http://collier.ifas.ufl.edu/CommHort...urfDrought.pdf
Above is a short how-to article published during a past drought for those wanting to promote root growth and develop a lawn that needs less water.
Summary:
Mow high
Water deeply and less frequently (after slight wilt to stimulate roots)
Fertilize with potassium for roots, use less nitrogen and more often for grass blade growth.
Relationship between mowing and root growth:
"Proper mowing practices are essential for good quality and drought-tolerant turf. Every time a lawn is
mowed, the metabolic activities of the grass are stressed, which reduces root growth. Mowing frequency
and cutting height need to be carefully considered for a healthy lawn. Using the highest cutting height on
the mower will facilitate turfgrass drought-conditioning. This will increase the grass leaf area, allowing for
more photosynthesis. This results in more carbohydrates, which are then available for use in recovery
from stresses such as drought. The higher the mowing height, the deeper and more extensive the root
system will be."