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id	title	status	source_sections	related_topics	key_equations	key_terms	images	examples	open_questions
thermal-physics	Thermal Physics of Spark Channels	established	spark-physics.txt: Part 5 Section 5.4 (lines 281-313)	[streamers-and-leaders energy-and-growth field-thresholds empirical-scaling power-optimization qcw-operation branching-physics equations-and-bounds open-questions]	[thermal-diffusion-time-constant thermal-diffusivity conductivity-from-electron-density power-to-sustain-plasma air-heating-efficiency leader-energy-balance conductance-relaxation]	[tau_thermal alpha d QCW burst_mode streamer leader thermal_ionization vibrational_relaxation ionization_energy_cost power_budget non_equilibrium_plasma aborted_leader dark_period thermal_ratcheting associative_ionization thermal_instability electron_attachment_time conductance_relaxation leader_energy_balance sword_spark driven_leader burst_ceiling frequency_threshold]	[thermal-diffusion-vs-diameter.png qcw-vs-burst-timeline.png]	[spark-growth-timeline.md]	[What are the exact contributions of convection, radiation, and ionization memory to observed persistence? How does radial temperature profile evolve during and after a pulse? Can thermal persistence be modeled with a single effective time constant, or is a multi-exponential required? What is the quantitative role of nitrogen vibrational relaxation in ionization memory? How does altitude (reduced pressure) affect thermal diffusion and persistence times? How many aborted leader attempts typically precede stable inception in QCW mode?]

Thermal Physics of Spark Channels

The thermal behavior of spark channels determines how long a conductive path persists after energy injection ceases, whether a streamer can transition to a leader, and why QCW mode is fundamentally more efficient than burst mode for producing long sparks. Thermal physics bridges the gap between the instantaneous electrical properties described by circuit-topology and the time-evolving behavior that distinguishes operating modes.

Pure Thermal Diffusion

The simplest model of channel cooling is radial heat diffusion from a hot cylinder into ambient air. The characteristic time constant for this process is:

tau_thermal = d^2 / (4 * alpha_thermal)

Where:

tau_thermal is the thermal diffusion time constant [s]
d is the channel diameter [m]
alpha_thermal is the thermal diffusivity of air [m^2/s]

The thermal diffusivity of air at standard conditions:

alpha_thermal = k / (rho_air * c_p) = 2 * 10^-5 m^2/s

Where:

k is the thermal conductivity of air [W/(m*K)]
rho_air is the density of air [kg/m^3]
c_p is the specific heat capacity at constant pressure [J/(kg*K)]

Diffusion Time Constants by Diameter

The quadratic dependence on diameter produces enormous variation:

Channel Diameter	Type	tau_thermal
10 um	Very thin streamer	~1.3 us
100 um	Typical streamer	0.1-0.2 ms
500 um	Thick streamer / thin leader	3 ms
1 mm	Thin leader	12.5 ms
3 mm	Typical leader	110 ms
5 mm	Thick leader	300-600 ms
10 mm	Very thick leader / arc	1.25 s

The image thermal-diffusion-vs-diameter.png plots this relationship, showing the dramatic range from microseconds for thin streamers to seconds for thick leaders.

Key Insight: Diameter Squared

The d^2 dependence is critically important. A channel that is 10x thicker has a thermal time constant that is 100x longer. This creates a powerful positive feedback loop: thicker channels (leaders) persist longer, allowing more energy injection, which further heats and expands the channel, increasing persistence even more. This runaway process is central to the streamers-and-leaders transition.

Beyond Pure Diffusion: Observed Persistence

Actual spark channel persistence is significantly longer than predicted by pure thermal diffusion. Three mechanisms contribute to this extended lifetime:

1. Buoyancy and Convection

Hot gas in the channel is less dense than surrounding air. Buoyancy forces create an upward convection column that:

Maintains a coherent hot gas structure above the initial channel position
Continuously replaces cooled gas at the channel edges with hot gas from the core
Creates a self-sustaining thermal plume that persists well after the electrical discharge ends
Effective for thick channels (leaders) where buoyancy forces exceed viscous drag

For vertical or upward-angled sparks, convection can maintain a hot column for seconds. For horizontal sparks, the column rises and eventually disconnects, reducing persistence.

2. Ionization Memory

Even after the gas temperature drops below the thermal ionization threshold (~5000 K), significant free electron density persists because:

Recombination is slow: Electron-ion recombination in air at moderate densities has time constants of milliseconds to tens of milliseconds
Metastable states: Nitrogen molecules excited to metastable electronic states (lifetimes ~ms) can participate in Penning ionization
Vibrational relaxation: Nitrogen vibrational modes store energy for milliseconds, slowly releasing it to sustain partial ionization
Electron attachment/detachment: Electrons attach to O2 to form O2^- (fast), but thermal detachment returns them when temperature is still elevated (slow)

The net result: a partially ionized channel with moderate conductivity persists longer than the thermal profile alone would suggest. This is especially important in the temperature range 2000-4000 K where thermal ionization is negligible but residual ionization from previous heating still exists.

Quantitative Data: Vibrational Relaxation and Recombination

The qualitative mechanisms above now have quantitative timescales from the gas discharge literature:

Nitrogen vibrational relaxation time at 1 atm: >100 us [Becker et al. 2005, Ch 5, p. 231]

This is much longer than the electron-ion recombination time (~50 us at n_e ~ 10^13 cm^-3; see streamers-and-leaders) and comparable to the thermal diffusion time for thin streamers (~100-200 us for d ~ 100 um). The vibrational energy reservoir in N2 acts as a "battery" that slowly releases energy back into the electron population through superelastic collisions, maintaining a higher effective electron temperature (and hence lower attachment rate) than the translational gas temperature alone would suggest.

Recombination rates for major atmospheric ions: ~2 * 10^-7 cm^3/s at 300 K [Becker et al. 2005, Ch 4, p. 174]

For a streamer with n_e ~ 10^13 cm^-3, the recombination time constant is tau_recomb ~ 1/(2e-7 * 1e13) ~ 50 us. This confirms that recombination is indeed "slow" relative to the attachment time (16 ns) but "fast" relative to observed persistence (1-5 ms). The gap between the recombination time (~50 us) and the observed persistence (~1-5 ms) is filled primarily by the vibrational relaxation mechanism and by metastable states.

This partially answers the open question "What is the quantitative role of nitrogen vibrational relaxation in ionization memory?": vibrational relaxation sustains partial ionization for at least 100 us beyond the cessation of direct energy input, which is comparable to the QCW inter-cycle gap at typical repetition rates.

3. Broadened Effective Channel Diameter

During the discharge, the channel heats and expands. The hot region is broader than the initial conducting core:

During active discharge: conducting core may be 1 mm, but heated region extends to 3-5 mm
After discharge: the broader heated region defines the effective cooling diameter
This increases the effective tau_thermal by a factor of (d_effective/d_core)^2, which can be 10-25x

Effective Persistence Times

Combining all three mechanisms, observed persistence times are:

Thin Streamers: ~1-5 ms

Pure thermal diffusion: 0.1-0.2 ms (for d ~ 100 um)
Ionization memory extends to ~1-3 ms
Minimal buoyancy effect (too thin)
Persistence dominated by ionization/metastable memory
Significance: this is comparable to QCW inter-cycle gaps at 200-1000 Hz repetition rates

Thick Leaders: Seconds

Pure thermal diffusion: 300-600 ms (for d ~ 5 mm)
Buoyancy/convection extends to multiple seconds
Ionization memory further extends conductivity window
Broadened diameter adds another factor of several
Significance: once a leader forms, it can persist through multiple QCW ramp cycles or between closely spaced bursts

Temperature Ranges by Channel Type

The temperature of the conducting channel determines its electrical properties and the dominant ionization mechanism:

Channel Type	Temperature Range	Ionization Mechanism	Plasma Conductivity
Cold streamer	300-1000 K	Photoionization (external UV)	sigma ~ 0.01 S/m
Warm streamer	1000-3000 K	Residual + impact ionization	sigma ~ 0.1-1 S/m
Transition	3000-5000 K	Mixed thermal/residual	sigma ~ 1-10 S/m
Leader	5000-20000 K	Thermal (Saha equation)	sigma ~ 10-100 S/m
Arc	>10000 K	Full thermal equilibrium	sigma ~ 100-10000 S/m

Corresponding Resistivities

Hot leader plasma: rho ~ 1-10 ohm*m
Warm streamer plasma: rho ~ 10-100 ohm*m
Cold streamer: rho ~ 100-1000 ohm*m

These resistivity ranges connect directly to the resistance bounds used in the lumped-model and distributed-model:

R_segment = rho * L_segment / A_cross_section
         = rho * L_segment / (pi * (d/2)^2)

For a 10 cm segment of 1 mm diameter leader at rho = 5 ohm*m:

R = 5 * 0.1 / (pi * (0.5e-3)^2) = 637 kohm

This is within the expected range (see equations-and-bounds).

Critical Temperature for Leader Inception

The temperature tables above show the transition zone at 3000-5000 K. However, the minimum gas temperature for stable leader inception is a nuanced question. Liu (2017) demonstrates through detailed kinetic modeling that:

The gas temperature must significantly exceed 2000 K for stable leader inception, not merely reach it.

The reason is gas dynamics during the transition process:

Streamer heating raises temperature to ~2000-3000 K in the stem (the short channel connecting the streamer base to the electrode)
Heated gas expands, causing pressure-driven outflow that reduces density
Convection losses during expansion can drop the temperature back below the critical threshold
If temperature falls back below ~1500 K during expansion, the stem cools to ambient and the leader attempt aborts

This means the traditional criterion of "T > 2000 K" is necessary but not sufficient. The gas must be heated to a temperature high enough that even after expansion and convection losses, the resulting channel remains above the critical ionization threshold. In practice, this requires initial heating to significantly above 2000 K (perhaps 3000-4000 K).

[Liu 2017, Ch 3, "Streamer-to-leader transition"]

Three-Tier Temperature Threshold

Bazelyan & Raizer (2000) provide quantitative clarity on what happens at each temperature stage, resolving the apparent contradiction between Liu's "2000 K onset" and the "5000 K for leader" figure in the Temperature Ranges table above:

Temperature	What happens	Channel status
>2000 K	Thermal ionization begins; V-T relaxation accelerates; eta_T -> 1.0	Fragile — expansion/convection can abort
>4000 K	Associative ionization (N + O -> NO+ + e) provides field-free electron source; n_e ~ 7*10^12 cm^-3 at equilibrium	Robust — survives without applied field
>5000 K	Electron attachment to O2 virtually nonexistent; complex ion decay + associative ionization fully compensate recombination	Fully self-sustaining — channel persists indefinitely

[Bazelyan & Raizer 2000, Physics-Uspekhi 43(7), pp. 703, 715-716]

The 4000 K threshold is particularly significant: it marks where the channel gains a field-independent ionization source. Below 4000 K, ionization depends on the applied field (which may be intermittent at TC frequencies). Above 4000 K, the channel generates its own electrons through N+O collisions regardless of external conditions.

For TC sparks, the practical criterion is: the stem/channel must reach 4000-5000 K to survive as a leader. The 2000 K onset (Liu) is where thermal runaway begins; 5000 K (Bazelyan) is where the channel becomes truly persistent. The heating efficiency bottleneck (eta_T ~ 10% below 1000 K) is what makes crossing this range so slow.

Thermal Instability Contraction Time

The physical mechanism converting a streamer into a leader channel is ionization-overheating (thermal) instability — current from many streamers contracts into a thin filament:

Contraction time: tau_ins ~ 1 us [Bazelyan & Raizer 2000, p. 704]

This is derived from: tau_ins ~ l/v_s, where l ~ v_s/nu_a ~ 1 cm (conducting streamer length), v_s ~ 10^7 cm/s (streamer velocity near leader tip), and nu_a ~ 10^7 s^-1 (electron attachment frequency).

Critical for TC frequencies: At 200 kHz (period = 5 us), the contraction instability can build up within a single half-cycle. At 50 kHz (period = 20 us), it completes well within one cycle. This means the thermal instability is not the bottleneck for leader formation at TC frequencies — the bottleneck is accumulating enough total energy to heat gas from 300 K to 5000 K, which takes ~1-5 ms even at MW/m power densities (due to the eta_T ~ 10% efficiency).

Connection to aborted leaders: Before stable leader inception, Schlieren photography shows a sequence of:

Streamer burst propagates from electrode
Dark period (~1-5 ms duration) where space charge from the streamer shields the electrode field
Recovery as ions drift and field rebuilds
Next streamer burst (possibly stronger if residual heating persists)
Cycle repeats until one burst deposits enough energy for successful leader inception

Multiple "aborted leaders" (streamer bursts that heat the stem to near-critical temperature but fail to sustain it through expansion) typically precede the first stable leader. Each aborted attempt pre-heats the gas slightly, making the next attempt more likely to succeed — a form of thermal ratcheting.

[Liu 2017, Ch 2-3; Les Renardieres Group 1977, 1981]

Implication for Tesla coils: In QCW mode, the initial 1-3 ms of the ramp may produce several aborted leader attempts before the first stable leader forms at the base. This is consistent with the observation that the first few milliseconds of QCW growth are inefficient (high epsilon) before the leader "catches" and efficiency improves.

Why the Transition Takes So Long: Air Heating Efficiency

A puzzle in the streamer-to-leader transition is why it takes milliseconds despite the enormous power densities in thin streamer channels (up to MW/m — see the Step 2 calculation in streamers-and-leaders). The answer is that most of the electrical energy does not heat the gas:

eta_T = 0.1 + 0.9 * [tanh(T/T_amb - 4) + 1] / 2

[da Silva et al. 2019, after Flitti & Pancheshnyi 2009]

Gas Temperature	eta_T	Meaning
300 K (ambient)	~0.10	Only 10% heats gas; 90% goes to N2 vibrational modes
600 K	~0.10	Still mostly vibrational excitation
1200 K	~0.55	Transition: V-T relaxation accelerating
2000 K	~1.0	Full thermalization: all power heats gas

At ambient temperature, 90% of the electrical energy deposited in a streamer channel goes into exciting N2 vibrational modes rather than raising the translational (gas) temperature. These vibrational modes relax slowly (>100 us at 1 atm, see Section "Ionization Memory" above), so the energy is temporarily "trapped" in internal degrees of freedom.

Physical consequence: The effective heating power in a cold streamer is only ~10% of I^2*R. The 64 MW/m calculated for a 100 um streamer carrying 100 mA (see streamers-and-leaders) produces only ~6.4 MW/m of actual gas heating initially. As the gas warms past ~1000 K, vibrational relaxation accelerates, eta_T rises toward 1.0, and the heating becomes self-reinforcing — this is the thermal runaway that triggers leader formation.

Connection to thermal ratcheting: During aborted leader attempts, each streamer burst deposits energy at only 10% efficiency into gas heating. But the vibrational energy reservoir (90% of the input) slowly thermalizes over ~100 us, providing residual heating that persists into the dark period. Successive bursts benefit from this accumulated vibrational energy, making each attempt more likely to succeed.

QCW Mode: Exploiting Thermal Persistence

QCW (Quasi-Continuous Wave) mode operates with long ramp times of 5-20 ms. This duration is carefully chosen relative to thermal time constants:

Why QCW Works

Ramp duration exceeds streamer persistence (~1-5 ms): The continuous ramp keeps feeding energy before streamers can cool and deionize. Unlike burst mode where channels cool between pulses, QCW never gives the channel time to die.
Continuous energy injection maintains E_tip: The voltage ramp compensates for the capacitive-divider effect, keeping the tip field above the propagation threshold for a longer growth period.
Promotes streamer-to-leader transition: Sustained current through the same channel causes Joule heating (I^2*R). Over several milliseconds, the channel temperature rises from ~1000 K (streamer) through 3000 K (transition) to 5000+ K (leader). See streamers-and-leaders for the detailed mechanism.
Leader channels enable further growth: Once formed, leaders have low resistance, high conductivity, and long persistence. They act as efficient "wires" conducting energy to the tip where new streamers form and themselves transition to leaders.
Thermal accumulation reduces epsilon: As described in energy-and-growth, the accumulated thermal energy makes subsequent extension cheaper: epsilon(t) = epsilon_0 / (1 + alpha * integral(P dt)). QCW's long ramp allows significant accumulation.

QCW Timing Analysis

Typical QCW ramp: 12 ms at 190 kHz

0-2 ms: Voltage builds toward inception. No spark yet.
2-4 ms: Streamers form and begin growing. High epsilon (~15 J/m). Fast propagation but energy-expensive.
4-8 ms: Sustained current heats channels. Transition zone. Temperature rises past 5000 K at base. Epsilon dropping (10 J/m and decreasing).
8-12 ms: Leader-dominated base, streamer tips. Low epsilon (5-8 J/m) for base extension. Overall growth slowing as capacitive-divider attenuates V_tip.

The image qcw-vs-burst-timeline.png shows side-by-side comparison of power, length, and temperature evolution for QCW vs burst modes.

Burst Mode: Fighting Thermal Physics

Burst mode operates with short pulses (50-500 us) separated by gaps that allow significant cooling:

Why Burst Mode Is Inefficient for Length

Channel cools between pulses: At typical burst repetition rates (100-1000 Hz), the gap between pulses is 1-10 ms. Thin streamers (tau ~ 0.1-0.2 ms) are completely cold by the next pulse. Even nascent leaders cool significantly.
Must re-ionize repeatedly: Each burst pulse must re-establish the conductive channel from scratch (or from residual ionization). This re-ionization energy is "wasted" from a length perspective -- it rebuilds what was already created.
High peak current but no thermal accumulation: Burst mode delivers high instantaneous power, creating bright, thick channels. But the energy goes into heating and radiation rather than forward propagation because there is no persistent leader to channel it efficiently.
Voltage collapse limits length: During a single burst, the spark extends until the capacitive-divider drops E_tip below threshold. Because the burst is short (< 1 ms), there is no time for leader formation to mitigate the voltage division. The spark is streamer-dominated throughout.
Net result: high epsilon: All the inefficiencies compound. Burst mode epsilon of 30-100+ J/m means 3-20x more energy per meter of spark compared to QCW.

Burst Mode Advantages

Despite being length-inefficient, burst mode has applications:

Visual impact: High peak current produces bright, thick, visually impressive sparks
Audio modulation: Short bursts enable musical Tesla coils
Simpler control: No voltage ramping required
Lower average power: Shorter duty cycle reduces thermal stress on components

Connection to Plasma Conductivity

The temperature-dependent conductivity of the spark channel plasma connects thermal physics to the circuit models in lumped-model and distributed-model:

sigma(T) varies from ~0.01 S/m (cold) to ~100 S/m (hot arc)
rho(T) = 1/sigma(T) varies from ~100 ohm*m (cold) to ~0.01 ohm*m (hot)

The resistance of a channel segment:

R = rho(T) * L_segment / (pi * (d/2)^2)

As temperature rises:

rho decreases (more conductive)
d increases (thermal expansion)
Both effects decrease R
R moves toward R_opt_power (the hungry streamer self-optimization, see power-optimization)

This temperature-resistance coupling is the physical mechanism behind the hungry streamer principle: the plasma adjusts its temperature (and hence conductivity and diameter) to maximize power extraction.

Conductivity from First Principles

The plasma conductivity can be calculated directly from the electron density:

sigma = n_e * e^2 / (m_e * nu_e-air)

where:
  nu_e-air = N_air * sigma_collision * v_e
  sigma_collision = 1.5 * 10^-15 cm^2    (electron-air collision cross section)
  N_air ~ 2.5 * 10^19 cm^-3 at STP
  v_e ~ 10^6 m/s (mean electron speed at ~1 eV)

[Becker et al. 2005, Ch 5, p. 229]

Example: For n_e = 10^13 cm^-3 in air at STP:

nu_e-air = 2.5e19 * 1.5e-15 * 1e8 ~ 3.75 * 10^12 s^-1
sigma = (1e13 * (1.6e-19)^2) / (9.1e-31 * 3.75e12) ~ 0.075 S/m

This is consistent with the "cold streamer" conductivity range (0.01-0.1 S/m) in the table above, providing an independent cross-check from first principles.

Leader Channel Energy Balance

For a well-developed leader channel at atmospheric pressure, the channel state is quasi-stationary and determined primarily by current. The energy balance between Joule heating and heat conduction gives:

P_L = i * E ~ 4*pi * lambda_m * delta_T

where:
  P_L = power per unit length [W/m]
  lambda_m = thermal conductivity at channel boundary temperature [W/(cm*K)]
  delta_T = T_axis - T_boundary ~ 2*k*T / I_eff (small due to exponential sigma(T))

[Bazelyan & Raizer 2000, "Lightning Physics and Lightning Protection," IOP, Ch 2, pp. 87-88, Eq. 2.46-2.47]

Key result: At i = 1 A, T = 5000 K: lambda_m = 0.02 W/(cm*K), giving P_L ~ 130 W/cm = 13 kW/m and E ~ 130 V/cm. These are within a factor of 2 of experimental leader field measurements.

The physical insight is that the channel temperature is only weakly dependent on power (T ~ P_L^(1/2)) because thermal conductivity rises rapidly with temperature. This means the channel self-regulates: large changes in current (and hence power) produce only modest changes in temperature, which is why the "current channel with a fixed boundary" model works so well.

E/N variation with temperature: As the channel heats from cold, the reduced field E/N drops dramatically:

Gas Temperature	E/N	Dominant Ionization	Implication
1000 K	55 Td	O2 electron impact	High field needed
2500 K	~15 Td	NO electron impact	Threshold drops
4000-4500 K	~3 Td	Associative (N+O->NO+)	Very low field
6000 K	1.5 Td	Thermal equilibrium	Near-zero external field

[Bazelyan & Raizer 2000, Ch 2, p. 86, after calculations in reference 34]

This confirms the three-tier temperature threshold: below 2500 K, high fields are needed for ionization; above 4000 K, the channel maintains itself with minimal field; by 6000 K, external field requirements are negligible.

Connection to TC sparks: A TC leader at 1-3 A carrying current through a 1 m channel requires P_L ~ 13-40 kW/m. At 5 kW total spark power (typical mid-range DRSSTC), only about 0.1-0.4 m of channel can be maintained at full leader conditions simultaneously. This is consistent with the observation that TC sparks have a short leader base transitioning to streamer tips.

Thermal conductivity of air at 5000 K: lambda_m = 0.02 W/(cmK) = 2 W/(mK). This is ~80x higher than at ambient (0.025 W/(m*K)), which is why the leader channel self-regulates: the strong temperature dependence of conductivity acts as negative feedback — higher temperature increases heat losses, stabilizing T_axis.

Maximum Heatable Channel Radius

The energy stored in the electrostatic field of the tip sets a hard upper limit on the channel radius that can be heated to leader temperatures:

pi * r_0max^2 * rho_0 * h(T) = pi * epsilon_0 * U^2 / 2

r_0max = U * sqrt(epsilon_0 / (2 * rho_0 * h(T)))

[Bazelyan & Raizer 2000, "Lightning Physics and Lightning Protection," IOP, Ch 2, p. 67, Eq. 2.34]

where rho_0 = 1.2 * 10^-3 g/cm^3 is cold air density, h(T) is specific enthalpy:

h(5000 K) = 12 kJ/g
h(10000 K) = 48 kJ/g (roughly h ~ T^2)

Tip Voltage	r_0max (cold air)	After expansion to 5000 K	Channel area
100 kV	5.4 um	26 um	minuscule
500 kV	27 um	130 um	thin streamer
1 MV	54 um	260 um	thick streamer

Physical meaning: The channel that the tip's energy can heat to 5000 K is extremely thin — even at 1 MV, only ~0.05 mm cold radius. This is the fundamental reason leaders are thin: the available energy per meter constrains the heatable volume. The channel can only thicken later through sustained current from the circuit (not from tip charge alone).

After thermal expansion to 5000 K (density drops ~5x), the channel expands by ~5x in cross-section, giving the post-expansion radii in the table. These are consistent with measured leader radii of ~0.1-0.3 mm.

Minimum leader radius from diffusion: The ambipolar diffusion coefficient D_a ~ 4 cm^2/s sets a floor on channel filament size: r_0min ~ 30 um. Perturbations smaller than this are smoothed by diffusion before the contraction instability can grow. The probable pre-expansion leader radius is ~100 um. [Bazelyan & Raizer 2000, Ch 2, pp. 71-72]

Conductance Relaxation and Thermal Hysteresis

The channel conductance does not respond instantaneously to current changes. The relaxation model from return stroke physics applies to any spark channel:

dG/dt = [G_st(i) - G(t)] / tau_g

tau_g = 40 us   (current rising, channel heating)
tau_g = 200 us  (current falling, channel cooling)

[Bazelyan & Raizer 2000, Ch 4, pp. 194-195]

See equations-and-bounds Section 14.19 for the full model.

Thermal hysteresis for TC sparks: The 5:1 asymmetry between heating (40 us) and cooling (200 us) time constants creates a ratcheting effect over many RF cycles:

During the high-current half-cycle: conductance increases toward G_st(i_peak) with tau_g = 40 us
During the low-current half-cycle: conductance decreases toward G_st(0) = 0 with tau_g = 200 us
Net effect: conductance ratchets upward over ~10-50 RF cycles (total time ~50-250 us at 200 kHz)

This is the microsecond-timescale mechanism underlying the millisecond-timescale thermal ratcheting described in the "Aborted Leaders" section above. The asymmetric tau_g provides the per-cycle bias that accumulates over many cycles to drive the streamer-to-leader transition.

Community-Validated QCW Thermal Physics

The thermal physics framework above makes specific predictions about how QCW spark behavior should depend on timing, frequency, and power delivery mode. A comprehensive survey of community builder data [Phase 6 QCW community survey, 2026-02-10] provides strong empirical validation of these predictions and reveals several quantitative relationships not previously documented in the framework.

Frequency Threshold for Sword-Like Sparks: 300-600 kHz

Six or more independent builders have converged on a frequency range for producing straight "sword" sparks:

Observer	Observation	Source
Mads Barnkob	"Sword characteristic shows above 400 kHz"	HVF
LabCoatz (Zach Armstrong)	Below 300 kHz: "chaotic and less straight"; above 600 kHz: "more curvy"	Hackaday
Kaizer DRSSTC IV	QCW at ~100 kHz: "swirling" sparks, NOT straight	HVF
Fat Coil builder	"Above 350 kHz, plasma exhibits growth in straight segments"	TCML
Loneoceans SSTC3	Straight sparks at 380-420 kHz	loneoceans.com
Multiple QCW builders	All successful sword-spark builds operate 300-500 kHz	Build survey

Thermal physics explanation: The RF half-period at 400 kHz is 1.25 us. The thermal diffusion time for a 100 um streamer is ~125 us — 100x longer than the RF period. The channel experiences effectively continuous heating with negligible cooling between RF half-cycles. The conductance relaxation time constant (tau_g = 40 us for heating) spans ~16 RF cycles at 400 kHz, ensuring smooth, monotonic conductance increase.

At 50-100 kHz (half-period 5-10 us), thinner streamers (10-50 um, tau ~ 1-30 us) experience significant cooling between cycles. The preferred conductive path diffuses and branches — the channel cannot maintain a single straight track. At >600 kHz, the observation of "curvy" sparks may relate to different physics (skin effect, displacement current dominance, or switching artifacts at extreme frequencies).

Quantitative prediction: At frequency f, the Joule heating rate scales as ~f (more half-cycles per unit time at the same peak current). A channel at 400 kHz receives ~4x more thermal energy per millisecond than at 100 kHz, for the same peak current. This accelerates the temperature ratchet through the critical 2000-5000 K zone.

Steve Ward 80 us Burst Ceiling

Steve Ward's DRSSTC-0.5 measurements provide a clean test of the burst-mode thermal limit:

ON Time	Spark Length	Input Power
~70 us	10-18 inches	33-180 W
>80 us	No additional length	Diminishing returns

Key observation: "Gained almost no spark length after about 80 us of ON period." [Steve Ward, stevehv.4hv.org/DRSSTC.5.htm]

This directly measures the burst-mode streamer growth saturation. The 80 us ceiling is strikingly consistent with the thermal time constant for 100 um streamers: tau_thermal ~ d^2/(4alpha) = (100e-6)^2 / (42e-5) ~ 125 us. After approximately one thermal time constant, channels are cooling as fast as they are being heated — additional energy goes into re-heating decaying channels rather than new forward growth. This is the fundamental wall that QCW overcomes by sustaining drive beyond this timescale.

Connection to framework: Steve Conner independently confirmed this finding: short bursts of high peak power grow sparks more efficiently than long bursts of low peak power (100 us burst outperforms 150 us at the same total energy). This is consistent with the power optimization framework — higher peak power pushes the initial streamer further before the 80 us thermal ceiling is hit.

Three Ramp Regimes

Loneoceans documented three distinct QCW ramp outcomes through controlled variation of ramp duration on his QCW v1.5 build:

Ramp Duration	Visual Result	Framework Interpretation
Too short (<5 ms)	"Gnarly, segmented sparks"	Insufficient time for leader transition; spark operates mostly as streamer
Optimal (~10-20 ms)	Straight sword sparks	Leader forms within first few ms; grows continuously during remainder
Too long (>25 ms)	"Really hot and fat but bushy" without extra length	Leader reaches voltage-limited L_max; excess energy causes branching

The "too long" regime is revealing: Once the leader reaches its maximum length (set by the capacitive divider — see capacitive-divider), additional energy cannot extend it further. The leader channel becomes very hot and thick (more C_sh → more voltage division → further from E_propagation threshold). The excess power must dissipate somewhere, and lateral breakouts from the superheated leader trunk become the path of least resistance. This naturally produces the "fat and bushy" appearance.

The "too short" regime confirms the 0.5-2 ms transition time: Ramps shorter than ~5 ms do not allow the full streamer-to-leader transition. The "segmented" appearance suggests the spark advances as disconnected leader segments that don't merge into a continuous trunk before the ramp ends — consistent with the thermal ratcheting model requiring multiple dark period cycles.

Pulse-Skip Modulation Does Not Produce Full Sword Sparks

Multiple experimenters (Steve Ward, Steve Conner, others on HVF circa 2011) attempted pulse-skip approaches to achieve QCW-like behavior and could not produce full sword sparks.

Steve Ward's explanation: Smoothing ripples from missing pulses would require the coil to store excessive energy between cycles. Sword sparks need "relatively smooth/continuous modulation of the spark power with little ripple."

Implementation detail: In a DRSSTC, pulse-skip is a bridge current-limiting method, not a power-off state. During skip cycles, the H-bridge shorts the primary tank (via GDT inversion or leg inhibit) while IGBTs continue switching synchronized to feedback. Primary current does not drop to zero — it decays gradually through the resonant system's loaded Q. Phase coherence is maintained throughout. The resulting current envelope is a sawtooth bounded by the OCD threshold, delivering approximately constant average power rather than a smoothly ramping profile.

Thermal physics connection: The sawtooth power envelope has per-cycle jitter from the on-off-on switching pattern. True QCW delivers a smooth quadratic power envelope (from a linear voltage ramp: P ~ V^2) — the natural profile for growing a spark against increasing capacitive loading. The jitter from pulse-skip disrupts the clean, monotonic conductance buildup in a single dominant channel. This is a continuum, not a binary threshold: Bresenham-algorithm pulse-density modulation (distributing skip events optimally for a linear ramp) produces sparks that are noticeably more sword-like but still branch — intermediate between coarse pulse-skip and true analog QCW. See qcw-operation Section 2.3 for full details including implementation methods and the distinction from staccato operation.

QCW Timing Summary vs Framework Predictions

Quantity	Framework Value	Community Measurement	Agreement
Streamer-to-leader transition time	0.5-2 ms	Ramp must be >5 ms for swords	Consistent (first few ms spent on aborted leaders)
Burst ceiling (thermal saturation)	tau_thermal ~ 125 us (100 um)	Steve Ward: 80 us ceiling	Good (within factor 1.5)
Optimal QCW ramp	>5x tau_thermal, <L_max/v_growth	10-20 ms	Consistent
Frequency for continuous heating	f >> 1/(2*tau_thermal) ~ 4 kHz	300-600 kHz (well above minimum)	Consistent
Conductance ratcheting period	~10-50 RF cycles at 200 kHz	Sword sparks at >300 kHz, not at <100 kHz	Consistent
QCW growth rate	Not previously predicted	~170 m/s (derived from tau_g)	New data; derivable from framework

Power Budget for Sustaining Plasma

The conductivity-temperature relationship above can be connected to the power required to sustain the plasma at each density level. These power budgets place fundamental constraints on the minimum power per unit length needed to maintain a spark channel.

Average Ionization Energy

The average energy required to produce one electron-ion pair in air is:

E_ionization_avg ~ 14 eV

[Becker et al. 2005, Ch 7, p. 440]

This is higher than the first ionization potential of either N2 (15.6 eV) or O2 (12.1 eV) because some electron energy goes into excitation, dissociation, and vibrational/rotational modes rather than ionization. The effective cost per ionization event includes all these "waste" channels.

Power to Sustain a Given Electron Density

The power required to maintain a steady-state electron density depends on the loss mechanism (attachment or recombination) and the ionization energy cost:

Condition	T_gas	Power to sustain n_e = 10^13	Dominant loss
Cold air	~300 K	1.4 kW/cm^3	Three-body attachment
Hot air	~2000 K	14 kW/cm^3	Thermal dissociation + radiation

[Becker et al. 2005, Ch 7, p. 440; Ch 5, p. 230]

The factor of 10 difference reflects the different loss regimes: in cold air, electron attachment to O2 is the primary loss; in hot air, thermal processes (dissociation, recombination, radiation) dominate.

Linear Power Density for Spark Channels

Converting volumetric power to linear power (per unit length of channel):

For a streamer channel (d = 100 um, A = 7.85 * 10^-6 cm^2):

P_linear = 1.4 kW/cm^3 * 7.85e-6 cm^2 = 0.011 W/cm = 1.1 W/m

For a leader channel (d = 3 mm, A = 0.071 cm^2):

P_linear = 14 kW/cm^3 * 0.071 cm^2 = 1.0 kW/m

Independent Check on Epsilon

These linear power densities provide an independent check on the epsilon values from energy-and-growth. If the channel must sustain ~1 W/m (streamer) to ~1000 W/m (leader) just to maintain ionization, and the channel grows at ~10^6 m/s (streamer) to ~10^3 m/s (leader), then the energy per meter of forward propagation for maintenance alone is:

epsilon_maintenance ~ P_linear / v_propagation

Streamer: ~1 W/m / 10^6 m/s ~ 10^-6 J/m
Leader:   ~1000 W/m / 10^3 m/s ~ 1 J/m

These are lower bounds (maintenance only, not initial ionization, heating, expansion, radiation, or branching). The actual observed epsilon values (5-100 J/m) are 5-100x higher than the leader maintenance minimum, which is consistent: most energy goes into initial channel heating and losses, not steady-state maintenance.

Equilibrium vs. Non-Equilibrium Electron Density

Condition	T_gas (K)	n_e (cm^-3)	Regime
Equilibrium air at 2900 K	2900	4 * 10^10	Very low ionization
Non-equilibrium DC discharge	700-2000	> 10^12	Discharge-sustained
Streamer body	300-1000	10^11 - 10^13	Non-equilibrium
Fully developed spark	~5000+	~10^16	Approaching LTE

[Becker et al. 2005, Ch 5, p. 229; Ch 2, pp. 23, 38]

This table illustrates a critical point: at temperatures below ~3000 K, thermal equilibrium produces negligible ionization (n_e ~ 10^10). The streamer electron densities (10^11-10^13) are sustained entirely by the applied electric field, not by temperature. Only above ~5000 K does thermal ionization become significant, marking the transition to leader behavior and the regime where the hungry streamer principle operates via temperature-conductivity feedback.

Practical Design Implications

For QCW Coil Designers

Ramp time should exceed 5 ms: This ensures enough time for streamer-to-leader transition at the base of the spark
Longer ramps (10-20 ms) are more efficient: But require more total energy and may exceed component thermal limits
Frequency tracking is essential: During the long QCW ramp, the loaded pole shifts significantly as C_sh grows. Driving at the wrong frequency can reduce power delivery by 3-5x (see coupled-resonance)
Match at 50-70% of target length: Because impedance changes dramatically during growth, matching at the midpoint provides best average efficiency

For Burst Mode Coil Designers

Repetition rate affects effective epsilon: Faster repetition (> 500 Hz) allows some thermal memory between bursts, reducing effective epsilon
Single-shot length follows sqrt(E): For isolated bursts with no thermal carryover, Freau's scaling applies (see empirical-scaling)
Peak current determines brightness, not length: Increasing peak current makes brighter sparks but hits the capacitive-divider voltage limit at the same length

Key Relationships

Derives from: First principles of heat transfer (Fourier's law applied to cylindrical geometry)
Interacts with: streamers-and-leaders (thermal physics governs the transition between these regimes)
Interacts with: energy-and-growth (thermal accumulation modifies epsilon over time)
Enables: power-optimization (thermal self-adjustment is the mechanism for hungry streamer optimization)
Constrains: distributed-model (resistance bounds depend on temperature/conductivity ranges)
Explains: empirical-scaling (different scaling laws for QCW vs burst arise from thermal persistence differences)
Connects to: field-thresholds (temperature affects local gas properties and thus field requirements)

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