= Solution
\b[<Homologous temperature> and deformation.] <Homologous temperature> is $\Theta=T/T_m$, with both temperatures in <kelvin> and $T_m$ the appropriate pressure- and composition-dependent melting <temperature>. It measures proximity to melting more meaningfully than an absolute <temperature> alone. Pressure, composition and water can alter both $T_m$ and the flow law, so equal $\Theta$ does not imply identical rheology.
A useful thermally activated <material creep> law is
$$
\dot\epsilon=A\sigma^n d^{-p}
\exp\left[-\frac{E^*+PV^*}{RT}\right],
$$
with <stress> $\sigma$, <grain size> $d$, <activation energy> $E^*$ and <activation volume> $V^*$. Water activity can add another factor. Increasing <temperature> raises the rate at fixed <stress> when the activation enthalpy is positive, greatly lowering effective <viscosity>. Solid material can therefore flow on geological timescales well below melting.
<Diffusion creep> is approximately <linear> in <stress>: bulk-diffusion <Nabarro-Herring creep> has grain-size scaling $d^{-2}$, while grain-boundary <Coble creep> has $d^{-3}$ in their ideal regimes. <Dislocation creep> usually has a <stress> exponent greater than one and weaker grain-size dependence. Its effective <viscosity> scales as $\eta_{\rm eff}\propto\sigma^{1-n}$ at fixed <temperature>, so stronger forcing changes the apparent <viscosity>. At lower <homologous temperature>, brittle/frictional deformation or high-stress dislocation mechanisms can become important. The dominant process also depends on pressure and confinement; low <temperature> is not by itself a guarantee of brittle failure deep in a planet.
A <deformation mechanism map> compares the competing rates or strengths as functions of <stress> and <homologous temperature> at specified <grain size>, pressure and composition. Fine grains favour <diffusion creep>; larger <stress> favours nonlinear <dislocation creep>. There is no universal <temperature> dividing all these regimes. This sensitivity helps explain a strong outer <lithosphere> above a slowly flowing solid <mantle> and why modest thermal or water variations can reorganize deformation.
\b[Determining the actual planform of <mantle convection>.] The task is to infer the geometry of upwellings, downwellings and connecting flow in the present <Earth>, rather than to draw a preferred laboratory convection cell. Plate velocities and the distribution of subduction zones directly constrain surface motion and likely cold downwellings. Ridges, volcanism and hotspots provide further clues, but a ridge need not mark the centre of a deep plume, nor is every hotspot uniquely explained by one plume geometry.
<Seismic tomography> supplies three-dimensional <velocity> structure from body-wave travel times, waveforms and surface-wave <dispersion>. Fast slab-like structures can often be traced beneath subduction zones into the lower <mantle>. The 1997 global P-wave study reported long narrow lower-mantle structures connected with convergence and interpreted some slab segments as reaching the deepest <mantle>; that is evidence for deep circulation, not proof that every slab follows one route.<Evidence for deep mantle circulation from global tomography by R. D. van der Hilst and others>[Nature 386, pages 578–584; abstract on lower-mantle structures and mantle-wide flow]{ref} <Temperature> affects seismic speed, but composition, phase changes, water and anisotropy also matter, so <velocity> is not itself a flow map.
Gravity and the <geoid> constrain the spatially integrated <density> field and its induced boundary deformation. <Dynamic topography> supplies information about vertical stresses, after separating crustal structure, isostatic loads and thermal subsidence. <Seismic anisotropy> can constrain deformation and fabric orientations, subject to a mineral-physics relation between fabric and flow. Discontinuity topography and the <mantle>'s heat budget provide further tests. Each observation has a different sensitivity to <density>, <temperature> and <viscosity>, making their combination more useful than any one image.
Compare these observations with three-dimensional geodynamic models that solve creeping-flow force balance, mass conservation and thermal transport, including buoyancy from <temperature> and composition, realistic plate motions, <viscosity> structure and phase boundaries. Predict seismic observables, gravity, <geoid> and <dynamic topography> from the same model, and assess data coverage and resolution before interpreting a feature. Whole-mantle exchange, impeded or deflected slabs near transition zones, and broad deep anomalies can coexist; the evidence does not require identical stacked or single-layer cells everywhere. \b[The actual planform is constrained by joint seismic, gravitational, surface-kinematic and thermal evidence, with substantial nonuniqueness in its detailed flow and history.]
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