This study investigates the predictability of two weather regimes, atmospheric blocking and zonal flows, and their transitions using a simplified coupled atmosphere-land model implemented in Python. The model simulates midlatitude atmospheric dynamics based on a two-layer quasi-geostrophic channel system on a beta-plane, incorporating simplified land effects. The research analyzes the model's responses to environmental parameters and confirms its ability to replicate Earth-like flow regimes.

Through Gaussian mixture clustering, distinct blocking, zonal, and transition flow regimes are identified, revealing their dependence on surface friction. To assess predictability and persistence, the study computes averaged local Lyapunov exponents for each regime. Results indicate the coexistence of zonal, blocking, and transition regimes, with predictability varying based on surface friction conditions. The presence of periodic behavior and the impact of model resolution on predictability are also explored. The study emphasizes that when all three regimes coexist,  zonal patterns exhibit longer predictability compared to blocking patterns that are associated with instability, and transition patterns have reduced predictability when coexisting with other regimes. Additionally, specific surface friction values influence the predictability and characteristics of blocking regimes in different geographic locations.

While this study shows promise in enhancing our comprehension of predictability within multiple regime systems, certain crucial issues need to be resolved before considering its publication.

Major comments:

One major concern pertains to the classification of three weather regimes using a machine learning (ML) method and the correlation of each identified regime with stability or instability, determined by averaged local Lyapunov exponents. While Charney and Devore's (1979) model is traditionally used for studying non-chaotic weather regimes, Lorenz's (1963a) model is renowned for illustrating chaotic features. It's crucial to note that Lorenz applied a similar approach in the early 1960s to study chaotic and nonlinear oscillatory solutions in a two-layer quasi-geostrophic system (e.g., Lorenz 1962, 1963b; Shen, Pielke Sr., and Zeng, 2023, https://www.mdpi.com/2073-4433/14/8/1279 ). A brief review of Lorenz's contributions is necessary, and a comparison of the QG-based models should address the variation in the number of Fourier modes concerning chaotic features.

Several issues need attention:

1. Gaussian Mixture Clustering (GMC) was employed for classification, assuming each cluster has a Gaussian component. The suitability of this assumption for chaotic regimes should be addressed, especially considering the regular spatial patterns that appear in the classified regimes. Associating different weather regimes with components of the leading Lyapunov vector, particularly in the presence of multiple positive Lyapunov exponents, should be addressed. The challenge is heightened by the time varying components of each Lyapunov vector along the solution orbit, making it difficult to link specific components to zonal or blocking regimes.
2. The use of a fixed cluster number (3) in GMC for models with different numbers of positive Lyapunov exponents (LEs) (e.g., 3 positive LEs in Figure 7 and 12 positive LEs in Figure 12) raises concerns. Various clustering values should be explored to illustrate the relationship between the number of weather regimes and the number of positive LEs. For example, can we observe similar weather regimes in the 30 and 165 variable systems? If this is the case, does it imply a consistent number of multiple regimes or equilibrium points across both systems?
3. Is it feasible to compute Lyapunov exponents for individual weather regimes? Could the time evolution of each weather regime be graphed for comparative analysis?
4. Concerning predictability in typical dynamical systems, characterized by systems of ordinary differential equations (ODEs), a positive Lyapunov exponent (LE) usually signifies temporal chaos (distinct from spatial-temporal chaos). In estimating predictability horizons under different conditions, a higher positive LE, on average, implies a greater averaged growth rate, indicating faster error growth and thus diminished predictability horizons. However, when applying this concept to assess predictability in spatial-temporal systems, it becomes imperative to account for errors related to spatial movement. The analogy of whether more intense hurricanes (with higher growth rates) are less predictable encourages authors to contemplate the influence of spatial movement on error predictions. Therefore, in contrast to a zonal flow, although a blocking regime is linked to instability manifested by a larger LE, the consideration of spatial movement is essential when comparing errors in zonal and blocking cases in order to compare their predictability horizons.

Specific Comments.

Page 5: Please indicate whether and how Eqs. (1) and (2) are coupled with Eqs. (3) and (4).

Page 7: Please indicate the equation(s) that contains Cg.

Page 7, Figure 1c with ACC.  Is it possible to identify a time scale of approximately 36 days within the timeframe spanning 20 to 80 days?

Page 7: Can a figure analogous to Figure 1 be generated for each of the categorized regimes?

Page 9: The discussions on the Oseledec method should be expanded to incorporate insights from Lorenz's contributions. (e.g., Lorenz 1965; please see a review by Shen, Pielke Sr, and Zeng, 2023).

Page 11: Please add Figure B3 to include flows for Cg = 400 or kd = 0.12, for periodic flows.

Pages 12 and 13: while lambda_1 and lambda_2 are used for representing the 1st and 2nd LEs, respectively, the symbol lambda indicates heat exchange. Please consider making changes to reduce confusion.

Page 12, line 280. Does the selection of lambda = 0 result in an uncoupled model? How can this be contrasted with the models proposed by Charney and Devore, Lorenz (1962) and/or Lorenz (1963b)?

Pages 12 & 17, (in Figures 7 & 12), please offer perspectives on whether the presence of the plateau suggests the existence of singular eigenvalues with higher multiplicity.

Reviewer’s Comment on

Variability and Predictability of a reduced-order land atmosphere coupled model

by Anupama K Xavier, Jonathan Demaeyer, and Stéphane Vannitsem

General. This paper presents a coupled land-atmosphere model that is applied to midlatitude atmospheric variability and predictability. The model itself is formulated in a flexible Python framework called qgs (Demaeyer et al., J. Open Source Software, 2020) that adds surface features, oceanic or terrestrial, to a two-layer quasi-geostrophic model in a periodic beta-channel. It seems to be the first application of qgs to coupling with a land surface. The Vannitsem group in Brussels has earned already quite a reputation with its modular arbitrary-order ocean-atmosphere model (MAOOAM: De Cruz et al., Geoscientific Model Development, 2016) that has been used worldwide for nonlinear studies of the climate in several configurations and on different time scales.

    The present paper examines the effects of the surface features on the number and stability of distinct atmospheric midlatitude regimes, including zonal, blocking and transient ones. The key tools are the backward Lyapunov spectrum and the associated vectors, abbreviated as BLVs, which are studied as a function of the coupling coefficients between the atmosphere and the ground, namely those that scale the momentum and heat exchanges. The model has a fairly low order, with a total of 30 spectral variables, ten each for the barotropic and baroclinic components of the atmosphere, and ten more for the ground temperature field. Despite this low order, the spatial features of the blocking and zonal regimes in model simulations are rather realistic; see Appendix B.

     The most interesting finding of the paper, to my mind, is the greater predictability of zonal flows, when this regime coexists with the blocked one for the same parameter values. This finding seems to contradict both the prevalence of zonal flows over blocked ones in observations and the widely held belief in the community that it is blocked flows that are stabler. More on this in the first major comment below.

     Overall, the paper should be accepted and published pretty much in the present form. Two major comments follow and addressing them is recommended.

Major Comments.

1. Given the importance of the finding on the relative predictability of the blocked and zonal regimes, I would suggest giving a bit of history on it. Overall, the review on low-frequency variability (LFV) of the midlatitude atmosphere in the paper’s introduction is quite careful and complete. But the authors might wish to emphasize the fact that Legras & Ghil (JAS, 1985) were the first to find the greater stability and hence predictability of zonal flows in their 25-variable barotropic model on the sphere. This result was followed in the experimental paper of Weeks et al. (Science, 1997), using a barotropic rotating annulus, by a study of the variability and persistence of the laboratory blocked flow that essentially confirmed the findings of Legras & Ghil (1985); see especially Fig. 5 in Weeks et al. (1997). So did the Lucarini & Gritsun ( Dyn., 2020) paper, which used the three-layer quasi-geostrophic model of Marshall & Molteni (JAS, 1993) and the methodology of unstable periodic orbits (UPOs). The fact that Lucarini & Gritsun (2020) used a baroclinic model removes the doubts about the greater stability of zonal flows being exclusively due to the barotropic character of the Legras & Ghil (1985) model and of the Weeks et al. (1997) apparatus.

    It would be of particular interest if the authors of the present paper could take a closer look at baroclinic vs. barotropic effects in their model, with respect to this question of the relative stability and persistence of blocked vs. zonal flows, when the two types of regimes coexist. See also the discussion in Ghil & Lucarini (Rev. Mod. Phys., 2020, p. 035002-36).

1. The authors refer to using a “machine learning algorithm called Gaussian Mixture Clustering (GMC),” which is described in Appendix A. While machine learning and AI are all the rage these days, I’d be curious to know how this algorithm differs from the one that was used on observational data by Smyth et al. (JAS, 1999).

Michael Ghil