Editing Apps/GoldreichWeber80 (section)

===Governing Equations===
{{ GW80 }} begin with the identical set of [[PGE#Principal_Governing_Equations|principal governing equations]] that serves as the foundation for all of the discussions throughout this H_Book.  In particular, as is documented by their equation (1), their adopted equation of state is adiabatic/polytropic,
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<math>~P = \kappa \rho^\gamma \, ,</math>
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&#8212; where both <math>~\kappa</math> and <math>~\gamma</math> are constants &#8212; and therefore satisfies what [[PGE#PGE:AdiabaticFirstLaw|we have referred to]] as the

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<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />
('''Specific Entropy Conservation''')

{{Math/EQ_FirstLaw02}} .
</div>
 
their equation (3) is what [[PGE/Euler#in_terms_of_the_vorticity:|we have referred to]] as the

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Euler Equation<br />
<span id="ConservingMomentum:Lagrangian"><font color="#770000">'''in terms of the Vorticity'''</font></span>,

{{Math/EQ_Euler04}}
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where, <math>~\vec\zeta \equiv \nabla\times \vec{v}</math> is the fluid vorticity; their equation (4) is the

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<font color="#770000">'''Poisson Equation'''</font><br />

{{Math/EQ_Poisson01}}
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and their equation (2) is what [[PGE/ConservingMass#Eulerian_Representation|we have referred to]] as the

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<span id="ConservingMass:Eulerian"><font color="#770000">'''Eulerian Representation'''</font></span><br />
or<br />
<span id="ConservingMass:Conservative"><font color="#770000">'''Conservative Form'''</font></span><br />
of the Continuity Equation,

{{Math/EQ_Continuity02}}
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although, for the derivation, below, we prefer to start with what [[PGE/ConservingMass#Lagrangian_Representation|we have referred to]] as the
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<span id="ConservingMass:Lagrangian"><font color="#770000">'''Standard Lagrangian Representation'''</font></span><br />
of the Continuity Equation,

{{Math/EQ_Continuity01}}
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Tweaking the set of principal governing equations, as we have written them, to even more precisely match equations (1) - (4) in Goldreich &amp; Weber (1980), we should replace the state variable <math>~P</math> (pressure) with <math>~H</math> (enthalpy), keeping in mind that, <math>~\gamma = 1 + 1/n</math>, and, [[SR#Barotropic_Structure|as presented in our introductory discussion of barotropic supplemental relations]],
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<math>~H = \biggl( \frac{\gamma}{\gamma-1} \biggr) \kappa \rho^{\gamma-1} \, ,</math>
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and,
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<math>~\nabla H = \frac{\nabla P}{\rho} \, .</math>
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