α
Figure 4.11: Zonal winds averaged over a mid-latitude band. The solid lines are for the Northern Hemisphere while the dotted lines are for the Southern Hemisphere. Two flows are shown, with different initial conditions. The Southern hemisphere flows converge to two very finely spaced equilibria while the Northern Hemisphere ones have two distinct equilibria.
The values of the winds used in this experiment are probably larger than what would be expected in the atmosphere. However, we should note that there are many unknowns and features that have only been crudely represented in our model such as the effect of zonally asymmetric heating, the precise latitudinal profile of the zonal jets, and the appropriate relaxation time for the jet structure as compared to the Ekman dissipation time-scale, to mention just a few. Ambiguous interpretations of the barotropic vorticity equation as the vertically-averaged flow or as the flow at a particular vertical level (Tung, 1985) means that the range of realistic parameters is potentially large, which could easily account for the higher than expected winds.
4.2
Parameterization of Non-linear Interactions
We found in Sections 4.1.3 and 4.1.4 that the severely truncated three-component system (STM) is a very useful device for understanding and predicting the behaviour of the DNS. However, we also found some significant differences between the two models. These differences arise because of the absence of non-linear interactions in the STM. In this section, we shall attempt to improve the performance of the STM, so that it is better able to simulate the broad results of the DNS while still maintaining its simple structure.
54 Multiple Equilibria and Atmospheric Blocking
Figure 4.12: Time series of wavenumber 2 kinetic energy in a (statistically) steady state. The top curve is the ‘blocked’ flow while the bottom one is ‘unblocked’.
Fig. 4.1. The drag, Im(zk), on the large scale flow in the three-component system has
a higher peak, which is more than three times the corresponding drag in the C16 DNS. Moreover, the peak in the STM occurs for higher values ofU than that for the C16 DNS. The differences between the two models are only apparent for 1< U <5 ms−1; outside of this range, the two models are in excellent agreement. The physical reason for this is not hard to understand. The stationary wave amplitude is only significant in the previously mentioned range; hence, it is only in this range that a topographic instability emerges once the amplitude reaches some critical value. Outside of this range, there is no other source of instability in this problem, hence the STM is sufficient for describing the system as other modes are not excited. The phenomenology of topographic turbulence also tells us that once transient eddies are excited, they will contribute to the drag on the large scale flow via the topography. Hence, the curve in Fig. 4.1 (from the C16 DNS) is the net result of two processes: the reduction of the drag due to non-linear instability of the stationary wave, and its enhancement due to interaction between transient eddies and topography. The net effect is clearly a reduction of the drag. The reduction of the topographic drag may be parameterized by imposing a stronger dissipation coefficient α (see Eq. 4.14). However, this alone will not shift the peak of the curve to lower values of U. To accomplish this we need to adjust the frequency ωU
k as well. It is convenient to think of the shift in ωUk as
arising due to a shift in β. Thus, a lower value of β will result inωUk vanishing at a lower value of U, and hence the peak of the curve will also shift in the same direction. This suggests that it may be possible to parameterize the effect of the non-linear interactions
§4.2 Parameterization of Non-linear Interactions 55
Figure 4.13: Zonal wind contour plot in ‘blocked’ state in the presence of latitudinally dependent jets with underlying topography of figure 4.10.
Figure 4.14: Eddy Streamfunction contour plot in ‘blocked’ state in the presence of latitudinally dependent jets with underlying topography of figure 4.10.
on the drag by increasing the dissipation coefficient, α, and reducingβ.
It is possible to calculate effective coefficients αr and βr by inverting the three- component expressions, Eqs. 4.14 and 4.15, and using the values of Re(zk) and Im(zk)
from the C16 DNS. Hence,
αr(U) = − kx|hk|2U Im(zk) |z|2 . (4.31) and ωr(U) = − kx|hk|2U Re(zk) |z|2 . (4.32)
Thus, withωr(U) from Eq. 4.32, we obtain from Eq. 4.8:
βr(U) =k2 U− ωr(U) kx −k20U. (4.33) The effective parameters αr and βr are shown in Figs. 4.17 and 4.18, respectively, as functions of U. The effective dissipation coefficient,αr, as expected, is stronger than the actual Ekman drag coefficient, α, by as much as a factor of 5 at U = 2.3 ms−1. This can be regarded as the parameterization for the net effect of stationary wave amplitude reduction, due to non-linear instability, and its enhancement due to coupling between transient eddies and the topography. Clearly, the loss of amplitude is the dominant effect. The effective change in the Coriolis parameter,βr, is on the other hand weaker, reducing
56 Multiple Equilibria and Atmospheric Blocking
Figure 4.15: Zonal wind contour plot in ‘unblocked’ state in the presence of latitudinally depen- dent jets with underlying topography of figure 4.10.
Figure 4.16: Eddy Streamfunction contour plot in ‘unblocked’ state in the presence of latitudi- nally dependent jets with underlying topography of figure 4.10.
to a minimum value about 2 times smaller than the actual value at 2.8 ms−1. This can be regarded as a parameterization for the shift of resonance to lower values of U of the stationary wave.
The replacementsα→αr(U) and ω→ωr(U) in Eqs. 4.14 will yield the drag Im(zk)
in the STM identical to the drag in the C16 DNS. However, the use of U dependent parameters may not be a convenient solution to the problem. It is possible, instead, to treat the problem by least squares minimization as follows. We seek to minimize the function f(α, β) = N X i=1 yi−Zi(α, β, Ui)2, (4.34) where Z(α, β, Ui) = − kx|hk|2αUi α2+k2 x Ui−β+k 2 0Ui k2 . (4.35)
Here, the yis are the values of Im(zk) calculated from the C16 DNS at distinct values of
Ui. Z is the functional form of the topographic drag as calculated from the STM.α and
β are parameters that minimize the total error. The function f(α, β) has been minimized numerically using Brent’s Principal Axis (PRAXIS) method (Brent, 1973). The non- dimensional values α = 4.38×10−2 and β = 0.15 were found; these are consistent with the expectation that the actual value of α = 1.82×10−2 would increase and that of
β = 0.26 would decrease. The values of −Im(zk) using the STM, Eq. 4.14, with α →α
§4.2 Parameterization of Non-linear Interactions 57
4.19. Clearly, there is an underestimation of the drag at the peak (1.5 ≤U ≤2.5), and mostly overestimation in other areas. This is the price we have to pay for working with
U independent parameters. In many respects, however, the use of the parameters α and
β improves the profile of the drag. The difference in amplitude between the peaks of the three-component system and the C16 DNS is only about 25% as opposed to the factor of 3 difference when using the parameters α and β. The peak is also now correctly shifted to about 1.5 ms−1 from the original 2.0 ms−1 calculated withα and β.
Figure 4.17: The effective dissipation coefficientαr(U) in the three-component system (STM).
Figure 4.18: The effective variation of Coriolis parameterβr(U) in the three-component system (STM).
58 Multiple Equilibria and Atmospheric Blocking
Figure 4.19: A comparison of the drag calculated with αandβ in the three-component system (STM) (solid curve) and the corresponding drag calculated in the C16 DNS (asterisks).