Both Cartesian tensor and vector notation will be used in this text. The notation xi means the i-component of the vector x = (x1 x2, x3). When used in the argument of a function [e.g., f(xi)], xi, represents the whole vector, so that f(xi) stands for f(xl,x2,x3). Repeated indices indicate summations over all coordinate directions, (uiui = u2i = u21 + u22 + u23). Two special and frequently used tensors are the unit tensor ∂ij and the alternating tensor εijk. The unit tensor has components equal to unity for i = j and zero for i ≠ j. The alternating tensor has components equal to +1 when the indices are in cyclical sequence 1, 2, 3 or 2, 3, 1 or 3, 1,2; equal to -1 when the indices are not cyclical; and equal to zero when two indices are the same. The vorticity vector is defined by the relation The symbol = is used throughout to represent a definition or identity. Conditions near the sea surface are usually very anisotropic. It is often desirable to distinguish between the horizontal and vertical directions. We shall do so by using an x, y, z coordinate system with the origin at mean sea level and the z -axis pointing upward. Unless otherwise specified, the x and y directions will be toward east and north. The vertical velocity will be denoted by W; the horizontal velocity by the vector U with components U and V. Unity vectors in the x, y, z directions are denoted by i, j, k. The usual vector operation symbols will be used only to represent operations within the horizontal plane. For example, In a fluid one has to distinguish between local changes and changes that are experienced by an individual fluid element as it moves about. The former can be recorded by a fixed sensor and is represented by the partial time differential. The individual change could only be recorded by a sensor that would float with the element. It is denoted by the total time differential In a treatise that covers such a variety of topics, some use of the same symbols for different properties is inevitable.
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