Thesis. 1976. Ph.D.--Massachusetts Institute of Technology. Dept. of Meteorology. / Microfiche copy available in Archives and Science. / Vita. / Bibliography: leaves 241-244. / by Dale B. Haidvogel. / Ph.D.

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This investigation is concerned with the determination of the sea state in terms of wave spectra. The phenomenon was calculated for two different bathymetries. The purpose is to develop a finite difference method with an upwind differencing scheme to g solve several formulations of the wave action conservation equation. The computations were done in the wave number space and the frequency direction space. For the case of a beach with constant slope the results were compared with the analytical solution. For the case of an elliptical submerged shoal, they were compared with experimental data. The results of the computer code showed a fairly good qualitative agreement with the actual values for a smooth distribution of input energy. / Master of Science

The Spectral Ocean Wave Model (SOWM) is a numerical wave prediction model which calculates directional wave spectra from input wind fields. As do the majority of wave models, it uses a point spectral growth mechanism, i.e., it applies the energy balance equation in a directionally integrated form. The directionality of its growth is obtained from an assumed spreading function on the wind direction. In this study, the energy balance equation is applied in directional form using directional atmospheric energy source functions. The B function of Miles’ instability mechanism is derived following the analysis of Phillips and it is tuned to the directionally integrated form used in the SOWM. Two infinite ocean wave models are used to compare the behavior of the point and directional growth mechanisms under various wind conditions. The directional form shows more flexibility in responding to directionally varying winds while the point spectral form creates excess energy spread widely over direction when operating in the presence of swell. / Ph. D. / incomplete_metadata

The complexity of numerical ocean circulation models requires careful checking with a variety of test problems. The purpose of this paper is to develop a test problem involving Rossby and gravity waves in a two-layer fluid in a channel. The goal is to compute very accurate solutions to this test problem. These solutions can then be used as a part of the checking process for numerical ocean circulation models. Here, Chebychev pseudo-spectral methods are used to solve the governing equations with a high degree of accuracy. Chebychev pseudo-spectral methods can be described in the following way: For a given function, find the polynomial interpolant at a particular non-uniform grid. The derivative of this polynomial serves as an approximation to the derivative of the original function. This approximation can then be inserted to differential equations to solve for approximate solutions. Here, the governing equations reduce to an eigenvalue problem with eigenvectors and eigenvalues corresponding to the spatial dependences of modal solutions and the frequencies of those solutions, respectively. The results of this method are checked in two ways. First, the solutions using the Chebychev pseudo-spectral methods are analyzed and are found to exhibit the properties known to belong to physical Rossby and gravity waves. Second, in the special case where the two-layer model degenerates to a one-layer system, some analytic solutions are known. When the numerical solutions are compared to the analytic solutions, they show an exponential rate of convergence. The conclusion is that the solutions computed using the Chebychev pseudo-spectral methods are highly accurate and could be used as a test problem to partially check numerical ocean circulation models. / Graduation date: 2012

The instabilities of shallow shear flows are analyzed to study exchanges processes across shear flows in inland and coastal waters, coastal and ocean currents, and winds across the thermal-and-moisture fronts. These shear flows observed in nature are driven by gravity and governed by the shallow water equations (SWE). A highly accurate, and robust, computational scheme has been developed to solve these SWE. Time integration of the SWE was carried out using the fourth-order Runge-Kutta scheme. A third-order upwind bias finite difference approximation known as QUICK (Quadratic Upstream Interpolation of Convective Kinematics) was employed for the spatial discretization. The numerical oscillations were controlled using flux limiters for Total Variation Diminishing (TVD). Direct numerical simulations (DNS) were conducted for the base flow with the TANH velocity profile, and the base flow in the form of a jet with the SECH velocity profile. The depth across the base flows was selected for the' balance of the driving forces. In the rotating flow simulation, the Coriolis force in the lateral direction was perfectly in balance with the pressure gradient across the shear flow during the simulation. The development of instabilities in the shear flows was considered for a range of convective Froude number, friction number, and Rossby number. The DNS of the SWE has produced linear results that are consistent with classical stability analyses based on the normal mode approach, and new results that had not been determined by the classical method. The formation of eddies, and the generation of shocklets subsequent to the linear instabilities were computed as part of the DNS. Without modelling the small scales, the simulation was able to produce the correct turbulent spreading rate in agreement with the experimental observations. The simulations have identified radiation damping, in addition to friction damping, as a primary factor of influence on the instability of the shear flows admissible to waves. A convective Froude number correlated the energy lost due to radiation damping. The friction number determined the energy lost due to friction. A significant fraction of available energy produced by the shear flow is lost due the radiation of waves at high convective Froude number. This radiation of gravity waves in shallow gravity-stratified shear flow, and its dependence on the convective Froude number, is shown to be analogous to the Mach-number effect in compressible flow. Furthermore, and most significantly, is the discovery from the simulation the crucial role of the radiation damping in the development of shear flows in the rotating earth. Rings and eddies were produced by the rotating-flow simulations in a range of Rossby numbers, as they were observed in the Gulf Stream of the Atlantic, Jet Stream in the atmosphere, and various fronts across currents in coastal waters.

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There are three main wave types present in the Sun’s atmosphere: Alfvén waves and fast and slow magnetoacoustic waves. Alfvén waves are purely magnetic and would not exist if it was not for the Sun’s magnetic field. The fast and slow magnetoacoustic waves are so named due to their relative phase speeds. As the magnetic field tends to zero, the slow wave goes to zero as the fast wave becomes the sound wave. When a resonance occurs energy may be transferred between the different modes, causing one to increase in amplitude whilst the other decreases. This is known as mode conversion. Mode conversion of fast and slow magnetoacoustic waves takes place when the characteristic wave speeds, the sound and Alfvén speeds, are equal. This occurs in regions where the ratio of the gas pressure to the magnetic pressure, known as the plasma β, is approximately unity. In this thesis we investigate the conversion of fast and slow magnetoacoustic waves as they propagate from low- to high-β plasma. This investigation uses a combination of analytical and numerical techniques to gain a full understanding of the process. The MacCormack finite-difference method is used to model a wave as it undergoes mode conversion. Complementing this analytical techniques are employed to find the wave behaviour at, and distant from, the mode-conversion region. These methods are described in Chapter 2. The simple, one-dimensional model of an isothermal atmosphere permeated by a uniform magnetic field is studied in Chapter 3. Gravitational acceleration is included to ensure that mode conversion takes place. Driving a slow magnetoacoustic wave on the upper boundary conversion takes place as the wave passes from low- to high-β plasma. This is expanded upon in Chapter 4 where the effects of a non-isothermal temperature profile are examined. A tanh profile is selected to mimic the steep temperature gradient found in the transition region. In Chapter 5 the complexity is increased by allowing for a two-dimensional model. For this purpose we choose a radially-expanding magnetic field which is representative of a coronal hole. In this instance the slow magnetoacoustic wave is driven upwards from the surface, again travelling from low to high β. Finally, in Chapter 6 we investigate mode conversion near a two-dimensional, magnetic null point. At the null the plasma β becomes infinitely large and a wave propagating towards the null point will experience mode conversion. The methods used allow conversion of fast and slow waves to be described in the various model atmospheres. The amount of transmission and conversion are calculated and matched across the mode-conversion layer giving a full description of the wave behaviour.