A convolution-based method of spectral nudging of atmospheric fields is developed in the Australian Community Climate and Earth Systems Simulator (ACCESS) version 1.3 which uses the UK Met Office Unified Model version 7.3 as its atmospheric component. The use of convolutions allow for flexibility in application to different atmospheric grids. An approximation using one-dimensional convolutions is applied, improving the time taken by the nudging scheme by 10–30 times compared with a version using a two-dimensional convolution, without measurably degrading its performance. Care needs to be taken in the order of the convolutions and the frequency of nudging to obtain the best outcome. The spectral nudging scheme is benchmarked against a Newtonian relaxation method, nudging winds and air temperature towards ERA-Interim reanalyses. We find that the convolution approach can produce results that are competitive with Newtonian relaxation in both the effectiveness and efficiency of the scheme, while giving the added flexibility of choosing which length scales to nudge.

Atmospheric modelling is a discipline that has impacts in many fields
of scientific study as well as everyday life. For example, numerical
weather prediction

Nudging is useful for model development and scientific studies, where
a more realistic atmospheric circulation can help determine errors or
feedbacks in particular components of the model. Nudging in
atmospheric models has been used to reduce the size of transport
errors of trace gases for atmospheric chemistry

This paper describes an efficient method for implementing
a convolution-based spectral nudging scheme in atmospheric models,
which is demonstrated using the Australian Community Climate and Earth
System Simulator (ACCESS;

ACCESS is a numerical model designed to simulate Earth's weather
and climate systems. ACCESS is used for a wide range of applications
from climate change scenarios and numerical weather prediction, to
targeted scientific studies into areas such as atmospheric chemistry
and aerosols, and the carbon cycle. ACCESS is composed of a number of
different submodels, of which the atmospheric component is the UK Met
Office Unified Model (UM;

Nudging was originally implemented in the UM at the University of
Cambridge, UK

An alternate approach to Newtonian relaxation is spectral nudging

The paper is structured as follows: Sect.

The performance of the spectral nudging is analysed in
Sect.

The process of nudging aims at perturbing prognostic variables

The standard Newtonian relaxation is applied by taking the difference
between

The code used for the relaxation nudging is based on code from

Spectral nudging extends the Newtonian relaxation method by taking the
correction term and applying a spectral (low-pass) filter so that
large spatial wavelengths are adjusted while smaller wavelengths are
left essentially unperturbed. The method chosen to do this is based
on

The correction for spectral nudging is applied as follows:

The ACCESS grid is horizontally decomposed into domains that are
assigned to individual processors. The calculation of the convolution
at any point requires a global sum. Global information is not stored
on individual processors, so the message passing interface (MPI) is
used to gather the

The naive implementation of the spectral filter involves a large
computational effort (of order

To improve the computational efficiency of the spectral nudging
scheme, the 2-D convolution can be separated into two 1-D
convolutions, thereby reducing the computational effort to order

Using this 1-D approximation, there is a choice in which convolution
to apply first (i.e. either the zonal or meridional
directions). Swapping the order of the integrals (convolutions)
results in numerically different solutions. It is found that to reduce
the error it is best to apply the convolution first along the
latitudinal direction, then longitudinally. This is discussed in
Sect.

It also needs to be noted that the 1-D spectral filter is dependent on
the model grid and the way the grid is decomposed into domains for
each processor. The configuration of the ACCESS grid allows for the
convolution to be computed along rows of equal latitude or longitude
and for those results to be efficiently distributed to rows or columns of
processors. This approach needs to be modified for grids which do not
have these symmetries. See

This paper uses simulations of ACCESS, in the ACCESS1.3 atmosphere-only configuration

A series of 1-year simulations were run, starting from 1 January 1990, each initialized in the same state, from a previous climate simulation, i.e. with an initial state unrelated to any historical synoptic patterns. The only differences between simulations were in the nudging configuration. These short experiments were chosen to evaluate the performance of different nudging methods and choice of nudging parameters. Longer climate simulations may also provide more in depth insight into biases in the nudging scheme, but this evaluation is beyond the scope of this paper.

The nudging component used the ERAI reanalysis product as the host
model, provided at 6 hourly intervals. The ERAI data was linearly interpolated temporally to each time
step. It was interpolated horizontally using bi-cubic interpolation, from its
native 0.75

The ERAI data set was interpolated vertically to the ACCESS1.3 model levels,
using the vertical interpolation developed in

Nudging was applied to potential temperature (

The nudging adjustment was only applied from vertical level 7, corresponding to
about 1

The parameters varied in the experiments were the nudging method,
nudging period, maximum nudging strength, and spectral filter
length. The relaxation nudging is always applied every time step, and
the spectral filter can be applied at frequencies that are multiples
of the time step and divide into 6

Simulations presented use a maximum nudging strength corresponding to either
a 1 h

Spatial distributions of the RMSE in air temperature of ACCESS simulations.
This is measured in Kelvin on a 2-D horizontal plane at 250

To determine the performance of the spectral filter, we look at the
nudged runs compared with ERAI, as well as comparing with a control
simulation without nudging. The control simulation also gives an indication
of the behaviour of the nudging tendencies that were required to change the
evolution of the simulation. The analysis was conducted on the nudged air
temperature and wind fields, measured on planes of constant pressure at 250,
500 and 850

After describing the impact of nudging in Sect.

Figure

Spatial distributions of the RMSE in air temperature of ACCESS simulations.
This is measured in Kelvin on a 2-D horizontal plane at 500

Spatial distributions of the RMSE in air temperature of ACCESS simulations.
This is measured in Kelvin on a 2-D horizontal plane at 850

Spatial distributions of the difference in variance of air temperature
between ACCESS simulations and ERAI. This is measured in Kelvin squared,
on a 2-D horizontal plane at 500

Figures

The

Spatial distributions of the RMSE for MSLP in hectopascal, between ACCESS
simulations and ERAI. This is averaged over daily mean values for 1-year simulations excepting the first 10 days.

Spatial distributions of the RMSE of monthly mean precipitation
in millimetres per day, between ACCESS simulations and ERAI. This is averaged
over a 1-year simulation.

Comparison of RMSE and GAE in air temperature measured in Kelvin, for 1-year simulations excepting the first 10 days, using different nudging methods. Spectral nudging experiments use nudging applied once an hour.

Comparison of RMSE and GAE in

Comparison of RMSE and GAE in

Since we intend to use the nudging in the simulation over climate timescales
(i.e. decades), it is useful to determine how well the simulation predicts
the variance as well as the mean air temperature.
Figure

It is also useful to compare the performance of the model between small and
large spatial scales. The RMSE gives the error grid point by grid point, at
the smallest length scale. To evaluate the error at the largest length scale
(the whole globe), the global mean of the difference between ACCESS and ERAI
can be used. We refer to this as the global average error (GAE). Values for
the GAE are included in Tables

Tables

In addition to constraining the nudged parameters, it is important that the
nudging does not have a detrimental effect on other atmospheric processes.
For this study, we examine the simulated MSLP and precipitation. We have
chosen to concentrate on these fields since they can be readily compared to
ERAI results and can also potentially be tested by observational data. For
simplicity, in this paper we will compare the simulated results to ERAI
predictions, noting that ERAI also produces an imperfect simulation of
rainfall. A more detailed discussion of how nudging can effect model physics
can be found in

In Fig.

The majority of results presented in this manuscript are for nudging simulations using the 1-D filter. To justify this choice of spectral filter method, in this section we compare the results of different configurations of the 1-D filter to those obtained using the 2-D filter. There are two ways to order the convolutions in the 1-D filter, with the zonal convolution followed by the meridional convolution (1-D filter, long–lat), or the meridional convolution followed by the zonal convolution (1-D filter, lat–long).

The RMSE of air temperature at 500

RMSE of air temperature at 500

RMSE of temperature at 500

To more closely compare the different ordering of the 1-D
convolutions, Fig.

The difference between the long–lat and the lat–long version of the 1-D filter occurs because the grid points near the pole are physically close together in the longitudinal direction. A small error at the pole could be spread zonally across multiple grid points. In the long–lat case, this error will remain after the initial zonal convolution. On the other hand, when the meridional convolution is applied first, the error near the poles can be reduced. This is because the values at grid boxes close to the poles have a smaller weighting in the meridional convolution, as they have a smaller area.

As the 1-D filter constrains the model to a similar extent as the 2-D filter, with much reduced computational effort, it is clearly the preferred choice. The 1-D filter with the meridional convolution applied first has better performance at the poles, so it is the optimum configuration. All simulations using spectral nudging refer to this configuration, except where specified otherwise.

Figure

Tables

As seen in Sect.

To further show the effect of the spectral filter at different length
scales, the simulation output was re-gridded to a range of coarser
resolutions. Re-gridding to coarser resolutions removes the fine-scale
detail in a similar way to the spectral filter, so the performance of
the spectral filter should improve at coarser resolutions. This is
shown in Fig.

At the highest resolutions, the relaxation nudging has a smaller RMSE,
showing that it constrains the small length scales more tightly than
the spectral nudging. For the spectral nudging, decreasing

Plot of average RMSE of temperature at 500

Figure

All valid choices of nudging period are able to sufficiently constrain
the model, given a comparable

Examining Fig.

The spectra when nudging every time step is qualitatively similar to the control simulation but shifted down in magnitude. The spectral nudging at every time step has a spectrum in between the curves for the control and relaxation nudging. The spectrum for the 2-D filter is indistinguishable to the equivalent simulations using the 1-D spectral filter with the same filter length scale (2-D filter not shown).

Considering the speed benefits of different nudging frequencies, the
1-D spectral filter nudged every 6

Nudging at hourly intervals can be used as a compromise between speed of computation and reducing the distortions in the spectra, and is the standard period of nudging used for spectral nudging in this paper.

Temporal Fourier spectra for temperature at 500

This paper has introduced the use of spectral nudging in the UM and ACCESS. This is achieved through a novel convolution method, first described by Thatcher and McGregor (2009), but generalized in this paper for use with latitude–longitude grids as used by the ACCESS atmospheric model. Analysis of the different configurations of nudging shows that the nudging schemes effectively constrain the nudged fields to follow the host model (ERAI). We have surveyed the spectral filter across a range of filter length scales. The spectral nudging scheme approaches the Newtonian relaxation nudging when small length scales are nudged, but allows the flexibility to nudge only large spatial structures when the filter length scale is increased.

Our results show that simulation errors in air temperature are greater near the surface for all nudging methods, which is expected due to the different representation of land-surface parametrizations. Although our objective was to compare the Newtonian relaxation with the spectral filter in ACCESS, we note that differences occur where there is a mismatch in the orographic height between the ACCESS simulation and ERAI, suggesting a problem with the vertical interpolation to the ACCESS grid used by the nudging. We intend to address this problem in future work.

We have also considered the implications of nudging on MSLP and precipitation, which are not directly perturbed by the nudging. MSLP is a reasonably smoothly varying field and is well constrained by the nudging in all simulations to agree with ERA-Interim. There are some differences under high orography, although this may be more related to the method used for calculating MSLP under orography rather than the nudging method. The nudged simulation improved the monthly mean rainfall compared to the control simulation. Furthermore, the spectral nudging simulations predicted rainfall that was in closer agreement with ERAI than the relaxation nudging simulations. This provides an example of where the spectral filter can have an advantage over the Newtonian relaxation approach, particularly for physical processes that are sensitive to the local behaviour of the atmosphere.

The 1-D spectral filter is shown to perform as well as the 2-D filter, while producing a speedup of 10–30 times. This is achieved by the approximation of separating the 2-D convolution into 1-D convolutions and by using symmetries of the model grid to reduce communication between processors. We also identified that, due to the geometry of our grid, the order of convolutions in the 1-D filter was important. To reduce error in the approximation, the meridional convolution is applied first.

Nudging with different frequencies was also investigated, showing that nudging every 6 h is still able to constrain the model, but introduces distortions to the spectra. Nudging once an hour produces a speedup in comparison to nudging every time step, while introducing minimal distortions, so it was used for the majority of simulations.

The approach used to implement the 2-D and 1-D spectral filters is applicable to many other models. The 2-D convolution method can be implemented on any grid, though it suffers from being computationally expensive. The 1-D filter can be applied to irregular or more complex grids, but would require modification to separate the 2-D Gaussian function using an approximation that is appropriate for the particular grid.

Future work on spectral nudging in ACCESS will involve generalizing the spectral nudging to limited area and stretched grid configurations. Another potential approach to gaining a speedup in the convolution-based spectral filter is to compute the convolutions over a small neighbourhood, rather than the whole globe, ignoring areas where the Gaussian function has values close to 0. The ability to extend the convolution-based spectral filter within the ACCESS/UM framework and in other modelling systems is an advantage of this approach.

Due to intellectual property right restrictions, CSIRO cannot publish the full source code for ACCESS or the UM. The Met Office Unified Model (UM) with the spectral nudging source code and configuration described in this paper can be obtained under an end-user license agreement (EULA) from CSIRO for educational and non-commercial research use for specific projects. To request a EULA for the modified UM, and/or to obtain the ACCESS1.3 model configuration used in this paper, please contact Tony Hirst (tony.hirst@csiro.au).

Thanks to Peter Dobrohotoff, John McGregor and Tony Hirst for their feedback in the preparation of the manuscript and the anonymous reviewers for suggested revisions to improve the manuscript. This research was undertaken with the assistance of resources from the National Computational Infrastructure (NCI), which is supported by the Australian Government. This work included funding by the Australian Government through the Australian Climate Change Science Programme. ERA-Interim data, from the European Centre for Medium-Range Weather Forecasts (ECMWF) was used in this research. The UM was made available to CSIRO under the consortium agreement Met Office's Unified Model Earth System Modelling software (Met Office Ref. L1587). Edited by: G. Mann