---
id: phys-05
title: "Thermal Memory and Channel Persistence"
section: "Spark Growth Physics"
difficulty: "intermediate"
estimated_time: 40
prerequisites: ["phys-03", "phys-04"]
objectives:
  - Understand thermal diffusion time constants for plasma channels
  - Calculate channel persistence times for different diameters
  - Recognize the role of convection in extending channel lifetime
  - Apply thermal memory concepts to QCW vs burst mode operation
tags: ["thermal-diffusion", "convection", "channel-persistence", "time-constants", "ionization-memory"]
---

# Thermal Memory and Channel Persistence

Once formed, a plasma channel doesn't instantly disappear. It has **thermal memory** - the channel stays hot and partially ionized for some time after power is removed. Understanding these timescales is crucial for optimizing operating modes.

## Temperature Regimes

Plasma channels exist in different temperature regimes depending on current and power density:

### Streamers (Cold Plasma)

```
Temperature: T ≈ 1000-3000 K
- Weakly ionized (few % ionization)
- Mostly neutral gas with some ions/electrons
- Purple/blue color (N₂ molecular emission)
- Low conductivity
```

### Leaders (Hot Plasma)

```
Temperature: T ≈ 5000-20,000 K
- Fully ionized plasma
- White/orange color (blackbody + line emission)
- High conductivity
- Approaching temperatures of stellar photospheres!
```

**Temperature comparison:**
- Room temperature: 300 K
- Candle flame: 1500 K
- Thin streamers: 1000-3000 K
- Thick leaders: 5000-20,000 K
- Sun's photosphere: 5800 K

Leaders are literally as hot as the surface of the Sun!

## Thermal Diffusion Time

Heat diffuses radially outward from the hot channel core according to:

```
τ_thermal = d² / (4α_thermal)

where:
  d = channel diameter [m]
  α_thermal ≈ 2×10⁻⁵ m²/s (thermal diffusivity of air)
```

**Physical meaning:** Time for heat to diffuse a distance d through air by conduction.

### Examples for Different Channel Sizes

**Thin streamer (d = 100 μm):**

```
τ = (100×10⁻⁶)² / (4 × 2×10⁻⁵)
  = 10⁻⁸ m² / (8×10⁻⁵ m²/s)
  = 1.25×10⁻⁴ s
  = 0.125 ms
  ≈ 0.1-0.2 ms
```

**Medium channel (d = 2 mm):**

```
τ = (2×10⁻³)² / (4 × 2×10⁻⁵)
  = 4×10⁻⁶ m² / (8×10⁻⁵ m²/s)
  = 0.05 s
  = 50 ms
```

**Thick leader (d = 5 mm):**

```
τ = (5×10⁻³)² / (4 × 2×10⁻⁵)
  = 25×10⁻⁶ m² / (8×10⁻⁵ m²/s)
  = 0.3125 s
  = 312 ms
  ≈ 0.3-0.6 s
```

**Key insight:** Thermal diffusion time scales as d² - thicker channels persist much longer!

## Why Observed Persistence is Longer

Pure thermal diffusion predicts cooling in 0.1-300 ms, but channels persist longer due to additional effects:

### 1. Convection (Buoyancy)

Hot gas is less dense and rises:

```
Buoyancy velocity: v ≈ √(g × d × ΔT/T_amb)

where:
  g = 9.8 m/s² (gravity)
  d = channel diameter
  ΔT = temperature excess above ambient
  T_amb = ambient temperature (≈300 K)
```

**Example: 2 mm channel at ΔT = 10,000 K**

```
v ≈ √(9.8 × 0.002 × 10000/300)
  ≈ √(9.8 × 0.002 × 33.3)
  ≈ √(0.653)
  ≈ 0.81 m/s
```

The hot channel rises at ~0.8 m/s, creating a continuously renewing hot column!

**Effect on persistence:**
- Rising column remains coherent (doesn't diffuse sideways as fast)
- Maintains hot gas path for seconds
- Why Tesla coil sparks leave visible "smoke trails"
- Enhances thermal memory significantly

### 2. Ionization Memory

Even after thermal cooling begins, ions and electrons persist:

```
Recombination time: τ_recomb = 1/(α_recomb × n_e)

where:
  α_recomb ≈ 10⁻¹³ m³/s (recombination coefficient)
  n_e = electron density [m⁻³]

Typical: τ_recomb ≈ 10 μs to 10 ms
```

**Effect on persistence:**
- Channel remains partially ionized after cooling
- Lower resistance than cold air
- Easier to re-ionize than virgin air
- "Memory" of previous discharge path

### 3. Broadened Effective Diameter

Turbulence and mixing increase effective channel size:

```
d_effective > d_initial (due to turbulence)

Larger diameter → longer τ_thermal
```

## Effective Persistence Times

Combining all effects:

**Thin streamers:**
```
Pure thermal: ~0.1-0.2 ms
With convection: ~1-5 ms
Ionization memory: ~0.1-1 ms
Effective persistence: ~1-5 ms
```

**Thick leaders:**
```
Pure thermal: ~50-300 ms
With convection: seconds (buoyant column maintained)
Ionization memory: ~1-10 ms
Effective persistence: seconds
```

**Visual evidence:** High-speed photography shows spark channels glowing and rising for seconds after power is removed.

{image:spark-channel-persistence-sequence}

## QCW Advantage

QCW ramp times (5-20 ms) are designed to exploit channel persistence:

### Timeline of QCW Growth

```
t = 0 ms:
  - Initial streamers form from topload
  - Thin, fast, purple channels
  - Temperature: ~2000 K

t = 0.5-1 ms:
  - Current begins flowing through streamers
  - Joule heating: P = I²R
  - Temperature rising

t = 1-2 ms:
  - Channel heats to 5000+ K
  - Thermal ionization becomes dominant
  - Leader formation begins at base

t = 2-5 ms:
  - Leader established and growing
  - Hot channel maintained by continuous power
  - New growth builds on existing ionization
  - Temperature: 10,000-20,000 K

t = 5-20 ms:
  - Leader continues extending
  - Persistence time >> growth time
  - Channel stays hot entire duration
  - Efficient energy use: no re-ionization needed

t > 20 ms (after ramp ends):
  - Power removed
  - Channel begins cooling
  - Buoyancy carries hot gas upward
  - Visible glow for seconds
```

**Key advantage:** The ramp duration (5-20 ms) is shorter than thermal diffusion time (50+ ms for leaders), so the channel NEVER cools during growth!

### Energy Efficiency Mechanism

**QCW flow:**
```
Energy → Initial ionization (startup cost)
       → Heating to leader temperature
       → Maintaining hot channel (low cost)
       → Extending length (efficient)

Result: Most energy after startup goes into extension
ε_QCW ≈ 5-15 J/m (low, efficient)
```

## Burst Mode Problem

Burst mode pulses are short (50-500 μs) with long gaps (ms):

### Timeline of Burst Mode

```
t = 0 μs:
  - High voltage, cold air
  - Streamer inception

t = 0-100 μs:
  - First pulse (high peak power)
  - Bright streamers form
  - Some heating but limited
  - Temperature reaches ~3000-5000 K

t = 100 μs (pulse ends):
  - Power removed
  - Channel begins cooling immediately
  - Thermal diffusion time ~0.1-0.5 ms for thin channels

t = 0.1-1 ms:
  - Channel cools significantly
  - Temperature drops to ~1000 K
  - Ionization recombines
  - Channel approaching cold air

t = 1-10 ms (between pulses):
  - Next pulse arrives
  - Must re-ionize mostly cold gas
  - Energy wasted on re-heating
  - Little thermal memory remains

Result: Each pulse restarts from nearly cold conditions!
```

**Energy inefficiency mechanism:**

```
Energy → Initial ionization (EVERY pulse)
       → Heating (REPEATED)
       → Brief brightening
       → Cooling (wasted)
       → Re-ionization overhead (high)

Result: Energy into repeated startup, not cumulative growth
ε_burst ≈ 30-100+ J/m (high, inefficient)
```

### Analogy: Boiling Water

**QCW (efficient):**
```
Turn stove on and keep it on
Water heats up once
Maintain boiling continuously
Minimal energy to sustain
```

**Burst (inefficient):**
```
Pulse stove on/off rapidly
Water heats briefly
Water cools between pulses
Must reheat repeatedly
High energy for little sustained boiling
```

---

## WORKED EXAMPLE: Thermal Time Constants

**Given:**
- Channel diameter: d = 2 mm (typical leader)
- Air thermal diffusivity: α = 2×10⁻⁵ m²/s
- Temperature excess: ΔT = 8000 K
- Ambient temperature: T_amb = 300 K

**Find:**
(a) Pure thermal diffusion time
(b) Convection velocity
(c) QCW ramp time recommendation

### Solution

**Part (a): Thermal diffusion time**

```
τ_thermal = d² / (4α)
          = (2×10⁻³)² / (4 × 2×10⁻⁵)
          = 4×10⁻⁶ m² / (8×10⁻⁵ m²/s)
          = 0.05 s
          = 50 ms
```

**Part (b): Convection velocity**

```
v ≈ √(g × d × ΔT/T_amb)
  ≈ √(9.8 × 0.002 × 8000/300)
  ≈ √(9.8 × 0.002 × 26.67)
  ≈ √(0.523)
  ≈ 0.72 m/s
```

Upward velocity of ~0.7 m/s helps maintain hot column.

**Part (c): QCW ramp recommendation**

```
τ_thermal = 50 ms

For efficient QCW operation:
T_ramp << τ_thermal (finish before significant cooling)

Recommended: T_ramp = 0.1 × τ to 0.4 × τ
           = 5-20 ms

Sweet spot: ~10 ms (20% of τ_thermal)
```

**Reasoning:**
- If T_ramp >> τ_thermal (e.g., 200 ms):
  - Channel cools during ramp
  - Must reheat repeatedly
  - Loses QCW efficiency advantage

- If T_ramp << τ_thermal (e.g., 1 ms):
  - May not form thick leaders
  - Closer to burst behavior
  - Doesn't exploit full persistence

- Optimal: T_ramp ≈ 10-20 ms
  - Channel stays hot throughout
  - Leaders form and persist
  - Maximum efficiency

---

## WORKED EXAMPLE: Burst vs QCW Timing

**Given:**
- Burst pulse: 200 μs every 5 ms (5 ms period)
- QCW ramp: 15 ms continuous
- Both use same average power

**Find:**
(a) Why burst is inefficient for thin channels (d = 100 μm)
(b) Why QCW is efficient for thick channels (d = 3 mm)

### Solution

**Part (a): Burst with thin streamers**

```
Channel diameter: d = 100 μm
Thermal time: τ = (100×10⁻⁶)² / (8×10⁻⁵) = 0.125 ms

Timeline:
t = 0: Pulse starts, channel forms
t = 200 μs: Pulse ends (0.2 ms)
           Channel cooling for: 0.125 ms ≈ τ/1
t = 5 ms: Next pulse
         Channel has cooled for: 5 ms = 40 × τ
         COMPLETELY COLD

Result: Each pulse re-ionizes from scratch
        High ε (inefficient)
```

**Part (b): QCW with thick leaders**

```
Channel diameter: d = 3 mm
Thermal time: τ = (3×10⁻³)² / (8×10⁻⁵) = 112 ms

Timeline:
t = 0: Ramp starts, initial streamers
t = 2 ms: Heating → leader formation begins
t = 5 ms: Leader well-established (hot)
t = 15 ms: Ramp ends
          Total time elapsed: 15 ms = 0.13 × τ

Cooling fraction: exp(-15/112) ≈ exp(-0.13) ≈ 0.88

Result: Channel stays at 88% of peak temperature!
        Leader persists throughout ramp
        Low ε (efficient)
```

---

## Key Takeaways

- **Thermal diffusion time**: τ = d²/(4α), scales quadratically with diameter
- **Thin streamers**: τ ≈ 0.1-0.2 ms (fast cooling)
- **Thick leaders**: τ ≈ 50-600 ms (slow cooling)
- **Convection**: Hot gas rises at ~0.5-1 m/s, maintains hot column for seconds
- **Ionization memory**: Partial ionization persists 0.1-10 ms after thermal cooling
- **Effective persistence**: 1-5 ms for streamers, seconds for leaders
- **QCW advantage**: Ramp time (5-20 ms) << leader thermal time (~50+ ms)
- **Burst problem**: Gap between pulses (ms) >> streamer thermal time (~0.1 ms)

## Practice

{exercise:phys-ex-05}

**Problem 1:** A streamer has d = 150 μm. Calculate τ_thermal. If burst pulse width is 500 μs with 10 ms between pulses, does the channel cool significantly?

**Problem 2:** Why do thick leaders persist longer than thin streamers? Give two physical reasons with approximate timescales.

**Problem 3:** A QCW coil uses 25 ms ramps. For a 3 mm diameter leader (τ ≈ 100 ms), estimate the fraction of peak temperature remaining at end of ramp (use exponential cooling approximation).

---
**Next Lesson:** [Streamers vs Leaders](06-streamers-vs-leaders.md)