Table 1 Summary of physical quantities and reference values used in the model

Velocity

\(\varvec{u}\)

\(u_\text {conv} = {5}\,\hbox {cm}\,\hbox {yr}^{-1}\)

Dynamic pressure

p

\(\hbox {Pa}\)

Temperature

T

\(T_0 = {273}\,\hbox {K}\), \(T_\text {max} = {1573}\,\hbox {K}\)

Time

t

\(t_\text {slab} = {11}\,\hbox {Myr}\)

Position

\(\varvec{x}\)

\(\hbox {km}\)

Radial distance

r

\(r = \Vert \varvec{x} \Vert _2\)

Radius of the Earth

\(r_0\)

\({6371}\,\hbox {km}\)

Depth

d

\(d = r_0 - r\)

Plate depth

\(d_\text {plate}\)

\({50}\,\hbox {km}\)

Coupling depth

\(d_c\)

\({75}\,\hbox {km}\)

Slab thickness

\(d_\text {slab}\)

\({200}\,\hbox {km}\)

Dynamic viscosity

\(\eta\)

\(\text {Pa}\,\text {s}\) (Eq. (5))

Stress tensor

\(\varvec{\sigma }\)

\(\hbox {Pa}\)

Density

\(\rho\)

\({\left\{ \begin{array}{ll} {2700}\,\hbox {kg}\,\hbox {m}^{-3} &{} \varvec{x} \in {\Omega }_\text {plate} \text { and } {0}\,\hbox {km} \le d < {40}\,\hbox {km} \\ {3300}\,\hbox {kg}\,\hbox {m}^{-3} &{} \text {otherwise} \end{array}\right. }\)

Thermal conductivity

k

\({\left\{ \begin{array}{ll} {2.5}\,\hbox {W}\,\hbox {m}^{-1}\,\hbox {K}^{-1} &{} \varvec{x} \in {\Omega }_\text {plate} \text { and } {0}\,\hbox {km} \le d < {40}\,\hbox {km} \\ {3}\,\hbox {W}\,\hbox {m}^{-1}\,\hbox {K}^{-1} &{} \text {otherwise} \end{array}\right. }\)

Heat capacity

\(c_p\)

\({1250}\,\hbox {J}\,\hbox {kg}^{-1}\,K^{-1}\)

Radiogenic heat source

Q

\({\left\{ \begin{array}{ll} {1.3}\,\upmu \hbox {W}\,\hbox {m}^{-3} &{} \varvec{x} \in {\Omega }_\text {plate} \text { and } {0}\,\hbox {km} \le d < {15}\,\hbox {km} \\ {0.27}\,\upmu \hbox {W}\,\hbox {m}^{-3} &{} \varvec{x} \in {\Omega }_\text {plate} \text { and } {15}\,\hbox {km} \le d \le {40}\,\hbox {km} \\ {0} &{} \text {otherwise} \\ \end{array}\right. }\)

Surface heat flux

\(q_\text {surf}\)

\({65}\,\hbox {mW}\,\hbox {m}^{-2}\)