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Upper-tropospheric inflow layers in tropical cyclones
Upper-tropospheric inflow layers in tropical cyclones
Three-dimensional numerical simulations of tropical-cyclone intensification with sufficient vertical resolution have shown the development of a layer of strong inflow just beneath the upper-tropospheric outflow layer as well as, in some cases, a shallower layer of weaker inflow above the outflow layer. Here I provide an explanation for such inflow jets in the context of the prototype problem for tropical-cyclone intensification, which considers the evolution of a vortex on an f-plane in a quiescent environment, starting from an initially-symmetric, moist, cloud-free vortex over a warm ocean. I attribute the inflow layers to a subgradient radial force that exists through much of the upper troposphere beyond a certain radius. Some effects of the inflow layers on the storm structure are discussed. An alternative explanation that invokes classical axisymmetric balance theory is found to be problematic. The consequences of regularizing the Sawyer-Eliassen equation to calculate the streamfunction for the axisymmetric secondary circulation of a tropical cyclone are explored. Regularization is an ad hoc procedure in which the coefficients of the equation are suitably modified to replace negative values of the discriminant by small positive values, thereby ensuring that the equation is globally elliptic. The consequences of the procedure may be understood in terms of the analog behaviour of a stretched membrane subject to a particular force distribution. Several regularization procedures are assessed by comparing the azimuthally-averaged radial flow from a three-dimensional numerical simulation of a tropical cyclone with that from an axisymmetric balance calculation of the Sawyer-Eliassen equation, forced by diabatic and frictional terms diagnosed from the simulation. The comparison shows that the largest challenge for regularization occurs in regions of inertial instability, especially when the diagnosed forcing overlaps with such regions. In the example shown, the diagnosed balanced flow is sensitive to the particular regularization procedure and none of the procedures examined gave a flow that was structurally and quantitatively close to that obtained from the numerical solution in and near the region of regularization. The flow in regions of large vertical shear that are common in the lower part of the boundary layer is less sensitive to the regularization procedure. Nevertheless, there are comparatively large differences between the low-level inflow in the azimuthally-averaged numerical solution and the axisymmetric balance solution. These differences can be attributed to the intrinsic lack of balance in the boundary layer. This finding, together with the issues associated with regularization, is further confirmation that balance dynamics is unable to adequately capture the flow in the boundary layer, contrary to recent claims. Two methods for solving the Sawyer-Eliassen equation for the corresponding balanced secondary circulation of a numerically-simulated, high-resolution tropical cyclone vortex are compared. In idealized calculations for a symmetrically stable vortex, both methods (successive over-relaxation and multi-grid) converge and the solutions are broadly similar. In more typical cases, where the vortex has regions of inertial or symmetric instability, it is necessary to coarsen the data from the numerical simulation to determine the balanced secondary circulation. A convergent solution can be obtained with the multi-grid method for a finer grid spacing than with the successive over-relaxation method. However, the multi-grid method fails to converge when the vertical grid spacing is similar to that of the numerical simulation. Results using both methods confirm the inability of the balance formulation in capturing the strong inflow and resulting tangential wind spin up in the frictional boundary layer during a period of rapid intensification. The balanced secondary circulation may show such an inflow layer. However, caution is called for in attributing this inflow layer to a balanced flow response driven by the distribution of diabatic heating and tangential momentum forcing. This study suggests that it is likely an artifact of the ad hoc regularization procedure that is necessary to keep the Sawyer-Eliassen equation globally elliptic in regions of inertial and/or symmetric instability. Lagrangian air parcel trajectories emanating from the inflow layer that develops be- neath the upper-tropospheric outflow layer show that about a half of these trajectories end up in the outflow layer, itself. The other half slowly subside to the mid- to upper troposphere, below the outflow layer, and drift slowly outwards as a result of a relatively weak overturning circulation in that region. Calculations show that pseudo-equivalent potential temperature is not approximately conserved along the air parcel trajectories indicating that the turbulent diffusion of heat and moisture along the trajectories is appreciable in the middle and upper troposphere.
tropical cyclone, typhoon, vortex intensification, hurricane
Wang, Shanghong
2021
English
Universitätsbibliothek der Ludwig-Maximilians-Universität München
Wang, Shanghong (2021): Upper-tropospheric inflow layers in tropical cyclones. Dissertation, LMU München: Faculty of Physics
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Abstract

Three-dimensional numerical simulations of tropical-cyclone intensification with sufficient vertical resolution have shown the development of a layer of strong inflow just beneath the upper-tropospheric outflow layer as well as, in some cases, a shallower layer of weaker inflow above the outflow layer. Here I provide an explanation for such inflow jets in the context of the prototype problem for tropical-cyclone intensification, which considers the evolution of a vortex on an f-plane in a quiescent environment, starting from an initially-symmetric, moist, cloud-free vortex over a warm ocean. I attribute the inflow layers to a subgradient radial force that exists through much of the upper troposphere beyond a certain radius. Some effects of the inflow layers on the storm structure are discussed. An alternative explanation that invokes classical axisymmetric balance theory is found to be problematic. The consequences of regularizing the Sawyer-Eliassen equation to calculate the streamfunction for the axisymmetric secondary circulation of a tropical cyclone are explored. Regularization is an ad hoc procedure in which the coefficients of the equation are suitably modified to replace negative values of the discriminant by small positive values, thereby ensuring that the equation is globally elliptic. The consequences of the procedure may be understood in terms of the analog behaviour of a stretched membrane subject to a particular force distribution. Several regularization procedures are assessed by comparing the azimuthally-averaged radial flow from a three-dimensional numerical simulation of a tropical cyclone with that from an axisymmetric balance calculation of the Sawyer-Eliassen equation, forced by diabatic and frictional terms diagnosed from the simulation. The comparison shows that the largest challenge for regularization occurs in regions of inertial instability, especially when the diagnosed forcing overlaps with such regions. In the example shown, the diagnosed balanced flow is sensitive to the particular regularization procedure and none of the procedures examined gave a flow that was structurally and quantitatively close to that obtained from the numerical solution in and near the region of regularization. The flow in regions of large vertical shear that are common in the lower part of the boundary layer is less sensitive to the regularization procedure. Nevertheless, there are comparatively large differences between the low-level inflow in the azimuthally-averaged numerical solution and the axisymmetric balance solution. These differences can be attributed to the intrinsic lack of balance in the boundary layer. This finding, together with the issues associated with regularization, is further confirmation that balance dynamics is unable to adequately capture the flow in the boundary layer, contrary to recent claims. Two methods for solving the Sawyer-Eliassen equation for the corresponding balanced secondary circulation of a numerically-simulated, high-resolution tropical cyclone vortex are compared. In idealized calculations for a symmetrically stable vortex, both methods (successive over-relaxation and multi-grid) converge and the solutions are broadly similar. In more typical cases, where the vortex has regions of inertial or symmetric instability, it is necessary to coarsen the data from the numerical simulation to determine the balanced secondary circulation. A convergent solution can be obtained with the multi-grid method for a finer grid spacing than with the successive over-relaxation method. However, the multi-grid method fails to converge when the vertical grid spacing is similar to that of the numerical simulation. Results using both methods confirm the inability of the balance formulation in capturing the strong inflow and resulting tangential wind spin up in the frictional boundary layer during a period of rapid intensification. The balanced secondary circulation may show such an inflow layer. However, caution is called for in attributing this inflow layer to a balanced flow response driven by the distribution of diabatic heating and tangential momentum forcing. This study suggests that it is likely an artifact of the ad hoc regularization procedure that is necessary to keep the Sawyer-Eliassen equation globally elliptic in regions of inertial and/or symmetric instability. Lagrangian air parcel trajectories emanating from the inflow layer that develops be- neath the upper-tropospheric outflow layer show that about a half of these trajectories end up in the outflow layer, itself. The other half slowly subside to the mid- to upper troposphere, below the outflow layer, and drift slowly outwards as a result of a relatively weak overturning circulation in that region. Calculations show that pseudo-equivalent potential temperature is not approximately conserved along the air parcel trajectories indicating that the turbulent diffusion of heat and moisture along the trajectories is appreciable in the middle and upper troposphere.