A simplified model of the atmospheric boundary layer (ABL)
of intermediate complexity between a bulk parameterization and a three-dimensional
atmospheric model is developed and integrated to the Nucleus for European Modelling of the Ocean (NEMO) general circulation model.
An objective in the derivation of such a simplified model, called ABL1d, is
to reach an apt representation in ocean-only numerical simulations of some of the
key processes associated with air–sea interactions at the characteristic scales of
the oceanic mesoscale. In this paper we describe the formulation of the
ABL1d model and the strategy to constrain this model with large-scale
atmospheric data available from reanalysis or real-time forecasts. A particular
emphasis is on the appropriate choice and calibration of a turbulent closure scheme
for the atmospheric boundary layer. This is a key ingredient to properly represent
the air–sea interaction processes of interest. We also provide a detailed description
of the NEMO-ABL1d coupling infrastructure and its computational efficiency.
The resulting simplified model is then tested for several boundary-layer regimes
relevant to either ocean–atmosphere or sea-ice–atmosphere coupling. The coupled
system is also tested with a realistic 0.25∘ resolution global configuration.
The numerical results are evaluated using standard metrics
from the literature to quantify the wind–sea-surface-temperature
(a.k.a. thermal feedback effect),
wind–current (a.k.a. current feedback effect), and ABL–sea-ice couplings.
With respect to these metrics, our results show very good agreement with observations
and fully coupled ocean–atmosphere models for a computational overhead of about
9 % in terms of elapsed time compared to standard uncoupled simulations.
This moderate overhead, largely due to I/O operations, leaves room for further
improvement to relax the assumption of horizontal homogeneity behind ABL1d
and thus to further improve the realism of the coupling while keeping the flexibility
of ocean-only modeling.
Introduction
Owing to advances in computational power, global oceanic models used for research or
operational purposes are now configured with increasingly higher horizontal and vertical
resolution, thus resolving the baroclinic deformation radius in the tropics
(e.g., ).
Meanwhile fine-scale local models are routinely used to simulate submesoscales,
which occur on scales on the order of 0.1–20 km horizontally, and their
impact on larger scales e.g.,.
By increasing the oceanic model resolution, small-scale features are explicitly resolved, but an apt representation of the associated processes also requires the relevant scales
to be present in the surface forcings including the proper interaction with
the low-level atmosphere.
Historical context
Historically, oceanic general circulation models (OGCMs) were forced by
specified wind stress and thermal boundary conditions (from observations or
reanalysis) independent from the oceanic state, thus often leading to important
drifts in model sea surface properties. To minimize such drifts, a flux correction
in the form of a restoration of sea-surface temperature and salinity toward climatological
values can be added e.g.,.
To overcome the shortcomings of the forcing with specified flux,
proposed to use a parameterization of the atmospheric surface layer (ASL) constrained by
large-scale meteorological data and by the sea state (essentially the sea-surface temperature
and sometimes the surface currents) to compute the turbulent components of air–sea fluxes.
Currently, whatever the target applications, such a technique is widely used in the absence
of a concurrently running atmospheric model.
Such parameterization of the ASL (known as bulk parameterization, e.g., ), which corresponds to a generalization of the
classical neutral wall law to stratified conditions , is expected to
be valid in the first few tens of meters in the atmosphere. In practice, unless a fully coupled
ocean–atmosphere model is used, atmospheric quantities at 10 m, either from existing
numerical simulations of the atmosphere or from observations, are prescribed as input to the
bulk parameterization. Throughout the paper, this approach will be referred to as
“ASL forcing strategy”.
A problem with such methodology is that the fast component of the system
(the atmosphere) is specified to force the slow component (the ocean),
whereas the inertia is in the latter. Indeed, a change in wind stress or heat flux
will affect 10 m winds and temperature more strongly than sea surface
currents and temperature. In the “ASL forcing strategy”, the key marine atmospheric
boundary layer (MABL) processes are not taken into account, and thus feedback loops
between the MABL and the upper ocean are not represented.
Air–sea interactions at oceanic mesoscales
An increasing number of studies based either on observational studies and/or on air–sea
coupled simulations have unambiguously shown the existence of air–sea interactions at
oceanic mesoscales
(e.g., ). Those interactions affect the mass, heat,
and momentum exchange between the atmosphere and the ocean. We focus in this
work on the dynamical response of the surface wind stress to the sea-surface
properties (sea surface temperature (SST) and
currents) which directly affects low-level winds, temperature, and humidity.
Several mechanisms responsible for the surface wind-stress
response to SST and oceanic currents can be invoked:
Downward momentum mixing. SST-induced changes in the
stratification produce significant changes of wind speed and turbulent fluxes
throughout the MABL with an increase (decrease) in wind speed over warm
water (cold water). As the wind blows over warm water, the MABL becomes
more unstable, which leads to an increased vertical mixing, resulting in a
downward mixing of momentum from the upper atmosphere to the surface
strengthening surface winds on the warm side of an SST front
e.g.,. This mechanism results in a proportional
relationship between wind-stress intensity and SST mesoscale anomalies
which has been identified in observations and coupled simulations
(e.g., ).
Considering spatial derivatives of this proportional relationship leads
to a correlation between wind-stress divergence (curl) and downwind (crosswind) SST–gradient e.g.,.
Atmospheric pressure adjustment. This mechanism corresponds to an
adjustment of the atmospheric pressure gradient to the underlying SST, which
manifests itself as a linear relation between horizontal wind divergence and
the Laplacian of SST .
Oceanic current feedback. The momentum exchange between the
ocean and the atmosphere is also largely affected by a dynamical coupling through
the dependence of surface wind stress on oceanic surface currents
e.g.,. This coupling results in a drag exerted
by the air–sea interface on the ocean which leads to a systematic reduction
of the wind power input to the oceanic circulation.
Even if these three mechanisms are mainly active at oceanic eddy scales, they can induce
significant effects at larger scales in regions with large SST gradients and/or surface
currents .
They jointly leave their imprint on the wind divergence, and
identifying the relative importance of each mechanism on the momentum balance
is difficult because it depends on the dynamical regime and on the spatial and
temporal scales of interest .
In the ASL coupling strategy the pressure adjustment mechanism is absent, and only a
small fraction of the downward momentum mixing mechanism is accounted for through the
modification of the surface drag coefficient depending on the ASL stability
. As far as the current feedback
is concerned,
showed that the reduction of wind power input to the ocean
is systematically overestimated in oceanic simulations based on an ASL forcing strategy
compared to air–sea coupled simulations. A simulation that neglects the MABL adjustment
to the current feedback cannot represent the partial re-energization of the ocean by the atmosphere and
hence overestimates the drag effect by more than 30 %
(e.g., ). The ASL
forcing strategy used in most oceanic models will thus overestimate the current feedback
effect and underestimate the downward momentum mixing.
The proposed approach and focus for this paper
The various aspects discussed so far suggest that a relevant coupling at the characteristic
scales of the oceanic mesoscales requires nearly the same horizontal resolution in the
ocean and the atmosphere (since the atmosphere must “see” oceanic eddies and fronts)
as well as an atmospheric component more complete than a simple ASL parameterization to
estimate air–sea fluxes. This assessment raises numerous questions on current practices
to force oceanic models across all scales
This remark is supported by the
conclusions of the CLIVAR Working Group on Model Development following the Kiel meeting
in April 2014: http://www.clivar.org/sites/default/files/documents/exchanges65_0.pdf (last access: 20 January 2021)
in the absence of an interactive atmospheric model. The computational cost associated with the
systematic use of fully coupled ocean–atmosphere models of similar horizontal resolution is
generally unaffordable and comes with practical issues like the proper definition of
initial conditions via data assimilation techniques
e.g., and the proper choice of a
parameterization set. Moreover, in the fully coupled case at basin or global
interannual scales the temporal consistency with the observed variability is generally
lost unless a nudging toward observations or reanalysis is done in the atmosphere
above the MABL e.g.,.
There is thus clearly room for improvement in the methodology to compute the surface boundary
conditions for an ocean model. Alternatives to the ASL forcing strategy have already
been suggested by , , and . They proposed
a vertically integrated thermodynamically active and dynamically passive MABL model where the wind and the MABL height are specified as in the current practices. Such a model allows a better feedback between
SSTs and low-level air temperature
and humidity because the latter two are prognostic .
However, by construction, such models do not reproduce the various
aforementioned coupling mechanisms affecting the surface wind stress.
Their focus is on the improvement of the large-scale thermodynamics while ours is on the
improvement of the eddy-scale momentum exchanges.
In the present study we propose an alternative methodology to
improve the representation of the downward momentum mixing and
of the current feedback effect in ocean-only simulations,
leaving aside the pressure gradient adjustment from now on.
Our aim is to account for the modulation of atmospheric turbulence
by anomalies in sea-surface properties in the air–sea flux computation,
which is thought to be the main coupling mechanism at the characteristic
scales of the oceanic mesoscales.
As a step forward beyond the ASL forcing strategy we propose to
complement the ASL parameterization with an ABL parameterization while keeping a
single-column frame. By construction our approach excludes horizontal advection
whose effect can be important in the vicinity of strong SST fronts
e.g.,. However, we considered
that finding a simple and efficient MABL parameterization is the top priority
to start investigating the viability of our approach in terms of practical
implementation and computational cost. Indeed there exists a large variety of
parameterization schemes to represent the effects of subgrid-scale turbulent
mixing in the ABL seeand references therein.
The schemes based on a diagnostic or prognostic turbulent kinetic
energy (TKE) are very popular for operational and research purposes
despite well-identified shortcomings e.g.,.
For our purposes we do not need the full complexity of the schemes
used in practice in atmospheric models because aspects like
cloud processes and complex terrains are outside our scope.
For this reason, the guideline in this paper is the development
and the testing of a simplified version of the TKE-based scheme
proposed by for over-water and over-sea-ice conditions.
Note that the single-column approximation for our simplified model
selected in this study is only a temporary choice to provide evidence
on the viability of the whole approach. More advanced formulations
allowing a more realistic momentum balance
(i.e., including advection) to be recovered will be studied in future work.
Content
The objective of the present study is to introduce a simplified model of the MABL
of intermediate complexity between a bulk parameterization and a full three-dimensional
atmospheric model and to describe its integration to the Nucleus for European Modelling
of the Ocean (NEMO) general circulation model .
This approach will be referred to as the “ABL coupling strategy”.
A constraint in the conception of such a simplified model is to allow an apt representation
of the downward momentum mixing mechanism and partial re-energization of the ocean by the
atmosphere while keeping the computational efficiency and flexibility inherent to ocean-only modeling. The paper is organized as follows. In Sect. , we describe the
continuous formulation of the simplified model called ABL1d, including the
parameterization scheme used to represent vertical turbulent mixing in the MABL and
the strategy to constrain this model with large-scale atmospheric conditions.
Section provides the description of the dicretization and of the practical
implementation of the ABL1d model in the NEMO framework. In Sects.
and numerical results obtained for some atmosphere-only simplified test cases
available in the literature and for a coupled NEMO-ABL1d
simulation in a global configuration are shown. Finally, our conclusions and perspectives
are summarized and discussed in Sect. .
Model equations
In this section we first provide some basic elements on model reduction
to motivate our approach and mention possible alternatives
(Sect. ). Then we detail
the continuous formulation of the ABL1d model and discuss the
assumptions made. In particular the governing equations and necessary boundary
conditions are given in Sect. and the turbulence closure
scheme for the MABL in Sect. . Finally in Sect. we discuss the methodology to relax the ABL1d
prognostic quantities toward large-scale data.
Motivations and proposed approach
Global oceanic models can be run at higher resolution than global atmospheric models
because of their affordable computational cost. From an oceanic perspective, we generally
simulate at high resolution (in space and time) ocean fields
ϕHRoce(x,y,z,t) over a time interval t∈[0,T]
over which only large-scale atmospheric data
ϕLSatm(x,y,z,t)
are known from the integration of a model Matm using lower-resolution surface oceanic data ϕLSoce(x,y,z=0,t) to compute its surface boundary
conditions, namely
ϕLSatm(x,y,z,t)=Matm(ϕLSoce(x,y,z=0,t)),t∈[0,T].
Instead of directly using ϕLSatm(x,y,z=10m,t) to
constrain the oceanic model as in the ASL forcing strategy, our objective is to estimate (without running the full atmospheric model again) the correction to the 10m
large-scale atmospheric data associated both with the
fine resolution in the oceanic surface fields and with the two-way air–sea coupling.
Somehow we aim to find a methodology to get a cheap estimate
ϕ̃HRatm(x,y,z=10m,t) of the solution
that would have been obtained using a coupling of Matm and the oceanic model
at high resolution. To do so we could imagine several approaches:
(i) estimate
∂Matm∂ϕLSoce (i.e., the derivatives of the atmospheric
solution with respect to the oceanic parameters) via sensitivity analysis which would require to have the possibility to operate Matm;
(ii) build a surrogate model via learning strategies which would require a huge amount of data
and computing time;
(iii) select the feedback loops of interest and define a
simplified model to mimic the underlying physical mechanisms. Following the terminology of
, the first two approaches enter the class of statistical or empirical
data-driven models emulating the original model responses while the third one enters the class
of low-fidelity physically based surrogates which are built on a simplified version of the
original system of equations. In the present study we consider this latter approach, in the
spirit of , who derived a simplified oceanic model by degenerating
the primitive equation system and prescribing geostrophic currents into the momentum equation
in substitution of the horizontal pressure gradient. In this model, a simple 1D oceanic mixed
layer is three-dimensionalized via advective terms to couple the vertical columns with each other.
The idea here is to translate this idea to the MABL context.
In the rest of this section we describe the continuous formulation of our simplified MABL
model which will be referred to as ABL1d.
Formulation of a single-column approach
The formulation of the ABL1d model is derived under the following assumptions:
(i) horizontal homogeneity (i.e., ∂x⋅=∂y⋅=0);
(ii) the atmosphere in the computational domain being transparent (i.e., ∂zI=0 with I the radiative flux) meaning that cloud physics is ignored and solar radiation
and precipitations at the air–sea interface are specified as usual from observations e.g.,;
(iii) vertical advection being neglected. Such assumptions prevent the model from prognostically
accounting for the SST-induced adjustment of the atmospheric horizontal pressure gradient and for
horizontal advective processes associated with a higher resolution boundary condition at the air–sea interface.
The focus here is on the proper representation of the modulation of the MABL turbulent mixing
by the air–sea feedback, which is thought to be the main coupling mechanism at the characteristic
scales of the oceanic mesoscales impacting ϕatm(z=10m,t) and hence air–sea fluxes.
This mechanism is expected to explain most of the eddy-scale
wind–SST and wind–current interactions and is key to properly downscaling
large-scale atmospheric data produced by a coarse-resolution GCM to the oceanic resolution.
At a given location in space, the ABL1d model for the Reynolds-averaged
profiles of horizontal velocities uh(z,t), potential temperature θ(z,t),
and specific humidity q(z,t), given a suitable initial condition, is
∂tuh=-fk×uh+∂zKm∂zuh+RLS∂tθ=∂zKs∂zθ+λs(θLS-θ)∂tq=∂zKs∂zq+λs(qLS-q)
for the height z between a lower boundary zsfc and an upper boundary ztop,
which will be considered horizontally constant because only the ocean and sea-ice-covered areas are
of interest. In Eq. (), k=(0,0,1)t is a vertical unit vector, f is the Coriolis parameter,
Km and Ks are the eddy diffusivity for momentum and scalars respectively, the subscript LS is
used to characterize large-scale quantities known a priori, λs(z,t) is the inverse of a relaxation timescale,
and RLS denotes a large-scale forcing for the momentum equation. RLS can either
represent a forcing by geostrophic winds uG (i.e., RLS=fk×uG)
or equivalently by a horizontal pressure gradient
(i.e., RLS=1ρa∇hpLS) combined with a standard Newtonian
relaxation (i.e., RLS=λm(uLS-uh)).
Because of the simplifications made to derive the ABL1d model the RLS term
and a nonzero λs are necessary to prevent the prognostic variables from drifting very far away
from the large-scale values used to “guide” the model. By itself a relaxation term does
not directly represent any real physical process, but the rationale is that it accounts for the
influence of large-scale three-dimensional circulation processes not explicitly
represented in a simple 1D model. Note that this methodology is currently used to evaluate GCM parameterizations in 1D column models. Once the turbulent mixing and the Coriolis term have
been computed to provide a provisional prediction ϕn+1,⋆ at time n+1 for any
ABL1d prognostic variable ϕ, the relaxation term provides a weighting between
this prediction and the large-scale quantities:
ϕn+1=ΔtλϕLS+(1-Δtλ)ϕn+1,⋆,
with Δt the increment of the temporal discretization. Above the boundary layer,
the ABL1d formulation is unable to properly represent the physics; therefore the λ
parameter should be large, while in the first tens of meters near the surface
we expect the ABL1d model to accurately represent the interaction with the
fine-resolution oceanic state, and thus the relaxation toward ϕLS should
be small. The exact form of the λs and λm coefficients
is discussed later in Sect. . Note that because of the relaxation
term, three-dimensional atmospheric data for uLS, θLS,
qLS, and possibly 1ρa∇hpLS
sampled between zsfc and ztop must be provided to the oceanic model
instead of the two-dimensional data (usually at 10m) necessary for an ASL forcing
strategy. Since the ABL1d model does not include any representation of radiative
processes and microphysics, the radiative fluxes and precipitation at the air–sea interface
are similar to the one provided for a standard uncoupled oceanic simulation.
The model requires boundary conditions for the vertical mixing terms which are computed
via a standard bulk formulation:
3Km∂zuhz=zsfc=CD‖uh(zsfc)-uoce‖(uh(zsfc)-uoce),4Ks∂zϕz=zsfc=Cϕ‖uh(zsfc)-uoce‖(ϕ(zsfc)-ϕoce),withϕ=θ,q.
For the sake of consistency, it is preferable to use a bulk formulation
as close as possible to the one used to compute the three-dimensional large-scale
atmospheric data ϕLSatm. Because in the present study
the plan is to use a large-scale forcing from ECMWF reanalysis products, we use the
IFS (Integrated Forecasting System: https://www.ecmwf.int/en/forecasts/documentation-and-support/changes-ecmwf-model/ifs-documentation, last access: 20 January 2021) bulk formulation such as implemented in the
AeroBulk (https://brodeau.github.io/aerobulk/, last access: 20 January 2021) package
to compute CD, Cθ, and Cq in realistic simulations (see Sect. ).
Note that for an ASL forcing strategy uh(zsfc) and ϕ(zsfc) in Eq. ()
would be equal to uLS(z=10m) and ϕLS(z=10m) respectively, while
in the ABL coupling strategy those variables are provided prognostically by an ABL1d model.
As far as the boundary conditions at z=ztop are concerned, Dirichlet boundary conditions
uh(ztop)=uLS(ztop) and
ϕ(ztop)=ϕLS(ztop) are prescribed.
Model () is a first step before evolving toward a more advanced surrogate model including
horizontal advection and fine-scale pressure gradients in the future. A particular focus of the present study is
on the appropriate choice of a closure scheme to diagnose the eddy diffusivities Km and Ks.
This is a key step to properly represent the downward mixing process.
Turbulence closure scheme
This subsection describes the turbulence scheme used to compute
the eddy diffusivity for momentum and scalars. Those eddy diffusivities
are responsible for a vertical mixing of atmospheric variables due to
turbulent processes. The turbulence scheme we have implemented in
our ABL1d model is very similar to the so-called CBR-1d scheme of
, which is used operationally at Meteo France .
We chose to recode the parameterization from scratch for several reasons:
computational efficiency, consistency with the NEMO
coding rules, use of a geopotential vertical coordinate, and flexibility
to add elements specific to the marine atmospheric boundary layer.
CBR-1d is a one-equation turbulence closure model based on a prognostic
TKE and a diagnostic computation of appropriate
length scales. The prognostic equation for the TKE
e=12u′u′+v′v′+w′w′
(with ⋅ the Reynolds averaging operator) is
∂te=-uh′w′⋅∂zuh+gθvrefw′θv′-∂ze′w′+1ρap′w′-ε,
where horizontal terms and vertical advection are neglected, as usually done in
mesoscale atmospheric models. Here θv is the virtual potential temperature,
ρa is atmospheric density, and ε is a dissipation term. In order to express the evolution of e
in terms of Reynolds-averaged atmospheric variables
we consider the standard closure assumptions
for the first order turbulent fluxes to obtain
the classical TKE prognostic equation
∂te=Km‖∂zuh‖2-KsN2+∂zKe∂ze-cεlεe3/2,
where lε is a dissipative length scale, cϵ a constant,
and N2 is the moist Brunt–Väisälä frequency computed as
N2=(g/θvref)∂zθ+0.608∂z(θq) with θvref=288K.
The eddy diffusivities for momentum Km, TKE Ke, and scalars Ks all
depend on e and on a mixing length scale lm:
(Km,Ks,Ke)=(Cm,Csϕz,Ce)lme,
with (Cm,Cs,Ce) a triplet of constants and ϕz a stability function
proportional to the inverse of a turbulent Prandtl number, given by
ϕz(z)=1+maxC1lmlεN2/e,-0.5455-1.
The ϕz function is bounded not to exceed ϕzmax=2.2 as done in the Arpege model of Meteo France e.g.,.
Assuming that the minimum of ϕz is attained in the linearly stratified limit
(i.e., for lm=lε=2e/N2),
values of the maximum Prandtl number Prt=Cm/(Csϕz)
are given in Table . Constant values for Cm, Cs, Ce,
cε, and C1 can be determined from different methods, leading
to nearly similar values. The traditional way is to use the inertial–convective
subrange theory of locally isotropic turbulence .
Another way relies on a theoretical turbulence
model partly based on renormalization group methods see.
For the present study, the sets proposed by and
will be considered (Table ). A major difference between the two sets
concerns the value of Cm. This difference is explained by a reevaluation of the
energy redistribution among velocity components by pressure fluctuations, whose
magnitude is assumed to be proportional to the degree of energy anisotropy as
initially introduced by . Note that the constant set of
is now used by default in both research and operational Meteo France models.
Set of turbulence scheme constants from
and . Prt=Cm/(Csϕz) is the turbulent Prandtl number.
CmCsCecεC1PrtminPrtmaxlmmin (CBR00)0.06670.16670.40.70.1390.1820.5111.5 m (CCH02)0.1260.1430.340.8450.1430.1820.5150.79 m
The Dirichlet boundary condition for TKE applied
at the top z=ztop is
e(z=ztop)=emin=10-6 m2 s-2
and at the bottom z=zsfc we have
e(z=zsfc)=esfc=u⋆2Cmcε+0.2w⋆2,
with u⋆ and w⋆ the friction and convective velocities given by
the bulk formulation. The value for emin has been chosen empirically as
well as background values Kmmin=10-4 m2 s-1
and Ksmin=10-5 m2 s-1 for eddy diffusivities.
The minimum value for lm is simply set as
lmmin=KmminCmemin. There are multiple
options to compute the mixing lengths lm and lε (this point will
be discussed later in Sect. ), but all options have identical
boundary conditions lm(z=ztop)=lmmin and
lm(z=zsfc)=Lsfc=κ(Cmcε)1/4Cm(zsfc+z0).
The value of Lsfc results from the similarity theory in the
neutrally stratified surface layer (Sect. 4.1 in , and Appendix ).
In Eq. (), κ is the von Karman constant and z0 a roughness length
computed within the bulk algorithm.
The way esfc and Lsfc are obtained is detailed in Appendix .
Our current implementation of boundary layer subgrid processes is an
eddy-diffusivity approach which does not include any explicit representation
of boundary-layer convective structures. This could be done via a mass-flux
representation e.g., or the
introduction of a countergradient term e.g.,.
This point is left for future developments of the ABL1d model.
Processing of large-scale forcing and Newtonian relaxation
As mentioned earlier, the ABL1d model () requires three-dimensional (x,y,z)
large-scale atmospheric variables ϕLSatm, while existing uncoupled oceanic forcing strategies
require only two-dimensional (x,y) atmospheric variables. This is a difficulty
for efficiency reasons since it substantially increases the number of I/Os but also
for practical reasons because it requires the development of a dedicated tool to extract
large-scale atmospheric data and interpolate them on prescribed geopotential heights
from their native vertical grid, which can be either pressure based or arbitrary Lagrangian
Eulerian. Such tools have been developed specifically to work with ERA-Interim, ERA5, and
operational IFS datasets and are described in Appendix .
Beyond the particular values of ϕLSatm, the form of the
relaxation timescale has a great impact on model solutions. The vertical profile for the
λm and λs coefficients in Eq. ()
is chosen to nudge strongly above the MABL and moderately in the MABL
with a smooth transition between its minimum and maximum value
to avoid large vertical gradients in λm and λs,
which would result in artificially large vertical gradients in atmospheric variables.
In practice the λm(z) and λs(z) functions depend on the
following parameters:
(λmmax,λmmin) and (λsmax,λsmin),
which define the maximum and minimum of the nudging coefficient for momentum and scalars respectively.
Following Eq. (),
a guideline to set reasonable values for those parameter values would be to make sure that
Δtλsmax≈1 (i.e., the large-scale value is imposed above the
boundary layer) and choose λsmin based on the typical adjustment timescale
of the ABL to surface perturbations. Broadly speaking the ABL can be defined as the
region that responds to surface forcings with a timescale of about an hour
e.g.,. In the realistic numerical experiments
shown in Sect. , we used λsmin=190[min],
which, for an oceanic dynamical time step Δt=1080s, would lead to
Δtλsmin=0.2.
(i.e., the boundary layer values
are the result of a weighting with a weight 0.8 for the ABL1d prediction
and 0.2 for the large-scale value).
(βmin,βmax), which defines the extent of the transition zone separating the maximum
and minimum of the nudging coefficient
We considered the following general form for λs(z) and λm(z), with hbl
the boundary layer height whose value is diagnosed using an integral Richardson number criteria
(Sect. 3.2 and 3.3 in )
with a critical value equal to C1:
λs(z)=λsmin,z≤βminhbl,∑m=03αmzhblm,z∈]βminhbl;βmaxhbl[,λsmax,z≥βmaxhbl,
where four αm coefficients are necessary to guarantee the continuity of λs(z)
and its derivative ∂zλs at z=βminhbl and z=βmaxhbl.
We easily find
α0=(3βmax-βmin)βmin2λsmax+(βmax-3βmin)βmax2λsmin(βmax-βmin)3,α1=-6βmaxβmin(λsmax-λsmin)(βmax-βmin)3,α2=3(βmax+βmin)(λsmax-λsmin)(βmax-βmin)3,α3=-2(λsmax-λsmin)(βmax-βmin)3.
The value of hbl is bounded beforehand to guarantee that at least 3 grid points
are such that z≤βminhbl and z≥βmaxhbl. A typical profile
of the λs(z) is shown in Fig. a.
(a) Typical profile of the nudging coefficient λs(z) with respect to the parameters λsmax,λsmin,βmin,βmax,hbl. (b) Equatorial restoring function req with respect to the Coriolis frequency f.
When the model is forced by the large-scale pressure gradient (or the geostrophic winds),
the parameter λm(z) should be theoretically zero at high and middle latitudes.
However, for the equatorial region, a Newtonian relaxation toward the large-scale winds
should be maintained. To do so, the coefficient λm(z) is multiplied by a coefficient req,
which is a function of the Coriolis parameter f. The req coefficient is equal to zero for large values
of |f| and increases to 1 when approaching the Equator.
The following form satisfies those constraints (see also Fig. b):
req(f)=sinπ2f-fmaxfmax6,fmax=2π12×3600s-1.
Numerical discretization and implementation within NEMO
We have introduced so far the continuous formulation of the
ABL1d model. In this section we describe the discretization
methods used and how this model is included in the NEMO modeling
framework. In particular, the discretization of the Coriolis term
and of the TKE Eq. () and associated
mixing lengths are described in Sect.
and respectively. Details about the practical
implementation in NEMO are given in Sect.
for the coupling aspects and for the
computational aspects.
The ABL1d model () is discretized
in time with an Euler backward scheme for the vertical diffusion terms,
semi-implicitly for the Coriolis term, and explicitly for the relaxation term,
which means that the model is stable as long as λsΔt≤1.
The variables are defined on a non-staggered grid in the horizontal
(a.k.a Arakawa A-grid). Because we consider a computational
domain exclusively over water or sea ice, topography is not considered and
vertical levels are flat and fixed in time which, among other things, allows
the large-scale data ϕLS to be interpolated on the vertical grid offline.
The position of the various quantities introduced so far on the computational
grid is given in Fig. .
Vertical grid variable arrangements and important notations.
Coriolis term treatment
Since in our implementation the horizontal velocity components are collocated,
the discretization of the Coriolis term is straightforward and is energetically neutral.
In the event the ABL1d is integrated with a time step much larger than the oceanic
time step, specific care must be given to the stability of the Coriolis term time stepping. A semi-implicit scheme with weighting parameter
γ reads
uhn+1,⋆=-(fΔt)k×(1-γ)uhn+γuhn+1,⋆,
where the exponent ⋆ is used here to emphasize that uhn+1,⋆
is a temporary value at time n+1 before vertical diffusion and Newtonian relaxation are applied. For a given grid cell with index (i,j), the semi-implicit scheme can be written in a more compact way as
ui,jn+1,⋆=(1-γ(1-γ)(fi,jΔt)2)ui,jn+(fi,jΔt)vi,jn1+(fi,jΔt)2γ2,vi,jn+1,⋆=(1-γ(1-γ)(fi,jΔt)2)vi,jn-(fi,jΔt)ui,jn1+(fi,jΔt)2γ2.
The associated amplification factor modulus is |Acor|=1+(1-γ)2(fΔt)21+γ2(fΔt)2
meaning that unconditional stability is obtained as long as γ≥1/2.
For the numerical results obtained below in Sects. and we used
γ=0.55, which is deliberately slightly dissipative.
Discretization of TKE equation
In Sect. we have presented the continuous formulation of the
TKE-based turbulence closure of the ABL1d model. In the following
we describe how the positivity of TKE can be preserved and how the mixing lengths
lm and lε are computed. We provide a substantial discussion on
the latter aspect because numerical results are very sensitive to the choices made.
TKE positivity preservation
The TKE equation is discretized
using a backward Euler scheme in time with a linearization of the dissipation term
cεlεe3/2, which is discretized
as cεlεenen+1.
However, such discretization is not unconditionally positivity-preserving for
TKE, which could give rise to unphysical solutions e.g.,.
Ignoring the diffusion term, the TKE prognostic Eq. () can be written
as an ordinary differential equation (ODE) of the form
∂te=S(uh,N2)-D(e,t)e,withS(uh,N2)=Km‖∂zuh‖2-KtN2,D(e,t)=cεlεen,
where the last term can be seen as a damping term. For ODEs like Eq. ()
it can be shown that for an initial condition e(0)≥0 and S(uh,N2)≥0,
the solution e(t) keeps the same sign as e(0) whatever the sign of the damping coefficient D(e,t).
Assuming that S(uh,N2) and D(e,t) are positive, a backward Euler discretization of the
damping term in Eq. () would lead to
en+1=en+ΔtS(uh,N2)1+ΔtD(e,t),
which preserves positivity since for en≥0 we obtain en+1≥0.
However, there is no guarantee that the forcing term S(uh,N2) is positive,
in particular when the shear is weak and the stratification is large. When
S(uh,N2) is negative a specific treatment (known as the “Patankar trick”; see )
is required.
In the event of a negative S(uh,N2), the idea is to move the buoyancy
term from S to D after dividing it by en, such that
S(uh,N2)=Km‖∂zuh‖2 is now strictly positive
and D(e,t)=cεlεen+KsN2en.
Such a procedure is a sufficient condition to preserve the positivity of the
TKE without ad hoc clipping of negative values. Moreover our discretization
of the shear and buoyancy terms in the TKE equation is done in an
energetically consistent way following .
Mixing length computation
Another challenging task when implementing a TKE scheme is the discretization of the
mixing lengths. As mentioned earlier, four different discretizations of lm
(lε) have been coded. All discretizations consider the boundary
conditions given in Eq. (). The values of lm and lε
are traditionally computed from two intermediate length scales lup and ldwn,
which respectively correspond to the maximum upward and downward displacement
of a parcel of air with a given initial kinetic energy. Once lup and ldwn
have been estimated by one of the methods described below, the dissipative and mixing length
scales lm and lε are computed as
12alm=12lup1a+ldwn1aa,12blε=minlup,ldwn,
where a≈-32 for CBR00 and a≈-67 for CCH02 (see Appendix ). The impact of the weighting
between lup and ldwn to compute lm
can be significant for idealized experiments like the ones
presented in Sect. but for more realistic
cases results are weakly sensitive and equivalent to the ones
obtained with the simpler weighting lm=lupldwn.
In the following we provide the continuous form of the various ways
to compute lup and ldwn implemented in the ABL1d model.
The discretization aspects are detailed in Appendix .
length scale. A classical approach in atmospheric models is the use of the
mixing length
see also
which defines
lup and ldwn as∫zz+lupN2(s)(s-z)ds=e(z),∫z-ldwnzN2(s)(z-s)ds=e(z).By construction such mixing lengths are bounded by the distance
to the bottom and the top of the computational domain. It is worth noting
that for constant values of N2, Eq. () gives lup2N22=e(z) and
ldwn2N22=e(z) respectively, which is equivalent
to the length scale.
In the remainder we will note lBL89, the mixing length obtained
from Eq. ().
Adaptation of NEMO's length scale. The standard NEMO algorithm Sect. 10.1.3 in is simple and
efficient compared to Eq. (). This algorithm is based on the
length scale
lD80=2e(z)/N2.
lup and ldwn are first initialized to
lup=ldwn=lD80.
The resulting length scales are then limited not only by the distance
to the surface and to the top but also by the distance to a strongly
stratified portion of the air column. This limitation amounts to control of
the vertical gradients of lup(z) and ldwn(z) such that
they are not larger than the variations of altitude. The resulting mixing
length will be simply referred to as lD80.
Note that the Taylor expansion of the integral in Eq. () is∫zz+lupN2(s)(s-z)ds≈N2(z)lup22+dN2dzlup33+O(lup4),which shows that the lD80 mixing length is an approximation of
lBL89, which is obtained by retaining only the leading order term
in the Taylor expansion.
length scale. Recently, proposed a modification of the
mixing length. This modification
turns out to improve results for stably stratified boundary layers
typical of areas covered by ice. They propose to add a shear-related
term to Eq. () such that the definition of lup and
ldwn becomes∫zz+lupN2(s)(s-z)+c0e(s)‖∂suh‖ds=e(z),∫z-ldwnzN2(s)(z-s)+c0e(s)‖∂suh‖ds=e(z),where c0 is a parameter whose value should be smaller than
Cm/cε. The value of c0 will be chosen
based on numerical experiments presented in Sect. .
In the following this mixing length will be referred to as lR17.
A local buoyancy- and shear-based length scale. For the sake of computational efficiency, we have derived a local
version of the length scale which is original
to the present paper. Under the assumption that lup (ldwn)
is small compared to the spatial variations of N2, e, and ‖∂zuh‖,
we end up with the following second-order equation for lup:N2(z)2lup2+c0e(z)‖∂zuh‖lup=e(z),whose unique positive solution islD80⋆(z)=2e(z)c0‖∂zuh‖+c02‖∂zuh‖2+2N2(z).We easily find that
lD80⋆=lD80 for ‖∂zuh‖=0,
and
lD80⋆=e(z)c0‖∂zuh‖
for N2=0, which is consistent with the shear-based length scale of .
Once lD80⋆ has been computed we apply the same algorithmic
approach as in the lD80 case.
The performance of those four length scales for various physical flows
is discussed in Sect. .
Coupling with ocean and sea ice
For the practical implementation of the ABL coupling strategy within a global
oceanic model, a proper coupling method is required for stability and consistency
purposes e.g.,,
and the ABL1d must have the ability to handle grid cells partially covered
by sea ice. For the coupling strategy, a so-called implicit flux coupling which is
unconditionally stable Appendix B in and asymptotically
consistent for Δt→0 is used.
Because vertical diffusion in ABL1d is handled implicitly in time,
the boundary conditions (Eqs. and ) should be provided at time n+1.
The implicit flux coupling amounts to discretize the boundary conditions
Eqs. () and () as
15Km∂zuhz=zsfcn+1=CD‖uhn(zsfc)-ũoce‖(uhn+1(zsfc)-ũoce),16Ks∂zϕz=zsfcn+1=Cϕ‖uhn(zsfc)-ũoce‖(ϕn+1(zsfc)-ϕ̃oce),
where ũoce and ϕ̃oce
are either the instantaneous values at time n if NEMO and ABL1d
have the same time step or an average over the successive oceanic substeps otherwise.
Particular care has also been given to the compatibility between the ABL1d model
and SI3 (Sea Ice model Integrated Initiative) the sea-ice component of NEMO.
SI3 is a multi-category model whose state variables relevant for our study are the
ice surface temperature Tlice with associated fractional area al
(for the lth category), and the ice velocity uice (same for all
categories). Note that the values of the exchange coefficients over sea ice CDice,
Cθice, and Cqice are different from their oceanic counterparts but
are the same over all sea-ice categories. At this point there are several strategies for the
ABL1d/SI3 coupling:
Run the ABL1d model over the whole ABL for each
category l and then average atmospheric variables weighted by al.
Run a single ABL1d model with a category-averaged surface flux.
In the current version of NEMO Cθice is a function of the
averaged temperature Tice which means that it is equivalent to compute
a flux over each category before averaging them
and to compute a single flux using the averaged surface temperature, indeed∑lalCθice‖uh(zsfc)-uice‖(θ(zsfc)-Tlice)=Cθice‖uh(zsfc)-uice‖θ(zsfc)-∑lalTlice.
The second option has been
preferred because it is much easier to implement and more computationally efficient. It amounts
to consider an ice surface temperature averaged over all categories
Tice=∑l=1ncatalTlice for the computation
of ice–atmosphere turbulent fluxes (Tice also enters in the computation of
qice). Noting Foce the fraction of open water (lead), the boundary condition
() and () are modified in
Km∂zuhz=zsfcn+1=FoceCD‖uhn(zsfc)-ũoce‖(uhn+1(zsfc)-ũoce)+(1-Foce)CDice‖uhn(zsfc)-ũice‖(uhn+1(zsfc)-ũice),Ks∂zϕz=zsfcn+1=FoceCϕ‖uhn(zsfc)-ũoce‖(ϕn+1(zsfc)-ϕ̃oce)+(1-Foce)Cϕice‖uhn(zsfc)-ũice‖(ϕn+1(zsfc)-ϕ̃ice).
Because the dynamics of sea ice is computed before the thermodynamics see Fig. 1 in,
the ABL1d–SI3 coupling follows these different steps:
compute surface fluxes over ice and ocean and integrate the ABL1d model for given values Focen and aln,
compute the dynamics of sea ice,
update Focen and aln in Foce⋆ and al⋆ because of step 2,
distribute the fluxes over each ice category considering the updated values al⋆Sect. 3.6 in
compute the thermo-dynamics of sea ice.
Computational aspects
As described in , the NEMO source code is organized
to separate the ocean routines on one side and the routines responsible for
the surface boundary
conditions computation (including sea ice and the coupling interfaces)
on the other side.
This makes a clear separation between the standard ocean model (OCE component)
and the so-called surface module (SAS component). As schematically described in
Fig. , the ABL1d model has been implemented within the SAS
component, which allows the following useful features:
The ABL1d model can be run in standalone mode (coupled or not with sea ice)
with prescribed oceanic surface fields.
The ABL1d model can be run in detached mode; i.e., the OCE and SAS
components run on potentially separate processors and computational grids communicating via the
OASIS3-MCT coupling library .
Schematic representation of the ASL forcing strategy (left) and ABL coupling strategy (right) in terms of code organization and required external data. The OCE and SI3 components represent the oceanic and sea-ice dynamics and thermodynamics respectively while the ASL
component is in charge of providing boundary conditions related to atmospheric conditions.
In the NEMO computational framework the so-called surface module (SAS component),
delineated by dashed line polygons, is virtually separated from the OCE component, which allows SAS to be run in standalone or detached mode (see Sect. ).
An other capability implemented within the NEMO modeling framework
is the possibility to interpolate forcing fields on the fly. This is particularly useful
for the ABL coupling strategy since three-dimensional atmospheric data must be interpolated
on the ABL1d computational grid. As the current implementation of the on-the-fly
interpolation only works in the horizontal, the vertical interpolation of large-scale
atmospheric data on the ABL1d vertical grid is done offline. Nevertheless
it means that the size of input data compared to an ASL forcing strategy is N
times larger with N the number of vertical levels in the ABL. A possibility to
improve the efficiency for the reading of input data would be to
take advantage of the parallel I/O capabilities provided by the XIOS library
XML-IO-Server; which
is currently used in NEMO only for writing output data. This technical development is
left for future work. This is a key aspect because, as discussed later in Appendix , the main source of computational overhead associated
with the ABL coupling strategy is due to the time spent waiting for input files
to be read.
Description of the idealized experiments performed in Sect. .
LMO is the Monin–Obukhov length.
To check the relevance of our ABL1d model for idealized
atmospheric situations typical of the atmospheric boundary layer
over water or sea ice, we performed a set of single-column experiments.
Each of those experiments are evaluated with benchmark large eddy
simulations (LESs). Moreover, we use standardized test cases from the
literature to allow our results to be cross-compared with other
well-established ABL schemes. In the following we consider a neutrally
stratified (Sect. ) and a stably
stratified (Sect. ) case as well as a case
with a transition from stable to unstable stratification representative
of an atmospheric flow over an SST front (Sect. ).
All ABL1d simulations presented here have been performed directly
within the SAS component of the NEMO modeling framework
and can be reproduced using the code available at
https://zenodo.org/record/3904518 (last access: 20 January 2021) ,
which also includes the scripts to generate the figures.
An objective of the present section is to illustrate the type of sensitivity
we can expect from the ABL1d model and discriminate between the various
options available in the code. The experiments showed in Sect.
and are meant to investigate the impact of
(i) the set of constant coefficients (CBR00 vs. CCH02),
(ii) the various formulations of lm and lε among
the algorithms described in Sect. , and
(iii) the parameter value c0 in the
lR17 and lD80⋆ mixing length computation.
Those experiments will allow several options to be discarded. The ability
of the remaining options to represent the downward mixing mechanism,
discussed in Sect. , is then evaluated in
Sect. . The robustness of the results to the
bulk formulation and to the nudging coefficient is also checked.
For each experiment we explicitly provide the initial and boundary
conditions as well as all the necessary parameter values (see Table )
so that the experiments can be reproduced easily by other modeling groups.
Results obtained for the neutral boundary-layer
case of with the
CBR00 model constants (left panels, a to d) and CCH02 model constants (right panels, d to e)
for different parameter values for c0 and different
mixing-length formulations (lD80⋆ for black lines
or lR17 for gray lines). Results are shown for u(a, e),
v(b, f), e(c, g), and lm(d, h). In the top
four panels results are compared with LESs from (their Fig. 16). As in ,
simulations were run over a period of 10/f and results are averaged
over the last 3/f period.
Neutral turbulent Ekman layer
We first propose to investigate the simulation of a neutrally stratified atmosphere
analogous to a classical turbulent Ekman layer. The selected case is based on
the setup described in . The initial conditions for this
experiment are not defined analytically; they are given by Table A1 in
However, we did not find significant differences
in numerical solutions when using the following initial conditions:
uh(z,t=0)=uG,e(z,t=0)=emin.
.
This test case is mainly used to check the adequacy of our surface boundary
conditions with similarity theory and the proper calibration of the parameter
c0 in the lD80⋆ and lR17 formulations of the mixing
lengths. In theory, the lD80 and lBL89 mixing lengths do not
support the asymptotic limit N2=0 but for the integrity of numerical results
a minimum threshold Nε2 on the stratification is imposed in the code.
In this case the procedure to compute those mixing lengths as described in Appendix will provide identical results,
namely lup=ztop-z and ldwn=z-zsfc (i.e., the distance from the top and from the bottom of the computational domain).
We test here the lD80⋆ and lR17 introduced
in Sect. . The reference solution is taken from
(panels a and b in their Fig. 16). Results are obtained using the ABL1d
model with either the CBR00 (Fig. a–d)
or the CCH02 (Fig. e–h) set of parameters.
All experiments have been done with c0=0.15 and c0=0.2.
All simulations are able to reproduce the overall behavior
of the LES case. The main outcomes are as follows:
The best agreement is obtained when using the
CCH02 constants along with lD80⋆ mixing length
and c0=0.2.
The results obtained for lD80 and lBL89 are
identical and close to the lR17 results with c0=0.15 (not shown).
All simulations with the CCH02 set of parameters show
reasonable results.
Results obtained for the stably stratified boundary-layer
case of for the parameter values CBR00
(left panels, a to c) and CCH02 (right panels, d to f) with different
mixing-length formulations: lD80 for black solid lines, lD80⋆ with c0=0.15 for dashed black lines (c0=0.2 for dotted black lines), lBL89 for solid gray lines, and lR17 with c0=0.15 for dashed gray lines (c0=0.2 for dotted gray lines). Results are shown for potential temperature θ(a, d), wind speed (b, e), and lm(c, f). Dotted red lines represent LES results from . Instantaneous profiles after 9 h are shown.
Stably stratified boundary layer (GABLS1)
Within the Global Energy and Water Exchanges (GEWEX) atmospheric boundary
layer study (GABLS), idealized cases for stable surface boundary
layers have been investigated e.g.,. Such conditions
are typical of areas covered with sea ice. Here we consider the GABLS1
case, whose technical description is available at
http://turbulencia.uib.es/gabls/gabls1d_desc.pdf (last access: 20 January 2021).
This experiment is particularly interesting as
significant differences generally exist between solutions obtained from LES
and single-column simulations, for example when the
length scale is used
e.g.,.
A large-scale geostrophic wind is imposed as well as
a cooling of the surface temperature θs(t) given by
θs(t)=263.5-0.25(t/3600s).
The parameter values for this test are reported
in Table and the initial conditions
are uh(z,t=0)=uG, and
θ(z,t=0)=265z≤100m265+0.01(z-100)otherwise,e(z,t=0)=emin+0.4(1-z/250)3z≤250meminotherwise.
The solutions after 9 h of simulation are shown
in Fig. (left panels) for CBR00 parameter values
and in Fig. (right panels) for CCH02 parameter values.
The reference solution is taken from
LESs. As expected, solutions based on a mixing length
ignoring the contribution from the vertical shear exhibit a boundary layer that is too thick and a wind speed maximum located too high
in altitude. Using a buoyancy- and shear-based mixing length mitigates
the issue and provides very good agreement with reference solutions
when the CCH02 model constants are used. The best results are
obtained for lD80⋆ with c0=0.2 and
lR17 with c0=0.15. Solutions obtained with the
CBR00 model constants systematically predict larger
turbulent kinetic energy and mixing lengths, resulting
in large values of Ks in the first 100 m near
the surface (not shown). The mismatch in terms of TKE is partially
explained by the difference in boundary conditions since
with CBR00 constants we have esfc=4.628u⋆2
while with CCH02 constants we get esfc=3.065u⋆2 from Eq. ().
Note that the proper calibration of the c0 constant jointly with the
cε is the subject of several ongoing studies. Since our
simulations reproduce the known sensitivity to those parameters,
the ABL1d model could directly benefit from new findings
on that topic.
The main outcomes are as follows:
The CCH02 set of parameters provides results of better quality than
the CBR00 constants. For the sake of simplicity, we will retain only the
CCH02 parameters for the numerical results shown in the remainder.
The buoyancy- and vertical-shear-based mixing lengths lR17
and lD80⋆ are superior to the buoyancy-based mixing lengths
lD80 and lBL89 for stable boundary layers.
Winds across a midlatitude SST frontSetup and reference solutions
An idealized experiment particularly relevant for the coupling of the MABL
with mesoscale oceanic eddies (and potentially submesoscale fronts) was initially suggested by and then revised
by . More recently
derived an analytical model based on a similar
setup. The geometry of the problem is two-dimensional x–z with an SST
front along the x axis:
θs(x)=288.95+Δθ2tanhxLθ,
where Δθ=3K, Lθ=100 km, and
x∈[-1800km,1800km].
As indicated in Table , a zonal geostrophic wind of
15ms-1 is prescribed, balanced by
a vertically homogeneous meridional pressure gradient.
The wind thus flows over cold water before reaching a
warm SST anomaly, which is 3K warmer. We consider a dry case, in which
the model is initialized ∀x with
θ(z,t=0)=288.95+N2θref/gz,q(z,t=0)=0,
where N2=10-4s-2 and θref=288K.
The velocities are systematically initialized with geostrophic winds.
All simulations are run for 36 h when the flow reaches a quasi-equilibrium
state.
For this configuration the reference solution is obtained from the
mesoscale non-hydrostatic model (MesoNH) v5.3.0 ,
where microphysics and radiation packages have not been activated. The horizontal resolution is
Δx=1 km and the model is discretized with 91 vertical levels from the
surface to 20 km height. The vertical resolution near the surface is
Δz=10m and around Δz=100m at 2000m height.
The turbulence scheme is the 1.5-order closure of in
its one-dimensional form with the lBL89 mixing length and CCH02 set
of parameters.
Sea surface fluxes are computed using the bulk parameterization COARE3.0,
which is also available in NEMO from the AeroBulk package.
As far as the ABL1d model is concerned,
the top of the computational domain is ztop=2000m
and the vertical grid is stretched with a typical resolution
of 20m near the surface and 100m near z=ztop
with a first grid point located at z=10m.
In the horizontal, the resolution is Δx=6 km.
Zonal (a, b, c) and meridional (d, e, f)
components of atmospheric winds for the reference MesoNH simulation
(c, f), for ABL1d simulations with lD80
mixing length and CCH02 model constants (a, d), and with
lBL89 mixing length and CCH02 model constants (b, e).
Temperature contours are shown in white with a contour every
0.5∘C between 15 and 17.5∘C.
The SST front is centered at x=0 km.
Zonal (a) and meridional (b) components of
10m winds and 10m temperature (c) for
the reference MesoNH simulation (dashed red)
and for ABL1d simulations with different mixing-length
formulations for the winds across a midlatitude SST front experiment.
Numerical results
For this configuration, results will be mostly evaluated in terms of
10m winds u10 and temperature θ10.
As an illustration of the type of result we get, we first compare the
MesoNH solution and the ABL1d solution obtained with the
lD80 and lBL89 mixing lengths in Fig. .
It is worth noting that the MesoNH solution closely compares with the
solution of (their Fig. 2) with a shallow
boundary layer height (around 400m) before the front and
a thicker one (around 800m) after the front where momentum mixing
is enhanced. Over the front, as noted by with
a similar setup, the effect of advection is predominant for meridional winds,
thus explaining the differences seen with the ABL1d simulations.
Indeed with ABL1d, whatever the numerical options, the atmospheric
column will locally adjust to the underlying oceanic conditions since
horizontal advection is neglected. This explains the absence of horizontal
lag when passing over the front in the ABL1d solution compared to
the MesoNH solution. However, away from the SST front the solutions are
very similar in terms of boundary layer height and vertical wind structure.
In anticipation of a coupling with an oceanic model, the most important
quantities to look at are the 10 m atmospheric variables rather
than the full 3D vertical structure of the MABL. In Fig. ,
the 10 m wind components and temperature when the ABL1d model
reaches a quasi-equilibrium state are shown for different mixing length options,
as well as the MesoNH results. First, the results obtained with the lR17
are very different from the expected behavior, and we will focus the discussion
on other options. In terms of zonal 10 m wind the buoyancy-based lBL89
and lD80 mixing lengths provide a good agreement with the MesoNH solution,
which could be expected as the MesoNH solution has been generated using the
lBL89 mixing length. As soon as the mixing length is a function of
buoyancy and vertical shear (as is the case for lD80⋆) the simulated
winds are weaker because the boundary layer is thinner. This leads to improved results
in the stably stratified case shown earlier, but in the present case, which is more representative
of realistic configuration in the MABL, it leads to a mixing that is too weak. However,
compared to the lR17 mixing length the lD80⋆ still performs
reasonably well but the winds on the warm side of the front are about
1ms-1 weaker than the MesoNH winds for c0=0.15 and
become weaker and weaker as c0 increases.
The main outcomes are as follows:
In the frontal region the effect of horizontal advection is predominant and the
ABL1d model cannot reproduce the horizontal lag seen in the reference solution
when passing over the front.
The ABL1d model reproduces the downward momentum mixing mechanism correctly.
The best results are obtained with the buoyancy-based lBL89 and lD80 mixing lengths.
The lR17 mixing length will be discarded from the comparison.
Although relevant for the present study this 2D x–z setup is not fully
representative of realistic conditions because the air column has time to adjust
to the underlying oceanic state, which is kept frozen in time.
2D time vs. height sections representing the temporal evolution of the
zonal (a, b, c) and meridional (d, e, f) components of atmospheric winds for ABL1d simulations
of an air column crossing an SST front with COARE bulk formulation (a, d)
and IFS bulk formulation (middle and right panels). For the case presented in the
right panels, a Newtonian relaxation toward the initial temperature profile was
added with λsmin=148[h] and
λsmax=16[h].
The simulations were performed with lD80 mixing length and CCH02
model constants.
Temporal evolution of the zonal (a, c, e) and meridional
(b, d, f) components of 10 m atmospheric winds for
ABL1d simulations of an air column crossing an SST front.
The temporal evolution of SST (solid red lines) is also shown.
For each panel the results from four different
simulations are shown: with COARE bulk formulation (solid lines) or
IFS bulk formulation (dashed lines), with Newtonian relaxation on temperature
such that λsmin=148[h] and
λsmax=16[h] (gray lines)
or no relaxation (black lines).
Panels (a, b) are obtained from simulations performed with lD80 mixing length, (c, d) with lD80⋆ (c0=0.15), and (e, f)lBL89.
A single-column version
An alternative to the x–z setup would be to formulate the test case
as a Lagrangian advection of an air column over an SST front by
prescribing a temporal evolution of sea surface temperature
θs(t) as
θs(t)=288.95+312tanh3(t-144×103s)20000,t∈[0,80×3600s].
In this case the air column does not necessarily have time to adjust to
the underlying oceanic conditions. Initial conditions are the same as
the ones of the 2D x–z case. For this test case we do not have a reference
solution, but it is expected that the temporal evolution of the solution
should be relatively similar to the spatial evolution in the MesoNH 2D
x–z case studied in previous subsection. This can be seen from Fig. , where there is a clear similarity between the time vs. height
sections obtained with the ABL1d simulations and the x vs. height
sections shown for MesoNH in Fig. . The ABL1d solution
shows a temporal lag analogous to the horizontal lag in the reference solution
for the 2D x–z case. In addition to that, we also use this test case to
investigate the sensitivity of the solutions to the bulk formulation and to
the Newtonian relaxation which was absent in simulations discussed so far.
We consider the COARE and IFS bulk formulations, which are relatively close to each
other, to check the robustness of the results to small perturbations in
surface fluxes. We also consider simulations with a relaxation of the temperature
variable toward the initial condition with a fast relaxation timescale
λs=16[h] above the ABL and a slower one
λs=148[h] in the ABL. This is meant to check
that the relaxation does not completely overwrite the physics of the coupling
we aim to represent with the ABL1d model. Results from those sensitivity
experiments are shown in Figs. and .
In particular in Fig. the evolution of the 10m
winds across the SST front closely resembles the one shown in
Fig. (dashed red lines) for MesoNH. Moreover, the results
in Fig. are robust to a change of bulk formulation to compute
the surface fluxes. Reassuringly, adding a relaxation toward large-scale data
which did not see the SST front does not deteriorate the realism of the solutions,
as can be seen from Fig. c and f
and (gray lines).
The main outcomes are as follows:
The response of the ABL1d model to evolving oceanic conditions
is not local in time (it shows a temporal lag).
The good representation of the downward momentum mixing process
is not sensitive to the bulk formulation.
Adding a relaxation term toward large-scale data does not deteriorate
the realism of the solutions significantly.
Based on the results reported in this section, the best balance between
efficiency and physical relevance is obtained when using the parameter
values from CCH02 and the modified mixing-length
formulation lD80 or lD80⋆. In particular we
could imagine using the lD80 formulation over water and the
lD80⋆ formulation over sea ice.
Coupled numerical experiments
Using atmosphere-only experiments, we have been focused so far on the good
representation of the downward momentum mixing mechanism and of the stable
boundary layers typical of areas covered with sea ice. In the following we
check that those two aspects are still adequately represented in a realistic
coupled NEMO-ABL1d simulation. This simulation will also be used
to look at the wind–current interaction, which was left aside so far.
We performed a 5-year global simulation using the ORCA025 configuration.
Details and illustrations are given hereafter.
Coupled NEMO-ABL1d configuration
We use here a global ORCA025 configuration at a 0.25∘ horizontal resolution
with 75 vertical z levels forced by the ECWMF ERA-Interim 6 h
analysis . This configuration is identical to the one described in
(see their Sect. 4.1.1).
The ABL1d-NEMO coupled simulation is carried out with the same numerical options
as in a standard ASL forcing strategy. However, in the ABL coupling strategy,
the two-dimensional near-surface air temperature, humidity, and winds used in the
usual ASL forcing are replaced by three-dimensional atmospheric variables sampled
between the surface and 2000m preprocessed following the different steps
described in Appendix . The large-scale pressure gradient computed
during the preprocessing is used as a geostrophic forcing for the ABL1d
model dynamics. Three-dimensional atmospheric variables are generated over the 2014–2018
period and vertically interpolated on 50 levels between 10 and
2000m with a vertical resolution increasing with height. Grid resolution
is about 20m near the air–sea interface and reaches 70m at the
top of the ABL1d domain. The choice of a vertical extent of 2000m
and 50 vertical levels in the ABL1d model is somewhat arbitrary and the
robustness of numerical results to these choices will be investigated in future studies.
For the simulations presented here, the same horizontal grid and time step
(Δt=1200s) are chosen in the ABL1d and NEMO models.
The options associated with the ABL coupling
available through the NEMO standard name list are reported in
Table , and a detailed profiling of the code is presented
in Appendix in order to assess the overhead associated
with the ABL coupling strategy vs. ASL coupling strategy.
This profiling shows that the overhead associated with the ABL1d
(when using the lD80 mixing length) is on the order of 4 %
and the one associated with the input part of the I/O operations is 5 %.
Overall there is an increase of 9 % in elapsed time compared to the standard
ASL forcing strategy.
Name list parameters in the NEMO(v4.0) to set in the name list section namsbc_abl before running a simulation coupled with ABL1d.
Name list parameterTypeDescriptionln_hpgls_frcbooleantrue if RLS=1ρa∇hpLS in Eq. ()ln_geos_windsbooleantrue if RLS=fk×uG in Eq. ()nn_dyn_restoreinteger(=0) no wind relaxation (=1) wind relaxation scaled by req(f) as in (=2) wind relaxation everywherern_ldyn_minrealinverse of λmmax in hours (see Sect. )rn_ldyn_maxrealinverse of λmmin in hours (see Sect. )rn_ltra_minrealinverse of λsmax in hours (see Sect. )rn_ltra_maxrealinverse of λsmin in hours (see Sect. )nn_amxlinteger(=0) lD80 mixing length (=1) lD80⋆ mixing length (=2) lBL89 mixing length (=3) lR17 mixing lengthrn_CmrealCm parameter in TKE schemern_CtrealCs parameter in TKE schemern_CerealCe parameter in TKE schemern_Cepsrealcε parameter in TKE schemern_Rodrealc0 parameter in lD80⋆ and lR17 mixing lengthsrn_RicrealC1 parameter in the definition of ϕzln_smth_pblhbooleanhorizontal smoothing of ABL heightNumerical results
In this section, we evaluate the ABL coupling strategy in a realistic context for a set
of relevant metrics. The objective is not to conduct a thorough physical analysis
of the numerical results but to illustrate the potential of the ABL coupling strategy
and its proper implementation in NEMO. To evaluate our numerical results, we use
standard metrics from the literature to quantify the wind–SST (a.k.a. thermal feedback
effect), wind–currents (a.k.a. current feedback effect), and MABL–sea-ice couplings e.g.,.
Thermal feedback effect
To quantify the surface wind response to SST, we show in Fig. a
a global map of the temporal correlation between the high-pass-filtered 10m
wind speed from the first vertical level in the ABL1d model and the SST.
The same correlation is shown in from satellite observations
(their Fig. 1d) and from coupled numerical experiments between a
0.1∘ ocean and a 0.25∘ atmospheric model (their Fig. 1c).
Consistent with observations and fully coupled models, the correlation obtained from the coupled
NEMO-ABL1d simulation shows large positive correlations
over regions like the Southern Ocean, Kuroshio, and Gulf Stream extensions
as well as in the Gulf of Guinea. Correlations are, however, weaker than observations in the northern and equatorial Pacific between 90 and 180∘W. As the thermal feedback strength is related to the ocean model resolution , we can expect a better agreement with observations using a higher resolution configuration such as ORCA12 (1/12∘ resolution). This coupling sensitivity to the oceanic resolution will be addressed in a future study.
(a) Global map of temporal correlation of high-pass-filtered wind speed
at the first vertical level of the ABL1d model with SST from NEMO.
Both NEMO and ABL1d are configured at 0.25∘ resolution.
(b, c) Global maps of the coupling coefficient between the surface current
vorticity and the wind curl (sw, b) and between the surface current
vorticity and the wind-stress curl (sτ, c) estimated from a 0.25∘
resolution coupled NEMO-ABL1d global simulation.
The fields are first temporally averaged using a 29 d running mean
and spatially high-pass filtered.
Current feedback effect
Other processes of interest are those related to the coupling between oceanic
surface currents, wind stress, and wind. Such coupling is responsible for a dampening
of the eddy mesoscale activity in the ocean. In ,
two coupling coefficients called sw and sτ are defined to quantify
this effect. sτ is a measure of the sink of energy from the eddies
and fronts to the ABL and sw quantifies the partial re-energization of
the ocean by the wind response to the wind–current coupling. This re-energization
is absent in the ASL forcing strategy, which results in an excessive dampening of
the oceanic eddy mesoscale activity. In practice, sτ (sw)
corresponds to the slope of the linear relationship between high-pass-filtered surface
current vorticity and surface wind-stress (wind) curl. Global maps
of sτ and sw computed from our coupled NEMO-ABL1d
global simulation are shown in Fig. . Large negative values of
sτ indicate an efficient dampening of the eddy mesoscale activity by
the current feedback (i.e., a large sink of energy from the ocean to the atmosphere),
and the large positive values of sw indicate an efficient wind response and re-energization
of the mesoscale currents. Our numerical experiment provides results very
consistent with the results obtained from coupled simulations between NEMO
and the WRF shown in
(their Fig. 1b for sτ and 2c for sw).
As mentioned earlier, with an ASL forcing strategy we would systematically
have sw=0 and stronger sτ values.
Yearly average of sea-ice cover (contours) and atmospheric
boundary layer height (shaded) over the antarctic (b) and
arctic (a) regions.
MABL and sea-ice coupling
The last illustration of our implementation presented in this section is the
coupling of ABL1d with sea ice. As described in Sect. ,
sea ice generally induces a shallow stably stratified boundary layer due to the
near-surface air cooling. This increased vertical stability tends to reduce
atmospheric turbulence, producing shallower ABL heights over sea ice. This
relationship between sea-ice concentration and ABL height is clearly visible
from Fig. on both Arctic and Antarctic domains, where the ABL
height follows a progressive decrease from about 800 to 200m
in the transition zone between the open ocean and fully ice-covered regions.
This coupling between the ABL and sea ice have important effects on near-surface
wind, temperature, and humidity, and consequently on sea-ice concentration
evolution, which will need to be specifically assessed in future ABL-based studies.
ConclusionsSummary
A simplified atmospheric boundary layer (ABL) model has been
developed and integrated to an oceanic model. This development is made with
the objective to improve the representation of air–sea interactions in eddying
oceanic models compared to the standard forcing strategy where the 10 m
height atmospheric quantities are prescribed. For this preliminary study, the
simplified ABL model takes the form of a single-column model including a
turbulence scheme coupled to each oceanic grid point. A crucial hypothesis
is that the dominant process at the characteristic scale of the oceanic mesoscale
is the so-called downward mixing process which stems from a modulation
of atmospheric turbulence by sea surface temperature (SST) anomalies.
Our approach can be seen as an extended bulk approach: instead of
prescribing atmospheric quantities at 10 m to compute
air–sea fluxes via an atmospheric surface layer (ASL) parameterization,
atmospheric quantities in the first few hundred meters are used to
constrain an ABL model which provides 10 m atmospheric values
to the ASL parameterization.
An important point is that our modeling
strategy keeps the computational efficiency and flexibility inherent
to ocean-only modeling. Indeed, the overhead generally observed in terms of computational cost compared to the usual ASL forcing strategy
is roughly 10 %, and half of this overhead is due to I/O operations since
the ABL model is constrained by 3D atmospheric data.
In this paper the key components of such an approach have been
described. This includes the large-scale forcing strategy, the
coupling with the ocean and sea ice and last but not least
the ABL turbulence closure scheme
based on a prognostic equation for the turbulence kinetic energy.
The resulting simplified model, called ABL1d, has been
tested for several boundary-layer regimes relevant to either
ocean–atmosphere or sea-ice–atmosphere coupling. Results have
systematically been evaluated against large eddy simulations (LESs).
Furthermore we have investigated
the behavior of the model to several parameters including the
formulation of the mixing length and the turbulence model constants.
First results from a global ABL1d-NEMO configuration
show an excellent behavior in terms of wind–SST two-way coupling.
A first analysis of the impact of the coupling with ABL1d
from a physical viewpoint is presented in .
Future work
Now that an adequate computational framework and an efficient
turbulent scheme that can be operated for a reasonable computational
overhead have been developed, the next step is to
investigate the relevance of the single-column representation of
the ABL selected for the present paper. Indeed, several studies have
already shown that momentum vertical turbulent mixing, pressure gradient,
Coriolis, and nonlinear advection are all important to the momentum balance
in the marine atmospheric boundary layer at the vicinity of oceanic fronts
(see for example ).
It is well known that the relative importance of those terms depends on
the wind regime: for strong winds the vertical mixing is the dominant
mechanism while for weak winds the pressure adjustment mechanism dominates.
The current single-column approximation is based on the assumption that
the balance is dominated by vertical turbulent mixing, and the effect of
other terms is roughly represented by the geostrophic guide and/or the
nudging term. The test case presented in Sect. clearly
illustrates the limitations of a single-column approach ignoring advective
effects. However, before moving to more advanced formulations,
our rationale was that the two main bottlenecks in terms of computational
cost inherent to the ABL coupling strategy are the reading of 3D atmospheric
data and the choice of ABL turbulent scheme. As a first step, we focused
on those two aspects to assess whether or not our approach can be a viable
option. Even if the justification of our model is not beyond reproach,
it already brings an improvement compared to the ASL forcing strategy.
Several ways to improve the methodology presented here are currently
under investigation. At a practical level, ways to lower the computational
overhead due to I/O operations will be investigated using the parallel
I/O capabilities provided by the XIOS library which is currently used in NEMO
only for outputs. At a more fundamental level, the continuous formulation of the
ABL1d model will be completed to improve the representation of the
momentum balance by integrating the effect of horizontal advection and
fine-scale pressure gradients. Increasing the complexity of the model should
allow the impact of the nudging term on the ABL solutions to be lowered.
In the event our approach turns out to be physically sound for a reasonable
complexity it could be useful not only for offline oceanic simulations but
also in coupled simulations to downscale the information from a low-resolution
atmospheric component to a high-resolution oceanic component. A standalone ABL
model of intermediate complexity could also play a role in coupled data assimilation
where the current practice is generally to assimilate data separately in the ocean
and the atmosphere, ignoring the air–sea interactions, which results in inconsistencies
at the air–sea interface in the initial conditions, causing initial shocks in the
coupled forecasts e.g.,.
We wish to conclude this study by clarifying that the framework we have
developed within NEMO is general enough to allow alternative
approaches (e.g., via model-driven empirical models) to be seamlessly
tested and confronted with the ABL coupling strategy.
Surface boundary conditions for TKE and mixing lengths
In this appendix, following the methodology of
re-expressed with our notations, we quickly recall how the surface boundary
conditions for the turbulent kinetic energy (TKE) and for the mixing
lengths lm and lε are determined via a matching between
the subgrid turbulence scheme and the surface-layer theory. The simplest
form of atmospheric surface layer (ASL) theory, namely for neutral
stratification, is considered. Under the quasi-equilibrium hypothesis
the evolution Eq. () for the TKE reduces to the equilibrium between turbulence production and
dissipation Km‖∂zuh‖2=cεlεe3/2 which,
combined with Km=Cmlme, leads to
e=Cmcε(lmlε)‖∂zuh‖2.
The similarity theory for the ASL in the neutral case is such that
‖∂zuh‖=u⋆κ(z+z0)
with κ the Von Karman constant and z0 the roughness length
which can be combined with Eq. () to get that in the surface layer:
e=Cmcεlmlεκ2(z+z0)2u⋆2,
with u⋆2=‖uh′w′‖
the friction velocity. Moreover, enforcing the consistency
between the eddy diffusivity for momentum given by the ASL
theory (Km=κu⋆(z+z0))
and the one given by the TKE closure (Km=Cmlme)
leads to
lm=κ(z+z0)Cmu⋆e→lm3lε=cεCm-3κ(z+z0)4.
We thus have two relations () and ()
for three unknowns (e, lm, and lε).
At this point our derivation will differ from
as we will assume that lm=lε=Lsfc in the ASL.
Under this assumption, combining Eqs. () and
() we easily obtain
e=u⋆2Cmcε,Lsfc(z)=κ(Cmcε)1/4Cm(z+z0),
for z≤δasl, where δasl
corresponds to the extent of the ASL.
The expression of Lsfc(z) for z≤δasl is also used as a constraint to define the weighted average needed to determine lm from lup and ldwn:
lm=12lup1/a+ldwn1/aa
(equivalent to Eq. ).
In the ASL we further assume that ldwn≈0 and lup≈δasl; for
lm(z=δasl) to be consistent with
Lsfc(z=δasl) we should have
12δasl1/aa=κ(Cmcε)1/4Cm(δasl+z0).
Considering that δasl≫z0,
Eq. () is satisfied for
a=-log(cε)-3log(Cm)+4log(κ)log(16).
Considering the CBR00 model constants we obtain
a=-1.4796≈-3/2 and
a=-0.860834≈-6/7 for the CCH02 constants (see Table ).
Preprocessing of atmospheric data from IFSAltitude of IFS vertical levels
The ABL1d model is discretized on fixed in time and space
geopotential levels while the IFS model uses a pressure-based sigma
coordinate. A first step is to recover the altitude associated with
each sigma level. The pressure pk+12 defined at cell
interfaces between two successive vertical layers is given by
pk+12=Ak+12+Bk+12ps,k∈[[1,Nifs]],
where Ak+12 (Pa) and
Bk+12 (dimensionless) are constants given by a smooth
analytical function defining the vertical grid stretching.
Typical values of the altitude of grid points in the vertical for
a standard 60-level grid (L60) and a surface pressure of 1013hPa
are given in Table .
Once the values of pk+12 and ps are known,
the altitude of cell interfaces can be computed by integrating
the hydrostatic equilibrium
∂zϕ=-RdTvp∂zp
vertically. In Eq. (), ϕ is the geopotential,
Tv the virtual temperature, and Rd the specific gas constant
for dry air. At a discrete level we get
∫zk-12zk+12∂sϕds=-RdTv(zk)∫zk-12zk+12∂sppds,
which gives
e3tkifs=-RdTv(zk)glnpk+12pk-12.
Once the layer thicknesses e3tkifs are known,
horizontal wind components, potential temperature, and specific humidity
can be interpolated on the ABL1d vertical levels. Under the
constraint that
∫zsfcztopψifs(z)dz=∫zsfcztopψ(z)dz for any IFS
quantity ψifs to be interpolated. Wind components are
interpolated using a fourth-order compact scheme while tracers
are interpolated using a WENO-like PPM scheme
(Alexander Shchepetkin, personal communication, 2001) which is monotonic.
Altitude zk and layer thickness e3tk of the IFS L60
vertical grid in the first 2000 m with respect to the parameter
values Ak and Bk of a surface pressure ps=1013hPa.
IndexAkBkAltitudeLayer thickness[Pa]zk[m]e3tkcep[m]10.0000001.00000010.0020.0020.0000000.99763034.9729.9437.3677430.99401971.8943.92465.8892440.988270124.4861.305210.3938900.979663195.8581.496467.3335880.967645288.55104.017855.3617550.951822404.72128.4381385.9125980.931940546.06154.4092063.7797850.907884713.97181.61102887.6965330.879657909.57209.81113850.9133300.8473751133.73238.78124941.7783200.8112531387.12268.33136144.3149410.7715971670.26298.31147438.8032230.7287861983.49328.58Filtering in the presence of boundaries
Because of the IFS numerical formulation and of the post-processing of output data,
the solutions sometimes contain high-frequency oscillations at the
vicinity of the land-sea interface. This problem is further compounded
when the nearshore topography is steep. The atmospheric fields over water
thus need to be smoothed horizontally to specifically remove the 2Δx noise.
We use a standard two-dimensional Shapiro filter which, in the absence of lateral boundaries, can be formulated as
ψi,j⋆=ψi,j+14δi+1/2,j(x)-δi-1/2,j(x),ψi,jf=ψi,j⋆+14δi,j+1/2(y,⋆)-δi,j-1/2(y,⋆),
where δi+1/2,j(x)=ψi+1,j-ψi,j
and δi,j+1/2(y,⋆)=ψi,j+1⋆-ψi,j⋆.
The amplification factor associated with this filter is
Ashap(θx,θy)=141+cosθx1+cosθy,θx=kxΔx,θy=kyΔy,
which guarantees that one iteration of the filter is sufficient to
remove the grid-scale noise since
Ashap(π,π)=Ashap(π,θy)=Ashap(θx,π)=0
and that Ashap≤1 (i.e., no waves are amplified).
In the presence of solid boundaries we would like to retain those properties
as much as possible. A straightforward approach would be to impose a no-gradient
condition at the coast, i.e., δi+1/2,j(x)=0 as soon as
tmaski+1,j×tmaski,j=0
(δi,j+1/2(y,⋆)=0 as soon as
tmaski,j+1×tmaski,j=0),
with tmask the indicator function equal to 1 over water and 0 over land.
Let us also consider the following alternative boundary conditions
δi+1/2,j(x)=-δi-1/2,j(x),iftmaski+1,j=0,δi-1/2,j(x)=-δi+1/2,j(x),iftmaski-1,j=0,
and similar in the y direction. We do not elaborate on this choice but it can be shown
theoretically that boundary conditions () provide a better control of
grid-scale noise near the coast. To illustrate this point,
in Fig. the surface pressure gradients are shown for different
boundary conditions. In particular it can be seen near the coast that the
no-gradient boundary condition (panels b and e) leaves some artificial patterns
in gradients, especially in the Peru–Chile current system, while the boundary condition
() efficiently mitigates this issue. Note that it is particularly
essential to make sure that the surface pressure field is sufficiently smooth
because gradients of this field are used to compute geostrophic winds which
are important for the large-scale forcing of the ABL1d model.
Atmospheric surface pressure horizontal gradients in x(a, b, c)
and y(d, e, f) directions obtained from the original IFS data (a, d),
after a Shapiro filtering with no-gradient boundary conditions (b, e), and
after a Shapiro filtering with boundary conditions () (c, f).
The area in red is covered by land.
Large-scale pressure gradient computation
The last aspect of the pre-processing of atmospheric data we would
like to discuss is the computation of the large-scale pressure
gradient (or equivalently of the geostrophic wind components)
The objective is to estimate the following terms:
RLSu=1ρa(∂xp)z,RLSv=1ρa(∂yp)z,
where ⋅z denotes a gradient along constant geopotential height.
Using the hydrostatic balance we have
1ρa=-g(∂zp)-1,
which leads to
RLSu=-g(∂zp)-1(∂xp)z,RLSv=-g(∂zp)-1(∂yp)z.
Assuming a generalized vertical coordinate s=s(x,y) the computation
of gradients along constant height is not straightforward since
(∂xp)z=(∂xp)s-(∂zp)(∂xz)s
leading to
(∂xp)z(∂zp)-1=(∂zp)-1(∂xp)s-(∂xz)s.
In the particular case of the IFS coordinate s we have
(∂zp)-1(∂xp)s=B(z)∂xps(∂zA)+(∂zB)ps,
and (∂xz)s can be estimated after integrating the hydrostatic balance.
Starting from the layer interface height zi,j,k+1/2ifs,
surface pressure (ps)i,j, and parameter values Ak,Bk,Ak+1/2,Bk+1/2
the different steps are the following:
compute Δxi,j and Δyi,j from latitudes and longitudes;
compute horizontal gradients ∂xps and ∂yps
for surface pressureFXi+1/2,j=2(ps)i+1,j-(ps)i,jΔxi,j+Δxi+1,j,FYi,j+1/2=2(ps)i,j+1-(ps)i,jΔyi,j+Δyi,j+1;
compute horizontal gradients (∂xz)s and (∂yz)sdZxi+1/2,j,k=zi+1,j,k+1/2cep-zi,j,k+1/2cep+zi+1,j,k-1/2cep-zi,j,k-1/2cepΔxi,j+Δxi+1,j,dZyi,j+1/2,k=zi,j+1,k+1/2cep-zi,j,k+1/2cep+zi,j+1,k-1/2cep-zi,j,k-1/2cepΔyi,j+Δyi,j+1;
compute (∂zp)-1(∂xp)s via ()wrkXi,j,k=12Bkzi,j,k+1/2cep-zi,j,k-1/2cepFXi+1/2,j+FXi-1/2,j(ps)i,j(Bk+1/2-Bk-1/2)+(Ak+1/2-Ak-1/2),wrkYi,j,k=12Bkzi,j,k+1/2cep-zi,j,k-1/2cepFYi+1/2,j+FYi-1/2,j(ps)i,j(Bk+1/2-Bk-1/2)+(Ak+1/2-Ak-1/2);
finalize (we get a minus sign in RLSu because the grid in the y direction is flipped in the raw data)(RLSu)i,j,k=-gwrkYi,j,k-12(dZyi,j+1/2,k+dZyi,j-1/2,k),(RLSv)i,j,k=-gwrkXi,j,k-12(dZxi+1/2,j,k+dZxi-1/2,j,k).
Discrete algorithms to compute lup and ldwn
In the following we describe the discrete algorithms used to provide
the mixing lengths lup and ldwn given in Sect. .
Four different ways to compute those quantities have been implemented
in the ABL1d model.
Bougeault and Lacarrère (1989) length scale
The mixing length defines
lup and ldwn as
∫zz+lupN2(s)(s-z)ds=e(z),∫z-ldwnzN2(s)(z-s)ds=e(z).
By construction such mixing lengths are bounded by the distance
to the bottom and the top of the computational domain and revert
to the length scale
(i.e., lup=ldwn=2e(z)/N2) for
N2=cste. An objective is to also satisfy this
last property at a discrete level. Considering a simple trapezoidal
rule to approximate the integral in Eq. () over each grid cell,
the procedure for the computation of lup(zk+1/2) is given in Algorithm 1.
In the case N2(zp+1/2)=N2(zp-1/2)=Ncst2(∀p), Algorithm 1 gives the following sequence:
FC(zk+1/2)=-e(zk+1/2),FC(zk+3/2)=-e(zk+1/2)+Ncst2e3t(zk+1)22,FC(zk+5/2)=-e(zk+1/2)+Ncst2e3t(zk+1)+e3t(zk+2)22…
As soon as FC(zp+1/2) changes sign we stop the procedure
because lup such that
-e(zk+1/2)+Ncst2lup2=0, which corresponds
to the length scale, has been found.
We note lBL89 the mixing length corresponding to the
algorithm.
Adaptation of NEMO's length scale
The standard NEMO algorithm Sect. 10.1.3 in is much
easier to discretize. As a first step the length
scale lD80 is
computed at cell interfaces, such that
lD80k+1/2=max2ek+1/2maxN2,Nε2,lmin,
with Nε2 the minimum stratification allowed whose value is set to
the smallest positive real computer value. The vertical gradients of lD80 are
then limited such that they stay smaller than the variations of height. This amounts
to compute lup and ldwn as
C2lupk-1/2=minlupk+1/2+e3tk,lD80k-1/2,C3ldwnk+1/2=minldwnk-1/2+e3tk,lD80k+1/2,
with e3tk the thickness of vertical layer k (Fig. ).
The resulting mixing length is simply referred to as lD80.
Rodier et al. (2017) length scale
Recently, proposed to add a shear-related term
to Eq. () such that the definition of lup and
ldwn becomes
∫zz+lupN2(s)(s-z)+c0e(s)‖∂suh‖ds=e(z),∫z-ldwnzN2(s)(z-s)+c0e(s)‖∂suh‖ds=e(z),
where c0 is a parameter whose value should be smaller than
Cm/cε.
At a discrete level, the FC function in Algorithm 1
is replaced by
FC(zp+1/2)=FC(zp-1/2)+e3t(zp)2N2(zp+1/2)(zp+1/2-zk+1/2)+N2(zp-1/2)(zp-1/2-zk+1/2)+c0e3t(zp)2e(zp+1/2)‖∂zuh(zp+1/2)‖+e(zp-1/2)‖∂zuh(zp-1/2)‖.
This mixing length will be referred to as lR17.
A local buoyancy- and shear-based length scale
For the sake of computational efficiency, we have derived a local
version of the length scale
(denoted as lD80⋆) which is original to the present paper:
lD80⋆(z)=2e(z)c0‖∂zuh‖+c02‖∂zuh‖2+2N2(z).
Once lD80⋆ has been computed at cell interfaces z=zk+1/2
we apply the limitations () and () as in
the NEMO algorithm.
Code performance and profiling
To finalize our description of the implementation of the simplified atmospheric boundary layer model in NEMO, we assess in this appendix the computational efficiency of our
approach. We compare the performance of two simulations: one with a coupling
with the ABL1d model (with 50 vertical levels) which requires reading
3D atmospheric data in input files, and one with a standard ASL forcing strategy
which necessitates reading only 2D atmospheric data. For the coupling with
ABL1d, we consider the lD80 mixing length which gave robustly
good results across the different numerical tests investigated earlier in the
paper. The simulations are performed with NEMO version 4.0 for the ORCA025
configuration previously described on 128 cores (Intel(R) Xeon(R) E5 processors
2.6GHz) compiled with ifort (v13.0.1) using the
“-i4-r8-O3-fp-modelprecise-fno-alias” options.
The I/Os are handled via the Lustre file system.
Each MPI subdomain has 80×130 points in the horizontal and 75 points
in the vertical. The various reports given below have been obtained from a
built-in NEMO code instrumentation dedicated to calculation measurement
e.g.,. As mentioned earlier, the outputs
are done using the parallel I/O capabilities provided by the XIOS library.
Thanks to XIOS, we do not expect any significant difference between the
two simulations regarding the cost of output operations. However, the use
of XIOS to handle input operations is still under development, and because
of the significant amount of data to read in the ABL coupling strategy
it makes sense to assess the associated overhead. We ran the
ASL forced and ABL coupled NEMO simulations for 20d such that
the cost of the initialization step is no longer visible in the averaged
cost per time step. Moreover, for the two simulations, the atmospheric data
necessary for the computation of the turbulent components of air–sea fluxes
are provided every 6h.
We first show in Fig. the elapsed time for each
time step over the first 48h of the simulations with different ways
to specify the surface fluxes. For most time steps, the overhead associated
with the ABL1d when using the lD80 mixing length is very small
(on the order of 4 %); however, every 18 time steps (i.e., every 6 h),
there is a larger overhead due to the input part of the I/O operations. To further
refine our assessment, we report in Table the elapsed and
CPU time spent on average over all the processors in the 11 most expansive
sections of the code. As expected, the CPU time is not significantly affected
by the ABL1d model (increase of 4 %), but the elapsed time is increased
by about 9 % because of the time spent in waiting for I/O operations.
The overhead associated with input operations could be mitigated
by reducing the number of vertical levels in the ABL1d model
(we used 50 levels here to get an upper bound on the computational overhead)
and either by using XIOS to handle input operations or by running ABL1d
in detached mode as explained in Sect.
such that the time spent reading input files is covered by actual computations.
Nonetheless the small increase in CPU time leaves room for further improvements
of the ABL model to relax the horizontal homogeneity assumption.
Elapsed time for each time step of a 48h simulation
with standard ASL forcing strategy (black circles) and ABL forcing strategy
using the lD80 mixing length (gray diamonds). For the two simulations
the time step size is Δt=1200s.
Report of the elapsed time and CPU time
in different sections of the NEMO (v4.0) code for the ASL forcing strategy
(left portion of the table) and the ABL coupling strategy (right portion of the table).
The timing is averaged on all processors.
The right-most column provides a quick description of the task handled by the corresponding section.
On top of the timing in seconds the percentage of the total CPU and elapsed time associated
with each section is reported in parentheses. The computational overhead associated with the ABL coupling strategy can be estimated from the sbc section and the
elapsed and CPU time.
The changes to the NEMO code have been made on the
standard NEMO code (release 4.0). The code can be downloaded from the NEMO website (http://www.nemo-ocean.eu/, last access: 23 June 2020).
The NEMO code modified to include the ABL1d model is available in the Zenodo archive (10.5281/zenodo.3904518, ).
The name lists and data used to produce the figures are also available in
the Zenodo archive.
Author contributions
FL wrote the paper with the help of all the coauthors.
FL, GS, and GM designed and developed
a preliminary version of the ABL1d model within the NEMO 3.6 stable version.
This original code was then ported to NEMO release 4.0. JLR and HG provided inputs in the design of the TKE closure scheme and of the numerical
experiments. FL carried out the idealized numerical experiments,
GS the realistic experiments, and JLR the
MesoNH simulations.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
We thank Anton Beljaars and one anonymous reviewer whose efforts helped to
improve earlier versions of this paper. We also thank Pascal Marquet and
Sébastien Masson for useful discussions.
Florian Lemarié and Jean-Luc Redelsperger
also acknowledge the support by Mercator-Ocean and the Copernicus Marine
Environment Monitoring Service (CMEMS) through contract
22-GLO-HR – Lot 2 (High-resolution ocean, waves, atmosphere interaction).
Financial support
This research has been supported by the H2020 European Institute of Innovation and Technology (IMMERSE (grant no. 821926)).
Review statement
This paper was edited by Christina McCluskey and reviewed by Anton Beljaars and one anonymous referee.
References
Abel, R.: Aspects of air-sea interaction in atmosphere-ocean models, PhD
thesis, Kiel University, 2018.
Andren, A., Brown, A. R., Mason, P. J., Graf, J., Schumann, U., Moeng, C.-H.,
and Nieuwstadt, F. T. M.: Large-eddy simulation of a neutrally stratified
boundary layer: A comparison of four computer codes, Q. J. Roy. Meteor.
Soc., 120, 1457–1484, 1994.Ayet, A. and Redelsperger, J.-L.: An analytical study of the atmospheric
boundary layer flow and divergence over a SST front, Q. J. Roy. Meteor.
Soc., 145, 2549–2567, 10.1002/qj.3578, 2019.Baklanov, A. A., Grisogono, B., Bornstein, R., Mahrt, L., Zilitinkevich, S. S.,
Taylor, P., Larsen, S. E., Rotach, M. W., and Fernando, H. J. S.: The
Nature, Theory, and Modeling of Atmospheric Planetary Boundary Layers, B.
Am. Meteorol. Soc., 92, 123–128, 10.1175/2010BAMS2797.1,
2011.Barnier, B., Siefridt, L., and Marchesiello, P.: Thermal forcing for a
global ocean circulation model using a three-year climatology of ECMWF
analyses, J. Mar. Res., 6, 363–380,
10.1016/0924-7963(94)00034-9, 1995.Barnier, B., Madec, G., Penduff, T., Molines, J.-M., Treguier, A.-M., Le
Sommer, J., Beckmann, A., Biastoch, A., Böning, C., Dengg, J., Derval,
C., Durand, E., Gulev, S., Remy, E., Talandier, C., Theetten, S., Maltrud,
M., McClean, J., and De Cuevas, B.: Impact of partial steps and momentum
advection schemes in a global ocean circulation model at eddy-permitting
resolution, Ocean Dynam., 56, 543–567, 10.1007/s10236-006-0082-1,
2006.Bazile, E., Marquet, P., Bouteloup, Y., and Bouyssel, F.: The Turbulent Kinetic
Energy (TKE) scheme in the NWP models at Meteo France, in: Workshop on
Workshop on Diurnal cycles and the stable boundary layer, 7–10 November 2011,
ECMWF, Shinfield Park, Reading, 127–135,
available at: https://www.ecmwf.int/node/8006 (last access: 20 January 2021), 2012.
Beljaars, A.: The parametrization of surface fluxes in large-scale models
under free convection, Q. J. Roy. Meteor. Soc., 121, 255–270, 1995.Beljaars, A., Dutra, E., Balsamo, G., and Lemarié, F.: On the numerical stability of surface–atmosphere coupling in weather and climate models, Geosci. Model Dev., 10, 977–989, 10.5194/gmd-10-977-2017, 2017.Bielli, S., Douville, H., and Pohl, B.: Understanding the West African monsoon
variability and its remote effects: An illustration of the grid point nudging
methodology, Clim. Dynam., 35, 159–174, 10.1007/s00382-009-0667-8,
2009.Bougeault, P. and André, J.-C.: On the Stability of the THIRD-Order
Turbulence Closure for the Modeling of the Stratocumulus-Topped Boundary
Layer, J. Atmos. Sci., 43, 1574–1581,
10.1175/1520-0469(1986)043<1574:OTSOTT>2.0.CO;2, 1986.
Bougeault, P. and Lacarrère, P.: Parameterization of orography-induced
turbulence in a mesobeta-scale model, Mon. Weather Rev., 117, 1872–1890,
1989.Bourras, D., Reverdin, G., Giordani, H., and Caniaux, G.: Response of the
atmospheric boundary layer to a mesoscale oceanic eddy in the northeast
Atlantic, J. Geophys. Res., 109, 10.1029/2004JD004799, 2004.Brivoal, T., Samson, G., Giordani, H., Bourdallé-Badie, R., Lemarié, F., and Madec, G.: Impact of the current feedback on kinetic energy over the North-East Atlantic from a coupled ocean/atmospheric boundary layer model, Ocean Sci. Discuss. [preprint], 10.5194/os-2020-78, in review, 2020.Brodeau, L., Barnier, B., Gulev, S. K., and Woods, C.: Climatologically
Significant Effects of Some Approximations in the Bulk Parameterizations of
Turbulent Air–Sea Fluxes, J. Phys. Oceanogr., 47, 5–28,
10.1175/JPO-D-16-0169.1, 2017.Bryan, F. O., Tomas, R., Dennis, J. M., Chelton, D. B., Loeb, N. G., and
McClean, J. L.: Frontal Scale Air–Sea Interaction in High-Resolution
Coupled Climate Models, J. Climate, 23, 6277–6291,
10.1175/2010JCLI3665.1, 2010.Burchard, H.: Energy-conserving discretisation of turbulent shear and buoyancy
production, Ocean Modell., 4, 347–361,
10.1016/S1463-5003(02)00009-4,
2002a.
Burchard, H.: Applied Turbulence Modelling in Marine Waters, Lecture Notes in
Earth Sciences, Springer Berlin Heidelberg, 2002b.
Businger, J. and Shaw, W.: The response of the marine boundary layer to
mesoscale variations in sea-surface temperature, Dynam. Atmos. Oceans, 8,
267–281, 1984.
Chelton, D. B. and Xie, S.-P.: Coupled ocean-atmosphere interaction at oceanic
mesoscales, Oceanography, 23, 52–69, 2010.
Cheng, Y., Canuto, V. M., and Howard, A. M.: An improved model for the
turbulent PBL, J. Atmos. Sci., 59, 1550–1565, 2002.Couvelard, X., Lemarié, F., Samson, G., Redelsperger, J.-L., Ardhuin, F., Benshila, R., and Madec, G.: Development of a two-way-coupled ocean–wave model: assessment on a global NEMO(v3.6)–WW3(v6.02) coupled configuration, Geosci. Model Dev., 13, 3067–3090, 10.5194/gmd-13-3067-2020, 2020.Craig, A., Valcke, S., and Coquart, L.: Development and performance of a new version of the OASIS coupler, OASIS3-MCT_3.0, Geosci. Model Dev., 10, 3297–3308, 10.5194/gmd-10-3297-2017, 2017.
Cuxart, J., Bougeault, P., and Redelsperger, J.-L.: A turbulence scheme
allowing for mesoscale and large-eddy simulations, Q. J. Roy. Meteor.
Soc., 126, 1–30, 2000.
Cuxart, J., Holtslag, A. A. M., Beare, R. J., Bazile, E., Beljaars, A., Cheng,
A., Conangla, L., Ek, M., Freedman, F., Hamdi, R., Kerstein, A., Kitagawa,
H., Lenderink, G., Lewellen, D., Mailhot, J., Mauritsen, T., Perov, V.,
Schayes, G., Steeneveld, G.-J., Svensson, G., Taylor, P., Weng, W., Wunsch,
S., and Xu, K.-M.: Single-Column Model Intercomparison for a Stably
Stratified Atmospheric Boundary Layer, Bound.-Lay. Meteorol., 118, 273–303,
2006.
Deardorff, J. W.: Three-dimensional numerical study of turbulence in an
entraining mixed layer, Bound.-Lay. Meteorol., 7, 199–226, 1974.
Deardorff, J. W.: Stratocumulus-capped mixing layers derived from a three
dimensional model, Bound.-Lay. Meteorol., 18, 495–527, 1980.Dee, D. P., Uppala, S. M., Simmons, A. J., Berrisford, P., Poli, P., Kobayashi,
S., Andrae, U., Balmaseda, M. A., Balsamo, G., Bauer, P., Bechtold, P.,
Beljaars, A. C. M., van de Berg, L., Bidlot, J., Bormann, N., Delsol, C.,
Dragani, R., Fuentes, M., Geer, A. J., Haimberger, L., Healy, S. B.,
Hersbach, H., Hólm, E. V., Isaksen, L., Kållberg, P., Köhler, M.,
Matricardi, M., McNally, A. P., Monge-Sanz, B. M., Morcrette, J.-J., Park,
B.-K., Peubey, C., de Rosnay, P., Tavolato, C., Thépaut, J.-N., and Vitart,
F.: The ERA-Interim reanalysis: configuration and performance of the data
assimilation system, Q. J. Roy. Meteor. Soc.,
137, 553–597, 10.1002/qj.828, 2011.
Deleersnijder, E., Beckers, J.-M., Campin, J.-M., El Mohajir, M., Fichefet, T.,
and Luyten, P.: Some mathematical problems associated with the development
and use of marine models, in: The Mathematics of Models for Climatology and
Environment, edited by: Díaz, J. I., Springer Berlin
Heidelberg, Berlin, Heidelberg, 39–86, 1997.
Deremble, B., Wienders, N., and Dewar, W. K.: CheapAML: A Simple, Atmospheric
Boundary Layer Model for Use in Ocean-Only Model Calculations, Mon. Weather
Rev., 141, 809–821, 2013.Deshayes, J., Tréguier, A.-M., Barnier, B., Lecointre, A., Sommer, J. L.,
Molines, J.-M., Penduff, T., Bourdallé-Badie, R., Drillet, Y., Garric,
G., Benshila, R., Madec, G., Biastoch, A., Böning, C. W., Scheinert, M.,
Coward, A. C., and Hirschi, J. J.-M.: Oceanic hindcast simulations at high
resolution suggest that the Atlantic MOC is bistable, J. Geophys. Res., 40,
3069–3073, 10.1002/grl.50534, wOS:000321951300034, 2013.
Dewar, W. K. and Flierl, G. R.: Some Effects of the Wind on Rings, J. Phys.
Oceanogr., 17, 1653–1667, 1987.
Frenger, I., Gruber, N., Knutti, R., and Munnich, M.: Southern Ocean Eddies
Affect Local Weather, Nat. Geosci., 6, 608–612, 2013.
Giordani, H., Planton, S., Bénech, B., and Kwon, B.-H.: Atmospheric boundary
layer response to sea surface temperatures during the Semaphore experiment,
J. Geophys. Res., 103, 25047–25060, 1998.Giordani, H., Caniaux, G., and Prieur, L.: A Simplified 3D Oceanic Model
Assimilating Geostrophic Currents: Application to the POMME Experiment, J.
Phys. Oceanogr., 35, 628–644, 10.1175/JPO2724.1, 2005.Haney, R. L.: Surface Thermal Boundary Condition for Ocean Circulation Models,
J. Phys. Oceanogr., 1, 241–248,
10.1175/1520-0485(1971)001<0241:STBCFO>2.0.CO;2, 1971.
Hogg, A., Dewar, W., Berloff, P., Kravtsov, S., and Hutchinson, D. K.: The
effects of mesoscale ocean-atmosphere coupling on the large-scale ocean
circulation, J. Climate, 22, 4066–4082, 2009.Hourdin, F., Couvreux, F., and Menut, L.: Parameterization of the Dry
Convective Boundary Layer Based on a Mass Flux Representation of Thermals, J.
Atmos. Sci., 59, 1105–1123,
10.1175/1520-0469(2002)059<1105:POTDCB>2.0.CO;2, 2002.
Kilpatrick, T., Schneider, N., and Qiu, B.: Boundary Layer Convergence Induced
by Strong Winds across a Midlatitude SST Front, J. Climate, 27, 1698–1718,
2014.Kleeman, R. and Power, S.: A Simple Atmospheric Model of Surface Heat Flux for
Use in Ocean Modeling Studies, J. Phys. Oceanogr., 25, 92–105,
10.1175/1520-0485(1995)025<0092:ASAMOS>2.0.CO;2, 1995.Lac, C., Chaboureau, J.-P., Masson, V., Pinty, J.-P., Tulet, P., Escobar, J., Leriche, M., Barthe, C., Aouizerats, B., Augros, C., Aumond, P., Auguste, F., Bechtold, P., Berthet, S., Bielli, S., Bosseur, F., Caumont, O., Cohard, J.-M., Colin, J., Couvreux, F., Cuxart, J., Delautier, G., Dauhut, T., Ducrocq, V., Filippi, J.-B., Gazen, D., Geoffroy, O., Gheusi, F., Honnert, R., Lafore, J.-P., Lebeaupin Brossier, C., Libois, Q., Lunet, T., Mari, C., Maric, T., Mascart, P., Mogé, M., Molinié, G., Nuissier, O., Pantillon, F., Peyrillé, P., Pergaud, J., Perraud, E., Pianezze, J., Redelsperger, J.-L., Ricard, D., Richard, E., Riette, S., Rodier, Q., Schoetter, R., Seyfried, L., Stein, J., Suhre, K., Taufour, M., Thouron, O., Turner, S., Verrelle, A., Vié, B., Visentin, F., Vionnet, V., and Wautelet, P.: Overview of the Meso-NH model version 5.4 and its applications, Geosci. Model Dev., 11, 1929–1969, 10.5194/gmd-11-1929-2018, 2018.
Lafore, J. P., Stein, J., Asencio, N., Bougeault, P., Ducrocq, V., Duron, J.,
Fischer, C., Héreil, P., Mascart, P., Masson, V., Pinty, J. P.,
Redelsperger, J. L., Richard, E., and Vilà-Guerau de Arellano, J.: The
Meso-NH Atmospheric Simulation System. Part I: adiabatic formulation and
control simulations, Ann. Geophys., 16, 90–109, 1998.
Lambaerts, J., Lapeyre, G., Plougonven, R., and Klein, P.: Atmospheric response
to sea surface temperature mesoscale structures, J. Geophys. Res., 118,
9611–9621, 2013.
Large, W. G.: Surface Fluxes for Practitioners of Global Ocean Data
Assimilation, in: Ocean Weather Forecasting. An Integrated View of
Oceanography, edited by: Chassignet, E. P. and Verron, J., chap. 9,
Springer, 229–270, 2006.Large, W. G. and Yeager, S. G.: The global climatology of an interannually
varying air-sea flux data set, Clim. Dynam., 33, 341–364,
10.1007/s00382-008-0441-3, 2009.Lemarié, F. and Samson, G.: A simplified atmospheric boundary layer model for
an improved representation of air-sea interactions in eddying oceanic models:
implementation and first evaluation in NEMO (v4.0)), Zenodo,
10.5281/zenodo.3904518, 2020.Lemarié, F., Kurian, J., Shchepetkin, A. F., Molemaker, M. J., Colas, F., and
McWilliams, J. C.: Are there inescapable issues prohibiting the use of
terrain-following coordinates in climate models?, Ocean Modell., 42, 57–79, 10.1016/j.ocemod.2011.11.007,
2012.LeMone, M. A., Angevine, W. M., Bretherton, C. S., Chen, F., Dudhia, J.,
Fedorovich, E., Katsaros, K. B., Lenschow, D. H., Mahrt, L., Patton, E. G.,
Sun, J., Tjernström, M., and Weil, J.: 100 Years of Progress in Boundary
Layer Meteorology, Meteorol. Monogr., 59, 9.1–9.85,
10.1175/AMSMONOGRAPHS-D-18-0013.1, 2019.
Lilly, D.: The representation of small-scale turbulence in numerical simulation
experiments, in: Proc. IBM Sci. Comput. Symp. on Environmental Sci., 14–16 November 1966, Thomas J. Watson Res. Center, Yorktown Heights, N. Y., IBM
Form 320–1951, 195–210, 1967.
Lindzen, R. S. and Nigam, S.: On the role of sea surface temperature gradients
in forcing low-level winds and convergence in the tropics, J. Atmos. Sci.,
44, 2418–2436, 1987.
Madec, G.: NEMO ocean engine, in: Note du Pole de modélisation No. 27,
Institut Pierre-Simon Laplace (IPSL), France, 2012.Maisonnave, E. and Masson, S.: Ocean/sea-ice macro task parallelism in NEMO,
in: Technical report, TR/GMGC/15/54, available at:
https://www.cerfacs.fr/~maisonna/Reports/opa_sas_tr.pdf (last access: 20 January 2021),
2015.Maisonnave, E. and Masson, S.: NEMO 4.0 performance: how to identify and
reduce unnecessary communications, in: Technical report, TR/CMGC/19/19,
available at:
https://cerfacs.fr/wp-content/uploads/2019/01/GLOBC-TR_Maisonnave-Nemo-2019.pdf (last access: 20 January 2021),
2019.Marchesiello, P., Capet, X., Menkes, C., and Kennan, S.: Submesoscale dynamics
in tropical instability waves, Ocean Modell., 39, 31–46,
10.1016/j.ocemod.2011.04.011, 2011.McWilliams, J. C., Gula, J., and Molemaker, M. J.: The Gulf Stream North Wall:
Ageostrophic Circulation and Frontogenesis, J. Phys. Oceanogr., 49,
893–916, 10.1175/JPO-D-18-0203.1, 2019.Metzger, E. J., Smedstad, O. M., Thoppil, P. G., Hurlburt, H. E., Cummings,
J. A., Wallcraft, A. J., Zamudio, L., Franklin, D. S., Posey, P. G., Phelps,
M. W., Hogan, P. J., Bub, F. L., and DeHaan, C. J.: US Navy Operational
Global Ocean and Arctic Ice Prediction Systems, Oceanogr., 27, 32–43,
10.5670/oceanog.2014.66, 2014.Meurdesoif, Y., Caubel, A., Lacroix, R., Dérouillat, J., and Nguyen, M.: XIOS
Tutorial, available at:
http://forge.ipsl.jussieu.fr/ioserver/raw-attachment/wiki/WikiStart/XIOS-tutorial.pdf (last access: 20 January 2021),
2016.
Minobe, S., Kuwano-Yoshida, A., Komori, N., Xie, S.-P., and Small, R. J.:
Influence of the Gulf Stream on the troposphere, Nature, 452, 206–209, 2008.
Monin, A. S. and Obukhov, A. M.: Basic laws of turbulent mixing in the surface
layer of the atmosphere, Trudy Akademii Nauk SSSR Geofizicheskogo Instituta,
24, 163–187, 1954.Mulholland, D. P., Laloyaux, P., Haines, K., and Balmaseda, M. A.: Origin and
Impact of Initialization Shocks in Coupled Atmosphere-Ocean Forecasts, Mon.
Weather Rev., 143, 4631–4644, 10.1175/MWR-D-15-0076.1, 2015.
Oerder, V., Colas, F., Echevin, V., Masson, S., Hourdin, C., Jullien, S.,
Madec, G., and Lemarié, F.: Mesoscale SST–wind stress coupling in the
Peru–Chile current system: Which mechanisms drive its seasonal variability?,
Clim. Dynam., 47, 2309–2330, 2016.O'Neill, L. W., Esbensen, S. K., Thum, N., Samelson, R. M., and Chelton,
D. B.: Dynamical Analysis of the Boundary Layer and Surface Wind Responses
to Mesoscale SST Perturbations, J. Climate, 23, 559–581,
10.1175/2009JCLI2662.1, 2010.Razavi, S., Tolson, B. A., and Burn, D. H.: Review of surrogate modeling in
water resources, Water Resour. Res., 48, W07401, 10.1029/2011WR011527,
2012.
Redelsperger, J. L., Mahé, F., and Carlotti, P.: A Simple And General
Subgrid Model Suitable Both For Surface Layer And Free-Stream Turbulence,
Bound.-Lay. Meteorol., 101, 375–408, 2001.Renault, L., Molemaker, M. J., Gula, J., Masson, S., and McWilliams, J. C.:
Control and Stabilization of the Gulf Stream by Oceanic Current Interaction
with the Atmosphere, J. Phys. Oceanogr., 46, 3439–3453,
10.1175/JPO-D-16-0115.1, 2016a.
Renault, L., Molemaker, M. J., McWilliams, J. C., Shchepetkin, A. F.,
Lemarié, F., Chelton, D., Illig, S., and Hall, A.: Modulation of Wind
Work by Oceanic Current Interaction with the Atmosphere, J. Phys.
Oceanogr., 46, 1685–1704, 2016b.Renault, L., Lemarié, F., and Arsouze, T.: On the implementation and
consequences of the oceanic currents feedback in ocean–atmosphere coupled
models, Ocean Modell., 141, 101 423,
10.1016/j.ocemod.2019.101423,
2019a.Renault, L., Masson, S., Oerder, V., Jullien, S., and Colas, F.: Disentangling
the Mesoscale Ocean-Atmosphere Interactions, J. Geophys. Res., 124,
2164–2178, 10.1029/2018JC014628, 2019b.Rodier, Q., Masson, V., Couvreux, F., and Paci, A.: Evaluation of a Buoyancy
and Shear Based Mixing Length for a Turbulence Scheme, Front. Earth Sci.,
5, 65, 10.3389/feart.2017.00065,
2017.
Rotta, J.: Statistische theorie nichthomogener turbulenz, Z.
Physik, 129, 547–572, 1951.Rousset, C., Vancoppenolle, M., Madec, G., Fichefet, T., Flavoni, S., Barthélemy, A., Benshila, R., Chanut, J., Levy, C., Masson, S., and Vivier, F.: The Louvain-La-Neuve sea ice model LIM3.6: global and regional capabilities, Geosci. Model Dev., 8, 2991–3005, 10.5194/gmd-8-2991-2015, 2015.Schneider, N. and Qiu, B.: The Atmospheric Response to Weak Sea Surface
Temperature Fronts, J. Atmos. Sci., 72, 3356–3377,
10.1175/JAS-D-14-0212.1, 2015.
Seager, R., Blumenthal, M. B., and Kushnir, Y.: An advective atmospheric mixed
layer model for ocean modeling purposes: global simulation of surface heat
fluxes, J. Climate, 8, 1952–1964, 1995.Small, R. J., deSzoeke, S. P., Xie, S. P., O'Neill, L., Seo, H., Song, Q.,
Cornillon, P., Spall, M., and Minobe, S.: Air-sea interaction over ocean
fronts and eddies, Dynam. Atmos. Oceans, 45, 274–319,
10.1016/j.dynatmoce.2008.01.001, 2008.Soares, P. M. M., Miranda, P. M. A., Siebesma, A. P., and Teixeira, J.: An
eddy-diffusivity/mass-flux parametrization for dry and shallow cumulus
convection, Q. J. Roy. Meteor. Soc., 130, 3365–3383,
10.1256/qj.03.223, 2004.Spall, M.: Midlatitude Wind Stress–Sea Surface Temperature Coupling in the
Vicinity of Oceanic Fronts, J. Climate, 20, 3785–3801, 10.1175/JCLI4234.1, 2007.
Takano, K., Mintz, Y., and Han, J.-Y.: Numerical simulation of the world ocean
circulation, Second Conf. on Numerical Weather Prediction, Monterey, CA,
Amer. Meteor. Soc., 121–129, 1973.Troen, I. B. and Mahrt, L.: A simple model of the atmospheric boundary layer;
sensitivity to surface evaporation, Bound.-Lay. Meteorol., 37, 129–148,
10.1007/BF00122760, 1986.von Schuckmann, K., Traon, P.-Y. L., Smith, N., Pascual, A., Brasseur, P.,
Fennel, K., Djavidnia, S., Aaboe, S., Fanjul, E. A., Autret, E., Axell, L.,
Aznar, R., Benincasa, M., Bentamy, A., Boberg, F., Bourdallé-Badie, R.,
Nardelli, B. B., Brando, V. E., Bricaud, C., Breivik, L.-A., Brewin, R. J.,
Capet, A., Ceschin, A., Ciliberti, S., Cossarini, G., de Alfonso, M.,
de Pascual Collar, A., de Kloe, J., Deshayes, J., Desportes, C., Drévillon,
M., Drillet, Y., Droghei, R., Dubois, C., Embury, O., Etienne, H., Fratianni,
C., Lafuente, J. G., Sotillo, M. G., Garric, G., Gasparin, F., Gerin, R.,
Good, S., Gourrion, J., Grégoire, M., Greiner, E., Guinehut, S., Gutknecht,
E., Hernandez, F., Hernandez, O., Høyer, J., Jackson, L., Jandt, S., Josey,
S., Juza, M., Kennedy, J., Kokkini, Z., Korres, G., Kõuts, M., Lagemaa, P.,
Lavergne, T., le Cann, B., Legeais, J.-F., Lemieux-Dudon, B., Levier, B.,
Lien, V., Maljutenko, I., Manzano, F., Marcos, M., Marinova, V., Masina, S.,
Mauri, E., Mayer, M., Melet, A., Mélin, F., Meyssignac, B., Monier, M.,
Müller, M., Mulet, S., Naranjo, C., Notarstefano, G., Paulmier, A., Gomez,
B. P., Gonzalez, I. P., Peneva, E., Perruche, C., Peterson, K. A., Pinardi,
N., Pisano, A., Pardo, S., Poulain, P.-M., Raj, R. P., Raudsepp, U., Ravdas,
M., Reid, R., Rio, M.-H., Salon, S., Samuelsen, A., Sammartino, M.,
Sammartino, S., Sandø, A. B., Santoleri, R., Sathyendranath, S., She, J.,
Simoncelli, S., Solidoro, C., Stoffelen, A., Storto, A., Szerkely, T., Tamm,
S., Tietsche, S., Tinker, J., Tintore, J., Trindade, A., van Zanten, D.,
Vandenbulcke, L., Verhoef, A., Verbrugge, N., Viktorsson, L., von Schuckmann,
K., Wakelin, S. L., Zacharioudaki, A., and Zuo, H.: Copernicus Marine Service
Ocean State Report, J. Oper. Oceanogr., 11, S1–S142, 2018.
Wallace, J. M., Mitchell, T. P., and Deser, C.: The Influence of Sea-Surface
Temperature on Surface Wind in the Eastern Equatorial Pacific: Seasonal and
Interannual Variability, J. Climate, 2, 1492–1499,
10.1175/1520-0442(1989)002<1492:TIOSST>2.0.CO;2, 1989.Wilson, J. M. and Venayagamoorthy, S. K.: A Shear-Based Parameterization of
Turbulent Mixing in the Stable Atmospheric Boundary Layer, J. Atmos. Sci.,
72, 1713–1726, 10.1175/JAS-D-14-0241.1, 2015.