Scientific Methods & References

1. Definitions & Symbols

2. Saturation Vapor Pressure Models

The app uses a Magnus–Tetens–type expression (FAO-56 form) for saturation vapor pressure over liquid water:

This parameterization (Allen et al., 1998, FAO-56) is standard in agronomy and evapotranspiration modeling. Alternative high-accuracy formulations include Buck (1981, 1996) and Sonntag (1990). For horticultural VPD ranges \(T \in [0, 40]^\circ\mathrm{C}\), the FAO-56 expression tracks these within ~0.1–0.3 %, well below sensor and environmental variability in grow environments.

2.1 Alternative formulations (cross-check)

• Buck (1981, 1996): Piecewise empirical equations for \(e_s\) over water and ice; widely cited in meteorology.
• Sonntag (1990): Revision of Magnus equation coefficients, accurate across 0–50 °C.
• Alduchov & Eskridge (1996): Improved constants for the August–Roche–Magnus dew-point relationship; also used here for dew point (Section 4).

3. Vapor Pressure Deficit (Air & Leaf)

Ambient water vapor partial pressure is \( e(T,RH) = \frac{RH}{100}\,e_s(T) \). Following standard definitions (FAO-56; Jones, Plants and Microclimate), VPD is the difference between saturation at a reference surface and the ambient vapor pressure.

3.1 Air VPD

3.2 Leaf VPD (with leaf–air ΔT)

When a leaf is warmer than the surrounding air by \(\Delta T\), the evaporative demand at the leaf surface is modeled as:

This assumes (i) the leaf surface is near saturation at \(T_\ell\), and (ii) ambient vapor pressure is governed by air \(T\) and \(RH\). It captures the first-order effect described in plant microclimate literature (Monteith & Unsworth; Jones). In high-advection or boundary-layer-limited regimes, canopy models add aerodynamic resistances; the app intentionally uses the simpler diagnostic definition for speed and clarity in grow-room use.

4. Dew Point & Absolute Humidity

4.1 Dew point (August–Roche–Magnus form)

With \( a = 17.27 \) and \( b = 237.3^\circ\mathrm{C} \) (Alduchov & Eskridge, 1996 / FAO-style), define \( \gamma(T,RH) = \ln(RH/100) + \dfrac{aT}{b+T} \).

4.2 Absolute humidity

Using ideal-gas relationships with \(R_d/R_v \approx 0.622\) yields a convenient engineering form for absolute humidity (mass concentration of water vapor):

Here \(e\) is vapor pressure in hPa (1 kPa = 10 hPa) and \(T\) is absolute temperature in Kelvin. The constant 216.7 follows from the universal gas constant and molecular weights (WMO/ASHRAE tables).

5. Station Pressure & Altitude

For reference, the app computes an approximate station pressure from altitude using the US Standard Atmosphere barometric formula:

Earlier versions (pre-1.4.2) multiplied \(e_s\) by \(P(h)/101.325\) as a heuristic scaling factor. This was removed after review confirmed it is not required by the standard VPD definition and introduced a small, unnecessary bias at higher elevations.

6. Analytic Solvers & Numerics

6.1 Solve RH for a target VPD (fixed T)

From \( VPD = e_s(T)\,(1 - RH/100) \) it follows directly that \( RH = 100\,\bigl(1 - VPD/e_s(T)\bigr) \).

6.2 Solve T for a target VPD (fixed RH)

There is no closed form for \(T\) in the Magnus expression. The app uses a stable bounded search on \(T\in[5,45]^\circ\mathrm{C}\) with a small step for widget responsiveness. A Newton–Raphson variant was profiled but the bounded sweep is more robust on-device and avoids rare divergence at extreme RH.

6.3 Numerics & stability

• Precision. All exponentials are evaluated in double precision; results are clamped to physically meaningful ranges (e.g., \(RH\in[0,100]\)).
• Guarding. To avoid division by zero, \(b+T\) and color-scale denominators use a small \(\varepsilon\) floor (e.g., \(10^{-6}\)).
• Widget lifecycle. Timeline updates are debounced and snapshots are written to an App Group store to minimize energy and prevent black widgets during rapid interactions.

7. Visualization & Color Mapping

Heatmaps use the perceptually uniform Viridis palette with green near the active stage's ideal band midpoint. The app computes a band-relative signed distance:

and maps \(s\) into Viridis parameter \(t\) such that \(t\approx 0.72\) (green) at \(s=0\), shifting toward indigo (\(s\ll0\), too humid) and yellow (\(s\gg0\), too dry).

The heatmap image is pre-rendered to an off-screen RGBA buffer once per size/stage/ΔT combination (not every frame). Isolines (every 0.2 kPa) and the cursor halo are vector overlays drawn cheaply on top.

Why Viridis? Unlike rainbow maps, Viridis is near-perceptually-uniform in lightness and color, improves readability for color-vision-deficient users, and preserves ordinal structure (van der Walt & Smith, 2015).

8. Validation Strategy

• Unit tests compare \(e_s(T)\) against Buck (1996) values at 0–40 °C; max relative error < 0.3 %.
• Spot checks of \(T_d\) vs Alduchov–Eskridge yield < 0.2 °C absolute differences across RH 10–90 %.
• VPD isolines at integer tenths (0.6, 0.8, 1.0 … kPa) match analytic \(RH\) inversion within ±0.2 % RH over the plotted domain.

9. Limits & Uncertainties

• Sensor error dominates. Typical low-cost RH sensors have ±2–3 % RH and temp ±0.2–0.5 °C; this maps to ~±0.03–0.08 kPa VPD around 1 kPa.
• Leaf energy balance. The simple \(T_\ell=T+\Delta T\) approach is diagnostic. Detailed canopy models would add stomatal, boundary-layer, and radiative terms (Monteith & Unsworth).
• Pressure independence. VPD is computed without scaling \(e_s\) by station pressure, consistent with the standard thermodynamic definition. The barometric formula is retained in the codebase for psychrometric context only.
• Domain. Plots are limited to \(T\in[10,35]^\circ\mathrm{C}\), \(RH\in[30,95]\%\), covering typical controlled-environment horticulture.

10. Code Mapping

Traceability from equation to implementation. Function names refer to the shared helpers in the widget and app targets.

11. References

• Allen, R.G., Pereira, L.S., Raes, D., Smith, M. (1998). FAO Irrigation and Drainage Paper 56: Crop Evapotranspiration (FAO-56). UN-FAO. — Provides the 0.6108–17.27–237.3 Magnus-Tetens form used here for \(e_s(T)\) and standard VPD/ET conventions.
• ASHRAE (2017/2021). Handbook — Fundamentals. American Society of Heating, Refrigerating and Air-Conditioning Engineers. — Psychrometric relations, absolute humidity conversions, and standard properties.
• Alduchov, O.A., Eskridge, R.E. (1996). "Improved Magnus form approximation of saturation vapor pressure." J. Appl. Meteor. 35:601–609. — Coefficients often used for dew point; consistent with the implementation above.
• Buck, A.L. (1981, updated 1996). "New equations for computing vapor pressure and enhancement factor." J. Appl. Meteor. 20:1527–1532. — High-accuracy empirical \(e_s(T)\) formulas over water/ice; used for cross-validation.
• Sonntag, D. (1990). "Important new values of the physical constants of 1986, vapor pressure formulations based on the ITS-90, and psychrometric formulae." Z. Meteorologie 70:340–344.
• WMO (2008/2018). Guide to Meteorological Instruments and Methods of Observation (WMO-No. 8). World Meteorological Organization. — Definitions and recommended practice for humidity measurements and conversions.
• Jones, H.G. (2013). Plants and Microclimate (3rd ed.). Cambridge Univ. Press. — Leaf energy balance, stomatal control, and the interpretation of leaf-air VPD.
• Monteith, J.L., Unsworth, M.H. (2013). Principles of Environmental Physics (4th ed.). Academic Press. — Canonical text for surface energy balance and psychrometrics in plant environments.
• U.S. Standard Atmosphere (1976). NOAA/NASA/USAF. — Basis for the barometric \(P(h)\) approximation used for station pressure.
• van der Walt, S., Smith, N. (2015). "A better default colormap for Matplotlib." (SciPy 2015). — Rationale for the Viridis palette (perceptually uniform, CVD-friendly).