1-D model for providing initial vertical velocity profiles#
The initial profiles of the horizontal wind components in PALM can be prescribed by the user by piecewise linear gradients or by directly using observational data. Alternatively, a 1-D model can be employed to calculate stationary boundary-layer wind profiles as a better first guess of initial conditions. The arrays of the 3-D variables are then initialized with the (stationary) solution of the 1-D model. These variables are
The 1-D model assumes the profiles of
The default turbulence parameterization is based on the turbulence kinetic energy
The dissipation rate is parametrized by
after Detering and Etling (1985). The mixing length is calculated after Blackadar (1997) as
The turbulent fluxes are calculated using a gradient approach (first-order closure):
where
with the similarity functions
Note that the distinction of cases in the Eq. above is done using the value of
As for the 3-D model, a turbulence model based on
Moreover, a Rayleigh damping can be switched on to speed up the damping of inertial oscillations. The 1-D model is discretized in space using finite differences. Discretization in time is achieved using the 3rd-order Runge--Kutta time-stepping scheme (Williamson, 1980). In order to avoid very small time steps forced by the diffusion time step criterion, the implicit Crank-Nicolson time step scheme can be used instead for treating all diffusion terms that appear in the prognostic equations. As a part of the Crank-Nicolson scheme, the algorithm of Stone (1973) is used to solve the tridiagonal systems of equations.
Dirichlet boundary conditions are used at the top and bottom boundaries of the model, except for
References#
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Blackadar, A.K. (1997): Turbulence and Diffusion in the Atmosphere. Springer. Berlin, Heidelberg, New York, 185 pp.
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Detering, H.W., Etling D. (1985): Application of the E-epsilon turbulence model to the atmospheric boundary layer. Boundary-Layer Meteorol., 33, 113–133.
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Stone, H.S. (1973): An Efficient Parallel Algorithm for the Solution of a Tridigonal Linear System of Equations. Journal of the Association for Computing Machinery, 20, 27-38.
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Williamson, J.H. (1980). Low-storage Runge–Kutta schemes. J. Comput. Phys., 35, 48–56.