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**Solid Earth**
An interactive open-access journal of the European Geosciences Union

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- Abstract
- Introduction
- Dynamic topography driven by a rising sphere: analytical and numerical solutions
- The impact of nonlinear viscosity on dynamic topography
- Discussion and conclusion
- Code and data availability
- Author contributions
- Competing interests
- Special issue statement
- Acknowledgements
- Financial support
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SE | Volume 10, issue 6

Solid Earth, 10, 2167–2178, 2019

https://doi.org/10.5194/se-10-2167-2019

© Author(s) 2019. This work is distributed under

the Creative Commons Attribution 4.0 License.

https://doi.org/10.5194/se-10-2167-2019

© Author(s) 2019. This work is distributed under

the Creative Commons Attribution 4.0 License.

Special issue: Understanding the unknowns: the impact of uncertainty in the...

**Research article**
20 Dec 2019

**Research article** | 20 Dec 2019

The impact of rheological uncertainty on dynamic topography predictions

^{1}EarthByte Group and Basin Genesis Hub, School of Geosciences, The University of Sydney, Sydney, NSW 2006, Australia^{a}now at: GeoQuEST Research Centre, School of Earth and Environmental Sciences, University of Wollongong, Wollongong, NSW 2522, Australia

^{1}EarthByte Group and Basin Genesis Hub, School of Geosciences, The University of Sydney, Sydney, NSW 2006, Australia^{a}now at: GeoQuEST Research Centre, School of Earth and Environmental Sciences, University of Wollongong, Wollongong, NSW 2522, Australia

**Correspondence**: Ömer F. Bodur (omer.bodur@sydney.edu.au)

**Correspondence**: Ömer F. Bodur (omer.bodur@sydney.edu.au)

Abstract

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Much effort is being made to extract the dynamic components of the Earth's topography driven by density heterogeneities in the mantle. Seismically mapped density anomalies have been used as an input into mantle convection models to predict the present-day mantle flow and stresses applied on the Earth's surface, resulting in dynamic topography. However, mantle convection models give dynamic topography amplitudes generally larger by a factor of ∼2, depending on the flow wavelength, compared to dynamic topography amplitudes obtained by removing the isostatically compensated topography from the Earth's topography. In this paper, we use 3-D numerical experiments to evaluate the extent to which the dynamic topography depends on mantle rheology. We calculate the amplitude of instantaneous dynamic topography induced by the motion of a small spherical density anomaly (∼100 km radius) embedded into the mantle. Our experiments show that, at relatively short wavelengths (<1000 km), the amplitude of dynamic topography, in the case of non-Newtonian mantle rheology, is reduced by a factor of ∼2 compared to isoviscous rheology. This is explained by the formation of a low-viscosity channel beneath the lithosphere and a decrease in thickness of the mechanical lithosphere due to induced local reduction in viscosity. The latter is often neglected in global mantle convection models. Although our results are strictly valid for flow wavelengths less than 1000 km, we note that in non-Newtonian rheology all wavelengths are coupled, and the dynamic topography at long wavelengths will be influenced.

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Bodur, Ö. F. and Rey, P. F.: The impact of rheological uncertainty on dynamic topography predictions, Solid Earth, 10, 2167–2178, https://doi.org/10.5194/se-10-2167-2019, 2019.

1 Introduction

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The Earth's mantle is continuously stirred by hot upwellings from the core–mantle boundary and by subduction of colder plates from the surface into the deep mantle (Pekeris, 1935; Isacks et al., 1968; Molnar and Tapponnier, 1975; Stern, 2002). This introduces temperature and density anomalies that stimulate mantle flow and forces dynamic uplift or subsidence at the plates' surfaces (Gurnis et al., 2000; Braun, 2010; Moucha and Forte, 2011; Flament et al., 2013). Dynamic topography can affect the entire planet's surface with varying magnitudes. Because it is typically a low-amplitude and long-wavelength transient signal, it is often dwarfed by isostatic topography associated with variations in the thickness and density of sediments, crust and mantle lithosphere.

For the present day, the observational constraints on dynamic topography come from residual topography measurements (Hoggard et al., 2016). Residual topography is calculated by removing the isostatically compensated topography from the Earth's topography (Crough, 1983; Cazenave et al., 1989; Davies and Pribac, 1993; Steinberger, 2007, 2016). The comprehensive work from Hoggard et al. (2016) revealed that residual topography varies between ±500 m at very-long wavelengths (i.e. ∼ 10 000 km) and can increase up to ±1000 m at shorter wavelengths (i.e. ∼1000 km). However, these residuals depend on our knowledge of the thermal and mechanical structure of the lithosphere and therefore may not be an accurate estimation of the deeper mantle contribution to the Earth's topography. Another approach to constrain present-day Earth's dynamic topography involves numerical modelling of present-day mantle flow using seismically mapped density anomalies as an input (Steinberger, 2007; Moucha et al., 2008; Conrad and Husson, 2009). However, this method requires a detailed knowledge of the viscosity structure in the Earth's interior (Parsons and Daly, 1983; Hager, 1984; Hager et al., 1985; Hager and Clayton, 1989), and translating seismic velocities to physical properties (e.g. temperature) of the mantle introduces further uncertainties (Cammarano et al., 2003). The problem is that dynamic topography predictions derived from mantle convection models are generally larger by a factor of 2 (more significant at the very-large scales) than estimates from residual topography (Hoggard et al., 2016; Cowie and Kusznir, 2018; Davies et al., 2019; Steinberger et al., 2019). We hypothesise that this could be related to an oversimplification of the mantle rheology. In this paper, we explore how, at wavelengths <1000 km, the magnitude of dynamic topography changes when we use a rheological model in which the viscosity depends on strain rate, temperature, pressure and fluid content. We first summarise the well-established analytical solution for calculating dynamic topography induced by a spherical density anomaly embedded into an isoviscous fluid (Morgan, 1965a; Molnar et al., 2015). Then, assuming isoviscous rheology, we illustrate that the amplitude of dynamic topography depends on the viscosity structure of the Earth's interior as shown by Morgan (1965a) and Molnar et al. (2015). Finally, we use 3-D coupled thermo–mechanical numerical experiments of the Stokes flow to assess the dependence of dynamic topography on nonlinear rheology using viscosity which depends on temperature, pressure, strain rate and fluid content. We show that plausible nonlinear rheologies can induce local variations in viscosity and result in dynamic topography of lower amplitude compared to those derived from models using isoviscous rheology.

2 Dynamic topography driven by a rising sphere: analytical and numerical solutions

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We assume here a simple 2-D model representing a very-viscous spherical
density anomaly embedded into a semi-infinite isoviscous fluid bounded by an
upper free surface. Earliest analytical investigations revealed that, albeit
counter-intuitive, the magnitude of the induced surface deflection due to
the rising sphere is independent of the viscosity of the fluid. The dynamic
topography is a function of the vertical total stress (*σ*_{zz})
applied to the surface which is proportional to the size and depth of the
density anomaly according to Eq. (1) (Morgan, 1965a,
b):

$$\begin{array}{}\text{(1)}& {\mathit{\sigma}}_{zz}\left(x,\mathrm{0}\right)=\left[\mathrm{2}g\mathit{\delta}\mathit{\rho}{r}^{\mathrm{3}}\right]{\displaystyle \frac{{D}^{\mathrm{3}}}{{\left({D}^{\mathrm{2}}+{x}^{\mathrm{2}}\right)}^{\mathrm{5}/\mathrm{2}}}},\end{array}$$

where *g* is the gravitational acceleration, *δ**ρ* is density
difference between the anomaly and the ambient material, *r* is radius of the
sphere, and *D* is distance from the surface to the centre of the anomaly
(modified from Morgan, 1965a, see Fig. 1a). The dynamic topography *e* is
given by the following:

$$\begin{array}{}\text{(2)}& e\left(x\right)={\displaystyle \frac{{\mathit{\sigma}}_{zz\phantom{\rule{0.125em}{0ex}}\left(x,\phantom{\rule{0.125em}{0ex}}\mathrm{0}\right)}}{g\phantom{\rule{0.125em}{0ex}}\mathrm{\Delta}\mathit{\rho}}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\mathrm{at}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}z=\mathrm{0},\end{array}$$

where Δ*ρ* is the density difference between the mantle and air
(or water assuming a sea-load when *e*<0)
(Morgan, 1965a; Houseman and Hegarty, 1987). In
Fig. 1a, we plot the dynamic topography induced by a sphere of 1 %
density anomaly, whose centre is at 372 km depth (*D*=372 km) below the free
surface. We calculate the vertical total stress and convert it to dynamic
topography by using Eq. (2) for different values of the radius of the
sphere. The amplitude of dynamic topography shows an accelerating increase
by cubic dependence on the radius of the spherical density anomaly (Fig. 1a, solid black line). For the same problem, Molnar et al. (2015) provided a
solution by considering a higher-order term, resulting in a slight difference
from the solution of Morgan (1965a) (see Appendix A3 in Molnar et al., 2015),
which allows the consideration of density anomalies of finite viscosity (*η*_{sphere}) (Eq. 3):

$$\begin{array}{}\text{(3)}& \begin{array}{rl}{\mathit{\sigma}}_{zz}\left(C,\mathrm{0}\right)& ={\displaystyle \frac{-\mathit{\delta}\mathit{\rho}{r}^{\mathrm{3}}D}{\mathrm{3}f}}[{\displaystyle \frac{\mathrm{3}-\mathrm{2}f}{{C}^{\mathrm{3}}}}+{\displaystyle \frac{\mathrm{18}\left(f-\mathrm{1}\right){r}^{\mathrm{2}}}{{C}^{\mathrm{5}}}}\\ & +{\displaystyle \frac{\mathrm{6}f{D}^{\mathrm{2}}}{{C}^{\mathrm{5}}}}-{\displaystyle \frac{\mathrm{30}(f-\mathrm{1}){r}^{\mathrm{2}}{D}^{\mathrm{2}}}{{C}^{\mathrm{7}}}}],\end{array}\end{array}$$

where $C=\sqrt{{D}^{\mathrm{2}}+{x}^{\mathrm{2}}}$, and $f=({\mathit{\eta}}_{\mathrm{1}}\phantom{\rule{0.125em}{0ex}}+\frac{\mathrm{3}{\mathit{\eta}}_{\mathrm{sphere}}}{\mathrm{2}})/({\mathit{\eta}}_{\mathrm{1}}+{\mathit{\eta}}_{\mathrm{sphere}})$. One can find that *f*=1.5 if
the sphere is very viscous (*η*_{sphere}≫*η*_{1}), and *f*<1.5 for any other case. In Fig. 1a, we present two more plots of dynamic
topography where *f*=1.5 for hard sphere and *f*=1.25 for *η*_{sphere}=*η*_{1} by using Eqs. (2) and (3). Figure 1a shows that a
rising deformable sphere creates higher dynamic topography compared to a
very-viscous sphere. These show that the viscosity contrast between the
spherical anomaly and the surrounding material can affect the dynamic
topography. In the section that follows, we explore how dynamic topography
varies when there is layering in viscosity such as the presence of a strong
lithosphere above the convective mantle.

A more generalised solution has been put forward to accommodate the presence
of a stronger upper layer representing a lithosphere with viscosity *η*_{2} above a weaker layer with viscosity *η*_{1}, and with *η*_{1}<*η*_{2} representing the convective mantle (Fig. 1b). In
this case, Morgan (1965a) showed (Eq. 4) that the total normal stress
induced by the density anomaly is dependent on the mass anomaly per unit
length (*M*_{u}, for point sources integrated along a continuous line), the
depth of the centre of the sphere (*D*) and marginally on the ratio of the
viscosity of the convective mantle to the viscosity of the lithosphere
($R={\mathit{\eta}}_{\mathrm{1}}/{\mathit{\eta}}_{\mathrm{2}}$). The 2-layer problem is treated in Fourier
domain with the resulting total normal stress as below:

$$\begin{array}{}\text{(4)}& {\mathit{\sigma}}_{zz}\left(x,\mathrm{0}\right)=\underset{\mathrm{0}}{\overset{\mathrm{\infty}}{\int}}{\mathit{\sigma}}_{n}\mathrm{cos}nx\phantom{\rule{0.125em}{0ex}}\mathrm{d}n,\end{array}$$

where

$$\begin{array}{rl}{\mathit{\sigma}}_{n}& ={\displaystyle \frac{{M}_{u}g{e}^{-n(D-d)}}{\mathrm{2}\mathit{\pi}(R{S}_{h}+{C}_{h})}}\mathit{\{}\mathrm{1}+n\left(D-d\right)\\ & +nd\left[{\displaystyle \frac{\mathrm{1}-nD+n(D-d)(R{C}_{h}+{S}_{h})/(R{S}_{h}+{C}_{h})}{\mathrm{1}+nd(\mathrm{1}-{R}^{\mathrm{2}})/(R{S}_{h}+{C}_{h})(R{S}_{h}+{S}_{h})}}\right]\mathit{\}},\end{array}$$

*C*_{h}=cosh (*n**d*), *S*_{h}=sinh (*n**d*) and *d* is the upper-layer
thickness (modified from Morgan, 1965a). Following Morgan (1965a), Fig. 1b
illustrates the relative importance of *R* as well as the ratio of the thickness
of the upper layer to the depth of the anomaly (*d*∕*D*). As long as the
lithosphere is more viscous than the asthenosphere, the vertical total
stress at the surface has a minor dependence on the viscosity of the
lithosphere (see solid lines with *R*=1 and *R*=0.01 in Fig. 1b). Figure 1b
also shows that the magnitude of dynamic topography increases as the density
anomaly is brought closer to the surface (compare *R*=1, the solid black
line and the dashed black line). Moreover, its sensitivity on the relative
viscosity of the lithosphere also increases. Although an unrealistic
proposition for the Earth, when the lithosphere is less viscous than the
asthenosphere, the normal stress is much reduced and is strongly dependent
on the viscosity of the lithosphere (Fig. 1b). These demonstrate that
layering in viscosity can have a strong impact on the amplitude of dynamic
topography (Sembroni et al., 2017). In the next
section, we use the analytical solutions above to benchmark a numerical
model, which we will then extend to nonlinear viscosity.

For comparison with analytical solutions (Morgan, 1965a;
Molnar et al., 2015), we consider 3-D numerical models involving 1, 2 and 3
isoviscous layers. These benchmark experiments will be used as references
for non-isoviscous models discussed in Sect. 3. We use the open-source
code *Underworld* which solves the Stokes equation at insignificant Reynolds number
(Moresi et al., 2003, 2007). The 3-D
computational grid represents a domain 3840 km × 3840 km × 576 km with a
resolution of 6 km along the vertical *z* axis and 10 km along the *x* and *y* axes
(Fig. 2). In all experiments, we include a 42 km thick continental crust
above the upper mantle. The density structure is sensitive to the geotherm
via a coefficient of thermal expansion and compressibility (see Table 1 for
all parameters). The geotherm is defined using a radiogenic heat production
in the crust, a constant temperature of 20 ^{∘}C at the
surface and a constant temperature of 1350 ^{∘}C at 150 km.
We disregard the adiabatic heating, and the asthenosphere is kept at
1350 ^{∘}C. We embed a positive spherical temperature anomaly
of +324 ^{∘}C at a depth of 372 km below the surface, which
delivers a 1 % volumetric density difference. The radius of the sphere is
96 km. In all experiments, we impose free slip velocity boundary conditions
at all walls, such as *V*_{x} and *V*_{y} are set to be free, but *V*_{z}=0 cm yr^{−1} at the top wall. Taking advantage of the symmetry of the
experimental setup, we extract viscosity and velocity fields along a 2-D
cross section passing through the centre of the thermal anomaly, from which
we derive the streamlines and vertical velocity profiles along the vertical
axis at the centre of the models. We calculate the instantaneous dynamic
topography from the normal stress computed at the surface.

n/a – not applicable.
We use the rheological
parameters from ^{a} quartzite (Ranalli, 1995), ^{b} dry or wet
olivine (Hirth and Kohlstedt, 2003).

In the first experiment (Fig. 3a Experiment 1), we assign the same
depth-independent viscosity of 10^{21} Pa s to the crust, mantle and the
density anomaly. The streamlines for Experiment 1 (Fig. 3a) show formation
of two convective cells at the sides of the sphere covering the entire crust
and mantle. The vertical velocity profile indicates that the thermal anomaly
rises with a peak velocity of ∼2.4 cm yr^{−1}, which is
faster than the 2.0 cm yr^{−1} predicted by the analytical solution
(Fig. 4a). Experiment 1 predicts dynamic topography of 114 m (Fig. 4b) which
is lower than 132 m predicted by the analytical solution of Molnar et al. (2015).
We have verified that increasing the depth of our model from 576 to 864 km increases the dynamic topography from 114 to 122 m. Therefore, we
attribute the misfit in amplitude of dynamic topography to the finite space
in our numerical experiments. Our numerical experiment using isoviscous
material delivers a result globally consistent with the analytical solutions
of Morgan (1965a) and Molnar et al. (2015).

In Experiment 2, we assign to the lithosphere a constant viscosity 100 times
larger (10^{23} Pa s) than that of the asthenosphere (10^{21} Pa s, Fig. 3b) between *z*=150 km and base of the model. The convective cells become
narrower by the induced viscosity contrast (Fig. 3b). The streamlines are
deflected across the lithosphere–asthenosphere boundary due to the large
viscosity contrast (Fig. 3b), and there is a sharp variation in vertical
velocity at the base of the lithosphere (Fig. 4a, solid red line). The
maximum vertical velocity ∼2.1 cm yr^{−1} is attained near
the centre of the anomaly. When compared to Experiment 1, the dynamic
topography (Fig. 4b, solid red line) shows a significant increase from
∼114 to ∼174 m. This increase is consistent
with analytical estimations showing an increase in dynamic topography when
viscosity increases toward the surface (Fig. 1b, *R*<1). In
Experiment 2a (not shown here), we tested a different ratio of thickness of
the lithosphere to the depth of the anomaly (see *d*∕*D* in Eq. 4) by
increasing the lithospheric thickness from 150 to 200 km, while keeping
all parameters identical to those of Experiment 2. As predicted by Eq. (4),
Exp. 2 predicted dynamic topography of ∼191 m, being the
largest among all experiments (Fig. 4b, dashed red line). Overall, and
perhaps counter-intuitively, the presence of a thick viscous lithosphere
enhances the dynamic topography. Interestingly, in analogue experiments
where density anomaly is allowed to rise and interact with the lithosphere,
the amplitude of the dynamic topography is inversely correlated with the
thickness of the lithosphere (e.g. Griffiths et al.,
1989; Sembroni et al., 2017).

In Experiment 3 (Fig. 3c), we introduce a third 60 km thick low-viscosity
layer (i.e. 10^{19} Pa s) beneath the base of the lithosphere. The
existence of a low-viscosity layer has been discussed in several studies
(e.g. Craig and McKenzie, 1986; Phipps Morgan et al., 1995; Stixrude and
Lithgow-Bertelloni, 2005; Becker, 2017). In this experiment, in order to
prevent large viscosity contrast that can impede the numerical convergence,
the viscosities of the lithosphere and that of the asthenosphere are set to
10^{22} and 10^{21} Pa s, respectively. When compared to Experiment
1, streamlines indicate a further decrease in size of the convective cells
and more importantly, a strong horizontal divergence of the streamlines
within the low-viscosity layer (Fig. 3c). The vertical velocities are also
enhanced in the asthenosphere reaching up to ∼ 2.8 cm yr^{−1} slightly above the centre of the anomaly (Fig. 4a, solid orange
line). When compared to Experiment 1, we observe a strong reduction in
dynamic topography (Fig. 4b, solid orange line) from 114 to 88 m. This is
due to the damping effect of the low-viscosity channel that acts as a
decoupling layer, which reduces the deviatoric stress through its ability to
flow.

Until now, the viscosities were assumed to be constant. However, results from experimental deformation on mantle rocks strongly suggest that the viscosity is highly nonlinear (Hirth and Kohlstedt, 2003). In what follows, we explore the influence of more realistic viscosities on dynamic topography.

3 The impact of nonlinear viscosity on dynamic topography

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The Earth's mantle is not isoviscous. Geological records of relative sea level changes related to postglacial rebound, geophysical observations of density anomalies inferred from seismic velocity variations in the mantle and satellite measurements of the longest wavelength components of the Earth's geoid have been used to infer the radial viscosity profile of the Earth's interior (Hager et al., 1985; Forte and Mitrovica, 1996; Mitrovica and Forte, 1997; Kaufmann and Lambeck, 2000). Henceforward, beneath the lithosphere, a variation in viscosity up to 2 orders of magnitude has been proposed (e.g. Kaufmann and Lambeck, 2000). Investigations of the rheological properties of crustal and mantle rocks via rock deformation experiments revealed a nonlinear dependence of viscosity on applied deviatoric stress, pressure, temperature, grain size and the presence of fluids (Post and Griggs, 1973; Chopra and Paterson, 1984; Karato, 1992; Karato and Wu, 1993; Gleason and Tullis, 1995; Ranalli, 1995; Hirth and Kohlstedt, 2003; Korenaga and Karato, 2008). These experiments lead to the following relationship:

$$\begin{array}{}\text{(5)}& {\mathit{\eta}}_{\mathrm{eff}}\left(\dot{\mathit{\epsilon}},P,T\right)={A}^{\left({\scriptscriptstyle \frac{-\mathrm{1}}{n}}\right)}{f}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}^{\left({\scriptscriptstyle \frac{-r}{n}}\right)}{\dot{\mathit{\epsilon}}}^{\left({\scriptscriptstyle \frac{\mathrm{1}}{n}}-\mathrm{1}\right)}{e}^{\left({\scriptscriptstyle \frac{Q+PV}{nRT}}\right)},\end{array}$$

where $\dot{\mathit{\epsilon}}$ and *A* stands for strain rate and pre-exponential
factor; *r* and *n* are exponents for water fugacity (${f}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$) and deviatoric
stress, respectively; *V* and *Q* are the volume and energy of activation.

In the case where mantle flow is driven by the temperature difference at the
boundary of the convective layer or by internal heating, the dominant strain
mechanism is diffusion creep because low deviatoric stresses are expected in
the weak convective mantle (Karato and Wu, 1993;
Turcotte and Schubert, 2014). However, mantle flow in the vicinity of a
moving density anomaly is likely driven by deviatoric stresses that exceed
the threshold for dislocation creep. In this case, nonlinear viscosities
lead to strong local variation in viscosity. Are those local variations in
viscosity important for dynamic topography? To answer this question, we need
reasonable constraints on the rheological parameters controlling the
viscosity of mantle rocks. However, the extrapolation from laboratory strain
rates typically in the range of 10^{−6} s^{−1} to 10^{−4} s^{−1} to
mantle conditions where strain rates are typically on the order of
10^{−13} s^{−1} results in significant uncertainties on the activation
volume, activation energy and stress exponent (Hirth and
Kohlstedt, 2003; Korenaga and Karato, 2008). In what follows, we explore how
nonlinear viscosity impacts the dynamic topography and address how the
uncertainties on the activation volume can affect dynamic topography.

In Experiments 4 and 5 (Fig. 5), the viscosity depends on temperature, pressure and strain rate as indicated by Eq. (5), using published visco–plastic rheological parameters for the crust and mantle. Specifically, we use quartzite rheology for the crust (Ranalli, 1995), and we test both dry and wet olivine rheologies for the mantle (Hirth and Kohlstedt, 2003). Other parameters are identical to those in Experiments 1–3. We give all the rheological and thermal parameters in Table 1. For a given olivine rheology (i.e. dry or wet) we vary the activation volume by using the minimum and maximum reported values (Hirth and Kohlstedt, 2003).

In the numerical models, the plastic (i.e. brittle) deformation is described via

$$\begin{array}{}\text{(6)}& \mathit{\tau}=\mathit{\mu}{\mathit{\sigma}}_{n}+{C}_{\mathrm{0}},\end{array}$$

where *τ* is the 2nd invariant of the deviatoric stress tensor,
which varies with the coefficient of friction (*μ*), and depth via
lithostatic pressure (*σ*_{n}), as well as the cohesion (*C*_{0}).
Due to strain weakening, the cohesion and coefficient of friction decrease
from *C*_{0}=10 MPa and *μ*_{0}=0.577 to *C*_{0}=2 MPa and
*μ*_{1}=0.017 at which the maximum plastic strain (*ϵ*_{max}) is reached (i.e. 0.2, Table 1). The effective density (*ρ*) of
rocks is determined by the pressure and temperature using the following
equation:

$$\begin{array}{}\text{(7)}& \mathit{\rho}={\mathit{\rho}}_{\mathrm{0}}[\mathrm{1}-\mathit{\alpha}(T-{T}_{\mathrm{0}}\left)\right][\mathrm{1}+\mathit{\beta}(P-{P}_{\mathrm{0}}\left)\right],\end{array}$$

where *ρ*_{0}, *T*_{0}, *α* and *P*_{0} signify the reference
density, reference temperature, thermal expansion coefficient and the
compressibility, respectively.

In Experiments 4a and 4b, we consider dry dislocation creep for olivine (*n*>1, *p*=0, *r*=0). The reported activation volume for this rheology varies
between $\mathrm{6}\times {\mathrm{10}}^{-\mathrm{6}}$ and $\mathrm{27}\times {\mathrm{10}}^{-\mathrm{6}}$ m^{3} mol^{−1}
(Hirth and Kohlstedt, 2003). In Experiment 4a (Fig. 4b), we test
the lower value. The streamlines show a similar a pattern to Experiment 2.
Interestingly, the maximum vertical velocity peaks at 75 cm yr^{−1}, near
the upper boundary of the sphere (Fig. 6a, dashed black line). This is due
to the formation of a low-viscosity region above the rising sphere (Fig. 5a,
Experiment 4a). This experiment gives a dynamic topography of
∼149 m (Fig. 6b, dashed black line). It confirms that a
strong contrast in viscosity between the lithosphere and asthenosphere
enhances the dynamic topography signal. We note that the viscosity contrast
is attained by smoother transition between the lithosphere and asthenosphere
(Fig. 7a, dashed black line). We infer the mechanical thickness of the
lithosphere from the viscosity profiles plotted in Fig. 7a, along which
the lithosphere–asthenosphere transition zone shows a rapid decrease in
viscosity (Conrad and Molnar, 1997). We observe that the effective
mechanical thickness of the lithosphere is reduced to 140 km, compared to
the thickness of the thermal lithosphere (Fig. 7c).

When we increase the activation volume to $\mathrm{27}\times {\mathrm{10}}^{-\mathrm{6}}$ m^{3} mol^{−1},
the convection cells grow much larger and show continuity through the
lithosphere (Fig. 5a, Experiment 4b). The sphere has a very-low rising speed
of ∼ 0.25 cm yr^{−1} (Fig. 6a, solid black line). Compared
to Experiment 4a, the dynamic topography shows a strong decrease from
∼149 to ∼105 m (Fig. 6b, solid black line).
This is an example where the system behaves nearly as a single layer with
homogenous viscosity. The near absence of viscosity contrast between the
lithosphere and asthenosphere explains the smaller magnitude of the dynamic
topography. Moreover, the formation of moderately low-viscosity channel
(Fig. 7a, solid black line) also contributes to the decrease of the dynamic
topography.

In Experiments 5a and 5b, we consider dislocation creep of wet olivine. The
reported activation volume varies between $\mathrm{11}\times {\mathrm{10}}^{-\mathrm{6}}$ and $\mathrm{33}\times {\mathrm{10}}^{-\mathrm{6}}$ m^{3} mol^{−1} (Hirth and Kohlstedt, 2003).
In Experiment 5a, we test the lower value. The streamlines show a pattern
similar to Experiment 4a but with slightly larger convective cells (Fig. 5b, Experiment 5a). The rising velocity of the anomaly exceeds 140 cm yr^{−1} (Fig. 6a, dashed orange line), promoted by the low-viscosity
region sitting above the rising anomaly. The dynamic topography is
∼110 m (Fig. 6b, dashed orange line). This is a bit
surprising given the strong contrast in viscosity (3 orders of magnitude)
between the lithosphere and asthenosphere. However, Fig. 7a shows that the
thickness of the mechanical lithosphere is reduced by about 30 km in
comparison to Experiment 4a (e.g. 10 km reduction from thermal thickness)
which resulted in lower dynamic topography with similar viscosity contrast
(Fig. 7b, c).

In Experiment 5b, we increase the activation volume from $\mathrm{11}\times {\mathrm{10}}^{-\mathrm{6}}$ to $\mathrm{33}\times {\mathrm{10}}^{-\mathrm{6}}$ m^{3} mol^{−1}. The vertical
velocities show significant decrease from 140 to 0.34 cm yr^{−1} (Fig. 6a, solid orange line). This is due to an increase in
viscosity above the rising sphere. Compared to Experiment 5a, the dynamic
topography decreases from ∼110 to ∼90 m
(Fig. 6b, solid orange line). Compared to Experiment 4b, we expect the
dynamic topography to be higher due to slight increase in viscosity contrast
(Fig. 7a, b). However, the increase in thickness of the low-viscosity channel
(Fig. 7a, d) is more effective and thereby causes a greater reduction in
magnitude of the dynamic topography.

In summary, experiments using nonlinear rheology generally give lower amplitudes of dynamic topography compared to experiments using isoviscous rheology (Fig. 8). When we use dry olivine rheology for the upper mantle, the dynamic topography varies between ∼105 and ∼149 m, whereas under wet conditions, the dynamic topography varies between ∼90 and ∼110 m (Fig. 8). These variations are due to uncertainties in the activation volume as well as fluid content in olivine rheologies.

4 Discussion and conclusion

Back to toptop
Using coupled 3-D thermo-mechanical numerical experiments, we have modelled
the dynamic topography driven by a rising sphere of 1 % density anomaly,
having 96 km radius and emplaced at 372 km depth. In line with analytical
studies (Morgan, 1965a; Molnar et al., 2015), the
experiments show that dynamic topography is sensitive to viscosity contrast
between the lithosphere and asthenospheric mantle, and the thickness of the
lithosphere (Fig. 7). Higher viscosity contrasts amplify the dynamic
topography (Fig. 7a, b), whereas formation of a low-viscosity channel just
below the lithosphere has the opposite effect (Fig. 7a, d). The experiments
using nonlinear rheologies show local variations in viscosity, which
contribute to the dynamic thinning of the mechanical lithosphere and causes
reduction in dynamic topography. In addition, models using high-activation
volume creates a low-viscosity channel above the density anomaly, which
contributes to decreasing the dynamic topography. Using a larger viscosity
range in the models (10^{18} Pa s ≤ $\mathit{\eta}\left(PT\dot{\mathit{\epsilon}}\right)\le {\mathrm{10}}^{\mathrm{23}}$ Pa s) resulted in ∼5 % variation in the
amplitude of dynamic topography, indicating that the effects of nonlinear
rheology are reasonably captured in our models with smaller viscosity range
(10^{19} Pa s ≤ $\mathit{\eta}\left(PT\dot{\mathit{\epsilon}}\right)\le {\mathrm{10}}^{\mathrm{22}}$ Pa s).

Predictions of dynamic topography derived from mantle convection models are
compared against residual topography which is the component of Earth's
topography that is not compensated by isostasy (Flament et al.,
2013; Hoggard et al., 2016). In a recent work (Cowie and
Kusznir, 2018), it has been argued that dynamic topography predictions
require scaling of amplitudes by ∼0.75 to match the residual
topography, and when density anomalies shallower than 220 km are included,
the misfit requires a scaling factor of ∼0.35. It is also
important to consider that this misfit depends on the flow wavelength and is
suggested to be highest at lowest spherical harmonic degrees (*l*=2) or very-long wavelengths (Steinberger, 2016). Our numerical experiments
show that amplitude of dynamic topography can be nearly halved (e.g. from
∼174 m in Exp. 2 to ∼90 m in Exp. 5b) when we
consider nonlinear mantle rheology. Therefore, we propose that, at shorter
wavelengths (i.e. less than 1000 km), part of the misfit between the
dynamic topography extracted from mantle convection models and dynamic
topography estimated from residual topography can be attributed to the
Newtonian mantle viscosity used in convection models. If the density sources
are shallower, the dynamic topography becomes more sensitive to the
viscosity and density structure (Morgan, 1965a; Hager
and Clayton, 1989; Osei Tutu et al., 2018), and Newtonian viscosity may lead
to higher misfits.

Our models suggest that for shallow density anomalies in the mantle, nonlinear rheologies not only produce lateral variations in viscosity (Richards and Hager, 1989; Moucha et al., 2007) but also additional vertical variations in viscosity that impacts a relatively large area compared to the size of the anomaly in the mantle. We show that this impacts on the thickness of the mechanical lithosphere and predictions of the amplitude of dynamic topography.

As shown in Fig. 8, uncertainties on the activation volume result in variations in dynamic topography which are higher in experiments using dry olivine rheology (i.e. 17 %) compared to experiments using wet olivine rheology (10 %). The comparison between numerical experiments using dry olivine (Exp. 4a) and wet olivine (Exp. 5b) indicates that the variation in dynamic topography can be as much as 25 %. These variations can be lessened if we have better constraints on the mantle rheology, which will advance the dynamic topography models as well as our understanding of the interaction between deep mantle and the Earth's surface.

Code and data availability

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Code and data availability.

In our experiments we used Underworld, a free open-source code developed under the Australian AuScope initiative.

The version of Underworld code we used in our study can be found at: https://github.com/OlympusMonds/EarthByte_Underworld (last access: 25 October 2019).

To follow an open-source philosophy and promote reproducible science, we provide our input scripts (a suite of XML files) on the GitHub and EarthByte's freely accessible server: https://github.com/ofbodur/Bodur_and_Rey_EGU_SE_2019_Files (last access: 25 November 2019), https://www.earthbyte.org/ (last access: 19 December 2019).

Author contributions

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Author contributions.

ÖFB designed the experiments and wrote the paper. PFR contributed to the analysis of numerical modelling results and improved the paper.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Special issue statement

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Special issue statement.

This article is part of the special issue “Understanding the unknowns: the impact of uncertainty in the geosciences”. It is not associated with a conference.

Acknowledgements

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Acknowledgements.

This research was undertaken with the assistance of resources from the National Computational Infrastructure (NCI) through the National Computational Merit Allocation Scheme supported by the Australian Government, the Pawsey Supercomputing Centre with funding from the Australian Government and the government of Western Australia, and support from the Australian Research Council through the Industrial Transformation Research Hub grant ARC-IH130200012. We benefited from discussions with Gregory Houseman and Nicolas Flament about the Earth's dynamic topography.

Financial support

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Financial support.

This research has been supported by the Australian Research Council (grant no. IH130200012).

Review statement

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Review statement.

This paper was edited by Lucia Perez-Diaz and reviewed by Bernhard Steinberger and Mark Hoggard.

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Short summary

Convection in the deep Earth dynamically changes the elevation of plates. Amplitudes of those vertical motions predicted from numerical models are significantly higher than observations. We find that at small wavelengths (< 1000 km) this misfit can be due to the oversimplification in viscosity of rocks. By a suite of numerical experiments, we show that considering the non–Newtonian rheology of the mantle results in predictions in amplitude of dynamic topography consistent with observations.

Convection in the deep Earth dynamically changes the elevation of plates. Amplitudes of those...

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