正壓渦旋能量頻散之冪次律

Scaling Laws for Barotropic Vortex Energy Dispersion

吳華富(Hua-Fu Wu)

指導教授 ： 郭鴻基

國立臺灣大學/理學院/大氣科學研究所/碩士(2009年)

https://doi.org/10.6342/NTU.2009.00825

摘要

本研究延續Smith(1993, 1997)的工作，在正壓模式中，植入初始渦旋為DC渦旋(DeMaria and Chan, 1984)及修正阮肯渦旋(modified Rankine vortex)進行模擬，探討渦旋能量頻散的冪次律以及渦旋能量頻散與移行速度間的冪次律，以及渦旋結構與水平風切對冪次律的影響。所有需要考慮的變數有七個：渦旋移行速度 、渦旋能量頻散 、渦旋最大風速 、最大風速半徑 、行星渦度梯度 、積分時間 ，背景水平風切 。利用因次分析將七個參數減少至五個，再加上最大風速半徑外的風速遞減率共六個無因次的參數：無因次移行速度 、無因次能量頻散 、羅士比數 、無因次時間 、背景風切 以及最大風速半徑外風速遞減率 ，找出重要的無因次參數後，進一步探討：
1. 渦旋能量頻散與渦旋移行速度的冪次律
2. 不同的渦旋結構對渦旋能量頻散與渦旋移行速度的冪次律之影響
3. 不同的水平風切對渦旋能量頻散與渦旋移行速度的冪次律之影響

本研究中共三組實驗，在實驗一中令背景風切 ，並假設有以下函數關係： 、 與 。實驗二與實驗一類似，但將 。實驗三中，只考慮 時，將實驗一中的 。

由實驗結果我們得知：
1. 沒有風切，無因次時間相同時，無因次能量頻散隨羅士比數增加而減少。這表示當核心渦旋增強時，頻散能量與渦旋初始能量的比例會減少。
2. 沒有風切，無因次時間、行星渦度梯度、風速遞減率、環流相同時，半徑較大的渦旋有較明顯的能量頻散。
3. 沒有風切，積分時間、行星渦度梯度、風速遞減率、環流相同時，能量頻散隨渦旋最大風速增加而增加。
4. 沒有風切，無因次時間、行星渦度梯度、渦旋半徑、最大風速相同時，渦旋外圍渦度裙帶愈大，渦旋能量頻散也愈明顯。
5. 沒有風切，無因次時間、行星渦度梯度、渦旋半徑、最大風速相同時，阮肯渦旋比DC渦旋( )的能量頻散較少、β移行速度較快。
6. b=1，無因次時間、行星渦度梯度、渦旋半徑、最大風速相同時，渦旋能量頻散與移行速度隨風切值增加而增加。
7. 沒有風切，無因次時間相同時，渦旋能量頻散正比於β移行速度平方。

關鍵字

冪次律 ； 渦旋能量頻散 ； 渦旋移行速度 ； 羅士比數 ； 水平風切

並列摘要

In this research, we extend the work in Smith(1993, 1997) to generalize the scaling laws for vortex energy dispersion and drift speed to include the influences of variable vortex profile and meridional shear in the zonal environmental wind. The vortex profiles used in barotropic model are DC vortex (DeMaria and Chan, 1984) and modified Rankine vortex. The physical problem of beta-drift and vortex energy dispersion we consider here has 7 parameters; drift speed , vortex energy dispersion , maximum tangential wind speed in the vortex , the radius of maximum wind , planetary vorticity gradient , the integration time and the shear in the zonal environmental wind 。The use of dimensional analysis reduces these 7 parameters to 5, and a non-dimensional exponent related to the decrease of wind speed with radius in the outer part of the vortex: non-dimensional drift speed , non-dimensional vortex energy dispersion , Rossby number , non-dimensional time , non-dimensional horizontal shear and non-dimensional exponent and . In this research, we attempt to obtain the following scaling laws.
1. Scaling laws for vortex energy dispersion and drift speed.
2. The influences of vortex structure on the scaling law.
3. The influences of horizontal shear on the scaling laws.
There are three groups of experiments in this research. The non-dimensional drift speed and non-dimensional vortex energy dispersion are given by 、 and . In experiment 1, we examine the case with no horizontal shear ( ) and investigate the scaling laws for a range of Rossby number, vortex profile factor and non-dimensional time . In experiment 2, we replace in experiment 1 with . In experiment 3, we fix the vortex profile ( ) and investigate the scaling laws for a range of Rossby number, horizontal shear and non-dimensional time .
From the scaling laws we have the following conclusions:
1. When non-dimensional time are fixed with no shear, non-dimensional energy dispersion is large with small Rossby number.
2. When non-dimensional time, beta, vortex profile and circulation are fixed with no shear, energy dispersion is large with large radius of maximum wind speed.
3. When integration time, beta, vortex profile and circulation are fixed with no shear, energy dispersion is large with large vortex maximum tangential wind speed.
4. When non-dimensional time, beta, maximum tangential wind speed and radius of maximum wind speed are fixed with no shear, energy dispersion is large with small vortex profile factor.
5. When non-dimensional time, beta, maximum tangential wind speed and radius of maximum wind speed are fixed with no shear, Rankine vortex has less energy dispersion and larger drift speed than DC vortex with .
6. When we fix non-dimensional time, beta, maximum tangential wind speed, radius of maximum wind speed and , energy dispersion is large with large horizontal shear.
7. When non-dimensional time with are fixed no shear, non-dimensional energy dispersion is proportional to the square of drift speed.

並列關鍵字

scaling law ； vortex energy dispersion ； drift speed ； Rossby number ； horizontal shear

參考文獻

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Chang, C. P., J. M. Chen Chen, P. A. Harr, and L. E. Carr, 1996: Northwestward-propagating wave patterns over the tropical Western North Pacific during summer. Mon. Wea. Rev., 124, 2245-2266.

DeMaria, M., and J. C. L. Chan, 1984: Comments on “A numerical study of the interactions betweens two tropical cyclones.”Mon. Wea. Rev., 112, 1643-1645.

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