Mathematical treatment of atmospheric gravity waves
This topic is an enormous one and a student who seriously wants to get to know the literature must read a number of text books. The best one to start with is probably ‘Atmospheric Gravity Waves’ by Carmen J. Nappo (Academic Press) which presents an overview of the subject without too much detail. The most authoritative on internal waves in fluids is probably Lighthill’s classic ‘Waves in Fluids’ which helps place atmospheric gravity waves in the wider context of fluid mechanics.
However, most text books on atmospheric dynamics contain some introduction to gravity waves. The treatment here is very simple and assumes that the student is familiar with the fundamental equations of atmospheric dynamics as applied to the atmosphere.
As usual with mathematical physics the problem facing the student has as much to do with understanding the symbols used for different variables as it has with understanding the maths itself. This short document is no exception. So, here is a brief summary of my symbol conventions:
duuuuu))))Full differential: D Duuvw！！？？？dttxyz))))
duPartial differential: subscript Duuuuvuwu！！？？？txyzdt
Fundamental equations of adiabatic, inviscid atmospheric dynamics. These can be found in any dynamics text book but are written down slightly differently in each of them:
1Momentum equation: 1-3 DpfgVkVk！，?，?， ；
Continuity equation 4 D；！，；?.V
Adiabatic equation Dθ = 0 5
Equation of state p = ρrT 6
where V is the 3-D velocity vector, p is pressure, f the Coriolis parameter, g the acceleration due to gravity, k a unit vertical vector, ρ is density, T is temperature and θ potential temperature.
Gravity wave solutions
Equations 1-6 can describe all possible motions in the atmosphere, from sound waves and turbulence at the very small scales to the general circulation at the largest. In seeking wave solutions to a dynamical system the standard method in theoretical physics is to define a basic, slowly-varying or constant state for the system and linearise the equations about that state – i.e:
a) define each quantity where the overbar denotes the background (or mean) state and qqq！？'
the prime denotes departures from this state
b) expand equations 1-6 and omit all terms with a product of prime terms
c) solve the resulting set of linear equations by postulating wave-like solutions. Fourier analysis allows any arbitrary disturbance to be expressed as a linear superposition of these waves.
Note that the monochromatic wave solutions derived here and in all the text books do not correspond to actual atmospheric disturbances, which must necessarily consist of a superposition of these elementary components (monochromatic waves exist at all points in space and time; real gravity waves obviously do not).
The gravity-wave solutions we find depend on the basic state we choose to expand around – one
reason different text books give different expressions for gravity waves. Furthermore, this method, applied to the fundamental equations, describe more than just gravity waves – they describe sound
waves and a few other types as well. We will therefore make approximations to our equations to remove some of these undesirable solutions.
Simplest case: stationary, hydrostatic, non-rotating horizontally uniform basic state.
are all zero, so DV in equations 1-3 simply becomes the vector Stationary means uvw, and
???. Non-rotating means we drop the Coriolis terms and hydrostatic means . (,,)uvwp！，；gzttt
Horizontally uniform means are functions only of z. We also make two further p, and ；：
assumptions: that the flow is incompressible (?.V = 0) and . The latter assumption ：：；；'/ = -'/
removes the sound wave solutions from our equations: compressibility of air is a phenomenon associated with sound and the assumption that density fluctuations are caused by temperature rather
?than pressure suppresses the physics of sound waves. (Formally, this, and the neglect of in the ；
momentum equations except when combined with g, is known as the Boussinesq approximation).
Armed with all these assumptions we can write the five fundamental equations as follows:
1 …………………….. 1a ?? up + = 0tx；
1 …………………….. 2a ?? vp + = 0ty；
?；1; on substituting the hydrostatic equation and we get: ：：；；'/ = -'/?? wp + - p = 0tzz2；；
?：1…………………..3a ??wp + - g = 0tz；：
???……………………….4a uvw + + = 0xyz
??…………………………..5a + w = 0：：tz
We have a system of five linear equations in five unknowns. In fact, we have chosen such a simple basic state that there is no need to retain both horizontal directions: we can choose x to lie in
??whichever direction we like. Setting to zero reduces the set to four equations. Since these vp and y
are homogeneous (no constants or forcing functions) we seek harmonic solutions:
????= (U, W, P, ，) exp i(kx + mz - ~t) (,,,)uwp：
where the capitalised variables are the amplitudes of the corresponding primed variable, and k and m are the horizontal and vertical wavenumbers (components of the wave vector). Thus:
，~； 0 k/ 0，，U，，????W 0 -i im/ -g/~；：???? = 0????Pk m 0 0??????，0 0 -i：~：?z：?
Setting the determinant of the matrix to zero gives us the dispersion relation:
?Nk ~ = 22 ()km？
g2where ; N is the Brunt-Vaisala frequency. The amplitudes U, W, P and ， are related by = N：z：
the so-called polarisation relations:
2~/k)W; ， = (i/~)W U = -(m/k)W; P = -(m；：z
For a wave travelling towards the east, k and ~ are both positive. m can be either positive or negative, depending on whether the energy is propagating up or down. This in turn follows the
group velocity c = ?~. We can find this by differentiating the dispersion equation: gk
2，，)~)~，NmNmk，， = , = ,c??33g??2222??22k))m()()kmkm？？：?：?
For k,m>0, the group propagation is eastward and downward, so negative m corresponds to upward
energy propagation. Note that the wave vector k = (k,m) is perpendicular to the group velocity (k.c=0), confirming that the group propagation is along the phase lines. g
In the case of an upward-propagating wave (i.e. energy propagation upwards), m<0 and the phase
fronts move downwards. (Group and phase propagation are in opposite directions – exactly opposite to most waves we encounter, e.g. surface water waves). Then U and W are in phase – the phase
fronts tilt upwards and to the east. Pressure is in phase with U and W while ： leads by 90?. The plot
below shows a gravity wave pattern of this kind, with variation plotted as a function of x for fixed
values of z, and all at the same time.
u, w and p
： variation corresponding
to bold dashed line
Vertical distance (arbitrary units)-5.0
Horizontal distance (arbitrary units)
Individual air parcels execute slantwise positive – hence ： leads u and v in the motion as shown opposite, where the line diagram above.
depicts the path of the parcel. Maximum u
and w are then at the centre of the line. As the
motion is adiabatic, ： is conserved for the
parcels – but a horizontal section through the