The effects of the rheology of slabs are considered using finite element convection calculations with Newtonian (linear) and non-Newtonian (power law) temperature dependent rheology. However, the long-wavelength components of the geoid and topography are independent of the lateral variations in viscosity from the slab.
Background
- Numerical Background
- Geophysical Background
- Convection Background
- Summary
Internal heating will also increase the dynamical importance of the upper thermal boundary layer relative to the lower one. Subduction of the oceanic lithosphere clearly plays a major role in convection in the Earth's mantle.
ConMan: A Vectorized Finite Element Code for Mantle Convection
Introduction
Although we illustrate vectorization with the finite element code Con Man, the concepts we present are general. Because we use a general formulation of the finite element method, it would be particularly easy to generalize ConMan to solve other equations or other geometries.
Vectorization
There must be no subroutine calls or I/O statements inside the innermost do loop, because this inllbit vectorization of the loop (as compilers become more sophisticated, this statement will no longer be true). LM(IEL,1) is the global equation number of the first node of the IELth element.
Equations and Implementation
2.5) where u is the dimensionless velocity, 0 is the dimensionless temperature, p is the dimensionless pressure, k is the unit vector in the vertical direction, H is the expression of the heat source and t is the dimensionless time. N,(i) is the derivative x of the shape function evaluated at node i. The Ij refer to the 2 by 2 matrix location in the 8 by 8 element stiffness matrix).
Numerical Benchmarks
Hardware monitoring for CVBM delivers a total speed of 65 MFLOPS on the X-MP (95 MFLOPS on the faster Y-MP). Note that the acceleration, the ratio of scalar to vector speed, on the Convex is not as good as on the Crays.
Summary
Taking the 129 by 129 result as the exact solution, the difference graph (Figure 2.4b) shows the error in the SUPG method. The SUPG method has localized the error in the corner regions where the flow is most advective, and compares exactly with the sequential SUPG method code used in the benchmark paper (see Travis et al. for more details of the error associated with the method be associated).
Additional Notes
We have implemented these ideas in a new finite element code, ConMan, designed to study problems in mantle convection that takes advantage of the speed available in today's vector supercomputers. The CVBM solutions were provided by Brian Travis and were done as part of the IGPP Mantle Convection Workshop at Los Alamos National Laboratory.
On Rheology: The Relationship
Between Plate Velocity and 'french
Viscosity in Newtonian and Power-Law Subduction Calculations
Introduction
We investigate a range of weak zone viscosities using Newtonian rheologies to understand the effect of this parameterization of the weak zone on the plate velocity. We investigate the effect of increasing and decreasing the total variation of the viscosity for steady flows. We also investigate the effect of the viscosity of the weak zone with temperature-dependent rheology.
Newtonian versus Non-Newtonian Rheology
Increasing the slab viscosity up to 10,000 times the mantle viscosity does not change the solution significantly. The effect of plate/plate viscosity on steady surface velocities is shown in Figure 3.2. For viscosities much smaller than the mantle viscosity (on the left of Figure 3.3), the plate and slab velocities tend to asymptotic values and large shear drags.
Plate Viscosity - 1000
Temperature-Dependence
Richter, Nataf and Daly (1983) and Christensen (1984) showed that for many relationships the appropriate viscosity is the average viscosity of the system. T/o is the preexponential viscosity, T/(T) is the effective viscosity, T is the dimensionless temperature, E' is the dimensional activation energy divided by RllT, where R is the gas constant and llT is the scaling temperature (for reference E' / RllT ~ 60 for olivine). The only parameter allowed to vary in the calculations shown in Figure 3.7 is the maximum allowable viscosity.
Weak Zones
We investigate the effect of varying the size and viscosity of the weak zone for several temperature-dependent models. For lower activation energies, uniform plate velocities and plate-like downward wells are possible, depending on the strength and size of the trench weak zone. If the trench weak zone is too large (Figure 3.10b), the deformation rate under the slab is unrealistically high and too much of the boundary layer subducts.
Power-law Rheology
In Figure 3.11 the weak zone is on the left side of the initial sheet, and in Figure 3.12 the weak zone is in the middle of the initial sheet. When the weak zone is directly in the middle (Figure 3.12) pulls more from the boundary layer than when the weak zone is on one side or the other (Figure 3.11). The non-Newtonian calculation shown in Figure 3.14 shows that without a weak zone at the trench, non-Newtonian calculations can produce plate-like, as well as plate-like, solutions.
Summary
For weak zone viscosities close to the intrinsic viscosity, changing the weak zone viscosity by a factor of ~10 changes the plate velocity by a factor of ~3. Only when the weak zone viscosity is much smaller than the intrinsic one, velocities insensitive to the actual viscosity of the weak zone are used. A much weaker constitutive law than the dry olivine diffusive flow mechanism of Ashby and Verrall (1977, i.e. Newtonian viscosity, dependent on temperature and pressure) is needed in the vicinity of the channel, regardless of the size or weakness of the zone weak. .
Subduction and the Geoid
Slabs and the Deformation of a Chemical Boundary
The controlling factor in slab-density boundary interactions is the ratio of the density contrast along the boundary to the density contrast in the slab. For density discontinuities comparable to the plate density (0 > R > -1), the plate deforms the boundary, perhaps by a few hundred kilometers, but remains in the upper layer, and the amount of boundary deformation depends on time. For density discontinuities larger than the slab (R < -1), the slab remains in the upper layer with little or no detectable boundary deformation.
Geoid and Topography for Subduction Calculations
- Constant Viscosity Calculations
In Figure 4.6, only the long wavelength components of the geoid are plotted, for comparison with Richards and Hager (1984). In the subsequent figures, the "long wavelength" component of the geoid calculations are the degree 1 and 2 components, for comparison with Richards and Hager. Also, the size of the geoid will be affected by the depth of the discontinuity.
Distance (kilometers)
Temperature-Dependent Calculations
The plate was constrained to the side of the computational grid, the same as the geometry used in Chapter 3. In the long-wavelength geoid (Figure 4.17), local channel effects do not affect the long-wavelength geoid. Also, the effect of the density boundary, due to the lateral density variation, is small.
Surface Topography Contribution Bottom Topography Contribution
Chemical Boundary Contribution Total Geoid
Summary
To understand the dynamical effects of the 670-kilometer discontinuity, more investigations are needed to understand the wide range of interesting, time-dependent, and complex phenomena we have observed with deformable plates and boundaries. The relationship between peak stress, principal stress direction, and flow direction becomes complicated near a density discontinuity.
The long-wavelength geoid above subduction zones is insensitive to the parameterization of the slab. Jean1oz, Reflection properties of phase transitions and compositional change models of the 670 km discontinuity, J. Richter, Convection experiments in highly temperature-dependent viscosity fluid and the thermal evolution of the planets, Phys.
ConMan on Various Computers
This means that 4.3 million operations are needed to factorize the matrix and 133,000 operations are required for forward reduction and back substitution. For real-time comparisons, CVBM on a Cray X-MP scalar takes 2869.3 seconds and TDBM takes 6094.7 seconds.
Non-Newtonian Implementation and Verification
The initial condition was a steady temperature field from a non-Newtonian calculation and the effective viscosity reset to one. The initial condition was a steady temperature field from a non-Newtonian calculation and the effective viscosity reset to one. The reverse case, apparently time-dependent flow that becomes steady with increased resolution, has not previously been reported in non-Newtonian calculations, to our knowledge, although Machetel and Yuen (1987) find a similar behavior for spherical-shell convection using a finite difference scheme.
Calculation of Geoid, Topography and Deformed Boundaries
The stresses can be calculated from the density fields using the propagation matrix technique (e.g., Cathes, 1975), or the stresses can be calculated directly from the velocity field in a finite element formulation. Stresses from the finite element method are calculated inside the element and must be "smoothed" to nodes and boundaries. The smoothing of internal nodes is well understood; Hughes (1987) discusses smoothing in some detail, as do Ho-Liu, Raefsky, and Hager (1987), although they discuss smoothing in the context of the Nusselt number.
Surface
A point buoyancy force of 1 unit acts on a node in column 1, and the stresses within the element and at the nodes are shown: nl analytic is the analytic response for surface stress (node 1), el FEM is the stress value within element 1, e2 FEM is the stress value within of element 2, nl is the balanced voltage value at node 1 (after smoothing), n2 is the balanced voltage value at node 2 (after smoothing) and. The correction is a linear projection of the stress in element 2 and element 1 projected onto an imaginary element above the boundary. The buoyancy stress on the surface (line 1) is underestimated by 3/4, but the surface is usually a constant boundary condition.).
Geoid
It is most convenient to separate the horizontal and vertical dependencies of a given density field when considering the resulting geoid anomalies of that field. Hager (1983) showed that the geoid of this flat earth model is not very different from that calculated from a spherical model for all but the longest wavelengths. Various methods to calculate the chemical buoyancy were implemented and tested, but to calculate the geoid we always projected the element density to the nodes using a "smoothing" technique and calculated the chemical buoyancy contribution to the geoid in the same way as the temperature contribution. bution.
Chemical Composition Methods
The effect of thermal buoyancy is determined by the Rayleigh number, Ra = apD.Td3 / 1-''', where a is the coefficient of thermal expansion, p is the density, D.T is the temperature drop across the box, d is the thickness of the box , I-' is the viscosity, and " is the thermal diffusivity. The contrast of the chemical density is determined by the chemical Rayleigh number, Rb = D.pcd3 / 1-''', where D.pc is the density If for example, if there are 100 elements in the bottom fluid and 2,000 particles are used, for a 1% density contrast between the top and bottom fluids, each particle is given a density of 0.2 x Po, where po is the Boussinesque density.
ConMan: Example Subroutine
Additional Calculations
