Cq = Sound speed at sea surface za = Depth of channel = Sound speed gradient in channel (+) = Sound speed gradient below channel (-)

Figure 5.2 Geometry of bilinear-gradient model (Pedersen and Gordon, 1965).

Qualitatively, the relationship in Equation (5.2) predicts cylindrical spreading as a function of range combined with the effects of modal interactions. These modes may be thought of as damped sinusoidal waves. There are two basic factors that determine the degree to which a particular mode contributes to the result. First, the product |un(t)un(to)| depends on the source and receiver depths, but not on range. Specifically, an exponential damping factor implicit in the formulation causes the relative contributions of a mode to decrease with increasing range and with increasing mode number.

By invoking simplifying approximations, Pedersen and Gordon (1965) were able to generate analytical solutions using a variation of Equation (5.2) that agreed favorably with both ray-theoretical solutions and with experimental data. There are three important aspects of this approach that differ from other normal mode solutions: (1) the solution is valid for short ranges because the branch-cut integral is zero for this model; (2) the modes are damped since the wavenumbers are complex; and (3) there are no cutoffs in the frequency domain. Thus, higher-order modes are highly damped and only the lower-order modes need be considered for most practical problems.

5.2.3 Oceanographic mixed-layer models

New developments in numerical modeling in oceanography now permit the depth of the mixed layer to be forecast in both time and location. These predicted values can then be incorporated into surface duct models to predict the corresponding acoustic transmission loss. Such interfacing of models is referred to as "coupled ocean acoustic modeling" (Mooers et al., 1982). Models of the mixed layer have largely been restricted to the treatment of 1D approximations, which have proved useful when horizontal advection can be neglected. In many ocean areas, however, the 1D approximations appear to be inappropriate for estimating mixed layer depths (Garwood, 1979).

Mixed-layer models are of two basic types: differential and bulk. Differential models use the equations for conservation of momentum, heat, salt and turbulent kinetic energy (TKE) in their primitive form, and are not integrated over the mixed layer. The region where the local TKE is large enough to provide a certain minimum level of vertical mixing defines the mixed layer for these models. Bulk, or integrated, models assume that the mixed layer is a well-defined layer that is uniform in temperature and salinity. The governing equations for these models are obtained by integrating the primitive equations over the depth of the mixed layer.

Mixed-layer models respond to three basic types of forcing conditions: wind deepening, heating and cooling. Wind deepening is defined to occur when the mixed layer deepens due to the erosion of the stably stratified region at its base by wind-generated turbulence. The depth of mixing is governed by a balance between the stabilizing effect of surface heating, or a positive surface buoyancy flux, and the effect of mixing due to wind-generated turbulence. This balance governs the mixed layer depth during periods when the mixed layer is shallowing. Under conditions of cooling, a net surface heat loss, or negative surface buoyancy flux, causes the mixed layer to deepen due to convection. Convection usually occurs at night and is the dominant mechanism for deepening the mixed layer in fall and winter, especially under conditions of reduced solar heating and increased evaporative cooling due to increased wind speeds.

A number of mixed-layer models have been developed and some have been coupled with underwater acoustic transmission loss models to provide input (and feedback) for the parameters important for prediction of sound propagation in the surface duct. One system that is operational for fleet applications is the thermodynamical ocean prediction system, or TOPS (Clancy and Martin, 1979). TOPS is categorized as a differential mixed-layer model. Forecasts appear to agree well with measurements in those ocean area where the layer depth is dependent primarily on local conditions and not on advection from neighboring regions. A comparison of observations and predictions generated by TOPS (Figure 5.3) shows reasonable agreement. The top panel illustrates the observed wind speed. The middle panel compares the observed and predicted mixed layer temperatures while the bottom panel compares the observed and predicted depths over the same period.

Clancy and Pollak (1983) advocated coupling the TOPS synoptic mixed-layer model to an objective ocean thermal analysis system in order to produce a continuously updating, real-time, analysis-forecast-analysis system. The forecast component (TOPS) employs the Mellor and Yamada (1974) level-2 turbulence parameterization scheme. It includes advection by instantaneous wind drift and climatologically averaged geostrophic currents, and is forced by surface fluxes supplied by atmospheric models.

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