Computational Vector Mechanics in Atmospheric and Climate Modeling

The mathematical underpinnings of vector analysis are reviewed as they are applied in the development of the ensemble of numeric statements for subsequent matrix solution. With the continued advances in computational power, there is increased interest in the field of atmospheric modelling to decrease the computational scale to a micro‐scale. This interest is partially motivated by the ability to solve large scale matrix systems in the number of occasions to enable a small‐scale time advancement to be approximated in a finite‐difference scheme. Solving entire large scale matrix systems several times a modelling second is now computationally feasible. Hence the motivation to increase computational detail by reducing modelling scale

Mathematical Model of Cryospheric Response to Climate Changes

Abstract: This paper focuses on the development of simplified mathematical models of the cryosphere which may be useful in further understanding possible global climate change impacts and in further assessing future impacts captured by global circulation models (GCMs). The mathematical models developed by leveraging the dominating effects of freezing and thawing within the cryosphere to simplify the relevant heat transport equations are tractable to direct solution or numerical modeling. In this paper, the heat forcing function is assumed to be a linear transformation of temperature (assumed to be represented by proxy realizations). The output from the governing mathematical model is total ice volume of the cryosphere. The basic mathematical model provides information as a systems modeling approach that includes sufficient detail to explain ice volume given the estimation of the heat forcing function. A comparison between modeling results in the estimation of ice volume versus ice volume estimates developed from use of proxy data are shown in the demonstration problems presented

Cumulative Departure Model of the Cryosphere During the Pleistocene

Abstract: A mathematical model is developed to describe changes in ice volume in the cryosphere. Modeling the cryosphere may be useful in assessing future climate impacts currently captured by global circulation models (GCMs) by providing an opportunity to validate GCMs. Leveraging the dominating effects of freezing and thawing in the cryosphere to simplify relevant heat transport equations allows for the derivation of a mathematical model that can be solved exactly. Such exact solutions are useful in investigating other climatic components that may be similarly analyzed for possible GCM validation. The current trend in GCM advancement is to increase the complexity and sophistication of the various heat transport effects that are represented in the governing mathematical model in cumulative form as the heat forcing function. In this paper, simplified models are developed whose solution can be directly compared with available data forms representing temperature and ice volume during the Pleistocene. With careful integration of the Pleistocene temperature term in the mathematical solution, the well-known cumulative departure method can be resolved from the mathematical solution using a two-term expansion of the corresponding Taylor series. This simplification is shown to be a good approximation of the Pleistocene ice volume for given Pleistocene temperatures

Characterization of K-Complexes and Slow Wave Activity in a Neural Mass Model

NREM sleep is characterized by two hallmarks, namely K-complexes (KCs) during sleep stage N2 and cortical slow oscillations (SOs) during sleep stage N3. While the underlying dynamics on the neuronal level is well known and can be easily measured, the resulting behavior on the macroscopic population level remains unclear. On the basis of an extended neural mass model of the cortex, we suggest a new interpretation of the mechanisms responsible for the generation of KCs and SOs. As the cortex transitions from wake to deep sleep, in our model it approaches an oscillatory regime via a Hopf bifurcation. Importantly, there is a canard phenomenon arising from a homoclinic bifurcation, whose orbit determines the shape of large amplitude SOs. A KC corresponds to a single excursion along the homoclinic orbit, while SOs are noise-driven oscillations around a stable focus. The model generates both time series and spectra that strikingly resemble real electroencephalogram data and points out possible differences between the different stages of natural sleep