How Accurate are Turbulence Forecasts?

Turbulence modeling remains as one of physics biggest unresolved problems. There are workarounds on how to deal with it, some better than others, but no universally accepted model has yet been found. Atmospheric turbulence has the added challenge of the large scale of the atmosphere, which requires an enormous computational power to simulate its physics.

Before going into details, let us say that despite the challenges, atmospheric turbulence forecasts have reached quite a good prediction skill. The skill is measured by comparing the foretasted turbulence to aircraft measurements. This data is then put together to obtain the so called AUC factor (Area Under the Curve). A value of 1 AUC means a perfect prediction skill and 0.5 AUC what a random prediction model would give.

Comparison of 6 months of data from aircraft measurements shows that the prediction skill of NOAA’s latest GTG turbulence forecasts (the one we use in turbli) was about 0.85 AUC. For the same period, the previous NOAA turbulence forecasts for clear air turbulence shows a lower 0.71 AUC.

But what is the main limitation for higher accuracy?

Atmospheric turbulence forecasts rely on global weather forecasts, similar to those you check in your phone to see if its going to be sunny or rainy. Despite the complexity of these models, their working method can be summarized as follows:

1. Build a computational model of the atmosphere

2. Split the model into many small cells

3. At each cell, solve the conservation equations of mass, momentum and energy

4. Obtain the pressure, temperature and wind velocities at each cell.

Turbulence is a fluctuation of the wind velocity, so resolving the wind in full detail will directly give us the turbulence. However, doing so requires cell sizes in the order of millimeters or less, and we can safely say that no supercomputer will ever be able to run such fine model for the entire atmosphere. Right now, NOAA’s global model has reached a 13 km resolution, an impressive feat, but far from the requirements to resolve turbulence.

With a 13 km cell size, the conservation laws of mass, momentum and energy cannot be used in their original form. They won’t be able to predict the fine details of turbulence, so new correlations need to be added to compensate for this. This is the main problem. What once was a universal law, such as the conservation of mass or momentum, is combined with new correlations which make it loose it’s universal applicability.

Laws owe their name to being universally valid, applicable to any condition. Correlations, on the other hand, are only valid for the limited set of experimental data with which they were obtained and therefore don’t have the capability to predict all types of turbulence at all types of configurations.

The predictions of turbulence obtained global forecasts is actually so far from reality that they are not used for flight turbulence forecasting. This is also because the 13 km scales of turbulence predicted by the forecasts have little to do with the scales of a few meters which are of importance for the plane.

Here is where turbulence forecasting models come in.

What they do is take the wind velocities, pressures and temperatures obtained by the global weather models, and use a better set of correlations to predict the amount of turbulence which should be caused by such conditions.

The latest turbulence forecast from NOAA, the GTG, combines 10 different correlations to predict turbulence. It also makes a daily comparison to turbulence measurements from aircrafts to determine the best possible combination of these correlations.

But this is more of the same: a correlation which is only valid for a limit set of experimental data applied to the entire atmosphere. Using 10 different correlations might be a good idea, but it also highlights the attempt to compensate for the known weakness of each correlation alone.

Also, the physics of turbulence don’t change over a day. If we were using a perfect high resolution model, its parameters would always be kept the same. The fact that the combination of the 10 correlations needs to be changed daily using aircraft measurements is another attempt to get the best out of the non-universal correlations.

What to do to improve the prediction then?

Accepting that we will never see a full resolution model of the atmosphere, the only way forward is to further understand turbulence and use this knowledge to continue improving the correlations.

A direction not yet explored in flight turbulence forecasting would be to use the concept of the energy cascade. This is perhaps the most well-established theory of turbulence: large turbulent eddies forming due to shear, splitting to smaller eddies through vortex stretching and finally dissipated through viscous forces. In some cases, we also see an inverse energy cascade.

Research from several authors show that atmospheric turbulence follows that energy cascade laws. Therefore, the correlations used to predict turbulence could be oriented towards predicting the energy cascade profile at each cell. If found, we would have information on the entire spectrum of turbulence sizes and their energy and pay spacial attention to the ones which can affect the plane the most.

There are many other paths and theories currently being pushed to improve turbulence models. Since its start in the 1800s, the road for the perfect turbulence model is far from over. We should keep in mind what Heisenberg once said: “When I meet God, I am going to ask him two questions: Why relativity? And why turbulence? I really believe he will have an answer for the first.”