The stellar thermal wind as a consequence of oblateness

The lowest-order force balance in rotating stars is between gravity, pressure, and the centrifugal force (here referred to as 'GPR' balance). GPR balance determines both the stellar oblateness and the aspherical thermal anomalies. Here we emphasize a subtle point. Stellar thermal wind balance is simply the curl of GPR balance and the stellar thermal wind should be regarded as the baroclinic component of the oblateness. The thermal wind thus determines only part of the aspherical thermal anomalies, which have both baroclinic and barotropic contributions. Here we treat the full oblateness, including the thermal wind, using pressure coordinates. We derive the generalised stellar thermal wind equation and identify the parameter regime for which it holds. In the case of the Sun, not including the oblateness has resulted in conflicting calculations of the theoretical aspherical temperature anomaly. We provide new calculation here and find that the baroclinic anomaly from the thermal wind is ~3-60 times smaller than the barotropic anomaly and may not be measurable helioseismically. If measurement were possible, this would potentially yield a new way to bracket the depth of the solar tachocline.