Last time we derived a simple PDE describing the process of non-diffusive advection in a constantly flowing stream of water. That PDE looked like this: , where was the constant speed of the water. The solution of this PDE is a function of the form .
This PDE follows a more general form that looks like this: , where and are constants. If you’ve studied multivariable calculus, you might recognize this kind of equation. It is, in fact, a directional derivative.
We can rewrite , where . We can read this equation as “the gradient (or derivative) of in the direction of is zero”. Which means that in the direction , the solution attains the same constant value. It may be easier to see by graphing it:
If you think of as being a surface graphed above the plane of the image above, then on each of the light grey lines, has the same “height” above the screen. These lines are called characteristics. There’s no reason to suppose, however, that doesn’t vary across the different characteristics. As long as we follow one particular characteristic, however, will remain the same.
The mathematical interpretation of this line of reasoning is that is a function of the characteristics, since each characteristic determines a specific value of the solution. Therefore if we let represent the value of the characteristic, then , for some function . From the graph, we can tell that the characteristics are in fact lines. If we let be the value of the x-intercept, then the equation for the characteristic looks like: . Well, is a constant value, so we need to rearrange this expression to: . Since is constant for each line, we can let that be our .
Therefore we conclude that , for all functions . We can actually rewrite this as by just manipulating the equation at the end of the preceding paragraph. This tells us that the solution is moving in the positive direction, which matches well with our understanding of the advective equation. Note that to actually solve for specific solutions we need an initial condition of the form . Otherwise there’s no way to tell what the function actually is!