423
The smallest possible truncation of the expansion (29) that still retains some nonlinear interactions between the modes and coupling to
the topography is a triangular one that includes only the wavenumbers
(m, n) = (0,1), (0,2), (1, 1), (-1,1). The dynamics of the mode (0,1) is
trivial and uncoupled with the other modes, so we end up with a smallest
model that contains the nine degrees offreedom 1>0,2,1>1,1,80,2,81,1, U, V and
c (remember that 1>u and 8u are complex). An extended model with 21
degrees of freedom - which however is not accessible by analytical means
- is presented in Volker (1995).
The model equations are now obtained by projection. We simplify the
notation by defining some parameters
2To
T02= -
31T
The Emn and /-lmn are geometrically modified friction parameters and I
is the phase speed of the baroclinic Rossby waves. By projecting the
barotropic balance (20) on the wavenumbers (0,2) and (1,1) we obtain
three barotropic equations (the first is real, the second is complex)
o
bo
-E02 (1)02 - he02 ) + £l'R [1>u - heu] + 4T02
(30)
o
.f3
bo
-EU (1)11 - heu) + 2 g 1>U - 2£ [1>02 + U - h(802 + V)] (31)
where bu = -ibo/2 has been used for the expansion coefficient of the
topography. In the same way we obtain from the baroclinic balance (21)
the following three equations
1 d
- - (802 + V)
O"dt
1 d
--8 11
O"dt
8
2 0.< [A. 8*]
1 + h
- /-l02 02 + £O":.s 'Pu U + 4 - h - T02
(32)
-/-lu8u + i £: (1)11 + 811) -
2
- £0" [(1)02 + U)811 -1>11(802 + V)]
(33)