Development of a Rossby Wave Critical Level

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Abstract
We discuss the time-dependent behavior of a Rossby wave on a latitudinally varying flow, near the point where the steady-state wave equation is singular. The wave is forced by the switch-on of a steady forcing. Analytic solutions are obtained for the latitudinal propagation of nondivergent Rossby waves in a linear shear flow and for a large longitudinal wavelength. It is shown that the north–south eddy velocity v′ approaches the steady-state solution everywhere when nondimensional time >1, this time being a few days or less for atmospheric planetary waves. The east–west eddy velocity u′ takes much longer to approach a steady state near the singularity. One-half the steady-state amplitude of u′ is approached in a time inversely proportional to the square root of the distance from the singularity. The solution for u′ near the singularity settles down to the steady solution only after a time large compared to the inverse of the distance from the singularity. The steady-state solution for u′ is logar... Abstract We discuss the time-dependent behavior of a Rossby wave on a latitudinally varying flow, near the point where the steady-state wave equation is singular. The wave is forced by the switch-on of a steady forcing. Analytic solutions are obtained for the latitudinal propagation of nondivergent Rossby waves in a linear shear flow and for a large longitudinal wavelength. It is shown that the north–south eddy velocity v′ approaches the steady-state solution everywhere when nondimensional time >1, this time being a few days or less for atmospheric planetary waves. The east–west eddy velocity u′ takes much longer to approach a steady state near the singularity. One-half the steady-state amplitude of u′ is approached in a time inversely proportional to the square root of the distance from the singularity. The solution for u′ near the singularity settles down to the steady solution only after a time large compared to the inverse of the distance from the singularity. The steady-state solution for u′ is logar...

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