TransWikia.com

Advection-diffusion with periodic boundary conditions

Physics Asked by Quillo on October 7, 2020

Context: Consider the advection-diffusion equation with periodic boundary conditions (PBC) over a flat square domain $L times L$.
The scalar density $rho $ is transported by a prescribed field $mathbf{v}=-nabla U$, where $U(mathbf{x})$ is a scalar potential that has the periodicity imposed by the PBC. The density $rho$ evolves as

$$
partial_t rho(mathbf{x},t) = -nabla cdot [ mathbf{v}(mathbf{x}) rho(mathbf{x},t) – nabla rho(mathbf{x},t) ] = 0
$$

The steady-state solution is found by imposing $partial_t rho(mathbf{x},t) =0$ and has the usual Gibbs form:

$$
rho(mathbf{x}) , propto , e^{-U(mathbf{x}) }
$$

The problem: I am wondering how to find the steady-state the case in a slightly more general case, where

$$mathbf{v} = -nabla U + mathbf{q}$$

The potential $U$ has the periodicity imposed by the PBC and $mathbf{q} =(q_x,q_y)$ is a constant vector field. Hence, the equation we have to solve is

$$
nabla cdot [ , rho(x,y) , mathbf{q} – rho(x,y) nabla U(x,y) – nabla rho(x,y) , ] = 0
$$

with the periodic conditions
$rho(0,y) = rho(L,y)$,
$rho(x,0) = rho(x,L)$,
$U(0,y) = U(L,y)$,
$U(x,0) = U(x,L)$. For simplicity, I tried to consider the case $mathbf{q}=(q,0)$, but the problem still seems non-trivial.

Question: Any idea or reference about the diffusion-advection equation in periodic boundary conditions (in particular about the steady state)? Which is the "Gibbs-like solution" in this case?

Further considerations: I have the feeling that finding a solution is not easy because the potential that generates the constant field $mathbf{q}$ is $-mathbf{x}cdot mathbf{q}$, that is not periodic (i.e. it does not satisfy the PBC contitions).

Moreover, define the total current in the steady state as

$$
mathbf{J}(x,y) = rho(x,y) , [mathbf{q} – nabla U(x,y)] – nabla rho(x,y) , ,
$$

so that we have to find the $mathbf{J}$ such that

$$
nabla cdot mathbf{J} = 0 quad Rightarrow quad mathbf{J} = R nabla g
$$

where $R$ is a 90-degrees rotation and $g$ an unknown scalar potential. Note that $g$ does not have to respect the PBC, but $mathbf{J}$ does: (probably) the most general form of $g$ is

$$
g(x,y) = G(x,y) + a x + b y
$$

where $G$ respects the PBC and $a$ and $b$ are constants. Even though this problem is more likely to be studied by physicists, I have the feeling that the problem is intimately related to the topology of the 2D torus, so I posted also a similar question on math SE.

One Answer

Any $rho$ which solves the equation on the whole torus must also be a solution locally on every subset. In particular, it must be solution on the (non-toroidal) open $L times L $ square. Since solutions on the torus are a subset of the solutions on the square, the question becomes: Do there exist solutions on the square which happen to match at the boundaries?

On this square, we can define $V = U - mathbb{x} cdot mathbb{q}$, and we have an ordinary advection-diffusion equation. We know there exist solutions of the form $alpha e^{-V(mathbb{x})}$. We also know that $U$ is periodic, so $V$ can only be periodic if $mathbb{q} = mathbb{0}$. However $e^{-V}$ could still be periodic if $mathbb{q}$ is imaginary. Specifically, we have periodic solutions for $mathbb{q} = frac{2pi i}{L}mathbb{n}, mathbb{n} in mathbb{Z}^2$.

For other $mathbb{q}$, solutions proportional to $ e^{-V(mathbb{x})} $ cannot extend to solutions on the whole torus. The remaining question: Are such solutions the whole solution space?

Now, Matthew Kvalheim points to Zeeman, 1988. Theorem 3 reads

Let $U$ be a vector field on a compact manifold $X$ without boundary, and let $epsilon$ > 0. Then the Fokker-Planck equation for $U$ with $epsilon$-diffusion has a unique steady state, and all solutions tend to that steady state.

The torus is a compact manifold without boundary, Zeeman's $U$ is our $-nabla V$, and we have $epsilon = 1$, so the theorem tells us a solution $rho$ must exist and is unique (up to an overall scalar). Unfortunately, this proof is not constructive.

In one dimension, variation of parameters gives the solution $$rho = C_1 e^{-V}left(C_2 + int_0^x e^Vright)$$ and the requirement $rho(0) = rho(L)$ fixes $C_2$. We can try to extend this to two dimensions as follows: Assume $rho$ is of the form $alpha(x)e^{-V}$. Then the equation becomes $$ nabla cdot [-nabla V alpha(x)e^{-V} - nabla (alpha(x) e^{-V})] = 0 $$ which simplifies to $$ nabla cdot (nablaalpha(x) e^{-V}) = 0 $$ Solutions are $$ nablaalpha(x) e^{-V} = nabla times mathbf{psi} $$ for $mathbf{psi} = mathbf{e}_z $ and $g$ some scalar function. Then $$ nablaalpha(x) = e^V(nabla times mathbf{psi}) $$ If $$ nabla times (e^V(nabla times mathbf{psi})) = 0 $$ then this has solution $$ alpha(x,y) = C + left(int_0^x -e^V g_y dxright) + left(int_0^y e^V g_x dyright) $$ The requirement of periodic boundary conditions picks out some unique $g$, $C$ up to an overall constant. We need $$ alpha(x,0) = alpha(x,L)e^{-Lq_y} $$ or $$ C + left(int_0^x -e^V g_y dxright) = Ce^{-Lq_y} + left(int_0^x -e^V g_y dxright)e^{-Lq_y} + left(int_0^L e^V g_x dyright)e^{-Lq_y} $$ At $x = 0$ this simplifies to $$ C = frac{1}{e^{Lq_y} - 1}int_0^L e^V g_x(0,y) dy $$ It remains to find $g$.

I'm not sure that there is a nice expression for the solution in general. Some miscellaneous thoughts:

  • When $U = 0$, $rho = C$ is a solution, which corresponds to $alpha = e^V, g = xq_y-yq_x$. This shows that $g$ may be defined only on the square, not on the torus.
  • When $nabla U gg q$ or $q gg nabla U $, we can start with the nearby known solution and series expand.

Correct answer by Daniel on October 7, 2020

Add your own answers!

Ask a Question

Get help from others!

© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP