With the Boussinesq approximation, the incompressible, nonhydrostatic equations in a Cartesian reference frame become,
for momentum,
for continuity,
for density
ρ(x,y,z,t)=ρ0+ρ'(x,y,z,t) for for constant ρ0, with
U=(u,v,w) the components of the velocity, and the vector
g=(0,0,-1)g with constant gravity acceleration g.
The initial model density
ρ(x,y,z,t=0) = ρ0 + ρ'(x,y,z) is set
with
ρ0 = 1000.0 kg/m^3, and
where,
where x is the distance from the left hand edge of the domain.
Fig. 1 Experimental and model set-up. Horizontal dashed line is initial undisturbed
centre of the pycnocline, with solid line illustrating initial conditions for model
perturbed pycnocline. Slope gradient is
"s" , such that s= H/LS= d-/L. For reference,
horizontal distance from slope bottom to left hand domain edge is LR.
To be able to explore the parameter space and to be able to provide comparison with other model solutions, we defined a "standard run" . A "standard run" will use constant viscosity and diffusivity values of 1.0 * 10-6 m2s-1 . In this case, with H= 0.15m.
The standard model test runs are initialised using the parameters describing model experiment (d) in Table 1, in particular having a slope of 0.217, and an initial pycnocline depth of zi=0.024m below the upper rigid lid of the model.
Table 1 Initial conditions for eight test model experiments. For each of these model
experiments H = 0.15 m, LR = 0.05 m (see Fig. 1),
ρ0 = 1000 kg/m^3, and ai = 0.027 m, Δρ= 47.0kg/m^3, Δh=0.0035m, d+ = zi, and d-= 0.15 - zi . Pairs of experiments differ only in their initial zi value, which changes
subsequent a0 and λ model values.
For example, 
In this case, Δρ=
47.0kg/m^3, zi=0.024m, Δh=0.0035m, d+ = zi, and d-= 0.15 - zi,
H = 0.15 m, LR = 0.05 m. (MATLAB file)
Benchmark outputs
Table 2 Model conditions for five selected experiment numbers (in brackets) from Michallet
and Ivey (1999). For each of these model experiments H=0.15 m, LR
=1.0 m, ρ0 =1000 kg/ m^3.
Parameter definition:
Slope = H/Ls = d-/L (see the sketch in Fig. 1); zi: a distance below the top of the domain
(see the sketch in Fig. 1); Δρ: the
change in density across the pycnocline; a0: the amplitude of ISW; Lw: characteristic
length; ξ(x)= 2aisech2(x/2W)
Table 3 Experimental data for five selected experiment numbers from Michallet and Ivey
(1999). Variation in experimental Iribaren number ξ is based on range of internal
wave amplitudes a0, and a 5% error estimate in Lw by Michallet and Ivey (1999).
Parameter definition: a0: internal wave amplitude,
Lw: characteristic length, ξ(x)= 2aisech2(x/2W)
Fig.2 Decrease of wave energy (E/Ei) against the normalized traveled distance x/H(h2/H)2 for waves propagating at constant depth.
The instantaneous wave energy E relative to the initial wave engergy Ei, as a functioin of the normalized traveled distance that is, x/H(h2/H)2 [note that the waves were free to reflect at each end of the tank and that one flume length corresponds to x/H(h2/H)2 ≒10]. The average energy lost across one flume length is roughly 0.01 J/m^2. Figure 2 also shows the wave energy lost in the model during the propagation of an ISW in a channel of constant depth using free-slip sidewalls (run 2a in Table 1). In this model setup, the energy loss is caused by bottom friction and interfacial shear. The wave energy lost in the model during wave propagation over the length of one flume is roughly 10 times less than in the laboratory. Since the model has been set up to mimic the laboratory scales, we infer that there must be a dissipative mechanism in the laboratory that is not included in this model configuration.
Fig.3 Breaking location (criterion) as a function of λ.
Parameter definition:
λ=(kL)-1, k-1= sqrt( 4(d+d-)^2
/ 3(d+ - d-)a0 )
Fig.4 Reflection coefficient R against Iribarren number ξ for the dots (model) and Michallet and Ivey (1999) experiments (crosses). Size of boxes and crosses at each point indicate estimated error bounds. The bold curve is the best fit to model by Bourgault and Kelley (2007) for the free slip sidewalls.
