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.

Global available wind power for renewable energy production ultimately relies
on geophysical considerations of the Earth system, as shown by

Recent studies by

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?

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.

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.

We use the box-model proposed by

The Earth atmosphere is divided horizontally into

Therefore, the rate of entropy production in an atmospheric cell is given by:

The last equation of the model is the global energy balance, which constrains
the maximum of entropy production rate:

The solution of our MEP problem is to find the temperature field

Finally, such minima are the solutions of the following system of

Given the solution temperature field

Then, we can compute

Because this is a box-model, we need a discrete Laplace operator. Converting
our grid to a mathematical entity known as a

There are

We use the IPSL-CM4 grid at the

The positive discrete Laplacian operator on the graph

We are now able to compute the pseudo-potential

From this pseudo-potential

From this point on, we will only use the temperature and fluxes that occur
within the atmosphere layer. Therefore, we will write

All unresolved processes that lead to energy exchange (and entropy
production) are supposed to be included in the term

Under the hypotheses that

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.The mean winds are then computed as a two-dimensional quasi-geostrophic flow

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

Unless surface currents were accounted for, but this is not the case here.

.Finally, the

We present and discuss the results obtained with 2 vertical levels (ground
and atmosphere) and a

Figure

Figure

Figure

The annual mean wind in the upper horizontal grid (atmospheric level) is
shown in Fig.

The zonal mean of the annual mean wind speed, shown in
Fig.

The global average of the annual mean wind speed

The global average is consistent with most models and observations: for
instance, available data from NCEP-R2 reanalysis by

NCEP Reanalysis data
provided by the NOAA/OAR/ESRL PSD, Boulder, Colorado, USA, from their Web
site at

Historical simulation, part of the CMIP5 project,
provided by IPSL:

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

The global mean kinetic energy of the atmosphere in the present model is in
the range

For comparison, the Lorenz cycles depicted in

When adding zonal and eddy kinetic energy, and averaging the four seasonal values for each quantity.

:The total kinetic energy dissipation rate in our model is given by

It corresponds to

Referring to

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.

.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

We obtained

A simple

These main energetic features are illustrated in Fig.

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

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

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.

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.

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.