A 3D Sigma Coordinate Atmospheric Primitive Equation Model
Matt Onderlinde
MPO-662 Project - 2013
1) Introduction
When developing a 3-dimensional numerical model for forecasting in the atmosphere,
the choice of a vertical coordinate is critical. A standard x-y-z coordinate system becomes quite
complicated in regions of terrain because lower model levels may intersect the ground. To
simplify the impacts of terrain on the numerical schemes, a “sigma” coordinate system is
employed in which the vertical coordinate sigma is defined as:
𝜎=
𝑝𝑧
(𝑝𝑠 − 𝑝𝑡 )
(1)
where 𝑝𝑧 is the pressure at model level z, 𝑝𝑠 is the pressure at the surface and 𝑝𝑡 is the pressure
at the top of the model. This definition assures that σ ranges between 0 and 1 and no model
levels intersect the ground.
The model developed takes its initial and boundary conditions from the Global Forecast
System (GFS). A flexible system was developed such that the user need only define a central
latitude and longitude, grid spacing, and number of grid points in the x and y direction. In this
way the model can easily be tested over varying regions of the globe including regions of terrain
or ocean regions with no terrain variance. The model integrates a flux form of the governing
equations and the details of this procedure are discussed in the following sections.
2) Methods
The initialization routine reads user inputs from a namelist file. These inputs include
central latitude and longitude, grid spacing (dx and dy), number of vertical levels, and the top
pressure level that defines the model top. The routine defines a grid such that dx and dy
remain constant along the earth. This initialization means that no map factors are required to
correct for changing grid spacing across the grid. A staggered Arakawa-C grid is used with u, v,
and w points staggered by one half of a grid cell in the x, y, and z directions respectively. Sigma
vertical levels are defined to be monotonically decreasing from a value of 1 at the surface to a
value of 0 on the model top. Once the grid is created the GFS data is interpolated to each grid
point using an inverse-distance weighted average of the 4 nearest GFS data points. Boundary
conditions are also created from the GFS data. The boundary data are defined over the 7 rows
and columns along the grid boundaries.
The model integrates a conservative form of the compressible governing equations for
atmospheric flows. These equations come directly from the Weather Research and Forecast
(WRF) Model technical note (Skamarock et al. 2008). The equations are for dry air though a
specific humidity equation is included. Specific humidity is simply advected as a tracer in the
model and has no impact on the dynamics. The 8 governing equations are:
where v = (U,V,W), Ω = μη, and Θ = μѲ. U, V, and W are the mass weighted velocity components
defined as μu, μv, and μw. u, v, and w are the actual velocity components. Ω is the mass weighted
coordinate vertical velocity with η being the sigma vertical velocity. Θ is the mass weighted potential
temperature with Ѳ being the potential temperature itself. μ is defined as ps – pt such that the
capitalized quantities (U,V,W,Θ,Ω) are defined on a ‘per unit mass’ basis. φ is geopotential, α is inverse
density, p is pressure, p0 is a reference pressure (1000 hPa), Rd is the dry gas constant, γ is the ratio of
heat capacities for dry air ( = 1.4), and g is gravity. With these definitions, equations 2-6 are
conservative while equations 7-9 are not. The equation for specific humidity is equivalent to
equation (5) with Q and q replacing Θ and Ѳ respectively. In these equations, subscripts denote
partial differentiation in the x, y, η, and t dimensions. For generic variable a, the advection and
differential operators are given by:
A Runge-Kutta 3rd order (RK3) time integration scheme is used to handle the time
integration and 2nd ordered centered differencing is used for all spatial derivatives. The stability
of this scheme requires that the Courant number (defined as
𝑢𝛥𝑡
𝛥𝑥
) remain less than
approximately 1.7. Because no time splitting is done, very small time steps are required to
maintain stability when grid spacing becomes small. Also of concern are vertically propagating
sound waves due to the relatively small grid spacing in the vertical direction compared to the
horizontal direction.
Boundary conditions are applied at each of the 3 stages during the RK3 scheme. A
weighted relaxation scheme is used to relax the model domain to the GFS boundary data. This
relaxation scheme uses exponentially decaying weights so that the GFS boundary data
contributes totally to the values on the outermost rows and columns and minimally on the 8 th
row and column away from the boundary. In addition to the relaxation scheme, a smoothing
routine was used around the boundaries to aid in maintaining stability. In fact, smoothing was
done frequently throughout the solver routines in an effort to maintain stability. The
implications of the boundary condition and smoothing schemes are discussed in the results
section. Initially the value of μ was allowed to vary dynamically as described in equation (6).
However, this became a significant limitation on model stability. In an order to stabilize
simulations, μ was redefined as simply the time interpolated value derived from the GFS
surface pressure field. This created gradually varying surface pressure that allowed for longer
simulations.
3) Results
The simulation used to demonstrate the model described above is integrated on a
domain over the Atlantic Ocean to eliminate the impacts and complications of topography. The
grid spacing is 18 km in both the x and y directions and the domain is 91 by 91 grid points in the
horizontal direction with 41 vertical levels. The boundary condition data are from the 1200 UTC
1 January 2012 run of the GFS model. The simulation covers the 3 h period from 1200 UTC to
1500 UTC.
Figure (1) shows the wind field (vectors) overlaid on the surface pressure field (color
filled) for times 45 min, 90 min, 135 min, and 180 min. Because a time step (dt) of 20 sec is
used, this represents the solution after 540 time steps. The surface pressure field varies
smoothly because of the choice in lower boundary condition. The wind vectors do a fairly good
job responding to the low-level pressure field and this is best seen as the high pressure system
centered near the northeast corner of the domain expends westward. Figures 2-4 show similar
time plots for u-component of wind, temperature, and height. Each of these fields is shown on
model vertical level 6 which is approximately at the 850 hPa level. While u-wind and especially
height vary relatively smoothly through time, the temperature field acquires “waviness” which
is very apparent in the center of the domain where boundary smoothing is not applied. These
waves develop along the tighter wind and pressure gradient in the northwest corner of the
domain and then propagate southeastward rapidly across the domain. It is unclear whether
the waves develop from baroclinic instability in this region or if they are simply a result of
unphysical gradients at the edge of the boundary condition smoothing region. It appears most
likely that it is the former because no such waves occur in the GFS model during this period.
This type of “waviness” was a theme throughout all the simulations that were tested including
different domain sizes and simulations over topography. Even with extensive smoothing of
pressure gradients and boundary conditions these waves occurred. The waves eventually lead
to model destabilization and solutions go to infinity. This significantly limits the model’s
forecast range as stability beyond 6 h was not achieved.
Another consideration is the accuracy of the solutions calculated by the model. Because
only dry dynamics are considered and no time-splitting, divergence damping, sponge layers,
etc. are used, accuracy is not expected to be high. One method to check the accuracy of the
model is to simply compare the solution at 3 h to that of the GFS model. If the GFS solution is
assumed to be the “truth” then the difference between the forecasts will represent the error.
While quantitative measures of the error are not calculated, comparisons of skew-t diagrams
demonstrate the differences between GFS forecasts and forecasts generated by this model.
Figure (5) shows a comparison of skew-t diagrams at forecast time 3 h for latitude 27.2N and
longitude 44.2W (near the center of the domain). The most noticeable difference is in the low
level temperature and moisture profile which is much noisier in the sigma model than in the
GFS model. Outside of these differences the two profiles are fairly similar suggesting that the
advection of momentum and moisture is fairly consistent with the GFS solution.
Animations of various model fields were generated to show evolution during this 3 h
simulation. These animations along with the model code can be viewed at the following
website:
http://www.weathernerds.org/mjonder/MPO662_project.html .
4) Conclusions
A 3 dimensional sigma coordinate compressible primitive equation model is developed
to simulate the flow in the atmosphere. Fast moving acoustic and gravity waves severely limit
the stability of the model making simulations beyond 6 h seemingly impossible even with very
small time steps. These waves are most evident in the pressure and temperature fields but
gradually become notable in the kinematic fields as well. Comparisons with the GFS forecast
during this period suggest that some accuracy is achieved in the forecast though low-level
temperature and moisture forecasts contain significant errors. Additional development of
boundary relaxation, divergence damping, w-damping, and sponge layers at upper levels along
with time-splitting methods would surely improve the utility of this model.
5) References
Skamarock, W. C., J. B. Klemp, J. Dudhia, D. O. Gill, D. M. Barker, M. G. Duda, X. Y. Huang, W.
Wang, J. G. Powers, 2008: A Description of the Advanced Research WRF Version 3.
NCAR/TN–475+STR.
Fig 1. Wind vectors and surface pressure (Pa) for simulation times 45, 90, 135, and 180
minutes.
Fig. 2. ~850 hPa u-component of wind (m/s) at simulation times 45, 90, 135, 180 minutes.
Fig. 3. ~850 hPa temperature (K) at simulation times 45, 90, 135, 180 minutes.
Fig. 4. ~850 hPa height (m) at simulation times 45, 90, 135, 180 minutes.
Fig. 5. Comparison of GFS skew-t diagram (left) and sigma model skew-t diagram (right).
Valid time is 1500 UTC 1 Jan 2012.