Through two numerical experiments, a 1-D vertical model called NEMO1D was
used to investigate physical and numerical turbulent-mixing behaviour. The
results show that all the turbulent closures tested (

Copernicus is a multidisciplinary programme of the European Union for
sustainable services providing information on monitoring of the atmosphere,
climate change, and land and marine environments. The services also address the
management of emergency and security-related situations (

NEMO is a numerical-modelling tool developed and operated within a European consortium. Its code enables investigation of oceanic circulation (OPA) and its interactions with the atmosphere. Ice models (LIM) and biogeochemical models (TOP – PISCES or LOBSTER) could also be coupled. This primitive equation model offers a wide range of applications from short-term forecasts (Mercator Océan and MyOcean/Copernicus) and climate projections (Voldoire et al., 2013) to process studies (e.g. Chanut et al., 2008; Bernie et al., 2007).

OPA, the oceanic component of NEMO, is built from the Navier-Stokes equations applied to the earth referential system, in which the ocean is subjected to the Coriolis force. The prognostic variables are the horizontal components of velocity, the sea surface height and two active tracers (temperature and salinity).

These equations allow description of a wide range of processes at any spatial and temporal scales. This involves interpreting complex numerical results, which means that it is difficult to improve the model due to the multiple non-linear interactions between the different equation terms. Fortunately, there are a lot of numerical test cases in the literature that make it possible to isolate each term of the equations (advection, diffusion or coordinate systems) and improve it independently of the others.

Vertical mixing plays an essential role in ocean dynamics and it must therefore be correctly estimated. In particular, it creates the mixed layer, the homogeneous ocean layer that interacts directly with the atmosphere, which can then be modelled as the mixed layer depth (MLD). The MLD plays a very important role in the energetic exchanges between the ocean and the atmosphere and may have very high spatiotemporal variations (diurnal, seasonal, synoptical). In addition, MLD variability plays a crucial role in biogeochemical processes. The deepening episodes of the MLD during winter inject nutrients into the euphotic layer, with a strong impact on primary production (Flierl and Davis, 1993). Vertical mixing is also responsible for convection or seasonal stratification. Lastly, vertical mixing must be able to conserve the water masses.

To focus on vertical turbulent mixing, we use NEMO1D, a feature included in NEMO, to consider only one column of water. NEMO1D is a simple, robust, useful and powerful tool that enables quick and easy investigation of the physical processes affecting the vertical component of the ocean state variables: turbulence, surface and bottom boundary conditions, penetration radiation schemes, etc.

In this paper, based on the 1-D configuration, we assess the effectiveness of
different turbulent schemes available in NEMO. Section 2 describes the
equations of motions, the turbulent closures discussed in this paper and the
numerical contexts (vertical grids and time steps). Section 3.1 contains a
discussion of how the turbulence models performed in relation to the
empirical law which states that the mixed layer deepens as a function of
time, as described by Kato and Phillips 1969) and based on a laboratory
experiment. The sensitivity of the spatial and time discretization of each
turbulent scheme is also discussed. Section 3.2 reports on some experiments
performed under realistic conditions and these turbulent closures are
compared with data measurements taken at the PAPA station
(

NEMO is based on the 3-D primitive equations resulting from Reynolds
averaging of the Navier–Stokes equations and transport equations for
temperature and salinity. The set of equations is:

The turbulence issue raises the question of how to compute accurately the
vertical turbulent viscosity

Other more complex methods use the turbulent kinetic energy

The main objective of the present paper is to highlight and to understand the performance of a two-equation model compared to the standard turbulent model of NEMO referred to as the one-equation model. Simpler models than those described in Eq. (3a)–(3c) or other kinds of models such as such as K-Profile Parameterization (commonly called KPP suggested by Large et al., 1994) are not considered here, and could be considered in a further study.

The different ways of estimating

For the one-equation as for the two-equation model, the turbulent kinetic
energy (

However,

Finally,

The mixing length

The turbulent model, most frequently used by the NEMO community and commonly
called the TKE model, is a one-equation model suggested by Blanke and
Delecluse (1993). The parameterization of the mixing length is a
simplification of the formulation of Gaspar et al. (1990) by considering
only one mixing length established under the assumption of a linear
stratified fluid. This mixing length is then simply computed as a ratio
between the turbulent velocity fluctuations and the
Brunt–Väisälä frequency:

In addition, this mixing length is bounded by the distances between the calculation point position and the physical boundaries (surface and bottom).

The two-equation models, most frequently used by the ocean modelling
community, are:

Turbulence model constants.

The transport equation for the variable

The stability functions appear in Eq. (4a) and (4b).

For the TKE model, they are:

For GLS models, the stability functions are derived from the Reynolds stress
equations and depend on the shear and buoyancy numbers, defined in Eq. (3c),
called respectively

There are several articles on stability functions in the literature. The
main functions commonly used by the research community are Galperin et al. (1988), Kantha
and Clayson (1994), Hossain (1980) and Canuto A and B (Canuto et al., 2001).
This study does not discuss any sensitivity tests for these functions. This
has in fact been done by Burchard and Bolding (2001), who claim that better
results are obtained with the Canuto A stability functions. Thus, they were
naturally chosen for the study presented in this paper and are expressed as:

For information, the

The ocean surface is subjected to atmospheric forcing and eventually surface wave breaking that could induce significant mixing.

In the absence of an explicit wave description, the surface turbulent
quantities can be described as a logarithmic boundary layer with the
following properties:

Inside this logarithmic boundary layer, the turbulent kinetic energy is
constant and its flux is zero. The stability functions of Canuto A used in
the GLS closures (Eq. 16a and 16b) converge to the value

Both of these values are closed to the typical value of

Breaking waves induce significant mixing inside the surface layer through
the diffusion term, as proposed by Craig and Banner (1994):

For the TKE model, the simplification of Eq. (21) leads to a value of
turbulent kinetic energy at the surface (

In the GLS model, if the surface breaking waves mixing is considered, then
shear-free turbulence may be assumed. This special case is characterized by
a spatial decay of turbulence away from a source without mean shear. In
these conditions, the turbulence solution can be written as:

Note that the value of

For the TKE model, the surface roughness is computed using Charnock's
relation (1955):

For GLS models,

For all models, a background value of the surface roughness is set to

The values of constants are condensed in Table 1 and depend on the selected closure model.

For the TKE model, the constants of Eqs. (13a), (13b) and (8) are similar to that of the model of Gaspar et al. (1990).

The GLS closures are defined by the set of parameters

The well-known failure of

Umlauf and Burchard (2003) also suggested an optimal calibration of the
model using the polymorphic nature of the generic model. Thus, the
parameters

The results obtained with the GLS closures (

NEMO is based on the 3-D primitive equations (Eq. 1a–1c) but it offers the possibility of reducing the complexity of the system by limiting the domain to just one water column. This is the 1-D approach (NEMO1D).

The set of equations, after simplification of all horizontal gradients in the
primitive equations, then reduces to:

The C-grid (Arakawa and Lamb, 1977) is commonly used in NEMO. For purely numerical reasons, the A-grid is henceforth used by NEMO1D as the velocity components and the scalar values are calculated at the same point.

Due to the equation simplifications and the low computational cost, NEMO1D is an ideal tool for studying the vertical component of NEMO.

The choice of the vertical grid is crucial. A compromise should be found between the resolution needed for a good representation of the oceanic processes and the computational cost, linked to the number of cells and the Courant–Friedrichs–Lewy CFL criteria (Courant et al., 1928) which induce the time-step constraint.

Currently, for 3-D realistic applications using NEMO, there are two classes of vertical grids:

Thickness of vertical levels as a function of depth for two vertical grids: L31 (red) and L75 (black). Each plotting point (squares and stars) corresponds to the depth of levels.

low-resolution vertical grids with the thickness of first levels between 6 and 10 m. The grid with 31 cells (Fig. 1), used in the standard ORCA2 configuration of NEMO and specifically designed for climate applications (Dufresne et al., 2013), is a good example. This is the first grid selected for this study and will be referred to as L31.

high-resolution grids that typically have a thickness of 1 m in the first levels. The grid with
75 vertical cells described in Fig. 1 and referred to as L75 represents the second class of vertical
grids in our study. This grid is used in the global

Several time steps are used in 3-D simulations and these are fixed according to the spatial resolutions and their associated CFL conditions.

NEMO1D has no restriction on the time step, as there is no vertical advection and hence no CFL condition. However, we have tested the sensitivity of each turbulence model to the following time steps: 360, 1200 and 3600 s. These time steps correspond to those used in the global configurations at Mercator Océan and more generally in the research community. So we can easily regroup some pairs [vertical grid, time step] to retrieve some known configurations:

[L75, 360 s]: Global configurations at

[L75, 1200 s]: Global configurations at

[L31, 3600 s]: Global configurations at 1

[L75, 3600 s]: Global configurations at 1

For the TKE model,

For the GLS models, firstly,

To ensure a minimum level of mixing, background values are applied to the
following turbulence quantities:

All the terms of the differential equations are solved explicitly except the diffusive terms appearing in Eqs. (5), (11) and (29a)–(29c), which are solved implicitly.

It is obvious that the TKE and GLS closures are very different from a purely physical aspect but also in the way they are implemented. If we were to perform a relevant comparison of these turbulence models, we should, for example, use the same boundary conditions or the same stability functions. The problem then should simply involve comparing a parameterization of the mixing length to that obtained with a differential equation.

But the aim of this paper is to provide feedback to NEMO users on the one- and two-equation turbulent closures available in the model.

The Kato–Phillips (1969) experiment is classically used in the literature to
test and calibrate turbulence models (e.g. Burchard, 2001a;
Galperin et al., 1988). This laboratory experiment deals with the measurement of
the mixed layer deepening of an initial, linear, stratified fluid,
characterized by the Brunt–Väisälä frequency

The model MLDs are then computed with criteria linked to the depth of the
maximum of

The MLDs calculated with Eq. (30) are reliable on timescales of the order of 30 h, which determines the simulation duration.

To estimate the performance of each turbulence model, depending on the choice of the vertical grid and the time step, the metrics selected are the correlation coefficient, the standard deviation and the root mean square error (RMSE) of the simulated MLD compared to the analytic one given by Eq. (30).

The writing frequency for outputs is set by the highest time step considered in this study, i.e. 3600 s.

The surface boundary conditions are those described in Eqs. (18)–(20). Obviously we did not consider the wave breaking effect on the mixing. Nevertheless, the background value for the surface roughness was retained.

As NEMO1D has a low computational cost, a new vertical grid was adopted and
a water column of 100 m considered. This new vertical grid has
1000 levels (hereafter called L1000) at 10 cm intervals. Coupled with a time
step of 36 s, this numerical framework is ideal for checking the ability of
all the turbulence models considered in this study to reproduce
satisfactorily the empirical time-dependent relation (Eq. 30). Figure 2 shows
that all the models gave very similar results close to the empirical
solution. However, the numerical MLDs were slightly underestimated by
approximately 1 m: the RMSEs were between 0.9 m for

Time evolution of the MLD obtained as a function of the different turbulent closures with the 1000 levels vertical grid and a time step of 36 s.

In realistic 3-D global or regional configurations, the numerical framework
is less favourable for obvious computing time or storage reasons. The
vertical grids are then coarser (typically L31 or L75) with greater time
steps. Nevertheless, these time steps are limited by the CFL condition,
mainly on the vertical, and thus set by the choice of the grid. This
numerical restriction does not occur in NEMO1D due to the no-advection
assumption. Consequently, we have taken into account the 60 possibilities (5
turbulence models

Time evolution of the MLD obtained with the different turbulence
models for the pairs [L75, 360 s]

For the pair [L75, 360s] (Fig. 3a), all the models yielded suitable results
with associated RMSEs of the order of 3 m. For the pair [L75, 1200s] (Fig. 3b),
all the models were close with RMSEs around 4 m, except for

The numerical context of the pair [L31, 3600 s] (Fig. 3d) is obviously the
most difficult. The L31 vertical grid is not well adapted for this test case
due to its coarse surface layer resolution with a 10 m separation of the
first levels. The thresholds linked to the vertical grid are such that the
RMSE should not be compared to those obtained with the L75 grid. It should
be noted that (i) all closures represent the deepening of the maximum of

Taylor diagram of the 60 Kato–Phillips experiments.

To represent synthetically the 60 possibilities of our test cases, the
performed statistics (RMSE and correlation) were condensed in a Taylor
diagram (Fig. 4). Note that the red and blue cloud points, corresponding to
the tests done respectively with L1000 and L75 grids, are, for most of them,
statistically close to the reference solution with a standard deviation
lower than 25 % and a correlation greater than 95 %. However,

The green points, corresponding to tests using the L31 grid, show that all the turbulence models yielded very similar results at this resolution. The dispersion was only due to the normalized standard deviation while the RMSEs are similar. This kind of scatter is only influenced by the fact that the solution could be above or below the step of the coarse vertical discretization near the surface (Fig. 3d).

The PAPA station, located west of Canada in the Pacific Ocean
(50

The temperature and salinity time series exhibits a well-marked seasonal variability (Fig. 5). The temperature field shows the formation of stratification in summer followed by a homogenization of the water column in winter. The salinity field exhibits a stationary halocline at a depth of around 120 m while the surface variability is strongly correlated to the temperature field. Indeed, during strong stratification events, the MLD of only several tens of metres is isolated from the rest of the water column and when subjected to the precipitation rate, the salinity decreases. Next, during water column homogenization events, the salinity increases due to mixing with the deeper saltier waters. Note that the halocline depth can vary significantly during these several years of measurements (between 100 and 150 m in depth). This interannual variability is mostly attributed to variations in horizontal salt advection.

In this section, we compare all the turbulence model results with the data collected at the PAPA station between 15 June 2010 and 15 June 2011. For this period, there is no significant gap in the time series and the halocline depth remains stationary (120 m depth), i.e. advection is less influential.

A NEMO1D simulation is easily set up. The bathymetry considered is the value
of the global bathymetry file at

Temperature (top) and salinity (bottom) measured at the PAPA station covering the period 2009–2013.

Comparisons of ECMWF atmospheric values and measurements at the PAPA station.

The 3 h atmospheric forcing came from the European Centre for Medium-Range Weather Forecasts (ECMWF) analysis and forecasting
operational system. Although atmospheric measurements are taken at the PAPA
station, they are only available once a day. However, these data are useful
for checking that the ECMWF atmospheric fields are relevant enough to be
used with confidence. Statistical comparisons (mean of the model minus
observations, correlation coefficients and RMSEs) are
shown in Table 2. Mean errors for wind velocities and air temperature are
very small (respectively 1 m s

Regarding initial conditions, it would be ideal to initialize the model with temperature and salinity measurements taken at the PAPA station on 15 June 2010. However, the data cover only the first 200 m for salinity and the first 300 m for temperature. Moreover, the salinity and temperature data are not collected on the same vertical grid: 24 levels for the temperature as opposed to 18 levels for the salinity. Thus, below a depth of 200 or 300 m, data from the WOD09 climatology (Levitus et al., 2013) were considered. Fortunately, there is a close match between the measurements and the climatology data below a depth of 150 m (Fig. 6).

Initial conditions (black line and star) of temperature (left) and salinity (right) from measurements at the PAPA station (blue square) on 15 June 2010 and Levitus 2009 climatology data (red).

Observed temperature at PAPA station

Observed salinity at PAPA station

Daily vertical profiles of density during the stratified period (12 September 2010) on the left and at the beginning of the delayering period (12 October 2010) on the right.

Evolution of the temperature RMSE computed along the vertical
(0–120 m) for the pairs [L75, 360 s]

Daily vertical profiles of temperature for 20 August 2010:
observed (black), simulated with

For simulations under realistic conditions, the turbulence models should take
into consideration mixing caused by breaking waves. Thus, the surface
boundary conditions are those described in Sect. 2.2.4 (Eqs. 24–26 for the
GLS models and Eq. 22 for the TKE model). Moreover, the TKE model takes into
account the injection of surface energy inside the water column as described
in Eq. (23). This parameterization depends on the parameters

The bulk formulae used have been described in Large and Yeager (1994). The
albedo coefficient is set to 6 %, the atmospheric pressure to 100 800 Pa
and the air density to 1.22 kg m

For these experiments, the two vertical grids (L31 and L75) and the three different time steps (360, 1200, 3600 s) described in Sect. 2 have been taken into account. All possible pairs with all closures (four issued from GLS formulation and three different TKEs) have been performed. This involved 42 simulations. The next section gives the results obtained with the pair with the highest resolution [L75, 360 s]. This pair is the selected setting for the standard configuration PAPA1D. The last section discusses the spatio-temporal sensitivity.

Figures 7 and 8 represent the observed temperature and the observed salinity respectively at the PAPA station and the biases (model minus observation) obtained with different closures.

During the year of simulation, the summer stratification is well represented
with an increase of temperature of 6

In all simulations, this annual cycle is found and the general behaviour is
the same, except for the simulation TKE_30m in which the
differences between simulated temperature and measurements exhibit, during
summer, a vertical dipole, with a colder temperature than that observed,
reaching

The biases obtained for the experiments with the other turbulence models
(

As the precipitation rate has little influence on MLD dynamics (see Sect. 3.2.1),
the salinity biases (Fig. 8) are directly linked to the MLD
thickness by mixing and by evaporation. For the period from June to
September, all the models exhibit the same weak salinity biases, as the MLD
deepening is similar in each case. In October, for generic,

Regarding the sensitivity of the TKE model to the

As in Sect. 3.1.3, this section covers the sensitivity of the different closures to the vertical discretization (grid L75 and L31) and to the time step (360, 1200 and 3600 s). Figure 10 shows the evolution of the temperature RMSE for the different closures, computed for the first 120 m of the water column (depth corresponding to the halocline depth, Fig. 5).

In all cases, two periods can be distinguished: (i) from June to
November 2010 corresponding to the stratified period, for which high
variations of the RMSE have been observed (between 0.03 to 0.35

In both periods, the three TKE closures (red, orange and pink lines) do not show any significant sensitivity to the vertical discretization or the time step. This result is in agreement with the Kato–Phillips results presented in Sect. 3.1.

For the GLS closures, the two periods should be studied separately:

During the first period (June to November 2010), the

During the second period (December 2010 to June 2011), all the RMSEs are similar and weak (0.05

The RMSEs for

To focus on this point, vertical temperature profiles for both vertical grids were plotted for this date (Fig. 11). L75 tends to over-stratify and this result is in agreement with the previous section (Fig. 7). The profile obtained with L31 agrees better with the observations (dark blue line). This is due to the numerical dilution effect of the coarse grid in the surface layer. Indeed, this grid has only three levels in the first 30 m and is not able to create a strong stratification. In this case, a low vertical resolution becomes an advantage simply for numerical reasons.

This paper has described the 1-D model version available in NEMO (NEMO1D). This model is very useful for isolating and studying vertical processes, and for improving their representation before switching to a 3-D model.

The present study focused on the behaviour of two types of turbulence closures available in NEMO, i.e. TKE and GLS. There are many differences between these two approaches, with respect to both the mixing length estimate and the constants used or boundary conditions. For this reason, we did not concentrate on comparing the closures but rather on their different strengths and weakness. Two test cases were selected, an “idealized” one, based on the experiment described in Kato–Phillips, and a “realistic” one, reproducing one year of salinity and temperature measured at the PAPA buoy.

The first test case was based on observations performed in a laboratory
experiment. This experiment deals with the monitoring of the MLD deepening
of an initially linear stratified fluid only subjected to a stationary
surface stress. Because of its simplicity, this test case offers a perfect
context for validating the numerical assumptions and implementations.
Simulations were performed under favourable numerical conditions (grid with
a resolution of 0.1 m and a time step of 36 s). All the turbulence closures
correctly reproduced the experimental results described even if the TKE
closure slightly underestimates the MLD. This validates their implementation
in NEMO. However, we found some dependence on numerical conditions (ratio
time step/vertical discretization) for the GLS closures. This dependence is
strong for

The data collected at the PAPA station are typically used to perform studies with 1-D models. This mooring was naturally chosen to create a new reference configuration for NEMO. This new configuration was described, and then used to complement our study of TKE and GLS closures. The results show that these closures are largely able to reproduce the stratification/homogenization cycle observed at the station.

We have also demonstrated the major impact of a particular aspect of the
boundary conditions on the TKE closure, via the parameter

A key process has also been highlighted, namely the representation of
salinity, occurring during the homogenization event in mid-October. All the
models, except

The results obtained with NEMO1D were successfully compared with laboratory observations or in-situ measurements through a turbulent closure sensitivity study. The 1-D approach, applied at the PAPA station (new NEMO reference configuration PAPA1D) or at another location, could be useful for further investigation of the turbulent mixing or some other physical component affecting vertical processes. Indeed, the MLD is subject to complex interactions between the turbulence, the surface forcing (including atmospheric fluxes and waves) and the boundary treatments.

Some of the following questions could be investigated using this numerical tool.

What is the behaviour of the other turbulence models of NEMO (KPP) (Large et
al., 1994) and Paconowski and Philander (1981)? What is a more optimal value
of some parameters (

Finally, this 1-D model could also be useful for studies on coupling with atmospheric, ice or biogeochemical models.

This research was supported by the MyOcean2 European project and Mercator Océan. The authors wish to thank collaborators who have contributed to development of the NEMO ocean code as part of the NEMO consortium. We thank Yann Drillet, Jean-Michel Lellouche and Marie Drévillon for their useful comments and both reviewers for their very constructive suggestions and questions.Edited by: R. Marsh