Conductive heat diffusion and a thermal length scale
For stationary homogeneous material with constant positiveρ,c_p andk, no internal heat production and one-dimensional conduction, ρc_pT_t=kT_zz, with thermal diffusivityκ=k/(ρc_p).
Key knowledge
For stationary homogeneous material with constant positiveρ,c_p andk, no internal heat production and one-dimensional conduction, ρc_pT_t=kT_zz, with thermal diffusivityκ=k/(ρc_p). A boundary-temperature disturbance has a characteristic diffusion length of order√(κt). This is a scale, not a sharp thermal front: the diffusion model has nonzero tails.
Values and equations
ρc_pT_t=kT_zz; κ=k/(ρc_p); L_diff~√(κt)
- T
- temperature K
- t
- elapsed time s
- z,L_diff
- depth and characteristic diffusion length m
- ρ
- density kg m⁻³
- c_p
- specific sensible-heat capacity in the stated material model J kg⁻¹ K⁻¹
- k
- thermal conductivity W m⁻¹ K⁻¹
- κ
- thermal diffusivity m² s⁻¹
Conditions and scope
- Specify initial temperature and boundary temperature/flux conditions; constant coefficients, local sensible-heat model and no advection, phase change or radiogenic source in the displayed equation.
- Rock conductivity/heat capacity vary and groundwater/advection or mantle convection can transport heat. The length scale alone does not determine mantle temperature, lithosphere thickness or a unique age.
Sources · 1
References
Sources retain their recorded scope. A source link alone does not establish verification of every statement.
Evidence and review
Read actual printed29–31 control-volume balance, conduction PDE, κ units, scaling and steady solution. Clarified order-of-magnitude rather than a front and excluded universal numeric crustalκ, absolute planetary-cooling assertions and the source's suggestion that erf alone, without the similarity variable, solves the time-dependent PDE.
2026-10-10 · Independent AI editorial review: All73 exact supplement compact statements, conditions, displayed equations and247 origin semantic links read; all109 combined original36+supplement73 author-source rows, exact locators, hypotheses, exclusions and disclosed derivations assessed. Original36 exact scientific content is independently approved in the linked receipt. All18 inherited Earth compact references and shared Math/Physics/Measurement methods were considered in the full practical Earth inventory. Targeted actual primary sections listed below; not every source body/proof or every possible university elective independently reread. AI editorial reference review, not human certification, full-degree proficiency, new experimental validation or a claim of whole combined Earth-and-Cosmos approval. Receipt: earth-completion-independent-content-inventory-review.1.
Entry details
Permanent link to this entry ↗- Record identity
- uni-earth-conductive-heat-diffusion
- Revision
- core.1
- Structured reference record
- University core expansion