4.1.1 Single baroclinic mode disturbance of internal solitary waves propa- gation
We compute a large-amplitude solitary wave solution of the DJL equation using a code based on the algorithm of [99] (hereafter referred to as TEW). To determine a
particular solitary wave we set the unperturbed stratification ¯ ρ= ρ2+ρ1 2 + ρ1−ρ2 2 tanh[(y−y0)/δ], (4.1)
and select the wave amplitude. (The Boussinesq approximation would requireρ2−ρ1 ≪ ρ1,2, a condition verified in salt-stratifed lab experiments as well as in field conditions.)
For the equilibrium density (4.1) with sufficiently sharp pycnoclines δ ≪ y0, the am- plitude of nonlinear solitary waves (measurable, e.g., by isopycnal displacement from equilibrium) is known to have a limiting value (see, e.g., Lamb & Wan [58]) where the wave flattens and tends to the limit of a propagating front (i.e., a ‘conjugate state’). In this work we will simply refer to thex-position of the maximum isopycnal displacement as the wave maximum, and its y-position as the wave amplitude. We will be mostly concerned with waves whose amplitude is close to the limiting value. The velocity and density fields of such background travelling wave solution in the wave reference frame are denoted byU(x) andR(x) respectively. In an ideal unbounded domain one would have R→ ρ¯and U →(cW,0) as |x| → ∞. This is approximately true for our numer- ical implementation, as the horizontal extent of the domain is large (cf. Section 4.2 for details.) We denote the extrema of the horizontal velocity component in the wave self-induced shear U by U2(x) andU1(x), respectively, withU2(x)> cW > U1(x) in the wave frame.
Fluctuations with respect to the basic solution will be denoted by u′ and ρ′,
u=U +u′, ρ=R+ρ′, (4.2)
and similarly for any other variable.
In the absence of shear (U = 0) and in the lab frame (where the fluid is at rest at infinity), the unperturbed stratification reduces toR = ¯ρand supports neutrally-stable
linear baroclinic modes, governed by the eigenvalue problem
ˆ
ψyy+ gρ¯y/(¯ρc2)−k2
ˆ
ψ = 0 (4.3)
for the sinusoidal perturbation streamfunction ψ′ =ei(kx−ωt)ψˆ(y, k), with phase speed c(k)≡ω/k, supplemented by the slip-wall boundary conditions ˆψ(0) = ˆψ(1) = 0. For ¯
ρ given by (4.1), and assuming the Boussinesq approximation, whereby
¯ ρy ¯ ρ ≃ ρ2−ρ1 ρ2+ρ1 sech2[(y−y0)/δ] δ ,
eigenvaluescform a countable sequence, which for thin pycnoclinesδ≪1 can be shown (see, e.g., Banks et al. 1976) to be approximated by
c2n=g ρ2−ρ1
ρ2+ρ1
4δ
(1 + 2n+ 2δk)2 −1, n= 0,1, . . . ,
for any fixed wavenumber k. Such modes give rise to a system of waves occurring in pairs with the same phase speed |c| but opposite direction of propagation. Thus, eigenvalue pairs can be ordered in a sequence |c1|> |c2| >· · · >0 accumulating onto zero, i.e., cn→0 as n → ∞. This remains true for more general density backgrounds, as one can show by writing the eigenvalue problem in nonlocal symmetric form, whence the above property, as well as others (like completeness of eigenfunctions), follow from the classical spectral theory of integral operators (see, e.g., [97]).
In the upstream region we superimpose a small perturbation based on the first baroclinic mode of the undisturbed stratification for a wavenumber k =kB:
ψ ′(x, y) ρ′(x, y) = ˆ ψ(y, kB) ˆ ρ(y, kB) acos(kBx) exp −(x−x0)2/S at t= 0. (4.4)
16.2 2 4 6 8 10 12 14 0 1 -0.001 0 0.001 22.3 28.3 32.4 34.4
Figure 4.1: Simulation of an internal wave interacting with a train of upstream perturb- ing mode-1 baroclinic waves. The horizontal velocity component of the perturbation is depicted, internal labels denote the time instants. This is case W1R in table 4.1, full discussion in §4.2. Labels correspond to time, and the pycnocline is referenced by two isolines (R = 1.005,1.015) of the basic state density (varying in the range 1≤R ≤1.02). The domain shown is full-extent in the vertical, but only a center sub- section of the domain is shown in the horizontal. The area inside the box is magnified in figure 4.5.
Notice that since the baroclinic modes are a complete basis any upstream disturbance can be reconstructed by their superposition. Also, notice that in the wave frame all perturbations travel to the right, as the speed of the background wave is larger than the speed of the linear modes.
Figure 4.1 depicts an example of a numerical simulation based on the setup we have described. The parameters we consider for this and other numerical simulations are reported in table 4.2. The snapshots show the perturbation entering the wave and being swept away by the internal shear layer. In this case the pycnocline thickness is not small enough to achieve Ri<1/4, and no growth appears. Even though instability could be introduced by simply decreasingδ, we choose its value in order to stay slightly away from instability and emphasize the perturbation dynamics. As one can see, the structure of
the perturbation changes substantially as it starts to experience the background shear. The full discussion of this and similar phenomena encountered during the perturbation evolution is the focus of the rest of the paper. In particular, the numerical simulations are the subject of§4.2. We now turn our attention to the details of the spectral stability and modal analysis of stratified shear flows next, beginning with the simplest case of parallel flows.