The foregoing behavior depends in a delicate way on the relative importance of various force terms in the radial and tangential components of the momentum equation. For the purpose of interpretation it is helpful to rewrite Eqs.(3.20) and (3.21) in the following form: dus ds = wδ−+wsc δ − (vgr −vb) us (vgr+vb) R−s +f − CD δ q (u2 s+vb2), (3.41) dvb ds = wδ−+wsc δ (vb −vgr) us + vb R−s +f − CD δ q (u2 s+v2b) vb us . (3.42)
Here us = −ub is the radial inflow velocity and s = R −r, with s ≤ R, measures
distance inwards from the starting radius, R. In addition the flux terms on the far right of Eqs.(3.20) and (3.21) have been replaced with the formulation described in section (3.1). In this form the equations show how the (inward) radial and tangential components of flow change with decreasing radius.
angular momentum, rv+1 2f r
2, and spin faster. However, in the boundary layer these
rings of air are still spinning faster. But of course now frictional torque is acting and therefore the rate at which vb increases is reduced significantly. This effect is
represented by the last term in Eq.(3.42):
CD δ q (u2 s+vb2) vb us .
For the development of supergradient winds it is now necessary to have a sufficiently large radial displacement of air parcels in the boundary layer. This displacement is on the other hand just possible if the radially-inward wind speeds are large enough. From a Lagrangian viewpoint one may think of air parcels spiralling inwards. As they move slower inwards the tracks they follow become longer. This means that they have a longer way to go along where friction can act and reducevb.
An inspection of Eq.(3.42) shows that this effect is contained in the terms proportional to the inverse of us.
The foregoing discussion makes clear that the development of supergradient winds depends on the radial gradient of absolute angular momentum in the boundary layer and hence on that above the layer. This feature is also explored in the context of a linear boundary-layer model by Kepert (2001) and Kepert and Wang (2001). It is also consistent with the results in the context of the linear model in the foregoing chapter. Equation (3.41) shows that the only term that can cause a radially-inward acceleration in the slab model is the net pressure gradient. The effect of this pressure gradient is contained in the second term on the right-hand-side of Equation (3.41):
(vgr−vb) us (vgr+vb) R−s +f .
This term describes the net inward force which is due to the difference between the radial pressure gradient and the centrifugal and Coriolis forces. The first term in Eq.(3.41)
wδ−+wsc
δ
stands for the effects of the downward transport of radial momentum. This is zero in the present model. Finally the third term
CD
δ q
(u2
3.2. The new calculations - dynamical aspects
represents the frictional stress. Both of these act to reduce the radial inflow. If the flow is supergradient, i.e. if vb > vgr, the net pressure gradient acts radially outwards
also. The net inward force increases with the degree to which the tangential flow in the boundary layer is subgradient (i.e. to vgr−vb), which in turn increases as the effective
frictional torque becomes larger. Equation (3.42) shows that this torque is the only term that leads to a reduction of vb with decreasing radius as long as the flow in the
boundary layer remains supergradient. The friction terms are inversely proportional to the boundary-layer depth. This means that shallower boundary layers favour lower tangential wind speeds. However shallower boundary layers lead to larger radial wind speeds. This is because, at least in the outer part of the vortex for large radii, they cause a larger net pressure gradient. If one approaches the core region of the vortex and the radii are becoming smaller the situation is a little different. Now the term
vb/(R −s) in Eq.(3.42) becomes large and contributes to an increase in vb with s.
Thus larger radial wind speeds favor larger tangential wind speeds. This is because air parcels may move rapidly to smaller radii, where this effect is important. In addition they suffer less total frictional torque on the way (note that the frictional term in Eq.(3.42) decreases as us increases). The key to what determines the two flow regimes
depends on which of the foregoing processes dominates and boils down to whether or not the flow can become subgradient again before ub becomes zero. In the calculation
with δ = 680 m, the tangential flow just manages to become subgradient before ub
becomes zero, whereupon the inflow begins strengthen again with decreasing radius. Because the tangential wind speed at the top of the boundary layer decreases also, the flow again becomes supergradient so that ub decreases rapidly and vb−vgr decreases
until ub becomes subgradient again. These fluctuations are a kind of damped inertial
oscillation as described in Smith (2003). These waves are not very significant in reality. It is more realistic to interpret them as an artifact which is descended of the prescription of the tangential wind field at the top of the boundary layer. The radial scale of the waves is on the order of a few kilometers and decreases with radius. Thus such waves would not be resolvable by most numerical models of hurricanes. Moreover the implied radial gradients associated with them would stretch the assumptions of boundary layer theory, which assumes radial gradients of quantities to be small compared with vertical gradients. It turns that these oscillations have much smaller amplitudes in calculations that allow the boundary depth to decline with radius (see section (3.2.6)).
ences in the radial location where, wδ changes sign in Fig.(3.4). The larger effective
friction for the shallower boundary layer implies a larger net radial pressure gradient, which, in turn, leads to a larger acceleration of the radial flow and a decrease in the radius at which the radial gradient of inward mass flux changes sign.