Effects of realistic propagation of gravity waves (GWs) on distribution of GW pseudomomentum fluxes are explored using a global ray-tracing model for the 2009 sudden stratospheric warming (SSW) event.
Four-dimensional (4D;

Atmospheric gravity waves (GWs) play an important role in the momentum and energy budgets of global circulations in the middle and upper atmosphere.
GW pseudomomentum fluxes can induce large-scale momentum forcing, which can substantially change ambient winds, either when transience related to unsteady propagation or dissipation due to breaking or damping occurs

GWs may also affect global thermal structure through adiabatic vertical motions and heat deposition.
GW momentum forcing induces the meridional and vertical mass circulations that contribute to the temperature structure related to the Brewer–Dobson circulation in the stratosphere

In general, excitation of GWs is unsteady, and GWs propagate at finite group velocities in the form of localized packets or wave trains.
Hence, studies on propagation of GW packets in slowly varying large-scale flows have been carried out using ray-tracing modeling based on the spatial ray theory

However, GW parameterizations (GWPs) for global climate and numerical prediction models have dealt with propagation of GWs under simplifying assumptions that steady GWs propagate instantaneously only in the vertical direction from tropospheric sources to the model top.
To consider horizontal and time propagation of GWs,

There have been studies to understand the effects of horizontal and transient propagation of GWs on interactions between GWs and slowly varying mean flows.

Planetary-scale flows in the middle atmosphere, through which GWs propagate, can exhibit substantial spatial inhomogeneity and transience.
Substantially disturbed large-scale flows are often found during sudden stratospheric warming (SSW) events in association with large planetary wave (PW) activity

There have been various modeling studies on the roles of GWs in SSWs.

Satellite observations have presented evidence of substantial variations of GW activity around SSW onset dates.
GW activity is often found to be enhanced in the upper stratosphere before SSW onsets and in high-altitude regions where ESs form in the recovery phase of SSWs

The present study explores effects of the 4D (

The paper is organized as follows.
Section 2 presents formulations of the ray-tracing model.
Section 3 describes specification of large-scale flow from the ground to the lower thermosphere.
Ensembles for parameterized orographic and nonorographic GWs are presented in Sect. 4.
In Sect. 5, ray-tracing simulation results for the 2009 SSW are demonstrated by comparing the 4D (

A wave packet is defined by a group of phase surfaces over the distance of the order of a dominant wavelength.
A ray is a curve for which tangents coincide with a sequence of wave propagation directions

Kinematic wave theory

At each (

Time evolutions of position, wave number, and observed frequency of a wave packet are described as follows:

Here,

Equation (4) is isomorphic to the Hamilton equation for a physical system characterized by a Hamiltonian denoted by

In the computation of Eqs. (4) and (5), component forms are used (see Appendix A1), and the shallow atmosphere approximation (

The dispersion function

In the model, the anelastic dispersion relation for IGWs

The large-scale flow variables (

Time evolution of wave amplitude in slowly varying large-scale flows is described by the wave action conservation law

For computation of wave amplitude along a ray, the conservation law (Eq. 7) is changed into an equation for vertical action flux

Here,

In the wave–mean interaction, the vertical flux of IGW horizontal pseudomomentum (

The action conservation equation (Eq. 7) is related to conservation of the angular pseudomomentum of IGWs.
Combining the component form equation for

For dissipation of GWs, two separate processes are employed, namely nonlinear wave saturation and molecular diffusion.

Nonlinear saturation is computed by forcing

Molecular diffusion is important above the upper mesosphere.
Dividing the total GW energy equation by

Kinematic viscosity (

Time integrations of the ray-tracing equations (Eqs. 4–5 or Eqs. A10–A16) are carried out using the Livermore solver for ordinary differential equation (ODE) with automatic method switching (LSODA) based on the stiffness of an ODE system

Solutions (

The solver requires interpolation of

Action flux equation (Eq. 8) is actually an equation for

The dissipation timescale (

After

Time-varying large-scale flows during the 2009 SSW are specified by combining 6 h reanalysis data sets and empirical model results.
The reanalysis data are linearly interpolated at a 1 h interval (

The four vertical layers are (i)

Figure 1 shows latitude–height cross sections of ground-to-space (G2S) zonal wind and temperature at 60

Latitude–height cross sections of

For ray simulations, G2S data at hourly intervals are spatially smoothed using the vertical 1–2–1 smoother and horizontal moving averaging on the spherical surface.
Horizontal averaging is done using variables within the area of a spherical cap centered at every latitude–longitude grid point.
A spherical cap is defined by an angle between two lines from the sphere center to the center of the spherical cap's surface and to the cap's boundary.
The angle is set equal to about 2.7

In this study, orographic and nonorographic GWs are considered separately.
Properties of orographic GWs are given by the M87 scheme.
Nonorographic GWs are specified based on SA11 and

Orographic GWs (OGWs) are assumed to be excited when low-level horizontal winds are strong, and vertical parcel displacements, due to subgrid topography, are large (M87).

The vertical displacement (

Directions of horizontal wave number vectors are set opposite to horizontal wind vectors averaged within source layer. OGWs are launched at the top of the source layers.

For nonorographic GWs (NOGWs), three 14-discrete-wave schemes, as in SA11, are considered. One is a modified version of SA11, and the other two are derived from the empirical spectra of WM96.

The empirical GW energy spectrum (

Figure 2 illustrates angular histograms of spectral properties and Reynolds stress in the three 14-wave NOGW schemes.
In these schemes, horizontal propagation directions (

Each GW has an identical amount of Reynolds stress in the three schemes.
For this, the stresses for GWs with

Angular histograms of

In ray simulations, a single GW packet, stochastically chosen from GW source ensembles, is launched at a horizontal grid point where GWs are supposed to be generated. This approach is computationally efficient, allowing for statistical significance tests for differences between ray simulations.

The OGW scheme (M87) launches a single OGW at a horizontal grid point, but NOGW schemes usually specify multiple GWs.
Hence, GW ensembles are separately generated for OGWs and NOGWs.
For OGWs, the vertical displacement

Figure 3 demonstrates horizontal distributions of zonal OGW and NOGW pseudomomentum fluxes (

For OGWs,

Longitude–latitude distributions of zonal pseudomomentum fluxes (

For NOGWs, wave IDs of 1–14 are randomly spread around the globe.
The magnitude of ensemble-averaged zonal NOGW

GW ray simulations are carried out for the time period of 25 d from 00:00 UTC on 8 January 2009 to 00:00 UTC on 2 February 2009 for the 20 OGW and NOGW ensemble members.

For each GW ensemble member, two kinds of simulations are carried out, namely four-dimensional (4D;

Figure 4 shows latitude–height distributions of zonal mean zonal wind and ensemble averages of zonal mean zonal

Statistically significant differences between the 4D and 2D are found in five regions.
(i) In the latitude–height region of 40–70

Additional 4D OGW and NOGW

Latitude–height cross sections of

Figure 5 shows latitude–height distributions of zonal mean zonal wind and ensemble means of zonally averaged zonal wave numbers (

Latitude–height cross sections of

Statistically significant differences of zonal mean

The time rate of change of

Latitude–height cross sections of total and three major forcing terms – the zonal shear terms of the large-scale zonal and meridional winds (

Figure 6 shows latitude–height distributions of the total and three major forcing terms in the

The structure of the three major forcing terms in the

Latitude–height cross sections of zonal mean zonal wind (ZU) and ensemble means of zonally averaged meridional wave numbers (

Figure 7 shows latitude–height distributions of zonal mean zonal wind and ensemble means of zonally averaged meridional wave numbers (

In the SH, large positive

The structure of

Same as Fig. 7 except for the zonally averaged number of GW packets.

Figure 8 shows latitude–height distributions of zonal mean zonal wind and ensemble means of the zonally averaged number of GW packets for OGWs and NOGW

As the eastward jet in the stratosphere moves towards the North Pole on 20 January, the number of GW packets in the 4D increases in the NH polar mid- to upper stratosphere (50–80

GWs generally propagate more to the thermosphere in the 4D.
Even though the eastward winds are still large in the NH middle atmosphere on 15 and 20 January, both OGWs and NOGWs with eastward

Convergence (or focusing) of GW packets may have some effects on distribution of the GW pseudomomentum fluxes.

The spatial distribution of GW packets in the 4D is generally more contiguous in the latitudinal direction than that in the 2D in which latitudinal discontinuity is clear.
This difference indicates more than improvement in the smoothness of the GW pseudomomentum fluxes in the 4D.
Since

Longitude–latitude cross sections of

Figure 9 shows longitude–latitude distributions of zonal and meridional winds and ensemble averages of zonal pseudomomentum fluxes (

The horizontal structure of zonal

The increase in the magnitudes of zonal

Same as Fig. 9 except for NOGW

Horizontal distributions of zonal

Longitude–latitude cross sections of

Figure 11 shows longitude–latitude distributions of horizontal wind velocity (

Over the Tibetan Plateau and Rocky Mountains, the horizontal intrinsic group velocity is not close to zero.
Therefore, in these regions, OGWs propagate relative to the horizontal winds.
Meanwhile, in the meridionally elongated regions near 60

Equation (13) shows the magnitude of the 3D intrinsic group velocity as a function of wave numbers and intrinsic frequency under the Boussinesq approximation (

It is already seen that the magnitude of zonal and vertical wave numbers (Fig. 9) are substantially increased in the narrow and elongated regions around 60

The significant enhancement of zonal

As shown in Figs. 4–5, zonally averaged eastward

Time–height cross sections of

Figure 12 shows time–height cross sections of zonal mean zonal wind and ensemble averages of zonal

Enhanced westward

Close inspection of Fig. 12 for the early stage of the SSW evolution indicates that the differences between the 4D and 2D begin from the middle stratosphere around 10–11 January 2009.

Relative vorticity at 5 hPa and zonal pseudomomentum fluxes (

Figure 13 shows time evolutions of relative vorticity at 5 hPa and zonal pseudomomentum fluxes (

Same as Fig. 13 except for meridional pseudomomentum fluxes (

It is clear from Figs. 13 and 14 that time evolution of the ZWN2 structure of the

On 11 January, between 40 and 60

For an initially given aggregate of OGWs with the eastward

The OGWs with the southeastward

On 15 January, an aggregate of OGWs with the large southeastward

Effects of realistic propagation of parameterized GWs on GW pseudomomentum fluxes are investigated using a global ray-tracing model for the 2009 SSW event.
Two kinds (4D –

The global ray-tracing model used in this study is composed of two parts, namely ray-tracing and amplitude equations. Ray-tracing equations are formulated considering the curvature effects on the spherical Earth, and they compute the trajectory of GW packets and refraction due to spatiotemporal variations of the large-scale flow. The time evolution of the vertical flux of GW action flux is computed using the amplitude equation. In the amplitude equation, ray-tube effects associated with the geometry of neighboring rays are considered by evaluating group velocity components of GW packets at grid points. For dissipative processes, nonlinear saturation and molecular viscosity are computed along ray trajectories. These dissipations only act on the action flux without affecting GW propagation.

In realistic 4D propagation, horizontal refractions related to large-scale wind shear and curvature effects are essential compared to spatial gradients of thermodynamic large-scale properties such as stability and density.
Latitude–height structure of the zonal pseudomomentum fluxes (

In the NH upper stratosphere, westward

Enhancement of GW

In the early stage of the 2009 SSW evolution, it is interesting that the spatial distribution of the OGW

Interpretation of results shown in this paper may depend on the gridding method designed to generate gridded model outputs.
In this method, the spatial size of GW packets is assumed to be as large as horizontal and vertical grid spacings used in this study.
This implicit assumption may lead to overestimation of the magnitude of the GW

In the present study we have not discussed how GW momentum forcing can be estimated from the ray-simulation results. As described above, consideration of realistic propagation of localized GW packets in the slowly varying large-scale flows requires GWPs to compute influences of GWs in more nonlocal ways in space and time, which violates the basic assumptions of current modeling frameworks. In SSW cases considered in this study, large-scale flows can vary rapidly in space and time, and the nonlocal approach may particularly be more important since GWs can change vortex structure located around the GWs. However, at this point, it is not straightforward to present how to estimate the nonlocal influences of GWs in a clear way. In order to consider the nonlocality in models, one might either somehow extend columnar GWPs or explicitly implement ray-tracing formulations. One way or another, further theoretical developments of GW processes seem to be necessary as long as physically based methods with minimal ad hoc treatments are preferred.

Local time change of

From the definition of

Local time change of

Since

Here,

The constraint

Note that these relations are derived from spatial variations of the basis vectors (i.e.,

Substituting

From Eqs. (A7)–(A9), it can be shown that the magnitude of a 3D wave number vector is invariant with respect to the Earth's curvature by multiplying Eq. (A7) by

In the model, Eqs. (4) and (5) are approximated for the shallow atmosphere, and for the dispersion relation Eq. (6), they can be written in a component form as follows:

In Eqs. (A13)–(A16), terms starting with

Under the shallow atmosphere approximation

Using Eqs. (A13) and (A14), it can be proved that the magnitude of the horizontal wave number vector is invariant with respect to curvature effects as in Eqs. (A7)–(A9).

Viscous damping and thermal diffusion terms for GWs can be obtained by linearizing the viscosity term (derived from the symmetric stress tensor) in the Navier–Stokes equation and the diffusion term in thermodynamic energy equation

In the viscosity terms in Eq. (A22),

Viscous damping and thermal diffusion may affect propagation of GWs through modification of the dispersion relation

After substituting plane wave solutions such as

Therefore,

The LSODA solver employs sub-time stepping within each

In the gridding method, the horizontal projection of a 3D ray trajectory during

Using this gridding method, three components of group velocity are stored at the vertices of chosen grid cells between initial and final positions.
In addition, various ray properties such as

In the model, rays are eliminated when some criteria are satisfied after time integration for

The HWM14 and DWM07 model codes (Fortran) are included in the supporting information of

The ERA-Interim data are obtained using the Meteorological Archival and Retrieval System (MARS) of the European Centre for Medium-Range Weather Forecasts

The supplement related to this article is available online at:

ISS and CL planned this study. ISS developed the ray-tracing model, designed ray-tracing simulations, carried out model experiments, and analyzed model results. HYC provided advice on interactions between gravity waves and planetary waves. BGS provided information about previous studies on the 2009 SSW. All coauthors commented on the paper. ISS wrote the paper.

The authors declare that they have no conflict of interest.

The authors thank the three anonymous reviewers and the editor for their careful reading of the paper. Their comments substantially improved the original paper. This study was supported by the research funds (grant nos. PE19020 and PE20100) from the Korea Polar Research Institute and funded by the Korea Meteorological Administration/National Meteorological Satellite Center (KMA/NMSCs) project (grant no. NMSC-2016-3137). Also, this work was supported by the National Institute of Supercomputing and Network/Korea Institute of Science and Technology Information with supercomputing resources, including technical support (grant no. KSC-2016-C2-0034). Hye-Yeong Chun was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT; grant no. 2017R1A2B2008025).

This research has been supported by the Korea Polar Research Institute (grant nos. PE19020 and PE20100), the National Meteorological Satellite Center of the Korea Meteorological Administration (grant no. NMSC-2016-3137), and the National Research Foundation of Korea (NRF) funded by the Korea government (MSIT) (grant no. 2017R1A2B2008025).

This paper was edited by Amanda Maycock and reviewed by three anonymous referees.