  Though the Boltzmann-Gibbs framework of equilibrium statistical mechanics has
been successful in many arenas, it is clearly inadequate for describing many
interesting natural phenomena driven far from equilibrium. The simplest step
towards that goal is a better understanding of nonequilibrium steady-states
(NESS). Here we focus on one of the distinctive features of NESS, persistent
probability currents, and their manifestations in our climate system. We
consider the natural variability of the steady-state climate system, which can
be approximated as a NESS. These currents must form closed loops, which are odd
under time reversal, providing the crucial difference between systems in
thermal equilibrium and NESS. Seeking manifestations of such current loops
leads us naturally to the notion of probability angular momentum and
oscillations in the space of observables. Specifically, we will relate this
concept to the asymmetric part of certain time-dependent correlation functions.
Applying this approach, we propose that these current loops give rise to
preferred spatio-temporal patterns of natural climate variability that take the
form of climate oscillations such as the El-Ni\~{n}o Southern Oscillation
(ENSO) and the Madden-Julien Oscillation (MJO). In the space of climate
indices, we observe persistent currents and define a new diagnostic for these
currents: the probability angular momentum. Using the observed climatic time
series of ENSO and MJO, we compute both the averages and the distributions of
the probability angular momentum. These results are in good agreement with the
analysis from a linear Gaussian model. We propose that, in addition to being a
new quantification of climate oscillations across models and observations, the
probability angular momentum provides a meaningful characterization for all
statistical systems in NESS.
