Viscosity stratification and the aspect ratio of convection rolls

To clarify a mechanism by which earth's low--viscosity layer may increase the wavelength of mantle convection cells, we analyse the clockwise isothermal cellular motion driven by a uniform shear stress of magnitude $\tau$ applied at each end of a rectangle of height $2D$ and length $L$. The viscosity $\mu$ is a given piecewise-constant function of depth; within a low--viscosity channel of thickness $d$ located at the top of the layer, $\mu=m\mu_1$; elsewhere, within the `core', $\mu=\mu_1$. We show that in the double limit $d/D\to 0$, $m\to 0$, this two--layer flow is equivalent to one in single layer of viscosity $\mu_1$ with a new boundary condition at its top representing the interaction of the channel and core flows. Let $ x=x_*/L, $ $ y=y_*/D $ and $ \psi= \mu_1\psi_*/ \tau D^2. $ Then the stream function $\psi$ for the core motion satisfies the b.v.p. $ \psi_{yyyy}+2\alpha^2\psi_{xxyy} + \alpha^4\psi_{xxxx}=0; $ at $ |x|=1 $ , $\psi=0$, $ \alpha ^2\psi_{xx}=-1;$ at $ y=0 $, $\psi=0=\psi_{yy}; $ at $ y=1, $ $ \psi_{yy}- \alpha^2\psi_{xx}=0 $, and $ \psi_{yyy} +3\alpha^2\psi_{yxx} = 3\varepsilon\psi. $ Here $ \alpha=D/L $ and $ \varepsilon=mD^3/d^3. $ We find that for $\varepsilon\to 0$, the motion has two horizontal scales, namely $D$ and $L_1= D/\varepsilon^{1/2}\gg D$. If the rectangle length $L\sim L_1$, fluid sinks at one end and rises at the other; those end flows occur on the scale $D$, and are connected by a long--wave flow on the scale $L_1$. The cellular motion is closed within the low--viscosity layer. We have extended this method to treat convection rolls in a fluid of infinite Prandtl number. Our predicted heat flows agree well with those found in numerical simulations by Lenardic, Richards \& Busse {\it et al} (2005) ({\it J. Geophys. Res.}, to appear).