Upper thermal boundary layer (UTBL) parameterisation

When the surface boundary uses the grey-body atmosphere mode (outer_boundary_condition = 1), the radiating surface temperature can differ from the temperature of the topmost mantle staggered node. Aragog's UTBL parameterisation accounts for the unresolved thermal boundary layer between the magma ocean interior and the radiating surface by solving a cubic relation derived in Bower et al. (2018)¹ (Eq. 18). The implementation is in src/aragog/solver/boundary.py::_utbl_tsurf and uses Cardano's formula for the analytic real root.

When to enable

The parser default is param_utbl = false (see src/aragog/parser.py). Set it explicitly to true only in the standalone grey-body case.

How it works

The grey-body radiator emits at the surface radiating temperature \(T_\mathrm{surf}\), not at the interior staggered-node temperature \(T_\mathrm{int}\). Bower et al. (2018) (Eq. 18) parameterise the temperature drop across the UTBL as

where \(b\) has units of \(\mathrm{K^{-2}}\) and is supplied by param_utbl_const. Rearranged into standard cubic form,

which has one real root by Cardano's formula:

with \(p = 1/b\), \(q = -T_\mathrm{int}/b\). The result is always \(T_\mathrm{surf} < T_\mathrm{int}\), so the radiator emits less flux than the interior temperature alone would imply.

When param_utbl = false, the parser zeroes param_utbl_const and the surface flux uses \(T_\mathrm{surf} = T_\mathrm{int}\) directly.

Picking param_utbl_const

The default value param_utbl_const = 1.0e-7 \(\mathrm{K^{-2}}\) matches the SPIDER standalone reference for Earth-mass magma oceans. The parameter trades off as follows:

• Larger \(b\): bigger temperature drop across the UTBL at fixed \(T_\mathrm{int}\); the radiator is colder, so the planet cools more slowly.
• Smaller \(b\): the UTBL drop shrinks; in the limit \(b \to 0\) the parameterisation collapses to \(T_\mathrm{surf} = T_\mathrm{int}\) and the result agrees with param_utbl = false.

For super-Earth or sub-Earth standalone runs, calibrate \(b\) against a higher-resolution surface-layer simulation; the default is not validated outside the 1 \(M_\oplus\) regime.

Configuration example

[boundary_conditions]
outer_boundary_condition = 1      # grey-body atmosphere (UTBL applies here only)
outer_boundary_value     = 280.0  # W/m^2 reference flux
emissivity               = 1.0
equilibrium_temperature  = 273.0
param_utbl               = true
param_utbl_const         = 1.0e-7

Common pitfalls

• Enabling param_utbl with PROTEUS coupling: in coupled runs the atmosphere module supplies \(F_\mathrm{atm}\) and the boundary mode is 4 (prescribed flux). The UTBL parameterisation is bypassed silently in modes 4 and 5; turning it on does no harm but also does nothing.
• Negative param_utbl_const: not validated by the parser, but produces non-physical solutions (\(T_\mathrm{surf} > T_\mathrm{int}\)) when the cubic discriminant flips sign.
• Comparing UTBL on / off across runs: the diagnostic T_magma returned by SolverOutput is the topmost basic-node interior temperature, not the UTBL-corrected radiator temperature. To extract \(T_\mathrm{surf}\) for diagnostics, recompute it from T_magma and param_utbl_const using the same Cardano formula.

Cross-references

• Configuration: [boundary_conditions]: the schema rows for param_utbl and param_utbl_const.
• Heat-transport explainer: grey-body BC alongside the prescribed-flux and prescribed-temperature modes.