Modeling substorm dynamics of the magnetosphere: from self-organization and self-organized criticality to nonequilibrium phase transitions

Earth's magnetosphere during substorms exhibits a number of characteristic features such as the signatures of low effective dimension, hysteresis, and power-law spectra of fluctuations on different scales. The largest substorm phenomena are in reasonable agreement with low-dimensional magnetospheric models and in particular those of inverse bifurcation. However, deviations from the low-dimensional picture are also quite considerable, making the nonequilibrium phase transition more appropriate as a dynamical analog of the substorm activity. On the other hand, the multiscale magnetospheric dynamics cannot be limited to the features of self-organized criticality (SOC), which is based on a class of mathematical analogs of sandpiles. Like real sandpiles, during substorms the magnetosphere demonstrates features, that are distinct from SOC and are closer to those of conventional phase transitions. While the multiscale substorm activity resembles second-order phase transitions, the largest substorm avalanches are shown to reveal the features of first-order nonequilibrium transitions including hysteresis phenomena and a global structure of the type of a temperature-pressure-density diagram. Moreover, this diagram allows one to find a critical exponent, that reflects the multiscale aspect of the substorm activity, different from the power-law frequency and scale spectra of autonomous systems, although quite consistent with second-order phase transitions. In contrast to SOC exponents, this exponent relates input and output parameters of the magnetosphere. Using an analogy to the dynamical Ising model in the mean-field approximation, we show the connection between the data-derived exponent of nonequilibrium transitions in the magnetosphere and the standard critical exponent beta of equilibrium second-order phase transitions.