Column-Integrated Diabatic Heating

Overview

Diabatic heating encompasses all non-adiabatic processes that add or remove heat from an air parcel, including latent heat release during condensation, radiative heating/cooling, and sensible heat exchange. This diagnostic estimates the column-integrated diabatic heating rate as a residual of the thermodynamic energy equation: where ascending air cools adiabatically, any departure from that cooling rate implies diabatic heating (e.g. latent heat release in convective storms).

Input Variables

Constants

Computation

1. Static Stability Parameter

The static stability on pressure surfaces measures how temperature changes with pressure relative to an adiabatic profile:

$$\sigma_p = \frac{\partial T}{\partial p} - \frac{\kappa \, T}{p}$$

The temperature derivative $\partial T / \partial p$ is computed using centered finite differences along the pressure axis (one-sided at boundaries).

2. Adiabatic Cooling Integrand

The integrand combines the vertical motion with the static stability:

$$f(p) = c_p \, \omega \, \sigma_p$$

3. Column Integration

The column-integrated diabatic heating is obtained by integrating over pressure using the trapezoidal rule and reversing the sign:

$$\text{DH} = -\frac{1}{g} \int_{p_{\text{top}}}^{p_{\text{sfc}}} c_p \, \omega \, \sigma_p \; dp$$

Sign Convention

• Positive values indicate net diabatic heating (e.g. latent heat release in deep convection or warm advection exceeding adiabatic cooling).
• Negative values indicate net diabatic cooling (e.g. radiative cooling or evaporative cooling).

Units

Column-integrated diabatic heating has units of W m$^{-2}$.

References

• Diabatic process (Wikipedia)
• Thermodynamic energy equation (Wikipedia)
• Yanai, M., Esbensen, S. and Chu, J.-H. (1973). Determination of bulk properties of tropical cloud clusters from large-scale heat and moisture budgets. J. Atmos. Sci., 30, 611--627.

IVT Convergence (Moisture Flux Convergence)

Overview

IVT convergence measures where more moisture flows into a region than out of it. Positive convergence is directly associated with precipitation generation: the excess moisture that converges in the column must be removed, primarily through condensation and rainfall.

Input Variables

Computation

The moisture flux convergence is the negative divergence of the IVT vector field, computed on a spherical Earth with radius $R = 6\,371\,000$ m:

$$\text{MFC} = -\nabla \cdot \overrightarrow{\text{IVT}} = -\left( \frac{\partial \text{IVT}_u}{\partial x} + \frac{\partial \text{IVT}_v}{\partial y} \right)$$

The partial derivatives use the spherical metric terms:

$$\frac{\partial}{\partial x} = \frac{1}{R \cos\varphi} \frac{\partial}{\partial \lambda}$$

$$\frac{\partial}{\partial y} = \frac{1}{R} \frac{\partial}{\partial \varphi}$$

where $\varphi$ is latitude and $\lambda$ is longitude (both in radians).

For the meridional component, the spherical divergence form is used:

$$\frac{\partial \text{IVT}_v}{\partial y} = \frac{1}{R \cos\varphi} \frac{\partial (\text{IVT}_v \cos\varphi)}{\partial \varphi}$$

Derivatives are approximated using central finite differences on the regular lat/lon grid.

Units

IVT convergence has units of kg m$^{-2}$ s$^{-1}$, equivalent to a precipitation rate (mm s$^{-1}$). Positive values indicate moisture convergence (net inflow), negative values indicate divergence (net outflow).

References

• Moisture advection (Wikipedia)
• Banacos, P.C. and Schultz, D.M. (2005). The use of moisture flux convergence in forecasting convective initiation. Wea. Forecasting, 20, 351--366.

Integrated Vapour Transport (IVT)

Overview

Integrated Vapour Transport quantifies the total horizontal moisture flux through the atmospheric column. It is the primary diagnostic for identifying atmospheric rivers and is directly related to orographic and frontal precipitation.

Input Variables

Computation

1. Component-wise Moisture Flux Integration

The eastward and northward IVT components are computed by vertically integrating the moisture flux from the top of the atmosphere ($p_{\text{top}}$) to the surface ($p_{\text{sfc}}$):

$$\text{IVT}u = \frac{1}{g} \int{p_{\text{top}}}^{p_{\text{sfc}}} q \, u \; dp$$

$$\text{IVT}v = \frac{1}{g} \int{p_{\text{top}}}^{p_{\text{sfc}}} q \, v \; dp$$

where $g = 9.80665$ m s$^{-2}$. The integration uses the trapezoidal rule over the available pressure levels.

2. IVT Magnitude

The total IVT magnitude is:

$$|\text{IVT}| = \sqrt{\text{IVT}_u^2 + \text{IVT}_v^2}$$

Units

IVT components and magnitude are in kg m$^{-1}$ s$^{-1}$. Values exceeding approximately 250 kg m$^{-1}$ s$^{-1}$ are commonly used as a threshold for atmospheric river detection.

References

• Atmospheric river (Wikipedia)
• Zhu, Y. and Newell, R.E. (1998). A proposed algorithm for moisture fluxes from atmospheric rivers. Mon. Wea. Rev., 126, 725--735.
• Ralph, F.M. et al. (2004). Satellite and CALJET aircraft observations of atmospheric rivers over the Eastern North Pacific. Mon. Wea. Rev., 132, 1721--1745.

Maximum Updraft Strength

Overview

Maximum updraft strength estimates the strongest upward air motion at each grid point and timestep. It is derived from the model's pressure-level vertical velocity (omega), temperature, and specific humidity.

Input Variables

Computation

1. Virtual Temperature

The virtual temperature accounts for the density effect of water vapour in air:

$$T_v = T \, (1 + 0.61 \, q)$$

2. Air Density

Air density follows from the ideal gas law applied to moist air:

$$\rho = \frac{p}{R_d \, T_v}$$

where $R_d = 287.05$ J kg$^{-1}$ K$^{-1}$ is the specific gas constant for dry air and $p$ is the pressure in Pa.

3. Vertical Velocity

In pressure coordinates, upward motion corresponds to negative omega. The conversion to geometric vertical velocity $w$ (m s$^{-1}$) uses the hydrostatic approximation:

$$w = -\frac{\omega}{\rho \, g}$$

where $g = 9.80665$ m s$^{-2}$. This relationship follows directly from the hydrostatic equation $dp = -\rho \, g \, dz$ by taking the material derivative of both sides ($\omega = dp/dt$, $w = dz/dt$).

4. Maximum Across Pressure Levels

The updraft index is the maximum upward velocity across all available pressure levels at each grid point and timestep:

$$\text{updraft} = \max_p \, w(p)$$

Units

The output has units of m s$^{-1}$. Values above approximately 1 m s$^{-1}$ indicate strong convective updrafts.

References

• Omega equation (Wikipedia)
• Virtual temperature (Wikipedia)
• Primitive equations (Wikipedia)

Mixed-Layer CIN

Overview

Convective Inhibition (CIN) quantifies the energy barrier that an air parcel must overcome before it can rise freely and convect. It corresponds to the negatively buoyant region between the parcel's starting level and the Level of Free Convection (LFC). Larger (more negative) CIN values indicate a stronger "cap" that suppresses convection initiation.

The "mixed-layer" variant averages temperature and moisture over the lowest 100 hPa of the atmosphere to define the starting parcel, representing the well-mixed boundary layer rather than a single surface observation.

Input Variables

Computation

1. Dewpoint Temperature

The dewpoint temperature $T_d$ is derived from the specific humidity and pressure at each level using the inverse of the Clausius--Clapeyron relation (via MetPy's dewpoint_from_specific_humidity).

2. Mixed-Layer Parcel

MetPy's mixed_layer_cape_cin averages $T$ and $T_d$ over the lowest 100 hPa (configurable) to define a representative boundary-layer parcel, then lifts it adiabatically.

3. CIN Integration

CIN is the integral of negative buoyancy between the parcel start and the LFC:

$$\text{CIN} = -\int_{p_{\text{start}}}^{p_{\text{LFC}}} R_d \, (T_{\text{parcel}} - T_{\text{env}}) \, d\ln p$$

where $R_d$ is the specific gas constant for dry air. CIN is negative by convention (the parcel is cooler than the environment in this layer).

Units

CIN is in J kg$^{-1}$ (negative values). Typical thresholds:

References

• Convective inhibition (Wikipedia)
• MetPy: mixed_layer_cape_cin
• MetPy: dewpoint_from_specific_humidity

Mixed-Layer Equilibrium Level (EL)

Overview

The Equilibrium Level (EL) is the pressure level at which a rising convective parcel becomes neutrally buoyant again — its temperature equals the environmental temperature above the Level of Free Convection. The EL marks the approximate top of the convective cloud; overshooting tops may extend somewhat higher due to momentum.

A low EL pressure (e.g. 200 hPa, high altitude) indicates the potential for very tall, deep convection and is associated with more severe thunderstorms. A higher EL pressure (e.g. 400 hPa) indicates shallower convection.

The "mixed-layer" variant uses a parcel whose temperature and moisture are averaged over the lowest 100 hPa of the atmosphere, representing the well-mixed boundary layer.

Input Variables

Computation

1. Dewpoint Temperature

The dewpoint temperature $T_d$ is derived from the specific humidity and pressure at each level using the inverse of the Clausius--Clapeyron relation (via MetPy's dewpoint_from_specific_humidity).

2. Mixed-Layer Parcel

mixed_parcel averages $T$ and $T_d$ over the lowest 100 hPa to produce a representative boundary-layer parcel $(p_{ml}, T_{ml}, T_{d,ml})$.

3. Parcel Temperature Profile

parcel_profile lifts the mixed-layer parcel dry-adiabatically to the LCL, then moist-adiabatically above, producing $T_{\text{parcel}}(p)$.

4. EL Determination

MetPy's el finds the pressure where $T_{\text{parcel}}$ drops back below $T_{\text{env}}$ above the LFC. If no crossing exists (e.g. no convection or the parcel never becomes buoyant), the result is NaN.

Units

EL is reported as a pressure in hPa.

References

• Equilibrium level (Wikipedia)
• MetPy: el
• MetPy: mixed_parcel
• MetPy: parcel_profile

Mixed-Layer Level of Free Convection (LFC)

Overview

The Level of Free Convection (LFC) is the pressure level at which a lifted parcel first becomes warmer than the surrounding environment and begins to accelerate upward on its own. Below the LFC the parcel is negatively buoyant (held back by CIN); above it the parcel is positively buoyant and free convection occurs.

A low LFC (high pressure, e.g. 850 hPa, close to the surface) means convection can initiate with relatively little lift. A high LFC (low pressure, e.g. 600 hPa) means a parcel must be forced through a deep stable layer before it can convect freely.

The "mixed-layer" variant uses a parcel whose temperature and moisture are averaged over the lowest 100 hPa of the atmosphere, representing the well-mixed boundary layer.

Input Variables

Computation

1. Dewpoint Temperature

The dewpoint temperature $T_d$ is derived from the specific humidity and pressure at each level using the inverse of the Clausius--Clapeyron relation (via MetPy's dewpoint_from_specific_humidity).

2. Mixed-Layer Parcel

mixed_parcel averages $T$ and $T_d$ over the lowest 100 hPa to produce a representative boundary-layer parcel $(p_{ml}, T_{ml}, T_{d,ml})$.

3. Parcel Temperature Profile

parcel_profile lifts the mixed-layer parcel dry-adiabatically to the LCL, then moist-adiabatically above, producing $T_{\text{parcel}}(p)$.

4. LFC Determination

MetPy's lfc finds the pressure where $T_{\text{parcel}}$ first exceeds $T_{\text{env}}$ above the LCL. If no crossing exists (stable profile), the result is NaN.

Units

LFC is reported as a pressure in hPa.

References

• Level of free convection (Wikipedia)
• MetPy: lfc
• MetPy: mixed_parcel
• MetPy: parcel_profile

Most Unstable CAPE and CIN

Overview

Convective Available Potential Energy (CAPE) quantifies the buoyant energy available to an air parcel rising from its level of free convection to the equilibrium level. Higher CAPE values indicate greater potential for intense convective updrafts and severe thunderstorms. Convective Inhibition (CIN) measures the energy barrier that must be overcome before free convection can begin.

The "most unstable" variant selects the parcel from the pressure level with the highest equivalent potential temperature, representing the most convectively favourable starting point in the column.

Input Variables

Computation

1. Dewpoint Temperature

The dewpoint temperature $T_d$ is derived from the specific humidity and pressure at each level using the inverse of the Clausius--Clapeyron relation (via MetPy's dewpoint_from_specific_humidity).

2. Most Unstable CAPE and CIN

For each vertical column, MetPy's most_unstable_cape_cin routine:

1. Scans all pressure levels to find the one with the highest equivalent potential temperature $\theta_e$.
2. Lifts that parcel adiabatically and computes the positive and negative buoyancy areas on a skew-T diagram:

$$\text{CAPE} = -\int_{p_{\text{LFC}}}^{p_{\text{EL}}} R_d \, (T_{\text{parcel}} - T_{\text{env}}) \, d\ln p$$

$$\text{CIN} = -\int_{p_{\text{start}}}^{p_{\text{LFC}}} R_d \, (T_{\text{parcel}} - T_{\text{env}}) \, d\ln p$$

where $R_d$ is the specific gas constant for dry air, $p_{\text{LFC}}$ is the level of free convection, and $p_{\text{EL}}$ is the equilibrium level.

Units

Both CAPE and CIN are in J kg$^{-1}$. Typical thresholds:

References

• Convective available potential energy (Wikipedia)
• MetPy: most_unstable_cape_cin
• MetPy: dewpoint_from_specific_humidity

Precipitable Water (PW)

Overview

Precipitable water is the total amount of water vapour contained in a vertical column of atmosphere. It represents the depth of liquid water that would result if all the vapour in the column were condensed and collected at the surface. PW is an important predictor for heavy precipitation events and is routinely used in weather forecasting.

Input Variables

Computation

PW is the mass-weighted vertical integral of specific humidity over the atmospheric column:

$$\text{PW} = \frac{1}{g} \int_{p_{\text{top}}}^{p_{\text{sfc}}} q \; dp$$

where $g = 9.80665$ m s$^{-2}$. The integration uses the trapezoidal rule over the available pressure levels.

Units

PW has units of kg m$^{-2}$, which is numerically equivalent to mm of liquid water depth. Typical values range from a few mm in cold, dry polar air to over 60 mm in tropical environments.

References

• Precipitable water (Wikipedia)