One of the key theoretical tools used to study these
structural aspects of reefs is the “Richardson plot” (Hearn,
2008). This is based on the length, S, of a chain which is
composed of links of size s, and made to fit, as tightly as
possible, over the surface of a reef between two points
on the reef with a straight-line (not necessarily horizontal)
separation s 0 . The maximum link size that we could
choose is evidently s 0 . Experiments with chains are widely
used on reefs (e.g., Purkis and Kohler, 2008) and we can
also emulate these field experiments with numerical procedures if we have data on the three-dimensional structure
of a reef. Numerically, we can also repeat the chain experiment using a net with individual mesh elements each having a surface area of s, fitted tightly over the reef, and
compare the total area of the net, S, with the flat projection
of that area s 0 . These procedures give the linear (or areal)
“rugosity” S/s 0 from scale s 0 to s. The decadal rugosity r 10
refers to the case s 0 /s = 10.
The Richardson plot is dependent on the position on the
reef that we choose for our chain, or net, and will differ
greatly between the various zones of the reef. So, we think
of the rugosity as a strictly local measure. We can increase
the length of the chain, or the total area of the net, until we
cover the entire reef. As we increase the value of s 0 , in this
way, we span a larger part of the reef and so S(s) represents
an average over an increasing length, or area, of the reef.
In so doing, we do loose strictly local information through
this averaging process.
Hydrodynamic models, whether conceptual, analytic,
or numerical, use some sort of spatial network of cells
which can have a variety of shapes which may not necessarily be the same for all cells. A bathymetry of the reef
can be defined at the centers of these cells and this would
be the basis of a solid reef model. We can, in principle,
make a Richardson plot within each cell with s 0 spanning
that cell so that S(s) represents the cell’s “individual topography” within a larger scale “bathymetry” of resolution s 0 .
The “dimensional roughness” s from scale s 0 to s, is
defined as log 10 (r 10 ), or as the mean negative slope of
log(S) against log(s). It provides a “filling factor,” 0 < s
< 1 for the reef structure at scale s within the space defined
by s 0 . As an example, a line can be made to completely
“fill” a two dimensional space. This is called a “Peano
curve” after Giuseppe Peano (1858–1932). The length of
a chain fitted over such a line doubles whenever we half
the length of its individual links so that s = 1. Values of
s at scales of 10–100 m for many coastal structures are
of order 0.1–0.3, i.e., r 10 = 1.3–2.0. At scales from
a few meters to kilometers, filling factors, s, for reefs of
at least 50% (r 10 > 3.2) are reported by Purkis and Kohler
(2008) using Lidar data (“entry roughness”). Note that if s
approached unity at s = 10 m, any “holes” in the structure
would have a size less than 10 m. Increasing the filling factor increases the surface area and so assists the geochemical
processes but does increase the friction and impede flow; it
also increases stress, erosion and severity of storm damage.
Rugosity of natural surfaces tends to increase indefinitely as we reduce the spatial scale and this marks their
fundamental difference from geometric surfaces. For
example, a geometric surface formed from a flat bottom
covered with hemispheres fitted tightly together (so that
it looks rather like the back of one of the preformed sheets,
or boxes, used to pack hens’ eggs) is a geometric surface
and has a limiting areal rugosity (at all spatial scales less
than the diameters of the hemispheres) of 1 + p $ 4 (relative to a flat plane). This limiting rugosity is otherwise
independent of the diameter of the hemispheres. By contrast, natural surfaces do not reach a limiting rugosity at
small sales although they may have distinct zones, or
ranges, of spatial scales in which limits are effectively
reached. In reality, our “egg box” example shows the same
phenomenon since at small enough spatial scales we start
to see an increase in surface area due to the small-scale
structure of the actual material used in its construction.
In the same way, most coastal systems are represented
by a bathymetry, or topography, which represents the system down to scales of meters. There is then a separation of
scales until we see variability at scales that represent sand
ripples and sand grains. For these cases, the Richardson
plot naturally divides into distinct regions on the s axis
and the important hydrodynamic lengths correspond to
values of s that separate these regions. For example, the
roughness lengths used in most coastal systems is the scale
at which sand ripples, or sand grains, start to change the
Richardson plot. This separation of scales is important to
the structure of hydrodynamic processes.
Topography complexity in reefs stretches from scales
of 100’s of meters down to millimeters. One very important question is whether there are changes in the Richardson plot which allow some separation of scales. This is
likely to occur at the size of individual coral colonies. This
scale varies with position in the reef. The basic approach
used by Hearn (2010) is to devise an unstructured spatial
grid in which one colony occurs in each cell of the grid
and cells do not span parts of colonies. Solid reef models
incorporate high levels of rugosity at the sub-grid cell
scales through increased values of the roughness length.
Although simple intuitive estimates can be made of roughness lengths for experimental models, the best approach is
to fit the Richardson plot inside a cell to a model of the
roughness elements. This involves the form of S(s) averaged over a numerical cell (s 0 = planar area of cell) with
minimum s one to two orders of magnitude smaller than
s 0 . Critical to this fitting procedure is the spatial scales at
which the roughness elements start to affect the Richardson
plot. For, example, data for individual polyps have been
obtained by magnetic resonance imaging from scales of
hundreds of microns to centimeters (Zawada et al., 2006).
The Richardson plot for a linear chain can be used to
probe the anisotropy of the reef structure. Anisotropy
exists in “spur and grove” structures and is common on
the windward side of reefs as part of the wave dissipation
processes (“entry Spur and groove”). This is important to
hydrodynamic processes and is accommodated either by
using anisotropic friction or spatially resolving the structure within the bathymetry mesh.
570
HYDRODYNAMICS OF CORAL REEF SYSTEMS
structural aspects of reefs is the “Richardson plot” (Hearn,
2008). This is based on the length, S, of a chain which is
composed of links of size s, and made to fit, as tightly as
possible, over the surface of a reef between two points
on the reef with a straight-line (not necessarily horizontal)
separation s 0 . The maximum link size that we could
choose is evidently s 0 . Experiments with chains are widely
used on reefs (e.g., Purkis and Kohler, 2008) and we can
also emulate these field experiments with numerical procedures if we have data on the three-dimensional structure
of a reef. Numerically, we can also repeat the chain experiment using a net with individual mesh elements each having a surface area of s, fitted tightly over the reef, and
compare the total area of the net, S, with the flat projection
of that area s 0 . These procedures give the linear (or areal)
“rugosity” S/s 0 from scale s 0 to s. The decadal rugosity r 10
refers to the case s 0 /s = 10.
The Richardson plot is dependent on the position on the
reef that we choose for our chain, or net, and will differ
greatly between the various zones of the reef. So, we think
of the rugosity as a strictly local measure. We can increase
the length of the chain, or the total area of the net, until we
cover the entire reef. As we increase the value of s 0 , in this
way, we span a larger part of the reef and so S(s) represents
an average over an increasing length, or area, of the reef.
In so doing, we do loose strictly local information through
this averaging process.
Hydrodynamic models, whether conceptual, analytic,
or numerical, use some sort of spatial network of cells
which can have a variety of shapes which may not necessarily be the same for all cells. A bathymetry of the reef
can be defined at the centers of these cells and this would
be the basis of a solid reef model. We can, in principle,
make a Richardson plot within each cell with s 0 spanning
that cell so that S(s) represents the cell’s “individual topography” within a larger scale “bathymetry” of resolution s 0 .
The “dimensional roughness” s from scale s 0 to s, is
defined as log 10 (r 10 ), or as the mean negative slope of
log(S) against log(s). It provides a “filling factor,” 0 < s
< 1 for the reef structure at scale s within the space defined
by s 0 . As an example, a line can be made to completely
“fill” a two dimensional space. This is called a “Peano
curve” after Giuseppe Peano (1858–1932). The length of
a chain fitted over such a line doubles whenever we half
the length of its individual links so that s = 1. Values of
s at scales of 10–100 m for many coastal structures are
of order 0.1–0.3, i.e., r 10 = 1.3–2.0. At scales from
a few meters to kilometers, filling factors, s, for reefs of
at least 50% (r 10 > 3.2) are reported by Purkis and Kohler
(2008) using Lidar data (“entry roughness”). Note that if s
approached unity at s = 10 m, any “holes” in the structure
would have a size less than 10 m. Increasing the filling factor increases the surface area and so assists the geochemical
processes but does increase the friction and impede flow; it
also increases stress, erosion and severity of storm damage.
Rugosity of natural surfaces tends to increase indefinitely as we reduce the spatial scale and this marks their
fundamental difference from geometric surfaces. For
example, a geometric surface formed from a flat bottom
covered with hemispheres fitted tightly together (so that
it looks rather like the back of one of the preformed sheets,
or boxes, used to pack hens’ eggs) is a geometric surface
and has a limiting areal rugosity (at all spatial scales less
than the diameters of the hemispheres) of 1 + p $ 4 (relative to a flat plane). This limiting rugosity is otherwise
independent of the diameter of the hemispheres. By contrast, natural surfaces do not reach a limiting rugosity at
small sales although they may have distinct zones, or
ranges, of spatial scales in which limits are effectively
reached. In reality, our “egg box” example shows the same
phenomenon since at small enough spatial scales we start
to see an increase in surface area due to the small-scale
structure of the actual material used in its construction.
In the same way, most coastal systems are represented
by a bathymetry, or topography, which represents the system down to scales of meters. There is then a separation of
scales until we see variability at scales that represent sand
ripples and sand grains. For these cases, the Richardson
plot naturally divides into distinct regions on the s axis
and the important hydrodynamic lengths correspond to
values of s that separate these regions. For example, the
roughness lengths used in most coastal systems is the scale
at which sand ripples, or sand grains, start to change the
Richardson plot. This separation of scales is important to
the structure of hydrodynamic processes.
Topography complexity in reefs stretches from scales
of 100’s of meters down to millimeters. One very important question is whether there are changes in the Richardson plot which allow some separation of scales. This is
likely to occur at the size of individual coral colonies. This
scale varies with position in the reef. The basic approach
used by Hearn (2010) is to devise an unstructured spatial
grid in which one colony occurs in each cell of the grid
and cells do not span parts of colonies. Solid reef models
incorporate high levels of rugosity at the sub-grid cell
scales through increased values of the roughness length.
Although simple intuitive estimates can be made of roughness lengths for experimental models, the best approach is
to fit the Richardson plot inside a cell to a model of the
roughness elements. This involves the form of S(s) averaged over a numerical cell (s 0 = planar area of cell) with
minimum s one to two orders of magnitude smaller than
s 0 . Critical to this fitting procedure is the spatial scales at
which the roughness elements start to affect the Richardson
plot. For, example, data for individual polyps have been
obtained by magnetic resonance imaging from scales of
hundreds of microns to centimeters (Zawada et al., 2006).
The Richardson plot for a linear chain can be used to
probe the anisotropy of the reef structure. Anisotropy
exists in “spur and grove” structures and is common on
the windward side of reefs as part of the wave dissipation
processes (“entry Spur and groove”). This is important to
hydrodynamic processes and is accommodated either by
using anisotropic friction or spatially resolving the structure within the bathymetry mesh.
570
HYDRODYNAMICS OF CORAL REEF SYSTEMS