Directional derivatives are derivatives of a function of several variables along a specific direction; the partial derivatives are special cases of these, in which one differentiates in the direction of one of the coordinate axes.
Let $n, m\in \mathbb {N}$ with $n, m\geq 1$ and $f:\mathbb {R}^n\to \mathbb {R}^m$ differentiable. Then the derivative is a function $f':\mathbb{R}^n\to\mathbb{R}^{m\times n}$ and one has
\[ \begin{align} \left(f'\right)_{i, j} = \left(\frac{\partial f_i}{\partial x_j}\right). \end{align} \]
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 $\frac{d\tau}{dt} = \frac{d}{dt}T\left(r\left(t\right)\right)$, one has
\[ \begin{align} \frac{d\tau}{dt} = T'\frac{dr}{dt}\tag{B.4}\label{eq:mehr_dim_kette}. \end{align} \]
This again corresponds to the mnemonic outer derivative times inner derivative. If one assumes, for example, 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, one has
\[ \begin{align} \frac{dT}{dt} = \frac{\partial T}{\partial t} + \frac{dx}{dt}\frac{\partial T}{\partial x} + \frac{dy}{dt}\frac{\partial T}{\partial y} + \frac{dz}{dt}\frac{\partial T}{\partial z}. \end{align} \]
This is called the material derivative or total derivative, because here the property of a fixed particle is considered. 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 one is given, more generally, 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 defined by
\[ \begin{align} \rot\left(\mathbf{w}\right) \coloneqq \nabla\times\mathbf{w} = \sum_{i, j, k = 1}^3\epsilon_{i, j, k}\mathbf{e}_k\frac{\partial}{\partial x_i}w_j \end{align} \]
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 angular part of the Laplace operator, see Eq. (B.96), and $r$ the radius of the sphere. Since the $i-$component of the rotation is the rotation of the vector field under consideration in the $x_{j,k}-$plane, one can interpret
\[ \begin{align} \mathbf{k}\cdot\mathbf{w} \end{align} \]
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
\[ \begin{align} -\mathbf{v}_h\cdot\nabla. \end{align} \]
If one is given a differential balance equation for the quantity in the form
\[ \begin{align} \md{\psi} = \sum_iF_i \end{align} \]
with physical forcings $F_i$, then the equation for the local time change of $\psi$ follows:
\[ \begin{align} \frac{\partial\psi}{\partial t} = -\mathbf{v}\cdot\nabla T + \sum_iF_i = -\mathbf{v}_h\cdot\nabla - w\frac{\partial L}{\partial z} + \sum_iF_i, \end{align} \]
The advection is thus the part of the local time change that is brought about by transport.
Imagine a trajectory
\[ \begin{align} \mathbf{r}:I\subseteq\mathbb{R}\to\mathbb{R}^2;\alpha\mapsto\left(x, y\right)^T \end{align} \]
where $I$ is an interval. Here one considers only a small section
\[ \begin{align} \mathbf{r}:I'\subseteq I\to\mathbb{R}^2;\tau\mapsto\left(x, y\right)^T, \end{align} \]
where $I'$ is also an interval, and define
\[ \begin{align} \tau_1 &\coloneqq \sup\left(I'\right),\\ \tau_2 &\coloneqq \sup\left(I'\right),\\ \mathbf{r}_1 = \left(x_1, y_1\right)^T &\coloneqq f\left(\tau_1\right),\\ \mathbf{r}_2 = \left(x_2, y_2\right)^T &\coloneqq f\left(\tau_2\right).\\ \end{align} \]
Without loss of generality, one can place the origin of the coordinate system at $\mathbf{r}_1$. Now one seeks a point $\mathbf{r}_0 \coloneqq \left(x_0, y_0\right)^T$ that is required to have, as nearly as possible, 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 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 written out:
\[ \begin{align} x_2 = x'\Delta\tau + \frac{1}{2}x''\Delta\tau^2 & {} & y_2 = y'\Delta\tau + \frac{1}{2}y''\Delta\tau^2 \end{align} \]
The derivatives are evaluated at $\tau = \tau_1$. The second-order term must be retained here, because curvature is determined by how the derivative of a trajectory changes. Substituting this into Eq. (B.29) yields
\[ \begin{align} 0 &= x'^2\Delta\tau^2 + \frac{1}{4}x''^2\Delta\tau^4 + x'x''\Delta\tau^3 + y'^2\Delta\tau^2 + \frac{1}{4}y''^2\Delta\tau^4 + y'y''\Delta\tau^3\nonumber\\ & -2x'\Delta\tau x_0 - x''\Delta\tau^2x_0-2y'\Delta\tau y_0 - y''\Delta\tau^2y_0. \end{align} \]
The first-order term in $\Delta\tau$ implies:
\[ \begin{align} 2x'x_0 + 2y'y_0 = 0\tag{B.34}\label{eq:deriv_curv_2} \end{align} \]
The second-order term in $\Delta\tau$ 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) it follows that
\[ \begin{align} x_0 = -\frac{y'y_0}{x'}.\tag{B.36}\label{eq:deriv_curv_4} \end{align} \]
Inserting Eq. (B.35) yields
\[ \begin{align} -x''y'\frac{y_0}{x'} + y''y_0 = x'^2 + y'^2 \Rightarrow y_0 = \frac{x'^2 + y'^2}{y''-x''\frac{y'}{x'}}. \end{align} \]
Using Eq. (B.36), one obtains
\[ \begin{align} x_0 = -\frac{y'}{x'}\frac{x'^2 + y'^2}{y''-x''\frac{y'}{x'}}. \end{align} \]
Hence the magnitude of the radius of curvature is
\[ \begin{align} \left|r\right| &= \sqrt{1 + \frac{y'^2}{x'^2}}\frac{x'^2+y'^2}{\left|y''-x''\frac{y'}{x'}\right|}. \end{align} \]
Without loss of generality, one can set $\tau = x$, which gives
\[ \begin{align} x' = 1, & {} & x'' = 0. \end{align} \]
Therefore,
\[ \begin{align} \left|r\right| = \sqrt{1+y'^2}\frac{1+y'^2}{\left|y''\right|} & {} & \Rightarrow\left|r\right| = \frac{\left(1+y'^2\right)^{3/2}}{\left|y''\right|}. \end{align} \]
Regarding the sign of $r$, it is chosen to 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 a linear approximation for $1/r$ is needed, one usually uses
\[ \begin{align} \frac{1}{r} \approx y''\tag{B.43}\label{eq:curv_approx}. \end{align} \]
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 $\lambda\in \mathbb{C}$ a scalar. One defines
\[ \begin{align} \Delta\mathbf{v} \coloneqq \sum_{i = 1}^{3}\Delta v_i\mathbf{e}_i. \end{align} \]
The following hold
\[ \begin{align} \nabla\left(\psi + \chi\right) = \nabla\psi + \nabla\chi, & {} & \nabla\times\left(\mathbf{v} + \mathbf{w}\right) = \nabla\times\mathbf{v} + \nabla\times\mathbf{w}, \end{align} \] \[ \begin{align} \nabla\left(\lambda\psi\right) = \lambda\nabla\psi, & {} & \nabla\times\left(\lambda\mathbf{v}\right) = \lambda\nabla\times\mathbf{v} \end{align} \]
according to the observations in Sect. A.8. Furthermore, the following hold
\[ \begin{align} \nabla\times\nabla\psi &= \sum_{i, j, k = 1}^{3}\epsilon_{i, j, k}\frac{\partial}{\partial x_i}\frac{\partial\psi}{\partial x_j}\mathbf{e}_k = \sum_{i, j, k = 1}^{3}\epsilon_{i, j, k}\frac{\partial^2\psi}{\partial x_i\partial x_j}\mathbf{e}_k = \mathbf{0}, \tag{B.47}\label{eq:diff_op_rule_1}\\ \nabla\cdot\nabla\times\mathbf{v} &= \left(\sum_{l = 1}^{3}\mathbf{e}_l\frac{\partial}{\partial x_l}\right)\cdot\left(\sum_{i, j, k = 1}^{3}\epsilon_{i, j, k}\frac{\partial}{\partial x_i}v_j\mathbf{e}_k\right) = \sum_{i, j, k = 1}^{3}\epsilon_{i, j, k}\frac{\partial^2v_j}{\partial x_k\partial x_i} = 0, \tag{B.48}\label{eq:diff_op_rule_2}\\ \nabla\cdot\left(\psi\mathbf{v}\right) &= \sum_{i = 1}^{3}\frac{\partial}{\partial x_i}\left(\psi v_i\right) = \sum_{i = 1}^{3}v_i\frac{\partial\psi}{\partial x_i} + \psi\frac{\partial v_i}{\partial x_i} = \mathbf{v}\cdot\nabla\psi + \psi\nabla\cdot\mathbf{v}, \tag{B.49}\label{eq:diff_op_rule_3}\\ \frac{\partial}{\partial x_i}\nabla\times\mathbf{v} &= \frac{\partial}{\partial x_i}\sum_{j, k.l = 1}^{3}\epsilon_{j, k, l}\frac{\partial}{\partial x_j}v_k\mathbf{e}_l = \sum_{j, k, l = 1}^{3}\epsilon_{j, k, l}\frac{\partial}{\partial x_j}\frac{\partial v_k}{\partial x_i}\mathbf{e}_l = \nabla\times\frac{\partial}{\partial x_i}\mathbf{v}, \tag{B.50}\label{eq:diff_op_rule_4}\\ \nabla\times\psi\mathbf{v} &= \sum_{i, j, k = 1}^{3}\epsilon_{i, j, k}\frac{\partial}{\partial x_i}\left(\psi v_j\right)\mathbf{e}_k = \sum_{i, j, k = 1}^{3}\epsilon_{i, j, k}\left(v_j\frac{\partial\psi}{\partial x_i} + \psi\frac{\partial v_j}{\partial x_i}\right)\mathbf{e}_k\nonumber\\ &= \sum_{i, j, k = 1}^{3}\epsilon_{i, j, k} \frac{\partial\psi}{\partial x_i}v_j\mathbf{e}_k + \sum_{i, j, k = 1}^{3}\psi\frac{\partial}{\partial x_i}v_j\mathbf{e}_k = \left(\nabla\psi\right)\times\mathbf{v} + \psi\nabla\times\mathbf{v}\nonumber\\ &= -\mathbf{v}\times\nabla\psi + \psi\nabla\times\mathbf{v},\tag{B.51}\label{eq:diff_op_rule_5} \end{align} \] \[ \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} \] \[ \begin{align} \nabla\cdot\left(\mathbf{v}\times\mathbf{w}\right) &= \sum_{i = 1}^3\frac{\partial}{\partial x_i}\left(\mathbf{v}\times\mathbf{w}\right)_i = \sum_{i = 1}^{3}\frac{\partial}{\partial x_i}\left(\epsilon_{k, j, i}v_kw_j + \epsilon_{j, k, i}v_jw_k\right)\nonumber\\ &= \sum_{i = 1}^{3}\left(\epsilon_{k, j, i}\frac{\partial v_k}{\partial x_i}w_j + \epsilon_{k, j, i}v_k\frac{\partial w_j}{\partial x_i} + \epsilon_{j, k, i}\frac{\partial v_j}{\partial x_i}w_k + \epsilon_{j, k, i}v_j\frac{\partial w_k}{\partial x_i}\right)\nonumber\\ &= \sum_{i = 1}^{3}w_i\left(\epsilon_{k, j, i}\frac{\partial v_j}{\partial x_k} + \epsilon_{j, k, i}\frac{\partial v_k}{\partial x_j}\right) - v_i\left(\epsilon_{k, j, i}\frac{\partial w_j}{\partial x_k} + \epsilon_{j, k, i}\frac{\partial w_k}{\partial x_j}\right)\nonumber\\ &= \sum_{i = 1}^{3}w_i\left(\nabla\times\mathbf{v}\right)_i - v_i\left(\nabla\times\mathbf{w}\right)_i = \mathbf{w}\cdot\left(\nabla\times\mathbf{v}\right) - \mathbf{v}\cdot\left(\nabla\times\mathbf{w}\right), \tag{B.55}\label{eq:diff_op_rule_9}\\ \mathbf{u}\cdot\left[\left(\mathbf{v}\cdot\nabla\right)\nabla\psi\right] &= \sum_{i=1}^3u_i\left[\left(\mathbf{v}\cdot\nabla\right)\nabla\psi\right]_i = \sum_{i=1}^3u_i\left[\left(\mathbf{v}\cdot\nabla\right)\frac{\partial\psi}{\partial x_i}\right] = \sum_{i=1}^3u_i\sum_{j=1}^3v_j\frac{\partial^2\psi}{\partial x_j\partial x_i}\nonumber\\ &= \sum_{i, j=1}^3u_iv_j\frac{\partial^2\psi}{\partial x_j\partial x_i} = \sum_{i, j=1}^3v_iu_j\frac{\partial^2\psi}{\partial x_j\partial x_i} = \sum_{i=1}^3v_i\sum_{j=1}^3u_j\frac{\partial}{\partial x_j}\left(\nabla\psi\right)_i\nonumber\\ &= \sum_{i=1}^3v_i\left[\left(\mathbf{u}\cdot\nabla\right)\nabla\psi\right]_i = \mathbf{v}\cdot\left[\left(\mathbf{u}\cdot\nabla\right)\nabla\psi\right].\tag{B.56}\label{eq:diff_op_rule_12} \end{align} \]
Furthermore,
\[ \begin{align} \left(\mathbf{v}\cdot\nabla\right)\mathbf{v} & \stackrel{\text{Eq. }\href{#eq:diff_op_rule_6}{(B.52)}}{=} & \frac{1}{2}\nabla\left(\mathbf{v}\cdot\mathbf{v}\right) - \mathbf{v}\times\left(\nabla\times\mathbf{v}\right).\tag{B.57}\label{eq:diff_op_rule_10} \end{align} \]
This is called the Lamb transformation. Furthermore, for the advection of a scalar product, one has
\[ \begin{align} \left(\mathbf{u}\cdot\nabla\right)\left(\mathbf{v}\cdot\mathbf{w}\right) &= \left(\sum_{i = 1}^{3}u_i\frac{\partial}{\partial x_i}\right)\left(\sum_{j = 1}^{3}v_jw_j\right) = \sum_{i, j = 1}^{3}u_i\left[\frac{\partial v_j}{\partial x_i}w_j + \frac{\partial w_j}{\partial x_i}v_j\right]\nonumber\\ &= \sum_{j = 1}^{3}w_j\sum_{i = 1}^{3}u_i\frac{\partial v_j}{\partial x_i} + \sum_{j = 1}^{3}v_j\sum_{i = 1}^{3}u_i\frac{\partial w_j}{\partial x_i} = \mathbf{w}\cdot\left[\left(\mathbf{u}\cdot\nabla\right)\mathbf{v}\right] + \mathbf{v}\cdot\left[\left(\mathbf{u}\cdot\nabla\right)\mathbf{w}\right].\tag{B.58}\label{eq:diff_op_rule_11} \end{align} \]
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 the following hold
\[ \begin{align} \nabla\varphi\left(\mathbf{r}\right) &= \sum_{i = 1}^{3}\frac{\partial}{\partial x_i}\varphi\left(\mathbf{r}\right)\mathbf{e}_i = \sum_{i = 1}^{3}\frac{\partial}{\partial x_i}\exp\left(\mathbf{k}\cdot\mathbf{r}\right)\mathbf{e}_i = \sum_{i = 1}^{3}\frac{\partial}{\partial x_i}\exp\left(\sum_{j = 1}^{3}k_jx_j\right)\mathbf{e}_i\nonumber\\ &= \sum_{i = 1}^{3}k_i\exp\left(\mathbf{k}\cdot\mathbf{r}\right)\mathbf{e}_i = \varphi\left(\mathbf{r}\right)\mathbf{k},\\ \nabla\times\left[\mathbf{A}\varphi\left(\mathbf{r}\right)\right] &= \sum_{i, j, k = 1}^3\epsilon_{i, j, k}\frac{\partial\left[A_j\exp\left(i\mathbf{k}\cdot\mathbf{r}\right)\right]}{\partial x_i}\mathbf{e}_k = i\sum_{i, j, k = 1}^3\epsilon_{i, j, k}A_jk_i\exp\left(i\mathbf{k}\cdot\mathbf{r}\right)\mathbf{e}_k\nonumber\\ &= i\exp\left(i\mathbf{k}\cdot\mathbf{r}\right)\sum_{i, j, k = 1}^3\epsilon_{i, j, k}A_jk_i\mathbf{e}_k = i\exp\left(i\mathbf{k}\cdot\mathbf{r}\right)\mathbf{A}\times\mathbf{k} = i\left[\mathbf{A}\varphi\left(\mathbf{r}\right)\right]\times\mathbf{k},\tag{B.60}\label{eq:diff_op_harmonic_rule_1}\\ \Delta\varphi\left(\mathbf{r}\right) &= \nabla\cdot\nabla\varphi\left(\mathbf{r}\right) = \nabla\cdot\left(\varphi\left(\mathbf{r}\right)\mathbf{k}\right) = \mathbf{k}\cdot\nabla\varphi = \mathbf{k}^2\varphi\left(\mathbf{r}\right). \end{align} \]
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 basis $\mathbf{c}_1, \mathbf{c}_2, \mathbf{c}_3$ yield 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, a vector can be imagined as an arrow. In order to solve problems with a certain geometry more elegantly, one introduces generalized coordinates $q_i$ (see also Sect. 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 remark, the functions $f = f\left(x\right)$ and $f = f\left(q\right)$ are not the same; if one inserts the same arguments, one generally obtains different values. Usually, however, this is not distinguished in the notation, since both functions express the same thing.
If one wishes to express a vector field $\mathbf{v}$ in generalized coordinates, the simplest idea would be to also express the values of $\mathbf{v}$ in generalized coordinates. If $\mathbf{v}$ is a velocity field, one could express the values of $\mathbf{v}$ in the forms $\newdot{q}$ or $\mathbf{v} \equiv \mathbf{r}\left(q\right)$. Since this is often unintuitive, 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 normalized. If
\[ \begin{align} \mathbf{e}_1\times\mathbf{e}_2 = \mathbf{e}_3\tag{B.64}\label{eq:gen_coords_orth_criterion} \end{align} \]
is achieved by permuting the $\mathbf{e}_i$, the coordinate system $q$ is called orthogonal; coordinate systems that violate this condition 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 $\mathbf{v}$ and $\mathbf{w}$ be two vector fields. Then the scalar 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, one has
\[ \begin{align} \mathbf{v}\times\mathbf{w} = \left|\begin{array}{ccc} \mathbf{q}_2\times\mathbf{q}_3 & \mathbf{q}_3\times\mathbf{q}_1 & \mathbf{q}_1\times\mathbf{q}_2\\ v^{(1)} & v^{(2)} & v^{3}\\ w^{(1)} & w^{(2)} & w^{3} \end{array}\right|\tag{B.68}\label{eq:vector_product_gen} \end{align} \]
Let $f$ be a scalar field. Then the gradient of $f$ in generalized coordinates is
\[ \begin{align} \nabla f = \mathbf{q}^{(n)}\frac{\partial}{\partial q^{(n)}} = \mathbf{q}^{(n)}\nabla_nf.\tag{B.69}\label{eq:grad_gen} \end{align} \]
Thus, in generalized coordinates,
\[ \begin{align} \nabla\cdot\mathbf{v} = \frac{1}{\sqrt{g}}\frac{\partial}{\partial q^{(n)}}\left(\mathbf{q}^{(n)}\cdot\mathbf{v}\sqrt{g}\right)\tag{B.70}\label{eq:div_gen}. \end{align} \]
Thus,
\[ \begin{align} \nabla\times\mathbf{v} = \frac{1}{\sqrt{g}}\left|\begin{array}{ccc} \mathbf{q}_1 & \mathbf{q}_2 & \mathbf{q}_3 \\ \frac{\partial}{\partial q^{(1)}} & \frac{\partial}{\partial q^{(2)}} & \frac{\partial}{\partial q^{3}} \\ v_1 & v_2 & v_3 \end{array}\right|\tag{B.71}\label{eq:rot_gen} \end{align} \]
In this section, $\mathbf{v}$ is a vector field and $f$ is a scalar field; both are assumed to have all necessary smoothness properties. In spherical coordinates,
\[ \begin{align} \mathbf{r} = r\left(\begin{array}{c} \sin\left(\theta\right)\cos\left(\phi\right)\\ \sin\left(\theta\right)\sin\left(\phi\right)\\ \cos\left(\theta\right) \end{array}\right). \end{align} \]
Let $\mathbf{r} = \mathbf{r}\left(r, \theta, \phi\right)$ denote the transformation from spherical to Cartesian coordinates. Its Jacobian is
\[ \begin{align} J &= \left(\begin{array}{ccc} \frac{\partial x}{\partial r}&\frac{\partial x}{\partial\theta}&\frac{\partial x}{\partial\phi}\\ \frac{\partial y}{\partial r}&\frac{\partial y}{\partial\theta}&\frac{\partial y}{\partial\phi}\\ \frac{\partial z}{\partial r}&\frac{\partial z}{\partial\theta}&\frac{\partial z}{\partial\phi} \end{array}\right) = \left(\begin{array}{ccc} \sin\left(\theta\right)\cos\left(\phi\right)&r\cos\left(\theta\right)\cos\left(\phi\right)& -r\sin\left(\theta\right)\sin\left(\phi\right)\\ \sin\left(\theta\right)\sin\left(\phi\right)&r\cos\left(\theta\right)\sin\left(\phi\right)&r\sin\left(\theta\right)\cos\left(\phi\right)\\ \cos\left(\theta\right)& -r\sin\left(\theta\right)&0 \end{array}\right). \end{align} \]
The normalized columns of this matrix 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 yields 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 gives the contravariant basis vectors
\[ \begin{align} \mathbf{q}^{(r)} &= \left(\begin{array}{c} \sin\left(\theta\right)\cos\left(\phi\right)\\ \sin\left(\theta\right)\sin\left(\phi\right)\\ \cos\left(\theta\right) \end{array}\right),\\ \mathbf{q}^{(\theta)} &= \frac{1}{r}\left(\begin{array}{c} \cos\left(\theta\right)\cos\left(\phi\right)\\ \cos\left(\theta\right)\sin\left(\phi\right)\\ -\sin\left(\theta\right) \end{array}\right),\\ \mathbf{q}^{(\phi)} &= \frac{1}{r\sin\left(\theta\right)}\left(\begin{array}{c} -\sin\left(\phi\right)\\ \cos\left(\phi\right)\\ 0 \end{array}\right). \end{align} \]
For a vector field, one writes
\[ \begin{align} \mathbf{v} &= v^{(r)}\mathbf{q}_r + v^{(\theta)}\mathbf{q}_\theta + v^{(\phi)}\mathbf{q}_\phi = v_r\mathbf{q}^{(r)} + v_\theta\mathbf{q}^{(\theta)} + v_\phi\mathbf{q}^{(\phi)}\nonumber\\ &= \newtilde{v}^{(r)}\mathbf{e}_r + \newtilde{v}^{(\theta)}\mathbf{e}_\theta + \newtilde{v}^{(\phi)}\mathbf{e}_\phi = \newtilde{v}_r\mathbf{e}^{(r)} + \newtilde{v}_\theta\mathbf{e}^{(\theta)} + \newtilde{v}_\phi\mathbf{e}^{(\phi)}. \end{align} \]
It should be clear at this point that the dimensions of the components of $\mathbf{v}$ appearing in the first line are not uniform, since the basis elements behave the same way. Because the spherical coordinates are orthogonal,
\[ \begin{align} \newtilde{v}^{(r)} = \newtilde{v}_r, & {} & \newtilde{v}^{(\theta)} = \newtilde{v}_\theta, & {} & \newtilde{v}^{(\phi)} = \newtilde{v}_\phi. \end{align} \]
The component conversion 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 consider the gradient; Eq. (B.69) gives
\[ \begin{align} \nabla f &= \mathbf{q}^{(r)}\frac{\partial f}{\partial r} + \mathbf{q}^{(\theta)}\frac{\partial f}{\partial\theta} + \mathbf{q}^{(\phi)}\frac{\partial f}{\partial\phi} = \mathbf{e}^{(r)}\frac{\partial f}{\partial r} + \frac{1}{r}\mathbf{e}^{(\theta)}\frac{\partial f}{\partial\theta} + \frac{1}{r\sin\left(\theta\right)}\mathbf{e}^{(\phi)}\frac{\partial f}{\partial\phi}.\tag{B.88}\label{eq:grad_sphere} \end{align} \]
The divergence is now considered. In order to apply Eq. (B.70), one first computes
\[ \begin{align} \mathbf{q}^{(r)}\cdot\mathbf{v} = v^{(r)}, & {} & \mathbf{q}^{(\theta)}\cdot\mathbf{v} = v^{(\theta)}, & {} & \mathbf{q}^{(\phi)}\cdot\mathbf{v} = v^{(\phi)}. \end{align} \]
Therefore,
\[ \begin{align} \nabla\cdot\mathbf{v} &= \frac{1}{r^2\sin\left(\theta\right)}\left(\frac{\partial\left(v^{(r)}r^2\sin\left(\theta\right)\right)}{\partial r} + \frac{\partial\left(v^{(\theta)}r^2\sin\left(\theta\right)\right)}{\partial\theta} + \frac{\partial\left(v^{(\phi)}r^2\sin\left(\theta\right)\right)}{\partial\phi}\right)\nonumber\\ &= \frac{\partial v^{(r)}}{\partial r} + \frac{\partial v^{(\theta)}}{\partial\theta} + \frac{\partial v^{(\phi)}}{\partial\phi} + \frac{2v^{(r)}}{r} + \cot\left(\theta\right)v^{(\theta)}\nonumber\\ &= \frac{\partial\newtilde{v}^{(r)}}{\partial r} + \frac{1}{r}\frac{\partial\newtilde{v}^{(\theta)}}{\partial\theta} + \frac{1}{r\sin\left(\theta\right)}\frac{\partial\newtilde{v}^{(\phi)}}{\partial\phi} + \frac{2\newtilde{v}^{(r)}}{r} + \frac{\cot\left(\theta\right)}{r}\newtilde{v}^{(\theta)}.\tag{B.90}\label{eq:div_sphere} \end{align} \]
Combining Eqs. (B.88) and (B.90) gives 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 rewritten slightly. For this, use the diffeomorphism
\[ \begin{align} \theta\leftrightarrow\mu \coloneqq \cos\left(\theta\right), \end{align} \]
then one has
\[ \begin{align} \frac{d}{d\theta} = \frac{d\mu}{d\theta}\frac{d}{d\mu} = -\sin\left(\theta\right)\frac{d}{d\mu}. \end{align} \]
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 normalized basis elements, one obtains
\[ \begin{align} \nabla\times\mathbf{v} &= \frac{1}{r^2\sin\left(\theta\right)}\Bigg[\mathbf{e}_r\left(\frac{\partial\left(r\sin\left(\theta\right)\newtilde{v}_\phi\right)}{\partial\theta} - \frac{\partial\left(r\newtilde{v}_\theta\right)}{\partial\phi}\right) + r\mathbf{e}_\theta\left(\frac{\partial\newtilde{v}_r}{\partial\phi} - \frac{\partial\left(r\sin\left(\theta\right)\newtilde{v}_\phi\right)}{\partial r}\right)\nonumber\\ & + r\sin\left(\theta\right)\mathbf{e}_\phi\left(\frac{\partial\left(r\newtilde{v}_\theta\right)}{\partial r} - \frac{\partial\newtilde{v}_r}{\partial\theta}\right)\Bigg]\nonumber\\ &= \frac{\newtilde{v}_\theta}{r}\mathbf{e}_\phi + \frac{\newtilde{v}_\phi}{r\tan\left(\theta\right)}\mathbf{e}_r - \frac{\newtilde{v}_\phi}{r}\mathbf{e}_\theta + \mathbf{e}_r\left(\frac{1}{r}\frac{\partial\newtilde{v}_\phi}{\partial\theta} - \frac{1}{r\sin\left(\theta\right)}\frac{\partial\newtilde{v}_\theta}{\partial\phi}\right)\nonumber\\ & + \mathbf{e}_\theta\left(\frac{1}{r\sin\left(\theta\right)}\frac{\partial\newtilde{v}_r}{\partial\phi} - \frac{\partial\newtilde{v}_\phi}{\partial r}\right) + \mathbf{e}_\phi\left(\frac{\partial \newtilde{v}_\theta}{\partial r} - \frac{1}{r}\frac{\partial\newtilde{v}_r}{\partial\theta}\right)\tag{B.98}\label{eq:rot_sphere} \end{align} \]
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
\[ \begin{align} \frac{1}{2}\nabla\left(\mathbf{v}\cdot\mathbf{v}\right) &= \frac{1}{2}\mathbf{q}^{(r)}\frac{\partial}{\partial r}\left(\left(v^{(r)}\right)^2 + r^2\left(v^{(\theta)}\right)^2 + r^2\sin^2\left(\theta\right)\left(v^{(\phi)}\right)^2\right)\nonumber\\ & + \frac{1}{2}\mathbf{q}^{(\theta)}\frac{\partial}{\partial\theta}\left(\left(v^{(r)}\right)^2 + r^2\left(v^{(\theta)}\right)^2 + r^2\sin^2\left(\theta\right)\left(v^{(\phi)}\right)^2\right)\nonumber\\ & + \frac{1}{2}\mathbf{q}^{(\phi)}\frac{\partial}{\partial\phi}\left(\left(v^{(r)}\right)^2 + r^2\left(v^{(\theta)}\right)^2 + r^2\sin^2\left(\theta\right)\left(v^{(\phi)}\right)^2\right)\nonumber\\ &= \mathbf{q}^{(r)}\left(v^{(r)}\frac{\partial v^{(r)}}{\partial r} + r^2v^{(\theta)}\frac{\partial v^{(\theta)}}{\partial r} + r\left(v^{(\theta)}\right)^2 + r\sin^2\left(\theta\right)\left(v^{(\phi)}\right)^2 + r^2\sin^2\left(\theta\right)v^{(\phi)}\frac{\partial v^{(\phi)}}{\partial r}\right)\nonumber\\ & + \mathbf{q}^{(\theta)}\left(v^{(r)}\frac{\partial v^{(r)}}{\partial\theta} + r^2v^{(\theta)}\frac{\partial v^{(\theta)}}{\partial\theta} + r^2\sin\left(\theta\right)\cos\left(\theta\right)\left(v^{(\phi)}\right)^2 + r^2\sin^2\left(\theta\right)v^{(\phi)}\frac{\partial v^{(\phi)}}{\partial\theta}\right)\nonumber\\ & + \mathbf{q}^{(\phi)}\left(v^{(r)}\frac{\partial v^{(r)}}{\partial\phi} + r^2v^{(\theta)}\frac{\partial v^{(\theta)}}{\partial\phi} + r^2\sin^2\left(\theta\right)v^{(\phi)}\frac{\partial v^{(\phi)}}{\partial\phi}\right). \end{align} \]
In order to evaluate Eq. (B.68), one first computes
\[ \begin{align} \mathbf{q}_r\times\mathbf{q}_\theta &= \frac{1}{\sin\left(\theta\right)}\mathbf{q}_\phi = r^2\sin\left(\theta\right)\mathbf{q}^{(\phi)},\\ \mathbf{q}_\theta\times\mathbf{q}_\phi &= r^2\sin\left(\theta\right)\mathbf{q}_r = r^2\sin\left(\theta\right)\mathbf{q}^{(r)},\\ \mathbf{q}_\phi\times\mathbf{q}_r &= \sin\left(\theta\right)\mathbf{q}_\theta = r^2\sin\left(\theta\right)\mathbf{q}^{(\theta)}. \end{align} \]
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} \]
It 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 derivative one has
\[ \begin{align} \md{\mathbf{v}} &= \mathbf{e}_r\left(\md{\newtilde{v}_r} - \frac{\newtilde{v}_\phi^2 + \newtilde{v}_\theta^2}{r}\right) + \mathbf{e}_\theta\left(\md{\newtilde{v}_\theta} + \frac{\newtilde{v}_r\newtilde{v}_\theta}{r} - \frac{\newtilde{v}_\phi^2}{r\tan\left(\theta\right)}\right)\nonumber\\ & + \mathbf{e}_\phi\left(\md{\newtilde{v}_\phi} + \frac{\newtilde{v}_r\newtilde{v}_\phi}{r} + \frac{\newtilde{v}_\theta\newtilde{v}_\phi}{r\tan\left(\theta\right)}\right)\tag{B.107}\label{eq:mat_derivative_kugel} \end{align} \]
In order to transform to geographical coordinates, one considers the definition of this coordinate system 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 last identity transforms into the ordinary spherical coordinates. This holds analogously for second derivatives. For the representation of the Laplace operator, Eq. (B.91), it 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
\[ \begin{align} \nabla\times\mathbf{v} &= \left(\frac{\partial w}{\partial y} - \frac{\partial v}{\partial z}\right)\mathbf{i} + \left(\frac{\partial u}{\partial z} - \frac{\partial w}{\partial x}\right)\mathbf{j} + \left(\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}\right)\mathbf{k} + \frac{u\tan\left(\varphi\right)}{r}\mathbf{k} + \frac{u}{r}\mathbf{j} - \frac{v}{r}\mathbf{i}.\tag{B.115}\label{eq:rot_local} \end{align} \]
Furthermore,
\[ \begin{align} \zeta \coloneqq\mathbf{k}\cdot\left(\nabla\times\mathbf{v}\right) = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} + \frac{u\tan\left(\varphi\right)}{r}. \end{align} \]
It should be noted that
\[ \begin{align} \nabla_h \times \mathbf{v}_h &= \left(\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}\right)\mathbf{k} + \frac{u\tan\left(\varphi\right)}{r}\mathbf{k} + \frac{u}{r}\mathbf{j} - \frac{v}{r}\mathbf{i} \not= \zeta\mathbf{k}. \end{align} \]
For the material derivative it follows with Eq. (B.107)
\[ \begin{align} \md{\mathbf{v}} &= \mathbf{i}\left(\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w\frac{\partial u}{\partial z} - \frac{uv\tan\left(\varphi\right)}{a + z} + \frac{uw}{a + z}\right)\nonumber\\ & + \mathbf{j}\left(\frac{\partial v}{\partial t} + u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} + w\frac{\partial v}{\partial z} + \frac{u^2\tan\left(\varphi\right)}{a + z} + \frac{vw}{a + z}\right)\nonumber\\ & + \mathbf{k}\left(\frac{\partial w}{\partial t} + u\frac{\partial w}{\partial x} + v\frac{\partial w}{\partial y} + w\frac{\partial w}{\partial z} - \frac{u^2 + v^2}{a + z}\right)\tag{B.118}\label{eq:mat_deriv_momentum_local}\\ &= \mathbf{i}\left(\md{u} - \frac{uv\tan\left(\varphi\right)}{a + z} + \frac{uw}{a + z}\right)\nonumber\\ & + \mathbf{j}\left(\md{v} + \frac{u^2\tan\left(\varphi\right)}{a + z} + \frac{vw}{a + z}\right)\nonumber\\ & + \mathbf{k}\left(\md{w} - \frac{u^2 + v^2}{a + z}\right)\nonumber\\ &=: \md{\mathbf{v}}\vert_\text{comp} + \md{\mathbf{v}}\vert_\text{met}. \end{align} \]
The material derivative of the velocity is therefore composed of the material derivatives of the components
\[ \begin{align} \md{\mathbf{v}}\vert_\text{comp} \coloneqq \mathbf{i}\md{u} + \mathbf{j}\md{v} + \mathbf{k}\md{w}. \end{align} \]
and the metric terms
\[ \begin{align} \md{\mathbf{v}}\vert_\text{met} \coloneqq \mathbf{i}\left(-\frac{uv\tan\left(\varphi\right)}{a + z} + \frac{uw}{a + z}\right) + \mathbf{j}\left(\frac{u^2\tan\left(\varphi\right)}{a + z} + \frac{vw}{a + z}\right) \textcolor{red}{- \mathbf{k}\frac{u^2 + v^2}{a + z}}. \end{align} \]
The red term is the centrifugal acceleration.
Another form of the advective part of Eq. (B.118) is
\[ \begin{align} \left(\mathbf{v}\cdot\nabla\right)\mathbf{v} &= \left[\left(\mathbf{v}_h + w\mathbf{k}\right)\cdot\nabla\right]\left(\mathbf{v}_h + w\mathbf{k}\right) = \left(\mathbf{v}_h\cdot\nabla\right)\mathbf{v}_h + w\frac{\partial\mathbf{v}_h}{\partial z} + \left(\mathbf{v}\cdot\nabla\right)\left(w\mathbf{k}\right)\nonumber\\ & \stackrel{\href{#eq:diff_op_rule_10}{\text{Eq. (B.57)}}}{=} \left(\nabla\times\mathbf{v}_h\right)\times\mathbf{v}_h + \nabla\frac{\mathbf{v}_h^2}{2} + w\frac{\partial\mathbf{v}_h}{\partial z} + \left(\mathbf{v}\cdot\nabla\right)\left(w\mathbf{k}\right)\nonumber \end{align} \]
\[ \begin{align} \Leftrightarrow \left(\mathbf{v}\cdot\nabla\right)\mathbf{v} &= \left(\nabla_h\times\mathbf{v}_h\right)\times\mathbf{v}_h + \nabla_h\frac{\mathbf{v}_h^2}{2} + w\frac{\partial\mathbf{v}_h}{\partial z} + \left(\mathbf{v}\cdot\nabla\right)\left(w\mathbf{k}\right).\tag{B.122}\label{eq:2dvector_invariant_u_advection} \end{align} \]
In the last step
\[ \begin{align} \left(\mathbf{k}\frac{\partial}{\partial z}\times\mathbf{v}_h\right)\times\mathbf{v}_h &= \left( \begin{array}{c} -\frac{\partial v}{\partial z}\\ \frac{\partial u}{\partial z}\\ 0 \end{array}\right)\times \left( \begin{array}{c} u\\ v\\0 \end{array}\right) = -\left(u\frac{\partial u}{\partial z} + v\frac{\partial v}{\partial z}\right)\mathbf{k} = -\mathbf{k}\frac{\partial\mathbf{v}_h^2}{\partial z} \end{align} \]
was 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
\[ \begin{align} \left(\nabla\times\mathbf{v}\right)\times\mathbf{v}_h &= \left[\nabla\times\left(\mathbf{v}_h + w\mathbf{k}\right)\right]\times\mathbf{v}_h = \left(\nabla\times\mathbf{v}_h\right)\times\mathbf{v}_h + \left(\nabla\times w\mathbf{k}\right)\times\mathbf{v}_h\nonumber\\ & \stackrel{\text{Eq. }\href{#eq:rot_local}{(B.115)}}{=} \left(\nabla\times\mathbf{v}_h\right)\times\mathbf{v}_h + \left(\frac{\partial w}{\partial y}\mathbf{i} - \frac{\partial w}{\partial x}\mathbf{j}\right)\times\mathbf{v}_h = \left(\nabla\times\mathbf{v}_h\right)\times\mathbf{v}_h + \mathbf{k}\left(\mathbf{v}_h\cdot\nabla w\right)\nonumber\\ \Rightarrow\left(\nabla\times\mathbf{v}_h\right)\times\mathbf{v}_h &= \left(\nabla\times\mathbf{v}\right)\times\mathbf{v}_h - \mathbf{k}\left(\mathbf{v}_h\cdot\nabla w\right) \end{align} \]
and
\[ \begin{align} \left(\mathbf{v}_h\cdot\nabla\right)\left(w\mathbf{k}\right) &= \mathbf{k}\left(\mathbf{v}_h\cdot\nabla w\right) + w\left(u\frac{\partial\mathbf{k}}{\partial x} + v\frac{\partial\mathbf{k}}{\partial y}\right) = \mathbf{k}\left(\mathbf{v}_h\cdot\nabla w\right) + w\left(\frac{u}{r\cos\left(\phi\right)}\frac{\partial\mathbf{k}}{\partial\lambda} + \frac{v}{r}\frac{\partial\mathbf{k}}{\partial\phi}\right)\nonumber\\ &= \mathbf{k}\left(\mathbf{v}_h\cdot\nabla w\right) + w\left(\frac{u}{r}\mathbf{i} + \frac{v}{r}\mathbf{j}\right) = \mathbf{k}\left(\mathbf{v}_h\cdot\nabla w\right) + \frac{w}{r}\mathbf{v}_h \end{align} \]
the following holds
\[ \begin{align} \left(\mathbf{v}\cdot\nabla\right)\mathbf{v} &= \left(\nabla\times\mathbf{v}\right)\times\mathbf{v}_h + \nabla\frac{\mathbf{v}_h^2}{2} + w\frac{\partial\mathbf{v}}{\partial z} + \frac{w}{r}\mathbf{v}_h. \end{align} \]
By comparing with Sect. 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) are
\[ \begin{align} \md{\mathbf{v}}\vert_\text{met} &= \left(\begin{array}{c} -\frac{uv\tan\left(\varphi\right)}{a + z} + \frac{uw}{a + z}\\ \frac{u^2\tan\left(\varphi\right)}{a + z} + \frac{vw}{a + z}\\ -\frac{u^2 + v^2}{a + z} \end{array}\right).\tag{B.128}\label{eq:mat_deriv_momentum_met} \end{align} \]
The metric terms in Eq. (B.115) are
\[ \begin{align} \nabla\times\mathbf{v}\vert_\text{met} &= \left(\begin{array}{c} -\frac{v}{r}\\ \frac{u}{r}\\ \frac{u\tan\left(\varphi\right)}{r} \end{array}\right). \end{align} \]
If one multiplies this vectorially by $\mathbf{v}$, one obtains Eq. (B.128):
\[ \begin{align} \nabla\times\mathbf{v}\vert_\text{met}\times\mathbf{v} &= \left(\begin{array}{c} -\frac{v}{r}\\ \frac{u}{r}\\ \frac{u\tan\left(\varphi\right)}{r} \end{array}\right)\times\left(\begin{array}{c} u\\ v\\ w \end{array}\right) = \left(\begin{array}{c} -\frac{uv\tan\left(\varphi\right)}{a + z} + \frac{uw}{a + z}\\ \frac{u^2\tan\left(\varphi\right)}{a + z} + \frac{vw}{a + z}\\ -\frac{u^2 + v^2}{a + z} \end{array}\right) = \md{\mathbf{v}}\vert_\text{met}.\tag{B.130}\label{eq:mat_deriv_momentum_met_prop_0} \end{align} \]
The metric terms in momentum advection are therefore consequences of the metric terms in rotation.
Let $M, N\subseteq\mathbb{R}^3$, $f:M\to\mathbb{R}$ be continuous, $T:N\to M$ be a diffeomorphism, then one has
this corresponds to the multidimensional substitution formula. The determinant $\det\left(T'\right)$ is called the 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
\[ \begin{align} T' = \left(\begin{array}{ccc} \sin\left(\vartheta\right)\cos\left(\varphi\right)&r\cos\left(\vartheta\right)\cos\left(\varphi\right)& -r\sin\left(\vartheta\right)\sin\left(\varphi\right)\\ \sin\left(\vartheta\right)\sin\left(\varphi\right)&r\cos\left(\vartheta\right)\sin\left(\varphi\right)&r\sin\left(\vartheta\right)\cos\left(\varphi\right)\\ \cos\left(\vartheta\right)& -r\sin\left(\vartheta\right)&0 \end{array}\right). \end{align} \]
For the determinant, one has
\[ \begin{align} \det\left(T'\right) &= -r\sin\left(\vartheta\right)\cos\left(\varphi\right)\left(-r\sin\left(\vartheta\right)^2\cos\left(\varphi\right) - r\cos\left(\vartheta\right)^2\cos\left(\varphi\right)\right)\nonumber\\ & - r\sin\left(\vartheta\right)\sin\left(\varphi\right)\left(-r\sin\left(\vartheta\right)^2\sin\left(\varphi\right) - r\cos\left(\vartheta\right)^2\sin\left(\varphi\right)\right)\nonumber\\ &= r^2\sin\left(\vartheta\right)\cos\left(\varphi\right)^2 + r^2\sin\left(\vartheta\right)\sin\left(\vartheta\right)^2 = r^2\sin\left(\vartheta\right), \end{align} \]
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 continuously differentiable. Then
This is the Gaussian theorem.
A sphere $K_N\left(R\right)$ in $\mathbb{R}^N$ with radius $R\geq0$ can be defined by
\[ \begin{align} K_N\left(R\right) \coloneqq \left\{\mathbf{x} = \left(x_1, \dotsc, x_N\right)^T\in\mathbb{R}^N\newvline\sum_{i = 1}^{N}x_i^2\leq R^2\right\}. \end{align} \]
Its volume is
\[ \begin{align} V_N\left(R\right) = \int_{K_N}dx_1\dotsc dx_N. \end{align} \]
One has
\[ \begin{align} \int_{\mathbb{R}^N}\exp\left(-\sum_{i = 1}^{N}x_i^2\right)dx_1\dotsc dx_N &= \int_{0}^{\infty}e^{-R^2}\frac{dV_N}{dR}dR, \tag{B.138}\label{eq:n_dim_kugel_hilfe_1} \end{align} \]
this corresponds to a transformation to spherical coordinates. One 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 over the surface. The $N-$dimensional spherical coordinates consist of $N - 1$ angles and one 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 $N - 1$, and one can write
\[ \begin{align} f_N\left(R\right) = NC_NR^{N - 1}. \end{align} \]
By differentiating Eq. (B.139) with respect to $R$ and using Eq. (B.138), one obtains
\[ \begin{align} \frac{dV_N}{dR} &= NC_NR^{N - 1} \Rightarrow NC_N\int_{0}^{\infty}e^{-R^2}R^{N - 1}dR = \left(\int_{ - \infty}^{\infty}e^{-x^2}dx\right)^N = \pi^{N/2}. \end{align} \]
The last step follows with Eq. (A.107). Using the substitution rule, one has
\[ \begin{align} \int_{0}^{\infty}e^{-R^2}R^{N - 1}dR &= \int_{0}^{\infty}e^{-R}R^{\frac{N - 1}{2}}\frac{1}{2}R^{-\frac{1}{2}}dR = \frac{1}{2}\int_{0}^{\infty}e^{-R}R^{N/2 - 1}dR = \frac{1}{2}\Gamma\left(\frac{N}{2}\right). \end{align} \]
It follows with Eq. (A.113)
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.97).
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:
\[ \begin{align} \frac{d\mathbf{F}\left(t\right)}{dt} = \int_{\Omega\left(t\right)}\frac{\partial\mathbf{f}\left(x, y, z, t\right)}{\partial t}dx + \int_{\partial\Omega}\mathbf{f}\left(x, y, z, t\right)\left(\mathbf{v}\cdot d\mathbf{n}\right) \end{align} \]