# What is a typical value of the Péclet number for internal waves?

Defining the Péclet number as:

$Pe = \frac{U \lambda}{\kappa}$

Where $U$ is the characteristic fluid velocity, $\lambda$ is the wavelength of the internal wave and $\kappa$ is the turbulent eddy diffusivity. I found in Moum et al. (2003) that measurements show the mean of turbulent eddy diffusivity in Internal Solitary Waves is $\kappa = 0.005$ m$^2$ s$^{-1}$, and a typical velocity is $U = 0.6$ m s$^{-1}$. So for a $1000$ m wave I arrive at a Péclet number:

$Pe = 120,000$

In your experience is that a typical value that you have observed?

I don't have personal experience with this situation, but reading around suggests it depends what kind of mixing you are talking about, for example whether it is vertical or horizontal.

Looking around for recent examples, I see Chao et al. (2007) had a value of 4.0 m2/s for horizontal diffusivity. This larger value would give you a smaller Péclet number of 150. Your use of wavelength for the characteristic length suggests you might be more interested in that dimension?

Chao et al. used ~10–5 m2/s for vertical diffusivity, and I find Bogucki et al. and Lien et al. using ~10–2 m2/s and ~10–2 m2/s respectively. Those are not too distant from your number, and it looks like Moum et al. are talking about vertical mixing. However in this case I am not sure you want to use horizontal length and velocity; you might consider asking about this in Physics.SE, using the fluid-dynamics tag.

In any case, as we hone our intuition about such things, I like this quote from Durran (1999):

The Péclet number is completely analogous to the more familiar Reynolds number, which is the ratio of momentum advection to momentum diffusion. The difference between the Péclet and Reynolds numbers is due to the difference in the diffusivities of heat [or mass] and momentum. [My addition]

I like this because it suggests that some of the same intuition we have about Re may apply to Pe too, especially the fact that it can vary by several orders of magnitude (from here):

• Whale swimming at 10 m/s, Re = 3 × 108
• Duck flying at 20 m/s, Re = 3 × 105
• Copepod swimming at 0.2 m/s, Re = 3 × 102
• Larva swimming at 0.001 m/s, Re = 3 × 10–1
• Bacterium swimming at 0.00001 m/s, Re = 1 × 10–5

Reference
Shenn-Yu Chao et al., 2007, Assessing the West Ridge of Luzon Strait as an Internal Wave Mediator, Journal of Oceanography, Vol. 63, pp. 897 to 911, 2007. Available online.
Durran, D, 1999, Numerical Methods for Wave Equations in Geophysical Fluid Dynamics. Springer. Google Books.