Mixing
Green's function modified Helmholtz operator in square periodic domain
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My question is motivated by point vortex dynamics. The motion of $N$ point vortices, each of circulation $Gamma_i$ and located at $mathbf{x}_i$, is induced for a given elliptic operator (e.g. Laplace and modified Helmholtz operators) by its Green's function $G(mathbf{x}, mathbf{x}')$ for the domain under consideration. Point vortex motion is a Hamiltonian system with Hamiltonian
$$ H = frac{1}{2} sum_{i=1}^N sum_{j=1atop jneq i} ^N Gamma_i Gamma_j G(mathbf{x}_i, mathbf{x}_j), $$
$$Gamma_i frac{mathrm{d} x_i}{mathrm{d} t} = -frac{partial H } { partial y_i}; hspace{1cm} Gamma_i frac{mathrm{d} y_i}{mathrm{d} t} = frac{partial H } { partial x_i} . $$
Periodic domains are commonly studied in numerical simulations since the avoid boundary effects. The Green's function for the Laplacian operator $nabla^2$ in a square periodic domain $[0,2pi]^2$ (torus) can be computed by tiling $mathbb{R}^2$ with infinitely many copies of the domain and summing over the contributions from each copy, using that the Green's function of the Laplacian in $mathbb{R}^2$ is $G(mathbf{x},mathbf{x}') = frac {1}{2pi} log(|mathbf{x}-mathbf{x}'|)$.
The result is:
$$
G(mathbf{x},mathbf{x}') = frac{1}{4pi} left( left[sum_{m=-infty}^infty logleft( frac{cosh(x-x'-2pi m)-cos(y-y')}{cosh(2pi m)}right) right] - frac{(x-x')^2}{2pi} + G_{o} right),
$$
where $G_o$ is an additive constant (see this article by G. Esler, formula B1, for instance). This function is $2pi$-periodic in $x-x'$ and $y-y'$. The great advantage of this result is that the remaning infinite series is very rapidly converging and may be evaluated very cheaply numerically.
The way this has been computed for the first time, namely in this article by McWilliams and Weiss is by summing up the velocity field (simple ratios which may be summed relatively easily) and then integrating back up to get the Hamiltonian and the Green's function.
Now, if one considers the modified Helmholtz operator $nabla^2-lambda^2$ (corresponding to plasma flows, for instance), how can one calculate the Greens function for the periodic domain?
The Green's function in $mathbb{R}^2$ is $G(mathbf{x},mathbf{x}')=frac{1}{2pi} K_0(lambda |mathbf{x} -mathbf{x}'|)$. It seems like a relatively standard problem, but so far I haven't managed to do it. Thanks for your help!
greens-function
asked Jan 24 at 13:11
CycloneCyclone
1,301717
$endgroup$
add a comment |
$begingroup$
My question is motivated by point vortex dynamics. The motion of $N$ point vortices, each of circulation $Gamma_i$ and located at $mathbf{x}_i$, is induced for a given elliptic operator (e.g. Laplace and modified Helmholtz operators) by its Green's function $G(mathbf{x}, mathbf{x}')$ for the domain under consideration. Point vortex motion is a Hamiltonian system with Hamiltonian
$$ H = frac{1}{2} sum_{i=1}^N sum_{j=1atop jneq i} ^N Gamma_i Gamma_j G(mathbf{x}_i, mathbf{x}_j), $$
$$Gamma_i frac{mathrm{d} x_i}{mathrm{d} t} = -frac{partial H } { partial y_i}; hspace{1cm} Gamma_i frac{mathrm{d} y_i}{mathrm{d} t} = frac{partial H } { partial x_i} . $$
Periodic domains are commonly studied in numerical simulations since the avoid boundary effects. The Green's function for the Laplacian operator $nabla^2$ in a square periodic domain $[0,2pi]^2$ (torus) can be computed by tiling $mathbb{R}^2$ with infinitely many copies of the domain and summing over the contributions from each copy, using that the Green's function of the Laplacian in $mathbb{R}^2$ is $G(mathbf{x},mathbf{x}') = frac {1}{2pi} log(|mathbf{x}-mathbf{x}'|)$.
The result is:
$$
G(mathbf{x},mathbf{x}') = frac{1}{4pi} left( left[sum_{m=-infty}^infty logleft( frac{cosh(x-x'-2pi m)-cos(y-y')}{cosh(2pi m)}right) right] - frac{(x-x')^2}{2pi} + G_{o} right),
$$
where $G_o$ is an additive constant (see this article by G. Esler, formula B1, for instance). This function is $2pi$-periodic in $x-x'$ and $y-y'$. The great advantage of this result is that the remaning infinite series is very rapidly converging and may be evaluated very cheaply numerically.
The way this has been computed for the first time, namely in this article by McWilliams and Weiss is by summing up the velocity field (simple ratios which may be summed relatively easily) and then integrating back up to get the Hamiltonian and the Green's function.
Now, if one considers the modified Helmholtz operator $nabla^2-lambda^2$ (corresponding to plasma flows, for instance), how can one calculate the Greens function for the periodic domain?
The Green's function in $mathbb{R}^2$ is $G(mathbf{x},mathbf{x}')=frac{1}{2pi} K_0(lambda |mathbf{x} -mathbf{x}'|)$. It seems like a relatively standard problem, but so far I haven't managed to do it. Thanks for your help!
greens-function
asked Jan 24 at 13:11
CycloneCyclone
1,301717
$endgroup$
add a comment |
$begingroup$
My question is motivated by point vortex dynamics. The motion of $N$ point vortices, each of circulation $Gamma_i$ and located at $mathbf{x}_i$, is induced for a given elliptic operator (e.g. Laplace and modified Helmholtz operators) by its Green's function $G(mathbf{x}, mathbf{x}')$ for the domain under consideration. Point vortex motion is a Hamiltonian system with Hamiltonian
$$ H = frac{1}{2} sum_{i=1}^N sum_{j=1atop jneq i} ^N Gamma_i Gamma_j G(mathbf{x}_i, mathbf{x}_j), $$
$$Gamma_i frac{mathrm{d} x_i}{mathrm{d} t} = -frac{partial H } { partial y_i}; hspace{1cm} Gamma_i frac{mathrm{d} y_i}{mathrm{d} t} = frac{partial H } { partial x_i} . $$
Periodic domains are commonly studied in numerical simulations since the avoid boundary effects. The Green's function for the Laplacian operator $nabla^2$ in a square periodic domain $[0,2pi]^2$ (torus) can be computed by tiling $mathbb{R}^2$ with infinitely many copies of the domain and summing over the contributions from each copy, using that the Green's function of the Laplacian in $mathbb{R}^2$ is $G(mathbf{x},mathbf{x}') = frac {1}{2pi} log(|mathbf{x}-mathbf{x}'|)$.
The result is:
$$
G(mathbf{x},mathbf{x}') = frac{1}{4pi} left( left[sum_{m=-infty}^infty logleft( frac{cosh(x-x'-2pi m)-cos(y-y')}{cosh(2pi m)}right) right] - frac{(x-x')^2}{2pi} + G_{o} right),
$$
where $G_o$ is an additive constant (see this article by G. Esler, formula B1, for instance). This function is $2pi$-periodic in $x-x'$ and $y-y'$. The great advantage of this result is that the remaning infinite series is very rapidly converging and may be evaluated very cheaply numerically.
The way this has been computed for the first time, namely in this article by McWilliams and Weiss is by summing up the velocity field (simple ratios which may be summed relatively easily) and then integrating back up to get the Hamiltonian and the Green's function.
Now, if one considers the modified Helmholtz operator $nabla^2-lambda^2$ (corresponding to plasma flows, for instance), how can one calculate the Greens function for the periodic domain?
The Green's function in $mathbb{R}^2$ is $G(mathbf{x},mathbf{x}')=frac{1}{2pi} K_0(lambda |mathbf{x} -mathbf{x}'|)$. It seems like a relatively standard problem, but so far I haven't managed to do it. Thanks for your help!
greens-function
asked Jan 24 at 13:11
CycloneCyclone
1,301717
$endgroup$
My question is motivated by point vortex dynamics. The motion of $N$ point vortices, each of circulation $Gamma_i$ and located at $mathbf{x}_i$, is induced for a given elliptic operator (e.g. Laplace and modified Helmholtz operators) by its Green's function $G(mathbf{x}, mathbf{x}')$ for the domain under consideration. Point vortex motion is a Hamiltonian system with Hamiltonian
$$ H = frac{1}{2} sum_{i=1}^N sum_{j=1atop jneq i} ^N Gamma_i Gamma_j G(mathbf{x}_i, mathbf{x}_j), $$
$$Gamma_i frac{mathrm{d} x_i}{mathrm{d} t} = -frac{partial H } { partial y_i}; hspace{1cm} Gamma_i frac{mathrm{d} y_i}{mathrm{d} t} = frac{partial H } { partial x_i} . $$
Periodic domains are commonly studied in numerical simulations since the avoid boundary effects. The Green's function for the Laplacian operator $nabla^2$ in a square periodic domain $[0,2pi]^2$ (torus) can be computed by tiling $mathbb{R}^2$ with infinitely many copies of the domain and summing over the contributions from each copy, using that the Green's function of the Laplacian in $mathbb{R}^2$ is $G(mathbf{x},mathbf{x}') = frac {1}{2pi} log(|mathbf{x}-mathbf{x}'|)$.
The result is:
$$
G(mathbf{x},mathbf{x}') = frac{1}{4pi} left( left[sum_{m=-infty}^infty logleft( frac{cosh(x-x'-2pi m)-cos(y-y')}{cosh(2pi m)}right) right] - frac{(x-x')^2}{2pi} + G_{o} right),
$$
where $G_o$ is an additive constant (see this article by G. Esler, formula B1, for instance). This function is $2pi$-periodic in $x-x'$ and $y-y'$. The great advantage of this result is that the remaning infinite series is very rapidly converging and may be evaluated very cheaply numerically.
The way this has been computed for the first time, namely in this article by McWilliams and Weiss is by summing up the velocity field (simple ratios which may be summed relatively easily) and then integrating back up to get the Hamiltonian and the Green's function.
Now, if one considers the modified Helmholtz operator $nabla^2-lambda^2$ (corresponding to plasma flows, for instance), how can one calculate the Greens function for the periodic domain?
The Green's function in $mathbb{R}^2$ is $G(mathbf{x},mathbf{x}')=frac{1}{2pi} K_0(lambda |mathbf{x} -mathbf{x}'|)$. It seems like a relatively standard problem, but so far I haven't managed to do it. Thanks for your help!
greens-function
greens-function
asked Jan 24 at 13:11
CycloneCyclone
1,301717
asked Jan 24 at 13:11
CycloneCyclone
1,301717
edited Jan 24 at 13:18
Cyclone
asked Jan 24 at 13:11
CycloneCyclone
1,301717
asked Jan 24 at 13:11
CycloneCyclone
1,301717
asked Jan 24 at 13:11
CycloneCyclone
1,301717
1,301717
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