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id	title	status	source_sections	related_topics	key_equations	key_terms	images	examples	open_questions
open-questions	Open Questions and Future Research Directions	established	spark-physics.txt: Part 12 (lines 807-835), plus scattered notes throughout	[energy-and-growth thermal-physics streamers-and-leaders distributed-model field-thresholds empirical-scaling femm-workflow equations-and-bounds]	[Dynamic capacitance: d_eff(E) Branching current: I_branch proportional to d_branch^1.5 Time-dependent epsilon with thermal memory]	[epsilon variability branching dynamic capacitance radial temperature profile Monte Carlo transient simulation strike detection 3D FEA stochastic breakout Becker et al. 2005 Liu 2017 Yang et al. 2022 literature reference Gallimberti model aborted leader Phase 6 QCW survey sword_spark driven_leader QCW_measurement_gaps plasma_motion ion_wind bulk_gas_flow]	[]	[]	[This entire document is a catalog of open questions -- see section contents below.]

Open Questions and Future Research Directions

This document catalogs the known uncertainties, unexplored areas, and future enhancement possibilities in the Tesla coil spark modeling framework. The framework, as documented in equations-and-bounds, lumped-model, distributed-model, and related topics, provides a practical and accurate modeling approach. However, it makes deliberate simplifications and relies on empirical calibration in areas where the underlying plasma physics is too complex for closed-form treatment. This document makes those limitations explicit and identifies the most promising directions for future work.

1. Remaining Uncertainties in Current Framework

1.1 Energy per Meter (epsilon) Variability

The energy per meter epsilon is the most important empirical parameter in the framework (see energy-and-growth). It determines the growth rate dL/dt = P_stream / epsilon and the total energy requirement E_total = epsilon * L. Current knowledge:

What we know:

QCW-style growth: epsilon approximately 5-15 J/m
High duty cycle DRSSTC: epsilon approximately 20-40 J/m
Hard-pulsed burst mode: epsilon approximately 30-100+ J/m
epsilon decreases during heating: epsilon(t) = epsilon_0 / (1 + alpha * integral(P dt))

What we do not know:

How does epsilon depend on current density at the channel level? Higher peak current should improve leader formation efficiency, but the quantitative relationship is not established.
How does epsilon vary with operating frequency? The framework assumes epsilon is frequency-independent, but frequency affects the balance between displacement and conduction current, which may influence channel heating efficiency.
What is the precise effect of ambient conditions (temperature, humidity, altitude, barometric pressure) on epsilon? The field threshold E_propagation varies by +/-20-30% with altitude and humidity, but the corresponding variation in epsilon is not well characterized.
Can epsilon be predicted from first principles for a given set of operating conditions, or does it always require empirical calibration?

Impact of uncertainty: epsilon has +/-30-50% measurement tolerance. This propagates directly into growth rate and length predictions with the same fractional uncertainty. For design purposes, this is often acceptable (predicting spark length to within a factor of 1.5). For precision modeling, it is the dominant error source.

1.2 Field Threshold (E_propagation) Dependencies

The field threshold for sustained spark growth (see field-thresholds) has similar uncertainties:

Known dependencies:

Altitude: E_propagation scales approximately with air density (proportional to pressure/temperature)
Humidity: Higher humidity generally increases E_propagation (water molecules are electronegative)
- Literature update: Breakdown voltage in air has a minimum at ~1% water vapor content. Below ~1%, adding water vapor reduces breakdown voltage; above ~1%, the attachment effect dominates and raises it. For typical indoor conditions (0.5-2% water vapor), the effect on E_inception is modest and direction-dependent. The effect on E_propagation (which involves pre-conditioned channels) is less clear from the textbook data. [Becker et al. 2005, Ch 2, p. 30]
- Further update (Liu 2017): The conventionally cited mechanism by which humidity accelerates streamer-to-leader transition (faster V-T relaxation of N2 due to H2O collisions) is quantitatively weak. Liu's kinetic modeling shows the energy contribution from humidity-enhanced V-T relaxation is "several orders of magnitude smaller" than other energy sources during the transition process. This suggests humidity's effect on E_propagation may operate through a different mechanism than V-T relaxation (possibly attachment/detachment kinetics or changed recombination pathways). [Liu 2017, Ch 3]
Channel condition: A pre-heated channel from a previous pulse has lower E_propagation

Unknown dependencies:

How does tip geometry affect E_propagation? The framework uses a single value modified by the enhancement factor kappa, but branched or split tips may behave differently.
Does E_propagation change along the spark as the channel matures from streamer to leader?
What is the quantitative relationship between E_propagation and the repetition rate in burst mode? Faster bursts should reduce E_propagation due to residual ionization, but by how much?

Measurement tolerance: +/-20-30%. Combined with FEMM field accuracy of +/-10%, the total uncertainty in predicting the stall length is +/-30-40%.

1.3 Full Thermal Evolution

The framework uses a simplified thermal model (see thermal-physics):

tau_thermal = d^2 / (4 * alpha)
alpha = k / (rho_air * c_p) approximately 2e-5 m^2/s

What this captures: Pure thermal diffusion from a hot cylindrical channel into ambient air. This gives correct time constants for the initial cooling phase.

What this misses:

Convection: Hot air rises, creating buoyancy-driven convection that maintains the hot gas column longer than pure diffusion predicts. For thick leaders (d > 1 mm), convection dominates over diffusion for times longer than about 10 ms. This is why observed channel persistence (seconds for leaders) is much longer than the diffusion time constant (hundreds of milliseconds for 5 mm channels).
Radiation: At temperatures above 5000 K, radiative cooling from ionized gas becomes significant. The framework does not account for radiative losses, which could reduce the effective thermal memory at very high temperatures.
Convection enhancement of cooling: While buoyancy maintains the hot column structure, it also enhances convective heat transfer at the channel surface. The net effect depends on the balance between column maintenance and surface cooling.
Ionization energy: Thermal dissociation and ionization of air molecules absorb significant energy that is released upon recombination. This acts as an additional "thermal memory" beyond simple sensible heat.
- Literature update: The average ionization energy cost in air is ~14 eV per electron-ion pair [Becker et al. 2005, Ch 7, p. 440]. Nitrogen vibrational relaxation time at 1 atm is >100 us [Becker et al. 2005, Ch 5, p. 231], confirming that vibrational energy storage operates on the same timescale as thin streamer thermal diffusion (~100-200 us). Electron-ion recombination rates are ~2 * 10^-7 cm^3/s at 300 K [Becker et al. 2005, Ch 4, p. 174], giving tau_recomb ~ 50 us at n_e = 10^13 cm^-3. The gap between recombination decay (~50 us) and observed persistence (~1-5 ms) is partially explained by these vibrational and metastable energy reservoirs.
- Further update (Liu 2017): Detailed kinetic modeling (45 species, 192 reactions) confirms that N2 vibrational relaxation is not the sole or even dominant energy pathway during the critical transition phase. Direct electron impact heating becomes important in late-stage transition. The critical temperature for leader inception must significantly exceed 2000 K because convection losses during gas expansion can abort the leader if temperature only marginally exceeds the threshold. Multiple aborted leader attempts (thermal ratcheting) typically precede stable inception. [Liu 2017, Ch 3]

1.4 Branching

Real Tesla coil sparks branch extensively, especially in burst mode. The framework currently models a single unbranched channel.

Branching questions:

How is power divided among branches at a branch point? Is it proportional to branch impedance, or does some other mechanism apply?
Does the main channel (thickest branch) receive the majority of the power, with side branches being parasitic?
How does the total spark capacitance (C_sh) change with branching? Multiple branches increase the total conductor surface area, potentially increasing C_sh significantly beyond the single-channel 2 pF/foot estimate.
Can the distributed-model be extended to include branching by adding parallel R-C paths at branch nodes?

Proposed branching model (untested):

I_branch proportional to d_branch^1.5

This follows from the assumption that current density scales with cross-sectional area and conductivity scales with temperature (which scales with diameter for a given total power). The exponent 1.5 is intermediate between the area scaling (exponent 2) and the linear scaling. This model has not been validated against measurements.

1.5 Bulk Plasma Motion Inside the Channel

The framework models the spark channel as a stationary conductor with time-varying resistance. It does not address the motion of the plasma itself — either the charged particle drift or the bulk neutral gas flow. Several distinct motions are known or expected:

What we know:

Radial expansion [T1]: When the channel heats rapidly, the gas expands outward against surrounding cold air. Bazelyan measured initial expansion rate ~100 m/s, dropping to ~2 m/s at pressure equilibrium. The channel reaches pressure balance within microseconds; after that, hot gas is less dense but at ambient pressure. For lightning return strokes, this expansion launches a cylindrical shock wave (thunder). [Bazelyan & Raizer 2000, Ch 4, p. 167]
Buoyancy-driven convection [T1]: Hot channel gas rises. This is the primary mechanism extending leader persistence beyond pure thermal diffusion (see thermal-physics Section 1). Vertical or upward-angled sparks maintain a hot column for seconds; horizontal sparks lose coherence as the column rises and disconnects.
Electron and ion drift [T0]: Electrons drift toward the positive end of the field, ions toward the negative end. In a TC's AC field, directions reverse every half-cycle. Electron mobility is ~400x higher than ion mobility (mu_e = 600 cm^2/(Vs) vs mu_i = 1.5 cm^2/(Vs) [Bazelyan & Raizer 2000]). On the RF timescale (1-10 us half-period), ions are essentially stationary while electrons carry the current.
Ion wind / electric wind [T1 general; T4 for TC channels]: Ions colliding with neutrals transfer momentum, driving bulk gas flow at ~100 m/s in the drift direction. Well-documented in corona/ESP applications [Becker et al. 2005, Ch 9, pp. 581-583]. In a TC streamer crown, positive ions left behind by the fast-moving electron front drive neutral gas outward from the topload.

What we do not know:

Net axial bulk flow in a leader channel during AC operation [T4]: The AC reversal complicates the picture. Ion wind reverses every half-cycle — does it cancel to zero net axial flow, or does some asymmetry (positive vs negative streamer dynamics, polarity-dependent attachment rates) create a net drift? No measurements or models address this for TC sparks.
Gas flow at the propagating tip [T3]: As each new streamer segment heats and expands to become leader, the expansion should displace gas forward (outward from the topload). This creates a transient outward flow at the leader-streamer boundary. Whether this contributes meaningfully to propagation or is negligible compared to the photoionization-driven advance is unknown.
Acoustic effects of channel motion [T3]: The radial expansion produces acoustic signatures. Whether axial flows (if any) contribute to the characteristic sound of different operating modes is unexplored.
Effect on channel straightness [T4]: If there is net axial gas flow, it could contribute to (or detract from) the straightness of QCW sword sparks by stabilizing (or destabilizing) the thermal column.

Why this matters: If significant bulk gas motion exists inside the channel, it could affect heat transport (convective enhancement of axial thermal conduction), mass transport (fresh gas swept into hot regions), and channel stability. The current framework implicitly assumes the gas is stationary except for radial diffusion and buoyancy, which may miss important physics.

2. Future Physics Enhancements

2.1 Dynamic Capacitance

The current framework assumes constant channel diameter for the FEMM simulation. In reality, the effective diameter depends on the local electric field and plasma conditions:

d_eff(E) = d_0 * (1 + beta * ln(E / E_threshold))

where d_0 is the zero-field diameter, beta is a dimensionless expansion coefficient, and E_threshold is the field at which expansion begins. As the field increases, the ionization boundary expands outward, increasing the effective conductor diameter and hence the capacitance.

Impact: Dynamic capacitance would cause C_mut and C_sh to vary with the applied voltage, making the circuit nonlinear. At high voltages (near inception), the effective diameter could increase by a factor of 2-5, changing capacitances by 10-30% (logarithmic dependence). This is a second-order effect for most applications but could matter for precision growth modeling.

2.2 Radial Temperature Profiles

The current framework treats the channel as having a uniform temperature across its cross-section. In reality:

Hot core: The center of the channel is hottest (5000-20000 K for leaders), with the highest ionization and lowest resistivity.
Cool edges: Temperature drops off radially, with a transition zone where the plasma transitions from fully ionized to neutral air.
Effective radius: The "electrical diameter" (the radius within which most current flows) is smaller than the "optical diameter" (the radius within which the gas is visibly luminous).

A radial temperature profile would enable:

More accurate resistance estimates (integrate conductivity over the cross-section)
Better thermal time constant predictions (the cool outer shell cools faster than the hot core)
Modeling of the step-leader mechanism (hot core grows forward while cool edges lag)

2.3 Time-Dependent Epsilon with Thermal Memory

The framework already includes a first-order correction:

epsilon(t) = epsilon_0 / (1 + alpha * integral(P_stream dt))

A more sophisticated model could track the thermal state of the channel explicitly:

T_channel(t) = T_ambient + (1 / (m * c_p)) * [integral(P_heat dt) - integral(P_cool dt)]
epsilon(T) = epsilon_max * exp(-T / T_scale)

where P_heat is the Joule heating rate, P_cool is the combined conductive/convective/radiative cooling rate, and T_scale is a characteristic temperature above which leader formation becomes efficient. This couples the thermal evolution to the growth rate, creating a nonlinear system that must be solved iteratively.

2.4 Branching Models

A quantitative branching model would include:

Probability of branching at each time step (proportional to local E_tip and current)
Power division rule at branch points (proposed: I_branch proportional to d_branch^1.5)
Independent growth of each branch
Total C_sh as sum of all branch capacitances
Competition for current among branches

The main difficulty is that branching is inherently stochastic, making deterministic prediction impossible. Statistical approaches (ensemble averages, probability distributions of spark length) may be more appropriate.

3. Simulation Improvements

3.1 Full Transient with L(t) Evolution

The current workflow uses a sequence of static (AC) analyses: set up the spark at length L, compute steady-state power, step L forward. A true transient simulation would:

Start with zero spark length
At each time step, check E_tip against E_propagation
If E_tip > E_propagation: advance L by (P_stream / epsilon) * dt
Update the spark model (C_mut, C_sh, R) for the new length
Continue the SPICE transient without restarting

This requires a SPICE model with time-varying elements, which is not directly supported by standard SPICE. Possible implementations:

Python-controlled SPICE (update model parameters between time steps)
Behavioral modeling in SPICE using voltage-controlled parameters
Custom simulator with integrated circuit and growth equations

3.2 3D FEA for Complex Geometries

FEMM's axisymmetric solver cannot handle:

Horizontal or angled sparks
Multiple breakout points on asymmetric toploads
Sparks in the presence of nearby grounded objects (walls, equipment)
Branched sparks

3D FEA tools (Elmer, COMSOL, ANSYS Maxwell) can solve these cases but at significantly higher computational cost (minutes to hours per run instead of seconds). A practical approach is to use FEMM for the baseline vertical case and apply correction factors derived from 3D simulations for non-ideal geometries.

3.3 Monte Carlo for Stochastic Breakout and Branching

Spark formation and branching are stochastic processes influenced by:

Local surface field variations on the topload (surface roughness, sharp features)
Random seed electron availability (cosmic rays, photoionization)
Turbulent mixing affecting local gas composition and temperature

A Monte Carlo approach would:

Generate random initial conditions (breakout point, initial angle)
Propagate the spark with stochastic branching events
Repeat many times to build statistical distributions
Report mean and variance of spark length, total energy, branching pattern

This is computationally expensive but would provide uncertainty quantification that the deterministic model cannot.

3.4 Strike Detection

When a spark reaches a grounded object, the channel transitions from a high-impedance plasma load to a near-short-circuit:

R_spark -> R_strike approximately 1-10 ohm (arc contact resistance)

This transition happens in microseconds and causes:

Massive current surge (limited only by source impedance and primary circuit)
Rapid heating of the contact point
Potential damage to the target and the coil

Modeling strike events requires:

A distance criterion: when E_tip at a grounded surface exceeds inception threshold
A rapid R transition model (exponential decay from R_spark to R_strike)
Protection circuit modeling (fuses, IGBTs, current limiters)

4. Validation Needs

4.1 Systematic Measurements Across Coil Types

The current empirical calibration (epsilon and E_propagation) is based on a limited number of coils and operating conditions. A comprehensive validation program would include:

Multiple coil types: SSTCs, DRSSTCs, QCW-DRSSTCs, classical spark gap coils
Multiple frequencies: 50, 100, 200, 400 kHz
Multiple power levels: 0.5, 1, 5, 10, 15 kW primary input
Multiple topload sizes: Small (10 cm), medium (30 cm), large (50+ cm)
Controlled environments: Indoor (dry, known temperature) and outdoor (varying humidity, wind)

For each combination, measure:

Final spark length (mean and standard deviation over many shots)
Topload voltage (calibrated probe)
Primary and secondary current (Rogowski/CT)
Input power (current probe times voltage probe, integrated over burst)
Ringdown frequency and Q (with and without spark)

4.2 High-Speed Photography for Growth Rate Validation

Growth rate dL/dt = P_stream / epsilon is a central prediction of the model. Validating it requires:

High-speed camera (10,000+ fps) to track spark tip position versus time
Simultaneous electrical measurement (V_top, I) for instantaneous power
Frame-by-frame spark length extraction (image processing)
Comparison of measured dL/dt to predicted dL/dt from the model

This would directly calibrate epsilon as a function of time and operating conditions, rather than relying on final-length-only measurements.

4.3 RF Current Distribution Measurements

The distributed-model predicts that current decreases from base to tip. Validating this requires:

Multiple current sensors placed along the spark channel (extremely challenging)
Alternatively: multiple magnetic field sensors near the spark at different heights
Optical spectroscopy to infer local temperature and electron density (and hence local conductivity and current)

4.4 Parameter Database

A community database correlating spark parameters to operating conditions would enormously benefit the field:

Parameter	Coil	Frequency	Power	Mode	Environment	Measured Value
epsilon	DRSSTC-1	200 kHz	5 kW	QCW	Indoor, 25C	8.3 J/m
E_prop	DRSSTC-1	200 kHz	5 kW	QCW	Indoor, 25C	0.62 MV/m
Length	DRSSTC-1	200 kHz	5 kW	QCW	Indoor, 25C	1.8 +/- 0.3 m
...	...	...	...	...	...	...

Such a database does not currently exist in a systematic form.

4.5 Bayesian Model Calibration (Phase 8 — Active)

A focused experimental program to constrain the dynamic threshold parameters via Bayesian inference. See phases/phase-8-bayesian-model-calibration.md for the full research plan. Key measurements:

Ramp duration sweep at fixed power → constrains delta_T, tau_buildup
Power level sweep at fixed ramp → constrains coupled voltage-power limit
Frequency comparison (if feasible) → constrains thermal ratchet rate

This directly addresses the T3/T4 claims in field-thresholds Section 4.7 (dynamic E_propagation). Even 10-20 data points with physics-informed priors would dramatically constrain the model parameters. The fitting pipeline uses MCMC with model comparison (Bayes factor) to test whether the dynamic threshold model outperforms a fixed-threshold model.

5. Framework Limitations: Honest Assessment

5.1 What the Framework Does Well

Predicts spark impedance and power transfer with accuracy sufficient for coil design
Captures the essential physics: capacitive topology, power optimization, field-limited growth
Provides both simple (lumped) and detailed (distributed) models with clear trade-offs
Identifies measurable calibration parameters (epsilon, E_propagation) that separate coil-dependent from physics-dependent quantities

5.2 What the Framework Cannot Do

Predict absolute spark length from first principles: The framework requires empirical calibration of epsilon and E_propagation. Without measurements on the specific coil (or a similar one), length predictions have a factor-of-2 uncertainty.
Model branching quantitatively: The single-channel assumption is adequate for main-channel length but cannot predict branch structure, total luminous volume, or branch-related power losses.
Capture sub-microsecond dynamics: The framework operates at the RF cycle timescale (microseconds) and above. Nanosecond-scale phenomena (streamer head propagation, individual ionization events) are below its resolution.
Handle strike events: The transition from free spark to grounded arc is outside the framework's scope.

5.3 Where Empirical Calibration Fills Physics Gaps

The framework explicitly acknowledges that complex plasma physics (ionization kinetics, radiation transport, turbulent mixing, streamer branching statistics) is replaced by calibrated empirical parameters. This is a deliberate engineering choice:

epsilon replaces a detailed model of energy deposition, ionization, heating, and leader formation
E_propagation replaces a detailed model of streamer inception, photoionization, and space charge effects
R bounds replace a detailed model of plasma conductivity as a function of temperature, composition, and pressure
- Literature update (da Silva et al. 2019): The equilibrium resistance per unit length is now quantified as R = A/I^b (ohm/m), with fitted parameters for three current regimes. For TC-relevant currents (1-10 A): R = 12,400/I^1.84 ohm/m. This provides a physics-based resistance model that could replace or complement the empirical R bounds, connecting channel resistance directly to the current flowing through it. The steep b=1.84 exponent quantifies the positive feedback driving the streamer-to-leader transition. See equations-and-bounds Section 14.11.

The calibration approach works because the plasma self-optimizes (hungry streamer principle): the detailed microphysics adjusts itself to match the circuit constraints. The circuit constraints (topology, capacitances, source impedance) are well-characterized, so the macroscopic behavior is predictable even though the microscopic mechanism is complex.

6. Partial Answers from Literature

Several open questions in this framework now have partial answers from the gas discharge physics literature, specifically from Becker, Kogelschatz, Schoenbach & Barker, "Non-Equilibrium Air Plasmas at Atmospheric Pressure" (IOP, 2005). These do not close the questions but narrow the uncertainty range and provide quantitative anchors.

Can epsilon be predicted from first principles?

Partial answer: The minimum volumetric energy density for spark channel formation is 0.6-1 J/cm^3 [Becker et al. 2005, Ch 2, p. 59]. Combined with channel cross-sections, this gives epsilon_min ~ 0.001-0.07 J/m (streamer to leader diameters). Observed epsilon (5-100 J/m) is 100-10,000x higher, with the multiplier explained by branching, radiation, heating overhead, and incomplete energy utilization. A first-principles prediction requires modeling all these loss channels, which remains intractable for general conditions. Status: lower bound established; full prediction still requires empirical calibration.

Further update (da Silva et al. 2019): A major factor in the high observed epsilon is now quantified: the air heating efficiency eta_T is only ~10% at ambient temperature. 90% of electrical energy goes into N2 vibrational modes rather than gas heating. This means the "effective" power available for channel heating and leader formation is only 1/10 of the total electrical power at early stages. As the channel warms past ~1000-2000 K, eta_T rises to ~1.0 and energy utilization improves dramatically. This heating efficiency factor, combined with branching losses, partially closes the gap between epsilon_min and observed epsilon. Status: heating efficiency quantified; combined with branching and radiation models could enable semi-empirical epsilon prediction.

What is the role of nitrogen vibrational relaxation in persistence?

Partial answer: N2 vibrational relaxation time at 1 atm is >100 us [Ch 5, p. 231]. This is comparable to thin streamer thermal diffusion times (~100-200 us) and explains why ionization memory extends streamer persistence from ~0.1 ms (pure diffusion) to ~1-5 ms (observed). Status: timescale confirmed; quantitative contribution vs. other mechanisms (metastables, attachment/detachment) remains to be partitioned.

How does humidity affect E_propagation quantitatively?

Partial answer: Breakdown voltage has a minimum at ~1% water vapor content [Ch 2, p. 30]. For typical indoor conditions (0.5-2% water vapor), the humidity effect on E_inception is modest (+/-10%). The effect on E_propagation (which involves a pre-conditioned channel rather than initial breakdown) is less clear from the textbook data, which focuses on initial breakdown. Status: inception effect quantified; propagation effect still uncertain.

What electron densities exist in Tesla coil streamers?

Answered: Streamer body electron density is 10^11-10^13 cm^-3 [Ch 2, p. 37], with fully developed spark channels reaching ~10^16 cm^-3 [Ch 2, p. 38]. These values enable direct calculation of plasma conductivity and recombination rates, connecting the microscopic plasma state to the macroscopic resistance values used in lumped-model and distributed-model.

What power is needed to sustain a spark channel?

Answered: Power to sustain n_e = 10^13 cm^-3 ranges from 1.4 kW/cm^3 (cold air, attachment-limited) to 14 kW/cm^3 (2000 K air, equilibrium losses) [Ch 5, p. 230; Ch 7, p. 440]. For a 3 mm leader channel, this corresponds to ~1 kW/m linear power density, providing an independent check on power delivery requirements. See thermal-physics for the full analysis.

What determines the frequency threshold for straight (sword) QCW sparks?

Answered: Community data converges on 300-600 kHz for sword-like sparks [Phase 6 QCW survey, 6+ independent observers]. Below 300 kHz, QCW sparks are "chaotic and less straight"; above 600 kHz, "more curvy." The physical mechanism is the ratio of RF half-period to streamer tau_thermal: at 400 kHz (half-period = 1.25 us), the channel experiences effectively continuous heating (tau_thermal ~ 125 us is 100x longer). At 100 kHz (half-period = 5 us), thin streamers (10-50 um, tau ~ 1-30 us) experience significant cooling between cycles, allowing the preferred path to diffuse and branch. See thermal-physics for full analysis.

Does QCW require high voltage for leader formation?

Answered: No. The Bazelyan 300-400 kV leader formation threshold applies to single-shot impulse discharges, not sustained-drive QCW. QCW forms leaders at only 40-70 kV topload voltage (measured by Steve Ward, davekni, Loneoceans). The 15:1 voltage ratio (600 kV burst vs 40 kV QCW for same spark length) is the single most important quantitative result from the community survey. See streamers-and-leaders for updated leader formation discussion.

What is the QCW spark growth rate?

Partially answered: Community estimate of ~170 m/s (half the speed of sound), which is consistent with observed spark lengths over measured ramp durations. This implies a driven leader step time of ~60 us, close to the conductance relaxation tau_g = 40 us. Definitive measurement requires high-speed imaging synchronized with electrical waveforms (see measurement gaps below).

QCW Measurement Gaps

The community survey [Phase 6] identified these critical unmeasured quantities:

No direct arc current measurement on any QCW coil — the current flowing in the spark channel during QCW operation has never been measured
No spectroscopic temperature measurement of QCW sparks — the ~5000 K estimate is inferred from conductivity analysis, not measured
No time-resolved impedance measurement during QCW ramp — the impedance trajectory during growth is unknown
No high-speed imaging correlated with electrical waveforms in QCW mode
No measurement of epsilon for QCW sparks — only bounded from total input and estimated efficiency
No systematic frequency sweep — same coil tested at 100, 200, 300, 400 kHz to isolate frequency effect
Voltage gradient in TC sparks disputed — estimates range from 1.5 to 3 kV/cm

Key references for further investigation

The following primary sources from [Becker et al. 2005] are particularly relevant for deepening this framework:

Gallimberti (1972) -- Streamer propagation simulation methodology; early computational approach
Morrow & Lowke (1997) -- Ionization/attachment coefficients for air, used in most modern air discharge simulations
Kulikovsky (1998) -- Detailed positive streamer simulation with electron density profiles
Kunhardt (2000) -- Frequency dependence of breakdown voltage in gases
Raether (1964), Meek & Craggs (1978) -- Classical textbooks on spark discharge physics; more focused on spark/leader physics than the Becker et al. book
Babaeva & Naidis (2000) -- Review of 2D streamer simulation developments

These are recommended as follow-up reading, particularly Raether (1964) and Meek & Craggs (1978), which cover the spark and leader physics central to Tesla coil modeling in much greater depth than the Becker et al. book (which focuses on cold non-equilibrium plasmas for industrial applications).

Additional references integrated since initial literature review:

Liu (2017) -- "Electrical Discharges: Streamer-to-Leader Transition and Positive Leader Inception," KTH Doctoral Thesis. Detailed kinetic modeling of streamer-to-leader transition with 45 species, 192 reactions. Key findings: leader inception requires T >> 2000 K; Gallimberti model assumptions flawed; humidity V-T relaxation effect weak; multiple stems share current. Extensive experimental basis from Les Renardieres Group (1977, 1981).
Yang, Meng, Niu et al. (2022) -- "Arc Modeling Approaches: A Comprehensive Review," Frontiers in Physics. Reviews Mayr, Cassie, and hybrid arc models with parameter sensitivity analysis. Key finding: TC sparks are in the pure Mayr regime; tau_m ~ 10-100 us; P_0 ~ 1-100 W.
Les Renardieres Group (1977, 1981) -- Comprehensive experimental studies of long spark formation in air gaps. Schlieren photography of dark periods, aborted leaders, and stem physics. Primary experimental data used in Liu (2017) kinetic validation.
da Silva, C.L. et al. (2019) -- "The Plasma Nature of Lightning Channels and the Resulting Nonlinear Resistance," JGR Atmospheres, 10.1029/2019JD030693. Self-consistent plasma model yielding R = A/I^b power law for channel resistance; air heating efficiency eta_T formula; channel expansion dynamics. Rate coefficient Matlab code on Zenodo (10.5281/zenodo.2597562). Key finding: channel resistance is determined by current, not initial conditions — supports hungry streamer self-optimization.
Bazelyan, E.M. & Raizer, Yu.P. (2000) -- "The mechanism of lightning attraction and the problem of lightning initiation by lasers," Physics-Uspekhi 43(7), 701-716. Review paper. Key content: leader velocity formula v_L = 1500sqrt(Delta_U) cm/s; V-I characteristic iE=300 VA/cm; three-tier temperature thresholds (2000 K onset, 4000 K associative ionization, 5000 K self-sustaining); energy ceiling from tip capacitance W_max = piepsilon_0*U^2; electron attachment time ~100 ns; thermal instability contraction time ~1 us; electron/ion mobility in air. Cross-validates da Silva resistance values within factor ~2 for TC-relevant currents.
Bazelyan, E.M. & Raizer, Yu.P. (2000) -- "Lightning Physics and Lightning Protection," IOP Publishing, 328 pages. Comprehensive textbook by the same authors. Key additional content beyond the review paper: conductance relaxation model (dG/dt = [G_st(i)-G(t)]/tau_g, tau_g = 40 us heating / 200 us cooling); channel transmission line parameters (L_1 ~ 2.5-2.7 uH/m, C_1 ~ 10 pF/m, Z ~ 500 ohm); leader formation threshold (300-400 kV); leader channel energy balance (P_L = iE ~ 130 W/cm at 1 A, 5000 K); corona shielding rate limit (3.6 kV/us, far exceeded by TC toploads); stepped vs continuous leader propagation; E/N dependence on temperature (55 Td at 1000 K to 1.5 Td at 6000 K); dart leader velocity (1-4)*10^7 m/s; return stroke physics (35,000 K, wave at 0.4c).

7. Connection to Other Topics

Key Relationships

Motivates improvement of: energy-and-growth -- Better understanding of epsilon variability would improve growth predictions.
Motivates improvement of: thermal-physics -- Full thermal evolution including convection and radiation would improve epsilon modeling.
Motivates improvement of: distributed-model -- Branching extensions, time-varying parameters, and optimal segmentation are all open areas.
Motivates improvement of: field-thresholds -- Better characterization of E_propagation dependencies would reduce prediction uncertainty.
Motivates improvement of: femm-workflow -- 3D FEA and dynamic geometry updates would extend the framework's applicability.
Bounded by: equations-and-bounds -- All proposed improvements must remain consistent with established physical bounds and measurement tolerances.
Informed by: empirical-scaling -- Community observations of spark length versus power provide independent validation of model predictions.
Informed by: streamers-and-leaders -- Understanding the physical differences between streamers and leaders motivates the branching and transition models.

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