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**the isoviscous reference model**

Mantle convection and global plate motions are driven by thermal buoyancy forces due to density heterogeneities in the deep mantle. Two widely recognized lines of evidence suggest that these density heterogeneities are characterized by long wavelength structure: First, the Earth's non-hydrostatic gravity field, or geoid, has a very "red" spectrum, i.e., the shape of the geoid is dominated by its longest wavelength components, spherical harmonic degrees 2 and 3, representing wavelengths of 10,000 - 20,000 km. The second line of evidence is found in maps of seismic velocity heterogeneities in the deep mantle, which also exhibit red spectra.

Isoviscous Mantle Convection Simulation

The dominance of mantle convection by large-scale structure is a fundamental constraint in geodynamics, because simple fluid dynamical models yield a convection planform (the geometry of convection cells) characterized by much shorter lengthscales of the order of the mantle depth, or about 3000 km, as evidenced by the isoviscous reference simulation in the figure below.

Shown to the right is the temperature field at a snapshot in time for an incompressible (boussinesq) and only internally heated mantle flow (zero Reynolds number) simulation. The Rayleigh number based on internal heating is 4*10exp7. Blue is cold and red is hot. The upper 200 km (roughly the depth of the upper thermal boundary layer) are not shown, to permit a view on the convection planform below the boundary layer. The isoviscous reference model shows pointlike downwellings from the upper boundary layer and relatively short wavelength convection cells, quite unlike the Earth.

**depth-dependent viscosity**

Convection with Depth-Increasing Viscosity

An increase in the viscosity of the lower mantle relative to the upper mantle, as suggested among others by studies of post-glacial rebound, rotational dynamics, and models of the Earth's geoid results in a dramatic change of the planform. In the figure to the right we see that isolated, pointlike downwellings give way to long planar, sheet-like downwellings, much alike the stretched-out subduction zone systems on Earth.

Shown here is the temperature field at a snapshot in time for a boussinesq and purely internally heated mantle flow (zero Reynolds number) simulation. The upper 200 km are not shown. Blue is cold and red. All parameters are identical to the isoviscous reference model above, except that we have increased the viscosity of the lower mantle by a factor of 30, as suggested by studies of the geoid. The planform is dominated by sheet-like downwellings.

One may ask, if the planform change depends on the vigor of convection, or Rayleigh number. It can be shown that this is not the case. Convection with depthwise increasing viscosity like in the Earth, thus, is radically different from convection in an isoviscous model mantle.

**other effects**

Compressible Mantle Convection Simulations

There are many other effects that influence mantle convection : brittle failure in the surface plates, strongly variable viscosity, mineral phase changes, and both internal heating (radioactivity) and bottom heating from the core. Unfortunately, many of these effects are not well-constrained at present by either experimental or observational data. It is therefore important in geodynamic modeling of mantle convection to isolate these effects as clearly as possible before comparing elaborate geodynamic models to observations of seismic velocity heterogeneity, the geoid, geochemical data, etc.

Three potentially important effects are systematically studied in the in the four simulations shown below. 1) depth-dependent viscosity, 2) an endothermic phase change, and 3) bottom vs. internal heating. We model 3-D spherical, compressible (in the anelastic liquid approximation) convection at Rayleigh number 10exp8, thus approaching the dynamical regime of the mantle. An isoviscous, internally heated reference model displays point-like downwellings from the cold upper boundary layer, a blue (short spatial scales) spectrum of thermal heterogeneity and small but rapid time variations in flow diagnostics. Bottom heating has the predictable effect of adding a thermal boundary layer at the base of the mantle. We use a Clapeyron slope of gamma = -4MPa/K for the 670 km phase transition, resulting in a phase buoyancy parameter of P = -0.112. This phase change causes upwellings and downwellings to pause in the transition zone, but has little influence on the inherent time-dependence of flow and only a modest reddening effect on the heterogeneity spectrum. Larger values of P result in stronger effects but our choice of P is likely already too large to be representative of the mantle transition zone. By contrast, a modest factor 30 increase in lower mantle viscosity results in a planform dominated by long, linear downwellings, a red spectrum, and great temporal stability. Combinations of all three effects are remarkably predictable in terms of the single-effect models, and the effect of depth-dependent viscosity is found to be dominant.

Shown here is (a) the superadiabatic temperatures for the compressible, purely internally heated, isoviscous reference mantle convection calculation with Ra=1.1 x 10exp8. Blue is cold, and red is hot. The uppermost 200 km of the mantle is removed to show the temperatures beneath the boundary layer. Isolated downwelling plumes from the upper boundary layer dominate the planform. (b) Same as (a) except for the addition of 38 percent bottom heating from an isothermal core. Convection is influenced by largely axisymmetric upwellings (plumes) from the lower thermal boundary layer. (c) Same as (a) except for the addition of an endothermic phase change at 670 km depth with gamma = -4MPa/K. Downwellings pause in the transition zone, before entering the lower mantle. (d) Stratified viscosity case: Same as (a) except that the viscosity of the lower mantle has been increased by a factor of 30. The planform is dominated by long, linear downwelling sheets.

Printed 30. Mar 2020 13:22