From AMS Glossary
The operator (written ∇) is 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 the chosen function is σ, then
where differentiation with respect to x and y is understood as carried out on 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 is the density. Here
with differentiation carried out in isobaric surfaces, and ∇z is the horizontal gradient operator in Cartesian-altitude coordinates.
Term updated 8 March 2017.