ASP testing
Note that more test cases and references will be continuously added.
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Test simulations
Background Information
Here one can find the results of ASP for different commonly used
benchmarks for both non-hydrostatic and hydrostatic
dynamic cores.
But note that, as pointed out by many researchers, it is
important to keep in mind that these
simulations are sometimes highly idealized and bear
little resemblance to real atmospheric flow (notably for
certain non-hydrotatic benchmark cases). And, slight
differences in published results can be seen owing to
differences in numerical aspects (the order of
horizontal advection for example), or certain other
aspects of the solutions. Nonetheless, they are
extremely useful for evaluating the equations and
numerics of such cores and are currently considered to
be essential for evaluating and inter-comparing the
dynamic cores of different models.
Non-hydrostatic models typically solve for w or the
non-hydrostatic pressure using an elliptic equation
(horizontally implicit, vertically explicit: models such
as MesoNH for example) or using a
horizontally explicit and vertically implicit (HEVI)
approach (models such as WRF or MPAS). The former generally
enables use of a
larger time step, but the elliptic equation must be solved
using an iterative technique at each time step. The HEVI
approach usually involves the solution of a simple
tri-diagnoal matrix (single step) and is usually embedded within a time-split of a
Runge-Kutta 3 or 4 step method, thus it implies a lower
and fixed number of iterations per time step (albeit, a
likely smaller time step). In any case, both time steps
are obviously limited by the Courant number.
But when looking at the literature, the two very
different solution methods tend to give very similar
results (as one would hope!).
The results herein are using ASP, which uses a 3rd order
Runge-Kutta
scheme with split time steps for certain terms (treatment
of gravity and acoutic waves), and a HEVI approach is used
when the add-on non-hydrostatic module is actived.
Baroclinic wave (dry)
over the globe
This test consists in simulating the development of a baroclinic
wave after a certain number of days (when the wave
breaks) on a sphere (using the dimensions of the earth
here). The atmosphere is dry adiabatic, and the wave
grows from an imposed initial perterbation to the wind field.
This case is described in detail by Jablonowski and Williamson
(2006, Quart. J. Roy. Met. Soc.)and thus will not be
described in detail herein. It has become widely
used to benchmark atmospheric model dynamic cores.
The surface pressure field after 9 days of integration
on a 0.7500 x 0.9375 degree lat-lon grid is shown
below. ASP has been run with the non-hydrostatic module
turned off. The minimum and maximum surface pressure
values at day 9 are 945.551845 and 1018.214779 mb, respectively.
The surface pressure field with the non-hydrostatic
module on
(again on a 0.7500 x 0.9375 degree lat-lon grid) is shown
below. The minimum and maximum surface pressure
values at day 9 are
945.296892 and 1018.256570 mb, respectively. Differences
with the hydrostatic simulation (less than 1 mb) are, as one should
expect at these horizontal resolutions, exceedinly small.
The same field is shown below but for the hydrostatic
simulation
on a 1.40625 x 1.875 degree lat-lon grid. Consistent
with other studies, the amplification is damped
(the minimum and maximum surface pressure
values at day 9 are
949.838403 and 1017.841544
mb, respectively) owing to
the more coarse horizontal resolution.
The non-hydrostatic case is not shown since the impact
is even less than for the case on the higher resolution grid.
Non-hydrostatic Warm bubble test
Kilometric scale
dry Gaussian initial perterbation
The warm bubble evolution after 540s of
integration. The X-axis and Y-axis correspond to distance in km.
An animation
using ASP for the classic warm bubble test (plotted using
standard contour intervals)
used for benchmarking non-hydrostatic
dynamic cores is shown. The simulation lasts 2500
seconds with a rigid upper lid at about 23.9 km height, uses an initial
Gaussian potential temperature perterbation
of 6.6 C with calm winds, the upper boundary condition is no flux and w=0, a constant atmospheric
potential temperature distribution
in both the vertical and the horizontal at 30 C, the pressure at the model top is 50 mb, the horizontal
grid spacing is 100 m (20 km length), the domain consists in 200x200 grid cells thus the vertical resolution is also
approximately 100m. The bubble is itialized at 2750m above the ground, and extends 2500m in both the
horizontal and vertical directions. This case has been widely used (references to be added here) in
the atmospheric modeling community for decades. Concerning ASP options, for this case
the only diffusion active is 6th order in the z and
x-directions. The diffusivity is 0.10 X the maximum
value (note, tests have also been done with factors of
0.01, 0.001, 0.0001....: as the factor decreases, more
rotars form but solutions become slightly more noisy
also as the factor becomes small).
All other damping
(divergence damping for example, Raleigh sponge layer relaxation or diffusion, etc.) and
filtering (other diffusion options) are off. For this
simulation, the model
is using 5th order horizontal advection and 4th order
vertical advection.
There is also an animation
for the same configuration using continuous
contours (more
fluid-like) for 2400s of simulation with a rigid top at
175 mb or about 17.5 km.
Metric scale
dry Gaussian initial perterbation: Robert bubble test
These tests were inspired by the work of Robert et
al. (1993) who did dry adiabatic tests using an initial
Gaussian temperature perterbation of 0.5 C in a neutral atmosphere,
similar to the previous experiments but at a 10 m
resolution (with a vertical domain extending up to about
1500 m).
The
bubble
resembles quite closely the bubble they
presented after 18 minutes of integration, with slight
differences likely arising to differences in numerics
(they used a semi-implicit scheme time integration scheme,
while ASP uses
an explicit scheme with high-order
(5th in the horizontal, 4th in the vertical)
finite differences.
An animation of the
bubble using same initial
perterbation
but at a 5 m resolution,
is characterized by more shear-induced rotars.
This illustrates the well-known structural-dependence
of the convective bubbles on resolution.
Finally, an additonal simulation has been done using a
WENO 5th order scheme for horizontal advection of u, w,
potential temperature and geopotential, and also for the
vertical advection of u, w,
potential temperature, while geopotential is vertically
advected using a 4th order centered scheme.
The rigid lid is set at about 1500m, thus the bubble hits
the upper boundary and spreads out (the simulation is
twice as long as the original one shown by Robert et al.).
All
explicit numerical diffusion is off for this run. Note
that for
this simulation, no negative temperature values arise
which suggests the other schemes lead to oscillatory behavior
thereby enducing negative temperature values.
The main bubble structrure is similar, however, wake structures
are a bit different.
Non-hydrostatic Cold bubble test
The cold bubble evolution after 900s of
integration.
An animation using ASP for the classic cold bubble test
used for benchmarking non-hydrostatic
dynamic cores is shown
(there is also
a corresponding
animation over a zoomed-in part of the domain).
The simulation lasts 900 seconds, uses an initial
potential temperature perterbation
of -15 K with calm winds, the upper boundary condition is no flux and w=0, a constant atmospheric
potential temperature distribution
in both the vertical and the horizontal at 300 K, the pressure at the model top is 50 mb, the horizontal
grid spacing is 100 m (40 km length), the domain consists in
400x200 (x,z) grid cells thus the vertical resolution is also
approximately 100m. The bubble is itialized at
3000m above the ground, and extends 4000m in the
horizontal and 2000m in the vertical directions. This case has been widely used (references to be added here) in
the atmospheric modeling community for decades. Concerning ASP options, for this case
the only diffusion active is 2nd order in the x
and z directions with a
constant diffusivity of 75 m2/s: this is used for comparison
with other studies. All other damping
(divergence damping for example, Raleigh sponge layer relaxation or diffusion, etc.) and
filtering (other diffusion options) are off.
Snapshot of the potential temperature perterbation
at 900s using the standard color scale seen in
several publications.
There is also an animation
for the same configuration using continuous
contours (more fluid-like).
When this cold bubble case was originally defined years
ago, 2nd
order diffusion was required to maintain numerical
stability. However, now improved numerical techniques can
permit cold bubble simulations using higher order
more-scale selective diffusion. The above
simulation has been rerun using 6th in
place of 2nd order diffusion with an appropriately scaled
diffusivity, which is defined as 75/16 m6/s. The
corresponding
animation over a zoomed-in part of the domain
reveals an additional rotar which was damped out in the
baseline benchmark simulation, along with some other
interesting features. In addition, the simulation is run
4x longer than the baseline simulation, which permits
the density current to hit the lateral rigid
boundaries. Note, the model equations are in flux form,
so that both the dynamics and horizontal diffusion
prescribe zero flux at the lateral boundaries. Also note
that in the 3 points closest to each lateral boundary,
the order of the horizontal advection and diffusion are
reduced to second order schemes. Tests have shown that
the domain average potential temperature and the mass
are very highly conserved (for example, for these
simulations, domain averaged potential temperature
is conserved to within 0.0006 K and mass is conserved to
within 5x10-11 Pa).
This is purely an academic test,
but it illustrates the impact of diffusion (perhaps in
the future, a similar more modern
benchmark can be envisioned).
Mountain wave tests using the new non-hydrostaic core
Here are a set of standard mountain wave benchmark
tests for a stably stratified atmosphere and constant horizontal
flow entering the domain laterally. Mainly the height and slope
of the surface topography varies among the tests.
NOTE: More
information and references will soon be provided.
Linear hydrostatic
The vertical velocity, w, in m/s. Y corresponds
to height (km), and X corresponds to horisontal distance (km).
Linear non-hydrostatic
The vertical velocity, w, in m/s. Y corresponds
to height (km), and X corresponds to horisontal distance (km).
Non-linear non-hydrostatic
The vertical velocity, w, in m/s. Y corresponds
to height (km), and X corresponds to horisontal distance (km).
Schaer-type
mountain range
The vertical velocity, w, in m/s. Y corresponds
to height (km), and X corresponds to horisontal distance (km).
Feel free to send any comments (aaron.a.boone@gmail.com).
Disclaimer
Please note that this page has nothing to do with
my employer, CNRS, or where I am employed, at the National
Center for Meteorological Research (Centre National de
Recherches Meteorologiques: CNRM) at Meteo-France.
The opinions expressed herein are my own, and are not a
reflection of those where I work or of my employer.
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