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Ground Temperature Model

Surface temperature parameterization for ground longwave emission calculations.

Primary References:

  • Lindberg F, Onomura S, Grimmond CSB (2016) "Influence of ground surface characteristics on the mean radiant temperature in urban areas." International Journal of Biometeorology 60(9):1439-1452.
  • Lindberg F, Grimmond CSB (2011) "The influence of vegetation and building morphology on shadow patterns and mean radiant temperatures in urban areas." Theoretical and Applied Climatology 105:311-323.

Overview

Ground surface temperature directly affects upwelling longwave radiation (Lup), which contributes significantly to mean radiant temperature in urban environments. The model accounts for:

  1. Solar heating - Direct and diffuse radiation absorption
  2. Thermal inertia - Delayed response due to material heat capacity
  3. Surface properties - Albedo, emissivity, thermal conductivity

TsWaveDelay Model

The thermal delay model simulates ground temperature response to changing radiation conditions using an exponential decay function.

Equation

T_ground(t) = T_current × (1 - w) + T_previous × w

where:
    w = exp(-33.27 × Δt)
    Δt = time since last update (fraction of day)

Parameters

Parameter Value Description
Decay constant 33.27 Thermal response rate (day⁻¹)
Time threshold 59/1440 Minimum time step (~59 minutes)

Physical Interpretation

The decay constant (33.27 day⁻¹) corresponds to a thermal time constant of approximately:

τ = 1 / 33.27 ≈ 0.030 days ≈ 43 minutes

This represents the characteristic time for surface temperature to respond to changes in radiative forcing. After one time constant:

  • 63% of adjustment to new equilibrium
  • After 3τ (~2 hours): 95% adjustment

Algorithm

In Rust: ts_wave_delay(gvf_lup, firstdaytime, timeadd, timestepdec, tgmap1) and ts_wave_delay_batch_pure for batched 6-in-1 processing (center + 4 directional + ground).

def TsWaveDelay(T_current, firstdaytime, time_accumulated, timestep, T_previous):
    """
    Apply thermal delay to ground temperature.

    Args:
        T_current: Current radiative equilibrium temperature
        firstdaytime: True if first timestep after sunrise
        time_accumulated: Time since last full update (fraction of day)
        timestep: Current timestep duration (fraction of day)
        T_previous: Previous delayed temperature

    Returns:
        T_delayed: Temperature with thermal inertia applied
        time_accumulated: Updated time accumulator
        T_previous: Updated previous temperature for next iteration
    """
    if firstdaytime:
        T_previous = T_current

    if time_accumulated >= 59/1440:  # ~59 minutes threshold
        weight = exp(-33.27 * time_accumulated)
        T_previous = T_current * (1 - weight) + T_previous * weight
        T_delayed = T_previous
        time_accumulated = timestep if timestep > 59/1440 else 0
    else:
        time_accumulated += timestep
        weight = exp(-33.27 * time_accumulated)
        T_delayed = T_current * (1 - weight) + T_previous * weight

    return T_delayed, time_accumulated, T_previous

Surface Temperature Parameterization

For computing the instantaneous radiative equilibrium temperature, SOLWEIG uses a linear parameterization based on solar altitude.

Sinusoidal Diurnal Model

The ground temperature deviation from air temperature follows a sinusoidal diurnal phase (rust/src/ground.rs):

Tgamp = TgK × altmax + Tstart

if dectime > sunrise_frac:
    phase = (dectime - sunrise_frac) / (TmaxLST_frac - sunrise_frac)
    Tg = Tgamp × sin(phase × π/2)
else:
    Tg = 0    (pre-sunrise: no deviation from air temp)

Where:

  • Tgamp = maximum temperature amplitude (°C above air temp)
  • TgK = temperature increase rate (°C per degree of max solar altitude)
  • altmax = maximum solar altitude during the day (°)
  • Tstart = temperature offset at sunrise (°C)
  • dectime = current time as fraction of day
  • sunrise_frac = sunrise time as fraction of day
  • TmaxLST_frac = time of maximum surface temperature as fraction of day

Clearness Index Correction

After computing the sinusoidal Tg, a clearness index correction is applied to account for non-clear sky conditions:

corr = 0.1473 × ln(90 - zenith_deg) + 0.3454
CI_TgG = (radG / radG0) + (1 - corr)
CI_TgG = min(CI_TgG, 1.0)
Tg = max(Tg × CI_TgG, 0.0)

Where radG is measured global radiation and radG0 is theoretical clear-sky radiation. Under clear skies CI_TgG ≈ 1.0; under overcast conditions CI_TgG < 1.0, reducing the ground temperature response.

Land Cover Parameters

Surface Type Tstart (°C) k (°C/°) TmaxLST Source
Cobblestone -3.41 0.37 15:00 Lindberg et al. (2016)
Dark asphalt -9.78 0.58 15:00 Lindberg et al. (2016)
Grass -3.38 0.21 14:00 Lindberg et al. (2016)
Bare soil -3.01 0.33 14:00 Estimated
Water 0.0 0.00 12:00 Estimated

Note: Tstart is the temperature offset from air temperature at sunrise. Negative values indicate surfaces cooler than air at dawn.

Water temperature override: When land cover is active (the normal path), water pixels (lc_grid == 3) bypass this table entirely — their ground temperature is set to Twater - Ta from the weather file. The TgK/TmaxLST values only apply in the rare no-landcover fallback. With TgK=0.00, Tgamp=0 making TmaxLST irrelevant.

Properties

Thermal Inertia Effects

  1. Morning lag - Surfaces warm slower than instantaneous equilibrium
  2. Afternoon persistence - Surfaces remain warm after solar maximum
  3. Evening cooling - Gradual temperature decrease after sunset

Material Dependence

  1. High thermal mass (concrete, stone): Slower response, τ > 1 hour
  2. Low thermal mass (thin asphalt): Faster response, τ < 30 minutes
  3. Vegetation: Complex due to evapotranspiration

Diurnal Pattern

Morning:  T_ground < T_equilibrium (heating lag)
Midday:   T_ground ≈ T_equilibrium (near steady state)
Afternoon: T_ground > T_equilibrium (cooling lag)
Night:    T_ground slowly approaches T_air

Implementation Notes

State Management

The thermal delay model requires state to be carried between timesteps:

  • 6 directional tgmap1 arrays (center, E, S, W, N, ground)
  • tgout1 — ground temperature output history
  • firstdaytime flag — reset on first timestep after sunrise
  • timeadd accumulator — tracks time since last full update
  • timestep_dec — current timestep as fraction of day

For accurate results, use calculate() with a timeseries of weather data, which automatically manages thermal state. Single-timestep calculations will not capture thermal inertia effects.

Directional Components

Ground temperature affects directional Lup components (Lup_E, Lup_S, Lup_W, Lup_N) which are computed using Ground View Factors in each direction. The ts_wave_delay_batch_pure function processes all 6 directional channels in a single call.

Nighttime Behavior

Pre-sunrise (dectime <= sunrise_frac):

  • Ground temperature deviation Tg = 0 (no deviation from air temperature)
  • The TsWaveDelay model handles smooth transitions via thermal inertia
  • Emissivity assumed constant (typically 0.95)

Validation Status

The TsWaveDelay model parameters (decay constant 33.27) require validation against:

  • [ ] In-situ surface temperature measurements
  • [ ] Comparison with force-restore energy balance models
  • [ ] Sensitivity analysis for different surface types

The current parameterization is empirical and may need adjustment for specific climates or surface materials.

References

Primary UMEP Citation:

  • Lindberg F, Grimmond CSB, Gabey A, Huang B, Kent CW, Sun T, Theeuwes N, Järvi L, Ward H, Capel-Timms I, Chang YY, Jonsson P, Krave N, Liu D, Meyer D, Olofson F, Tan JG, Wästberg D, Xue L, Zhang Z (2018) "Urban Multi-scale Environmental Predictor (UMEP) - An integrated tool for city-based climate services." Environmental Modelling and Software 99, 70-87. doi:10.1016/j.envsoft.2017.09.020

Ground Temperature Model:

  • Lindberg F, Holmer B, Thorsson S (2008) "SOLWEIG 1.0 - Modelling spatial variations of 3D radiant fluxes and mean radiant temperature in complex urban settings." International Journal of Biometeorology 52(7), 697-713.
  • Lindberg F, Onomura S, Grimmond CSB (2016) "Influence of ground surface characteristics on the mean radiant temperature in urban areas." International Journal of Biometeorology 60(9), 1439-1452.
  • Offerle B, Grimmond CSB, Oke TR (2003) "Parameterization of net all-wave radiation for urban areas." Journal of Applied Meteorology 42(8), 1157-1173.