This page describes a series of 3D numerical simulation of gravity waves
generated by large a thunderstorm. The waves propagate in a realistic
thermodynamic environment and encounter significant wind shears associated
with the tides in the Mesosphere and lower Thermosphere. The simulations
not only reveal significant filtering of the wave packet due to the tidal
wind shear, but also localized regions of wave breaking.
Following Vadas and Fritts (2009) the thunderstorm is modeled as via a
vertical body force that generates a localized region of rising air. The
column of rising air (or plume) is axisymmetric in horizontal plane and
has a Gaussian cross-section with standard deviation h. The plume also
has a Gaussian distribution in the vertical direction, centered at an
altitude of z_c and having a vertical standard deviation of z_w. The forcing
functions are carefully constructed to be divergence-free. This feature
is critical to the success of the method in a fully-compressible environment
where non-solenoidal regions act as efficient sources for acoustic waves.
A natural consequence of the divergence-free constraint in conjunction with
the condition that the velocity perturbation from the plume vanish at
large horizontal distance is that there is no net mass flow in the vertical
direction. Thus, unlike Vadas and Fritts (2009), we also force the required
return (downward) flow in order to achieve zero net vertical mass flux on any
horizontal plane. The return flow takes the form of an axisymmetric ring
of Coward flow outside of the main plume. The divergence-free constraint
also requires that we force the flow in the horizontal directions. The
forcing functions are
where A is the maximum forcing amplitude and τ is the forcing period. The plume is centered at zc=10 km, has depth (standard deviation measure) of zw=2.5 km and a horizontal width (standard deviation) of h=4 km. The thunderstorm duration is τ=10 minutes and the maximum forcing amplitude is A=0.33 m/s2. These values are similar to Vadas and Fritts (2009) who use zc=7 km, zw=2.2 km, h=4.44 km, and τ=12 minutes. The choice A=0.33 yields a maximum updraft velocity of 15 m/s, which is considerably smaller than the 40 m/s considered by Vadas and Fritts (2009).
The gravity waves propagate into atmosphere with realistic thermodynamic profiles representative of solar mean conditions (after Vadas and Fritts 2006). The waves also encounter wind shears due to tides as computed by Han Li using the WACCM model. Four phases of the tidal winds over one diurnal cycle are considered for a region over Brazil having coordinates of 20 S 60 W on Jan 01 2015. The wind profiles are shown in the following four graphs.
The simulation code advances the full compressible equations cast in strong conservation law form and discritized using a second order finite volume scheme. Special care is taken with the interpolations used to compute the advective fluxes at the cell faces so that the scheme conserves kinetic and thermal energy independently. A consequence of this feature is that the scheme has no artificial dissipation.
Case 1 - Tides at 01:00 using mesh version 1.0
where A is the maximum forcing amplitude and τ is the forcing period. The plume is centered at zc=10 km, has depth (standard deviation measure) of zw=2.5 km and a horizontal width (standard deviation) of h=4 km. The thunderstorm duration is τ=10 minutes and the maximum forcing amplitude is A=0.33 m/s2. These values are similar to Vadas and Fritts (2009) who use zc=7 km, zw=2.2 km, h=4.44 km, and τ=12 minutes. The choice A=0.33 yields a maximum updraft velocity of 15 m/s, which is considerably smaller than the 40 m/s considered by Vadas and Fritts (2009).
The gravity waves propagate into atmosphere with realistic thermodynamic profiles representative of solar mean conditions (after Vadas and Fritts 2006). The waves also encounter wind shears due to tides as computed by Han Li using the WACCM model. Four phases of the tidal winds over one diurnal cycle are considered for a region over Brazil having coordinates of 20 S 60 W on Jan 01 2015. The wind profiles are shown in the following four graphs.
The simulation code advances the full compressible equations cast in strong conservation law form and discritized using a second order finite volume scheme. Special care is taken with the interpolations used to compute the advective fluxes at the cell faces so that the scheme conserves kinetic and thermal energy independently. A consequence of this feature is that the scheme has no artificial dissipation.
Case 1 - Tides at 01:00 using mesh version 1.0
The computations are performed on a domain having dimensions 1200 x 1200 x 200
km in the zonal, meridional, and vertical directions. The grid contains
600 x 600 x 288 mesh points and clustering is used to produce a locally
refined region to contain wave breaking with 400 m spacing in all three
coordinate directions. Two orthogonal cross sections of the grid along
with the vertical velocity field are shown in the following figures.
This grid arrangement allows for adequate resolution in the breaking zone
with a modest number of points. Simulations with a uniform grid spacing
would be prohibitively expensive.
The lower boundary is treated as a slip wall whereas radiation conditions in
concert with a sponge is used at all other boundaries. The sponge/radiation
condition produces solutions where little or no wave energy reflects (or
enters) at the boundaries.
Click on the image below to see an animation of the flow in the zonal-vertical
plane.
By a time of 1800 sec the leading edge of the wave packet has propagated
rather symmetrically to the upper position of the domain, while the bulk
of the wave energy is confined to lower altitudes on the upwind side
of the thunderstorm. The energy-containing region is limited to an altitude
of about 90 km, the location of the most significant positive tidal wind
shear. The pattern remains similar out to about 3000 sec with a focusing
of the high altitude waves on the upwind side and a defocusing on the
downwind side. The amplitude of the waves trapped below 90 km increases
and initial signs of instability are present by t=2500. At a time of
4000 sec the upper altitude waves have largely dispersed on the downwind
side and have just passed through maximum amplitude on the upwind side.
Almost no wave activity is seen above the energy-containing region of
the packet, which is now in the late stages of instability. Wave breaking
begins at a time of about 4500 sec and radiation of secondary waves begins
immediately afterward. The secondary waves are distributed almost
symmetrically in the upwind and downwind directions. Turbulence subsides
at about t=7000 sec but secondary radiation continues for the next 1000
sec or so. The pattern is dominated by secondary waves by the end of
the simulation at t=9000 sec.
Case 2 - Tides at 01:00 using mesh version 2.0
The computations are performed on a domain having dimensions 2500 x 2500 x 218
km in the zonal, meridional, and vertical directions. The grid contains
725 x 725 x 384 mesh points and clustering is used to produce a locally
refined region to contain wave breaking with 400 m spacing in all three
coordinate directions. Two orthogonal cross sections of the grid along
with the vertical velocity field are shown in the following figures.
This grid arrangement allows for adequate resolution in the breaking zone
with a modest number of points. Simulations with a uniform grid spacing
would be prohibitively expensive.
The lower boundary is treated as a slip wall whereas radiation conditions in
concert with a sponge is used at the upper boundary. Periodic conditions
are used at the lateral boundaries.
Click on the image below to see an animation of the vertical velocity
imaged in the zonal-meridional plane at an altitude of 95 km.
Click on the image below to see an animation of the vorticity magnitude
imaged in a zoomed-in view in the zonal-meridional plane at an altitude
of 95 km.
Click on the image below to see an animation of the vertical velocity
imaged in the zonal-meridional plane at an altitude of 175 km.
In order to assess the modifications to the mean winds and mean thermodynamic
state, the wave fields were averaged over a scale of 75 km in both the
zonal and meridional directions. Click on any of the next several images
to see an animation of the averaged flowfields.
The computations are performed on a domain having dimensions 2500 x 2500 x 218 km in the zonal, meridional, and vertical directions. The grid contains 725 x 725 x 384 mesh points and clustering is used to produce a locally refined region to contain wave breaking with 400 m spacing in all three coordinate directions. Two orthogonal cross sections of the grid along with the vertical velocity field are shown in the following figures.
This grid arrangement allows for adequate resolution in the breaking zone with a modest number of points. Simulations with a uniform grid spacing would be prohibitively expensive.
The lower boundary is treated as a slip wall whereas radiation conditions in concert with a sponge is used at the upper boundary. Periodic conditions are used at the lateral boundaries.
Click on the image below to see an animation of the vertical velocity imaged in the zonal-meridional plane at an altitude of 95 km.
Click on the image below to see an animation of the vorticity magnitude imaged in a zoomed-in view in the zonal-meridional plane at an altitude of 95 km.
Click on the image below to see an animation of the vertical velocity imaged in the zonal-meridional plane at an altitude of 175 km.
In order to assess the modifications to the mean winds and mean thermodynamic state, the wave fields were averaged over a scale of 75 km in both the zonal and meridional directions. Click on any of the next several images to see an animation of the averaged flowfields.