Commit 0335a833 authored by Jonathan Lambrechts's avatar Jonathan Lambrechts
Browse files

add bottom friction doc

parent 6dae3b12
Pipeline #1759 passed with stage
in 35 minutes and 3 seconds
\documentclass{paper}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\begin{document}
We suppose a logarithmic profile :
\[
u(z) = \frac{u^*}{\kappa}\,\ln\left({\frac z {z_0}}\right)
\]
where $z=0$ at the bottom of the sea and $z=H$ at the surface.
The mean 2D value can be obtained :
\[
u_{2d} = \frac 1 H \int_0^H u(z)\, dz = \frac{u^*}{\kappa}\left(\ln\left({\frac H {z_0}}\right)-1\right)
\]
which gives
\[
u^* = \frac{\kappa\,u_{2d}}{\ln\left({\frac H {z_0}}\right)-1}
\]
and
\[
u(z) = \frac{u_{2d}\ln\left({\frac z {z_0}}\right)}{\ln\left({\frac H {z_0}}\right)-1}
\]
in the 3D model, the bottom drag is parametrized as
\[
d = -\left(\frac \kappa {\ln\frac{z_{\text{bot}}}{z_0}}\right)^2
|u(z_{\text{bot}})|\,u(z_{\text{bot}})
\]
by inserting $u(z)$
\[
d = -\left(\frac{\kappa}{\ln\frac{H}{z_0}-1}\right)^2|u_{\text{2d}}|\,u_{\text{2d}}
\]
\end{document}
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment