Why Allen's Discriminant Uses a 1.67 Exponent: The Science Behind the Math

The Allen et al. (2011) severe weather discriminant for Australia uses the formula CAPE × (Shear)^1.67. But why 1.67 specifically? The answer is more fascinating than you might think.

The Formula

For severe weather forecasting in Australia, we use the Allen discriminant:

CAPE × (0-6km Shear)^1.67 > 115,000 = Significant severe potential

This formula tells us whether an environment can produce large hail (>5cm), destructive winds (>120 km/h), or severe microbursts.

But why raise shear to the power of 1.67? Why not just multiply CAPE and shear directly?

Three Reasons: Data, Physics, and Turbulence

1. Empirical Optimization (Data-Driven)

Allen et al. analyzed thousands of Australian severe weather events and tested various mathematical formulas to find what best separated severe from non-severe environments.

They likely tried: - Linear: CAPE × Shear^1.0 (just multiply them) - Square root: CAPE × Shear^0.5 - Squared: CAPE × Shear^2.0 - Various fractional exponents: 1.3, 1.5, 1.67, 1.8, etc.

1.67 won because it: - Maximized "hits" (correctly identifying severe storms) - Minimized "false alarms" (incorrectly warning for non-severe conditions) - Worked across diverse Australian environments (coastal, inland, tropical, temperate)

Think of it like tuning a radio - they adjusted the "dial" until they got the clearest signal separating severe from non-severe.

2. Theoretical Physics (Storm Dynamics)

The exponent around 5/3 ≈ 1.67 appears throughout fluid dynamics and turbulence theory.

Kolmogorov's -5/3 Law

In turbulent flows (like thunderstorms!), energy cascades from large eddies to small eddies following:

Energy Spectrum E(k) ∝ k^(-5/3)

Where k is the wavenumber (size scale). This is Kolmogorov's famous -5/3 law from 1941.

What does this mean for storms?

Wind shear creates turbulent eddies that help organize thunderstorms. The way energy transfers across different scales in these eddies follows this 5/3 power law. Using the same exponent in the discriminant captures how shear's organizing influence scales with intensity.

Updraft Dynamics

For a simplified thunderstorm updraft:

Updraft speed from buoyancy: w ∝ √(CAPE)
Rotation from shear-induced vorticity: ω ∝ Shear
Dynamic pressure perturbation from rotation: p' ∝ ω² ∝ Shear²

But the effective enhancement of updraft strength from shear is somewhere between linear and squared - roughly Shear^(3/2 to 2).

The 1.67 exponent sits perfectly in this range, suggesting it captures the nonlinear interaction between buoyancy and shear.

3. Why NOT 2.0? (The Goldilocks Principle)

If shear were squared (^2.0), the formula would be:

CAPE × Shear²

Problems with this: - Over-penalizes low shear environments - Too sensitive to measurement errors in shear - Misses some Australian severe events that occur with moderate CAPE + moderate shear

The 1.67 exponent is the Goldilocks value: - Strong enough to emphasize shear's critical importance - Not so strong that it dominates over CAPE entirely - Balances both ingredients appropriately

Comparison: U.S. vs Australian Approaches

U.S. Approach: Significant Tornado Parameter (STP)

STP = (MLCAPE/1500) × (ESRH/150) × (EBWD/20) × ((2000-MLLCL)/1000) × ((200+MLCIN)/150)

This is a product of normalized ratios - essentially multiplying several factors together with different weights.

Australian Approach: Allen Discriminant

CAPE × Shear^1.67 > 115,000

Much simpler! Just two ingredients with one tuned exponent.

Why the difference?

Australian severe weather: - Lower CAPE thresholds (1000-2000 J/kg vs U.S. 2500+ J/kg) - Shear more discriminating than absolute CAPE values - Coastal initiation mechanisms - Dry continental air masses

In Australia, shear plays a disproportionately large role compared to the U.S., hence the superlinear exponent.

What Does "Superlinear" Mean?

A linear relationship means doubling one thing doubles the result: - If Shear^1.0: Double shear = double the discriminant

A superlinear relationship (exponent > 1) means the effect is amplified: - If Shear^1.67: Double shear = 2^1.67 = 3.17× the discriminant

Why does this matter?

Imagine two environments:

Environment A: - CAPE = 1500 J/kg - Shear = 10 m/s - Discriminant = 1500 × 10^1.67 = 65,000 (below severe threshold)

Environment B: - CAPE = 1500 J/kg (same) - Shear = 20 m/s (doubled) - Discriminant = 1500 × 20^1.67 = 190,000 (well above severe threshold!)

Doubling the shear increased the discriminant by 2.9× - that's the power of the 1.67 exponent!

Real-World Example: Brisbane

Looking at a typical Brisbane severe weather day:

Morning (10 AM local): - MLCAPE = 471 J/kg (weak) - 0-6km Shear = 27.2 m/s (very strong) - Discriminant = 471 × 27.2^1.67 = 117,123 - Result: ✅ EXCEEDS significant severe threshold (115,000)

What this tells us: Even though CAPE is modest (471 J/kg would be considered "weak" in the U.S.), the exceptional shear compensates. The 1.67 exponent ensures we don't miss this setup.

If we used a linear relationship (Shear^1.0): - 471 × 27.2 = 12,811 ❌ (would miss this severe event!)

The superlinear exponent correctly captures the organizing power of strong shear in Australian conditions.

The Dimensional Analysis Problem

Here's something interesting: The units don't make physical sense!

CAPE [J/kg] × Shear^1.67 [m^1.67/s^1.67] = ???

This gives us a weird mixed unit with no direct physical meaning (joules per kilogram times meters to the 1.67 per seconds to the 1.67).

But that's okay!

The discriminant is a diagnostic tool, not a conserved physical quantity like energy or momentum. It's designed to separate severe from non-severe environments, not to represent a fundamental physical property.

Think of it like the "heat index" or "wind chill" - mathematically derived values that don't represent real physical quantities but are useful for decision-making.

Why This Matters for Forecasting

Understanding the 1.67 exponent helps you interpret forecasts better:

1. Shear is MORE Important Than You Think

Because of the superlinear exponent, strong shear can compensate for weak CAPE in Australian conditions. Don't ignore a setup just because CAPE is only 1000 J/kg - check the shear!

2. Forecast Errors in Shear Are Critical

A 20% error in shear measurement creates a 35% error in the discriminant (because 1.2^1.67 ≈ 1.35).

Lesson: Shear forecasts need to be accurate!

3. Australian vs U.S. Climatologies Are Different

The 1.67 exponent was optimized for Australian severe weather events. Using U.S.-based thresholds (which often use linear or squared relationships) would miss many Australian severe storms.

The Bottom Line

The 1.67 exponent in Allen's discriminant is:

✅ Empirically derived from thousands of Australian severe weather events ✅ Physically meaningful - aligns with turbulence theory and storm dynamics ✅ Optimally tuned - better than linear, square root, or squared relationships ✅ Australian-specific - captures the unique role of shear in Australian severe weather

It's a beautiful example of how data-driven optimization can align with fundamental physics, resulting in a simple yet powerful forecasting tool.