Streamfunction: Difference between revisions

Latest revision as of 08:40, 30 March 2024

A parameter of two-dimensional, nondivergent flow, with a value that is constant along each streamline.

For flow in the (x,y) plane, the streamfunction ψ is related to the respective coordinate velocities u and v by the equations

In meteorology the most common application of the streamfunction is in the assumption of geostrophic equilibrium. If variations in the Coriolis parameter f are ignored, the streamfunction in a constant-pressure surface is proportional to the geopotential gz, that is, ψ = gz/f; in an isentropic surface, it is ψ = (gz + c_pT)/f, called the Montgomery streamfunction, where c_p is the specific heat at constant pressure and T the Kelvin temperature.

Stokes's streamfunction (
also called current function): If the flow is three-dimensional but is axisymmetric (i.e., the same in every plane containing the axis of symmetry), a Stokes's streamfunction ψ will exist such that
where v_s is the speed in an arbitrary direction s, r is the distance from the axis of symmetry, and n is normal to the direction s, increasing to the left.
Note that Stokes's streamfunction has dimensions of volume per time. Streamfunctions can also be defined for more complex three-dimensional flows.

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+{{Term
+|Display title=streamfunction
-{{TermHeader}}
+|Definitions={{Definition
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+|Num=1
+|Meaning=#A [[parameter]] of two-dimensional, nondivergent flow, with a value that is constant  along each [[streamline]].
-<div class="termentry">
+|Explanation=For flow in the (''x'',''y'') plane, the streamfunction &#x003c8; is related to the respective coordinate velocities  ''u'' and ''v'' by the equations  <blockquote>[[File:ams2001glos-Se75.gif|link=|center|ams2001glos-Se75]]</blockquote> In meteorology the most common application of the streamfunction is in the assumption of  [[geostrophic equilibrium]]. If variations in the [[Coriolis parameter]] ''f'' are ignored, the streamfunction  in a [[constant-pressure surface]] is proportional to the [[geopotential]] ''gz'', that is, &#x003c8; = ''gz''/''f''; in an  [[isentropic surface]], it is &#x003c8; = (''gz'' + ''c''<sub>''p''</sub>''T'')/''f'', called the [[Montgomery streamfunction]], where ''c''<sub>''p''</sub> is  the [[specific heat]] at constant [[pressure]] and ''T'' the Kelvin [[temperature]].<br/>
-  <div class="term">
+#Stokes's streamfunction (<br/>''also called'' current function): If the flow is three-dimensional but  is [[axisymmetric]] (i.e., the same in every plane containing the axis of symmetry), a Stokes's streamfunction  &#x003c8; will exist such that  <blockquote>[[File:ams2001glos-Se76.gif|link=|center|ams2001glos-Se76]]</blockquote> where ''v''<sub>''s''</sub> is the speed in an arbitrary direction ''s'', ''r'' is the distance from the axis of symmetry, and  ''n'' is normal to the direction ''s'', increasing to the left.<br/> Note that Stokes's streamfunction has [[dimensions]] of volume per time. Streamfunctions can  also be defined for more complex three-dimensional flows.
-== streamfunction ==
+}}
-  </div>
+}}
-#<div class="definition"><div class="short_definition">A [[parameter]] of two-dimensional, nondivergent flow, with a value that is constant  along each [[streamline]].</div><br/> <div class="paragraph">For flow in the (''x'',''y'') plane, the streamfunction &#x003c8; is related to the respective coordinate velocities  ''u'' and ''v'' by the equations  <div class="display-formula"><blockquote>[[File:ams2001glos-Se75.gif|link=|center|ams2001glos-Se75]]</blockquote></div> In meteorology the most common application of the streamfunction is in the assumption of  [[geostrophic equilibrium]]. If variations in the [[Coriolis parameter]] ''f'' are ignored, the streamfunction  in a [[constant-pressure surface]] is proportional to the [[geopotential]] ''gz'', that is, &#x003c8; = ''gz''/''f''; in an  [[isentropic surface]], it is &#x003c8; = (''gz'' + ''c''<sub>''p''</sub>''T'')/''f'', called the [[Montgomery streamfunction]], where ''c''<sub>''p''</sub> is  the [[specific heat]] at constant [[pressure]] and ''T'' the Kelvin [[temperature]].</div><br/> </div>
-#<div class="definition"><div class="short_definition">Stokes's streamfunction (<br/>''also called'' current function): If the flow is three-dimensional but  is [[axisymmetric]] (i.e., the same in every plane containing the axis of symmetry), a Stokes's streamfunction  &#x003c8; will exist such that  <div class="display-formula"><blockquote>[[File:ams2001glos-Se76.gif|link=|center|ams2001glos-Se76]]</blockquote></div> where ''v''<sub>''s''</sub> is the speed in an arbitrary direction ''s'', ''r'' is the distance from the axis of symmetry, and  ''n'' is normal to the direction ''s'', increasing to the left.</div><br/> <div class="paragraph">Note that Stokes's streamfunction has [[dimensions]] of volume per time. Streamfunctions can  also be defined for more complex three-dimensional flows.</div><br/> </div>
-</div>
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-[[Category:Terms_S]]