An upper gravity-wave absorbing layer for NWP applications

It is well known that internal gravity waves play an important role in the dynamics of mesoscale circulations, particularly those occurring over mountainous terrain. However, their treatment in numerical simulation models has been problematic due to artificial influences of the upper boundary in the model domain. Conventional upper boundary conditions, such as a rigid lid or constant pressure surface, are totally reflective to upward propagating gravity-wave energy. Although both gravity-wave radiation boundary conditions and upper absorbing layers have been developed, these techniques have, in practice, proven more successful in idealized research simulations than in real atmospheric NWP applications. In this paper, we propose a simple approach for implementing an upper absorbing layer in split-explicit models that may prove more effective in many mesoscale NWP applications.

One approach for reducing the artificial reflection of gravity-wave energy at the top of a model domain is to employ a gravity-wave radiation condition along the upper boundary. A radiation condition for linear gravity waves can be derived that is independent of both the wave frequency and vertical wavelength for hydrostatic waves in the absence of rotation. The derivation of this radiation condition assumes that an infinite layer of constant wind and stability exists above the top boundary where the condition is applied. While these restrictions may be appropriate for many applications involving idealized flow, they are not as well suited for the horizontal inhomogeneities and wide range of scales encountered in numerical weather prediction. An alternative approach for mitigating gravity-wave energy reflection at the upper boundary is to include a damping (sponge) layer in the upper portion of the model domain. This layer may employ either horizontal diffusion or Rayleigh damping terms with damping coefficients that increase with height over a depth sufficient to achieve good absorption characteristics. Although a horizontal diffusion layer is conceptually well suited for both idealized and real-data simulations, for many practical applications, the maximum stable diffusion coefficient is significantly smaller than the values required to achieve effective absorption of gravity-wave energy. The implementation of an absorbing layer using Rayleigh damping has been quite successful in idealized simulations with a known background environmental state. However, difficulties arise in using this method for NWP applications since the entire atmospheric state is evolving as part of the simulation.

Here, we present a new approach for a gravity-wave absorbing layer that is simple to implement in split-explicit time-integration schemes and appears to work well for both idealized and real-data (NWP) applications. With this approach, an implicit Rayleigh damping term for the vertical velocity is added as a final adjustment at the end of each small (acoustic) time step. Analysis of the linear wave equation quantifies the absorption characteristics of this absorbing layer as a function of horizontal scale as well as the depth and strength of the damping region, and demonstrates that this approach can be effectively applied across a wide range of scales, from nonhydrostatic to synoptic. Numerical mountain-wave simulations are conducted to quantitatively evaluate the accuracy of this approach through comparison with linear analytic solutions. We further illustrate the utility of this proposed absorbing layer in an idealized squall simulation and in an NWP forecast for mountain waves over the eastern slope of the Colorado Rocky Mountains.