The nonlinear Charney equation. The atmospheric circulation is subject to a
complex system of differential equa- tions. A simplification is achieved by making
...
The nonlinear Charney equation The atmospheric circulation is subject to a complex system of differential equations. A simplification is achieved by making assumptions about the scales of motion which are in focus. This scaling approach yields balance relationships for the atmospheric motion. Balance relationships are needed in data assimilation where the global atmospheric state is estimated from observations and model forecasts. For the large-scale horizontal flow on the spherical Earth, the equations can be simplified to the so-called nonlinear Charney balance equation: ¶ µ 1 ∂Ψ ∂Ψ 2 (1) , ∇2 φ = ∇ · (f ∇Ψ) + 2 J a cos φ ∂λ ∂φ with the two variables geopotential φ and the horizontal streamfunction Ψ. The coordinates on the sphere are longitude λ and latitude φ. The parameter f is the Coriolis parameter. The function J designates the Jacobian with J(F, G) =
∂F ∂G ∂F ∂G − . ∂λ ∂φ ∂φ ∂λ
(2)
Equation (1) enables to determine the geopotential field on the sphere from a given streamfunction field or vice versa. This project aims at the application of the nonlinear Charney balance equation to the sphere. For application in meteorology, also the linearized and adjoint version is required. The linearization should be carried out around an arbitrary state.
