Solid Earth

The Solid Earth comprises a planet's solid inner core, liquid outer core, viscously deforming mantle, and mechanically stronger crust and lithosphere.

A long-term carbon cycle coupling silicate weathering, carbonate formation and interior recycling with volcanic carbon dioxide supply. Its temperature- and water-dependent weathering can provide a climate feedback. Active plate tectonics can support recycling, but the effectiveness of the feedback also depends on rock exposure, water and reaction rates.

Transport and eruption or intrusion of molten rock and associated gases. Subduction, spreading and mantle upwellings can supply melt; volcanism does not require globally active plate tectonics. Outgassing couples interior evolution to atmospheric inventories.

A transient release of stored mechanical energy in the solid planet, often associated with fault motion. Plate tectonics is a major terrestrial driver, but earthquakes can also occur without an active global plate system.

The mechanically strong outer shell of a rocky planet, including crust and the colder upper mantle. It can form mobile plates in plate tectonics or a relatively stagnant lid. Its temperature, water content and damage affect its ability to yield.

Magma is molten or partially molten rock within a planet. Its dynamic viscosity, cooling and interaction with the surrounding elasticity influence transport and storage.

A geological dike is a sheet-like body or conduit formed when magma enters a fracture cutting across the surrounding rock. A vertical narrow dike can be modeled as a two-dimensional channel for coupled flow, heat transfer and wall melting.

With equal solid and liquid densities, magma entering a dike at constant flux $Q_0$ has $bw=Q_0$: melt added by the walls exactly supplies the increasing cross-sectional storage. If wall heat flux is ${\alpha }_T\rho c_{p}w(T-T_m)$, both walls give $b_t=2{\alpha }_T{c}_{p}w(T-T_m)/L$. Combining this with sensible-heat conservation admits $b=B(t)f(\eta )$, $\eta =2{\alpha }_{T}z/B$, $B^2=4{\alpha }_T{Q}_{0}t/S$. The shape equations are $f(f-\eta f^{\prime })=\vartheta$ and $(S-\eta f){\vartheta }^{\prime }=-(S+\vartheta )\vartheta /f$.

New wall melt enters at the wall melting temperature $T_m$. Besides supplying its latent heat, the flowing magma must warm this added melt. Relative to $T_m$, conservative heat storage is $b(T-T_m)$, so differentiating it retains $(T-T_m)b_t$ and gives the displayed dilution term. Omitting that term requires an additional approximation, such as a small sensible-to-latent heat ratio; it does not follow from equal densities alone.

A lava lake is a persistent exposed body of molten or partly molten magma. Surface radiative cooling, thermal convection, latent heat, and internal radiogenic heating can jointly control formation and melting of its crust.

For a well-mixed lava lake of depth $H$ and a thin conductive crust of thickness $a$, put $\Delta T=T_m-T_s$, and use the convective flux $K(\overline{T}-T_m)$ with constant heat transfer coefficient $K>0$. The Stefan condition and liquid heat balance give, to leading order in $a/H$,Here latent heat $L_f$ is per unit mass and $k=\rho c_p\kappa$. Crust heating and changes of liquid depth are neglected. The linear conductive profile uses a quasi-steady approximation; thinness alone does not justify neglecting crust thermal storage. During initial growth, a small Stefan number supplies a sufficient rapid-adjustment regime. Define $h=a/H$, $\theta =(\overline{T}-T_m)/\Delta T$, and $\tau =\kappa t/H^2$. Scaling by the thermal diffusion time $H^2/\kappa$ and $\Delta T$ gives $h^{\prime }=S(1/h-B\theta )$ and ${\theta }^{\prime }=R-B\theta$, where $B=KH/k$, $R=Q{H}^2/(c_p\kappa \Delta T)$, and $S=c_p\Delta T/L_f$ is the Stefan number. Thus $\theta =R/B+({\theta }_0-R/B)e^{-B\tau }$ exactly. The reduction applies while $a/H\ll 1$, the conductive profile is approximately linear, and the thermal convection closure remains valid; it does not model complete freezing of a deep lake.

In the radiogenically heated lava-lake crust model, the thickness derivative vanishes at $h_e=1/(B\theta )$ when the liquid superheat $\theta$ is positive. This moving curve balances thermal conduction and convective heat flux; it is not itself the evolving crust. Cooling raises the curve and keeps an initially absent crust below it. Warming lowers the curve, allowing lava-crust overshoot under warming. At zero superheat the stationary thickness is infinite, so its finite formula applies only after positive superheat develops.

In the radiogenically heated lava-lake crust model, initially cool liquid warms exponentially and its instantaneous stationary lava-crust thickness $1/(B\theta )$ decreases. The actual crust can cross this moving value, reach a maximum, and melt back to $h_{\ast }=1/R$. Overshoot is not automatic. Linearizing about equilibrium givesIf $B<S{R}^2$ and ${\theta }_0<R/B$, the slowly decaying forced response has positive sign, so the crust eventually approaches equilibrium from above; an initially absent crust must then have overshot. In contrast, initially hot liquid cools, its instantaneous stationary thickness increases, and an initially absent crust cannot cross it from below: at a proposed first crossing its own derivative is zero but the stationary thickness has positive derivative. The hot-start crust therefore grows monotonically. These comparisons assume nonnegative liquid superheat.

In the radiogenically heated lava-lake crust model, temperature stays constant precisely when ${\theta }_0=R/B$. Then $h^{\prime }=S(1/h-R)$, and an initially absent crust satisfies the displayed law for $0\le h<1/R$. It follows by integrating $h/(1-Rh)$ from zero to $h$. The crust starts as $\sqrt{2S\tau }$ and tends to $1/R$ only at infinite time. The equilibrium remains thin only if $R\gg 1$.

A magma intrusion is the emplacement of magma into surrounding rock. A shallow, approximately axisymmetric intrusion can lift an elastic plate roof, with spreading determined by viscous peeling of an elastic plate or fracture toughness.

With the axisymmetric clamped-plate deflection and fixed front curvature ${\kappa }_f>0$, a magma intrusion hasThese relations describe the advancing fracture branch, with roof bending rather than hydrostatic pressure controlling the interior load.

Let $V_T$ be total magma volume shared between a reservoir and an advancing toughness-controlled magma intrusion, $p_r=E(V_T-V)$, and conduit conductance $\beta >0$. With $A=\sqrt{8\pi /3}B{\kappa }_f^{3/2}$,Equilibria satisfy $\sqrt{V}(V_T-V)=A/E$. Two positive equilibria exist when $A/E<2V_T^{3/2}/(3\sqrt{3})$; the smaller is unstable and the larger is stable under this volume equation. An advancing solution between them approaches the larger. The singular pressure as $V\to 0$ prevents this late-time model from nucleating an intrusion by itself.

If a positive vapour-tip pressure magnitude $p_T$ sets $h_f\sim p_T{l}_p^4/B$, matching $h_f/l_p^2$ to the interior curvature ${\kappa }_i$ gives $l_p\sim (B{\kappa }_i/p_T{)}^{1/2}$. The viscous peeling of an elastic plate law then yields $\dot{R}\sim B^{3/2}{\kappa }_i^{7/2}/(\mu p_T^{1/2})$. For constant injected volume flux $Q$, $V=Qt$ and ${\kappa }_i\sim V/R^4$, henceOrder-one coefficients require solving the local peeling profile. The scaling assumes a front region much shorter than the intrusion radius.

Mantle convection is the slow buoyancy-driven motion of a rocky planetary mantle. On geological timescales the solid mantle behaves as a highly viscous Newtonian fluid to first approximation.

A mobile-lid mode of planetary evolution in which coherent pieces of the lithosphere move, form at spreading regions and can recycle through subduction. Slab pull, ridge push and mantle flow compete with deformation resistance. It affects heat transport, volcanism, earthquakes and the carbonate-silicate cycle. It is not guaranteed solely by an exoplanet's mass.

Gravitational sliding of lithospheric material away from elevated spreading regions. It supplies a driving contribution to plate tectonics together with slab pull and mantle stresses.

Downward recycling of a lithospheric plate into the mantle. A sufficiently cold, dense slab can provide slab pull, while bending and friction resist motion. The resulting material and volatile recycling connects plate tectonics with volcanism and long-term climate.

A gravitational driving force associated with a dense descending slab. The slab's buoyancy and ability to remain coherent matter, so increased planetary gravity alone does not determine whether subduction occurs.

The Rayleigh number compares buoyancy driving with viscous and thermal diffusion. For a layer of thickness $h$, temperature drop $\Delta T$, density $\rho$, thermal expansion coefficient ${\alpha }_T$, gravitational acceleration $g$, thermal diffusivity $\kappa$, and dynamic viscosity $\mu$,Thermal convection begins when this exceeds a geometry-dependent critical value.

A local Rayleigh-number closure estimates the thickness $\delta$ of a turbulent thermal boundary layer by settingwhere ${\mathrm{Ra}}_c$ is an order-one or geometry-dependent critical value. Combining this with Fourier's law converts the boundary-layer temperature and density differences into a heat flux law.

Penetrative convection occurs when an unstable convecting region extends into, or is capped by, a stably stratified region. Water near its temperature of maximum density provides a common example: a cold stable layer can lie above a warmer turbulent convecting layer.

Fresh water reaches maximum mass density near $4{}^{\circ }\mathrm{C}$ at atmospheric pressure. A useful local equation of state isso water becomes less dense when its temperature moves either above or below $T_m$.

In an ice-covered freshwater lake, a stagnant layer colder than the density-maximum temperature can cap a warmer turbulent region. Maximizing the boundary-layer heat flux obtained from a Local Rayleigh-number closure gives an interface temperature $T_h$ satisfyingand a maximum upward heat flux proportional to $(T_w-T_m{)}^{5/3}$.