applied this model to study the flow over an idealized, bell-shaped mountain under different environments.

3.3. Large eddy simulation (LES)

With the use of an open lateral boundary and a very large number of grid points in the NTU/Purdue Nonhydrostatic Model, Hsu et al. (2004) were able to simulate the development of stratocumulus in a heterogeneous environment during a cold air outbreak event. A convective boundary layer (CBL) develops as very cold air originating from Siberia and China flows over the Japan Sea, the Yellow Sea, and the East China Sea during winter seasons. The CBL quickly deepens away from the coastline, with increasing fetch length and sea surface temperature. As the depth of the CBL changes, the embedded roll vortices (cloud streets) grow in size. The convection eventually becomes three-dimensional in the downstream region. The simulated convection also shows both 2D rolls and 3D cells in our simulation (Fig. 15). The increase of the CBL's depth and strength may result in changing cloud shapes.

4. Other Topics

4.1. Turbulence, pollution and PBL

In a higher-order turbulence parametrization, Sun and Ogura (1980) introduced a turbulence length scale in the stable layer that is related to the atmospheric stratification. They also included the equation for the potential temperature-humidity covariance (O'q') and were able to simulate the observed (O'q') and some third-order terms. Sun and Chang (1986a,b) and Sun (1986, 1988, 1989) extended Sun and Ogura's work by including the temperature-concentration covariance and using the observed turbulence length scale in the CBL (Caughey and Palmer, 1979) as a length scale in the model, successfully simulating the transport and dispersion of plumes in a con-vective boundary layer. Simulated results (Sun, 1988, 1989) were compared with the laboratory study by Deardorff and Willis (1975) for pollution released from point sources at different heights. Both laboratory and model simulations show that the plumes move upward from the point source near the surface, but the plumes from the elevated point sources descend quickly and form a high concentration zone near the

Figure 15. The simulated stratocumulus after 2-h integration during a cold air outbreak over a warm ocean. The background color shows the sea surface temperature distribution (scale indicated to the right of the figure; units in K). The thin red and orange lines identify the lowest cloud level at z = 300 m. Cold air with temperature of 280 K near surface comes from the left with wind speed of 10 m s~1. Cloud streets broke into three-dimensional cells in the downstream region. The grid intervals in x, y and z directions are 200, 100, and 50m, respectively (Hsu et al., 2004).

Figure 15. The simulated stratocumulus after 2-h integration during a cold air outbreak over a warm ocean. The background color shows the sea surface temperature distribution (scale indicated to the right of the figure; units in K). The thin red and orange lines identify the lowest cloud level at z = 300 m. Cold air with temperature of 280 K near surface comes from the left with wind speed of 10 m s~1. Cloud streets broke into three-dimensional cells in the downstream region. The grid intervals in x, y and z directions are 200, 100, and 50m, respectively (Hsu et al., 2004).

surface before they move upward. The modeled mean plume height and the surface concentration were also comparable with observations (Willis and Deardorff, 1978, 1981). Sensitivity tests show that the temperature-concentration covariance is crucial to turbulence-pollution modeling, which had been ignored before. The simplified version of this turbulence parametri-zation is also used in the PRCM (Chern, 1994). Currently, MacCall (2006) is working on the third-order turbulence closure to study the turbulence in a stable boundary layer.

A comprehensive snow-vegetation-soil module has been developed for the PRCM (Wu and Sun, 1990a,b; Sun and Wu, 1992; Sun, 1993a,b; Bosi-lovich and Sun, 1995; Sun and Bosilovich, 1996; Bosilovich and Sun, 1998; Chern and Sun, 1998; Sun and Chern, 2005). It handles diffusion and transport of heat and water substance inside snow, vegetation, and soil as well as the processes of melting and freezing and the fluxes at the interfaces, etc. Figure 16 shows a comparison between the observations and model simulations of the soil temperature during summer. Sun and Chern (2005) simulated the changes of the snow depth and soil temperature in the Sleepers Watershed Experiment during 19691974. The simulations are in good agreement with observations.

Kuo (1963) applied the linear theory to explain the cloud streets observed in a trough in the easterly wave (Malkus and Riehl, 1964). He successfully simulated the cloud streets forming along the wind shear in a dry, unstable atmosphere with a constant lapse rate, but failed to produce the larger cloud streets forming perpendicular to the wind shear. Using the real atmospheric stratification and including the effect of latent heat release, Sun (1978) demonstrated that observed clouds may be explained by the coexistence of two different types of clouds: the shallow convective-type cloud streets form along the wind shear, and the deep wave-type cloud streets develop perpendicular to the wind shear.

Sun and Orlanski (1981a,b) solved both linearized and nonlinear equations as initial value problems and confirmed that the two-day waves can be easily excited by the diurnal

Time

Figure 16. Comparison of model-simulated surface temperature (solid line) and observations (symbol x) during FIFE campaign, 25 June-25 July 1987 (Bosilovich and Sun, 1998).

Time

Figure 16. Comparison of model-simulated surface temperature (solid line) and observations (symbol x) during FIFE campaign, 25 June-25 July 1987 (Bosilovich and Sun, 1998).

oscillation of the land-sea contrast at lower latitudes (<15°). On the other hand, one-day and two-day waves may coexist at latitudes of up to 30°. These waves may correspond to the mesoscale cloud bands observed along coastlines with a space interval of a few ten to a few hundred kilometers (Fig. 1 in Sun and Orlanski, 1981a).

Integrating the linearized equations as a wet-symmetric instability problem, Sun (1984b, 1987) showed that the small-sized rainbands can organize into larger ones with strong narrow upward motions interspersed with weak and widespread downward motion, because the ascending air should be warmer than the descending air parcel to maintain the positive circulation. Those rain bands also propagated toward the warm side, which is the source of moisture in a rotating fluid. Sun (1995b) further proved that the y component of the earth rotation (2Q cos $; where $ is the latitude) can significantly impact the growth rate of symmetric instability in the lower latitudes, because the x component wind u can either enhance or decrease the z component acceleration through the u 2Q cos $ term in the momentum equation.

4.4. Cumulus parametrization scheme (CPS) and cloud models

Haines and Sun (1994) developed a simple steady 1D cloud model to be used in the cumulus parametrization scheme (Sun and Haines, 1996). This CPS also includes the Weisman and Klemp (1984) storm intensity according to the wind shear and buoyancy environment and allows three sizes, of clouds: small, medium, and large sizes, with different radius and cloud depth. The population of each cloud type is determined by the conservation of mass and moisture flux, and the ability to generate the fastest heating rate from the combined clouds. The CPS has been tested against the SESAME V storm-scale analysis from 2000 UTC to 2300 UTC on 20 May 1979, when the precipitation was almost exclusively convective in nature. They are in good agreement in the apparent heat source (Q1) and the apparent moisture sink (Q2), as shown in Fig. 17 for both strong and weak convections. The CPS was also successful in diagnosing the apparent heat source (Q1) and the apparent moisture sink (Q2) from purely con-vective sources even later in the SESAME V case, when a significant amount of stable type precipitation developed. The CPS has been incorporated in the PRCM to simulate the squall line observed during SESAME V. The simulated squall line formed about when and where the observed squall line did. Additionally, later deep convection to the south along the dry line was successfully simulated. However, it is very important and challenging to develop a better CPS with more rigorous assumptions and formation.

In order to avoid using a steady state cloud model as in Haines and Sun (1994), Chen and Sun (2002) have developed a time-dependent one-dimensional cloud model, which represents well the average properties of the clouds generated by the WRF model (Fig. 18). Furthermore, Chen and Sun (2004) developed a one-dimensional time-dependent tilting cloud model to represent the effect of wind shear. Hopefully, those cloud models can be incorporated into the CPS in the near future.

Sun and Chang (1992) applied a nonlinear baro-tropic model to study the barotropic instability (Kuo, 1949) of the modified hyper-tangent shear flows, which may represent the flow along a Mei-yu front or a cyclone family in the middle-to-high latitudes. In a ¡3 plane, the waves initially propagate as Rossby waves through the various basic states following linearized equations. Those waves move into the most unstable zone and, thus, grow more rapidly than in the weak shear zone. In the later stage, the nonlinear wave interaction becomes dominant and the transition among the modes becomes very difficult to predict. The simulations confirm that the results are very sensitive to external forcing, the basic wind field, the effect of 3, as well as the initial perturbation field. This may suggest that predictability is very limited even in this simple 2D barotropic flow. The evolution of 2D shear flow has also been studied by Oh (2007) using the semi-Lagrangian scheme on the characteristic shallow water equations.

The results of a numerical prediction model depend upon the model equations, the physics, the initial and boundary conditions, and the numerical methods. Therefore, we have also been working on numerical schemes in order to provide more accurate results. In addition to diffusion equations (Sun, 1982) and a forward-backward scheme for inertial-internal gravity waves (Sun, 1980, 1984a), Sun developed an advection scheme which is more accurate than the popular Crowley fourth-order advection scheme (Crowley, 1969). Sun et al. (1996), Sun and Yeh (1997), and Sun and Sun (2004) also developed a mass-conserved, positive-definite semi-Lagrangian scheme. A forward trajectory and simple mass correction are applied in this scheme. The procedure includes: (a) constructing the Lagrangian network induced by the motion of the fluid from the Eulerian network and finding the intersections of the networks by a general interpolation from the irregularly distributed Lagrangian grid to the regularly distributed Eulerian grid,

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