In this paper, a generic implementation approach is presented, with the aim of transforming a deterministic ocean model (like NEMO) into a probabilistic model. With this approach, several kinds of stochastic parameterizations are implemented to simulate the non-deterministic effect of unresolved processes, unresolved scales and unresolved diversity. The method is illustrated with three applications, showing that uncertainties can produce a major effect in the circulation model, in the ecosystem model, and in the sea ice model. These examples show that uncertainties can produce an important effect in the simulations, strongly modifying the dynamical behaviour of these three components of ocean systems.

The first requirement of an ocean model is the definition of the
system that the model is going to represent. As illustrated in
Fig.

Even if the union of the two systems

To obtain a reliable predictive model for

Schematic of the separation between resolved
and unresolved processes (systems

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

Stochastic parameterizations explicitly simulating model uncertainty
were first applied to ensemble weather forecasting by

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.

The ocean model used in this study is NEMO (Nucleus for a European
Model of the Ocean), as described in

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

for order 1 processes (AR(1)),

for order

Second, a spatial dependence between the processes can easily be
introduced by applying a spatial filter to the

Third, the marginal distribution of the stochastic processes can also
be easily modified by applying a nonlinear change of variable
(anamorphosis transformation) to the

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. (

Referring to the sketch presented in Fig.

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.

A first way of explicitly simulating uncertainties in meteorological
weather forecast was introduced about 15

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.

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

Obviously, the main difficulty with this method is to generate
fluctuations

This particular case corresponds to the stochastic parameterization
proposed in

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

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:

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

The purpose of this section is now to illustrate the impact of the
stochastic parameterizations presented 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

As a result of the nonlinearity of the seawater equation of state,
unresolved potential temperature (

It is interesting to note (as a complement to what is explained in

The first impact of the stochastic

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

To further explore the effect of these uncertainties, we are currently
applying the same stochastic parameterization to
a

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.

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

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

Surface phytoplankton concentration (in mmol N

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

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

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

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.

All examples of stochastic parameterizations described in this paper have been performed with the same generic tool that we have implemented in NEMO. The purpose of this appendix is to describe this tool, and to show that it could be easily adapted to work in any other modelling system.

The computer code is made up of one single FORTRAN module, with three public
routines to be called by the model (in our case, NEMO).

The first routine (sto_par, see Algorithm 1) is
a direct implementation of Eq. (

The second routine (sto_par_init, see Algorithm 2) is an
initialization routine mainly dedicated to the computation of parameters

The third routine (sto_rst_write) writes a “restart file”
with the current value of all autoregressive processes to allow
restarting of a simulation from where it has been interrupted. This file
also contains the current state of the random number generator. In
case of a restart, this file is then read by the initialization
routine (

sto_par

Save map from previous time step:

Draw new map of random numbers

Apply spatial filtering operator

Apply precomputed factor

Use previous process (one order lower) instead of white noise:

Multiply by parameter

Update map of autoregressive processes:

sto_par_init

Initialize number of maps of autoregressive processes to 0:

Set

Increase

Set order of autoregressive processes

Set mean (

Compute parameters

Define filtering operator

Compute factor

Initialize seeds for random number generator

Draw new map of random numbers

Apply spatial filtering operator

Apply precomputed factor

Initialize autoregressive processes to

Read maps of autoregressive processes and seeds for the random number generator form restart file (thus overriding the initial seed)

This module has been used to produce the three examples of stochastic
parameterization given in the paper, with the parameters given in
Table

This work has received funding from the European Community's Seventh Framework Programme FP7/2007-2013 under grant agreements 283367 (MyOcean2) and 283580 (SANGOMA), with additional support from CNES. The calculations were performed using HPC resources from GENCI-IDRIS (grant 2014-011279). Edited by: S. Valcke