Part VII : ECMWF WAVE - MODEL DOCUMENTATION ( CY 23 R 4 )

This document is partly based on Chapter III of "Dynamics and Modelling of Ocean Waves" by Komen et al, 1994. For more background information on the fundamentals of wave prediction models this book comes highly recommended. Here, after a historical introduction, we will describe the basic evolution equation, including a discussion of the parametrization of the source functions. This is then followed by a brief discussion of the ECMWF wave data assimilation scheme. The document closes with a presentation of the ECMWF version of the numerical scheme and of the structure of the software. 1.1 HISTORICAL REVIEW The principles of wave prediction were already well known at the beginning of the sixties. Yet, none of the wave models developed in the 1960s and 1970s computed the wave spectrum from the full energy balance equation. Additional ad hoc assumptions have always been introduced to ensure that the wave spectrum complies with some preconceived notions of wave development that were in some cases not consistent with the source functions. Reasons for introducing simplifications in the energy balance equation were twofold. On the one hand, the important role of the wave–wave interactions in wave evolution was not recognized. On the other hand, the limited computer power in those days precluded the use of the nonlinear transfer in the energy balance equation. The first wave models, which were developed in the 1960s and 1970s, assumed that the wave components suddenly stopped growing as soon as they reached a universal saturation level (Phillips, 1958). The saturation spectrum, represented by Phillips’s one-dimensional frequency spectrum and an empirical equilibrium directional distribution, was prescribed. Nowadays it is generally recognized that a universal high-frequency spectrum (in the region between 1.5 and 3 times the peak frequency) does not exist because the high-frequency region of the spectrum not only depends on whitecapping but also on wind input and on the low-frequency regions of the spectrum through nonlinear transfer. Furthermore, from the physics point of view it has now become clear that these so-called first generation wave models exhibit basic shortcomings by overestimating the wind input and disregarding nonlinear transfer. The relative importance of nonlinear transfer and wind input became more evident after extensive wave growth experiments (Mitsuyasu, 1968, 1969; Hasselmann et al., 1973) and direct measurements of the wind input to the waves (Snyder et al., 1981, Hasselmann et al., 1986). This led to the development of second generation wave models which attempted to simulate properly the so-called overshoot phenomenon and the dependence of the high-frequency region of the spectrum on the low frequencies. However, restrictions resulting from the nonlinear transfer parametrization effectively required the spectral shape of the wind sea spectrum to be prescribed. The specification of the wind sea spectrum was imposed either at the outset in the formulation of the transport equation itself (parametrical or hybrid models) or as a side condition in the computation of the spectrum (discrete models). These models therefore suffered basic problems in the treatment of wind sea and swell. Although, for typical synoptic-scale wind fields the evolution towards a quasi-universal spectral shape could be justified by two scaling arguments (Hasf 5 – 1 (Printed 19 September 2003) Part VII: ‘ECMWF Wave-model documentation’ selmann et al., 1976), nevertheless complex wind seas generated by rapidly varying wind fields (in, for example, hurricanes or fronts) were not simulated properly by the second generation models. The shortcomings of first and second generation models have been documented and discussed in the SWAMP (1985) wave-model intercomparison study. The development of third generation models was suggested in which the wave spectrum was computed by integration of the energy balance equation, without any prior restriction on the spectral shape. As a result the WAM group was established, whose main task was the development of such a third generation wave model. In this document we shall describe the ECMWF version of the so-called WAM model. In Komen et al. (1994) an extensive overview is given of what is presently known about the physics of wave evolution, in so far as it is relevant to a spectral description of ocean waves. Thus, in detail knowledge of the generation of ocean waves by wind and the impact of the waves on the air flow is described, a discussion of the importance of the resonant nonlinear interactions for wave evolution is given and the state of the art knowledge on spectral dissipation of wave energy by whitecapping and bottom friction is given. 1.2 OVERVIEW OF THIS DOCUMENT In this document we will try to make optimal use of the knowledge of wave evolution in the context of numerical modelling of ocean waves. However, in order to be able to develop a numerical wave model that produces forecasts in a reasonable time, compromises regarding the functional form of the source terms in the energy balance equation have to be made. For example, a traditional difficulty of numerical wave models has been the adequate representation of the nonlinear source term . Since the time needed to compute the exact source function expression greatly exceeds practical limits set by an operational wave model, some form of parametrization is clearly necessary. Likewise, the numerical solution of the momentum balance of air flow over growing ocean waves, as presented in Janssen (1989), is by far too time consuming to be practical for numerical modelling. It is therefore clear that a parametrization of the functional form of the source terms in the energy balance equation is a necessary step to develop an operational wave model. The remainder of this document is organised as follows. In Chapter 2 we discuss the kinematic part of the energy balance equation, that is, advection in both deep and shallow water, refraction due to currents and bottom topography. The next section, Chapter 3, is devoted to a parametrization of the input source term and the nonlinear interactions. The adequacy of these approximations is discussed in detail, as is the energy balance in growing waves. In Chapter 4 a brief overview is given of the method that is used to assimilate Altimeter wave height data. This method is called Optimum Interpolation (IO) and is a more or less one to one copy obtained from the work of Lorenc (1981). A detailed description of the method that is used at ECMWF, including extensive test results is provided by Lionello et al. (1992). SAR data may be assimilated in a similar manner. Next, in Chapter 5 we discuss the numerical implementation of the model. We distinguish between a prognostic part of the spectrum (that part that is explicitly calculated by the numerical model) (WAMDI , 1988), and a diagnostic part. The diagnostic part of the spectrum has a prescribed spectral shape, the level of which is determined by the energy of the highest resolved frequency bin of the prognostic part. Knowledge of the unresolved part of the spectrum allows us to determine the nonlinear energy transfer from the resolved part to the unresolved part of the spectrum. The prognostic part of the spectrum is obtained by numerically solving the energy balance equation. The choice of numerical schemes for advection, refraction and time integration is discussed. The integration in time is performed using a fully-implicit integration scheme in order to be able to use large time steps without incurring numerical instabilities in the highfrequency part of the spectrum. For advection and refraction we have chosen a first order, upwinding flux scheme. Advantages of this scheme are discussed in detail, especially in connection with the so-called garden sprinkler effect (see SWAMP, 1985, p144). Alternatives to first order upwinding, such as the semi-Lagrangian scheme which Snl 2 IFS Documentation Cycle CY23r4 (Printed 19 September 2003) Chapter 1 ‘Introduction’ is gaining popularity in meteorology, will be discussed as well. Chapter 6 is devoted to software aspects of the WAM model code with emphasis on flexibility, universality and design choices. A brief summary of the detailed manual accompanying the code is given as well (Günther et al. , 1992). Finally, in Chapter 7 we give a list of applications of wave modelling at ECMWF, including the two-way interaction of winds and waves. 3 IFS Documentation Cycle CY23r4 (Printed 19 September 2003) Part VII: ‘ECMWF Wave-model documentation’ 4 IFS Documentation Cycle CY23r4 (Printed 19 September 2003) IFS Documentation Cycle CY23r4 Part VII: ECMWF WAVE-MODEL DOCUMENTATION CHAPTER 2 The kinematic part of the energy balance equation Table of contents 2.1 The energy balance equation 2.2 Spherical coordinates 2.2.1 Great circle propagation on the globe 2.2.2 Shoaling 2.2.3 Refraction 2.2.4 Current effects 2.2.5 Concluding remark 2.1 THE ENERGY BALANCE EQUATION In this section we shall briefly discuss some properties of the energy balance equation in the absence of sources and sinks. Thus, shoaling and refraction—by bottom topography and ocean currents—are investigated in the context of a statistical description of gravity waves. Let and be the spatial coordinates and , the wave coordinates, and let (2.1) be their combined four-dimensional vector. The most elegant formulation of the "energy" balance equation is in terms of the action density spectrum which is the energy spectrum divided by the so-called intrinsic frequency . The action density plays the same role as the particle density in quantum mechanics. Hence there is an analogy between wave groups and particles, because wave groups with action have energy and momentum . Thus, the most fundamental form of the transport equation for the action density spectrum without the source term can be written in the flux form