Differential operators in cylindrical coordinates

$ \renewcommand{\vec}[1]{ {\bf #1} } \newcommand{\bnabla}{ \vec{\nabla} } \newcommand{\Rey}{Re} \def\vechat#1{ \hat{ \vec{#1} } } \def\mat#1{#1} $

Gradient, divergence, curl

$(\bnabla f)_r = \partial_{r}f, \quad (\bnabla f)_\theta = \frac1{r}\, \partial_{\theta} f, \quad (\bnabla f)_z = \partial_{z} f .$

$\bnabla \cdot \vec{A} = (\frac1{r} + \partial_{r}) A_r + \frac1{r}\,\partial_{\theta}A_\theta + \partial_{z} A_z .$

$(\bnabla \wedge \vec{A})_r = \frac1{r}\,\partial_{\theta} A_z - \partial_{z} A_\theta ,$

$(\bnabla \wedge \vec{A})_\theta = \partial_{z} A_r - \partial_{r} A_z ,$

$(\bnabla \wedge \vec{A})_z = (\frac1{r}+\partial_{r}) A_\theta - \frac1{r}\,\partial_{\theta} A_r .$

Laplacian

$\nabla^2 f = (\frac1{r}\partial_{r} + \partial_{rr} ) f +\frac1{r^2}\,\partial_{\theta\theta} f + \partial_{zz} f .$

$(\nabla^2 \vec{A})_r = \nabla^2 A_r - \frac{2}{r^2}\,\partial_{\theta}A_\theta - \frac{A_r}{r^2} ,$

$(\nabla^2 \vec{A})_\theta = \nabla^2 A_\theta + \frac{2}{r^2}\,\partial_{\theta}A_r - \frac{A_\theta}{r^2} ,$

$(\nabla^2 \vec{A})_z = \nabla^2 A_z .$

Advective operator

$(\vec{A}\cdot\bnabla\,\vec{B})_r = A_r\,\partial_{r}B_r + \frac{A_\theta}{r}\,\partial_{\theta}B_r + A_z\,\partial_{z}B_r - \frac{A_\theta B_\theta}{r} ,$

$(\vec{A}\cdot\bnabla\,\vec{B})_\theta = A_r\,\partial_{r}B_\theta + \frac{A_\theta}{r}\,\partial_{\theta}B_\theta + A_z\,\partial_{z}B_\theta + \frac{A_\theta B_r}{r} ,$

$(\vec{A}\cdot\bnabla\,\vec{B})_z = A_r\,\partial_{r}B_z + \frac{A_\theta}{r}\,\partial_{\theta}B_z + A_z\,\partial_{z}B_z .$

Velocity gradient tensor

$(\bnabla\vec{u})^T ~=~ \left[ \begin{array}{ccc} \partial_r u_r & \partial_r u_\theta & \partial_r u_z \\ \frac{1}{r}(\partial_\theta u_r-u_\theta) & \frac{1}{r}(\partial_\theta u_\theta + u_r) & \frac{1}{r}\partial_\theta u_z \\ \partial_z u_r & \partial_z u_\theta & \partial_z u_z \end{array} \right] $

Toroidal-poloidal decompositions

Axial form:

$\begin{eqnarray*} & \vec{A} = \bnabla \wedge(\vechat{z}\psi) + \bnabla \wedge\bnabla \wedge(\vechat{z}\phi) , & \\ & \psi=\psi(r,\theta,z), \quad \phi=\phi(r,\theta,z). \nonumber & \end{eqnarray*}$

i.e.

$\vec{A}_r = \frac1{r}\,\partial_{\theta} \psi + \partial_{rz} \phi ,$

$\vec{A}_\theta = -\partial_{r} \psi + \frac1{r}\,\partial_{\theta z} \phi ,$

$\vec{A}_z = - \nabla^2_h \phi \, ,$

where

$\nabla^2_h f = (\frac1{r}\,\partial_{r}+\partial_{rr}) f + \frac1{r^2}\,\partial_{\theta\theta} f .$

Radial form:

$\begin{eqnarray*} & \vec{A} = \psi_0\,\vechat{\theta} + \phi_0\,\vechat{z} + \bnabla \wedge(\vec{r}\psi) + \bnabla \wedge\bnabla \wedge(\vec{r}\phi) , & \\ & \psi_0=\psi_0(r), \quad \phi_0=\phi_0(r), \quad \psi=\psi(r,\theta,z), \quad \phi=\phi(r,\theta,z). \nonumber & \end{eqnarray*}$

i.e.

$\vec{A}_r = -r \nabla^2_c \phi ,$

$\vec{A}_\theta = \psi_0 + r \partial_{z} \psi + \partial_{r\theta} \phi ,$

$\vec{A}_z = \phi_0 - \partial_{\theta} \psi + (2+r\partial_{r})\partial_{z} \phi \, ,$

where

$\nabla^2_c f = \frac1{r^2}\,\partial_{\theta\theta} f + \partial_{zz} f .$