Week 4: The minimalist models for studying chaos and predictability II: Lorenz 96
Contents
Week 4: The minimalist models for studying chaos and predictability II: Lorenz 96¶
Following previous week, this week we will walk through the Lorenz 96 model, which is another widely used toy model in data assimilation and chaos studies. It is one of the first few models mimicking the nonlinear wave propagation in a narrow latitudinal band. A fun fact about Lorenz 96 model is that it was never published in a peer-reviewed scientific journal. Instead, this model was only presented by Ed. Lorenz at a conference in 1995 and was documented by Tim Palmer in a textbook later on in 2004. The story behind the model can be found in week 3.
The physical and mathematical background¶
The formula of Lorenz 96 model (25) is a set of prognostic equations which predicts the evolution of
One can easily find that two nonlinear terms,
The figure below shows a simulation from Lorenz 96 model with

Fig. 7 An example of LR 96 model. The initial condition is spatially autocorrelated with
Single scale¶
Now we can go a step further by generating large ensemble simulations (e.g., 250 different members) and see when the error across different members starts to saturate. Here we define the error as the averaged variance of simulated

Fig. 8 The error growth of LR96 model. The error is defined as the averaged variance across all ensemble members.¶
From FIG8, we can find the error almost grows linearly at the very beginning and start to saturate around
In the case above, the error doubling time is around
By perturbing the initial condition with infinitesimal error and inspecting the error growth of a given dynamical system, this is so-called perfect model experiment, which was first proposed in a conference at Boulder, Colorado, organized by World Meteorological Organization in 1966, where Jule Charney was the head committee. Since then, it has become a rule of thumb for estimating the predictability limit across different dynamical systems.
Note
In 1966, the numerical modeling of the complete global circulation was just leaving its infancy according to Lorenz’s words. The conference held in Boulder collected the three state-of-art climate models at that time: Leith (1965), Arakawa-Mintz (1965) and Smagorinsky (1965). During the break between sessions, Charney persuaded the developers of the three forecast models to do something similar to what we showed in FIG8. At that given period, the estimated error doubling time is around 5 days. However, as time passed by and more sophisticated models were developed, the estimated Lyapunov time is getting shorter and shorter. (e.g., 3 days in Smagorinsky 1969) (Can you answer why?)
Multi-scale interactions¶
In real atmosphere, the nonlinear interaction usually involved the energy cascade over different scales. Thus, Lorenz proposed another model, which involves the interaction between two scales.
(27) can be visualized as FIG9, which is from [RDubenN+17]. In a similar narrow latitudinal band, we implement ten additional small grids,

Fig. 9 A schematic diagram of 2-scale Lorenz 96 model from Russell et al. (2017)¶
Another way to visualize the connection between large and small scales is making a cross section (either with time or space fixed) such as the one shown below (FIG10). From Fig. 10, we can see that when

Fig. 10 A cross section (x-fixed) of LR96 2-scale model.¶
With (27), we repeat the same perfect model experiment and estimate the error growth rate for both small and large scales. The result is shown in FIG11. After introducing the small-scale to the system, we find the error doubling time is about an half of the error doubling time shown previously. It’s because the 2-scale version of LR96 model is more likely to resolve the small scale process. Thus, the early stage of error is dominated by the error growth in small scales. We can also see the error accumulated in large scales. The error growth in large scale goes through a linear regime before

Fig. 11 Similar to Fig. 8, except for the 2-scale LR96 model. The chosen parameters are c=b=10, h=1, K=100, and J=1000.¶
The early and late stages of error growth in a multi-scale flow.¶
From FIG11, we can find that the error growth in large scale goes through two stages. At the very beginning ,
where
By observing (28) and (29), one can find that
References¶
- Lor82
Ed N Lorenz. Atmospheric predictability experiments with a large numerical model. Tellus, 34(6):505–513, 1982.
- RDubenN+17
Francis P Russell, Peter D Düben, Xinyu Niu, Wayne Luk, and Tim N Palmer. Exploiting the chaotic behaviour of atmospheric models with reconfigurable architectures. Computer Physics Communications, 221:160–173, 2017.