folds provide quantitative estimates of the regularity of folding. The arc length is commonly, but
incorrectly, called the fold wavelength (Fig. 4.10b).
Our analysis considers the folded quartz vein to be
“isolated.” What we principally mean by this is
that the folding is independent of that in nearby
layers. That is, we do not see other nearby layers
that are folded in concert with them.
Note that the original vein thickness was not
uniform and that this has affected the regularity
of the folding, but the arc length to thickness
ratios fall within a modest range. What might
this regularity mean? Clearly, it cannot easily be
ascribed to something “built into” the rock prior
to folding, since the regularity is present only in
this layer. We must then ascribe it to some mechanism inherent in the process of folding itself.
Before discovering an explanation, and, as an aid
to this, we consider a perfectly regular, periodic
structure of the same sort (Fig. 4.10c). This consists of a layer embedded in a uniform medium,
with the upper and lower surfaces of the layer in
the form of in-phase sinusoidal surfaces with
wavelength, L, and amplitude, A. The infinite, perfectly periodic fold train is drawn with a low limb
dip.
The truly periodic fold train is an idealization
of the geometric form of a fold train and is used
here to isolate a small volume of rock containing
a single fold from the remainder of the layer.
Because of the periodicity, the two vertical planes
in Fig. 4.10c are mirror planes of symmetry. For
such a plane, two conditions apply. First, in the
deformation, which will be idealized to uniform
layer-parallel shortening, except for the flow associated with the folding, a particle cannot pass
through a mirror plane. Second, the shear stress
must vanish at a mirror plane.
We model the case of a segment of the layer
containing a single trough–crest–trough fold
with a horizontal span given by the wavelength, L,
and an arc length, L a Ͼ L. To remind ourselves of
the conditions that must apply at them, we
replace the bounding mirror planes by rigid
platens with smooth, frictionless, vertical surfaces. We suppose that these approach each other
at a rate corresponding to a rate of deformation,
D xx . As a natural starting point, we shall be concerned only in the fields of velocity and stress
within this region at an instant of time, with the
aim of analyzing the rate of change of quantities
of interest.
The contrasting possibilities of interest are: (i)
a negligible positive (or negative) value of the rate
of change of fold amplitude, dA/dt, in which case
the layer will undergo nearly uniform thickening;
and (ii) a large positive value of dA/dt, corresponding to the marked folding or buckling of the layer.
To think about that, we have to imagine, in a concrete fashion, what the properties of the rocks
involved are under the conditions that the folds
formed. Clearly, they have deformed in a more or
less continuous fashion. Although it takes a
rather large leap, we might assume, for simplicity,
they behave like other stiff, but still deformable,
fluid-like media with which we are familiar, and
treat them as viscous fluids. We then suppose the
layer and medium have viscosities and 1 ,
respectively.
We now have a reasonably clear idea of a
model that might address some aspect of the
folding process so we turn to dimensional analysis. The quantities involved in the model, with
their dimensions, are:
rate of change in fold amplitude,
(4.72)
(4.73)
(4.74)
(4.75)
(4.76)
(4.77)
(4.78)
There are seven physical quantities involving
three dimensions, M, L, and T, so the Buckingham
⌸ Theorem indicates that there are four dimensionless groups. Since M occurs only in the viscosities, one dimensionless group must be the
viscosity ratio
(4.79)
It is useful to choose groups that are relatively
simple and that have a concrete physical or geometrical interpretation. An appealing choice is
the aspect ratio, layer thickness to wavelength,
R ϭ
1
medium viscosity, 1 {ϭ}M L Ϫ1 T Ϫ1
layer viscosity, {ϭ}M L Ϫ1 T Ϫ1
wavelength, L{ϭ}L
layer thickness, H{ϭ}L
fold amplitude, A{ϭ}L
bulk rate of shortening, D xx {ϭ}T Ϫ1
dAրdt{ϭ}L T Ϫ1
4.3 DIMENSIONLESS GROUPS AND SCALING
141
incorrectly, called the fold wavelength (Fig. 4.10b).
Our analysis considers the folded quartz vein to be
“isolated.” What we principally mean by this is
that the folding is independent of that in nearby
layers. That is, we do not see other nearby layers
that are folded in concert with them.
Note that the original vein thickness was not
uniform and that this has affected the regularity
of the folding, but the arc length to thickness
ratios fall within a modest range. What might
this regularity mean? Clearly, it cannot easily be
ascribed to something “built into” the rock prior
to folding, since the regularity is present only in
this layer. We must then ascribe it to some mechanism inherent in the process of folding itself.
Before discovering an explanation, and, as an aid
to this, we consider a perfectly regular, periodic
structure of the same sort (Fig. 4.10c). This consists of a layer embedded in a uniform medium,
with the upper and lower surfaces of the layer in
the form of in-phase sinusoidal surfaces with
wavelength, L, and amplitude, A. The infinite, perfectly periodic fold train is drawn with a low limb
dip.
The truly periodic fold train is an idealization
of the geometric form of a fold train and is used
here to isolate a small volume of rock containing
a single fold from the remainder of the layer.
Because of the periodicity, the two vertical planes
in Fig. 4.10c are mirror planes of symmetry. For
such a plane, two conditions apply. First, in the
deformation, which will be idealized to uniform
layer-parallel shortening, except for the flow associated with the folding, a particle cannot pass
through a mirror plane. Second, the shear stress
must vanish at a mirror plane.
We model the case of a segment of the layer
containing a single trough–crest–trough fold
with a horizontal span given by the wavelength, L,
and an arc length, L a Ͼ L. To remind ourselves of
the conditions that must apply at them, we
replace the bounding mirror planes by rigid
platens with smooth, frictionless, vertical surfaces. We suppose that these approach each other
at a rate corresponding to a rate of deformation,
D xx . As a natural starting point, we shall be concerned only in the fields of velocity and stress
within this region at an instant of time, with the
aim of analyzing the rate of change of quantities
of interest.
The contrasting possibilities of interest are: (i)
a negligible positive (or negative) value of the rate
of change of fold amplitude, dA/dt, in which case
the layer will undergo nearly uniform thickening;
and (ii) a large positive value of dA/dt, corresponding to the marked folding or buckling of the layer.
To think about that, we have to imagine, in a concrete fashion, what the properties of the rocks
involved are under the conditions that the folds
formed. Clearly, they have deformed in a more or
less continuous fashion. Although it takes a
rather large leap, we might assume, for simplicity,
they behave like other stiff, but still deformable,
fluid-like media with which we are familiar, and
treat them as viscous fluids. We then suppose the
layer and medium have viscosities and 1 ,
respectively.
We now have a reasonably clear idea of a
model that might address some aspect of the
folding process so we turn to dimensional analysis. The quantities involved in the model, with
their dimensions, are:
rate of change in fold amplitude,
(4.72)
(4.73)
(4.74)
(4.75)
(4.76)
(4.77)
(4.78)
There are seven physical quantities involving
three dimensions, M, L, and T, so the Buckingham
⌸ Theorem indicates that there are four dimensionless groups. Since M occurs only in the viscosities, one dimensionless group must be the
viscosity ratio
(4.79)
It is useful to choose groups that are relatively
simple and that have a concrete physical or geometrical interpretation. An appealing choice is
the aspect ratio, layer thickness to wavelength,
R ϭ
1
medium viscosity, 1 {ϭ}M L Ϫ1 T Ϫ1
layer viscosity, {ϭ}M L Ϫ1 T Ϫ1
wavelength, L{ϭ}L
layer thickness, H{ϭ}L
fold amplitude, A{ϭ}L
bulk rate of shortening, D xx {ϭ}T Ϫ1
dAրdt{ϭ}L T Ϫ1
4.3 DIMENSIONLESS GROUPS AND SCALING
141