A steady, purely azimuthal flow model for the antarctic circumpolar current (Q1795169)

The author presents a model accounting for the flow around Antarctica of the Antarctic Circumpolar Current. Applying the \(f\)-plane approximation to the Euler equations for three-dimensional divergence-free ocean flows, he starts from the system of equations \(u_{t}+uu_{x}+vu_{y}+wu_{z}+2(\Omega ^{y}w-\Omega ^{z}v)=-\frac{1}{\rho }P_{x}\), \(v_{t}+uv_{x}+vv_{y}+wv_{z}+2( \Omega ^{z}u-\Omega ^{x}w)=-\frac{1}{\rho }P_{y}\), \( w_{t}+uw_{x}+vw_{y}+ww_{z}+2(\Omega ^{x}v-\Omega ^{y}u)=-\frac{1}{\rho } P_{z}-g\), \(u_{x}+v_{y}+w_{z}=0\), where \(u,v,w\) are the velocity components of the flow field in direction of increasing azimuth, latitude and elevation respectively, \(P\) is the pressure and \(\Omega ^{x}=0\), \(\Omega ^{y}=\Omega \cos\theta \), \(\Omega ^{z}=\Omega \sin\theta \), with \(\Omega =7.29\times 10^{-5}\operatorname{rad} s^{-1}\). The author assumes \(\theta =\frac{\pi }{4}\) and \(v=w=0 \) and he adds an a priori unknown forcing term \(F(z)\). He thus ends with the problem \(0=-\frac{1}{\rho }P_{x}+(vu_{z})_{z}\), \(-\sqrt{2}\Omega u=- \frac{1}{\rho }P_{y}+F\), \(-\sqrt{2}\Omega u=-\frac{1}{\rho }P_{z}+G\), \( u_{x}=0\). The boundary conditions \(P=Patm\), \(\tau :=\rho (vu_{z})_{z}=\tau _{0}\) at \(z=0\) (upper surface) and \(u=0\) at \(z=-d\) (lower surface) are imposed. The main result of the paper builds the expression of an exact solution \((u,P,F)\) to this problem.