1.2 Wave Equations in Geophysical Fluid Dynamics
19
Sinee the eoefficient matriees for the first-order derivatives in (1.47) are symmetric, the linearized Euler equations are a hyperbolie system. The eigenvalues of AI
are U. U. U + Cs. and U - Cs; those of A2 are O. O. cS. and -cs . The eigenvalues
involving C s give the speed at which sound waves in an unstratified fluid propagate parallel to the x and z eoordinate axes. As will be demonstrated below, sound
waves in an isothermally stratified atmosphere aetually propagate at slightly different speeds due to the influenee of the zero-order term in (1.47). The remaining
eigenvalues relate to the speed at which fluid pareels are advected horizontally
and vertieally by the mean flow. These eigenvalues have no relation to the propagation of gravity (or buoyaney) waves, which are the second type of fundamental
wave motion supported by (1.47).
When the basic state is isotherrnal, S is constant, and wave solutions to (1.47)
exist in the form
( u , w.
- - (J- .17: -) =
m
.n
{( Uo. Wo. o 0,17:0 e i (kX+lz - wt ) } •
provided that w, k, and (. satisfy the dispersion relation
(w-Uk)2= c; (k2 +(.2 + N
2+S2)
4N _
2 k 2 ] 1/2
2
± c2
...!. [ ( k2 + (.2 + N 2 + S2)2
(1.48)
(1.49)
(w - Uk)
2 - -:-;:;-----;;----=:----:::----;::
- k 2 + (.2 + (sl +
(1.50)
2
whieh is obtained by substituting (1.48) into (1.47) . As will be diseussed in Seetion 7.2.4, the seeond term inside the square root is mueh smaller than the first
term in most applieations, so (1.49) ean be separated into a pair of approximate
dispersion relations for the sound waves and the gravity waves. The dispersion
relation for the sound waves,
(w - Uk)2 = c; (k 2 + (.2) + S2 + N 2•
is obtained by taking the positive root in (1.49). In a manner analogous to the effeet of the Coriolis force on gravity waves in the shallow-water systern, the terms
involving the produet of N or S with the zero-order derivatives of the unknown
variables introduee a slight diserepaney between the phase speeds and group velocities of the aetual sound waves and those that might be suggested by the eigenvalues of AI and A2.
The dispersion relation for the gravity waves is obtained by taking the negative
root in (1.49), which to a good approximation yields
N
2k2
Neither the phase speeds nor the group veloeities of these waves have any relation to the eigenvalues of AI and A2. Unlike sound waves, gravity waves do
19
Sinee the eoefficient matriees for the first-order derivatives in (1.47) are symmetric, the linearized Euler equations are a hyperbolie system. The eigenvalues of AI
are U. U. U + Cs. and U - Cs; those of A2 are O. O. cS. and -cs . The eigenvalues
involving C s give the speed at which sound waves in an unstratified fluid propagate parallel to the x and z eoordinate axes. As will be demonstrated below, sound
waves in an isothermally stratified atmosphere aetually propagate at slightly different speeds due to the influenee of the zero-order term in (1.47). The remaining
eigenvalues relate to the speed at which fluid pareels are advected horizontally
and vertieally by the mean flow. These eigenvalues have no relation to the propagation of gravity (or buoyaney) waves, which are the second type of fundamental
wave motion supported by (1.47).
When the basic state is isotherrnal, S is constant, and wave solutions to (1.47)
exist in the form
( u , w.
- - (J- .17: -) =
m
.n
{( Uo. Wo. o 0,17:0 e i (kX+lz - wt ) } •
provided that w, k, and (. satisfy the dispersion relation
(w-Uk)2= c; (k2 +(.2 + N
2+S2)
4N _
2 k 2 ] 1/2
2
± c2
...!. [ ( k2 + (.2 + N 2 + S2)2
(1.48)
(1.49)
(w - Uk)
2 - -:-;:;-----;;----=:----:::----;::
- k 2 + (.2 + (sl +
(1.50)
2
whieh is obtained by substituting (1.48) into (1.47) . As will be diseussed in Seetion 7.2.4, the seeond term inside the square root is mueh smaller than the first
term in most applieations, so (1.49) ean be separated into a pair of approximate
dispersion relations for the sound waves and the gravity waves. The dispersion
relation for the sound waves,
(w - Uk)2 = c; (k 2 + (.2) + S2 + N 2•
is obtained by taking the positive root in (1.49). In a manner analogous to the effeet of the Coriolis force on gravity waves in the shallow-water systern, the terms
involving the produet of N or S with the zero-order derivatives of the unknown
variables introduee a slight diserepaney between the phase speeds and group velocities of the aetual sound waves and those that might be suggested by the eigenvalues of AI and A2.
The dispersion relation for the gravity waves is obtained by taking the negative
root in (1.49), which to a good approximation yields
N
2k2
Neither the phase speeds nor the group veloeities of these waves have any relation to the eigenvalues of AI and A2. Unlike sound waves, gravity waves do