Introduction
The first requirement of an ocean model is the definition of the
system that the model is going to represent. As illustrated in
Fig. , this usually amounts to defining an
appropriate separation between the system (A) and the
environment (B). For instance, in this study, we always use
a stand-alone ocean model, which means that the atmosphere is not
included in the system (A), but in the environment (B). A key property of any ocean model is also the separation
between resolved scales (in A) and unresolved scales
(in B), defining the spectral window that the model is going
to represent. In a similar way, marine ecosystems are too complex to
be entirely included in A. They can only be represented by
a limited number of variables Ci,i=1,…,n, providing
a synthetic picture of the ecosystem, while the remaining
biogeochemical diversity is included in B.
Even if the union of the two systems A and B could
be assumed deterministic, this is in general not true for
system A alone. The future evolution of A does not
only depend on its own dynamics and initial condition, but also on the
interactions between A and B. This means that the
only two ways of obtaining a deterministic model for A are
either to assume that the evolution of B is known (as is usually
done for the atmosphere in stand-alone ocean models) or to assume that
the effect of B can be parameterized as a function of what
happens in A (as is usually done for unresolved scales and
unresolved diversity). It is however important to recognize that this
is always an approximation and that B is often an important
source of uncertainty in the predictions made for A.
To obtain a reliable predictive model for A (in the sense
given in , and ), a consistent description of this
uncertainty should be embedded in the model itself. This transforms
the deterministic model into a probabilistic model, which fully
characterizes the quantity of information that the model contains
about A. Two important advantages of this probabilistic
approach are (i) to allow objective statistical comparison between
model and observations (by providing sufficient conditions to
invalidate the model; see for instance ), and (ii) to
provide a coherent description of model uncertainty
to data assimilation systems. The objective
of the modeller also changes: instead of designing a deterministic
model as close as possible to observations, a probabilistic model that
is both reliable (not yet invalidated by observations) and as
informative as possible about A must be designed.
Schematic of the separation between resolved
and unresolved processes (systems A and B).
Even if A∪B can be assumed deterministic,
system A alone is not deterministic in general,
because of the interactions with system B.
In practice, for a complex system, it is usually impossible to compute
explicitly the probability distribution describing the forecast. In
general, only a limited size sample of the distribution can be
obtained through an ensemble of model simulations, as is routinely done
in any ensemble data assimilation system (see ).
Ensemble simulations are produced by randomly sampling the various
kinds of uncertainty (in the dynamical laws, in the forcing, in the
parameters, in the initial conditions, etc.) in their respective
probability distribution (Monte Carlo simulations). To allow
objective comparison with observations or to correctly deal with model
uncertainties in data assimilation problems, non-deterministic models
are thus needed in many ocean applications. The most direct approach
to introducing an appropriate level of randomness in ocean models is to
use stochastic processes to mimic the effect of uncertainties.
In the discussion above (summarized in Fig. ), a specific
focus was given to uncertainties resulting from the effect that unresolved
processes (in B) produce on the system (A). However, there
is a variety of other sources of uncertainty in ocean models (e.g. numerical
schemes, machine accuracy, etc.) that do not enter this particular sketch,
and that may also require a stochastic approach .
Stochastic parameterizations explicitly simulating model uncertainty
were first applied to ensemble weather forecasting by
about 15 years ago. Since then,
stochastic parameterizations have emerged as a quickly developing area
of research in meteorology . In oceanography, however,
most state-of-the-art dynamical models are still deterministic. Up to
now, the development of stochastic dynamical equations has been mainly
focused on stochastic parameterization of Reynolds stresses in
idealized ocean modelling systems (see , and , for
a review). Only a few exploratory studies have attempted to
explicitly simulate uncertainties in realistic dynamical ocean models:
this has been done for the ocean circulation , for the
ocean ecosystem , and for the sea ice dynamics
. These preliminary studies nonetheless already show
that uncertainties can play a major role in dominant dynamical
behaviours of marine systems.
In line with these studies, the objective of this paper is to propose
a generic implementation of these stochastic parameterizations, and to
investigate several applications in which the randomness of the ocean
system may be an important issue. This is synthetically implemented
in the ocean model (see Sect. ) by adding one additional
module providing appropriate random processes to any non-deterministic
component of the system (circulation, ecosystem, sea ice). The method
is designed to be simple enough to allow a quick check of the effect
of uncertainties in the system, and flexible enough to apply to
various sources of uncertainty (atmosphere, unresolved scales,
unresolved diversity, etc.). Three applications are then illustrated
in Sect. , showing that the explicit simulation of
uncertainty can be important in a wide variety of ocean systems, by
stimulating important non-deterministic dynamical behaviours. The
first application (circulation model) is the same application as
in , but this previous paper only presented the average
effect of the stochastic parameterization, whereas the focus is here
on the randomness that is produced in the large-scale ocean
circulation. The second application (ecosystem model) is a first
attempt to apply stochastic parameterizations and to explicitly
simulate randomness in a basin-scale ocean ecosystem model. The third
application (sea ice model) is an attempt to reproduce the
parameterization developed in in our ocean model using
the generic implementation presented in Sect. , and to
illustrate the randomness that is generated in the interannual
variability of sea ice thickness.
Stochastic formulation of NEMO
The ocean model used in this study is NEMO (Nucleus for a European
Model of the Ocean), as described in . NEMO is the
European modelling framework for oceanographic research, operational
oceanography, seasonal forecast and climate studies. This model
system embeds various model components (see
http://www.nemo-ocean.eu/), including a circulation model (OPA,
Océan PArallélisé), ecosystem models, with various levels of
complexity (e.g. LOBSTER, LOCEAN Simulation Tool for Ecosystem and
Resources), and a sea ice model (LIM, Louvain-la-Neuve Ice Model).
The purpose of this section is to shortly describe the three kinds of
stochastic parameterizations that have been implemented in NEMO, and
to show that, from a technical point of view, they can be unified in
one single new module in NEMO, feeding the various sources of
randomness in the model. (More technical details about this module
can be found in the Appendix.)
Order n autoregressive processes
The starting point of our implementation of stochastic
parameterizations in NEMO is to observe that many existing
parameterizations are based on autoregressive processes, which are
used as a basic source of randomness to transform a deterministic
model into a probabilistic model. A generic approach is thus to add
one single new module in NEMO, generating processes with appropriate
statistics to simulate each kind of uncertainty in the model (see
examples in Sect. ).
In practice, at every model grid point, independent Gaussian
autoregressive processes ξ(i),i=1,…,m are first
generated using the same basic equation:
ξk+1(i)=a(i)ξk(i)+b(i)w(i)+c(i),
where k is the index of the model time step; and a(i),
b(i), c(i) are parameters defining the mean (μ(i)), SD
(σ(i)) and correlation timescale (τ(i)) of each
process:
for order 1 processes (AR(1)), w(i) is a Gaussian white noise, with
zero mean and SD equal to 1, and the parameters a(i), b(i),
and c(i) are given bya(i)=φb(i)=σ(i)1-φ2withφ=exp-1/τ(i)c(i)=μ(i)1-φ
for order n>1 processes (AR(n)), w(i) is an order n-1
autoregressive process, with zero mean, and SD equal to σ(i),
correlation timescale equal to τ(i), and the parameters
a(i), b(i), and c(i) are given bya(i)=φb(i)=n-12(4n-3)1-φ2withφ=exp-1/τ(i)c(i)=μ(i)1-φ
In this way, higher-order processes can be easily generated
recursively using the same piece of code implementing
Eq. (), and using successively processes from order 0
to n-1 as w(i). The parameters in Eq. () are
computed so that this recursive application of Eq. ()
leads to processes with the required SD and correlation timescale,
with the additional condition that the n-1 first derivatives of the
autocorrelation function are equal to zero at t=0, so that the
resulting processes become smoother and smoother as n is increased.
AR(2) processes (with other specifications) have already been applied
in several studies , and will be used in this
paper in the sea ice model application (see Sect. ).
Second, a spatial dependence between the processes can easily be
introduced by applying a spatial filter to the ξ(i). This can
be done either by applying a simple filter window to the ξ(i)
2-D or 3-D matrices ξ̃(i)=F[ξ(i)], or by
solving an elliptic equation: L[ξ̃(i)]=ξ(i).
In both cases, the filtering operator could be made flow dependent, or
more generally, the filter characteristics could be modified according
to anything that is resolved by the ocean model (in system A
in Fig. ). Technically, this only requires that the
description of the ocean model is made available to the filtering
routines. This filtering option (using a simple Laplacian filter) is
used in the sea ice application (see Sect. ).
Third, the marginal distribution of the stochastic processes can also
be easily modified by applying a nonlinear change of variable
(anamorphosis transformation) to the ξ(i) before using them in
the model ξ^(i)=T[ξ(i)]. This idea is similar
to what is done in ensemble data assimilation methods to transform
variables with non-Gaussian marginal distribution into Gaussian
variables . For instance, this method can
be very useful if the description of uncertainties in the model
requires positive random numbers. In this case, anamorphosis
transformation can be applied to transform the Gaussian ξ(i)
into positive ξ^(i) with lognormal or gamma distribution.
This anamorphosis option (using a gamma distribution) is used in the
sea ice application (see Sect. ).
Overall, this method provides quite a simple and generic way of
generating a wide class of stochastic processes. However, this also
means that new model parameters are needed to specify each of these
stochastic processes. As in any parameterization of lacking physics,
a very important issue is then to tune these new parameters using
either first principles, model simulations, or real-world
observations. This key problem of assessing the parameters involved
in Eq. () cannot be addressed in the present paper,
and we can only provide a very brief overview of the nature of the
problem. Many existing studies (e.g. )
already addressed the problem of choosing the coefficients of the AR(n)
processes to simulate the Reynolds stresses in atmospheric and oceanic
flows. Considerable progress has been made for this important
problem, but not all unresolved processes have received so much
attention, and it is often still difficult to figure out how to derive
the parameters of the AR(n) processes.
Referring to the sketch presented in Fig. , the general idea
to tune the parameters is to obtain reliable probabilistic information on
what happens in system B, and to reduce this information to a simple
statistical model (e.g. the autoregressive model described above). More
precisely, the probability distribution simulating the effect of B
should also be conditioned on what happens in system A. For
instance, it can be very important that the probability distribution for the
state of the atmosphere (e.g. surface winds) be conditioned on the state of
the ocean model (e.g. mesoscale eddies), to simulate the interaction between
A and B. Similarly, the probability distribution for
unresolved scales or unresolved diversity usually depends on what happens in
system A. This need to correctly simulate conditional probability
distributions explains why the tuning of the parameters is not easy, and why
an extensive database to learn the statistical behaviour of the coupling
between A and B is often necessary. In practice, this
learning information can be obtained either from observations of the two
systems or from other models explicitly simulating the coupling between
A and B. For instance, high-resolution observations or
high-resolution models can be used to tune a statistical model for unresolved
scales; a model of the atmospheric boundary layer can be used to learn the
statistical dependence of the state of the atmosphere on the ocean
conditions; a generic biogeochemical model involving a large number of
species can be used to understand the statistical effect that unresolved
diversity can produce in a simple ecosystem model.
The identification of an appropriate statistical model is thus an important
intermediate step that is far from straightforward, and for which it is
difficult to provide very precise guidelines. Despite these difficulties, our
point of view is that the tuning of the system is usually even more
problematic with a deterministic parameterization of unresolved processes,
since no deterministic simulation could exactly fit the real behaviour of the
system.
By explicitly simulating uncertainties, we can describe the actual random
behaviour of the system (see Fig. ); ensemble
simulations can be objectively compared to observations (using
probabilistic methods, see ); and the
model (including the stochastic parameters) can be rejected as soon as
the ensemble is not reliable. Unknown parameters could also be tuned
by solving inverse problems, until ensemble reliability is achieved.
Stochastic perturbed parameterized tendency
A first way of explicitly simulating uncertainties in meteorological
weather forecast was introduced about 15 years ago in the
ECMWF ensemble forecasting system . Their basic idea
was to separate the model tendency (M) into non-parameterized
(NP) and parameterized (P) tendencies (M=NP+P). The non-parameterized tendency (NP)
contains all processes that are fully resolved by the model, and can
be assumed free of uncertainties. The parameterized tendency (P) contains the parameterization of the effect of unresolved
processes (system B in Fig. ), which is
essentially uncertain. The stochastic parameterization is then
introduced by multiplying the parameterized tendency (P) by
a random noise, explicitly simulating the uncertainties in P.
The basic motivation was to produce ensemble forecasts with enhanced
dispersion to improve their reliability (i.e. their consistency with
available observations). This SPPT (for stochastic
perturbed parameterized tendency) parameterization is still used today in the
ECMWF ensemble forecasting system .
This kind of stochastic parameterization is also meaningful in ocean
models, and it can be directly applied in the model using the generic
implementation described in Sect. . This can be done
by using one or several of the ξ(i) given by
Eq. () as multiplicative noise for the various terms
of the parameterized tendency:
dxdt=NP(x,u,p,t)+∑i=1mP(i)(x,u,p,t)ξ(i)(t)with∑i=1mP(i)=P
where t is time; x, the model state vector; u,
the model forcing; and p, the vector of model parameters.
In this case, the mean of the ξ(i) must be set to 1, assuming
that the model parameterized tendencies are unbiased, and the other
statistical parameters (SD, time and space correlation structure,
marginal distribution) are free to be adjusted to any reasonable
assumption about the uncertainties. In ocean models, this stochastic
parameterization can be applied to any parameterization of unresolved
processes (see Fig. ), as for instance the diffusion
operators, simulating the effect of unresolved scales, the air–sea
turbulent fluxes, the parameterization of the various functions of the
ecosystem dynamics, usually describing the unresolved biologic
diversity, etc. An example of this SPPT parameterization is given
in the ecosystem application (see Sect. ).
Stochastic parameterization of unresolved fluctuations
Another way of explicitly simulating uncertainties in ocean models is
to directly represent the effect of unresolved scales in the model
equations using stochastic processes. Unresolved scales can indeed
produce a large-scale effect as a result of the nonlinearity of the
model equations. Important nonlinear terms in ocean models are for
instance the advection term, the seawater equation of state, the
functions describing the behaviour of the ecosystem, etc.
Concerning the advection term, the effect of unresolved scales is
usually parameterized as an additional diffusion, while for the other
terms it is most often ignored. However, in many cases, a direct way
of simulating this effect would be to generate an ensemble of random
fluctuations δx(i) with the same statistical
properties as the unresolved scales, and to average the model operator
over the ensemble:
dxdt=1m∑i=1mMx+δx(i),u,p,twith∑i=1mδx(i)=0.
This corresponds to an averaging of the model equations over a set of
fluctuations δx(i) representing the unresolved scales.
The zero mean fluctuations δx(i) can produce an
average effect (corresponding to an interaction between A
and B in Fig. ) as soon as the model M is nonlinear. In this parameterization, the number of independent
fluctuations (m) and the statistics for each of them should be
chosen to simulate the properties of the unresolved scales as
accurately as possible.
Obviously, the main difficulty with this method is to generate
fluctuations δx(i) with the right statistics to
faithfully correspond to the statistics of unresolved processes. As
a first very simple approach, this can be done using one or several of
the ξ(i) given by Eq. (), either by assuming
that the statistics of δx(i) can be directly
approximated by the simple statistical structure of autoregressive
processes ξ(i), or by assuming that δx(i) can
be computed as a joint function of the model state x and
the autoregressive processes ξ(i). For example, if the
fluctuations can be assumed proportional to the large-scale
gradient ∇x of the state vector, the
fluctuations δx(i) could be computed as the scalar
product of ∇x with random walks ξ(i):
δx(i)=ξ(i)⋅∇x.
This particular case corresponds to the stochastic parameterization
proposed in to simulate the effect of unresolved scales
in the computation of the horizontal density gradient because of the
nonlinearity of the seawater equation of state. Examples of this
parameterization are given in the circulation model application
(Sect. ) and in the ecosystem application
(Sect. ).
Before concluding this section, it is important to remember that the above
discussion only provides one possible framework for simulating the effect of
unresolved fluctuations, and that other approaches can be imagined. For
instance, a specific stochastic parameterization is already routinely applied
at ECMWF to simulate the backscatter of kinetic energy from unresolved scales
to the smaller scales that are resolved by the model . This
scheme has been developed for atmospheric applications but might also be
applicable to ocean models. On the other hand, the external forcing u
(e.g. atmospheric data, river runoff, open-sea boundary conditions) can also
be a major source of uncertainty in the model, which can be explicitly
simulated using a formulation similar to Eq. ():
dxdt=1m∑i=1mM(x,u+δu(i),p,t),
where the fluctuations δu(i) must be tuned
to correctly reproduce the effect of uncertainties in the forcing.
Introducing appropriate perturbations of the atmospheric data can for instance be useful
to include them in the control vector of ocean data assimilation systems
.
Stochastic parameterization of unresolved diversity
Another general source of uncertainty in ocean models is the simplification
of the system by aggregation of several system components
using one single state variable and one single set of parameters.
For instance, marine ecosystems always contain a wide
diversity of species, which cannot be described separately by the
model, and which must be aggregated in a limited number of state
variables. In a similar way, sea ice can display a wide variety of
dynamical behaviours, which cannot always be resolved by ocean models.
As unresolved scales, unresolved diversity generates uncertainties in
the evolution of the system, which can be explicitly simulated using
a similar approach:
dxdt=1m∑i=1mMx,u,p+δp(i),t,
where δp(i) are random parameter fluctuations
representing the various possible dynamical behaviours that are
simultaneously present in the system.
The application of this method requires a statistical description of
the uncertainties in the parameters; and again, as a first approach,
this can be parameterized using one or several of the ξ(i)
given by Eq. (). As a particular case, this method
includes the stochastic parameterization proposed in to
explicitly simulate uncertainties in ice strength in a finite element
ocean model. It was thus very easy to apply the same scheme in the
ice component of NEMO, as an example of this parameterization (see
Sect. ).
Impact on model simulations
The purpose of this section is now to illustrate the impact of the
stochastic parameterizations presented in Sect. in
various components of NEMO: in the ocean circulation component in
Sect. , in the ocean ecosystem in
Sect. , and in the sea ice dynamics in
Sect. . The focus of the discussion will be on the
probabilistic behaviour of the system (A) as a result of the
uncertainties (the interaction with B in
Fig. ). All applications have been performed using
the same generic code implementing the stochastic formulation of NEMO
described in Sect. .
Parameters of autoregressive processes for all applications
described in this paper. The number of processes is the number of
autoregressive processes used in each stochastic parameterization (sometimes
multiplied by 3 to produce one process for each component of the random
walks). The mean, SD and correlation timescale are the parameters
μ(i), σ(i) and τ(i) used in Eqs. () and
(). For the stochastic parameterization of the equation of state
(circulation model), the SD values are multiplied by sinϕ for ORCA2,
and by sin 2ϕ for NATL025, where ϕ is latitude.
Circulation model
Ecosystem
Sea ice
unresolved
unresolved
unresolved
ORCA2
NATL025
diversity
scales
diversity
Number of processes
6×3
1×3
6
1×3
1
Order of processes
1
1
1
1
2
Mean value
0
0
1
0
0
SD
σxy=4.2
σxy=1.4
0.5
σxy=3
1
σz=1
σz=0.7
σz=1
Correlation timescale
12 days
10 days
3 days
12 days
30 days
Spatial filtering
No
No
No
No
Laplacian
Anamorphosis
No
No
No
No
gamma
Stochastic circulation model
As a result of the nonlinearity of the seawater equation of state,
unresolved potential temperature (T) and salinity (S) fluctuations
(in system B) have a direct impact on the large-scale density
gradient (in system A), and thus on the horizontal pressure
gradient through the thermal wind equation. As shown in
, this effect can be simulated using the scheme
described in Sect. 2.3,
by applying Eq. () to the equation of state:
ρ stoch(T,S)=1m∑i=1mρT+δT(i),S+δS(i)with∑i=1mδT(i)=0,∑i=1mδS(i)=0,
where δT(i) and δS(i) explicitly simulate the
unresolved fluctuations of potential temperature and salinity. These
fluctuations are generated using random walks following
Eq. (), with parameters for the ξ(i) given in
Table (i.e. the same parameterization as
in ). This stochastic parameterization simulates the
exchange of potential energy between resolved and unresolved scales,
which results from the nonlinearity of the equation of state
(see , for more details). As for the Reynolds stresses,
this should be strongly constrained by physical principles, but we
will stick here to the parameters proposed in , which
were derived from a comparison
with higher-resolution reanalysis data.
It is interesting to note (as a complement to what is explained in
) that there is a close similarity between this
stochastic correction of the large-scale density and the
semi-prognostic method proposed in and . In both
cases, indeed the only correction applied to the model occurs in the
thermal wind equation through a direct correction of density, while
the conservation equation driving the evolution of potential
temperature, salinity and horizontal velocity are all kept unchanged.
We can thus be certain that the stochastic parameterization displays
the same nice conservation properties as the semi-prognostic method;
in particular, there is no direct modification of the T and S
properties of the water masses, no enhanced diapycnal mixing and thus
no compromise with the fact that the ocean interior should primarily
flow close to the neutral tangent plane. The modification of the
thermohaline structure of the ocean is only produced indirectly
through a modification of the main currents.
The first impact of the stochastic T and S fluctuations is indeed on the
mean circulation simulated by the model. This mean effect in a low-resolution global configuration of NEMO (the ORCA2 configuration, see
, for more detail) has been described in detail in
. In summary, the density correction is important (and
quite systematically negative because of the convexity of the equation
of state) along the main fronts separating the subtropical and
subpolar gyres. The mean pathway of the mean current is thus
modified, significantly reducing the biases of the deterministic
model. In particular, the Gulf Stream pathway no longer overshoots
and the structure of the northwestern corner becomes more realistic. The
impact on the mean circulation is similar to what can be obtained with
the semi-prognostic method , in which the density
correction is diagnosed from observations, whereas the stochastic
model behaves as an autonomous dynamical system.
Sample of sea surface height patterns (in meters), illustrating the
intrinsic interannual variability generated by the stochastic
parameterization of the equation of state in a low-resolution global ocean
model configuration (ORCA2): northwestern corner of the North Atlantic drift
(top panels), Brazil–Malvinas Confluence Zone (middle panels), and Agulhas
Current retroflection (bottom panels). For each region, the left panel
represents the non-stochastic simulation, and the other panels are 3
different years of the stochastic simulation.
The second effect of the stochastic T and S fluctuations is to generate
random variability in the system. Because of the nonlinearity of the
equation of state, the small scales constantly modify the structure of
the large-scale density, and thus the pathway of the large-scale
circulation. There is a constant flux of information from
system B (small scales) to system A (large scales),
which is represented in the stochastic model by the random
processes ξ(i), and which is totally absent in the
deterministic model. This effect is illustrated in
Fig. , which displays the pattern of sea
surface height (SSH) in several key regions of the Atlantic: the
northwestern corner (top panels), the Brazil–Malvinas Confluence Zone
(middle panels), and the Agulhas Current retroflection (bottom
panels). In the non-stochastic simulation, in the absence of interannual
variability of the atmospheric forcing (as in ), the
interannual variability is extremely weak (see for
a precise quantification): this is why only 1 typical year is shown,
since all years would appear identical. In the stochastic
simulation however, not only the mean SSH pattern is modified (as
shown in ), the interannual variability is also
strongly enhanced, and thus becomes more compatible with the intrinsic
large-scale SSH variability that is obtained from higher-resolution
models or from satellite altimetric measurements (as diagnosed
in ). This intrinsic variability (produced in the absence
of any interannual variability in the atmospheric forcing) is a good
proxy to the dispersion that would be observed in a truly
probabilistic ensemble forecast. In a high-resolution model, this
dispersion in the large-scale behaviour can only result from the
interaction with the mesoscale (as explained in ). In
the low-resolution ORCA2 configuration, this unpredictable and
intrinsically variable behaviour of the large scales is here (at least
partially) restored by a stochastic parameterization of the effect of
the mesoscale (which is in system B) on the large-scale
density. It must be mentioned however that such a small size sample
is not sufficient to provide accurate quantitative information on
the magnitude of this effect. To give more precise quantitative
results, further tuning and validation of the stochastic
parameterization are required.
To further explore the effect of these uncertainties, we are currently
applying the same stochastic parameterization to
a 1/4∘ resolution model configuration of the North Atlantic
(NATL025). The results (obtained with the parameters listed in
Table ) indicate that the stochastic parameterization
tends to produce a mean effect on the Gulf Stream pathway, and to
decorrelate the mesoscale patterns produced in different members of
the ensemble. The first questions that we would like to address with
this kind of simulation are whether it is possible to better tune the
stochastic parameterization using reference data, whether the ensemble
dispersion can explain a substantial part of the misfit with
altimetric observations, and thus whether this kind of ensemble can be
used to assimilate SSH measurements in NATL025. And then, as a longer-term perspective, maybe the stochastic processes ξ(i) can be
used as a control vector for data assimilation, which would therefore
display the same nice conservation properties as the semi-prognostic
method .
Stochastic ecosystem model
There are many sources of uncertainty in marine ecosystem models. To
simplify the discussion, only two classes of uncertainty will be
considered here: uncertainties resulting from unresolved biologic
diversity and uncertainties resulting from unresolved scales in
biogeochemical tracers (see Fig. ). On the one hand,
the most common simplification in biogeochemical models is to
aggregate the biogeochemical components of the ocean in a limited number of
categories (defining system A in Fig. ). This
reduces the number of state variables and parameters, and introduces
uncertainties in the model equations since the various components included in
one single category (unresolved diversity, in system B) do not
usually display the same dynamical behaviour. To simulate this first class of
uncertainty, we will
use the SPPT scheme described in Sect. 2.2 and
multiply the “source minus sink” terms (SMSk) of the ecosystem
model by a multiplicative noise:
SMSk stoch(Cl)=SMSk ref(Cl)×ξ(k),
where Cl are the biogeochemical tracer concentrations, and
ξ(k) are autoregressive processes obtained from
Eq. (), with parameters given in
Table . To simulate unresolved diversity, the scheme
described in Sect. 2.4 would probably have
been more natural, but in view of the large number of parameters in
the ecosystem model, the SPPT scheme is much easier to implement as
a first approach. On the other hand, to simulate uncertainties
resulting from unresolved scales, we will use the scheme described in
Sect. 2.3, by applying Eq. () to
the SMS terms:
SMSk stoch(Cl)=1m∑i=1mSMSkrefCl+δCl(i)with∑i=1mδCl(i)=0,
where δCl(i) explicitly simulate the unresolved
fluctuations of biogeochemical tracer concentrations. These
fluctuations are generated using random walks following
Eq. (), with parameters for the ξ(i) given in
Table . (Since little is known about uncertainties in
the ecosystem model, we just used here reasonable values for the
parameters.)
As a first approach, the impact of these two stochastic
parameterizations has been studied in a low-resolution global ocean
model, based on the ORCA2 configuration coupled to the LOBSTER
ecosystem model (using exactly the same model settings as in the
previous section). The ecosystem model (see for more
details) is a simple model including only six compartments
(Ck,k=1,…, 6): phytoplankton, zooplankton, nitrate, ammonium,
dissolved organic matter, and detritus. The behaviour of this model
is here illustrated in Fig. by the surface
phytoplankton in the North Atlantic for 15 June (in the second year of
simulation). As compared to the deterministic simulation (top left
panel), the stochastic simulation with the SPPT scheme (top right
panel) does not modify very strongly the general behaviour of the
system (despite the 50 % SD multiplicative noise), but
substantially increases the patchiness of the surface phytoplankton
concentration. This suggests the conjecture that uncertainties (in
particular, unresolved diversity) may partly explain the patchiness of
satellite ocean colour images. Conversely, the stochastic simulation
of unresolved scales (bottom left panel) does not increase patchiness,
but can significantly affect the local behaviour of the system,
sometimes increasing or decreasing the production (whether the second
derivative of the SMS term is positive or negative). At first sight,
these two sources of uncertainty are thus insufficient to explain the
considerable misfit between model simulation and ocean colour data.
As an additional experiment, the two stochastic parameterizations have
then been used together (bottom right panel), by simply generating
a sufficient number of autoregressive processes (corresponding to the
two columns together in Table ) to feed the two
schemes. This result shows that there is a strong interaction between
the two schemes, leading to a deep modification of the general
behaviour of the system, and to enhanced patchiness as compared to the
SPPT scheme alone. In our view, this directly leads to the idea that
uncertainties may be an important ingredient to understand the
dynamical behaviours of marine ecosystems, and to make the model
distribution consistent with ocean colour observations (in magnitude
and pattern). It must be noted however that these experiments only
represent a first attempt to explicitly simulate uncertainties in the
ecosystem component of NEMO, and that further studies are needed
before any meaningful quantitative result can be obtained.
Surface phytoplankton concentration (in mmol N m-3) for
15 June as obtained with various stochastic parameterizations of
uncertainties in the ecosystem model: no stochastic parameterization (top
left panel), stochastic simulation of unresolved diversity (top right panel),
stochastic simulation of unresolved scales (bottom left panel), and
stochastic simulation of unresolved diversity and unresolved scales (bottom
right panel).
Stochastic sea ice model
One of the main difficulties of sea ice models is to correctly
simulate the wide diversity of ice dynamical behaviours. Among ice
characteristics, the most sensitive parameter is certainly the ice
strength P∗. In simple ocean models (as in LIM2 in NEMO),
P∗ is assumed constant, whereas, in more complex models (as in
LIM3 in NEMO), the variations of P∗ can be explicitly resolved
as a function of the various types of ice simultaneously present at
every model grid point. The impact of uncertainties in P∗ has
already been studied in the work of
using a finite element ocean model (FESOM), coupled to
a simple sea ice model similar to LIM2. The purpose of this section
is to reproduce their parameterization in NEMO/LIM2 using the generic
technical approach described in Sect. . This can be
done very easily, almost without any additional implementation effort,
using the scheme described by Eq. () with m=1 and
P∗+δP∗=P∗×ξ,
where ξ is one of the autoregressive processes given by
Eq. (), with parameters given in
Table . The parameters are chosen to be close to the
stochastic parameterization in . Specificities are the
use of order 2 instead of order 1 autoregressive processes, and the
use of a gamma marginal distribution instead of another kind of
positive distribution in .
This stochastic parameterization has been applied to a low-resolution
global ocean configuration of NEMO, again without interannual
variability in the atmospheric forcing (using the same model settings
as in ). The behaviour of the model is here
illustrated in Fig. by the ice thickness in
the Arctic at the end of March (when the ice extension is close to its
maximum). As compared to the deterministic simulation (top left
panel), the first impact of the stochastic parameterization is to
systematically increase ice thicknesses, especially in the regions of
old ice (north of Greenland and western Canada), and to slightly
decrease the ice extension.
This mean effect results from the nonlinearity of the model response to P∗:
during the periods of small P∗, the ice thickness has the opportunity
to increase, and this increase is not counterbalanced by a symmetric decrease
of thickness during the periods of large P∗. This behaviour is very similar
to what is described in , and cannot be reproduced by
a simple uniform modification of P∗.
Sample of ice thickness patterns (in meters) in winter (end of
March), illustrating the intrinsic interannual variability generated by the
stochastic parameterization of ice strength in a low-resolution global ocean
model (ORCA2). The top left panel represents the non-stochastic simulation,
and the other panels are 3 different years of the stochastic simulation.
On the other hand, the stochastic fluctuations of P∗ also
generate random variability in the system. As for SSH in
Sect. , the interannual variability of the ice thickness
pattern is extremely weak in ORCA2 without interannual variability of
the atmospheric forcing (which is why only 1 typical year is shown
in Fig. ). In the stochastic simulation
however, not only is the mean ice thickness pattern modified (as for
SSH in Fig. ), but the interannual variability
(which is again a good proxy to ensemble dispersion as explained in
Sect. ) is also strongly enhanced. What is expected
from these results is thus that the explicit simulation of
uncertainties can provide us with an adequate basis for probabilistic
comparison with sea ice observations and help us in producing reliable
ensemble forecasts for sea ice data assimilation problems.
Consequently, it might also be that this stochastic approach
represents a worthwhile alternative to explicit resolutions of sea ice
diversity (as in LIM3).
Conclusions
In this paper, a simple and generic implementation approach has been
presented, with the purpose of transforming a deterministic ocean
model (like NEMO) into a probabilistic model. With this method, it is
possible to easily implement various kinds of stochastic
parameterizations mimicking the non-deterministic effect of unresolved
processes, unresolved scales, unresolved diversity, etc. It has
been shown indeed that ocean systems can often display a random
behaviour, which needs to be explicitly represented in ocean models.
Ensemble simulations are then required to sample all possible
behaviours of the system. Getting a reliable overview of all dynamical
possibilities is necessary to objectively compare models to
observations, and to correctly apply the model constraint in ocean data
assimilation problems.
Technically, what is proposed here is a very simple algorithmic
solution that is easy to adapt to many kinds of models, and that is generic enough
to deal with many different sources of uncertainty. This is obviously not
intended to be the final theoretical or technical solution for
simulating uncertainties. The algorithms and framework proposed in
this study only provide a first-guess solution, which is simple enough
to make a first quick evaluation of the effect of a given source of
uncertainty, and flexible enough to easily evolve as a better
understanding of the problem is progressively obtained.
This technique has been applied to several applications, showing that
randomness is ubiquitous in ocean systems: in the large-scale
circulation (e.g. because of the effect of unresolved scales through
the nonlinear equation of state), in the ecosystem model (e.g.
because of the effect of unresolved scales and unresolved
biogeochemical diversity), and in the sea ice dynamics (e.g.
because of the unresolved diversity of sea ice characteristics). In
each of these applications, uncertainty can be viewed as an essential
dynamical characteristic of the system, which can modify our
understanding of the ocean behaviour. As for any complex system,
constructing ocean models using optimal (but imperfect) components can
often be worse (less robust) than using unreliable components dealing
explicitly with their respective inaccuracy. The ocean is like dice
rolling on the table of a casino: we are unable to grasp all
subtleties of their movements, and we can only sample from all possible
outcomes of the game using probabilistic models.