Energy2green Wind And Solar Power System

We developed a simple theoretical model for the boundary layer (BL) of translating tropical cyclones (TCs). The model estimates the tangential, radial and vertical wind fields from a given radial profile of the tangential wind speed under geostrophic conditions (e.g. Holland's profile in Eq. (1)) and surface stress boundary conditions.

The model is based on Smith's (1968) formulation, which is corrected for the general case of a stress surface boundary (see Proposed Model) and modified to account for motion-induced asymmetries. The governing equations are solved using the momentum integral method, resulting in a very efficient computational scheme (less than a minute to run on a conventional computer). Contrary to the Shapiro (1983) and Kepert (2001) BL formulations, the present modified Smith (MS) model is stable over a wide range of storm parameters and produces results that are in good agreement with MM5 simulations and observations; see Figs. 5, 6 and 7.

In a parametric analysis, we have examined how the symmetric wind components and motion-induced asymmetries generated by the MS model depend on various storm characteristics. For TCs that translate in the northern hemisphere, the vertical and storm-relative radial wind velocity fields intensify at the right-front quadrant of the vortex, whereas the storm-relative tangential winds intensify at the left-front of the storm. The asymmetry is higher for faster-moving TCs. These findings are in accordance with field observations; see for example Shapiro (1983).

As the intensity of the TC (expressed through the maximum tangential velocity at gradient balance Vmax) increases, the horizontal and vertical velocity components also increase. Larger values of Holland's B parameter and lower values of the radius of maximum winds Rmax produce horizontal and vertical wind profiles that are more picked and concentrated closer to the TC center.

The effect of the frictional drag coefficient CD is more complex. As CD increases, the frictional drag force also increases resulting in lower tangential winds and higher radial convergence (stronger vertical winds). Another effect of increasing CD is higher asymmetry of the velocity field.

As described above, the model considers CD to be spatially constant. In reality, the drag coefficient varies with the sea state and hence with the near surface wind speed (Large and Pond 1981; Donelan et al. 2004). We are currently extending the model to include the coupling between CD and wind speed.

The combination of accuracy, stability and computational efficiency make the present model a good instrument for applications, like risk analysis, for which one must assess tropical cyclone winds under many parameter combinations. In this respect, a limitation of the model is that it produces smooth solutions, since it does not account for features like rainbands and local convective activity. It is the authors' belief that these features are best represented statistically.

Acknowledgments This work was supported by the Alexander S. Onassis Public Benefit Foundation under Scholarship No. F-ZA 054/2005-2006 and by ONR research grants N00014-03-1-0479 and N00014-04-6-0524. The authors are grateful to Melicie Desflots for providing the MM5 numerical simulations, to Robert Rogers and Jason Dunion for making available a code for Shapiro's (1983) model and to Anastasios Tsonis and James Elsner for reviewing this manuscript.

Appendix A

Analytical expressions for the parameters in Eq. (15)

Let h = / f2dj, 12 = /(1 - g2)dj, I3 = / (1 - g)dj, I4 = f fgdj, 0 0 0 0 11 1 15 = J fdj, 16 = J (g2 — g)dj and 17 = J (1 — g) dj, where f and g are the 0 0 0

I1 — o (a2 + 2a1a2 + 3a2), 12 — (a1 + a2) — — (3ag + 2a1a2 + a2) 8 8

where a1 and a2 are calculated from Eq. (12). Under Eq. (A.1), the parameters A1—A20 in Eq. (15) are given by

A1 — E2 [v2 cos2 0 17 + (vt sin 0 — vgr )211 + (15 — 14)(v2 sin 20 — 2vtvgr cos 0)]

A2 — E [(v2 cos 20 + V(Vgr sin 0)15 + (/v2 sin 20 — v^gr cos 0) /1

+ (vgr — 2vtvgr sin 0 — v2 cos 20)14 + vtvgr16 cos 0 — / 17v2 sin 20]

A3 — —v^ cos2 0 + v^ sin 20 + (2vtvgr cos 0 — v2 sin 20)14 — v^ sin2 0 + v2 12 + 2viv„r/6 sin 0

A4 — v^ cos 0 — v,/5 sin 20 + (v, sin 20 — 2vtvgr cos 0)14 + v^ sin 0

tcos [email protected]

A10 — 2E[v2/7 cos2 0 +11 (vgr — vt sin 0)2 + (v2 sin 20 — 2vtvgr cos 0)(15 —14)] (A. 10)

2 2 - I7 -vgr 2 - I1 vt cos O —--2I1(v, sin O — vgrJ —--h (vgr — v, sin OJ --—

2v,{I5 —14) —— cos O + (v2 sin 2O — 2v,vgr cos O) -

A12 = {v° cos 2O + vtvgr sin O)I5 + {/v2 sin 2O — v,vgr cos O)I1

+ {vgr — 2vtvgr sin O — v2t cos 2O)I4 + v tvgrI6 cos O — /v^ sin 2O

(—2v2 sin 2O + v,vgr cos O) I5 + (v2 cos 2O + v,vgr sin O) Í --jy +11

+ (/v° sin 2O — v,vgr cos O) —y- + (2v2 sin 2O — 2v,vgr cos O) I4

+ [fgr — 2v,vgr sin O — v° cos 2OJ — —g^ sin O + v ,vgr cos O -O

A14 E

- I5 -vg v, sin O ~q~I5 + (v2 cos 2O + v,vgr sin O) —--v, -^-h cos O

+ (/v2 sin 2O — v,vgr cos O) ^■ + 2I4 (vgr — v, sin O) °+ I6v,

^s— -r cos O -^ + fv2 — 2v,v„r sin O — v° cos 20^) —4 + v,v„r cos O —6

A15 = — v2I1 sin2O + v2 cos2 O^- — 2v2I5cos2O — v2 sin 20^-5

H 2v2 cos 2O H 2v vgr sin O I4 H v2 sin 2O — 2v vgr cos O

+ v2I7 sin 2O + v2 sin2 O —y- — 2v,vgr I6 cos O — 2v,vgr sin O

A18 = v,I5 sin O — v, cos O—- + v,I3 cos O + v, sin O

In Eqs. (A.2)-(A.19), the derivatives and for j = 1,..,7 can be calculated analytically using the chain rule dlj dlj da1 dlj da2 , ds @a1 ds da2 ds ' j ' : ' :

where s is either r or

0 and and ^a2 can be analytically derived from Eqs. (12)

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