4.1 Frequency and scales

Oceanic Rossby waves with a frequency of $\omega_r=2\cdot 10^{-8}\mbox{~s}^{-1}$ have a wave period of 10 years or so and thus could be implicated in decadal variability. This then implies a zonal wavenumber of $k=\pi /5500\mbox{~km}$ (for $L_o=45 \mbox{~km}$ and $\beta =1.8\cdot 10^{-11}\mbox{~s}^{-1}\mbox{~m}^{-1}$), a scale comparable to that of an ocean basin, and commensurate with, for example, the scales of the leading modes of variability found by Deser & Blackmon (1993) and Cayan (1992). It turns out that the modification of the real part of the phase speed associated with coupling (the second term in (43) is comparatively small (see below) and does not make a significant difference to the phase speed. Our advection and entrainment coupled modes propagate at essentially the speed of internal oceanic Rossby waves.

In figure 7, $\mu$ is plotted as a function of $\widehat{U}$ and $\widetilde{U}$ for a wave of size comparable to the NAO; $k=\pi /5500\mbox{~km}$ and $l=\pi /3200\mbox{~km}$. For $\widehat{U}>\frac{\widehat{\beta }}{\kappa ^{2}}=28 \mbox{~m/s}$, $\mu$ is positive, implying an atmospheric response which switches sign betwen upper and lower levels, leading to a decaying mode. In the lower left part of the figure, $0>\mu >-2$, again implying damping. An equivalent barotropic response (and therefore a growing mode) will occur if the zonal winds fall in the central triangular region. This can readily be achieved by typical middle-latitude tropospheric winds.

  

Figure: Contours of $\mu$ from equation (31) as a function of barotropic ( $\widehat{U}= u_1 + u_2$) and baroclinic ( $\widetilde{U}= u_1 - u_2$) wind speed, for a particular choice of wavenumber ( $k=\frac{\pi}{5500\mbox{km}}$, $l = \frac{\pi}{3200\mbox{km}}$.) A growing coupled mode is possible when $\mu < -2$ (the shaded region of the figure).