Using the simulation parameters in Table 1, the linear stability analysis was insensitive to setting νvνv to this smaller value, Selleck GKT137831 so for the purpose of this modeling exercise the smaller viscosity/diffusivity sufficed. One consequence of varying N2N2 and M2M2 is that the dynamics may become sensitive to whether the hydrostatic approximation is employed. Because the balanced Richardson number can be tuned by adjusting
the values of M2,N2M2,N2, and f , the individual parameters for each set are chosen to fix the hydrostatic parameter ( Marshall et al., 1997) equation(25) η=γ2Ri,where γ=h/Lγ=h/L is the aspect ratio of the motion. For η≪1η≪1 it is appropriate to use the hydrostatic approximation to the vertical momentum equation. The parameter γγ is estimated according to the initial M2M2 and N2N2 from the simulations. Because the unstable modes lie in an arc symmetric about the isopycnal, the mean aspect ratio of the motions can be taken as γ=M2/N2γ=M2/N2, and simple algebra gives equation(26) η=f2N21Ri2.The parameter choices in Table 1 are chosen so that η=0.1η=0.1 for the “hydrostatic” parameters and η=10η=10 for the “nonhydrostatic” parameters. Note that in both cases, the fully nonhydrostatic equations are solved. To check whether the results are sensitive to whether a model is run in hydrostatic mode, a parallel Apoptosis inhibitor set of the η=0.1η=0.1 simulations was
run using the MITgcm (Marshall et al., 1997) in hydrostatic mode and with identical initial conditions. The hydrostatic MITgcm gave nearly identical results (not shown) as long as the grid spacing ΔxΔx was less than half the wavelength of the most unstable mode; when ΔxΔx was set above this threshold the MITgcm was prone to numerical instability which eventually led to the simulation crashing. This numerical instability influenced TCL the choice to use the nonhydrostatic solver for these simulations over the MITgcm. Nonetheless, previous work by Mahadevan (2006) suggests that the average vertical fluxes at the length scales in these simulations should be similar regardless of whether the model is run hydrostatically or nonhydrostatically, so it is likely that the results from
the nonhydrostatic solver are robust for the η=0.1η=0.1 simulations at all resolutions. The simulation parameters in Table 1 were chosen specifically to demonstrate cases of grid-arrested restratification (Sets A and C) and completed restratification (B and D) by varying νhνh. The amount of restratification that takes place is not uniquely dependent on the parameter choices in each set; all of the parameters can be varied in relation to one another to change the anticipated final value of Ri . Fig. 4 shows the growth rate plots for each parameter set. In each case the horizontal viscosity damps the highest wavenumber modes, so that increasing the resolution beyond a certain point does not permit extra modes to become resolved or further restratification to occur.