Horizontal diffusive flux discretization

The horizontal diffusion for an arbitrary variable $\varphi$ (corresponding to the wind components or the scalar perturbations) is expressed in flux form in Eq. 92 where the diffusive flux is evaluated using a forward in space discretization

$\displaystyle {\frac{\partial F^o}{\partial x}} = {\frac{F_{i+1}^o - F_{i}^o }{d}}$ (157)

and the flux terms for the 1st, 3rd and 5th order differences are expressed (in one dimension) from Eq.s 93-94, respectively, as

$\displaystyle F^{1}_{\varphi\,x}$ $\displaystyle = \kappa_\varphi {\frac{\partial \varphi}{\partial x}}$ (158)
$\displaystyle F^{3}_{\varphi\,x}$ $\displaystyle = - \kappa_\varphi {\frac{\partial^{3} \varphi}{\partial x^{3}}}$ (159)
$\displaystyle F^{5}_{\varphi\,x}$ $\displaystyle = \kappa_\varphi {\frac{\partial^{5} \varphi}{\partial x^{5}}}$ (160)

where the diffusion coefficients are defined from Eq.s 101-102. Note that for the scalar fluxes, the diffusion coefficients are averaged to the $u$ and $v$ points using the same transform as in Eq.s 128-129. The odd-ordered difference schemes are off-centered (upwind), so that the third order flux is expressed as

$\displaystyle {\frac{\partial^3\varphi}{\partial x^3}}\bigg\vert_i =
{\frac{- \varphi_{i+1} + \varphi_{i-2} + 3 \left( \varphi_{i} - \varphi_{i-1}\right)
}{d^3}}$ (161)

and the 5th order flux is discretized as

$\displaystyle {\frac{\partial^5\varphi}{\partial x^5}}\bigg\vert_i =
{\frac{\va...
...1} - \varphi_{i-2}\right) +
10 \left( \varphi_{i} - \varphi_{i-1}\right)}{d^5}}$ (162)

The first order flux uses the same difference scheme as in Eq. 158.