Precipitation minus Evaporation (P-E)

To calculate Precipitation minus Evaporation (P-E) in numerical weather simulations using mixing ratio, the process involves analyzing atmospheric moisture fluxes and thermodynamic relationships.

地表可用水量 (降水減蒸發，P-E) 是反映氣候乾濕程度的關鍵指標，對區域水資源、農業生產和生態系統穩定性有重要影響。熱力學理論指出，隨著全球變暖，大氣中水汽含量上升，水汽輸送增強，這會促使乾燥的地區/季節變得更乾，而濕的地區/季節變得更濕。地表可用水量季節變率增強將使得水資源在年內分配更不均衡，導致濕季更容易出現洪澇災害，而乾季更容易發生乾旱事件，給區域水資源管理和調度帶來嚴峻挑戰和壓力

• Definition: P-E is the difference between the amount of precipitation (P) and evaporation (E) over a given area and time period. It represents the net supply of freshwater available for runoff, infiltration, and other hydrological processes.
• Role in Hydrology: P-E is crucial for understanding water availability, as it influences surface water and groundwater recharge. A positive P-E indicates a surplus of water, potentially leading to increased runoff and flooding, while a negative P-E suggests a deficit, often associated with drought conditions.

To calculate P-E, you need data on both precipitation and evaporation. Here's a general approach:

1. Precipitation Data: Obtain precipitation data from sources like rain gauges, satellite imagery, or reanalysis datasets (e.g., ERA5).

2. Evaporation Data: Evaporation can be estimated using models that incorporate factors like temperature, humidity, wind speed, and solar radiation. Common methods include the Penman-Monteith equation or using reanalysis datasets.

3. Compute P-E: Subtract the evaporation from the precipitation over the desired time period (e.g., monthly or annually). \[ P - E = \text{Precipitation} - \text{Evaporation} \]

Another Calculation Methods

1. Moisture Budget Equation

The fundamental relationship in atmospheric water balance is: \[ P - E = -\left(\frac{\partial q}{\partial t} + \nabla \cdot (\mathbf{v}q)\right) \]

Where:

• \(q\) = mixing ratio (kg water vapor/kg dry air)
• \(\mathbf{v}\) = horizontal wind vector
• \(\frac{\partial q}{\partial t}\) = local moisture tendency
• \(\nabla \cdot (\mathbf{v}q)\) = horizontal moisture advection

This residual method calculates P-E as the negative sum of:

• Local moisture changes over time
• Net horizontal moisture flux divergence

2. Direct Surface Flux Approach

For evaporation calculation using mixing ratio: \[ E = \rho_{air} \cdot C_E \cdot |\mathbf{v}| \cdot (q_{sat} - q_{air}) \] Where:

• \(\rho_{air}\) = air density
• \(C_E\) = bulk transfer coefficient (~1.5×10⁻³ over water)
• \(q_{sat}\) = saturation mixing ratio at surface temperature
• \(q_{air}\) = actual near-surface mixing ratio

Precipitation (P) is typically output from microphysics parameterizations in the model.

Key Implementation Steps

Step 1: Compute Saturation Mixing Ratio
Use Tetens formula for saturation vapor pressure: \[ e_s = 6.112 \exp\left(\frac{17.67T}{T + 243.5}\right) \quad [\text{hPa}] \]

Then calculate:

\[ q_{sat} = \frac{\varepsilon e_s}{P - e_s} \quad (\varepsilon = 0.622) \]

Step 2: Vertical Integration
For column-integrated P-E: \[ P - E = -\frac{1}{g} \int_{sfc}^{top} \left(\frac{\partial q}{\partial t} + \nabla \cdot (\mathbf{v} q)\right) dp \]

Step 3: Numerical Implementation
In finite difference models:

• Calculate moisture tendency between time steps
• Use centered differencing for advection terms
• Apply mass-corrected transport schemes to prevent numerical dispersion

Practical Considerations

Data Requirements

• 3D fields of mixing ratio and winds at model levels
• Surface pressure and temperature
• Precipitation outputs from convective/resolved schemes

Validation
Compare results against:

1. Direct surface flux measurements
2. Reanalysis-derived P-E products
3. River basin water balance estimates

The moisture budget method generally shows <10% error for monthly means in midlatitudes but requires careful handling of tropical convection systems^[3][5]. Recent NWP systems like ECMWF's IFS and NASA's GEOS use hybrid approaches combining both methods^[1][7].

MPAS: Registry.xml

About mixing ratio,

<var_array name="scalars" type="real" dimensions="nVertLevels nCells Time" packages="met_stage_out">
        <var name="qv" array_group="moist" units="kg kg^{-1}"
                description="Water vapor mixing ratio"/>

        <var name="qc" array_group="moist" units="kg kg^{-1}"
                description="Cloud water mixing ratio"/>

        <var name="qr" array_group="moist" units="kg kg^{-1}"
                description="Rain water mixing ratio"/>
</var_array>

NWP-AI

NeuralGCM

• NWP | AI | NeuralGCM

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• To diagnose precipitation minus evaporation rate (P − E)
• In this work, we only diagnosed precipitation minus evaporation from NeuralGCM. However, in future work, we plan to develop a scheme to reformulate NeuralGCM to predict precipitation and evaporation separately. This could be achieved, for example, by using conventional parameterization to estimate evaporation (and then calculating precipitation by adding it to P-E). Another approach could involve training a neural network (NN) specifically to predict evaporation, potentially optimized to predict the same fluxes as those in ERA5 data.

References

1. GLOBAL CHANGES IN PRECIPITATION MINUS EVAPORATION | 2023
2. Scott, Robert W., and James R. Scoggins. The moisture budget in relation to convection. No. M-216. 1977.
3. Trenberth, Kevin E., and Christian J. Guillemot. "Evaluation of the global atmospheric moisture budget as seen from analyses." Journal of Climate 8.9 (1995): 2255-2272. (Recommand)
4. Seager, Richard, and Naomi Henderson. "Diagnostic computation of moisture budgets in the ERA-Interim reanalysis with reference to analysis of CMIP-archived atmospheric model data." Journal of Climate 26.20 (2013): 7876-7901.
5. Moisture budget equation and application to Sahel Drought

1. https://www.osti.gov/servlets/purl/6807588 ↩
2. https://www.eoas.ubc.ca/books/Practical_Meteorology/prmet/Ch04-Moist.pdf ↩
3. https://journals.ametsoc.org/downloadpdf/journals/clim/28/20/jcli-d-15-0369.1.pdf ↩
4. https://weather.cod.edu/notes/materials/1110/Unit2_1110.pdf ↩
5. https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2022GL097725 ↩
6. https://www.weather.gov/epz/wxcalc_mixingratio ↩
7. https://atmos.uw.edu/~dennis/321/321_Lecture_15.pdf ↩
8. https://www.inscc.utah.edu/~krueger/5130/precip_rate.pdf ↩