Information Length as a Useful Index to Understand Variability in the Global Circulation

Abstract

With improved measurement and modelling technology, variability has emerged as an essential feature in non-equilibrium processes. While traditionally, mean values and variance have been heavily used, they are not appropriate in describing extreme events where a significant deviation from mean values often occurs. Furthermore, stationary Probability Density Functions (PDFs) miss crucial information about the dynamics associated with variability. It is thus critical to go beyond a traditional approach and deal with time-dependent PDFs. Here, we consider atmospheric data from the Whole Atmosphere Community Climate Model (WACCM) and calculate time-dependent PDFs and the information length from these PDFs, which is the total number of statistically different states that a system passes through in time. Time-dependent PDFs are shown to be non-Gaussian in general, and the information length calculated from these PDFs shed us a new perspective of understanding variabilities, correlation among different variables and regions.Specifically, we calculate time-dependent PDFs and information length and show that the information length tends to increase with the altitude albeit in a complex form. This tendency is more robust for flows/shears than temperature. Also, much similarity among flows and shears in the information length is found in comparison with the temperature. This means a stronger correlation among flows/shears because of a strong coupling through gravity waves in this particular WACCM model. We also find the increase of the information length with the latitude and interesting hemispheric asymmetry for flows/shears/temperature, a stronger anti-correlation (correlation) between flows/shears and temperature at a higher (low) latitude. These results also suggest the importance of high latitude/altitude in the information budge in the Earth's atmosphere, the spatial gradient of the information as a useful proxy for the transport of physical quantities.

Time-dependent probability density functions and information geometry of the low-to-high confinement transition in fusion plasma

“…ELMs, eruptions) well before other methods can, so that the occurrence of such events can be avoided or else controlled to some degree. It will also be of great interest to apply this methodology to understand the temporal-spatial dynamics in other L-H transition turbulence models as well as experimental data to quantify correlations at different spatial positions [26,28].…”

Causal Information Rate

“…Being very sensitive to evolving dynamics, it enables us to compare different far-from-equilibrium processes using the same dimensionless distance, as well as quantifying the relation (correlation, self-regulation, etc.) among variables (e.g., References [27][28][29][30]).…”

“…This is because Equations (3)-(5) can be calculated from any (numerical or experimental) data as long as time-dependent (marginal, joint) PDFs can be constructed. For instance, we used a time-sliding window method to construct time-dependent PDFs of different variables and then calculated E and L to analyze numerically generated time-series data for fusion turbulence [26], time-series music data [20], and numerically generated time-series data for global circulation model [28]. However, it is not always clear how many hidden variables are in a given data set.…”

“…We have recently proposed information-geometric theory as a powerful tool to understand non-equilibrium stochastic processes that often involve high temporal variabilities and large fluctuations [20][21][22][23][24][25][26][27][28][29][30][31][32], as often the case of rare, extreme events. This is based on the surprisal rate, r(x, t) = ∂ t s(x, t) = −∂ t ln p(x, t), where p(x, t) is a probability density function (PDF) of a random variable x at time t, and s(x, t) = − ln p(x, t) is a local entropy.…”

“…Furthermore, L is a path-dependent dimensionless distance and is uniquely defined as a function of time for fixed parameters and initial condition. These properties are advantageous for quantifying correlation in time-varying data and understanding selforganization, long memory, and hysteresis involved in phase transitions [20,24,[27][28][29][30]32]. In particular, we recently investigated a non-autonomous Kramer equation by including a sudden perturbation to the system to mimic the onset of a sudden event [32], demonstrating that our information rate predicts the onset of a sudden event better than one of the entropybased measures (information flow) (see Section 5.4 for details).…”