Del operator

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del operator

The operator (written ) used to transform a scalar field into the ascendent (the negative of the gradient) of that field.

In Cartesian coordinates the three-dimensional del operator is
and the horizontal component is
Expressions for in various systems of curvilinear coordinates may be found in any textbook of vector analysis. In meteorology it is often convenient to use a thermodynamic function of state, such as pressure or potential temperature, as the vertical coordinate. If σ be this parameter, then
where differentiation with respect to x and y is understood as carried out in surfaces of constant σ (the subscript usually being omitted). The horizontal component is now
If the quasi-hydrostatic approximation is justified, as in most meteorological contexts, pressure is a useful coordinate, and
where g is the acceleration of gravity and ρ the density. Here
with differentiation carried out in isobaric surfaces.