Capet et al. (2008c) find a decreasing downscale kinetic energy flux for an increase in the model resolution. This emphasizes the necessity of an adequate resolution for the ageostro-phic processes. Dissipation either due to numerical effects or due to any explicit diffusion acts predominantly at the grid scale. If the ageostrophic processes have a comparable scale, they are damped by the dissipation. Especially, the divergent velocities featured by the ageostrophic dynamics are immediately subject to dissipation, if they occur close to the
grid scale. In this case, a low resolution yields a strong interference between dissipation and the ageostrophic dynamics that might dampen a downscale energy flux. As described above, we aim to circumvent this problem by adjusting the resolution to the underlying dynamics and choose the resolution (and domain size) with respect to the length scale of the fastest growing unstable wave for each individual background state. Thus, we assume that the important dynamics which are responsible for the energy flux occur close to this spatial scale. However, in a series of sensitivity experiments, we aim to assure that the results described so far are robust with respect to different resolutions.
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102 103
EAPE [m3/s2]
k [1/m]
a)
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EKE [m3/s2]
k [1/m]
b)
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−2
−1 0 1 2
x 10−4
∂tEAPE [m3/s3]
k [1/m]
c)
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−2
−1 0 1 2
x 10−4
∂tEKE [m3/s3]
k [1/m]
d)
Figure 4.9: APE spectra (a), KE spectra (b) and tendencies for EAPE (c) and EKE (d) for simulations with different horizontal and vertical resolution for Ri0 = 200. The colors are the same as in Fig. 4.4. Solid lines indicate the standard simulation with a resolution of 120 x 120 x 40, the dashed lines indicate a doubled vertical resolution 120 x 120 x 80 (hardly distinguishable from standard simulation) and the dashed-dotted lines a doubled horizontal resolution 240 x 240 x 40.
Fig. 4.9 shows the kinetic and available potential energy spectraEKEandEAPE, respec-tively, as well as their tendencies in wavenumber space for different vertical and horizontal resolutions and an intermediate Ri (the same is found for large and small Ri). Doubling the vertical resolution from 40 to 80 layers yields hardly any changes of the spectra and tendencies. For a doubled horizontal resolution of 240 x 240 grid points, the main
charac-teristics are also similar to the counterparts with lower resolution. However, some minor differences can be observed. Higher horizontal resolution also allows to use smaller hori-zontal viscosity and we choose this in order to achieve a comparable grid Reynolds number for all simulations. This smaller viscosity yields a shift of the dissipation towards smaller scales as can be inferred from Fig. 4.9d. Therefore, the decrease of the spectral kinetic energy, that is induced by dissipation at small scales, is also shifted towards smaller scales for the simulations with higher horizontal resolution. In the higher resolved simulation, the slope ofEAPEis steeper at the large and intermediate scales in comparison to the standard simulation.
Fig. 4.10 shows the small-scale dissipation rate Ds normalized by the total amount of buoyancy production B for the standard simulations shown in Fig. 4.8b as well as for simulations with higher resolution. As for the local balances shown in Fig. 4.9, a doubling of the vertical resolution has nearly no effect on Ds. By contrast, doubling the horizontal resolution yields a stronger decay ofDs with larger Ri. While there is hardly any difference for the simulations with Ri≈20, a doubling of the resolution yields roughly 30% smallerDs at Ri≈1000. Thus, we find some quantitive implications on the small-scale dissipation rate by doubling the horizontal resolutions, even though there are hardly any major qualitative differences.
Another parameter that might influence the dissipation rate is the meridional buoyancy gradient M02 of the restoring target. So far, we only varied N02 for a fixed M02 = 4f2 to obtain a restoring buoyancy target in correspondence to the background Richardson number Ri0 =N02f2/M04. Fig. 4.10 also shows simulations for one set of different Ri0 with M02 = 0.25f2 and another set with M02 = 16f2. Although a change of M02 changes the total energy content and energy dissipation (not shown), there are hardly any changes concerning the small-scale dissipation rate normalized by the buoyancy production. Thus, for the simulations performed within this study, the meridional shear is only important for the absolute magnitude of the energy dissipation, but as long as it does not change the Richardson number (e.g. a smaller M02 is compensated by a smaller N02), it has no qualitative influence on the dynamics.
As discussed above, the vigorous re-stratification of the eddies is responsible for a larger mean Richardson number Ri in comparison to the initial Richardson number Ri0. For instance, a simulation in the standard configuration with Ri0 = 1 yields a mean Ri of Ri = 20. To also obtain simulations with Ri = O(1), we increase the restoring time scale λb fromλb = 2σmax to λb = 64σmax (λb is still roughly 40 times larger than the time
101 102 103 10−0.5
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Ds / B
Ri
a)
Figure 4.10: Globally averaged small-scale energy dissipation normalized by the globally averaged buoyancy production as a function of Ri for different sensitivity experiments. The blue dots indicate the simulations for different Ri0 in the standard configuration with (120 x 120 x 40) grid points, M02/f2 = 4 and λb = 2σmax. Simulations indicated by red and green dots deviate from the standard simulation by a doubled vertical (120 x 120 x 80) and a doubled horizontal (240 x 240 x 40) resolution, respectively. Crosses indicate simulations in standard configuration but with M02/f2 = 0.25and circles withM02/f2 = 16. Diamonds denote non-hydrostatic simulations with a larger restoring time scale of λb = 64σmax and restoring buoyancy targets corresponding to Ri0 between 0.1 and 4 and M02/f2 = 4. The black solid line indicates a power law fit to all simulation for which a linear regression yields Ds/B = Ri−0.13. Blue, green and red solid lines indicate fits to the simulations with standard (Ds/B = Ri−0.15), doubled vertical (Ds/B =Ri−0.19) and doubled vertical
(Ds/B =Ri−0.25) resolution, respectively.
step of the model). This larger restoring time scale yields a stronger compensation of the buoyancy production by a larger source of potential energy and thus reduces the deviations between Ri0 and Ri. A set of experiments with Ri0 varying from Ri0 = 0.1 up to Ri0 = 4 indeed features much smaller mean Richardson numbers, ranging between Ri = 1.2 and Ri= 6.8.
The dissipation rate throughout these experiments is nearly constant as can be inferred from Fig. 4.10. This can be explained with a saturation of the small-scale energy flux at a critical Richardson number. At this Ri, nearly all energy is transferred towards smaller scales where it is dissipated. For instance, for the simulations with increased λb, roughly 90% of the energy injected by buoyancy production is dissipated by the small-scale
dissipation. Since we find a downscale energy flux on all scales for these experiments (not shown), only such an amount of energy can be dissipated at large scales which is directly injected at these scales. Note that the baroclinic production has its maximum at the largest scales in the experiments with smaller Ri as can be inferred from Fig. 4.8. Therefore, it might not be astonishing that still roughly 10% of the baroclinic production is dissipated by the large-scale dissipation.