plasma-expert/context/streamers-and-leaders.md

id	title	status	source_sections	related_topics	key_equations	key_terms	images	examples	open_questions
streamers-and-leaders	Streamer and Leader Discharge Physics	established	spark-physics.txt: Part 5 Section 5.5 (lines 314-337)	[thermal-physics energy-and-growth field-thresholds capacitive-divider power-optimization qcw-operation branching-physics empirical-scaling distributed-model equations-and-bounds open-questions]	[growth-rate epsilon-thermal-refinement]	[streamer leader transition thermal_ionization photoionization Joule_heating epsilon hungry_streamer QCW burst_mode electron_density ionization_front recombination corona_to_spark_transition specific_energy_density microdischarge dark_period aborted_leader Gallimberti_model stem thermal_ratcheting leader_velocity electron_attachment_time Bazelyan_VI stepped_leader continuous_leader dart_leader leader_formation_threshold conductance_relaxation]	[streamers-vs-leaders-photos.png streamer-to-leader-transition-sequence.png]	[spark-growth-timeline.md]	[What is the exact current threshold for streamer-to-leader transition? How does branching factor differ between streamers and leaders? What determines the number of streamer branches at a leader tip? Can the transition be modeled as a sharp threshold or is it gradual? How does ambient humidity quantitatively affect the streamer-leader transition current? What is the quantitative role of photo-ionization vs. electron drift in positive streamer propagation in Tesla coil sparks? How many dark period cycles typically precede stable leader inception at TC voltages and electrode geometries? Can Gallimberti's transition model be corrected with empirical factors for TC conditions, or is a full kinetic model required?]

Streamer and Leader Discharge Physics

Tesla coil sparks are not a single phenomenon but a composite of two fundamentally different discharge types: streamers and leaders. Understanding the distinction between these types, and the transition from one to the other, is essential for explaining why different operating modes produce dramatically different spark lengths and efficiencies. This topic connects the microscopic plasma physics to the macroscopic circuit behavior described by energy-and-growth and power-optimization.

Streamer Discharges

Streamers are the initial, ephemeral discharge channels that form when the electric field at the topload exceeds the inception threshold (see field-thresholds).

Physical Properties

Property	Value	Notes
Diameter	10-100 um	Very thin filaments
Propagation speed	~10^6 m/s	Extremely fast (fraction of speed of light)
Current	milliamperes (mA)	Low current per channel
Propagation mechanism	Photoionization	UV photons ionize gas ahead of channel
Temperature	300-3000 K	Minimal heating (non-thermal plasma)
Resistance	Very high	rho ~ 10-100 ohm*m
Persistence	Microseconds (thermal)	Pure diffusion tau ~ 1-100 us for d ~ 10-100 um
Effective persistence	1-5 ms	Extended by ionization memory
Visual appearance	Purple/blue	Nitrogen second positive band emission
Branching	Highly branched	Many fine filaments
Energy per meter (epsilon)	High (30-100+ J/m)	Inefficient for forward propagation

Electron Density and Internal Structure

The properties table above gives macroscopic observables. At the microscopic level, streamers have well-characterized internal structure from both simulation and measurement:

Ionization front at the streamer head: The active ionization zone at the leading edge of a streamer has a thickness of approximately 0.015 cm (~150 um). This thin front is where electron avalanche multiplication is occurring -- it is the "engine" of the streamer. [Becker et al. 2005, Ch 2, p. 37]

Electron density in the streamer body:

Region	n_e (cm^-3)	Notes
Outer boundary (visible edge)	~10^11	Diffuse boundary of ionized region
Inner body (conducting core)	>10^13	Main current-carrying region
Fully developed spark channel	~10^16	After corona-to-spark transition

[Becker et al. 2005, Ch 2, pp. 37-38]

These densities are in the non-equilibrium regime: the electron temperature (~3 eV, ~35,000 K) is far above the gas temperature (~300-1000 K). See field-thresholds Section 1.4 for the breakdown physics behind this non-equilibrium state.

Connection to conductivity: The conductivity of the streamer body can be estimated from these electron densities using:

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

For n_e ~ 10^13 cm^-3 in warm air, this yields sigma ~ 0.01-0.1 S/m, consistent with the "cold streamer" range in thermal-physics. For a fully developed spark channel at n_e ~ 10^16, sigma reaches ~10-100 S/m (leader/arc range). See equations-and-bounds Section 14.6 for the full conductivity calculation.

Microdischarge reference properties (individual streamer filaments resemble atmospheric microdischarges):

Property	Value	Notes
Duration	1-10 ns	Single filament lifetime
Filament radius	~100 um	Consistent with streamer diameter range above
Peak current	0.1 A	Per individual filament
Current density	100-1000 A/cm^2	High due to small cross-section
Electron density	10^14 - 10^15 cm^-3	Higher than sustained streamer body
Electron energy	1-10 eV	Non-equilibrium
Degree of ionization	~10^-4	Very weakly ionized

[Becker et al. 2005, Ch 6, Table 6.2.1]

These microdischarge properties are relevant because individual Tesla coil streamers resemble atmospheric microdischarges in many respects -- similar diameters, current densities, and electron densities.

Streamer Velocity and Tip Physics

Streamer velocity is set by the ionization wave dynamics at the tip. The tip maintains a nearly constant maximum field of E_m ~ 150-170 kV/cm through a self-regulation mechanism, independent of the applied voltage.

V_s = v_im * r_m / [(2k-1) * ln(n_c/n_0)]

where:
  v_im = 1.1 * 10^10 s^-1     (ionization frequency at E_m)
  r_m = U_t / (2 * E_m)        (tip radius, grows with tip potential)
  k = 2.5                      (power index for ionization rate vs field)
  n_c = 9 * 10^13 cm^-3        (initial plasma density, INDEPENDENT of U_t)
  n_0 ~ 10^5 - 10^6 cm^-3      (seed electron density)

[Bazelyan & Raizer 2000, "Lightning Physics and Lightning Protection," IOP, Ch 2, pp. 41-43, Eq. 2.3, 2.6]

Key result: V_s is proportional to U_t. Since r_m ~ U_t at constant E_m, streamer velocity scales linearly with tip potential. Higher secondary voltage = faster streamers. Numerical example: at U_t = 34 kV, r_m = 0.1 cm, V_s = 1.7 * 10^6 m/s.

Minimum streamer velocity: V_s_min = (1.5-2) * 10^5 m/s, occurring at U_t = 5-8 kV. Streamers slower than this have never been observed — they cannot sustain propagation against attachment losses.

Voltage-independent initial plasma density: n_c = 9 * 10^13 cm^-3 is a fundamental constant of air breakdown at atmospheric pressure (set by E_m alone). This means a streamer channel has the same electron density regardless of whether it was created by a 10 kV or a 500 kV potential. What changes with voltage is the streamer velocity, diameter, and length — not the local plasma density.

Maximum Streamer Length

The maximum length a streamer can reach is set by the balance between tip potential and the critical propagation field:

l_max = (U_t - U_0) / E_cr ~ U_t / E_cr   (when external potential U_0 is small)

E_cr(+) = 4.5-5 kV/cm  (positive streamers in air)
E_cr(-) ~ 10 kV/cm     (negative streamers)

[Bazelyan & Raizer 2000, Ch 2, pp. 58-59, Eq. 2.32]

Tip Voltage	l_max (positive)	l_max (no attachment)	l_max (no losses)
250 kV	0.39 m	—	—
500 kV	0.94 m	1.25 m	3.0 m
750 kV	1.42 m	—	—

The "no attachment" and "no losses" columns show the enormous potential for longer streamers in pre-heated channels where electron attachment is suppressed. A TC spark re-using a thermally persistent channel (where T > 2000 K reduces attachment by orders of magnitude) can extend streamers far beyond the cold-air limit — this is the fundamental reason thermal persistence matters for TC spark length.

TC implication: At V_top = 400 kV, maximum cold-air streamer length is ~0.8-0.9 m. Beyond this, leader formation is required. This is consistent with the observation that burst-mode DRSSTCs plateau at ~1 m regardless of power.

Single Streamer Heating: Negligible

The energy deposited by a single streamer passage is fundamentally limited:

Energy density per passage: W = epsilon_0 * E_m^2 / 2 = 2.6 * 10^-2 J/cm^3
Temperature rise: Delta_T < W / c_v = 3 K

[Bazelyan & Raizer 2000, Ch 2, pp. 49-50, Eq. 2.17]

This is an essential result: a single streamer deposits only enough energy to raise the gas temperature by ~3 K. Leader formation requires heating to 5000+ K — which means the channel must accumulate energy from hundreds of streamer passages or from sustained current flow. This is why the leader mechanism (concentrating many streamers' worth of current through a single contracted filament) is necessary for TC spark growth beyond the streamer limit.

Increasing the applied voltage does NOT increase the specific energy deposition because the channel cross-section grows as U^2 while the energy scales as U^2 — the energy density (J/cm^3) remains ~epsilon_0 * E_m^2 / 2 regardless of voltage.

Propagation Mechanism: Photoionization

Streamers propagate via a fundamentally non-thermal mechanism:

1. Strong field at tip: The thin, pointed streamer tip concentrates the electric field to very high values (10-100 kV/cm)
2. Electron avalanche: Free electrons in the high-field region accelerate and ionize gas molecules through impact
3. UV emission: Excited nitrogen molecules emit UV photons (primarily in 98-102.5 nm range)
4. Photoionization: These UV photons ionize oxygen molecules up to ~1 mm ahead of the streamer tip, creating seed electrons
5. New avalanche: Seed electrons start new avalanches, extending the streamer
6. Self-propagating: Steps 1-5 repeat at ~10^6 m/s

Key physics: the propagation is electromagnetic (photon-mediated), not thermal. The gas behind the streamer tip is barely heated. This is why streamers can propagate so fast -- they do not wait for thermal processes.

The Photo-Ionization Debate

The role of photo-ionization in positive streamer propagation is well-established experimentally but quantitatively debated in the simulation literature:

• Positive streamers (propagating away from the anode/topload) require a source of seed electrons ahead of the streamer tip. Photo-ionization by UV from excited N2 molecules ionizing O2 molecules is the most widely accepted mechanism. In simulations, positive streamers will not propagate if both photo-ionization and background ionization are set to zero. [Becker et al. 2005, Ch 2, pp. 51-52; Morrow & Lowke 1995]

• Negative streamers (propagating toward the anode) can propagate without photo-ionization because electrons naturally drift ahead of the streamer tip. Simulations have reproduced negative streamer propagation and even branching from a single initial electron without photo-ionization. [Becker et al. 2005, Ch 2, p. 52; Arrayas et al. 2002]

• Simulation workaround: Because photo-ionization cross sections are poorly known, many simulation models substitute a uniform seed electron density of 10^7 - 10^8 cm^-3 instead of explicit UV transport, which produces similar results. [Becker et al. 2005, Ch 6, p. 281]

For Tesla coil sparks, which operate on AC waveforms, both positive and negative half-cycles contribute to propagation. The photo-ionization mechanism is most critical during the positive half-cycle when streamers must advance into virgin (unperturbed) air.

Why Streamers Are Inefficient

Despite their speed, streamers are poor at creating lasting conductive channels:

• Low current: Insufficient Joule heating (I^2 * R) to raise temperature significantly
• Thin channels: Cool quickly (tau ~ microseconds for d ~ 10 um)
• High resistance: Poor conductors, most of the voltage drops across the channel rather than reaching the tip
• Branching: Energy splits among many branches, diluting the current in each
• No thermal memory: Each streamer pulse must re-ionize fresh gas

The energy "wasted" in creating a streamer that immediately cools and deionizes is the fundamental reason burst mode (streamer-dominated) has high epsilon.

Leader Discharges

Leaders are the hot, persistent, highly conductive channels that form when sufficient sustained current flows through a streamer channel.

Physical Properties

Property	Value	Notes
Diameter	mm to cm	100-1000x thicker than streamers
Propagation speed	~10^3 m/s	Much slower than streamers
Current	Amperes (A)	High current, intense Joule heating
Propagation mechanism	Thermal ionization	Saha equilibrium at T > 5000 K
Temperature	5000-20000 K	Fully thermalized plasma
Resistance	Low	rho ~ 1-10 ohm*m
Persistence	Seconds	With convection maintaining hot column
Visual appearance	White/orange/yellow	Blackbody + line emission
Branching	Relatively straight	Few major branches
Energy per meter (epsilon)	Low (5-15 J/m)	Efficient for forward propagation

Propagation Mechanism: Thermal Ionization

Leaders propagate by a fundamentally different mechanism:

1. Hot conducting core: The leader channel is a thermalized plasma at 5000-20000 K
2. Current flows to tip: The low-resistance leader conducts current efficiently from the topload to its tip
3. Tip launches streamers: At the leader tip, the concentrated field creates new streamers
4. Streamers carry current: Some streamer branches carry enough current (fed from the leader) to undergo Joule heating
5. Heated streamers become leader: The heated channel transitions to a new leader segment
6. Leader extends: Steps 3-5 repeat, advancing the leader at ~10^3 m/s

The leader propagation speed is much slower than streamer speed because it is limited by thermal processes (heating gas from ~300 K to ~5000+ K takes time). But the leader is vastly more efficient because each meter of leader channel, once formed, persists and conducts efficiently.

Leader Velocity

Bazelyan & Raizer provide an empirical formula for leader velocity:

v_L = 1500 * sqrt(|Delta_U_t|)    [cm/s, with Delta_U_t in volts]

[Bazelyan & Raizer 2000, Physics-Uspekhi 43(7), p. 709, Eq. 5]

For a TC with 300 kV topload voltage: v_L = 1500 * sqrt(300,000) ~ 820,000 cm/s = 8.2 km/s. This is intermediate between laboratory sparks (~10 km/s) and lightning leaders (~100 km/s), consistent with observed TC spark growth.

The physical basis: the leader advance rate is set by the conducting streamer length l ~ 1 cm (limited by electron attachment at ~100 ns) divided by the thermal instability contraction time tau_ins ~ 1 us, giving v_L ~ 10^6 cm/s. The square root voltage dependence arises because higher tip voltage increases streamer vigor (length and density), expanding the zone available for contraction.

Electron attachment time in cool air: ~100 ns [Bazelyan & Raizer 2000, p. 703]

This is the fundamental timescale that limits streamer channel lifetime without heating. At TC frequencies of 50-400 kHz (half-periods of 1.25-10 us), a cold streamer goes through 12-100 attachment times per half-cycle. Without heating to >5000 K (where attachment becomes negligible), the streamer plasma dies between every half-cycle and must be re-created — this is why streamers are so energy-inefficient.

Stepped vs Continuous Leaders

Lightning observations reveal two distinct leader propagation modes, which have direct analogs in TC spark behavior:

Positive leaders (ascending from grounded objects, carrying positive charge) propagate continuously: the bright tip moves smoothly upward with gradually varying velocity. This is the dominant mode for TC sparks, where positive streamers/leaders propagate from the positive-going topload.

Negative leaders (descending from cloud, carrying negative charge) propagate in steps: discrete jumps of 10-200 m (average 30 m for lightning), separated by pauses of 30-90 us. Each new step briefly re-illuminates the entire channel behind it. The stepped pattern arises because negative streamers require a different mechanism (electron drift ahead of the tip rather than photoionization), leading to an intermittent advance.

[Bazelyan & Raizer 2000, "Lightning Physics and Lightning Protection," IOP, Ch 1, pp. 17-18]

Leader Type	Polarity	Pattern	Step size	Pause	Average velocity
Positive	+	Continuous	N/A	N/A	10^5 - 10^6 m/s
Negative	-	Stepped	10-200 m (avg 30 m)	30-90 us	10^5 - 10^6 m/s (averaged)
Dart (re-strike)	either	Continuous (fast)	N/A	N/A	(1-4) * 10^7 m/s

Average velocities are the same for stepped and continuous leaders when averaged over the total development time: 10^5-10^6 m/s (100-1000 km/s), with an average of ~3*10^5 m/s.

Dart leaders (subsequent strokes following existing hot channels) are always continuous and much faster: (1-4)*10^7 m/s. This is because they propagate through pre-heated, pre-ionized gas where the ionization front moves as a thermal wave rather than requiring fresh ionization.

TC relevance: TC sparks on the positive half-cycle behave as continuous leaders. On the negative half-cycle, stepped behavior could occur but is masked by the rapid AC reversal (half-period of 1.25-10 us at 100-400 kHz is shorter than the 30-90 us step pause). The result is that TC sparks effectively propagate as continuous leaders on both half-cycles, though with different microscopic mechanisms.

Leader Formation Voltage Threshold

A minimum potential difference is required to excite and develop a leader in air:

Delta_U_min ~ 300 - 400 kV

[Bazelyan & Raizer 2000, Ch 5, p. 271]

This is the total potential drop from electrode to the surrounding space needed to provide enough field energy for streamer formation, heating through the 2000-5000 K transition zone, and establishment of a self-sustaining hot channel.

TC implications for burst mode: Most burst-mode DRSSTCs operate with topload voltages of 100-600 kV. At the low end (100-200 kV), leader formation is marginal — the coil produces primarily streamers. At 300+ kV, leaders form readily, consistent with the dramatic improvement in spark length efficiency observed when coils cross the ~300 kV threshold. This provides physical backing for the common builder observation that "bigger coils are disproportionately more impressive" — they cross the leader formation threshold.

Critical caveat for QCW mode: The 300-400 kV threshold applies to single-shot impulse discharges where the entire streamer-to-leader transition must occur from a single event. QCW coils form leaders at dramatically lower topload voltages — typically only 40-70 kV — because the resonant circuit continuously injects energy over thousands of RF cycles (5-20 ms ramp). Multiple independent builders have confirmed this: davekni measured ~40 kV peak at 450 kHz producing 2-2.5 m sword sparks, while Steve Ward measured 40 kV rising to 55 kV over ~5000 RF cycles for 50+ inch arcs. The comparison is stark: a burst DRSSTC at 80 kHz needs ~600 kV for the same spark length a QCW achieves at 40 kV — a 15:1 voltage ratio. [Phase 6 QCW community survey, multiple sources]

The physical explanation: in QCW mode, the thermal ratcheting mechanism (see thermal-physics) accumulates energy from many RF cycles. Each cycle deposits a small increment via Joule heating, and the 5:1 asymmetry in conductance relaxation time constants (40 us heating vs 200 us cooling) ensures the temperature ratchets upward. Over ~1-5 ms, the stem temperature crosses through the critical thresholds (2000 K → 4000 K → 5000 K) without ever requiring the full 300-400 kV instantaneous voltage. The voltage merely needs to exceed the inception threshold and maintain current flow.

Why Leaders Are Efficient

• Low resistance: The hot, ionized channel conducts well, delivering most of the source voltage to the tip
• Persistence: Long thermal time constants (see thermal-physics) mean the channel stays hot and conductive for seconds
• Self-maintaining: As long as current flows, Joule heating maintains the temperature
• Focused energy: Less branching means energy is concentrated in the main propagation path
• Thermal accumulation benefit: epsilon(t) decreases as the channel accumulates thermal energy (see energy-and-growth)

The Streamer-to-Leader Transition

The transition from streamer to leader is the critical process that determines spark efficiency. It is the reason QCW mode (which promotes transition) produces longer sparks than burst mode (which cannot sustain it) for the same energy input.

Transition Sequence

The six-step transition process:

Step 1: High E-field creates streamers

• Topload voltage exceeds inception threshold (see field-thresholds)
• Multiple streamer branches form simultaneously
• Streamers propagate rapidly outward (~10^6 m/s)
• Channel is cold, thin, high-resistance

Step 2: Sufficient current causes Joule heating

• The resonant circuit continues driving current through the streamer channels
• Current distributes among branches, but some branches carry more than others
• Joule heating power per unit length: P_linear = I^2 * R_linear [W/m]
• For a 100 um streamer at rho = 50 ohm*m carrying 100 mA:

  R_linear = rho / A = 50 / (pi * (50e-6)^2) = 6.4 * 10^9 ohm/m
  P_linear = (0.1)^2 * 6.4e9 = 64 MW/m (!!)
  

  This enormous linear power density (even at low total current) is what drives the transition. The thin channel concentrates the heating.

Step 3: Heated channel undergoes thermal ionization

• Temperature rises from ~300 K through ~3000 K to 5000+ K over ~1 ms
• At ~3000-4000 K: significant thermal dissociation of N2 and O2 begins
• At ~5000 K: Saha equation predicts substantial ionization fraction
• Conductivity increases by orders of magnitude
• Resistance drops, allowing more current to flow (positive feedback)

Step 4: Leader forms from base

• The transition proceeds from the base (near topload) outward
• Base segments see the most current (no branching losses yet) and transition first
• Leader formation is progressive, not instantaneous
• Base becomes a bright, thick, low-resistance channel

Step 5: Leader tip launches new streamers

• The leader acts as an extension of the topload electrode
• At the leader tip, the electric field is enhanced (see field-thresholds)
• New streamers propagate from the leader tip into fresh air
• These are "fed streamers" -- receiving current from the leader

Step 6: Fed streamers convert to leader

• Current from the leader flows through the new streamers
• Higher current than free streamers (sustained by leader's low resistance)
• The same Joule heating process converts these streamers to leader
• The leader extends by one "step" as each generation of fed streamers transitions

This cycle repeats: leader -> streamers -> Joule heating -> new leader -> more streamers. The spark grows as a composite structure with a leader trunk and streamer crown.

Transition Threshold

The transition requires sufficient current density and duration. Approximate criteria:

• Current density: j > ~10^6 A/m^2 in the streamer (equivalently, ~10 mA in a 100 um channel)
• Duration: Must sustain heating for ~0.5-2 ms (long enough to raise temperature through ~3000 K to 5000+ K)
• Power density: P_volume > ~10^10 W/m^3 approximately, sustained for milliseconds

These thresholds explain why:

• QCW succeeds: Continuous drive for 5-20 ms provides ample time and current
• Burst mode fails: Short pulses (50-500 us) may not sustain heating long enough, especially if gaps allow cooling

Aborted Leaders and Dark Periods

The transition sequence above is idealized. In practice, multiple failed attempts typically precede stable leader inception. High-speed photography and Schlieren imaging reveal a characteristic cycle [Liu 2017; Les Renardieres Group 1977, 1981]:

Dark Period Cycle:

1. Streamer burst: Positive streamers propagate from electrode tip into virgin air (~10^6 m/s)
2. Space charge shielding: Positive ions left behind by the fast-moving electron front create a space charge cloud near the electrode that reduces the local electric field
3. Dark period: Field at electrode drops below inception threshold. No new streamers form. Duration ~1-5 ms (depends on gap geometry and voltage)
4. Ion drift recovery: Positive ions slowly drift outward under the applied field (mu_ion ~ 2 * 10^-4 m^2/(V*s)), gradually restoring the electrode field
5. Next burst: When the field recovers above inception, a new streamer burst occurs

Each burst deposits energy into the stem region (the short channel connecting the streamer base to the electrode). If the energy deposition from a single burst is insufficient to raise the gas temperature past the critical threshold for leader inception, the stem cools during the dark period and the leader attempt aborts.

Aborted leader progression:

• First burst: stem heats to ~1000-1500 K, cools back to ~500 K during dark period
• Second burst: residual warmth means less energy needed; stem reaches ~1800-2500 K, cools to ~800-1200 K
• Third/fourth burst: thermal ratcheting pushes temperature past critical threshold -> stable leader inception

Critical temperature requirement: The gas temperature must significantly exceed 2000 K for stable leader inception, not merely reach it. During gas expansion following heating, convection losses can drop the temperature back below the critical ionization threshold. The gas must be heated enough to survive this expansion cooling. See thermal-physics for the detailed mechanism.

Multiple stems share current: Schlieren photography shows that current from the electrode distributes among multiple streamer stems simultaneously, not just the strongest branch. This reduces the heating per individual stem and delays the transition. The stem that transitions to a leader first is typically the one that received the most cumulative energy across multiple burst cycles. [Liu 2017, Ch 2, Schlieren observations]

Gallimberti Model Critique

The widely-cited Gallimberti (1972) model for streamer-to-leader transition assumes:

1. Constant electric field in the stem during the transition process
2. Simplified V-T relaxation: Uses a simplified nitrogen vibrational-translational energy transfer model
3. Single stem: Assumes all current flows through one dominant stem

Liu (2017, Ch 3) demonstrates through detailed kinetic modeling (45 species, 192 reactions) that these assumptions do not hold:

• Stem field varies significantly as space charge evolves and the stem heats/expands
• V-T relaxation is not the dominant heating mechanism in the late stages of transition; direct electron impact heating becomes important
• Humidity effect on V-T relaxation is weak: The conventionally cited acceleration of V-T relaxation by water vapor is "several orders of magnitude smaller" than other energy sources during the transition [Liu 2017, Ch 3]
• Multiple stems share current, invalidating the single-stem assumption

Despite these limitations, Gallimberti's model captures the correct qualitative physics (energy accumulation in stem -> thermal runaway -> leader) and gives order-of-magnitude correct transition times. It remains useful as a conceptual framework but should not be trusted for quantitative predictions without correction factors.

Transition Energy Density Threshold

In addition to the current density and duration criteria above, the corona-to-spark transition can be characterized by a volumetric energy density threshold:

Minimum specific energy for spark channel formation: 0.6 - 1 J/cm^3

[Becker et al. 2005, Ch 2, p. 59]

This is the energy density that must be deposited in the streamer channel before it can transition to a self-sustaining spark (leader). For a 100 um diameter streamer channel, the energy per unit length to reach 1 J/cm^3 is:

E_per_length = 1 J/cm^3 * pi * (50 um)^2 = 7.85 * 10^-6 J/cm = 0.000785 J/m

This is a very small energy per meter compared to the observed epsilon values (5-100 J/m), confirming that the transition from streamer to leader is not primarily limited by total energy -- it is limited by the rate of energy deposition (power density). The current density criterion (j > 10^6 A/m^2) and the duration criterion (~0.5-2 ms) are the operative constraints. See energy-and-growth for how this connects to the physical origin of epsilon.

Spark Formation Dynamics

Once the corona-to-spark transition begins, two stages of spark formation are observed in high-speed photography:

1. Primary streamer: Fast propagation at ~10^8 cm/s (10^6 m/s) from the anode toward the cathode
2. Secondary streamer: Slower propagation at ~10^5 - 10^6 cm/s along the same trajectory, after a delay that depends on overvoltage

[Becker et al. 2005, Ch 2, pp. 59-60]

The secondary streamer propagates not by direct ionization in a strong field (like the primary) but by energy deposition into the existing channel (gas heating, vibrational excitation). This is the physical precursor to leader formation.

Upon bridging of the gap by the secondary streamer, the discharge current increases abruptly:

Spark current rise rate: dI/dt ~ 10^7 A/s

[Becker et al. 2005, Ch 2, p. 60]

Ion mobility governs how fast the positive space charge left behind by the fast-moving streamer tip can rearrange:

mu_ion ~ 2 * 10^-4 m^2/(V*s)    (in air at STP)

[Becker et al. 2005, Ch 2, p. 60]

This is much slower than electron mobility (~0.03 m^2/(V*s)), which is why the positive space charge from a streamer takes a relatively long time to redistribute -- contributing to the delay between primary and secondary streamer stages.

The Hungry Streamer Connection

The streamer-to-leader transition is intimately connected to Steve Conner's hungry streamer principle (see power-optimization):

The self-optimization feedback loop drives the system toward leader formation when sufficient power is available:

1. Streamer forms with high R (above R_opt_power)
2. Hungry streamer principle: plasma tries to reduce R toward R_opt_power to maximize power extraction
3. Mechanism: increased current -> Joule heating -> higher temperature -> higher conductivity -> lower R
4. As R decreases through the optimization, temperature rises, and the channel transitions from streamer to leader
5. Leader equilibrium: R stabilizes near R_opt_power at a temperature that maintains the required conductivity
6. If R_opt_power is below R_min (physical lower bound for plasma), the system is constrained and operates sub-optimally

The hungry streamer principle and the streamer-to-leader transition are two descriptions of the same physical process viewed from different perspectives: one from the circuit (impedance optimization) and one from the plasma physics (thermal evolution).

Quantitative Resistance vs Current: The Power Law

The qualitative hungry streamer picture now has a quantitative foundation from self-consistent plasma modeling. The equilibrium resistance per unit length follows a power law in current:

R = A / I^b    (ohm/m)

[da Silva et al. 2019, JGR Atmospheres]

Regime	Current Range	A (ohm * A^b / m)	b	TC Context
Region I	1-10 A	12,400	1.84	TC streamer/early leader
Region II	10-1,000 A	2,820	1.16	DRSSTC burst mode
Region III	1,000-10,000 A	180	0.75	Lightning return strokes

Worked examples for TC sparks:

• Streamer at 1 A: R ~ 12,400 ohm/m -> 12.4 kohm for 1 m spark (consistent with QCW/leader range)
• Early streamer at 0.1 A: R ~ 12,400 / (0.1)^1.84 ~ 860,000 ohm/m (very high, as expected for cold streamer)
• Leader at 10 A: R ~ 179 ohm/m -> 179 ohm for 1 m spark (hot leader, approaching arc)

The steep exponent in Region I (b = 1.84) means resistance drops nearly as the square of current — this is the quantitative expression of the positive feedback that drives the streamer-to-leader transition. Doubling the current reduces resistance by ~3.6x, which increases current further, driving the thermal runaway.

Why the transition is slow despite this positive feedback: The air heating efficiency eta_T is only ~10% at ambient temperature (see thermal-physics). 90% of the electrical energy goes into N2 vibrational excitation rather than gas heating. The thermal runaway only accelerates after the gas reaches ~1000-2000 K where eta_T approaches 1.0.

Cross-validation with Bazelyan V-I characteristic: Bazelyan & Raizer provide two formulas of increasing precision:

• Simple: iE = 300 VA/cm (arc approximation, valid for quick estimates)
• Precise (measured CVC): E = 32 + 52/i V/cm (Eq. 2.48 in the full textbook)

[Bazelyan & Raizer 2000, "Lightning Physics and Lightning Protection," IOP, Ch 2, p. 90]

The precise CVC reveals that at high currents, the field plateaus at 32 V/cm (not zero) — representing irreducible radiation and convection losses. The 52/i term dominates at TC-relevant currents (1-10 A). At i = 1 A: E = 84 V/cm (CVC) vs 300 V/cm (simple i*E=b formula). The simple formula overpredicts the field at low currents.

Both Bazelyan formulas agree with da Silva's power law within a factor of ~2 for 1-10 A — see equations-and-bounds Section 14.14 for a detailed comparison table. The three independent approaches (Bazelyan experimental, Bazelyan measured CVC, da Silva plasma modeling) converging to similar values is strong evidence that these resistance values are reliable for TC spark modeling.

Connection to Mayr equation: The R = A/I^b law describes where the resistance wants to go (equilibrium). The Mayr equation dG/dt = (1/tau) * (P/P_0 - 1) * G describes how fast it gets there. See equations-and-bounds Sections 14.8 and 14.11 for both models.

Composite Spark Structure

A fully developed Tesla coil spark is not purely streamer or purely leader. It is a composite:

Topload
  |
  [Leader trunk]  -- Hot, thick, low-R, persistent
  |                  Conducts efficiently from topload to crown
  |
  [Transition zone]  -- Intermediate properties
  |                     Recently converted streamers
  |
  [Streamer crown]  -- Cool, thin, high-R, ephemeral
                       Actively propagating into fresh air
                       Highly branched

Position-Dependent Properties

This composite structure maps directly to the distributed-model where segments near the base (leader) have low R and segments near the tip (streamer) have high R:

• Base segments (near topload): R ~ 1-10 kohm, hot, thick, leader
• Middle segments: R ~ 10-100 kohm, transition zone
• Tip segments: R ~ 100 kohm - 100 Mohm, cold, thin, streamer

The convergence behavior of the distributed model's iterative optimization naturally reproduces this structure:

• Base segments converge to low R (sharp power peak, well-coupled)
• Tip segments converge to high R (flat power curve, poorly coupled)

Practical Implications by Operating Mode

QCW Mode: Leader-Dominated

• Long ramp (5-20 ms) allows full transition at base within first few milliseconds
• Leader trunk grows progressively during ramp
• Low effective epsilon (5-15 J/m) because leader extension is efficient
• Leader persistence (seconds) means channel stays alive throughout ramp
• Streamer crown at tip is continuously fed by leader current
• Result: longest sparks per unit energy

Measured QCW growth rate: ~170 m/s (approximately half the speed of sound). This is estimated from community observations of spark growth during QCW ramps. [Phase 6 QCW survey, HVF topic 973]

Self-consistency check: at 170 m/s over a 10 ms ramp, a spark grows 1.7 m. Over a 20 ms ramp, 3.4 m. These match observed QCW spark lengths (1-2 m for standard builds, 3.35 m for the Fat Coil QCW build).

This 170 m/s rate is intermediate between free streamers (10^6 m/s) and natural lightning leaders (~10^4 m/s for stepped leaders, averaged). It represents a driven leader propagation mode unique to QCW: the leader advances continuously, fed by the circuit, at a rate limited by the thermal conversion of streamer-to-leader at the tip.

Driven leader step time: From the growth rate and Bazelyan's typical leader step length (~1 cm):

step_time ~ step_length / growth_rate ~ 0.01 m / 170 m/s ~ 60 us

This 60 us step time is close to the conductance relaxation heating time constant (tau_g = 40 us from Bazelyan, see thermal-physics). The channel needs approximately one tau_g to heat each new segment to leader temperature, so the leader advance rate is limited by how fast each new streamer can be thermally converted. The 1.5x ratio (60 us observed vs 40 us tau_g) is reasonable given that the transition also requires crossing the eta_T efficiency bottleneck (10% at ambient → 100% above 2000 K).

Contrast with Bazelyan leader velocity: The Bazelyan formula v_L = 1500*sqrt(|Delta_U_t|) gives ~4.7-8.2 km/s at 100-300 kV. This is 25-50x faster than the observed 170 m/s QCW growth rate. The discrepancy is explained by the fundamental difference between the two quantities: Bazelyan's v_L is the instantaneous leader step velocity (the speed of the thermal instability contraction within a single step), while the QCW 170 m/s is the net growth rate averaged over many steps including the time to heat each new streamer segment. The QCW leader advances in rapid micro-steps at ~km/s but spends most of its time waiting for each new segment to thermalize.

Burst Mode: Streamer-Dominated

• Short pulse (50-500 us) may not allow transition to complete
• Channel remains mostly streamer throughout the pulse
• High effective epsilon (30-100+ J/m) because streamer propagation is inefficient
• Channel cools between bursts (gap >> streamer persistence of ~1-5 ms)
• Each burst must re-ionize from scratch
• Result: bright but short sparks, energy-inefficient for length

High Duty Cycle DRSSTC: Hybrid

• Closely spaced bursts (gaps < 5 ms) allow some thermal memory
• Base may partially transition to leader between closely spaced pulses
• Intermediate epsilon (20-40 J/m)
• Neither fully leader-dominated nor fully streamer-dominated
• Result: moderate length efficiency, intermediate spark character

Observable Differences

The streamer/leader distinction is directly observable:

Visual

• Streamers: Purple/blue, fine filaments, highly branched, flickering
• Leaders: White/orange/yellow, thick trunk, straighter, more stable
• Composite: Purple crown with white/yellow base is characteristic of healthy QCW growth

Audio

• Streamers: Hissing/crackling sound (many small discharges)
• Leaders: Louder snap/crack (single powerful channel)
• QCW ramp: Tone that rises in pitch as ramp progresses

Electrical Signatures

• Streamer loading: High impedance, small frequency shift, small Q reduction
• Leader loading: Low impedance, large frequency shift, large Q reduction
• Transition: Impedance drops abruptly during transition (resistance falls by orders of magnitude)

These electrical signatures can be observed in the loaded pole behavior described in coupled-resonance.

Connection to Energy per Meter

The fundamental reason epsilon differs by mode comes down to this topic:

• Streamer epsilon is high because: thin channels cool fast, energy is wasted on branching, re-ionization overhead is large, high resistance means poor voltage delivery to tip
• Leader epsilon is low because: thick channels persist, energy is focused in main path, no re-ionization needed (already hot), low resistance delivers voltage efficiently to tip
• Mode determines which type dominates: QCW promotes leaders (low epsilon), burst maintains streamers (high epsilon)

This is the physical basis for the epsilon values used in energy-and-growth and the scaling relationships in empirical-scaling.

Recombination and Plasma Decay

When current ceases flowing through a spark channel, the plasma decays primarily through electron-ion recombination. The dominant recombination processes in air and their rate coefficients are:

Reaction	Rate Coefficient (cm^3/s)	Notes
O2+ + e-	1.9 * 10^-7 * (300/T_e)^0.5	Dominant in dry air
N2+ + e-	1.8 * 10^-7 * (300/T_e)^0.39	Fast at low T_e
NO+ + e-	4.3 * 10^-7 * (300/T_e)^0.37	Important in warm channels (NO forms above ~2000 K)
H3O+ + e-	6.3 * 10^-7 * (300/T_e)^0.5	Relevant in humid air, T_e < 1000 K

[Becker et al. 2005, Ch 4, pp. 170-175]

Key takeaway: All major simple atmospheric ion species recombine with electrons at approximately 2 * 10^-7 cm^3/s at 300 K electron temperature. [Becker et al. 2005, Ch 4, p. 174]

At high pressures, three-body recombination can increase rates to as high as 10^-4 cm^3/s. [Becker et al. 2005, Ch 4, p. 175]

Complex and cluster ions — much faster recombination:

Simple ions (O2+, N2+) quickly convert to complex and hydrated cluster ions in atmospheric air. These cluster ions recombine with electrons 5-25x faster than simple ions:

Ion	Rate (cm^3/s)	Formation timescale	Notes
O2+ (simple)	2.7 * 10^-7 * (300/T_e)^0.5	N/A (initial ion)	Baseline rate
O4+ (complex)	1.4 * 10^-6 * (300/T_e)^0.5	~10-100 ns	5x faster
H3O+(H2O)3 (hydrated)	6.5 * 10^-6 * (300/T_e)^0.5	~1 us (humid air)	25x faster

[Bazelyan & Raizer 2000, "Lightning Physics and Lightning Protection," IOP, Ch 2, pp. 52-56, Eq. 2.24-2.28]

Critical for TC sparks: Complex ions form within 10-100 ns, so the enhanced recombination rate applies almost immediately after the ionization front passes. In humid air (outdoor TC operation), the full hydration chain completes in ~1 us, giving recombination rates 25x the simple-ion value. This explains why:

• Outdoor TC sparks in humid conditions are noticeably shorter than indoor sparks
• The effective plasma lifetime in cold streamer channels is ~100-300 ns (not the ~800 ns from simple attachment alone)
• Channel conductivity drops by ~100x within 300 ns behind the streamer tip, leaving only a ~30 cm "alive" section at typical streamer velocities

Thermal decomposition of complex ions: At T > ~2000 K, the O4+ cluster breaks apart (k_decomp = 3.3 * 10^-6 * (300/T)^4 * exp(-5040/T) cm^3/s), reverting to simple ions with their slower recombination rate. This is another mechanism by which channel heating dramatically improves plasma persistence.

Connection to channel persistence: For a streamer with n_e ~ 10^13 cm^-3, the recombination time constant is:

tau_recomb = 1 / (alpha_recomb * n_e) = 1 / (2e-7 * 1e13) ~ 50 us

This is comparable to the pure thermal diffusion time for a 100 um streamer (~0.1-0.2 ms), confirming that ionization memory and thermal cooling compete on similar timescales. For leaders with n_e ~ 10^15 - 10^16 cm^-3, recombination is faster (~0.5-5 us), but continuous Joule heating maintains the ionization against recombination losses.

These quantitative recombination rates provide the microphysical foundation for the "ionization memory" mechanism described in thermal-physics, and explain why effective streamer persistence (~1-5 ms) significantly exceeds the pure thermal diffusion time -- the plasma decays slower than the thermal profile.

Key Relationships

• Derives from: field-thresholds (inception field creates initial streamers; propagation field sustains leader growth)
• Derives from: thermal-physics (thermal diffusion and persistence determine transition feasibility)
• Enables: energy-and-growth (channel type determines epsilon, the key parameter for growth prediction)
• Enables: empirical-scaling (different mode efficiencies explain different scaling exponents)
• Implements: power-optimization (hungry streamer self-optimization is the circuit-level view of the thermal transition)
• Structures: distributed-model (leader-base / streamer-tip composite maps to position-dependent R)
• Constrained by: capacitive-divider (voltage division limits current delivery to tip, affecting transition feasibility)