Why does wind energy become less efficient when used at larger scales? Basic physics explains this effect, starting with a very limited ability of the atmosphere to generate wind energy from radiation, as described in my new review just published.

Wind energy plays an important role in the transition to a carbon-neutral, sustainable energy system and is rapidly expanding. So it is a good time to ask how much wind energy there actually is, whether we get close to the limits anytime soon, and why the efficiency of wind energy must decline when used at larger scales. These are basic science questions: How, and why, does the atmosphere actually generate motion, how much does it generate, and how much of it can at most be used? These questions I address in a review paper just published in which I show that it does not take much physics to answer these.

How much wind can at most be generated? What happens to the efficiency of wind turbines when more and more wind energy use gets eventually to its limits? A few physical principles provide answers that are transparent and consistent with modelling.

In my group, we do science by looking at Earth system processes in terms of the energy they convert, in terms of the limits that constrain these conversions, and how these conversions are connected to the broader system functioning.  It is quite a rewarding view, very general, dealing with entropy and questions about life, but also with very practical and relevant implications when dealing with the limits of wind energy.  We have published on the limits of wind energy over the past 10 years (Kleidon 2010, Miller et al. 2011a, Miller et al. 2011b, Gans et al. 2012, Miller et al. 2015, Kleidon et al. 2016, Miller and Kleidon 2016, Miller and Kleidon 2017, Kleidon 2019, Kleidon and Miller 2020, Agora Energiewende et al. 2020), with an approach that deviates quite a bit from the mainstream view of wind energy, with some of the implications not being quite so popular, particularly that the limits to wind energy are rather low at large scales.  It uncomfortably placed us somewhere in between the dogmatic extremes of both ends of the renewable energy scale.  So when I was asked by a colleague from the German Meteorological Society if I wanted to submit a review paper of my choice to their Meteorologische Zeitschrift, this was the perfect occasion to provide a summary of the whole story.

From sunlight to heat and atmospheric motion

The starting point is how the atmosphere generates wind.  Most of the motion that is used for wind energy is generated by differences in solar radiation between the tropics and the poles.  This involves energy conversions:  Incoming solar radiation, energy in form of electromagnetic waves, is absorbed, which turns this energy into heat and which can then be measured by temperature.  Heat makes air less dense, so it expands the atmosphere upward, raising its center of mass and generating potential energy.  Then motion kicks in.  It converts differences in potential energy into kinetic energy.  This kinetic energy is further converted into heat, mostly due to friction near the Earth’s surface, where the spatial gradients in wind speed are the greatest.  Or it is converted by wind turbines into renewable energy, and converted into heat later when the electricity is being consumed by human activity.

How can we use this picture of energy conversion and determine what limits the conversion rates? A simple, yet profound way to look at this is to view the process of generating motion as the result of an atmospheric heat engine. The difference in solar radiation between tropics and poles sets up a heating difference that the heat engine uses to generate motion. This motion then feeds the engine with heat. In doing so, it depletes the heating difference, making the tropics cooler and the poles warmer. This effect is well known and documented, as it results in an imbalance in the radiative fluxes at the top of the atmosphere, with the tropics absorbing more solar radiation than they emit, and the reverse imbalance outside the tropics.


The atmosphere generates kinetic energy out of the difference in absorbed solar radiation between the tropics and the extratropics like a heat engine. Figure 3 in Kleidon (2021).

This picture of generating motion can be put into numbers by using energy balances for the tropics and the polar regions in combination with the so-called Carnot limit, a thermodynamic limit that describes how much work can be derived from a heat flux and temperature difference. The Carnot limit then sets the upper bound on the power, that is, the maximum for the generation rate of kinetic energy within the atmosphere. This limit is formed by the product of the heat transport from the tropics to the polar regions, and the temperature difference, which describes the efficiency of the heat engine. This temperature difference is derived from the energy balances, with the difference being smaller the more heat is transported. As a result, there is a maximum in the power that can be inferred using observations of the radiative fluxes, e.g., from the NASA CERES radiation dataset. The difference in absorption in the climatological mean turns out to be about 65 W m-2 of surface area, that is, the tropics absorb 65 W m-2 more than the global mean, and the regions outside the tropics absorb 65 W m-2 less than the global mean. So if 65 W m-2 were transported by the atmosphere from the tropics to the poles, there would be no difference in heat, or temperature, being left, and no power could be generated because the efficiency of the heat engine would drop to zero. The maximum in power is achieved when about half this value, about 32 W m-2, of heat is transported out of the tropics.

To get to the power, however, we need to multiply this heat flux with the efficiency. Because the temperature difference on Earth is comparatively small (about 30 K) compared to the mean temperature (about 300 K), this efficiency is also small, about 10%. And the last factor we have to account for is that heat is transported from the tropical half of the globe to the polar half, which yields an additional factor of one half. Taken together, this yields 32 W m-2 x 10% x 1/2 = 1.6 W m-2 that the atmosphere can at best generate at the planetary scale.

While this calculation does not include the contribution by the Hadley circulation or by local circulations, such as sea breezes, this magnitude agrees very well with a long established magnitude of 2 W m-2 for the generation of kinetic energy of the large-scale atmospheric circulation.  While the 2 W m-2 seem to be very little power compared to the 240 W m-2 of absorbed solar radiation, this is the best the atmosphere can do.  A direct implication can already be derived at this point: That the direct use of solar radiation as renewable energy, for instance by photovoltaics, has a much, much greater potential than wind energy at the planetary scale.

From atmospheric motion to renewable energy

So the 2 W m-2 set the starting point for kinetic energy – how much of it can now be used as renewable energy?  Without wind turbines, this energy is dissipated by friction, that is, it is converted into heat.  What friction does is that it wants to deplete differences in velocity, and these are greatest near the surface, and in regions where the wind speeds are high.  This dissipation is thus rather non-uniform, with peaks over the oceans in the mid latitudes, and greater in the winter season than in the summer, times and places that are known to be rather windy. 


The atmosphere generates about 2 W m-2 in the global mean, but its dissipation by friction near the surface varies quite substantially across regions and seasons.  Shown is the annual mean of frictional dissipation within the atmospheric boundary layer, taken from the ERA-5 Reanalysis dataset. Figure 6d in Kleidon (2021). 

But let’s leave aside these regional differences and aim for a global magnitude.  As it turns out, it is only a relatively small fraction of the kinetic energy that can be converted further.  Here’s why:  Wind, or, more precisely, its momentum, is transported from the so-called free atmosphere in about 1 – 2 km height and above down to the surface by differences in wind speed.  This dissipates kinetic energy already.  When wind turbines tap into the kinetic energy near the surface, some energy must still be left behind.  Otherwise, there would be no wind, and the rotors of the turbines would not turn.  And this leftover wind fuels some further dissipation by surface friction below the turbines.  Also the wakes that the turbines leave behind in the flow downwind reflect differences in wind speed, and the mixing to replenish the wind field consumes kinetic energy.  These processes represent losses that eat up kinetic energy and cannot be avoided when aiming to use wind as renewable energy.

These different effects can be quantified using the momentum balance of the lower atmosphere, with momentum being one of those physical quantities that is conserved.  When we now want to figure out how much at most can be used, and increase the number of wind turbines at large scales, then these would grab more and more of the wind energy near the surface.  This would slow down wind speeds, reduce yields and the dissipation by surface friction, but more of the kinetic energy is dissipated above the turbines (see purple area in the Figure below).  Hence, there is a maximum in how much wind can at best be used, which turns out to be about 26% of the natural dissipation rate.  With the global mean generation rate of 2 W m-2, this yields about 0.5 W m-2 as a potential resource estimate for wind energy at the global scale.  This estimate and its reasoning is consistent with estimates derived from much more detailed climate model simulations.


Processes that consume kinetic energy in the atmospheric boundary layer near the surface at large scale: boundary layer mixing above the turbines (purple), frictional dissipation below the turbines near the surface (red), wake dissipation (orange) and conversion into renewable energy by the turbines (black).  More wind turbines can grab a greater share of the kinetic energy near the surface, but they also result in greater mixing losses above the turbines. Based on Figure 7d in Kleidon (2021).

How can the potential be so low?  Current wind farms are quite productive!

This global picture is pretty grim in terms of how efficient wind energy as a form of renewable energy is.  Its current use seems much more promising than these estimates, so what’s wrong?  Actually, nothing is wrong.  The apparent discrepancy between the low, large-scale limit and the higher actual yields of small wind farms is due to the difference in scale.  At its extreme, a single, isolated wind turbine does not feel the reduction in wind energy that it causes in the flow downstream.  And the wind speeds at about 100m above the surface, where wind turbines currently operate, are quite high, because the intensity of friction, as characterized by the so-called drag coefficient, is rather low.  So for a single wind turbine, it is the horizontal transport of kinetic energy that determines how much electricity it generates.  

When we get to larger and larger scales, we need to consider the replenishment of kinetic energy from the free atmosphere above, and this replenishment is very low compared to the horizontal fluxes.  Hence, wind speeds need to decline with more extensive wind energy use within a region.  This makes turbines less efficient when more turbines are present in a region, the yields per unit area drop to eventually reach the large-scale limit.  What this means is that we will need to deal with these reduction effects in the decades to come when wind energy expands in scale, as we have recently shown in the study with Agora Energiewende on the German offshore wind energy potential.  And some states within Germany appear as if they reach this scale in the coming decades as well (see a previous blog post on this topic).  It does not mean that wind energy can still provide a lot of renewable energy, but we need to anticipate these yield reductions when looking at the future expansion of wind power.

Are other forms of renewable energy equally inefficient?

Are we going to see similar reduction effects in other forms of renewable energy?  Not necessarily.  When we look at biofuels, for instance, these are generated by photosynthesis.  This process operates quite differently, yet it observed efficiency is also very low, for good reasons (see this blog post and Figure below).  When the conversion efficiency from sunlight to photosynthesis to biomass is taken together, and biomass is further converted to electricity, e.g., in a power plant, we get an overall efficiency of 0.2% as well.  So it is not any better than wind, although for somewhat different reasons.


Comparing the total conversion efficiencies from sunlight to electricity for wind, biomass, and photovoltaics.  Photovoltaics has by far the highest efficiency because it converts sunlight directly into electricity without an intermediate step of converting it into heat. Figure 9 in Kleidon (2021).

Yet, there is one form of renewable energy that truly stands out: Solar energy.  Photovoltaics uses the low entropy of sunlight and converts it directly into electricity.  The efficiency of commercially available solar panels has already reached 20%.  The theoretical limit is much higher, slightly above 70% (see e.g. here).  What this means is that photovoltaics is able to convert much more of the energy contained of sunlight into useful, renewable energy.  Why?  Because it avoids the losses of converting sunlight into heat first.  This conversion into heat loses much of the low entropy of sunlight, and results in the low conversion efficiency of turning sunlight into wind.  That’s why wind power is quite constrained in its potential at large scales, because the efficiency of generating winds must be low because the temperature differences are low.  Since photovoltaics does not need such temperature differences for the conversion process, its conversion efficiency can be much higher, resulting in a much greater renewable energy potential.

Summary

To sum up, it does not require much physics to understand how and how much the atmosphere generates kinetic energy, and how much of this can at most be further converted into renewable energy.  The key ingredient is to follow the energy, through the conversion from solar radiation to heating differences, to kinetic energy and to the conversion to electricity.  Add thermodynamics to it, account for the interactions, and one gets at the resource potential for wind energy at the planetary scale.  

At a more general level the surprising result is that the atmosphere indeed operates around this maximum, working as hard as it can. This is something that is not obvious at all, although it has been formulated before in the context of the more spooky sounding hypothesis of Maximum Entropy Production (see e.g., this review article).  This serves as an example for a general evolutionary direction of Earth system processes, but that’s a different story.  At a practical level, we need to anticipate reduced efficiencies and yields of wind turbines when wind power is expanded in scale in the decades to come to make the transition to a sustainable energy system a success.


Reference:

Kleidon, A (2021) Physical limits of wind energy within the atmosphere and its use as renewable energy: From the theoretical basis to practical implications. Meteorol. Z., preprint availabile online, https://doi.org/10.1127/metz/2021/1062.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s