This page describes a numerical simulation of wave breaking produced by
a three-dimensional gravity wave packet. The packet is initialized at
a low altitude and is allowed to propagate upward through an isothermal
atmosphere. The wave amplitude grows with altitude until instability sets
in and the wave breaks. The packet then dissipates under the action of
turbulence. Secondary waves are generated by the breaking process and
these propagate to higher altitude.
The underlying wave has a horizontal wavelength of 30 km and a vertical
wavelength of 20 km. It is enveloped with Gaussian functions in all three
directions having standard deviations of 30 km in the horizontal directions
and 10 km in the vertical direction. The entire visible width of the packet
is about 180 km in the horizontal directions and 60 km in the vertical. The
intrinsic wave period is 10.65 minutes.
The packet is initialized at an altitude of 30 km with a non-dimensional
amplitude of 0.05 (5% of the amplitude required for overturning). The
atmosphere is isothermal with a buoyancy period of 6 minutes and an
opposing uniform zonal wind of half the zonal wave phase speed is applied.
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 numerical dissipation.
The computations are performed on a domain having dimensions 440 x 400 x 205 km in the zonal, meridional, and vertical directions. The grid contains 800 x 675 x 256 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 considerably more 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 zonal velocity perturbation in the zonal-vertical plane.
The early, mainly linear, part of the evolution is not shown. The animation begins at an elapsed time of 11.2 buoyancy periods (T_b), when the center of the wave packet has propagated to about 170 km and the wave amplitude is close to the critical value for overturning. Due to momentum flux divergence within the high-amplitude wave packet, there is a pronounced deficit in the zonal wind. This effect is visible even at the start of the animation as a pronounced bias towards low velocity values (blue colors). The mean wind deficit results in a refractive increase in the wave zonal phase speed. This so-called "self acceleration" provides a mechanism for wave breaking.
As time increases from 11.2 T_b, the wave overturns and produces turbulence in a zone that spreads both horizontally and vertically. There is also a pronounced descent of the turbulent zone to lower altitudes. Starting at a time of about 19 T_b, secondary waves become visible emanating from the head and tail of the packet. The mean wind deficit remains, although is is spread over a larger region of space.
The secondary waves are highlighted more clearly in the following animation of the potential temperature perturbation. Note that the colormap is changed periodically during the animation in order to visualize the diminishing perturbations with time.
Finally, the following animation of the vorticity magnitude highlights the turbulent structures.
Animations of the flow in horizontal planes at z=150 and 170 km are shown below.
Compressed directories of vtk files
The computations are performed on a domain having dimensions 440 x 400 x 205 km in the zonal, meridional, and vertical directions. The grid contains 800 x 675 x 256 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 considerably more 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 zonal velocity perturbation in the zonal-vertical plane.
The early, mainly linear, part of the evolution is not shown. The animation begins at an elapsed time of 11.2 buoyancy periods (T_b), when the center of the wave packet has propagated to about 170 km and the wave amplitude is close to the critical value for overturning. Due to momentum flux divergence within the high-amplitude wave packet, there is a pronounced deficit in the zonal wind. This effect is visible even at the start of the animation as a pronounced bias towards low velocity values (blue colors). The mean wind deficit results in a refractive increase in the wave zonal phase speed. This so-called "self acceleration" provides a mechanism for wave breaking.
As time increases from 11.2 T_b, the wave overturns and produces turbulence in a zone that spreads both horizontally and vertically. There is also a pronounced descent of the turbulent zone to lower altitudes. Starting at a time of about 19 T_b, secondary waves become visible emanating from the head and tail of the packet. The mean wind deficit remains, although is is spread over a larger region of space.
The secondary waves are highlighted more clearly in the following animation of the potential temperature perturbation. Note that the colormap is changed periodically during the animation in order to visualize the diminishing perturbations with time.
Finally, the following animation of the vorticity magnitude highlights the turbulent structures.
Animations of the flow in horizontal planes at z=150 and 170 km are shown below.