I accidentally posted this to the "Calculus & Beyond" forum when I meant to post it to the physics forum. If someone can tell me how to move this post, I will get rid of it here! 1. The problem statement, all variables and given/known data Consider a property, for example temperature θ, that is conserved during advection (i.e. Dθ/Dt = 0 where D/Dt represents the total derivative). Suppose the velocity of the field advecting the fluid is: u = ax, v = -ay, w = 0 Find the form of the temperature for t > 0 when at t=0, θ(t=0) = θ0y/L. 2. Relevant equations Dθ/Dt = ∂θ/∂t + u*(∂θ/∂x) + v*(∂θ/∂y) 3. The attempt at a solution So far, I've calculated the trajectories: x(t) = C1e^(at) y(t) = C2e^(-at) I also know that since θ is conserved, θ is a function of where in space the parcel originated. In other words, θ=θ(Y) where Y is the Lagrangian variable. I've expanded the total derivative and set it equal to zero since θ is conserved: (1) Dθ/Dt = ∂θ/∂t + u*(∂θ/∂x) + v*(∂θ/∂y) = 0 Then, I thought to say that θ(t)=(θ0y(t)/L)*f(t), where f(t) is some function of time that we want to find out so we can see what the function θ(t) looks like. I took this θ(t)=(θ0y(t)/L)*f(t) and plugged it back into equation (1), but don't think I'm getting anywhere. Can anyone tell me if I'm on the right track or if I'm going about this completely the wrong way?