Half-space cooling of an ideal lithosphere
For z>0 initially at uniform T_i and a surface held at T_s for t>0, constant-κ conduction gives T(z,t)=T_s+(T_i−T_s)erf[z/(2√(κt))].
Key knowledge
For z>0 initially at uniform T_i and a surface held at T_s for t>0, constant-κ conduction gives T(z,t)=T_s+(T_i−T_s)erf[z/(2√(κt))]. For T_i>T_s, upward surface heat flux is q_s=k(T_i−T_s)/√(πκt); a fixed temperature fraction lies at a depth proportional to√t. This idealization helps interpret oceanic lithosphere cooling.
Values and equations
T=T_s+(T_i−T_s)erf[z/(2√(κt))]; q_s=k(T_i−T_s)/√(πκt)
- T,T_s,T_i
- temperature, boundary temperature and initial interior temperature K
- z
- depth m
- t
- time since boundary cooling began s
- κ
- constant thermal diffusivity m² s⁻¹
- k
- thermal conductivity W m⁻¹ K⁻¹
- q_s
- upward surface heat flux W m⁻²
Conditions and scope
- Stationary semi-infinite homogeneous conductor,k>0,κ>0, no heat source, T→T_i at infinite depth and fixed surface temperature; the ideal discontinuous boundary switch produces a divergent flux as t→0⁺.
- Finite plate thickness, hydrothermal circulation, variable properties, internal heat and tectonic evolution require alternatives. A√age scaling is not a universal ocean-depth law or an exact thickness inferred without defining a temperature criterion.
Sources · 2
References
Sources retain their recorded scope. A source link alone does not establish verification of every statement.
Massachusetts Institute of Technology2 sources
- 12.002 Lecture15, printed31, error-function thermal solution and initial/boundary conditionsLeigh Royden and Benjamin Weiss — MIT OpenCourseWare (2008)Reference guide · PDF
- 12.002 Lecture13, printed29–31, conductive conservation, diffusivity, length scale and steady geothermLeigh Royden and Benjamin Weiss — MIT OpenCourseWare (2008)Reference guide · PDF
Evidence and review
Read actual Lecture15 entire thermal solution, erf definition and boundary/initial conditions. Restored poorly extracted square-root glyphs by similarity-variable/PDE substitution; surface heat flux is explicitly derived by differentiating erf and applying the inspected conductive law. Excluded claiming erf(z) itself solves time-dependent conduction or deriving actual ocean subsidence without buoyancy/plate assumptions.
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