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.

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

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

Another three-dimensional spectral analysis method is the 3D
Stockwell transform (3-D ST)

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.

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

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

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.

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

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

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.

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

In wave number space a rectangular bandpass filter reduces the complex spectrum to the user-predefined wave number limits

To get back from wave number space an inverse DFT is
performed.

The constructed complex-valued function

The phase gradient is a measure of the wave number
modulus:

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

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

Steps 4 to 11 are repeated for the other dimensions (

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:

For a comparison of wave characteristics obtained with different methods we
choose the test case presented in

One-dimensional test function (

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

The method showing the best agreement
with the theoretical value is UWaDi (Fig.

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

Vertical-propagating GWs are damped in ECMWF IFS products from 10

From the diagnosed fields of amplitude and wave number we calculate the
kinematic wave energy

A minor SSW occurred on 30 January 2016. Figure

Synoptical situation of the Northern Hemisphere from ECMWF analysis
at 10

In zonal mean the horizontal wavelength varies between 120 and 200

Zonal mean profiles at 60

We next inspect local profiles in different background wind conditions.
Longitude–height sections of zonal wind (Fig.

Zonal wind

To highlight the advantage of a local wave analysis we show profiles at
selected longitudinal positions (Fig.

During a local increase in wind speed above northern Europe the vertical
profiles of (1) show that the zonal wind meanders around
50

Above eastern Siberia a displaced stratospheric jet streak appears jointly
with high wave action (Fig.

Vertical profiles at 60

The topic of selective wave transmission was first modelled by

Comparing cases (1), (2), and
(3) with respect to their wave action profiles we diagnose
at 42

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

In the displaced stratospheric jet case (2)
(Fig.

Intrinsic frequency (bold, dark blue), horizontal wavelength (dashed, orange), and horizontal phase speed (bold, pink)
for profile (2) (Fig.

In the low-wind case (3) the high wave action in the upper
troposphere up to the height of the wind reversal at 23

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

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

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:

The total wave energy is composed of kinetic and potential
energy (

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.

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