PGE/RotatingFrame

NOTE to Eric Hirschmann & David Neilsen... I have moved the earlier contents of this page to a new Wiki location called Compressible Riemann Ellipsoids.

Rotating Reference Frame

Coordinate Transformation

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 ${\mathrm{\Omega }}_f$. In order to transform any one of the 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, ${\overrightarrow{\mathrm{\Omega }}}_f$; and the $d/dt$ operator, which denotes Lagrangian time-differentiation in the inertial frame, must everywhere be replaced as follows:

$[\frac{d}{dt}{]}_{inertial}\to [\frac{d}{dt}{]}_{rot}+{\overrightarrow{\mathrm{\Omega }}}_f\times .$

Performing this transformation implies, for example, that

${\overrightarrow{v}}_{inertial}={\overrightarrow{v}}_{rot}+{\overrightarrow{\mathrm{\Omega }}}_f\times \overrightarrow{x},$

and,

$[\frac{d\overrightarrow{v}}{dt}{]}_{inertial}=[\frac{d\overrightarrow{v}}{dt}{]}_{rot}+2{\overrightarrow{\mathrm{\Omega }}}_f\times {\overrightarrow{v}}_{rot}+{\overrightarrow{\mathrm{\Omega }}}_f\times ({\overrightarrow{\mathrm{\Omega }}}_f\times \overrightarrow{x})$

$=[\frac{d\overrightarrow{v}}{dt}{]}_{rot}+2{\overrightarrow{\mathrm{\Omega }}}_f\times {\overrightarrow{v}}_{rot}-\frac{1}{2}\mathrm{\nabla }[|{\overrightarrow{\mathrm{\Omega }}}_f\times \overrightarrow{x}{|}^2]$

(If we were to allow ${\overrightarrow{\mathrm{\Omega }}}_f$ to be a function of time, an additional term involving the time-derivative of ${\overrightarrow{\mathrm{\Omega }}}_f$ also would appear on the right-hand-side of these last expressions; see, for example, Eq.~1D-42 in BT87.) Note as well that the relationship between the fluid vorticity in the two frames is,

$[\overrightarrow{\zeta }{]}_{inertial}=[\overrightarrow{\zeta }{]}_{rot}+2{\overrightarrow{\mathrm{\Omega }}}_f.$

We begin by restating the Lagrangian representation of the intertial-frame Euler equation: $\frac{d{U}_i}{dt}{|}_{\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}$ $=$ $-\frac{1}{\rho }\frac{\partial p}{\partial x_i}-\frac{\partial \mathrm{\Phi }}{\partial x_i}.$ Drawing from Chapter 4, §25 of [EFE] — where the Cartesian components of the inertial-frame velocity $({\overrightarrow{v}}_{\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}})$ are represented by $U_i$ and the Cartesian components of the rotating-frame velocity $({\overrightarrow{v}}_{\mathrm{r}\mathrm{o}\mathrm{t}})$ are represented by $u_i$ — the LHS of the Euler equation transform as follows: $\frac{d{U}_i}{dt}{|}_{\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}$ $\to$ $\frac{d{U}_i}{dt}{|}_{\mathrm{r}\mathrm{o}\mathrm{t}}-{ϵ}_{imk}{\mathrm{\Omega }}_k{U}_m,$ where we also recognize that, $U_i$ $\to$ $u_i+{ϵ}_{ijk}{\mathrm{\Omega }}_j{x}_k.$ Both of these expressions make use of the three-element Levi-Civita tensor, ${ϵ}_{ijk}$. Its six nonzero component values are … $ijk$ ${ϵ}_{ijk}$   $ijk$ ${ϵ}_{ijk}$ 123 +1 132 -1 312 321 231 213 Hence, for example, transforming the x-component $(i=1)$ of $\overrightarrow{U}$ gives, $U_1$ $\to$ $u_1+{ϵ}_{1jk}{\mathrm{\Omega }}_j{x}_k=u_1+{ϵ}_{123}{\mathrm{\Omega }}_2{x}_3+{ϵ}_{132}{\mathrm{\Omega }}_3{x}_2=u_1+{\mathrm{\Omega }}_{2}z-{\mathrm{\Omega }}_{3}y;$ transforming the y-component $(i=2)$ gives, $U_2$ $\to$ $u_2+{ϵ}_{2jk}{\mathrm{\Omega }}_j{x}_k=u_2+{ϵ}_{231}{\mathrm{\Omega }}_3{x}_1+{ϵ}_{213}{\mathrm{\Omega }}_1{x}_3=u_2+{\mathrm{\Omega }}_{3}x-{\mathrm{\Omega }}_{1}z;$ and transforming the z-component $(i=3)$ gives, $U_3$ $\to$ $u_3+{ϵ}_{3jk}{\mathrm{\Omega }}_j{x}_k=u_3+{ϵ}_{312}{\mathrm{\Omega }}_1{x}_2+{ϵ}_{321}{\mathrm{\Omega }}_2{x}_1=u_3+{\mathrm{\Omega }}_{1}y-{\mathrm{\Omega }}_{2}x.$ These are the same three components that arise from the vector expression (from above), ${\overrightarrow{v}}_{inertial}={\overrightarrow{v}}_{rot}+\overrightarrow{\mathrm{\Omega }}\times \overrightarrow{x};$ we therefore recognize that, $\overrightarrow{\mathrm{\Omega }}\times \overrightarrow{x}={ϵ}_{ijk}{\mathrm{\Omega }}_j{x}_k$. We note as well that, $\overrightarrow{\mathrm{\Omega }}\times \overrightarrow{x}=-{ϵ}_{ijk}{\mathrm{\Omega }}_k{x}_j$. As viewed from the rotating frame of reference, then, the Euler equation becomes, $\frac{d{U}_i}{dt}{|}_{\mathrm{r}\mathrm{o}\mathrm{t}}-{ϵ}_{imk}{\mathrm{\Omega }}_k{U}_m$ $=$ $-\frac{1}{\rho }\frac{\partial p}{\partial x_i}-\frac{\partial \mathrm{\Phi }}{\partial x_i}$ $\Rightarrow \frac{d}{dt}[u_i+{ϵ}_{ijk}{\mathrm{\Omega }}_j{x}_k]$ $=$ ${ϵ}_{imk}{\mathrm{\Omega }}_k[u_i+{ϵ}_{ijk}{\mathrm{\Omega }}_j{x}_k]-\frac{1}{\rho }\frac{\partial p}{\partial x_i}-\frac{\partial \mathrm{\Phi }}{\partial x_i}$ $\Rightarrow \frac{d{u}_i}{dt}+{ϵ}_{ijk}\left[\right(\frac{d{\mathrm{\Omega }}_j}{dt})x_k+{\mathrm{\Omega }}_j(\frac{d{x}_k}{dt}\left)\right]$ $=$ ${ϵ}_{imk}{\mathrm{\Omega }}_k[u_i+{ϵ}_{ijk}{\mathrm{\Omega }}_j{x}_k]-\frac{1}{\rho }\frac{\partial p}{\partial x_i}-\frac{\partial \mathrm{\Phi }}{\partial x_i}$

Continuity Equation (rotating frame)

Applying these transformations to the standard, inertial-frame representations of the continuity equation presented elsewhere, we obtain the:

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

$[\frac{d\rho }{dt}{]}_{rot}+\rho \mathrm{\nabla }\cdot {\overrightarrow{v}}_{rot}=0$ ;

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

$[\frac{\partial \rho }{\partial t}{]}_{rot}+\mathrm{\nabla }\cdot (\rho {\overrightarrow{v}}_{rot})=0$ .

Euler Equation (rotating frame)

Applying these transformations to the standard, inertial-frame representations of the Euler equation presented elsewhere, we obtain the:

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

$[\frac{d\overrightarrow{v}}{dt}{]}_{rot}=-\frac{1}{\rho }\mathrm{\nabla }P-\mathrm{\nabla }\mathrm{\Phi }-2{\overrightarrow{\mathrm{\Omega }}}_f\times {\overrightarrow{v}}_{rot}-{\overrightarrow{\mathrm{\Omega }}}_f\times ({\overrightarrow{\mathrm{\Omega }}}_f\times \overrightarrow{x})$ ;

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

$[\frac{\partial \overrightarrow{v}}{\partial t}{]}_{rot}+({\overrightarrow{v}}_{rot}\cdot \mathrm{\nabla }){\overrightarrow{v}}_{rot}=-\frac{1}{\rho }\mathrm{\nabla }P-\mathrm{\nabla }[\mathrm{\Phi }-\frac{1}{2}|{\overrightarrow{\mathrm{\Omega }}}_f\times \overrightarrow{x}{|}^2]-2{\overrightarrow{\mathrm{\Omega }}}_f\times {\overrightarrow{v}}_{rot}$ ;

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

$[\frac{\partial \overrightarrow{v}}{\partial t}{]}_{rot}+({\overrightarrow{\zeta }}_{rot}+2{\overrightarrow{\mathrm{\Omega }}}_f)\times {\overrightarrow{v}}_{rot}=-\frac{1}{\rho }\mathrm{\nabla }P-\mathrm{\nabla }[\mathrm{\Phi }+\frac{1}{2}{v}_{rot}^2-\frac{1}{2}|{\overrightarrow{\mathrm{\Omega }}}_f\times \overrightarrow{x}{|}^2]$ .

Centrifugal and Coriolis Accelerations

Following along the lines of the discussion presented in Appendix 1.D, §3 of [BT87], in a rotating reference frame the Lagrangian representation of the Euler equation may be written in the form,

$[\frac{d\overrightarrow{v}}{dt}{]}_{rot}=-\frac{1}{\rho }\mathrm{\nabla }P-\mathrm{\nabla }\mathrm{\Phi }+{\overrightarrow{a}}_{fict}$,

where,

${\overrightarrow{a}}_{fict}\equiv -2{\overrightarrow{\mathrm{\Omega }}}_f\times {\overrightarrow{v}}_{rot}-{\overrightarrow{\mathrm{\Omega }}}_f\times ({\overrightarrow{\mathrm{\Omega }}}_f\times \overrightarrow{x}).$

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,

Coriolis Acceleration

${\overrightarrow{a}}_{Coriolis}\equiv -2{\overrightarrow{\mathrm{\Omega }}}_f\times {\overrightarrow{v}}_{rot},$

(see the related Wikipedia discussion) and the

Centrifugal Acceleration

${\overrightarrow{a}}_{Centrifugal}\equiv -{\overrightarrow{\mathrm{\Omega }}}_f\times ({\overrightarrow{\mathrm{\Omega }}}_f\times \overrightarrow{x})=\frac{1}{2}\mathrm{\nabla }[|{\overrightarrow{\mathrm{\Omega }}}_f\times \overrightarrow{x}{|}^2]$

(see the related Wikipedia discussion).

Nonlinear Velocity Cross-Product

In some contexts — for example, our discussion of Riemann ellipsoids or the analysis by Korycansky & Papaloizou (1996) of nonaxisymmetric disk structures — 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,

$\overrightarrow{A}\equiv ({\overrightarrow{\zeta }}_{rot}+2{\overrightarrow{\mathrm{\Omega }}}_f)\times {\overrightarrow{v}}_{rot}.$

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 ${\overrightarrow{\mathrm{\Omega }}}_f$ with z-axis

Without loss of generality we can set ${\overrightarrow{\mathrm{\Omega }}}_f=\hat{k}{\mathrm{\Omega }}_f$, that is, we can align the frame rotation axis with the z-axis of a Cartesian coordinate system. The Cartesian components of $\overrightarrow{A}$ are then,

$\hat{i}:A_x={\zeta }_y{v}_z-({\zeta }_z+2\mathrm{\Omega })v_y,$

$\hat{j}:A_y=({\zeta }_z+2\mathrm{\Omega })v_x-{\zeta }_x{v}_z,$

$\hat{k}:A_z={\zeta }_x{v}_y-{\zeta }_y{v}_x,$

where it is understood that the three Cartesian components of the vorticity vector are,

${\zeta }_x=[\frac{\partial v_z}{\partial y}-\frac{\partial v_y}{\partial z}],{\zeta }_y=[\frac{\partial v_x}{\partial z}-\frac{\partial v_z}{\partial x}],{\zeta }_z=[\frac{\partial v_y}{\partial x}-\frac{\partial v_x}{\partial y}].$

In turn, the curl of $\overrightarrow{A}$ has the following three Cartesian components:

$\hat{i}:[\mathrm{\nabla }\times \overrightarrow{A}{]}_x=\frac{\partial }{\partial y}[{\zeta }_x{v}_y-{\zeta }_y{v}_x]-\frac{\partial }{\partial z}[({\zeta }_z+2\mathrm{\Omega })v_x-{\zeta }_x{v}_z],$

$\hat{j}:[\mathrm{\nabla }\times \overrightarrow{A}{]}_y=\frac{\partial }{\partial z}[{\zeta }_y{v}_z-({\zeta }_z+2\mathrm{\Omega })v_y]-\frac{\partial }{\partial x}[{\zeta }_x{v}_y-{\zeta }_y{v}_x],$

$\hat{k}:[\mathrm{\nabla }\times \overrightarrow{A}{]}_z=\frac{\partial }{\partial x}[({\zeta }_z+2\mathrm{\Omega })v_x-{\zeta }_x{v}_z]-\frac{\partial }{\partial y}[{\zeta }_y{v}_z-({\zeta }_z+2\mathrm{\Omega })v_y].$

When $v_z=0$

If we restrict our discussion to configurations that exhibit only planar flows — that is, systems in which $v_z=0$ — then the Cartesian components of $\overrightarrow{A}$ and $\mathrm{\nabla }\times \overrightarrow{A}$ simplify somewhat to give, respectively,

$\hat{i}:A_x=-({\zeta }_z+2\mathrm{\Omega })v_y,$

$\hat{j}:A_y=({\zeta }_z+2\mathrm{\Omega })v_x,$

$\hat{k}:A_z={\zeta }_x{v}_y-{\zeta }_y{v}_x,$

and,

$\hat{i}:[\mathrm{\nabla }\times \overrightarrow{A}{]}_x=\frac{\partial }{\partial y}[{\zeta }_x{v}_y-{\zeta }_y{v}_x]-\frac{\partial }{\partial z}[({\zeta }_z+2\mathrm{\Omega })v_x],$

$\hat{j}:[\mathrm{\nabla }\times \overrightarrow{A}{]}_y=-\frac{\partial }{\partial z}[({\zeta }_z+2\mathrm{\Omega })v_y]-\frac{\partial }{\partial x}[{\zeta }_x{v}_y-{\zeta }_y{v}_x],$

$\hat{k}:[\mathrm{\nabla }\times \overrightarrow{A}{]}_z=\frac{\partial }{\partial x}[({\zeta }_z+2\mathrm{\Omega })v_x]+\frac{\partial }{\partial y}[({\zeta }_z+2\mathrm{\Omega })v_y],$

where, in this case, the three Cartesian components of the vorticity vector are,

${\zeta }_x=-\frac{\partial v_y}{\partial z},{\zeta }_y=\frac{\partial v_x}{\partial z},{\zeta }_z=[\frac{\partial v_y}{\partial x}-\frac{\partial v_x}{\partial y}].$

Related Discussions

• Wikipedia discussion of vorticity.
• Wikipedia discussion of Coriolis Effect.
• Wikipedia discussion of Centrifugal acceleration.