αdsin2h
π 2
1 − zzspongetop−z
i
if z ≥ (ztop− zsponge) 0 if z < (ztop− zsponge)
, (4.65)
where αd is the implicit damping coefficient, ztop is the height of the computational domain, and zsponge is the height of the sponge layer. Implicit Rayleigh damping effectively absorbs gravity waves without the requirement of an initial state to relax back to, as done in traditional sponge layers. This is advantageous because it allows upper portions of the domain to evolve more freely. Setting αd to 0.2 appears to be sufficient for preventing gravity waves from reflecting back down. The sponge layer is well above the tropopause, with a thickness that is approximately 20% the height of the computational domain. Additionally, W is set to zero at the upper boundary.
4.3 Initialization and Spin Up
The model is initialized with a sounding, shown in Fig. 4-5, that is neutral to undilute ascent of subcloud layer parcels. In this study, the sounding is derived from a surface parcel with an initial pressure of 1015 mb, an initial temperature of 28◦C, and an initial relative humidity of 75%. The parcel is lifted such that it conserves sp. In order
Temperature (C)
Pressure (mb)
−30 −20 −10 0 10 20 30
100
200
300 400 500 600 700 800 900 1000
Figure 4-5: An initial, neutral sounding derived from a surface temperature of 28◦C and surface relative humidity of 75%. The pseudoadiabats (red lines) are calculated using (4.19).
to have a tropopause and stratosphere, the Jordan mean hurricane season temperature profile is used above the height where the lifted parcel first becomes cooler than the Jordan sounding. The water vapor mixing ratio below the parcel’s lifting condensation level is retained. Above the lifting condensation level, the relative humidity is set at 50% so that evaporation and downdrafts can occur.
A tropical-storm strength vortex is inserted into the domain, with a radial tan-gential wind profile from the parametric formula of Emanuel (2004). A maximum tangential wind of 20 m s−1 at a radius of 100 km is used, and the constants in the parametric formula are the default ones given in App. B of Emanuel et al. (2006).
The vortex is assumed to decay with height as the square root of a quarter-cosine profile, such that the maximum tangential wind is initially at the surface and decays to zero at the tropopause.
The temperature and mass fields are then adjusted to be in thermal wind balance
with the initial vortex following the anelastic approximation from Smith (2006). The anelastic form of the equation for gradient wind balance, given by
∂Π
∂r = 1 cpdθv
f v + v2 r
, (4.66)
is integrated inward from the outer boundary in order to initialize the Exner function.
θv is the virtual potential temperature of the initial sounding and is only a function of height. Subsequently, θv is initialized to be in hydrostatic balance with the pressure field:
θv = − g
cpd∂Π∂z . (4.67)
The potential temperature and water vapor mixing ratio are then calculated such that the relative humidity and θv remain constant. Thereafter, sp can be calculated from (4.18), and ρd can be calculated from the ideal gas law.
The TC is spun up from its initial state until it reaches a steady state, as shown in Fig. 4-6 at seven days. The maximum tangential winds are 67 m s−1 at a height of 1 km, with the radius of maximum winds sloping outward with height. In comparison, the theoretical potential intensity, which is calculated from an algorithm using the initial sounding and the model’s surface exchange coefficients, is 85-93 m s−1. It is important to emphasize that the model’s potential intensity and the theoretical potential intensity are like two different species of the same genus. Differences between the two may arise due to differences in the model’s definition of entropy with the exact pseudoadiabatic or reversible form. Additionally, the model intensity’s sensitivity to resolution and the turbulence parameterization, particularly the horizontal mixing length (lh), plays a role. One could possibly tune lh in order to match the theoretical potential intensity more closely, but this would be ad-hoc. In lieu of any tuning, the model’s potential intensity is treated as the baseline that will be used to compare to additional experiments.
The secondary circulation, outlined by the mass streamfunction in Fig. 4-6a, con-sists of 20-25 m s−1 inflow in the boundary layer just radially outward from the radius
of maximum wind, 3-5 m s−1 vertical motion through the eyewall, and 15-20 m s−1 outflow around a height of 15 km. The waviness of the flow as it rises up the eyewall is indicative of an inertial oscillation caused by unbalanced flow. Additionally, the eyewall is characterized by slantwise neutrality, as indicated by the congruence of an-gular momentum, streamfunction, and entropy contours (not shown). At low levels, the entropy increases with decreasing radius. In the eyewall, there exists a column of high, nearly constant entropy. Much lower values of entropy are found at midlevels outside the eyewall, with the lowest values occurring at a height of 2-3 km. At this level, relatively dry air from aloft has subsided and cooled radiatively. Low-entropy air at midlevels does impinge a bit on the eyewall, especially as it begins to flare out above 5 km, causing evaporation and a downdraft of approximately 0.5-1 m s−1. This downdraft is hinted at by the closed contour in the mass streamfunction just inside a radius of 50 km. However, the downdraft is too spatially limited and weak to have much of an effect on the TC energy budget. Elsewhere, there is shallow convection occurring outside the eyewall, but the general subsidence from the secondary circu-lation and lack of convective available potential energy inhibits deep convection from forming.