IMPLICIT NONLINEAR NORMAL MODE INITIALIZATION FOR A BAROTROPIC PRIMITIVE EQUATION LIMITED AREA MODEL

A simple version of implicit nonlinear normal mode initialization is applied to a limited area one-level primitive equation model over a tropical domain. The model formulation is based on shallow water equations in spherical co-ordinate and potential enstrophy conserving finite difference scheme is employed. The model is used for predicting the movement of a typical monsoon depression formed over the Bay of Bengal. The above scheme is found to be very effective as it requires only three iterations for attaining balance between the mass and wind tields. However this model is not able to predict the movement of the depression very ac-curately due to the limitations of such a one-level model.

Challenges of Increased Resolution for the Local Ensemble Tangent Linear Model

An ensemble-based linearized forecast model has been developed for data assimilation applications for numerical weather prediction. Previous studies applied this local ensemble tangent linear model (LETLM) to various models, from simple one-dimensional models to a low-resolution (~2.5°) version of the Navy Global Environmental Model (NAVGEM) atmospheric forecast model. This paper applies the LETLM to NAVGEM at higher resolution (~1°), which required overcoming challenges including 1) balancing the computational stencil size with the ensemble size, and 2) propagating fast-moving gravity modes in the upper atmosphere. The first challenge is addressed by introducing a modified local influence volume, introducing computations on a thin grid, and using smaller time steps. The second challenge is addressed by applying nonlinear normal mode initialization, which damps spurious fast-moving modes and improves the LETLM errors above ~100 hPa. Compared to a semi-Lagrangian tangent linear model (TLM), the LETLM has superior skill in the lower troposphere (below 700 hPa), which is attributed to better representation of moist physics in the LETLM. The LETLM skill slightly lags in the upper troposphere and stratosphere (700–2 hPa), which is attributed to nonlocal aspects of the TLM including spectral operators converting from winds to vorticity and divergence. Several ways forward are suggested, including integrating the LETLM in a hybrid 4D variational solver for a realistic atmosphere, combining a physics LETLM with a conventional TLM for the dynamics, and separating the LETLM into a sequence of local and nonlocal operators.

Balance dynamics in rotating stratified turbulence

If classical quasigeostrophic (QG) flow breaks down at smaller scales, it gives rise to questions of whether higher-order nonlinear balance can be maintained, to what scale and for how long. These are naturally followed by asking how this is affected by stratification and rotation. To address these questions, we perform non-hydrostatic Boussinesq simulations where the initial data is balanced using the Baer–Tribbia nonlinear normal mode initialization scheme (NNMI), which is accurate to second order in the Rossby number, as the next-order improvement to first-order QG theory. The NNMI procedure yields an ageostrophic contribution to the energy spectrum that has a very steep slope. However, as time passes, a shallow range emerges in the ageostrophic spectrum when the Rossby number is large enough for a given Reynolds number. It is argued that this shallow range is the unbalanced part of the motion that develops spontaneously in time and eventually dominates the energy at small scales. If the initial flow is not nonlinearly balanced, the shallow range emerges at even lower Rossby number and it appears at larger scales. Through numerous simulations at different rotation and stratification, this study gives a clear picture of how energy is cascaded in different initially balanced regimes of rotating stratified flow. We find that at low Rossby number the flow mainly consists of a geostrophic part and a balanced ageostrophic part with a steep spectrum. As the Rossby number increases, the unbalanced part of the ageostrophic energy increases at a rate faster than the balanced part. Hence, the total energy spectrum displays a shallow range above a transition wavenumber. This wavenumber evolves to smaller values as rotation weakens.

A Quasi-Variational Algorithm for Nonlinear Normal-Mode Initialization

Balance equations of normal-mode initialization are nonlinear time-independent partial differential equations solved by iterative methods. For the given geopotential, there are regions where these equations are not elliptic, which is reflected in the divergence of iterative algorithms. Variational approaches used to minimize the geopotential changes are more expensive than conventional methods. In this study a simple quasi-variational algorithm is proposed based on different forms of normal-mode initialization equations, which achieves a good balance of atmospheric fields and ensures small changes of geopotential analysis values.

The equatorial counterpart of the quasi-geostrophic model

A uniformly valid balanced model that represents the quasi-geostrophic model's counterpart in the equatorial region is derived. The quasi-geostrophic model itself fails in the equatorial region because it is only valid where the dominant balance is geostrophic, i.e. where the Rossby number is small. The smallness of the Rossby number is assumed in the quasi-geostrophic model's standard derivation and therefore this derivation cannot be repeated for the equatorial region. An alternative derivation of the quasi-geostrophic model that is independent of the Rossby number was presented by Leith in 1980, using the geometric framework of nonlinear normal mode initialization. Its independence of the Rossby number allows it to be repeated for the equatorial region, leading to an equatorial balanced model that thus represents the equatorial counterpart of the quasi-geostrophic model. As such it also coincides with the quasi-geostrophic model sufficiently far away from the equator. Its dispersion relation can be expressed in an explicit analytic form and, compared to that of other balanced models of similar simplicity, approximates that of the shallow water equations strikingly well.

Ellipticity Conditions of the Shallow Water Balance Equations for Atmospheric Data

Nonlinear normal mode initialization equations, which provide required balance relations for atmospheric data, are considered in the generalized case of shallow water equations in arbitrary orthogonal coordinates. Using the concept of ellipticity in the sense of Douglis–Nirenberg, the conditions of well posedness of boundary value problems for balance equations are derived in the cases of constrained streamfunction, constrained potential vorticity, and constrained pressure fields.

Initialization Procedures

In this chapter we describe two of the most commonly used initialization procedures. These are the dynamic normal mode initialization and the physical initialization methods. Historically, initialization for primitive equation models started from a hierarchy of static initialization methods. These include balancing the mass and the wind fields using a linear or nonlinear balance equation (Charney 1955; Phillips 1960), variational techniques for such adjustments satisfying the constraints of the model equations (Sasaki 1958), and dynamic initialization involving forward and backward integration of the model over a number of cycles to suppress high frequency gravity oscillations before the start of the integration (Miyakoda and Moyer 1968; Nitta and Hovermale 1969; Temperton 1976). A description of these classical methods can be found in textbooks such as Haltiner and Williams (1980). Basically, these methods invoke a balanced relationship between the mass and motion fields. However, it was soon realized that significant departures from the balance laws do occur over the tropics and the upperlevel jet stream region. It was also noted that such departures can be functions of the heat sources and sinks and dynamic instabilities of the atmosphere. The procedure called nonlinear normal mode initialization with physics overcomes some of these difficulties. Physical initialization is a powerful method that permits the incorporation of realistic rainfall distribution in the model’s initial state. This is an elegant and successful initialization procedure based on selective damping of the normal modes of the atmosphere, where the high-frequency gravity modes are suppressed while the slow-moving Rossby modes are left untouched. Williamson (1976) used the normal modes of a shallow water model for initialization by setting the initial amplitudes of the high frequency gravity modes equal to zero. Machenhauer (1977) and Baer (1977) developed the procedure for nonlinear normal mode initialization (NMI), which takes into account the nonlinearities in the model equations. Kitade (1983) incorporated the effect of physical processes in this initialization procedure. We describe here the normal mode initialization procedure. Essentially following Kasahara and Puri (1981), we first derive the equations for vertical and horizontal modes of the linearized form of the model equations.