The buoyancy equation


   subroutine buoyancy(nlev,dt,cnpar,nub,gamb)

This subroutine solves a transport equation for the mean potential buoyancy,

$\displaystyle B=-g\frac{ {\langle \rho \rangle} -\rho_0}{\rho_0} \; , \quad$ (33)

where $ g$ is the accelaration of gravity, and $ {\langle \rho \rangle} $ and $ \rho_0$ are the mean potential density and the reference density, respectively. A simplified transport equation for $ B$ can be written as

$\displaystyle \dot{B} = {\cal D}_B \; , \quad$ (34)

where $ \dot{B}$ denotes the material derivative of $ B$, and $ {\cal D}_b$ is the sum of the turbulent and viscous transport terms modelled according to

$\displaystyle {\cal D}_B = \dfrac{\partial}{\partial {z}} \left( (\nu^B_t+\nu^B) \dfrac{\partial {B}}{\partial {z}} - \tilde{\Gamma}_B \right) \quad .$ (35)

In this equation, $ \nu^B_t$ and $ \nu^B$ are the turbulent and molecular diffusivities of buoyancy, respectively, and $ \tilde{\Gamma}_B$ denotes the non-local flux of buoyancy, see section 4. In the current version of GOTM, we set $ \nu^B_t = \nu^\Theta_t$ for simplicity. Source and sink terms are completely disregarded, and thus (34) mainly serves as a convenient tool for some idealized test cases in GOTM.

Diffusion is treated implicitly in space (see equations (7)- (9)), and then solved by a simplified Gauss elimination. Vertical advection is included, and it must be non-conservative, which is ensured by setting the local variable adv_mode=0, see section 8.5 on page [*].


   use meanflow,      only: h,w,buoy,T,avh,init_buoyancy
   use meanflow,      only: w_grid,grid_method
   use observations,  only: b_obs_NN,b_obs_surf,b_obs_sbf
   use observations,  only: w_adv_discr,w_adv_method
   use util,          only: Dirichlet,Neumann
   use util,          only: oneSided,zeroDivergence
   number of vertical layers
   integer, intent(in)                 :: nlev
    time step (s)
   REALTYPE, intent(in)                :: dt
   numerical "implicitness" parameter
   REALTYPE, intent(in)                :: cnpar
   diffusivity of buoyancy (m^2/s)
   REALTYPE, intent(in)                :: nub(0:nlev)
   non-local buoyancy flux (m^2/s^3)
   REALTYPE, intent(in)                :: gamb(0:nlev)
   Original author(s): Hans Burchard & Karsten Bolding

Karsten Bolding 2012-12-28