The Effect of Deformational Flows on Tracer Advection in CAM-FV!

Figure 5: Tracer mixing ratios for the Nair and Lauritzen (2010) horizontal deformation test using CAM-FV. The test employs a deformational flow that is reversed (at time t=T/2) back to the initial state so the initial conditions are the final analytic solution.!

Figure 6: Extending the concept of the Nair and Lauritzen test; rather than returning to the initial state, the deformation is continued until the tracers are carried below the scale of the grid, and are no longer ‘resolved’. This allows us to test how adequately the sub-grid characteristics of the dycore imitate reality (in the form of a highly resolved run).!

Figure 7: This test stretches the tracer below the grid scale of a 2ºx2º grid. The left plot shows the integral of tracer variance using a reference solution (0.125ºx0.125º) averaged onto coarser grids. Decrease in tracer variance with time shows the downscale cascade. The right hand plot compares the integral of tracer variance for a host of different 1-D operators that are available in CAM-FV on a coarse 2ºx2º grid. The operators are (1st) upwind, (VL) van Leer limited 2nd order, (LW) Lax-Wendroff dispersive, unlimited 2nd order, and (PPM) 3rd order piecewise parabolic method with appropriate limiter. LW appears better, but stores the tracer variance in the form of grid-scale noise, and in reality is highly inaccurate. See Kent et. al. 2011b for more details. !

!""#""$%&'()#'!**+,-*.'/0'1,-*#,'1,-%"2/,('3*)#4#"'$%'()#'5.%-4$*-6'7/,#"'/0'8#%#,-6'7$,*+6-9/%':/;#6"!James Kent, Jared P. Whitehead, Christiane Jablonowski, Richard B. Rood !

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Overview of the Dynamical Cores !We focus our attention on the finite-volume dynamical core, however, we also show snapshots of the diffusion characteristics of the other dycores in CAM (described in Neale et al. 2010): !1) NCAR CAM 5.0 Finite-Volume (FV): finite-volume approach with monotonicity constraints (PPM), latitude-longitude grid, D/C-grid staggering, floating Lagrangian vertical coordinate with hybrid coordinate as reference grid, explicit time-stepping with explicitly added divergence damping (diffusion of divergence). Advection is treated similarly.!2) NCAR CAM 5.0 Semi-Lagrangian (SLD): "#4$\U-%&,-%&$-%'"2#*(,-6'(,-%"0/,4'4/;#6'E$()'(,$-%&+6-,'(,+%*-9/%, Gaussian grid, hybrid (!) vertical coordinate, 2-time level, semi-implicit time-stepping with explicitly -;;#;']()\/,;#,').2#,\;$f+"$/%. Advection is treated similarly.!3) NCAR CAM 5.0 Eulerian (EUL): spectral transform method with triangular truncation, Gaussian grid, hybrid (!) vertical coordinate, 3-time level, semi-implicit time-stepping with explicitly added 4th-order hyper-diffusion. Advection is treated via a diffusive semi-Lagrangian algorithm.!4) NCAR CAM 5.0 Spectral Element (SE): spectral element based method, cubed-sphere gird, explicit Runge-Kutta type time-stepping with explicitly added 4th-order hyper-diffusion to dynamics and tracers.!

All dynamical cores are hydrostatic and based on the Primitive Equations set. They are run in their operational configurations which includes their typical diffusion mechanisms (e.g. horizontal divergence damping, diffusion and hyperdiffusion, digital filters, monotonicity or flux limiting constraints, and Asselin time filtering for the 3-time level in EUL).!

Conclusions and Future Work:"• '[#,9*-6',#"/6+9/%')-"'-'"$&%$A*-%('$42-*('/%'()#'-**+,-(#'(,-%"2/,('/0'(,-*#,"?'-%;'$%*,#-"#"'$%')/,$O/%(-6',#"/6+9/%'")/+6;''''C#'*-,#0+66.'E#$&)#;'-&-$%"('$%*,#-"#"'$%'()#'B#,9*-6>''g+,()#,'$%B#"9&-9/%'$%(/'()#'$42-*('/0'B#,9*-6',#"/6+9/%'/%'/()#,''''4/;#6"'-%;'/%';.%-4$*-66.',#6#B-%('(#"('*-"#"'E$66'C#'2+,"+#;>'

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Measurements of the consistency between the dynamical core and tracer transport algorithms via potential vorticity!

Figure 3: A passive tracer is initialized with the potential vorticity (PV) in the baroclinic wave test case (test case 2 of Jablonowski et. al. 2008). PV is conserved (in the absence of friction and diabatic effects) so this tracer (tracer PV) can be compared to PV computed via dynamical variables like wind and temperature fields (dynamic PV). This plot is the l4 inconsistency (dynamic PV-tracer PV). CAM-EUL and CAM-SLD have a higher effective resolution (T85) than SE (ne16p4) and FV (2ºx2º) in this plot, but once the wave develops around day 7, they perform worse in terms of consistency. See Whitehead et. al. 2011 for more in depth comparisons.!

Figure 4: Contour plots in PV units (PVU) at day 12.!

Dynamic PV has steeper gradients than tracer PV for all the dycores. Even at very high resolutions (1/4ºx1/4º with 208 vertical levels), these differences in the nonlinear regime are significant (greater than 25% l4 inconsistency). Hence, is consistency a viable goal in designing a dycore, and if so, how can these differences be acounted for? !

Figure 1: Latitude-height cross sections of a smooth tracer, showing the effect of increases in vertical resolution. The tracer is advected vertically (no horizontal motion) in three wave cycles over a period of 12 days, and returns to its starting point (the initial state is the analytical solution).'With only 30 vertical"levels the tracer is extremely diffused.!

Figure 2: Snapshots at day 4 of a tracer advected by a gravity wave (test 6 in Jablonowski et al. 2008). Plot b) is a high resolution reference solution. Results at 1ºx1º resolution are shown in c) and e), results at 2ºx2º resolution are shown in d) and f). These results illustrate the importance of vertical resolution in comparison to horizontal resolution for the gravity wave test, see Kent et. al. 2011a for more details.!

Impact of vertical resolution on CAM-FV!

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