Stratification
191
tan m, i.e. m 1 ∈ (π/2, 3π/2).F o rS → 0 and fixed wavevector k,t h e r ei sa n
intersection for m =0 . From (8.59c), we see that this is exactly the barotropic
mode. For S = O(1) and γ → 0,wefindm i = iπ, i.e. the Rossby waves for a
flat bottom, including the barotropic mode.
For m 2 < 0,w ed e fi n eμ 2 = −m 2 > 0 and the solution for Φ(z) and the
dispersion relation (directly from (8.60) through substitution m = iμ)are
Φ(z)=A cosh μz,
(8.61a)
μ tanh μ = −
γkS
σ
⇒ tanh μ = −
γ
β
(μ −
S(k 2 + l 2 )
μ
). (8.61b)
There is now only a single eigenvalue (Fig. 8.7b), which has special properties.
The eigenfunction has a maximum amplitude at the bottom and decreases exponentially upwards; this type of mode is a so-called “bottom trapped” mode. For
S → 0 and fixed k, the dispersion relation becomes
σ = −(γ + β)
k
k 2 + l 2 .
(8.62)
This is exactly the Rossby wave frequency for which bottom slope and β-effect
are additive.
191
tan m, i.e. m 1 ∈ (π/2, 3π/2).F o rS → 0 and fixed wavevector k,t h e r ei sa n
intersection for m =0 . From (8.59c), we see that this is exactly the barotropic
mode. For S = O(1) and γ → 0,wefindm i = iπ, i.e. the Rossby waves for a
flat bottom, including the barotropic mode.
For m 2 < 0,w ed e fi n eμ 2 = −m 2 > 0 and the solution for Φ(z) and the
dispersion relation (directly from (8.60) through substitution m = iμ)are
Φ(z)=A cosh μz,
(8.61a)
μ tanh μ = −
γkS
σ
⇒ tanh μ = −
γ
β
(μ −
S(k 2 + l 2 )
μ
). (8.61b)
There is now only a single eigenvalue (Fig. 8.7b), which has special properties.
The eigenfunction has a maximum amplitude at the bottom and decreases exponentially upwards; this type of mode is a so-called “bottom trapped” mode. For
S → 0 and fixed k, the dispersion relation becomes
σ = −(γ + β)
k
k 2 + l 2 .
(8.62)
This is exactly the Rossby wave frequency for which bottom slope and β-effect
are additive.