章 増田

Analytical formulas representing the idealized growth of wind-waves, duration-limited and fetch-limited

The SMB method is an historical wave-forecasting method, which is based on fetch-limited and durationlimited growth of wind-waves. However, wind-wave growth envisaged by the words fetch-limited and duration-limited has been ambiguous, and clear explanation seems to have been absent. A simplest conceptual model is proposed here for the idealized growth of wind-waves, where the model equation is based on several assumptions: (1) a well-established fetch relation for small to moderate fetches, (2) ubiquitous equilibrium governed by the 3/2-power law of Toba, (3) propagation of wind-waves as a whole at the group velocity of waves with the significant wave period, (4) presence of the upper bound of the significant wave period, and (5) tanh-type fetch dependence of wave energy. In particular, the fetch relation of (5) is a modified Wislons formula I, where variables are normalized by a single scale of fectch and upper bounds of wind-waves. The model yields analytical formulas that clearly illustrate the growth of wind-waves, both fetch-limited and duration-limited. The result suggests that wave growth is better described by substantial duration tsd than the conventional discrimination of duration-limited and fetch-limited; tsd allows a unified description of wave-growth. Related problems are examined and discussed as well: fetch relations at large fetches, normalized and approximate formulas of fetch relations, momentum and energy input, and so on.

Vertical modes of quasi-geostrophic flows in an ocean with bottom topography : evolution equation and energetics

Quasi-geostrophic current is expanded in terms of vertical modes such as barotropic and baroclinic ones. Then the evolution of quasi-geostrophic motion is understood from the behavior of each vertical mode. There are some subtle issues, however, as regards vertical modes: boundary conditions, difference between a level model and a layer model, and so on. A comprehensive formulation is given of the expanSiOn of the quasi-geostrophic flows in terms of vertical modes both for a level model and for a layer model. Vertical modes are defined in almost the same manner for a level model and a layer model. Evolution equation and energetics of each mode are derived and argued. In addition to the ordinary nonlinear advective three-mode interaction, a gentle relief of bottom topography allows inter-modal energy transfer via bottom topography; in particular the so-called JEBAR effect and topographic torque is classified into one of such two-mode interaction (coupling). Under the rigid lid surface condition, the barotropic mode does not make a triplet with two different baroclinic modes, in the three-mode nonlinear advective interaction. Detailed balance turns out to hold for inter-modal energy transfer with respect to three-mode nonlinear advective interaction and two-mode coupling via topography. Potential enstrophy transfer among vertical modes is formulated also. It is shown that detailed balance is assured for inter-modal enstrophy transfer as well, if potential enstrophy is chosen properly. On the other hand detailed balance is not expected for inter-modal enstrophy transfer by two-mode coupling via bottom topography.