Linear Model Description

The linear model applied here was originally developed by Smith (1980) and generalized to a 3-layer model to include any wind direction and partial absorption at the lower boundary by Smith and Skubis (Smith et al. 2002). Beginning in late 2000, Doyle and Smith (2002) added a critical layer formulation to allow turning winds and Coriolis effects. The current version is quite general. It includes non-hydrostatic, rotational, wind-turning and lower boundary absorption effects with arbitrary 2-D terrain.

The linear model solves the three-dimensional nonhydrostatic equations of linear mountain wave theory for 3 vertical layers using the Fast Fourier Transform (FFT) method of Smith (1980) as discussed in Smith (2001) and Smith (2002). The linearized equations for the horizontal velocity, vertical velocity, and pressure perturbation are reduced to a single equation for the vertical displacement, h (x, y, z). A Fourier Transform is used from (x, y) to (k, l) coordinates so that the equation for the transformed vertical displacement (k, l, z) becomes

, (1)

where s is the intrinsic frequency such that s=Uk+Vl. The solution to the 3-layer system, which contain constant wind and stability (i=1, 2, 3) is

, (2)

where the vertical wavenumber is given by m²=(k²+l²)(N_i²-s)/s_i² and A_i and B_i are the amplitude coefficients for the upward and downward propagating waves, respectively. A radiation condition is applied at the top boundary. At the lower boundary, a linear boundary condition is formulated using a reflection coefficient, q, so that the boundary in Fourier space is

, (3)

where (k, l) is the transformation of the terrain, h (x, y). In this formulation, the dissipation of downward propagating waves due to critical level absorption within the boundary layer is represented when q<1 (e.g., see Smith 2001). A second free parameter of the linear model is the reference height, Z_ref, and is defined as the depth of the upstream stagnant or topographically blocked layer. The effective terrain height is then defined as h(x, y)-Z_ref, where h(x, y) is based on the DMA 100-m resolution data set. The terrain is reduced around the edges of the computational domain to avoid spurious wave wrapping associated with the FFT method.

Linear Model Initialization

The linear model is initialized from NRL's Coupled Ocean Atmospheric Mesoscale Prediction System (COAMPS) operational runs at FNMOC. The COAMPS output for a particular grid point upstream of the topography is interpolated and then averaged to create a 3-layer sounding. This 3-layer sounding of the static stability and wind field is then used to initialize the linear model based a particular operational forecast time.

References

Doyle, J.D., and R.B. Smith, 2003: Mountain waves over the Hohe Tauern. Quart. J. Roy. Meteor. Soc.., 129, 799-823.

Smith, R. B., 1980: Linear theory of stratified hydrostatic flow past an isolated mountain. Tellus, 32, 348-364.

Smith, R.B, 2001. Stratified flow over topography. Stratified Flows in the Environment, Ed. R. Grimshaw, Kluwer Publ.

Smith, R.B, S.T. Skubis, J.D. Doyle, A. Broad, C. Kiemle and H. Volkert, 2002.  Mountain waves over Mt. Blanc: Influence of a stagnant boundary layer. J. Atmos. Sci, 59, 2073-2092.