[en] We present results from an MHD model for baroclinic instability in the solar tachocline that includes rotation, effective gravity, and toroidal field that vary continuously with height. We solve the perturbation equations using a shooting method. Without toroidal fields but with an effective gravity declining linearly from a maximum at the bottom to much smaller values at the top, we find instability at all latitudes except at the poles, at the equator, and where the vertical rotation gradient vanishes (32.°3) for longitude wavenumbers m from 1 to >10. High latitudes are much more unstable than low latitudes, but both have e -folding times that are much shorter than a sunspot cycle. The higher the m and the steeper the decline in effective gravity, the closer the unstable mode peak to the top boundary, where the energy available to drive instability is greatest. The effect of the toroidal field is always stabilizing, shrinking the latitude ranges of instability as the toroidal field is increased. The larger the toroidal field, the smaller the longitudinal wavenumber of the most unstable disturbance. All latitudes become stable for a toroidal field exceeding about 4 kG. The results imply that baroclinic instability should occur in the tachocline at latitudes where the toroidal field is weak or is changing sign, but not where the field is strong.

[en] We solve the nongeostrophic baroclinic instability problem for the tachocline for a continuous model with a constant vertical rotation gradient (the Eady problem), using power series generated by the Frobenius method. The results confirm and greatly extend those from a previous two-layer model. For effective gravity G independent of height, growth rates and ranges of unstable longitudinal wavenumbers m and latitudes increase with decreasing G. As with the two-layer model, the overshoot tachocline is much more unstable than the radiative tachocline. The e-folding growth times range from as short as 10 days to as long as several years, depending on latitude, G, and wavenumber. For a more realistic effective gravity that decreases linearly from the radiative interior to near zero at the top of the tachocline, we find that only m = 1, 2 modes are unstable, with growth rates somewhat larger than for constant G, with the same value as at the bottom of the tachocline. All results are the same whether we assume that the vertical velocity or the perturbation pressure is zero at the top of the layer; this is a direct consquence of not employing the geostrophic assumption for perturbations. We explain most of the properties of the instability in terms of the Rossby deformation radius. We discuss further improvements in the realism of the model, particularly adding toroidal fields that vary in height, and including latitudinal gradients of both rotation and toroidal fields

[en] We build a hydrodynamic model for computing and understanding the Sun's large-scale high-latitude flows, including Coriolis forces, turbulent diffusion of momentum, and gyroscopic pumping. Side boundaries of the spherical 'polar cap', our computational domain, are located at latitudes ≥ 60°. Implementing observed low-latitude flows as side boundary conditions, we solve the flow equations for a Cartesian analog of the polar cap. The key parameter that determines whether there are nodes in the high-latitude meridional flow is ε = 2ΩnπH²/ν, where Ω is the interior rotation rate, n is the radial wavenumber of the meridional flow, H is the depth of the convection zone, and ν is the turbulent viscosity. The smaller the ε (larger turbulent viscosity), the fewer the number of nodes in high latitudes. For all latitudes within the polar cap, we find three nodes for ν = 10¹² cm² s^–1, two for 10¹³, and one or none for 10¹⁵ or higher. For ν near 10¹⁴ our model exhibits 'node merging': as the meridional flow speed is increased, two nodes cancel each other, leaving no nodes. On the other hand, for fixed flow speed at the boundary, as ν is increased the poleward-most node migrates to the pole and disappears, ultimately for high enough ν leaving no nodes. These results suggest that primary poleward surface meridional flow can extend from 60° to the pole either by node merging or by node migration and disappearance.

[en] We show that simple two- and three-layer flux-transport dynamos, when forced at the top by a poloidal source term, can produce a widely varying amplitude of toroidal field at the bottom, depending on how close the meridional flow speed of the bottom layer is to the propagation speed of the forcing applied above the top layer, and how close the amplitude of the α-effect is to two values that give rise to a resonant response. This effect should be present in this class of dynamo model no matter how many layers are included. This result could have implications for the prediction of future solar cycles from the surface magnetic fields of prior cycles. It could be looked for in flux-transport dynamos that are more realistic for the Sun, done in spherical geometry with differential rotation, meridional flow, and α-effect that vary with latitude and time as well as radius. Because of these variations, if resonance occurs, it should be more localized in time, latitude, and radius.

[en] Extensive studies over the past decade showed that HD and MHD nonaxisymmetric instabilities exist in the solar tachocline for a wide range of toroidal field profiles, amplitudes, and latitude locations. Axisymmetric instabilities (m = 0) do not exist in two dimensions, and are excited in quasi-three-dimensional shallow-water systems only for very high field strengths (2 mG). We investigate here MHD axisymmetric instabilities in a three-dimensional thin-shell model of the solar/stellar tachocline, employing a hydrostatic, non-Boussinesq system of equations. We deduce a number of general properties of the instability by use of an integral theorem, as well as finding detailed numerical solutions for unstable modes. Toroidal bands become unstable to axisymmetric perturbations for solar-like field strengths (100 kG). The e-folding time can be months down to a few hours if the field strength is 1 mG or higher, which might occur in the solar core, white dwarfs, or neutron stars. These instabilities exist without rotation, with rotation, and with differential rotation, although both rotation and differential rotation have stabilizing effects. Broad toroidal fields are stable. The instability for modes with m = 0 is driven from the poleward shoulder of banded profiles by a perturbation magnetic curvature stress that overcomes the stabilizing Coriolis force. The nonaxisymmetric instability tips or deforms a band; with axisymmetric instability, the fluid can roll in latitude and radius, and can convert bands into tubes stacked in radius. The velocity produced by this instability in the case of low-latitude bands crosses the equator, and hence can provide a mechanism for interhemispheric coupling.

[en] We show that systematic differences between surface Doppler and magnetic element tracking measures of solar meridional flow can be explained by the effects of surface turbulent magnetic diffusion. Feature-tracking speeds are lower than plasma speeds in low and mid latitudes, because magnetic diffusion opposes poleward plasma flow in low latitudes whereas it adds to plasma flow at high latitudes. Flux-transport dynamo models must input plasma flow; the model outputs yield estimates of the surface magnetic feature tracking speed. We demonstrate that the differences between plasma speed and magnetic pattern speed in a flux-transport dynamo are consistent with the observed difference between these speeds.

[en] Evidence of the existence of hydrodynamic and MHD Rossby waves in the Sun is accumulating rapidly. We employ an MHD Rossby wave model for the Sun in simplified Cartesian geometry, with a uniform toroidal field and no differential rotation, to analyze the role of each force that contributes to Rossby wave dynamics, and compute fluid particle trajectories followed in these waves. This analysis goes well beyond the traditional formulation of Rossby waves in terms of conservation of vorticity. Hydrodynamic Rossby waves propagate retrograde relative to the rotation of the reference frame, while MHD Rossby waves can be both prograde and retrograde. Fluid particle trajectories are either clockwise or counterclockwise spirals, depending on where in the wave pattern they are initiated, that track generally in the direction of wave propagation. Retrograde propagating MHD Rossby waves move faster than their hydrodynamic counterparts of the same wavelength, becoming Alfvén waves at very high field strengths. Prograde MHD Rossby waves, which have no hydrodynamic counterpart, move more slowly eastward than retrograde MHD Rossby waves for the same toroidal field, but with a speed that increases with toroidal field, in the high field limit again becoming Alfvén waves. The longitude and latitude structures of all these waves, as seen in their velocity streamlines and perturbation field lines as well as fluid particle trajectories, are remarkably similar for different toroidal fields, rotation, longitudinal wavelength, and direction of propagation.

[en] In this work, we derive magnetic toroids from surface magnetograms by employing a novel optimization method, based on the trust region reflective algorithm. The toroids obtained in this way are combinations of Fourier modes (amplitudes and phases) with low longitudinal wavenumbers. The optimization also estimates the latitudinal width of the toroids. We validate the method using synthetic data, generated as random numbers along a specified toroid. We compute the shapes and latitudinal widths of the toroids via magnetograms, generally requiring several m's to minimize residuals. A threshold field strength is chosen to include all active regions in the magnetograms for toroid derivation, while avoiding non-contributing weaker fields. Higher thresholds yield narrower toroids, with an m = 1 dominant pattern. We determine the spatiotemporal evolution of toroids by optimally weighting the amplitudes and phases of each Fourier mode for a sequence of five Carrington Rotations (CRs) to achieve the best amplitude and phases for the middle CR in the sequence. Taking more than five causes “smearing” or degradation of the toroid structure. While this method applies no matter the depth at which the toroids actually reside inside the Sun, by comparing their global shape and width with analogous patterns derived from magnetohydrodynamic (MHD) tachocline shallow water model simulations, we infer that their origin is at/near the convection zone base. By analyzing the “Halloween” storms as an example, we describe features of toroids that may have caused the series of space weather events in 2003 October–November. Calculations of toroids for several sunspot cycles will enable us to find similarities/differences in toroids for different major space weather events.