Rotating and magnetized stratified horizontal-shear turbulence at low Prandtl number: regime boundaries, scaling laws, and the sign of turbulent angular-momentum transport in the solar tachocline

The solar tachocline, the thin shear layer at the base of the solar convection zone, is expected to be turbulent because of the latitudinal differential rotation communicated downward from the convection zone. Simulations and asymptotic theory published in 2020–2024 established that non-rotating, hydrodynamic, strongly stratified turbulence driven by horizontal shear at low Prandtl number ($Pr\ll1$) falls into distinct regimes with scaling laws for the vertical eddy scale, velocity, and turbulent diffusivity, and these scalings have been numerically validated. However, the tachocline rotates (small large-scale Rossby number) and is likely magnetized, and it is not known how rotation and magnetic fields alter the existence, boundaries, and scalings of this stratified turbulent regime, nor whether the resulting large-scale turbulent transport of angular momentum is diffusive (down-gradient) or anti-diffusive/non-diffusive (e.g. off-diagonal stress). This sign and tensor-structure question decides whether purely hydrodynamic tachocline confinement models are self-consistent or whether a magnetic field is required, and it remains open as of 2026.

Background

The solar tachocline is a thin (a few percent of the solar radius, inferred from helioseismology — the inversion of solar oscillation frequencies) shear layer straddling the base of the convection zone, across which the latitudinal differential rotation of the envelope relaxes to the nearly uniform rotation of the radiative interior. Spiegel and Zahn (1992) modeled it as a steady state of thermal-wind balance (a balance between Coriolis terms and buoyancy in the momentum equation), angular-momentum balance, and thermal equilibrium, assuming the turbulence present there acts as a strongly anisotropic viscosity, dominant in the horizontal direction because the stable stratification (quantified by the buoyancy or Brunt–Väisälä frequency $N$, the oscillation frequency of a displaced fluid parcel) suppresses vertical motion. In that model the tachocline 'burrows' into the interior by radiative spreading on secular timescales; a dominant horizontal turbulent viscosity is invoked to stop the burrowing. Gough and McIntyre (1998) argued instead that only an interior fossil magnetic field can confine the tachocline, partly because two-dimensional rotating turbulence tends to transport angular momentum anti-diffusively (up-gradient, sharpening rather than smoothing shear), as in the Earth's atmosphere; Tobias, Diamond and Hughes (2007) similarly showed, in β-plane magnetohydrodynamics (MHD on the rotating tangent plane with the latitudinal variation of the Coriolis parameter), that tachocline turbulence need not transport angular momentum diffusively at all. Whether the real, strongly stratified, thermally diffusive tachocline turbulence acts diffusively or anti-diffusively is the crux separating hydrodynamic from magnetic confinement scenarios.

The relevant fluid dynamics is controlled by the Prandtl number $Pr=\nu/\kappa_T$ (ratio of viscosity to thermal diffusivity; about $10^{-6}$ in the solar tachocline), the Reynolds number $Re$ (inertia versus viscosity; huge in stars), the Péclet number $Pe=Pr\,Re$ (advection versus thermal diffusion), and the stratification strength. Because stellar $Pr$ is tiny, radiative-zone turbulence differs fundamentally from geophysical stratified turbulence. In a 2020 series of works (Garaud's review-chapter 'The tachocline revisited', the companion ApJ paper, and Cope, Garaud and Caulfield's Journal of Fluid Mechanics paper), direct numerical simulations (DNS) of idealized, triply periodic, uniformly stratified flows driven by a horizontal-shear body force — deliberately ignoring rotation, magnetic fields, curvature, and compositional stratification — showed that the flow develops vertically modulated meanders whose thin vertical scales are thermally diffusive, triggering secondary vertical (Kelvin–Helmholtz-type) shear instabilities and producing small-scale, nearly isotropic turbulence with anisotropy only at the large horizontal scales. Several regimes exist depending on the relative importance of stratification and thermal diffusion, with theoretically motivated scaling laws for the vertical eddy scale $l_z$, vertical velocity $u_z^{rms}$, temperature fluctuations, and the vertical turbulent diffusion coefficient $D$; the tachocline was argued to lie in the (high-Péclet) stratified turbulence regime, and the implied turbulent diffusivity, taken at face value, is large enough to threaten the assumptions of the Spiegel–Zahn model.

The 2020 chapter explicitly listed what remained open: (1) the simulations ignored rotation and magnetic fields, so how these alter the regime's existence, boundaries, and scalings — and whether they flip the sign of the net large-scale angular-momentum transport — was unknown; (2) the regime boundaries and scaling prefactors at truly astrophysical parameters ($Pr\ll10^{-6}$-level, very large $Pe$ and $Re$), including possible logarithmic corrections in $Pr$, were conjectural; (3) the nonlinear interaction between the tachocline's large-scale vertical shear and the horizontal shear instabilities was unexplored. Substantial progress followed: Chini et al. (2022) derived a multiscale, quasilinear reduced model of strongly stratified turbulence in which fast small-scale shear instabilities are slaved to marginally stable slow large-scale fields; Shah et al. (2024) extended this to low $Pr$ with a formal asymptotic regime map, recovering the Cope et al. scalings in the fully diffusive regime and Chini et al.'s in the non-diffusive regime, and showing that new diffusive regimes exist at low $Pr$ (also implying Zahn's standard stellar-evolution prescription is valid only in part of parameter space); Garaud et al. (2024) numerically validated these scaling laws; Skoutnev (2023) derived complementary critical-balance scalings for the regime $Pr\,Rb<1$ ($Rb$ the buoyancy Reynolds number built on the turbulent vertical scales). On the vertical-shear question, Garaud, Khan and Brown (2024) showed by DNS that in the non-diffusive regime the mean vertical shear does not control the turbulence unless its Richardson number $Ri$ (stratification versus shear; below unity a vertical shear is unstable) is below order one, as it rarely is in stars. On the rotation side, only linear stability analyses with the full Coriolis acceleration (Park, Prat, Mathis and Bugnet 2020) and phenomenological relaxation-time (τ-approximation) transport scalings (Prat and Mathis 2021) exist; no nonlinear simulations of the low-$Pr$ stratified turbulent regime with rotation or magnetic fields have been published, and Shah et al. explicitly note their regime map holds only when rotation and magnetic fields are absent. Meanwhile Garaud, Gough and Matilsky (2025) used the validated non-rotating scalings to build a new self-consistent hydrodynamic tachocline model (with thickness scaling as the ratio of the solar angular velocity to the buoyancy frequency, $h\sim\Omega_\odot/N_m$), replacing Spiegel–Zahn under a diffusively approximated, anisotropic, horizontally isotropic turbulent transport — but the paper itself states that whether the tachocline turbulence indeed acts as a diagonal diffusive tensor, and how rotation and magnetization modify it, remains an open question that must be settled to complete the theory. The 2023 review by Strugarek et al. confirms that confinement of this 'doubly sheared, highly stratified, low-Pr, magnetized' layer remains a central unsolved problem.

Problem Statement

For stably stratified, thermally diffusive turbulence driven by horizontal shear at low Prandtl number, in the parameter regime of the solar tachocline ($Pr\sim10^{-6}$, $Re\gg1$, strong stratification, concurrent large-scale latitudinal shear, and likely magnetization), determine how rotation and magnetic fields modify the problem's nonlinearly saturated state. Specifically: (i) establish how rotation (including the full Coriolis acceleration at the small large-scale Rossby numbers relevant to the tachocline) and magnetic fields alter the existence, regime boundaries, and scaling laws of the stratified turbulent regime identified in the non-rotating, hydrodynamic case — including whether the primary large-scale horizontal-shear instability survives, is modified, or is suppressed; (ii) determine the resulting effective turbulent transport of angular momentum and heat at large scales, in particular the sign and tensor structure of the turbulent stress — whether it is well represented by a diagonal, down-gradient (diffusive) viscosity/diffusivity as assumed in hydrodynamic tachocline models, or contains anti-diffusive or off-diagonal (non-diffusive) contributions as two-dimensional rotating and β-plane MHD turbulence would produce; and (iii) quantify how the concurrent large-scale vertical shear of the tachocline, in combination with the latitudinal shear, modifies this nonlinear saturation and the resulting transport coefficients. These three parts are jointly required: a complete treatment of the open problem must address all of (i)–(iii), and a result on any single part does not by itself resolve the task.

References

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