GMDGeoscientific Model DevelopmentGMDGeosci. Model Dev.1991-9603Copernicus PublicationsGöttingen, Germany10.5194/gmd-10-827-2017OZO v.1.0: software for solving a generalised omega
equation and the Zwack–Okossi height tendency equation using WRF model outputRantanenMikamika.p.rantanen@helsinki.fihttps://orcid.org/0000-0003-4279-0322RäisänenJounihttps://orcid.org/0000-0003-3657-1588LentoJuhaStepanyukOlegRätyOlleSinclairVictoria A.https://orcid.org/0000-0002-2125-4726JärvinenHeikkihttps://orcid.org/0000-0003-1879-6804Department of Physics, University Of Helsinki, Helsinki, FinlandCSC – IT Center for Science, Espoo, FinlandMika Rantanen (mika.p.rantanen@helsinki.fi)21February201710282784119August201629September20163February20176February2017This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://gmd.copernicus.org/articles/10/827/2017/gmd-10-827-2017.htmlThe full text article is available as a PDF file from https://gmd.copernicus.org/articles/10/827/2017/gmd-10-827-2017.pdf
A software package (OZO, Omega–Zwack–Okossi) was developed to diagnose the
processes that affect vertical motions and geopotential height tendencies in
weather systems simulated by the Weather Research and Forecasting (WRF)
model. First, this software solves a generalised omega equation to calculate
the vertical motions associated with different physical forcings: vorticity
advection, thermal advection, friction, diabatic heating, and an imbalance
term between vorticity and temperature tendencies. After this, the
corresponding height tendencies are calculated with the Zwack–Okossi tendency
equation. The resulting height tendency components thus contain both the
direct effect from the forcing itself and the indirect effects (related to
the vertical motion induced by the same forcing) of each physical mechanism.
This approach has an advantage compared with previous studies with the
Zwack–Okossi equation, in which vertical motions were used as an independent
forcing but were typically found to compensate the effects of other forcings.
The software is currently tailored to use input from WRF simulations with
Cartesian geometry. As an illustration, results for an idealised 10-day
baroclinic wave simulation are presented. An excellent agreement is found
between OZO and the direct WRF output for both the vertical motion and the
height tendency fields. The individual vertical motion and height tendency
components demonstrate the importance of both adiabatic and diabatic
processes for the simulated cyclone. OZO is an open-source tool for both
research and education, and the distribution of the software will be
supported by the authors.
Introduction
Today, high-resolution atmospheric reanalyses provide a three-dimensional
(3-D)
view on the evolution of synoptic-scale weather systems . On the other hand, simulations by atmospheric models allow for exploring
the sensitivity of both real-world and idealised weather systems to factors
such as the initial state e.g., lower boundary conditions
e.g., and representation of
sub-grid scale processes e.g.. Nevertheless, the complexity of atmospheric dynamics often makes
the physical interpretation of reanalysis data and model output far from
simple. Therefore, there is also a need for diagnostic methods that help to
separate the effects of individual dynamical and physical processes on the
structure and evolution of weather systems.
Two variables that are of special interest in the study of synoptic-scale
weather systems are the geopotential height tendency and vertical motion
. Height tendencies are directly related to the
movement and intensification or decay of low- and high-pressure systems.
Vertical motions affect atmospheric humidity, cloudiness, and precipitation.
They also play a crucial role in atmospheric dynamics by inducing adiabatic
temperature changes, by generating cyclonic or anticyclonic vorticity, and by
converting available potential energy to kinetic energy
.
For the need of diagnostic tools, some software packages have been developed
to separate the contributions of each forcing to the vertical motion and
height tendency. DIONYSOS , a tool for analysing
numerically simulated weather systems, provides currently online daily
diagnostics for the output of numerical weather prediction models. RIP4
can calculate Q-vectors and a quasi-geostrophic (QG) vertical motion
from Weather Forecast and Research (WRF) model output, but the division of
ω into contributions from various atmospheric processes is not
possible in RIP4. Furthermore, many research groups have developed tools for
their own needs but do not have resources to distribute the software.
Here, we introduce a software package Omega–Zwack–Okossi (OZO) that can be used for diagnosing
the contributions of different dynamical and physical processes to
atmospheric vertical motions and height tendencies. OZO calculates vertical
motion from a quasi-geostrophic and a generalised omega equation
while height tendencies are calculated using the
Zwack–Okossi tendency equation . The
current, first version of OZO has been tailored to use output from the
WRF model
simulations run with idealised Cartesian geometry. Due to the wide use of the
WRF model, we expect OZO to be a useful open-source tool for both research
and education.
In the following, we first introduce the equations solved by OZO: the two
forms of the omega equation in Sect. 2.1 and the Zwack–Okossi height tendency
equation in Sect. 2.2. The numerical techniques used in solving these
equations are described in Sect. 3. We have tested the software using output
from an idealised 25 km resolution WRF simulation described in Sect. 4.
Section 5 provides some computational aspects of the software. The next two
sections give an overview of the vertical motion (Sect. 6) and height
tendency (Sect. 7) calculations for this simulation. Software limitations and
plans for future development are presented in Sect. 8, and the conclusions
are given in Sect. 9. Finally, information about the data and code
availability is given in Sect. 10.
EquationsOmega equation
The omega equation is a diagnostic tool for estimating atmospheric vertical
motions and studying their physical and dynamical causes. Its well-known QG
form, obtained by combining the QG vorticity and thermodynamic equations,
infers vertical motion from geostrophic advection of absolute vorticity and
temperature :
LQG(ω)=FV(QG)+FT(QG),
where
LQG(ω)=σ0p∇2ω+f2∂2ω∂p2
and the two right-hand side (RHS) terms are
FV(QG)=f∂∂pVg⋅∇ζg+f,FT(QG)=Rp∇2Vg⋅∇T.
(the notation is conventional, see Table for an explanation
of the symbols.)
List of mathematical symbols.
cp = 1004 J kg-1specific heat of dry air at constant volumefCoriolis parameterFforcing in the omega equationFfriction force per unit massg = 9.81 m s-2gravitational accelerationkunit vector along the vertical axisLlinear operator in the left-hand side of the omega equationppressureQdiabatic heating rate per massR = 287 J kg-1gas constant of dry airS=-T∂lnθ∂pstability parameter in pressure coordinatesttimeTtemperatureVhorizontal wind vectorVggeostrophic wind vectorαrelaxation coefficientσ=-RTpθ∂θ∂phydrostatic stabilityσ0isobaric mean of hydrostatic stabilityζvertical component of relative vorticityζgrelative vorticity of geostrophic windζagrelative vorticity of ageostrophic windω=dpdtisobaric vertical motion∇horizontal nabla operator∇2horizontal laplacian operator
Qualitatively, the QG omega equation indicates that cyclonic (anticyclonic)
vorticity advection increasing with height and a maximum of warm (cold)
advection should induce rising (sinking) motion in the atmosphere. However,
when deriving this equation, ageostrophic winds, diabatic heating, and
friction are neglected. In addition, hydrostatic stability is treated as a
constant and several terms in the vorticity equation are omitted. Although
Eq. () often provides a reasonable estimate of synoptic-scale
vertical motions at extratropical latitudes , these
approximations inevitably deteriorate the accuracy of the QG omega equation
solution.
The omega equation can be generalised by relaxing the QG approximations
e.g..
Here we use the formulation
L(ω)=FV+FT+FV+FQ+FA,
where
L(ω)=∇2(σω)+f(ζ+f)∂2ω∂p2-f∂2ζ∂p2ω+f∂∂pk⋅∂V∂p×∇ω
is a generalised form of Eq. () and the RHS terms have the
expressions
FV=f∂∂pV⋅∇ζ+f,FT=Rp∇2V⋅∇T,FF=-f∂∂pk⋅∇×F,FQ=-Rcpp∇2Q,FA=f∂∂p∂ζ∂t+Rp∇2∂T∂t.
Apart from the reorganisation of the terms in Eq. (), this
generalised omega equation is identical with the one used by
. It follows directly from the isobaric primitive
equations, which assume hydrostatic balance but omit the other approximations
in the QG theory. The first four terms on the RHS represent the effects of
vorticity advection (FV), thermal advection (FT),
friction (FF), and diabating heating (FQ). The last
term (FA) describes imbalance between the temperature and
vorticity tendencies. For constant f and constant R, the imbalance term
is directly proportional to the pressure derivative of the ageostrophic
vorticity tendency .
Because the operators LQG and L are linear, the contributions of
the various RHS terms to ω can be calculated separately if homogeneous
boundary conditions (ω=0 at horizontal and vertical boundaries) are
used. These contributions will be referred to as ωX, where X
identifies the corresponding forcing term. The contribution of orographic
vertical motions could be added by using a non-zero lower boundary condition
but is not included in the current version
of OZO.
Zwack–Okossi height tendency equation
In the Zwack–Okossi method , height
tendencies are calculated from the geostrophic vorticity tendency. Neglecting
the variation of the Coriolis parameter,
ζg=gf∇2Z
and hence
∂Z∂t=∇-2fg∂ζg∂t.
The geostrophic vorticity tendency at level pL is obtained from
the equation
∂ζg∂t(pL)=1ps-pt∫ptps∂ζ∂t-∂ζag∂tdp-Rf∫ptps∫ppL∇2∂T∂tdppdp,
where ps (here 1000 hPa) and pt (here 100 hPa) are the lower and the
upper boundaries of the vertical domain. The vorticity tendency
∂ζ∂t is calculated from the vorticity
equation (Eq. 2 in ) and the temperature tendency
∂T∂t from the thermodynamic equation (Eq. 3.6 in
). The ageostrophic vorticity tendency
∂ζag∂t is estimated from time series
of vorticity and geostrophic vorticity (Eq. ) using centred
time differences
∂ζag∂t≈Δ(ζ-ζg)2Δt.
For the calculations shown in this paper, Δt=1800 s was used.
In Eq. (), the first integral gives the mass-weighted vertical
average of the geostrophic vorticity tendency between the levels ps and
pt. The difference between the geostrophic vorticity tendency at level
pL and this mass-weighted average is obtained from the second
double integral. In this latter integral, hydrostatic balance is assumed to
link the pressure derivative of the geostrophic vorticity tendency to the
Laplacian of the temperature tendency.
Analogously with the vertical motion, the height tendency is divided in OZO
to the contributions of different physical and dynamical processes as
∂Z∂t=∂Z∂tV+∂Z∂tT+∂Z∂tF+∂Z∂tQ+∂Z∂tA.
By substituting the vorticity equation and the thermodynamic equation into
Eq. (), and then combining the result with
Eq. (), the expressions for the RHS components of
Eq. () are derived as follows:
Vorticity advection (V) and friction (F) have a direct effect on the vorticity tendency in Eq. ().
Thermal advection (T) and diabatic heating (Q) have a direct effect on the temperature tendency in Eq. ().
The ageostrophic vorticity tendency in Eq. () is attributed to the imbalance term (A).
All five terms also affect the vorticity and temperature tendencies
indirectly via vertical motions, which are calculated for each of them
separately with the generalised omega equation.
This results in the following new expressions:
∂Z∂tV=fg(ps-pt)∇-2[∫ptps-V⋅∇(ζ+f)-ωV∂ζ∂p.+(ζ+f)∂ωV∂p+k⋅∂V∂p×∇ωVdp.-Rf∫ptps∫ppL∇2SωVdppdp],∂Z∂tT=fg(ps-pt)∇-2[∫ptps-ωT∂ζ∂p+(ζ+f)∂ωT∂p.+k⋅∂V∂p×∇ωTdp.-Rf∫ptps∫ppL∇2-V⋅∇T+SωTdppdp],∂Z∂tF=fg(ps-pt)∇-2[∫ptpsk⋅∇×F-ωF∂ζ∂p.+(ζ+f)∂ωF∂p+k⋅∂V∂p×∇ωFdp.-Rf∫ptps∫ppL∇2SωFdppdp],∂Z∂tQ=fg(ps-pt)∇-2[∫ptps-ωQ∂ζ∂p+(ζ+f)∂ωQ∂p.+k⋅∂V∂p×∇ωQdp.-Rf∫ptps∫ppL∇2Qcp+SωTdppdp],∂Z∂tA=fg(ps-pt)∇-2[∫ptps-∂ζag∂t-ωA∂ζ∂p.+(ζ+f)∂ωA∂p+k⋅∂V∂p×∇ωAdp.-Rf∫ptps∫ppL∇2SωAdppdp].
The equation system used in this study has been adopted partly from
. However, whereas used
the vorticity equation and the non-linear balance equation to obtain height
tendencies, the Zwack–Okossi equation is used here. The main advantage of
this choice is its smaller sensitivity to numerical errors. Our method
produces quite smooth vertical profiles of height tendencies because the
tendencies at neighbouring levels are bound to each other by the vertical
integration in Eq. (). On the other hand, our method differs from
earlier applications of the Zwack–Okossi equation
e.g. because the use of the
generalised omega equation eliminates vertical motion as an independent height
tendency forcing. This is an important advantage because these earlier
studies have shown a tendency of compensation between vertical motions and
the other forcing terms.
Earlier diagnostic tools have come close to our technique. The most similar
approach is probably used in the DIONYSOS tool.
Regardless of many similarities, there are still three major differences
between DIONYSOS and OZO. First, DIONYSOS eliminates the ageostrophic
vorticity tendency as an independent forcing using an iterative procedure.
Second, DIONYSOS uses simple parameterisations of diabatic heating and
friction, whereas OZO relies directly on model output. Third, DIONYSOS uses
the method of to convert vorticity tendencies to
height tendencies, whereas the Zwack–Okossi method is used in OZO.
Vorticity and temperature advection by non-divergent and divergent winds
Following the Helmholtz theorem, the horizontal wind can be divided to
non-divergent (Vψ) and divergent (Vχ) parts.
Their contributions to vorticity advection and temperature advection can be
separated as
-V⋅∇ζ+f=-Vψ⋅∇ζ+f-Vχ⋅∇ζ+f,-V⋅∇T=-Vψ⋅∇T-Vχ⋅∇T,
and the same applies to the corresponding ω and height tendency
contributions. This separation between Vψ and Vχ
contributions is included in OZO because found it
to be important for the height tendencies associated with vorticity
advection.
OZO calculates the divergent part of the wind (Vχ) from the
gradient of the velocity potential χ (Eq. ), which
is derived from the horizontal divergence by inverting the Laplacian in
Eq. ().
Vχ=∇χ∇2χ=∇⋅V
The non-divergent wind is obtained as the difference between V and
Vχ. The output of OZO explicitly includes the vorticity advection
and temperature advection terms of the ω and height tendency equations
due to both the full wind field V and the divergent wind
Vχ. The contributions associated with the non-divergent wind
Vψ can be calculated as their residual.
Numerical methods
The first version of the OZO software package is tailored to use output from
WRF simulations in idealised Cartesian geometry. The computational domain is
periodic in the zonal direction, whereas symmetric boundary conditions are
used at the northern and southern boundaries. Before the calculation, the WRF
data need to be interpolated to pressure coordinates.
Calculation of right-hand side terms
All of the right-hand side terms of the omega equation
(Eq. ) and the Zwack–Okossi equation (Eq. )
are calculated in grid point space. Horizontal and vertical derivatives are
approximated with two-point central differences with the exception at the
meridional and vertical boundaries, where one-sided differences are used. In
the calculation of the imbalance term of the omega and Zwack–Okossi equations
(Eqs. and ), also tendencies of T, ζ and Z
are needed. These tendencies are approximated by central differences of the
corresponding variables.
Because the calculations are done in pressure coordinates, the lower boundary
of the domain does not correspond to the actual surface. To mitigate the
impact of this, vorticity and temperature advection, friction, diabatic
heating, and the ageostrophic vorticity tendency are all attenuated below the
actual surface by multiplying them by a factor varying from 0 to 1. The
multiplication factor at each level depends on how far down the level is
below the ground. For example, for a surface pressure of 950 hPa and
vertical mass-centred grid spacing of 50 hPa, the multiplication
factor is 0 at 1000 hPa, 0.5 at 950 hPa, and 1 at
900 hPa and all higher levels.
Inversion of left-hand side operators
The omega equation is solved using a multigrid algorithm
. Each multigrid cycle starts from the original
(finest) grid, denoted below with the superscript (1). A rough solution in
this grid (ω̃(1)) is found using ν1 iterations of
simultaneous under relaxation, starting either from the previous estimate of
ω or (in the first cycle) ω=0. The residuals,
Res(1)=F-L(ω̃(1)),
are then upscaled to a coarser grid (superscript (2)), in which the number of points is halved in all three dimensions. In this grid, the equation
L(ω(2))=Res(1)
is then roughly solved
replicating the method used in the first grid. The residual of this
calculation is fed to the next coarser grid, and the procedure is continued
until the grid only has five points on its longest (meridional) axis. Thus,
for an idealised baroclinic wave simulation with horizontal resolution of
25 km, the meridional axis has 320 grid points. That makes six coarser
resolutions (160, 80, 40, 20, 10, and 5 points on the meridional axis) in
addition to the original one.
Having obtained the estimate ω̃(N) for the coarsest grid, a new estimate for the second coarsest grid is formed as
ω̃NEW1(N-1)=ω̃(N-1)+αω̃(N),
where α is a relaxation coefficient. To reduce the effect of
regridding errors, this estimate is refined using ν2 iterations of
simultaneous under relaxation. The result,
ω̃NEW2(N-2), is then substituted to the next finer
grid
ω̃NEW1(N-2)=ω̃(N-2)+αω̃NEW2(N-1)
and the sequence is repeated until the original grid is reached.
After each multigrid cycle, the maximum difference between the new and the
previous estimate of ω in the original (finest) grid is computed. If
this difference exceeds a user-defined threshold (by default 5×10-5Pas-1), the multigrid cycle is repeated. Typically,
several hundreds of cycles are required to achieve the desired convergence.
OZO has four parameters for governing the multigrid algorithm, with the
following default values: the under relaxation coefficient (α=0.017),
the number of sub-cycle iterations in the descending (ν1=30) and
ascending phases of the multigrid cycle (ν2=4), and the threshold for
testing convergence (toler =5×10-5Pas-1). All these
parameters can be adjusted via a name list. The mentioned default values of
α, ν1 and ν2 have been optimised for a 25 km
grid resolution. At lower resolution, α can be increased and ν1
and ν2 reduced to speed up the algorithm.
In the Zwack–Okossi equation, geostrophic vorticity tendencies are converted
to geopotential height tendencies using Eq. (), which is a
2-D Poisson's equation. To solve this equation computationally
effectively, we utilise Intel's MKL (Math Kernel Library) fast Poisson solver
routines, which employ the DFT (discrete Fourier transform) method. Intel's
MKL is widely available and free to download, although registration is
required.
Boundary conditions
In the omega equation, homogeneous boundary conditions (ω=0) at both
the meridional and the lower and the upper boundaries are used. For the
Zwack–Okossi equation, a slightly more complicated procedure is used to
ensure that the area means of the individual height tendency components are
consistent with the corresponding temperature tendencies. First, for all of
V, T, F, Q and A, the height tendency is initially solved from
Eq. () using homogenous boundary conditions
(∂Z∂t=0) at the northern and southern
boundaries. Then, the area mean temperature tendencies for these five terms
are calculated, taking into account both the direct effect of temperature
advection (for T) and diabatic heating (for Q) and the adiabatic
warming–cooling associated with the corresponding omega component. These
temperature tendencies are substituted to the hypsometric equation to
calculate the corresponding area mean height tendencies. In the vertical
integration of the hypsometric equation, the area mean height tendency at the
lower boundary (1000 hPa) is set to 0, following the expectation
that the total atmospheric mass in the model domain is conserved.
Horizontally homogeneous adjustments are then made to the initial height
tendencies for V, T, F, Q and A, forcing their area means to agree with
those derived from the hypsometric equation.
The WRF set-up
WRF is a non-hydrostatic model and can generate atmospheric simulations using
real data or idealised configurations . The
calculations presented in this paper used input data from an idealised moist
baroclinic wave simulation, which simulates the evolution of a baroclinic
wave within a baroclinically unstable jet in the Northern Hemisphere under
the f-plane approximation . The value of the
Coriolis parameter was set to 10-4s-1 in the whole model
domain.
The simulation presented in Sects. 6–7 was run for 10 days with a 30 min
output interval, in a domain of 4000 km×8000 km×16 km, with 25 km
horizontal grid spacing. The horizontal grid was Cartesian and staggered, and
64σ levels were used in the vertical direction. The boundary
conditions were symmetric in the meridional direction and periodic in the
zonal direction. Cloud microphysics is parameterised using the WRF single-moment 3-class scheme . Cumulus convection is
treated with the Kain–Fritsch scheme and boundary
layer turbulence with the YSU scheme . The radiation scheme was switched off in our model integration.
After running the simulation, data were interpolated from model levels to 19
evenly spaced pressure levels (1000, 950, …, 100 hPa). The
interpolation was done with WRF utility wrf_interp, which is freely
available from the WRF website. During the interpolation, the horizontal data
grid was unstaggered to mass points, thus having 160 grid points in the zonal
and 320 grid points in the meridional direction.
The model output data contained all the variables needed in solving the
generalised omega equation and the Zwack–Okossi equation: temperature, wind
components, geopotential height, surface pressure, and parameterised diabatic
heating and friction components. Diabatic heating and friction in WRF
included contributions from various physical processes, such as cumulus
convection, boundary layer physics and microphysics. The physical tendencies
are not in the default WRF output, and need to be added by modifying the WRF
registry file.
To study the performance of OZO at a resolution that is more similar to that
used in several earlier diagnostic studies of synoptic-scale dynamics
,
WRF was also run in the same domain at 100 km grid spacing. Results for this
simulation are presented in the supplementary material. An intermediate
simulation at 50 km resolution was also made to study the resolution
dependence of the computational performance.
Computational aspects
OZO can be run on a basic laptop with Linux environment, provided that
standard NetCDF library, Intel's MKL and some Fortran compiler, preferably
GNU's gfortran, are available. The source code of OZO is written in Fortran
90 standard and can be currently compiled only for a serial version.
The inversion of the left-hand side operator of the omega equation
(Eq. ) is computationally quite a heavy process. Hence, in
our previously described test runs, the calculation of the five plus two
ω components took almost all of the total computing time. For the
height tendencies, the inversion of the horizontal Laplacian
(Eq. ) is much more straightforward to do and, moreover, MKL
fast Poisson solver routines are employed. For that reason, only a small
fraction of the computing time is spent for the height tendency calculation.
Table 2 provides information on the absolute computing times and how they
depend on the number of grid points in the model domain. It is our goal to
improve the computational performance by utilising better scalable solver
routines for the omega equation in a future version of the software.
The dependence of computing time on model resolution. Note that the numbers of grid points are per vertical level, whereas the computing times are per the whole 3-D domain.
DXNumber of grid pointsAverage computing timeper vertical levelper timestep100 km32003.1 s50 km12 8001.1 min25 km51 20019 minResults – vertical motionComparison between calculated and WRF-simulated vertical motions
(a) The sum of all ω components from
Eq. () (ωTOT), (b)ω from WRF
(ωWRF), (c) difference
(ωTOT-ωWRF) and (d) solution of the QG
omega equation (ωV(QG)+ωT(QG)) at 700 hPa
level at time 118 h. Unit is Pas-1. Contours show geopotential
height at 900 hPa level with an interval of 50 m. Labels on x and
y axes indicate grid point numbers. Note that the area covers only half of
the model domain in the meridional direction.
Figure compares the solution of the generalised omega
equation (ωTOT) with ω as being obtained directly from the
WRF output (ωWRF), at the 700 hPa level after 118 h
of simulation when the cyclone is approaching its maximum intensity. The
agreement is excellent. A strong maximum of rising motion (ω≈-3Pas-1) occurs near the occlusion point to the east of the
surface low in both cases, with a somewhat weaker ascent along the cold frontal
zone to the south-west and in the north-eastern sector of the low. Descent
prevails further east of the low and behind the cold front. However, lots of
mesoscale details are visible in both the simulated and the calculated
ω, although with some not perfect agreement between these two. The
difference between ωTOT and ωWRF
(Fig. c) is noisy, although it suggests slightly more
discrepancy behind the cold front where shallow convection takes place.
The QG omega equation (Eq. ) also captures the large-scale
patterns of rising and sinking motion reasonably well
(Fig. d). However, it substantially underestimates the ascent
to the east of the low and along the cold front, and there are two bands of
strong descent (behind the cold front and to the east of the low) that are
much weaker in ωWRF and ωTOT. Furthermore, many
of the mesoscale details shared by ωWRF and
ωTOT are missing in the QG solution.
The majority of the differences between ωTOT and
ωWRF most likely result from numerical errors. One source of
these errors is the approximation of the time derivatives in FA
in Eq. () with central differences. As shown in Fig. , the agreement between ωTOT and
ωWRF gradually improves when the half-span of these time
differences is reduced from 2 h to 1 min (i.e. the time step in
the WRF simulation). The 30 min output interval used for the other
calculations shown in this paper is a compromise between accuracy and the
need to control the data volume of the WRF output for the 10-day simulation.
The difference ωTOT-ωWRF at 700 hPa
level at time 118 h at (a) 2 h, (b) 1 h,
(c) 30 min and (d) 1 min time resolution in the
computation of the imbalance term. Unit is Pas-1.
A more comprehensive statistical evaluation of the calculated vertical
motions is given in Figs. and , by using
data from the whole model area and the 8 last days of the simulation. The
first 2 days, when both the cyclone and the vertical motions are still
weak, are neglected. Figure shows the time-averaged
spatial correlation between ωWRF and various omega equation
solutions. The correlation between ωTOT (line VTFQA)
and ωWRF is excellent, reaching 0.95 in the mid-troposphere and
exceeding 0.88 at all levels from 250 to 850 hPa. However, leaving
out ωA, which requires non-synoptic information from the
time derivatives of temperature and vorticity, deteriorates the correlation
substantially (line VTFQ). The solution of the QG omega equation
only correlates with ωWRF at r∼ 0.6 in the
mid-troposphere (line VT(QG)), although the correlation approaches 0.65 at
300 hPa where diabatic heating is less important than at lower
levels.
Correlation of the omega equation solutions with
ωWRF. VTFQA =ωTOT, VTFQ=ωTOT-ωA, VT(QG)=ωV(QG)+ωT(QG).
RMS amplitudes of ωWRF, ωTOT, and the
individual ω components from Eq. ().
reported excellent correlations (at most levels, from
0.9 to 0.96) between the vertical motions calculated by the DIONYSOS tool and
their original model output (their Figs. 2a and 3a), for two cases and nine
synoptic times at 3 h intervals for both. These correlations are
similar to or even higher than those shown in Fig. .
However, in these tests the horizontal resolution of DIONYSOS was 100 km,
whereas we used 25 km resolution in our WRF simulation. In fact, the
correlations for OZO are also improved and reach 0.97 in the mid-troposphere
for a 100 km resolution WRF simulation (see the Supplement, Fig. S3).
Naturally, the performance may also depend on the synoptic case studied.
Figure compares the root-mean-square (RMS) amplitudes of
ωWRF and ωTOT. RMS(ωTOT) is
typically about 5 % smaller than RMS(ωWRF), which is
presumably due to the truncation errors that occur when the derivatives in
the forcing terms are approximated with second-order central differences. An
exception to this is the lower troposphere where vertical motions calculated
by OZO(ωTOT) are slightly stronger compared to the direct WRF
output (ωWRF). The RMS amplitudes of the individual ω
components will be discussed in the next subsection.
Contributions of individual forcing terms to ω
Vertical motions induced by individual forcing terms at level
700 hPa at time 118 h. (a)ωV,
(b)ωT, (c)ωF,
(d)ωQ, (e)ωA and
(f)ωTOT. Unit is Pas-1 and contour
lines show 900 hPa geopotential height with 50 m interval.
Figure shows the contributions of the five individual
forcing terms to ω(700hPa) for the situation studied in
Fig. . Vorticity advection and thermal advection both
contribute substantially to the vertical motions
(Fig. a–b), but the maximum of ascent is further east
for ωT (Fig. b) than
ωV (Fig. a). Due to this phase shift,
there is a cancellation between rising motion from ωV and
sinking motion from ωT behind the cold front. This
cancellation effect is well-known and typically occurs just behind the cold
front, where cold advection and increasing cyclonic vorticity advection with
height take place . On the other hand, these
two terms both induce rising motion to the south-east of the centre of the
low. Diabatic heating, which is dominated by latent heat release, strongly
enforces the ascent along the main frontal zones of the cyclone
(Fig. d). Compensating subsidence prevails in the
surrounding areas, except for spots of localised ascent associated with
convective precipitation well behind the cold front. The imbalance term
ωA is remarkably large but noisy. Friction induces ascent
close to the centre of the low and descent around and to the north-east of the
surface high (Fig. c), but its contribution is quite
weak at the 700 hPa level. Figures S1 and S2 in the Supplement
present the actual forcing fields for the vertical motions fields in
Fig. a–e.
In terms of the RMS amplitudes evaluated over the whole model area and the
last 8 days of the simulation, temperature advection makes the largest
contribution to the calculated ω (line T in Fig. ).
Vorticity advection (line V), diabatic heating (line Q) and the imbalance
term (line A) are all similar in magnitude in the mid- and upper
troposphere. Near the surface, the imbalance term has a tendency to
compensate the effects of temperature advection, as their RMS values are even
larger than the RMS of the total vertical motion (line TOT).
RMS(ωQ) peaks at 850 hPa, which is mostly due to
shallow convection behind the cold front (not shown). This peak is visible
also in the RMS(ωTOT). Our sensitivity experiments with
different output intervals suggest that the half-hour temporal resolution is
too sparse for proper estimation of the imbalance term in the presence of
fast-moving convection cells, which causes the over-estimation of total
vertical motion in the lowest troposphere. RMS(ωF) is at its
maximum near the top of the boundary layer at 900 hPa but remains
weak even at this level.
These results are partly consistent with similar calculations made for
observation-based analysis data and for model data
at lower spatial resolution. However, the
imbalance term was relatively small in the study by Räisänen
(), because the 6 h time resolution of its
input data was not sufficient for a proper estimation of this term. In
addition, RMS(ωT) is more dominant in our study than in
and in . The main reason
for this is the improved horizontal resolution.
found that the influence of thermal advection compared to vorticity advection
increases when the horizontal resolution is improved. This is because
vertical motions induced by thermal advection typically have a smaller
horizontal scale than those induced by vorticity advection
. In this paper, the horizontal resolution is almost
10 times higher than in the study by Räisänen ().
Conversely, RMS(ωF) is smaller in Fig.
than found for the midlatitudes in and
. This may be at least in part because
and included both land
and sea areas, whereas our WRF simulation was made for an idealised ocean
surface.
A further division of ωV and ωT to contributions from
advection by rotational and divergent wind reveals that they both are largely
dominated by the rotational wind (not shown).
Results – height tendencyComparison between calculated and WRF-simulated height tendencies
In this subsection, the calculated total height tendencies are compared with
height tendencies from the WRF simulation. The latter were estimated as
central differences from the 30 min time series of the simulated
geopotential heights.
As Fig. , but for the height tendency at 900 hPa.
Unit is mh-1.
Figure shows the distributions of the calculated height
tendency, the WRF height tendency and their difference slightly before the
cyclone reaches its maximum intensity (t=118 h). The values are shown at
the 900 hPa level, which is sufficiently low to represent the
processes affecting the low-level cyclogenesis. Negative (positive) height
tendency over the low centre indicates deepening (weakening) of the low. In
general, a very close agreement between the fields is seen, although a somewhat
more positive bias is visible behind the cold front.
Time mean (a) rms error and (b) spatial
correlation coefficient between the calculated and WRF height tendency over
the last 8 days of the 10-day simulation.
Figure shows the time-averaged RMS error and correlation
coefficient between the calculated and WRF height tendency as a function of
height. The RMS error is quite constant from 950 hPa up to the
250 hPa level (Fig. a). Above this level, the error
grows rapidly towards the stratosphere. This error growth is accompanied by a
decrease of correlation coefficient at the same altitude. This deterioration
is presumably at least partly due to the 50 hPa vertical resolution
in the OZO, which is too coarse for an adequate representation of
stratospheric dynamics.
The correlation between the calculated and WRF height tendency is highest in
the upper troposphere, which is roughly 0.97. The correlation weakens
slightly closer the surface, but still exceeds 0.95. Thus, the calculated
height tendency is generally in very good agreement with the tendency
diagnosed directly from the WRF output. These correlations are comparable but
mostly slightly higher than those reported by for
DIONYSOS (their Fig. 6a).
Contributions of individual terms during the mature stage
As Fig. , but for height tendency components
at 900 hPa level. Unit is mh-1. Note that the colour scale of
(f) differs from Fig. a.
The contributions of the individual height tendency components at the
900 hPa level at 118 h are shown in Fig. .
Vorticity advection (Fig. a) produces a wide and strongly
positive height tendency behind the surface low (see Sect. 7.4 for further
analysis of this term). Thermal advection (Fig. b) causes
a positive height tendency in the area behind the surface low and a negative
height tendency at the opposite side. This large negative height tendency
ahead of the low is caused by warm air advection in the mid- and upper
troposphere. In this baroclinic life cycle simulation, thermal advection is
the main contributor to the movement of the cyclone, which is in agreement
with the study of .
Friction (Fig. c) always acts to damp synoptic-scale
weather systems and is thereby inducing a positive (negative) height tendency
over the surface low (high). Diabatic heating (Fig. d) is
causing uniformly negative lower tropospheric height tendencies in the
vicinity of the surface low. The largest negative height tendency due to
diabatic heating is located south-east from the low centre, where strong
latent heat release occurs in connection with frontal precipitation.
The imbalance term (Fig. e) shows more small-scale
structure than the other terms. In general, however, it is in phase with the
total height tendency near the centre of the low, with negative values to the
east and positive values to the west. The reason for this feature is most
probably the following. The conversion from geostrophic vorticity tendencies
to height tendencies for the other terms was done by assuming a geostrophic
balance according to Eq. (). However, in cyclones the wind
is typically sub-geostrophic e.g..
Therefore, the tendency of geostrophic vorticity exceeds the actual vorticity
tendency. This implies that height tendencies calculated from the actual
vorticity tendency under the geostrophic assumption will be too small. The
imbalance term takes care of this and makes the calculated total height
tendency to correspond better to the actual change of the geopotential height
field.
Time series of individual height tendency components at the 900 hPa
level from the cyclone centre during the deepening period. V is vorticity
advection, T is thermal advection, F is friction, Q is diabatic
heating, A is imbalance term, Tot is total and WRF is WRF height
tendency. Height tendencies at the cyclone centre were averaged over all grid
boxes in which the 900 hPa geopotential height was less than 5 m above its
minimum. In addition, the values are 2.5 h moving averages.
Height tendencies in the cyclone centre
Figure shows the 900 hPa level height tendencies
induced by the five individual terms in the cyclone centre during the
deepening period. The low deepens vigorously roughly between 72 and 120 h of
simulation, as shown by the negative total height tendency (black line)
during this period. The total height tendency is also in a good agreement
with the WRF height tendency (dotted line). The deepening is mostly due to
vorticity advection (blue line) and the imbalance term (grey line). Later on,
roughly from 96 h onward, diabatic heating (red line) and thermal advection
(orange line) together with the imbalance term make the largest contributions
to maintaining the intensity of the surface low. Friction (green line)
systematically destroys cyclonic vorticity over the cyclone centre and thus
produces a slightly positive height tendency during the whole life cycle.
After circa 96 h, vorticity advection also acts a damping mechanism for the
surface low (see also the next subsection).
The effect of vorticity and thermal advection by divergent and non-divergent winds
Height tendency associated with vorticity advection by the
(a) divergent and (b) nondivergent winds at 900 hPa at
time 118 h. Unit is mh-1.
Figure presents the height tendencies associated with
vorticity advection by Vχ and Vψ separately. The
divergent wind vorticity advection causes widespread and strong positive
height tendency over and around the surface low (Fig. a).
According to , the divergent wind transports
anticyclonic vorticity from the surroundings of the surface low, and is thus
acting to reduce the cyclonic vorticity at the centre of the low. In the case
of the non-divergent wind component (Fig. b), positive
(negative) height tendencies behind (ahead of) the low originate from the
upper troposphere, where the non-divergent wind and thereby vorticity
advection is the strongest. Cyclonic vorticity advection ahead of the trough
produces negative height tendency in the same area, while anticyclonic
vorticity advection ahead of the ridge does the opposite. Furthermore, the
non-divergent vorticity advection is substantially contributing to the
deepening of the cyclone, since the area of the negative height tendency
reaches the centre of the low as well.
In contrast to vorticity advection, thermal advection by divergent winds was
found to cause a negligible height tendency (not shown).
Limitations and plans for future developments
The idealised baroclinic wave simulation provides an effective and widely
used tool for studying cyclone dynamics in an easily controlled model
environment. For this reason, we chose to begin the development work of OZO
from a relatively simple Cartesian implementation. Nevertheless, the
idealised Cartesian geometry obviously reduces the number of potential
applications of OZO. We aim to extend OZO to more complex spherical grid
applications in the future. However, in principle the use of OZO is possible
with some limited-area real cases where spherical geometry has been used, if
the data are regridded to Cartesian geometry afterwards. However, in this
case, one must change the lateral boundary conditions of OZO since they are
tailored for periodic model domain by default.
Another limiting factor in the current version of OZO is the weak scalability
in the multigrid omega equation solver together with the lack of
parallelisation of the source code. For high-resolution runs, this means
significant slowing of the calculations (Table 2). To reduce the lengthy
computing times, we are currently developing a parallel version of OZO, which
uses a different solving method for the omega equation. We aim to release
this parallel version by the end of the year 2017.
An issue associated with the physical interpretation of high-resolution
simulations should also be noted. In OZO, the vertical motion and height
tendency contributions of vorticity advection, thermal advection, friction,
and diabatic heating are calculated assuming geostrophic balance between the
vorticity and temperature tendencies, while the deviations from this balance
are attributed to the imbalance term. As the balance assumption is
increasingly violated at higher resolution, the imbalance term grows larger.
This complicates the interpretation of the results particularly when the
imbalance term opposes the other terms. Such compensation is indeed apparent
in our results for vertical motions in the lower troposphere, where the
contributions of thermal advection and diabatic heating are strongly opposed
by the imbalance term (Fig. ). At resolutions much higher
than 25 km, issues like this are likely to become increasingly problematic.
Conclusions
In this paper, a software package called OZO is introduced. OZO is a tool for
investigating the physical and dynamical factors that affect atmospheric
vertical motions and geopotential height tendencies, tailored for WRF
simulations with idealised Cartesian geometry. As input to OZO, the output of
the WRF model interpolated to evenly spaced pressure levels is required.
The generalised omega equation diagnoses the contributions of different
physical and dynamical processes to vertical motions: vorticity advection,
thermal advection, friction, diabatic heating, and imbalance between
temperature and vorticity tendencies. Then, analogously with the vertical
motion, the height tendencies associated with these forcings are calculated.
As an advance over traditional applications of the Zwack–Okossi equation
, the use of the generalised omega
equation allows OZO to eliminate vertical motion as an independent forcing in
the calculation of height tendencies.
The calculated total vertical motions and height tendencies in the test case
are generally in excellent agreement with the vertical motions and height
tendencies diagnosed directly from the WRF simulations. The time-averaged
correlation between the calculated and the WRF height tendency was 0.95–0.97
in the troposphere. For the vertical motion as well, a correlation of 0.95
was found in the mid-troposphere. Our analysis further illustrates the
importance of both adiabatic and diabatic processes to atmospheric vertical
motions and the development of the simulated cyclone.
The OZO software is applicable to different types of WRF simulations, as far
as Cartesian geometry with periodic boundary conditions in the
zonal
direction is used. One example of potential applications are simulations with
increased sea surface temperatures as the lower boundary condition. Combined
with OZO, such simulations provide a simple framework for studying the
changes in cyclone dynamics in a warmer climate.
Data and code availability
The source code of OZO is freely available under MIT licence in GitHub
(https://github.com/mikarant/ozo). OZO v.1.0 described in this
manuscript is also archived at http://doi.org/10.5281/zenodo.157188. In
addition to the source code, the package also includes a makefile for
compiling and running the program, a small sample input dataset for testing
the functionality, and two README files containing the instructions for both
generating input data with WRF and running the OZO program. The WRF model as
well as the interpolation utility (wrf_interp) are downloadable
from the WRF users page
(http://www2.mmm.ucar.edu/wrf/users/downloads.html). Intel's MKL can be
downloaded after registration from their web page
(https://software.intel.com/en-us/articles/free-mkl). OZO v1.0 is
guaranteed to work with WRF version 3.8.1.
The Supplement related to this article is available online at doi:10.5194/gmd-10-827-2017-supplement.
The authors declare that they have no conflict of
interest.
Acknowledgements
We thank the Doctoral Programme of Atmospheric Sciences, University of
Helsinki for financially supporting the work of M. Rantanen. The work of O.
Stepanyuk was supported by the Maj and Thor Nessling foundation (project
201600119) and the work of O. Räty was supported by the Vilho, Yrjö, and
Kalle Väisälä Foundation. V. A. Sinclair was supported by the Academy
of Finland Centre of Excellence Program (grant
272041).Edited by: D. Ham
Reviewed by: two anonymous referees
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