Supplement 2.6: Differentials and derivatives (3/4)

The temperature as an example ... (cont.)

The total derivative of the temperature can be given if the spatial variables resp. the position vector are functions of the time:

\vec{r} = f (t)

This is obviously the case for moving air or flowing water if the position vector marks the timewise changing position of a defined air or water element. It follows:

\begin{array}{l} \frac{d T}{d t} = \frac{\partial T}{\partial x} \frac{d x}{d t} + \frac{\partial T}{\partial y} \frac{d y}{d t} + \frac{\partial T}{\partial z} \frac{d z}{d t} + \frac{\partial T}{\partial t} \\ = \frac{\partial T}{\partial x} u + \frac{\partial T}{\partial y} v + \frac{\partial T}{\partial z} w + \frac{\partial T}{\partial t} \end{array}

Hereby,

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

are the velocity components of the flow in the directions $x$ , $y$ and $z$ .

In vector notation:

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

or with the velocity vector $\vec{v} = (u, v, w)$ :

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

In hydro- and aerodynamics such as in meteorology and oceanography, specific words have been formed for these terms:

$\frac{d T}{d t}$ is the Lagrangian form: the spatial coordinates and the time are variable, i.e., an air or water element is timewise observed while moving through space and its change in temperature is described.
In experiments for example, this can be carried out through a thermometer within a buoyancy-neutral balloon which follows the wind like an air element; in waters or the sea, it is a buoyancy-neutral drifter which follows the water current.
$\frac{\partial T}{\partial t}$ is the Eulerian form: the time is variable while the position is fixed. The temperature of the passing flow of air or water is measured this time. This is why it is also called local derivative.
In experiments, this can be put into practice by the help of sensors mounted at a fixed spot, for example with an installed weather station or a data buoy secured on the ground. Examples of temperature data of the sea which has been collected at an installed measuring station in lower Saxony's Wadden Sea close to Spiekeroog island are vigorously discussed in the SEOS Tutorial about time series analysis.
$\vec{v} \cdot \nabla T$ is called the convective term: it includes the temperature gradient, i.e., the spatial change of temperature which is multiplied with the velocity vector. Added with the Eulerian form the convective term yields the Lagrangian form of the temperature field.

Weather balloons, which are important for the weather forecast, do not behave according to the Lagrangian approach since they do not float in the air but rise up to about 30 km into the sky in order to record vertical profiles of the air temperature and other parameters.

Drifters in the ocean fulfil by far better the requirements: they float as shown in the Global Drifter Program with the current on the water surface or they float in 1000 to 2000 m depth as the Argo Floats do. These so called Lagrangian drifters can be operated autonomously for one to two years before requiring maintenance.

The picture below shows drifting paths (also called trajectories) that were recorded in the south of Puerto Rico with drifters. The measurements serve the examination of ocean current, temperature data was not collected. They show how variable the current may be even in a small coastal area.

Trajectories of drifters who started at the same time from several locations in the south of Puerto Rico on the surface (rectangles) and in 18 m depth (stars). The coloured scale shows the time in hours after release of the drifters.
Source: Caribbean Coastal Ocean Observation System (Caribbean IOOS).

The following global maps of the land surface temperature are snapshots as explained in the infobox 'Fewer variables' at the end of the previous page. The time does not appear as a variable. Instead, the temperatures measured during one day or several days are accumulated in one map. Such snapshots are called synoptic representation, i.e., as an overall representation of several data measured at the same time; in case of global data sets this goal cannot be fully met.

22 Feb 2017 18-26 Feb 2017 Feb 2017

Global land surface temperatures. Left: 22 February 2017. Middle: weekly mean of 18 to 26 February 2017. Right: monthly mean of February 2017. Mosaic representation from data of the MODIS Spectrometer on NASA's Terra satellite at daylight. It becomes clear that data from one single day is not enough to depict the entire earth's surface; as a cause, the incomplete coverage of the satellite and clouding covers thwart temperature measurement on the surface.
Source: NEO - NASA Earth Observations.

When preparing a video from chronological snapshots, one obtains a Lagrangian presentation since the spatial coordinates (here: in two dimensions) as well as the time is considered. On the Links page you find a reference leading to a land surface temperature video over several years published in NASA's Earth Observatory as an example of a Lagrangian presentation of global temperatures.

Le spectre de la Terre

Supplement 2.6: Differentials and derivatives (3/4)

The temperature as an example ... (cont.)