J. Adv. Model. Earth Syst., Vol. 2, Art. #11, 31 pp.
Development of a Quasi-3D Multiscale Modeling Framework: Motivation, Basic Algorithm and Preliminary results Joon-Hee Jung1 and Akio Arakawa2 1 2
Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado Department of Atmospheric and Oceanic Sciences, University of California Los Angeles, California
Manuscript submitted 2 August 2010; in final form 13 October 2010 A new framework for modeling the atmosphere, which we call the quasi-3D (Q3D) multi-scale modeling framework (MMF), is developed with the objective of including cloud-scale three-dimensional effects in a GCM without necessarily using a global cloud-resolving model (CRM). It combines a GCM with a Q3D CRM that has the horizontal domain consisting of two perpendicular sets of channels, each of which contains a locally 3D grid-point array. For computing efficiency, the widths of the channels are chosen to be narrow. Thus, it is crucial to select a proper lateral boundary condition to realistically simulate the statistics of cloud and cloud-associated processes. Among the various possibilities, a periodic lateral boundary condition is chosen for the deviations from background fields that are obtained by interpolations from the GCM grid points. Since the deviations tend to vanish as the GCM grid size approaches that of the CRM, the whole system of the Q3D MMF can converge to a fully 3D global CRM. Consequently, the horizontal resolution of the GCM can be freely chosen depending on the objective of application, without changing the formulation of model physics. To evaluate the newly developed Q3D CRM in an efficient way, idealized experiments have been performed using a small horizontal domain. In these tests, the Q3D CRM uses only one pair of perpendicular channels with only two grid points across each channel. Comparing the simulation results with those of a fully 3D CRM, it is concluded that the Q3D CRM can reproduce most of the important statistics of the 3D solutions, including the vertical distributions of cloud water and precipitants, vertical transports of potential temperature and water vapor, and the variances and covariances of dynamical variables. The main improvement from a corresponding 2D simulation appears in the surface fluxes and the vorticity transports that cause the mean wind to change. A comparison with a simulation using a coarse-resolution 3D CRM is also made. DOI:10.3894/JAMES.2010.2.11 1. Introduction While it is well recognized that cloud and cloud-associated processes play crucial roles in the climate system, the progress of our ability to represent these processes in climate models has been very slow (Randall et al. 2003). In particular, it remains extremely challenging to formulate the net effects of the complicated interactions of various cloud-scale processes for use in climate models. There are even some conceptual problems in the conventional formulation of model physics, as emphasized by Arakawa (2004). As far as the representation of deep moist convection is concerned, we have only two kinds of model This work is licensed under a Creative Commons Attribution 3.0 License.
physics: highly parameterized and explicitly simulated. Correspondingly, besides those models that explicitly simulate turbulence (e.g., direct numerical simulation and large eddy simulation models), there have been only two families of three-dimensional models for the atmosphere as shown in Fig. 1: one represented by conventional general circulation models (GCMs) and the other by cloud-resolving models (CRMs). In Fig. 1, the abscissa is the horizontal resolution and the ordinate is a measure for the degree of parameterization increasing downwards. Here, ‘‘the degree of parameterization’’ may be interpreted as a reduction in the degrees of freedom. Due to the difference in the degree To whom correspondence should be addressed. Joon-Hee Jung, Department of Atmospheric Science, Colorado State University, Fort Collins, CO 80523-1371, USA
[email protected]
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Jung and Arakawa
Figure 1. Two families of atmospheric models classified with their degrees of parameterization and horizontal resolutions.
of parameterization, the two families of models have qualitatively different model physics (Arakawa 2004). For a given observed large-scale condition, we can identify the apparent heat source, Q1, and the apparent moisture sink, Q2, from the residuals in large-scale heat and moisture budgets as pioneered by Yanai et al. (1973). Here the heat source and moisture sink refer to the source of the sensible heat, cpT, and sink of latent heat, Lq, respectively. Following Yanai et al. (1973), a number of authors extended this type of analysis to a variety of regions (see Yanai and Johnson 1993 for a review). From these studies we see that the relation between vertical profiles of Q1 and Q2 are cloud-regime dependent. When clouds are not strongly convective, there is a tendency toward Q12 QR > > < at the boundary at y~SB ðA:12Þ Ð NB Ð NB > y~{ SB SuTdy and x~ SB SvTdy > > : at the boundary at y~NB ðwSB Þ where STdenotes the average over the whole channel. The horizontal velocities in the shaded area in Fig. 8(b) can be obtained either from (A.8) or from the continuity equation (A.6) and/or the definition of vorticity (A.1) with the known values inside the channel. The latter procedure guarantees that the horizontal velocities outside the channel are dynamically consistent regardless of the lateral boundary condition we use. Let q be one of the scalar prognostic variables such as potential temperature and mixing ratios of various species of water. It is predicted by
Lq L L L r0 ~{ ðr0 uqÞz ðr0 vqÞz ðr0 wqÞ zr0 Sq , ðA:13Þ Lt Lx Ly Lz where Sq is the source of q per unit mass. Appendix B Solving a relaxed elliptic equation for w In the Q3D CRM, vertical velocity w is formally predicted using a relaxed version of the elliptic equation (A.7), through which the algorithm of solving an elliptic equation is localized. The localization of the procedure for solving the elliptic equation is compatible with the Q3D algorithm, in which the w-equation is not solved for the entire 3D domain. This method has been validated through a comparison of simulated realizations of convectively active regimes using the anelastic 3D CRM of Jung and Arakawa (2008) with a direct method to solve an elliptic equation and the method described here. Based on the comparison, it is concluded that the error due to the relaxation of anelastic balance can be confined to an acceptable level even with a relatively small number of iterations at each time step. The below briefly describes how we formally predict w using a relaxed version of the elliptic equation. In the anelastic system, the continuity equation is given as (A.6) and the vertical velocity w is determined by solving (A.7). By relaxing the anelastic approximation, the continuity equation becomes
adv-model-earth-syst.org
Development of a Q3D MMF Lr’ L L L z ðr0 uÞz ðr0 v Þz ðr0 w Þ~0 Lt Lx Ly Lz
ðB:1Þ
where the prime in the superscript denotes the deviation from the reference state and the w equation (A.7) becomes
L2 L2 L 1 L L L r’ z ðr w Þ z wz Lz r0 Lz 0 Lz Lt r0 Lx 2 Ly 2 Lg Lj { ~{ : Lx Ly
ðB:2Þ
The last term on the left hand side is omitted when the anelastic balance is assumed, so that its nonzero value is primarily due to sound waves, for which L r’ 1 L p’ & Lt r0 c 2 Lt r0
backward-implicit scheme for the vertical derivative term. Then, the final discrete form of w equation can be written as " # 1 m 2 2 1 1 1 z z z z r0,k{1=2 Dt Dx 2 Dy 2 Dz r0,k Dz r0,k{1 Dz 1 1 ðr0 w Þnz1 { ðr w Þnz1 i,j,k{1=2 Dz r0,k Dz 0 i,j,kz1=2 1 ðB:6Þ z ðr0 w Þnz1 i,j,k{3=2 r0,k{1 Dz m n 1 n n ~ wi,j,k{1=2 z 2 wi{1,j,k{1=2 zwiz1,j,k{1=2 Dt Dx 1 n n nz1 z 2 wi,j{1,k{1=2 zFi,j,k{1=2 zwi,jz1,k{1=2 Dy
ðB:3Þ
where c is the speed of sound. Rewriting the vertical derivative of the right hand side of (B.3) using the vertical component of the momentum equation, we obtain, approximately,
2 1 L2 w L L2 L 1 L ~ z ðr w Þ wz c 2 Lt 2 Lz r0 Lz 0 Lx 2 Ly 2 Lg Lj z { : Lx Ly
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ðB:4Þ
Since our interest is in the near-equilibrium state, we can replace the wave equation above by the following diffusion equation that shares the same equilibrium solution:
2 Lw L L2 L 1 L Lg Lj ðr w Þ z { ~ z ðB:5Þ wz m Lt Lz r0 Lz 0 Lx Ly Lx 2 Ly 2 where m defines the time scale for adjustment toward anelastic balance and is empirically chosen. When a small but non-zero value of m is used, a small imbalance can exist in the right hand side of (B.5) and such an imbalance is diffused throughout the time integration. In solving the incompressible Navier-Stokes equations, the idea of converting the elliptic equation into another form of equation is not unusual. For example, Chorin (1967) transformed an elliptic problem to a hyperbolic problem by including the local pressure change rate as the effect of artificial compressibility, which is called the method of artificial compressibility. The hyperbolic problem was then solved by standard, time-marching methods with an empirically chosen artificial speed of sound wave. Since Chorin’s introduction, this method has been widely used for steady or unsteady problems (for a short review, see Madsen and Schaffer 2006). To discretize (B.5), we use a partially backward-implicit scheme for the horizontal derivative term and a fully
where F is the last term in the right hand side of (B.5), the superscript n denotes the time step and the subscripts indicate the location of each variable following the grid index in Fig. 2 of Jung and Arakawa, (2008). Here, Dx, Dy and Dz are grid sizes of x-, y-, and z-directions, respectively. For simplicity, a constant vertical grid size is assumed for the description of the method presented here. Applying nz1 this to all heights with ðr0 w Þnz1 i,j,K z1=2 ~0 and ðr0 w Þi,j,1=2 ~ 0, where k5K+K and K indicate upper and lower boundaries, respectively, (B.6) can be written as zb^k ðr0 w Þnz1 zc^k ðr0 w Þnz1 ~A^k a^k ðr0 w Þnz1 i,j,^k {1 i,j,^k i,j,^k z1
ðB:7Þ
where ^k :k{1=2, 8 1 1 > > a^k :{ > > Dz r0,k{1 Dz > > > " > # > > 1 m 2 2 1 1 1 > >b : > z z z z ^k > > r0,k{1=2 Dt Dx 2 Dy 2 Dz r0,k Dz r0,k{1 Dz > > < 1 1 c : > ^k > ðB:8Þ > Dz r0,k Dz > > > > m 1 > n n n > > z 2 wi{1,j,k{1=2 zwiz1,j,k{1=2 A^k : wi,j,k{1=2 > > Dt Dx > > > > 1 n n nz1 > :z w zw i,jz1,k{1=2 zFi,j,k{1=2 Dy 2 i,j{1,k{1=2 This tri-diagonal system for the vertical velocity is solved by a Gaussian elimination method at a point (i, j). Since we use an explicit and local algorithm, iterations are necessary within each time step to solve (B.6). Through various pre-tests with the 3D CRM, we found that it generally requires only about 10 iterations to reach an acceptable level of convergence when the value of m is selected as
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ðB:9Þ
and the initial guess for iteration is obtained from a linear extrapolation of past values. This is because the initial guess can be fairly close to the expected solution if a relatively short integration time step is used for the model. As the number of iteration increases the number of neighboring points involved increases, i.e., the domain of influence expands. As long as the number of iteration is finite, the domain of influence is finite so that the calculation is local. References Arakawa, A., 2004: The cumulus parameterization problem: Past, present, and future. J. Climate, 17, 2493–2525, doi: 10.1175/1520-0442(2004)017,2493:RATCPP.2.0.CO;2. Arakawa, A., and J.-M. Chen, 1987: Closure assumptions in the cumulus parameterization problem. Short- and medium-range Numerical Prediction, Collection of papers presented at the WMO/IUGG NWP Symposium, Tokyo, T. Matsuno, Ed., Meteor. Soc. Japan, 107–131. Arakawa, A., and V. R. Lamb, 1977: Computational design of the basic dynamical process of the UCLA general circulation model. Methods in Computational Physics, 17, Academic Press, 173–265. Benedict, J. J., and D. A. Randall, 2009: Structure of the Madden-Julian oscillation in the superparameterized CAM. J. Atmos. Sci., 66, 3277–3296, doi: 10.1175/ 2009JAS3030.1 Buizza, R., 2010: Horizontal resolution impact on shortand long-range forecast error. Q. J. R. Meteorol. Soc., 136, 1020–1035, doi: 10.1002/qj.613. Cheng, M.-D., and M. Yanai, 1989: Effects of downdrafts and mesoscale convective organization on the heat and moisture budgets of tropical cloud clusters. Part III: Effects of mesoscale convective organization. J. Atmos. Sci, 46, 1566–1588, doi: 10.1175/1520-0469(1989)046, 1566:EODAMC.2.0.CO;2. Chorin, A. J., 1967: A numerical method for solving incompressible viscous flow problems. J. Comput. Phys, 2, 12– 26, doi: 10.1016/0021-9991(67)90037-X. Deardorff, J. W., 1972: Parameterization of the planetary boundary layer for use in general circulation models. Mon. Wea. Rev, 100, 93–106, doi: 10.1175/15200493(1972)100,0093:POTPBL.2.3.CO;2. E, W., B. Engquist, X. Li, W. Ren, and E. Vanden-Eijnden, 2007: Heterogeneous multiscale methods: A review. Commun. Comput. Phys, 2, 367–450. Fu, Q., S. K. Krueger, and K. N. Liou, 1995: Interactions of radiation and convection in simulated tropical cloud clusters. J. Atmos. Sci, 52, 1310–1328, doi: 10.1175/ 1520-0469(1995)052,1310:IORACI.2.0.CO;2. Grabowski, W. W., 2001: Coupling cloud processes with the large-scale dynamics using the cloud-resolving convectJAMES
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