Dimensional analysis
Dimensional analysis is a powerful tool for understanding the interpretation of hydrologic variables, and the reporting of dimensions is fairly common in the hydrologic literature. Also, dimensional analysis is a powerful tool for checking the validity of mathematical calculations, which is why carrying units through calculations is so important in applied mathematics. If adept at dimensional analysis, mistakes in mathematical derivations or calculations can quickly become obvious.
Through this class, attention to the dimensionality of variables will greatly enhance the logic of the quantitative material we will review. Dimensional analysis allows equations to simply make sense, rather than having to memorize them. The goal is to take your mathematical ability to the next step, where equations and algebra are a language for communicating theory, and not just tools for calculating numbers.
Contents of this module
Dimensions and units
Water is physical matter, which means it has mass and occupies a volume of space. Let's start with the dimensions that describe these fundamental properties (6:33 min).
Units do not define dimensions. In fact, the relationship is quite the opposite. You have to know the dimensions of a variable before you can decide on the appropriate units for a number to evaluate it. The following reviews the relationship between units and dimensions (5:05 min).
Many hydrologic variables are normalized to area for various reasons. Let's review how we make sense of area-specific precipitation volumes and intensities, leaving us with length or length per time as the simplified dimensions of area-normalized volume or volumetric flux (3:34 min).
Dimensional analysis
A combination of variables with interrelated dimensions compose the reference frames that characterize our conceptual models of hydrologic systems. We will need some more fundamental dimensions that ultimately allow us to think about the energy states that allow us to predict how water moves through these systems (4:43 min).
The derivation of Sir Isaac Newton's definition of energy is critical to understanding how matter affects other matter adjacent to it, due to phenomena like gravitational mechanics or thermodynamics. A review of basic Newtonian mechanics will help clarify how more complex dimensions of velocity, acceleration, and force are derived from the fundamental dimensions of mass, length, and time (5:09 min).
In combination with a firm grasp on algebra, dimensional analysis is critical to intuitive understanding of applied mathematics. With practice, these skills make it easy to spot mistakes both in calculations (working units through the algebra) as well as derivations. (2:00 min).
Many scientists are first introduced to dimensional analysis via training in the manipulation of units. Let's review the "unit factor" method of unit conversion to understand how it is related to dimensional analysis. Unit factors are basically dimensionless ratios that allow unambiguous organization of the translation of linearly related scales. This video also gives a brief practical example of using the equation editor in Word to show your mathematical work with a professional presentation (8:02 min).
Dimensions of concentrations and volumes
Finally, let's dive deeper into an example of dimensional thinking by reviewing the mathematics of solutes and solutions that are quite commonly applied across many applications in hydrology. Also we can review the nature of the physical property of electrical conductivity of water that is frequently used to assess the concentration of dissolved ions.
The following video references the dimensions of molarity as [L^-3] or molality as [M^-1]. This practice is outdated and my current preference is to recognize variables quantifying numbers of molecules (in units of moles) specifically as the dimension of count, symbolized as [N]. This is to clearly differentiate variables that are counts of entities from variables that are dimensionless ratios, which have very different interpretations. My current preference is to think of the dimensionality of molarity as [N L^-3] and molality as [N M^-1], rather than the dimensionless counts suggested in this video (9:53 min).
We can put understanding of solutions to work by thinking through the estimation of the volume of a solvent by dilution with a known mass of solute. This is good preparation for understanding the principles underlying the dilution gauging method of stream flow measurement (7:55 min).
Summary and supporting materials
Study guide
Click this link to download the MS Word file
Study guides are designed to summarize the vocabulary, concepts, and mathematics learned in this module.
study_guide_dimensions.pdfReadings from Dingman (3rd ed)
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A list of associated readings from Physical Hydrology by S. Lawrence Dingman (3rd edition)
dingman_3ed_dimensions.pdfSlides for dimensional analysis
Click this link to download the MS PowerPoint file
The embedded Google viewer below sometimes provides poor renderings of Microsoft files. Use the link above to download the original file with proper formatting.
Slides for concentration and volume
Click this link to download the MS PowerPoint file
The embedded Google viewer below sometimes provides poor renderings of Microsoft files. Use the link above to download the original file with proper formatting.
Examples of unit conversion using unit factors
This is the MS Word document with a brief example of unit factors from the video.
Click this link to download the MS Word document
The embedded Google viewer below sometimes provides poor renderings of Microsoft files. Use the link above to download the original file with proper formatting.
In particular, the Google viewer may show the units in an italicized font, which is not proper form.
Useful materials for further study or skill development
Laboratory preparation materials for this module
A deeper dive from another class
A table of examples of dimensions commonly used in hydrology
dimension_table.pdfAnother video on dimensional analysis
A video from Sixty Symbols on the flexibility and power of dimensional analysis techniques
Algebraic treatment of units and dimensions
Another review of treating units algebraically and dimensional analysis from Khan Academy
Unit factor method
Another web page with a review of the unit factor method
Note that this web page perhaps mistakenly refers to the unit-factor method of unit conversion as "dimensional analysis". Dimensional analysis is a much broader and more powerful tool than unit conversion, so I do not consider this an accurate designation. Nevertheless, dimensional analysis is required to understand how the unit-factor method works and this web page is still quite useful.
A web page with a useful description of the detailed steps of the unit factor method
Another video with MS Word Equation Editor keyboard shortcuts
A useful video on basic Microsoft Word equation editor shortcuts. Note that the links in this video description are broken. A link to a reference for keyboard shortcuts is provided below.
A quick reference for keyboard shortcuts in Microsoft equation editor
Equation Editor.pdf