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To verify the performance of the proposed transport model, several widely used benchmark tests, including solid body rotation, moving vortices, and deformational flow tests are performed on the spherical mesh.
The normalized errors proposed by Williamson et al. (1992) are used:
where
$\Omega $ is the whole computational domain and$q$ and${q_t}$ refer to numerical solutions (volume-integrated average in our paper) and exact solutions, respectively. -
The solid-body rotation test (Williamson et al., 1992) is widely used in two-dimensional spherical transport modeling to evaluate the performance of a transport model. The wind components in the latitude-longitude coordinates
$ \left(\lambda,\theta \right) $ are defined as:where
$\left({{u_{\rm{s}}},{v_{\rm{s}}}} \right)$ is the velocity vector,${u_0} = {{2\pi R} / {1036800}}$ (1036800 s equals 12 days), which means it takes 12 days to complete a full revolution on the sphere, R is the radius of the sphere, and$ \alpha $ is a parameter which controls the rotation angle. In this test, two kinds of initial conditions are used, including a cosine bell and a step cylinder. -
The initial condition of a cosine bell test is specified as:
where
${r_{\rm{d}}}$ is the great circle distance between$\left({\lambda,\,\theta } \right)$ and the center of the cosine bell, located at$\left({{{3\pi }/ 2},\,0} \right)$ ,${r_0} = {{7\pi R} / {64}}$ is the radius of the cosine bell, and${h_0} = 1$ .The normalized errors on
$32 \times 32 \times 6$ meshes and with 256 time steps compared with other existing published semi-Lagrangian schemes, the PPM-M scheme (Zerroukat et al., 2007) and CSLAM-M (Lauritzen et al., 2010), are presented in Table 1. The result shows that CSLR1 and CSLR1-M get almost the same result. And our scheme is comparable to the PPM-M scheme, and the result in the near-pole flow direction ($\alpha = {\pi / 2}$ and$\alpha = {\pi / 2} - 0.05$ ) is better than the CSLAM-M scheme.Scheme l1 l2 l∞ α=0 CSLR1(CSLR1-M) 0.116 0.097 0.114 PPM-M 0.101 0.095 0.115 CSLAM-M 0.075 0.075 0.141 α=π/4 CSLR1(CSLR1-M) 0.083 0.081 0.139 PPM-M 0.078 0.086 0.159 CSLAM-M 0.048 0.060 0.130 α=π/2 CSLR1(CSLR1-M) 0.077 0.067 0.080 PPM-M 0.109 0.102 0.118 CSLAM-M 0.075 0.075 0.141 α=π/2−0.05 CSLR1(CSLR1-M) 0.078 0.068 0.088 PPM-M 0.109 0.102 0.124 CSLAM-M 0.070 0.069 0.133 Table 1. Comparison of the normalized errors of rotation of a cosine bell after one revolution with other published schemes.
To check the influence of the weak singularities at the eight vertices of the cubed-sphere gird, this test is conducted with
$\alpha = {\pi / 4}$ to pass through four vertices. The history of normalized errors (CSLR1 and CSLR1-M are almost the same, so we only present the result of CSLR1-M here) are shown in Fig. 7. We can see that the normalized errors have little fluctuations (except the${l_\infty }$ errors at around day 4 and day 10) when the flow passes four weak singularities.Figure 7. History of normalized errors of the solid body rotation of a cosine bell for one revolution on grid N = 32 (number of cells in one direction on each cell), 256 time steps and with α = π/4.
To demonstrate the ability of the CSLR1-M scheme using a large Courant number to transport, we use 72 time steps (local maximum Courant number is about 1.78) with rotation angle
$\alpha = {\pi / 2}$ to complete one revolution. The normalized errors are${l_1} = 0.052$ ,${l_2} = 0.046$ , and${l_\infty } = 0.061$ . -
A non-smooth step cylinder is calculated to evaluate the non-oscillatory property. The initial distribution is specified as
where
${r_{\rm{d}}}$ is the great circle distance between$\left({\lambda,\theta } \right)$ and$\left({{{3\pi } / 2},0} \right)$ , which is the center of the step cylinder,${r_1} = {2 / 3}R$ and${r_2} = {1 / 3}R$ .In this test, we set
$\alpha = {\pi / 4}$ , which is the most challenging case of the rotation test where the step cylinder moves through four vertices and along two boundary edges of the cubed-sphere grid to complete a full revolution. Here, we use 90×90×6 meshes and 720 time steps to conduct this test. The numerical results after 12 days are shown in Fig. 8, and we can see that the CSL2 scheme will generate obvious oscillations around the discontinuities. By using the CLSR1 and CSLR1-M approaches, these nonphysical oscillations are effectively removed. The maximum and minimum value of CSL2 are${q_{\max }} = 1034.23$ and${q_{\min }} = - 2.45$ , and for CLSR1 and CSLR1-M they are${q_{\max }} = 1001.85$ and${q_{\min }} = 0$ . The history of relative mass errors is given in Fig. 9, which shows that the relative mass errors are up to the tolerance of machine precision, therefore the proposed global transport model is exactly mass conservative during the simulation procedure. -
The second benchmark test we used is the moving vortices test proposed by Nair and Jablonowski (2008). The wind component of this test is a combination of the solid body rotation test and two vortices, and it is much more complicated than the solid body rotation test. The velocity fields on the sphere are specified as:
where
${u_{\rm{s}}}$ and${v_{\rm{s}}}$ are calculated by Eqs. (37) and (38), and the rotation angle of this test is set to be$\alpha = {\pi / 4}$ .${\rho _0} = 3$ ,${\lambda _{\rm{c}}}\left(t \right)$ and${\theta _{\rm{c}}}\left(t \right)$ are the center of the moving vortex at time$ t $ , and the calculation procedure of${\lambda _{\rm{c}}}\left(t \right)$ and${\theta _{\rm{c}}}\left(t \right)$ can be found in (Nair and Jablonowski, 2008).The tracer field is defined as:
where
$\gamma $ is a parameter to control the smoothness of the tracer field,$\left({\lambda ',\theta '} \right)$ is the rotated spherical coordinates, which can be calculated by:and
$\left({{\lambda _{\rm{p}}},\,{\theta _{\rm{p}}}} \right) = \left({\pi,\,{\pi / 2} - \alpha } \right)$ is the North Pole of the rotated spherical coordinate. In this test, we followed Norman and Nair (2018) to set$\gamma = {10^{ - 2}}$ to conduct a large gradient in tracer distribution to check the non-oscillatory property and the performance of positivity preserving. When$t = 0$ in Eq. (46), we get the initial condition.This test is conducted on 80 × 80 × 6 meshes and uses 400 time steps to move forward 12 days. The contour plots in Fig. 10 show that compared with the exact solution, our proposed scheme can simulate this complicated procedure well. The plot along the equator is presented in Fig. 11, and it shows that there are no obvious oscillations around large gradients. The normalized errors of CSLR1 and CSLR1-M are almost the same, being
${l_1} = 5.295 \times {10^{ - 2}}$ ,${l_2} = 0.1295$ , and${l_\infty } = 0.5667$ , respectively. The histories of minimum values are shown in Fig. 12, where we can see that the CSLR1 scheme would produce negative values during the simulation procedure, while the minimum values of CSLR1-M can completely preserve positivity (the minimum values are within the machine precision). -
The last benchmark test used in our paper is the deformational flow test proposed by Nair and Lauritzen (2010), which is the most challenging test case. The nondivergent and time-dependent flow fields are defined as:
where
$\kappa = 2$ ,$T = 5$ , and$\lambda ' = \lambda - \left({{{2\pi t} /T}} \right)$ .Two kinds of initial conditions are checked here, including the twin slotted cylinders case to evaluate the positivity preserving property and correlated cosine bells to evaluate the nonlinear correlations between tracers (Lauritzen and Thuburn, 2012). By the given flow fields, the initial distributions will be deformed into thin bars during the first half period, then return to its initial state during the second half period.
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The initial condition is defined as:
where
${r_0} = 0.5$ and${r_i}\left({i = 1,2} \right)$ represent the great circle distances between the center of the two slotted cylinders and a given point. The centers of the two slotted cylinders are located at$\left({{\lambda _1},{\theta _1}} \right) = \left({{{5\pi } / 6},0} \right)$ and$\left({{\lambda _2},{\theta _2}} \right) = \left({{{7\pi } /6},0} \right)$ , respectively.The numerical results of deformational flow of the CSLR1-M scheme with 90 × 90 × 6 meshes and with 390 time steps (local maximum Courant number is about 3) are shown in Fig. 13. As shown in Fig. 13b, the two slotted cylinders are deformed into two thin filaments by the background flow field during the first half period. Figure 13c gives the counters of the slotted cylinders at the final time, and it is indicated that the proposed scheme can correctly reproduce this complicated deformational flow and does not produce oscillations. The histories of minimum values are shown in Fig. 14, which indicates that the CSLR1 scheme would produce negative values, while the CSLR1-M scheme keeps minimum values within the tolerance of machine precision, which can be viewed as non-negativity. The Normalized errors are
${l_1} = 0.3287$ ,${l_2} = 0.3321$ , and${l_\infty } = 0.9415$ for both the CSLR1 and CSLR1-M schemes. -
To check the ability of preserving nonlinearly correlated relations between two tracers, we used two kinds of tracers. One is the quasi-smooth twin cosine bells:
where
${h_i} = \dfrac{1}{2}\left[ {1 + \cos \left({\dfrac{{\pi {r_i}}}{{{r_0}}}} \right)} \right]$ for$i = 1,2$ .The other one is the correlated cosine bells:
where
$ \psi \left(q \right) = - 0.8{q^2} + 0.9$ .This test is conducted on
$90 \times 90 \times 6$ meshes with 1800 time steps. The scatter plot of numerical result at$t = {T / 2}$ is shown in Fig. 15. The solution of cosine bells is in the x-direction, and the correlated cosine bells is in the y-direction. The mixing diagnostics are${l_r} = 1.05 \times {10^{ - 3}}$ ,${l_u} = 2.40 \times {10^{ - 5}}$ , and${l_0} = 5.57 \times {10^{ - 4}}$ , respectively (see Lauritzen and Thuburn, 2012) for the detail definition of these three parameters). The CSLR1-M scheme is built using a monotone rational polynomial with modest accuracy, which always overly flattens the maximum and minimum values, as shown in the bottom-right corner of Fig. 15. In the whole, the scattering points of the CSLR1-M scheme are almost located inside the convex hull which means that the CSLR1-M scheme can preserve nonlinearly correlated relations between tracers well.
Scheme | l1 | l2 | l∞ | |
α=0 | ||||
CSLR1(CSLR1-M) | 0.116 | 0.097 | 0.114 | |
PPM-M | 0.101 | 0.095 | 0.115 | |
CSLAM-M | 0.075 | 0.075 | 0.141 | |
α=π/4 | ||||
CSLR1(CSLR1-M) | 0.083 | 0.081 | 0.139 | |
PPM-M | 0.078 | 0.086 | 0.159 | |
CSLAM-M | 0.048 | 0.060 | 0.130 | |
α=π/2 | ||||
CSLR1(CSLR1-M) | 0.077 | 0.067 | 0.080 | |
PPM-M | 0.109 | 0.102 | 0.118 | |
CSLAM-M | 0.075 | 0.075 | 0.141 | |
α=π/2−0.05 | ||||
CSLR1(CSLR1-M) | 0.078 | 0.068 | 0.088 | |
PPM-M | 0.109 | 0.102 | 0.124 | |
CSLAM-M | 0.070 | 0.069 | 0.133 |