The dynamic kb-equation


   subroutine kbeq(nlev,dt,u_taus,u_taub,z0s,z0b,h,NN,SS)

The transport equation for (half the) buoyancy variance, $ k_b= {\langle b'^2 \rangle} /2$, follows from the equation for the buoyancy fluctations (see Sander (1998)). In the case of a Boussinesq-fluid, this equation can be written as

$\displaystyle \dot{k_b} = {\cal D}_b + P_b - \epsilon_b \; , \quad$ (160)

where $ \dot{k_b}$ denotes the material derivative of $ k_b$. $ P_b$ is the production of $ k_b$ be mean density gradients, and $ \epsilon_b$ the rate of molecular destruction. $ {\cal D}_b$ represents the sum of the viscous and turbulent transport terms. It is presently evaluated with a simple down gradient model in GOTM.

The production of buoyancy variance by the vertical density gradient is

$\displaystyle P_b = - {\langle w'b' \rangle} \dfrac{\partial {B}}{\partial {z}} = - {\langle w'b' \rangle} N^2 \quad .$ (161)

Its computation is discussed in section 4.8.

The rate of molecular destruction, $ \epsilon_b$, can be computed from either a transport equation or a algebraic expression, section 4.7.10.


   use turbulence,   only: Pb,epsb,nuh
   use turbulence,   only: kb,kb_min
   use turbulence,   only: k_ubc, k_lbc, ubc_type, lbc_type
   use util,         only: Dirichlet,Neumann
   number of vertical layers
   integer,  intent(in)                :: nlev
   time step (s)
   REALTYPE, intent(in)                :: dt
   surface and bottom
   friction velocity (m/s)
   REALTYPE, intent(in)                :: u_taus,u_taub
   surface and bottom
   roughness length (m)
   REALTYPE, intent(in)                :: z0s,z0b
   layer thickness (m)
   REALTYPE, intent(in)                :: h(0:nlev)
   square of shear and buoyancy
   frequency (1/s^2)
   REALTYPE, intent(in)                :: NN(0:nlev),SS(0:nlev)
   Original author(s): Lars Umlauf

Karsten Bolding 2012-12-28