during calibration of the model (see Seelig, Harris, and Herchenfoder, 

 in preparation, 1977) . 



For M inlets connecting the bay to the sea, the total discharge 

 for all inlets, Qy, is: 



M 



Qr = Ji Qm • " (8) 



The continuity equation is written as: 



at 



^ay 



(9) 



Bay levels and inlet current velocities are determined by solving the 

 simultaneous differential equations (7) and {9) using a Runge-Kutta-Gill 

 fourth order finite-difference technique in conjunction with initial con- 

 ditions and the time history of water levels in the sea. Derivation and 

 sample applications of this model are given in Seelig, Harris, and 

 Herchenroder (in preparation, 1977) . 



To obtain response characteristics similar to Figure 2 for a specific 

 inlet-bay system, the model can be run by assuming sinusoidal seawater 

 level fluctuations with a typical amplitude; e.g., 3 centimeters (0.1 

 foot) at periods covering the anticipated range of lake oscillation modes. 

 Each run of the model will give predicted bay levels and inlet current 

 velocities for the wave period used. Sample model results for Pentwater 

 inlet are shown in Figure 4. The sea level, predicted bay level, and 

 inlet velocity are shown in the lower part of Figure 4; the importance 

 of each of the terms in the equation of motion, normalized by dividing 

 by the magnitude of the largest term at each time step, is shown in the 

 upper part of the figure. For this condition, the bay level fluctuation 

 is larger than the sea level fluctimtion due to inertia in the system 

 and the bay level lags the sea level by 84° (Fig. 4) . Plotting results 

 from many runs similar to Figure 4, but with many different forcing 

 periods, will give the response characteristics of the inlet-bay system. 

 These curves for Pentwater (Fig. 5) show that the Helmholtz period with 

 friction, Tg, is 1.8 hours, waves with periods of 1 to 3 hours will be 

 amplified by the system, and waves with a period of 1.4 hours will gen-' 

 erate the highest inlet current velocities (3-centimeter wave amplitude 

 assumed) . The effect of friction on lu is demonstrated by the dash- 

 line in Figure 2. The frictionless Helmholtz period is also coinciden- 

 tally 1.4 hours (from eqs. 4 and 5). 



The calculated bay amplification and channel velocity in Figure 5 

 are for a Manning's n = 0.036. The numerical model usually had to be 

 run for three or four cycles for the bay response to build to equilibrium. 

 In the prototype harbor it is likely that equilibriiim (full amplification) 

 is never fully achieved. Thus, the calibration curve in Figure 5 forms 

 the upper envelope of measured prototype data. 



20 



