Oct 20, 1994 - Printed in Great Britain. All rights reserved ..... in a good agreement with general nonlinear theory according to ..... 44, 1775. 1993. J. atmos. Sci.
JournalofAtmospheric and TerrestrialPhysics, Vol. 57, No. 11, pp. 1221 1231, 1995
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The nonlinear mechanism of gravity wave generation by meteorological motions in the atmosphere Alexander S. Medvedev *l and Nikolai M. Gavrilov2 ~York University North York, Ontario, Canada, M3J 1P3 2St Petersburg University, St. Petersburg-Petrodvorets, 198904, Russia (Received 20 October 1994; accepted in revised form 22 November t994)
Abstract--Using asymptotic expansions of the hydrodynamic equations in the Rossby number and the method of multiple time scales, we derive approximate expressionsfor the inhomogeneous "forcing" terms which describe the continuous generation of inertio-gravity waves by quasi-geostrophic motions. As a result of numerical modelling applied to the evolution of tropospheric meso- and macro-scale wave sources, the values of these forcing terms are estimated. A three-dimensionalnumerical simulation of wave propagation from a mesometeorologicaltropospheric eddy into the upper atmosphere was done to estimate the gravity wave response to the sources described. The results of the calculations show that the most part of the wave energy propagates quasi-horizontally carried by two-dimensionalinertio-gravity waves. At the same time, a part of the energy is transported into the upper atmosphere by internal-gravity waves which can create regions of wave disturbance in the upper atmosphere at considerable distances from the source site. The amplitudes of these waves increase with increasing intensity and decreasing time scales of the wave sources and can reach the values observed in the upper atmosphere.
1. INTRODUCTION It is widely accepted now that internal gravity waves (IGW) play an essential role in the general circulation of the middle atraosphere. Their ability to transfer upward momentum and energy and to deposit them in the upper layers through dissipation and/or breaking is well recognized. Progress has also been made in the development of parameterization schemes for gravity wave drag and in their applications to general circulation models of the atmosphere (Andrews et al., 1987; Fritts, 1989). All these parameterizations require knowledg,~ of gravity wave spectrum at the lower boundary. Due to our lack of knowledge of gravity wave sources as well as their morphology in the lower atmosphere, the characteristics of wave sources are often specified arbitrarily and may be considered as "tunablle parameters". Possible exceptions are parameterizations of orographically generated IGW, the physics of which is rather well substantiated (e.g. Palmer et al., 1986 ; McFarlane, 1987). A significant part of the gravity waves is generated
* On leave from St. Petersburg University, Atmospheric Physics Dept., St. Petersburg-Petrodvorets, 198904, Russia.
in the troposphere due to various mechanisms. The waves are observed in the atmosphere at almost all times and everywhere (Nastrom and Fritts, 1992). Hence, the problem arises of finding permanently active sources of wave motions in the atmosphere. This problem is important also because there are indications that wave sources other than flow over mountains are required to explain the momentum balance in the middle atmosphere (Rind et al., 1988 ; McLandress and McFarlane, 1993). Observations of the gravity wave spectrum in the mesosphere and lower thermosphere (Fritts, 1984) show that the considerable portion of IGW energy is due to nonzero-frequency waves which are associated with sources other than topography. One possible candidate for such a mechanism is the nonlinearity of atmospheric motions. Numerous experiments on the detection of atmospheric waves give evidence of wave excitation by turbulent motions of different scales. Wavelengths and periods of generated waves depend on the scales of synoptic and turbulent motions. Small-scale turbulence excites acoustic waves. Meso-scale turbulence is a source of IGW, while large-scale synoptic motions generate low-frequency internal gravity and planetary waves. A theory of sound generation by small-scale tur-
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A. S. Medvedev and N. M. Gavrilov
bulence was given by Lighthill (1952, 1978). Excitation of internal gravity waves by meso-scale turbulence was considered by Stein (1967). Numerical simulations of gravity waves generation by different kinds of atmospheric motions have been made recently (Fovell et al., 1992 ; Snyder et al., 1993). These models are based on the full set of primitive hydrodynamical equations and require as much computations as numerical models of atmospheric general circulation. At the same time, analysis of experimental data and parameterization of the waves influence of gravity on the general circulation require simplified approaches and formulae for description of the main processes leading to gravity wave generation. There is the dynamical classic theory based on the works by Obukhov (1949) and Monin (1958) and reviewed by Blumen (1972) which describes adaptation of large-scale pressure and wind fields to the geostrophic equilibrium state. According to this theory the process of "geostrophic adjustment" is accompanied by gravity wave generation. In the linear theory the source of gravity wave motions is the initial geostrophic imbalance of wind and pressure fields, and the reason for this imbalance is not considered in the classic theory and may be treated as being external. Zhu and Holton (1987) have applied this linear theory to show the possibility of wave excitation due to local forcing caused by breaking gravity waves. Fritts and Luo, 1992 have considered the I G W generation due to initial geostrophic imbalance in jet streams. Besides "external" reasons there is the universal "internal" one. This is the nonlinearity of atmospheric motions. The qualitative consequence of the classic theory (Monin, 1958) is that wave motions appear as a result of continuous competition between the disturbance of mass and momentum geostrophic balance due to nonlinear advection and the tendency of the atmosphere to establish a quasi-geostrophic equilibrium. Recent works (Gavrilov, 1987; Medvedev, 1987) show that a useful tool for study of the processes of gravity wave generation could be the multiple time scale expansions into series in Rossby or Mach numbers. In this paper the method is applied to develop approximate expressions for the inhomogeneous "forcing" terms that describe the continuous generation of inertio-gravity waves by background quasi-geostrophic motions. In Section 3 a brief description of the numerical model for tropospheric meso- and macro-scale sources and the results of numerical calculations of real and model wave sources are given. In order to estimate the typical amplitudes of gravity waves in the middle atmosphere created by non-linear wave sources we have carried out threedimensional numerical modelling of the wave propa-
gation from a typical tropospheric vortex into the upper atmosphere using simplified background conditions. The results are described in Section 4. 2. THEORY
The starting point of our consideration is the set of primitive equations for three-dimensional atmosphere taken in the form (Pedlosky, 1987) :
du dt-fv= dw
~_+g=
_p_j Op dv 3p ~x' dt +fu= _p-1 @'
_p-1
Op dp Oz' dt
I-p
dO
d~-t= O, ® = T
(Ou
3v
Ow)
~xx + ~yy + ~-z
p=pnT,
=0"
' (1)
where d / d t = O/Ot+ uO/Ox+ vO/Oy+ w6/Oz; t is time ; u, v and w are horizontal and vertical velocity components along axes x, y and z, respectively ; p, p and T are pressure, density and temperature; O is the potential temperature; Ps0 is the globally averaged pressure at the surface level; g is the acceleration of gravity; cp is the specific heat capacity at constant pressure; R is the gas constant ; f i s the Coriolis parameter which is considered here to be a constant. Then we introduce dimensionless variables as follows [cf. Pedlosky, 1987]:
(x,y)=L(x',y'), z=Dz', ¢ = f It', D (u, v) = U(u', v'), w = T Uw', p =p~(z) + ps(z) UfLp', p = ps(z) (1 + Ro Fp'), (2) where the prime denotes dimensionless variables, p, and p~ are globally averaged pressure and density for which the relation dp~(z)/dz = -p~(z)g is valid, L and D are horizontal and vertical scales of length, U is the scale of horizontal velocity, and f - t is the "natural" temporal scale. We assume all the scales are chosen in such a way that corresponding dimensionless variables do not exceed unity. In (2) the dimensionless Rossby Ro and Froude F numbers were introduced :
U fZL2 Ro =7£, F= gD "
(3)
Equations (1) in dimensionless form are as follows (hereafter we omit the prime for dimensionless variables for simplicity)
g÷e°~U~x4-Vfffy"FW Oz
I+Ro Fp ~3x (4)
Nonlinear mechanism of gravity wave generation
dv
dv
dv W~z
d~ + Ro U~x +rOy +
[
+u=
1+ Ro Fp dy
(
w)l
(5)
dw + Ro U~x dw + V~yy + W~z 62(1+ Ro Fp) ~z
=-pg
%
+.o
jd ~zp,p-p,
.( dp, % -~-(1-~ Ro Fp)~{~x ~- ~y -~-ps l ljzpsW) : 0
oo
; do
dO+wdO\ ,
-~z + R°~U~x +t'~y
dz ]
+~
w dos(1 dz + Ro FO) = O,
1 dos
S=®,F dz
1223 N2D2 N2= Y d®s
f2LZ'
DOs dz'
(7)
where S is the static stability parameter and N is Brunt-V~iis~ilfi frequency. The nonlinear advective terms in (4)-(7) have the factor Ro, which is small for synoptic scale motions. If one searches for the solution to the equation set (7) as a conventional series in Ro, terms of the series will grow in time proportional to powers of (zRo)~oo at z-~oo. So, the conventional method of expansion of hydrodynamic variables into asymptotic series over the perturbation parameter Ro converges only for small z, when zRo < 1. In order to overcome this difficulty with secular terms, we shall use the method of multiple timescales (Blumen, 1972; Jeffrey and Kawahara, 1982) where variables in the hydrodynamic equations are presented as the asymptotic series
r~r + Ro-~T+ Ro-2T,+... ; O/dz~ d/dz + Ro d/d T+ Ro2d/d7".+... ;
f2L2 l n ( l + R o F~)=~ln(l + R o p ~ p ) - l n ( l + Ro Fp).
F= ~ (Ro)kFk(x,y,z,z,T,T.,...), k=0
(6) In the above set of equations the 6 = D/L and scaling for potential temperature in dimensional form ® . = O s ( l + R o F~9) were introduced, where In ®s(z) = y-llnp~(z) -lnps(z) + const and y is the ratio of specific heat capacities. As usual, we have to choose definite scales thus restricting our consideration only to a certain class of atmospheric motions. For synoptic scale motions in the atmosphere with L ~ 1000 km, U ~ 10 ms -J, D ~ 10 km and f ~ 10 -4 s -J, we have Ro
