Exact solution to advection equation in spherical coordinates

1. Sep 23, 2010

lostidentity

I've been trying to find the exact solution to the advection equation in spherical coordinates given below

$$\frac{\partial{\phi}}{\partial{t}} + \frac{u}{r^2}\frac{\partial{}}{\partial{r}}r^2\phi = 0$$

Where the velocity u is a constant. First I tried to expand the second term using product rule and then apply the separation of variables, which gives me the following.

Separation of variables

$$\phi = R(r)T(t)$$

$$\frac{1}{T}\frac{dT}{dt} = -\frac{u}{R}\frac{dR}{dr} - \frac{2u}{r} = -\lambda^2$$

$$\phi = Ae^{\frac{\lambda}{u}(r-ut)-2}$$
Wonder if this is correct? For Cartsesian coordinates I know the solution is very simple just $$\phi=\phi_0(x-ut)$$ where $$\phi_0$$ is the initial profile for phi.