# Difference between revisions of "Del operator"

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== del operator == | == del operator == | ||

− | <div class="definition"><div class="short_definition">The [[operator]] (written '''∇''') is used to transform a [[scalar]] field into the [[ascendent]] (the negative of the [[gradient]]) of that [[field]].</div><br/> <div class="paragraph">In [[Cartesian coordinates]] the three-dimensional del operator is <div class="display-formula"><blockquote>[[File:ams2001glos-De16.gif|link=|center|ams2001glos-De16]]</blockquote></div> and the horizontal component is <div class="display-formula"><blockquote>[[File:ams2001glos-De17.gif|link=|center|ams2001glos-De17]]</blockquote></div> 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|thermodynamic function of state]], such as [[pressure]] or [[potential temperature]], as the vertical coordinate. If the chosen function is σ, then <div class="display-formula"><blockquote>[[File:ams2001glos-De18.gif|link=|center|ams2001glos-De18]]</blockquote></div> where differentiation with respect to ''x'' and ''y'' is understood as carried out on surfaces of constant | + | <div class="definition"><div class="short_definition">The [[operator]] (written '''∇''') is used to transform a [[scalar]] field into the [[ascendent]] (the negative of the [[gradient]]) of that [[field]].</div><br/> <div class="paragraph">In [[Cartesian coordinates]] the three-dimensional del operator is <div class="display-formula"><blockquote>[[File:ams2001glos-De16.gif|link=|center|ams2001glos-De16]]</blockquote></div> and the horizontal component is <div class="display-formula"><blockquote>[[File:ams2001glos-De17.gif|link=|center|ams2001glos-De17]]</blockquote></div> 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|thermodynamic function of state]], such as [[pressure]] or [[potential temperature]], as the vertical coordinate. If the chosen function is ''σ'', then <div class="display-formula"><blockquote>[[File:ams2001glos-De18.gif|link=|center|ams2001glos-De18]]</blockquote></div> 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 <div class="display-formula"><blockquote>[[File:ams2001glos-De19.gif|link=|center|ams2001glos-De19]]</blockquote></div> If the [[quasi-hydrostatic approximation]] is justified, as in most meteorological contexts, pressure is a useful coordinate, and <div class="display-formula"><blockquote>[[File:ams2001glos-De20_revised2.gif|link=|center|ams2001glos-De20_revised2]]</blockquote></div> where ''g'' is the [[acceleration of gravity]] and ''ρ'' is the [[density]]. Here <div class="display-formula"><blockquote>[[File:ams2001glos-De21.gif|link=|center|ams2001glos-De21]]</blockquote></div> with differentiation carried out in [[isobaric surfaces]], and '''∇'''<sub>''z''</sub> is the horizontal gradient operator in Cartesian-altitude coordinates.</div><br/> </div> |

<p>''Term updated 8 March 2017.''</p> | <p>''Term updated 8 March 2017.''</p> |

## Latest revision as of 12:23, 8 March 2017

## del operator

**∇**) is used to transform a scalar field into the ascendent (the negative of the gradient) of that field.

**∇**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

*x*and

*y*is understood as carried out on surfaces of constant

*σ*(the subscript usually being omitted). The horizontal component is now

*g*is the acceleration of gravity and

*ρ*is the density. Here

**∇**

_{z}is the horizontal gradient operator in Cartesian-altitude coordinates.

*Term updated 8 March 2017.*

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