Derivations for computable Quantities

This page collects some derivations for computable quantities:

Throughout this document we use \(\alpha,\beta\) as generic indices for the coordinates \(\rho,\theta,\zeta\) unless otherwise specified.

Flux aligned coordinates

\[\begin{split} \begin{align} \vec{e}_\alpha &= \pdv{\vec{x}}{\alpha} \\ \vec{e}^\alpha &= \grad \alpha \\ \Jac &= \vec{e}_\rho \cdot (\vec{e}_\theta \times \vec{e}_\zeta) \\ \Jac &= \qty(\grad \rho \cdot \grad \theta \times \grad \zeta)^{-1} \\ \end{align} \end{split}\]

\[\begin{split} \begin{align} \vec{e}_\rho &= \Jac \grad\theta \times \grad\zeta \\ \vec{e}_\theta &= \Jac \grad\zeta \times \grad\rho \\ \vec{e}_\zeta &= \Jac \grad\rho \times \grad\theta \\ \end{align} \end{split}\]

\[\begin{split} \begin{align} \vec{k}_{\alpha\beta} &:= \pdv[2]{\vec{x}}{\alpha}{\beta} \\ &= \pdv{\vec{e}_\alpha}{\beta} = \pdv{\vec{e}_\beta}{\alpha} \\ \end{align} \end{split}\]

Magnetic field \(\vec{B}\)

\[\begin{split} \begin{align} B^\rho &= 0 \\ B^\thet &= \frac{1}{\Jac} \dPhidr \left(\iota-\pdv{\lambda}{\zeta} \right) \\ B^\zeta &= \frac{1}{\Jac} \dPhidr \left(1+\pdv{\lambda}{\thet} \right) \\ \end{align} \end{split}\]

\(\Rightarrow\) B_contra_t, B_contra_z

Derivatives of \(\modB\)

\[\begin{split} \begin{align} \modB^2 &= B_\thet B^\thet + B_\zeta B^\zeta\\ \modB^2 &= B^\thet B^\thet g_{\thet\thet} + 2 B^\thet B^\zeta g_{\thet\zeta} + B^\zeta B^\zeta g_{\zeta\zeta} \\ \pdv{\modB^2}{\alpha} &= 2 B^\thet \pdv{B^\thet}{\alpha} g_{\thet\thet} + B^\thet B^\thet \pdv{g_{\thet\thet}}{\alpha} \\ &+ 2 \left(\pdv{B^\thet}{\alpha} B^\zeta + B^\thet \pdv{B^\zeta}{\alpha}\right) g_{\thet\zeta} + B^\thet B^\zeta \pdv{g_{\thet\zeta}}{\alpha} \\ &+ 2 B^\zeta \pdv{B^\zeta}{\alpha} g_{\zeta\zeta} + B^\zeta B^\zeta \pdv{g_{\zeta\zeta}}{\alpha} \\ \pdv{\modB}{\alpha} &= \frac{1}{2\modB} \pdv{\modB^2}{\alpha} = \frac{\vec{B}}{\modB} \cdot \pdv{\vec{B}}{\alpha} \\ \nabla \modB &= \pdv{\modB}{\rho} \nabla\rho + \pdv{\modB}{\thet} \nabla\thet + \pdv{\modB}{\zeta} \nabla\zeta \\ \end{align} \end{split}\]

for \(\alpha\in\left\{\rho,\thet,\zeta\right\}\).

\(\Rightarrow\) dmodB_dr, dmodB_dt, dmodB_dz, grad_modB

Derivatives of \(\vec{B}\)

As \(\vec{B}\) is a vector, its derivative \(\grad\vec{B}\) is a matrix, which complicates things a bit, as the matrix product is not commutative. We denote the outer product with \(\otimes\).

\[\begin{split} \begin{align} \grad \vec{B} &:= \grad \otimes \vec{B} \\ &= \sum_\alpha \grad\alpha \otimes \pdv{\vec{B}}{\alpha} \\ &= \grad\rho \otimes \pdv{\vec{B}}{\rho} + \grad\thet \otimes \pdv{\vec{B}}{\thet} + \grad\zeta \otimes \pdv{\vec{B}}{\zeta} \\ \pdv{\vec{B}}{\alpha} &:= \sum_\beta \pdv{\qty(B^\beta \vec{e}_\beta)}{\alpha} \\ &= \sum_\beta \pdv{B^\beta}{\alpha} \vec{e}_\beta + B^\beta \vec{k}_{\beta\alpha} \\ &= \cancel{\pdv{B^\rho}{\alpha}} \erho + \cancel{B^\rho} \vec{k}_{\rho\alpha} + \pdv{B^\thet}{\alpha} \ethet + B^\thet \vec{k}_{\thet\alpha} + \pdv{B^\zeta}{\alpha} \ezeta + B^\zeta \vec{k}_{\zeta\alpha} \\ \nabla \vec{B} &= \sum_{\alpha\beta} \grad\alpha \otimes \left(\pdv{B^\beta}{\alpha} \vec{e}_\beta + B^\beta \vec{k}_{\beta\alpha} \right) \end{align} \end{split}\]

for \(\alpha,\beta\in\left\{\rho,\thet,\zeta\right\}\).

Due to xarray’s limitations with multiple dimensions of the same name, \(\grad\vec{B}\) is not directly available as a computable quantity, and its components should be used instead.

\(\Rightarrow\) dB_dr, dB_dt, dB_dz

---

From

\[\begin{split} \begin{align} B^\thet &= \frac{1}{\Jac} \dPhidr \left(\iota-\pdv{\lambda}{\zeta} \right) \\ B^\zeta &= \frac{1}{\Jac} \dPhidr \left(1+\pdv{\lambda}{\thet} \right) \\ \end{align} \end{split}\]

it follows

\[\begin{split} \begin{align} \pdv{B^\thet}{\rho} &= \frac{1}{\Jac} \frac{d^2\Phi}{d\rho^2} \left(\iota-\pdv{\lambda}{\zeta} \right) + \frac{1}{\Jac} \dPhidr \left(\frac{d\iota}{d\rho}-\pdv{^2\lambda}{\rho\partial\zeta} \right) - \pdv{\Jac}{\rho} \frac{1}{\Jac^2} \dPhidr \left(\iota-\pdv{\lambda}{\zeta} \right) \\ \pdv{B^\thet}{\thet} &= - \frac{1}{\Jac} \dPhidr \pdv{^2\lambda}{\thet\partial\zeta} - \pdv{\Jac}{\thet} \frac{1}{\Jac^2} \dPhidr \left(\iota-\pdv{\lambda}{\zeta} \right) \\ \pdv{B^\thet}{\zeta} &= - \frac{1}{\Jac} \dPhidr \pdv{^2\lambda}{\zeta^2} - \pdv{\Jac}{\zeta} \frac{1}{\Jac^2} \dPhidr \left(\iota-\pdv{\lambda}{\zeta} \right) \\ \pdv{B^\zeta}{\rho} &= \frac{1}{\Jac} \frac{d^2\Phi}{d\rho^2} \left(1+\pdv{\lambda}{\thet} \right) + \frac{1}{\Jac} \dPhidr \pdv{^2\lambda}{\rho\partial\thet} - \pdv{\Jac}{\rho} \frac{1}{\Jac^2} \dPhidr \left(1+\pdv{\lambda}{\thet} \right) \\ \pdv{B^\zeta}{\thet} &= \frac{1}{\Jac} \dPhidr \pdv{^2\lambda}{^2\thet} - \pdv{\Jac}{\thet} \frac{1}{\Jac^2} \dPhidr \left(1+\pdv{\lambda}{\thet} \right) \\ \pdv{B^\zeta}{\zeta} &= \frac{1}{\Jac} \dPhidr \pdv{^2\lambda}{\thet\partial\zeta} - \pdv{\Jac}{\zeta} \frac{1}{\Jac^2} \dPhidr \left(1+\pdv{\lambda}{\thet} \right) \\ \end{align} \end{split}\]

\(\Rightarrow\) dB_contra_t_dr, dB_contra_t_dt, dB_contra_t_dz, dB_contra_z_dr, dB_contra_z_dt, dB_contra_z_dz

---

The gradient \(\nabla\vec{B}\) can be used to compute the magnetic gradient length scale \(L_{\nabla\vec{B}}\) (L_gradB). Details are found in John Kappel et al 2024 PPCF 66 025018 DOI:10.1088/1361-6587/ad1a3e.

Normalized magnetic field

The normalized magnetic field (i.e. the unit vector along the magnetic field) is defined as

\[\begin{split} \begin{align} \vec{b} &:= \frac{\vec{B}}{\modB} \\ \end{align} \end{split}\]

Its gradient (a matrix) is then:

\[\begin{split} \begin{align} \grad\vec{b} &= \sum_\alpha \grad\alpha \otimes \pdv{\vec{b}}{\alpha} \\ \pdv{\vec{b}}{\alpha} &= \frac{1}{\modB}\pdv{\vec{B}}{\alpha} - \frac{\vec{B}}{\modB^2}\pdv{\modB}{\alpha} \\ &= \frac{1}{\modB}\qty(\pdv{\vec{B}}{\alpha} - \vec{b} \qty(\vec{b} \cdot \pdv{\vec{B}}{\alpha})) \\ \end{align} \end{split}\]

\(\Rightarrow\) db_dr, db_dt, db_dz

Geodesic curvature

The fieldline curvature vector \(\vec{\kappa}_B\) is defined as

\[\begin{split} \begin{align} \vec{\kappa}_B &:= \vec{b} \cdot \grad\vec{b} \\ &= \sum_\alpha \qty(\frac{\vec{B}}{\modB} \cdot \grad\alpha) \pdv{\vec{b}}{\alpha} \\ \end{align} \end{split}\]

The geodesic curvature \(\kappa_G\) is defined as

\[\begin{split} \kappa_G &:= \vec{\kappa}_B \cdot \qty(\grad\rho \times \vec{b}) \\ \end{split}\]

Which could be rewritten as

\[\begin{split} \begin{align} \kappa_G &= \vec{b} \cdot \qty(\grad \vec{b}) \cdot \qty(\grad\rho \times \vec{b}) \\ &= \sum_\alpha \qty(\frac{\vec{B}}{\modB} \cdot \grad\alpha) \pdv{\vec{b}}{\alpha} \cdot \qty(\grad\rho \times \vec{b}) \\ \end{align} \end{split}\]

\(\Rightarrow\) kappa_B, kappa_G

Effective geometric quantities

The plasma volume \(V\), surface are \(A_\text{surface}\) and length of the magnetic axis \(L_\text{axis}\) are defined as

\[\begin{split} \begin{align} V &= \int_0^1 d\rho \int_0^{2\pi} d\theta \int_0^{2\pi} d\zeta \Jac \\ A_\text{surface} &= \left. \int_0^{2\pi} d\theta \int_0^{2\pi} d\zeta \left|\vec{e}_\theta \times \vec{e}_\zeta\right| \right|_{\rho=1}\\ L_\text{axis} &= \left. \int_0^{2\pi} d\zeta \left|\vec{e}_\zeta\right| \right|_{\rho=0,\theta=0}\\ \end{align} \end{split}\]

All three are independent of the coordinate frame and parametrization, and \(V,A_\text{surface}\) only depend on the boundary shape and not the full equilibrium.

From these we can compute the effective minor radius, effective major radius, effective aspect ratio and effective elongation, by relating them to an equivalent torus with elliptical cross section. In particular we define the equivalent torus as a torus with circular axis and constant elliptical cross-section, which has the same volume, surface area and axis length as the configuration of interest.

\[\begin{split} \begin{align} V &= 2\pi^2 r_\text{minor,eff}^2 r_\text{major,eff} \\ A_\text{surface} &= 4\pi^2 r_\text{minor,eff} r_\text{major,eff} \tilde{C}(E_\text{eff}) \\ L_\text{axis} &= 2\pi r_\text{major,eff} \end{align} \end{split}\]

with the effective elongation \(E_\text{eff} := \frac{a}{b}\) defined as the ratio of the cross-sections semi-major axis and semi-minor axis (\(a \geq b\)). The circumference of the ellipse does not have a closed form. We use Ramanujan’s approximation

\[\begin{split} \begin{align} C &= 2\pi r_\text{minor,eff} \tilde{C} \\ \tilde{C} &= \frac{E_\text{eff} + 1}{2\sqrt{E_\text{eff}}} \left[ 1 + 3 \frac{h}{10 + \sqrt{4 - 3h}} \right] \\ h &:= \frac{(E_\text{eff} - 1)^2}{(E_\text{eff} + 1)^2}. \end{align} \end{split}\]

We invert these formulas to obtain

\[\begin{split} \begin{align} r_\text{major,eff} &= \frac{L_\text{axis}}{2\pi} \\ r_\text{minor,eff} &= \sqrt{\frac{V}{\pi L_\text{axis}}} \\ a_\text{eff} &= \frac{r_\text{major,eff}}{r_\text{minor,eff}} \\ \tilde{C}(E_\text{eff}) &= \frac{A_\text{surface}}{2\sqrt{\pi V L_\text{axis}}}, \end{align} \end{split}\]

where we find \(E_\text{eff}\) from the value of \(\tilde{C}\) using the Newton-Raphson method.

\(\Rightarrow\) V, A_surface, L_axis, r_minor, r_major, aspect_ratio, elongation

Vacuum magnetic well depth

We define the vacuum magnetic well depth as

\[ \begin{align} d_\text{well} &= \frac{\frac{dV}{d\Phi_n}(\rho=0) - \frac{dV}{d\Phi_n}(\rho=1)}{\frac{dV}{d\Phi_n}(\rho=0)} \end{align} \]

positive values of \(d_\text{well}\) indicate \(\frac{d^2V}{d\Phi_n^2} < 0\) which is favorable for stability.

\(\Rightarrow\) vacuum_magnetic_well_depth

Mercier criterion

We follow the formulas reported in Landreman & Jorge (2020), given in Bauer et al. (1984); Ichiguchi et al. (1993).

\[ D_\text{Merc} = D_\text{M,Shear} + D_\text{M,Curr} + D_\text{M,Well} + D_\text{M,Geod}, \]

where a positive value of \(D_\text{Merc}\) indicates stability and

\[\begin{split} \begin{align} D_\text{M,Shear} &= \frac{1}{16\pi^2} \pqty{\dv{\iota}{\Phi}}^2 \\ D_\text{M,Curr} &= -\frac{s_\chi}{\pqty{2\pi}^4} \dv{\iota}{\Phi} \int\dd S \frac{\vec{\Xi} \cdot \vec{B}}{\abs{\grad\Phi}^3} \\ D_\text{M,Well} &= \frac{\mu_0}{\pqty{2\pi}^6} \dv{p}{\Phi} \pqty{s_\Phi \dv[2]{V}{\Phi} - \mu_0 \dv{p}{\Phi} \int dS \frac{1}{\modB^2 \abs{\grad\Phi}}}\int\dd S \frac{\modB^2}{\abs{\grad\Phi}^3} \\ D_\text{M,Geod} &= \frac{1}{\pqty{2\pi}^6} \bqty{ \pqty{\int\dd S \frac{\mu_0 \vec{J}\cdot\vec{B}}{\abs{\grad\Phi}^3}}^2 -\pqty{\int\dd S \frac{\modB^2}{\abs{\grad\Phi}^3}} \pqty{\int\dd S \frac{\pqty{\mu_0 \vec{J}\cdot\vec{B}}^2}{\modB^2 \abs{\grad\Phi}^3}} } \\ \end{align} \end{split}\]

with

\[\begin{split} \begin{align} s_\chi &= \text{sgn}{\chi} \\ s_\Phi &= \text{sgn}{\Phi} \\ \dd S &= \abs{\grad\rho} \abs{\Jac} \dd\theta \dd\zeta = \abs{\vec{e}_\theta \times \vec{e}_\zeta} \dd\theta \dd\zeta\\ \vec{\Xi} &= \mu_0 \vec{J} - \dv{\aqty{B_\theta}}{\Phi} \vec{B} \\ \frac{\vec{\Xi} \cdot \vec{B}}{\abs{\grad\Phi}^3} &= \frac{\mu_0 \vec{J}\cdot\vec{B}}{\abs{\grad\Phi}^3} - \dv{\aqty{B_\theta}}{\Phi} \frac{\modB^2}{\abs{\grad\Phi}^3} \\ \grad\Phi &= \dv{\Phi}{\rho} \grad\rho \\ \dv{\iota}{\Phi} &= \dv{\iota}{\rho} \pqty{\dv{\Phi}{\rho}}^{-1} \\ \dv{p}{\Phi} &= \dv{p}{\rho} \pqty{\dv{\Phi}{\rho}}^{-1} \\ \dv{\aqty{B_\theta}}{\Phi} &= \dv{\aqty{B_\theta}}{\rho} \pqty{\dv{\Phi}{\rho}}^{-1} \\ \dv[2]{V}{\Phi} &= \dv[2]{V}{\rho} \pqty{\dv{\Phi}{\rho}}^{-2} - \dv{V}{\rho} \dv[2]{\Phi}{\rho} \pqty{\dv{\Phi}{\rho}}^{-3} \\ \end{align} \end{split}\]

Note that all four terms scale with \(\pqty{\dv{\Phi}{\rho}}^{-2}\).

\(\Rightarrow\) D_Merc, D_Merc_Shear, D_Merc_Curr, D_Merc_Well, D_Merc_Geod

Transformation to Boozer coordinates

In terms of the GVEC coordinates \((\rho,\thet,\zeta)\) the Boozer transform is given as

\[\begin{split} \begin{align} \rho_B &= \rho \\ \thet_B &= \thet +\lambda(\rho,\thet,\zeta) +\iota(\rho)\nu_B(\rho,\thet,\zeta) \\ \zeta_B &= \zeta \nu_B(\rho,\thet,\zeta) \\ \end{align} \end{split}\]

The derivatives are

\[\begin{split} \begin{align} \pdv{\rho_B}{\rho} &= 1, & \pdv{\thet_B}{\rho} &= \pdv{\lambda}{\rho}+\pdv{\iota}{\rho}\nu_B + \iota\pdv{\nu_B}{\rho} ,& \pdv{\zeta_B}{\rho} &= \pdv{\nu_B}{\rho} \\ \pdv{\rho_B}{\thet} &= 0 ,& \pdv{\thet_B}{\thet} &= 1 + \pdv{\lambda}{\thet} + \iota\pdv{\nu_B}{\thet} ,& \pdv{\zeta_B}{\thet} &= \pdv{\nu_B}{\thet} \\ \pdv{\rho_B}{\zeta} &= 0 ,& \pdv{\thet_B}{\zeta} &= \pdv{\lambda}{\zeta} + \iota\pdv{\nu_B}{\zeta} ,& \pdv{\zeta_B}{\zeta} &= 1 + \pdv{\nu_B}{\zeta} \\ \end{align} \end{split}\]

We can compute the ratio of the Jacobian determinants

\[\begin{split}\begin{align} \frac{\Jac}{\Jac_B} &= \frac{\erho \cdot(\ethet \times \ezeta)}{\vec{e}_{\rho_B} \cdot(\vec{e}_{\thet_B} \times \vec{e}_{\zeta_B})} \\ &= \pdv{\thet_B}{\thet}\pdv{\zeta_B}{\zeta} - \pdv{\thet_B}{\zeta}\pdv{\zeta_B}{\thet} \\ \end{align} \end{split}\]

with that we can express the inverse derivatives

\[\begin{split} \begin{align} \pdv{\rho}{\rho_B} &= 1, & \pdv{\rho}{\thet_B} &= 0 ,& \pdv{\rho}{\zeta_B} &= 0 \\ \pdv{\thet}{\rho_B} &= \frac{\Jac_B}{\Jac}\left(\pdv{\thet_B}{\zeta} \pdv{\zeta_B}{\rho} - \pdv{\zeta_B}{\zeta} \pdv{\thet_B}{\rho}\right ),& \pdv{\thet}{\thet_B} &= \frac{\Jac_B}{\Jac} \pdv{\zeta_B}{\zeta},& \pdv{\thet}{\zeta_B} &= -\frac{\Jac_B}{\Jac} \pdv{\thet_B}{\zeta} \\ \pdv{\zeta}{\rho_B} &= \frac{\Jac_B}{\Jac}\left(\pdv{\zeta_B}{\thet} \pdv{\thet_B}{\rho} - \pdv{\thet_B}{\thet} \pdv{\zeta_B}{\rho} \right ),& \pdv{\zeta}{\thet_B} &= -\frac{\Jac_B}{\Jac}\pdv{\zeta_B}{\thet},& \pdv{\zeta}{\zeta_B} &= \frac{\Jac_B}{\Jac} \pdv{\thet_B}{\thet}\\ \end{align} \end{split}\]

The basis vectors are then computed as:

\[\begin{split} \begin{align} \vec{e}_{\rho_B} &=& \erho + &\pdv{\thet}{\rho_B}\ethet + \pdv{\zeta}{\rho_B}\ezeta \\ \vec{e}_{\thet_B} &=& &\pdv{\thet}{\thet_B}\ethet + \pdv{\zeta}{\thet_B}\ezeta \\ \vec{e}_{\zeta_B} &=& &\pdv{\thet}{\zeta_B}\ethet + \pdv{\zeta}{\zeta_B}\ezeta \\ \end{align} \end{split}\]

and equivalently

\[\begin{split} \begin{align} \nabla \rho_B &= \nabla \rho \\ \nabla \thet_B &= \pdv{\thet_B}{\rho}\nabla \rho + \pdv{\thet_B}{\thet}\nabla \thet + \pdv{\thet_B}{\zeta}\nabla \zeta \\ \nabla \zeta_B &= \pdv{\zeta_B}{\rho}\nabla \rho + \pdv{\zeta_B}{\thet}\nabla \thet + \pdv{\zeta_B}{\zeta}\nabla \zeta \\ \end{align} \end{split}\]

Transformation to PEST coordinates

In terms of the GVEC coordinates \((\rho,\thet,\zeta)\) the PEST transform is given as

\[\begin{split} \begin{align} \rho_P &= \rho \\ \thet_B &= \thet + \lambda(\rho,\thet,\zeta) \\ \zeta_P &= \zeta \\ \end{align} \end{split}\]

The derivatives are

\[\begin{split} \begin{align} \pdv{\rho_P}{\rho} &= 1, & \pdv{\thet_P}{\rho} &= \pdv{\lambda}{\rho},& \pdv{\zeta_P}{\rho} &= 0,\\ \pdv{\rho_P}{\thet} &= 0,& \pdv{\thet_P}{\thet} &= 1 + \pdv{\lambda}{\thet},& \pdv{\zeta_P}{\thet} &= 0,\\ \pdv{\rho_P}{\zeta} &= 0,& \pdv{\thet_P}{\zeta} &= \pdv{\lambda}{\zeta},& \pdv{\zeta_P}{\zeta} &= 1.\\ \end{align} \end{split}\]

The reciprocal basis vectors are then

\[\begin{split} \begin{align} \grad \rho_P &= \grad \rho \\ \grad \thet_P &= \pdv{\thet_P}{\rho}\grad \rho + \pdv{\thet_P}{\thet}\grad \thet + \pdv{\thet_P}{\zeta}\grad \zeta \\ \grad \zeta_P &= \grad \zeta \\ \end{align} \end{split}\]

The Jacobian determinant is therefore

\[\begin{split} \begin{align} \Jac_P &= \frac{1}{\grad\rho_P \cdot \grad\theta_P \times \grad\zeta_P} \\ &= \frac{1}{\grad\rho \cdot \pdv{\thet_P}{\thet}\grad\thet \times \grad\zeta} \\ &= \frac{\Jac}{\pdv{\thet_P}{\thet}} \\ &= \frac{\Jac}{1 + \pdv{\lambda}{\thet}}. \\ \end{align} \end{split}\]

with that we can express the inverse derivatives

\[\begin{split} \begin{align} \pdv{\rho}{\rho_P} &= 1, & \pdv{\rho}{\thet_P} &= 0 ,& \pdv{\rho}{\zeta_P} &= 0 \\ \pdv{\thet}{\rho_P} &= -\pdv{\thet_P}{\rho}\left(\pdv{\thet_P}{\thet}\right)^{-1},& \pdv{\thet}{\thet_P} &= \left(\pdv{\thet_P}{\thet}\right)^{-1}, & \pdv{\thet}{\zeta_P} &= -\pdv{\thet_P}{\zeta}\left(\pdv{\thet_P}{\thet}\right)^{-1} \\ \pdv{\zeta}{\rho_P} &= 0,& \pdv{\zeta}{\thet_P} &= 0,& \pdv{\zeta}{\zeta_P} &= 1\\ \end{align} \end{split}\]

The basis vectors are then computed as:

\[\begin{split} \begin{align} \vec{e}_{\rho_P} &=& \erho + &\pdv{\thet}{\rho_P}\ethet \\ \vec{e}_{\thet_P} &=& &\pdv{\thet}{\thet_P}\ethet \\ \vec{e}_{\zeta_P} &=& &\pdv{\thet}{\zeta_P}\ethet + \ezeta \\ \end{align} \end{split}\]