A good surface forecast is often critically dependent on accurate estimates of surface fluxes, and in turn, on reasonably accurate soil moisture and temperature estimates. For example, the RUC-2 uses equations to represent all of the processes in the figure below that take place in the PBL to make its forecasts of low level conditions.
FIGURE 1
The Nested Grid Model, or NGM, develops a mixed layer near the ground
in response to buoyancy produced from heating and moistening of the earth's
surface and mechanical stirring by the wind. The mixed layer that
forms is adiabatic with a uniform specific humidity. The NGM allows
the mixed layer to extend up to 300 or 200 mb under extreme conditions
such as very strong solar insolation or a cold air outbreak over warm water.
The model also accounts for the conservation of energy of the mixed layer
(Hoke et al., 1989). For example, in dry convection, which is just
convection without precipitation, the temperature profile is adjusted to
conserve the layer's energy (Glickman, 2000).
In the Global Spectral Model, or GSM, the height of the PBL is calculated using the bulk-Richardson number (Troen and Mahrt, 1986). The bulk-Richardson number is the ratio of convective available potential energy (CAPE) to the magnitude of low-level shear; it measures the intensity of turbulence. The GSM also calculates the Richardson number to indicate the presence of turbulence, using it to determine the stability of a level of the atmosphere. The Richardson number is similar to the bulk-Richardson number and is the ratio of buoyant suppression of turbulence to the shear generation of turbulence. Values less than 0.25 are considered unstable (Glickman, 2000). Since the PBL is on average less stable than the free atmosphere, the model is able to determine the height of the PBL by noting the point where the Richardson number changes significantly.
The ETA and Meso-ETA models use turbulent kinetic energy as a prognostic variable and do not differentiate between the PBL and the rest of the atmosphere (Black, 1994). A prognostic variable is one that contains a time derivative and can therefore be used to predict the variable at a later time (Glickman, 2000).
The European Centre for Medium-Range Weather Forecasts Model, ECMWF, determines the PBL diagnostically. Diagnostic means it does not contain a time derivative. The PBL is typically represented by the first five vertical levels above the surface (at about 996, 983, 955, 909, and 846 mb for a surface pressure of 1000 mb, or at approximate elevations of 30 m, 150 m, 400 m, 850 m, and 1450 m, respectively). The PBL height is determined as the greater of these two heights: that predicted from Ekman theory or the convective height that depends on dry static energy in the vertical (Phillips, 1990). Ekman theory expresses a layer of air in a statically neutral atmosphere surrounding a rotating planet where surface friction and small eddies influence the PBL. Dry static energy in the vertical (also called the Montgomery streamfunction) is a thermodynamic concept similar to potential temperature except that any kinetic energy is locally dissipated into heat (Glickman, 2000).
The Rapid Update Cycle Model, RUC, added the PBL height as a diagnostic variable in October 1999. The RUC finds the height aloft at which virtual potential temperature, theta-v, exceeds the potential temperature at the surface.
The Mesoscale Model, MM5, has several options for parameterizing the PBL. These will be discussed in more detail in the vertical mixing section.
The MRF (Medium Range Forecast) and the WRF (Weather Resaerch and
Forecasting) both employ the same scheme to determine the PBL. The method used
is a first-order vertical diffusion scheme. There exists a diagnosticaly
determined PBL height that uses the Bulk-Richardson approach to iteratively
estimate a PBL height starting from the ground upward. Once the PBL height is
determined, the profile of the coefficient of diffusivity is specified as a
cubic function of the PBL height. The actual values of the coefficients are
determined by matching with the surface-layer fluxes. There is also a
counter-gradient flux parameterization that is based on the fluxes at the
surface and the convective velocity scale. (Hong and Pan(1996))
Mechanical mixing is also known as turbulence, and the whirlpools of air that result are called eddies. Three factors are critical in determining the magnitude and location of eddies:
To parameterize boundary layer vertical mixing into weather prediction models, various methods apply.
In the NGM, the following relationships are observed in parameterizing vertical mixing:
For the RUC, the PBL level is estimated using a Richardson number approach. Based on the PBL level, the model allows for two separate grids; a high-resolution atmospheric boundary layer grid and a low-resolution grid above it. The two grids are layered vertically, and physics calculations are done on the high-resolution grid, while dynamics calculations are done on the coarse-resolution grid. Within the PBL, turbulent fluxes are calculated from the Burk and Thompson (BT) level 3.0 method. A benefit of this method is that it allows improvements to be made to it easily.
The ECMWF was improved in August 1993 with a change in its calculation
of vertical mixing (Brankovic and Molteni, 1997). The old scheme
was based on a Richardson number approach. The change allows vertical
mixing to occur above the PBL in conditions of static instability.
The MM5 is a relatively new mesoscale model that aims to take advantage
of the greater speed of today's computers. A higher-order boundary
layer structure could therefore be included in the MM5, and there are two
main advanced options for boundary layer schemes and four other options.
The advanced options include the 2.5-level boundary layer parameterization
and the more widely used Blackadar scheme. The 2.5-level option can
simulate both saturated and sub-saturated boundary layers, and as a result
has outperformed the Blackadar scheme with regards to the convective PBL
(NRL, 1998). The four other options are a Richardson number approach,
the same method the ETA uses (turbulent kinetic energy), the same method
the GSM models use (mechanical mixing on the sigma level), and the Gayno-Seaman
scheme.
MRF and WRF Models: The WRF Model uses the same vertical mixing scheme in the
boundary layer as the Medium Range Forecast Model. The coefficient of
diffusivity within the PBL is based on a cubic function constructed from the PBL
height. The magnitude of the K within PBL is scaled to match the surface layer
by the use of the convective velocity scale. Prandtl number is used to scale
the thermal diffusivity from the momentum diffusivity coefficients. A base value
of 1 m**2/sec is added for numerical stability. The coefficient of
diffusivity above the PBL is constructed from the local richardson number used
to scale the coefficient with a limiting length scale specified (currently
lambda=150 m).
Vertical diffusion is calculated by a fully implicit time
integration scheme used to calculate the update of prognostic variables due to
vertical diffusion.
Before we examine how each model takes into account horizontal mixing and diffusion, we need to understand the terms parameterization and smoothing:
-A parameterization is the representation in a dynamic model of the physical effects in terms of admittedly oversimplified parameters, rather than requiring such effects to be consequences of the dynamics of the system. In other words, a model that uses a parameterization is estimating physical quantities at different points based on simplified dynamics of nearby points.
-A smoothing function, such as one the NGM uses, simply averages data in space and time between nearby points. It is designed to compensate for random errors at small scales and data points that are not representative of the area around it (Glickman, 2000).Forhorizontal mixing, the models use various-order parameterizations. The higher the order a parameterization is, the lesser the sharpness of boundaries will be.
The GSM uses a second-order scheme taken from Lieth, 1971.
The ECMWF uses a fourth-order scheme applied on vertical surfaces to vorticity, divergence, and moisture, and on pressure surfaces to temperature (Brankovic and Molteni, 1997).
The two ETA models use a second-order scheme calculated from wind and turbulent kinetic energy.
The MM5 uses two different schemes for horizontal diffusion. A second-order scheme is used for the grid points that are next to lateral boundary points. The shore of Lake Michigan and a cold front are examples of lateral boundary points. For the interior grid points, a fourth-order scheme is used.
The NGM is different in that it does not use a parameterization scheme, but instead uses a form of horizontal smoothing. This horizontal smoothing is applied every 30 minutes in an attempt to control noise and is performed on the horizontal wind component, potential temperature, and specific humidity for selected levels in the model (Hoke et. al, 1989).
In 1998, a study was published analyzing the performance of numerical models during a return flow event along the United States Gulf of Mexico coastal regions. The air mass transformation (AMT) model was employed, which uses a high-resolution one-dimensional PBL model in addition to a two-layer soil model. Due to the lack of good soil data, however, the soil type and moisture content values created by the ETA model are used rather than actual measurements, which limits the useful results (Kara et. al, 1998).
In conclusion, an increasing interest is being developed in studying complex PBL processes that interact with clouds and precipitation, or that occur over nonhomogeneous surfaces. The understanding of these complicated regimes is still primitive, and development of parameterization schemes for these regimes is slow. As can be seen from the 1998 analysis of the performance of numerical models during a return flow event along the Gulf Coast, obtaining good data for PBL processes is difficult. We probably need a much denser data set (which is too expensive) before PBL processes could accurately be represented in models. It is therefore unlikely to generalize a simple scaling relationship because external forcing (in the form of precipitation, convection, etc.) is too varied. Another constraint is that the PBL often undergoes rapid temporal or spatial changes, which makes it difficult to average by smoothing or parameterizing over time or space.
In the future, work on more complex PBL parameterizations will continue. Field experiments conducted by research models will exploit new technology to document PBL's in the equatorial Pacific, the North American winter, and the sub-Arctic summer. Increasingly sophisticated models will combine PBL physics with different types of forcing. Improved computers and instrumentation will make these studies more feasible. But researchers would still have to deal with various constraints such as the hard-to-predict natural variabilities of PBL regimes.
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