A Theoretical Estimate of the Pole-Equator Temperature Difference and a Possible Origin of the Near-Surface Shear Layer

Abstract

Convective motions in the deep layers of the solar convection zone are affected by rotation, making the convective heat transport latitude-dependent, but this is not the case in the top layers near the surface. We use the thermal wind balance condition in the deeper layers to estimate the pole–equator temperature difference. Surface observations of this temperature difference can be used for estimating the depth of the near-surface layer within which convection is not affected by rotation. If we require that the thermal wind balance holds in this layer also, then we have to conclude that this must be a layer of strong differential rotation and its characteristics which we derive are in broad agreement with the observational data of the near-surface shear layer.

Appendix A: On Differentiating Entropy Along an Ioschore

The difference of specific entropy $\Delta S$ given by Equation 5 refers to an isochore over which $\rho$ does not vary. If $l$ is the length measured along an isochore, then the entropy difference between two points on an isochore should be given be

$$ \Delta S = \frac{d S}{dl} \Delta l. $$

(A.1)

From the chain rule of partial differentiation, it follows that

$$ \frac{d S}{dl} = \left (\frac{\partial S}{\partial r} \right )_{ \theta } \frac{d r}{dl} + \left (\frac{\partial S}{\partial \theta } \right )_{r} \frac{d \theta }{dl}. $$

It is easy to argue that the first term in this equation is going to be negligible. Efficient convection tends to homogenize $S$ in the radial direction so that $(\partial S/ \partial r)_{\theta } \approx 0$. Since the rotational flattening of the Sun is very small, we expect the isochoric surfaces to be very nearly spherical so that $d r/dl$ is also expected to be very small. Since the first term is a product of two small terms, we can write

$$ \frac{d S}{dl} \approx \left (\frac{\partial S}{\partial \theta } \right )_{r} \frac{d \theta }{dl}. $$

Substituting this in Equation A.1, we get

$$ \Delta S \approx \left (\frac{\partial S}{\partial \theta } \right )_{r} \Delta \theta . $$

(A.2)

If we take $\Delta \theta = \pi /2$ for the pole–equator difference, then Equation A.2 leads to Equation 6.