LCL -CAPE -CIN -z 0w 0

Rising Air Parcel Lab — 2D Sounding and Phase Plane

The 2D companion of the 3D valley scene. Here you see what 3D hides: the parcel's cross-section with its forces, the temperature sounding with CAPE and CIN shaded, the buoyancy profile and the height-versus-updraft phase plane.

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Educational parcel model. One undiluted parcel in a hydrostatic atmosphere: no mixing, no water loading, no wind shear. Real cloud tops and updrafts are lower than parcel theory says (updrafts about half). Not a forecast.

Weather

Air and ground

Live readout

Buoyancy of the parcel (T parcel - T air)

Orange area above zero = CAPE (energy gained), blue below zero = CIN (energy to be paid first). Dot = parcel now.

Phase plane: height and updraft speed

Kinetic energy w²/2 = w0²/2 (w0 = trigger kick) + ∫B dz. The curve ends where it hits w = 0: that is where the thermal stops. Dashed: the CAPE-only upper bound sqrt(2 CAPE) from the 3D page.

Experiment: condensation level against dew point

Sweeps the surface dew point from 0 C to just below the air temperature and compares the numerical parcel lifting with the textbook rule 125 m per degree of dew-point depression and with the Bolton closed form.

Independent numerical checks

Controls

Sliders change the air (temperature, dew point, lapse rate) and the parcel (ground heating excess, optional inversion lid). Presets load the same sets as the 3D page.
The thermal starts with the Trigger kick and then follows the buoyancy; Playback speed is simulated seconds per real second.
Arrows on the parcel: red = buoyancy (x8), blue down = weight (1 g, for scale).

The science

Dry ascent. An unsaturated parcel keeps its potential temperature, so T = T₀ (p/p₀)^R/c_p, about 9.8 K/km, while its mixing ratio r is fixed. Pressure follows dp/dz = −p g/(R T_env).

Lifting condensation level. The parcel saturates when r = r_s(T,p) = 0.622 e_s/(p − e_s) with e_s = 6.112 exp(17.67 T/(T + 243.5)) hPa (Bolton 1980). Rule of thumb: LCL ≈ 125 m × (T − T_d).

Moist ascent. Latent heat shrinks the lapse rate to Γ_m = g (1 + L r_s/(R T)) / (c_p + L² r_s ε/(R T²)) (3–6 K/km low down). The equivalent potential temperature (Bolton 1980, eq. 43) stays constant: one of the checks.

Buoyancy, CAPE, CIN. B = g (T_parcel − T_env)/T_env. LFC: where B first turns positive above the LCL; EL: where it turns negative again. CAPE = ∫_LFC^EL B dz; CIN is the negative area below the LFC. Upper bound w_max = √(2 CAPE).

Animated thermal [illustrative]. The moving parcel obeys w dw/dz = B from an adjustable starting kick (default 10 m/s, i.e. 50 J/kg), so it needs kinetic energy ½w0² larger than the CIN to get through to the LFC; with too small a kick the thermal stalls and no deep cloud forms. Its on-screen rise speed uses the true w (at least 1 m/s so it keeps moving). The cloud is drawn from the LCL up to the highest point the parcel reached; cloud shape and rain streaks (cloud deeper than 4 km) are artistic.

Simplifications. Reversible moist adiabat (no fallout), virtual temperature and water loading neglected, constant parcel moisture, piecewise-linear environment with an optional inversion and an isothermal stratosphere above 11 km. Real updrafts are about half of the undiluted value because of mixing.