make a visual model of this outcrop that honors
the geometry of the prototype using image processing software (Fig. 4.11b). Here the wavelength
and height are halved in value, so L m ϭ 106 mm and
H m ϭ 6 mm. The model ratio, L r , for the lengths is:
(4.88)
Any other length that we measure on this outcrop
is related to the corresponding length in the
model by this same ratio. For example, for the
height of the gray band we have:
(4.89)
The prototype and model are geometrically similar.
The image of the outcrop shown in Fig. 4.11a is
scaled down to a smaller size to fit on the page of
this book. Thus, the lens cap displayed on the
figure is not 50 mm in diameter, but it represents
50 mm on the actual exposure. The photographic
image is, itself, a geometrically scaled model of
the exposure and the process of reproduction of
this image maintains geometric similarity.
Another “model” of the exposure is shown in
Fig. 4.11c. Is this geometrically similar? The field
of view has changed somewhat, but the image
seems to have many similarities to the prototype
shown in Fig. 4.11a. On the other hand, a careful
inspection reveals that the fold shape is distorted
relative to the prototype. We measure the wavelength in this model as L m ϭ 106 mm and the
height as H m ϭ 12 mm. Comparing the model
ratios we find:
(4.90)
The scaling in the horizontal direction is the same
as that used to produce Fig. 4.11b, but lengths
measured in the vertical direction are the same as
those in the prototype. Thus, the height of the
layer in the prototype and “model” are identical.
This “model” does not preserve geometric similarity. Comparing the lens cap in the three figures
confirms the scaling relations.
Tectonic processes may be very slow to develop,
but they are not static. Thus, the relative time
scales for the model, T m , and the prototype, T p , must
be considered carefully when designing a model
L m
L p
ϭ
1
2
ϭ L r ,
H m
H p
ϭ
1
1
϶ L r
H m
H p
ϭ L r ϭ
1
2
L m
L p
ϭ L r ϭ
1
2
experiment. The time is measured from some arbitrary moment, often at the initiation of the
process, and the model ratio for time is defined as:
(4.91)
Given this ratio, one can compare the prototype
and a model at corresponding times during the
development of the process. We say that the prototype and the model are kinematically similar if
they are geometrically similar at every corresponding time over the duration of the process. By
corresponding time we mean a time for the prototype process and a time for the model process
that are related by the model ratio for time.
Kinematic similarity can be understood in
terms of two motion pictures, one of the model
and the other of the prototype. Let’s say the
model ratio for time is T r ϭ 3.1 ϫ 10
Ϫ11 , so each
model second represents one thousand prototype
years. The camera recording the model process
shoots at a speed of one frame per second and the
camera recording the prototype process shoots at
a speed of one frame per thousand years. If each
successive pair of frames of the two motion pictures is geometrically similar, the two processes
also are kinematically similar. The corresponding
frames may have different length scales, but
lengths throughout the prototype and model
obey the model ratio for lengths.
Turning to a geological example, it is likely
that the layers shown in Fig. 4.11a had lesser
amplitudes and greater wavelengths at an earlier
time in the folding process. In Fig. 4.12 three different stages in the hypothetical one million year
development of the prototype fold are illustrated,
assuming that the deformation conserved
volume and that the shortening in the direction
of the measured wavelength is simply the reciprocal of the elongation perpendicular to this
direction. We start to record the process at an
arbitrary time (0 s) shown in Fig. 4.12a. Five
hundred thousand years into the process the prototype fold would look like the image in Fig.
4.12b, and at the end of the one million years the
fold would have attained the shape observed in
outcrop today (Fig. 4.12c).
For a kinematically similar model of this
folding process using the model ratio proposed
T m
T p
ϭ T r
4.4 SCALED LABORATORY MODELS
145
the geometry of the prototype using image processing software (Fig. 4.11b). Here the wavelength
and height are halved in value, so L m ϭ 106 mm and
H m ϭ 6 mm. The model ratio, L r , for the lengths is:
(4.88)
Any other length that we measure on this outcrop
is related to the corresponding length in the
model by this same ratio. For example, for the
height of the gray band we have:
(4.89)
The prototype and model are geometrically similar.
The image of the outcrop shown in Fig. 4.11a is
scaled down to a smaller size to fit on the page of
this book. Thus, the lens cap displayed on the
figure is not 50 mm in diameter, but it represents
50 mm on the actual exposure. The photographic
image is, itself, a geometrically scaled model of
the exposure and the process of reproduction of
this image maintains geometric similarity.
Another “model” of the exposure is shown in
Fig. 4.11c. Is this geometrically similar? The field
of view has changed somewhat, but the image
seems to have many similarities to the prototype
shown in Fig. 4.11a. On the other hand, a careful
inspection reveals that the fold shape is distorted
relative to the prototype. We measure the wavelength in this model as L m ϭ 106 mm and the
height as H m ϭ 12 mm. Comparing the model
ratios we find:
(4.90)
The scaling in the horizontal direction is the same
as that used to produce Fig. 4.11b, but lengths
measured in the vertical direction are the same as
those in the prototype. Thus, the height of the
layer in the prototype and “model” are identical.
This “model” does not preserve geometric similarity. Comparing the lens cap in the three figures
confirms the scaling relations.
Tectonic processes may be very slow to develop,
but they are not static. Thus, the relative time
scales for the model, T m , and the prototype, T p , must
be considered carefully when designing a model
L m
L p
ϭ
1
2
ϭ L r ,
H m
H p
ϭ
1
1
϶ L r
H m
H p
ϭ L r ϭ
1
2
L m
L p
ϭ L r ϭ
1
2
experiment. The time is measured from some arbitrary moment, often at the initiation of the
process, and the model ratio for time is defined as:
(4.91)
Given this ratio, one can compare the prototype
and a model at corresponding times during the
development of the process. We say that the prototype and the model are kinematically similar if
they are geometrically similar at every corresponding time over the duration of the process. By
corresponding time we mean a time for the prototype process and a time for the model process
that are related by the model ratio for time.
Kinematic similarity can be understood in
terms of two motion pictures, one of the model
and the other of the prototype. Let’s say the
model ratio for time is T r ϭ 3.1 ϫ 10
Ϫ11 , so each
model second represents one thousand prototype
years. The camera recording the model process
shoots at a speed of one frame per second and the
camera recording the prototype process shoots at
a speed of one frame per thousand years. If each
successive pair of frames of the two motion pictures is geometrically similar, the two processes
also are kinematically similar. The corresponding
frames may have different length scales, but
lengths throughout the prototype and model
obey the model ratio for lengths.
Turning to a geological example, it is likely
that the layers shown in Fig. 4.11a had lesser
amplitudes and greater wavelengths at an earlier
time in the folding process. In Fig. 4.12 three different stages in the hypothetical one million year
development of the prototype fold are illustrated,
assuming that the deformation conserved
volume and that the shortening in the direction
of the measured wavelength is simply the reciprocal of the elongation perpendicular to this
direction. We start to record the process at an
arbitrary time (0 s) shown in Fig. 4.12a. Five
hundred thousand years into the process the prototype fold would look like the image in Fig.
4.12b, and at the end of the one million years the
fold would have attained the shape observed in
outcrop today (Fig. 4.12c).
For a kinematically similar model of this
folding process using the model ratio proposed
T m
T p
ϭ T r
4.4 SCALED LABORATORY MODELS
145