Difference between revisions of "User:Tohline/PGE/RotatingFrame"

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(→‎Steady-State Governing Relations: Completing set of equations and setting time-derivatives to zero.)
(Expand on discussion/presentation)
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=Compressible Analogs of Riemann S-Type Ellipsoids=
Here we attempt to develop a self-consistent-field type, iterative technique that will permit the construction of steady-state structures that are compressible analogs of Riemann S-Type (incompressible) ellipsoids.  We will build upon the recent work of [http://adsabs.harvard.edu/abs/2006ApJ...639..549O Ou (2006)].


==Standard Steady-State Governing Relations==
==Standard Steady-State Governing Relations==
Line 16: Line 19:
<math>
<math>
\nabla^2 \Phi = 4\pi G \rho
\nabla^2 \Phi = 4\pi G \rho
</math>
</div>
==Potential Solution Strategy==
===Preamble:===
Specify the three "polar" boundary locations, <math>a, b,</math> and <math>c</math>; specify the <i>direction</i> but not the magnitude of the rotating frame's angular velocity, for example, <math>(\vec{\omega}/\omega) = \hat{k}</math>; pin the central density to the value <math>\rho_c = 1</math>.  Define the following dimensionless density and velocity:
<div align="center">
<math>
\Lambda \equiv \frac{\rho}{\rho_c} ;
</math>
<math>
\vec{\zeta} \equiv \frac{\vec{v}}{a \omega} .
</math>
</div>
===Step #1:===
Guess a 3D density distribution &#8212; such as a uniform-density ellipsoid, or one of the converged models from Ou (2006) &#8212; that conforms to a selected set of <i>positional</i> boundary conditions, that is, where the density goes to zero along the three principal axes at <math>x=a</math>, <math>y = b</math>, and <math>z = c</math>.  Solve the Poisson equation in order to derive values for <math>\Phi</math> everywhere inside and on the surface of the 3D configuration.
===Step #2:===
Use the curl of the Euler equation and the continuity equation to numerically derive the <i>structure</i> but not the overall magnitude of the velocity flow-field throughout the 3D configuration.  Take advantage of the fact that the direction, <math>(\vec{\omega}/\omega)</math>, has been specified; and assume that the 3D density distribution is known.  Here are the relevant equations:
<div align="center">
<math>
\nabla\cdot(\Lambda \vec{\zeta}) = 0
</math>
<math>
\nabla\times [(\vec{\zeta}\cdot \nabla)\vec{\zeta}] = -2 \nabla\times \biggl[\hat{k}\times\vec{\zeta} \biggr]
</math>
</div>
The first of these is a scalar equation; the second is a vector equation and it will presumably provide two useful scalar equations (constraining the two components of <math>\vec{\zeta}</math> that are perpendicular to <math>\hat{k}</math> ?).  Since the left-hand-side of the second equation is obviously nonlinear in the velocity, we may have to linearize this set of equations and look for small "corrections" <math>\delta\vec{v}</math> to an initial "guess" for the velocity field, such as the flow field in Riemann S-type ellipsoids, which is also the flow-field adopted by Ou (2006).  The relevant linearize set of equations is:
===Step #3:===
Take the divergence of the Euler equation and use it to solve for <math>H</math> throughout the configuration, given the structure of the flow-field obtained in Step #2.  Boundary conditions at the three "poles" of the configuration may suffice to uniquely determine <math>\omega</math>, the overall normalization factor for the flow-field <math>\vec\zeta</math> &#8212; hopefully this is analogous to solving for the vorticity parameter <math>\lambda</math> in Ou (2006) &#8212; and the Bernoulli constant (or something equivalent).  The relevant "Poisson"-like equation is:
<div align="center">
<math>
\nabla^2 \biggl[H + \Phi -\frac{1}{2}\omega^2 R^2  \biggr] = - \nabla\cdot [(\vec{v}\cdot \nabla)\vec{v} + 2\vec{\omega}\times\vec{v} ] .
</math>
</math>
</div>
</div>

Revision as of 05:36, 10 March 2010


Whitworth's (1981) Isothermal Free-Energy Surface
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Compressible Analogs of Riemann S-Type Ellipsoids

Here we attempt to develop a self-consistent-field type, iterative technique that will permit the construction of steady-state structures that are compressible analogs of Riemann S-Type (incompressible) ellipsoids. We will build upon the recent work of Ou (2006).

Standard Steady-State Governing Relations

As viewed from a rotating frame of reference and written in Eulerian form, the steady-state version of the three-dimensional principal governing equations are:

<math> \nabla\cdot(\rho \vec{v}) = 0 </math>

<math> (\vec{v}\cdot \nabla)\vec{v} = -\nabla \biggl[H + \Phi -\frac{1}{2}\omega^2 R^2 \biggr] -2\vec{\omega}\times\vec{v} </math>

<math> \nabla^2 \Phi = 4\pi G \rho </math>

Potential Solution Strategy

Preamble:

Specify the three "polar" boundary locations, <math>a, b,</math> and <math>c</math>; specify the direction but not the magnitude of the rotating frame's angular velocity, for example, <math>(\vec{\omega}/\omega) = \hat{k}</math>; pin the central density to the value <math>\rho_c = 1</math>. Define the following dimensionless density and velocity:

<math> \Lambda \equiv \frac{\rho}{\rho_c} ; </math>

<math> \vec{\zeta} \equiv \frac{\vec{v}}{a \omega} . </math>

Step #1:

Guess a 3D density distribution — such as a uniform-density ellipsoid, or one of the converged models from Ou (2006) — that conforms to a selected set of positional boundary conditions, that is, where the density goes to zero along the three principal axes at <math>x=a</math>, <math>y = b</math>, and <math>z = c</math>. Solve the Poisson equation in order to derive values for <math>\Phi</math> everywhere inside and on the surface of the 3D configuration.

Step #2:

Use the curl of the Euler equation and the continuity equation to numerically derive the structure but not the overall magnitude of the velocity flow-field throughout the 3D configuration. Take advantage of the fact that the direction, <math>(\vec{\omega}/\omega)</math>, has been specified; and assume that the 3D density distribution is known. Here are the relevant equations:

<math> \nabla\cdot(\Lambda \vec{\zeta}) = 0 </math>

<math> \nabla\times [(\vec{\zeta}\cdot \nabla)\vec{\zeta}] = -2 \nabla\times \biggl[\hat{k}\times\vec{\zeta} \biggr] </math>

The first of these is a scalar equation; the second is a vector equation and it will presumably provide two useful scalar equations (constraining the two components of <math>\vec{\zeta}</math> that are perpendicular to <math>\hat{k}</math> ?). Since the left-hand-side of the second equation is obviously nonlinear in the velocity, we may have to linearize this set of equations and look for small "corrections" <math>\delta\vec{v}</math> to an initial "guess" for the velocity field, such as the flow field in Riemann S-type ellipsoids, which is also the flow-field adopted by Ou (2006). The relevant linearize set of equations is:

Step #3:

Take the divergence of the Euler equation and use it to solve for <math>H</math> throughout the configuration, given the structure of the flow-field obtained in Step #2. Boundary conditions at the three "poles" of the configuration may suffice to uniquely determine <math>\omega</math>, the overall normalization factor for the flow-field <math>\vec\zeta</math> — hopefully this is analogous to solving for the vorticity parameter <math>\lambda</math> in Ou (2006) — and the Bernoulli constant (or something equivalent). The relevant "Poisson"-like equation is:

<math> \nabla^2 \biggl[H + \Phi -\frac{1}{2}\omega^2 R^2 \biggr] = - \nabla\cdot [(\vec{v}\cdot \nabla)\vec{v} + 2\vec{\omega}\times\vec{v} ] . </math>


Whitworth's (1981) Isothermal Free-Energy Surface

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