\documentclass[hessd, online, hvmath]{copernicus}

\begin{document}\hack{\sloppy}

\title{The importance of year-to-year variation of meteorological and runoff forcings for water quality of a temperate, dimictic lake}

\Author[1,**]{A.~T.}{Romarheim}
\Author[2,3,*]{K.}{Tominaga}
\Author[1]{G.}{Riise}
\Author[2]{T.}{Andersen}

\affil[1]{Department of Environmental Sciences, Norwegian University of Life Sciences, {\AA}s, Norway}
\affil[2]{Department of Biosciences, University of Oslo, Oslo, Norway}
\affil[3]{Norwegian Institute for Water Research, Oslo, Norway}
\affil[*]{now at: Centre for Ecological and Evolutionary Synthesis (CEES), Department of Biosciences, University of Oslo, Oslo, Norway}
\affil[**]{deceased}

\correspondence{K.~Tominaga (koji.tominaga@ibv.uio.no)}

\runningtitle{Natural stochasticity vs. management effort}
\runningauthor{A.~T.~Romarheim et~al.}

\received{10 August 2014}
\accepted{20 October 2014}
\published{}

\firstpage{1}

\maketitle

\begin{abstract}
  Natural stochasticity can pose challenges in managing the quality of
  the environment, or hinder understanding of the system structure. It
  is problematic because unfavourable stochastic events cancel the
  costly management effort and because favourable stochastic event
  overestimates success of the management effort. This paper presents
  a~variance-based modelling method that can be used to quantify the
  extent to which the natural stochasticity can affect the target
  environment. We use a~case study of a~eutrophication assessment
  in a~Norwegian lake of {\AA}rungen, together with a~lake model
  MyLake, in order to present the method, and how this method could
  assist in answering scientific and managerial
  questions. Specifically, the case study's goal was to disentangle
  the respective significance of nutrient loading in runoff and
  meteorology on the lake. Many scientifically
  and managerially relevant understandings have been revealed. For
  example, variation in runoff volume was most prevalent during autumn
  and winter, while variation in phosphorus inflow was most extensive
  from late winter to early spring. Thermal related properties in the
  lake were well predicted by the model, and showed that the time of
  thermocline formation varied among years by more than one month,
  from mid-April to mid-May, whereas loading
  was the most important factor for phytoplankton biomass and water
  transparency. Mild winters and greater inputs of suspended matter
  and phosphorus were followed by increased phytoplankton biomass and
  light attenuation. These findings suggest also that future changes
  in the global climate may have important implications for local
  water management decision-making. The present method of
  disentangling mutually confounding factors is not limited to lake
  water quality studies and therefore should provide certain utility
  in other types of ecosystem modelling.
\end{abstract}

\introduction

Natural stochasticity sometimes presents challenges in maintaining the
quality of the environment. Such is the case for reducing nutrient
loading in the hope for making the quality of downstream lake
better. Natural variation in weather for example may confound costly
abatement efforts by counteracting the abatement of nutrient
loading. 
In particular, nutrient loading is determined both by hydrology
(partially determined by meteorological forcing on land) and by the
management effort (agriculture and urban related nutrient
loading). Furthermore, weather may also be directly consequential in
lake processes such as
algal growth. 
Under this kind of challenge, it is paramount to evaluate to
what extent the confounding variable can actually make significant
differences in lake water quality . The present study illustrates how
a~variance-based modelling method could disentangle two major factors
affecting a lake, with a~test case study of eutrophication
recovery of
a~Norwegian lake.

Scientific community has learnt that nutrient enrichment of lakes may
lead to high phytoplankton mass development, low water transparency,
and fish mortality due to oxygen depletion
\citep{smith_eutrophication:_1999}. Mainly two factors affect the
nutrient loading to lakes: (1) the soil and land use in the lake
catchment, and (2) the hydrology of the watershed. Phosphorus is
generally regarded as the limiting nutrient for phytoplankton
production in freshwater lakes \citep{schindler_evolution_1977}. Much
effort has therefore been given to reduce phosphorus input to aquatic
ecosystems, which has demonstrably led to reduced phytoplankton
production and increased water transparency in many lakes in Europe
and North America \citep{jeppesen_lake_2005}. On the other hand, many
lakes have showed delayed or negligible improvements in water quality
despite reduced nutrients loading \citep{jeppesen_restoration_2007}.

Year-to-year weather variations have also been recognised to affect
physical, chemical and biological processes in lakes
\citep{bailey-watts_poor_1999,blenckner_large-scale_2007,jeppesen_climate_2009,jeppesen_shallow_2007,whitehead_review_2009}. Increase
in air temperature has been shown to increase the water temperature
\citep{george_impact_2007} and the stability of thermal stratification
\citep{straile_complex_2003}, change the phytoplankton community
towards dominance of species adapted to warmer water
\citep{weyhenmeyer_response_2002}, and possibly lead to earlier and
higher phytoplankton production
\citep{huber_phytoplankton_2008,weyhenmeyer_response_2002}. The
changes in thermal conditions and mixing regime can in turn influence
the light, oxygen and nutrient dynamics in the lake, and thereby
impact the phytoplankton primary production and community structure
\citep{tirok_effect_2007,wilhelm_impact_2008}. Precipitation is also
deemed as an important factor determining the water transparency,
runoff intensity, and suspended matter discharge
\citep{arheimer_climate_2005,noges_role_2007,ulen_agriculture_2007}.

Norway has, generally, a~low fraction of arable land (3\,{\%}) and low
population density (12\,\unit{persons\,km^{-2}}), so eutrophication is
mainly recognizable in intensive agricultural districts at low
altitudes. Lake {\AA}rungen is situated in a~developed agricultural
area south-east in Norway, and is one of the most nutrient rich lakes
in the country. Geological studies suggest that the natural phosphorus
concentration of lakes in this area is 7--8\,\unit{mg\,m^{-3}}
\citep{borch_titaksplan_2007}. Eutrophication became a~problem in the
lake during the 1960s, with phosphorus concentration exceeding
400\,\unit{mg\,m^{-3}} in the 1980s
\citep{lovstad_arungen_1993}. Algal blooms, low water transparency,
malodorous water, reduced fish stocks, and occasional mass mortality
of fish were observed in the lake in this period
\citep{ensby_arungen:_1984}. Despite
investments in sewage treatment and extensive changes in agricultural
practices since the 1970s to reduce nutrient leaching and erosion from
the catchment, algal growth is still high.

Predicting eutrophication responses to nutrient loading is a~complex
task due to the stochastic dynamics of lake's response to weather, the
confounding factor. Here we used one of the lake models (MyLake) \citep{saloranta_mylake_2007} based on system of
processes that have been identified as a~primary tool for improving
our understanding of recovery and progression of eutrophication
\citep{mooij_challenges_2010}. More precisely, we made combinations of
forcing data (meteorological and nutrient loading inputs) for the
model, with
\begin{enumerate}
  \item some forcing data being the actual records and
  \item others being the average day-to-day record for the period of 16\,\unit{years}, repeated 16 times.
\end{enumerate}
This approach criss-cross the day-to-day and year-to-year variability
in a~systematic way, and simulation outputs based on dissimilar
combinations of these ``actual'' and ``repeated avarage year'' can be
analysed once again from the day-to-day and year-to-year
variability. In principle, the actual data are more variable than the
repeated average year. If a~particular forcing variable is significant
in affecting the lake water quality, switching from the actual data to
repeated average year reduces the year-to-year variation in lake water
quality variables. Combinations of these two types of forcing data are
described in Sect.~2.5. This approach of comparing to average year by
opting in the average year for a selected variable is used in a~slightly
different context by \citet{johnk_summer_2008}, with a~difference
being that the ``average year'' was taken from one candidate year in
\citeauthor{johnk_summer_2008}, while the ``average year'' in the
current study was indeed day-of-year-specific year-to-year
average. More importantly, the current study demonstrates the use of
year-to-year variance for each day of year, which is a~powerful visual
and quantitative tool in detecting the critical season for a~variable.

The separation of two temporarily varied factors affecting the same
environmental receptor is not only useful in the lake water quality
modeling. For example, agricultural yield and forestry are affected by weather, soil
conditions, diseases, and tilling and
fertilisation amount and timing. Other examples may be climate change
impacts on physical landscape, such as glacial extent or surface water
ice cover, which are affected by stochastic meteorological conditions
and warming forcing which are mostly anthropogenic but also of natural
origin (e.g., volcanic activities), as well as regional multi-year
fluctuation such as the North Atlantic Oscillation or El Nino.


The main aim of the current study is to evaluate the relative
importance of year-to-year variation of two major factors, namely
meteorological forcing and nutrient loading, contributing to lake's
physical, chemical and biological conditions. We also visually inspect
how their respective significance varies over the season. To this aim,
(1) the MyLake model \citep{saloranta_mylake_2007} was first calibrated
against the lake data, (2) then various meteorological and nutrient loading
scenarios combining variation of these two major factors were
applied, and finally (3) the predicted lake responses were compared
among different scenarios, in particular, the variance among years,
rather than the absolute value of the predictions.

\section{Material and methods}

\subsection{Study site}

Lake {\AA}rungen is a~dimictic lake with maximum and average depths of
13 and 8\,\unit{m}, respectively. The lake is located in south-east
Norway ($59{\degree}41^{\prime}18^{\prime\prime}\,\mathrm{N}$,
$10{\degree}44^{\prime}38^{\prime\prime}\,\mathrm{E}$; Fig.~1),
25\,\unit{km} south of Oslo, and has a~surface area of
1.2\,\unit{km^2}. The catchment area covers 51\,\unit{km^2}, where
53\,{\%} is agricultural land, 34\,{\%} forestry, 10\,{\%} densely
populated and 3\,{\%} open water surfaces. The lake is highly exposed
to agricultural runoff that causes high nutrient and particle
loading. Runoff is mainly through 6 streams of 1.5 to 5\,\unit{km}
length. The outlet connects the lake to the marine environment as Lake
{\AA}rungen enters the Oslofjord through a~3\,\unit{km} long stream.

\subsection{Model}

MyLake is a~one-dimensional lake model, adapted from MINLAKE
\citep{riley_minlake:_1988}, which simulates daily changes in physical
and chemical dynamics over the depth gradient
\citep{saloranta_mylake_2007}. The model simulates ice and snow
dynamics in a~mechanistic manner and it has been applied to
winter-freezing lakes in Norway and Finland
\citep{kankaala_methanotrophic_2006,lydersen_thermos-projektet:_2003,saloranta_impacts_2009,saloranta_highlighting_2006, Couture2014}. It
was therefore considered as a~suitable model for Lake {\AA}rungen.

\subsection{Inputs and outputs}

MyLake requires input of meteorological forcing, runoff volume and
temperature, and fluxes of suspended inorganic particles and total
phosphorus (TP) to model phosphorus and phytoplankton dynamics in the
lake (Table~1). Meteorological data for daily air temperature, global
radiation, cloud cover, precipitation, relative humidity and wind
speed were obtained from the nearby meteorological station located at
the Norwegian University of Life Sciences
($59{\degree}39^{\prime}37^{\prime\prime}\,\mathrm{N}$,
$10{\degree}46^{\prime}54^{\prime\prime}\,\mathrm{E}$). 
Direct measurements of daily runoff volume, runoff water temperature,
and fluxes of suspended inorganic particles and total phosphorus to
the study lake {\AA}rungen was not available. However these values were
estimated using the Skuterud monitoring station (Fig. 1) with a
hydrovolumetric weir at which these runoff variables were monitored (1994--2010),
providing accurate flux at this subcatchment. In order to account for
runoff contributions of different types of landuse in other
subcatchments, such as agriculture and urban build up, we used
previously determined scaling factor that both correct for
flow and nutrient factors \citep{askilsrud_nutrient_2010}. 
The monitoring station is located at an inlet
stream to {\O}stensj{\o}vann
($59{\degree}41^{\prime}18^{\prime\prime}\,\mathrm{N}$,
$10{\degree}49^{\prime}45^{\prime\prime}\,\mathrm{E}$), a~small lake
of 0.4\,\unit{km^2} which drains into the Lake {\AA}rungen
\citep{deelstra_jord-_2007}. 
A~separate MyLake model was set up
for Lake {\O}stensj{\o}vann to account for the buffering effects of
this lake in the largest sub-catchment of Lake {\AA}rungen. The
simulated water properties of Lake {\O}stensj{\o}vann were combined
with runoff from the other sub-catchments as an estimate of the total
runoff to Lake {\AA}rungen.

Six variables including whole-lake average TP pool, mean surface
chlorophyll concentration, light attenuation coefficient, thermocline
depth, epilimnion temperature, and ice thickness were calculated from
unprocessed model outputs (Table~1) in order to ease interpretation of
the statistical analyses for scenario experiment described below.

\subsection{Model calibration}

Water temperature, TP, soluble reactive phosphorus (SRP), and
chlorophyll~$a$~concentration from the deepest location in the lake
were used to calibrate the model (Table~1). Vertical water temperature
profile was continuously logged every hour at eight depths between 0.7
and 12.6\,\unit{m} by Hobo pendant temperature loggers (model
64K-UA-002-64; Onset Computer Corporation, Bourne, MA, USA) in the
period from November~2008 to August~2010. Water samples for chemical
and biological analyses were collected with a~modified Ruttner water
sampler at seven depths twice a~month or monthly ($n=49$) from
January~2008 to September~2010. TP, SRP and chlorophyll~$a$~were
determined spectrophotometrically (UV-VIS Spectrophotometer UV-1201,
Shimadzu, Kyoto, Japan).

We deployed the Markov chain Monte Carlo (MCMC) method
\citep{andrieu_introduction_2003, saloranta_impacts_2009} to calibrate the model. The
calibration consisted of two stages. The first MCMC calibration stage
involved three physical parameters (see Table~2) that only affect heat
dynamics, in particular thermocline depth. This first calibration was
run against daily temperature measurements, using 2000 MCMC steps with
the first 1000 for burn-in. The second MCMC calibration stage
involving eight parameters (see Table~2) that affect phosphorus and
chlorophyll dynamics, but not temperature, was run against
measurements of TP, SRP, and chlorophyll~$a$~in 30\,000 MCMC steps
with the first 10\,000 for burn-in. 
For example, algal growth in the MyLake model is a function of
nutrient concentration, light availability and water temperature, and
amplitude of these factors were controlled by parameters. In the
present study, runoff was given as external input to the model, and
water temperature and underwater light conditions were determined in
the first stage of MCMC. Therefore, for the example of algal growth,
the second stage of MCMC only changed the amplitude of algal growth in
response to these external factors. 
For these MCMC applications,
convergence was monitored visually. Linear interpolation was used to
match model outputs on a~0.5\,\unit{m} vertical grid to the actual
measurement depths. Although it was not used directly during the MCMC
calibration, model goodness of fit was informally assessed by root
mean square error (RMSE). The medians of the posterior parameter
distributions generated from both stages of the MCMC calibrations were
used for the scenarios experiments described in the following.

\subsection{The variance-based method for disentangling confounding factors}

The model was run under four  scenarios for nutrient loading and weather
in order to quantify the respective impacts of weather variation and
loading conditions on phosphorus and phytoplankton dynamics
(Table~3). 
Runoff volume (in the runoff group) is predominantly influenced by
precipitation (in the meteorology group), but runoff volume was kept
together with runoff concentrations because the most important
influence the runoff has is the total amount of nutrients, which we
obtain by multiplying concentration by volume. 
These scenarios were based on input combination of observed
data (original data, 1994--2010) and synthetic data, where the
synthetic data were created by taking the year-to-year mean ($n=16$)
of each of the day of the year. Synthetic data repeats the calculated
mean year with 365 days sixteen times. The 29 February is removed in
year-to-year mean calculation, and 28 February was repeated to account
for the 29 in leap years.

This approach combines the actual forcing data for some variables
together with average year data for the other forcing data
variables. Comparing such a~synthetic scenario against the scenario
using full actual forcing data elucidates importance of year-to-year
variation of the second set of variables. For example, scenario C
(repeated average-year runoff) will necessarily have lower
year-to-year variation ($n = 16$) in output variables compared with
scenario A~(full original data) (Table~3); the difference in the
scenario input (i.e.~runoff input) accounts for the reduction in the
year-to-year varition in the output variables. By examining which
output variables are most reduced in year-to-year variation among all
output variables, we obtain crucial information on the importance of
runoff forcing data in the output. Thus we achieved the the goal of the
present study, namely disentangling the relative importance of
year-to-year variation of two major factors (meteorological and loading
inputs). The idea of mixing actual and average-year forcing data were
derived from the study conducted by \citet{johnk_summer_2008}, in
which the authors assessed the relative importance of various
meteorological variables (air temperature, cloudiness, wind speed) in
affecting various lake responses such as water temperature and algal
cell counts of 3 phytoplankton groups. The present study compares
standard deviation of the 16\,\unit{years} of simulation outputs for
each day of year for 6 output variables among several sceniarios.

Combining the repeated average input together with actual stochastic
input (such as in the scenarios B and C, Table 3) causes unrealistic
input for two reasons. One reason is the inconsistency among variables
on the daily basis. For example, because runoff is controlled by
precipitation, the scenario C for instance (original weather +
averaged runoff) on a daily basis may suffer from a potentially
undesirable situation such as high precipitation with no cloud on a
certain day. This is unrealistic, but from the lake ecosystem
perspective, the runoff's role is mostly as the source of nutrient,
and weather as the source of energy, and the fact that the lake water
is accumulation of old water from runoff introduced many days ago, the
influence of this inconsistency in combination is minimal. The second
reason is that the intra-variability or within-year variability that
naturally stochastic variables such as meteorological inputs should
have is lost in the present study's design. This can cause problems
such as not providing extreme wind events due to averaging despite
such wind events may be crucial for the on-set of ice formation, or
determination of thermocline. Hydrodynamic models are therefore
usually driven by stochastically generated time series (Semenov et
al. 1998 or Schlabing et al. 2012). Alternatively, most average
looking year could have been chosen as in\citet{johnk_summer_2008}. However, the current study requires the average to be done
on multiple criteria or variables, making choosing one example year
will also be not average year on all the variables. Due to the purpose
of reducing the year-to-year variation and not to be particularly
bound by a selected single year or randomly generated year, repeated
average year was used despite these potential problems. In the present
study, year-to-year variation in outputs of these four scenarios will
be discussed but not the actual values on the daily basis, and the
results are interpreted with care that scenario A is the most variable
year-to-year, that, either B or C is the next variable, and finally
that D is the least variable year-to-year. Although this care does not
safeguard the unrealistic intra-year variation, potential
inconsistency is detected. 


\subsection{Post-simulation methods}

All the data analysis and statistical analyses were done using R
(version 3.1.2).
A~two-way analysis of variance (ANOVA) was run on the 16\,\unit{years}
of water year based simulation statistics (water year mean, see
Table~1), among scenarios A, B, C and D (two weather factors by two
loading factors, see Table~3). All annual averages are computed over
the period from 1 October to 30 September, commonly used in Europe to
refer on a~hydrological year, or a~water year
\citep{otnes_hydrologi_1978}. Since treatment contrasts are nested
within water years we factored out the between-year variances to gain
a~greater power in the statistical tests. Principal component analysis
(PCA) was used to explore the relationships between meteorological and
land-related forcing and their relevance for the simulated lake
response. Four water years with extreme PCA scores were selected for
studying contrasting lake responses in closer details.

\section{Results}

\subsection{Calibration}

The simulated water temperature and thermal dynamics of the lake were
in agreement with the lake observed data. Observed water temperature
measurements were well predicted by simulation and the RMSE was less
than 2\,{\degree}C at all lake depths (Fig.~2). After the water
temperature calibration, parameters controlling TP, SRP, and
chlorophyll~$a$~were calibrated against observed data for the period
from January~2008 to September~2010. The epilimnion TP, SRP, and
chlorophyll~$a$~concentrations were well predicted by the model,
although their prediction was less successful than the prediction of
the water temperature. 
The TP and SRP were better predicted by the model in pelagic surface
water than in deep water whereas the chlorophyll~$a$~showed the opposite
pattern. 
In general, the model
simulated well TP and SRP, although both phosphorus forms were
overestimated in early spring and autumn at shallow depths, while
underestimated in bottom water. Simulated SRP concentrations were also
somewhat higher than observed in winters. However, the simulation
succeeded in showing a~decreasing trend of lake phosphorus in spring
and midsummer, and in mimicking its increase during the autumn mixing
of water. Although the simulated chlorophyll concentrations were lower
than the measured values, the model was able to predict seasonal
variation in phytoplankton primary production and to simulate high
phytoplankton biomasses in the lake epilimnion during midsummer.

\subsection{Variability in forcing data}

Inter-annual variation was expressed as the standard deviation in
inputs and outputs between the years. All weather inputs varied
between years (Fig.~3), with air temperature and global radiation
having the strongest seasonal pattern in inter-annual variation
(i.e. greater 16\,year variation as compared to year-to-year variation
on a~day-of-year basis). The inter-annual variation in air temperature
was strongest in the winter period, whereas global radiation varied
most during the summer months. The variation in cloud cover,
precipitation and relative humidity was generally similar across
seasons. The year-to-year variation for precipitation was particularly
high in December and in the period from July to September, the latter
reflecting extreme precipitation events. Wind speed varied most in
winters.

Runoff input data on water flow and concentrations of TP and suspended
matter all varied seasonally and between years. The variation in
runoff volume was greatest in the period from October to May. No clear
seasonal pattern in the degree of variability could be found for
suspended matter and TP fluxes, although the variation of TP influx
seemed to peak in February--March.

\subsection{Variability in simulation outputs: disentangled significance of confounding \hack{\newline} factors}

Differences in year-to-year variation among the scenarios (Fig.~4) and
the annual statistics (Table~4) illustrate the seasonal influence of
the external forcing on the thermal regime and the phosphorus and
phytoplankton dynamics in the lake. 
There is large amount of N in {\AA}rungen, as N is applied in surplus at
the agricultural fields. Even though there is some reduction in N
during summer, N is excluded as the limiting nutrient (Romarheim, 2012). 
The lake responded differently
between years; all simulated outputs, except ice thickness, showed
large variation in the beginning and at the end of the phytoplankton
growing season (Fig.~4). All simulated output variables were
influenced by external forcing as they varied
inter-annually for all model scenarios (see variance decomposition in Table 4). Ice thickness was
significantly affected by weather ($P < 0.001$) as both air
temperature and winter precipitation highly contributed to its
variation between years (Table~4; Fig.~4). The variation in
thermocline depth in May and October was well revealed by the model,
and seemed to be equally dependent on weather and loading. The
epilimnion temperature during the whole growing season was largely
controlled by weather. The TP content in the lake was most variable in
the period from November to January, and in April and July. Loading
could mostly explain the inter-annual TP variation in the lake,
whereas precipitation contributed to TP variation only in the spring,
and air temperature only during the winter period. Loading was the
overall most important factor in controlling the light attenuation
coefficient ($P < 0.001$) and surface chlorophyll
concentration ($P < 0.001$). Weather seemed to be important
in controlling chlorophyll and light attenuation in early spring
whereas loading was the most important factor controlling the both
variables from June to September.
The year-to-year variation in these two variables from June to September under the scenario C was higher than scenario A, despite that scenario C is less variable year-to-year than scenario A in loading. This might be because scenario C may distribute the autumn runoff events that is still significant after spreading over the years, and this might have caused the light-related variables to be unstable on a day-to-day scale.

The years 1996, 2000, 2006 and 2007 were the four most extreme years
determined on the basis of the PCA analysis (Fig.~5). The year 1996
was characterized by relatively low average annual air temperature,
a~thin cloud cover and low precipitation, which resulted in low
epilimnion water temperature, short lasting thermocline, low runoff
volume, and TP in the lake. The year 2007 represents an opposite to
1996, regarding weather characteristics and resulting model simulation
with relatively high average annual air temperature and
precipitation. Increase in wind speed, decrease in air temperature,
and decrease in precipitation coincided with increase in ice thickness
and glocal radiation, such as in 2003--2006, 2009, and 2010. These
weather conditions resulted in lower suspended inorganic particles and
TP in runoff which coincided with lower surface chlorophyll
concentration and light attenuation. The year 2006 was identified as
the extreme during this period, with a~cold winter followed by a~warm
summer. In contrast, the year 2000 was characterized as a~year with
less global radiation, lower summer air temperature, and higher wind
speed, but with higher winter temperature and precipitation. Such
weather conditions pronounced higher TP and suspended particle in
runoff compared with an average year, resulting in a~high surface
chlorophyll concentration and lower water transparency.

The methodological choice of using a repeating average year (see
Section 2.5) may have reduced the impact of extreme weather events but
probably caused little difference int he overall interpretation and
conclusion. 
But the result that the daily year-to-year
variation for either scenario B or C was closely following scenario A,
depending on the modelled variable (TP content, surface chlorophyll,
light attenuation coefficient for scenario B, and ice thickness,
thermocline depth, and epilimnion temperature for scenario C) provides
confidence that the conclusions will not change due to the
methodological limitation. 


\conclusions[Discussion]

\subsection{Year-to-year variability in lake responses}

Year-to-year weather variations, as well as the influence of catchment
land-use and hydrology, hinder our understanding of how individual
stressors may affect the lake response
\citep{blenckner_conceptual_2005}. Our model experiment, which
involved the input of weather and loading data for 16\,\unit{years},
was able to outline the respective importance of year-to-year
variation in external forcing on physical, chemical and biological
response in Lake {\AA}rungen. The combination of high forcing
variability and high lake response sensitivity made the inter-annual
variation most dramatically expressed in spring and autumn.

Air temperature, precipitation, and wind speed are the principal
factors influencing freshwater ecosystems in a~changing climate
\citep{nickus_direct_2010}. The lake thermal regime was to a~large
extent affected by weather conditions, particularly by air
temperature. Variable winter air temperatures were an important factor
influencing the heating and mixing processes during spring. A~dynamic
physical environment at the beginning of the growing season has
considerable influence on the phytoplankton community structure and
its dynamics \citep{weyhenmeyer_response_2002}. For example, increased
surface water temperatures in the English Lake District
\citep{george_impact_2007} and incomplete water mixing in Lake
Constance \citep{straile_complex_2003} have earlier been associated
with mild winters. High inter-annual variation in winter air
temperatures in Lake {\AA}rungen was reflected in the simulated ice
thickness and phenology of ice formation, with ice forming in
December--January and disappearing in March--April. Also in other lake
studies, thinner and shorter ice cover has been related to mild
winters \citep{nickus_direct_2010}. In this study, the time of
thermocline formation varied among years by more than one month, from
mid-April to mid-May. The large year-to-year variation in thermocline
depth and duration could lead to changes in temperature, light, and
nutrient condition in the lake, which further shape the phytoplankton
community and determine its total biomass
\citep{padisak_convergence_2010,zohary_phytoplankton_2010}. For
instance, early disappearance of diatoms and high development of
cyanobacteria in European lakes has been related to winter warming and
increased water temperature
\citep{weyhenmeyer_response_2002}. Furthermore, increase in water
stability favours the buoyant phytoplankton species such as bloom
forming cyanobacteria
\citep{reynolds_effects_1983,winder_temporal_2008}.

Year-to-year variation in phosphorus content in the lake was highly
influenced by nutrient loading. This indicates that the external
nutrient supply remains an important source of phosphorus in the
lake. Although the changes in nutrient loading are primarily linked to
anthropogenic activities in the catchment, in particular to practices
in agriculture, the short-term variations in weather and runoff can
also influence the nutrient supply from external sources. Lake
{\AA}rungen is surrounded by agricultural land, and is especially
sensitive to variable weather conditions that promote nutrient loading
from the catchment. More precisely, air temperature and rainfall
frequency and intensity affect the runoff and the soil erosion
pattern, particularly during the winter period. Increase in winter
temperatures occuring together with frequent freezing and melting
events increases the risk of erosion, which again will increase the
nutrient loading to the lake
\citep{bechmann_phosphorus_2005,jeppesen_climate_2009,noges_role_2007}. Although
not statistically demonstrated in the present study, the indirect
impacts of weather \mbox{conditions} on discharge may still be important in
regulating the nutrient dynamics. Variable winter weather conditions,
and the time of ice out were most important cause of year-to-year
variable phosphorus content in the period from November to January and
in April in Lake {\AA}rungen. Enhanced phosphorus concentrations in
streams during winter, and high phosphorus loading in early spring,
both contribute to the total lake phosphorus concentration during the
following summer in two Norwegian lakes with agricultural catchments
\citep{bechmann_phosphorus_2005}. In addition, the variation in summer
TP content could also be the result of between-year variation in
rainfall, especially due to extreme precipitation events as observed
for particular days in July. High inter-annual variation in TP content
can consequently result in variable phytoplankton biomass between
years.

Phytoplankton biomass and light were significantly affected by
loading, although atmospheric forcing also contributed to their
inter-annual variation in the lake. The effect of loading was
pronounced during the whole algal growing season, whereas weather had
the strongest effect in early spring and from mid-August to the end of
the growing season. Thermal stratification is highly dependent on
weather and may further influence water mixing as well as light and
nutrient regimes, which are important in controlling the phytoplankton
dynamics
\citep{padisak_convergence_2010,zohary_phytoplankton_2010}. Similar
pattern of year-to-year variation in water temperature, chlorophyll
concentration, and light attenuation indicates a~close relationship
between these variables. Higher air temperature promoted higher water
temperature and higher stability of the thermal stratification which
enhanced phytoplankton production of bloom forming cyanobacteria
\citep{reynolds_effects_1983,weyhenmeyer_response_2002}. Runoff and
soil erosion, caused by intense precipitation and frequent melting of
snow and ice during mild winters, affect eutrophication and water
turbidity \citep{bechmann_phosphorus_2005,jeppesen_climate_2009}. Thus
light may limit phytoplankton growth more than nutrients in highly
turbid lakes such as Lake {\AA}rungen
\citep{dokulil_environmental_1994}. Reduced light availability may be
crucial for the competitive success of cyanobacteria which are
functionally adapted to low light conditions
\citep{litchman_population_1998}. Particularly high dominance of
cyanobacteria has been \mbox{observed} in the Lake {\AA}rungen after mild
winters followed by low light conditions in spring (\mbox{Romarheim} et~al.,
unpublished). Therefore, additional measures to control soil erosion
may need to be implemented in water management, not only to reduce the
supply of nutrients, but also to avoid low water transparency which
may favour development of potentially toxic cyanobacteria.

\subsection{Implications for lake management}

According to our PCA analysis, most of the 1990s were categorized by
winters with higher temperatures and more rainfall.  The mild winters
were related to a~positive North Atlantic Oscillation (NAO) phase
which has been shown to strongly influence physicochemical and
biological responses in western European lakes
\citep{george_impact_2007,straile_response_2003,weyhenmeyer_response_2002}. The
effect of climate condition on water ecosystems, however, should be
considered individually as the lake response is also determined by the
lake's geographical position, landscape topography, and the lake's
morphometry and mixing regime \citep{nickus_direct_2010}. Our model
experiment indicated that greater inflow of suspended matter and
phosphorus to the lake {\AA}rungen is expected after mild winters with
high precipitation. Consequently, higher chlorophyll concentrations
and greater light attenuation were predicted after mild and wet
winters such in the year of 2000. Mild winters may thus counteract
measures aimed to reduce external nutrient supply and to control
phytoplankton production in cold temperate lakes, appearing to have
contributed to little improvement. On the contrary, cold winters were
related to thicker ice layer, less inflow of suspended matter and
phosphorus, and low chlorophyll and light attenuation. This was
consistent with the observed increase in water transparency and
reduction of phytoplankton biomass, particularly of cyanobacteria, in
Lake {\AA}rungen after the cold winter in 2010 (Romarheim et~al.,
unpublished). Special attention must therefore be given to management
practices, which should minimize the use of fertilizers and reduce the
risk of nutrient runoff and soil erosion, especially in areas that
drains directly into the lake. Higher annual air temperature coincided
with warmer epilimnion, shallower thermocline and extended summer
\mbox{stratification} such as for the year of 2007. In addition, high annual
precipitation and runoff volume, particularly in summer, coincided
with warmer years.

Globally, all years in the period from 1995 to 2006, with exception of
1996, were among the warmest since 1850
\citep{trenberth_observations:_2007}. Likewise, the year 1996 was
characterised with low average annual air temperature in our model
experiment. Lower annual air temperature and low rainfall in 1996 led
to low epilimnion temperature, and a~deep and short lasting
thermocline. According to future climate predictions for Scandinavia,
warmer winters and increased winter precipitation are expected in
south-eastern Norway \citep{hanssen-bauer_statistical_2005}. If so, we
should also expect more intensive soil erosion, higher phosphorus
loading, lower water transparency, and greater phytoplankton
biomasses, primarily of cyanobacteira in the lakes. Global climate
changes and inter-annual variations in the local weather both
directly, and indirectly through an impact on the catchment, influence
the physicochemical and biological processes in lakes. 
The limnological and biogeochemical knowledge of this lake identified
by decomposing year-to-year variation of the two factors, carries
potential in connecting future management. Runoff is partially
controlled by precipitation which in turn is predicted to change, and
so are air temperature and global radiation. 
Therefore, the
effect of climate should be considered in future decision-making
concerning water management.

\begin{acknowledgements}
We thank the Norwegian Institute for Agricultural and Environmental Research
(Bioforsk) for providing runoff data from Skuterud monitoring station. We also
thank Johnny~Kristiansen at the Department of Environmental Sciences, Norwegian
University of Life Sciences for excellent assistance during the field and
laboratory work. PURA, an EU water framework directive project within the Follo,
Oslo region, Norway, is also thanked for financial support during the study. This
work was partly carried out with support from the Research Council of Norway
(Project Eutropia~190028/S30). We thank Tuomo Saloranta for providing an example
code for parameter estimation. The calculations presented in this manuscript was
made possible with freely available software, in particular, Octave and R, and we
are grateful for efforts made by the authors of the free software. This open-access
publication was made possible with the Open Access Publication Fund at the University of Oslo.
\end{acknowledgements}

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\begin{table}
\caption{Input and output data, and observed lake data for the calibration of the MyLake model and statistics for the ANOVA and PCA analyses.}
\scalebox{.85}[.85]
{\begin{tabular}{lll}
\tophline
MyLake inputs &MyLake outputs (selected) &Observed lake data \\
\middlehline
\textit{Meteorological data}$^{\text{a, f}}$ &\textit{Calibration purpose}$^{\mathrm{a}}$ &\textit{Calibration purpose} \\
Air temperature &(every 0.5\,\unit{m} by depth) &(at 7 depths) \\
Global radiation &Water temperature &Water temperature$^{\mathrm{a}}$ \\
Cloud cover &TP concentration &TP concentration$^{\mathrm{b}}$ \\
Precipitation &SRP concentration &SRP concentration$^{\mathrm{b}}$ \\
Relative humidity &Chlorophyll concentration &Chlorophyll~$a$~concentration$^{\mathrm{b}}$ \\
Wind speed &&\\
 &\textit{Statistics calculated for PCA} &\\
\textit{Runoff}$^{\text{a, g}}$ &(volume weighted mean 0--3.0\,\unit{m}) &\\
Flow volume &TP content &\\
Water temperature &Mean surface chlorophyll &\\
Suspended matter flux &Light attenuation coeficient &\\
TP flux &Thermocline depth &\\
 &Mean epilimnion temperature$^{\mathrm{e}}$ &\\
\textit{Statistics calculated for PCA} &Ice thickness &\\
Global radiation$^{\mathrm{c}}$ &&\\
Cloud cover$^{\mathrm{c}}$ &&\\
Air temperature$^{\mathrm{c}}$ &&\\
Wind speed$^{\mathrm{c}}$ &&\\
Precipitation$^{\mathrm{c}}$ &&\\
Flow volume$^{\mathrm{c}}$ &&\\
Winter air temperature$^{\text{d, h}}$ &&\\
Summer air temperature$^{\text{d, i}}$ &&\\
Winter precipitation$^{\text{d, h}}$ &&\\
Summer precipitation$^{\text{d, i}}$ &&\\
Suspended matter flux$^{\mathrm{d}}$ &&\\
TP flux$^{\mathrm{d}}$ &&\\
\bottomhline
\end{tabular}}
\scalebox{.85}[.85]
{\belowtable{\hack{\vspace*{2mm}}
$^{\mathrm{a}}$~Daily data.
$^{\mathrm{b}}$~Biweekly data.
$^{\mathrm{c}}$~Annual mean.
$^{\mathrm{d}}$~Water year basis (October through September).
$^{\mathrm{e}}$~Volume weighted above thermocline depth.
$^{\mathrm{f}}$~Inferred with data from {\AA}s meteorological station.
$^{\mathrm{g}}$~Inferred with data from Skuterud monitoring station and land use.
$^{\mathrm{h}}$~December through March, mean.
$^{\mathrm{i}}$~June through September, mean.}}
\end{table}

\begin{table}%T2
\caption{Parameters involved in calibration based on two-stage Markov chain Monte Carlo (MCMC) application (first stage for three parameters using 2000 MCMC steps with 1000 steps for burn-in and second stage for eight parameters using 30\,000 MCMC steps with 10\,000 for burn-in). MyLake equation numbers refer to the original model description \citep{saloranta_mylake_2007}. Median values were chosen among the posterior parameter distribution.}
\scalebox{.60}[.60]
{\begin{tabular}{lllll}
\tophline
Parameter &Value &Equation &Unit &Prior distribution\\
\middlehline
\textit{Physical parameters} &&&&\\
Open-water vertical diffusion coefficient &$7.56 \times 10^{-3}$&Eq.~(10) &\unit{m^2\,day^{-1}} &$0.00706 \times (1.18 \times (10^x) \times 10^{-6})^{0.56}$ where $x\sim \textrm{N}(0, 1)$  \\
Wind sheltering coefficient &$7.96 \times 10 ^{-2}$ &Eq.~(13) &-- &
$1-e^{-0.3\times1.18\times 10^x\times 10^{-6}}$ where $x\sim \textrm{N}(0, 1)$\\
Minimum possible stability frequency &$9.31 \times 10^{-5}$ &Eq.~(10)&\unit{s^{-2}} & $1.14 \times 10^{-4} \times 10^x$ where $x\sim \textrm{N}(0, 1)$\\
\cline{1-5}
\textit{Biological and chemical parameters} &&&&\\
PAR saturation level for photosynthesis &$2.04 \times 10^{-4}$&Eq.~(29) &mol\,quanta\,\unit{m^{-2}\,s^{-1}} & $3.00\times10^{-5}\times 10^x$ where $x\sim \textrm{N}(0, 0.5)$\\
Particle resuspension mass transfer coefficient &$2.94 \times 10^{-5}$ &{\S} 2.7 &\unit{m\,day^{-1}}, dry &$3.63\times10^{-7}\times 10^x$ where $x\sim \textrm{N}(0, 0.5)$\\
Settling velocity for suspended matter &$1.38$ &Eq.~(20) &\unit{m\,day^{-1}} &$0.25 \times 10^x$ where $x\sim \textrm{N}(0, 0.5)$\\
Settling velocity for chlorophyll &$7.31 \times 10^{-2}$ &Eq.~(20)&\unit{m\,day^{-1}} &$0.200 \times 10^x $ where $x\sim \textrm{N}(0, 0.5)$\\
Specific mortality rate of phytoplankton &$1.86 \times 10^{-1}$ &Eq.~(26) &\unit{day^{-1}} &$0.200 \times 10^x $ where $x\sim \textrm{N}(0, 0.5)$\\
Max specific growth rate of phytoplankton &$1.76$ &Eq.~(27) &\unit{day^{-1}} &$1.50 \times 10^x $ where $x\sim \textrm{N}(0, 0.5)$\\
Half saturation inorganic phosphorus &$9.99 \times 10^2$ &Eq.~(24) &\unit{mg\,m^{-3}}& $2500 \times 10^x $ where $x\sim \textrm{N}(0, 0.5)$\\
concentration for Langmuir isotherm &&&&\\
Saturation level for inorganic phosphorus isotherm &$4.96 \times 10^4$ &Eq.~(24) &mg \unit{kg^{-1}} &$8000 \times 10^x $ where $x\sim \textrm{N}(0, 0.5)$\\
\bottomhline
\end{tabular}}
\end{table}

\begin{table}%T3
\caption{Model scenarios. The scenarios comprise either original input data (denoted O), pseudo repeated average year based on 16\,years of input data (denoted R), or a~combination of O and R.}
\begin{tabular}{lcccc}
\tophline
Model scenarios &A &B &C &D \\
\middlehline
Model inputs &&&&\\
\cline{1-5}
\textit{Weather} &&&&\\
Global radiation &O &R &O &R \\
Cloud cover &O &R &O &R \\
Relative humidity &O &R &O &R \\
Wind speed &O &R &O &R \\
Air pressure &O &R &O &R \\
Air temperature &O &R &O &R \\
Precipitation &O &R &O &R \\
\cline{1-5}
\textit{Runoff} &&&&\\
Flow volume &O &O &R &R \\
Suspended matter flux &O &O &R &R \\
Inflow water temperature &O &O &R &R \\
TP flux &O &O &R &R \\
\bottomhline
\end{tabular}
\end{table}

\begin{table}
\caption{Summary results for six two-factor within-subject ANOVA ($n =
  16 \times 2 \times 2$). Significance of additive and interactive
  effects of weather (two levels, original O or repeated average R)
  and loading (two levels, original O or repeated average R) inputs on
  the six selected model outputs are shown using $P$~value. High
  $P$~values for interactive effects for all six tests indicate pure
  additive two-factor model and test for each factor
  separately. The variance decomposition shows the relative
  contributions of each factor and interaction to the total sum of
  squares with the between-year error term factored out.}
\scalebox{.70}[.70]{\begin{tabular}{lrrrrrrrrr}
\tophline
& F-value & & & P-value &&& Variance decomposition &&\\
\middlehline
Model outputs &W &L &W\,$\times$\,L &W &L &W\,$\times$\,L &W &L &W\,$\times$\,L\\
\middlehline
Ice thickness    & 31.93 & 0.13  & 0.34 & $ < 0.000 $ & 0.723 & 0.565 & 0.413 & 0.002 & 0.004 \\
Thermocline depth  & 0.27  & 0.91  & 1.93 & 0.605 & 0.346 & 0.172 & 0.006 & 0.019 & 0.040 \\
Epilimnion temperature   & 3.19  & 4.91  & 0.39 & 0.081 & 0.032 & 0.537 & 0.060 & 0.092 & 0.007 \\
TP content   & 0.81  & 1.26  & 0.02 & 0.374 & 0.268 & 0.888 & 0.017 & 0.027 & 0.000 \\
Surface chlorophyll   & 0.05  & 18.76 & 0.14 & 0.827 &$ < 0.000 $& 0.713 & 0.001 & 0.293 & 0.002 \\
Light attenuation coefficient & 0.78  & 14.82 & 0.23 & 0.382 & $ < 0.000 $& 0.631 & 0.013 & 0.244 & 0.004 \\
\bottomhline
\end{tabular}}
\end{table}

\begin{figure}
\includegraphics[width=8.3cm]{Fig1.pdf}
\caption{Map of catchment draining into (1) Lake {\AA}rungen, with the (2) weather station at {\AA}s, (3) Lake {\O}stensj{\o}vannet and (4) the Skuterud monitoring station. Runoff data from the Skuterud sub-catchment (indicated by dark shading) are scaled up according to land area and usage of the rest of the catchment to estimate the total loading to Lake {\AA}rungen.}
\end{figure}

\begin{figure}
\includegraphics[width=9cm]{Fig2.pdf}
\caption{Simulated (line) and observed (circles) lake state variables for water temperature, TP, SRP, and chlorophyll~$a$~concentrations at seven depths. RMSE values are in their respective original units, and they are shown only for the presentation purpose and were not used during the calibration. See text for the details of calibration procedure.}
\end{figure}

\begin{figure}
\includegraphics[width=8cm]{Fig3.pdf}
\caption{Input variability shown as standard deviations on a~water year scale (day-by-day, year-to-year variation, $n =16$, curves), with the overall 16\,year standard deviations indicated by horizontal lines.}
\end{figure}

\begin{figure}
\includegraphics[width=14cm]{Fig4.pdf}
\caption{Output variability shown as standard deviations on a~water year 
scale (day-by-day, year-to-year variation, $n = 16$) for 
scenarios A~(black solid line or top solid line), 
B (red dotted line), 
C (blue dashed line), and 
D (green solid line or bottom solid line). 
Consequently, each panel illustrates $4 \times 365$ standard 
deviation values, and each standard deviation is based 
on sample size $n = 16$. See Table~3 for scenario configurations.}
\end{figure}

\begin{figure}
\includegraphics[width=14cm]{Fig5.pdf}
\caption{Principal component analysis (PCA) loadings for the two
  greatest components (explaining 39.0 \% and 16.5 \% of variance) and
  scores for the two components for 16 water
  years (letters). Black coding for PC loading indicates the weather input,
  brown the runoff input, and blue the lake simulation.}
\end{figure}

\end{document}
