|Wosnitza, E. 2002. Data Analysis in Thermomechanical Analogue Modelling. Schellart, W. P. and Passchier, C. 2002. Analogue modelling of large-scale tectonic processes. Journal of the Virtual Explorer.|
Data Analysis in Thermomechanical Analogue Modelling
Velocity and Length
governing heat transfer in thermally isotropic materials, without internal
heat production, is
During deformation, thermal and mechanical velocities must be treated equally. Thus, it must be ensured that the time needed by material particles or heat to move over a certain distance is scaled by the same factor. For the scaling factors of length l, time t and thermal diffusivity κ,
can be obtained, where the scaling factors SX of a property X represent the ratio of laboratory values to natural values. Equation 2.2 couples the length and time scales to the material-dependent scale of thermal diffusivity.
The length and time
scales are also coupled by the request for kinematic similarity. All velocities
in the model must be scaled by the same factor
The thermal diffusivity of rocks is about 10-6m2s-1 (e.g. Ranalli, 1987; Fowler, 1990, ch. 7), whereas that of paraffin wax is about 8x10-8m2s-1 (Rossetti et al., 1999). This results in a scaling factor Sκ of 8x10-2. Orogenic convergence rates are in the range 1-5cm/a (e.g. Pfiffner & Ramsay, 1982). The experimental convergence velocities 8-10µm/s chosen for the experiments are therefore scaled by 2.5x104. From the resulting scaling factor for time of 1.26x10-10, and using equation 2.3, the scaling factor for length is found to be 3.17x10-6. As a consequence, the initial model length of 45cm corresponds to a length in nature of about 140km, and one hour of experiment time corresponds to a geological time of 0.9Ma.
The scaling of length is limited by the size of the smallest particles used in the experiments. The grains of the brittle material used are smaller than 200µm, which scale to 63m in nature. The process of the data analysis (Part 3) uses markers with a spacing of the order of centimetres, limiting the interpretation of the experiments to lengths larger than 3 km.
the model is only
scaled to gravity when
This also implies the scaling factor for stresses to be
The mean densities of the upper and lower crust are 2800 kg/m3 and 3300 kg/m3, respectively (Fowler, 1990). The densities of the according analogue materials Jet-Plast and paraffin wax are 736 kg/m3 and 882 kg/m3. The resulting scaling factor for density Sρ = 0.26 applies for both upper and lower crust. Using equation 2.6 and 2.7, scaling factors for viscosity and stress are found to be Sη=10-16 and Sσ=8x10-7.
The scaling factor for viscosity can be used to determine the viscosities needed in the analogue model. For this, the viscosities of natural rocks must be calculated from published flow parameters and an assumed geothermal gradient. Using a geotherm of 20°C/km typical for a convergent collisional orogenic setting (e. g. Decker et al., 1988), a wide range of viscosities can be found. Basing on parameters from Carter & Tsenn (1987) for "wet" dunite and a temperature of 720°C at a depth of 35km, a viscosity of 3.1x1022 Pa s is obtained for the top of the upper mantle. Scaling this value to the laboratory using Sη = 10-16, the viscosity for the wax needs to be 3.5MPa s. This viscosity can be reached using the paraffin wax P57 at a temperature of 42.3°C, which is close to the melting point of the other paraffin wax (P43). In order to have the base of the upper mantle as weak as the base of the lower crust, a temperature of around 56°C must be set at the model base. Since the above depth of the Moho scales to 11 cm in the model and an overall lithospheric thickness of 60km scales to a model height of 19 cm, the required thermal gradient in the experiments turns out to be 1.7°C/cm. These arguments leads to the viscosity profile for the model shown in Figure 3.
According to Weijermars & Schmeling (1986), rheological similarity also demands equality of the dimensionless stress exponents for natural and analogue materials. This exponent is between 1.8 and 5.1 for the materials shown in Table 2. For the paraffin waxes used, the measured data is consistent with n = 1. Rossetti et al. (1999) obtained similar low (n < 1.3) stress exponents, while Mancktelow (1988) gave values between 2.4 and 4.1. This value is only significant in scenarios with large gradients in the strain rate.
or more commonly as
(Means, 1990). This equation does not influence the scaling factors for stress or strain rate, but it requires the exponent
to be dimensionless.
Because of SR = 1,
Using materials with lower activation energies such as colophony (Cobbold & Jackson, 1992, H = 255 kJ/mol) would improve the quality of scaling. Materials with a larger activation energy such as the paraffin used by Rossetti et al. (1999) result in a required scaling factor for temperature larger than one. In that case, the temperatures in the laboratory should be larger than those in nature, in order to provide correct scaling.
However, following an argument of Cobbold & Jackson (1992, p. 257), only the first two or three orders of magnitude of the overall strength variation are likely to have significant mechanical consequences. Whether the weaker layers are weaker by four or by five or even more orders of magnitude does not influence the stiffness of the overall scenario. Therefore, an attempt to perfectly scale the experiments for temperature is probably unnecessarily rigorous.
may be given for the model. These temperatures can be used in PTt
paths (see sections 3.3.3 and 4.2.5). To scale a model temperature TL
to nature, the following "work-around" for the scaling of temperature
Here, the thermal gradient in the laboratory TL(zL) is used to calculate a corresponding "depth" zL . Using the scaling factor for length SL, zL is converted to a natural "depth" zN. From the latter, the natural temperature can be reconstructed using TN(zN). "Depth" in this context is not to be taken literally since it describes the depth in a scenario with a homogeneous geothermal gradient. Instead of a simple scaling factor, the proposed work-around provides the linear relationship
Therefore, the temperatures obtained this way should be used with caution. Nevertheless, at the present stage of this work, this is the only way to gain temperature predictions from the analogue experiments.
The angle of internal friction is dimensionless, therefore it must be equal in the model and in nature. Jet-Plast, the brittle material chosen has an angle of internal friction of (37±2)°. Natural brittle rocks show angles of internal friction of 25° - 35° (e. g. Lallemand et al., 1994). Although the scaling in this case is not perfect, the error bars overlap.
Cohesion is a stress, consequently it needs to be scaled with the scaling factor for stresses. According to Hoshino et al. (1972), crustal rocks show cohesions of <20 MPa. In fact, these values were measured for rather small samples (some centimetres height). Due to pre-existing fractures, the bulk cohesion of the upper crust might be lower. Sand has cohesions of 20 -170 Pa (Lallemand et al., 1994). Due to the composition of the Jet-Plast, electrostatic forces can be assumed to lead to a lower cohesion. The upper limit of the natural cohesions scales down to 17 Pa which is well in the plausible range for the model material.