Baroclinic Instability in the Solar Tachocline for Continuous Vertical Profiles of Rotation, Effective Gravity, and Toroidal Field
Abstract
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 themore »
 Authors:
 High Altitude Observatory, National Center for Atmospheric Research, 3080 Center Green, Boulder, CO 803073000 (United States)
 Publication Date:
 OSTI Identifier:
 22663277
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Astrophysical Journal; Journal Volume: 842; Journal Issue: 2; Other Information: Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 79 ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; DISTURBANCES; EQUATIONS; GRAVITATION; INSTABILITY; MAGNETIC FIELDS; MAGNETOHYDRODYNAMICS; PERTURBATION THEORY; ROTATION; SUN; SUNSPOTS
Citation Formats
Gilman, Peter A., Email: gilman@ucar.edu. Baroclinic Instability in the Solar Tachocline for Continuous Vertical Profiles of Rotation, Effective Gravity, and Toroidal Field. United States: N. p., 2017.
Web. doi:10.3847/15384357/AA7682.
Gilman, Peter A., Email: gilman@ucar.edu. Baroclinic Instability in the Solar Tachocline for Continuous Vertical Profiles of Rotation, Effective Gravity, and Toroidal Field. United States. doi:10.3847/15384357/AA7682.
Gilman, Peter A., Email: gilman@ucar.edu. 2017.
"Baroclinic Instability in the Solar Tachocline for Continuous Vertical Profiles of Rotation, Effective Gravity, and Toroidal Field". United States.
doi:10.3847/15384357/AA7682.
@article{osti_22663277,
title = {Baroclinic Instability in the Solar Tachocline for Continuous Vertical Profiles of Rotation, Effective Gravity, and Toroidal Field},
author = {Gilman, Peter A., Email: gilman@ucar.edu},
abstractNote = {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.},
doi = {10.3847/15384357/AA7682},
journal = {Astrophysical Journal},
number = 2,
volume = 842,
place = {United States},
year = 2017,
month = 6
}

The solar tachocline is likely to be close to a geostrophic 'thermal wind', for which the Coriolis force associated with differential rotation is closely balanced by a latitudinal pressure gradient, leading to a tight relation between the vertical gradient of rotation and the latitudinal entropy gradient. Using a hydrostatic but nongeostrophic spherical shell model, we examine baroclinic instability of the tachocline thermal wind. We find that both the overshoot and radiative parts of the tachocline should be baroclinicly unstable at most latitudes. Growth rates are roughly five times higher in middle and high latitudes compared to low latitudes, and muchmore »

BAROCLINIC INSTABILITY IN THE SOLAR TACHOCLINE. II. THE EADY PROBLEM
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 twolayer 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 twolayer model, the overshoot tachocline is much more unstable than the radiative tachocline. The efolding growth times range from as short as 10 days to as long as several years, depending on latitude,more » 
NONLINEAR EVOLUTION OF GLOBAL HYDRODYNAMIC SHALLOWWATER INSTABILITY IN THE SOLAR TACHOCLINE
We present a fully nonlinear hydrodynamic 'shallowwater' model of the solar tachocline. The model consists of a global spherical shell of differentially rotating fluid, which has a deformable top, thus allowing motions in radial directions along with latitudinal and longitudinal directions. When the system is perturbed, in the course of its nonlinear evolution it can generate unstable lowfrequency shallowwater shear modes from the differential rotation, highfrequency gravity waves, and their interactions. Radiative and overshoot tachoclines are characterized in this model by high and low effective gravity values, respectively. Building a semiimplicit spectral scheme containing very low numerical diffusion, we performmore » 
Magnetic field gradient effects on RayleighTaylor instability with continuous magnetic field and density profiles
In this paper, the effects of magnetic field gradient (i.e., the magnetic field transition layer effects) on the RayleighTaylor instability (RTI) with continuous magnetic field and density profiles are investigated analytically. The transition layers of magnetic field and density with two different typical profiles are studied and the analytic expressions of the linear growth rate of the RTI are obtained. It is found that the magnetic field effects strongly reduce the linear growth rate of the RTI, especially when the perturbation wavelength is short. The linear growth rate of the RTI increases with the thickness of the magnetic field transitionmore » 
A neoclassical calculation of toroidal rotation profiles and comparison with DIIID measurements
Momentum and particle balance and neoclassical viscosity were applied to calculate the radial profile of toroidal rotation velocity in several DIIID [J. Luxon, Nucl. Fusion 42, 614 (2002)] discharges in a variety of energy confinement regimes (lowmode, lowmode with internal transport barrier, highmode, and highmode with quiescentd double barrier). Calculated toroidal rotation velocities generally were found to (over) predict measured values to well within a factor of 2.