ESCI 343 – Atmospheric Dynamics II Lesson 9 – Internal Gravity Waves References: An Introduction to Dynamic Meteorology (3rd edition), J.R. Holton Atmosphere-Ocean Dynamics, A.E. Gill Waves in Fluids, J. Lighthill Reading: Holton, 7.4.1 THE BRUNT-VÄISÄLÄ FREQUENCY Before progressing with an analysis of internal waves we should review the important concept of the Brunt-Väisälä frequency. The vertical acceleration on an air parcel is 1Dw pgDt zρ∂= − −∂ (1) where quantities with a ‘~’ character are properties of the air parcel, while those with an overbar are for the surrounding environment. Assuming that the atmosphere is in hydrostatic balance we can write pgzρ∂= −∂ (2) and so ( )1Dwg g gDtρ ρρρ ρ −= − − − = −   . (3) Defining the perturbation density as the difference in density between the parcel and its surrounding air at the same level, ρ ρ ρ′= −, (4) then (3) can be written as DwgDtρρ′= −. (5) If the parcel starts out at level 0z and has the same density as its environment, 0 0( ) ( )z zρ ρ= (6) and is displaced adiabatically a small vertical distance z, then its new density will can be expressed as a Taylor series expansion 0 0 0( ) ( ) ( )pz z z z z zz p zρ ρρ ρ ρ∂ ∂ ∂+ ≅ + = +∂ ∂ ∂ . (7) As parcels rise and expand their pressure instantaneously adjusts to be equal to that of the surrounding environment, so that2p p= (8) and therefore p pgz zρ∂ ∂= = −∂ ∂. (9) Furthermore, we know that 21sp cθρ ∂= ∂ . (10) Using (9) and (10), (7) becomes 0 02( ) ( )sgz z z zcρρ ρ+ ≅ −. (11) The density of the environment can also be expanded using Taylor series, 0 0( ) ( )z z z zzρρ ρ∂+ ≅ +∂. (12) From (11) and (12) the perturbation density, (4), becomes 2sgzz cρ ρρ ∂′= − + ∂  (13) and so (5) becomes 22sgg zz cDwDtρ ρρ ∂+ ∂ =. (14) For small displacements the denominator can be approximated as ρ ρ≅ (15) and so (14) is now 22sDw g gzDt z cρρ ∂= + ∂ . (16) Equation (16) has the form of 2220D zN zDt+ = (17) where 222sg gNz cρρ ∂≡ − + ∂ . (18) Equation (17) is a homogeneous 2nd-order differential equation. Although N 2 is a function of z, in those cases were is a constant then (17) has solutions given by 2 2( )N t N tz t Ae Be− − −= + . (19)3These solutions are fundamentally different depending on whether N 2 is positive or negative. If N 2 is positive, N itself if real, and solutions to (19) are ( )iNt iNtz t Ae Be−= + (20) which are oscillations having an angular frequency of N. N is therefore a fundamental frequency of the oscillation, and is referred to as the Brunt-Väisälä frequency (or buoyancy frequency). If N 2 is negative, then N itself is imaginary, and solutions to (19) are ( )N t N tz t Ae Be−= +. (21) These solutions are exponential with time, and are not oscillatory. The Brunt-Väisälä frequency frequency is directly related to the static stability of the atmosphere. N2 N Solutions for z(t) Static Stability positive real oscillations stable negative imaginary exponential growth unstable For the atmosphere 22sg gz cρρ∂>>∂ (22) and so we can get away with defining N 2 as 2gNzρρ∂≅ −∂. (23) Also, for an ideal gas, the Brunt-Väisälä frequency can be written in terms of potential temperature as 2gNzθθ∂=∂. (24) Equation (24) is valid for an ideal gas only, whereas (18) is true for any fluid. For an ideal gas, (18) and (24) are equivalent (see exercises). An additional result that will be of use in the next section is that from (13) and (18) we can derive a direct relationship between the Brunt-Väisälä frequency and the perturbation density, 2Nzgρρ′= −. (25) DISPERSION RELATION FOR PURE INTERNAL WAVES For the present discussion we will ignore changes in density due to local compression or expansion, which is a valid assumption as long as the waves are short compared to the scale at which the density changes with height (large values of wave4number). We will therefore use the linearized, anelastic continuity equation, so that the governing equations are xptu∂′∂−=∂′∂ρ1 (26) yptv∂′∂−=∂′∂ρ1 (27) gzptwρρρ′−∂′∂−=∂′∂1 (28) ∂′∂+∂′∂+∂′∂−=′zwyvxudzdwρρ (29) which when written in flux form are ( )xput∂′∂−=′∂∂ρ (30) ( )ypvt ∂′∂−=′∂∂ρ (31) ( )gzpwtρρ′−∂′∂−=′∂∂ (32) ( ) ( ) ( )0=′∂∂+′∂∂+′∂∂wzvyuxρρρ. (33) Equations (30) thru (33) are four equation in five unknowns (u′, v′, w′, p′, and ρ′). The fifth equation is found by taking 22t∂∂ of (25) to get ( )wtgNtwNgt′∂∂=∂′∂=∂′∂ρρρ2222 (34) Though we could write sinusoidal solutions for the five dependent variables and solve a 5×5 determinant to get the dispersion relation, this would be tedious. We can eliminate u′, v′, and w′ from the equations and reduce the number of equations to two as follows: • Take t∂∂of (33), and combining it with x∂∂of (30), y∂∂of (31), and z∂∂of (32) to get pgz′∇−=∂′∂21ρ. (35) • Eliminate w′ between (32) and (34) to get zpgNNt∂′∂−=′+∂′∂2222ρρ. (36)