## The (real) butterfly effect: the impact of resolving the mesoscale range

What does the ‘butterfly effect’ exactly mean? Many people would attribute the butterfly effect to the famous 3-dimensional non-linear model of Lorenz (1963) whose attractor looks like a butterfly when viewed from a particular angle. While it serves as an important foundation to chaos theory (by establishing that 3 dimensions are not only necessary for chaos as mandated in the Poincaré-Bendixson Theorem, but are also sufficient), the term ‘butterfly effect’ was not coined until 1972 (Palmer et al. 2014) based on a scientific presentation that Lorenz gave on a more radical, more recent work (Lorenz 1969) on the predictability barrier in multi-scale fluid systems. In this work, Lorenz demonstrated that under certain conditions, small-scale errors grow faster than large-scale errors in such a way that the predictability horizon cannot be extended beyond an absolute limit by reducing the initial error (unless the initial error is perfectly zero). Such limited predictability, or the butterfly effect as understood in this context, has now become a ‘canon in dynamical meteorology’ (Rotunno and Snyder 2008). Recent studies with advanced numerical weather prediction (NWP) models estimate this predictability horizon to be on the order of 2 to 3 weeks (Buizza and Leutbecher 2015; Judt 2018), in agreement with Lorenz’s original result.

The predictability properties of a fluid system primarily depend on the energy spectrum, whereas the nature of the dynamics per se only plays a secondary role (Rotunno and Snyder 2008). It is well-known that a slope shallower than (equal to or steeper than) -3 in the energy spectrum is associated with limited (unlimited) predictability (Lorenz 1969; Rotunno and Snyder 2008), which could be understood through analysing the characteristics of the energy spectrum of the error field. As shown in Figure 1, the error appears to grow uniformly across scales when predictability is indefinite, and appears to ‘cascade’ upscale when predictability is limited. In the latter case, the error spectra peak at the small scale and the growth rate is faster there.

The Earth’s atmospheric energy spectrum consists of a -3 range in the synoptic scale and a $-\frac{5}{3}$ range in the mesoscale (Nastrom and Gage 1985). While the limited predictability of the atmosphere arises from mesoscale physical processes, it would be of interest to understand how errors grow under this hybrid spectrum, and to what extent do global numerical weather prediction (NWP) models, which are just beginning to resolve the mesoscale $-\frac{5}{3}$ range, demonstrate the fast error growth proper to the limited predictability associated with this range.

We use the Lorenz (1969) model at two different resolutions: $K_{max}=11$, corresponding to a maximal wavenumber of $2^{11}=2048$, and $K_{max}=21$. The former represents the approximate resolution of global NWP models (~ 20 km), and the latter represents a resolution about 1000 times finer so that the shallower mesoscale range is much better resolved. Figure 2 shows the growth of a small-scale, small-amplitude initial error under these model settings.

In the $K_{max}=11$ case where the $-\frac{5}{3}$ range is not so much resolved, the error growth remains more or less up-magnitude, and the upscale cascade is not visible. The error is still much influenced by the synoptic-scale -3 range. Such behaviour largely agrees with the results of a recent study using a full-physics global NWP model (Judt 2018). In contrast, with the higher resolution $K_{max}=21$, the upscale propagation of error in the mesoscale is clearly visible. As the error spreads to the synoptic scale, its growth becomes more up-magnitude.

To understand the dependence of the error growth rate on scales, we use the parametric model of Žagar et al. (2017) by fitting the error-versus-time curve for every wavenumber / scale to the equation $E\left ( t \right )=A\tanh\left ( at+b\right )+B$, so that the parameters $A, B, a$ and $b$ are functions of the wavenumber / scale. Among the parameters, a describes the rate of error growth, the larger the quicker. A dimensional argument suggests that $a \sim (k^3 E(k))^{1/2}$, so that $a$ should be constant for a $-3$ range $(E(k) \sim k^{-3})$, and should grow $10^{2/3}>4.5$-fold for every decade of wavenumbers in the case of a $-\frac{5}{3}$ range. These scalings are indeed observed in the model simulations, except that the sharp increase pertaining to the $-\frac{5}{3}$ range only kicks in at $K \sim 15$ (1 to 2 km), much smaller in scale than the transition between the $-3$ and $-\frac{5}{3}$ ranges at $K \sim 7$ (300 to 600 km). See Figure 3 for details.

This explains the absence of the upscale cascade in the $K_{max}=11$ simulation. As models go into very high resolution in the future, the strong predictability constraints proper to the mesoscale $-\frac{5}{3}$ range will emerge, but only when it is sufficiently resolved. Our idealised study with the Lorenz model shows that this will happen only if $K_{max} >15$. In other words, motions at 1 to 2 km have to be fully resolved in order for error growth in the small scales be correctly represented. This would mean a grid resolution of ~ 250 m after accounting for the need of a dissipation range in a numerical model (Skamarock 2004).

While this seems to be a pessimistic statement, we have observed that the sensitivity of the error growth behaviour to the model resolution is itself sensitive to the initial error profile. The results presented above are for an initial error confined to a single small scale. When the initial error distribution is changed, the qualitative picture of error growth may not present such a contrast between the two resolutions. Thus, we highlight the need of further research to assess the potential gains of resolving more scales in the mesoscale, especially for the case of a realistic distribution of error that initiates the integrations of operational NWP models.

A manuscript on this work has been submitted and is currently under review.

This work is supported by a PhD scholarship awarded by the EPSRC Centre for Doctoral Training in the Mathematics of Planet Earth, with additional funding support from the ERC Advanced Grant ‘Understanding the Atmospheric Circulation Response to Climate Change’ and the Deutsche Forschungsgemeinschaft (DFG) Grant ‘Scaling Cascades in Complex Systems’.

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