Supplement 2.6: Differentials and derivatives (4/4)

The current velocity as an example of a vector

The formalism given so far is valid not only for scalar quantities such as the temperature but also for vectors, for example the velocity vector. When replacing in the equation

\frac{d T}{d t} = \frac{\partial T}{\partial t} + \vec{v} \cdot \nabla T

$T$ by $\vec{v}$ , it becomes:

\frac{d \vec{v}}{d t} = \frac{\partial \vec{v}}{\partial t} + (\vec{v} \cdot \nabla) \vec{v}

The notations 'Lagrangian and Eulerian form' as well as 'convective term' are used in the same way. Special attention has to be payed to the bracketing, though. What does it mean?

The term $\vec{v} \cdot \nabla T$ in the example of temperature is the scalar product of the vectors $\vec{v}$ and $\nabla T$ . The order of multiplication is not of importance in this case: is is also correct to calculate $\vec{v} \cdot \nabla$ at first and further multiply the scalar result with $T$ .

However, the term $\nabla \vec{v}$ is not defined, which is why the order of multiplication is of importance here. This is clarified by the brackets in the term $(\vec{v} \cdot \nabla) \vec{v}$ . The dot product in brackets becomes

\vec{v} \cdot \nabla = u \frac{\partial}{\partial x} + v \frac{\partial}{\partial y} + w \frac{\partial}{\partial z}

and the result - a scalar - is multiplied with $\vec{v}$ ,

(\vec{v} \cdot \nabla) \vec{v} = u \frac{\partial \vec{v}}{\partial x} + v \frac{\partial \vec{v}}{\partial y} + w \frac{\partial \vec{v}}{\partial z}

its Cartesian components:

\begin{array}{l} (\vec{v} \cdot \nabla) u = u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} \\ (\vec{v} \cdot \nabla) v = u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z} \\ (\vec{v} \cdot \nabla) w = u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y} + w \frac{\partial w}{\partial z} \end{array}

Task 5: Nabla operator and velocity vector ↓

The Nabla operator can be linked to a vector, for example the velocity vector.

a) Depict the terms $\nabla \cdot \vec{v}$ and $\nabla \times \vec{v}$ ` in Cartesian coordinates. Are they scalars or vectors?

b) What is the meaning of these terms? What does the term $\vec{v} \cdot \nabla$ mean?

You may find the solutions at this site.

Hence, the total derivative of the velocity in components becomes:

\frac{d u}{d t} = \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z}

\frac{dv}{d t} = \frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z}

\frac{dw}{d t} = \frac{\partial w}{\partial t} + u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y} + w \frac{\partial w}{\partial z}

These are the fundamental kinematic relations used for the physical description of currents: they specify the acceleration of deformable media such as air and water.

When adding other acceleration terms which result from forces as the cause and effect of acceleration, the hydrodynamic momentum equation will be obtained. If friction belongs to the forces, the result will be the Navier Stokes equation of hydrodynamics.

Mechanics of mass points and of deformable media ↓

If the movement of an object is depicted with the focus on "How" it happens, this is called kinematics of motion. For describing the motion of the mass point of an object which is the centre of gravity, its position $\vec{r} (t)$ , its velocity $\vec{v} (t) = \frac{d \vec{r} (t)}{d t}$ and its acceleration $\vec{a} (t) = \frac{d \vec{v} (t)}{d t} = \frac{d^{2} \vec{r} (t)}{d t^{2}}$ are used.

Considering the "Why" of motion, the forces influencing the movement come into play; this represents the dynamics of motion. Newton's equation of motion

\vec{F} = m \frac{d \vec{v}}{d t} = m \vec{a}

is an example for an object having the mass $m$ which is accelerated by $\vec{a}$ due to an external force $\vec{F}$ .

For deformable media, i.e., gases and liquids the movement of the centre of gravity rarely leads to the desired result. Moreover, the details of the movement of this spatially extended and in most cases complex structured object including the internal processes are of interest. The equation of motion is analogously formulated; using the mass per volume, i.e., the density at the place of the mass, $ρ = \frac{d m}{d V}$ , which may vary in space and time. Correspondingly, the external force density is introduced replacing the external forces:

\frac{d \vec{F}}{d V} = ρ \frac{d \vec{v}}{d t} = ρ \vec{a}

An infinitesimal mass element is accelerated just as the centre of gravity of a solid body according to $\frac{d \vec{v}}{d t}$ . In this case, it is the summation of the local acceleration $\frac{\partial \vec{v}}{\partial t}$ and the influence of the surrounding elements of the medium $(\vec{v} \cdot \nabla) \vec{v}$ , i.e., the convective term.

The following graph shows the trajectories of surface drifters, with a colour bar in this example, which shows the drifting speed in m/s. It is revealed that the drifting speed is very variable and additionally can take on high values in the open sea. This is an example of the Lagrangian representation of currents.

Drifting routes near the Dominican Republic and Puerto Rico

Trajectories of 18 surface drifters in the surrounding of the Dominican Republic and Puerto Rico. The colour bar shows the drifting speed in m/s.
Source: Caribbean Coastal Ocean Observation System (Caribbean IOOS).

The next graphics shows a time series of ocean currents over two days in Lower Saxony's Wadden Sea near Spiekeroog island. The diagramme shows the current speed, which is characterised by the half-day cycle of the tides, and the current direction which is dominated by the north-south direction through Spiekeroog's tidal inlet. This is an example of the Eulerian representation of flow velocity.

Current data taken at a time-series station in the tidal inlet of Spiekeroog island. Centre and bottom: amount and direction of the current. Top right: vectors of the current velocity in a polar coordinate diagramme.

With satellite-borne Radar Altimetry it is possible to determine global ocean currents from the measured topography of the ocean surface. For this aspect you may find a video of several years realised by NASA's Scientific Visualization Studio with additional links as an example of a Lagrangian representation of ocean currents.

Le spectre de la Terre

Supplement 2.6: Differentials and derivatives (4/4)

The current velocity as an example of a vector