Revision as of 22:12, 4 September 2020

Implications of Hybrid Scheme

Background

Key H_Book Chapters

[Ref01] Inertial-Frame Euler Equation

[Ref02] Traditional Description of Rotating Reference Frame

[Ref03] Hybrid Advection Scheme

[Ref04] Riemann S-type Ellipsoids

[Ref05] Korycansky and Papaloizou (1996)

Principal Governing Equations

Quoting from [Ref01] … Among the principal governing equations we have included the inertial-frame,

Lagrangian Representation
of the Euler Equation,

<math>\frac{d\vec{v}}{dt} = - \frac{1}{\rho} \nabla P - \nabla \Phi</math>

[EFE], Chap. 2, §11, p. 20, Eq. (38)
[BLRY07], p. 13, Eq. (1.55)

Shifting into a rotating frame characterized by the angular velocity vector,

<math>~\vec{\Omega}_f \equiv \hat\mathbf{k} \Omega_f \, ,</math>

and applying the operations that are specified in the first few subsections of [Ref02], we recognize the following relationships …

<math>~\vec{v}_\mathrm{inertial}</math>	<math>~=</math>	<math>~\vec{v}_\mathrm{rot} + {\vec\Omega}_f \times \vec{x} \, ,</math>
<math>~\biggl[ \frac{d \vec{v}}{dt} \biggr]_\mathrm{inertial}</math>	<math>~=</math>	<math>~ \biggl[ \frac{d \vec{v}}{dt} \biggr]_\mathrm{rot} + 2{\vec\Omega}_f \times {\vec{v}}_\mathrm{rot} + {\vec\Omega}_f \times ({\vec\Omega}_f \times \vec{x}) </math>
	<math>~=</math>	<math>~ \biggl[ \frac{d \vec{v}}{dt} \biggr]_\mathrm{rot} + 2{\vec\Omega}_f \times {\vec{v}}_\mathrm{rot} - \frac{1}{2} \nabla \| {\vec\Omega}_f \times \vec{x}\|^2 </math>
	<math>~=</math>	<math>~ \biggl[ \frac{\partial \vec{v}}{\partial t} \biggr]_\mathrm{rot} + ({\vec{v}}_\mathrm{rot} \cdot \nabla){\vec{v}}_\mathrm{rot} + 2{\vec\Omega}_f \times {\vec{v}}_\mathrm{rot} - \frac{1}{2} \nabla \| {\vec\Omega}_f \times \vec{x}\|^2 \, .</math>

Making this substitution on the left-hand-side of the above-specified "Lagrangian Representation of the Euler Equation," we obtain what we have referred to also in [Ref02] as the,

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

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

This form of the Euler equation also appears early in [Ref05], where we set up a discussion of the paper by Korycansky & Papaloizou (1996, ApJS, 105, 181; hereafter KP96). But, for now, let's back up a couple of steps and retain the total time derivative on the left-hand-side. That is, let's select as the foundation expression the,

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

<math>~\biggl[ \frac{d \vec{v}}{dt} \biggr]_\mathrm{rot} </math>	<math>~=</math>	<math>~- \frac{1}{\rho} \nabla P - \nabla \Phi - 2{\vec\Omega}_f \times {\vec{v}}_\mathrm{rot} - {\vec\Omega}_f \times ({\vec\Omega}_f \times \vec{x}) \, ,</math>
[EFE], Chap. 2, §12, p. 25, Eq. (62)

which also serves as the foundation of most of our [Ref03] discussions.

Exercising the Hybrid Scheme

Focus on Tracking Angular Momentum

Let's begin by using <math>~\mathbf{u'}</math>, instead of <math>~{\vec{v}}_\mathrm{rot}</math>, to represent the fluid velocity vector as viewed from the rotating frame of reference. Our foundation expression becomes,

<math>~- \frac{1}{\rho} \nabla P - \nabla \Phi - 2{\vec\Omega}_f \times \bold{u}' - {\vec\Omega}_f \times ({\vec\Omega}_f \times \vec{x}) \, ,</math>

where we appreciate that we can move from the Lagrangian to an Eulerian representation by employing the operator substitution,

<math>~\rightarrow</math>

<math>~\frac{\partial}{\partial t} + \mathbf{u'} \cdot \nabla </math>

Next, using [Ref03] as a guide, let's focus on tracking angular momentum. We need to break the vector momentum equation, as well as the velocity vectors, into their <math>~(\bold{\hat{e}}_\varpi, \bold{\hat{e}}_\varphi, \bold{\hat{k}})</math> components.

NOTE: For the time being, we will write the velocity vector in terms of generic components, namely,

<math>~\bold{u}' = \bold{\hat{e}}_\varpi u'_\varpi + \bold{\hat{e}}_\varphi u'_\varphi + \bold{\hat{k}}u'_z \, .</math>

But, eventually, we want to explicitly insert the rotating-frame velocity that underpins the equilibrium properties of Riemann S-type ellipsoids. In Chap. 7, §47, Eq. 1 (p. 130) of [EFE], this is given in Cartesian coordinates, so we will need to convert his expressions to the equivalent cylindrical-coordinate components.

The time-derivative on the left-hand-side of our foundation expression becomes,

<math> \frac{d\mathbf{u'}}{dt} </math>	<math>~=~</math>	<math> \frac{d}{dt} [ \mathbf{\hat{e}}_\varpi u'_\varpi + \mathbf{\hat{e}}_\varphi u'_\varphi + \mathbf{\hat{k}} u'_z ] </math>
	<math>~=~</math>	<math> \mathbf{\hat{e}}_\varpi \frac{d u'_\varpi}{dt} + \mathbf{\hat{e}}_\varphi \frac{d u'_\varphi}{dt} + \mathbf{\hat{k}} \frac{d u'_z}{dt} + ( u'_\varpi) \frac{d}{dt}\mathbf{\hat{e}}_\varpi + ( u'_\varphi) \frac{d}{dt}\mathbf{\hat{e}}_\varphi </math>
	<math>~=~</math>	<math> \mathbf{\hat{e}}_\varpi \frac{d u'_\varpi}{dt} + \mathbf{\hat{e}}_\varphi \frac{d u'_\varphi}{dt} + \mathbf{\hat{k}} \frac{d u'_z}{dt} + \mathbf{\hat{e}}_\varphi(u'_\varpi) \frac{u'_\varphi}{\varpi} - \mathbf{\hat{e}}_\varpi(u'_\varphi) \frac{u'_\varphi}{\varpi} \, . </math>

We also recognize that, when expressed in cylindrical coordinates,

<math> ~{\vec{\Omega}}_f \times \vec{x} </math>	<math>~=~</math>	<math> {\hat\mathbf{k}} \Omega_f\times (\mathbf{\hat{e}}_\varpi \varpi + \mathbf{\hat{k}}z) = \mathbf{\hat{e}}_\varphi \Omega_f \varpi \, , </math>
<math> {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x}) </math>	<math>~=~</math>	<math> \hat{\mathbf{k}} \Omega_f \times ( \mathbf{\hat{e}}_\varphi \Omega_f \varpi ) = - \mathbf{\hat{e}}_\varpi \Omega_f^2 \varpi \, , </math>
<math> {\vec{\Omega}}_f \times {\mathbf{u'}} </math>	<math>~=~</math>	<math> {\hat\mathbf{k}} \Omega_f\times (\mathbf{\hat{e}}_\varpi u'_\varpi + \mathbf{\hat{e}}_\varphi u'_\varphi + \mathbf{\hat{k}}u'_z) = \mathbf{\hat{e}}_\varphi \Omega_f u'_\varpi - \mathbf{\hat{e}}_\varpi \Omega_f u'_\varphi \, , </math>
<math> {\vec{v}}_\mathrm{inertial} </math>	<math>~=~</math>	<math> \mathbf{u'} + \mathbf{\hat{e}}_\varphi \Omega_f \varpi \, . </math>

The set of scalar momentum-component equations is obtained by "dotting" each unit vector into the vector equation.

<math>\mathbf{\hat{e}}_\varpi:</math>	<math>~\frac{d u'_\varpi}{dt} - \frac{(u'_\varphi)^2}{\varpi} </math>	<math>~=</math>	<math>~- \mathbf{\hat{e}}_\varpi \cdot \frac{\nabla P}{\rho} - \mathbf{\hat{e}}_\varpi \cdot \nabla \Phi + 2 \biggl[ \Omega_f u'_\varphi \biggr] + \Omega_f^2 \varpi </math>
<math>~\Rightarrow ~~~ \frac{d u'_\varpi}{dt} </math>		<math>~=</math>	<math>~- \mathbf{\hat{e}}_\varpi \cdot \frac{\nabla P}{\rho} - \mathbf{\hat{e}}_\varpi \cdot \nabla \Phi + \frac{1}{\varpi} \biggl[ (u'_\varphi)^2 + 2 \Omega_f u'_\varphi \varpi + \Omega_f^2 \varpi^2 \biggr]</math>
		<math>~=</math>	<math>~ - \mathbf{\hat{e}}_\varpi \cdot \frac{\nabla P}{\rho} - \mathbf{\hat{e}}_\varpi \cdot \nabla \Phi + \frac{1}{\varpi} (u'_\varphi + \Omega_f \varpi)^2 \, ; </math>
<math>\mathbf{\hat{e}}_\varphi:</math>	<math>~\frac{d u'_\varphi}{dt} + \frac{u'_\varpi u'_\varphi}{\varpi} </math>	<math>~=</math>	<math>~- \mathbf{\hat{e}}_\varphi \cdot \frac{\nabla P}{\rho} - \mathbf{\hat{e}}_\varphi \cdot \nabla \Phi - 2\biggl[ \Omega_f u'_\varpi \biggr] </math>
(mult. thru by ϖ) <math>~\Rightarrow ~~~\frac{d (\varpi u'_\varphi )}{dt} </math>		<math>~=</math>	<math>~- \mathbf{\hat{e}}_\varphi \cdot \frac{\varpi \nabla P}{\rho} - \mathbf{\hat{e}}_\varphi \cdot \varpi \nabla \Phi - 2 \Omega_f \varpi u'_\varpi \, ; </math>
<math>\mathbf{\hat{k}}:</math>	<math>~\frac{d u'_z}{dt} </math>	<math>~=</math>	<math>~- \mathbf{\hat{k}} \cdot \frac{\nabla P }{\rho} - \mathbf{\hat{k}} \cdot \nabla \Phi \, . </math>

Now, recalling that <math>~\mathbf{u'} = (\mathbf{v} - \mathbf{\hat{e}}_\varphi \varpi \Omega_f)</math>, let's make the substitutions …

<math>~u'_\varpi \rightarrow v_\varpi \, ,</math>

<math>~u'_\varphi \rightarrow (v_\varphi - \varpi\Omega_f) \, ,</math> and,

<math>~u'_z \rightarrow v_z \, .</math>

This mapping gives,

<math>\mathbf{\hat{e}}_\varphi:</math>	<math>~\frac{d [\varpi v_\varphi - \varpi^2 \Omega_f]}{dt} </math>	<math>~=</math>	<math>~- \mathbf{\hat{e}}_\varphi \cdot \frac{\varpi \nabla P}{\rho} - \mathbf{\hat{e}}_\varphi \cdot \varpi \nabla \Phi - 2 \Omega_f \varpi v_\varpi \, ; </math>
<math>~\Rightarrow ~~~ \frac{d (\varpi v_\varphi )}{dt} </math>		<math>~=</math>	<math>~- \mathbf{\hat{e}}_\varphi \cdot \frac{\varpi \nabla P}{\rho} - \mathbf{\hat{e}}_\varphi \cdot \varpi \nabla \Phi \, ; </math>
<math>~\Rightarrow ~~~ \frac{1}{\varpi} ~\frac{d (\varpi v_\varphi )}{dt} </math>		<math>~=</math>	<math>~- \mathbf{\hat{e}}_\varphi \cdot \biggl[ \frac{\nabla P}{\rho} + \nabla \Phi \biggr] \, ; </math>
<math>\mathbf{\hat{k}}:</math>	<math>~\frac{d v_z}{dt} </math>	<math>~=</math>	<math>~- \mathbf{\hat{k}} \cdot \biggl[ \frac{\nabla P }{\rho} + \nabla \Phi \biggr] \, . </math>
<math>\mathbf{\hat{e}}_\varpi:</math>	<math>~\frac{d v_\varpi}{dt} </math>	<math>~=</math>	<math>~ - \mathbf{\hat{e}}_\varpi \cdot \biggl[ \frac{\nabla P}{\rho} + \nabla \Phi \biggr] + \frac{v_\varphi^2}{\varpi} \, ; </math>

Steady-State Velocity Field for Jacobi Ellipsoids

In steady-state, the (Lagrangian time-derivative) operator on the left-hand-side of all three component equations maps to the following operator:

<math>~\mathbf{u'} \cdot \nabla</math>	<math>~=</math>	<math>~\sum_{i=1}^3 u'_i \frac{\partial}{\partial x_i} \, ,</math>	(in Cartesian coordinates);
<math>~\mathbf{u'} \cdot \nabla</math>	<math>~=</math>	<math>~ u'_\varpi \frac{\partial}{\partial \varpi} + \frac{u'_\varphi}{\varpi} \frac{\partial}{\partial \varphi} + u'_z \frac{\partial}{\partial z} \, ,</math>	(in cylindrical coordinates);

We know, as well, that,

<math>~u'_\varpi = u'_x \cos\varphi + u'_y \sin\varphi \, ,</math>

and,

<math>~u'_\varphi = u'_y \cos\varphi - u'_x \sin\varphi \, .</math>

Hence, the cylindrical-coordinate-based operator may be rewritten as,

<math>~\mathbf{u'} \cdot \nabla</math>

<math>~ ( u'_x \cos\varphi + u'_y \sin\varphi ) \frac{\partial}{\partial \varpi} + ( u'_y \cos\varphi - u'_x \sin\varphi )\frac{1}{\varpi} \frac{\partial}{\partial \varphi} + u'_z \frac{\partial}{\partial z} \, .</math>

Drawing from [ Ref04 ] … As Ou(2006) has pointed out, the velocity field of a Riemann S-type ellipsoid as viewed from a frame rotating with angular velocity <math>~{\vec{\Omega}}_f = \boldsymbol{\hat{k}} \Omega_f</math> takes the following form:

<math>~{\mathbf{u'}}</math>

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

Ou(2006), p. 550, §2, Eq. (3)

where <math>~\lambda</math> is a constant that determines the magnitude of the internal motion of the fluid, and the origin of the x-y coordinate system is at the center of the ellipsoid. This velocity field, <math>~\mathbf{u'}</math>, is designed so that velocity vectors everywhere are always aligned with elliptical stream lines by demanding that they be tangent to the equi-effective-potential contours, which are concentric ellipses. Hence, for Riemann S-type ellipsoids, we have,

<math>~u'_x = \lambda\biggl(\frac{a}{b}\biggr)y = \lambda\biggl(\frac{a}{b}\biggr)\varpi \sin\varphi \, ;</math>

<math>~u'_y = -\lambda\biggl(\frac{b}{a}\biggr)x = -\lambda\biggl(\frac{b}{a}\biggr)\varpi \cos\varphi \, ;</math>

So, for the velocity flow that underpins Riemann S-type ellipsoids, the cylindrical-coordinate-based operator is

<math>~\mathbf{u'} \cdot \nabla</math>	<math>~=</math>	<math>~ \biggl[ \lambda\biggl(\frac{a}{b}\biggr)\varpi \sin\varphi \cos\varphi -\lambda\biggl(\frac{b}{a}\biggr)\varpi \cos\varphi \sin\varphi \biggr] \frac{\partial}{\partial \varpi} + \biggl[ -\lambda\biggl(\frac{b}{a}\biggr)\varpi \cos\varphi \cos\varphi - \lambda\biggl(\frac{a}{b}\biggr)\varpi \sin\varphi \sin\varphi \biggr] \frac{1}{\varpi} \frac{\partial}{\partial \varphi} </math>
	<math>~=</math>	<math>~ \biggl[ \biggl(\frac{a}{b}\biggr) - \biggl(\frac{b}{a}\biggr) \biggr]\lambda \varpi \sin\varphi \cos\varphi \frac{\partial}{\partial \varpi} - \biggl[ \biggl(\frac{b}{a}\biggr) \cos^2\varphi + \biggl(\frac{a}{b}\biggr) \sin^2\varphi \biggr]\lambda \frac{\partial}{\partial \varphi} \, .</math>

And, given that,

<math>~\mathbf{\hat{e}}\Omega_f \varpi</math>

<math>~ \Omega_f \varpi \biggl[ \boldsymbol{\hat{\jmath}} \cos\varphi - \boldsymbol{\hat{\imath}} \sin\varphi \biggr] \, , </math>

the inertial-frame velocity components are,

<math>~ v_x = \lambda\biggl(\frac{a}{b}\biggr)\varpi \sin\varphi - \Omega_f \varpi \sin\varphi = \biggl[ \lambda\biggl(\frac{a}{b}\biggr) - \Omega_f \biggr]\varpi\sin\varphi \, ;</math>

<math>~ v_y = -\lambda\biggl(\frac{b}{a}\biggr) \varpi\cos\varphi + \Omega_f\varpi \cos\varphi = \biggl[\Omega_f -\lambda\biggl(\frac{b}{a}\biggr) \biggr]\varpi\cos\varphi \, ;</math>

That is,

<math>~v_\varpi = v_x \cos\varphi + v_y \sin\varphi</math>	<math>~=</math>	<math>~ \biggl[ \lambda\biggl(\frac{a}{b}\biggr) - \Omega_f \biggr]\varpi\sin\varphi \cos\varphi + \biggl[\Omega_f -\lambda\biggl(\frac{b}{a}\biggr) \biggr]\varpi\cos\varphi \sin\varphi </math>
	<math>~=</math>	<math>~ \biggl[ \biggl(\frac{a}{b}\biggr) - \biggl(\frac{b}{a}\biggr) \biggr]\lambda \varpi\sin\varphi \cos\varphi \, ; </math>
<math>~v_\varphi = v_y \cos\varphi - v_x \sin\varphi</math>	<math>~=</math>	<math>~ \biggl[\Omega_f -\lambda\biggl(\frac{b}{a}\biggr) \biggr]\varpi\cos^2\varphi - \biggl[ \lambda\biggl(\frac{a}{b}\biggr) - \Omega_f \biggr]\varpi\sin^2\varphi </math>
	<math>~=</math>	<math>~ \Omega_f \varpi -\lambda \varpi \biggl[ \biggl(\frac{b}{a}\biggr) \cos^2\varphi +\biggl(\frac{a}{b}\biggr) \sin^2\varphi \biggr] \, . </math>

Note, as well, that,

<math>~\frac{v_\varphi^2}{\varpi}</math>	<math>~=</math>	<math>~ \frac{1}{\varpi} \biggl\{ \Omega_f \varpi -\lambda \varpi \biggl[ \biggl(\frac{b}{a}\biggr) \cos^2\varphi +\biggl(\frac{a}{b}\biggr) \sin^2\varphi \biggr] \biggr\}^2 </math>
	<math>~=</math>	<math>~ \varpi \biggl\{ \Omega_f^2 - 2\lambda \Omega_f \biggl[ \biggl(\frac{b}{a}\biggr) \cos^2\varphi +\biggl(\frac{a}{b}\biggr) \sin^2\varphi \biggr] + \lambda^2 \biggl[ \biggl(\frac{b}{a}\biggr) \cos^2\varphi +\biggl(\frac{a}{b}\biggr) \sin^2\varphi \biggr]^2 \biggr\} \, . </math>

Finally, then, we find that the left-hand-side of the momentum-component expressions are,

<math>\mathbf{\hat{k}}:</math>	<math>~\frac{d v_z}{dt} </math>	<math>~=</math>	<math>~0 \, ; </math>
<math>\mathbf{\hat{e}}_\varpi:</math>	<math>~\frac{d v_\varpi}{dt} </math>	<math>~=</math>	<math>~ \biggl\{ \biggl[ \biggl(\frac{a}{b}\biggr) - \biggl(\frac{b}{a}\biggr) \biggr]\lambda \varpi \sin\varphi \cos\varphi \frac{\partial}{\partial \varpi} - \biggl[ \biggl(\frac{b}{a}\biggr) \cos^2\varphi + \biggl(\frac{a}{b}\biggr) \sin^2\varphi \biggr]\lambda \frac{\partial}{\partial \varphi} \biggr\}\biggl[ \biggl(\frac{a}{b}\biggr) - \biggl(\frac{b}{a}\biggr) \biggr]\lambda \varpi\sin\varphi \cos\varphi </math>
		<math>~=</math>	<math>~ \lambda \biggl[ \biggl(\frac{a}{b}\biggr) - \biggl(\frac{b}{a}\biggr) \biggr]\biggl\{ \biggl[ \biggl(\frac{a}{b}\biggr) - \biggl(\frac{b}{a}\biggr) \biggr]\lambda \varpi \sin\varphi \cos\varphi \frac{\partial}{\partial \varpi} \biggl[ \varpi\sin\varphi \cos\varphi \biggr] - \biggl[ \biggl(\frac{b}{a}\biggr) \cos^2\varphi + \biggl(\frac{a}{b}\biggr) \sin^2\varphi \biggr]\lambda \frac{\partial}{\partial \varphi}\biggl[ \varpi\sin\varphi \cos\varphi \biggr] \biggr\} </math>
		<math>~=</math>	<math>~ \lambda \biggl[ \biggl(\frac{a}{b}\biggr) - \biggl(\frac{b}{a}\biggr) \biggr]\biggl\{ \biggl[ \biggl(\frac{a}{b}\biggr) - \biggl(\frac{b}{a}\biggr) \biggr]\lambda \varpi \sin^2\varphi \cos^2\varphi + \biggl[ \biggl(\frac{b}{a}\biggr) \cos^2\varphi + \biggl(\frac{a}{b}\biggr) \sin^2\varphi \biggr] \lambda\varpi \biggl[ \sin^2\varphi - \cos^2\varphi \biggr] \biggr\} </math>
		<math>~=</math>	<math>~ \lambda^2 \varpi \biggl( \frac{a}{b} - \frac{b}{a} \biggr) \biggl[ \biggl( \frac{a}{b}\biggr) \sin^4\varphi - \biggl(\frac{b}{a}\biggr) \cos^4\varphi \biggr] \, ; </math>
<math>\mathbf{\hat{e}}_\varphi:</math>	<math>~\frac{1}{\varpi} ~\frac{d (\varpi v_\varphi )}{dt} </math>	<math>~=</math>	<math>~ \frac{1}{\varpi}\biggl\{ \biggl[ \biggl(\frac{a}{b}\biggr) - \biggl(\frac{b}{a}\biggr) \biggr]\lambda \varpi \sin\varphi \cos\varphi \frac{\partial}{\partial \varpi} - \biggl[ \biggl(\frac{b}{a}\biggr) \cos^2\varphi + \biggl(\frac{a}{b}\biggr) \sin^2\varphi \biggr]\lambda \frac{\partial}{\partial \varphi} \biggr\} \biggl\{ \biggl[\Omega_f -\lambda\biggl(\frac{b}{a}\biggr) \biggr]\varpi^2\cos^2\varphi - \biggl[ \lambda\biggl(\frac{a}{b}\biggr) - \Omega_f \biggr]\varpi^2\sin^2\varphi \biggr\} </math>
		<math>~=</math>	<math>~ \biggl[ \biggl(\frac{a}{b}\biggr) - \biggl(\frac{b}{a}\biggr) \biggr] 2\varpi \lambda \sin\varphi \cos\varphi \biggl\{ \biggl[\Omega_f -\lambda\biggl(\frac{b}{a}\biggr) \biggr]\cos^2\varphi - \biggl[ \lambda\biggl(\frac{a}{b}\biggr) - \Omega_f \biggr]\sin^2\varphi \biggr\} </math>
			<math>~+ \biggl[ \biggl(\frac{b}{a}\biggr) \cos^2\varphi + \biggl(\frac{a}{b}\biggr) \sin^2\varphi \biggr] \biggl[ \frac{a}{b} - \frac{b}{a} \biggr] 2\varpi \lambda^2 \sin\varphi \cos\varphi </math>
		<math>~=</math>	<math>~ \biggl( \frac{a}{b} - \frac{b}{a} \biggr) 2\varpi \lambda \sin\varphi \cos\varphi \biggl\{ \biggl[\Omega_f -\lambda\biggl(\frac{b}{a}\biggr) \biggr]\cos^2\varphi - \biggl[ \lambda\biggl(\frac{a}{b}\biggr) - \Omega_f \biggr]\sin^2\varphi +2\lambda\biggl[ \biggl(\frac{b}{a}\biggr) \cos^2\varphi + \biggl(\frac{a}{b}\biggr) \sin^2\varphi \biggr] \biggr\} </math>
		<math>~=</math>	<math>~ \biggl( \frac{a}{b} - \frac{b}{a} \biggr) 2\varpi \lambda \sin\varphi \cos\varphi \biggl\{ \biggl[\Omega_f +\lambda\biggl(\frac{b}{a}\biggr)\biggr]\cos^2\varphi + \biggl[ \Omega_f + \lambda\biggl( \frac{a}{b}\biggr) \biggr]\sin^2\varphi \biggr\} \, . </math>

Try Again

An example equation of motion is,

<math>~\bold{a} = - \frac{1}{\rho}\bold\nabla P - \bold\nabla \Phi_\mathrm{grav} \, .</math>

Here we will focus only on the left-hand-side of this equation, examining various ways the vector acceleration may be mathematically expressed. We will consider, for example, building a model in a curvilinear (cylindrical), rather than a Cartesian, coordinate base; and viewing a model's evolution in a rotating, rather than inertial, frame of reference.

Inertial Frame

As viewed from a cylindrical-coordinate-based <math>~(\varpi, \varphi, z)</math> inertial reference frame, we are interested in specifying the location,

<math>~\bold{x} = \mathbf{\hat{e}}_\varpi \varpi + \bold{\hat{k}} z \, ,</math>

[BT87], p. 646, Appendix §1.B.2, Eq. (1B-18)

of a Lagrangian fluid element at time <math>~t = 0</math> — hereafter denoted by the subscript, <math>~0</math> — as well as at later times. Although the position vector, <math>~\bold{x}</math>, does not explicitly display a dependence on the azimuthal coordinate angle, <math>~\varphi</math>, it is important to realize that the orientation in space of the unit vector, <math>~\bold{\hat{e}}_\varpi</math>, does depend on the value of this coordinate angle.

At any point in time, the instantaneous velocity of this Lagrangian fluid element will correspond precisely with the (total) time-derivative of its instantaneous position vector, that is,

<math>~\bold{v}</math>	<math>~\equiv</math>	<math>~\frac{d\bold{x}}{dt}</math>	<math>~=</math>	<math>~\bold{\hat{e}}_\varpi \frac{d\varpi}{dt} + \bold{\hat{k}} \frac{dz}{dt} + \varpi \frac{d \bold{\hat{e}}_\varpi}{dt}</math>
			<math>~=</math>	<math>~\bold{\hat{e}}_\varpi \frac{d\varpi}{dt} + \bold{\hat{k}} \frac{dz}{dt} + \varpi \biggl[ \bold{\hat{e}}_\varphi \frac{d\varphi}{dt} \bigg] \, .</math>
[BT87], p. 647, Appendix §1.B.2, Eq. (1B-23)

In carrying out this time differentiation, the last term on the right-hand-side accounts for the aforementioned dependence of <math>~\bold{\hat{e}}_\varpi</math> on <math>~\varphi</math>. Similarly, the following component breakdown of the Lagrangian fluid element's acceleration takes into account the dependence of <math>~\bold{\hat{e}}_\varphi</math> on <math>~\varphi</math>:

<math>~\bold{a}</math>	<math>~\equiv</math>	<math>~\frac{d\bold{v}}{dt}</math>	<math>~=</math>	<math>~ \bold{\hat{e}}_\varpi \frac{d^2\varpi}{dt^2} + \bold{\hat{k}} \frac{d^2z}{dt^2} + \bold{\hat{e}}_\varphi \biggl[\frac{d\varpi}{dt} \cdot \frac{d\varphi}{dt} + \varpi \frac{d^2\varphi}{dt^2}\biggr] + \varpi \frac{d\varphi}{dt} \biggl[ \frac{d\bold{\hat{e}}_\varphi}{dt} \biggr] + \frac{d\varpi}{dt} \biggl[ \frac{d\bold{\hat{e}}_\varpi}{dt} \biggr] </math>
			<math>~=</math>	<math>~ \bold{\hat{e}}_\varpi \frac{d^2\varpi}{dt^2} + \bold{\hat{k}} \frac{d^2z}{dt^2} + \bold{\hat{e}}_\varphi \biggl[\frac{d\varpi}{dt} \cdot \frac{d\varphi}{dt} + \varpi \frac{d^2\varphi}{dt^2}\biggr] + \varpi \frac{d\varphi}{dt} \biggl[- \bold{\hat{e}}_\varpi \frac{d\varphi}{dt} \biggr] + \frac{d\varpi}{dt} \biggl[ \bold{\hat{e}}_\varphi \frac{d\varphi}{dt} \biggr] </math>
			<math>~=</math>	<math>~ \bold{\hat{e}}_\varpi \biggl[\frac{d^2\varpi}{dt^2} - \varpi \biggl(\frac{d\varphi}{dt}\biggr)^2 \biggr] + \bold{\hat{e}}_\varphi \biggl[ 2 \biggl( \frac{d\varpi}{dt} \cdot \frac{d\varphi}{dt} \biggr) + \varpi \frac{d^2\varphi}{dt^2}\biggr] + \bold{\hat{k}} \frac{d^2z}{dt^2} \, . </math>
[BT87], p. 647, Appendix §1.B.2, Eq. (1B-24)

Let's rewrite the velocity vector as,

<math>~\bold{\hat{e}}_\varpi \dot\varpi + \bold{\hat{e}}_\varphi \varpi \dot\varphi + \bold{\hat{k}} \dot{z} \, ,</math>

and (the second line of) this acceleration expression as,

<math>~~\bold{a} \equiv \frac{d\bold{v}}{dt} = \bold{\hat{e}}_\varpi \frac{d \dot\varpi}{dt} + \bold{\hat{e}}_\varphi \frac{d}{dt}\biggl[\varpi \dot\varphi \biggr] + \bold{\hat{k}} \frac{d \dot{z}}{dt} + \underbrace{ \dot\varpi \biggl[ \bold{\hat{e}}_\varphi \frac{d\varphi}{dt} \biggr] - \varpi \dot\varphi \biggl[\bold{\hat{e}}_\varpi \frac{d\varphi}{dt} \biggr] }_\text{curvature terms}\, . </math>

Now, if <math>~\bold{B}</math> is a vector quantity that characterizes some property of a fluid element — such as momentum density, velocity, or vorticity — the difference between the Lagrangian and Eulerian time-derivatives of that vector quantity is given by the expression,

<math>~\frac{d\bold{B}}{dt} - \frac{\partial \bold{B}}{\partial t}</math>

<math>~(\bold{v} \cdot \bold\nabla)\bold{B} \, ,</math>

where the various elements of this right-hand-side mathematical operator can be obtained by replacing <math>~\bold{A}</math> with <math>~\bold{v}</math> in the so-called convective operator.

Convective Operator in Cylindrical Coordinates

<math>~(\bold{A} \cdot \bold\nabla) \bold{B}</math>	<math>~=</math>	<math>~ \bold{\hat{e}}_\varpi \biggl[ A_\varpi \frac{\partial B_\varpi}{\partial \varpi} + \frac{A_\varphi }{\varpi}\frac{\partial B_\varpi}{\partial \varphi} + A_z \frac{\partial B_\varpi}{\partial z} - \frac{A_\varphi B_\varphi}{\varpi} \biggr] </math>
		<math>~ + \bold{\hat{e}}_\varphi \biggl[ A_\varpi \frac{\partial B_\varphi}{\partial \varpi} + \frac{A_\varphi }{\varpi}\frac{\partial B_\varphi}{\partial \varphi} + A_z \frac{\partial B_\varphi}{\partial z} + \frac{A_\varphi B_\varpi}{\varpi} \biggr] </math>
		<math>~ + \bold{\hat{e}}_z \biggl[ A_\varpi \frac{\partial B_z}{\partial \varpi} + \frac{A_\varphi }{\varpi}\frac{\partial B_z}{\partial \varphi} + A_z \frac{\partial B_z}{\partial z} \biggr] \, . </math>
[BT87], p. 651, Appendix §1.B.3, Eq. (1B-54)

We will adopt the following, more compact notation:

<math>~(\bold{A} \cdot \bold\nabla) \bold{B}</math>

<math>~ \bold{\hat{e}}_\varpi \biggl[ \mathcal{L}_A B_\varpi- \frac{A_\varphi B_\varphi}{\varpi} \biggr] + \bold{\hat{e}}_\varphi \biggl[ \mathcal{L}_A B_\varphi + \frac{A_\varphi B_\varpi}{\varpi} \biggr] + \bold{\hat{e}}_z \biggl[ \mathcal{L}_A B_z \biggr] \, , </math>

where the operator,

<math>~\mathcal{L}_A</math>

<math>~\equiv</math>

<math>~ \biggl[ A_\varpi \frac{\partial }{\partial \varpi} + \frac{A_\varphi }{\varpi}\frac{\partial }{\partial \varphi} + A_z \frac{\partial }{\partial z} \biggr] \, . </math>

In particular, if we are examining the behavior of the fluid velocity <math>~(\bold{B} \rightarrow \bold{v} )</math>, we find that,

<math>~\frac{d\bold{v}}{dt} - \frac{\partial \bold{v}}{\partial t}</math>		<math>~=~(\bold{v} \cdot \bold\nabla)\bold{v} </math>
		<math>~=~ \bold{\hat{e}}_\varpi \biggl[ \mathcal{L}_v \dot\varpi \biggr] + \bold{\hat{e}}_\varphi \biggl[ \mathcal{L}_v (\varpi \dot\varphi) \biggr] + \bold{\hat{e}}_z \biggl[ \mathcal{L}_v \dot{z} \biggr] ~+ \underbrace{\bold{\hat{e}}_\varphi (\dot\varphi \dot\varpi ) -~ \bold{\hat{e}}_\varpi ( \varpi {\dot\varphi}^2 ) }_\text{curvature terms} \, , </math>

where the operator,

<math>~\mathcal{L}_v</math>

<math>~\equiv</math>

<math>~ \biggl[ v_\varpi \frac{\partial }{\partial \varpi} + \frac{v_\varphi }{\varpi}\frac{\partial }{\partial \varphi} + v_z \frac{\partial }{\partial z} \biggr] \, . </math>

Notice that the pair of "curvature terms" that appear in this expression are identical to the pair of curvature terms that appear in the acceleration expression, above. We conclude, therefore, that for each of the three separate (cylindrical-coordinate-based) components of the vector acceleration, the relationship between the Lagrangian (total) and Eulerian (partial) time derivative is, respectively,

<math>~\bold{\hat{e}}_\varpi</math>:	<math>~\frac{d\dot\varpi}{dt} - \varpi {\dot\varphi}^2</math>	<math>~=</math>	<math>~ \frac{\partial \dot\varpi}{\partial t} + \biggl[\mathcal{L}_v \dot\varpi \biggr] - \varpi {\dot\varphi}^2 \, ; </math>
<math>~\bold{\hat{e}}_\varphi</math>:	<math>~\frac{d (\varpi \dot\varphi ) }{dt} + \dot\varpi \dot\varphi</math>	<math>~=</math>	<math>~ \frac{\partial (\varpi \dot\varphi ) }{\partial t} + \biggl[ \mathcal{L}_v (\varpi \dot\varphi ) \biggr] + \dot\varpi \dot\varphi \, ; </math>
<math>~\bold{\hat{k}}</math>:	<math>~\frac{d \dot{z} }{dt}</math>	<math>~=</math>	<math>~ \frac{\partial \dot{z} }{\partial t} + \biggl[ \mathcal{L}_v \dot{z} \biggr] \, . </math>

Rotating Frame

Drawing from an accompanying discussion of rotating reference frames, let's build our model in a cylindrical coordinate system that is spinning about its <math>~\bold{\hat{k}}</math>-axis with a time-independent angular velocity, <math>~\bold\Omega_f = \bold{\hat{k}} \Omega_f</math>. Furthermore, let's use <math>~\bold{u}</math> — instead of <math>~\bold{v}</math> — to represent the velocity as viewed in the rotating frame. We know that,

<math>~ \bold{u} + \bold\Omega_f \times \bold{x}_\mathrm{rot} </math>

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

and,

<math>~\bold{a} = \frac{d\bold{v}}{dt} = \frac{d\bold{u}}{dt} - \overbrace{ \biggl[ \underbrace{(- 2\bold\Omega_f \times \bold{u}) }_\text{Coriolis} ~+~ \underbrace{(- \bold\Omega_f \times (\bold\Omega_f \times \bold{x}_\mathrm{rot} )) }_\text{Centrifugal} \biggr] }^\text{Fictitious accelerations} \, . </math>

[BT87], p. 664, Appendix §1.D.3, Eq. (1D-43)

Note that, in the particular case being considered here,

<math>~\bold{a}_\mathrm{fict}</math>	<math>~=</math>	<math>~ - 2\bold\Omega_f \times \bold{u} - \bold\Omega_f \times (\bold\Omega_f \times \bold{x}_\mathrm{rot} ) </math>
	<math>~=</math>	<math>~ - 2\bold\Omega_f \times [ \bold{v} - \bold\Omega_f \times \bold{x}_\mathrm{rot} ] - \bold\Omega_f \times (\bold\Omega_f \times \bold{x}_\mathrm{rot} ) </math>
	<math>~=</math>	<math>~ - 2\bold\Omega_f \times \bold{v} + \bold\Omega_f \times (\bold\Omega_f \times \bold{x}_\mathrm{rot} ) </math>
	<math>~=</math>	<math>~ - 2\bold\Omega_f \times \bold{v} + \bold{\hat{k}}\Omega_f \times (\bold{\hat{e}}_\varphi \varpi \Omega_f ) </math>
	<math>~=</math>	<math>~ - 2\Omega_f [\bold{\hat{e}}_\varphi v_\varpi - \bold{\hat{e}}_\varpi v_\varphi] - (\bold{\hat{e}}_\varpi \varpi \Omega_f^2 ) \, . </math>
	<math>~=</math>	<math>~ + \bold{\hat{e}}_\varpi [2\Omega_f v_\varphi - \varpi \Omega_f^2 ] - \bold{\hat{e}}_\varphi 2\Omega_f v_\varpi \, . </math>

Let's examine how the Eulerian time-derivative of <math>~\bold{u}</math> differs from the Eulerian time-derivative of <math>~\bold{v}</math>. We can write,

<math>~\frac{\partial \bold{v}}{\partial t} + (\bold{v} \cdot \bold\nabla)\bold{v}</math>	<math>~=</math>	<math>~\frac{d\bold{u}}{dt} - \bold{a}_\mathrm{fict}</math>
<math>~\Rightarrow ~~~ \frac{\partial \bold{v}}{\partial t} </math>	<math>~=</math>	<math>~\frac{d\bold{u}}{dt} - \bold{a}_\mathrm{fict} - [ ( \bold{u} + \bold{\hat{e}}_\varphi \varpi \Omega_f ) \cdot \bold\nabla ] \bold{v} </math>
	<math>~=</math>	<math>~\frac{d\bold{u}}{dt} - \bold{a}_\mathrm{fict} - (\bold{u} \cdot \bold\nabla)\bold{v} - [ \bold{\hat{e}}_\varphi \varpi \Omega_f \cdot \bold\nabla ] \bold{v} \, . </math>

The right-hand-side of this expression contains two convective operators. The component elements of the one that is farthest to the right can be examined by replacing <math>~\bold{A}</math> with <math>~\bold{\hat{e}}_\varphi \varpi \Omega_f</math> and by replacing <math>~\bold{B}</math> with <math>~\bold{v}</math> in the generic convective operator shown above. This gives,

<math>~[ \bold{\hat{e}}_\varphi \varpi \Omega_f \cdot \bold\nabla ] \bold{v}</math>

<math>~ \bold{\hat{e}}_\varpi \biggl[ \Omega_f \frac{\partial v_\varpi}{\partial \varphi} - \Omega_f v_\varphi \biggr] + \bold{\hat{e}}_\varphi \biggl[ \Omega_f \frac{\partial v_\varphi}{\partial \varphi} + \Omega_f v_\varpi \biggr] + \bold{\hat{e}}_z \biggl[ \Omega_f \frac{\partial v_z}{\partial \varphi} \biggr] \, . </math>

The component elements of the second one from the right can be examined by replacing <math>~\bold{A}</math> with <math>~\bold{u}</math> and by replacing <math>~\bold{B}</math> with <math>~\bold{v}</math> in the generic convective operator. This gives,

<math>~(\bold{u} \cdot \bold\nabla) \bold{v}</math>	<math>~=</math>	<math>~ \bold{\hat{e}}_\varpi \biggl[ u_\varpi \frac{\partial v_\varpi }{\partial \varpi} + \frac{u_\varphi }{\varpi}\frac{\partial v_\varpi}{\partial \varphi} + u_z \frac{\partial v_\varpi}{\partial z} - \frac{u_\varphi v_\varphi}{\varpi} \biggr] </math>
		<math>~ + \bold{\hat{e}}_\varphi \biggl[ u_\varpi \frac{\partial v_\varphi}{\partial \varpi} + \frac{u_\varphi }{\varpi}\frac{\partial v_\varphi}{\partial \varphi} + u_z \frac{\partial v_\varphi}{\partial z} + \frac{u_\varphi v_\varpi}{\varpi} \biggr] + \bold{\hat{e}}_z \biggl[ u_\varpi \frac{\partial v_z}{\partial \varpi} + \frac{u_\varphi }{\varpi}\frac{\partial v_z}{\partial \varphi} + u_z \frac{\partial v_z}{\partial z} \biggr] \, . </math>

Hence, we can write,

<math>~\frac{d\bold{u}}{dt} - \frac{\partial \bold{v}}{\partial t}</math>	<math>~=</math>	<math>~ \bold{a}_\mathrm{fict} + [ \bold{\hat{e}}_\varphi \varpi \Omega_f \cdot \bold\nabla ] \bold{v} + (\bold{u} \cdot \bold\nabla)\bold{v} </math>
	<math>~=</math>	<math>~ \biggl\{ \bold{\hat{e}}_\varpi [2\Omega_f v_\varphi - \varpi \Omega_f^2 ] - \bold{\hat{e}}_\varphi 2\Omega_f v_\varpi \biggr\} + \biggl\{ \bold{\hat{e}}_\varpi \biggl[ \Omega_f \frac{\partial v_\varpi}{\partial \varphi} - \Omega_f v_\varphi \biggr] + \bold{\hat{e}}_\varphi \biggl[ \Omega_f \frac{\partial v_\varphi}{\partial \varphi} + \Omega_f v_\varpi \biggr] + \bold{\hat{e}}_z \biggl[ \Omega_f \frac{\partial v_z}{\partial \varphi} \biggr] \biggr\} </math>
		<math>~+ \bold{\hat{e}}_\varpi \biggl[ u_\varpi \frac{\partial v_\varpi }{\partial \varpi} + \frac{u_\varphi }{\varpi}\frac{\partial v_\varpi}{\partial \varphi} + u_z \frac{\partial v_\varpi}{\partial z} - \frac{u_\varphi v_\varphi}{\varpi} \biggr] </math>
		<math>~ + \bold{\hat{e}}_\varphi \biggl[ u_\varpi \frac{\partial v_\varphi}{\partial \varpi} + \frac{u_\varphi }{\varpi}\frac{\partial v_\varphi}{\partial \varphi} + u_z \frac{\partial v_\varphi}{\partial z} + \frac{u_\varphi v_\varpi}{\varpi} \biggr] + \bold{\hat{e}}_z \biggl[ u_\varpi \frac{\partial v_z}{\partial \varpi} + \frac{u_\varphi }{\varpi}\frac{\partial v_z}{\partial \varphi} + u_z \frac{\partial v_z}{\partial z} \biggr] </math>
	<math>~=</math>	<math>~ \bold{\hat{e}}_\varpi \biggl\{ [2\Omega_f v_\varphi - \varpi \Omega_f^2 ] + \biggl[ \Omega_f \frac{\partial v_\varpi}{\partial \varphi} - \Omega_f v_\varphi \biggr] + \biggl[ u_\varpi \frac{\partial v_\varpi }{\partial \varpi} + \frac{u_\varphi }{\varpi}\frac{\partial v_\varpi}{\partial \varphi} + u_z \frac{\partial v_\varpi}{\partial z} - \frac{u_\varphi v_\varphi}{\varpi} \biggr] \biggr\} </math>
		<math>~ + \bold{\hat{e}}_\varphi \biggl\{ - 2\Omega_f v_\varpi + \biggl[ \Omega_f \frac{\partial v_\varphi}{\partial \varphi} + \Omega_f v_\varpi \biggr] + \biggl[ u_\varpi \frac{\partial v_\varphi}{\partial \varpi} + \frac{u_\varphi }{\varpi}\frac{\partial v_\varphi}{\partial \varphi} + u_z \frac{\partial v_\varphi}{\partial z} + \frac{u_\varphi v_\varpi}{\varpi} \biggr] \biggr\} </math>
		<math>~ + \bold{\hat{k}} \biggl\{ \biggl[ \Omega_f \frac{\partial v_z}{\partial \varphi} \biggr] + \biggl[ u_\varpi \frac{\partial v_z}{\partial \varpi} + \frac{u_\varphi }{\varpi}\frac{\partial v_z}{\partial \varphi} + u_z \frac{\partial v_z}{\partial z} \biggr] \biggr\} </math>
	<math>~=</math>	<math>~ \bold{\hat{e}}_\varpi \biggl\{ \biggl[ u_\varpi \frac{\partial v_\varpi }{\partial \varpi} + \biggl( \Omega_f + \frac{u_\varphi }{\varpi} \biggr) \frac{\partial v_\varpi}{\partial \varphi} + u_z \frac{\partial v_\varpi}{\partial z}\biggr] + \Omega_f [ v_\varphi - \varpi \Omega_f ] - \frac{(v_\varphi -\varpi \Omega_f) v_\varphi}{\varpi} \biggr\} </math>
		<math>~ + \bold{\hat{e}}_\varphi \biggl\{ \biggl[ u_\varpi \frac{\partial v_\varphi}{\partial \varpi} + \biggl(\Omega_f + \frac{u_\varphi }{\varpi} \biggr) \frac{\partial v_\varphi}{\partial \varphi} + u_z \frac{\partial v_\varphi}{\partial z}\biggr] + \biggl( u_\varphi - \varpi \Omega_f\biggr) \frac{ v_\varpi }{\varpi} \biggr\} + \bold{\hat{k}} \biggl\{ u_\varpi \frac{\partial v_z}{\partial \varpi} + \biggl( \Omega_f + \frac{u_\varphi }{\varpi} \biggr) \frac{\partial v_z}{\partial \varphi} + u_z \frac{\partial v_z}{\partial z} \biggr\} </math>

Other (Old)

It can readily be appreciated that the first few terms in our next-to-last acceleration expression may be grouped together as the partial time-derivative of the velocity, that is, we can write,

<math>~\frac{\partial \bold{v}}{\partial t} - \bold{\hat{e}}_\varpi \biggl[\varpi \biggl(\frac{d\varphi}{dt}\biggr)^2 \biggr] + \bold{\hat{e}}_\varphi \biggl[ 2 \biggl( \frac{d\varpi}{dt} \cdot \frac{d\varphi}{dt} \biggr) + \varpi \frac{d^2\varphi}{dt^2}\biggr] </math>

the azimuthal coordinate location it is critically important to recognize that the orientation in space of the pair of cylindrical-coordinate-based unit vectors, <math>~\bold{\hat{e}}_\varpi</math> and <math>~\bold{\hat{e}}_\varphi</math>, depend on the fluid element's coordinate location. Specifically,

Let's begin by building a model in a cylindrical coordinate system that is spinning about its <math>~\bold{\hat{k}}</math>-axis with an angular velocity, <math>~{\vec{\Omega}}_f = {\hat\mathbf{k}} \Omega_f</math>. Furthermore, let's use <math>~\bold{v}</math> to represent the inertial-frame velocity and use <math>~\bold{u}</math> to represent the velocity as viewed in the rotating frame. We know that,

<math>~\frac{d}{dt} \bold{\hat{e}}_\varpi</math>	<math>~=</math>	<math>~\mathbf{\hat{e}}_\varphi \frac{u_\varphi}{\varpi} \, ,</math>
<math>~\frac{d}{dt} \bold{\hat{e}}_\varphi</math>	<math>~=</math>	<math>~ - \mathbf{\hat{e}}_\varpi \frac{u_\varphi}{\varpi} \, ,</math> and,
<math> \bold{v} - \bold{u} = {\vec{\Omega}}_f \times \vec{x} </math>	<math>~=~</math>	<math> {\hat\mathbf{k}} \Omega_f\times (\mathbf{\hat{e}}_\varpi \varpi + \mathbf{\hat{k}}z) = \mathbf{\hat{e}}_\varphi \Omega_f \varpi \, . </math>

Let's take the total time-derivative of an arbitrary vector, <math>~\bold{A}</math>, which when written in cylindrical coordinates takes the form,

<math>~\bold{\hat{e}}_\varpi A_\varpi + \bold{\hat{e}}_\varphi A_\varphi + \bold{\hat{k}}A_z \, .</math>

Associated with this Lagrangian fluid element will be various (generally, time-dependent) scalar attributes, <math>~q(\bold{x}_0, t)</math>, such as mass-density or pressure, and various vector attributes, <math>~\bold{A}(\bold{x}_0,t)</math>, such as velocity or vorticity.

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