CRR Thunderstorm System
1. Electric Field Accumulation and Coherence
Atmospheric electric fields accumulate through charge separation processes in thunderclouds. The CRR coherence operator represents this accumulation mathematically.
$$\mathbf{E}(x,t) = -\nabla \phi(x,t)$$ $$\frac{\partial \rho}{\partial t} = -\nabla \cdot \mathbf{J}$$ where \(\rho\) is charge density and \(\mathbf{J}\) is current density.
$$C(x,t) = \int_0^t L(x,\tau) \, d\tau$$ where \(L(x,t)\) represents the local charge accumulation rate:
$$L(x,t) = \mu(x,t) \cdot \sigma(x,t) \cdot \rho_{\text{moisture}}(x,t)$$
L_functional() = moisture × chargeRate × density × ageFactor
C += L_functional() × dt // Euler integration
2. Critical Breakdown Field and Rupture Threshold
Lightning occurs when the electric field exceeds the dielectric breakdown threshold of air. This is mapped to the CRR rupture condition.
$$E_{\text{breakdown}} \approx 3 \times 10^6 \, \text{V/m} \, (p/p_0)$$ where \(p\) is atmospheric pressure. Breakdown occurs when:
$$|\mathbf{E}(x,t)| \geq E_{\text{breakdown}}$$
$$C_{\text{crit}} = \Omega \ln\left(\frac{\Lambda}{\lambda_0}\right)$$ Rupture occurs deterministically when:
$$C(x,t) \geq C_{\text{crit}}$$
C_crit = Omega × log(Lambda / lambda0)
if (cloud.C >= cloud.C_crit) {
performRupture(); // Deterministic trigger
}
3. Leader Propagation and Field-Guided Pathfinding
Lightning leaders propagate through regions of high field strength, following the path of least resistance. This is implemented through field-based pathfinding.
$$\mathbf{v}_{\text{leader}} \propto \mathbf{E}_{\text{local}} + \nabla|\mathbf{E}|$$ Leader steps follow field gradients with stochastic perturbations.
Step selection weighted by:
$$S(x') = \alpha|\mathbf{E}(x')| + \beta C_{\text{field}}(x') + \gamma(y' - y) + \epsilon$$ where \(\epsilon\) represents stochastic branching.
score = fieldAttraction × 0.35 + coherenceAttraction × 0.45
+ groundAttraction × 0.035 + random() × 0.25
nextPos = argmax(score) // Select highest-scoring position
4. Positive Leaders from Ground
When downward leaders approach the ground, upward positive leaders can emerge from tall objects or field-enhanced points, propagating upward to meet the descending leader.
When downward leader reaches height threshold:
$h_{\text{leader}} \leq h_{\text{threshold}} \approx 100\text{px}$ Positive leader initiates with strong parent-seeking bias:
$S_{\text{up}}(x') = \alpha_p|\vec{r}_{\text{parent}}|^{-1} + \beta|\mathbf{E}(x')| + \gamma C_{\text{field}}(x') + \epsilon$ Maximum propagation: \(\Delta h_{\text{max}} \approx 120\text{px}\)
if (leaderY >= groundLevel - 100 && !hasPositiveLeader) {
// Probabilistic spawn (60-85% based on proximity & field)
probability = 0.6 + heightFactor×0.25 + fieldFactor;
if (random() < probability) createPositiveLeader();
}
// Strong bias toward parent: parentAttraction × 0.6
// Terminates if parent dies or distance > 120px
5. Post-Discharge Recovery and Regeneration
After discharge, the atmospheric system recovers through charge redistribution. The CRR regeneration operator weights this recovery by accumulated history.
$$\frac{\partial \mathbf{E}}{\partial t} = -\frac{\mathbf{E}}{\tau_{\text{relax}}} + \mathbf{S}_{\text{source}}$$ where \(\tau_{\text{relax}}\) is the relaxation timescale.
$$\mathcal{R}[\chi](x,t) = \int_{-\infty}^t \phi(x,\tau) \exp\left(\frac{C(x,\tau)}{\Omega}\right) d\tau$$ Weight function exponentially amplifies based on prior coherence.
rebirth_integral = 0;
for (hist of C_history) {
rebirth_integral += exp(hist.C / Omega) × dt;
}
rebirthWeight = rebirth_integral / normalization;
Summary: CRR as a Coarse-Grained Description
The CRR framework provides a coarse-grained mathematical description of thunderstorm electrodynamics. The coherence integral \(C(t) = \int_0^t L \, d\tau\) accumulates charge separation work. The rupture condition \(C \geq C_{\text{crit}}\) represents deterministic breakdown. The regeneration operator \(\mathcal{R} \propto \exp(C/\Omega)\) encodes memory effects in post-discharge recovery. Positive leaders from ground add physical realism to the bidirectional attachment process, spawning probabilistically (60-85% base chance) based on proximity and field strength, with strong parent-seeking behavior and realistic propagation limits.