Low-order models, initialization, and the slow manifold

A nine equation low-order model of the shallow water equations is used as a testbed to compare several initialization strategies and to find points satisfying conditions of “slowness.” Several methods are explored to initialize the low order model: (1) Lorenz's method of successively zeroing tendencies; (2) minimization of the sum of the squares of the tendencies; (3) use of a balance condition. We find that the balance condition produces the smoothest solution on a consistent basis. In addition, Lorenz initialization, the Newton-Kantorovich Theorem, and the Nested Interval Property are used to compute points devoid of gravity waves to order N for low values of the forcing (F1 ≤ 0.1). Several of these points are calculated.