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Weather regimes are recurrent and persistent large-scale atmospheric circulation patterns that modulate the occurrence of local impact variables such as extreme precipitation. In their capacity as mediators between long-range teleconnections and these local extremes, they have shown potential for improving sub-seasonal forecasting as well as long-term climate projections. However, existing methods for identifying weather regimes are not designed to capture the physical processes relevant to the impact variable in question while still representing the full atmospheric phase space. This paper introduces a novel probabilistic machine learning method, RMM-VAE, for identifying weather regimes targeted to a local-scale impact variable. Based on a variational autoencoder architecture, the method combines non-linear dimensionality reduction with a prediction task and probabilistic clustering in a coherent architecture. The new method is applied to identify circulation patterns over the Mediterranean region targeted to precipitation over Morocco and compared to three existing approaches, two established linear methods and another machine learning approach. The RMM-VAE method identifies regimes that are more predictive of the target variable compared to the two linear methods, and more robust and persistent compared to the alternative machine learning method, while also improving the reconstruction of the input space. The results demonstrate the potential benefit of the new method for use in various climate applications such as sub-seasonal forecasting, while also highlighting the trade-offs involved in targeted clustering.

The standard approach when studying atmospheric circulation regimes and their dynamics is to use a hard regime assignment, where each atmospheric state is assigned to the regime it is closest to in distance. However, this may not always be the most appropriate approach as the regime assignment may be affected by small deviations in the distance to the regimes due to noise. To mitigate this we develop a sequential probabilistic regime assignment using Bayes Theorem, which can be applied to previously defined regimes and implemented in real time as new data become available. Bayes Theorem tells us that the probability of being in a regime given the data can be determined by combining climatological likelihood with prior information. The regime probabilities at time $t$ can be used to inform the prior probabilities at time $t+1$, which are then used to sequentially update the regime probabilities. We apply this approach to both reanalysis data and a seasonal hindcast ensemble incorporating knowledge of the transition probabilities between regimes. Furthermore, making use of the signal present within the ensemble to better inform the prior probabilities allows for identifying more pronounced interannual variability. The signal within the interannual variability of wintertime North Atlantic circulation regimes is assessed using both a categorical and regression approach, with the strongest signals found during very strong El Ni\~no years., Comment: Accepted for publication in Journal of Climate

Atmospheric circulation is often clustered in so-called circulation regimes, which are persistent and recurrent patterns. For the Euro-Atlantic sector in winter, most studies identify four regimes: the Atlantic Ridge, the Scandinavian Blocking and the two phases of the North Atlantic Oscillation. These results are obtained by applying k-means clustering to the first several empirical orthogonal functions (EOFs) of geopotential height data. Studying the observed circulation in reanalysis data, it is found that when the full field data is used for the k-means cluster analysis instead of the EOFs, the optimal number of clusters is no longer four but six. The two extra regimes that are found are the opposites of the Atlantic Ridge and Scandinavian Blocking, meaning they have a low-pressure area roughly where the original regimes have a high-pressure area. This introduces an appealing symmetry in the clustering result. Incorporating a weak persistence constraint in the clustering procedure is found to lead to a longer duration of regimes, extending beyond the synoptic timescale, without changing their occurrence rates. This is in contrast to the commonly-used application of a time-filter to the data before the clustering is executed, which, while increasing the persistence, changes the occurrence rates of the regimes. We conclude that applying a persistence constraint within the clustering procedure is a superior way of stabilizing the clustering results than low-pass filtering the data., Comment: Submitted to Quarterly Journal of the Royal Meteorological Society (accepted)

The extent to which the ongoing decline in Arctic sea ice affects mid-latitude climate has received great attention and polarised opinions. The basic issue is whether the inter-annual variability in Arctic sea ice is the cause of, or the response to, variability in mid-latitude atmospheric circulation. A recent paper by Mori et al. (M19) claims to have reconciled previous conflicting studies by showing that a consistent mid-latitude climate response to inter-annual sea-ice anomalies can be identified in both the ERA-Interim reanalysis, taken as observations, and in an ensemble of atmosphere-only (AMIP) climate model simulations. We here demonstrate that such a conclusion cannot be drawn, due to issues with the interpretation of the maximum covariance analysis performed. After applying the M19 approach to the output from a simple statistical model, we conclude that a predominant atmospheric forcing of the sea-ice variability, rather than the converse, is the most plausible explanation of the results presented in M19.

We study the inverse energy transfer in forced two-dimensional (2D) Navier--Stokes turbulence in a doubly periodic domain. It is shown that an inverse energy cascade that carries a nonzero fraction of the injected energy to the large scales via a power-law energy spectrum $\propto k^{-\alpha}$ requires that $\alpha\ge5/3$. This result is consistent with the classical theory of 2D turbulence that predicts a $k^{-5/3}$ inverse-cascading range, thus providing for the first time a rigorous basis for this important feature of the theory. We derive bounds for the Kolmogorov constant $C$ in the classical energy spectrum $E(k)=C\epsilon^{2/3}k^{-5/3}$, where $\epsilon$ is the energy injection rate. Issues related to Kraichnan's conjecture of energy condensation and to power-law spectra as the quasi-steady dynamics become steady are discussed., Comment: 14 pages, submitted to JFM

This study is concerned with how the attractor dimension of the two-dimensional Navier--Stokes equations depends on characteristic length scales, including the system integral length scale, the forcing length scale, and the dissipation length scale. Upper bounds on the attractor dimension derived by Constantin--Foias--Temam are analysed. It is shown that the optimal attractor-dimension estimate grows linearly with the domain area (suggestive of extensive chaos), for a sufficiently large domain, if the kinematic viscosity and the amplitude and length scale of the forcing are held fixed. For sufficiently small domain area, a slightly ``super-extensive'' estimate becomes optimal. In the extensive regime, the attractor-dimension estimate is given by the ratio of the domain area to the square of the dissipation length scale defined, on physical grounds, in terms of the average rate of shear. This dissipation length scale (which is not necessarily the scale at which the energy or enstrophy dissipation takes place) can be identified with the dimension correlation length scale, the square of which is interpreted, according to the concept of extensive chaos, as the area of a subsystem with one degree of freedom. Furthermore, these length scales can be identified with a ``minimum length scale'' of the flow, which is rigorously deduced from the concept of determining nodes., Comment: No figures, 14 pages

We study two-dimensional turbulence in a doubly periodic domain driven by a monoscale-like forcing and damped by various dissipation mechanisms of the form $\nu_{\mu}(-\Delta)^{\mu}$. By ``monoscale-like'' we mean that the forcing is applied over a finite range of wavenumbers $k_\min \leq k \leq k_\max$, and that the ratio of enstrophy injection $\eta \geq 0$ to energy injection $\epsilon \geq 0$ is bounded by $k_\min^2 \epsilon \leq \eta \leq k_\max^2 \epsilon$. It is shown that for $\mu\geq 0$ the asymptotic behaviour satisfies (eqnarray) \norm u_1^2&\leq&k_\max^2\norm u^2,(eqnarray) where $\norm u^2$ and $\norm u_1^2$ are the energy and enstrophy, respectively. It is also shown that for Navier-Stokes turbulence ($\mu = 1$), the time-mean enstrophy dissipation rate is bounded from above by $2\nu_1 k_\max^2$. These results place strong constraints on the spectral distribution of energy and enstrophy and of their dissipation, and thereby on the existence of energy and enstrophy cascades, in such systems. In particular, the classical dual cascade picture is shown to be invalid for forced two-dimensional Navier--Stokes turbulence ($\mu=1$) when it is forced in this manner. Inclusion of Ekman drag ($\mu=0$) along with molecular viscosity permits a dual cascade, but is incompatible with the log-modified -3 power law for the energy spectrum in the enstrophy-cascading inertial range. In order to achieve the latter, it is necessary to invoke an inverse viscosity ($\mu<0$)., Comment: Accepted for publication in Physica D