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The advection equation is :

$$\frac{\partial f(x,y,t)}{\partial t} + \nabla_{(x,y)} \cdot (A f)= 0$$

With initial condition $f(x,y,0) = f_0(x,y)$.

If the vector $A$ is constant, ie. $A = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix}$ we have a linear advection. Using the characteristic method we found

$$\frac{dX}{dt} = a_1\implies X(t) = x_0 + a_1t \\ \frac{dY}{dt} = a_2\implies Y(t) = y_0 + a_2t$$

If $A = \begin{pmatrix} -x \\ y \end{pmatrix}$ the motion of the advection is a circle :

$$\frac{dX}{dt} = y \implies X(t) = c_0\sin(t+c_1) \\ \frac{dY}{dt} = -x\implies Y(t) = c_0 \cos(t+c_1)$$

So I was wondering what can I do to have a spiral advection ? Let's say I want to found :

$$X(t) = t\cos(t+\theta_0) \\ Y(t) = t\sin(t+\theta_0)$$

if we derivate this system :

$$\frac {dX(t)}{dt} = -t\sin(t+\theta_0) + \cos(t+\theta_0) = x/t - y = a_1(t,x,y)$$

$$\frac {dY(t)}{dt} = t\cos(t+\theta_0) + \sin(t+\theta_0) = y/t + x = a_2(t,x,y)$$

We can notice then that $\nabla \cdot A = \dfrac{2}{t}$

Then the equation is :

$$\frac{\partial f(x,y,t)}{\partial t} + A \cdot \nabla_{(x,y)} f = - \dfrac{2}{t} f$$

But I don't manage to solve it. Using the characteristic method we found the spiral equation for X and Y but for the last element I have :

$$\dfrac{dF(x,y,t)}{dt} = - \dfrac{2}{t}f \implies f = C_0 / t^2$$

which is not defined when $t=0$ so I don't know how to apply the initial condition.

How can I solve this equation ?

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