusoidal pattern of
irradiance throughout the day, with the calculation of light attenuation
using a piecewise Beer's law (Eq. 10). The irradiance leaving the base of
each layer is
Ibase,i=Ibase,i-1exp[-kPAR,i(zbase,i-zbase,i-1)],
where Ibase,0 is the irradiance immediately below the ocean surface and
zbase,i is the depth of the base of the layer i (where zbase,0=0).
The subroutine of Anderson (1993) also takes account of the fact that, in
reality, α (the initial slope of the P-I curve) depends on the
spectral properties of light and therefore varies with depth in the water
column. This parameter is the product of photosynthetic absorption cross
section ac(λ), which is spectrally dependent (λ
denotes wavelength), and quantum yield φA (Platt and Jassby,
1976; Platt, 1986):
α(λ)=ac(λ)φA.
Ordinarily (e.g. Table 2), α is presented as the initial slope of
the P-I curve for white light (i.e. spectral distribution as for irradiance
at the ocean surface). The corresponding value of α for the
wavelength at which absorption is maximum, αmax, is (Anderson,
1993)
αmax=2.602α.
The value of α for any given wavelength of PAR, α(λ), is then
α(λ)=αmaxa⋅(λ),
where a∗(λ) is the dimensionless chlorophyll absorption
cross section for wavelength λ. An additional complication,
however, is that a∗(λ) only applies when irradiance is
specified as a scalar flux (Morel, 1991). Irradiance in the model is a
downwelling flux and so Anderson (1993) converted between the two by
defining a new version of the chlorophyll absorption cross section (which can
be used in Eq. (C9) in place of a∗(λ), in combination
with downwelling irradiance):
α#(λ)=a∗(λ)kPAR(λ)/ac(λ).
Coefficients for use in the Anderson (1993) calculation of
photosynthesis.
Again using the piecewise three-layer scheme described above for kPAR,
an average value of a# can be calculated for each layer by deriving
an empirical approximation of Morel's (1988) full spectral model. As a first
step, a# at the ocean surface is calculated as
abase,0#=h0+h1C1/2+h2C+h3C3/2+h4C2,
where the polynomial coefficients are given in Table C1. The a# at
the base of each layer and the average a# in each layer are then
calculated as
abase,i#=αbase,i-1#+αcalc,i#,aav,i#=αbase,i-1#+0.5αcalc,i#,
where acalc,i# is a lengthy empirical calculation:
acalc,i#=f{zbase,i}-f{zbase,i-1},f{z}=(z+1)(g1+g2C1/2+g5C+g7C3/2)+f1{z+1}(g3+g4C1/2+g9C),+f2{z+1}(g6+g10C)+f3{z+1}g8f1{z+1}=(z+1)ln(z+1)-(z+1),f2{z+1}=(z+1)ln2(z+1)-2f1{z+1},f3{z+1}=(z+1)ln3(z+1)-3f2{z+1}.
The coefficients, gx, are provided in Table C1. With irradiance assumed
to vary sinusoidally through the day, the average rate of photosynthesis
within a layer i is
V¯P(H,τ)=DVPmax24HπkPAR∑j=15Ωj(V1j-V2j),V1=αmaxaav,i#Ibase,i-1/VPmax,V2=αmaxaav,i#Ibase,i/VPmax,
where D is day length (hours) and Ωj are the polynomial
coefficients (Platt et al., 1990; Table C1).
EMPOWER1.0 user guide
Installation and setup. The R programming language is
freeware and is readily downloaded from the Internet for use on personal
computers. For example, visit page http://www.r-project.org/.
After installation, set up a directory to hold the model code and associated
input and output files. We recommend also downloading an R editor, e.g,
Tinn-R (also freeware).
Running R. Open the R console. From the toolbar, select
“File” and “Change dir ...” and select the directory in which the model
code and input files have been placed. To run the model, type:
source(“EMPOWER1.R”)
Preparation of input files. The model reads in three input
files, each as ASCII text files.
File NPZD_parms.txt. This file includes a single line
header and then lists the value of each model parameter in turn, followed by
a text string for the purpose of annotation. When changing the parameter
list in the model, the corresponding section in the R code must be altered
accordingly.
File NPZD_extra.txt. This file holds initial values for
state variables, additional parameters, and various flags: choice of
station, choices for photosynthesis calculations (P-I curve, light
attenuation, etc.) and grazing formulation. The user is at liberty to add to
or remove from this list of flags as is desired. This file also contains
flags for core model functions: run duration, time step, output type (none,
last year, whole simulation), output frequency and integration method (Euler
or Runge Kutta). These latter functions are required by the core code and
should not be removed from this file.
File stations_forcing.txt. This file has a header line
for information and then holds monthly values for forcing, in our case
mixed layer depth and temperature, for each station. There are 13
entries in each case, the first and last being the same and corresponding to
the beginning and end of the year. A 366 unit array is set up in the model
code for each forcing variable, with unit 1 corresponding to t=0, and
linear interpolation carried out on the monthly values to fill each array.
Output files. These are generated automatically by the
model, on completion of each model simulation. The type of output generated
is controlled by flags (above). The output files are ASCII, comma separated
and do not have headers. They are readily imported into various software
packages, e.g. Microsoft Excel, for further analysis. The files are the following.
File out_statevars.txt. Outputs the state variables,
ordered as they are in array X in the code.
File out_fluxes.txt. Outputs the model fluxes, ordered
as they are in matrix flux (i,j) in function FNget_flux. Thus,
each line (corresponding to a point in time for output) has Nsvar*nfluxmax
entries where Nsvar is the number of state variables in the model and
nfluxmax is the maximum number of fluxes per state variable.
File out_aux.txt. This file stores the values of
auxiliary variables, as defined by the user in array Y (final section of
function FNget_flux). The maximum size of this array is set
by variable nDvar.
Altering the model structure. If the user wants to change
the number of state variables, nDvar or nfluxmax (above), adjustments
should first be made to the short section of code “Variables specific to
model: adjust accordingly”. Alter nSvar, the initialisation of array X
(which holds the state variables) and the text arrays Svarname and Svarnames
(which are used for output). Then go to function FNget_flux
and rewrite the line of code unpacking the state variables. Finally, specify
the terms associated with the new state variable(s) in matrix flux (i,j).
Altering model equations. The model equations are handled in
function FNget_flux and can be adjusted as desired by the
user, calling additional functions as necessary.
Graphical output. The model automatically generates
graphical output on the computer screen on completion of each simulation. An
advantage of R is that the syntax for generating plots is straightforward
and the user should have no problem, working from the plots provided, in
generating extra graphs as desired.
Light attenuation in MEDUSA
Light attenuation in the water column in the MEDUSA model (Yool et al.,
2011, 2013a) is calculated assuming that PAR at the ocean surface can be
divided equally into two wavebands, nominally red and green. The attenuation
of each is calculated through the water column using Beer's law. The average
light in a model layer can then be calculated on the basis of summing the
two wavebands, this average is then used in combination with a P-I curve to
calculate photosynthesis. The extinction coefficients for red and green
light, xkr and xkg, are
xkr=xkr0+xkrp⋅exp(xlrln(C)),xkg=xkg0+xkgp⋅exp(xlg.ln(C)),
where C is chlorophyll (mg m-3). Values for the coefficients are
xkr0 = 0.225, xkrp = 0.037, xlr = 0.674, xkg0 = 0.0232, xkgp = 0.074,
and xlg = 0.629.
The Supplement related to this article is available online at doi:10.5194/gmd-8-2231-2015-supplement.
Acknowledgements
T. R. Anderson and A. Yool acknowledge support from the Natural Environment Research
Council, UK, as part of the Integrated Marine Biogeochemical Modelling
Network to Support UK Earth System Research (i-MarNet) project (grant ref.
NE/K001345/1). W. C. Gentleman acknowledges support from the Natural Sciences and
Engineering Council of Canada. We wish to thank two anonymous referees for
their critique of the manuscript.Edited by: A. Ridgwell
ReferencesAlderkamp, A.-C., Kulk, G., Buma, G. J., Visser, R. J. W., Van Dijken, G. L.,
Mills, M. M., and Arrigo, K. R.: The effect of iron limitation on
photophysiology of Phaeocycstis Antarctica (Prymnesiophyceae) and Flagiariopsis cylindrus (Bacillariophyceae) under
dynamic irradiance, J. Phycol., 8, 45–59, 2012.Anderson, T. R.: A spectrally averaged model of light penetration and
photosynthesis, Limnol. Oceanogr., 38, 1403–1419, 1993.Anderson, T. R.: Relating C:N ratios in zooplankton food and faecal pellets
using a biochemical model, J. Exp. Mar. Biol. Ecol., 184, 183–199, 1994.Anderson, T. R.: Plankton functional type modelling: running before we can
walk?, J. Plankton Res., 27, 1073–1081, 2005.Anderson, T. R.: Progress in marine ecosystem modelling and the “unreasonable
effectiveness of mathematics”, J. Mar. Syst., 81, 4–11, 2010.Anderson, T. R. and Gentleman, W. C.: The legacy of Gordon Arthur Riley
(1911–1985) and the development of mathematical models in biological
oceanography, J. Mar. Res., 70, 1–30, 2012.Anderson, T. R. and Hessen, D. O.: Carbon or nitrogen limitation in marine
copepods?, J. Plankton Res., 17, 317–331, 1995.Anderson, T. R. and Mitra, A.: Dysfunctionality in ecosystem models: an
underrated pitfall?, Prog. Oceanogr., 84, 66–68, 2010.Anderson, T. R. and Pondaven, P.: Non-Redfield carbon and nitrogen cycling in
the Sargasso Sea: pelagic imbalances and export flux, Deep-Sea Res. Pt. I, 50,
573–591, 2003.Anderson, T. R., Gentleman, W. C., and Sinha, B.: Influence of grazing
formulations on the emergent properties of a complex ecosystem model in a
global general circulation model, Prog. Oceanogr., 87, 201–213, 2010.Anderson, T. R., Hessen, D. O., Mitra, A., Mayor, D. J., and Yool, A.:
Sensitivity of secondary production and export flux to choice of trophic
transfer formulation in marine ecosystem models, J. Mar. Syst., 125, 41–53,
2013.Anderson, T. R., Christian, J. R., and Flynn, K. J.: Modeling DOM
biogeochemistry, in: Biogeochemistry of marine dissolved organic matter, 2nd
Edn., edited by: Hansell, D. A. and Carlson, C. A., Academic Press, 635–667, 2014.Antonov, J. I., Seidov, D., Boyer, T. P., Locarnini, R. A., Mishonov, A. V.,
Garcia, H. E., Baranova, O.K., Zweng, M. M., and Johnson, D. R.: World Ocean
Atlas 2009, Volume 2: Salinity, edited by: Levitus, S., NOAA Atlas NESDIS 69, U.S.
Government Printing Office, Washington, DC, 184 pp., 2010.Arhonditsis, G. B., Adams-Vanharn, B. A., Nielsen, L., Stow, C. A., and Reckhow,
K. H.: Evaluation of the current state of mechanistic aquatic biogeochemical
modeling: Citation analysis and future perspectives, Environ. Sci. Technol.,
40, 6547–6554, 2006.Backhaus, J. O., Hegseth, E. N., Wehde, H., Irigoien, X., Hatten, K., and
Logemann, K.: Convection and primary production in winter, Mar. Ecol. Prog.
Ser., 251, 1–14, 2003.Bar-Yam, U.: Dynamics of Complex Systems, Addison-Wesley, Reading,
Massachusetts, 848 pp., 1997.Boushaba, K. and Pascual, M.: Dynamics of the “echo” effect in a
phytoplankton system with nitrogen fixers, Bull. Math. Biol., 67, 487–507,
2005.Blackford, J. C., Allen, J. I., and Gilbert, F. J.: Ecosystem dynamics at six
contrasting sites: a generic modelling study, J. Mar. Syst., 52, 191–215,
2004.Bouman, H. A., Platt, T., Kraay, G. W., Sathyendranath, S., and Irwin, B. D.:
Bio-optical properties of the subtropical North Atlantic. I. Vertical
variability, Mar. Ecol. Prog. Ser., 200, 3–18, 2000.Bratbak, G., Egge, J. K., and Heldal, M.: Viral mortality of the marine alga
Emiliania huxleyi (Haptophyceae) and termination of algal blooms, Mar. Ecol. Prog. Ser., 93,
39–48, 1993.Bratbak, G., Willson, W., and Heldal, M.: Viral control of Emiliania huxleyi blooms?, J. Mar.
Syst., 9, 75–81, 1996.Brock, T. D.: Calculating solar radiation for ecological studies, Ecol.
Modell., 14, 1–19, 1981.Chai, F., Lindley, S. T., Toggweiler, J. R., and Barber, R. T.: Testing the
importance of iron and grazing in the maintenance of the high nitrate
condition in the equatorial Pacific Ocean, a physical-biological model
study, in: The Changing
Ocean Carbon Cycle, edited by: Hanson, R. B., Ducklow, H. W., and Field, J. G., International Geosphere–Biosphere Programme (IGBP) Book
Series 5. Cambridge University Press, Cambridge, 156–186, 2000.Chuck, A., Tyrrell, T., Totterdell, I. J., and Holligan, P. M.: The oceanic
response to carbon emissions over the next century: investigation using
three ocean carbon cycle models, Tellus, 57B, 70–86, 2005.Coale, K. H., Johnson, K.S., Fitzwater, S. E., Gordon, R. M., Tanner, S., Chavez, F. P., Ferioli, L., Sakamoto, C., Rogers, P., Millero, F., Steinberg, P., Nightingale, P., Cooper, D., Cochlan, W. P., Landry, M. R.,
Constantinou, J., Rollwagen, G., Trasvina, A., and Kudela, R.: A massive phytoplankton bloom induced by an
ecosystem-scale iron fertilization experiment in the equatorial Pacific
Ocean, Nature, 838, 495–501, 1996.Cullen, J. J.: On models of growth and photosynthesis in
phytoplankton,
Deep-Sea Res., 37, 667–683, 1990.Danovaro, R., Corinaldesi, C., Dell'Anno, A., Fuhrman, J. A., Middelburg,
J. J., Noble, R. T. and Suttle, C. A.: Marine viruses and global climate
change. FEMS Microbiol. Rev., 35, 933–1034, 2011.Ducklow, H. W. and Harris, R. P.: Introduction to the JGOFS North Atlantic
Bloom Experiment, Deep Sea Res. Pt. II, 40, 1–8, 1993.Edwards, A. M. and Yool, A.: The role of higher predation in plankton
population models, J. Plankton Res., 22, 1085–1112, 2000.Eppley, R. W.: Temperature and phytoplankton growth in the sea, Fish. Bull.
Nat. Ocean Atmos. Adm., 70, 1063–1085, 1972.Eppley, R. W. and Peterson, B. J.: Particulate organic matter flux and
planktonic new production in the deep ocean, Nature, 282, 677–680, 1979.Evans, G. T. and Parslow, J. S.: A model of annual plankton cycles, Biol.
Oceanogr., 3, 327–347, 1985.Fasham, M. J. R.: Modelling the marine biota, in: The Global Carbon
Cycle, NATO ASI Series Vol. I15, edited by: Heimann, M., 457–504, 1993.Fasham, M. J. R.: Variations in the seasonal cycle of biological production in
subarctic oceans: A model sensitivity analysis, Deep-Sea Res. Pt. I, 42,
1111–1149, 1995.Fasham, M. J. R. and Evans, G. T.: The use of optimization techniques to model
marine ecosystem dynamics at the JGOFS station at 47∘ N
20∘ W, Phil. Trans. R. Soc. Lond. B, 348, 203–209, 1995.Fasham, M. J. R., Ducklow, H. W., and McKelvie, S. M.: A nitrogen-based model of
plankton dynamics in the oceanic mixed layer, J. Mar. Res., 48, 591–639,
1990.Fennel, K., Losch, M., Schröter, J., and Wenzel, M.: Testing a marine
ecosystem model: sensitivity analysis and parameter optimization, J. Mar.
Syst., 28, 45–63, 2001.Findlay, H. S., Yool, A., Nodale, M., and Pitchford, J. W.: Modelling of autumn
plankton bloom dynamics, J. Plankton Res., 28, 209–220, 2006.Fleming, R. H.: The control of diatom populations by grazing, J. Cons. Int.
Expl. Mer., 14, 210–227, 1939.Follows, M. J., Dutkiewicz, S., Grant, S., and Chisholm, S. W.: Emergent
biogeography of microbial communities in a model ocean. Science, 315,
1843–1846, 2007.Friedrichs, M. A. M., Dusenberry, J. A., Anderson, L. A., Armstrong, R. A., Chai,
F., Christian, J. R., Doney, S. C., Dunne, J., Fujii, M., Hood, R.,
McGillicuddy, D. J., Moore, K. J., Schartau, M., Spitz and Y. H., Wiggert,
J. D.: Assessment of skill and portability in regional marine biogeochemical
models: Role of multiple planktonic groups, J. Geophys. Res., 112, C08001,
10.1029/2006JC003852, 2007.Frost, B. W.: Grazing control of phytoplankton stock in the open subarctic
Pacific Ocean: a model assessing the role of mesozooplankton, particularly
the large calanoid copepods Neocalanus spp., Mar. Ecol. Prog. Ser., 39, 49–68, 1987.Fulton, E. A., Smith, A. D. M., and Johnson, C. R.: Mortality and predation in
ecosystem models: is it important how these are expressed?, Ecol. Model.,
169, 157–178, 2003a.Fulton, E. A., Smith, A. D. M., and Johnson, C. R.: Effect of complexity on
marine ecosystem models, Mar. Ecol. Prog. Ser., 253, 1–16, 2003b.Fulton, E. A., Parslow, J. S., Smith, A. D. M., and Johnson, C. R.: Biogeochemical
marine ecosystem models II: the effect of physiological detail on model
performance, Ecol. Model., 173, 371–406, 2004.Fussmann, G. F. and Blasius, B.: Community response to enrichment is highly
sensitive to model structure, Biol. Lett., 1, 9–12, 2005.Garcia, H. E., Locarnini, R. A., Boyer, T. P., Antonov, J. I., Zweng, M. M.,
Baranova, O. K., and Johnson, D. R.: World ocean atlas 2009, volume 4:
nutrients (phosphate, nitrate, silicate), in: NOAA Atlas NESDIS 71, edited
by: Levitus, S., US Government Printing Office, Washington, DC, 398 pp., 2010.Gentleman, W.: A chronology of plankton dynamics in silico: how computer models have
been used to study marine ecosystems, Hydrobiologia, 480, 69–85, 2002.Gentleman, W., Leising, A., Frost, B., Strom, S., and Murray, J.: Functional
responses for zooplankton feeding on multiple resources: a review of
assumptions and biological dynamics, Deep Sea Res. Pt. II, 50, 2847–2875, 2003.Gilbert, P. M., Allen, J. I., Artioli, Y., Beusen, A., Bouwman, L., Harle, J.,
Holmes, R., and Holt, J.: Vulnerability of coastal ecosystems to changes in
harmful algal bloom distribution in response to climate change: projections
based on model analysis, Global Change Biol., 20, 3845–3858, 2014.Gran, H. H.: Phytoplankton. Methods and problems, J. Conseil Int. Expl. Mer.,
7, 343–358, 1932.Gran, H. H. and Braarud, T.: A quantitative study of the phytoplankton in the
Bay of Fundy and the Gulf of Maine (including observations on hydrography,
chemistry and turbidity), J. Biological Bd. Canada, 1, 279–433, 1935.Grotzer, T. A. and Basca, B. B.: How does grasping the underlying causal
structures of ecosystems impact students' understanding?, J. Biol. Educat.,
38, 16–29, 2003.Harrison, W. G. and Platt, T.: Photosynthesis-irradiance relationships in
polar and temperate phytoplankton populations, Polar Biol., 5, 153–164,
1986.Hashioka, T., Vogt, M., Yamanaka, Y., Le Quéré, C., Buitenhuis, E. T., Aita,
M. N., Alvain, S., Bopp, L., Hirata, T., Lima, I., Sailley, S., and Doney, S.
C.: Phytoplankton competition during the spring bloom in four plankton
functional type models, Biogeosciences, 10, 6833–6850,
10.5194/bg-10-6833-2013, 2013.Hemmings, J. C. P., Srokosz, M. A., Challenor, P., and Fasham, M. J. R.:
Split-domain calibration of an ecosystem model using satellite ocean colour
data, J. Mar. Syst., 50, 141–179, 2004.Hinckley, S., Coyle, K. O., Gibson, G., Hermann, A. J., and Dobbins, E. L.: A
biophysical NPZ model with iron for the Gulf of Alaska: Reproducing the
differences between an oceanic HNLC ecosystem and a classical northern
temperate shelf ecosystem, Deep Sea Res. Pt. II, 56, 2520–2536, 2009.Holt, J., Allen, J. I., Anderson, T. R., Brewin, R., Butenschön, M.,
Harle, J., Huse, G., Lehodey, P., Lindemann, C., Memery, L., Salihoglu, B.,
Senina, I., and Yool, A.: Challenges in integrative approaches to modelling
the marine ecosystems of the North Atlantic: Physics to fish and coasts to
ocean. Prog. Oceanogr., 129, 285–313, 2014.Huisman, J., Arrayas, M., Ebert, U., and Sommeijer, B.: How do sinking
phytoplankton species manage to persist?, Am. Nat., 159, 245–254, 2002.Huot, Y., Babin, M., and Bruyant, F.: Photosynthetic parameters in the
Beaufort Sea in relation to the phytoplankton community structure,
Biogeosciences, 10, 3445–3454, 10.5194/bg-10-3445-2013, 2013.Hurtt, G. C. and Armstrong, R. A.: A pelagic ecosystem model calibrated with
BATS data, Deep-Sea Res., 43, 653–683, 1996.Iqbal, M.: An Introduction to Solar Radiation. Academic Press, Toronto, 390
pp., 1983.Josey, S. A., Pascal, R. W., Taylor, P. K., and Yelland, M. J.: A new formula for
determining the atmospheric longwave flux at the ocean surface at mid-high
latitudes, J. Geophys. Res., 108, 3108, 10.1029/2002JC001418, 2003.Kawamiya, M., Kishi, M., Yamanaka, Y., and Suginohara, N.: An
ecological-physical coupled model applied to Station Papa, J. Oceanogr., 51,
635–664, 1995.Kearney, K. A., Stock, C., Aydin, K., and Sarmiento, J. L.: Coupling planktonic
ecosystem and fisheries food web models for a pelagic ecosystem: Description
and validation for the subarctic Pacific, Ecol. Modell., 237–238, 43–62,
2012.Kidston, M., Matear, R., and Baird, M. E.: Phytoplankton growth in the
Australian sector of the Southern Ocean, examined by optimising ecosystem
model parameters, J. Mar. Syst., 128, 123–137, 2013.Kimball, H. H.: Amount of solar radiation that reaches the surface of the
earth on the land and on the sea, and methods by which it is measured, Mon.
Weather Rev., 56, 393–398, 1928.Knapp, A. K. and D'Avanzo, C.: Teaching with principles: toward more
effective pedagogy in ecology, Ecosphere, 1, 1–10, 2010.Kwiatkowski, L., Yool, A., Allen, J. I., Anderson, T. R., Barciela, R.,
Buitenhuis, E. T., Butenschön, M., Enright, C., Halloran, P. R., Le
Quéré, C., de Mora, L., Racault, M.-F., Sinha, B., Totterdell, I. J., and
Cox, P. M.: iMarNet: an ocean biogeochemistry model intercomparison project
within a common physical ocean modelling framework, Biogeosciences, 11,
7291–7304, 10.5194/bg-11-7291-2014, 2014.Landry, M. R., Barber, R. T., Bidigare, R. R., Chai, F., Coale, K. H., Dam, H. G.,
Lewis, M. R., Lindley, S. T., McCarthy, J. J., Roman, M. R., Stoecker, D. K.,
Verity, P. G., and White, J. R.: Iron and grazing constraints on primary
production in the central equatorial Pacific: An EqPac synthesis, Limnol.
Oceanogr., 42, 405–418, 1997.Landry, M. R., Selph, K. E., Taylor, A. G., Décima, M., Balch, W. M., and
Bidigare, R. R.: Phytoplankton growth, grazing and production balances in the
HNLC equatorial Pacific, Deep Sea Res. Pt. II, 58, 524–535, 2011.Le Quéré, C., Harrison, S. P., Prentice, I. C., Buitenhuis, E. T.,
Aumont, O., Bopp, L., Claustre, H., Cotrim Da Cunha, L., Geider, R., Giraud,
X., Klaas, C., Kohfeld, K. E., Legendre, L., Manizza, M., Platt, T., Rivkin,
R. B., Sathyendranath, S., Uitz, J., Watson, A. J., and Wolf-Gladrow, D.:
Ecosystem dynamics based on plankton functional types for global ocean
biogeochemistry models, Global Change Biol., 11, 2016–2040, 2005.Levy, M., Klein, P., and Treguier, A.-M.: Impacts of sub-mesoscale physics
on phytoplankton production and subduction, J. Mar. Res., 59, 535–565, 2001.Lewis, K. and Allen, J. I.: Validation of a hydrodynamic–ecosystem model
simulation with time-series data collected in the western English Channel, J.
Mar. Syst., 77, 296–311, 2009.Lewis, K., Allen, J. I., Richardson, A. J., and Holt, J. T.: Error
quantification of a high resolution coupled hydrodynamic-ecosystem
coastal-ocean model: Part3, validation with continuous plankton recorder
data, J. Mar. Syst., 63, 209–224, 2006.Llebot, C., Spitz, Y. H., Solé, J., and Estrada, M.: The role of
inorganic nutrients and dissolved organic phosphorus in the phytoplankton
dynamics of a Mediterranean bay. A modeling study, J. Mar. Syst., 83,
192–208, 2010.Locarnini, R. A., Mishonov, A. V., Antonov, J. I., Boyer, T. P., Garcia, H. E.,
Baranova, O. K., Zweng, M. M., and Johnson, D. R.: World Ocean Atlas 2009,
Volume 1: Temperature, edited by: Levitus, S., NOAA Atlas NESDIS 68, U.S.
Government Printing Office, Washington, DC, 184 pp., 2010.Lochte, K., Ducklow, H. W., Fasham, M. J. R., and Stienen, C.: Plankton
succession and carbon cycling at 47∘ N 20∘ W during the JGOFS
North Atlantic Bloom Experiment, Deep Sea. Res. Pt. II, 40, 91–114, 1993.
Mayor, D. J., Cook, K., Thornton, B., Walsham, P., Witte, U. F. M., Zuur, A. F., and Anderson, T. R.:
Absorption efficiencies and basal turnover of C, N and fatty acids in a marine Calanoid copepod, Funct. Ecol., 25, 509–518, 2011.Marañón, E. and Holligan, P.M.: Photosynthetic parameters of
phytoplankton from 50∘ N to 50∘ S in the
Atlantic Ocean. Mar. Ecol. Prog. Ser., 176, 191-203, 1999.Martin, J. H. and IronEx team: Testing the iron hypothesis in ecosystems of the
equatorial Pacific Ocean, Nature, 371, 123–129, 1994.Matear, R. J.: Parameter optimization and analysis of ecosystem models using
simulated annealing: A case study at Station P, J. Mar. Res., 53, 571–607,
1995.Mitra, A.: Are closure terms appropriate or necessary descriptors of
zooplankton loss in nutrient–phytoplankton–zooplankton type models?, Ecol.
Model., 220, 611–620, 2009.Mitra, A., Flynn, K. J., and Fasham, M. J. R.: Accounting for grazing dynamics
in nitrogen-phytoplankton-zooplankton models, Limnol. Oceanogr., 52,
649–661, 2007.Mitra, A., Castellani, C., Gentleman, W. C., Jónasdóttir, S. H.,
Flynn, K. J., Bode, A., Halsband, C., Kuhn, P., Licandro, P., Agersted, M.
D., Calbet, A., Lindeque, P. K., Koppelmann, R., Møller, E. F., Gislason,
A., Nielsen, T. G., and St John, M.: Bridging the gap between marine
biogeochemical and fisheries sciences; configuring the zooplankton link,
Prog. Oceanogr., 129, 176–199, 2014.Mongin, M., Nelson, D. M., Pondaven, P., and Tréguer, P.: Simulation of
upper-ocean biogeochemistry with a flexible-composition phytoplankton model:
C, N and Si cycling and Fe limitation in the Southern Ocean, Deep-Sea Res.
Pt. II, 53, 601–619, 2006.Moore, K. J., Doney, S. C., and Lindsay, K.: Upper ocean ecosystem dynamics and
iron cycling in a global three-dimensional model, Global Biogeochem. Cy., 18,
GB4028, 10.1029/2004GB002220, 2004.Morel, A.: Optical modelling of the upper ocean in relation to its biogenous
matter content (case 1 waters), J. Geophys. Res., 93, 10749–10768, 1988.Morel, A.: Light and marine photosynthesis: a spectral model with
geochemical and climatological implications, Prog. Oceanogr., 26, 263–306,
1991.Murray, A. G. and Parslow, J. S.: The analysis of alternative formulations
in a simple model of a coastal ecosystem, Ecol. Model., 119, 149–166, 1999.Natvik, L.-J., Eknes, M., and Evensen, G.: A weak constraint inverse for a
zero-dimensional marine ecosystem model, J. Mar. Syst., 28, 19–44, 2001.Neubert, M. G., Klanjscek, T., and Caswell, H.: Reactivity and transient
dynamics of predator-prey and food web models, Ecol. Model., 179, 29–38,
2004.Onitsuka, G. and Yanagi, T.: Differences in ecosystem dynamics between the
northern and southern parts of the Japan Sea: Analyses with two ecosystem
models, J. Oceanogr., 61, 415–433, 2005.O'Reilly, J. E., Maritorena, S., Mitchell, B. G., Siegal, D. A., Carder, K. L.,
Garver, S. A., Kahru, M., and McClain, C.: Ocean color chlorophyll algorithms
for SeaWiFS, J. Geophys. Res., 103, 24937–24953, 1998.Oschlies, A. and Garçon, V.: An eddy-permitting coupled
physical-biological model of the North Atlantic 1. Sensitivity to advection
numerics and mixed layer physics, Global Biogeochem. Cy., 13, 135–160,
1999.Oschlies, A. and Schartau, M.: Basin-scale performance of a locally
optimized marine ecosystem model, J. Mar. Res., 63, 335–358, 2005.Platt, T.: Primary production of the ocean water column as a function of
surface light intensity algorithms for remote sensing, Deep-Sea Res., 33,
149–163, 1986.Platt, T. and Jassby, A. D.: The relationship between photosynthesis and
light for natural assemblages of coastal marine phytoplankton, J. Phycol.,
12, 421–430, 1976.Platt, T., Gallegos, C. L., and Harrison, W. G.: Photoinhibition of
photosynthesis in natural assemblages in marine phytoplankton, J. Mar. Res.,
38, 687–701, 1980.Platt, T., Sathyendranath, S., and Ravindran, P.: Primary production by
phytoplankton: Analytic solutions for daily rates per unit area of water
surface, Proc. R. Soc. Lond. Ser. B, 241, 101–111, 1990.Popova, E. E., Fasham, M. J. R., Osipov, A. V., and Ryabchenko, V. A.: Chaotic
behaviour of an ocean ecosystem model under seasonal external forcing, J.
Plankton Res., 19, 1495–1515, 1997.Price, N. M., Ahner, B. A., and Morel, F. M. M.: The equatorial Pacific: Grazer
controlled phytoplankton populations in an iron-limited ecosystem, Limnol.
Oceanogr., 39, 520–534, 1994.Record, N. R., Pershing, A. J., Runge, J. A., Mayo, C. A., Monger, B. C., and
Chen, C.: Improving ecological forecasts of copepod community dynamics using
genetic algorithms, J. Mar. Syst., 82, 96–110, 2010.Reed, R. K.: On estimating insolation over the ocean, J. Phys. Oceanogr., 7,
482–485, 1977.Rey, F.: Photosynthesis-irradiance relationships in natural phytoplankton
populations of the Barents Sea, Polar Res., 10, 105–116, 1991.Riley, G. A.: Factors controlling phytoplankton populations on Georges
Bank, J. Mar. Res., 6, 54–73, 1946.Riley, G. A., Stommel, H., and Bumpus, D. F.: Quantitative ecology of the
plankton of the western North Atlantic, Bull. Bingham Oceanogr. Coll., 12,
1–169, 1949.Riley, J. S., Sanders, R., Marsay, C., Le Moigne, F. A. C., Achterberg, E.
P., and Poulton, A. J.: The relative contribution of fast and slow sinking
particles to ocean carbon export, Global Biogeochem. Cy., 26, GB1026,
10.1029/2011GB004085, 2012.Robinson, C. L. K., Ware, D. M., and Parsons, T. R.: Simulated annual plankton
production in the northeastern Pacific coastal upwelling domain, J. Plankton
Res., 15, 161–183, 1993.Rykiel Jr., E. J.: Testing ecological models: the meaning of
validation, Ecol. Modell., 90, 229–244, 1996.Salihoglu, B., Garçon, V., Oschlies, A., and Lomas, M. W.: Influence of
nutrient utilization and remineralization stoichiometry on phytoplankton
species and carbon export: A modeling study at BATS. Deep-Sea Res. Pt. I, 55,
73–107, 2008.Sathyendranath, S., Stuart, V., Nair, A., Oka, K., Nakane, T., Bouman, H.,
Forget, M.-H., Maass, H., and Platt, T.: Carbon-to-chlorophyll ratio and
growth rate of phytoplankton in the sea, Mar. Ecol. Prog. Ser., 383, 73–84,
2009.Schartau, M., Oschlies, A., and Willebrand, J.: Parameter estimates of a
zero-dimensional ecosystem model applying the adjoint method, Deep-Sea Res.
Pt. II, 48, 1769–1800, 2001.Shine, K. P.: Parametrization of the shortwave flux over high albedo surfaces
as a function of cloud thickness and surface albedo, Q. J. Roy. Meteorol.
Soc., 110, 747–764, 1984.Slezak, D. F., Suárez, C., Cecchi, G. A., Marshall, G., and Stolovitzky,
G.: When the optimal is not the best: Parameter estimation in complex
biological models, Plos ONE, 5, 1–10, 2010.Smith, S. D. and Dobson, F. E.: The heat budget at Ocean Weather Ship
Bravo, Atmos.-Ocean., 22, 1–22, 1984.Smith Jr., W. O. and Lancelot, C.: Bottom-up versus top-down control in
phytoplankton of the Southern Ocean, Antarctic Sci., 16, 531–539, 2004.Soetaert, K., Petzoldt, T., and Woodrow, S.: Solving differential equations
in R, The R Journal, 2, 5–15, 2010.Spitz, Y. H., Moisan, J. R., Abbott, M. R., and Richman, J. G.: Data assimilation
and a pelagic ecosystem model: parameterization using time series
observations, J. Mar. Syst., 16, 51–68, 1998.Spitz, Y. H., Moisan, J. R., and Abbott, M. R.: Configuring an ecosystem model
using data from the Bermuda Atlantic Time Series (BATS), Deep-Sea Res.
Pt. II, 48, 1733–1768, 2001.Steele, J. H.: Plant production on the Fladen Ground, J. Mar. Biol. Ass.
UK, 35, 1–33, 1956.Steele, J. H.: Plant production in the northern North Sea, Scottish Home
Dept., Mar. Res., 1958, 1–36, 1958.Steele, J. H.: Environmental control of photosynthesis in the sea, Limnol.
Oceanogr., 7, 137–150, 1962.Steele, J. H.: The Structure of Marine Ecosystems, Harvard Univ. Press, 128
pp., 1974.Steele, J. H.: Prediction, scenarios and insight: The uses of an end-to-end
model, Prog. Oceanogr., 102, 67–73, 2012.Steele, J. H. and Henderson, E. W.: A simple plankton model, Am. Nat., 117,
676–691, 1981.Steele, J. H. and Henderson, E. W.: The role of predation in plankton
models, J. Plankton Res., 14, 157–172, 1992.Steele, J. H. and Henderson, E. W.: The significance of interannual
variability. In: Towards a Model of Ocean Biogeochemical Processes, edited
by: Evans, G. T. and Fasham, M. J. R., Springer Verlag, Heidelberg, 237–360,
1993.Steele, J. H. and Henderson, E. W.: Predation control of plankton
demography, ICES J. Mar. Sci., 52, 565–573, 1995.Straile, D.: Gross growth efficiencies of protozoan and metazoan zooplankton
and their dependence on food concentration, predator-prey weight ratio, and
taxonomic group, Limnol. Oceanogr., 42, 1375–1385, 1997.Thekaekara, M. P. and Drummond, A. J.: Standard values for the solar constant
and its spectral components, Nature, 229, 6–9, 1971.Tsang, C.-F.: The modeling process and model validation, Ground Water, 29,
825–831, 1991.Tyrrell, T.: The relative influences of nitrogen and phosphorus on oceanic
primary production, Nature, 400, 525–531, 1999.Vallina, S. M., Simó, R., Anderson, T. R., Gabric, A., Cropp, R., and Pacheco,
J. M.: A dynamic model of oceanic sulfur (DMOS) applied to the Sargasso Sea:
Simulating the dimethylsulfide (DMS) summer paradox, J. Geophys. Res., 113,
G01009, 10.1029/2007JG000415, 2008.Vallina, S. M., Ward, B. A., Dutkiewicz, S., and Follows, M. J.: Maximal feeding
with active prey-switching: A kill-the-winner functional response and its
effect on global diversity and biogeography, Prog. Oceanogr., 120, 93–109,
2014.Ward, B. A. and Waniek, J. J.: Phytoplankton growth conditions during autumn
and winter in the Irminger Sea, North Atlantic, Mar. Ecol. Prog. Ser., 334,
47–61, 2007.Ward, B. A., Friedrichs, M. A. M., Anderson, T. R., and Oschlies, A: Parameter
optimisation techniques and the problem of underdetermination in marine
biogeochemical models, J. Mar. Syst., 81, 34–43, 2010.Ward, B. A., Schartau, M., Oschlies, A., Martin, A. P., Follows, M. J., and
Anderson, T. R.: When is a biogeochemical model too complex? Objective model
reduction and selection for North Atlantic time-series sites, Prog.
Oceanogr., 116, 49–65, 2013.Weinbauer, M. G.: Ecology of prokaryotic viruses, FEMS Microb. Rev., 28,
127–181, 2004.Wiggert, J. D., Murtugudde, R. G., and Christian, J. R.: Annual ecosystem
variability in the tropical Indian Ocean: Results of a coupled bio-physical
ocean general circulation model. Deep-Sea Res. Pt. II, 53, 644–676, 2006.Wilson, S. E., Steinberg, D. K., and Buesseler, K. O.: Changes in fecal pellet
characteristics with depth as indicators of zooplankton repackaging of
particles in the mesopelagic zone of the subtropical and subarctic North
Pacific Ocean, Deep-Sea Res. Pt. II, 55, 1636–1647,
10.1016/j.dsr2.2008.04.019, 2008.
Wollrab, S. and Diehl, S.: Bottom-up responses of the lower oceanic food web
are sensitive to copepod mortality and feeding behaviour, Limnol. Oceanogr.,
60, 641–656, 2015.Wood, S. N. and Thomas, M. B.: Super-sensitivity to structure in biological
models. Proc. Roy. Soc. Lond. B, 266, 565–570, 1999.Xiao, Y. and Friedrichs, M. A. M.: Using biogeochemical data assimilation to
assess the relative skill of multiple ecosystem models in the Mid-Atlantic
Bight: effects of increasing the complexity of the planktonic food web,
Biogeosciences, 11, 3015–3030, 10.5194/bg-11-3015-2014, 2014.Ye, Y., Völker, C., Bracher, A., Taylor, B., and Wolf-Gladrwo, D. A.:
Environmental controls on N2 fixation by Trichodesmium in the
tropical eastern North Atlantic Ocean – A model-based study, Deep Sea Res.
Pt. I, 64, 104–117, 2012.Yool, A., Popova, E. E., and Anderson, T. R.: Medusa-1.0: a new intermediate
complexity plankton ecosystem model for the global domain, Geosci. Model
Dev., 4, 381–417, 10.5194/gmd-4-381-2011, 2011.Yool, A., Popova, E. E., and Anderson, T. R.: MEDUSA-2.0: an intermediate
complexity biogeochemical model of the marine carbon cycle for climate change and ocean acidification studies, Geosci. Model Dev., 6, 1767–1811, 10.5194/gmd-6-1767-2013, 2013a.Yool, A., Popova, E. E., Coward, A. C., Bernie, D., and Anderson, T. R.: Climate change and ocean acidification impacts
on lower trophic levels and the export of organic carbon to the deep ocean, Biogeosciences, 10, 5831–5854, 10.5194/bg-10-5831-2013, 2013b.