Stochastic forcing can, sometimes, stabilise atmospheric regime dynamics, increasing their persistence. This counter-intuitive effect has been observed in geophysical models of varying complexity, and here we investigate the mechanisms underlying stochastic regime dynamics in a conceptual model. We use a six-mode truncation of a barotropic

Similar results were obtained in other simple models

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:

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

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.

The deterministic Charney–deVore model is obtained by spectral truncation of a barotropic flow in a

A 2000 MTU integration of the CdV79 system, showing the evolution of each mode amplitude. The dynamics are chaotic but weakly so, showing clear quasi-periodic behaviour. Two regimes of behaviour can be seen: fast chaotic oscillations, divided by periods of slow evolution with irregular duration. The mode values at the blocked and zonal fixed points are shown with dashed blue and red lines.

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.

A representative sample of the model's temporal evolution is shown in Fig.

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

When studying a multimodal system, bulk metrics such as the mean or variance can obscure the underlying dynamical structure. Therefore it is natural to use a regime framework as a way of partitioning phase space into dynamically distinct regions of interest. Following the approach of K14, we use a hidden Markov model (HMM) to identify qualitatively different dynamical regimes, with the minor methodological change of fitting the HMM to the full six-dimensional state vector, rather than restricting it to the

Left: time evolution of the

For non-zero noise amplitude,

The “pullback attractor” of

Figure

Figure

The distribution of blocking regime lifetimes, as a function of stochastic forcing amplitude. Shaded regions mark the interquartile range, with the thick black line showing the median and the red line the mean. Dotted black lines show the 0th and 95th percentiles of the distribution.

This is in excellent agreement with K14, which showed increasing regime persistence with noise amplitude that peaked at

A schematic showing one variable,

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

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.

Because

When considering the particular case of

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

Here we derived the CdV79 model introduced in Sect.

The momentum equation in a rotating frame is given by

We divide our fluid height

We consider a channel of dimensionless zonal length

To convert this infinite dimensional partial differential equation into a system of ordinary differential equations, we can use a Galerkin projection. This involves expanding our spatially dependent variables (

The real-valued basis functions used in the spectrally truncated CdV79 system, plotted as streamfunctions.

Although up to this point our approximations have been motivated well physically, we cannot rigorously justify introducing such a drastic truncation without resorting to self-interest: such a system is much easier to study and can be understood in more detail than a high-dimensional model. Nonetheless, we may reason that any phenomena observed in this truncated model, which will have to derive solely from the dynamics of the largest scales, may have analogues in the true atmospheric circulation, although there they will be shaped and influenced by coupling to smaller scales. The resulting basis functions are shown in Fig.

Concretely, while the true spectral equation for the largest wave modes, Eq. (

We choose our

The orographic profile used in the CdV79 system, with a trough in the centre of the domain and a ridge over the periodic boundary. Sub-panels show cross sections across the domain, following the dotted black lines.

The background state in the Crommelin 1904 formulation of the CdV79 system. It is zonally symmetric and represents the impact of the meridional temperature gradient. Again, sub-panels show cross sections across the domain, following the dotted black lines.

Obtaining the coefficients in Eq. (

There are now a number of dimensionless free parameters in the model, which we set equal to the values used in K14 and C04:

In Sect.

If, as in the main text, we assume

Python code for deriving Eq. (1) and Fortran code for integrating it are available at

No data sets were used in this article.

The supplement related to this article is available online at:

JD performed all simulations and drafted the manuscript, under the supervision of TP.

The contact author has declared that neither of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the special issue “Interdisciplinary perspectives on climate sciences – highlighting past and current scientific achievements”. It is not associated with a conference.

The authors would like to thank Glenn Shutts, Daan Crommelin, and Frank Kwasniok for enlightening discussions. We would like to thank one anonymous reviewer and Tamás Bódai for valuable feedback that has helped to strengthen the manuscript considerably.

This research has been supported by the Natural Environment Research Council, National Centre for Earth Observation (grant no. NE/L002612/1).The article processing charges for this open-access publication were covered by the Karlsruhe Institute of Technology (KIT).

This paper was edited by Lesley De Cruz and reviewed by Tamás Bódai and one anonymous referee.