4.1 Dynamics
4.1.4 Direct Numerical Simulations
We now carry a DNS of the spectral equations with the maximum circular truncation wavenumber of 16 (C16). We have found, through experimentation, that this truncation wavenumber is sufficient for exploring high resolution behaviour for our chosen parame- ters. Obviously, for different sets of parameters, this resolution may not be sufficient; for example, if the dissipation is much weaker. Unless otherwise stated, we keep the same set-up and parameters as stated in the previous section. In addition, we set the parameter
ν = 6.06×1013 m4 s−1 in (4.1). These values are plausible for the atmosphere and were
specified by HE, whose results we seek to confirm (although they did not specify the value of the parameterν and scale heightHsused). In our experiments, unlike HE, we have not added modes of random amplitude to the sinusoidal topography; this enables us to have
44 Multiple Equilibria and Atmospheric Blocking
the best possible comparison with the STM results. The full spectral equations
∂ζk ∂t = X p X q δ(k+p+q) [K(k,p,q)ζ−pζ−q+A(k,p,q)ζ−ph−q] +ikx k2(β+k 2 0U)ζk−ikxU ζk−ikxhkU −(α+νk4)ζk, (4.24) and ∂U ∂t =α(U −U)− X k ikxζkh∗k k2 . (4.25)
are stepped forward in time using a predictor-corrector algorithm. A more detailed de- scription of the numerical methodology for the DNS is given in the thesis of O’Kane (2003). The interaction coefficients K andA are defined in Eqs. 2.52 and 2.51, respectively.
We start off the flow simulations with both a low initial U = 2.0 m s−1 and a high
U = 10.0 m s−1 for a range of values of Hm: 0< Hm ≤2500 m. The initial small-scale field is set to zero. The timestep is 1/60 day. We shall, at times, use a scaled time, αt, for convenience (when αt = 1, t = 8.7 days). A bifurcation results as the parameter
Hm is varied, as shown in Fig. 4.2. As Hm is increased from zero, the flows exhibit one (unblocked) equilibrium. Then as Hm reaches a critical value (around 1000 m), the flows suddenly exhibit two equilibria (blocked and unblocked). WhenHm reaches another critical value (around 2300 m), the flows suddenly revert to one equilibrium (blocked). All the evolved final flow velocities are for t = 167 days. This curve confirms the result obtained by HE although the values for the topography are larger by roughly a factor of two. This might be due to a different scale height (not specified) used in that study. We can see that, at high resolution, the range of values of Hm for which multiple equilibria exist is reduced (now 1000 m< Hm<2300 m) although clearly this range remains significant.
H
Figure 4.2: Equilibrium values ofU for two flows with different initial conditions as a function of maximum topographic heightHm.
The first point to branch in Fig. 4.2, at Hm = 1000 m for initial U = 2.0 m s−1, is interesting because it enables us to see how adding extra modes affects the low-order system described in Section 3. This is because the flows have been initialized with no energy in the small-scales; thus, in the initial stages the DNS behaves exactly like the
§4.1 Dynamics 45
STM model (which has multiple states at this topographic height). When the other modes have picked up sufficient energy, as a result of non-linear interactions, the behaviour of the two systems starts to diverge. As can be seen in Fig. 4.3, the flow seems to settle in the blocked state until αt ≈ 45 and then suddenly makes a transition to the unblocked state. This brings up the question of whether all the flows in the blocked state eventually end up in the unblocked state. For this system at least, the answer is no: in the long run, the dual states seem to persist once the topography is high enough. We can see this in Fig. 4.4: when Hm= 1100 m, after αt≈60, the flow develops an instability but remains in the blocked state. We have evolved the flow for as long as αt ≈320 but no transition was observed; it appears to have settled permanently in this unsteady equilibrium state.
α
α α
α
Figure 4.3: Kinetic energy time series of (a) large scale flow, (b) kx=±3, (c)kx= 0, and (d) kx6= 0,±3 modes forHm= 1000 m
It is worthwhile to take a closer look at what happens to the different components of the flow for both Hm = 1000 m and Hm = 1100 m. The former being the case where the transient eddies actually ‘destroy’ the multiple states while the latter being a case where they are preserved. ForHm = 1000 m (Fig. 4.3), the small-scale flow is dominated by the kx = ±3, ky = 0 modes, as a result of interaction with the topography, until
αt≈40. When 40< αt <50, the flow suddenly jumps to the unblocked state; during this intermediate time, there is a dramatic drop in wavenumber 3 energy and a subsequent rise in the energies of the other modes while the large scale flow rapidly relaxes towards
U. Forαt >50, the wavenumber 3 energy has settled to a lower—but not insignificant— value, and the other modes have vanished. It is important to note that the energy in
46 Multiple Equilibria and Atmospheric Blocking
α
α α
α
Figure 4.4: Kinetic energy time series of (a) large scale flow, (b) kx=±3, (c)kx= 0, and (d) kx6= 0,±3 modes forHm= 1100 m
wavenumber 3 that remains in the unblocked state is not due to topographic drag in the flow, of which there is virtually none. We can see this in Fig. 4.5, which shows Re(zk)
and Im(zk) (zk has been defined in (4.12)). It is clear from equation (4.9) that Im(zk)
represents the topographic drag on the flow; this drag disappears once the transient eddies have perturbed the system and the flow settles to a higher value of U. The remaining energy in wavenumber 3 is due to the real part of zk.
We have also shown, in Fig. 4.1, the resonance curve (asterisks) for the DNS. When compared to the resonance curve for the STM (solid curve), it is clear that the transient eddies have a damping effect on the stationary wave amplitude; hence, this topographic height represents the boundary at which multiple equilibria start to appear in the high- order system whereas for the low-order system the boundary is for smaller values of Hm. The resonance curve for the DNS has been calculated by evolving the small-scale flow, equation (4.24), with U kept constant, for a range of values of U (0 ≤ U ≤ 15 m s−1).
Im(zk), fork= (3,0), was then averaged over a series of time-steps once the system had
reached a steady state, for each value of U. It is also worthwhile noting that the resonant wind has shifted to a somewhat lower value as compared to the low-order system (STM). A similar effect is seen in the article by Speranza (1986), for example, which discusses the effects of wave-wave interactions on the low-order system.
For the case whenHm = 1100 m (Fig. 4.4), the flow is dominated by wavenumber 3 until αt ≈60 after which there is an analogous drop in that wavenumber’s energy. The
§4.1 Dynamics 47
α α
Figure 4.5: Time-evolution of (a) real part, and (b) imaginary part of z for Hm = 1000 m, obtained by DNS.
α α
Figure 4.6: Same as in figure 4.5, but forHm= 1100 m.
difference is that the extraction of energy from wavenumber 3 is not sufficient to allow the large scale flow to relax towardsU. The flow instead becomes dominated by transient eddies of various scales. In Fig. 4.6, the diagnostics Re(zk) and Im(zk) show that the
mean drag remains more or less the same, even after instability sets in. The drop in wavenumber 3 energy is clearly due to the drop in the real part of zk.
It seems then that, with this simple model at least, the destabilizing effect of the transient eddies is not enough to limit the flow to just one equilibrium state (the unblocked state). Multiple equilibria as envisioned by CdVcanappear in complex (multi-component) systems like the DNS. In the following sections we explore further a number of issues related to the behaviour of models exhibiting multiple equilibria.