Atmosphere CCM2 model. For the range of anticipated variability of source spectra in the troposphere the scheme produces plausible results consistent with ...
The Parameterization of Gravity Wave Drag Based on the Nonlinear Diffusion of Wave Spectra
Alexander S. Medvedev and Gary P. Klaassen Department of Earth and Atmospheric Science York University North York, Ontario, M3J IP3 Canada Byron A. Boville National Center for Atmospheric Research Climate Modeling Section P.O. Box 3000, Boulder, CO 80307 USA Abstract We present a new parameterization of gravity wave drag based on the Medvedev and Klaassen [1995] theory of gravity v,,'ave spectral evolution and saturation. The scheme has been tested in a column model as well as with a mechanistic version of three-dimensional Middle Atmosphere CCM2 model. For the range of anticipated variability of source spectra in the troposphere the scheme produces plausible results consistent with observations and with theoretical estimates. Results also indicate that the specification of seasonal and latitudinal variability of wave sources is required for three-dimensional modeling. 1
Introduction
Gravity wave (GW) drag deposited by waves propagating from below substantially affects the circulation of the middle atmosphere. Parameterization of these processes in general circulation models requires the specification of an absorption mechanism which causes damping of waves, and therefore, a deposition of their momentum to the mean flow. A manifestation of systematic wave damping processes in the atmosphere is the apparent "universality" of observed gravity wave spectra. Perhaps the most significant feature of "universal" spectra is the powerlaw behavior of the high vertical wavenumber tails of their power spectral densities (PSD), S. The latter can be approximated by S(m) = A oN 2/ m 3 , where N is the Brunt-Vaisala frequency, m is the vertical wavenumber, and Ao is the spectral amplitude. It is this "saturation" of NATO ASI Series. Vol. I 50 Gravity Wave Processes Their Parameterization in Global Climate Models Edited by Kevin Hamilton © Springer.Verlag Berlin Heidelberg 1997
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wave amplitudes with height, despite the diminishing gas density, which implies some kind of wave damping mechanism. Many'theories have been devised to explain the observed spectral structure of the variance field. For example, Zhu [1994] invokes offresonant wave-wave interactions as well as radiative damping, while Gardner [1994] attributes damping to molecular and/or turbulent diffusion. Some theories assume that the saturated behavior is due to the convective and/or dynamical instability of wave harmonics or wave packets while they propagate upward independently [Dewan and Good, 1986; Smith et al., 1987]. Other theories attribute saturation to nonlinear interactions between harmonics within the broad spectra of gravity waves. In Weinstock's [1990] approach harmonics with smaller m are considered as a source of enhanced scale-dependent diffusion which can affect the growth of lower m harmonics with height. The Doppler spread theory [Hines, 1991b; 1993a] treats the irregular fluctuations of the wave field as an additional background which force harmonics to spread over the spectrum. Hines [1993a] argued that the principle effect of this spreading is to shift wave harmonics from the lower m part of the spectrum to the high-m portion where they dissipate due to instabilities and/or diffusion. Further critical assessment of these theories can be found in [Hines, 1991a; 1993b]. In [Medvedev and Klaassen, 1995] (hereafter referred to as MK95) we employed Weinstock's [1982] technique and Hines' physical insight regarding nonlinear Doppler effects to develop a more general approach for describing the evolution of GW spectra in the atmosphere. There we assumed that only longer period waves in the spectrum constitute an additional background wind for a given wave component. This lowfrequency, spectrum-induced mean wind produces Doppler shifting as well as modified critical levels for higher-frequency harmonics. Interactions near these spectrum induced critical lines result in wave damping and, eventually, in wave obliteration. We have shown in MK95 that these spectral interactions can alternatively be treated in terms of nonlinear enhanced diffusion in the spirit of Weinstock [1990]. The MK95 theory distinguishes between wave breaking and wave saturation, and gives the criteria for both processes in terms of wave spectra. It also predicts the development of a "universal" saturated shape for an arbitrary source spectrum of GW. The predictions of the MK95 theory for a single wave under saturation conditions coincide with those of Lindzen [1981]. The purpose of this paper is to present a practical gravity wave drag parameterization based on the MK95 theory, which can be used in large-
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scale circulation models. Parameterizations usually imply further simplifications of the theories they are based on as well as an introduction of some tunable parameters. The scheme we present will retain all physically important features of MK95, and relies on relatively few tunable parameters apart from the source spectrum. To date we are aware of three other spectral GW drag schemes which have been developed and tested in atmospheric models. One approach is to consider harmonics in the spectrum as propagating independently, with each harmonic attaining saturation according to Lindzen's [1981J scheme. Although this approach has been successfully used in a number of numerical models (e.g., [Garcia and Solomon, 1985]), it tends to generate step-function GW drag profiles where the momentum deposition is concentrated at and above the breaking levels. The spectral parameterization developed by Fritts and Lu [1993] combines theoretical aspects of the assumed instability mechanism of wave saturation, and the semi-empirical a priori form of GW spectra. Since observed instantaneous spectra usually deviate from the averaged "universal" shape, one may expect that this approach will be successful in reproducing wave drag profiles when GW spectra are close to the average condition, and will deviate when wave spectra differ from the "universal" one. The parameterization based on Doppler spreading theory [Hines, 1996], rather than tracking individual harmonics, calculates the evolution with height of average spectral parameters such as Rl\llS velocity. In this paper we present a gravity wave drag parameterization based on the MK95 theory. In Section 2 we review the relevant formulae from MK95, while in Section 3 we discuss numerical aspects of the.scheme. The range of variability of input source spectra are discussed in Section 4. In Section 5, we discuss the results of column-model tests based on representative distributions of mean fields from CIRA-86, and in Section 6 we present tests with a three-dimensional mechanistic version of the Middle Atmosphere CCM2 model (MA CCM2). 2
Theory
According to MK95, the evolution of gravity wave spectra with height is given by the following equation
dS(mR) = (_poz dz
. Po
+ mRz mR
_ f3)S(mR),
(1)
where S is the power spectral density (PSD) of horizontal wind associated with gravity waves, Po is the mean density, mR is the real part of
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the vertical wavenumber, and f3 is the coefficient of nonlinear damping due to interactions of the harmonic mR with other harmonics in the spectrum. The vertical wavenumber mR accounts for an additional nonlinear Doppler shift produced by lower frequency waves in the spectrum and is related to the "linear" vertical wavenumber mo by the expression
(2) In (2) the following notation for the error function of an imaginary argument is used 2 t> 2 erfi(a) = Vi Jo exp(x )dx, and the dimensionless parameter a is defined by N a =
c-u
...;2mR(7 = ...;2(7'
(72 =
{'Xl
JmR
S(m')dm',
(3)
where N is the buoyancy frequency, and (7 is the horizontal RMS wind created by all wave harmonics in the spectrum with vertical wavenumbers larger than that of a given harmonic. Note here that if the spectrum consists of a single monochromatic wave, the RMS velocity (7 coincides with the wave amplitude u', and the parameter a is proportional an inverse Froude number F r = mu' / N for this wave. The convective instability threshold for monochromatic waves is u' ---+1 c - u I, where c is the horizontal phase velocity of the wave, and u is the mean wind. In the case of a broad spectrum our theory gives the instability threshold (7 ---+1 c - u I, or a ---+ 1/...;2. Therefore, it is convenient to consider the parameter a as a straightforward generalization of the inverse Froude number (or equivalently as the square root of the Richardson number) for the case of a broad wave spectrum. The expression for the nonlinear damping coefficient has the form
(4) The set of equations (1)-(4) for calculating the evolution of the PSD S with height becomes closed if we add the expression for the dependence of the "linear" vertical wavenumber mo on background parameters in the form N(2)
(2f =
N(l)
W -
~u,
(5)
mo mo where superscripts 1 and 2 denote quantities at two height levels, and ~u is the mean wind shear between these two levels. (5) follows from the dispersion relation for gravity waves and represents the variations of
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the apparent vertical wavenumber of a harmonic with the constant phase velocity due to vertically inhomogeneous mean wind and temperature. Once the amplitude of the harmonic S(m) and the coefficient of nonlinear damping (3(m) are known, the momentum deposition associated with this wave can be found from a;z:(m) = S(m)(3(m)k h , m
(6)
where kh is the horizontal wavenumber. Note here, that the evolution of spectra is entirely described by (1 )-(5) only in terms of vertical wavenumber, while the intensity of wave drag a;z:(m) is proportional to k h • In our scheme, kh enters only as a scaling factor for the momentum deposition, and is consequently the only tunable parameter in the scheme apart the initial spectrum itself. The total wave drag created by the entire spectrum is calculated by adding up the contributions from all harmonics. MK95 have shown that gravity wave saturation in the atmosphere typically occurs at 0 have m > 0, and "intrawaves" with c-il < 0 are formally described by m < O. The resulting wave drag is calculated as a sum of two profiles (generally it is a difference), while the RMS ve-
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100
ZONAL MEAN WIND
50N50S ....
90 80 ~
:.:: W
70 60
0
50
f-
40
«
30
~
i= ..J
20 10
q U.
MIS
figure 2: Mean zonal wind for January at 500 N (solid line) and .50 0 g (dashed line) latitudes from the CIRA-86 model used in the parameterization tests.
locity is created by waves going in both directions. We now consider the sensitivity of the proposed scheme to spectral resolution by discretizing the same source spectrum with varying numbers of harmonics. Fig. 3 exhibits profiles obtained for the "modified Desaubies" source spectrum (12) with s = 1, m. = 0.006 rad m- 1, and the amplitude A o = 100 m 3 s-2, employing M = 200 (which we regard as an "exact" solution), 30, 15, and 5 harmonics. It is seen from the figure that below 90 km, the resulting wave drag is almost indistinguishable for M = 200, 30, and 15 harmonics, while the M = 5 case still provides reas0aable accuracy. The higher sensitivity of the results to variations of M above 90 km are related to an insufficient resolution of lower-m/higher-c portion of the spectrum which contains the most of wave energy at these heights. The accuracy of the parameterization above the mesopause could be increased if one were to employ variable 6.m-grids with resolution concentrated toward lower vertical wavenumbers. We next consider behaviour of the parameterization scheme when various source spectra are specified. Fig. 4a presents profiles of gravity wave drag for a "winter" distribution of mean wind employing the "modified Desaubies" source spectra (12) with s = 1 and m. = 0.006 rad m- 1 approximated by M = 15 harmonics. Profiles are shown for A o = 50, 100, and 500 m 3 s-2. It is seen from the figure that the factor of ten increase in the overall spectral amplitude increases the wave drag peak value from rv -70 to rv -180 m S-1 day-I, although we note that the response is not
319
95
JANUARY, SON
90
:E
85
.,'"
~
ui
M=200M=30···· M=15 M= 5···
80
0
::;)
I-
i=
oJ
q:
75 70 65
ACCELERATION, MlSlDAY
0
Figure 3: GW drag profiles calculated for "winter" conditions (50 0 N January) utilizing At = 200, 30, 15 and 5 harmonics in the model spectrum,
linear. The heights of drag maxima decrease from""'" 80 km for 04 0 = 50 to ,. . ., 70 km for A o = 500 m 3 s-2 reflecting the obvious fact that waves \'v'ith weaker amplitudes generally saturate at higher altitude. The RMS horizontal wind predicted by the parameterization can also be compared qualitatively to that deduced from observations. In the Fig. 4b we have plotted (j calculated over the entire spectrum as a function of altitude. At the lower boundary (10 km) the RMS velocity associated with the input spectra varies between 1 and 2 m s-1, which is in a good accordance with observations (e.g., [Sato, 1994]), and with values recommended for use in gravity wave drag models (e.g., [Fritts and Lu, 1993]). In the middle atmosphere the calculated profiles exhibit RMS velocity fluctuations up to a few tens m S-1, which are also consistent with observations [Tsuda et al., 1994]. However, the RMS wind reaches a maximum near z ,. . ., 60 - 70 km (slightly below the peaks of momentum deposition) and diminishes at higher altitudes. Observations, on the other hand, indicate the RMS wind should increase slowly above 60-70 km. The explanation for this difference may be found in the limited range of our spectrum. To produce stronger parameterized RMS winds at higher altitudes, one must decrease mmin of the input spectrum. It is natural to anticipate that upon doing so, the peaks in RMS velocity profiles will flatten, and RMS profiles will vary more slowly with height. However, we have chosen not to do this owing to the large uncertainty in the shape of gravity wave spectra at lower m.
320 JANUARY SON
95 90f
90
(a)
85
..
::; 80
"W 0
>- 70 ~ -' 65
(b)
80 70
60
0
~ ~
-'
