ESDDEarth System Dynamics DiscussionsESDDEarth Syst. Dynam. Discuss.2190-4995Copernicus GmbHGöttingen, Germany10.5194/esdd-6-407-2015Inferring global wind energetics from a simple Earth system model based on the principle of maximum entropy productionKarkarS.sami.karkar@epfl.ch PaillardD.didier.paillard@lsce.ipsl.frhttps://orcid.org/0000-0002-9995-2794Laboratoire des Sciences du Climat et de l'Environnement, CEA-CNRS-UVSQ, Orme des Merisiers, Bât. 701, 91191 Gif-sur-Yvette CEDEX, Francenow at: LEMA-EPFL, Station 11, 1015 Lausanne, SwitzerlandS. Karkar (sami.karkar@epfl.ch) and D. Paillard (didier.paillard@lsce.ipsl.fr)2March20156140743318December201420January2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://esd.copernicus.org/preprints/6/407/2015/esdd-6-407-2015.htmlThe full text article is available as a PDF file from https://esd.copernicus.org/preprints/6/407/2015/esdd-6-407-2015.pdf
The question of total available wind power in the atmosphere is highly
debated, as well as the effect large scale wind farms would have on the
climate. Bottom-up approaches, such as those proposed by wind turbine
engineers often lead to non-physical results (non-conservation of energy,
mostly), while top-down approaches have proven to give physically consistent
results. This paper proposes an original method for the calculation of mean
annual wind energetics in the atmosphere, without resorting to heavy
numerical integration of the entire dynamics. The proposed method is derived
from a model based on the Maximum of Entropy Production (MEP) principle,
which has proven to efficiently describe the annual mean temperature and
energy fluxes, despite its simplicity. Because the atmosphere is represented
with only one vertical layer and there is no vertical wind component, the
model fails to represent the general circulation patterns such as cells or
trade winds. However, interestingly, global energetic diagnostics are well
captured by the mere combination of a simple MEP model and a flux inversion
method.
Introduction
Global available wind power for renewable energy production ultimately relies
on geophysical considerations of the Earth system, as shown by
. Whatever technological progress could be made on the
design of wind turbines and wind farms, an upper limit is dictated, above
all, by the maximum possible rate of conversion of available potential
energy, which comes primarily from the Sun's differential heating, into
kinetic energy – that is to say, winds.
Recent studies by , and have
shown that the top-down approach, using Earth system models, despite an
oversimplified representation of wind turbines, is necessary to evaluate the
real maximum power extractable from winds. Whereas the bottom-up method
derived from the turbine engineering viewpoint lacks energy conservation on
the global scale.
A wide variety of Earth system models exist, ranging from zero dimensional
energy balance models, unidimensional to tridimensional box models with no
dynamics, intermediate complexity models, up to “IPCC-class” models. In the
latter, the dynamics of the atmosphere, ocean, ice and biosphere, as well as
the exchanges between these components, are computed by integration of the
dynamical equations for one part, ad hoc parameterizations for another part,
and coupling.
One important question remains: is it possible to devise the energetics of
atmospheric circulation, without computing the entire dynamics? Or, put
differently: how far can we go, in terms of wind energetics, with a simple
box model?
is the first to propose a simple box model that relies
on a maximization principle. Reformulated by , the
function to be maximized in this kind of model appears to be the entropy
production rate, though it strongly resembles the conjecture of maximum
generation of available potential energy of . This
hypothesis (rather than a physical “principle”), gave very good results
concerning the mean climate on Earth, especially concerning the surface
temperature distribution. Models based on the maximum entropy production
(MEP) have even shown to be in good agreement with the climate of other
planet-like systems, such as Mars and Titan, as reported by
. For a review on the subject of global climate and the
MEP principle, the reader is referred to . Recently,
revisited these ideas and proposed a model that is both
consistent with the MEP hypothesis and does not contain ad hoc
parameterizations of the clouds. In their model, the only unknowns are the
temperature and the convergence of energy fluxes in each cell, while they
prescribed the radiative budget, the surface albedo, and the composition of
the (dry) atmosphere. The only equations are energy balances (local and
global), and a maximization of the total entropy production rate related to
the diabatic heating.
In this paper, we propose an original method for the calculation of mean
annual wind energetics in the atmosphere, based on this MEP model.
The paper is organized as follows. In Sect. we present the
methods, hypotheses and equations of the model. In Sect. , we
present the results and discuss their relevance, in comparison with previous
results from the literature. We conclude the paper in a final section where
future developments of this model are proposed.
Methods
This section presents the necessary hypotheses and equations used
successively in: the MEP climate model, the inference from divergences to
fluxes, the computation of mean atmospheric winds.
MEP climate model
We use the box-model proposed by . This model is based
on the hypothesis that all exchanges of energy in the Earth system, apart
from the radiative exchanges, occur with the maximum possible rate of entropy
production.
The Earth atmosphere is divided horizontally into N=nlat×nlon cells, following a latitude–longitude regular grid. Along
the vertical, two levels are distinguished: the “ground” level, and the
“atmosphere” level. Each cell is assumed to reach a local thermodynamic
equilibrium. The ith atmospheric cell has a temperature Tia
(respectively Tig for a ground cell), and is subject to an
energy balance between the radiative fluxes budget
Ria(Tia,Tig) (resp.
Rig(Tia,Tig)) and the other forms of
energy transport (for instance sensible and latent heat exchanges with
neighboring cells) which are summed up into a term dia (resp.
dig) that stands for the divergence of energy fluxes in each
box (per unit area, in Wm-2). The local energy balance, assuming
stationary state, leads to 2N equations of the form:
RiaTia,Tig-dia=0RigTia,Tig-dig=0
where i runs from 1 to N, and where the radiative budget has the following form:
RiaTia,Tig=cia,0+cia,aTia4+cia,gTig4RigTia,Tig=cig,0+cig,aTia4+cig,gTig4
where the radiative coefficients cix,y are computed from the
insolation, albedo, atmospheric chemical composition, and atmospheric
standard humidity profile of each grid cell (details of the radiative code
can be found in Herbert et al., 2011).
Therefore, the rate of entropy production in an atmospheric cell is given by:
σia=-AidiaTia=-AiRia(Tia,Tig)Tia,
where Ai is the surface of the grid cell. (The same formula applies to
a ground cell, substituting the superscript a with g.) The total rate of
entropy production in the whole Earth system is then equal to:
σ=-∑i=1NAiRiaTia,TigTia+RigTia,TigTig.
The last equation of the model is the global energy balance, which constrains
the maximum of entropy production rate:
e=∑i=1NAidia+dig=0.
The solution of our MEP problem is to find the temperature field Ti (we
now omit the superscript a or g) that maximizes σ as given by
Eq. (), under the constrain e=0. It is
sought by finding the minima of the Lagrange function:
L(β,Ti)=-σ-βe
where β is the Lagrange parameter associated to the global energy
balance and has the dimension of the inverse of a temperature.
Finally, such minima are the solutions of the following system of 2N+1
nonlinear equations, that sums up the model's equations:
∂Ria∂Tia1/Tia-β-RiaTia,TigTia2+∂Rig∂Tia1/Tig=0∂Rig∂Tig1/Tig-β-RigTia,TigTig2+∂Ria∂Tig1/Tia=0∀i∈[1,N]∑i=1NAiRiaTia,Tig+RigTia,Tig=0
where the 2N+1 unknowns are (β,[Tia,Tig]1,N).
Energy transport: from divergence to heat fluxesPrinciple
Given the solution temperature field T (from now on, we will drop the
subscript i and the superscript a/g), we can compute the divergence field
d. Now, formally, given this scalar field d, the goal is to compute
a vector field F of energy fluxes, such that: d=divF. To ensure unicity of the solution, we assume only one hypothesis,
that F is irrotational:
rotF=0.
Note that this assumption does not imply that the velocity field be
irrotational. Assuming that F is irrotational is equivalent to
assuming that it is the gradient of a scalar quantity Φ. It corresponds
somewhat to a local version of the second law of thermodynamics: the heat
flux flows along the gradient of this scalar field (as it would flow from
warm to cold regions). A rotational part would transport heat along a loop,
and therefore necessarily from cold regions to warm regions at some point. At
this point, it seems a reasonable way to constrain F, knowing only
its divergence d. If additional constraints were to be added (e.g.
mechanical, hydrological…), this constraint could probably be released
as the degeneracy on F would disappear.
Then, we can compute F, via the pseudo-potential Φ, which is
obtained by inverting the Laplacian operator Δ:
F=gradΦ=gradΔ-1d.
Construction of the Laplace operator
Because this is a box-model, we need a discrete Laplace operator. Converting
our grid to a mathematical entity known as a graph, such a discrete
operator exists and has the properties of the continuous Laplacian. In this
equivalent graph, each grid cell of the model is a vertex (or node), and the
edges represent the connections between grid cells.
There are N atmospheric boxes in the model, each of which are connected to
4 adjacent atmospheric boxes from the same level and to one ground box
(except at the North and South poles where each box is connected to only 3
adjacent atmospheric boxes from the same level and one ground box). The
connection graph on the ground level is different as it depends on the
land-sea mask, because we assume that continents are not able to transport
energy (or that it is negligible). Thus, two adjacent boxes containing ocean
are connected, but there is no connection between a land box and an ocean
box, or between two land boxes. Given a land fraction for each grid
cell
We use the IPSL-CM4 grid at the 72×96 resolution and its
land fraction mask, or interpolated versions of these two for different
resolutions.
, and applying a threshold at 1/2, we obtain the land-sea
mask, from which we construct the connection graph.
The positive discrete Laplacian operator on the graph G is defined as:
Δ=AG-DG
where DG is the degree matrix of the graph
(a diagonal matrix containing the degree of each vertex, i.e. the number of
other vertices it is connected to), and AG the
adjacency matrix (a matrix defined such that
AG[i,j]=1 if vertices i and j are connected, and
AG[i,j]=0 otherwise).
Inversion of the divergence
We are now able to compute the pseudo-potential Φ=Δ-1d. Note
that inverting the Laplace operator Δ, in the case of a large number
of grid cells, can be a cumbersome operation. However, being a large but
sparse matrix, efficient algorithms exist to approximate the solution to d=ΔΦ with very good accuracy, without the need for a full matrix
inversion (minimum residuals, and other iterative methods alike).
From this pseudo-potential Φ, we can recover the flux field
F=gradΦ. However, the vector field F is only
defined on the existing edges of the graph, that is, between two adjacent and
connected cells of our grid. For two such cells, the flux of energy flowing
from cell i to cell j is:
Fi→j=Φ[j]-Φ[i].
MEP winds
From this point on, we will only use the temperature and fluxes that occur
within the atmosphere layer. Therefore, we will write Ti instead of
Tia (i∈[1,N]).
Mass exchange rate
All unresolved processes that lead to energy exchange (and entropy
production) are supposed to be included in the term d. Thus, the
interaction between two adjacent, connected cells i and j shall be
entirely described by the flux Fi→j. Assuming a dry
atmosphere, we use the hypothesis that all energy is exchanged as sensible
heat. Taking into account mass conservation within the exchanges between the
two cells, the same quantity of mass (per unit of time) fi,j is
exchanged from cell i, with temperature Ti, to cell j, and from cell
j, with temperature Tj to cell i, the net budget in (sensible) heat
exchange being Fi→j from the viewpoint of cell i. The mass
exchange rate coefficient for these two cells fi,j is then defined as:
Fi→j=cpfi→j(Ti-Tj).
Under the hypotheses that Ti≠Tj, comes the coefficient fi,j:
fi,j=Fi→jcp(Ti-Tj).
Note that fi,j should always be positive since it represents a mass
flux, and the corresponding energy should be directed from hot to cold. This
constrain is not enforced in this paper, and there are a few locations where
mass fluxes are negative (also known as negative diffusion). A more rigorous
approach would be to solve a constrained optimization problem, including the
positivity constraint on fi,j. Unfortunately, the associated numerical
problem is far more difficult to solve. Empirically, there are only very few
locations where fi,j<0. This paper mainly addresses globally integrated
energetic features, so the conclusions should not be affected by this
marginal inconsistency.
Quasi-geostrophic flowHypotheses
We assume that the atmosphere
Here, we only intend to represent the
vertical mean of the winds in the troposphere, however no hypothesis is made
on the vertical structure.
is composed of dry air, approximated as a perfect
gas. We use the hydrostatic approximation along the vertical axis, and assume
constant sea level pressure. Orography is neglected.
Equations
The mean winds are then computed as a two-dimensional quasi-geostrophic flow
u, expressed using the vertical pressure coordinate, such that its
zonal and meridional components (u,v) satisfy:
-2mΩsinθv=mRslogpp0∂T∂x+Fdissip,x2mΩsinθu=mRslogpp0∂T∂y+Fdissip,y
where θ is the latitude, m is the total mass of the air contained in
the current grid cell, p is the pressure at the barycenter of the cell,
p0 the reference pressure (here: the constant sea level pressure),
Rs the specific gas constant for the air, and
Fdissip is the dissipation term.
Given that we have assumed an exchange rate of mass for each pair of
atmospheric cells, the same coefficient links the exchange rate of momentum.
The dissipation term, for cell i, is then written:
Fi,dissip=-∑jfi,j(ui-uj)
where the sum excludes non-connected cells: fi,j=0 is assumed for every
pair of non-connected cells. The sum also excludes cells from the ground
layer: a null velocity field is assumed on the whole ground
layer
Unless surface currents were accounted for, but this is not
the case here.
.
Finally, the 2N unknowns (ui,vi) are solutions of the linear system of 2N equations:
-2miΩsinθivi=miRslogpip0∂Ti∂x-∑jfi,j(ui-uj)2miΩsinθiui=miRslogpip0∂Ti∂y-∑jfi,j(vi-vj).
Results
We present and discuss the results obtained with 2 vertical levels (ground
and atmosphere) and a 36×48 regular grid (latitude, longitude). Other
horizontal resolutions were tested, ranging from 18×24 to
96×144. Note that results do not always improve with finer grids, as
they do with most models, because cells too close to the equator lead to
numerical instabilities (see discussion below).
Temperature field
Figure shows the temperature field computed by the MEP model. As
pointed out by , the result is in good agreement with
the output of the IPSL-CM4 model, an IPCC-class Earth system model, except
for a global positive bias mainly due to the lack of representation of water
clouds in this model. The reader is referred to for
a more detailed review and discussion concerning the temperature field
results.
Heat flux
Figure shows the poleward meridional heat flux in the
atmosphere. It is in good qualitative agreement with ,
but it lacks a factor of approximately 1.5 for good quantitative agreement.
showed that the repartition between oceanic and
atmospheric transports in the MEP model suffers from the lack of
representation of explicit dynamics. For instance, the model fails to
represent the Atlantic meridional overturning circulation, which greatly
affects the overall ocean heat transport, and consequently the atmospheric
transport. However, given the simplicity of the MEP model, the qualitative
agreement with known data and more elaborated models is remarkable.
Figure shows a map of the vertical heat flux (positive
downward). It is similar to the output of the IPSL-CM5A model, shown in
Fig. , except in regions where the water cycle plays
a dominant role: essentially over oceans at low latitudes.
Annual mean winds
The annual mean wind in the upper horizontal grid (atmospheric level) is
shown in Fig. . The horizontal resolution shown here is
36×48. This annual mean wind is dominated by the geostrophic
component, which results in a general West-wind trend with small fluctuations
around land-ocean borders. The model fails to represent the main
characteristics of the atmospheric circulation (westerlies, trade winds).
This is not surprising since there is no vertical representation in our
1-layer atmosphere model, which therefore cannot represent Hadley or Ferrel
cells. The global pattern is quite independent of the resolution, though
strong variations occur near the equator.
The zonal mean of the annual mean wind speed, shown in
Fig. , shows three peaks. Two peaks are located
around 70∘ S (strong peak) and 70∘ N (moderate
peak). The third one, near the equator, is dominant. However, it is believed
to be due to spurious, unphysical, strong meridional components, as the
computed winds tend to diverge near the equator.
The global average of the annual mean wind speed 〈||u||〉
is in the range 11.8–13.0ms-1 (depending on the
resolution). It is remarkable that variations of both the global mean, and
the zonal mean distribution of the wind speed, are very small when the
resolution changes.
The global average is consistent with most models and observations: for
instance, available data from NCEP-R2 reanalysis by leads to
a global average, annual mean value of 10.9ms-1 for the wind
speed (weighted average in the three spatial dimensions, simple mean along
the time axis) on the period 1948–present
NCEP Reanalysis data
provided by the NOAA/OAR/ESRL PSD, Boulder, Colorado, USA, from their Web
site at http://www.esrl.noaa.gov/psd/,
ftp://ftp.cdc.noaa.gov/Datasets/ncep.reanalysis.derived/pressure/wspd.mon.mean.nc.
. IPSL-CM5A model historical simulation for the period
1950–2005
Historical simulation, part of the CMIP5 project,
provided by IPSL: http://icmc.ipsl.fr/. Files used:
http://dods.extra.cea.fr/store/p86caub/IPSLCM5A/PROD/historical/historical2/ATM/Analyse/TS_MO_YE/historical2_19500101_20051231_1Y_vitu.nc
(and “vitv”).
gives an annual mean, global average value of
10.1ms-1.
Winds energetics
The detailed figures of winds derived from this simple model are not very
reliable due to low resolution and lack of vertical representation, but some
global quantities are. It is interesting to check if global values that are
characteristics of wind energetics are also captured by the proposed model,
as it is based on energy exchanges and thermodynamics. We thus focus on
global means of three main energetic features: the global mean kinetic energy
contained in the atmosphere (expressed in energy per unit surface), the
global kinetic energy dissipation rate, and the kinetic energy dissipation
rate inside the atmospheric boundary layer. The latter is especially
interesting in the context of renewable energy: it is assumed to be the
maximal power available for energy harvesting using surface-based wind farms.
Note that the ranges given on the following values reflect the variations
obtained when the resolution of the model is swept from 18×24 to
96×144.
Global mean kinetic energy in the atmosphere
The global mean kinetic energy of the atmosphere in the present model is in
the range 〈KE〉=10–20×105Jm-2. It is
a rather broad range, though the mean value (14.3×105Jm-2) is comparable with the values that were
obtained using publicly available data of the NCEP-R2 reanalysis:
12.7×105Jm-2, or with that of the IPSL-CM5A historical
simulation: 9.2×105Jm-2.
For comparison, the Lorenz cycles depicted in based on
the 1979–2001 period for three different reanalyses also lead to similar
values
When adding zonal and eddy kinetic energy, and averaging the
four seasonal values for each quantity.
: 15.0×105Jm-2
for NCEP-R2, 16.2×105Jm-2 for ERA-40 (see
), and 15.9×105Jm-2 for JRA-25 (see
). On the other hand, found in their study
a very low value: 3.0×105Jm-2, while
did not provide a value for the global kinetic energy.
Kinetic energy dissipation rate in the whole atmosphere
The total kinetic energy dissipation rate in our model is given by
〈D〉=∑iui.∑jfi,j(ui-uj)
and is found to be: 〈D〉=1000–2500 TW
It corresponds to 1.96–4.91Wm-2.
.
Referring to , values from reanalyses are 1415 TW for
ERA-40, 1316 TW for NCEP-R2, and 1372 TW for JRA-25. For comparison,
found 950(±110) TW and reports
1600 TW. The IPSL-CM5A historical simulation does not give access to this
value
There is no track of the momentum drag coefficient, nor of the
turbulent kinetic energy dissipated by the numerical scheme and it is quite
complicated to reconstruct all these quantities.
.
Dissipation rate inside the atmospheric boundary layer
In the context of wind turbines, the dissipation rate of kinetic energy in
the whole atmosphere is not of great interest, considering that wind turbines
are surface structures, whether land-born or offshore. Moreover, the only
vertical component in our model is the distinction between the ground level
and the atmospheric level. Thus, it is interesting to check how the
associated momentum drag, and its corresponding kinetic energy dissipation
rate, compares with the dissipation rate of kinetic energy in the atmospheric
boundary layer (ABL) in more elaborated models.
In our model, the kinetic energy dissipated in the ABL is given by
DABL=∑iτi.ui
where τi=fi,jui when i is an atmospheric cell and
j is the corresponding ground cell below.
We obtained DABL=400–800 TW, to be compared with the
513 TW for the ERA-40 reanalysis, reported by , while
they found 425(±75) TW with their own model. Comparison is not possible
with Marvel et al. as they did not provide this value, or with the IPSL-CM5A
historical simulation
A simple τ.u10m
computation gave 142 TW, which seems very low. The authors believe that it
is not accurate. For the actual computation of what was dissipated in the ABL
during the simulation, one needs to know the momentum drag coefficient, whose
value is not constant and not easily accessible.
.
These main energetic features are illustrated in Fig. ,
where MEP refers to the present model, and is compared to the different
reanalyses and models cited above.
Discussion
The MEP model wind formulation suffers from several flaws that prevent it
from getting a good picture on a regional basis. Specifically, spurious,
strong meridional (and sometimes, zonal) components occur near the equator,
showing a great variability when the resolution changes. This high
variability seem to result from a numerical instability around the equator:
it is linked to our formulation for the flux fi,j (Eq. 11) which becomes
divergent when temperature gradients are very small. It is probably necessary
to represent explicitly latent heat fluxes in this region to avoid such
a difficulty. Note that the problem is even worse when a finer resolution is
used, as cells with the lowest latitude become closer to the equator. This
explains the lack of convergence of the model, with respect to the spatial
resolution.
Considering the various energetic diagnostics, we notice a rather broad range
for all values, with high center values. This overestimation of energy
content and, more importantly, energy dissipation rates may be caused by the
lack of water cycle in the model. All energy fluxes are supposed to be
sensible heat exchanges, which results in mass exchange rate coefficients
fi,j that could be overestimated up to a factor 2. This, in turn, would
overestimate the ageostrophic component of the winds, responsible for the
generation and dissipation of kinetic energy.
Nevertheless, given the spread of the values found in the literature, the
energetics of the atmosphere–ground interface and of the vertically
integrated winds seem to have been surprisingly well captured by the proposed
model, despite its simplicity.
Conclusions
Because the current version of the model is two-dimensional (horizontal), the
velocity field is unable to reproduce the well-known Hadley, Ferrel and Polar
cells that characterize the general circulation in the atmosphere. However,
the proposed model agrees with reanalysis data and IPCC-class ESMs in terms
of wind energetics.
Therefore, the general idea of the applying MEP principle to the earth system
seems most appropriate for climate studies.
The model could be improved, especially by using a 3-D atmospheric grid,
together with orography, a representation of a basic water cycle, and
a seasonal cycle. Ongoing works are directed toward these improvements.
Other refinements such as advection could also be added to this model.
However, the added value of including more explicit processes in the
equations will have to be balanced with the added complexity of the model,
given the general philosophy of this model which consists in modelling as few
processes as possible and computing the contribution of all unresolved
processes using the MEP principle.
From such a basic level of simplicity, it is difficult to extrapolate how far
this model can be used to compute realistic wind energetics on a continental
or regional basis. While there is a lot of room for improvement, it is worth
emphasizing that in our simple approach, and in contrast to more classical
GCM studies, our results do not depend on parameterizations of convection
(i.e. kinetic energy inputs) nor on parameterizations of the boundary layer
(i.e. energy dissipation through drag coefficients). It is therefore very
encouraging that our results are in agreement with these more detailed
models.
Future, more complex versions of this model could give even better results.
It would then provide an interesting tool for some climate studies such as
available wind power, and the influence of large scale wind farms on Earth
climate, for instance.
Acknowledgements
This work was funded by the CEA Energie program. The authors wish to thank
Corentin Herbert, Bérangère Dubrulle, François Daviaud,
Pierre Sepulchre, Masa Kageyama and Gilles Ramstein for fruitful discussions.
The authors also express their thanks to Hannah Kohrs for language
improvements.
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Left: vertical cut of the MEP model with an atmospheric cell
on top of a ground cell. Energy fluxes are represented with arrows.
Right: sample of the model grid, with full connectivity at the
atmospheric level (grey cells), and limited connectivity at the ground level
(blue cells: ocean, orange cells: continent). Connectivity between adjacent
cells is figured with light lines (grey: atmospheric, blue: oceanic, green:
vertical).
Temperature field (in ∘C) on the surface layer of the
grid (72×96), given by the MEP model. Input data (albedo, atmospheric
composition, and insolation) and grid resolution are similar to that of the
IPSL-CM4 model for pre-industrial conditions. Dotted lines are
0 ∘C contours, plain lines are positive 10 ∘C
spaced contours, and dashed lines are negative 10 ∘C spaced
contours.
Poleward meridional heat transported by the atmosphere, resulting
from the MEP model (72×96 grid).
Annual mean vertical heat flux from the atmosphere to the ground,
resulting from the MEP model (72×96 grid).
Annual mean vertical heat flux from the atmosphere to the ground,
resulting from the IPSL-CM5A model in pre-industrial conditions (96×95
grid, 39 vertical atm. levels).
Annual mean wind vectors resulting from the quasi-geostrophic model
and the flux inversion of the MEP model (36×48 grid).
Zonal mean of the annual mean wind speed, resulting from the
quasi-geostrophic model and the flux inversion of the MEP model
(36×48 grid).
Comparison of MEP model wind energetics with reanalyses and other
models.