first suggested the atmosphere may possess multiple large-scale quasi-equilibrium states to describe the empirically observed tendency for European weather patterns to take on a finite range of frequently recurrent configurations. These are now known in meteorology as atmospheric regimes, and in dynamical systems, such behaviour is known as crisis-induced intermittency . The first theoretical study of multiple atmospheric equilibria, carried out by (hereafter CdV79), showed that the flow in a low-order mid-latitude barotropic channel could equilibrate to either a zonally symmetric flow or a stationary blocking pattern. They suggested real-world equivalents of these equilibria states might be destabilised by perturbations from unresolved modes (in this idealised model, this included spatial scales of ∼O(1000) km), causing shifts from one state to another and introducing a vacillation between large-scale weather patterns.

Similar results were obtained in other simple models , but these equilibria were often absent in less truncated systems, rendered “unrealisable” by shifts in the model attractor or collapsing completely due to structural instability . Therefore later work moved away from analysis of fixed points, treating regimes as chaotic flows originating from attractor merging and exploring the role of homoclinic orbits in preferred transitions between regimes .

Despite a deepening mathematical understanding of regimes, the Earth system appears to be high-dimensional, and as a result linking the regime behaviour of simple models to that of realistic models has proved challenging, especially when further complicated by the introduction of stochastic forcing. Earth-system models often contain explicit stochastic parameterisation schemes and in addition a range of small-scale processes that can be thought of as randomly forcing the large-scale flow. Contrary to linear intuition, there is evidence that this fast variability can stabilise regimes: (hereafter K14) found additive noise increases regime lifetime in both the Lorenz 1963 system and a chaotic reformulation of the CdV79 model found by (hereafter C04). Further, stochastic forcing has been shown to improve regime predictability by increasing persistence in truncations of the Lorenz 1996 model and climate model regime representation , as well as the representation of other slow-varying climate modes .

Outside of geoscience, similar stochastic persistence effects have been observed and explained for transient chaotic maps. For such open systems, chaotic dynamics only persist for finite time, eventually “escaping”, often to a steady state . It has been shown for such systems how stochastic forcing can delay the escape of trajectories in the case of 1D and 2D maps via a “fattening” of the fractal structure. This is of interest for fully chaotic geophysical regime systems as the transition between regimes can be conceived of as an escape from one regime to another, and so increases in regime persistence correspond to decreases in escape rate. This approach was introduced in to understand the stochastically forced Duffing oscillator.

We will study the stochastically forced CdV79 model, as retuned by C04, and show that similar mechanisms explain stochastic persistence in this six-dimensional, continuous-time, non-hyperbolic regime system as in simpler transiently chaotic maps. We introduce the model and its regime structure in Sect. . We then discuss the impact of stochasticity on that regime structure in Sect.  and demonstrate that this can be understood as a result of the theory of hitting time problems in combination with a geometric mechanism. We summarise our results and discuss the implications for weather and climate in Sect. .

The deterministic Charney–deVore model is obtained by spectral truncation of a barotropic flow in a β-plane channel geometry. It can be expressed as a system of six ordinary differential equations, derived in full in Appendix : 1x˙1=-C(x1-x1*)+γ̃1x3x˙2=-Cx2+β1x3-α1x1x3-δ1x4x6x˙3=-Cx3-β1x2+α1x1x2+δ1x4x5-γ1x1x˙4=-C(x4-x4*)+ϵ(x2x6-x3x5)+γ̃2x6x˙5=-Cx5+β2x6-α2x1x6-δ2x4x3x˙6=-Cx6-β2x5+α2x1x5+δ2x4x2-γ2x4. The six variables {xi} represent spectral mode amplitudes of a streamfunction field, subject to a linear relaxation to a zonally symmetric background state, Coriolis and orographic forces, and quadratic advective non-linearities. To explore stochastic effects, we introduce an additive white noise vector to the tendency, given by a normal distribution ξi with standard deviation σ, as in K14: 2x˙i=x˙i,deterministic+σξi. The system is integrated using the Euler–Maruyama difference scheme with a time step of dt=2×10-4 model time units (MTU) and is sampled every 1 MTU. We compute integrations of length 200 000 MTU for a range of σ values.

To visualise the model phase space, we project the six-dimensional attractor into the space of the two leading empirical orthogonal functions (EOFs) of the deterministic system, as shown in Fig. , capturing 94 % of the system's variability (Fig. S1 in the Supplement shows the various projections of the six x modes). The attractor is highly structured, with a number of strongly preferred trajectories through phase space (note the logarithmic colour scheme in Fig. ). This low-dimensional attractor shows no obviously separable phase space regions such as the twin lobes of the Lorenz 1963 system, and we see that the trajectories are highly twisted together, producing a scroll-like geometry.

A representative sample of the model's temporal evolution is shown in Fig. . Intermittency is readily apparent, with predictable slowly evolving periods, with near-zero phase speed, corresponding to persistent “blocked” flow patterns, divided by highly chaotic, fast-evolving flows that contain both zonally symmetric states and transient wave activity. The regime structure is therefore quite asymmetrical, in contrast to the Lorenz 1963 or 1996 systems whose regimes are essentially equivalent to each other. Instabilities in the CdV system arise from two basic mechanisms: an orographically induced saddle-node bifurcation (a transition from one to multiple fixed-point equilibria), as identified in the original CdV79 paper, and a barotropic Hopf bifurcation (a transition from a fixed point to a periodic orbit) associated with the onset of a zonally propagating waves in the model domain. Both bifurcations are codimension 1, existing along a surface with one fewer dimension than the parameter space, and so can in general be found by tuning one model parameter. Stable steady-state and periodic wave solutions to Eq. () are common for arbitrary sets of model parameters (not shown), while chaotic – and especially intermittently chaotic – solutions are harder to find. In the original formulation of CdV79, two stable fixed points were found, corresponding to a zonally symmetric state, and a “blocked” flow, with strong wave activity. The model parameters used here, and in C04 and K14, are close to a parameter set where the barotropic Hopf and orographic saddle-node bifurcations merge: one fixed point bifurcates into two fixed points and a limit cycle. This fold-Hopf point has codimension 2, meaning two model parameters must be tuned to obtain it. High-codimension bifurcations seem to be of reduced relevance to the real world, representing exceptional states in parameter space. However, the fold-Hopf bifurcation is known to produce a range of lower codimension and so more relevant bifurcations at nearby parameter values, including paths to chaotic dynamics . Most relevant to us is the generation of homoclinic orbits, a type of bifurcation where orbits stretch from a fixed point back to itself, with an infinite period. These orbits can lead to fully fledged chaotic dynamics and the creation of a strange attractor through intersection with an unstable toroidal limit cycle. At the boundary crisis this creates a “whirlpool repeller” and exactly the scrolling structure seen in Fig. . The two fixed points, analogues of those in CdV79, remain but are now unstable, and at the parameter values chosen here the strange attractor moves between the vicinity of these two fixed points, as seen in Figs.  and , shadowing collapsed homoclinic orbits. The flow patterns corresponding to these fixed points are shown in Fig. S2. See and C04 for more details on the bifurcation structure of the model. As a result the regimes of the CdV system are rooted in the dynamics of unstable homoclinic orbits, with close approaches to the unstable fixed point leading to periods of slow evolution and then rapid “bursting” to chaotic, more turbulent flow. Such behaviour is seen in other simple atmospheric models including the Lorenz 1984 model and in the dynamics of non-linear circuits and neurons .

This type of regime behaviour is of a fundamentally different type to that of the archetypal Lorenz 1963 system. In Lorenz 1963, crisis-induced intermittency creates regime dynamics when the basins of attraction of two attractors merge together, triggering movement between two distinct regimes of fully chaotic behaviour. The Charney–deVore system, in contrast, experiences almost deterministic dynamics before drifting away and entering a transient chaotic regime. At an unpredictable later time, the system will once again return to the neighbourhood of the periodic orbit and re-enter the slow-evolving regime. For a recent discussion of the co-existence of predictable and chaotic flow states in atmospheric systems see .

Figure a shows Fourier transforms of the total mode amplitude ∑ixi2 for three representative amplitudes of stochastic forcing. The deterministic σ=0 spectrum possesses a clear peak consistent with the quasi-periodic nature of the dynamics. For the σ=0.01 case this peak has collapsed, producing a flatter spectrum with power shifted towards low frequencies. This is not a monotonic effect: for σ=0.04, the total power in the low-frequency region decreases, and the spectrum flattens. Figure b shows this non-monotonicity clearly, with a peak in low-frequency variability close to σ=0.01.

This is in excellent agreement with K14, which showed increasing regime persistence with noise amplitude that peaked at σ=0.008 and then decreased. We show here that this is associated with a genuine slowing of the system variables visible in the Fourier spectrum and is observable independent of adopting a regime perspective. With this reassurance, and given the HMM maintains the same qualitative characteristics for σ values ≤0.016, we judge that the regime assignment provides a sensible way to analyse the amplified low-frequency variability seen in Fig.  in more detail. Figure  shows the distribution of lifetimes for each of the three regimes, as a function of increasing σ. While all regimes show some increase in regime persistence for moderate noise amplitudes, the effect is by far the most significant in the blocking regime, showing a near trebling of mean regime lifetime that peaks at σ=0.008. As shown in Fig. S5, this peak is associated with a 150 % increase in the number of blocks lasting at least 100 MTU and a 600 % increase in those lasting longer than 175 MTU. Changes in the median blocking lifetime are smaller, and the distribution is clearly skewed, with the 95th percentile of blocking lifetime exceeding 600 MTU when persistence is maximum.

How can the ability of random forcing to stabilise regime dynamics be understood? What causes the non-monotonicity between stochastic amplitude and persistence, and why is the response here asymmetric, primarily in the blocking regime? As mentioned, non-monotonic changes of transient lifetime due to stochastic forcing are understood for low-dimensional transiently chaotic maps. Symmetric perturbations can decrease on average the region of phase space in which chaotic trajectories can “escape” (i.e. decay) for a large class of 1D concave maps, lengthening their lifetime . Additionally, stochasticity smooths the probability density of the deterministic system. If the escape region of the system is of higher than average density due to low dimensionality, then stochastic forcing can on average knock more trajectories out of this region than in to it, again increasing persistence .

In our system we first consider the mean state changes. The average distance between points in the stochastic simulations and their closest neighbour on the deterministic attractor increases, approximately linearly, as the amplitude of the stochastic forcing increases (Fig. a). Once again there is asymmetry between regimes, with the minimum distance during blocking growing far faster than during the other regimes. Figure b and c show that this is associated with a drift towards the neighbourhood of the unstable blocking fixed point, with no equivalent result seen for the zonal fixed point (also visible in Fig. S4). If we consider the dynamics in the neighbourhood of the blocking fixed point, we realise the deterministic tendency there is near zero, and so the additive stochastic terms dominate entirely: the dynamics approach six-dimensional Brownian motion. How long, in expectation, will it take for this Brownian motion to move the system away from the fixed point once again and return it to a dynamics-dominated flow state? This question is simply a reformulation of the classic hitting time problem: at what time will a random process first reach a distance a from its initial condition? For Brownian motion in three or more dimensions it is known that in fact the average time taken is infinite: the distribution of hitting times is fat-tailed, and so the mean diverges . Therefore the skewed distribution of blocking lifetimes seen in Fig.  can be understood in terms of the amount of time spent near the fixed point, as shown in Fig. b. As σ increases past 0.008, the average distance from both the attractor and the blocking fixed point increases. As the stochastic forcing becomes stronger than the deterministic damping along the stable manifold, the system increasingly explores the full six-dimensional phase space, and so the directionality of the forcing vanishes. There is a conceptual similarity here to the second stochastic persistence mechanism of found for mixed phase space systems, whereby trajectories in a 1D transiently chaotic map can scatter into islands of stability, which introduces a fat tail to the transient lifetime. However, the increased persistence we find is not entirely a result of excursions to the noise-dominated neighbourhood of the fixed point. There is also an important interaction between stochasticity and the low-dimensional, ordered flow of the blocking regime. We present a heuristic mechanism similar in character to that of but now considering a 1D flow as in the schematic shown in Fig. . We can think of this as a parameterisation along the 1D unstable manifold of our CdV79 system moving between a highly chaotic state and a slowly evolving, predictable state. Immediately upon entering the regime of slow-evolving dynamics, the system is at a point x=xi and will stay in the quasi-stationary regime until it reaches x=xc. We assume that x grows monotonically with the time t in this regime, so we can parameterise x in terms of t or vice versa: f:x→t, g:t→x. If we let x=f(t) be any convex, monotonic function defined over the interval [x0,xc], then t=g(x)=f-1(x), and so g is therefore a concave, monotonic function with negative curvature. We consider applying a single perturbation Δx to the system at a point xp, drawn from an even probability distribution P(Δx) with fast-decaying tails and support over [x0-xp, xc-xp]. Informally, this amounts to choosing a point xp far from the edge of the predictable regime, and so there is only a vanishing probability of Δx directly triggering a regime transition. When the standard deviation of P is small compared to xc-x0, this will apply to most values of x in the interval. From the evenness of P, we see that the impact of the perturbation on the variable x, in expectation, is zero: 3〈Δx〉=0. However this is not in general the case for 〈Δt〉: 4〈Δt〉=〈g(xp+Δx)-g(xp)〉=〈Gp(Δx)〉, where Gp, as a translation of g, is also concave. Using the definition of the expectation we find 5〈Δt〉=∫-∞∞Gp(Δx)⋅P(Δx)dΔx, where the limits are justified by our requirement for vanishing tails for P and our choice of xp.

Because P(Δx) is even, then if we Taylor-expand Gp, we will lose all odd powers on integration and be left with 6〈Δt〉=∫-∞∞∑n=1∞1(2n)!d2nGpd(Δx)2n⋅P(Δx)dΔx. And having taken Δx to be small, we can drop high-order terms: 7〈Δt〉=∫-∞∞12d2Gpd(Δx)2⋅P(Δx)dΔx. Because Gp is concave, this derivative is negative, and we obtain 8〈Δt〉=-∫-∞∞12d2Gpd(Δx)2⋅P(Δx)dΔx=-I, where I is positive. Therefore any convex perturbation growth will be delayed (i.e. intermittency persisted) by small symmetric random perturbations, and the average lifetime of the stable regime will increase. The case for a negatively oriented variable with concave curvature is precisely equivalent, while for a positively oriented variable with concave curvature or a negatively oriented variable with convex curvature, decreased persistence would be seen. For the linear case with zero curvature, there is of course no impact of the perturbation on persistence.

When considering the particular case of f(t)=x0eλt, and P(Δx)=N(0,σ), which is interesting for our system, we obtain analytically 〈Δt〉=-σ22λxp2 (see Appendix ). This quadratic dependence on σ2 is exactly what is found in and for the escape rate for 1D maps. However it is not clear that the two derivations are totally equivalent; we consider a convex monotonic instability and they a concave non-invertible map. Higher σ corresponds to longer persistence (remembering our overall weak noise approximation), as does small λ, indicating that nearly stable intermittent dynamics can be most substantially persisted. We also see in this case that smaller xp leads to larger drift, indicating that perturbations soon after the system has entered the intermittent state are most impactful. While we have considered only a single perturbation, the reasoning generalises to a continuous stochastic process, where the 〈Δt〉 term we have calculated is now a spatially dependent drift term. The crux of the argument can be understood solely from examination of Fig. . Any low-dimensional, convexly growing instability in the climate system should be susceptible to the same stochastic suppression. We can also see why only the blocking regime showed dramatic persistence increases, as it is the only regime where the variables evolve with a constant sign of curvature and where there is only one unstable dimension. The decrease of lifetime for larger noise values does not emerge naturally from this framework but can be understood as entering a noise-dominated regime where the stochastic terms are much larger than the deterministic tendency.

In this paper we have investigated the interaction between stochasticity and strongly non-linear dynamics in a six-equation conceptual model of atmospheric flow. We consider the case of additive stochastic forcing applied to a chaotic flow which transitions intermittently between chaotic zonal and quasi-stationary blocked regimes, which showed exhibited stochastically reinforced regime persistence for a range of noise amplitudes. We have investigated the mechanisms driving this behaviour and provide heuristic arguments based on the stochastic suppression of instability growth and random scattering close to an unstable fixed point. The low dimensionality and ordered nature of the blocking regime leads to a slow monotonically growing instability, which will on average be persisted by a random perturbation rather than accelerated. This effect is amplified, and a heavy tail introduced into the lifetime of blocking events, through chance excursions to the neighbourhood of a deterministically inaccessible fixed point, where the overall deterministic tendency is negligible. Here, the dynamics are almost entirely noise-dominated, and the system will remain close to the fixed point until it scatters away. Ultimately, if stochastic forcing is increased past a certain point, persistence decreases. This is not a resonance effect but is instead a distinct feature exhibited by transient systems and systems near boundary crises , where a balance must be struck between providing enough diffusion to escape the deterministic transition trajectories and providing so much diffusion that new purely stochastic routes out of the regime are created. The mechanisms driving stochastic regime dynamics have many similarities to the dynamics of stochastic transiently chaotic maps. Understanding regime transitions as escapes from the neighbourhood of one non-attracting set to the neighbourhood of another, which will itself eventually escape, provides a different perspective that makes the connection to the theory of stochastically perturbed escape rates clear. In atmospheric physics more attention is generally paid to continuous-time fully chaotic conceptual models such as the archetypal Lorenz 1963 system , but the study of regime systems may well benefit from a deeper consideration of the framework of transient chaos and the wealth of work done on discrete-time maps. Indeed, all chaotic systems can be understood in terms of transient shadowing of unstable periodic orbits , and such ideas have been recently applied to blocking flows and regime systems . Characterising the unstable periodic orbits associated with each regime could even allow the non-monotonic dependence of regime lifetime on stochasticity to be analytically computed using perturbative methods . Understanding the impacts of stochastic parameterisation on truly transient processes, such as mesoscale convective features, is also of great importance . It remains to be seen however how relevant these simple mechanisms are in more complex atmospheric models, and indeed the impact of stochastic physics on climate model regime structure is still unclear, complicated by high sampling variability in the Euro-Atlantic region . A natural next step is to ask how stochasticity impacts regimes in quasi-geostrophic settings and to explore how the mechanisms uncovered here evolve (or, indeed, are rendered extinct) as the complexity of the flow increases. Finally, intermittent dynamical systems such as those possessing circulation regimes are typically close to a bifurcation. Such near-crisis dynamics are particularly sensitive to stochastic forcing, and as we have seen, the response of the system to forcing amplitude can be complex. As a further example, demonstrated high sensitivity of the escape rate in a spatio-temporal system to the spatial correlation of the noise process. This highlights the potential importance of carefully selecting the amplitude and kind of stochastic parameterisations in complex models: not all representations of stochasticity in a system can be expected to be equivalently effective.