forces are used to define additional dimensionless
groups that may be useful in model design
(Ramberg, 1967, p. 40):
(4.104)
“Ramberg” Number ϭ Ra ϵ
(4.105)
“Smoluchowski” Number ϭ Sm ϵ
(4.106)
Ramberg has referred to the ratio of gravitation to
viscous forces as a “nameless ratio” but we
suggest the Ramberg Number in recognition of the
major contribution he made to tectonic modeling. Ramberg suggested the name for the
Smoluchowski Number because of the two papers
published by M. Smoluchowski in which a theory
for buckling of elastic layers on a viscous foundation is developed along with associated model
experiments (Smoluchowski, 1909). The Stokes
Number is named for one of the pioneers in fluid
dynamics who solved some of the classic problems for slow viscous flow. For strict dynamic similarity each of these numbers for the prototype
must be equal to the corresponding number for
the model.
The procedure for tectonic model design may
be further simplified when some of the characteristic forces are much smaller than others. For
example, inertial forces typically are much
smaller than either the viscous or gravitational
forces, so the Reynolds and Froude Numbers are
very small. For example, in the rise of a salt dome
the relevant parameters take on the following
ranges (Ramberg, 1967, pp. 48–9):
(4.107)
1 ϫ 10 6 N s m Ϫ2 Յ Յ 1 ϫ 10 16 N s m Ϫ2
3 ϫ 10 Ϫ10 m s Ϫ1 Յ v o Յ 3 ϫ 10 Ϫ7 m s Ϫ1
1.5 ϫ 10 3 m Յ w o Յ 3 ϫ 10 3 m
gravitational force
pressure force
g
⌬pրw o
,
gravitational force
viscous force
g
v o րw 2
o
,
pressure force
viscous force
Stokes Number ϭ St ϵ
⌬pրw o
v o րw 2
o
,
The radius of the prototype salt dome ranges
from 1.5 to 3 km, and the velocity ranges from
1 cm a
Ϫ1 to 1 m a
Ϫ1 . These values are inferred
from field observations and interpretations of
geological records. The density and viscosity are
values estimated from laboratory measurements.
Density is well known, but viscosity is poorly
constrained because it can vary over many orders
of magnitude depending upon temperature, pressure, and chemical environment. Based on the
ranges of values given above, the Reynolds
g ϭ 9.8 m s Ϫ2
ϭ 2.16 ϫ 10 3 kg m Ϫ3
148
PHYSICAL QUANTITIES, FIELDS, DIMENSIONS, AND SCALING
Fig 4.13 Centrifuge models and equipment (Ramberg,
1967). (a) Model of salt dome development. (b) Centrifuge
used to increase the body force: 1, model in centrifuge cup;
2, stroboscopic light reflector; 3, camera; 4, camera
electronics; 5, stroboscope; 6, temperature and speed
control unit; 7, motor; 8, refrigerator.
(b)
1
2
3
4
5
6
7
8
(a)
groups that may be useful in model design
(Ramberg, 1967, p. 40):
(4.104)
“Ramberg” Number ϭ Ra ϵ
(4.105)
“Smoluchowski” Number ϭ Sm ϵ
(4.106)
Ramberg has referred to the ratio of gravitation to
viscous forces as a “nameless ratio” but we
suggest the Ramberg Number in recognition of the
major contribution he made to tectonic modeling. Ramberg suggested the name for the
Smoluchowski Number because of the two papers
published by M. Smoluchowski in which a theory
for buckling of elastic layers on a viscous foundation is developed along with associated model
experiments (Smoluchowski, 1909). The Stokes
Number is named for one of the pioneers in fluid
dynamics who solved some of the classic problems for slow viscous flow. For strict dynamic similarity each of these numbers for the prototype
must be equal to the corresponding number for
the model.
The procedure for tectonic model design may
be further simplified when some of the characteristic forces are much smaller than others. For
example, inertial forces typically are much
smaller than either the viscous or gravitational
forces, so the Reynolds and Froude Numbers are
very small. For example, in the rise of a salt dome
the relevant parameters take on the following
ranges (Ramberg, 1967, pp. 48–9):
(4.107)
1 ϫ 10 6 N s m Ϫ2 Յ Յ 1 ϫ 10 16 N s m Ϫ2
3 ϫ 10 Ϫ10 m s Ϫ1 Յ v o Յ 3 ϫ 10 Ϫ7 m s Ϫ1
1.5 ϫ 10 3 m Յ w o Յ 3 ϫ 10 3 m
gravitational force
pressure force
g
⌬pրw o
,
gravitational force
viscous force
g
v o րw 2
o
,
pressure force
viscous force
Stokes Number ϭ St ϵ
⌬pրw o
v o րw 2
o
,
The radius of the prototype salt dome ranges
from 1.5 to 3 km, and the velocity ranges from
1 cm a
Ϫ1 to 1 m a
Ϫ1 . These values are inferred
from field observations and interpretations of
geological records. The density and viscosity are
values estimated from laboratory measurements.
Density is well known, but viscosity is poorly
constrained because it can vary over many orders
of magnitude depending upon temperature, pressure, and chemical environment. Based on the
ranges of values given above, the Reynolds
g ϭ 9.8 m s Ϫ2
ϭ 2.16 ϫ 10 3 kg m Ϫ3
148
PHYSICAL QUANTITIES, FIELDS, DIMENSIONS, AND SCALING
Fig 4.13 Centrifuge models and equipment (Ramberg,
1967). (a) Model of salt dome development. (b) Centrifuge
used to increase the body force: 1, model in centrifuge cup;
2, stroboscopic light reflector; 3, camera; 4, camera
electronics; 5, stroboscope; 6, temperature and speed
control unit; 7, motor; 8, refrigerator.
(b)
1
2
3
4
5
6
7
8
(a)