Assessing Tracer Transport Algorithms and the ... - Semantic Scholar

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Assessing Tracer Transport Algorithms and the Impact of Vertical Resolution in a Finite-Volume Dynamical Core JAMES KENT, CHRISTIANE JABLONOWSKI, JARED P. WHITEHEAD, AND RICHARD B. ROOD Department of Atmospheric, Oceanic and Space Sciences, University of Michigan, Ann Arbor, Ann Arbor, Michigan (Manuscript received 29 June 2011, in final form 19 October 2011) ABSTRACT Modeling the transport of trace gases is an essential part of any atmospheric model. The tracer transport scheme in the Community Atmosphere Model finite-volume dynamical core (CAM-FV), which is part of the National Center for Atmospheric Research’s (NCAR’s) Community Earth System Model (CESM1), is investigated using multidimensional idealized advection tests. CAM-FV’s tracer transport algorithm makes use of one-dimensional monotonic limiters. The Colella–Sekora limiter, which is applied to increase accuracy where the data are smooth, is implemented into the CAM-FV framework, and compared with the more traditional monotonic limiter of the piecewise parabolic method (the default limiter). For 2D flow, CAM-FV splits dimensions, allowing overshoots and undershoots, with the Colella–Sekora limiter producing larger overshoots than the default limiter. The impact of vertical resolution is also explored. A vertical Lagrangian coordinate is used in CAM-FV, and is periodically remapped back to a fixed Eulerian grid. For purely vertical motion, it is found that lessfrequent remapping of the Lagrangian coordinate in CAM-FV improves results. For full 3D tests, the vertical component of the tracer transport dominates the error and limits the overall accuracy. If the vertical resolution is inadequate, increasing the horizontal resolution has almost no effect on accuracy. This is because the vertical resolution currently used in CAM version 5 may not be sufficiently fine enough to resolve some atmospheric tracers and provide accurate vertical advection. Idealized tests using tracers in a gravity wave agree with these results.

1. Introduction Accurate representation of tracer transport is important to weather and climate models. In climate simulations with the Met Office’s Unified Model there are typically 25 individual tracers (Collins et al. 2008), whereas in the chemistry version of the National Center for Atmospheric Research (NCAR) Community Atmosphere Model (CAM), over 100 tracers are transported (Lamarque et al. 2008). The tracer transport scheme is considered part of the dynamical core—the fluid dynamics component of a general circulation model (GCM)—and is closely linked to physical parameterizations and other model components, such as the chemistry package. In fact, some errors in chemistry models are purely due to the numerical transport scheme (Prather et al. 2008) as opposed to errors in chemical reactions or physical

Corresponding author address: James Kent, Department of Atmospheric, Oceanic and Space Sciences, 2455 Hayward St., University of Michigan, Ann Arbor, Ann Arbor, MI 48109-2143. E-mail: [email protected] DOI: 10.1175/MWR-D-11-00150.1 Ó 2012 American Meteorological Society

parameterizations. Numerical effects in tracer transport schemes may also lead to errors in cloud microphysical parameterizations (Ovtchinnikov and Easter 2009). Therefore, it is important in the design of dynamical cores to ensure accurate tracer transport (Rasch et al. 2006). Advection algorithms are one of the main components in dynamical cores, as they can be used both in solving the governing dynamical equations and in the transport of tracers. A number of different types of numerical methods are used in transport schemes, such as finite-volume methods (Lin and Rood 1996), semi-Lagrangian methods (Zerroukat et al. 2002), and spectral element methods (Taylor et al. 1997; Dennis et al. 2005). This paper focuses on the characteristics of the transport scheme of NCAR’s Community Atmosphere Model finite-volume (CAMFV) dynamical core (Neale et al. 2010). Specifically, we analyze the tracer transport in CAM version 5.0, which is identical to the recently released version 5.1 from June 2011. In general, tracer mixing ratios must remain positive; therefore, it is essential that numerical schemes used for

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tracer transport do not produce negative tracer mixing ratios. A variety of different methods to maintain positivity can be used, such as filling algorithms, or the use of ‘‘limiters’’ (see Rood 1987). Finite-volume methods, such as those used in the horizontal discretization of CAM-FV, often make use of flux or slope limiters. These limiters are used in areas of steep gradients or extrema to limit the discretized numerical fluxes of the tracer constituents and prevent any nonphysical under- or overshoots from occurring (LeVeque 1992). One problem with limiters is that they can be overly diffusive, especially when transporting smooth data. In this paper we implement the Colella–Sekora limiter (Colella and Sekora 2008) for the piecewise parabolic method (PPM; see Colella and Woodward 1984) into CAM-FV, which is designed to retain accuracy for smooth data while remaining monotonic in one dimension. This new limiter is tested against other limiting options in the CAM-FV framework—using one-, two-, and three-dimensional tests—to show the effect that the choice of limiter has on the accuracy of tracer transport in CAM-FV. In atmospheric modeling the emphasis is often placed on the horizontal aspects of the flow. As computational power increases, the horizontal resolution is generally increased at a much faster rate than the vertical resolution. This may be justified when considering the aspect ratio of the atmosphere, and that stratification and the effects of rotation indicate that large-scale atmospheric flow is very similar to two-dimensional horizontal flow. However, the vertical component is important, and as atmospheric models aim to resolve more and more horizontal scales, it will be essential that the vertical aspects of the flow and vertical transport are modeled accurately. Just increasing the horizontal resolution in atmospheric models may not improve the overall accuracy and may not lead to convergence (Roeckner et al. 2006) as the vertical resolution should also be increased (Lindzen and FoxRabinovitz 1989; Persson and Warner 1991). A common choice of the vertical discretization in dynamical cores is based on low-order finite-difference methods (Staniforth and Wood 2008), although fluxlimiting schemes (Thuburn 1993) and finite-element methods (Untch and Hortal 2004) have been used. Some models even use different types of vertical discretization for dynamic variables and tracers, such as the CAM Eulerian spectral transform dynamical core (CAM-EUL), which uses spectral methods for the dynamics and a semiLagrangian scheme for the tracers (Neale et al. 2010). CAM-FV, however, uses a floating Lagrangian controlvolume discretization in the vertical (Lin 2004) that is regularly remapped back to an Eulerian reference frame. In this paper we will investigate the vertical transport in CAM-FV using different vertical remapping time steps of

the Lagrangian coordinate. We will also vary our grid spacings to determine the effect of changes in resolution (horizontal and vertical) on tracer transport in the CAMFV dynamical core. These investigations are accomplished through the use of idealized test cases. To establish the accuracy of any numerical scheme, testing must take place using idealized test cases (see Williamson et al. 1992; Jablonowski et al. 2008; Nair and Lauritzen 2010). Idealized test cases have either known solutions or known features that are easy to evaluate in model tests. With tracer transport tests it is possible to advect a tracer back to its starting point, giving an analytic reference solution (i.e., the initial conditions) to compare with. This can be done for simple passive advection tests with time-invariant velocities, or with more complicated deformation tests that prescribe a time-varying flow field (Nair and Lauritzen 2010). For more complicated and realistic flows, analytical solutions do not exist, and high-resolution simulations are often used to provide a reference solution. Most of the tests we consider in this paper will assess the advection scheme of CAM-FV in isolation. However, we also utilize the dynamical core in adiabatic mode (i.e., no physical parameterizations) to investigate tracer transport in flow generated by the gravity wave test (Jablonowski et al. 2008; Tomita and Satoh 2004). We repeat some of the tests using CAM-EUL to provide a comparison with CAM-FV. The paper is structured as follows. Section 2 briefly describes the advection algorithm in the CAM-FV dynamical core, and discusses the limiters and schemes that will be used for tracer transport. The results of oneand two-dimensional horizontal advection tests are given in section 3. Three-dimensional tests and purely vertical advection tests are in section 4. Both sections 3 and 4 are based on prescribed flow fields. Section 5 investigates tracer transport using internally generated three-dimensional flow fields that focus on the propagation of gravity waves. Conclusions are drawn in section 6, and the test case descriptions are contained in the appendix.

2. The tracer advection schemes in CAM Tracer transport is governed by the continuity equation and the tracer conservation equation, which are given respectively by ›r 1 $ (rv) 5 0, and ›t (›rq) 1 $ (rvq) 5 0, ›t

(1) (2)

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where r is the fluid density, q is the tracer mixing ratio, ($) is the divergence operator, and v is the threedimensional velocity vector. In advective form, the tracer equation yields ›q 1 v $q 5 0. ›t

(3)

Note that if the fluid is incompressible, $ v 5 0, then Eq. (3) can be written in flux form [Eq. (2)]. CAM-FV splits the governing equations into horizontal and vertical parts; the horizontal components are solved using finite-volume methods and the vertical component is solved using a floating Lagrangian coordinate and vertical remapping. This transforms the governing equations in CAM-FV into the form ›dp 1 $h (dpvh ) 5 0, and ›t (›dpq) 1 $h (dpvh q) 5 0, ›t

(4) (5)

where the pressure thickness dp between the boundary pressure interfaces of the finite-volume cell replaces the role of density. The subscript h restricts the divergence operator and the velocity to their horizontal components (Lin 2004). The two-dimensional horizontal tracer transport equation is solved using the Lin–Rood scheme (Lin and Rood 1996, 1997) with semi-Lagrangian extensions in the longitudinal direction on a latitude–longitude grid. This lessens the otherwise severe time step restriction near the poles and allows Courant–Friedrichs– Lewy (CFL) numbers greater than 1 in the zonal direction (although the meridional CFL number must remain less than 1). Vertical advection is modeled via a vertical remapping algorithm that remaps the prognostic variables at regular time intervals from the Lagrangian levels to a fixed Eulerian reference frame. The remapping algorithm is based on piecewise parabolic subgrid reconstructions, and conserves tracer mass (Lin 2004). The idea of a Lagrangian vertical coordinate and remapping algorithm, based on the work of Starr (1945), has also been used in other dynamical cores (Nair et al. 2009; Toy and Randall 2009; Bleck et al. 2010). In CAM-FV, the remap is performed every ‘‘remap’’ time step, which is generally 4–10 times larger than the dynamics time step. The remapping can introduce additional errors and diffusion into the vertical component of the solution. Therefore, changing the remapping time step should have an effect on the accuracy of the vertical tracer transport. The Lin–Rood scheme, used for the horizontal transport, makes use of multiple one-dimensional steps. Onedimensional methods are used to calculate the fluxes X

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and Y, in the longitudinal (x) and latitudinal (y) directions respectively, which in turn are used to calculate one-dimensional operators F and G. To solve the twodimensional problem, a half time step is performed in the y direction and then this is used to complete a full time step in the x direction. The same process is used to give a full step in the y direction. The F and G (and the advective forms of these operators, f and g; see Lin and Rood 1996) are used to solve for the tracer at the new time step as 1 1 qn11 5 qn 1 F qn 1 g(qn ) 1 G qn 1 f (qn ) , 2 2 (6) where n is the temporal index. The operators F and G are the differences of the fluxes at cell interfaces in their respective directions. A number of different onedimensional numerical schemes or limiters can be used with the Lin–Rood scheme in CAM-FV to calculate the fluxes X and Y. The default limiter in CAM5 is a modified version of the PPM limiter (Colella and Woodward 1984), and is given in appendix B of Lin (2004); we call this the ‘‘default’’ limiter throughout this paper. To show the effect of the one-dimensional limiters and schemes (used with the Lin–Rood scheme) on tracer transport, we also make use of other options contained in CAMFV: a first-order upwind scheme and the second-order van Leer scheme with the monotonized-central (MC) limiter (van Leer 1974, 1977). We call the latter the van Leer scheme throughout this paper. We will also apply the Colella–Sekora limiter (Colella and Sekora 2008) for PPM. The Colella–Sekora limiter is designed to maintain high-order accuracy at smooth extrema in one dimension. We use the version with improvements from McCorquodale and Colella (2011) and Almgren et al. (2010). As these limiters–schemes for the Lin–Rood method are one dimensional, the Lin–Rood scheme splits dimensions in terms of limiting. Therefore, the Lin–Rood scheme does not give a strictly monotone solution for two-dimensional flow because of this dimensional splitting (Rasch 1994). CAM-FV also uses a ‘‘filling’’ algorithm to ensure that the mixing ratio of any tracer cannot be negative, while also conserving the tracer (Neale et al. 2010). If the horizontal transport produces negative mixing ratios, then mass is borrowed from neighboring cells in the zonal direction. If negative mixing ratios are generated by the vertical advection then the tracer is capped at zero, and mass is borrowed from the layer above. If the layer above does not contain enough mass, then the mass is not conserved. It is worth noting that in its default operational configuration, CAM-FV uses different limiters for the horizontal and vertical directions in

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the tracer advection algorithm. The constraint for the vertical tracer remap is the ‘‘quasi-monotone’’ constraint of Huynh (1997), also given in appendix B of Lin (2004). For some of the tests, comparisons will be made with the spectral transform Eulerian dynamical core CAMEUL, which uses a monotonic semi-Lagrangian advection scheme that is subdivided into horizontal and vertical substeps (see Neale et al. 2010; Williamson and Rasch 1989, 1994). Previous studies show that for tracer tests in full atmospheric simulations (dynamics and physical parameterizations), CAM-FV and CAM-EUL exhibit different behavior (Rasch et al. 2006). CAM-FV utilizes a latitude–longitude grid. CAMEUL is based on a Gaussian grid that closely resembles an equidistant latitude–longitude grid. The horizontal resolution in CAM-FV is written in terms of degrees, and the corresponding number of grid points (latitude 3 longitude) is shown in Table 1. The CAM-EUL resolutions are denoted by the highest wavenumber used for the triangular truncation, as explained later in section 4a. For the three-dimensional tests the resolution is written as, for example, 18 3 18 L60, which signifies 18 latitude by 18 longitude with 60 vertical levels. For time stepping, CAM-FV has independent dynamics, tracer, and remap time steps. This means that different-sized time steps can be used for dynamic variables, tracer transport, and vertical remapping (see Neale et al. 2010). In contrast, the dynamics and tracer time steps in CAMEUL are identical.

3. Assessment of the horizontal advection in CAM Horizontal tests are used to determine the accuracy of the two-dimensional tracer transport scheme in CAMFV. The tests are devised such that the final tracer solution should equal the initial tracer distribution and therefore error norms can be calculated exactly. The normalized root-mean-square l2 error norm is calculated as ( )1/2 I[(q 2 qT )2 ] l2 5 , (7) I[(qT )2 ] and the normalized l‘ error norm is calculated as l‘ 5

maxjq 2 qT j , maxjqT j

(8)

where qT is the true (analytic) solution and I is the global two-dimensional integral. To perform purely horizontal tests using the CAM-FV framework, the vertical velocity is indirectly set to zero. This is achieved by guaranteeing that the pressure thickness dp of the Lagrangian layer stays constant at all times.

TABLE 1. Horizontal grid spacings (lat 3 lon) for CAM-FV. The number of latitudes includes both pole points. Degrees Grid points

28 3 28 91 3 180

18 3 18 181 3 360

1/28 3 1/28 361 3 720

1/48 3 1/48 721 3 1440

a. One-dimensional solid-body rotation To evaluate the accuracy of the different flux limiters in CAM-FV, a one-dimensional advection test is used. The tracers are advected once around the equator with a solid-body rotation zonal velocity over the course of 12 days [see test case 1 in Williamson et al. (1992) and Eqs. (A1) and (A2)]. This test demonstrates the accuracy and convergence of the numerical schemes in one dimension. The tracers are initialized first as a Gaussian bell and then as a slotted cylinder, described by Eqs. (A3)– (A6) in the appendix. The spatial resolution varies from 28 3 28 to 1/ 48 3 1/ 48, and a time step of Dt 5 450 s is used for each resolution. This gives maximum CFL numbers 0.0781, 0.1563, 0.3125, and 0.625 for the 28 3 28, 18 3 18, ½8 3 ½8, and 1/ 48 3 1/ 48 resolution grids, respectively. The first-order upwind scheme, van Leer, and PPM with the default and Colella–Sekora limiters are examined for the equatorial advection test. Log plots of the normalized l2 and l‘ error norms at day 12 are shown in Fig. 1. The first-order scheme is not included as its error is two orders of magnitude larger than the other schemes. For example, the first-order scheme has an l2 error norm of 0.2502 for the 18 3 18 Gaussian bell test, whereas the van Leer scheme has an l2 error norm of 0.0069. The horizontal axis is the number of longitudinal grid points (N). Different powers of the longitudinal resolution are shown to highlight convergence rates—that is, the rate at which the error of a numerical scheme approaches zero as the grid spacing tends to zero. For the Gaussian bell, PPM with the Colella–Sekora limiter is the most accurate, while PPM with the default limiter has a smaller error norm than the van Leer scheme (for all resolutions). The runs with the default limiter and the Colella–Sekora limiter appear to converge between second- and thirdorder accuracy for both the l2 and l‘ errors. For the slotted cylinder the default limiter outperforms the Colella– Sekora limiter and each of the schemes have a convergence rate less than order 1. The poor convergence rates are due to the slotted cylinder being discontinuous (Holdaway et al. 2008). All of the limiters and schemes used ensure that there are no overshoots or undershoots in the one-dimensional tests (even without the use of the filling algorithm). The maximum value of the initial tracers is 1. The first-order scheme is very diffusive, and after 12 days with 18 3 18 resolution reduces the peak of the Gaussian bell to

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FIG. 1. Normalized (a),(b) l2 and (c),(d) l‘ error norms in the equatorial advection test with CAM-FV [van Leer scheme (VL), PPM with the default limiter (df), and PPM with the Colella–Sekora limiter (CS)] for (a),(c) the Gaussian hill and (b),(d) the slotted cylinder tracer fields. The N denotes the number of longitudes, and powers of N show convergence rates.

0.7228. This is in contrast to the van Leer scheme and PPM with the default and Colella–Sekora limiters, which reduce the peak to 0.9878, 0.9923, and 0.9930, respectively. This is similar to the peaks for the slotted cylinder. The one-dimensional advection tests demonstrate that the Colella–Sekora limiter is the most accurate for smooth data, whereas the default limiter is the most accurate for steep and discontinuous data.

b. Horizontal cross-polar test Modeling cross-polar flow is important for any dynamical core. As CAM-FV makes use of a latitude– longitude grid, it is vital to test the advection scheme when there is cross-polar transport. The Lin–Rood scheme and CAM-FV have previously been tested for horizontal cross-polar advection (see Lin and Rood 1996; Jablonowski et al. 2006, 2009). To test the different choice of schemes/limiters used with the Lin–Rood method, we perform a solid-body rotation over the poles [see test case 1, with a 5 p/2 in Williamson et al. (1992),

and Eqs. (A7) and (A8) in the appendix]. As with the equatorial advection test, the tracers are initialized as both a Gaussian hill and slotted cylinder, the spatial resolution varies from 28 3 28 to 1/ 48 3 1/ 48, and a time step of Dt 5 450 s is used for each resolution. This ensures that the CFL number is less than 1 in the latitudinal direction for each resolution. Figure 2 shows contour plots of CAM-FV with the first-order scheme, the van Leer scheme, and PPM and the default and Colella–Sekora limiters at day 12 using the slotted cylinder initial conditions on the 18 3 18 grid. The first-order scheme is very diffusive, while the other schemes keep the shape of the slotted cylinder well. The normalized l2 and l‘ error norms for the first-order upwind scheme, van Leer, and PPM, with the default and Colella–Sekora limiters, are given in Table 2 for both the Gaussian hill and slotted cylinder initial conditions. As with the equatorial advection test, the error norms for the first-order scheme are significantly larger than those for the other schemes tested. For this test, PPM with the

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FIG. 2. Horizontal cross-polar test on the 18 3 18 grid after 12 days, using CAM-FV with (a) the first-order scheme, (b) the van Leer scheme, (c) PPM and the default limiter, and (d) PPM with the CS limiter, and the slotted cylinder initial conditions.

default limiter outperforms PPM with the Colella– Sekora limiter for both sets of initial conditions. This is because the cross-polar test is not one-dimensional because of the latitude–longitude grid (i.e., there is both zonal and meridional flow). This also leads to over- and undershoots with each scheme (apart from first order) for the slotted cylinder initial conditions.

c. Two-dimensional deformation test To test two-dimensional horizontal tracer transport in the CAM-FV dynamical core, the deformation test described in Nair and Lauritzen (2010) (test 4) is used. This

test is significantly more challenging than the solid-body rotation test and is therefore more suitable for testing atmospheric tracer transport schemes. This deformation test reverses the time-varying prescribed flow at the halfway point of the simulation, so that the final tracer distribution should equal the initial tracer distribution. This means that error norms can be calculated for each scheme/limiter used in CAM-FV; from Eq. (7), qT 5 q(t 5 0). Additionally, a zonal background velocity overlays the deformational flow field, so that the deformed tracers are advected once around the earth, preventing the possible cancellation of dispersion errors when the

TABLE 2. Normalized (top) l2 and (bottom) l‘ error norms for the horizontal cross-polar test after 12 days. The first number is for the Gaussian hill (q1) and the second for the slotted cylinder (q2). Scheme First order van Leer PPM with default PPM with Colella–Sekora First order van Leer PPM with default PPM with Colella–Sekora

28 3 28 0. 4194/0.5578 0.0176/0.2601 0.0094/0.2001 0.0108/0.2070 0.4712/0.6363 0.0373/0.6172 0.0206/0.6200 0.0284/0.6191

18 3 18 0.2608/0.4591 0.0039/0.1970 0.0023/0.1505 0.0025/0.1557 0.3043/0.6405 0.0121/0.6178 0.0058/0.5902 0.0071/0.5690

1/28 3 1/28 0.1326/0.3686 0.000 70/0.1476 0.000 62/0.1178 0.000 63/0.1212 0.1604/0.8589 0.0035/0.6230 0.0016/0.6040 0.0017/0.5944

1/48 3 1/48 0.0723/0.3077 0.000 18/0.1147 0.000 16/0.0910 0.000 16/0.0933 0.0890/0.6674 0.0013/0.6458 0.0007/0.6216 0.0007/0.6019

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FIG. 3. Deformation test on the 18 3 18 grid at (a),(b) half time (t 5 T/2) and (c),(d) final time (t 5 T) using CAM-FV with PPM and the (a),(c) default limiter and (b),(d) CS limiter and the cosine bell initial conditions.

original flow is reversed. Using this test also allows us to compare results with the numerical schemes presented in Nair and Lauritzen (2010). The initial conditions are given by Eqs. (A9)–(A13) in the appendix. In contrast to Nair and Lauritzen (2010), they are formulated with physical dimensions. Two cosine bells and two slotted cylinders are used as the tracer distributions. The tracer time step is Dt 5 1080 s for the 28 3 28 and 18 3 18 resolutions, and Dt 5 432 s for higher horizontal resolutions. The test is simulated for T 5 12 days. This gives maximum CFL numbers as 0.4859, 0.9718, 0.7775, and 1.5549 in the zonal direction, and 0.2984, 0.5968, 0.4775, and 0.9549 in the meridional direction for the 28 3 28, 18 3 18, ½8 3 ½8, and 1/ 48 3 1/ 48 resolution grids, respectively. During the deformation the tracer mixing ratios are ‘‘quasi resolved’’ at all resolutions. This means that despite the severe stretching of the initially compact tracer, the tracer does not break up into multiple cells. However, the representation of the deformed tracer on the coarse discretized grids is poor in comparison to the higher resolutions. The half time (t 5 T/2) and final time (t 5 T ) tracer mixing ratios on the 18 3 18 grid are shown in Fig. 3 for the cosine bell initial conditions and Fig. 4 for the slotted cylinder. PPM with the default and Colella–Sekora

limiters are shown. As stated above, at the half time and the point of greatest deformation, the features of the cosine bells are still large enough to be resolved on the 18 3 18 grid. For both sets of initial conditions, the plots are similar at t 5 T/2 for both limiters, although the Colella–Sekora limiter has allowed more undershoots. The normalized l2 and l‘ error norms for the van Leer scheme and PPM with the default and Colella–Sekora limiters on 28 3 28, 18 3 18, ½8 3 ½8, and 1/ 48 3 1/ 48 resolution grids at time t 5 T are shown in Table 3 for the cosine bells and Table 4 for the slotted cylinders. Whereas the Colella–Sekora limiter has a noticeably smaller error norm than the default limiter for smooth data in the one-dimensional tests (in section 3a), for the two-dimensional deformation tests the Colella–Sekora limiter has a larger error norm for the coarser resolution; this is consistent with the cross-polar test (section 3b). For higher resolution tests, the error norm of the Colella– Sekora limiter is very similar to that of the default limiter. This is partly because the cosine bells are not as smooth as the Gaussian bells used in the one-dimensional tests, and partly because of the dimensional splitting employed by the Lin–Rood scheme. This dimension splitting reduces the improvements seen by the Colella–Sekora limiter in one dimension, and confirms that the Lin–Rood scheme

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FIG. 4. As in Fig. 3, but with the slotted cylinder initial conditions.

is not completely monotonic in two dimensions. As the flow is two dimensional, overshoots and undershoots can occur for the van Leer scheme and the default and Colella–Sekora limiters, although for the cosine bells only undershoots occur. The Colella–Sekora limiter has the largest undershoots because it is the least diffusive; using 18 3 18 resolution, the minimum value of the cosine bells after 12 days is 0.074 for Colella–Sekora, 0.086 for the default limiter, and 0.096 for van Leer (it is 0.1 initially). These undershoots contribute to the Colella–Sekora limiter having a larger error norm than the default limiter. We compare the results for the cosine bells initial conditions with the normalized l2 error norms given in Nair and Lauritzen (2010) for a third-order discontinuous Galerkin (DG) transport scheme (Nair et al. 2005) and the conservative semi-Lagrangian multitracer transport scheme (CSLAM) of Lauritzen et al. (2010). Note, however, that both the DG transport scheme and CSLAM make use of the cubed-sphere grid, while CAMFV uses the orthogonal latitude–longitude grid—the type of grid also affects accuracy, and therefore this is not meant to be a direct comparison between CAM-FV and these numerical schemes. With grid spacing equivalent to 1.58 3 1.58, the DG scheme has an error norm of 0.0562, although the time step used is 180 s (for T 5 5 days). With the same grid spacing, CSLAM’s error varies between

0.1088 and 0.0328, depending on the time step. The results in Table 3 show that the errors for CAM-FV using PPM with the default or Colella–Sekora limiters are of a similar order of magnitude as those for the DG scheme and CSLAM on the cubed-sphere grid. Also, without monotonic limiters, the undershoots for the DG scheme and CSLAM are similar to those of CAM-FV using PPM with the default or Colella–Sekora limiters. For the slotted cylinder initial conditions, overshoots and undershoots occur for the van Leer scheme and PPM with the default and Colella–Sekora limiters on 18 3 18 and higher resolution. Once again the Colella– Sekora limiter has the largest over- and undershoots. On

TABLE 3. Normalized (top) l2 and (bottom) l‘ error norms for the 2D deformation test after 12 days using the cosine bell initial conditions (q3). Scheme van Leer PPM with default PPM with Colella–Sekora van Leer PPM with default PPM with Colella–Sekora

28 3 28 0.3445 0.2506 0.2854

18 3 18 0.1110 0.0585 0.0686

1/28 3 1/28 0.0401 0.0179 0.0177

1/48 3 1/48 0.0082 0.0046 0.0045

0.4878 0.4024 0.4275

0.2189 0.1290 0.1406

0.0926 0.0508 0.0480

0.0233 0.0119 0.0103

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TABLE 4. Normalized (top) l2 and (bottom) l‘ error norms for the 2D deformation test after 12 days using the slotted cylinder initial conditions (q4).

TABLE 5. Normalized (top) l2 and (bottom) l‘ error norms for the 3D passive advection test. The first number is for the smooth tracer (q5) and the second for the steep tracer (q6).

Scheme van Leer PPM with default PPM with Colella–Sekora van Leer PPM with default PPM with Colella–Sekora

Scheme First order van Leer PPM with default PPM with Colella–Sekora First order van Leer PPM with default PPM with Colella–Sekora

28 3 28 0.4253 0.3812 0.3927

18 3 18 0.3280 0.2839 0.2914

1/28 3 1/28 0.2391 0.2003 0.2102

1/48 3 1/48 0.1719 0.1581 0.1620

0.7744 0.8457 0.8697

0.8442 0.8415 0.8449

0.8440 0.8341 0.8347

0.8405 0.8339 0.8315

the 18 3 18 grid, the maximum after 12 days is 1.0594 for the Colella–Sekora limiter, 1.0337 for the default limiter, and 1.0100 for the van Leer scheme (the maximum is 1 initially). The undershoots never fall below zero (the minimum is 0.1 initially) and this is because of the filling algorithm preventing negative values. The extent of the over- and undershoots can be seen for the default and Colella–Sekora limiters in Fig. 4. We repeat the tests using a smaller tracer time step, Dt 5 300 s, for each spatial resolution. The results of these tests indicate that a smaller time step, and therefore more numerical tracer transport steps, can lead to a less dispersive (and therefore more diffused) solution. For example, the maximum of the slotted cylinder after 12 days is 1.0003 for the Colella–Sekora limiter, 0.9997 for the default limiter, and 0.9997 for the van Leer scheme when using Dt 5 300 s on the 18 3 18 resolution. These maxima are much smaller than those found when using Dt 5 1080 s. (Note that each of the limiters still produce undershoots for the slotted cylinder.) The smaller time step solutions can actually lead to larger error norms for the coarser resolution grids; the peaks of the cosine bells are reduced and the shape of the slotted cylinder is smoothed out. However, these solutions contain far fewer over- and undershoots than those with the larger time step, indicating that more time steps can lead to more diffusion in the solution for this type of advective scheme. This is consistent with results from the solid-body advection test with CAM-FV by Jablonowski et al. (2006).

4. Impact of the vertical resolution a. Three-dimensional assessments For tracer transport in GCMs, three-dimensional advection must be considered. Varying the grid size will show the impact of vertical and horizontal resolution on tracer transport in CAM-FV. The three-dimensional passive advection test uses the velocities and initial conditions described in the test case (3–3–56) from

28 3 28 L60 0.8219/0.8160 0.1848/0.4802 0.1340/0.4216 0.1191/0.4305

18 3 18 L60 1/28 3 1/28 L60 0.6670/0.7078 0.4132/0.5786 0.1258/0.3982 0.1190/0.3673 0.1213/0.3791 0.1179/0.3611 0.0939/0.3756 0.0888/0.3537

0.8065/0.9744 0.2732/0.9164 0.2239/0.8637 0.2018/0.8484

0.6595/0.9624 0.1870/0.8787 0.1717/0.8577 0.1515/0.8218

0.4070/0.9370 0.1859/0.8381 0.1924/0.8562 0.1204/0.8603

Jablonowski et al. (2008). In essence, the tracer is transported once around the sphere during a 12-day period at a 458 rotation angle, while following a wave-like trajectory in the vertical direction. The initial conditions are given by Eqs. (A14)–(A22) in the appendix. Two tracers are selected. One is a smooth tracer distribution [q5, given by Eq. (A19)] and the other contains sharp edges and resembles a slotted ellipse [q6, given by Eq. (A21)]. A time step of Dt 5 900 s is used for both the tracer time step and the vertical remap for each horizontal resolution. The maximum height of the domain—the model top—is set to 12 km. The tests were performed using the first-order and van Leer schemes and PPM was performed with the default and Colella–Sekora limiters. The vertical resolution was kept at 60 levels spaced by Dz 5 200 m, and the horizontal resolution was varied from 28 3 28 to ½8 3 ½8. The normalized l2 and l‘ error norms at day 12 are shown in Table 5. The error norms of PPM with the Colella–Sekora and the default limiters are similar to the errors in the van Leer scheme, whereas there is a large difference between them and the error norm of the first-order scheme. The first-order scheme performs poorly because it is far too diffusive to accurately advect the tracers; the maximum of the smooth tracer after 12 days using the first-order scheme is just 0.39 on the 18 3 18 grid (it is initially 1). These results can be compared with CAM-EUL with 60 vertical levels. Note that CAM-EUL uses different horizontal resolution designations. The normalized l2 error norms for CAM-EUL are (the first number is for the smooth tracer q5 and the second for the steep tracer, q6): 0.5359/0.6647 for the spectral triangular truncation T85 (128 3 256 grid points and a time step of 600 s), 0.5425/0.6512 for T170 (256 3 512 grid points and a time step of 300 s), and 0.5425/ 0.6442 for T340 (512 3 1024 grid points and a time step of 150 s). These three spectral resolutions correspond to the grid spacings 1.48 3 1.48, 0.78 3 0.78, and 0.358 3 0.358,

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FIG. 5. Latitude–height cross sections of the smooth tracer q5 at l 5 2708 for the three-dimensional advection test at day 12. Using 18 3 18 L60 resolution with Dz 5 200 m, (a) the initial tracer and reference solution at day 12, (b) CAM-FV with PPM and the default limiter, (c) PPM and the Colella–Sekora limiter, and (d) CAM-EUL with a T170 L60 resolution with the Gaussian grid spacing of 0.78 3 0.78.

and are thereby even finer than the corresponding CAM-FV grid spacings of 28, 18, and ½8. The error norms for CAM-EUL are much larger than for CAMFV. Latitude–height cross sections of the smooth tracer, q5, at the longitude l 5 2708 at day 12 are shown in Fig. 5 and can be compared to the initial conditions that serve as the reference solution. The figure depicts the CAMFV PPM scheme with the default limiter, PPM with the Colella–Sekora limiter, and CAM-EUL. The initial tracer and the solutions for the default and Colella–Sekora limiters are plotted for the 18 3 18 resolution, while the CAM-EUL solution is plotted for T170 resolution. Each plot is for 60 vertical levels. For CAM-FV, PPM with the default and Colella–Sekora limiters produces very similar results. There are slight overshoots in the center of the tracer, while the edges have been smoothed out. CAMEUL produces a heavily diffused solution. The tracer has been spread out and the maximum is 0.6225, compared to a maximum of 1 for the initial tracer. For the smooth tracer q5 PPM with Colella–Sekora is the most accurate, followed by the default limiter, then van Leer and then the first-order scheme. For the steepsided tracer q6 the error norms for Colella–Sekora and the default limiters are similar. However, although the accuracy improves for each scheme as the horizontal resolution increases, each of the schemes’ errors converge at a rate less than order 1, even for the smooth tracer q5.

b. Vertical advection in isolation To test the effects of the vertical transport in isolation, we repeat the advection test, but set the horizontal velocities to zero. The vertical advection tests use the vertical pressure velocity designed for the three-dimensional tests given in Eq. (A16). The tracer undergoes three oscillations in the vertical during the 12-day simulation. This tests the accuracy of the vertical component of tracer transport in the CAM-FV dynamical core. The initial tracer is the smooth tracer used in the three-dimensional tests, q5 given by Eq. (A19). The vertical remapping time step in CAM-FV can be changed. Theoretically, Lagrangian transport is exact (true Lagrangian transport is parcel based; in CAM-FV the vertical transport is really semi-Lagrangian transport with moving boundaries, and could destroy information in large cells), and the remapping to fixed Eulerian coordinates introduces errors. This implies that a larger remap time step, and therefore fewer remaps, would produce a more accurate solution; the results show that this is generally true, suggesting that more frequent remapping does lead to an accumulation of spatial errors. The normalized l2 and l‘ error norms for the smooth tracer q5 at day 12 for the vertical advection test using remapping time steps of 900 and 7200 s are shown in Table 6 for a range of vertical resolutions that

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TABLE 6. Normalized (top) l2 and (bottom) l‘ error norms for the 1D vertical advection test of the smooth tracer (q5) at the 28 3 28 (CAM-FV) and T85 (CAM-EUL) horizontal resolutions with varying vertical grid spacing. The L denotes the number of levels with equidistant grid spacing Dz and a model top at 12 km. The grid spacings are Dz 5 400, 200, 100, and 50 m; 900 s and 7200 s denote the vertical remap time steps. Model CAM-FV 900 s CAM-FV 7200 s CAM-EUL CAM-FV 900 s CAM-FV 7200 s CAM-EUL

L30 0.3566 0.2986 0.7307 0.2928 0.2488 0.6331

L60 0.0871 0.0492 0.5255 0.1071 0.0646 0.4380

L120 0.0427 0.0145 0.2740 0.0659 0.0234 0.3013

L240 0.0060 0.0017 0.2118 0.0109 0.0039 0.2609

vary the equidistant vertical grid spacing between Dz 5 400 m and Dz 5 50 m. The horizontal resolution is kept constant at 28 3 28. The model top is set to 12 km as before. Also shown are the results when using CAMEUL (with a horizontal resolution of T85 and a time step of 300 s). The results show that the errors decrease rapidly for CAM-FV as the vertical resolution is increased. The normalized l2 errors are converging much more rapidly for CAM-FV than CAM-EUL, with faster convergence for the larger remapping time step. Using a longer remap step does decrease the error, and produces a slightly smoother solution. This is in contrast to the results for the two-dimensional horizontal tests, where using a smaller time step resulted in a more diffused solution. Overshoots are possible with the vertical component of CAM-FV because the remap is not strictly monotone. Therefore, more frequent remaps allow small overshoots to occur more frequently than when using a longer remap step. CAM-FV with the Lagrangian vertical coordinate has a significantly smaller error norm than the Eulerian core at all resolutions tested. CAM-EUL produces a heavily diffused solution, with no overshoots or undershoots.

c. Convergence with horizontal and vertical resolution Using the three-dimensional advection test from Jablonowski et al. (2008), we can investigate error convergence with increases in both horizontal and vertical resolution. The normalized l2 error norms when using CAM-FV with the default limiter are presented in Table 7 for 28 3 28, 18 3 18, and ½8 3 ½8 horizontal resolution, and 30, 60, and 120 vertical levels for the smooth tracer q5. A time step of Dt 5 900 s is used for both the tracer time step and the vertical remap for each resolution. The results show that for this tracer test, the increases in vertical resolution improve accuracy more than increases in horizontal resolution. Convergence rates

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TABLE 7. Normalized l2 error norms for the 3D passive advection test with increases in both horizontal and vertical resolution for the smooth tracer (q5). CAM-FV with PPM and the default limiter is used. CAM-FV L30 L60 L120

28 3 28 0.3633 0.1340 0.0798

18 3 18 0.3579 0.1213 0.0544

1/28 3 1/28 0.3565 0.1179 0.0431

when keeping the vertical resolution fixed are much less than one. This is due to the vertical advection errors dominating. Convergence rates when keeping the horizontal resolution fixed are just below second order, although the highest rate of convergence is when both horizontal and vertical resolution are increased. The impact of increasing vertical resolution for this test case appear because with only 30 vertical levels there are an insufficient number of grid points to resolve the tracers in the vertical; increases in vertical resolution resolve the smoothness of the tracer, and therefore leads to large increases in accuracy. The results from this section demonstrate the importance of vertical resolution for tracer transport, which may be underappreciated in current GCMs (Senior et al. 2011). (Note that for many physical parameterizations it is not easy to change the vertical resolution, unlike with the adiabatic dynamical core, and therefore there are few studies that investigate sensitivities to vertical resolution using full GCMs of dynamics and physics.) The results indicate that the vertical component can provide an upper bound on the accuracy of the dynamical core for three-dimensional tracer transport. This can be seen when comparing the error norms for the threedimensional test with the tracer profile q5 using the resolutions ½8 3 ½8 L30 and 28 3 28 L60, which correspond to vertical grid spacings Dz 5 400 m and Dz 5 200 m, respectively. Both runs use the PPM scheme with the default limiter and a remap time step of 900 s. With the 28 3 28 L60 resolution the error norm is 0.1340, whereas the error norm for ½8 3 ½8 L30 is 0.3566. Note that for purely vertical advection the error is 0.0871 for 60 vertical levels and 0.3566 for 30. The ½8 3 ½8 L30 resolution has almost eight times more grid points than 28 3 28 L60, but the overall error is limited by the poor resolution in the vertical. The tests using CAM-EUL reinforce this concept. The purely vertical advection test for CAM-EUL at T85 resolution with 60 levels gives an error of 0.5274, and for the full three-dimensional test with 60 levels, the error never drops below this regardless of increases in horizontal resolution. Therefore, to increase the accuracy of the three-dimensional tests, and to see expected convergence at the order of accuracy of the numerical scheme, the vertical resolution must be

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increased along with the horizontal. Note that this is not a common practice in GCMs. There are other methods to improve accuracy in vertical advection without increased resolution. One way to increase accuracy would be to use a higher-order remapping algorithm [based, for example, on piecewise cubic or higher (Zerroukat et al. 2005, 2010)]. Another would be to have a fully three-dimensional transport scheme, as opposed to the horizontal–vertical split used by both CAM-FV and CAM-EUL.

levels with an equidistant grid spacing of Dz 5 500 m. It is positioned so that its horizontal center is aligned with the horizontal center of the temperature perturbation. This will test the ability of CAM-FV to advect a poorly resolved tracer with strong vertical velocity. The second tracer is a thin band around the equator, defined as

5. Three-dimensional advection in idealized dynamical core experiments

where l and f denote the longitude and latitude, and d1 is defined by Eq. (A20) with the distance r in Eq. (A20) now given as the great circle distance along the surface of the earth:

The previous advection tests made use of prescribed velocities. To consider test cases of more realistic, increased complexity we investigate three-dimensional tracer transport with dynamically induced velocities. In this section we investigate the propagation of gravity waves using the gravity wave test case described in detail in Jablonowski et al. (2008) (also see Tomita and Satoh 2004). In essence, the test case describes the evolution and propagation of gravity waves that are triggered by a large-scale temperature perturbation in a stratified atmosphere. We use the parameters for test case (6–0–0)— that is, no Coriolis force, the Brunt–Vaïsa¨la¨ frequency set to 0.01 s21, the maximum amplitude of the temperature perturbation set to 10 K, and the initial wind speeds set to zero. The model top is placed at 10 km. We perform a model experiment with three individual tracer fields to demonstrate the effects of vertical and horizontal resolution on tracer transport. Although the experiment is idealized, these velocities might be considered more realistic than those prescribed by the tracer advection test cases specified in previous sections. The CAM-FV dynamical core is utilized in an adiabatic mode (i.e., without physical parameterizations) and uses the fourth-order divergence damping as explained in Whitehead et al. (2011). We use the PPM tracer advection scheme with the default limiter in all gravity wave tests. For each set of horizontal and vertical resolutions, a time step of 900 s is used for the tracer transport and the vertical remap, while a time step of 90 s is used for the dynamics. Varying the horizontal and vertical resolutions will show the impact of resolution on tracer transport. The first tracer, q7, is a smooth profile, set up as for the three-dimensional advection test case using Eq. (A19). The tracer has the center position set to (lc, fc) 5 (p, 0), the vertical center at z0 5 3 km, and the horizontal and vertical half widths chosen to be R 5 a/3 and Z 5 0.8 km, where a is the earth’s radius. This tracer is designed so that it is not very well resolved on 20 vertical

1 q8(l, f, z) 5 [1 1 cos(pd1 )], 2

r 5 a arccos(sinfc sinf 1 cosfc cosf).

(9)

(10)

The latitudinal center of the strip is at fc 5 0 (the equator), the vertical center is at z0 5 5 km, and the horizontal and vertical half widths are chosen to be R 5 a/6 and Z 5 0.4 km. On 20 vertical levels the tracer is very poorly resolved, and the sides of the tracer are very steep. The band will be advected by the strong vertical velocity in the center of the domain, and this will show the effects of numerical diffusion on poorly resolved tracers. The final tracer q9 comprises two smooth profiles, each similar to q7, described by Eq. (A19). The magnitude of the tracer is 1, but the minimum value of the tracer is set to 0.1. This is so that any undershoots are not filled by the filling algorithm, as would be the case if the minimum was set to 0. The centers of the tracers are (lc, fc) 5 (7p/12, 0) and (lc, fc) 5 (17p/12, 0); both have the vertical center at z0 5 5 km and the horizontal and vertical half widths are R 5 a/3 and Z 5 0.8 km. The centers of the tracers are positioned away from the center of the temperature perturbation so that this tracer will not be affected by the strong vertical velocity in the center of the domain. This will test the tracer transport algorithm in areas of strong horizontal and weak vertical velocity. The initial tracer distributions along the equator of q7, q8, and q9 are shown in Figs. 6a, 7a, and 8a for the 18 3 18 L40 resolution. This vertical resolution corresponds to Dz 5 250 m. The tracers are designed so that they are not very well resolved on only 20 vertical grid points (with Dz 5 500 m). A high-resolution run of ½8 3 ½8 L125, with Dz 5 80 m, is used as the reference solution. Error norms are not computed, but qualitative differences between the reference solution and runs at coarser resolutions are discussed. The longitude–height cross sections of the tracer distributions along the equator after 90 h are shown in Figs. 6b–d, 7b–d, and 8b–d for tracers q7, q8, and q9,

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FIG. 6. Longitude–height cross sections of the tracer q7 distribution at the equator, (a) initially at the resolution 18 3 18 L40, and after 90 h at the resolutions (b) ½8 3 ½8 L125, (c) 18 3 18 L20, and (d) 28 3 28 L40.

respectively. The top right-hand plots (Figs. 6b, 7b, and 8b) show the results when using the high-resolution grid ½8 3 ½8 L125, and can be used as a reference solution. Initially there is a strong vertical velocity at the center of

the temperature perturbation in the middle of the domain, around l 5 1808 and z 5 5000 m, and this provides the updraft that advects and slightly deforms tracer q7. The tracer band around the equator, q8, is also advected

FIG. 7. As in Fig. 6, but for the tracer q8 distribution.

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FIG. 8. As in Fig. 6, but for the tracer q9 distribution.

upward at the center of the perturbation, but then the rest of the band follows the gravity wave as it travels out from the center. The two ellipses of q9 oscillate as the gravity wave passes through them. Figures 6c, 7c, and 8c show tracers q7, q8, and q9 after 90 h using 18 3 18 L20 resolution. The tracers are not very well resolved, and because of this the model is very diffusive. Tracer q7 has a similar shape to the reference solution, but much of the detail has been lost. Using 20 vertical levels, q8 in Fig. 7c initially appears as a steepsided band across the equator. As the tracer is advected along with the gravity wave the band is diffused out. At the center of the perturbation the band has been pushed too high; the top of the band is at approximately 8500 m for the reference solution, but at 9500 m for the 18 3 18 L20 solution. This is due to the poor representation of the tracer on 20 vertical levels. The two ellipses of q9 are diffused and spread out, even though the vertical velocity is much smaller than at the center of the temperature perturbation. Increasing the vertical resolution to L40 (not shown), while keeping the horizontal resolution at 18 3 18, improves the results significantly over the L20 solution for q7, q8, and q9. The shape of each of the tracers is more similar to the high-resolution reference solution, and the tracers are diffused much less. This is because more of the tracers are resolved on a 40-level resolution. To demonstrate the impact of vertical resolution on tracer transport using the gravity wave test, we decrease the horizontal resolution while increasing the vertical

resolution. Comparing the results for 18 3 18 L20 with 28 3 28 L40 shows that increasing the vertical resolution can be more beneficial than increasing the horizontal resolution, provided that the tracer is already at least marginally resolved on the horizontal grid; that is, starting with a 28 3 28 L20 grid, it is better to just increase the vertical resolution to L40 than to only increase the horizontal to 18 3 18. The results for 28 3 28 L40 are shown in Figs. 6d, 7d, and 8d. Tracer q7 is not as diffused and has a slightly better shape than for the 18 3 18 L20 resolution results. The tracer band, q8, is less spread out, and the maximum height of the band is closer to that of the reference solution than that of the 18 3 18 L20 solution. The final tracer q9 is less diffused and has larger maximums than the 18 3 18 L20 solution. Repeating the test with CAM-EUL gives similar results (not shown). The solution with T85 resolution and 40 vertical levels produces better results than that with T170 resolution and 20 vertical levels; the tracers have a better shape and are less diffused. These results demonstrate that just increasing horizontal resolution, as is often done in GCM convergence studies, might not improve the accuracy of tracer transport for three-dimensional flow. For the gravity wave test, the increase in vertical resolution has a larger impact on the tracers than an increase in horizontal resolution. Doubling the number of vertical levels from 20 to 40 significantly increases the tracer representation in the selected test case, even though the tracers are still not fully resolved with 40 levels.

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Increasing the vertical remapping time step for the gravity wave test from 900 to 7200 s, but keeping the other time steps (i.e., dynamics and tracer time steps) constant, has very little impact on the tracers. The larger remapping step (i.e., less-frequent remaps) produces a slightly smoother solution than the smaller remapping step. This is consistent with the results of changing the remap step in the vertical velocity tracer tests (section 4b).

6. Conclusions Tracer transport is an important component in dynamical cores of GCMs. We have tested the tracer transport in CAM-FV using a number of different idealized tests that clearly isolate cause and effect in one-, two-, and three-dimensional transport problems. Special attention has been paid to the impact of the inherent numerical diffusion in CAM-FV and the effects of different horizontal and vertical resolutions. Selected comparisons are also provided to the tracer transport scheme in the spectral transform dynamical core CAM-EUL. For simple one-dimensional tests, the choice of the scheme/limiter to calculate the numerical fluxes in CAM-FV is important. Using the PPM scheme with the Colella–Sekora limiter, the handling of smooth features is improved when compared with PPM with the default limiter. The default limiter, however, performs better for steep-sided and discontinuous data. To increase the difficulty of the advection test, both a cross-polar flow and a two-dimensional deformation test (Nair and Lauritzen 2010) were used. For twodimensional flow, the Lin–Rood method can lead to overshoots and undershoots; this is because the limiters used with the Lin–Rood scheme are not multidimensional, and may not be applied correctly to the cross terms. The less diffusive schemes, for example PPM with the Colella–Sekora limiter, have the largest over- and undershoots. The improvements seen with the Colella–Sekora limiter for one-dimensional flow are not replicated in the two-dimensional advection tests. For strictly vertical advection, the size of the vertical remap time step does have a small impact on the accuracy of CAM-FV. Using a larger remapping time step (i.e., less-frequent remaps) resulted in fewer overshoots and a more accurate solution. However, it is the vertical resolution that has the largest effect on accuracy. With poor vertical resolution, many tracer features are underresolved and the vertical advection in CAM-FV is suboptimal. With 30 vertical levels, which is a similar number of levels to that used in climate simulations (see, e.g., Hannay et al. 2009—although in practice GCMs typically

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have a higher model top, and therefore the physical grid spacing is even poorer than in the tests performed here), CAM-FV has a 35.66% error when vertically advecting the smooth data over 12 days. This leads to poor accuracy in the three-dimensional tests—regardless of increased horizontal resolution, the vertical resolution provides a lower bound on the error. The gravity wave test (Jablonowski et al. 2008), used to consider more realistic flow fields, produced similar results. This illustrates that the vertical transport errors may dominate three-dimensional tracer transport in some dynamical cores, and that for some tracer transport problems it would be better to increase vertical rather than just horizontal resolution. These results are reinforced by similar results using CAM-EUL. The results from these tests demonstrate the impact that poor vertical resolution has on tracer transport in CAM-FV and CAM-EUL, and potentially other dynamical cores. The Met Office’s Hadley Centre Global Environmental Model version 1 (HadGEM1) climate model uses 38 vertical levels in the atmosphere for seasonal and centennial climate simulations (Johns et al. 2006) [although 60 vertical levels are an option in HadGEM2 (Collins et al. 2008)], while NCAR’s CAM3 and CAM4 models used 26 vertical levels (Jablonowski and Williamson 2006), with 30 vertical levels as default in the recently released CAM5 (each model has the model top at approximately 40 km with nonequidistant vertical grid spacing). However, these are not enough vertical levels for accurate vertical tracer advection in the tests presented in this paper. For three-dimensional flow, just increasing the horizontal resolution will not lead to convergence of tracer errors; the vertical resolution must also be increased. Acknowledgments. Support for this research has been provided by the U.S. Department of Energy’s SciDAC program under Grant DE-FG02-07ER64446. We thank two anonymous reviewers for their helpful comments.

APPENDIX Test Case Descriptions a. One-dimensional advection test This appendix describes some of the test cases used in this paper. For the one-dimensional advection test, the velocities are prescribed as u(l, f) 5 u0 cosf, and

(A1)

y(l, f) 5 0,

(A2)

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where u is the zonal velocity, y is the meridional velocity, u0 5 2pa/(12 days), and a is the earth’s radius. The tracers are initialized as a Gaussian bell and a slotted cylinder. The Gaussian bell is described by (Levy et al. 2007) q1(l, f) 5 expf25[(X~ 2 Xc )2 1 (Y~ 2 Yc )2 1 (Z~ 2 Zc)2 ]g, (A3)

8 > > > and jl 2 lc j , and f 2 fc , 2 , 1 if r # > > 2 12 24 : 0:1 otherwise. if r #

This gives an ellipse with radius ½. Here, r denotes the great circle distance between two points along the surface of a unit sphere r 5 arccos[sinfc sinf 1 cosfc cosf cos(l 2 lc )],

(A6)

with the center of the cylinder at (lc, fc) 5 (p, 0). For the cross-polar horizontal advection test, the velocities are given as u(l, f) 5 u0 sinf cosl, and

(A7)

y(l, f) 5 2u0 sinl.

(A8)

The tracers are initialized as for the one-dimensional advection test, except the center is at (lc, fc) 5 (3p/2, 0).

b. Two-dimensional horizontal deformation test For the two-dimensional deformation test, the prescribed winds are given as (Nair and Lauritzen 2010) pt cos(f) , and u(l, f) 5 k sin2 (l9) sin(2f) cos 1 2ap T T (A9) pt , y(l, f) 5 k sin(2l9) cos(f) cos T

8 > > >1 > > > > < q4(l, f) 5 1 > > > > 1 > > > : 0:1

(A4)

(A10)

(A5)

where l9 5 l 2 (2pt/T ), k is the magnitude of the velocity, t is time, and T is the total time of the simulation. We will use k 5 10a/T and T 5 12 days. This gives maximum velocities as u ’ 100 m s21 and y ’ 61.5 m s21. We note that these velocities are by design rather extreme in comparison to typical atmospheric advection speeds. This tests the tracer advection algorithm under extreme conditions. Two cosine bells and two slotted cylinders will be used as the tracer distributions. The cosine bell tracer is given as 8 1 > > 0:1 1 0:9h1 if r1 # , > < 2 1 q3(l, f) 5 > 0:1 1 0:9h2 if r2 # , > > 2 : 0:1 otherwise,

(A11)

where, for i 5 1, 2, 1 hi 5 [1 1 cos(2pri )], 2

(A12)

and ri is the nondimensional great circle distance [Eq. (A6)] to the centers of the bells: (5p/6, 0) and (7p/6, 0). This gives two cosine bells with maximum height 1 and nondimensional radius ½. The slotted cylinder is similar to the advection around the equator case, but now with two cylinders:

1 1 and jl 2 lci j $ , 2 12 1 1 5 if r1 # and jl 2 lc1 j , and f 2 fc1 , 2 , 2 12 24 1 1 5 and jl 2 lc2 j , and f 2 fc2 . , if r2 # 2 12 24 otherwise. if ri #

(A13)

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The centers of the cylinders are the same as for the cosine bells: (5p/6, 0) and (7p/6, 0).

where )# " ( r 2 z 2 z 2 0 d1 5 min 1, . 1 R Z

c. Three-dimensional advection test The three-dimensional advection test (Jablonowski et al. 2008) prescribes a solid-body rotation in the horizontal, u(l, f) 5 u0 (cosf cosa 1 sinf cosl sina), and (A14) y(l, f) 5 2u0 sinl sina,

v5

h dp 2p pi 5 v0 cos t sin s(p) , dt t 2

(A16)

(A20)

Here r denotes the great circle distance given by Eq. (A6) multiplied by the radius of the earth, a. The second tracer is a slotted ellipse, given as 1 if d2 # 1 q6(l, f, z) 5 , 0 if d2 . 1

(A15)

for all heights, where u0 5 2pa/T, a is the angle of rotation, and a is the radius of the earth. The flow orientation angle is a 5 p/4, to ensure there is both zonal and meridional motion, and the length of the simulation is T 5 12 days. This provides one complete revolution of the tracer around the earth. The vertical pressure velocity is given as

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(A21)

where d2 5

r 2 R

1

z2z0 Z

2 .

(A22)

To cut the slot out of the ellipse, q6 5 0 if z . z0 and fc 2 1/ 8 , f , f 1 (1/ 8). The center positions are set to (l , c c fc) 5 (3p/2, 0), with the vertical center at z0 5 4.5 km and the horizontal and vertical half widths chosen to be R 5 a/3 and Z 5 1 km.

where

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!ffi # u u p 2 ptop s(p) 5 min 1, 2 tsin p . p0 2 ptop

REFERENCES

"

(A17)

Here p0 is a reference pressure, set to 1000 hPa, and ptop is the pressure at the model top, set to ptop 5 254 hPa. This corresponds to a height of z 5 12 000 m, when assuming isothermal conditions with the temperature set to 300 K. The vertical oscillation period is t 5 345 600 s (i.e., T/3) and the maximum vertical pressure velocity is set to v0 5 p4 3 104 Pa t 21. This gives three vertical cycles over the 12-day simulation. The function s(p) [Eq. (A17)] is used to make sure that the vertical velocity goes to zero at the surface and the model top, and ensures that nothing is advected into the surface of the earth. This is implemented in CAM-FV by updating pressure using v [Eq. (A16)]. Pressure at the future time t2 5 t1 1 Dt, where Dt denotes the length of a tracer time step, can be calculated as

p(t2 ) 5 p(t1 ) 1 Dtv0 cos

n 2p po (t1 1 Dt) sin s[p(t1 )] . t 2 (A18)

The first, smooth tracer is given by 1 q5(l, f, z) 5 [1 1 cos(pd1 )], 2

(A19)

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VOLUME 140

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