User:Tohline/ThreeDimensionalConfigurations/Challenges

Challenges Constructing Ellipsoidal-Like Configurations

First, let's review the three different approaches that we have described for constructing Riemann S-type ellipsoids. Then let's see how these relate to the technique that has been used to construct infinitesimally thin, nonaxisymmetric disks.

Riemann S-type Ellipsoids

Usually, the density, <math>~\rho</math>, and the pair of axis ratios, <math>~b/a</math> and <math>~c/a</math>, are specified. Then, the Poisson equation is solved to obtain <math>~\Phi_\mathrm{grav}</math> in terms of <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math>. The aim, then, is to determine the value of the central enthalpy, <math>~H_0</math> — alternatively, the thermal energy density, <math>~\Pi</math> — and the two parameters, <math>~\Omega_f</math> and <math>~\lambda</math>, that determine the magnitude of the velocity flow-field. Keep in mind that, as viewed from a frame of reference that is spinning with the ellipsoid (at angular frequency, <math>~\Omega_f</math>), the adopted (rotating-frame) velocity field is,

<math>~\bold{u}</math>

<math>~=</math>

<math>~\lambda \biggl[ \boldsymbol{\hat\imath} \biggl( \frac{a}{b}\biggr) y - \boldsymbol{\hat\jmath} \biggl( \frac{b}{a} \biggr) x \biggr] \, .</math>

Hence, the inertial-frame velocity is given by the expression,

<math>~\bold{v}</math>

<math>~=</math>

<math>~\bold{u} + \bold{\hat{e}}_\varphi \Omega_f \varpi \, .</math>

While we will fundamentally rely on the <math>~(\Omega_f, \lambda)</math> parameter pair to define the velocity flow-field, in discussions of Riemann S-type ellipsoids it is customary to also refer to the following two additional parameters: The (rotating-frame) vorticity,

<math>~\boldsymbol{\zeta} \equiv \boldsymbol{\nabla \times}\bold{u}</math>	<math>~=</math>	<math>~ \boldsymbol{\hat\imath} \biggl[ \frac{\partial u_z}{\partial y} - \frac{\partial u_y}{\partial z} \biggr] + \boldsymbol{\hat\jmath} \biggl[ \frac{\partial u_x}{\partial z} - \frac{\partial u_z}{\partial x} \biggr] + \bold{\hat{k}} \biggl[ \frac{\partial u_y}{\partial x} - \frac{\partial u_x}{\partial y} \biggr] </math>
	<math>~=</math>	<math>~\bold{\hat{k}} \biggl[ - \lambda \biggl(\frac{b}{a} + \frac{a}{b}\biggr) \biggr] \, ;</math>

and the dimensionless frequency ratio,

<math>~f</math>

<math>~\equiv</math>

<math>~\frac{ \zeta}{\Omega_f} \, .</math>

2^nd-Order TVE Expressions

As we have discussed in detail in an accompanying chapter, the three diagonal elements of the <math>~(3 \times 3)</math> 2^nd-order tensor virial equation are sufficient to determine the equilibrium values of <math>~\Pi</math>, <math>~\Omega_3</math>, and <math>~\zeta_3</math>.

Indices		2nd-Order TVE Expressions that are Relevant to Riemann S-Type Ellipsoids
<math>~i</math>	<math>~j</math>
<math>~1</math>	<math>~1</math>	<math>~0</math> <math>~=</math> <math>~ \biggl[ \frac{3\cdot 5}{2^2\pi a b c\rho} \biggr] \Pi +\biggl\{ \Omega_3^2 + 2 \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3 ~-~(2\pi G\rho) A_1 \biggr\} a^2 + \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 b^2 </math>
<math>~2</math>	<math>~2</math>	<math>~0</math> <math>~=</math> <math>~ \biggl[ \frac{3\cdot 5}{2^2\pi a b c \rho} \biggr]\Pi + \biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 a^2 + \biggl\{ \Omega_3^2 + 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3 ~-~( 2\pi G \rho) A_2 \biggr\}b^2 </math>
<math>~3</math>	<math>~3</math>	<math>~0</math> <math>~=</math> <math>~ \biggl[ \frac{3\cdot 5}{2^2\pi abc\rho} \biggr]\Pi - (2\pi G \rho)A_3 c^2 </math>

The <math>~(i, j) = (3, 3)</math> element gives <math>~\Pi</math> directly in terms of known parameters. The <math>~(1, 1)</math> and <math>~(2, 2)</math> elements can then be combined in a couple of different ways to obtain a coupled set of expressions that define <math>~\Omega_3</math> and <math>~f \equiv \zeta_3/\Omega_3</math>.

<math>~\biggl[ \frac{b^2 a^2}{b^2+a^2}\biggr] f \Omega_3^2 </math> <math>~=</math> <math>~ \pi G\rho \biggl[ \frac{(A_1 - A_2)a^2b^2}{ b^2 - a^2} - A_3 c^2\biggr] \, ; </math> [ EFE, Chapter 7, §48, Eq. (34) ] and, <math>~ \Omega_3^2 \biggl\{1 + \biggl[ \frac{a^2b^2}{(a^2 + b^2)^2}\biggr] f^2 \biggr\} </math> <math>~=</math> <math>~ \frac{2\pi G\rho}{ (a^2-b^2) } \biggl[ A_1 a^2 - A_2 b^2 \biggr] \, . </math> [ EFE, Chapter 7, §48, Eq. (33) ]

Ou's (2006) Detailed Force Balance

In a separate accompanying chapter, we have described in detail how Ou(2006) used, essentially, the HSCF technique to solve the detailed force-balance equations. Beginning with the,

Eulerian Representation
of the Euler Equation
as viewed from a Rotating Reference Frame

<math>\biggl[\frac{\partial\vec{v}}{\partial t}\biggr]_{rot} + ({\vec{v}}_{rot}\cdot \nabla) {\vec{v}}_{rot}= - \frac{1}{\rho} \nabla P - \nabla \Phi_\mathrm{grav}

- {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x}) - 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} \, ,</math>

it can be shown that, for the velocity fields associated with all Riemann S-type ellipsoids,

<math>~({\vec{v}}_{rot}\cdot \nabla) {\vec{v}}_{rot}</math>	<math>~=</math>	<math>~ -\nabla \biggl[ \frac{1}{2} \lambda^2(x^2 + y^2) \biggr] \, ; </math>
<math>~- {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x})</math>	<math>~=</math>	<math>~ +\nabla\biggl[\frac{1}{2} \Omega_f^2 (x^2 + y^2) \biggr] \, ; </math>
<math>~- 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} </math>	<math>~=</math>	<math>~ - \nabla\biggl[ \Omega_f \lambda\biggl( \frac{b}{a} x^2 + \frac{a}{b}y^2 \biggr) \biggr] \, . </math>

Hence, within each steady-state configuration the following Bernoulli's function must be uniform in space:

<math>~ H + \Phi_\mathrm{grav} - \frac{1}{2} \Omega_f^2(x^2 + y^2) - \frac{1}{2} \lambda^2(x^2 + y^2) + \Omega_f \lambda \biggl(\frac{b}{a}x^2 + \frac{a}{b}y^2 \biggr) </math>

<math>~=</math>

<math>~ C_B \, , </math>

Ou(2006), p. 550, §2, Eq. (6)

where <math>~C_B</math> is a constant. So, at the surface of the ellipsoid (where the enthalpy H = 0) on each of its three principal axes, the equilibrium conditions demanded by the expression for detailed force balance become, respectively:

1. On the x-axis, where (x, y, z) = (a, 0, 0):
   

   <math>~2\biggl[ \frac{C_B}{a^2} + (\pi G\rho)I_\mathrm{BT} \biggr]</math>

   <math>~=</math>

   <math>~ (2\pi G \rho) A_1 - \Omega_f^2 - \lambda^2 + 2\Omega_f \lambda \biggl(\frac{b}{a} \biggr) </math>

2. On the y-axis, where (x, y, z) = (0, b, 0):
   

   <math>~2\biggl[ \frac{C_B}{a^2} + (\pi G\rho)I_\mathrm{BT} \biggr]</math>

   <math>~=</math>

   <math>~ (2\pi G \rho) A_2 \biggl( \frac{b^2}{a^2}\biggr) - \Omega_f^2 \biggl( \frac{b^2}{a^2} \biggr) - \lambda^2\biggl( \frac{b^2}{a^2} \biggr) + 2\Omega_f \lambda \biggl(\frac{b}{a}\biggr) </math>

3. On the z-axis, where (x, y, z) = (0, 0, c):
   

   <math>~\Rightarrow ~~~ 2 \biggl[ \frac{C_B}{a^2} + (\pi G\rho)I_\mathrm{BT}\biggr]</math>

   <math>~=</math>

   <math>~ (2\pi G \rho) A_3 \biggl( \frac{c^2}{a^2}\biggr) </math>

This third expression can be used to replace the left-hand-side of the first and second expressions. Then via some additional algebraic manipulation, the first and second expressions can be combined to provide the desired solutions for the parameter pair, <math>~(\Omega_f, \lambda)</math>, namely,

<math>~\frac{\Omega_f^2}{(\pi G \rho)}</math>

<math>~=</math>

<math>~\frac{1}{2} \biggl[M + \sqrt{ M^2 - 4N^2} \biggr] \, ,</math>

and

<math>~\frac{\lambda^2}{(\pi G \rho)}</math>

<math>~=</math>

<math>~\frac{1}{2} \biggl[M - \sqrt{ M^2 - 4N^2} \biggr] \, ,</math>

Ou(2006), p. 551, §2, Eqs. (15) & (16)

where,

<math>~M</math>	<math>~\equiv</math>	<math>~ 2\biggl[ A_1 - A_2 \biggl( \frac{b^2}{a^2}\biggr) \biggr]\biggl[ \frac{a^2}{a^2 - b^2} \biggr] \, ,</math>     and,
<math>~N</math>	<math>~\equiv</math>	<math>~ \frac{1}{a b ( a^2 - b^2 )}\biggl[ A_3 ( a^2 - b^2 )c^2 - (A_2 - A_1) a^2 b^2 \biggr] \, . </math>

Hybrid Scheme

In a separate chapter we have detailed the hybrid scheme. For steady-state configurations, the three components of the combined Euler + Continuity equations give,

Hybrid Scheme Summary for Steady-State Configurations <math>~\boldsymbol{\hat{k}:}</math> <math>~ \bold\nabla \cdot (\rho v_z \bold{u}) </math> <math>~=</math> <math>~\bold{\hat{k}} \cdot (\rho \bold{a}) \, ;</math> <math>~\bold{\hat{e}_\varpi:}</math> <math>~ \bold\nabla \cdot (\rho v_\varpi \bold{u}) </math> <math>~=</math> <math>~\bold{\hat{e}}_\varpi \cdot (\rho \bold{a}) + \frac{v_\varphi^2}{\varpi} \, ;</math> <math>~\bold{\hat{e}_\varphi:}</math> <math>~ \bold\nabla \cdot (\rho \varpi v_\varphi \bold{u}) </math> <math>~=</math> <math>~\bold{\hat{e}}_\varphi \cdot (\rho \varpi \bold{a}) \, .</math>

In this context, the vector acceleration that drives the fluid flow is, simply,

<math>~\bold{a}</math>

<math>~=</math>

<math>~-\nabla(H + \Phi_\mathrm{grav} ) \, .</math>

Then, for the velocity flow-patterns in Riemann S-type ellipsoids, we have,

<math>~\nabla \cdot (\rho v_z \bold{u})</math>	<math>~=</math>	<math>~0</math>           (because <math>~v_z = 0</math>);
<math>~\nabla \cdot (\rho v_\varpi \bold{u})</math>	<math>~=</math>	<math>~\frac{\lambda^2}{\varpi^3} \biggl[\frac{a}{b} - \frac{b}{a} \biggr] \biggl\{ y^4 \biggl(\frac{a}{b}\biggr) - x^4 \biggl(\frac{b}{a}\biggr) \biggr\}\rho \, ; </math>
<math>~\nabla \cdot (\rho \varpi v_\varphi \bold{u})</math>	<math>~=</math>	<math>~ 2 \lambda xy \Omega_f \biggl[\frac{a}{b} - \frac{b}{a} \biggr]\rho \, ; </math>
<math>~\varpi v_\varphi</math>	<math>~=</math>	<math>~ - \biggl[ \lambda \biggl(\frac{b}{a}\biggr) - \Omega_f\biggr]x^2 - \biggl[ \lambda \biggl(\frac{a}{b}\biggr) - \Omega_f\biggr]y^2 \, . </math>

Vertical Component:   Given that <math>~\bold{\hat{k}}\cdot (\rho \bold{a}) = 0</math>, we deduce that,

<math>~H_0 </math>

<math>~=</math>

<math>~\pi G \rho c^2 A_3 \, . </math>

Azimuthal Component:   Irrespective of the <math>~(x, y, z)</math> location of each fluid element, this component requires,

<math>~ - a b \lambda \Omega_f </math>

<math>~=</math>

<math>~ \pi G \rho \biggl[ \frac{( A_1 - A_2 )a^2b^2}{b^2 - a^2} - c^2 A_3 \biggr] \, . </math>

Radial Component:   After inserting the "azimuthal component" relation and marching through a fair amount of algebraic manipulation, we find that Irrespective of the <math>~(x, y, z)</math> location of each fluid element, this component requires,

<math>~ \frac{2\pi G \rho }{(a^2 - b^2) } \biggl[ A_1 a^2 - A_2 b^2 \biggr] </math>

<math>~=</math>

<math>~ \biggl[ \lambda^2 + \Omega_f^2\biggr] \, . </math>

Compressible Structures

Here we draw heavily on the published work of Korycansky & Papaloizou (1996, ApJS, 105, 181; hereafter KP96) that we have reviewed in a separate chapter.

Returning to the above-mentioned,

Eulerian Representation
of the Euler Equation
as viewed from a Rotating Reference Frame

<math>\biggl[\frac{\partial\vec{v}}{\partial t}\biggr]_{rot} + ({\vec{v}}_{rot}\cdot \nabla) {\vec{v}}_{rot}= - \frac{1}{\rho} \nabla P - \nabla \Phi_\mathrm{grav}

- {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x}) - 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} \, ,</math>

we next note — as we have done in our broader discussion of the Euler equation — that,

<math> (\vec{v}\cdot\nabla)\vec{v} = \frac{1}{2}\nabla(\vec{v}\cdot\vec{v}) - \vec{v}\times(\nabla\times\vec{v}) = \frac{1}{2}\nabla(v^2) + \vec{\zeta}\times \vec{v} , </math>

where,

<math> \vec\zeta \equiv \nabla\times\vec{v} </math>

is commonly referred to as the vorticity. Making this substitution, but dropping the "rot" subscript, we obtain what is essentially equation (7) of KP96, that is, the,

Euler Equation
written in terms of the Vorticity and
as viewed from a Rotating Reference Frame

<math>\biggl[\frac{\partial\vec{v}}{\partial t}\biggr] + ({\vec{\zeta}}+2{\vec\Omega}_f) \times {\vec{v}}= - \frac{1}{\rho} \nabla P - \nabla \biggl[\Phi + \frac{1}{2}v^2 - \frac{1}{2}|{\vec{\Omega}}_f \times \vec{x}|^2 \biggr]</math> .

Hence, in steady-state, the Euler equation becomes,

<math> \nabla F_B + \vec{A} = 0 , </math>

where, the scalar "Bernoulli" function,

<math> F_B \equiv \frac{1}{2}v^2 + H + \Phi - \frac{1}{2}|\Omega\hat{k} \times \vec{x}|^2 ; </math>

and,

<math> \vec{A} \equiv ({\vec{\zeta}}+2{\vec\Omega}_f) \times {\vec{v}} . </math>

See Also

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation