ACPAtmospheric Chemistry and PhysicsACPAtmos. Chem. Phys.1680-7324Copernicus PublicationsGöttingen, Germany10.5194/acp-18-6971-2018A novel method for the extraction of local gravity wave parameters from gridded three-dimensional data: description,
validation, and applicationDiagnosis of local gravity wave propertiesSchoonLenaschoon@iap-kborn.deZülickeChristophLeibniz Institute of Atmospheric Physics, Theory and Modelling Department, Schlossstrasse 6, 18225 Kühlungsborn, GermanyLena Schoon (schoon@iap-kborn.de)17May20181896971698318May201730May201725April201825April2018This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://acp.copernicus.org/articles/18/6971/2018/acp-18-6971-2018.htmlThe full text article is available as a PDF file from https://acp.copernicus.org/articles/18/6971/2018/acp-18-6971-2018.pdf
For the local diagnosis of wave properties, we develop, validate, and apply a
novel method which is based on the Hilbert transform. It is called Unified
Wave Diagnostics (UWaDi). It provides the wave amplitude and
three-dimensional wave number at any grid point for gridded three-dimensional
data. UWaDi is validated for a synthetic test case comprising two different
wave packets. In comparison with other methods, the performance of UWaDi is
very good with respect to wave properties and their location. For a first
practical application of UWaDi, a minor sudden stratospheric warming on 30
January 2016 is chosen. Specifying the diagnostics for hydrostatic
inertia–gravity waves in analyses from the European Centre for Medium-Range
Weather Forecasts, we detect the local occurrence of gravity waves throughout the
middle atmosphere. The local wave characteristics are discussed in terms of
vertical propagation using the diagnosed local amplitudes and wave numbers.
We also note some hints on local inertia–gravity wave generation by the
stratospheric jet from the detection of shallow slow waves in the vicinity of
its exit region.
Introduction
The importance of gravity waves (GWs) for the dynamics of the Earth's
atmosphere is without controversy. They influence dynamics from planetary
scales to turbulent microscales and play an important role in the middle
atmosphere circulation . GWs typically appear as
packets localised in space and time. Hence, it is desirable to diagnose them
locally as precisely as possible. Here, we want to introduce a new method
called
Unified Wave Diagnosis (UWaDi). It provides phase-independent local wave
quantities like amplitude and wave number without any prior assumption. In
the following, we develop, validate, and apply this novel method. The
application concentrates on the analysis of GWs for locally varying
background wind conditions in the winter 2015–2016.
In the past, several methods were developed to estimate wave properties like
amplitudes and wave number vectors. All of them have to deal with the fact
that the data sampling procedure influences the results. A common approach to
obtaining the vertical wave numbers and GW frequency of high-pass-filtered wind
fluctuations is a method involving the Stokes parameters . This method is based
on polarisation relations and works for single-column measurements. It
provides the wave properties in preselected vertical height sections of
finite lengths. Next to its original application to radar measurements, it is
used for radiosonde data . A supplement to this method
called DIV was introduced by . It determines the dominating
harmonic wave in a box from the first zero crossing of the autocorrelation
function. The maximal detectable wavelength is restricted by the box size.
The analysed quantity is the horizontal divergence to get the ageostrophic
flow without numerical filtering. A further technique is based on sinusoidal
few wave fits (S3-D) . This method was created for the
analysis of three-dimensional data from remote sensing observations
but is also applicable to model data
. The first modes with the highest variance are taken from a
fit that minimises the variance-weighted squared deviations over all points
in a finite analysis box. Only a small number of sinusoidal curves are fitted
and there might remain uncovered variances in the analysis volume. These
methods have in common that the analysed spatial scales are dependent on the
predefined analysis box size and the assumption of the spatial homogeneity of the
wave field in these boxes is essential. Nevertheless, these methods are
superior to a classic Fourier transform in that they allow us to
search for waves with bigger wavelengths than the box size.
Another three-dimensional spectral analysis method is the 3D
Stockwell transform (3-D ST) . This method is capable of
analysing the full range of length scales sampled in satellite data and is
not restricted to fixed box sizes. At every grid point, a local wave spectrum
is estimated using a window function of frequency-dependent width. S3-D and 3-D ST assume homogeneity inside the analysis box. Furthermore, they use a small
number of the most prominent waves for the estimation of variances. This
leaves some variance unattributed and hence means a loss of information. We
search for a method which detects the full variance in each data point.
With UWaDi we find the dominating wave with the Hilbert transform at every
data point. This approach does not rely on choosing the size of an analysis
volume aforehand. The calculation of wave quantities at every grid point is
computationally cheap. There is no need to assume homogeneity and no
restriction on detectable wavelengths besides the Nyquist wavelength. Here,
the method is developed to work with three-dimensional equally gridded data.
In general, the Hilbert transform can be applied to data of any
dimensionality. Wave properties such as the amplitude and wave number are
estimated phase independently, while all variance is attributed to one wave
mode. Every variable including any kind of wave-like structure can be
diagnosed. used the method to obtain the envelope of a train
of Rossby waves in one dimension. A supplement was made for waves not in line
with grids by an extension of the formulation to stream lines to obtain
quasi-one-dimensional wave packets .
provide a three-dimensional application to Rossby waves and GWs.
use the Hilbert transform to identify Rossby wave trains on a longitude–time
plane and introduce their approach as an “objective identification method”.
Our method focuses on the local site of GW occurrence and the additional
provision of the wave number in every dimension. The latter was not presented
before. We aim to cover the retrieval of local wave properties from
arbitrarily orientated wave packets. Amplitude and wave number are returned
on the same grid as the input data. After the mathematical description of the
method and how it is implemented, it will be validated with synthetic data to
demonstrate its quality in comparison with other methods.
For a demonstration of a practical application in a geophysical context, we
will investigate GWs. Their sources are usually found in the troposphere
where waves are generated by flow over orography, by convection, frontal
systems, and jet imbalances. These waves propagate upwards with increasing
amplitudes and break in the middle atmosphere where they deposit their
momentum to the background flow. Strong influence is exerted on global
circulation patterns in the mesosphere and in the stratosphere
. GWs play crucial roles in modulating
the quasi-biennial oscillation (QBO) and the Brewer–Dobson circulation
. Another stratospheric
phenomenon in which GWs play a role is a sudden stratospheric warming (SSW). For
this phenomenon, a variety of definitions exist , but the
most common one is given by the World Meteorological Organization stating
that an SSW is characterised by a reversal of the 60 to 90∘ N temperature gradient. Major warmings are associated with a wind reversal at
10 hPa and 60∘ N, and minor SSWs (mSSWs) are associated with a wind deceleration
at 10 hPa and 60∘ N, where the prevailing westerlies are not
turned into easterlies. While planetary waves are the most important drivers
of SSWs , GWs are affected by the differing background
wind conditions during SSWs and are suspected to modulate the polar vortex in
the post-warming phase of an SSW . The behaviour of GWs and
planetary waves during an SSW was investigated through simulations and different
measurement techniques. It was found that these GWs are, next to selective
transmission, the subject of variable sources including unbalanced flow
adjustment . We are interested in the
longitude-dependent transmission of GWs during an SSW. Pioneering work was
done by . They analysed model data and found that
the selective transmission of GWs during an SSW is dependent on longitude
according to the planetary wave structures. Therefore, regions where vertical
wave propagation is inhibited exist, as do regions where waves can
propagate up to the mesosphere. The analysis of was
restricted to parameterised GWs of the “intermediate range” that they
defined between 50 and 200 km. They state that it remains
unclear in what way GWs of larger scale will act during SSWs. A study on a
self-generated SSW in a model showed that GWs reverse the circulation in the
mesosphere and lower thermosphere during an SSW by altering the altitude of GW
breaking. This altitude is highly dependent on the specification of GW
momentum flux in the lower atmosphere . In the
UWaDi application, we want to locate the occurrence of GWs precisely in space
and give first interpretations using the information on their changing
amplitude and wave number in local vertical profiles.
The northern winter 2015–2016 brought up several interesting features,
including specific GW patterns. The beginning of the winter was characterised
by an extraordinarily strong and cold polar vortex driven by a deceleration
of planetary waves in November–December 2015 .
Thereinafter, for the end of that winter a record Arctic ozone loss was
expected . Furthermore, the extraordinary polar vortex
caused a southward shift of planetary waves leading to anomalies in the QBO
. A joint field campaign of the research projects
METROSI, GW-LCYCLE2 (both part of ROMIC), and PACOG (MS-GWaves) took place in
Scandinavia in January 2016. found a summer-like zonal wind
reversal in the upper mesosphere lasting until the end of January 2016,
leading to different GW filtering processes in the mesosphere compared to
usual winter-like wind conditions. During the field campaign the first
tomographic observations of GWs by an infrared limb imager provided a full
three-dimensional picture of a GW packet above Iceland .
Additionally, a remarkable comparative study indicates that forecasts of the
current operational cycle (41r2) of the European Centre for Medium-Range
Weather Forecasts (ECMWF) Integrated Forecast System (IFS) shows good
accordance with space-borne lidar measurements while picturing large-scale
and mesoscale wave structures in polar stratospheric clouds
. We choose the mid-winter of 2016 for a first
application of UWaDi because it is very well sampled with observations of GW
properties. We intend to provide additional impulses to the evaluation of
observations and start with a study of ECMWF analyses. These gridded data are
suitable to analyse the local occurrence and their coupling as the analyses
resolve essential parts of GW dynamics in the stratosphere. Validation
studies with satellite measurements point out that ECMWF analyses capture GWs
well in the middle and high latitudes .
Mid-latitude GWs are captured especially well, being driven by orographic and
jet-stream-associated sources . Our approach
concentrates on fields of horizontal divergence in ECMWF IFS data. By
choosing horizontal divergence as the analysis variable, the calculation
of residuals is omitted. Hence, an explicit separation of a slowly varying
background wind, which may include further uncertainties, is avoided. The
horizontal divergence is a dynamical indicator for GWs because it is free
from geostrophically balanced structures .
Studying mountain waves, and have
shown that divergence values correlate with GW-induced temperature
anomalies. Furthermore, we use horizontal divergence to derive the total
wave energy in Appendix from the polarisation relations. In
this study we concentrate on vertical profiles of GW occurrence to give a
first impression of the functionality of this method.
Recently, the importance of the oblique propagation of GWs in a general context was discussed by
, , and
. We point out that the meridional propagation of GWs is important for the deposition of GW drag from the stratosphere up into
the mesosphere. For instance, discuss the role of oblique propagation for the redistribution of selective transmission of
GWs in the upper troposphere and lower stratosphere. These are phenomena which require a detailed analysis of the propagation of localised wave packets.
Here, the authors focus on the introduction of the novel method and give
first preliminary scientific results from a demonstrative application. We
show locally diagnosed GW properties and give some hints on physical
interpretation. A full three-dimensional spatial–temporal analysis of GWs
during the SSW 2015–2016 goes beyond the scope of this paper and will be made
the subject of subsequent publications.
The paper is organised as follows. After providing a step-by-step
introduction and validation of the novel method in Sect. , we
give a short overview of the estimation of wave quantities for synthetic data
and describe the analysis data. In Sect. we show our results
for the application of the mSSW on 30 January 2016 for which we study
longitude-dependent GW occurrence. The discussion of our results in
Sect. is followed by the “Summary and conclusion” section
(Sect. ).
Method and data
In this section we develop and validate an algorithm to
extract wave parameters from gridded three-dimensional data. For the local
diagnosis of waves, phase-independent estimates of wave amplitudes and
the wave vector are essential. For this, we employ the Hilbert transform
e.g.. The Hilbert transform shifts any sinusoidal
wave structure by a quarter phase, i.e. turning a sine into a cosine. By
constructing a new complex-valued data series consisting of the original
field as the real part and its Hilbert transform as the imaginary part, the
absolute value is always the amplitude (square root of squared real and
imaginary part). The amplitude is independent of the phase of the wave and
the wavelength of the underlying oscillation and there is no need for any
explicit fitting of a particular wave. In addition, the absolute wave number
in all three dimensions is determined from the phase gradient. The only
requirement for the method to work is that the data contain any harmonic
components. Thus, it works universally for any given variable, and hence it was
called Unified Wave Diagnostics.
Step-by-step outline of the method
In the following we introduce UWaDi in a step-by-step outline. Further, we
validate it with a well-defined test wave packet in comparison with other
methods. In general, UWaDi is a script package which allows the user to steer
data preprocessing, the main wave analysis, and data plotting from a set of
namelists.
UWaDi requires data from equidistant grids. For the ECMWF analyses on a longitude–latitude grid, the latitude dependence of grid
distance is taken into account by determining the longitudinal grid distance by dx=2πrθγ360 where rθ=Rcos(θ)
is the latitude-dependent Earth radius (R=6371km, Earth radius; θ, latitude in ∘) and γ denotes the resolution of the
gridded input data (e.g. γ=0.36∘).
To retrieve vertical equidistant levels, the hybrid levels of the ECMWF data are firstly transformed to pressure levels. Secondly,
these pressure levels are assigned to equidistant height levels. For this purpose, we assume hydrostatic conditions and consider the surface
geopotential and pressure as well as temperature and humidity. Both steps are performed with the help of common functions provided in the
NCAR command language (NCL). This might cause problems in areas of high orography and inside the planetary boundary layer. These areas are
not considered in the following analysis.
The method can handle any kind of variable. For the present application, we choose horizontal divergence. While this
quantity was available in the ECMWF analyses, other data sources might require its calculation from the wind fields. However, the required
preprocessing of the target variable is done in this step.
The underlying Hilbert transform is implemented with a discrete Fourier transform (DFT), which creates a complex
spectrum in wave number space from the real valued data in real space (e.g. ). The processing of the three-dimensional data
in the (x,y,z) space is begun with the x direction. The mathematics behind the Hilbert transform is described briefly for a one-dimensional
function fx originating in real space:
fk=DFT(fx).
The index k denotes the wave number space, and x describes the function f in real space.
DFTs can be biased by variance leakage through side lobes in spectral space. Tapering methods abandon this but can smear out nearby
wave numbers. A loss of absolute amplitude can be overcome by using normalised weights e.g.. For the present study,
however, the best results were obtained by turning the taper off.
In wave number space a rectangular bandpass filter reduces the complex spectrum to the user-predefined wave number limits kmin
and kmax. Here, we make sure that only waves of the considered range of wave numbers are used for the following analysis.
fk,filtered=F(kmin,kmax)fk
To get back from wave number space an inverse DFT is
performed.
fx^=2DFT-1(fk,filtered)
The constructed complex-valued function fx^ consists of the input data fx as the real part and the Hilbert transformed
function H(fx) as the imaginary part:
fx^=fx+iH(fx).
It provides the amplitude ax,
ax=|fx^|=fx2+H(fx)2,
and the phase estimate Φx,
Φx=atanH(fx)fx.
The phase gradient is a measure of the wave number
modulus:
kx=dΦxdx≈DFT-1(kDFTfx^)|fx^|.
Due to the finite character of the data series it may happen that high-frequency spurious fluctuations appear after the
Hilbert transform. We damp them by applying a low-pass filter. We smooth over a number of grid points determined by the lower wave number limit kmin.
The identification of outliers is taken care of by two different quality checks. Firstly, the amplitude and wave number are
checked for at least a half-undamped wave. Therefore, the packet length lx is essential. It is calculated by correlation function Ri:
lx=Δx∑i=0imaxRi,
with imax=N-15. This method goes back to
. The quality check is then defined by the inequality
kxlx>π.
Secondly, the retrieved signals are required to lie above the noise level of
the input data. An empirical threshold c checks the amplitude for being
valid considering the standard deviation of the input function σ(fx):
ax>cσ(fx).
Empirically, we use c=0.01. This idea follows . UWaDi uses
a quality flag q=1, which is set to false (q=0) if at least one quality
check is rejected.
Steps 4 to 11 are repeated for the other dimensions (y,z).
Amplitude and absolute wave number are saved on the same grid as the input data to create a full three-dimensional analysis of
local wave quantities. The amplitude is combined to a wave-number-weighted sum of the three spatial
dimensions:
a=∑d=x,y,zqdkd2ad2∑d=x,y,zqdkd212.
The absolute wave number is determined by
k=∑d=x,y,zqd×kd212,
with d denoting the spatial index.
The method provides an exact measure of the amplitude in the sense of the sum
of the squared amplitudes of the wave modes. The dominating wave number is the
amplitude-weighted sum of all. Spectrally wide dynamics can cause a
significant reduction of information (Appendix ). Applying UWaDi
with several narrow bandpass limits would provide information on spectrally
spread waves. The wave numbers are estimated as moduli, which means the
three-dimensional wave number (±kx, ±ky, ±kz) allows for
eight possible directions. However, the method is recommended for the first
guess of the dominant wave packet, including the derivation of the intrinsic
frequency from the dispersion relation.
Validation of the method
For a comparison of wave characteristics obtained with different methods we
choose the test case presented in (Fig. a). In
this exercise, a couple of localised wave packets with the wave numbers 4 and
9 are given in one dimension on the interval [0,4π] by
fx=exp-x-4.52cos(4x)+exp-x-7.52cos(9x).
One-dimensional test function (a, bold line) adapted from Zimin et
al. (2003) and its envelope (a, thin line). Comparison of amplitude (b) and
wave number (c) calculated by different methods: UWaDi (solid, red), DIV
(dotted, orange), S1-D (dashed, green), and 1-D ST (dash-dotted, blue). Valid
estimates are drawn in bold.
Here, the quality check (step 11) requires the amplitudes to
exceed half of the sample standard deviation.
UWaDi based on a Hilbert transform is a continuous method working without any
box parameter. 3-D ST, S3-D, and DIV need box width parameters to be adapted to
the corresponding scientific case. A compromise between accuracy in space and
wave number has to be found. For DIV the box length is set to
LDIV=8.0,
which covers the largest anticipated wavelength. For 1-D ST, which was
modified from 3-D ST, two steering parameters have to be adapted to the
present task. We find a box width factor of CL1DST=0.25 suiting our
requirements. Next to that, 1-D ST provides a correction factor for the
absolute amplitude value CA1DST. In three-dimensional data analyses
enormous amplitude reductions can occur. Artificially changing the amplitude
impacts the variance conservation of the technique. This is why we choose
CA1DST=1 for our example. For S1-D a fixed box length has to be
determined in advance. We find LS1-D=1.5 to give acceptable results. We
note that an extension to three dimensions would add information and
therefore accuracy. However, in order to realise comparability we stay with
the strictly one-dimensional set-up.
The method showing the best agreement
with the theoretical value is UWaDi (Fig. b). For the amplitude,
both wave packets are clearly distinguishable and the maximum peaks are
recovered exactly. 1-D ST and S1-D rebuild the wave packet shape as well. The
lack of an absolute amplitude value might be adjusted with empirical correction
factors provided for 1-D ST. Nevertheless, the amplitudes of both wave packets
differ from each other. DIV provides a smeared out wave-packet envelope. The
wave number calculation is best for UWaDi (Fig. c). The high peaks
at the beginning and end of the wave packets are sorted out by the quality
check. DIV meets the right wave numbers as well, but does not cover the whole
spatial range of the two wave packets. 1-D ST and S1-D show small deviations
from the expected values. Altogether, UWaDi shows the best agreement with
the theoretical expectations.
Analysis data
ECMWF data from the IFS operational cycle 41r1 is chosen for this analysis.
We performed comparison studies between IFS data on different grid sizes
(0.1, 0.36, 1∘). By considering our bandpass filter
conditions we found reliable results for the 0.1 and
0.36∘ grids. To compromise between computational costs and the stability
of the results we decided that data with a resolution of 0.36∘ (ca. 40 km) meet our requirements. They are retrieved from a resolution of
T511. We discuss resolved gravity waves of a horizontal scale between 100 and 1500 km. In the vertical direction we are interested in
gravity waves within the wavelength limits of 1 to 15 km.
These scales fulfil the assumption of hydrostatics and cover the range of
mid- and low-frequency GWs .
Vertical-propagating GWs are damped in ECMWF IFS products from 10 hPa
(≈ 30 km) upwards . At 10 hPa the
stratospheric sponge starts and a damping of wave propagation is expected
. The mesospheric sponge follows at 1 hPa,
acting on the divergence and therefore directly on the GW properties. We
restrict our analysis to a maximum altitude of 45 km and therefore
follow the advice of . We interpolate model levels to
equidistant height levels between 2 and 45 km with a distance
of 500 m and provide initial scientific analysis for a snapshot on 30
January 2016 at 00:00 UTC, corresponding to a minor SSW.
Gravity-wave-specific quantities
From the diagnosed fields of amplitude and wave number we calculate the
kinematic wave energy e and wave action A. In order to find the
ageostrophic GW motion we analyse fields of horizontal divergence. The
kinematic wave energy is derived from polarisation equations for GWs assuming
hydrostatics (Appendix ):
e=δ2kh2.
In this formula we need information on the divergence variance and the
horizontal wave number. Both are provided by UWaDi from the three-dimensional
divergence field.
δ2=a22kh2=kx2+ky2
The wave action is a conserved quantity as long as the slowly varying wave
packet does not interact with the mean flow, e.g. by dissipation, absorption,
or breaking . Wave action is defined by putting the
kinematic wave energy e in relation to the intrinsic (flow-relative)
frequency ω^:
A=ρeω^,
with ρ being the density. The intrinsic frequency ω^ is
calculated with the dispersion relation in mid- and low-frequency
approximation:
ω^2=f2+N2kx2+ky2kz2. From A=ρeω^= constant, one can see the following.
Density effect: e∝1ρ∝expzH. The above derived energy undergoes an exponential
increase according to the density with the scale height H in vertical direction z.
Wind effect: e∝uh. This relation holds for a stationary horizontally homogeneous mean flow
(u(z),v(z)), which implies
the invariance of the horizontal wave number (kx= const and ky= const) along with the apparent (ground-based) frequency
(ω=ω^+kxu+kyv=ω^+khuhcos(αk-αu)=const). The energy scaling is obtained with the
invariance of the wave action for an upwind wave (αk→αu+π) as e=Aρω^→Aρω+khuh due to the Doppler shift of the intrinsic frequency .
For the following analysis primarily wave action is used.
Results
A minor SSW occurred on 30 January 2016. Figure a
shows the wind speed of the Northern Hemisphere at 10 hPa at 00:00 UTC. A
vortex displacement from the pole is visible. The displaced vortex causes
areas of strongly curved winds. The horizontal divergence as a measure of GWs
shows high wave activity above two areas (Fig. b). Firstly, above
northern Europe horizontal divergence is aligned cross-stream. Secondly,
spiral-like patterns appear above eastern Siberia, corresponding to an area
of a curved jet streak. UWaDi applied to the field of horizontal divergence
provides GW amplitude and wave action (Fig. c, d). GW amplitudes
show patterns in regions of strongly alternating horizontal divergence. The
wave action shows the highest peak above northern Europe and lower values
above eastern Siberia.
Synoptical situation of the Northern Hemisphere from ECMWF analysis
at 10 hPa on 30 January 2016. Wind speed (a) and horizontal divergence (b).
Gravity wave amplitude (c) and wave action (d). Circled numbers along
60∘ N latitude indicate the positions of three vertical profiles for
later analysis.
In zonal mean the horizontal wavelength varies between 120 and 200 km (Fig a). In the mid-stratosphere between 18
and 40 km of altitude the horizontal wavelength remains nearly constant.
Thus, our assumption of a homogeneous background wind field is approximately
valid in the mid-stratosphere. The vertical wavelength scales from 2.2 to 5.2 km. It increases throughout the whole atmospheric
section with a slight change in gradient at the altitude of the tropopause
(10 km). The decrease in vertical wavelength above 35 km
of altitude is dubious. It occurs frequently, also for other temporal snapshots.
We suspect an influence of the IFS sponge layer, but do not exclude an
influence of the decreasing zonal wind, and therefore remain critical on
interpretations at these altitudes. The horizontal phase speed in the wave
direction
(ch=ωkh=ω^kh+uhcos(αk-αu))
is approximated for an upwind wave (ch→c^h-uh). It
remains unchanged for waves propagating passively through a stationary
horizontally homogeneous wind field. The zonal mean remains nearly constant
with a value of 5 ms-1 below the tropopause and then steadily
increases in the stratosphere (Fig. b). The indicated variations
in horizontal wave number and phase speed contradict assumptions of a passive
propagation through a stationary horizontally homogeneous wind field.
Zonal mean profiles at 60∘ N with (a) zonal wind (green,
dotted), energy (dark blue, dashed), and wave action (red, solid)
and (b)
apparent horizontal phase speed (pink, dotted), vertical wavelength (light
blue, dashed), and horizontal wavelength (orange, solid).
We next inspect local profiles in different background wind conditions.
Longitude–height sections of zonal wind (Fig. a) and wave action
(Fig. b) at 60∘ N help to find the location of interesting
vertical profiles. Three profiles are chosen that are representative for
regions of similar filter conditions. The first profile
(1) at 7.56∘ E is chosen to be in a longitudinal
range characterised by strong zonal eastward winds and lies in the
deceleration area of the jet stream above northern Europe. Profile
(2) is at 151.92∘ E in the area of a
strongly curved stratospheric jet associated with the displacement of the
polar vortex. In Fig. a it is visible as a wind intrusion in the
altitude range between 14 and 34 km. The wave action shows a
peak in that height range (Fig. b). For comparison we take a third
profile (3) at 240.12∘ E in a region of low wind
speed above Canada, which means weak tropospheric and weak stratospheric jets.
Zonal wind (a) and wave action (b) at 60∘ N on 30 January 2016
in longitude–height section. Numbered vertical profiles for further analysis
are highlighted.
To highlight the advantage of a local wave analysis we show profiles at
selected longitudinal positions (Fig. a).
During a local increase in wind speed above northern Europe the vertical
profiles of (1) show that the zonal wind meanders around
50 ms-1 (Fig. b). In the stratosphere, the vertical
wavelength is nearly constant with an average wavelength of 8 km, which
is higher in the troposphere with 11.5 km (not shown). The wave action
shows a high gradient changing from 104
to 103kgm-1s-1 and remains at this high level above 20 km.
Above eastern Siberia a displaced stratospheric jet streak appears jointly
with high wave action (Fig. ). The zonal wind vertical profile
(2) shows this in a height range of 14 to 30 km with an increase from 5 to maximal 30 ms-1 (Fig. c).
The wave action follows the
structure of the zonal wind. Notably, peak GW activity takes place in the
lower stratosphere, clearly above the tropospheric jet stream. At these
altitudes, GWs with a vertical wavelength of 2 km are found and the
horizontal wavelength is about 350 km (Fig. ).
The
last set of vertical profiles is located in an area of low zonal winds
(3) (Fig. d). In the troposphere eastward
winds and in the middle stratosphere westward winds occur with magnitudes
below 20 ms-1. Above the altitude of the wind reversal the wave
action remains constant.
Vertical profiles at 60∘ N on 30 January 2016. Local vertical
profiles at 7.56∘ E (a), 151.92∘ E (b), and
240.12∘ E (c)
with the wave action (solid, red) and zonal wind (dashed, green). Local
profiles according to markers in Fig. .
Discussion
The topic of selective wave transmission was first modelled by
. They highlighted the longitude-dependent gravity wave
propagation during an SSW by focussing on the impact on the mesosphere.
further point out that selective filtering by
anomalous winds during an SSW create a heavy impact on GW propagation through
the whole atmosphere. They point out theoretically that during the upward
propagation of GWs, these waves get attenuated or eliminated by distinct
specifications of background flows. Here, we compare local vertical profiles
of background wind and GW parameters from analysis data.
Comparing cases (1), (2), and
(3) with respect to their wave action profiles we diagnose
at 42 km values ranging over 3 orders of magnitude. This confirms
the high spatial variability of GWs during SSWs. Also the shapes of the
profiles differ clearly. One class is characterised by a steady decrease
until 25 km and constance above (such as the zonal mean and cases
1 and 3). Another class of profiles
shows a well-expressed peak in the stratosphere and a steady decrease above
(case 2). The detailed analysis-related GW dynamics goes
beyond the scope of this paper. However, some hypotheses are formulated.
In the high-wind case (1), showing the highest values of
wave action and weak in the vertical wave number above northern Europe, we
find the longest vertical wavelength of our study (8 km). These steep
waves hint at an orographic GW caused by the eastward flow above the
Scandinavian mountain ridge Kjølen. This is comparable to the findings of
, who showed that during the SSW 2009 westward-propagating GW packets emanate from key topographical features around the
polar edge and that these wave packets have long vertical wavelengths.
Further detailed analyses of this GW packet are expected in upcoming
publications according to the joint measurement campaign of ROMIC and
MS-GWaves at, amongst others, Kiruna, Sweden (67∘ N, 20∘ E).
In the displaced stratospheric jet case (2)
(Fig. c) we find a GW packet triggered off a curved
stratospheric jet streak. Firstly explained by , jet exit
regions in the troposphere are expected to emit GWs. The increase in wave
action in the middle stratosphere according to the intrusion of westerlies
seen in Fig. a and b leads to the assumption that the present
feature is associated with the stratospheric jet. The horizontal divergence
field supports this hypothesis with GW structures spiraling out of the curved
jet above eastern Siberia. These features are comparable to the simulations
of the troposphere by , which resolved shallow near-inertia
waves in jet exit regions. Further hints are the higher wave action and
the smallest found wavelength (1.9 km) along the largest found
horizontal wavelength (350 km).
Intrinsic frequency (bold, dark blue), horizontal wavelength (dashed, orange), and horizontal phase speed (bold, pink)
for profile (2) (Fig. c).
In the low-wind case (3) the high wave action in the upper
troposphere up to the height of the wind reversal at 23 km might be
caused by orographically induced GWs due to the position in the lee of the Rocky
Mountains. Above that, the overall lowest values of wave action are found,
agreeing with measurements in that height range . It
is interesting to note that the shape of the wave action profile is similar
to the zonal mean profile (compare Figs. a and c), while the wind above 25 km goes into another
direction.
Summary and conclusion
With UWaDi we provide a tool for the analysis of gridded
three-dimensional data to estimate amplitude and wave number
phase independently and locally. The method is based on a Hilbert transform
with which the use of predefined analysis boxes is avoided. It returns an
estimate for each data grid point but stays computationally cheap. With regard
to the locality it clearly shows its advantages in a method comparison for a
synthetic one-dimensional test case. Disadvantages may play a role when the
wave spectrum is broad and the attribution of the variance to one dominant
harmonic is not justified. The additional estimation of the wave numbers
completes the elements of a wave packet description. There is an ambiguity in
the sign of the wave number and in the direction of the wave
vector, which is the case for all spatial analysis methods considered
in this paper. Still, the method is recommended as a reliable local diagnosis
of medium complexity.
For the analysis of gravity waves, we estimated wave energy and wave action
from the horizontal divergence. This approach does not require an explicit
numerical filtering, which is a practical advantage. Other methods for the
analysis of unbalanced flow components are available, although more
complicated . While the chosen formula requires the
variance (or squared amplitude) and wave numbers, the Hilbert transform
method may also provide local estimates for more complex quantities as
included in the combined Rossby wave and gravity wave diagnostics of
. For our study, which is focused on hydrostatic
inertia–GW occurrence, the specific approach provides reliable results at low
complexity.
With the demonstrative analysis of the synoptic situation on 30
January 2016 we show the advantages of UWaDi: providing wave quantities on
every grid point. Longitude-dependent GW filter processes, known as selective
wave transmission, can be diagnosed spatially in detail. Local vertical
profiles show selective wave transmissions and generation processes. We found
cases with a steady decrease in the wave action through the tropopause up to
the mid-stratosphere and constant values above in contrast to a case with a
strong peak in the lower stratosphere and a steady decrease above. The latter
happened in an area where the wind field is affected by the mSSW,
characterised by a curved jet stream exit region in the stratosphere;
we discuss GW generation by spontaneous emission. The diagnosed long horizontal
and short vertical wavelengths support this hypothesis. With the present
method we plan to join the closer evaluation of observations and models with
respect to local features of GW generation and propagation.
All data and documented software are available on request from the
corresponding author. Details on ECMWF data can be found at www.ecmwf.int (ECMWF, 2017).
The ECMWF data used here were accessed through the
ECMWF MARS Archive on 19 October 2017.
Estimates for two-wave mixture
In this section we mathematically illustrate the amplitude
and wave number estimates for a superposition of waves. For simplicity,
imagine a mixture of two waves which have amplitudes changing on much larger
scales than the lengths of the carrier waves:
f=a1cos(k1x+ϕ1)+a2cos(k2x+ϕ2).
The Hilbert transform creates
H=a1sin(k1x+ϕ1)+a2sin(k2x+ϕ2).
The local amplitude is calculated by
a2=f2+H2.
It contains terms with equal wavelengths (the desired squared sines and
cosines) and mixed-wavelength terms which are either slow
(±(k1-k2)) or fast (±(k1+k2)). The application of the low-pass
filter (Step 10) is intended to eliminate the fast spurious
components which are expected to create the most fuzziness. With this
procedure we find from the equal-wave-number terms the sum of all squared
amplitudes:
a2=a12+a22.
This means that all variance is included in this estimate. For the wave numbers
we find from the definition
k2=k12a12+k22a22a12+a22.
This is the amplitude-weighted sum of squared wave numbers. The covariance
(or squared standard deviation) is the mean of squares.
s2=〈f2〉=〈a12cos2(k1x+ϕ1)+a22cos2(k2x+ϕ2)〉=a122+a222=a22
Hence, the ensemble average results in half of the squared amplitude.
Derivation of total wave energy
The total wave energy is composed of kinetic and potential
energy (etot=ekin+epot). The following considerations are related
to linearised equations in Boussinesq approximation in a resting environment
e.g.. We use the definitions of horizontal divergence
δ=i(kxu+kyv) and vorticity ζ=i(kxv-kyu) to rewrite the kinetic energy as
ekin=12(u2+v2)=12δ2+ζ2kh2.
The potential energy is expressed with the buoyancy tendency
-iωb=-N2w to yield
epot=12b2N2=12N2w2ω2.
In order to express the total energy in terms of the divergence, both
formulae are combined with the vorticity tendency
-iωζ=fδ and the continuity equation
(δ=-ikzw):
etot=12δ2kh21-f2ω2-N2ω2δ2kz2.
The final result is obtained with the incorporation of the dispersion relation
for hydrostatic inertia–GWs ω2=f2+N2kh2kz2 reading
etot=δ2kh2.
The authors declare that they have no conflict of
interest.
This article is part of the special issue “Sources, propagation, dissipation and
impact of gravity waves (ACP/AMT inter-journal SI)”. It is not associated with a
conference.
Acknowledgements
We acknowledge funding for the research unit Multiscale Dynamics of
Gravity Waves (project Spontaneous Imbalance) from the Deutsche
Forschungsgemeinschaft (DFG) through grant ZU 120/2-1. The publication of this article was funded by the
Open Access Fund of the Leibniz Association. Furthermore, the
authors want to thank the ECMWF for data supply. Useful comments on this
paper were given by Vivien Matthias and Steffen Hien. Special thanks go
to one of the reviewers for the provision of programme code and check values.
Edited by: Markus Rapp
Reviewed by: two anonymous referees
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