TENSOR ANALYSIS OF MAIN DIFFERENTIAL OPERATORS UTILIZED FOR DESCRIPTION OF OCEAN HYDRODYNAMICS IN CURVILINEAR ORTHOGONAL COORDINATE SYSTEM

Abstract

The paper presents an analysis of tensor expressions in different curvilinear orthogonal coordinate systems. The analysis reveals specific properties of a number of approximated coordinate systems widely used in the studies of ocean dynamics. The paper consists of two parts. The part 1 presents a brief overview of the key definitions and important relations of tensor analysis which are utilized in part 2 of the paper. The part 2 considers invariant representation of different types of vector products, divergence of vector field and divergence of symmetric tensor of rank 2, gradient of a scalar filed, curl of a vector filed. The part 2 also discusses specific properties of the rate-of-strain tensor, general form of the Laplace operator, properties of operator nabla, and general forms of material derivative for scalar and vector fields. The equations for the properties under consideration are derived for the physical components of the corresponding tensors.