Coronal Heating Damping Scale

τ-Formula

ℓ_heat = ι_τ² · R_☉ ≈ 0.117 R_☉

Derivation

F_τ \;=\; ρ\,v_A\,v_conv^2 \;(1 - e^-ι_τ^2\,L/λ_A),

where $ρ$ is the photospheric mass density, $v_A$ the Alfv'en speed, $v_conv ∼ 1$ km/s the convective velocity, $L$ the coronal loop length, and $λ_A$ the dominant Alfv'en wavelength. No free parameters enter the damping fraction $ι_τ^2$.

Numerical evaluation. For typical active-region parameters ($ρ ∼ 3 × 10^-15$ g cm$^-3$, $B_0 ∼ 100$ G, $v_conv ∼ 1$ km/s, $L ∼ 10^10$ cm, $λ_A ∼ 10^9$ cm):

• The Alfv'en speed is $v_A = B_0/4πρ ≈ 1.6 × 10^8$ cm s$^-1$.
• The wave energy flux is $F_A = ρ\,v_A\,v_conv^2 ≈ 4.9 × 10^5 \;erg\;cm^-2\;s^-1$.
• The damping fraction is $1 - (-ι_τ^2\,L/λ_A) = 1 - (-0.117 × 10) ≈ 0.69$.
• The heating flux is $F_τ ≈ 3.4 × 10^5 \;erg\;cm^-2\;s^-1$.

Source

This prediction is derived in the Numerical Physics Ledger (Chapter 65 — collective-dynamics), Books IV–V of Panta Rhei.