modelling Level 4

Thunderstorm Dynamics and Severe Weather

What atmospheric conditions produce severe thunderstorms? Convective Available Potential Energy (CAPE) and wind shear determine storm intensity and structure. This model derives CAPE equations, implements updraft velocity calculations, demonstrates supercell dynamics, and predicts severe weather occurrence.

Prerequisites: cape, updraft velocity, supercell dynamics, shear vorticity

27 Feb 2026 Updated 12 Mar 2026 11 min read

1. The Question

Will today’s atmospheric conditions produce tornadoes?

Thunderstorm requirements:

Moisture: High humidity (fuel)
Instability: Warm surface, cool aloft (buoyancy)
Lift: Trigger mechanism (front, convergence, terrain)
Shear: (For organization and severity)

Storm types:

Single-cell: Short-lived (30-60 min), pulse storms
Multicell: Cluster, longer-lived (2-4 hours)
Supercell: Rotating updraft, most severe (hours)

Severe criteria (USA):

Hail ≥1 inch (2.5 cm)
Wind ≥58 mph (93 km/h, 50 kt)
Tornado (any intensity)

Applications:

Severe weather forecasting
Warning lead time
Aviation safety
Agriculture (hail damage)
Insurance risk assessment

2. The Conceptual Model

CAPE (Convective Available Potential Energy)

Energy available for updrafts:

\[CAPE = g \int_{LFC}^{EL} \frac{T_v' - T_{v,env}}{T_{v,env}} dz\]

Where:

$g$ = 9.81 m/s²
$LFC$ = level of free convection (m)
$EL$ = equilibrium level (m)
$T_v’$ = virtual temperature of parcel (K)
$T_{v,env}$ = environmental virtual temperature (K)

Virtual temperature:

\[T_v = T (1 + 0.61 q)\]

Where $q$ = mixing ratio (kg/kg)

Units: J/kg (energy per unit mass)

Typical values:

CAPE < 1000: Weak instability
CAPE = 1000-2500: Moderate
CAPE = 2500-4000: Strong
CAPE > 4000: Extreme (supercell environment)

Maximum updraft velocity:

\[w_{max} = \sqrt{2 \times CAPE}\]

Example: CAPE = 3000 J/kg

\[w_{max} = \sqrt{6000} = 77 \text{ m/s}\]

Extreme updraft!

Wind Shear

Change in wind with height:

Bulk shear (0-6 km):

\[S = \sqrt{(u_6 - u_0)^2 + (v_6 - v_0)^2}\]

Where $u, v$ = wind components at surface (0) and 6 km.

Critical thresholds:

S < 10 m/s: Disorganized storms
S = 10-20 m/s: Organized multicells
S > 20 m/s: Supercells likely

Storm-relative helicity (SRH):

Measures streamwise vorticity:

\[SRH = \int_0^{z} (V - C) \cdot \frac{\partial V}{\partial z} dz\]

Where:

$V$ = environmental wind vector
$C$ = storm motion vector

SRH > 150 m²/s²: Tornadic supercells favored

Supercell Structure

Rotating updraft:

Mesocyclone (2-10 km diameter, rotation).

Key features:

Updraft: 20-50 m/s, tilted (shear)
Downdraft: Rear-flank, forward-flank
Hook echo: Radar signature (tornado possible)
Overshooting top: Penetrates tropopause

Vorticity sources:

Environmental shear → horizontal vorticity

Updraft tilts → vertical vorticity (rotation)

3. Building the Mathematical Model

Parcel Theory

Lifted parcel:

Starts at surface with temperature $T_0$, pressure $p_0$.

Dry adiabatic ascent:

\[T = T_0 \left(\frac{p}{p_0}\right)^{R/c_p}\]

Where:

$R/c_p = 0.286$ (dry air)

Condensation occurs at LCL (lifting condensation level):

\[LCL \approx 125 (T_0 - T_d)\]

Where $T_d$ = dew point temperature (°C).

Above LCL:

Moist adiabatic (slower cooling, ~6°C/km vs 10°C/km dry).

Buoyancy:

\[B = g \frac{T_v' - T_{v,env}}{T_{v,env}}\]

Positive B → acceleration upward.

Updraft Equation

Vertical momentum:

\[\frac{dw}{dt} = B - \frac{1}{\rho} \frac{dp}{dz} - \varepsilon w\]

Where:

$w$ = vertical velocity
$B$ = buoyancy
$\varepsilon$ = entrainment/drag coefficient

Simplified (neglecting pressure gradient, drag):

\[w^2 = 2 \times CAPE\]

More realistic (with entrainment):

\[w^2 = 2 \times CAPE \times (1 - \varepsilon)\]

Typical $\varepsilon = 0.3-0.5$

Actual updrafts: 50-70% of theoretical maximum.

Hail Growth

Embryo ascent in updraft:

Hailstone grows by accretion (collecting supercooled droplets).

Terminal velocity balance:

\[w = V_t\]

Where $V_t$ = hailstone fall speed.

For spherical hailstone:

\[V_t = \sqrt{\frac{8 r g \rho_h}{3 C_d \rho_a}}\]

Where:

$r$ = radius (m)
$\rho_h$ = 900 kg/m³ (ice density)
$C_d$ = 0.6 (drag coefficient)
$\rho_a$ = air density

Updraft required for large hail:

1 inch (2.5 cm): $w \approx 25$ m/s
2 inch (5 cm): $w \approx 35$ m/s
4 inch (10 cm): $w \approx 50$ m/s

Extreme CAPE enables giant hail.

4. Worked Example by Hand

Problem: Calculate CAPE and predict severe weather potential.

Sounding data (simplified):

Surface (1000 mb):

Temperature: 30°C
Dew point: 24°C

500 mb (5.5 km):

Temperature: -10°C

Environmental lapse rate: 7°C/km (average)

Wind profile:

Surface: 180°/10 kt
6 km: 240°/40 kt

Calculate LCL, CAPE, bulk shear, severe potential.

Solution

Step 1: LCL

\[LCL = 125 \times (30 - 24) = 125 \times 6 = 750 \text{ m}\]

Step 2: Parcel path

Assume moist adiabatic above LCL: ~6°C/km

At 5.5 km:

From surface (30°C):

Dry ascent to 0.75 km: $T = 30 - 10(0.75) = 22.5°C$
Moist ascent 4.75 km: $T = 22.5 - 6(4.75) = -6.0°C$

Parcel temperature at 5.5 km: -6°C
Environment: -10°C

Buoyancy: Parcel warmer by 4°C!

Step 3: CAPE (simplified)

Assume average buoyancy from LFC (1 km) to EL (12 km):

Average $\Delta T = 3°C$, depth = 11 km

\[CAPE = 9.81 \times \frac{3}{273} \times 11000 = 9.81 \times 0.011 \times 11000 = 1187 \text{ J/kg}\]

Moderate CAPE

Step 4: Maximum updraft

\[w_{max} = \sqrt{2 \times 1187} = \sqrt{2374} = 48.7 \text{ m/s}\]

Strong updrafts possible

Step 5: Bulk shear

Wind at surface: $u_0 = 10 \sin(180°) = 0$, $v_0 = 10 \cos(180°) = -10$ kt

Wind at 6 km: $u_6 = 40 \sin(240°) = -34.6$ kt, $v_6 = 40 \cos(240°) = -20$ kt

\[S = \sqrt{(-34.6 - 0)^2 + (-20 - (-10))^2} = \sqrt{1197 + 100} = \sqrt{1297} = 36 \text{ kt} = 18.5 \text{ m/s}\]

Moderate-strong shear

Step 6: Severe weather potential

CAPE = 1187 J/kg (moderate)
Shear = 18.5 m/s (organized storms)
Combination: Severe thunderstorms likely
Hail (updrafts support 1-2 inch)
Wind (downdrafts, shear)
Tornadoes possible (if SRH also elevated)

Severe thunderstorm watch warranted.

5. Computational Implementation

Below is an interactive severe weather parameter simulator.

Surface temperature (°C): 30 Surface dew point (°C): 22 Mid-level temperature (°C): -10 Bulk shear 0-6km (m/s): 15

CAPE: -- J/kg

Max updraft: -- m/s

Severe potential: --

Storm type: --

Observations:

CAPE increases with surface warmth and mid-level cooling
High moisture (small T-Td spread) increases CAPE
Shear determines storm organization and severity
Combination of high CAPE and strong shear = supercells
Hodograph shows wind turning with height (directional shear)
Straight hodograph = less favorable for rotation

Key insights:

Both instability (CAPE) and shear required for severe weather
Supercells need 2500+ J/kg CAPE and 20+ m/s shear
Maximum updraft velocity scales with square root of CAPE
Storm type predictable from environmental parameters

6. Interpretation

Tornado Forecasting

Ingredients:

CAPE: Energy for updrafts
Shear: Rotation potential (SRH)
LCL: Low cloud base (<1500 m favors tornadoes)
Capping: Inhibition layer (prevents early convection, stores energy)

Significant Tornado Parameter (STP):

\[STP = \frac{CAPE}{1500} \times \frac{SRH}{150} \times \frac{2000 - LCL}{1000} \times \frac{S}{20}\]

STP > 1: Significant tornado (EF2+) environment
STP > 3: Violent tornado possible

May 20, 2013 Moore, OK:

CAPE: 3500 J/kg
SRH: 400 m²/s²
Shear: 25 m/s
STP: ~6
Result: EF5 tornado

Hail Forecasting

MESH (Maximum Expected Size of Hail):

Based on radar-derived maximum reflectivity and height.

Environmental indicators:

CAPE > 2000 J/kg
Strong mid-level winds (advect hail)
Wet-bulb zero height ~2.5-3.5 km (growth zone)

Record hail: 8 inch diameter (20 cm), South Dakota 2010.

Required updrafts ~60+ m/s (CAPE >5000 J/kg).

Aviation Hazards

Thunderstorms dangerous for aircraft:

Turbulence: Updrafts/downdrafts exceed aircraft capability
Icing: Supercooled droplets
Lightning: Electronics damage, structural
Hail: Airframe damage, windscreen cracks

Avoidance: 20+ nautical miles from severe storms

Microbursts:

Localized downdraft (< 4 km diameter).

Surface wind divergence: 100+ kt possible.

Windshear: Fatal on takeoff/landing.

7. What Could Go Wrong?

Capping Inversion Too Strong

Warm layer aloft prevents lifting to LFC.

CAPE exists but storms never initiate.

Forecaster dilemma:

Severe environment, but no storms (false alarm).

Solution: Monitor for triggers (fronts, outflow boundaries).

Mesoscale Convective Systems

Organized complex of storms.

Different dynamics than isolated supercells.

MCS: Squall line, bow echo, derecho

Challenges:

Evolving structure
Complex outflow interactions
Rapid changes

Derechos: Widespread wind damage (>100 mph possible).

Storm Mergers

Multiple storms interact.

Can intensify or weaken depending on configuration.

Example - Fujiwara effect:

Storms orbit each other, merge.

Unpredictable evolution.

Tornadic Debris Signature

Radar detects lofted debris, not tornado itself.

TDS = tornado confirmed

But: Tornado may dissipate before reaching target.

Warning verification challenge.

8. Extension: Dual-Polarization Radar

Polarimetric variables:

ZDR (differential reflectivity):

Shape information
Large hail (ZDR < 0 dB, tumbling)

KDP (specific differential phase):

Rain rate, hail discrimination

ρHV (correlation coefficient):

Mixed hydrometeors
Debris (ρHV < 0.90)

Improved:

Hail detection
Heavy rain estimation
Tornado warning (TDS)

9. Math Refresher: Hydrostatic Balance

Pressure Gradient

Vertical pressure gradient:

\[\frac{dp}{dz} = -\rho g\]

Hydrostatic equation

Buoyancy:

Deviation from hydrostatic produces vertical acceleration.

\[\frac{dw}{dt} = -\frac{1}{\rho} \frac{dp}{dz} - g = B\]

Where $B$ = buoyancy force.

Summary

CAPE quantifies energy available for convective updrafts from surface to equilibrium level
Maximum updraft velocity scales as square root of 2×CAPE typically 50-70% of theoretical
Wind shear determines storm organization with >20 m/s favoring supercells
Severe weather requires combination of instability (CAPE >1500 J/kg) and shear (>15 m/s)
Supercells feature rotating updrafts capable of producing large hail and tornadoes
Significant Tornado Parameter combines CAPE, shear, LCL, and SRH for tornado forecasting
Hail growth requires strong updrafts balancing terminal velocity of growing stones
Applications span severe weather prediction, aviation safety, insurance risk assessment
Dual-polarization radar provides enhanced hydrometeor classification and debris detection
Critical tool for warning lead time and public safety during severe convective events