Geometric similarity

To be geometrically similar, lengths in the model and natural example must be proportional and angles must be equal.^[15] When the length of a natural prototype (p) is ${\displaystyle l_{n}^{p}}$ (n=1, 2, 3...) and the angle is ${\displaystyle \alpha _{n}^{p}}$. Correspondingly, the length in the model (m) is ${\displaystyle l_{n}^{m}}$ and the angle is ${\displaystyle \alpha _{n}^{m}}$. They need to conform to the following formulas:^[1]

${\displaystyle {\frac {l_{1}^{m}}{l_{1}^{p}}}={\frac {l_{2}^{m}}{l_{2}^{p}}}={\frac {l_{3}^{m}}{l_{3}^{p}}}={\frac {l_{n}^{m}}{l_{n}^{p}}}}$ & ${\displaystyle \alpha _{n}^{m}=\alpha _{n}^{p}}$

For example, 1 centimetre in the model represents 1 kilometre in nature.

Kinematic similarity

To be kinematically similar, they must be geometrically similar and the time needed for changes to occur must be proportional.^[9] When the required time for changing is ${\displaystyle t_{n}}$:^[1]

${\displaystyle {\frac {t_{1}^{m}}{t_{1}^{p}}}={\frac {t_{2}^{m}}{t_{2}^{p}}}={\frac {t_{3}^{m}}{t_{3}^{p}}}={\frac {t_{n}^{m}}{t_{n}^{p}}}}$

For example, 1 second in the model represents 1 thousand years in the nature.

As is known: ${\displaystyle v={\frac {l}{t}}}$, the velocities (${\displaystyle v}$) can be scaled by the following equation:^[1]

${\displaystyle v^{p}=v^{m}{\frac {l^{p}t^{m}}{l^{m}t^{p}}}}$

Dynamic similarity

When the models and the natural world are geometrically and kinematically similar, dynamic similarity additionally requires that the various forces acting on a point in the model are proportional to those at a corresponding point in nature.^[15] When the forces (${\displaystyle F_{n}}$) acting on the system are ${\displaystyle F_{g}}$ (gravity), ${\displaystyle F_{v}}$ (viscous force), and ${\displaystyle F_{f}}$ (friction):^[15]

${\displaystyle {\frac {F_{g}^{m}}{F_{g}^{p}}}={\frac {F_{v}^{m}}{F_{v}^{p}}}={\frac {F_{f}^{m}}{F_{f}^{p}}}={\frac {F_{n}^{m}}{F_{n}^{p}}}}$

However, since the forces acting in the nature are unmeasurable, it is impossible to scale the forces and stresses directly. Scientists have been using different formulas to convert forces into the parameters that can be measured. Cauchy momentum equation is usually used for showing the relationship between forces and densities (${\displaystyle \rho }$ is density):^[1]

${\displaystyle {\frac {F^{m}}{F^{p}}}={\frac {\rho ^{m}(l^{m})^{3}}{\rho ^{p}(l^{p})^{3}}}}$(Generating from Cauchy momentum equation^[16])

Stokes' law is usually used for showing the relationship between forces and density contrasts (${\displaystyle \Delta \rho }$ is density constant):^[1]

${\displaystyle {\frac {F^{m}}{F^{p}}}={\frac {\Delta \rho ^{m}(l^{m})^{3}}{\Delta \rho ^{p}(l^{p})^{3}}}}$(Generating from Stokes' law^[17])

(While the gravitational acceleration ${\displaystyle g^{m}=g^{p}}$)

Since the densities and density contrasts are proportional to forces and stresses, it is easy to scale densities or density contrasts instead of scaling forces and stresses.^[1]

However, these two equations can lead to different topography scales.^[1]

The simple analogue modelling of the fold-thrust belt system taken in the Nanjing University physical laboratory. The white and blue materials are quartz sands.

Systems

For experimental systems, the energy can be supplied externally (at the boundary) and internally (buoyancy forces). If the deformation is only caused by internal forces, it is a closed system. Conversely, if the deformations are caused by external forces or a combination of internal and external forces, it is an open system.^[1]

For the open system, the extrusion or stretching forces are imposed externally. However, the buoyancy forces can be generated both externally or internally. The materials and thermal energy can be added to or remove from the system. For the closed system, there is no energy and materials added to the system. Thus, all the deformations are caused by internal buoyancy forces. Only buoyancy-driven deformation can be simulated in a closed system.^[1]

Gravity field

The simple analogue modelling of a subduction zone. The materials this model uses are sand mixture and silicone putty for the continental crust (left in layered brown) and oceanic crust (right in layered brown), and glucose syrup for the asthenosphere (greenish-blue liquid in the glass tank). There is a heater in the tank for heating the liquid.^[2][19][20]

Because the major research object of analogue modelling is Earth, the gravity field that most experiments utilize is ordinarily the Earth's field of gravity. However, many models are carried out using simulated gravity, for example, by use of a centrifuge. These technologies are usually used in studying development of gravity-controlled structures, such as dome formation,^[21] and diapirism.^[1]