Improving Model Grid Resolution Solution Adaptive Modeling Technique

Even, given the improved model initial condition, predictions of hurricanes can not be improved without accurately resolving the fine scale structure of the eye (cloud scale) and the steering large-scale environment (synoptic scale). Numerical models of the atmosphere are derived fundamentally from the Taylor series approximation which relates the value of a continuous function f(x) to the known values of the function and its derivatives at a set of discrete points, x;.

where, f',f",f"denote the spatial derivatives , , @x3, respectively, evaluated at x = xi. An examination of the above equation reveals that the general accuracy (or "order") of the calculation depends upon the number of terms on the RHS retained in the calculation. This translates into greater requirements for the computational resources. Hence it is customary to strike a balance by truncating the series at a suitable order of accuracy. An examination of the above equation also reveals that the error can be minimized by reducing the magnitude of (x - xi). This is in fact attained by increasing the grid resolution.

Therefore, in a numerical model of the atmosphere, the spatial scales of motion that can be resolved are dependent on the grid spacing used to represent the computational domain. In general, to effectively represent any feature requires that the feature be defined over four grid cells. Traditional numerical models use constant grid spacing. This implies that these models have prescribed lower limits for the spatial scale. However, atmospheric motions occur over various spatial scales. Introducing high grid resolution to resolve the smallest scales of interest throughout a simulation domain is not always practical, since the size of the modeling domain, the interactions between the atmospheric processes, the complexity of the numerical algorithms, and the operational requirements place restrictions on the grid resolution that can be achieved using today's computers (clusters or supercomputers).

The alternative, then, is to use methodologies or techniques capable of providing local refinement only in necessary key regions. In recent years, new grid methods have been developed to deal with multiscale events in atmospheric motion. In general, these methods can be classified into four groups: grid stretching techniques, grid nesting techniques, grid refinement techniques, and unstructured grid techniques. In the following section, these grid techniques are discussed briefly.

Anthes 1970 introduced a nonlinear grid transformation function that mapped nonuniform physical coordinates onto orthogonal, uniform computational coordinates. This mapping, commonly known as grid stretching, provides increased spatial resolution in selected regions of the computational domain, and has been widely applied in convective and mesoscale modeling studies (Wilhelmson and Chen, 1982). A principal limitation of the grid stretching technique is that one must know a priori, and for the duration of the calculation, which regions of the domain will require high spatial resolution.

The grid nesting technique is another way to provide increased spatial resolution in model forecasts of small scale features without requiring a fine mesh throughout the entire domain. This technique involves the sequential placement of multiple fine-scale meshes in desired regions of the domain (Jones, 1977). One problem with this technique, however, is the interaction among multiple nested meshes, particularly the tendency for propagating dispersive waves to discontinuously change their speeds upon passing from one mesh to the next and to reflect off the boundaries of each nested grid (Clark and Farley, 1984).

The grid refinement technique is a relatively new and powerful method that attempts to change resolution locally in response to the evolving solution. This technique can be subdivided into two basic categories: adaptive grid refinement and continuous dynamic grid adaptation. The adaptive grid refinement technique refines the grid resolution either by adding grid points to an existing grid or by making use of separate fine grids that overlie existing coarser grids where the solution error is presumably high. Though similar in concept to the grid nesting technique, the adaptive grid refinement technique automatically determines the location, orientation, and resolution of the subgrids, and can be incorporated as a general solver in hydro-dynamical models without modifying the governing equations or their numerical representations. This technique has been successfully applied to atmospheric models by Skamarock et al. (1989). The continuous dynamic grid adaptation technique involves methods that redistribute a fixed number of grid points in a predetermined manner (Dietachmayer and Droegemeier, 1992). The criteria that determine how the grid points are redistributed with time are the critical element of this method, and are typically based on the physical characteristics of the evolving flow field. In this manner, the computational mesh structure is globally and dynamically coupled to the physics of the problem so that both are solved in a continuous manner (Thompson, 1984).

The unstructured grid technique (e.g., triangular prism computational mesh) is rather new to the atmospheric science community. In many fields of engineering applications (Baum et al., 1993), there is recognition that this method is more efficient and accurate than the structured logical grid approach used in more traditional codes. The primary benefit of the unstructured grid technique over a conventional structured grid lies in its ability to smoothly transition from high resolution where needed to low resolution elsewhere. This feature effectively removes the wave reflection and wave dispersion problems that are common in other grid techniques. The flexibility of unstructured grids also facilitates the gridding of arbitrary surfaces and volumes in three dimensions. In particular, unstructured grid cells in the horizontal dimension can increase local resolution to better capture the topography or important physical features of atmospheric flow and cloud dynamics. Therefore, this new grid technique can be suitable for studying the inherently multiscale nature of atmospheric motions (e.g., hurricanes).

For the first time, this technique has been applied to three dimensional atmospheric modeling by Bacon et al. (2000). Operational Multiscale Environment model with Grid Adaptivity (OMEGA) is a fully non-hydrostatic, three-dimensional prognostic model that is based on an adaptive, unstructured triangular prism grid. A full description of the system can be found in Bacon et al. (2000) and the details of validation studies can be found in Boybeyi et al. and Gopalakrishnan et al. 2002. The unstructured grid of the model provides flexibility to increase local grid resolution to better capture the important physical features of atmospheric flow and cloud dynamics (e.g., location, eye, eyewall, and spiral bands of a hurricane). This grid feature provides smooth transition from high resolution where needed to low resolution elsewhere and hence effectively removes the wave reflection and wave dispersion problems that are common in other grid techniques.

The model grid can adapt dynamically to a variety of user-specifiable adaptive criteria (e.g., precursors to convection, eye of a hurricane). In other words, the model focuses the model's horizontal grid resolution during the run on regions of complex and critical phenomena to improve the overall quality and efficiency of simulations. The criteria have been well tested for the adaptation to wind-speed and pressure minima for hurricane simulations (cf. Figs. 6 and 7).

The model permits horizontal grids of continuously varying spatial resolutions ranging from about 100 km down to about 1 km to resolve important physical features of atmospheric circulation and cloud dynamics (Figs. 6 and 7). This

Fig. 6 Shows 4-day forecasts of hurricane FRANCES (from 9/1/2004 00Z to 9/ 4/2004 00Z) using OMEGA model with its dynamic grid adaptation: 1) top figures shows the simulation domain and model grid adapting to the evolving hurricane. The model grid resolution ranges from 60 km down to 1 km, 2) middle figures show OMEGA model predicted flow field and hurricane track in a zoomed area, and 3) bottom figures show model predicted vertically integrated water vapor in a further zoomed area and corresponding satellite image

Fig. 6 Shows 4-day forecasts of hurricane FRANCES (from 9/1/2004 00Z to 9/ 4/2004 00Z) using OMEGA model with its dynamic grid adaptation: 1) top figures shows the simulation domain and model grid adapting to the evolving hurricane. The model grid resolution ranges from 60 km down to 1 km, 2) middle figures show OMEGA model predicted flow field and hurricane track in a zoomed area, and 3) bottom figures show model predicted vertically integrated water vapor in a further zoomed area and corresponding satellite image

Fig. 7 Shows a zoomed view of dynamically adapted grid for the hurricane Frances case presented in Fig. 4 (top right figure). Note that grid resolution continuously ranges from 60 km down to 1 km or less

provides not only a higher resolution in the region of evolving weather systems (dynamic adaptation), but also allows a natural interaction with and influence upon the larger scale flow avoiding the wave-reflecting problem. We are currently investigating the possible impact of using solution adaptive modeling technique on a better TC intensity forecast. This work is still under investigation.

Combining remote sensing data sets (i.e., 3D or 4D data assimilation) with the concept of solution adaptive modeling technique presented above may be the most beneficial in improving TC track and particularly intensity forecast. For example, SST is one of the most important data set in accurately determining the storm intensity. We have also investigated the impact of warm sea surface temperature anomaly (SSTA) position on hurricane intensification, using the following remote sensing and buoy data combined with a numerical model:

• SST from the Geostationary Operational Environmental System (GOES) at 6 km resolution from the NASA JPL (http://podaac.jpl.nasa.gov/).

• SST from the tropical rainfall measuring mission (TRMM) microwave imager (TMI) at 0.25° resolution from the Remote Sensing Systems (http://www.ssmi. com). The advantage of using SST from microwave observations, like the TMI, is that it provides retrievals even under intense cloudy conditions associated with TC's.

• Buoy observations of winds, SST, surface air temperature, and dew point from the National Data Buoy Center (NDBC) (http://www.ndbc.noaa.gov/).

For example, both the buoy observations and TMI measurements show SST increase in advance of about 2 days prior to the significant intensification of hurricane Katrina, while surface air temperature declined and reached minimum value at the time of the maximum hurricane intensity. This may be because it may need a period of time for a tropical cyclone to accumulate energy for developing into a hurricane, similar to the evaporation of water vapor, which may need some time to heat the water and make it evaporate, while evaporative cooling makes surface air temperature decrease.

Figure 8 (from Sun et al., 2007) compares the pre-storm SST (27 August 2005) with the post-storm SST (30 August 2005). Figure 8a shows that SST right before the storm development was above 31 °C along the Gulf coast and storm track and was much warmer than the long-term mean for August (Fig. 8c). Also, the SST right after the storm passage showed cooling at the right side of the storm track (Fig. 8b). As compared to the pre-storm SST (Fig. 8a), the strongest cooling in SST is over 6°C, and occurred in the right-front quadrant of the storm track, near the location of the strongest intensity (category 5 as represented by red circles), where higher, longer, and more developed ocean waves usually produce higher drag (Moon et al., 2004).

The difference between the SST prior to and after the storm is related to the storm intensity (Zhu and Zhang, 2006). The stronger the TC, the larger is the difference. For Katrina, the pre-storm and post-storm SST difference was up to 6°C.

The SST map shows that the entire Gulf of Mexico was almost uniformly ^30°C prior to Katrina's intensification (Fig. 8a), while SSTA data show clearly there existed a hot patch along the right side of the storm track, where Hurricane Katrina underwent quick intensification and reached the strongest intensity when it moved into the Gulf of Mexico. It has been found that most clouds and precipitation develop at the right side of the storm track. Desflots et al. (2004) indicate that it is the vertical wind shear that caused this wavenumber one rainfall asymmetry, while shear is due to change of direction of the upper-level wind.

Braun and Tao 2000 showed the significant sensitivity of Hurricane Bob (1991) to several planetary boundary layer (PBL) schemes in MM5 and suggested the dependence of simulated intensity on surface fluxes. We find that the maximum latent heat flux (LHF) occurs at the northeast quadrant or to the right side of storm track, at least at its mature stages. Although the wavenumber one asymmetry may not be caused by high SSTA, because high SST exists at the location of the maximum LHF, and increases the LHF, it enhances the effect of the LHF, and therefore may have played an important role in hurricane intensification.

In order to investigate the effect of the SSTA on hurricane intensification, we performed two control numerical experiments using the latest version of mesoscale model MM5 (Sun et al. 2007). The difference in the simulated tracks was found to be minor. However, as indicated by the pronounced difference in simulated minimum SLP, the simulated hurricane intensity shows remarkable sensitivity to the high SSTA. During the 36-84 h simulations, the control experiment with out the warm SSTA generated weaker intensity or higher minimum SLP than that with the higher SSTA. Although the SST reduction due to storm-induced upwelling and vertical mixing should result in a weaker-simulated hurricane intensity than that simulated when the SST was held constant during the simulations, the simulated

Fig. 8 TMI SST on: (a) 26 August 2005 for two days prior to the storm, and (b) 30 August 2005 for one day after the storm, and (c) 8-year (1998-2005) averaged monthly mean SST in August. The color bar in (a) and (b) is the same as in (c). The circles of different colors indicate the track and intensity of Hurricane Katrina (from Sun et al. 2007)

LHF from the experiment with the warm SSTA at the right side of the storm track is higher than that from the control experiment without higher SSTA, leading to the stronger deepening in the minimum SLP. These experiments further confirm the

TMI observations and show the important impacts of the warm SSTA on hurricane intensification.

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