Chapter 4 Stability of Rossby Waves
4.5 Comparison of the 3MT and the 4MT models with DNS
Traditionally, four modes have been employed in the study of the modulational instabil- ity [8; 20], the justification being that if one of the sidebands is not initially excited, it rapidly becomes so, driven by the instability [20]. It will be clarified here why a single triad and the 3MT is not commonly used to derive the modulational instability.
It has been stated that the modulational instability is a sum of the decay insta- bilities for the two resonant triads∆+ = 0and∆−= 0and that asM →0 it collapses
to to these resonant manifolds. These two curves are mostly disjoint from each other except at the origin. Thus the two unstable eigenvectors of the instability of the 4MT are equivalent to the eigenvectors of the two decay instability triads. For smallM, the maximum growth rate of both the instabilities become identical while for largerM values, that of the modulational instability is typically larger.
In the strong interaction limit, maximum instability is obtained when the primary and secondary waves are perpendicular [8] and the traditional approach is to select a meridional flow such that the primary wavevector,p is along thex-axis and the pertur- bation q along the x-axis. In this situation, looking at figure 4.2, q is equally close to
both branches of the resonant manifold suggesting that the 4MT should be used since interactions with both manifolds are likely. There is the possibility thatqis off-zonal in that it also has ax-component. An investigation of whether the fundamental mechanism of the modulational instability retains one or both sidebands will be carried out using the 3MT and 4MT models two weakly nonlinear (M = 0.1) scenarios:
(i) Pure meridional primary wave,p= (10,0)with a pure zonal modulation,q= (0,1).
(ii) Pure meridional primary wave,p= (10,0)with an off-zonal modulation,q= (9,6).
The actual modulational mode that gives the maximum of the instability in this case is
q = (9.43,5.35) but due to the discreteness of the wavenumbers in the periodic box, this exact wavenumber cannot be selected in thek-space domain and this chosen mode is actually the only excited mode in the vicinity of the theoretical value. This sparsity of active modes due to the discreteness of the domain can have a profound effect on numerical simulations of weakly nonlinear regimes [77] if not carefully avoided.
Referring now to case (i) for the zonal modulation, figure 4.3 compares the amplitude of this mode,|Ψq|obtained from DNS of Eq. 3.61 to solutions from the 3MT
in Eq. 4.3 and the those from the 4MT in Eq. 4.5 with the initial condition being based on the unstable eigenvector of the decay instability. It is clear from figure 4.3(a) that the DNS follow the growth rate predicted by the modulational instability rather than that of the decay instability. Looking more closely at early times in figure 4.3(b), it becomes apparent that while the second sideband is not initially excited, it quickly becomes so [20] with the result that in the time of the order of one inverse of the instability growth rate. Since no other modes are rapidly excited, the 4MT is a better model for this situation up to at least 10 instability times.
For scenario (ii), figure 4.4 compares the amplitude of this mode,|Ψq|obtained
from DNS of Eq. 3.61 to solutions from the 3MT in Eq. 4.3 and the those from the 4MT in Eq. 4.5 with the initial condition being based on the unstable eigenvector of the decay instability The growth rates of the decay and modulational instability are practically identical and both of the models agree well with DNS up to seven characteristic times. As well as the growth rate, they predict well the maximum amplitude of the zonal jet
10-5 10-4 10-3 10-2 10-1 100 101 0 2 4 6 8 10 12 14 | Ψq |/| Ψ0 | t/τ (a) PDE 3MT 4MT 10-5 10-4 10-3 10-2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 t/τ (b) |Ψq| (PDE) |Ψp -| (PDE) |Ψp +| (PDE) 3MT 4MT
Figure 4.3: Amplitude of the zonal mode with wavenumberqforM = 0.1, p = (10,0)
obtained from DNS and from solutions of 3MT and 4MT models. Case (i), pure zonal modulation, q = (0,1). (a) Long time evolution. (b) Zoomed view of early time evolution.
although the subsequent decrease in amplitude is not as well described as in case (i) by the 4MT model.
From these results it can be concluded that the three-wave interaction is indeed the basic nonlinear process when M 1 provided the triad is not degenerate, in the sense that it does not contain quasi-resonant modes which are equidistant from two different resonant manifolds, as happens when the vectorq is zonal. In these cases, the 3MT system is just as good as the 4MT and it describes well the full CHM system for over several characteristic times. On the other hand, the most relevant configuration with q zonal is in fact, degenerate. In this case, however, the 4MT model works well over many characteristic times whereas the 3MT fails almost immediately. Thus, to have a wider range of applicability, the 3MT model is abandoned and the 4MT model is employed in the study of the modulational instability.
10-5 10-4 10-3 10-2 10-1 100 101 0 5 10 15 20 25 30 35 40 | Ψq |/| Ψ0 | t/τ PDE 3MT 4MT
Figure 4.4: As for figure 4.3(a) but now for case (ii), off-zonal modulations,q = (9,6).