View source for Convection equations

Consider a fluid flow in a layer of uniform depth where the temperature difference, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110380/c1103801.png" />, between the upper and lower bounding planes is kept constant. Such a system has a steady-state solution in which there is no fluid motion and the temperature varies linearly. If this solution is unstable, convection should develop. When all motion is parallel to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110380/c1103802.png" />-plane, the governing equations are [[#References|[a1]]]:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110380/c1103803.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110380/c1103804.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110380/c1103805.png" /> is the height of the layer (in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110380/c1103806.png" />-direction),

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110380/c1103807.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110380/c1103808.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110380/c1103809.png" /></td> </tr></table>

stands for the Jacobian determinant, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110380/c11038010.png" /> is a [[Stream function|stream function]] for the two-dimensional fluid motion, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110380/c11038011.png" /> is the deviation of the temperature from the case where no convection occurs. The coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110380/c11038012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110380/c11038013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110380/c11038014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110380/c11038015.png" /> are, respectively, kinematic viscosity, gravity acceleration, thermal expansion, and thermal conductivity. The part of the first equation that does not depend upon <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110380/c11038016.png" /> is the third component of the vorticity equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110380/c11038017.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110380/c11038018.png" /> is the velocity vector and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110380/c11038019.png" /> is the vorticity.

See [[#References|[a4]]], [[#References|[a5]]], and also [[Curl|Curl]] and [[Vector product|Vector product]].

By expanding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110380/c11038020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110380/c11038021.png" /> in double [[Fourier series|Fourier series]] with coefficients depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110380/c11038022.png" /> and truncating to three terms, the [[Lorenz equations|Lorenz equations]] result [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Saltzman,   "Finite amplitude free convection as an initial value problem. I"  ''J. Atmos. Sci.'' , '''19'''  (1962)  pp. 329–341</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.N. Lorenz,   "Deterministic non-periodic flow"  ''J. Atmos. Sci.'' , '''20'''  (1963)  pp. 130–141</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.K. Batchelor,   "An introduction to fluid dynamics" , Cambridge Univ. Press  (1967)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  B.K. Shivamoggi,   "Theoretical fluid dynamics" , Martinus Nijhoff  (1985)  pp. 13–14</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  "Modern developments in fluid dynamics"  S. Goldstein (ed.) , Dover, reprint  (1965)  pp. 114</TD></TR></table>