The beta effect has been found to stabilize the large scale eddies as predicted by the phenomenology of QG turbulence. In general, the eddy kinetic energy produced for a given mean shear is reduced when compared to the equivalent layer case, which had no beta effect. This means that the eddies are baroclinically more stable—even in a linear sense— with the beta effect operating. The non-linear interactions are also suppressed, particularly the long range interactions between the large and small scales. This is apparent from the shapes of the subgrid fluxes; when compared to the equivalent layer case, the large scale contribution is significantly diminished. This is more noticeable in the oceanic simulations, which have a large separation between the large and injection scales.
The flux in the top level is greater than the flux in the bottom level in this problem. It is found that the flux of enstrophy plays an important role in determining the relative magnitudes of the subgrid matrix parameters. In the atmosphere, where the matrices are practically diagonal because the truncation scale is much greater than the deformation scale, the stochastic backscatter in the top level is found to be as much as an order of magnitude greater than that in the bottom level as a result of the greater flux there. The drain dissipation parameters, on the other hand, turn out to be relatively close in magnitude. In the atmospheric LES it is again found that, broadly, the deterministic and stochastic parameterizations perform comparably well. However, a closer inspection of the LES near the truncation scale reveals that the stochastic parameterization performs marginally better. Specifically, the deterministic parameterization tends to slightly lift the tail of the spectrum right at the truncation scale. This is particularly evident for the top level, which has a strong flux. The stochastic parameterization does not display this phenomenon when implemented in the LES.
For oceanic simulations, where the full matrix structure of the parameters needs to be considered, the structure of these matrices is as follows. The elements of the drain dissipation matrix have in general larger real than imaginary components, with the ex- ception of the M21 element. Both the diagonal elements are positive near the truncation
scale, with the top level contribution being slightly negative at the large scales. This is because the barotropic mode is predominantly the top level in this problem; that is
ψ = 12(ψ1+ψ2) ≈ 12ψ1, if ψ1 ≫ ψ2. In terms of magnitude, the top level diagonal con-
tribution tends to be less than that for the bottom level. The off-diagonal elements are dominated by the top level contribution (M12element), which is negative at all wavenum-
bers, and comparable in size to the diagonal elements. The bottom level off-diagonal contribution (M21element) is in general the smallest element, and its real part is negative
except near the truncation scale. In fact, for this element, it is the imaginary part that is dominant, and this has a similar structure to the real part. The backscatter matrix is dom- inated by theF11element; the other elements are somewhat smaller and have similar sizes.
Structurally, the diagonal elements are positive while the real parts of the off-diagonal ele- ments are negative; the imaginary parts of the off-diagonal elements are generally smaller in size. The deterministic formulation again displayed the difficulty identified in the previ- ous chapter; namely, that the diagonal barotopic dissipation matrix element in the BTBC formulation (−Mr
11) is significantly negative. In this case, this matrix element was found
to be negative at all scales at T63. In comparison, the corresponding matrix element in the stochastic formulation (the drain dissipation matrix element) was found to be only slightly negative at the large scales, but significantly, was positive near the truncation scale. The two matrix elements are shown side-by-side for comparison in Fig. 7.40. In the stochastic
§7.3 Discussion 183
Figure 7.40: A comparison of diagonal barotropic dissipation coefficients (−Mr
11) for: (a) net dissipation (−Mrn); (b) drain dissipation (−M
r
d) matrices in the BTBC formulation (T63).
formulation, the injection of barotropic energy from the subgrid scales, due to the inverse cascade, is captured by the stochastic forcing, rather than by the negative dissipation of the deterministic formulation. This is a more numerically stable configuration.
In running the oceanic LES, it as found that, as for the equivalent layer problem, it is impossible to use the deterministic parameterization because it is too numerically unstable. The stochastic parameterization generally gives good agreement with the higher resolution simulation. Furthermore, it is sufficient to use the isotropized (m averaged) matrix parameters in this problem despite the fact that the beta effect induces large scale anisotropy in the form of westward travelling Rossby waves. In the next chapter, instead of the (m, n) = (0,1) (solid body rotation) relaxation imposed in this and the previous chapter, we shall examine more complicated sources of baroclinic mean energy, namely, meridionally confined jets and currents.
Chapter 8
Flows with Jets and Differential
Rotation
In this chapter we shall continue to analyze two-layer flows in the presence of differential rotation. However, the mean field imposed here is of a more complicated nature than the (m, n) = (0,1) relaxation of the previous two chapters. We impose mean fields correspond- ing to the zonal circulation found in the atmosphere, with strong mid-latitude jet streams. For the ocean, we impose a strong current at 60o South, which roughly corresponds to the Atlantic Circumpolar Current (ACC). Spectrally, these mean fields corresponding to forcing on the modes (m, n) = (0, n), where nis typically confined to small wavenumbers (we have chosen n < 15). Importantly, n is no longer just one as in the previous two chapters. This means that there are several spectral sources of mean baroclinic energy which can be used to generate transient energy. Hence, the assumption that the change in potential energy is small compared to the total potential energy cannot be used in gen- eral to justify the exclusion of the mean subgrid forcing as was done in the previous two chapters. This makes the problem harder as we have to parameterize both the mean and transient fluxes. The motivation, of course, is that this situation is closer to what happens in real geostrophic flows. The governing equations for the DNS are again the spherical two layer QGPV equations, including differential rotation:
∂qi ∂t =−J(ψi, qi)−2 ∂ψi ∂λ −D 0 iζi+κ(qreli −qi), (8.1) where i=1,2. The zonal wind is relaxed towards ureli ; qreli in Eq. 8.1 is the potential vorticity corresponding to the imposed wind ureli = (urel
i ,0). The other terms in Eq. 8.1 are the same as defined in previous chapters.