Iterative Solutions to the Steady-state Axisymmetric Boundary-layer Equations under an Intense Pressure Gradient

Steady-state solutions to the complete axisymmetric Navier-Stokes equations are obtained by an iterative technique for an intense pressure-gradient force representative of the tropical cyclone. Solutions for the horizontal and vertical components of motion are compared for various horizontal and vertical mixing coefficients, drag coefficients, and Coriolis parameters. A multilevel model is used, and the results are compared with those from a simple onelevel model. l. INTRODUCTION In the experiments discussed here, the complete equaBecause of the intense frictional convergence in the boundary layer and the rapid decrease of mixing ratio with height, the major water vapor convergence in tropical cyclones occurs in the lowest few kilometers. Thus, the frictionally induced vertical motion a t the top of the planetary boundary layer becomes a close measure of this convergence (Miller 1962, Riehl and Malkus 1961). Ap proximate expressions relating this vertical motion to the mean steady-state tangential wind have been given by several investigators (e.g., Syofio 1951, Charney and Eliassen 1949, Ogura 1964, and Smith 1968). Kuo (1971) utilized an iterative procedure to obtain exact solutions to the equations of motion in the boundary layer of a maintain ed vortex without the Coriolis term. In a theoretical study of the hurricane boundary layer, Rosenthal (1962) neglected lateral mixing and vertical advection and obtained analytic solutions to the linearized equations. Rosenthal found a decreasing depth of inflow toward the center, with the consequence that the vertical velocities near the center appeared too small. Miller (1965) utilized a quasi-time-dependent numerical model to obtain steady solutions under various formulations of horizontal and vertical mixing and found that an inflow layer of constant depth could be obtained if the vertical mixing coefficient were made proportional to the pressure gradient. I n this paper, steady-state solutions to the axisymmetric boundary-layer equations are obtained by a numerical technique similar to Miller’s (1965) , with particular applicability to the boundary layer in hurricanes. The results are compared with those obtained by Kuo (1971) from a different approach. The results obtained from the equations of motion in primitive form are useful in interpreting results from more complex hurricane models, as well as providing lower boundary conditions on the mass transport required in diagnostic models such as Barrientos (1964) or Anthes (1970b). tions of motion are solved numerically for thehorizontal and vertical velocity components under a steady, intense pressure-gradient force to study the effects of the variations of horizontal and vertical mixing, drag coefficients, and Coriolis parameter on the vertical motion a t the top of the boundary layer. A multilevel model is used, and the resuIts are compared with those obtained with a simple one-level model. This is a relevant comparison in view of the use of low vertical resolution boundary layers in hurricane models. 2. A MULTILEVEL BOUNDARY LAYER MODEL