This study evaluates the sensitivity of wind turbine hub height wind speed forecasts to the planetary boundary layer (PBL) scheme, grid length, and initial condition selection in the Weather Research and Forecasting (WRF) Model over complex terrain. Eight PBL schemes available for the WRF-ARW dynamical core were tested with initialconditionssourcesfromtheNorthAmericanMesoscale(NAM)modelandGlobalForecastSystem(GFS)to produceshort-termwindspeedforecasts.Thelargestimprovementsinforecastaccuracyprimarilydependedonthe grid length or PBL scheme choice, although the most important factor varied by location, season, time of day, and bias-correction application. Aggregated over all locations, the Asymmetric Convective Model, version 2 (ACM2) PBL scheme provided the best forecast accuracy, particularly for the 12-km grid length. Other PBL schemes and grid lengths, however, did perform better than the ACM2 scheme for individual seasons or locations.

2.1            SWI – Severe Weather Indicator

Concept

The Severe Weather Indicator product (SWI) analyses the radar volume data to detect the following severe weather phenomena:

·      Storm areas and reflectivity cores

·      Mesocyclones and Meso-Anticyclones

·      Regions of divergence and convergence

·      Microbursts

The benefit is that SWI provides the user with an overview of hazardous weather phenomena within one image display. The SWI product needs radar data from a volume scan with Z, V, and W data scanned simultaneously.

For each type of weather phenomena a different kind of symbol is used. The visualization is done using a dynamic overlay onto every top projection product.

Product Definition

The SWI product worksheet consists of four tabs, one for each weather phenomena:

·      Microburst              Þ explained here

·      Storm                     Þ explained in chapter 6.1

·      Mesocyclone         Þ explained in chapter 6.2

·      Con-Divergence    Þ explained in chapter 6.4

The different weather phenomena are derived in different algorithm steps of the SWI product. Storm, Mesocyclone, and Con-Divergence algorithms are independent of each other, which is why they are available as stand-alone products (see corresponding chapters). The Microburst algorithm however is based on the output of the other three algorithms. Therefore, in the following only the Microburst algorithm is explained in detail. For the definition of the other algorithms, please refer to the before-mentioned chapters.

Preliminary Analysis

Westward moving disturbances are easily detected along the equator at most longitudes and during all season. Figure 1 is representative Hovmoller diagram of unfiltered, twice-daily 850 mb meridional wind at the equator over the pasific ocean from 1 november through 31 desember 1986. Numerous disturbances are evident. During this particular season they tend to originate near the date line and disappear around 130^o E. Their phase speeds are 5-10 m s^-1 with local periods of 4-10 days. similar disturbances are evident at other longitudes, particularly the africa-atlantic region. africa is though to be a region of genesis for westward propagating disturbances with periods of 3-5 days (e.g., Burpee 1972).

GAMBAR

Fig. 1. Hovmoller diagram twice-daily 850 mb meridional wind at the equator 1 november-31 desember 1986. Contours are plotted at intervals of 3 m s^-1 with negative contours dashed. Zero contour is omitted.

               The power spectrum of the 850 mb meridional wind at the equator as a function of longitude is presented in Fig. 2a. The power of calculated at 61 frequencies for each 120 day segment beginning on 1 september and then averaged over the eight years. There is subtantial power in the 3.5-6 day band (hereafter referred to as "synoptic -frequency") ar all longitudes except 20^o-30^o E¹ and 45^o -60^o W. In fact, at many longitudes there is as much power in this band as there is at periods greater than 10 days and, in particular, near the date line and over the eastern Atlantic there is more power at synoptic than at lower frequencies.

6.2 Observations used for model initialization

period is used to judge observation quality. For example, Hollingsworth et al. (1986) describe how the operational ECMWF data-assimilation system can be used to monitor observation quality. This automated and economical approach to the QC process allows suspect instruments to be identified and corrective action taken, without routinely visiting and inspecting every instrument.

6.2.4 Other observation processing

Whether winds are observed and reported in terms of the individual components or as speed and direction, the measurements may need to be converted to the model wind components. This is because the model u that is defined to be parallel to the grid-point rows, and the model v that is parallel to the grid-point columns, generally differ from the geocentric u and v that are defined relative to latitude and longitude lines. For every vertical column of grid points (the same i, j coordinate), the mathematical transformation will be slightly different. This necessity may be most easy to accidentally overlook when the model coordinates are Cartesian, and the grid-point rows and columns are approximately oriented east–west and north–south.
            Software that interpolates (analyzes) observations to a model grid operates in the framework
of the model’s horizontal coordinate system. Thus, because observation locations are typically defined in terms of latitude and longitude coordinates, there needs to be a transformation to the horizontal coordinates of the model, if it is x–y and not latitude–longitude based.
            Lastly, the units of the observations may need to be transformed to those employed by the model. For example, wind speeds are often reported in knots, but models generally use the meter–kilogram–second (mks) system. And it is common for humidity observations to require conversion as well.

6.2.5 Metadata

Metadata (also called meta-knowledge) accompany the observations themselves, and provide information necessary for their use. Essential types of metadata include the file structure, data format (e.g., NetCDF), the variable (e.g., wind speed), the units (e.g., mks), and the time and three-dimensional-spatial coordinates of the observation. Optional, but useful, information includes the instrument type, the date of the most-recent calibration, and a photo of the instrument site and surroundings. The concept of metadata also applies to model-generated data as well, although the relevant information will obviously be different.
          Conventions have been established for the format of metadata. For example, the NetCDF (Network Common Data Format) Climate and Forecast (CF) Metadata Convention is a welldocumented standard for observational and forecast metadata, which is designed to promote the processing and sharing of files created with the NetCDF Application Programmer Interface [NetCDF API]. The CF conventions generalize and extend the convention of the Cooperative Ocean/Atmosphere Research Data Service, a NOAA/university cooperative group





Model initialization
whose goal is the sharing and distribution of global atmospheric and oceanographic research
data sets.


6.2.6 Targeted or adaptive observations

Economic and other constraints limit the number of observations that are made of the atmosphere, and thus it is reasonable to want to obtain observations from locations where they will have the largest positive impact on model-forecast accuracy, for a particular prevailing weather situation. Methods have been developed to satisfy this need, where the measurements are referred to as adaptive or targeted observations. However, it is clearly not economically feasible to deploy mobile observation platforms on a day-to-day basis. But, there are high-impact weather events, such as hurricanes or severe extratropical cyclones, for which special aircraft observations are made. If the aircraft can be routed so as to provide observations from locations for which the forecast skill is very sensitive to the accuracy of the initial conditions, the procedure can save lives. The routine use of targeted aircraft observations may become more common with the continued development of
unmanned aerial vehicles.

      Various strategies for observation targeting have been evaluated as part of the following
field programs.

• Fronts and Atlantic Storm Tracks EXperiment (FASTEX; Emanuel and Langland 1998;
   Bergot 1999, 2001; Bishop and Toth 1999; Joly et al. 1999; and Bergot and Doerenbecher
   2002)
• NORth Pacific EXperiment (NORPEX, Langland et al. 1999, Majumdar et al. 2002a)
• Atlantic THORPEX (The Hemispheric Observing-system Research and Predictability
   EXperiment) Observing System Test (Langland 2005)
• Annual US NWS Winter Storm Reconnaissance (WSR) programs (Szunyogh et al.
   2000, 2002; Majumdar et al. 2002b)

       The following notational framework for viewing the adaptive-observation problem is provided by Berliner et al. (1999), Majumdar et al. (2006), and others. Let Xi , Xa, and Xv represent n dimensional vectors that define the state of the atmosphere at times ti , ta, and tv , respectively, in terms of the grid-point values of variables or spectral coefficients. The initial time, ti, is when the decision must be made, based on Xi information, about the types and locations of special observations to be collected at time ta (the targeted observation time, and the analysis (initial) time of the operational forecast), where the objective is to optimize the statistical properties of a forecast Xv at the verification time tv . Within the interval ta − ti, the observing platforms need to travel to the target locations so that observations can be made at ta for use in initializing the forecast. The time
interval ta − ti is chosen based on logistical considerations associated with planning the surveillance mission, launching the aircraft, and getting the aircraft to the necessary locations to make the observations. The data set Xa is the result of assimilating standard observations and the special targeted observations, and is used as the initial conditions for the forecast.

Parameterization

The representation of the effects of sub-grid scale processes in terms of grid-scale variables predicted by the model

• NWP models cannot resolve features and/or processes within a grid box realistically

• Parameterization has its greatest impact on predictions of sensible weather at the surface

• Physical processes typically parameterized

• Soil moisture/temperature

• Long wave radiation

• Solar isolation/reflection

• Evaporation

• Convection

• Cloud and precipitation processes

• Friction/turbulence

Convective Parameterization Schemes

• Most NWP models use these parameterization Schemes

• Designed to reduce atmospheric instability in the model

• Prediction of precipitation is a by-product of how the scheme reduces instability

• Expectations of schemes to accurately predict location and timing of convective precipitation is usually low

Planetary Boundary Layer in NWP

3 reasons processes need to be parameterized

1. Phenomena are too small or too complex to be resolved numerically – computers aren’t powerful enough to directly treat them

2. Processes are often not understood well enough to be represented by an equation

3. Effects profoundly impact model fields and are crucial for making realistic forecasts

Problems associated with using parameterizations result from:

1. Increasing complexity of parameterization

2. Interactions between parameterization schemes – these are harder to trace than errors occurring in a single scheme

Dynamics and Numerics of Shallow Water Flows

Outline:

• governing equations

• dimensionles s parameters

• wave propagation, Tsunamis

• hydraulic jumps

• vortex shedding

• Seiche waves

• numerical implementation

Shallow water equations

1. Commonly used in large-scale ocean models

2. Start with Euler’s equations

Phase speed of shallow-water waves

Hydraulic jumps

Shallow-Water Flow Past a Ridge

Atmospheric flow past a ridge

Shallow water equations

• Neglect vertical accelerations

→ Hydrostatic pressure in fluid

−        Assume no vertical variation in (u, v)

a) Advantages

• Allows variable depth in natural way

• Three coupled, hyperbolic PDEs

• The equations admit (weak) discontinuous solutions, which approximate breaking waves

b) Disdvantages

• Equations admit no wave dispersion and no smooth waves of permanent form

• Actual breaking waves create significant vertical variation in horizontal velocity. These equations give vertically averaged velocities, at best.

OR IN COMPUTER CODE:

>> Du/Dt = (f0 + beta*y)v - g*Deta/Dx

>> Dv/Dt = -(f0 + beta*y)u - g*Deta/Dy

>> Deta/Dt = -H*(Du/Dx + Dv/Dy)

Can be derived from primitive equations based on a number of assumptions:

1) The fluid is barotropic

2) The hydrostatic or shallow water assumption based on H<<L

3) Boussinesq assumption

4) neglect vertical component of Coriolis term

5) neglect small advection and diffusion terms

6) eta << H

Periods of seiche waves

Dimensionless Formulation of Shallow-Water Equations

Dimensional formulation                                                    dimensionless formulation

                                                                        With

Numerical Implementation

Staggered Grid

Array Structure

Numerical Integration of Momentum Equation

Dimensionless Formulation

Centered Differences in space and time

Solve for time level n+1

Numerical Integration of Mass Equation

Dimensionless Formulation

Centered Differences in space and time

Solve for time level n+1

tambahan :

1. Lower-tropospheric WRF sounding overlays from (a) nonlocal (YSU and MRF) and hybrid local and nonlocal (ACM2) schemes, and from (b) local (MYJ and QNSE) and hybrid local–nonlocal (ACM2) schemes, plotted around and below the 600-mb level at 0400 UTC 1 Jan 2011 for JAN, plotted beside soundings from the observed JAN sounding and corre-spondingRUC–SFCOA sounding (Cohen et al. 2015)

2. The velocity decreases with increasing distance from the point of impact (due to mass conservation).This provokes the transition from super critical to subcritical condition. The transition is accompanied by a Hydraucalic Jump, where some of the kinetic energy is dissipated in turbulence.

3. Constituents of the PBL and their evolution through the diurnal and nocturnal cycles [from Kis and Straka (2010) and Stull (1988)].

114
Numerical solutions to the equations
3.6 Upper-boundary conditions
 Artificial upper-boundary conditions are required in all atmospheric models because the model atmospheres do not extend to infinity. Indeed, for some historical applications the upper boundary has been located within the troposphere in order to save computational resources. An example of this approach is that Lavoie (1972) placed the upper model boundary, the “lid”, at the top of the boundary layer. Pielke (2002a) describes the location of the upper boundary in various historical model applications.
            Upward-propagating internal-gravity waves, for example generated by mountains or by deep convective storms, can extend to great heights in the atmosphere. Commonly used upper-boundary conditions (e.g., rigid lid, free surface) completely reflect these waves, which is a problem because no such reflection happens in nature, and erroneous downwardpropagating waves contaminate the model solution. There are a number of approaches for preventing this from happening. One involves the use in the model of a gravity-wave absorbing layer, or sponge layer, immediately below the model top, to prevent the wave from reaching the top and reflecting. Such wave absorption can be produced by employing a greatly enhanced, artificial horizontal and/or vertical diffusion (viscosity), where the viscosity increases from the standard value at the bottom of the layer to a maximum at the top boundary. A particular disadvantage of this approach is that the absorbing layer may need to be thick, spanning a large number of model layers and thus involving a large computational overhead. The overall effectiveness of the absorption depends on the wavelength of the gravity wave, the thickness of the absorbing layer, and the distribution of viscosity in the layer. Note that using a shallow absorbing layer with a very large, but computationally stable, viscosity will not be effective because large gradients in viscosity will also cause wave reflections. Klemp and Lilly (1978) defined the entire upper half of their computational domain as the absorbing layer. Figure 3.50 shows a two-dimensional model solution for idealized flow over a maximum in the orography, with and without the use of a viscous damping layer. The Gaussian obstacle had a 5-km half-width, and an amplitude of 1 km. Shown is the vertical motion field in the lowest 10 km of the 50-km-deep model. The model is described in Sharman and Wurtele (1983). The damping spanned the 20 km below a rigid lid that defined the model top. Without the damping, the reflected waves produce considerable noise in the troposphere, over 40 km below the model top. The waves in the solution for the experiment with the absorbing layer could be a result of imperfect damping, or more likely they could be a consequence of wave reflections from the lateral boundaries. An alternative approach for damping the waves before they reach the upper boundary is to use a Raleigh damping layer, again below the model top, where model variables are relaxed toward a predetermined reference state. For example, the Rayleigh damping term in a prognostic equation would be like


where α is any dependent variable, α is the reference value of that variable, and τ(z) increases upward within the damping layer and defines its vertical structure (e.g., see

Ierley et al. [13] solved a a class of nonlinear parabolic partial differential
equations with periodic boundary conditions using a Fourier representation in
space and a Chebyshev representation in time. Similarly as for the GWRM,
the Burger equation and other problems were solved with high resolution.
Tang and Ma [14] also used a spatial Fourier representation for solution of
parabolic equations, but introduced Legendre Petrov-Galerkin methods for
the temporal domain.
In 1994, Bar-Yoseph et al [15, 16] used space-time spectral element methods
for solving one-dimensional nonlinear advection-diffusion problems and
second order hyperbolic equations. Chebyshev polynomials were later employed
in space-time least-squares spectral element methods [17].
A theoretical analysis of Chebyshev solution expansion in time and onedimensional
space, for equal spectral orders, was given in [18]. The minimized
residuals employed were however different from those of the GWRM.
More recently Dehghan and Taleei [19] found solutions to the non-linear
Schr¨odinger equation, using a time-space pseudo-spectral method where the
basis functions in time and space were constructed as a set of Lagrange
interpolants.
Time-spectral methods feature high order accuracy in time. For implicit
finite difference methods, deferred correction may provide high order temporal
accuracy [20, 21]. A relatively recent approach to increase the temporal
efficiency of finite difference methods is time-parallelization via the parareal
algorithm [22]. This method, however, features rather low parallel efficiency
and improvements have been suggested, for example the use of spectral deferred
corrections [23].
An interesting Jacobian-free Newton-Krylov method for implicit timespectral
solution of the compressible Navier-Stokes equations has recently
been put forth by Attar [24].
A time-spectral method for periodic unsteady computations, using a
Fourier representation in time, was suggested in [25] and further developed
in [26] and [27]. A generalization to quasi-periodic problems was developed
in [28].
In summary, although time-spectral methods have been explored in various
forms by several authors during the last few decades, and were found
to be highly accurate, the GWRM as described in [3] has not been pursued.
The present work contributes to the evaluation of this method.
The structure of the paper is as follows. In section 2 we briefly review the
general GWRM formalism for solving a set of pde’s but subsequently restrict