B Vector analysis

B.1 Multidimensional derivatives

Directional derivatives are derivatives of a function of several variables along a specific direction, with the partial derivatives being special cases of these in which the derivative is derived in the direction of one of the coordinate axes.

Let $n, m\in \mathbb {N}$ be differentiable with $n, m\geq 1$ and $f:\mathbb {R}^n\to \mathbb {R}^m$. Then the derivative is a function $f':\mathbb{R}^n\to\mathbb{R}^{m\times n}$ and it holds

\[ \begin{align} \left(f'\right)_{i, j} = \left(\frac{\partial f_i}{\partial x_j}\right). \end{align} \]

The matrix $f'$ is called the Jacobi matrix of $f$. In particular, the definition of the gradient of a differentiable scalar function $g:\mathbb{R}^n\to\mathbb{R}$ results as the transpose of the derivative:

\[ \begin{align} \grad\left(g\right) \coloneqq \left(g'\right)^T = \left(\frac{\partial g}{\partial x_i}\right) \end{align} \]

Define the Nabla operator $\nabla$ by

\[ \begin{align} \nabla \coloneqq \sum_{i = 1}^{3}\mathbf{e}_i\frac{\partial}{\partial x_i}. \end{align} \]

Let $n, m\in \mathbb {N}$ with $n, m\geq 1$ and a differentiable function $T:\mathbb {R}^n\to \mathbb {R}^m$ be given and a differentiable curve $r: \mathbb {R}\to \mathbb {R}^n$ be given. One defines a function $\tau:\mathbb{R}\to\mathbb{R}^m$ by $\tau\left(t\right) \coloneqq T\left(r\left(t\right)\right)$. For the derivative $\frac{d\tau}{dt} = \frac{d}{dt}T\left(r\left(t\right)\right)$ applies

\[ \begin{align} \frac{d\tau}{dt} = T'\frac{dr}{dt}\tag{B.4}\label{eq:mehr_dim_kette}. \end{align} \]

This corresponds again to the saying external derivative times internal derivative. If you take e.g. For example, assuming that $T = T\left(x, y, z, t\right)$ is the temperature field and $\mathbf{r}\left(t\right) = \left(x(t), y(t), z(t), t\right)^T$ is a 4-particle trajectory, then $T(t) = T\left(\mathbf{r}\left(t\right)\right) = T\left(x(t), y(t), z(t), t\right)$ is the temperature at the location of the particle at time $t$. With the multidimensional chain rule applies

\[ \begin{align} \frac{dT}{dt} = \frac{\partial T}{\partial t} + \frac{dx}{dt}\frac{\partial T}{\partial x} + \frac{dy}{dt}\frac{\partial T}{\partial y} + \frac{dz}{dt}\frac{\partial T}{\partial z}. \end{align} \]

This is called the material derivative or total derivative, because the property of a solid particle is considered here. One defines a differential operator

\[ \begin{align} \md{} \coloneqq \frac{\partial}{\partial t} + u\frac{\partial}{\partial x} + v\frac{\partial}{\partial y} + w\frac{\partial}{\partial z} = \frac{\partial}{\partial t} + \mathbf{v}\cdot\nabla\tag{B.6}\label{eq:mat_derivative_op} \end{align} \]

with the components $\left(u, v, w\right)$ of the wind field. If you have been given three generalized coordinates $x_1, x_2, x_3$ (for example spherical coordinates or pressure coordinates), this can be written as

\[ \begin{align} \md{} = \frac{\partial}{\partial t} + \sum_{i = 1}^{3}u_i\frac{\partial}{\partial x_i} \end{align} \]

with the generalized velocities $u_i$. The divergence $\div$ of a vector field $\mathbf{w} = \left(w_1, w_2, w_3\right)^T$ is defined by

\[ \begin{align} \div\left(\mathbf{w}\right) \coloneqq \nabla\cdot\mathbf{w} = \sum_{i = 1}^3\frac{\partial w_i}{\partial x_i}. \end{align} \]

The Laplace operator $\Delta$ of a scalar field $\psi$ is the divergence of the gradient, i.e

\[ \begin{align} \Delta\psi \coloneqq \div\left(\grad\left(\psi\right)\right) = \nabla^2\psi = \sum_{i = 1}^3\frac{\partial^2\psi}{\partial x_i^2}. \end{align} \]

The Laplace operator applied to a vector field is defined by

\[ \begin{align} \Delta\mathbf{w} \coloneqq \sum_{i = 1}^3\mathbf{e}_i\Delta w_i. \end{align} \]

The rotation $\rot$ of a vector field is given by

\[ \begin{align} \rot\left(\mathbf{w}\right) \coloneqq \nabla\times\mathbf{w} = \sum_{i, j, k = 1}^3\epsilon_{i, j, k}\mathbf{e}_k\frac{\partial}{\partial x_i}w_j \end{align} \]

defined. One further defines the operator of the horizontal material derivative by

\[ \begin{align} \md{_h} \coloneqq \frac{\partial}{\partial t} + u\frac{\partial}{\partial x} + v\frac{\partial}{\partial y} = \frac{\partial}{\partial t} + \mathbf{v}_h\cdot\nabla\tag{B.12}\label{eq:mat_derivative_op_hor} \end{align} \]

Analogously, one defines the horizontal $\nabla-$operator by

\[ \begin{align} \nabla_h \coloneqq \nabla - \mathbf{k}\cdot\nabla. \end{align} \]

The horizontal Laplace operator $\Delta_h$ is defined by

\[ \begin{align} \Delta_h \coloneqq \frac{1}{r^2}\Delta_{\theta, \phi} = \left(\nabla_h\right)^2 \end{align} \]

with the angle component of the Laplace operator, see Eq. (B.96), and $r$ as the sphere radius. Since the i component of the rotation is the rotation of the vector field under consideration in the xj,k plane, one can

\[ \begin{align} \mathbf{k}\cdot\mathbf{w} \end{align} \]

as the horizontal component of the rotation of the vector field $\mathbf{w}$. The operator

\[ \begin{align} -\mathbf{v}\cdot\nabla \end{align} \]

is the advection operator, which is applicable to scalar and vector fields. The horizontal portion of this is

\[ \begin{align} -\mathbf{v}_h\cdot\nabla. \end{align} \]

You have a differential balance equation of the size in the form

\[ \begin{align} \md{\psi} = \sum_iF_i \end{align} \]

given with physical forcings $F_i$, the equation follows for the local time change of $\psi$

\[ \begin{align} \frac{\partial\psi}{\partial t} = -\mathbf{v}\cdot\nabla T + \sum_iF_i = -\mathbf{v}_h\cdot\nabla - w\frac{\partial L}{\partial z} + \sum_iF_i, \end{align} \]

The Advection is the proportion of the local temporal change that is caused by transport.

B.1.1 Radius of curvature

Imagine a trajectory

\[ \begin{align} \mathbf{r}:I\subseteq\mathbb{R}\to\mathbb{R}^2;\alpha\mapsto\left(x, y\right)^T \end{align} \]

before, where $I$ is an interval. We are only looking at a small section here

\[ \begin{align} \mathbf{r}:I'\subseteq I\to\mathbb{R}^2;\tau\mapsto\left(x, y\right)^T, \end{align} \]

$I'$ sei ebenfalls ein Intervall, und definiere

\[ \begin{align} \tau_1 &\coloneqq \sup\left(I'\right),\\ \tau_2 &\coloneqq \sup\left(I'\right),\\ \mathbf{r}_1 = \left(x_1, y_1\right)^T &\coloneqq f\left(\tau_1\right),\\ \mathbf{r}_2 = \left(x_2, y_2\right)^T &\coloneqq f\left(\tau_2\right).\\ \end{align} \]

O.B.d. A. one can place the origin of the KS in $\mathbf{r}_1$. Now you look for a point $\mathbf{r}_0 \coloneqq \left(x_0, y_0\right)^T$, which is required to have the same distance $\left|r\right|$ from $\mathbf{r}_1$ and $\mathbf{r}_2$, i.e

\[ \begin{align} \sqrt{\left(\mathbf{r}_1 - \mathbf{r}_0\right)^2} & \hastobe \sqrt{\left(\mathbf{r}_2 - \mathbf{r}_0\right)^2},\\ \Rightarrow r_1^2 + r_0^2 - 2\mathbf{r}_1\cdot\mathbf{r}_0 &= r_2^2 + r_0^2 - 2\mathbf{r}_2\cdot\mathbf{r}_0,\\ \Rightarrow 0 &= r_2^2 - 2\mathbf{r}_2\cdot\mathbf{r}_0.\tag{B.29}\label{eq:deriv_curv_1} \end{align} \]

Now one expands $\mathbf{r}_2$ to the second order,

\[ \begin{align} \mathbf{r}_2 &= \mathbf{r}_1 + \frac{d\mathbf{r}}{d\tau}\Delta\tau + \frac{1}{2}\frac{d^2\mathbf{r}}{d\tau^2}\Delta\tau^2 + \mathcal{O}\left(\Delta\tau^3\right) \end{align} \]

with

\[ \begin{align} \Delta\tau \coloneqq \tau_2 - \tau_1, \end{align} \]

where the higher order terms are no longer noted:

\[ \begin{align} x_2 = x'\Delta\tau + \frac{1}{2}x''\Delta\tau^2 & {} & y_2 = y'\Delta\tau + \frac{1}{2}y''\Delta\tau^2 \end{align} \]

The derivatives are to be calculated at the point $\tau = \tau_1$. The second order must be taken into account here, since curvature is about changing the derivative of a trajectory. Putting this into Eq. (B.29), follows

\[ \begin{align} 0 &= x'^2\Delta\tau^2 + \frac{1}{4}x''^2\Delta\tau^4 + x'x''\Delta\tau^3 + y'^2\Delta\tau^2 + \frac{1}{4}y''^2\Delta\tau^4 + y'y''\Delta\tau^3\nonumber\\ & -2x'\Delta\tau x_0 - x''\Delta\tau^2x_0-2y'\Delta\tau y_0 - y''\Delta\tau^2y_0. \end{align} \]

The first order of $\Delta\tau$ implies:

\[ \begin{align} 2x'x_0 + 2y'y_0 = 0\tag{B.34}\label{eq:deriv_curv_2} \end{align} \]

The second order of $\Delta\tau$ implies:

\[ \begin{align} x''x_0 + y''y_0 = x'^2 + y'^2\tag{B.35}\label{eq:deriv_curv_3} \end{align} \]

From Eq. (B.34) follows

\[ \begin{align} x_0 = -\frac{y'y_0}{x'}.\tag{B.36}\label{eq:deriv_curv_4} \end{align} \]

In Eq. Inserting (B.35) results in this

\[ \begin{align} -x''y'\frac{y_0}{x'} + y''y_0 = x'^2 + y'^2 \Rightarrow y_0 = \frac{x'^2 + y'^2}{y''-x''\frac{y'}{x'}}. \end{align} \]

With Eq. (B.36) follows

\[ \begin{align} x_0 = -\frac{y'}{x'}\frac{x'^2 + y'^2}{y''-x''\frac{y'}{x'}}. \end{align} \]

This follows for the amount of the radius of curvature

\[ \begin{align} \left|r\right| &= \sqrt{1 + \frac{y'^2}{x'^2}}\frac{x'^2+y'^2}{\left|y''-x''\frac{y'}{x'}\right|}. \end{align} \]

O.B.d. A. you can set $\tau = x$, it follows from this

\[ \begin{align} x' = 1, & {} & x'' = 0. \end{align} \]

Therefore applies

\[ \begin{align} \left|r\right| = \sqrt{1+y'^2}\frac{1+y'^2}{\left|y''\right|} & {} & \Rightarrow\left|r\right| = \frac{\left(1+y'^2\right)^{3/2}}{\left|y''\right|}. \end{align} \]

Regarding the sign of $r$, you specify that it should be positive if the trajectory curves to the left, i.e

\[ \begin{align} r &= \frac{\left(1+y'^2\right)^{3/2}}{y''}\tag{B.42}\label{eq:curv}. \end{align} \]

If you need a linear expression for $1/r$, you usually use

\[ \begin{align} \frac{1}{r} \approx y''\tag{B.43}\label{eq:curv_approx}. \end{align} \]

B.1.2 Calculation rules for differential operators

Let $\mathbf{u}, \mathbf{v}, \mathbf{w}:\mathbb{R}^3\to \mathbb{R}^3$ with $\mathbf{u} = \left(u_1, u_2, u_3\right)^T, \mathbf{v} = \left(v_1, v_2, v_3\right)^T$ and $\mathbf{w} = \left(w_1, w_2, w_3\right)^T$ three vector fields, $\psi, \chi:\mathbb{R}^3\to \mathbb{R}$ two scalar fields and $\lambda\in \mathbb{C}$ one scalar. It is defined

\[ \begin{align} \Delta\mathbf{v} \coloneqq \sum_{i = 1}^{3}\Delta v_i\mathbf{e}_i. \end{align} \]

They apply

\[ \begin{align} \nabla\left(\psi + \chi\right) = \nabla\psi + \nabla\chi, & {} & \nabla\times\left(\mathbf{v} + \mathbf{w}\right) = \nabla\times\mathbf{v} + \nabla\times\mathbf{w}, \end{align} \] \[ \begin{align} \nabla\left(\lambda\psi\right) = \lambda\nabla\psi, & {} & \nabla\times\left(\lambda\mathbf{v}\right) = \lambda\nabla\times\mathbf{v} \end{align} \]

according to the findings in section A.8. Still apply

\[ \begin{align} \nabla\times\nabla\psi &= \sum_{i, j, k = 1}^{3}\epsilon_{i, j, k}\frac{\partial}{\partial x_i}\frac{\partial\psi}{\partial x_j}\mathbf{e}_k = \sum_{i, j, k = 1}^{3}\epsilon_{i, j, k}\frac{\partial^2\psi}{\partial x_i\partial x_j}\mathbf{e}_k = \mathbf{0}, \tag{B.47}\label{eq:diff_op_rule_1}\\ \nabla\cdot\nabla\times\mathbf{v} &= \left(\sum_{l = 1}^{3}\mathbf{e}_l\frac{\partial}{\partial x_l}\right)\cdot\left(\sum_{i, j, k = 1}^{3}\epsilon_{i, j, k}\frac{\partial}{\partial x_i}v_j\mathbf{e}_k\right) = \sum_{i, j, k = 1}^{3}\epsilon_{i, j, k}\frac{\partial^2v_j}{\partial x_k\partial x_i} = 0, \tag{B.48}\label{eq:diff_op_rule_2}\\ \nabla\cdot\left(\psi\mathbf{v}\right) &= \sum_{i = 1}^{3}\frac{\partial}{\partial x_i}\left(\psi v_i\right) = \sum_{i = 1}^{3}v_i\frac{\partial\psi}{\partial x_i} + \psi\frac{\partial v_i}{\partial x_i} = \mathbf{v}\cdot\nabla\psi + \psi\nabla\cdot\mathbf{v}, \tag{B.49}\label{eq:diff_op_rule_3}\\ \frac{\partial}{\partial x_i}\nabla\times\mathbf{v} &= \frac{\partial}{\partial x_i}\sum_{j, k.l = 1}^{3}\epsilon_{j, k, l}\frac{\partial}{\partial x_j}v_k\mathbf{e}_l = \sum_{j, k, l = 1}^{3}\epsilon_{j, k, l}\frac{\partial}{\partial x_j}\frac{\partial v_k}{\partial x_i}\mathbf{e}_l = \nabla\times\frac{\partial}{\partial x_i}\mathbf{v}, \tag{B.50}\label{eq:diff_op_rule_4}\\ \nabla\times\psi\mathbf{v} &= \sum_{i, j, k = 1}^{3}\epsilon_{i, j, k}\frac{\partial}{\partial x_i}\left(\psi v_j\right)\mathbf{e}_k = \sum_{i, j, k = 1}^{3}\epsilon_{i, j, k}\left(v_j\frac{\partial\psi}{\partial x_i} + \psi\frac{\partial v_j}{\partial x_i}\right)\mathbf{e}_k\nonumber\\ &= \sum_{i, j, k = 1}^{3}\epsilon_{i, j, k} \frac{\partial\psi}{\partial x_i}v_j\mathbf{e}_k + \sum_{i, j, k = 1}^{3}\psi\frac{\partial}{\partial x_i}v_j\mathbf{e}_k = \left(\nabla\psi\right)\times\mathbf{v} + \psi\nabla\times\mathbf{v}\nonumber\\ &= -\mathbf{v}\times\nabla\psi + \psi\nabla\times\mathbf{v},\tag{B.51}\label{eq:diff_op_rule_5} \end{align} \] \[ \begin{align} \nabla\left(\mathbf{v}\cdot\mathbf{w}\right) &= \nabla\sum_{i = 1}^{3}v_iw_i = \sum_{i = 1}^{3}\frac{\partial}{\partial x_i}\left(\sum_{j = 1}^{3}v_jw_j\right)\mathbf{e}_i = \sum_{i = 1}^{3}\sum_{j = 1}^{3}\left(\frac{\partial v_j}{\partial x_i}w_j + v_j\frac{\partial w_j}{\partial x_i}\right)\mathbf{e}_i\nonumber\\ &= \sum_{i = 1}^{3}\sum_{j = 1}^{3}\left(\frac{\partial v_j}{\partial x_i}w_j + w_j\frac{\partial v_i}{\partial x_j} - w_j\frac{\partial v_i}{\partial x_j} + v_j\frac{\partial w_j}{\partial x_i} + v_j\frac{\partial w_i}{\partial x_j} - v_j\frac{\partial w_i}{\partial x_j}\right)\mathbf{e}_i\nonumber\\ &= \sum_{i = 1}^{3}\sum_{j = 1}^{3}\left(w_j\left(\frac{\partial v_j}{\partial x_i} - \frac{\partial v_i}{\partial x_j}\right) + v_j\left(\frac{\partial w_j}{\partial x_i} - \frac{\partial w_i}{\partial x_j}\right) + w_j\frac{\partial v_i}{\partial x_j} + v_j\frac{\partial w_i}{\partial x_j}\right)\mathbf{e}_i\nonumber\\ &= \sum_{i, j = 1}^{3}w_j\frac{\partial v_i}{\partial x_j}\mathbf{e}_i + \sum_{i, j = 1}^{3}v_j\frac{\partial w_i}{\partial x_j}\mathbf{e}_i + \sum_{i, j, k = 1}^{3}\epsilon_{i, j, k}^2\left(w_j\left(\frac{\partial v_j}{\partial x_i} - \frac{\partial v_i}{\partial x_j}\right) + v_j\left(\frac{\partial w_j}{\partial x_i} - \frac{\partial w_i}{\partial x_j}\right)\right)\mathbf{e}_i\nonumber\\ &= \sum_{j = 1}^{3}w_j\frac{\partial}{\partial x_j}\sum_{i = 1}^{3}v_i\mathbf{e}_i + \sum_{j = 1}^{3}v_j\frac{\partial}{\partial x_j}\sum_{i = 1}^{3}w_i\mathbf{e}_i + \sum_{i, j, k = 1}^{3}\epsilon_{i, j, k}^2\left(w_j\left(\frac{\partial v_j}{\partial x_k} - \frac{\partial v_k}{\partial x_j}\right) + v_j\left(\frac{\partial w_j}{\partial x_k} - \frac{\partial w_k}{\partial x_j}\right)\right)\mathbf{e}_k\nonumber\\ &= \left(\mathbf{w}\cdot\nabla\right)\mathbf{v} + \left(\mathbf{v}\cdot\nabla\right)\mathbf{w} + \sum_{i, j, k = 1}^{3}\epsilon_{i, j, k}^2v_j\left(\frac{\partial w_j}{\partial x_k} - \frac{\partial w_k}{\partial x_j}\right)\mathbf{e}_k + \sum_{i, j, k = 1}^{3}\epsilon_{i, j, k}^2w_j\left(\frac{\partial v_j}{\partial x_k} - \frac{\partial v_k}{\partial x_j}\right)\mathbf{e}_k\nonumber\\ &= \left(\mathbf{v}\cdot\nabla\right)\mathbf{w} + \left(\mathbf{w}\cdot\nabla\right)\mathbf{v} + \sum_{i, j, k = 1}^{3}\epsilon_{i, j, k} v_j\epsilon_{i, j, k}\left(\frac{\partial w_j}{\partial x_k} - \frac{\partial w_k}{\partial x_j}\right)\mathbf{e}_k + \sum_{i, j, k = 1}^{3}\epsilon_{i, j, k} w_j\epsilon_{i, j, k}\left(\frac{\partial v_j}{\partial x_k} - \frac{\partial v_k}{\partial x_j}\right)\mathbf{e}_k\nonumber\\ &= \left(\mathbf{v}\cdot\nabla\right)\mathbf{w} + \left(\mathbf{w}\cdot\nabla\right)\mathbf{v} + \sum_{i, j, k = 1}^{3}\epsilon_{i, j, k}v_i\epsilon_{i, j, k}\left(\frac{\partial w_i}{\partial x_k} - \frac{\partial w_k}{\partial x_i}\right)\mathbf{e}_k + \sum_{i, j, k = 1}^{3}\epsilon_{i, j, k}w_i\epsilon_{i, j, k}\left(\frac{\partial v_i}{\partial x_k} - \frac{\partial v_k}{\partial x_i}\right)\mathbf{e}_k\nonumber\\ &= \left(\mathbf{v}\cdot\nabla\right)\mathbf{w} + \left(\mathbf{w}\cdot\nabla\right)\mathbf{v} + \sum_{i, j, k = 1}^{3}\epsilon_{i, j, k}v_i\left(\nabla\times\mathbf{w}\right)_j\mathbf{e}_k + \sum_{i, j, k = 1}^{3}\epsilon_{i, j, k}w_i\left(\nabla\times\mathbf{v}\right)_j\mathbf{e}_k\nonumber\\ &= \left(\mathbf{v}\cdot\nabla\right)\mathbf{w} + \left(\mathbf{w}\cdot\nabla\right)\mathbf{v} + \mathbf{v}\times\left(\nabla\times\mathbf{w}\right) + \mathbf{w}\times\left(\nabla \times\mathbf{v}\right), \tag{B.52}\label{eq:diff_op_rule_6} \end{align} \] \[ \begin{align} \nabla\times\left(\mathbf{v}\times\mathbf{w}\right) &= \sum_{i, j, k = 1}^{3}\epsilon_{i, j, k}\frac{\partial}{\partial x_i}\left(\mathbf{v}\times\mathbf{w}\right)_j\mathbf{e}_k = \sum_{i, j, k = 1}^{3}\epsilon_{i, j, k}\frac{\partial}{\partial x_i}\left(\epsilon_{ikj}v_iw_k + \epsilon_{kij}v_kw_i\right)\mathbf{e}_k\nonumber\\ &= \sum_{i, j, k = 1}^{3}\epsilon_{i, j, k}^2\frac{\partial}{\partial x_i}\left(v_kw_i - v_iw_k\right)\mathbf{e}_k = \sum_{i, k = 1}^{3}\frac{\partial}{\partial x_i}\left(v_kw_i - v_iw_k\right)\mathbf{e}_k\nonumber\\ &= \sum_{i, k = 1}^{3}\left(\frac{\partial v_k}{\partial x_i}w_i + v_k\frac{\partial w_i}{\partial x_i} - v_i\frac{\partial w_k}{\partial x_i} - w_k\frac{\partial v_i}{\partial x_i}\right)\mathbf{e}_k = \sum_{i = 1}^{3}w_i\frac{\partial}{\partial x_i}\sum_{k = 1}^{3} v_k\mathbf{e}_k - \sum_{i = 1}^{3}v_i\frac{\partial}{\partial x_i}\sum_{k = 1}^{3}w_k\mathbf{e}_k\nonumber\\ & + \sum_{i = 1}^{3}\frac{\partial w_i}{\partial x_i}\sum_{k = 1}^{3}v_k\mathbf{e}_k - \sum_{i = 1}^{3}\frac{\partial v_i}{\partial x_i}\sum_{k = 1}^{3}w_k\mathbf{e}_k \nonumber\\ &= \left(\mathbf{w}\cdot\nabla\right)\mathbf{v} - \mathbf{w}\left(\nabla\cdot\mathbf{v}\right) + \mathbf{v}\left(\nabla\cdot\mathbf{w}\right) - \left(\mathbf{v}\cdot\nabla\right)\mathbf{w}, \tag{B.53}\label{eq:diff_op_rule_7} \end{align} \] \[ \begin{align} \nabla\left(\nabla\cdot\mathbf{v}\right) &= \nabla\sum_{i = 1}^{3}\frac{\partial v_i}{\partial x_i} = \sum_{i = 1}^{3}\sum_{j = 1}^{3}\frac{\partial^2 v_j}{\partial x_i\partial x_j}\mathbf{e}_i = \sum_{i = 1}^{3}\left(\sum_{j = 1}^{3}\left(\frac{\partial^2v_j}{\partial x_i\partial x_j}\right) + \Delta v_i - \Delta v_i\right)\mathbf{e}_i\nonumber\\ &= \Delta\mathbf{v} + \sum_{i = 1}^{3}\sum_{j = 1}^{3}\left(\frac{\partial^2 v_j}{\partial x_i\partial x_j} - \frac{\partial^2 v_i}{\partial x_j^2}\right)\mathbf{e}_i = \Delta\mathbf{v} + \sum_{i = 1}^{3}\sum_{j = 1}^{3}\frac{\partial}{\partial x_j}\left(\frac{\partial v_j}{\partial x_i} - \frac{\partial v_i}{\partial x_j}\right)\mathbf{e}_i\nonumber\\ &= \Delta\mathbf{v} + \sum_{i = 1}^{3}\sum_{\substack{j = 1,\\j\not = i}}^{3}\frac{\partial}{\partial x_j}\left(\frac{\partial v_j}{\partial x_i} - \frac{\partial v_i}{\partial x_j}\right)\mathbf{e}_i = \Delta\mathbf{v} + \sum_{i = 1}^{3}\sum_{\substack{j = 1,\\j\not = i}}^3\frac{\partial}{\partial x_j}\left[\sum_{k = 1}^{3}\epsilon_{i, j, k}^2\left(\frac{\partial v_j}{\partial x_i} - \frac{\partial v_i}{\partial x_j}\right)\right]\mathbf{e}_i\nonumber\\ &= \Delta\mathbf{v} + \sum_{i = 1}^{3}\sum_{\substack{j = 1,\\j\not = i}}^3\frac{\partial}{\partial x_j}\left[\sum_{k = 1}^{3}\epsilon_{i, j, k}\left(\nabla\times\mathbf{v}\right)_k\right]\mathbf{e}_i = \Delta\mathbf{v} + \sum_{i, j, k = 1}^{3}\epsilon_{i, j, k}\frac{\partial}{\partial x_j}\left(\nabla\times\mathbf{v}\right)_k\mathbf{e}_i\nonumber\\ &= \Delta\mathbf{v} + \sum_{i, j, k = 1}^{3}\epsilon_{i, j, k}\frac{\partial}{\partial x_i}\left(\nabla\times\mathbf{v}\right)_j\mathbf{e}_k\nonumber\\ &= \Delta\mathbf{v} + \sum_{i, j, k = 1}^{3}\left(\nabla\times\epsilon_{i, j, k}\frac{\partial}{\partial x_i}\mathbf{v}\right)_j\mathbf{e}_k = \Delta\mathbf{v} + \nabla\times \left(\sum_{i, j, k = 1}^{3}\epsilon_{i, j, k}\frac{\partial}{\partial x_i}\mathbf{v}\right)_j\mathbf{e}_k\nonumber\\ &= \Delta\mathbf{v} + \nabla\times\sum_{i, j, k = 1}^{3}\epsilon_{i, j, k}\frac{\partial}{\partial x_i}v_j\mathbf{e}_k = \Delta\mathbf{v} + \nabla\times\left(\nabla\times\mathbf{v}\right)\tag{B.54}\label{eq:diff_op_rule_8}, \end{align} \] \[ \begin{align} \nabla\cdot\left(\mathbf{v}\times\mathbf{w}\right) &= \sum_{i = 1}^3\frac{\partial}{\partial x_i}\left(\mathbf{v}\times\mathbf{w}\right)_i = \sum_{i = 1}^{3}\frac{\partial}{\partial x_i}\left(\epsilon_{k, j, i}v_kw_j + \epsilon_{j, k, i}v_jw_k\right)\nonumber\\ &= \sum_{i = 1}^{3}\left(\epsilon_{k, j, i}\frac{\partial v_k}{\partial x_i}w_j + \epsilon_{k, j, i}v_k\frac{\partial w_j}{\partial x_i} + \epsilon_{j, k, i}\frac{\partial v_j}{\partial x_i}w_k + \epsilon_{j, k, i}v_j\frac{\partial w_k}{\partial x_i}\right)\nonumber\\ &= \sum_{i = 1}^{3}w_i\left(\epsilon_{k, j, i}\frac{\partial v_j}{\partial x_k} + \epsilon_{j, k, i}\frac{\partial v_k}{\partial x_j}\right) - v_i\left(\epsilon_{k, j, i}\frac{\partial w_j}{\partial x_k} + \epsilon_{j, k, i}\frac{\partial w_k}{\partial x_j}\right)\nonumber\\ &= \sum_{i = 1}^{3}w_i\left(\nabla\times\mathbf{v}\right)_i - v_i\left(\nabla\times\mathbf{w}\right)_i = \mathbf{w}\cdot\left(\nabla\times\mathbf{v}\right) - \mathbf{v}\cdot\left(\nabla\times\mathbf{w}\right), \tag{B.55}\label{eq:diff_op_rule_9}\\ \mathbf{u}\cdot\left[\left(\mathbf{v}\cdot\nabla\right)\nabla\psi\right] &= \sum_{i=1}^3u_i\left[\left(\mathbf{v}\cdot\nabla\right)\nabla\psi\right]_i = \sum_{i=1}^3u_i\left[\left(\mathbf{v}\cdot\nabla\right)\frac{\partial\psi}{\partial x_i}\right] = \sum_{i=1}^3u_i\sum_{j=1}^3v_j\frac{\partial^2\psi}{\partial x_j\partial x_i}\nonumber\\ &= \sum_{i, j=1}^3u_iv_j\frac{\partial^2\psi}{\partial x_j\partial x_i} = \sum_{i, j=1}^3v_iu_j\frac{\partial^2\psi}{\partial x_j\partial x_i} = \sum_{i=1}^3v_i\sum_{j=1}^3u_j\frac{\partial}{\partial x_j}\left(\nabla\psi\right)_i\nonumber\\ &= \sum_{i=1}^3v_i\left[\left(\mathbf{u}\cdot\nabla\right)\nabla\psi\right]_i = \mathbf{v}\cdot\left[\left(\mathbf{u}\cdot\nabla\right)\nabla\psi\right].\tag{B.56}\label{eq:diff_op_rule_12} \end{align} \]

Also applies

\[ \begin{align} \left(\mathbf{v}\cdot\nabla\right)\mathbf{v} & \stackrel{\text{Glg. }\href{#eq:diff_op_rule_6}{(B.52)}}{=} & \frac{1}{2}\nabla\left(\mathbf{v}\cdot\mathbf{v}\right) - \mathbf{v}\times\left(\nabla\times\mathbf{v}\right).\tag{B.57}\label{eq:diff_op_rule_10} \end{align} \]

This is called Lamb transformation. This also applies to the advection of a scalar product

\[ \begin{align} \left(\mathbf{u}\cdot\nabla\right)\left(\mathbf{v}\cdot\mathbf{w}\right) &= \left(\sum_{i = 1}^{3}u_i\frac{\partial}{\partial x_i}\right)\left(\sum_{j = 1}^{3}v_jw_j\right) = \sum_{i, j = 1}^{3}u_i\left[\frac{\partial v_j}{\partial x_i}w_j + \frac{\partial w_j}{\partial x_i}v_j\right]\nonumber\\ &= \sum_{j = 1}^{3}w_j\sum_{i = 1}^{3}u_i\frac{\partial v_j}{\partial x_i} + \sum_{j = 1}^{3}v_j\sum_{i = 1}^{3}u_i\frac{\partial w_j}{\partial x_i} = \mathbf{w}\cdot\left[\left(\mathbf{u}\cdot\nabla\right)\mathbf{v}\right] + \mathbf{v}\cdot\left[\left(\mathbf{u}\cdot\nabla\right)\mathbf{w}\right].\tag{B.58}\label{eq:diff_op_rule_11} \end{align} \]

Let $\mathbf{k} = \left(k_1, k_2, k_3\right)^T\in \mathbb{C}^3$, $\varphi:\mathbb{R}^3\to \mathbb{C};\varphi\left(\mathbf{r}\right) = \exp\left(\mathbf{k}\cdot\mathbf{r}\right)$ and $\mathbf{A}\in \mathbb{R}^3$. Then apply

\[ \begin{align} \nabla\varphi\left(\mathbf{r}\right) &= \sum_{i = 1}^{3}\frac{\partial}{\partial x_i}\varphi\left(\mathbf{r}\right)\mathbf{e}_i = \sum_{i = 1}^{3}\frac{\partial}{\partial x_i}\exp\left(\mathbf{k}\cdot\mathbf{r}\right)\mathbf{e}_i = \sum_{i = 1}^{3}\frac{\partial}{\partial x_i}\exp\left(\sum_{j = 1}^{3}k_jx_j\right)\mathbf{e}_i\nonumber\\ &= \sum_{i = 1}^{3}k_i\exp\left(\mathbf{k}\cdot\mathbf{r}\right)\mathbf{e}_i = \varphi\left(\mathbf{r}\right)\mathbf{k},\\ \nabla\times\left[\mathbf{A}\varphi\left(\mathbf{r}\right)\right] &= \sum_{i, j, k = 1}^3\epsilon_{i, j, k}\frac{\partial\left[A_j\exp\left(i\mathbf{k}\cdot\mathbf{r}\right)\right]}{\partial x_i}\mathbf{e}_k = i\sum_{i, j, k = 1}^3\epsilon_{i, j, k}A_jk_i\exp\left(i\mathbf{k}\cdot\mathbf{r}\right)\mathbf{e}_k\nonumber\\ &= i\exp\left(i\mathbf{k}\cdot\mathbf{r}\right)\sum_{i, j, k = 1}^3\epsilon_{i, j, k}A_jk_i\mathbf{e}_k = i\exp\left(i\mathbf{k}\cdot\mathbf{r}\right)\mathbf{A}\times\mathbf{k} = i\left[\mathbf{A}\varphi\left(\mathbf{r}\right)\right]\times\mathbf{k},\tag{B.60}\label{eq:diff_op_harmonic_rule_1}\\ \Delta\varphi\left(\mathbf{r}\right) &= \nabla\cdot\nabla\varphi\left(\mathbf{r}\right) = \nabla\cdot\left(\varphi\left(\mathbf{r}\right)\mathbf{k}\right) = \mathbf{k}\cdot\nabla\varphi = \mathbf{k}^2\varphi\left(\mathbf{r}\right). \end{align} \]

B.2 Differential operators under coordinate transformations

In general, points in $\mathbb{R}^3$ are denoted by three real numbers $x = \left(x_1, x_2, x_3\right)$, which in linear combination with the standard base $\mathbf{c}_1, \mathbf{c}_2, \mathbf{c}_3$ produce a vector $\mathbf{r}$:

\[ \begin{align} \mathbf{r} = x_1\mathbf{c}_1 + x_2\mathbf{c}_2 + x_3\mathbf{c}_3 \end{align} \]

In this context, you can imagine a vector as an arrow. In order to be able to solve things with a certain geometry more elegantly, generalized coordinates $q_i$ are introduced (see also section 2.2). Scalar fields $f$ can then be written as $f = f\left(q\right)$. Here $q$ stands for the tuple $q = \left(q_1, q_2, q_3\right)$. As a note, the functions $f = f\left(x\right)$ and $f = f\left(q\right)$ are not the same, if you use the same arguments you generally get different values. Most of the time, however, there is no distinction between this in the notation because both functions express the same thing.

If you want to write down a vector field $\mathbf{v}$ in generalized coordinates, the simplest idea would be to also write down the values ​​of $\mathbf{v}$ in generalized coordinates. If $\mathbf{v}$ is a velocity field, one could write down the values ​​of $\mathbf{v}$ in the forms $\newdot{q}$ or $\mathbf{v} \equiv \mathbf{r}\left(q\right)$. Since this is often unclear, in general coordinate systems (also called curvilinear coordinates) a location-dependent basis is introduced, the elements of which are defined by

\[ \begin{align} \mathbf{e}_i \coloneqq \frac{1}{|\frac{\partial\mathbf{r}}{\partial q_i}|}\frac{\partial\mathbf{r}}{\partial q_i}. \end{align} \]

These elements are standardized. Can

\[ \begin{align} \mathbf{e}_1\times\mathbf{e}_2 = \mathbf{e}_3\tag{B.64}\label{eq:gen_coords_orth_criterion} \end{align} \]

achieved by swapping the $\mathbf{e}_i$, the $q$ is called orthogonal; Coordinates that violate this are not discussed in this book. One then writes for vector fields

\[ \begin{align} \mathbf{v} = \sum_{i = 1}^{3}v_i\mathbf{e}_i, \end{align} \]

where both the $v_i$ and the $\mathbf{e}_i$ depend on the location. Partial derivatives with respect to the local basis are defined by directional derivatives along the local coordinate axes:

\[ \begin{align} \frac{\partial}{\partial x_i} \coloneqq \mathbf{e}_i\cdot\nabla = \frac{1}{|\frac{\partial\mathbf{r}}{\partial q_i}|}\left(\frac{\partial x}{\partial q_i}\frac{\partial}{\partial x} + \frac{\partial y}{\partial q_i}\frac{\partial }{\partial y} + \frac{\partial z}{\partial q_i}\frac{\partial }{\partial z}\right) = \frac{1}{|\frac{\partial\mathbf{r}}{\partial q_i}|}\frac{\partial}{\partial q_i}.\tag{B.66}\label{eq:part_derivative_gen} \end{align} \]

Let $\mathbf{v}$, $\mathbf{w}$ be two vector fields, then the dot product is

\[ \begin{align} \mathbf{v}\cdot\mathbf{w} = v^{(m)}w^{(n)}g_{m,n}.\tag{B.67}\label{eq:inner_gen} \end{align} \]

For the vector product applies

\[ \begin{align} \mathbf{v}\times\mathbf{w} = \left|\begin{array}{ccc} \mathbf{q}_2\times\mathbf{q}_3 & \mathbf{q}_3\times\mathbf{q}_1 & \mathbf{q}_1\times\mathbf{q}_2\\ v^{(1)} & v^{(2)} & v^{3}\\ w^{(1)} & w^{(2)} & w^{3} \end{array}\right|\tag{B.68}\label{eq:vector_product_gen} \end{align} \]

Let $f$ be a scalar field, then the gradient of $f$ in generalized coordinates holds

\[ \begin{align} \nabla f = \mathbf{q}^{(n)}\frac{\partial}{\partial q^{(n)}} = \mathbf{q}^{(n)}\nabla_nf.\tag{B.69}\label{eq:grad_gen} \end{align} \]

This therefore applies in generalized coordinates

\[ \begin{align} \nabla\cdot\mathbf{v} = \frac{1}{\sqrt{g}}\frac{\partial}{\partial q^{(n)}}\left(\mathbf{q}^{(n)}\cdot\mathbf{v}\sqrt{g}\right)\tag{B.70}\label{eq:div_gen}. \end{align} \]

So it applies

\[ \begin{align} \nabla\times\mathbf{v} = \frac{1}{\sqrt{g}}\left|\begin{array}{ccc} \mathbf{q}_1 & \mathbf{q}_2 & \mathbf{q}_3 \\ \frac{\partial}{\partial q^{(1)}} & \frac{\partial}{\partial q^{(2)}} & \frac{\partial}{\partial q^{3}} \\ v_1 & v_2 & v_3 \end{array}\right|\tag{B.71}\label{eq:rot_gen} \end{align} \]

B.2.1 Spherical coordinates

In this section $\mathbf{v}$ is a vector field and $f$ is a scalar field, both are said to have all the necessary properties. In spherical coordinates applies

\[ \begin{align} \mathbf{r} = r\left(\begin{array}{c} \sin\left(\theta\right)\cos\left(\phi\right)\\ \sin\left(\theta\right)\sin\left(\phi\right)\\ \cos\left(\theta\right) \end{array}\right). \end{align} \]

Let $\mathbf{r} = \mathbf{r}\left(r, \theta, \phi\right)$ be the transformation from spherical to Cartesian coordinates, then the equation $J$ applies to their derivation

\[ \begin{align} J &= \left(\begin{array}{ccc} \frac{\partial x}{\partial r}&\frac{\partial x}{\partial\theta}&\frac{\partial x}{\partial\phi}\\ \frac{\partial y}{\partial r}&\frac{\partial y}{\partial\theta}&\frac{\partial y}{\partial\phi}\\ \frac{\partial z}{\partial r}&\frac{\partial z}{\partial\theta}&\frac{\partial z}{\partial\phi} \end{array}\right) = \left(\begin{array}{ccc} \sin\left(\theta\right)\cos\left(\phi\right)&r\cos\left(\theta\right)\cos\left(\phi\right)& -r\sin\left(\theta\right)\sin\left(\phi\right)\\ \sin\left(\theta\right)\sin\left(\phi\right)&r\cos\left(\theta\right)\sin\left(\phi\right)&r\sin\left(\theta\right)\cos\left(\phi\right)\\ \cos\left(\theta\right)& -r\sin\left(\theta\right)&0 \end{array}\right). \end{align} \]

The normalized columns of this are

\[ \begin{align} \mathbf{e}_r = \mathbf{e}^{(r)} &= \left(\begin{array}{c} \sin\left(\theta\right)\cos\left(\phi\right)\\ \sin\left(\theta\right)\sin\left(\phi\right)\\ \cos\left(\theta\right) \end{array}\right),\tag{B.74}\label{eq:kugel_zu_global_1}\\ \mathbf{e}_\theta = \mathbf{e}^{(\theta)} &= \left(\begin{array}{c} \cos\left(\theta\right)\cos\left(\phi\right)\\ \cos\left(\theta\right)\sin\left(\phi\right)\\ - \sin\left(\theta\right) \end{array}\right),\tag{B.75}\label{eq:kugel_zu_global_2}\\ \mathbf{e}_\phi = \mathbf{e}^{(\phi)} &= \left(\begin{array}{c} - \sin\left(\phi\right)\\ \cos\left(\phi\right)\\ 0 \end{array}\right).\tag{B.76}\label{eq:kugel_zu_global_3} \end{align} \]

The columns of $J$ are the covariant basis vectors

\[ \begin{align} \mathbf{q}_r &= \left(\begin{array}{c} \sin\left(\theta\right)\cos\left(\phi\right)\\ \sin\left(\theta\right)\sin\left(\phi\right)\\ \cos\left(\theta\right) \end{array}\right),\\ \mathbf{q}_\theta &= r\left(\begin{array}{c} \cos\left(\theta\right)\cos\left(\phi\right)\\ \cos\left(\theta\right)\sin\left(\phi\right)\\ -\sin\left(\theta\right) \end{array}\right),\\ \mathbf{q}_\phi &= r\sin\left(\theta\right)\left(\begin{array}{c} -\sin\left(\phi\right)\\ \cos\left(\phi\right)\\ 0 \end{array}\right). \end{align} \]

This results in the determinant $g$ of the metric tensor

\[ \begin{align} g = \left|\begin{array}{ccc} 1 & 0 & 0 \\ 0 & r^2 & 0 \\ 0 & 0 & r^2\sin^2\left(\theta\right) \end{array}\right| = r^4\sin^2\left(\theta\right). \end{align} \]

Since this matrix is ​​diagonal, spherical coordinates are orthogonal. This therefore applies to the contravariant basis vectors

\[ \begin{align} \mathbf{q}^{(r)} &= \left(\begin{array}{c} \sin\left(\theta\right)\cos\left(\phi\right)\\ \sin\left(\theta\right)\sin\left(\phi\right)\\ \cos\left(\theta\right) \end{array}\right),\\ \mathbf{q}^{(\theta)} &= \frac{1}{r}\left(\begin{array}{c} \cos\left(\theta\right)\cos\left(\phi\right)\\ \cos\left(\theta\right)\sin\left(\phi\right)\\ -\sin\left(\theta\right) \end{array}\right),\\ \mathbf{q}^{(\phi)} &= \frac{1}{r\sin\left(\theta\right)}\left(\begin{array}{c} -\sin\left(\phi\right)\\ \cos\left(\phi\right)\\ 0 \end{array}\right). \end{align} \]

Now you write for a vector field

\[ \begin{align} \mathbf{v} &= v^{(r)}\mathbf{q}_r + v^{(\theta)}\mathbf{q}_\theta + v^{(\phi)}\mathbf{q}_\phi = v_r\mathbf{q}^{(r)} + v_\theta\mathbf{q}^{(\theta)} + v_\phi\mathbf{q}^{(\phi)}\nonumber\\ &= \newtilde{v}^{(r)}\mathbf{e}_r + \newtilde{v}^{(\theta)}\mathbf{e}_\theta + \newtilde{v}^{(\phi)}\mathbf{e}_\phi = \newtilde{v}_r\mathbf{e}^{(r)} + \newtilde{v}_\theta\mathbf{e}^{(\theta)} + \newtilde{v}_\phi\mathbf{e}^{(\phi)}. \end{align} \]

It should be clear at this point that the dimensions of the components of $\mathbf{v}$ appearing here in the first line are not uniform, since the same is true for the bassi elements. Due to the orthogonality of the spherical coordinates,

\[ \begin{align} \newtilde{v}^{(r)} = \newtilde{v}_r, & {} & \newtilde{v}^{(\theta)} = \newtilde{v}_\theta, & {} & \newtilde{v}^{(\phi)} = \newtilde{v}_\phi. \end{align} \]

The conversion of the components is:

\[ \begin{align} v_r = \newtilde{v}_r, & {} & v_\theta = r\newtilde{v}_\theta, & {} & v_\phi = r\sin\left(\theta\right)\newtilde{v}_\phi,\\ v^{(r)} = \newtilde{v}_r, & {} & v^{(\theta)} = \frac{1}{r}\newtilde{v}_\theta, & {} & v^{(\phi)} = \frac{1}{r\sin\left(\theta\right)}\newtilde{v}_\phi. \end{align} \]

First, the gradient is considered, with Eq. (B.69) is obtained

\[ \begin{align} \nabla f &= \mathbf{q}^{(r)}\frac{\partial f}{\partial r} + \mathbf{q}^{(\theta)}\frac{\partial f}{\partial\theta} + \mathbf{q}^{(\phi)}\frac{\partial f}{\partial\phi} = \mathbf{e}^{(r)}\frac{\partial f}{\partial r} + \frac{1}{r}\mathbf{e}^{(\theta)}\frac{\partial f}{\partial\theta} + \frac{1}{r\sin\left(\theta\right)}\mathbf{e}^{(\phi)}\frac{\partial f}{\partial\phi}.\tag{B.88}\label{eq:grad_sphere} \end{align} \]

Now the divergence is considered. Vm Eq. To be able to use (B.70), you first calculate

\[ \begin{align} \mathbf{q}^{(r)}\cdot\mathbf{v} = v^{(r)}, & {} & \mathbf{q}^{(\theta)}\cdot\mathbf{v} = v^{(\theta)}, & {} & \mathbf{q}^{(\phi)}\cdot\mathbf{v} = v^{(\phi)}. \end{align} \]

Therefore applies

\[ \begin{align} \nabla\cdot\mathbf{v} &= \frac{1}{r^2\sin\left(\theta\right)}\left(\frac{\partial\left(v^{(r)}r^2\sin\left(\theta\right)\right)}{\partial r} + \frac{\partial\left(v^{(\theta)}r^2\sin\left(\theta\right)\right)}{\partial\theta} + \frac{\partial\left(v^{(\phi)}r^2\sin\left(\theta\right)\right)}{\partial\phi}\right)\nonumber\\ &= \frac{\partial v^{(r)}}{\partial r} + \frac{\partial v^{(\theta)}}{\partial\theta} + \frac{\partial v^{(\phi)}}{\partial\phi} + \frac{2v^{(r)}}{r} + \cot\left(\theta\right)v^{(\theta)}\nonumber\\ &= \frac{\partial\newtilde{v}^{(r)}}{\partial r} + \frac{1}{r}\frac{\partial\newtilde{v}^{(\theta)}}{\partial\theta} + \frac{1}{r\sin\left(\theta\right)}\frac{\partial\newtilde{v}^{(\phi)}}{\partial\phi} + \frac{2\newtilde{v}^{(r)}}{r} + \frac{\cot\left(\theta\right)}{r}\newtilde{v}^{(\theta)}.\tag{B.90}\label{eq:div_sphere} \end{align} \]

By combining the equations (B.88) and (B.90) we get for the Laplace operator

\[ \begin{align} \Delta f &= \frac{\partial^2f}{\partial r^2} + \frac{1}{r^2}\frac{\partial^2f}{\partial\theta^2} + \frac{1}{r^2\sin^2\left(\theta\right)}\frac{\partial^2f}{\partial\phi^2} + \frac{2}{r}\frac{\partial f}{\partial r} + \frac{\cot\left(\theta\right)}{r^2}\frac{\partial f}{\partial\theta}\nonumber\\ &= \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial f}{\partial r}\right) + \frac{1}{r^2\sin\left(\theta\right)}\frac{\partial}{\partial\theta}\left(\sin\left(\theta\right)\frac{\partial f}{\partial\theta}\right) + \frac{1}{r^2\sin^2\left(\theta\right)}\frac{\partial^2f}{\partial\phi^2}.\tag{B.91}\label{eq:laplace_klugel} \end{align} \]

This can be reshaped a bit. Diffeomorphism is used for this

\[ \begin{align} \theta\leftrightarrow\mu \coloneqq \cos\left(\theta\right), \end{align} \]

then applies

\[ \begin{align} \frac{d}{d\theta} = \frac{d\mu}{d\theta}\frac{d}{d\mu} = -\sin\left(\theta\right)\frac{d}{d\mu}. \end{align} \]

This means that the Laplace operator becomes

\[ \begin{align} \Delta &= \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial}{\partial r}\right) + \frac{1}{r^2\sin\left(\theta\right)}\left(-\sin\left(\theta\right)\frac{\partial}{\partial\mu}\right)\left(\sin\left(\theta\right)\left(-\sin\left(\theta\right)\frac{d}{d\mu}\right)\right) + \frac{1}{r^2\sin^2\left(\theta\right)}\frac{\partial^2}{\partial\phi^2}\nonumber\\ &= \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial}{\partial r}\right) + \frac{1}{r^2}\frac{\partial}{\partial\mu}\left(1 - \mu^2\right)\frac{\partial}{\partial\mu} + \frac{1}{r^2\left(1 - \mu^2\right)}\frac{\partial^2}{\partial\phi^2}\nonumber\\ &= \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial}{\partial r}\right) + \frac{1}{r^2}\Delta_{\mu, \phi}\tag{B.94}\label{eq:laplace_kugel_split} \end{align} \]

with

\[ \begin{align} \Delta_{\mu, \phi} = \frac{\partial}{\partial\mu}\left(1 - \mu^2\right)\frac{\partial}{\partial \mu} + \frac{1}{1 - \mu^2}\frac{\partial^2}{\partial\phi^2} \end{align} \]

as the angular component of the Laplace operator. This angular component is calculated in spherical coordinates by comparing with Eq. (B.91) to

\[ \begin{align} \Delta_{\theta, \phi} = \frac{1}{\sin\left(\theta\right)}\frac{\partial}{\partial\theta}\left(\sin\left(\theta\right)\frac{\partial}{\partial\theta}\right) + \frac{1}{\sin^2\left(\theta\right)}\frac{\partial^2}{\partial\phi^2}.\tag{B.96}\label{eq:laplace_winkelanteil} \end{align} \]

Now the rotation is calculated, using Eq. (B.71)

\[ \begin{align} \nabla\times\mathbf{v} &= \frac{1}{r^2\sin\left(\theta\right)}\left[\mathbf{q}_r\left(\frac{\partial v_\phi}{\partial\theta} - \frac{\partial v_\theta}{\partial\phi}\right) + \mathbf{q}_\theta\left(\frac{\partial v_r}{\partial\phi} - \frac{\partial v_\phi}{\partial r}\right) + \mathbf{q}_\phi\left(\frac{\partial v_\theta}{\partial r} - \frac{\partial v_r}{\partial\theta}\right)\right]. \end{align} \]

By rescaling to standardized basic elements you get

\[ \begin{align} \nabla\times\mathbf{v} &= \frac{1}{r^2\sin\left(\theta\right)}\Bigg[\mathbf{e}_r\left(\frac{\partial\left(r\sin\left(\theta\right)\newtilde{v}_\phi\right)}{\partial\theta} - \frac{\partial\left(r\newtilde{v}_\theta\right)}{\partial\phi}\right) + r\mathbf{e}_\theta\left(\frac{\partial\newtilde{v}_r}{\partial\phi} - \frac{\partial\left(r\sin\left(\theta\right)\newtilde{v}_\phi\right)}{\partial r}\right)\nonumber\\ & + r\sin\left(\theta\right)\mathbf{e}_\phi\left(\frac{\partial\left(r\newtilde{v}_\theta\right)}{\partial r} - \frac{\partial\newtilde{v}_r}{\partial\theta}\right)\Bigg]\nonumber\\ &= \frac{\newtilde{v}_\theta}{r}\mathbf{e}_\phi + \frac{\newtilde{v}_\phi}{r\tan\left(\theta\right)}\mathbf{e}_r - \frac{\newtilde{v}_\phi}{r}\mathbf{e}_\theta + \mathbf{e}_r\left(\frac{1}{r}\frac{\partial\newtilde{v}_\phi}{\partial\theta} - \frac{1}{r\sin\left(\theta\right)}\frac{\partial\newtilde{v}_\theta}{\partial\phi}\right)\nonumber\\ & + \mathbf{e}_\theta\left(\frac{1}{r\sin\left(\theta\right)}\frac{\partial\newtilde{v}_r}{\partial\phi} - \frac{\partial\newtilde{v}_\phi}{\partial r}\right) + \mathbf{e}_\phi\left(\frac{\partial \newtilde{v}_\theta}{\partial r} - \frac{1}{r}\frac{\partial\newtilde{v}_r}{\partial\theta}\right)\tag{B.98}\label{eq:rot_sphere} \end{align} \]

For the velocity advection one uses the Lamb transformation Eq. (B.57)

\[ \begin{align} \left(\mathbf{v}\cdot\nabla\right)\mathbf{v}&= \frac{1}{2}\nabla\left(\mathbf{v}\cdot\mathbf{v}\right) + \left(\nabla\times\mathbf{v}\right)\times\mathbf{v}. \end{align} \]

In preparation, one uses Eq. (B.67):

\[ \begin{align} \mathbf{v}^2 = \left(v^{(r)}\right)^2 + r^2\left(v^{(\theta)}\right)^2 + r^2\sin^2\left(\theta\right)\left(v^{(\phi)}\right)^2 \end{align} \]

From this it follows

\[ \begin{align} \frac{1}{2}\nabla\left(\mathbf{v}\cdot\mathbf{v}\right) &= \frac{1}{2}\mathbf{q}^{(r)}\frac{\partial}{\partial r}\left(\left(v^{(r)}\right)^2 + r^2\left(v^{(\theta)}\right)^2 + r^2\sin^2\left(\theta\right)\left(v^{(\phi)}\right)^2\right)\nonumber\\ & + \frac{1}{2}\mathbf{q}^{(\theta)}\frac{\partial}{\partial\theta}\left(\left(v^{(r)}\right)^2 + r^2\left(v^{(\theta)}\right)^2 + r^2\sin^2\left(\theta\right)\left(v^{(\phi)}\right)^2\right)\nonumber\\ & + \frac{1}{2}\mathbf{q}^{(\phi)}\frac{\partial}{\partial\phi}\left(\left(v^{(r)}\right)^2 + r^2\left(v^{(\theta)}\right)^2 + r^2\sin^2\left(\theta\right)\left(v^{(\phi)}\right)^2\right)\nonumber\\ &= \mathbf{q}^{(r)}\left(v^{(r)}\frac{\partial v^{(r)}}{\partial r} + r^2v^{(\theta)}\frac{\partial v^{(\theta)}}{\partial r} + r\left(v^{(\theta)}\right)^2 + r\sin^2\left(\theta\right)\left(v^{(\phi)}\right)^2 + r^2\sin^2\left(\theta\right)v^{(\phi)}\frac{\partial v^{(\phi)}}{\partial r}\right)\nonumber\\ & + \mathbf{q}^{(\theta)}\left(v^{(r)}\frac{\partial v^{(r)}}{\partial\theta} + r^2v^{(\theta)}\frac{\partial v^{(\theta)}}{\partial\theta} + r^2\sin\left(\theta\right)\cos\left(\theta\right)\left(v^{(\phi)}\right)^2 + r^2\sin^2\left(\theta\right)v^{(\phi)}\frac{\partial v^{(\phi)}}{\partial\theta}\right)\nonumber\\ & + \mathbf{q}^{(\phi)}\left(v^{(r)}\frac{\partial v^{(r)}}{\partial\phi} + r^2v^{(\theta)}\frac{\partial v^{(\theta)}}{\partial\phi} + r^2\sin^2\left(\theta\right)v^{(\phi)}\frac{\partial v^{(\phi)}}{\partial\phi}\right). \end{align} \]

Vm Eq. To be able to evaluate (B.68), you first calculate it

\[ \begin{align} \mathbf{q}_r\times\mathbf{q}_\theta &= \frac{1}{\sin\left(\theta\right)}\mathbf{q}_\phi = r^2\sin\left(\theta\right)\mathbf{q}^{(\phi)},\\ \mathbf{q}_\theta\times\mathbf{q}_\phi &= r^2\sin\left(\theta\right)\mathbf{q}_r = r^2\sin\left(\theta\right)\mathbf{q}^{(r)},\\ \mathbf{q}_\phi\times\mathbf{q}_r &= \sin\left(\theta\right)\mathbf{q}_\theta = r^2\sin\left(\theta\right)\mathbf{q}^{(\theta)}. \end{align} \]

It follows with Eq. (B.68)

\[ \begin{align} & \left(\nabla\times\mathbf{v}\right)\times\mathbf{v}\nonumber\\ &= \left|\begin{array}{ccc} \mathbf{q}^{(r)} & \mathbf{q}^{(\theta)} & \mathbf{q}^{(\phi)}\\ \frac{\partial v_\phi}{\partial\theta} - \frac{\partial v_\theta}{\partial\phi} & \frac{\partial v_r}{\partial\phi} - \frac{\partial v_\phi}{\partial r} & \frac{\partial v_\theta}{\partial r} - \frac{\partial v_r}{\partial\theta}\\ v^{(r)} & v^{(\theta)} & v^{(\phi)} \end{array}\right|\nonumber\\ &= \mathbf{q}^{(r)}\left[v^{(\phi)}\left(\frac{\partial v_r}{\partial\phi} - \frac{\partial v_\phi}{\partial r}\right) - v^{(\theta)}\left(\frac{\partial v_\theta}{\partial r} - \frac{\partial v_r}{\partial\theta}\right)\right]\nonumber\\ & + \mathbf{q}^{(\theta)}\left[v^{(r)}\left(\frac{\partial v_\theta}{\partial r} - \frac{\partial v_r}{\partial\theta}\right) - v^{(\phi)}\left(\frac{\partial v_\phi}{\partial\theta} - \frac{\partial v_\theta}{\partial\phi}\right)\right]\nonumber\\ & + \mathbf{q}^{(\phi)}\left[v^{(\theta)}\left(\frac{\partial v_\phi}{\partial\theta} - \frac{\partial v_\theta}{\partial\phi}\right) - v^{(r)}\left(\frac{\partial v_r}{\partial\phi} - \frac{\partial v_\phi}{\partial r}\right)\right]\nonumber\\ &= \mathbf{q}^{(r)}\left[v^{(\phi)}\left(\frac{\partial v^{(r)}}{\partial\phi} - \frac{\partial\left(r^2\sin^2\left(\theta\right)v^{(\phi)}\right)}{\partial r}\right) - v^{(\theta)}\left(\frac{\partial\left(r^2v^{(\theta)}\right)}{\partial r} - \frac{\partial v^{(r)}}{\partial\theta}\right)\right]\nonumber\\ & + \mathbf{q}^{(\theta)}\left[v^{(r)}\left(\frac{\partial \left(r^2v^{(\theta)}\right)}{\partial r} - \frac{\partial v^{(r)}}{\partial\theta}\right) - v^{(\phi)}\left(\frac{\partial\left(r^2\sin^2\left(\theta\right)v^{(\phi)}\right)}{\partial\theta} - \frac{\partial\left(r^2v^{(\theta)}\right)}{\partial\phi}\right)\right]\nonumber\\ & + \mathbf{q}^{(\phi)}\left[v^{(\theta)}\left(\frac{\partial\left(r^2\sin^2\left(\theta\right)v^{(\phi)}\right)}{\partial\theta} - \frac{\partial\left(r^2v^{(\theta)}\right)}{\partial\phi}\right) - v^{(r)}\left(\frac{\partial v^{(r)}}{\partial\phi} - \frac{\partial\left(r^2\sin^2\left(\theta\right)v^{(\phi)}\right)}{\partial r}\right)\right]\nonumber\\ &= \mathbf{q}^{(r)}\left[v^{(\phi)}\left(\frac{\partial v^{(r)}}{\partial\phi} - r^2\sin^2\left(\theta\right)\frac{\partial v^{(\phi)}}{\partial r} - 2r\sin^2\left(\theta\right)v^{(\phi)}\right) - v^{(\theta)}\left(2rv^{(\theta)} + r^2\frac{\partial v^{(\theta)}}{\partial r} - \frac{\partial v^{(r)}}{\partial\theta}\right)\right]\nonumber\\ & + \mathbf{q}^{(\theta)}\left[v^{(r)}\left(2rv^{(\theta)} + r^2\frac{\partial v^{(\theta)}}{\partial r} - \frac{\partial v^{(r)}}{\partial\theta}\right) - v^{(\phi)}\left(r^2\sin^2\left(\theta\right)\frac{\partial v^{(\phi)}}{\partial\theta} + 2r^2\sin\left(\theta\right)\cos\left(\theta\right)v^{(\phi)} - r^2\frac{\partial v^{(\theta)}}{\partial\phi}\right)\right]\nonumber\\ & + \mathbf{q}^{(\phi)}\bigg[v^{(\theta)}\left(r^2\sin^2\left(\theta\right)\frac{\partial v^{(\phi)}}{\partial\theta} + 2r^2\sin\left(\theta\right)\cos\left(\theta\right)v^{(\phi)} - r^2\frac{\partial v^{(\theta)}}{\partial\phi}\right)\nonumber\\ & - v^{(r)}\left(\frac{\partial v^{(r)}}{\partial\phi} - 2r\sin^2\left(\theta\right)v^{(\phi)} - r^2\sin^2\left(\theta\right)\frac{\partial v^{(\phi)}}{\partial r}\right)\bigg]. \end{align} \]

Thus follows

\[ \begin{align} & \left(\mathbf{v}\cdot\nabla\right)\mathbf{v}\nonumber\\ &= \mathbf{q}^{(r)}\left[v^{(r)}\frac{\partial v^{(r)}}{\partial r} + v^{(\theta)}\frac{\partial v^{(r)}}{\partial\theta} + v^{(\phi)}\frac{\partial v^{(r)}}{\partial\phi} - r\left(v^{(\theta)}\right)^2 - r\sin^2\left(\theta\right)\left(v^{(\phi)}\right)^2\right]\nonumber\\ & + \mathbf{q}^{(\theta)}\left[r^2\left(v^{(r)}\frac{\partial v^{(\theta)}}{\partial r} + v^{(\theta)}\frac{\partial v^{(\theta)}}{\partial\theta} + v^{(\phi)}\frac{\partial v^{(\theta)}}{\partial\phi}\right) - r^2\sin\left(\theta\right)\cos\left(\theta\right)\left(v^{(\phi)}\right)^2 + 2rv^{(\theta)}v^{(r)}\right]\nonumber\\ & + \mathbf{q}^{(\phi)}\bigg[r^2\sin^2\left(\theta\right)\left(v^{(r)}\frac{\partial v^{(\phi)}}{\partial r} + v^{(\theta)}\frac{\partial v^{(\phi)}}{\partial\theta} + v^{(\phi)}\frac{\partial v^{(\phi)}}{\partial\phi}\right)\nonumber\\ & + 2r\sin\left(\theta\right)v^{(\phi)}\left(\sin\left(\theta\right)v^{(r)} + r\cos\left(\theta\right)v^{(\theta)}\right)\bigg]\nonumber\\ &= \mathbf{e}_r\left[\newtilde{v}_r\frac{\partial\newtilde{v}_r}{\partial r} + \newtilde{v}_\theta\frac{1}{r}\frac{\partial\newtilde{v}_r}{\partial\theta} + \newtilde{v}_\phi\frac{1}{r\sin\left(\theta\right)}\frac{\partial\newtilde{v}_r}{\partial\phi} - \frac{1}{r}\left(\newtilde{v}_\theta\right)^2 - \frac{1}{r}\left(\newtilde{v}_\phi\right)^2\right]\nonumber\\ & + \frac{1}{r}\mathbf{e}_\theta\left[r^2\left(\newtilde{v}_r\frac{1}{r}\frac{\partial\newtilde{v}_\theta}{\partial r} - \frac{\newtilde{v}_r\newtilde{v}_\theta}{r^2} + \newtilde{v}_\theta\frac{1}{r^2}\frac{\partial\newtilde{v}_\theta}{\partial\theta} + \newtilde{v}_\phi\frac{1}{r^2\sin\left(\theta\right)}\frac{\partial\newtilde{v}_\theta}{\partial\phi}\right) - \frac{1}{\tan\left(\theta\right)}\left(\newtilde{v}_\phi\right)^2 + 2\newtilde{v}_\theta\newtilde{v}_r\right]\nonumber\\ & + \frac{1}{r\sin\left(\theta\right)}\mathbf{e}_\phi\bigg[r\sin\left(\theta\right)\left(\newtilde{v}_r\frac{\partial\newtilde{v}_\phi}{\partial r} + \frac{1}{r}\newtilde{v}_\theta\frac{\partial\newtilde{v}_\phi}{\partial\theta} + \newtilde{v}_\phi\frac{1}{r\sin\left(\theta\right)}\frac{\partial\newtilde{v}_\phi}{\partial\phi}\right) - \sin\left(\theta\right)\newtilde{v}_r\newtilde{v}_\phi - \cos\left(\theta\right)\newtilde{v}_\theta\newtilde{v}_\phi\nonumber\\ & + 2\newtilde{v}_\phi\left(\sin\left(\theta\right)\newtilde{v}_r + r\cos\left(\theta\right)\frac{1}{r}\newtilde{v}_\theta\right)\bigg]. \end{align} \]

So for the material derivation applies

\[ \begin{align} \md{\mathbf{v}} &= \mathbf{e}_r\left(\md{\newtilde{v}_r} - \frac{\newtilde{v}_\phi^2 + \newtilde{v}_\theta^2}{r}\right) + \mathbf{e}_\theta\left(\md{\newtilde{v}_\theta} + \frac{\newtilde{v}_r\newtilde{v}_\theta}{r} - \frac{\newtilde{v}_\phi^2}{r\tan\left(\theta\right)}\right)\nonumber\\ & + \mathbf{e}_\phi\left(\md{\newtilde{v}_\phi} + \frac{\newtilde{v}_r\newtilde{v}_\phi}{r} + \frac{\newtilde{v}_\theta\newtilde{v}_\phi}{r\tan\left(\theta\right)}\right)\tag{B.107}\label{eq:mat_derivative_kugel} \end{align} \]

B.2.1.1 Transformation to geographical coordinates

In order to transform to geographical coordinates, one considers the definition of this CS in Sect. D.1.3 and Eq. (B.66):

\[ \begin{align} \frac{\partial}{\partial x} &= \frac{1}{r\cos\left(\varphi\right)}\frac{\partial}{\partial\lambda} = \frac{1}{r\sin\left(\theta\right)}\frac{\partial}{\partial\phi}\\ \frac{\partial}{\partial y} &= \frac{1}{r}\frac{\partial}{\partial\varphi} = -\frac{1}{r}\frac{\partial}{\partial\theta}\\ \frac{\partial}{\partial z} &= \frac{\partial}{\partial r} \end{align} \]

The final identity transforms into the ordinary spherical coordinates. This applies analogously to second derivatives. For the representation of the Laplace operator Eq. (B.91) follows

\[ \begin{align} \Delta &= \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial }{\partial r}\right) + \frac{1}{r^2\cos^2\left(\varphi\right)}\frac{\partial ^2}{\partial\lambda^2} + \frac{1}{r^2\cos\left(\varphi\right)}\frac{\partial}{\partial\varphi}\left(\cos\left(\varphi\right)\frac{\partial}{\partial\varphi}\right)\nonumber\\ &= \frac{\partial^2}{\partial z^2} + \frac{\partial ^2}{\partial x^2} + \frac{\partial ^2}{\partial y^2} + \frac{2}{r}\frac{\partial}{\partial z} - \frac{\tan\left(\varphi\right)}{r}\frac{\partial}{\partial y}.\tag{B.111}\label{eq:laplace_geographische} \end{align} \]

Eq. (B.90) becomes

\[ \begin{align} \nabla\cdot\mathbf{v} &= \frac{\partial v_z}{\partial r} + \frac{1}{r}\frac{\partial v_y}{\partial\varphi} + \frac{1}{r\cos\left(\varphi\right)}\frac{\partial v_x}{\partial\lambda} + \frac{2v_z}{r} - \frac{v_y\tan\left(\varphi\right)}{r} = \nabla\cdot\mathbf{v}\nonumber\\ &= \frac{\partial v_z}{\partial r} + \frac{\partial v_y}{\partial y} + \frac{\partial v_x}{\partial x} + \frac{2v_z}{r} - \frac{v_y\tan\left(\varphi\right)}{r}.\tag{B.112}\label{eq:div_geo} \end{align} \]

If $\mathbf{v} = \left(u, v, w\right)^T$ is the wind vector, it follows

\[ \begin{align} \nabla\cdot\mathbf{v} &= \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} - \frac{v\tan\left(\varphi\right)}{r} + \frac{2w}{r}. \end{align} \]

Eq. (B.98) becomes

\[ \begin{align} \nabla\times\mathbf{v} &= -\frac{v_y}{r}\mathbf{i} + \frac{v_x\tan\left(\varphi\right)}{r}\mathbf{k} + \frac{v_x}{r}\mathbf{j} + \mathbf{k}\left(-\frac{1}{r}\frac{\partial v_x}{\partial\varphi} + \frac{1}{r\cos\left(\varphi\right)}\frac{\partial v_y}{\partial\lambda}\right)\nonumber\\ & -\mathbf{j}\left(\frac{1}{r\cos\left(\varphi\right)}\frac{\partial v_z}{\partial\lambda} - \frac{\partial v_x}{\partial r}\right) + \mathbf{i}\left(-\frac{\partial v_y}{\partial r} + \frac{1}{r}\frac{\partial v_z}{\partial\varphi}\right)\nonumber\\ &= -\frac{v_y}{r}\mathbf{i} + \frac{v_x\tan\left(\varphi\right)}{r}\mathbf{k} + \frac{v_x}{r}\mathbf{j} + \mathbf{k}\left(-\frac{\partial v_x}{\partial y} + \frac{\partial v_y}{\partial x}\right)\nonumber\\ & -\mathbf{j}\left(\frac{\partial v_z}{\partial x} - \frac{\partial v_x}{\partial r}\right) + \mathbf{i}\left(-\frac{\partial v_y}{\partial r} + \frac{\partial v_z}{\partial y}\right).\tag{B.114}\label{eq:rot_geo} \end{align} \]

If $\mathbf{v}$ is the wind vector again, it follows

\[ \begin{align} \nabla\times\mathbf{v} &= \left(\frac{\partial w}{\partial y} - \frac{\partial v}{\partial z}\right)\mathbf{i} + \left(\frac{\partial u}{\partial z} - \frac{\partial w}{\partial x}\right)\mathbf{j} + \left(\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}\right)\mathbf{k} + \frac{u\tan\left(\varphi\right)}{r}\mathbf{k} + \frac{u}{r}\mathbf{j} - \frac{v}{r}\mathbf{i}.\tag{B.115}\label{eq:rot_local} \end{align} \]

Still applies

\[ \begin{align} \zeta \coloneqq\mathbf{k}\cdot\left(\nabla\times\mathbf{v}\right) = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} + \frac{u\tan\left(\varphi\right)}{r}. \end{align} \]

It should be noted that this applies

\[ \begin{align} \nabla_h \times \mathbf{v}_h &= \left(\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}\right)\mathbf{k} + \frac{u\tan\left(\varphi\right)}{r}\mathbf{k} + \frac{u}{r}\mathbf{j} - \frac{v}{r}\mathbf{i} \not= \zeta\mathbf{k}. \end{align} \]

For the material derivative it follows with Eq. (B.107)

\[ \begin{align} \md{\mathbf{v}} &= \mathbf{i}\left(\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w\frac{\partial u}{\partial z} - \frac{uv\tan\left(\varphi\right)}{a + z} + \frac{uw}{a + z}\right)\nonumber\\ & + \mathbf{j}\left(\frac{\partial v}{\partial t} + u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} + w\frac{\partial v}{\partial z} + \frac{u^2\tan\left(\varphi\right)}{a + z} + \frac{vw}{a + z}\right)\nonumber\\ & + \mathbf{k}\left(\frac{\partial w}{\partial t} + u\frac{\partial w}{\partial x} + v\frac{\partial w}{\partial y} + w\frac{\partial w}{\partial z} - \frac{u^2 + v^2}{a + z}\right)\tag{B.118}\label{eq:mat_deriv_momentum_local}\\ &= \mathbf{i}\left(\md{u} - \frac{uv\tan\left(\varphi\right)}{a + z} + \frac{uw}{a + z}\right)\nonumber\\ & + \mathbf{j}\left(\md{v} + \frac{u^2\tan\left(\varphi\right)}{a + z} + \frac{vw}{a + z}\right)\nonumber\\ & + \mathbf{k}\left(\md{w} - \frac{u^2 + v^2}{a + z}\right)\nonumber\\ &=: \md{\mathbf{v}}\vline_\text{comp} + \md{\mathbf{v}}\vline_\text{met}. \end{align} \]

The material derivative of the speed is therefore composed of the material derivatives of the components

\[ \begin{align} \md{\mathbf{v}}\vline_\text{comp} \coloneqq \mathbf{i}\md{u} + \mathbf{j}\md{v} + \mathbf{k}\md{w}. \end{align} \]

and the metric terms

\[ \begin{align} \md{\mathbf{v}}\vline_\text{met} \coloneqq \mathbf{i}\left(-\frac{uv\tan\left(\varphi\right)}{a + z} + \frac{uw}{a + z}\right) + \mathbf{j}\left(\frac{u^2\tan\left(\varphi\right)}{a + z} + \frac{vw}{a + z}\right) \textcolor{red}{- \mathbf{k}\frac{u^2 + v^2}{a + z}}. \end{align} \]

The red term is the centrifugal acceleration.

Another form of the advective part of Eq. (B.118) is

\[ \begin{align} \left(\mathbf{v}\cdot\nabla\right)\mathbf{v} &= \left[\left(\mathbf{v}_h + w\mathbf{k}\right)\cdot\nabla\right]\left(\mathbf{v}_h + w\mathbf{k}\right) = \left(\mathbf{v}_h\cdot\nabla\right)\mathbf{v}_h + w\frac{\partial\mathbf{v}_h}{\partial z} + \left(\mathbf{v}\cdot\nabla\right)\left(w\mathbf{k}\right)\nonumber\\ & \stackrel{\href{#eq:diff_op_rule_10}{\text{Glg. (B.57)}}}{=} \left(\nabla\times\mathbf{v}_h\right)\times\mathbf{v}_h + \nabla\frac{\mathbf{v}_h^2}{2} + w\frac{\partial\mathbf{v}_h}{\partial z} + \left(\mathbf{v}\cdot\nabla\right)\left(w\mathbf{k}\right)\nonumber \end{align} \]

In the last step

\[ \begin{align} \left(\mathbf{k}\frac{\partial}{\partial z}\times\mathbf{v}_h\right)\times\mathbf{v}_h &= \left( \begin{array}{c} -\frac{\partial v}{\partial z}\\ \frac{\partial u}{\partial z}\\ 0 \end{array}\right)\times \left( \begin{array}{c} u\\ v\\0 \end{array}\right) = -\left(u\frac{\partial u}{\partial z} + v\frac{\partial v}{\partial z}\right)\mathbf{k} = -\mathbf{k}\frac{\partial\mathbf{v}_h^2}{\partial z} \end{align} \]

used. Eq. (B.122) is called 2D vector invariant form of velocity advection. Another useful form of this is

\[ \begin{align} \left(\mathbf{v}\cdot\nabla\right)\mathbf{v} &= \left(\nabla\times\mathbf{v}_h\right)\times\mathbf{v}_h + \nabla\frac{\mathbf{v}_h^2}{2} + w\frac{\partial\mathbf{v}_h}{\partial z} + \left(\mathbf{v}\cdot\nabla\right)\left(w\mathbf{k}\right)\nonumber\\ &= \left(\nabla\times\mathbf{v}_h\right)\times\mathbf{v}_h + \nabla\frac{\mathbf{v}_h^2}{2} + w\frac{\partial\mathbf{v}}{\partial z} + \left(\mathbf{v}_h\cdot\nabla\right)\left(w\mathbf{k}\right). \end{align} \]

Because of

\[ \begin{align} \left(\nabla\times\mathbf{v}\right)\times\mathbf{v}_h &= \left[\nabla\times\left(\mathbf{v}_h + w\mathbf{k}\right)\right]\times\mathbf{v}_h = \left(\nabla\times\mathbf{v}_h\right)\times\mathbf{v}_h + \left(\nabla\times w\mathbf{k}\right)\times\mathbf{v}_h\nonumber\\ & \stackrel{\text{Glg. }\href{#eq:rot_local}{(B.115)}}{=} \left(\nabla\times\mathbf{v}_h\right)\times\mathbf{v}_h + \left(\frac{\partial w}{\partial y}\mathbf{i} - \frac{\partial w}{\partial x}\mathbf{j}\right)\times\mathbf{v}_h = \left(\nabla\times\mathbf{v}_h\right)\times\mathbf{v}_h + \mathbf{k}\left(\mathbf{v}_h\cdot\nabla w\right)\nonumber\\ \Rightarrow\left(\nabla\times\mathbf{v}_h\right)\times\mathbf{v}_h &= \left(\nabla\times\mathbf{v}\right)\times\mathbf{v}_h - \mathbf{k}\left(\mathbf{v}_h\cdot\nabla w\right) \end{align} \]

and

\[ \begin{align} \left(\mathbf{v}_h\cdot\nabla\right)\left(w\mathbf{k}\right) &= \mathbf{k}\left(\mathbf{v}_h\cdot\nabla w\right) + w\left(u\frac{\partial\mathbf{k}}{\partial x} + v\frac{\partial\mathbf{k}}{\partial y}\right) = \mathbf{k}\left(\mathbf{v}_h\cdot\nabla w\right) + w\left(\frac{u}{r\cos\left(\phi\right)}\frac{\partial\mathbf{k}}{\partial\lambda} + \frac{v}{r}\frac{\partial\mathbf{k}}{\partial\phi}\right)\nonumber\\ &= \mathbf{k}\left(\mathbf{v}_h\cdot\nabla w\right) + w\left(\frac{u}{r}\mathbf{i} + \frac{v}{r}\mathbf{j}\right) = \mathbf{k}\left(\mathbf{v}_h\cdot\nabla w\right) + \frac{w}{r}\mathbf{v}_h \end{align} \]

applies

By comparing with section 13.3 one finds that the term $\frac{w}{r}\mathbf{v}_h$ does not appear under the shallow atmosphere approximation.

The metric terms in Eq. (B.118)

\[ \begin{align} \md{\mathbf{v}}\vline_\text{met} &= \left(\begin{array}{c} -\frac{uv\tan\left(\varphi\right)}{a + z} + \frac{uw}{a + z}\\ \frac{u^2\tan\left(\varphi\right)}{a + z} + \frac{vw}{a + z}\\ -\frac{u^2 + v^2}{a + z} \end{array}\right).\tag{B.128}\label{eq:mat_deriv_momentum_met} \end{align} \]

The metric terms in Eq. (B.115)

\[ \begin{align} \nabla\times\mathbf{v}\vline_\text{met} &= \left(\begin{array}{c} -\frac{v}{r}\\ \frac{u}{r}\\ \frac{u\tan\left(\varphi\right)}{r} \end{array}\right). \end{align} \]

If you multiply this vectorially by $\mathbf{v}$, you get Eq. (B.128):

\[ \begin{align} \nabla\times\mathbf{v}\vline_\text{met}\times\mathbf{v} &= \left(\begin{array}{c} -\frac{v}{r}\\ \frac{u}{r}\\ \frac{u\tan\left(\varphi\right)}{r} \end{array}\right)\times\left(\begin{array}{c} u\\ v\\ w \end{array}\right) = \left(\begin{array}{c} -\frac{uv\tan\left(\varphi\right)}{a + z} + \frac{uw}{a + z}\\ \frac{u^2\tan\left(\varphi\right)}{a + z} + \frac{vw}{a + z}\\ -\frac{u^2 + v^2}{a + z} \end{array}\right) = \md{\mathbf{v}}\vline_\text{met}.\tag{B.130}\label{eq:mat_deriv_momentum_met_prop_0} \end{align} \]

The metric terms in momentum advection are therefore consequences of the metric terms in rotation.

B.3 Multidimensional integral calculus

Let $M, N\subseteq\mathbb{R}^3$, $f:M\to\mathbb{R}$ be continuous, $T:N\to M$ be a diffeomorphism, then we have

this corresponds to the multidimensional substitution formula. The determinant $\det\left(T'\right)$ is called functional determinant. As an example, take $M = \mathbb{R}$, $N = \left[0, \infty\right)\times\left[0, \pi\right]\times\left[0, 2\pi\right)$ and

\[ \begin{align} T = r\left(\begin{array}{c} \sin\left(\vartheta\right)\cos\left(\varphi\right)\\ \sin\left(\vartheta\right)\sin\left(\varphi\right)\\ \cos\left(\vartheta\right) \end{array}\right), \end{align} \]

then applies

\[ \begin{align} T' = \left(\begin{array}{ccc} \sin\left(\vartheta\right)\cos\left(\varphi\right)&r\cos\left(\vartheta\right)\cos\left(\varphi\right)& -r\sin\left(\vartheta\right)\sin\left(\varphi\right)\\ \sin\left(\vartheta\right)\sin\left(\varphi\right)&r\cos\left(\vartheta\right)\sin\left(\varphi\right)&r\sin\left(\vartheta\right)\cos\left(\varphi\right)\\ \cos\left(\vartheta\right)& -r\sin\left(\vartheta\right)&0 \end{array}\right). \end{align} \]

applies to the determinant

\[ \begin{align} \det\left(T'\right) &= -r\sin\left(\vartheta\right)\cos\left(\varphi\right)\left(-r\sin\left(\vartheta\right)^2\cos\left(\varphi\right) - r\cos\left(\vartheta\right)^2\cos\left(\varphi\right)\right)\nonumber\\ & - r\sin\left(\vartheta\right)\sin\left(\varphi\right)\left(-r\sin\left(\vartheta\right)^2\sin\left(\varphi\right) - r\cos\left(\vartheta\right)^2\sin\left(\varphi\right)\right)\nonumber\\ &= r^2\sin\left(\vartheta\right)\cos\left(\varphi\right)^2 + r^2\sin\left(\vartheta\right)\sin\left(\vartheta\right)^2 = r^2\sin\left(\vartheta\right), \end{align} \]

in the case of geographical coordinates the functional determinant is $r^2\cos\left(\varphi\right)$.

Stokes' theorem Eq. (15.26) has already been introduced. Let $A\subseteq\mathbb{R}^3$ be connected and $\mathbf{v}:A\to\mathbb{R}^3$ be continuous-differentiable. Then applies

This is the Gaussian theorem.

A sphere $K_N\left(R\right)$ in $\mathbb{R}^N$ with radius $R\geq0$ can be defined by

\[ \begin{align} K_N\left(R\right) \coloneqq \left\{\mathbf{x} = \left(x_1, \dotsc, x_N\right)^T\in\mathbb{R}^N\newvline\sum_{i = 1}^{N}x_i^2\leq R^2\right\}. \end{align} \]

Their volume is

\[ \begin{align} V_N\left(R\right) = \int_{K_N}dx_1\dotsc dx_N. \end{align} \]

It applies

\[ \begin{align} \int_{\mathbb{R}^N}\exp\left(-\sum_{i = 1}^{N}x_i^2\right)dx_1\dotsc dx_N &= \int_{0}^{\infty}e^{-R^2}\frac{dV_N}{dR}dR, \tag{B.138}\label{eq:n_dim_kugel_hilfe_1} \end{align} \]

this corresponds to a transformation to spherical coordinates. You can also transform the integral $V_N\left(R\right)$ to spherical coordinates by writing

\[ \begin{align} V_N\left(R\right) = \int_{K_N}dV_N = \int_{0}^{R}f_N\left(R\right)dR, \tag{B.139}\label{eq:n_dim_kugel_hilfe_2} \end{align} \]

here $f_N\left(R\right)$ is the functional determinant integrated across the surface. The N-dimensional spherical coordinates consist of N - 1 angles and a distance. Therefore $f_N\left(R\right)$ is the determinant of a real $N\times N-$matrix in which the factor $R$ occurs in $N - 1$ columns. $f_N\left(R\right)$ is therefore of degree $N - 1$, you can write

\[ \begin{align} f_N\left(R\right) = NC_NR^{N - 1}. \end{align} \]

By differentiating Eq. (B.139) to $R$ and with Eq. (B.138) follows

\[ \begin{align} \frac{dV_N}{dR} &= NC_NR^{N - 1} \Rightarrow NC_N\int_{0}^{\infty}e^{-R^2}R^{N - 1}dR = \left(\int_{ - \infty}^{\infty}e^{-x^2}dx\right)^N = \pi^{N/2}. \end{align} \]

The last step follows with Eq. (A.105). It applies with the substitution rule

\[ \begin{align} \int_{0}^{\infty}e^{-R^2}R^{N - 1}dR &= \int_{0}^{\infty}e^{-R}R^{\frac{N - 1}{2}}\frac{1}{2}R^{-\frac{1}{2}}dR = \frac{1}{2}\int_{0}^{\infty}e^{-R}R^{N/2 - 1}dR = \frac{1}{2}\Gamma\left(\frac{N}{2}\right). \end{align} \]

It follows with Eq. (A.111)

B.3.1 Transport theorem

Let a time-dependent connected set $\Omega = \Omega\left(t\right) \subseteq \mathbb{R}^3$ be given, which moves with the continuously differentiable vector field $\mathbf{v} = \mathbf{v}\left(\mathbf{r}, t\right)$.

The so-called transport theorem is the three-dimensional generalization of the Leibniz rule Eq. (A.95).

This is the transport theorem or Reynolds transport theorem. $f$ can also be the component of a vector field. The derivation can thus be generalized vectorially: