Analytical approximations to spurious short-wave baroclinic instabilities in ocean models

Most community ocean models that use z- or s-coordinates stagger their variables in the vertical using a Lorenz grid. Spurious short-wave baroclinic instabilities have been shown to occur on that grid by Arakawa and Moorthi. As the vertical resolution of the grid is improved, the wavelength of the spurious modes decreases and they become more and more trapped near one of the boundaries but they continue to grow at almost the same rate as the deep Eady/Charney modes. The spurious instabilities in the case of the Eady problem are here shown to be accurately reproduced by an analytical calculation which reduces the stability problem to a quadratic equation for their complex phase speeds. The interpretation of these spurious instabilities as resulting from spurious sheets of potential vorticity is revisited. A new interpretation is presented using a finite difference analogue of the Charney-Stern-Pedlosky integral constraint. This indicates that the spurious instabilities result from a vertical averaging of the advection of the relative vorticity which leads to a spurious interior source term in the finite difference potential vorticity equation.