28.2.4 analytic_zonal_winds

F. Bryan (1987) introduced an idealized wind stress which has been used in many subsequent studies (e.g., Weaver and Sarachik 1991)

$$\tau^{\lambda} 0.2 - 0.8 \, \sin(6 \, \vert\phi\vert) - 0.5 \, [1 - \tanh(10 \, \vert\phi\vert)] - 0.5 \, \biggl( 1 - \tanh[10(\pi/2 - \vert\phi\vert)] \biggr)$$ =

$$-0.8 \, [\sin(6 \, \vert\phi\vert) + 1] + \frac{ \tanh(5\pi - 10\vert\phi\vert) + \tanh(10\vert\phi\vert)}{2}$$ = (28.7)

$$\tau^{\phi}$$ = 0. (28.8)

The units are dyne/cm² for the wind stress, and the latitude $\phi$ is in radians. The curl of this wind stress is given by

$$\nabla \wedge \vec{\tau} -\hat{z} \, a^{-1} \, \partial_{\phi} \, \tau^{\lambda}$$ =

$$-\frac{ \hat{z}}{a} \biggl[ \frac{24}{5} \, \cos(6\phi) + 5 \, \mbox{sech}^{2} (5\pi - 10\vert\phi\vert) - 5 \, \mbox{sech}^{2}(10\phi) \biggr].$$ = (28.9)

Figure 28.1 shows a plot of the wind stress and its curl.