An Implicitly Balanced Hurricane Model with Physics-Based Preconditioning

A numerical framework for simulating hurricanes based upon solving a nonlinear equation set with an implicitly balanced solution procedure is described in this paper. The physical model is the Navier–Stokes equations plus a highly simplified and differentiable microphysics parameterization package. Because the method is fully implicit, the approach is able to employ time steps that result in Courant–Friedrichs–Lewy (CFL) numbers greater than one for advection, gravity, and sound waves; however, the dynamical time scale of the problem must still be respected for accuracy. The physical model is solved via the Jacobian-free Newton–Krylov (JFNK) method. The JFNK approach typically requires the approximate solution of a large linear system several times per time step. To increase the efficiency of the linear system solves, a physics-based preconditioner has been employed. To quantify the accuracy and efficiency of the new approach against traditional approaches, the implicitly balanced solver was first compared against semi-implicit approaches for the simulation of a precipitating moist bubble. The moist-bubble simulations demonstrated the ability of the implicitly balanced approach to achieve a given level of accuracy in a more efficient manner than either a first-order semi-implicit approach or a traditional leapfrog semi-implicit approach. This behavior is further illustrated in first-of-a-kind three-dimensional implicitly balanced hurricane simulations that reveal the first-order-in-time semi-implicit algorithm needs to take a time step at least 60 times smaller than the implicitly balanced algorithm to produce a comparable accuracy.