Difference between revisions of "Del operator"

From Glossary of Meteorology
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== del operator ==
 
== 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|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;''') 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 '''&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 the chosen function is &#x003c3;,  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  &#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_revised.gif|link=|center|ams2001glos-De20_revised]]</blockquote></div> where ''g'' is the [[acceleration of gravity]] and &#x003c1; 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 '''&nabla;'''<sub>''z''</sub> is the horizontal gradient operator in Cartesian-altitude coordinates.</div><br/> </div>
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<p>''Term updated 8 March 2017.''</p>
 
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Revision as of 11:56, 8 March 2017



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
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 the chosen function is σ, then
ams2001glos-De18
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
ams2001glos-De19
If the quasi-hydrostatic approximation is justified, as in most meteorological contexts, pressure is a useful coordinate, and
ams2001glos-De20_revised
where g is the acceleration of gravity and ρ is the density. Here
ams2001glos-De21
with differentiation carried out in isobaric surfaces, and z is the horizontal gradient operator in Cartesian-altitude coordinates.

Term updated 8 March 2017.