A Constrained Tectonics Model for Coronal Heating

An analytical and numerical treatment is given of a constrained version of the tectonics model developed by Priest, Heyvaerts, & Title [2002]. We begin with an initial uniform magnetic field ${\bf B} = B_0 \hat{\bf z}$ that is line-tied at the surfaces $z = 0$ and $z = L$. This initial configuration is twisted by photospheric footpoint motion that is assumed to depend on only one coordinate ($x$) transverse to the initial magnetic field. The geometric constraints imposed by our assumption precludes the occurrence of reconnection and secondary instabilities, but enables us to follow for long times the dissipation of energy due to the effects of resistivity and viscosity. In this limit, we demonstrate that when the coherence time of random photospheric footpoint motion is much smaller by several orders of magnitude compared with the resistive diffusion time, the heating due to Ohmic and viscous dissipation becomes independent of the resistivity of the plasma. Furthermore, we obtain scaling relations that suggest that even if reconnection and/or secondary instabilities were to limit the build-up of magnetic energy in such a model, the overall heating rate will still be independent of the resistivity.

💡 Research Summary

The paper presents a highly simplified, analytically tractable version of the “tectonics” model originally proposed by Priest, Heyvaerts, and Title (2002) to address the long‑standing problem of coronal heating. The authors begin with a uniform vertical magnetic field B = B₀ ẑ that is line‑tied at two planar boundaries (z = 0 and z = L). The key geometric restriction is that the photospheric footpoint motions are assumed to depend on only one transverse coordinate, x, while being independent of y. Under the low‑β, reduced MHD (RMHD) approximation, this restriction reduces the full three‑dimensional RMHD equations to a pair of linear, one‑dimensional equations for the stream function φ(x,z,t) and the magnetic potential A(x,z,t). By expanding φ and A in Fourier modes along x, the problem separates into independent ordinary differential equations in z for each mode n, with resistivity η and viscosity ν appearing only as diffusion coefficients in the x‑direction.

Two distinct driving scenarios are examined. First, a constant footpoint velocity (time‑independent φₙᴸ) is imposed. In the steady‑state limit (∂/∂t → 0) the authors obtain closed‑form expressions for φₙ(z) and Aₙ(z). In the limit of very small η, the transverse magnetic field B⊥ becomes essentially uniform along z and scales as B⊥ ≈ v_L τ_r/L, where v_L is the typical photospheric speed and τ_r = w²/η is the resistive diffusion time based on the characteristic footpoint displacement scale w. Consequently, the Ohmic dissipation rate W_d ∝ η J² scales as η⁻¹, diverging as η → 0. This analytical solution serves as a benchmark for the numerical code.

Second, the authors introduce a more realistic, stochastic footpoint driver. The driver is constructed as a sum of many sinusoidal modes with random phases that evolve via a random walk with a prescribed coherence time τ_coh. The amplitude φ₀(t) follows φ₀ cos θ(t), where θ(t) increments by a random number multiplied by π Δt/τ_coh at each time step. This mimics the observed near‑random shuffling of magnetic elements in the solar magnetic carpet. Numerical integration is performed using spectral decomposition in x, a leap‑frog scheme in z, and an implicit time‑stepping algorithm that permits long time steps despite the stiff diffusion terms. Simulations are run for many resistive diffusion times to achieve a statistical steady state.

The central result is that when τ_coh ≪ τ_r (the coherence time of the footpoint motions is orders of magnitude shorter than the resistive diffusion time), the time‑averaged heating rate becomes essentially independent of η. The authors derive a scaling W_d ≈ B₀² v_L l_r/L, where l_r = v_L τ_r is the distance a footpoint travels during a resistive diffusion time. Because l_r itself scales as 1/η, the η‑dependence cancels, leaving a heating rate set solely by the photospheric velocity and the magnetic field strength. This independence persists even if reconnection or secondary instabilities were to limit the buildup of magnetic energy, as suggested by additional scaling arguments.

The paper discusses the implications for the solar corona, where the Lundquist number (S = v_A L/η) can reach 10¹²–10¹³, making τ_r astronomically large (10¹²–10¹³ s). In such a regime, the observed photospheric coherence times (∼10³ s) are indeed far shorter than τ_r, placing the corona squarely in the regime where the model predicts resistivity‑independent heating. The authors acknowledge that their model deliberately excludes nonlinear mode coupling, reconnection, and turbulence, but argue that the derived scaling relations are robust enough to suggest that even in a fully three‑dimensional, turbulent corona, the overall heating rate may remain governed primarily by the statistics of footpoint motions rather than by microscopic resistivity.

In summary, the study provides a clear analytical framework and supporting numerical evidence that, under realistic solar conditions, the coronal heating rate can be decoupled from plasma resistivity. This challenges the notion that the efficiency of nanoflare‑type heating must depend sensitively on η, and instead highlights the dominant role of photospheric footpoint dynamics in supplying the energy required to sustain the hot solar corona.