The dynamics of the atmosphere and oceans are complicated because of the vast range of length and time scales involved. Understanding how energy cascades from the large to small scales is an outstanding problem in the field and of great interest. In any attempt to do this it is always necessary to specify the physical structure of the basis functions. Perhaps the most popular choice are Fourier modes, which are desirable because they 1) can form a complete basis; 2) are well understood because of the richness of Fourier analysis; and 3) are a basis for high-order spectral methods. This is a convenient choice but numerous other possibilities exist, such as polynomials and wavelets. All of these choices are generic in that they do not arise from the underlying physics of the waves and can usually be applied to virtually any problem. The motivation for this work stems from the idea that a better choice for basis functions should be dictated by the model equations.

One relatively simple model that has often been used to looked at energy transfers between different length and time scales is the Rotating Shallow Water model (RSW). It is restrictive in that it only describes homogeneous fluids, however, because it can contain both fast gravity and slow Rossby waves it is a useful paradigm to study energy transfers between waves with vastly different scales. The pioneering work of Leith (1980) investigated the decomposition of the RSW into its linear modes and subsequently others have built on this to understand the modal structure of stratified flows. In these works the emphasis has been on f-plane and therefore the slow component was a vortical mode that does not propagate.

In his original paper Leith points out that it would be interesting to extend his methodology to a beta-plane and in this talk we present results from our preliminary work to do just that. This is done numerically using spectral methods to find the most accurate solutions possible for a given number of degrees of freedom. First, we determine the structure of the linear RSW modes on a beta-plane in meridional channel. In the continuous limit these waves form a complete basis and are a natural set of basis functions to study in this model and have extensions in other models. Second, we present results from a series of numerical experiments of freely-evolving flows to address how energy is transferred between the linear waves. This will consist of wave-wave interactions as well as geostrophic turbulent flows.

The fundamental computational challenge for climate and weather models is to efficiently and accurately resolve the vast range of space and time scales that characterize atmosphere and ocean flows. Not only do these scales span many orders of magnitude, the minimum dynamically active scale is also highly intermittent in both time and space. In this talk we introduce an innovative wavelet-based approach to dynamically adjust the local grid resolution to maintain a uniform specified error tolerance. The wavelet multiscale method is used to make dynamically adaptive the TRiSK model (Ringler et al. 2010) for the rotating shallow water equations on the sphere. We have carefully designed the inter-scale restriction and prolongation operators to retain the mimetic properties that are the main strength of this model. The wavelet method is computationally efficient and allows for straightforward parallelization using MPI. We will show verification results from the suite of smooth test cases proposed by Williamson (1991), and a more recent nonlinear test case suggested by Galewsky (2004): an unstable mid-latitude zonal jet. To investigate the ability of the method to handle boundary layers in ocean flows, we will also show an example of flow past an island using penalized boundary conditions. This adaptive "dynamical core" serves as the foundation on which to build a complete climate or weather model.

While the midlatitude weather and climate variability is associated with quasi-geostrophic planetary Rossby waves that are fairly easy to predict via simple dry dynamical models such the quasi-geosptrophic equations, the atmospheric variability in the tropics involves a hierarchy of equatorially trapped waves that interact strongly with moist convection. Equatorially trapped waves display a wide variety of physical and dynamical features including geostrophic balance and non-balance, dispersive and non dispersive behaviour, symmetric and anti-symmetric structure. They propagate in both directions along the equator and are responsible for a significant portion of weather variability in the tropics on the scales of a few days to months. The interactions between equatorially trapped and midlatitude Rossby waves play a major role in the lateral energy exchange between the tropics and extra-tropics. In fact, it is believed that progress in medium to long range weather forecasts, over a few weeks to months, and climate predictions in particular, depends on a proper representation of tropical convection and associated rain fall variability in numerical weather prediction and global climate models. In this talk, I will use some simplified primitive equation models to illustrate some scenarios of tropical and extra-tropical interactions of equatorially trapped waves and barotropic Rossby waves, which propagate north- and southward from the tropics to the midlatitudes.

The waveform tomography, or full waveform inversion (FWI), is in general better than the ray tomography, however a reliable initial model is usually required to ensure the success. In this work we designed a cascade-like hybrid tomography technology to solve the crosswell seismic inversion problem. The new method is a combination of several widely used tomography technologies. We start from the Radon-transform based back projection (BP) method, which produces a smooth initial velocity model. Next, the Linear Iterative Reconstruction (LIR) method such as SIRT is adopted to update this initial model, then it is further improved by the nonlinear Gradient-based Eikonal Equation Tomography (GEET) method. The velocity model reconstructed from the previous multi ray tomography methods is sufficiently reliable to serve as the initial model for the computational intensive waveform tomography, from which the accurate velocity model is obtained. The numerical example shows that this hybrid method has great potential in reconstructing accurate acoustic velocity or other high-resolution reservoir characterization between two wells. It is noticeable that this hybrid method is able to obtain accurate velocity even when the recorded seismic data is in poor coverage at spatial and the signal-to-noise ratio is low.