Geoscience Reference
In-Depth Information
Appendix C
Dimensional Analysis
Some problems in natural sciences are too complex to describe with fundamental
laws. In those cases, dimensional analysis is an important tool to ind the dependence
of a certain variable in a low (e.g., a concentration gradient) on other quantities.
Dimensional analysis consists of four steps:
1. Find the relevant physical quantities that (may) determine the quantity of interest.
2. Make dimensionless groups out of the quantities selected in step 1.
3. Do an experiment (or analyse existing data) in which all quantities selected in step 1 are
measured.
4. Find a relationship between the dimensionless groups made in step 2, and calculated
with the data of step 3. If all goes well, the dimensionless groups show a universal rela-
tionship that can also be used for other, similar situations.
Below, we briely focus on the four steps, and we take as an example the vertical
gradient of the mean horizontal wind speed
u
z
, under conditions where buoyancy
plays a role.
C.1 Choose Relevant Physical Quantities
The selection of relevant quantities requires insight into the problem, and some expert
judgement. But if one selects too few quantities, the relationships found in step 4 will
not be universal: they will differ from one experiment to another. On the other hand,
if too many quantities are selected, it will turn out that the relationships found in step
4 will not depend on the irrelevant quantities.
For the example at hand, the relevant quantities are:
u
The wind speed gradient
z itself
Height above the ground,
z (this determines the size of turbulent eddies)
356
Search WWH ::




Custom Search