The view adapted here is that the dominant mesoscale motions are due to internal gravity waves and show that previous and new vertical wave number spectra of horizontal winds are consistent with the notion of a saturation limit on wave amplitudes. It is also proposed that, at any height, only those vertical wave numbers m less than m sub asterisk are at saturation amplitudes, where m sub asterisk is the vertical wave number of the dominant energy-containing scale. Wave numbers m less than m sub asterisk are unsaturated, but experience growth with height due to the decrease of atmospheric density. The result is a saturated spectrum of gravity waves with both m sub asterisk decreasing and wave energy increasing with height. This saturation theory is consistent with a variety of atmospheric spectral observations and provides a basis for the notion of a universal spectrum of atmospheric gravity waves

Fritts, D. C.

Smith, S. A.

English

NASA Technical Reports Server

DN87-10467
3.7.8 EVIDENCE OF A SATURATED GRAVITY-WAVE SPECTRUM
THROUGHOUT THE ATMOSPHERE
David C. Fritts and Steven A. Smith
Institute and Department of Physics _I _/[I_ 'vGeophysical
University of Alaska fi_ _\,Fairbanks, AK
and
Thomas E. VanZandt
Aeronomy Laboratory, NOAA
Boulder, CO 80303
i. INTRODUCTION
A number of studies have addressed the frequency and wave number spectra
of atmospheric motions over the past two decades and have revealed a surprising
degree of consistency among different measurement techniques. This consistency
in spectral shape and power has motivated two competing theories concerning the
nature of atmospheric motions responsible for the spectral characteristics.
One theory attributes atmospheric motions primarily to two-dimensional
(essentially horizontal) turbulence (GAGE, 1979); the other identifies such
motions with internal gravity waves (VANZANDT, 1982), analogous to the motion
spectrum believed to exist in the oceans (GARRETT and I_NK, 1975).
In this paper, we adopt the view that the dominant mesoscale motions are
due to internal gravity waves and show that previous and new vertical wave
number spectra of horizontal winds are consistent with the notion of a
saturation limit on wave amplitudes. We also propose that, at any height, only
those vertical wave numbers m > m, are at saturation amplitudes, where m,
is the vertical wave number of the dominant energy-containing scale. Wave
numbers m < m, are unsaturated, but experience growth with height due to the
decrease of atmospheric density. The result is a saturated spectrum of gravity
waves with both m, decreasing and wave energy increasing with height. This
saturation theory is consistent with a variety of atmospheric spectral
observations and provides a basis for the notion of a "universal" spectrum of
atmospheric gravity waves. It should be noted that the saturation spectrum
argument has been advanced independently by DEWAN and GOOD (1985).
2. A SATURATED VERTICAL WAVE NUMBER SPECTRUM
Vertical wave number spectra of horizontal wind fluctuations from several
sources that span the height range from the ground to 130 km are plotted in
Figure i. The curve labelled "troposphere" is smoothed data obtained with the
use of Jimspheres by ENDLICH et el. (1969). This curve is an average of the
spectra from six wind profiles, acquired at roughly 2-hr intervals, from which
the average wind profile was subtracted, and has a slope at high wave numbers
of _-2.5. The best fit curve of DEWAN et el. (1984) to their stratospheric
spectra is displayed as the curve labelled "stratosphere". This curve follows
an _-2.7 power l_w (where m is vertical wave number) from a wave number of
10 -9 to 5 x I0 -_ cycles/m (corresponding to vertical wavelengths of 20 m to
I Pun). The curve labelled '_esosphere" is the inferred horizontal velocity
spectrum from the average of summer-time spectra obtained near the mesopause
with the Poker Flat MST radar as discussed by SMITH et el. (1985). The slope
of this velocity spectrum is -2.5. Finally, the thermospheric spectrum was
obtained from winds measured between 85 and 130 km with a three-axis
accelerometer sphere flown as part of the STATE campaign. This spectrum is of
https://ntrs.nasa.gov/search.jsp?R=19870001034 2020-03-23T14:34:47+00:00Z
245
w
Z
o.
0
WAVELENGTH (km)
,o.i ,o 5 ,2 ,
L
,°'t- "X-..
THERMOSPHERE _
10 5
10 4
10 3
102 -
101 --
10 °
.2 .1 .05
I i
MESOSPHERE_\ N2
TROPOSPHERE_\
STRATOSPHERE --__
1 I I
_o-4 lo-3 Id2
WAVENUMBER (cycle/m)
Figure i. Vertical wave number spectra obtained in four regions of the
atmosphere using a variety of different techniques. See text for
detail s.
the scalar wind speed measured by the falling sphere and was smoothed by two
passes of a three-point running average to increase the confidence level. It
was not possible to remove the mean wind from this velocity data, however.
Figure 1 shows clearly that the spectral shape and the spectral power are
very similar from the stratosphere to the lower thermosphere. This spectral
similarity can be extended to the troposphere as well by scaling the spectra to
a common value of the Brunt-Vaisala frequency, N. The data of DEWAN et al.
(1984) and the radar and falling sphere data were obtained from regions with N
% 0.02 rad/s (a Brunt-Vaisala period near 5 rain). However, a typical value of
N in the troposphere is half that in the stratosphere. The linear saturation
theory reviewed in the next section implies that spectral power should scale as
N2. Thus, an increase in the tropospheric data of ENDLICH et al. (1969) of
% 4 is required to permit its comparison with data obtained at greater heights.
Assuming that most of the spectral power is associated with vertically
propagating gravity waves, the consistency of the scaled vertical wave number
spectra with height clearly suggests that some process is acting to limit wave
amplitudes, particularly at high vertical wave numbers, throughout the
atmosphere. This is consistent with the suggestion, due to WEINSTOCK (1982),
that gravity waves are saturated throughout much of the atmosphere.
246
3. LINEAR SATORATION THEORY
Linear saturation theory assumes that a monochromatic gravity wave will be
limited to that amplitude at which the wave just reaches the point of
convective instability, i.e., where
@ = 0,
z
where 8 is the potential temperature.
FRITTS, 1984)
(i)
This is equivalent to the condition (see
u' = C - 3, (2)
where u' and u are the horizontal perturbation velocity and the mean flow in
the direction of wave motion. This may be written, using the dispersion
relation for gravity waves with intrinsic frequencies _ such that f2 << m2
<< N2, as
u' : N/m. (3)
Thus, the power spectral density for horizontal velocity fluctuations inferred
by assuming that each component of the gravity-wave spectrum is individually
saturated is
E(m) = N2/2m 3. (4)
The third power of m is due to the wave number bandwidth corresponding to each
component of the gravity-wave spectrum. This spectral power is shown for N =
0.02 rad/s with a dashed line in Figure I.
The theoretical curve exceeds the power spectral densities observed
throughout the atmosphere by about 2. We believe this difference may be
attributed to the superposition of gravity waves in the atmosphere, which seems
to restrict amplitudes to less than monochromatic saturation values (SMITH and
FRITTS, 1983; MEEK et al., 1985), in good agreement with the observed power
spectral densities. Recent numerical results by FRITTS (1985) suggest a
similar reduction in the wave amplitudes required for saturation due to super-
position.
4. VARIATION OF GRAVITY-WAVE SPECTRUM WITH HEIGHT
Both the data presented in Figure 1 and the form of the analytic gravity-
wave spectrum used to fit successfully previous oceanic and atmospheric
gravity-wave spectra (GARRETT and MU_<, 1975; VANZANDT, 1982) suggest that the
gravity-wave spectrum at any height departs from a saturated spectrum at
sufficiently small vertical wave numbers. Observatiorml evidence of this
departure is most evident in the tropospheric curve in Figure 1, but may be
present in the thermospheric data as well. The departure in the analytic
spectral description consistent with a saturated high wave number spectrum
enters as a spectral break at a wave number m, via a term of the form
(I + m/m,) -3, (5)
providing limiting slopes of -3 and 0 for high and low wave numbers,
respectively.
The departure from a saturated spectrum at small vertical wave numbers in
the lower atmosphere suggests that these motions are not excited at large
amplitudes. As these motions propagate upward, however, they are expected to
247
growwith height due to the decrease in atmospheric density. The results of
this process are a gravity-wave energy and a dominant vertical wavelength that
increase with height, as noted in a number of previous studies.
A model of the proposed vertical wave number spectrum and its variation
with height is presented in Figure 2. This spectrum assumes a vertical wave
number dependence given by (5). At vertical wave numbers m > m,. for which
the spectrum is assumed saturated, the energy levels are the same at all
heights. At smaller (unsaturated) wave numbers, however, the srowth of wave
amplitudes with height causes the dominant vertical scale (m, -_) to
increase accordingly. This is shown with values of the dominant vertical wave-
length (k = m_- ) of 1. 5. and 20 km in the troposphere, stratosphere.
Z x
and mesosphere, respectlvely.
Evidence of this behavior is provided by a variety of observations.
including gravity-wave energies that increase with height (BALSLEY and CARTER.
1982; BALSLEY and GARELLO. 1985) and detailed gravity-wave studies in the
stratosphere, mesosphere, and lower thermosphere showing gravity-wave scales
that increase with height (SATO and WOODMAN. 1982; BARAT. 1983; SMITH and
FRITTS. 1983; MEEK et al.. 1985; FRITTS et al.. 1985). We should expect, how-
ever. that the energy at low wave numbers, and hence m,. will be considerably
more variable than the energy at m > m,. because the gravity-wave energy at
m < m, is determined by the strenghts of various gravity-wave sources at
lower levels, whereas that at m > m, is limited _ saturation processes.
108 .05
10 7
'_ 10 6
o
_ 10 5
_ 10 4
W
Q
10 3
0
10 2
o
a. 1° I
10 0
WAVELENGTH (km)
20 10 5 2 1 .5 .2 .1
I I I I I J I I
\
\
J \\
MESOS_\
\\_ N2
s6 4 16 3 ld 2
WAVENUMBER (cycle/m)
Figure 2. Model saturated gravity-wave spectrum scaled to a common N2
at three different heights. Note the constant (saturated) energies
for large wave numbers and the increase in the dominant vertical
scale with height.
248
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Evidence of a saturated gravity-wave spectrum throughout the atmosphere

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