Scalars and Vectors

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Scalars and Vectors

Background

This article attempts to explain and illustrate the difference between two very familiar concepts, without introducing extraneous complexity. The mathematician or physicist requiring more complete information on these topics should refer to the references below.

Scalars

From the marine science point of view, a scalar (see reference below) is a measured quantity, such as temperature, salinity or current speed, that is completely specified by its magnitude and has no direction. Scalars are almost always associated with a particular unit of measurement, such as degrees Celsius, or cm/sec, but this is neither universal nor required.

Scalar Fields

In the physical sciences, a scalar field (see reference below) associates a numerical value (the scalar) with every point in space. A scalar field can be roughly represented graphically (or with a table), by specifying the spacing between visualized (or tabulated) values in the X, Y and Z directions. Graphical representations of scalar fields are often called rasters; tabular representations of scalar fields are often called grids. But it must be remembered that the scalar field itself has an infinite number of points, while the grid/raster includes only a sub-set of these point values, selected for convenience of display or storage.

Vectors

There are many definitions of the term vector, but only two basic vectors are of concern for marine data managers: Vector graphics, and Euclidean vectors. The single term "vector" is widely used without apparent confusion within the marine community, indicating the we are already tuned to contextual cues. This article will attempt to leave this happy circumstance untouched.

Vector Graphics

"Vector graphics is the use of geometrical primitives such as points, lines, curves, and shapes or polygon(s), which are all based on mathematical equations, to represent images in computer graphics." [From the Wikipedia reference below]. To this list, of course, should be added the ability to draw alphanumeric characters. The set of specific shapes of the alphanumeric characters is also known as a font. Two systems using vector graphics are HPGL and DXF. Encapsulated Post-Script formatting (EPS) is another example.

Euclidean Vectors

A Euclidean vector is a physical quantity having both speed and direction, such as wind velocity (usually m/sec) or current velocity (usually cm/sec). Physical vectors of this type almost always have units of measurement, as indicated. There are two different methods to specify any Euclidean vector: U and V component vectors, or speed and direction.

Euclidean Vectors Specified by U and V Components

This drawing shows how a vector (the "resultant vector") can be specified by two orthogonal vector components, usually designated U and V. By convention U is the east-west vector component, with east being positive. V is the north-south component, with north being positive. Notice that the location of the vector is taken to be the X, Y coordinates of its base point. So, to summarize, at every location X, Y there is a pair of vector components U and V which uniquely specify a physical vector.

Examples of U and V Vector Component Fields

For visualization and computations, we can treat the U and V components like scalar fields, and display them as grids/rasters, as you see here. These images show the U and V components of the surface currents offshore Namibia (from a shipdrift climatology developed by Richardson). The white areas are either over land or the analysis area.

Example of a Resultant Vector Field

If you calculate and display all the vectors represented by the above U and V values, then you arrive at the vector field shown here. A vector field is a parallel concept to a scalar field (see above). Note that figures like this are only qualitatively useful without some indication of the scale of the arrows.

Euclidean Vectors Specified by Speed and Direction

An alternate way to specify Euclidean vectors is by speed (a scalar quantity) and direction (a unit vector in the desired direction). There are two separate ways to accomplish this, as shown below.

Mathematical Direction - Mathematicians measure angles counter-clockwise from the eastward direction (zero degrees or radians). This is important to know, because when converting from U and V components, the usual trigonometric functions result in angles in this system.

Geographic Direction - Geographers and earth scientists more commonly use angular directions which increase in a clockwise sense from the north pole (zero degrees or radians). This results in the familiar canonical directions N=0°, E=90°, S=180° and W=270°.

Conversion from U/V to Speed/Direction - The program UV2SPDIR is available in OceanTeacher.

Oceanographic versus Meteorological Vector Directions

Oceanographers always depict ocean currents by vectors that point in the same direction as the flow ("toward which"), or exactly the same as the "geographic direction" in the preceding section. Thus, the cardinal directions can be listed as:

0 = toward north
90 = toward east
180 = toward south
270 = toward west
360 = toward north

Meteorologists, on the other hand, sometimes describe wind vectors by the direction "from which" they blow:

0 = toward south
90 = toward west
180 = toward north
270 = toward east
360 = toward south

This situation requires that the directional "sense" of the vectors, although it can usually be inferred from their context, must be checked in all cases for realistic flows. Plotting software usually has a software switch that allows correct depiction of either convention; if no switch is available, then the oceanographic, or "toward which," convention is the usual default.

Euclidean Vector Data Publication and Storage

Storage of Euclidean vector data today is almost universally accomplished with U and V component grids. This is especially true for model simulations, because U and V are always the calculated quantities. A few older datasets, including the earliest analyses of the global shipdrift data at the US National Oceanographic Data Center, were performed and stored using speed and direction. Although speed and direction are not entirely obsolete, for all practical purposes U and V component grids have replaced them.

Additional Resources

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Information about this article

Short title: Scalars and Vectors

Description: What are scalar and vector data, and how are they stored and represented?

Expertise level: beginner

Author: Murray.Brown

Approval status: approved

Approved by: Murray.Brown

Last change: 2012-2-7

Subsection of: Numerical Data

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