  Consideration of various hydrodynamic phenomena involves the study of the
Navier-Stokes (N-S) equations, what is hard enough for analytical and numerical
investigations since already in three-dimensional (3D) case it is a challenging
task to study the limit behavior of N-S solutions. The low-order models (LOMs)
derived from the initial N-S equations by Galerkin method allow one to overcome
difficulties in studying the limit behavior and existence of attractors. Among
the simple LOMs with chaotic attractors there are famous Lorenz system, which
is an approximate model of two-dimensional convective flow and
Glukhovsky-Dolzhansky model, which describes a convective process in
three-dimensional rotating fluid and can be considered as an approximate model
of the World Ocean. One of the widely used dimensional characteristics of
attractors is the Lyapunov dimension. In the study we follow a rigorous
approach for the definition of the Lyapunov dimension and justification of its
computation by the Kaplan-Yorke formula, without using statistical physics
assumptions. The exact Lyapunov dimension formula for the global attractors is
obtained and peculiarities of the Lyapunov dimension estimation for
self-excited and hidden attractors are discussed. A tutorial on numerical
estimation of the Lyapunov dimension on the example of the
Glukhovsky-Dolzhansky model is presented.
