2.1 The Geometric Logic of a Three-Station Rib

Bailey's algorithm is not simply "three stations in a line." It is a discrete approximation of a second-order spatial derivative (the Laplacian) oriented perpendicular to the mountain barrier. Every geometric property — bearing, segment lengths, balance ratio, station elevations — directly affects the accuracy of that approximation.

Four requirements for a valid rib:

• G1Mountain-normal orientation: Appalachian ridge axis ~030°–210°. Mountain-normal = roughly 100°–120°. PIT→MRB→ACY (101.6°) and ERI→IPT→LGA (103.3°) both qualify.
• G2Sufficient total length: Must span the full CAD perturbation — upstream reference, mesohigh core, downstream reference. Target ~490–550 km for the PA/Appalachian system.
• G3Balance ratio near 1.0: Bailey accepted up to ~1.65. Above ~2.0 the Laplacian becomes unreliable without explicit correction. CRW→RIC→ORF at 3.20 is definitively disqualified.
• G4Correct terrain positioning: A west of barrier crest (unperturbed upstream); B on eastern slope where mesohigh centers; C well east in coastal plain (unperturbed downstream).

2.2 Bailey's Original Southern Network

The spine at 55.1° is roughly perpendicular to the ribs (~120°). This orthogonality is intentional — ribs measure the cross-barrier gradient, the spine measures the along-barrier gradient. Together they give Bailey a 2D picture of the pressure field from surface stations alone.

2.3 Your Mid-Atlantic PA Extension

Why CRW→RIC→ORF is Disqualified (ratio 3.20)

With segments 386.3 km and 120.9 km, the symmetric Laplacian assigns equal weight to a synoptic-scale gradient (386 km) and a mesoscale gradient (121 km). These are physically incommensurable — averaging them produces a meaningless second derivative. The ratio-3.20 rib cannot reliably detect the CAD mesohigh.

Why RIC→BWI→ABE is a Spine (bearing 24.3°)

Despite having the most perfect balance ratio (1.01) in the entire dataset, this route runs nearly north-south — parallel to the Appalachian front, not perpendicular to it. All three stations (RIC, BWI, ABE) sit on the eastern slope and are likely inside the cold pool simultaneously during CAD. Use it as the PA spine for the along-barrier gradient check.

3.1 Why the Variable Choice Matters

Bailey uses dry potential temperature (θ) — temperature and pressure only. CADEX uses virtual potential temperature (θᵥ), which adds a moisture correction. In a typical late-autumn PA CAD event: the cold pool at MDT might have T=4°C, Td=1°C (nearly saturated), while ACY has T=12°C, Td=2°C (warmer but drier). The moisture correction increases the density contrast between the cold-moist pool and the warm-dry air at ACY — making the signal stronger in θᵥ than in dry θ by ~0.27 K. In summer CAD events this difference can be large enough to reverse the sign of the density difference if dry θ is used. This is why the PA algorithm adopts θᵥ.

3.2 Station Pressure vs. SLP

Both algorithms require station pressure (actual pressure at station elevation), not SLP. The SLP reduction introduces errors of 1–3 hPa during CAD events with sharp surface inversions — large enough to contaminate the Laplacian signal.

3.3 Potential Temperature (θ) — Full Derivation

From the first law of thermodynamics applied to a dry adiabatic process, entropy is conserved. Setting entropy equal at the parcel's actual state (T, p) and at the reference state (θ, p₀=1000 hPa):

3.4 The Full θᵥ Derivation Chain

Step 1 — Saturation Vapor Pressure (August-Roche-Magnus)

Step 2 — Actual Vapor Pressure from Dew Point

Step 3 — Mixing Ratio

Step 4 — Virtual Temperature

Step 5 — Virtual Potential Temperature

3.5 Full Variable Summary Table

3.6 Sensitivity Analysis

Combined θᵥ uncertainty: ±0.35–0.45 K per station. Across the rib, Laplacian uncertainty is roughly ±0.7–1.0 K from sensor noise alone — directly informing the threshold values in Part IV.

4.1 What the Laplacian Measures

The Laplacian is the second spatial derivative — it measures the curvature of a scalar field. On a three-station rib, it detects whether the midpoint station is a local maximum (hill-shaped = ridge) or minimum (bowl-shaped = trough) relative to the two endpoints.

4.2 Symmetric vs. Asymmetric Laplacian

The standard symmetric three-point Laplacian assumes equal segment lengths d:

For unequal segments d₁ (A→B) and d₂ (B→C), the asymmetric (generalized) discrete Laplacian is required:

4.3 Worked Example: Pressure Laplacian on PIT→MRB→ACY

Using the Part III CAD scenario: pₛ(PIT)=977.6, pₛ(MRB)=999.9, pₛ(ACY)=1017.5 hPa. Segments d₁=226.7 km, d₂=292.7 km.

4.4 Potential Temperature Laplacian (CPI)

4.5 The Line D Criterion

4.6 The Joint Trigger

4.7 Symmetric vs. Asymmetric — Quantifying the Bias

For PIT→MRB→ACY (ratio 1.29): using the symmetric formula with d = average segment = 259.7 km gives ∇²p = −0.697 hPa/(100km)². The correct asymmetric result is −1.473 hPa/(100km)².

5.1 CADEX Architecture Overview

Where Bailey builds upward from a second-order spatial operator (the Laplacian), CADEX builds from a simpler first-order difference applied to a more physically complete variable (θᵥ). The CADINX is fundamentally a weighted potential temperature difference between the upslope and downslope stations, computed across a two-station pair rather than a three-station rib.

This reflects the different terrain of the White Mountains. In New Hampshire, the ridges are shorter, more irregular, and more densely spaced than in the southern Appalachians. A three-station rib spanning 500 km would cross several independent terrain features. Miller et al. chose shorter, more localized two-point transects that each capture one specific terrain-trapping geometry.

The consequence for PA: CADINX alone is insufficient. It cannot distinguish between a deep, mesohigh-anchored CAD pool and simple cold air advection on a flat pressure gradient. Bailey's Laplacian detects the mesohigh explicitly. The PA algorithm needs both.

5.2 The CADINX Formula (PA Adaptation)

During a PA CAD event, the cold pool occupies the eastern slope. The midslope station (MRB or IPT) is the coldest. The PA midpoint-depression formulation is:

5.3 Intensity Classification

5.4 Worked Example: CADINX on JST→MDT→PHL

5.5 What Each Algorithm Detects — Comparison Matrix

6.1 Bailey's Six-Criteria System

Bailey requires all six criteria simultaneously satisfied for ≥ 6 consecutive hours. A single criterion failing at any timestep resets the event counter.

Criterion 1 — Pressure Laplacian Threshold

Criterion 2 — Potential Temperature Laplacian Threshold

Criterion 3 — Line D (SLP Gradient)

Criterion 4 — Parent Anticyclone

A closed 1024+ hPa isobar must be identifiable within ~2000 km of the CAD region. The center must be positioned north or northeast of the CAD region. For automated implementation: mean SLP over the northern box (42°–50°N, 65°–85°W) > 1020 hPa.

Criterion 5 — 850 hPa Flow Direction

The 850 hPa wind at the nearest radiosonde site (BUF, PIT, or IAD for PA) must not be from the SW quadrant (225°–315°) with speed > 15 knots. Prevents flagging residual cold pools during active warm frontal passages.

Criterion 6 — Six-Hour Persistence

All five criteria above must be simultaneously satisfied for six consecutive hourly observations. A single-hour gap resets the event counter. This criterion excludes transient radiative cold pools, frontal passages, and data gaps. It is Bailey's most conservative filter.

6.2 CADEX's Two-Threshold System

CADEX is structurally simpler: CADINX > minimum threshold AND duration ≥ 3 hours. No parent anticyclone check. No 850 hPa wind check. No pressure Laplacian. The simplicity aids operational use but prevents mechanism classification.

6.3 Head-to-Head: March 15 PA CAD Event

Classic late-season PA CAD event, Ontario high at ~1026 hPa, 18Z conditions:

Both agree on intensity. But CADEX cannot distinguish this from flat-gradient cold advection producing the same θᵥ contrast. Bailey's ∇²p = −1.88 hPa/(100km)² confirms the mesohigh — only Bailey can make that distinction.

7.1 Bailey's Six-Type Taxonomy

Classification is applied after the detection algorithm confirms a CAD event is occurring. It is based on the parent anticyclone characteristics and synoptic flow pattern.

CDEN Classic Dense Cold Air Damming

Parent high: ≥ 1030 hPa, centered north or northeast (New England, Maritime Canada). Strong ∇²p (Z_p < −1.5) and strong CPI (Z_θ > +1.5). Northeasterly surface winds at MRB ≥ 10 kts. Duration ≥ 12 hours.

The deepest, most persistent CAD type. Cold pool typically 1500–3000 m deep. At MRB, surface temperatures may be 10–15°C below the 850 hPa temperature, indicating a strong inversion. Winter precipitation is typically freezing rain or ice pellets — overrunning warm moist air cannot penetrate the cold pool. Occurs 5–15 times per year in the Mid-Atlantic.

CDRY Classic Dry Cold Air Damming

Parent high: ≥ 1025 hPa, centered Ohio Valley or Great Lakes (west of Appalachians). Moderate ∇²p (−1.5 < Z_p < −1.0). Dew point depression at midslope station > 5°C.

Cold air spills eastward over the Appalachians rather than being fed from the north. Cold pool is shallower and drier than CDEN. Winter precipitation is more likely to be snow. More common in early and late winter (October, November, March).

HYBR Hybrid Cold Air Damming

Intermediate between CDEN and CDRY. Parent high moves from northwest to northeast during the event. θᵥ contrast moderate (CADINX_PA −3 to −7 K). Both Laplacians marginal. Most complex type — precipitation type (mixed ice, snow, freezing rain) is notoriously difficult to forecast. Events often begin as CDRY and evolve toward CDEN as the high moves northeast.

WKDR Weak Dry Cold Air Damming

Both Laplacians marginal (Z values −0.5 to −1.0). No well-defined parent anticyclone. Duration 6–12 hours. Often post-frontal. Cold pool is shallow (<500 m), mesohigh barely detectable. Surface temperatures at MDT only 2–4°C below 850 hPa temperature. Erodes quickly with daytime heating.

INST Instantaneous (In-Situ) Cold Air Damming

CADINX_PA criterion met, but ∇²p criterion not met (no detectable mesohigh). Duration ≥ 3 hours. No parent anticyclone required. Cold pool forms in place through radiative cooling, orographic drainage, or evaporative cooling. Common in Pennsylvania's river valleys (Susquehanna, Juniata, Cumberland Valley corridors) on calm, clear nights. This is what CADEX detects and Bailey misses.

UNKN Unknown / Unclassifiable

Meets detection criteria but cannot be unambiguously placed in Types 1–5. Typically assigned when parent anticyclone data is missing, Laplacian values are contradictory across the two ribs, or event duration is exactly at the persistence threshold.

7.2 CADEX's Three-Type Scheme

7.3 Where PA Events Most Commonly Fall

8.1 Why Neither Algorithm Transfers Directly to PA

8.2 The PA-CADINX Algorithm: Complete Equation Set

For each rib (southern: PIT→MRB→ACY; northern: ERI→IPT→LGA) at each hour t:

8.3 Full Dual-Rib Worked Example

Warm-season analog scenario (Ontario high, 1026 hPa, June-type pattern, 06Z):

8.4 Threshold Calibration Strategy

8.5 Final Comparison: Bailey vs. CADEX vs. PA-CADINX