Diffusion Equation Question

[juan-park]

I see from a other papers that the diffusivity equation is for displacement. After derivation the equation I get is , but in tracker.py it is .

Am I missing something? Any guidance would be greatly appreciated!

[bjornaa]

Dear Juan,

I have double checked and am convinced that the formulation is correct. With a diffusion equation ct = D(c{xx} + c_{yy}) the variance in x-direction grows linearly with coefficient as 2D. With random walk, X[n+1] = X[n] + R, the variance grows linearly with coefficient <R^2>/dt. LADiM uses random walk velocity u = R/dt, giving 2D = <u^2>*dt, the formula used in tracker.py

Note that the particle tracking uses the variance in x and y directions separately. The total two-dimensional variance grows like 4D (and 6D for three-dimensional isotropic diffusion). This may be the reason different factors show up in the literature.

Bjørn

[gillibrandpa]

Hi Juan, Bjørn The reason lies in the choice of random number generator.

Traditionally, dx = Rsqrt(2D_h*dt) was used in particle tracking models, with the random number generator being R = +/- 1. This generator has a random number distribution with a mean of zero and a standard deviation of 1. Particles then diffused according to Fickian diffusion theory, with the particle variance proportional to 2.D_h.t

The downside to that approach was that all random steps were the same size: +/- sqrt(2. D_h. dt) in the x- and y-directions.

Choosing a random number generator of -1 <= R <= 1 (i.e. R = 2rnd -1) gives a range of random displacement steps. This generator has a mean of zero, but a standard deviation of 1/sqrt(3). The correction to get a standard deviation of 1 involves multiplying by sqrt(3). So dx = Rsqrt(3)*sqrt(2. D_h. dt) = Rsqrt(6.D_h.dt). We then have the correct representation of Fickian diffusion, such that particle location variance grows as 2.D_h.t again.

Hope that helps. Best wishes Phil Gillibrand

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Dr Philip Gillibrand Oceanographer/Modeller

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Dear Juan,

I have double checked and am convinced that the formulation is correct. With a diffusion equation ct = D(c{xx} + c_{yy}) the variance in x-direction grows linearly with coefficient as 2D. With random walk, X[n+1] = X[n] + R, the variance grows linearly with coefficient <R^2>/dt. LADiM uses random walk velocity u = R/dt, giving 2D = <u^2>*dt, the formula used in tracker.py

Note that the particle tracking uses the variance in x and y directions separately. The total two-dimensional variance grows like 4D (and 6D for three-dimensional isotropic diffusion). This may be the reason different factors show up in the literature.

Bjørn

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[bjornaa]

Thanks Phil,

I may have misinterpreted the Juan's R.

Attached is a pdf file that describes more closely how LADiM is handling horizontal diffusjon. random_walk_diffusion.pdf

Bjørn