Del operator

del operator

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.