Difference between revisions of "Del operator"

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<div class="definition"><div class="short_definition">The [[operator]] (written '''&nabla;''') 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 '''&nabla;''' 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 &#x003c3; be this [[parameter]],  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 in surfaces of constant  &#x003c3; (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.gif|link=|center|ams2001glos-De20]]</blockquote></div> where ''g'' is the [[acceleration of gravity]] and &#x003c1; 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]].</div><br/> </div>
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<div class="definition"><div class="short_definition">The [[operator]] (written '''&nabla;''') 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 '''&nabla;''' 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 &#x003c3; be this [[parameter]],  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 in surfaces of constant  &#x003c3; (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.gif|link=|center|ams2001glos-De20]]</blockquote></div> where ''g'' is the [[acceleration of gravity]] and &#x003c1; 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]].</div><br/> </div>
 
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Revision as of 15:46, 25 April 2012



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
ams2001glos-De16
and the horizontal component is
ams2001glos-De17
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
ams2001glos-De18
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
ams2001glos-De19
If the quasi-hydrostatic approximation is justified, as in most meteorological contexts, pressure is a useful coordinate, and
ams2001glos-De20
where g is the acceleration of gravity and ρ the density. Here
ams2001glos-De21
with differentiation carried out in isobaric surfaces.