3dVOM (3-D Velocities over Mountains) is a finite-difference numerical model designed for high-resolution simulations of lee waves generated by flow over complex terrain. The model is based on a set of simplified equations of motion, the linearised shallow Boussinesq equations of motion for a dry atmosphere. The equations of motion are linearised about an upwind profile of wind and temperature (assumed to be dependent on height only) and are integrated forward in time until a quasi-steady lee-wave field is obtained. The model includes a 1st-order (mixing-length) turbulence closure scheme. Because the equations of motion are greatly simplified (relative to those in a full NWP forecast model) solutions can be generated relatively quickly. The more technical aspects of the code can be viewed here

The 3dVOM code is used to generate high-resolution detailed forecasts of lee-wave fields and associated near-surface winds. Met Office Unified Model (UM) global forecast data are used to generate forecast profiles of wind and temperature for the Sierra Nevada range (at 118 W, 36.6 N) and the 3dVOM wave forecasts are based on these profiles. Note that over the lower portion of the forecast profiles (below 1500 m above ground level) the UM wind data are replaced by a wind profile which varies logarithmically with height. This approach was adopted to avoid uncertainties connected with the sub-grid flow blocking drag parametrisation applied in the UM, which may affect the quality of the low-level forecast winds in regions of significant orography.

The horizontal resolution is approx. 2 km. Terrain heights are shown in metres. The lines labelled W-E, S-N, NW-SE, SW-NE, Track A, Track B and Track C show the orientations of the available vertical cross sections. The latter three are the expected aircraft tracks of the NCAR Hiaper and FAAM BAe-146 aircraft during the T-REX campaign.

Technical details of 3dVOM

The model is based on the linearised shallow-Boussinesq equations of motion and is an extension of the inviscid model described by Vosper (2003) to include a representation of boundary-layer turbulence via a first-order mixing-length closure scheme (see King et al. 2005).

The model equations are linearised about upwind mean wind speed and potential temperature fields, assumed to be functions of height only. Perturbations to this horizontal background flow (due to gravity waves for example) in wind velocity and potential temperature are then calculated using the linearised equations. At each time step an elliptic equation is solved for the wave-induced pressure perturbation ensuring that the linearized continuity equation is satisfied. Turbulent mixing is represented by an eddy viscosity, which itself is a function of a mixing length (prescribed as a function of height and stability) and vertical wind shear. The eddy viscosity is partitioned into background and perturbation parts and terms in the perturbation momentum equations are linearised in the usual way. It should be noted that only the vertical gradients of the turbulent stresses are retained in the equations since these generally dominate in the boundary layer.

The model equations are discretised onto a staggered terrain-following mesh which is stretched in the vertical to enable stress gradients in the boundary layer to be resolved. The horizontal advection scheme is fourth-order accurate and the time integration scheme is semi-implicit based on an operator splitting technique. At the lateral boundaries a periodic condition is applied on all flow variables. In order to minimize wrap-around effects Rayleigh damping is applied to the velocity and potential temperature perturbations in thin columns adjacent to the upwind and downwind boundaries. A rigid-lid upper boundary condition is applied and damping is also applied near the upper boundary to absorb upwind-radiating wave motion. Near the ground, the flow is assumed to be neutrally stratified with a zero heat flux condition. A no-slip condition is applied via a log-law formulation for the wind speed following Beljaars et al. (1987). At the surface, the vertical gradient of perturbation potential temperature, is set to zero and the vertical velocity perturbation, w', is given by the linearized lower boundary condition w'=U*dh/dx+V*dh/dy where h(x,y) is the height of the topography and U and V are the respective background westerly and southerly wind components at the surface.