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id	title	status	source_sections	related_topics	key_equations	key_terms	images	examples	open_questions
branching-physics	Branching Physics and Multi-Channel Dynamics	provisional	spark-physics.txt: Part 5.5, Part 12 (lines 360-396, 1007, 1015)	[streamers-and-leaders thermal-physics power-optimization energy-and-growth capacitive-divider qcw-operation distributed-model equations-and-bounds open-questions]	[nonlinear-resistance-power-law conductance-relaxation]	[branching streamer leader fractal_dimension Laplacian_instability thermal_ratcheting nonlinear_resistance negative_differential_resistance]	[]	[]	[What is the quantitative current division rule at branch points in TC sparks? How much does branching increase effective C_sh beyond single-channel estimates? What fraction of total epsilon is attributable to branching losses? Can branch competition dynamics be measured with time-resolved imaging on a TC? Does the fractal dimension of TC sparks correlate with operating mode or frequency?]

Branching Physics and Multi-Channel Dynamics

Real Tesla coil sparks branch extensively — especially in burst mode. The current framework models a single unbranched channel, which is adequate for main-channel length prediction but cannot explain total luminous volume, power budget overhead, or the morphological differences between operating modes. This topic develops the physics of branching from established discharge science and connects it to the TC-specific observations documented in qcw-operation and thermal-physics.

1. Why Discharges Branch: Laplacian Instability

1.1 The Mechanism

A conducting channel propagating in a background electric field concentrates the field at its tip (see field-thresholds, tip enhancement factor kappa = 2-5). If the tip is slightly perturbed — widened or displaced — the concentrated field splits into two lobes, each of which can independently ionize the gas ahead of it. This is a Laplacian instability [T0], mathematically identical to the Saffman-Taylor instability in viscous fingering.

The instability is intrinsic to any conducting object growing in a Laplacian field. What determines whether and when a particular discharge branches is the source and amplitude of the initial perturbation.

1.2 Perturbation Sources

For streamers in air at atmospheric pressure, the dominant perturbation sources are:

Electron density fluctuations (stochastic particle noise) [T1]: At the ionization front, the electron density is finite (~10^13 cm^-3), meaning the number of electrons in a volume element at the streamer tip is not infinite. Poisson fluctuations create density variations that seed the instability. Luque & Ebert (2011) showed that this intrinsic noise is sufficient to trigger branching of positive streamers in air at atmospheric pressure, and that without noise, branching occurs much later if at all. [Luque & Ebert 2011, Phys Rev E 84, 046411]
Photoionization seed electrons [T1]: For positive streamers, propagation requires photoionization to create seed electrons ahead of the tip. The stochastic spatial distribution of these seed electrons creates field non-uniformities that can trigger branching. More photoionization means a denser, more uniform seed electron cloud, which actually reduces branching. This is why positive streamers in pure nitrogen (no O2 to provide photoionization) branch differently than in air.
Prior discharge remnants [T3]: In repetitive operation (TC burst mode), residual ionization and thermal channels from previous pulses create non-uniform starting conditions for each new pulse.

1.3 Characteristic Branching Length

A streamer branches after propagating approximately 10-20 diameters from its last branch point [T1]. This ratio of branching length to streamer diameter agrees within a factor of 2 with experimental measurements. [Luque & Ebert 2011]

Channel type	Diameter	Branching distance	Branching frequency	Tier
Thin streamer	10-50 um	0.1-1 mm	Very frequent	T1
Thick streamer	50-100 um	0.5-2 mm	Frequent	T1
Early leader	0.1-1 mm	1-20 mm	Occasional	T3
Mature leader	1-10 mm	10-200 mm	Rare	T3

The geometric branching distance alone predicts that streamers are highly branched and leaders are relatively straight — consistent with observation [T3]. But geometry is only half the story. Thermal feedback (Section 3) further suppresses branching in leaders.

1.4 Branching Angles

Experimental measurements of streamer branching in air show:

Average branching angle: ~90 degrees (full angle between daughter branches) in air at atmospheric pressure [T1]
The angle is normally distributed around this mean [T1]
In low-O2 mixtures (less photoionization), the average decreases to ~66 degrees [T1]
Branching is predominantly binary (two daughters); triple branching is rare and has been specifically documented as noteworthy [T1] [Nijdam et al., J Phys D]
Daughter branches are typically similar in diameter immediately after splitting, but quickly diverge due to competition (Section 3) [T3]

1.5 Fractal Dimension

The spatial structure of branched discharge trees has a measured fractal dimension:

D = 2.16 +/- 0.05 (needle-plane corona discharge) [T1] [Plasma Physics Reports, 2002]

For reference:

D = 1.0: a straight line (no branching)
D = 2.0: a structure that fills a plane
D = 3.0: a structure that fills a volume

D ~ 2.2 means the discharge tree is moderately space-filling — a 3D branched structure that is slightly denser than a flat tree. This is consistent with the bushy appearance of burst-mode TC sparks. QCW sword sparks, being nearly unbranched, would have D approaching 1.0 [T4 — no measurement exists].

The fractal dimension connects to total surface area and hence to C_sh [T3]: a more branched tree (higher D) has more total conductor surface area per unit of main-channel length, increasing its capacitive coupling to ground.

2. Streamer vs Leader Branching

The two discharge types branch for fundamentally different reasons and with very different consequences.

2.1 Streamers: Branching Is the Default

Streamers are cold, transient channels. They have:

No significant thermal inertia (tau_thermal ~ 1-100 us for d = 10-100 um)
No mechanism to preferentially sustain one path over another
Fast propagation (~10^6 m/s) that outpaces any thermal feedback

As a result, streamers branch readily at the rate predicted by the Laplacian instability (~every 10-20 diameters). Each branch propagates independently. The result is a highly branched tree of thin filaments — the characteristic purple/blue corona visible on burst-mode TC sparks.

Energy consequence [T3]: Each branch channel absorbs energy (ionization, heating, radiation) but contributes little to forward propagation. This is a major contributor to the high epsilon values observed in burst mode (30-100+ J/m) — most of the energy goes into side branches that don't extend the main channel. See energy-and-growth.

2.2 Leaders: Thermal Feedback Suppresses Branching

Leaders are hot (>5000 K), thermally self-sustaining channels (see streamers-and-leaders). They have:

Large thermal inertia (tau_thermal ~ 0.3-1+ seconds for d = 1-10 mm)
Nonlinear V-I characteristics that create competitive dynamics (Section 3)
Slower propagation (~10^3 m/s net growth) allowing thermal feedback to operate

The combination of large diameter (branching every 10-200 mm geometrically) and thermal competition (Section 3) means leaders branch far less frequently than streamers [T3]. The few branches that do form quickly lose the competition for current and extinguish, leaving a relatively straight main channel — the characteristic white/yellow sword of QCW operation.

3. Branch Competition: Nonlinear Resistance and Current Hogging

This section describes the central physical mechanism that determines whether a branched discharge consolidates into a single channel or remains multi-branched. It follows directly from the da Silva nonlinear resistance law documented in streamers-and-leaders and equations-and-bounds Section 14.11.

3.1 The Instability

The equilibrium resistance per unit length of a discharge channel follows a power law in current:

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

[da Silva et al. 2019]

For TC-relevant currents (Region I, 1-10 A): A = 12,400, b = 1.84 [T1].

The critical feature is that b > 1. This makes the V-I characteristic of the channel have negative slope (negative differential resistance) [T0]:

V = R * I = A * L * I^(1-b)

For b = 1.84: V proportional to I^(-0.84). Voltage drop decreases with increasing current.

3.2 Why This Causes Current Hogging

Consider two parallel branches of equal length at the same applied voltage V:

Branch 1: V = A * L * I_1^(1-b)
Branch 2: V = A * L * I_2^(1-b)

The equal-sharing solution (I_1 = I_2) exists but is unstable when b > 1 [T0 — mathematical consequence of b > 1]:

If branch 1 receives slightly more current (I_1 = I_0 + delta), its voltage drop decreases
Since both branches are at the same voltage, branch 1 can now carry even more current
This is positive feedback — the perturbation grows
Branch 1 heats up, becomes more conductive, draws more current
Branch 2 cools, becomes more resistive, loses current
Eventually branch 1 carries nearly all the current and branch 2 extinguishes

This is the same instability that causes parallel arcs to merge and arc attachment points to wander [T1]. It is well-established plasma physics, not a hypothesis.

3.3 Competition Timescale

The rate at which one branch "wins" is governed by the conductance relaxation time:

tau_g = 40 us (heating) / 200 us (cooling)

[Bazelyan & Raizer 2000; see thermal-physics]

After ~3-5 heating time constants (~120-200 us), the competition is largely resolved — one branch dominates [T3 — timescale inferred from tau_g]. This timescale is critical for understanding TC operating modes:

Operating mode	Characteristic time	tau_competition / time	Branching outcome
Single burst pulse	70-150 us	~1	Competition barely resolves; multiple branches coexist
QCW ramp	10-20 ms	~50-100	Competition fully resolves; single dominant channel
Burst repetition gap	5-10 ms (at 100-200 Hz)	N/A	Channels cool and decay between pulses

This single mechanism explains the morphological difference between burst and QCW sparks [T3]. Burst pulses are too short for the nonlinear competition to select a winner. QCW ramps are long enough for thermal feedback to consolidate current into one channel.

3.4 Frequency Dependence

At higher RF frequencies, the channel receives more heating cycles per unit time (at the same peak current). This accelerates the thermal ratchet that drives branch competition:

At 400 kHz: ~16 RF cycles per tau_g (40 us). Heating is effectively continuous. Competition resolves quickly.
At 100 kHz: ~4 RF cycles per tau_g. Heating is intermittent. Thin streamers may cool between cycles, resetting the competition.

This is the physical basis for the 300-600 kHz frequency threshold for QCW sword sparks documented in qcw-operation [T3 — mechanism inferred; the frequency threshold itself is T2]. The frequency threshold is not about breakdown physics — it is about whether the thermal competition can resolve fast enough to suppress branching during the QCW ramp. See thermal-physics Section 7 for community observations.

4. Branching Regimes in Tesla Coil Operation

4.1 Burst Mode: Branching Dominates

In burst mode (70-150 us ON time [T2], 200-600 kV topload [T2]):

Peak voltage exceeds leader formation threshold (~300-400 kV for single-shot) [T3]
But ON time is comparable to the competition timescale (~120-200 us) [T3]
Multiple streamer channels form simultaneously from the topload [T2]
Thermal competition begins but does not fully resolve before the pulse ends [T3]
Between pulses (5-10 ms gap), all channels cool and decay [T3]
Next pulse starts fresh — no accumulated thermal advantage for any channel [T3]

Result: highly branched, bushy sparks [T2]. High epsilon (30-100+ J/m) because energy feeds many branches [T3 — epsilon values are T2, mechanism is T3].

4.2 QCW: Competition Suppresses Branching

In QCW mode (10-20 ms ramp [T2], 300-600 kHz [T2], 40-70 kV topload [T2]):

Voltage starts low and ramps over many milliseconds [T2]
At inception, a few streamer channels form [T3]
Thermal competition begins immediately (tau_competition ~ 120-200 us) [T3]
Within 0.5-2 ms, one channel dominates via the current-hogging instability [T3]
The winning channel transitions to a leader (>2000 K → 5000 K via thermal ratcheting) [T3]
For the remaining 10-18 ms, the leader grows as a single, straight channel [T2]
Side branches are continuously suppressed: any new branch that forms quickly loses the competition to the established hot channel [T3]

Result: straight sword sparks [T2]. Low epsilon (5-15 J/m) because energy is concentrated in one channel [T3 — epsilon values are T2, mechanism is T3].

4.3 "Too Long" QCW Ramp: Branching Returns

When the QCW ramp exceeds ~25 ms [T2] (documented by Loneoceans, see qcw-operation):

The leader reaches its voltage-limited maximum length (set by the capacitive divider — see capacitive-divider) [T3]
Additional energy cannot extend the leader further (E_tip < E_propagation) [T3]
The leader channel becomes very hot and thick, increasing its C_sh [T3]
Excess power must dissipate somewhere [T0 — conservation of energy]
Lateral breakouts from the superheated leader trunk become the path of least resistance [T3]
These new branches compete with each other but not effectively with the main channel (which is already saturated) [T3]

Result: "hot and fat but bushy" sparks [T2] — a thick leader trunk with side branches. The main channel doesn't get longer, just fatter and more branched.

4.4 Pulse-Skip / Bresenham: Intermediate Behavior

The user's observation [T2] (documented in qcw-operation Section 2.3) that Bresenham pulse-density modulation produces sparks that are "more sword-like but still branches" is exactly what the competition model predicts [T3]:

Bresenham PDM delivers a coarse approximation of a linear envelope [T2]
The heating is less smooth than true analog QCW [T3]
Per-cycle current jitter means the thermal advantage of the winning channel fluctuates [T3]
Competition still operates but with more noise, so side branches persist longer [T3]
Result: intermediate morphology on the continuum between burst (fully branched) and analog QCW (unbranched) [T3]

5. Capacitive Loading of Branches

Each branch segment has its own shunt capacitance C_sh to ground. The total C_sh of a branched tree exceeds that of a single channel of the same main-channel length.

5.1 Physical Argument

A single channel of length L at height h above ground has [T0]:

C_sh ~ 2*pi*epsilon_0*L / ln(2h/d)    (thin-wire approximation)

A branched tree with total conducting length L_total > L (main channel length) has additional C_sh from side branches [T0]. The branches are laterally displaced from the main channel, reducing mutual shielding between them, so the capacitance does not simply scale with total length — it depends on the spatial extent of the tree [T3].

5.2 Qualitative Estimates

QCW sword (minimal branching): C_sh is close to the single-channel value [T3]. The empirical 2 pF/foot rule applies (or possibly overestimates, since it was likely calibrated against partially branched sparks).
Burst mode (heavy branching): C_sh may be 2-5x the single-channel value [T4 — no measurement], because the branched tree has much more total surface area exposed to ground.
This is consistent with Loneoceans' frequency tracking data [T2]: a 1.78 m QCW spark produced only 8.7% frequency shift, while a simulated solid wire of 1 m produced 24% shift. The QCW spark's low effective capacitance reflects both its plasma nature and its minimal branching [T3].

5.3 Consequence for Voltage Division

Higher C_sh from branching worsens the capacitive divider (see capacitive-divider):

V_tip = V_topload * C_mut / (C_mut + C_sh)

More branching → higher C_sh → lower V_tip → lower E_tip → harder to propagate → more stall and more branching. This is a positive feedback loop [T3] that drives burst-mode sparks toward heavily branched, voltage-limited configurations. It is the capacitive complement to the thermal competition mechanism.

QCW breaks this feedback loop by suppressing branching early, keeping C_sh low, maintaining V_tip high, and enabling continued forward propagation [T3].

6. Power Budget: Branching as Energy Overhead

The connection between branching and epsilon is direct:

6.1 Energy Accounting

Total energy delivered to the spark distributes among:

Main channel forward growth (useful work): ionization, heating to leader temperature
Side branch formation and heating (overhead): each branch absorbs energy but doesn't extend the main channel
Radiation and convection losses from all channels
Capacitive energy stored in C_sh (including branch contributions)

In burst mode, items 2-4 dominate [T3]. The ratio of useful work to total energy is low, explaining the high epsilon (30-100+ J/m).

In QCW mode, branch suppression eliminates most of item 2 early in the ramp [T3]. Energy concentrates in the main channel, keeping epsilon low (5-15 J/m).

6.2 Quantitative Estimate

The efficiency ratio between QCW and burst can be roughly estimated from the spark:secondary ratios documented in qcw-operation:

Burst: spark:secondary = 2.5-3.6:1
QCW: spark:secondary = 7-16.5:1

The QCW advantage is 3-5x [T2 — derived from community-measured ratios], which includes both branching reduction and the thermal efficiency gain from sustained leader operation [T3]. Separating these contributions requires measurements that do not yet exist (see Open Questions).

7. What We Do Not Know

7.1 Current Division Rule

The existing framework proposes I_branch proportional to d_branch^1.5 (see open-questions Section 1.4), but this is unvalidated. The physics of Section 3 suggests that current division is better understood through the nonlinear V-I instability than through a static power-law in diameter:

At the moment of splitting, daughter branches are similar and share current roughly equally
The equal-sharing equilibrium is unstable (b > 1 in the da Silva law)
Within ~100-200 us, one branch dominates via current hogging
The "steady-state current division" is therefore not a useful concept — the system is transient and winner-take-all

A static rule like I proportional to d^n misses this essential dynamics.

7.2 Fractal Dimension vs Operating Mode

No measurements exist of the fractal dimension of TC sparks as a function of frequency, power level, or operating mode. Such measurements (from high-resolution photographs) would directly test the competition model: D should decrease (approach 1.0) as frequency increases and as ramp duration increases.

7.3 Branching Fraction of Epsilon

What fraction of total epsilon is attributable to branching losses vs other overhead (radiation, convection, stored energy)? This requires either:

Time-resolved imaging correlated with electrical waveforms (not yet done on any TC)
Careful comparison of epsilon between highly branched and minimally branched sparks of the same length under controlled conditions

7.4 Branch Initiation from Leader Trunk

The "too long" QCW regime (Section 4.3) produces lateral breakouts from a superheated leader trunk. The physics of initiation from a hot, thick channel into cold air is different from streamer branching (which occurs at the propagating tip). This may involve thermal instabilities of the channel boundary rather than Laplacian field instabilities.

Key Relationships

Derives from: streamers-and-leaders (discharge types), thermal-physics (time constants), power-optimization (nonlinear R)
Explains: energy-and-growth (why epsilon differs by mode), qcw-operation (sword vs bushy morphology, frequency threshold), capacitive-divider (C_sh dependence on morphology)
Connects to: distributed-model (potential extension to branched networks), open-questions (Sections 1.4, 2.4)
Key data: da Silva R = A/I^b with b = 1.84 (equations-and-bounds Section 14.11), tau_g = 40/200 us (equations-and-bounds Section 14.19)

References

Luque & Ebert (2011), "Electron density fluctuations accelerate the branching of positive streamer discharges in air," Phys Rev E 84, 046411
da Silva et al. (2019), "The Plasma Nature of Lightning Channels and the Resulting Nonlinear Resistance," JGR Atmospheres
Bazelyan & Raizer (2000), "Lightning Physics and Lightning Protection," IOP Publishing
Nijdam et al. (2010), "Stereo-photography of streamers in air," J Phys D: Appl Phys
Phase 6 QCW Community Survey (2026)

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