Editing PGE/RotatingFrame

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<font color="red"><b>
NOTE to Eric Hirschmann &amp; David Neilsen...  
</b></font>
I have moved the earlier contents of this page to a new Wiki location called
[[Apps/RiemannEllipsoidsCompressible|Compressible Riemann Ellipsoids]].

=Rotating Reference Frame=
At times, it can be useful to view the motion of a fluid from a frame of reference that is rotating with a uniform (''i.e.,'' time-independent) angular velocity <math>~\Omega_f</math>.  In order to transform any one of the [[PGE#Principal_Governing_Equations|principal governing equations]] from the inertial reference frame to such a rotating reference frame, we must specify the orientation as well as the magnitude of the angular velocity vector about which the frame is spinning, <math>{\vec\Omega}_f</math>; and the <math>~d/dt</math> operator, which denotes Lagrangian time-differentiation in the inertial frame, must everywhere be replaced as follows:
<div align="center">
<math>
\biggl[\frac{d}{dt} \biggr]_{inertial} \rightarrow \biggl[\frac{d}{dt} \biggr]_{rot} + {\vec{\Omega}}_f \times .
</math>
</div>
Performing this transformation implies, for example, that
<div align="center">
<math>
\vec{v}_{inertial} = \vec{v}_{rot} + {\vec{\Omega}}_f \times \vec{x} ,
</math>
</div>
and,
<div align="center">
<math>
\biggl[ \frac{d\vec{v}}{dt}\biggr]_{inertial} = \biggl[ \frac{d\vec{v}}{dt}\biggr]_{rot} + 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} + {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x}) 
</math>

<math>
= \biggl[ \frac{d\vec{v}}{dt}\biggr]_{rot} + 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} - \frac{1}{2} \nabla \biggl[ |{\vec{\Omega}}_f \times \vec{x}|^2 \biggr] 
</math>
</div>
(If we were to allow <math>{\vec\Omega}_f</math> to be a function of time, an additional term involving the time-derivative of <math>{\vec\Omega}_f</math> also would appear on the right-hand-side of these last expressions; see, for example, Eq.~1D-42 in [[Appendix/References|BT87]].)  Note as well that the relationship between the fluid [[PGE/RotatingFrame#WikiVorticity|vorticity]] in the two frames is,
<div align="center">
<math>
[\vec\zeta]_{inertial} = [\vec\zeta]_{rot} + 2{\vec\Omega}_f .
</math>
</div>


==Continuity Equation (rotating frame)==
Applying these transformations to the standard, inertial-frame representations of the continuity equation presented [[PGE/ConservingMass#Continuity_Equation|elsewhere]], we obtain the:

<div align="center">
<font color="#770000">'''Lagrangian Representation'''</font><br />
of the Continuity Equation <br />
<font color="#770000">'''as viewed from a Rotating Reference Frame'''</font>

<math>\biggl[ \frac{d\rho}{dt} \biggr]_{rot} + \rho \nabla \cdot {\vec{v}}_{rot} = 0</math> ;
</div>


<div align="center">
<font color="#770000">'''Eulerian Representation'''</font><br />
of the Continuity Equation <br />
<font color="#770000">'''as viewed from a Rotating Reference Frame'''</font>

<math>\biggl[ \frac{\partial\rho}{\partial t} \biggr]_{rot} + \nabla \cdot (\rho {\vec{v}}_{rot}) = 0</math> .
</div>


==Euler Equation (rotating frame)==
Applying these transformations to the standard, inertial-frame representations of the Euler equation presented [[PGE/Euler#Euler_Equation|elsewhere]], we obtain the:

<div align="center">
<font color="#770000">'''Lagrangian Representation'''</font><br />
of the Euler Equation <br />
<font color="#770000">'''as viewed from a Rotating Reference Frame'''</font>

<math>\biggl[ \frac{d\vec{v}}{dt}\biggr]_{rot} = - \frac{1}{\rho} \nabla P - \nabla \Phi - 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} - {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x})</math> ;
</div>


<div align="center">
<font color="#770000">'''Eulerian Representation'''</font><br />
of the Euler Equation <br />
<font color="#770000">'''as viewed from a Rotating Reference Frame'''</font>

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


<div align="center">
Euler Equation<br />
written <font color="#770000">'''in terms of the Vorticity'''</font> and<br />
<font color="#770000">'''as viewed from a Rotating Reference Frame'''</font>

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


==Centrifugal and Coriolis Accelerations==

Following along the lines of the discussion presented in Appendix 1.D, &sect;3 of [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], in a rotating reference frame the Lagrangian representation of the Euler equation may be written in the form,
<div align="center">
<math>\biggl[ \frac{d\vec{v}}{dt}\biggr]_{rot} = - \frac{1}{\rho} \nabla P - \nabla \Phi + {\vec{a}}_{fict} </math>,
</div>
where,
<div align="center">
<math>
{\vec{a}}_{fict} \equiv - 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} - {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x}) .
</math>
</div>
So, as viewed from a rotating frame of reference, material moves as if it were subject to two ''fictitious accelerations'' which traditionally are referred to as the,
<div align="center">
<font color="#770000">'''Coriolis Acceleration'''</font> 

<math>
{\vec{a}}_{Coriolis} \equiv - 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} ,
</math>
</div>

(see the related [[PGE/RotatingFrame#WikiCoriolis|Wikipedia discussion]]) and the
<div align="center">
<font color="#770000">'''Centrifugal Acceleration'''</font> 

<math>
{\vec{a}}_{Centrifugal} \equiv - {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x})
= \frac{1}{2} \nabla\biggl[ |{\vec{\Omega}}_f \times \vec{x}|^2  \biggr] 
</math>
</div>
(see the related [[PGE/RotatingFrame#WikiCentrifugal|Wikipedia discussion]]).

==Nonlinear Velocity Cross-Product==

In some contexts &#8212; for example, our discussion of [[Apps/RiemannEllipsoidsCompressible|Riemann ellipsoids]] or the analysis by [[Apps/Korycansky_Papaloizou_1996|Korycansky &amp; Papaloizou (1996)]] of nonaxisymmetric disk structures &#8212; it proves useful to isolate and analyze the term in the "vorticity formulation" of the Euler equation that involves a nonlinear cross-product of the rotating-frame velocity vector, namely,
<div align="center">
<math>
\vec{A} \equiv ({\vec{\zeta}}_{rot}+2{\vec\Omega}_f) \times {\vec{v}}_{rot} .
</math>
</div>

NOTE: To simplify notation, for most of the remainder of this subsection we will drop the subscript "rot" on both the velocity and vorticity vectors.

===Align <math>{\vec\Omega}_f</math> with z-axis===
Without loss of generality we can set <math>{\vec\Omega}_f = \hat{k}\Omega_f</math>, that is, we can align the frame rotation axis with the z-axis of a Cartesian coordinate system.    The Cartesian components of <math>{\vec{A}}</math> are then,
<div align="left">
<math>
\hat{i}: ~~~~~~ A_x = \zeta_y v_z - (\zeta_z + 2\Omega) v_y  ,
</math><br />

<math>
\hat{j}: ~~~~~~ A_y = (\zeta_z + 2\Omega) v_x - \zeta_x v_z  ,
</math><br />

<math>
\hat{k}: ~~~~~~ A_z = \zeta_x v_y - \zeta_y v_x  ,
</math>
</div>
where it is understood that the three Cartesian components of the vorticity vector are,
<div align="center">
<math>
\zeta_x = \biggl[\frac{\partial v_z}{\partial y} - \frac{\partial v_y}{\partial z} \biggr] ,
 ~~~~~~ \zeta_y = \biggl[ \frac{\partial v_x}{\partial z} - \frac{\partial v_z}{\partial x} \biggr] ,
 ~~~~~~ \zeta_z = \biggl[ \frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y} \biggr] .
</math>
</div>
In turn, the curl of <math>\vec{A}</math> has the following three Cartesian components: 
<div align="left">
<math>
\hat{i}: ~~~~~~ [\nabla\times\vec{A}]_x = \frac{\partial}{\partial y}\biggl[ \zeta_x v_y - \zeta_y v_x \biggr] - \frac{\partial}{\partial z}\biggl[ (\zeta_z + 2\Omega) v_x - \zeta_x v_z \biggr],
</math><br />

<math>
\hat{j}: ~~~~~~ [\nabla\times\vec{A}]_y = \frac{\partial}{\partial z}\biggl[ \zeta_y v_z - (\zeta_z + 2\Omega) v_y \biggr] - \frac{\partial}{\partial x}\biggl[ \zeta_x v_y - \zeta_y v_x \biggr] ,
</math><br />

<math>
\hat{k}: ~~~~~~ [\nabla\times\vec{A}]_z = \frac{\partial}{\partial x}\biggl[ (\zeta_z + 2\Omega) v_x - \zeta_x v_z \biggr] - \frac{\partial}{\partial y}\biggl[ \zeta_y v_z - (\zeta_z + 2\Omega) v_y \biggr] .
</math>
</div>

===When <math>v_z = 0</math>===

If we restrict our discussion to configurations that exhibit only planar flows &#8212; that is, systems in which <math>v_z = 0</math> &#8212; then the Cartesian components of <math>{\vec{A}}</math> and <math>\nabla\times\vec{A}</math> simplify somewhat to give, respectively,
<div align="left">
<math>
\hat{i}: ~~~~~~ A_x = - (\zeta_z + 2\Omega) v_y  ,
</math><br />

<math>
\hat{j}: ~~~~~~ A_y = (\zeta_z + 2\Omega) v_x  ,
</math><br />

<math>
\hat{k}: ~~~~~~ A_z = \zeta_x v_y - \zeta_y v_x  ,
</math>
</div>

and,
<div align="left">
<math>
\hat{i}: ~~~~~~ [\nabla\times\vec{A}]_x = \frac{\partial}{\partial y}\biggl[ \zeta_x v_y - \zeta_y v_x \biggr] - \frac{\partial}{\partial z}\biggl[ (\zeta_z + 2\Omega) v_x \biggr],
</math><br />

<math>
\hat{j}: ~~~~~~ [\nabla\times\vec{A}]_y = - \frac{\partial}{\partial z}\biggl[(\zeta_z + 2\Omega) v_y \biggr] - \frac{\partial}{\partial x}\biggl[ \zeta_x v_y - \zeta_y v_x \biggr] ,
</math><br />

<math>
\hat{k}: ~~~~~~ [\nabla\times\vec{A}]_z = \frac{\partial}{\partial x}\biggl[ (\zeta_z + 2\Omega) v_x  \biggr] + \frac{\partial}{\partial y}\biggl[ (\zeta_z + 2\Omega) v_y \biggr] ,
</math>
</div>
where, in this case, the three Cartesian components of the vorticity vector are,
<div align="center">
<math>
\zeta_x = - \frac{\partial v_y}{\partial z} ,
 ~~~~~~ \zeta_y = \frac{\partial v_x}{\partial z} ,
 ~~~~~~ \zeta_z = \biggl[ \frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y} \biggr] .
</math>
</div>

=Related Discussions=
* <span id="WikiVorticity">Wikipedia discussion of [http://en.wikipedia.org/wiki/Vorticity vorticity].</span>
* <span id="WikiCoriolis">Wikipedia discussion of [http://en.wikipedia.org/wiki/Coriolis_effect#Formula Coriolis Effect].</span>
* <span id="WikiCentrifugal">Wikipedia discussion of [http://en.wikipedia.org/wiki/Centrifugal_force#Derivation_using_vectors Centrifugal acceleration].</span>

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