A five-dimensional Lorenz-type model near the temperature of maximum density

TL;DR

This paper develops a five-dimensional Lorenz-type model to analyze water convection near the temperature of maximum density, revealing new critical points and flow directions, with numerical phase trajectory analysis.

Contribution

It introduces a novel five-dimensional Lorenz-type model specifically for water near maximum density, extending previous models and analyzing stability and flow directions.

Findings

Nontrivial critical points emerge when the zero critical point becomes unstable.

The model exhibits nonlinear equations similar to the standard Lorenz model.

Numerical analysis shows phase trajectories akin to known five-dimensional Lorenz extensions.

Abstract

The current study formulates a convective model of the Lorenz type near the temperature of maximum density. The existence of this temperature actualizes water dynamics in temperate lakes. There is a conceptual interest what this feature induces in Lorenz-type models. The consideration starts with the zero coefficient of thermal expansion. Other steps are like famous Tritton's approach to derive the Lorenz model. This allows us to reduce difficulties with a selection of Galerkin functions. The analysis focuses on changes induced by zeroing the coefficient of thermal expansion. It results in a five-dimensional Lorenz-type model, whose equations are all nonlinear. The new model reiterates many features of the standard Lorenz model. The nontrivial critical points appear, when the zero critical point becomes unstable. The nontrivial critical points correspond to two possible directions of fluid flow. Phase trajectories of the new model were studied numerically. The results are similar to the known five-dimensional extensions of the Lorenz model.