੩ㇺᄢቇ㒐ἴ⎇ⓥᚲᐕႎ ╙ภ$ᐔᚑ㧣ᐕ᦬
Annuals of Disas. Prev. Res. Inst., Kyoto Univ., No. 48 B, 2005

㘑ᵄ਄ࠍㅢㆊߔࠆ᳇ᵹߩᰴర⋥ធᢙ୯⸘▚ᧁේ⋥ੱ ࡮ᬀ↰ᵗආ࡮⧎ፒ⑲ผ ੩ㇺᄢቇᄢቇ㒮ℂቇ⎇ⓥ⑼
੩ㇺᄢቇᄢቇ㒮ℂᎿቇ⎇ⓥ⑼

ⷐ ᣦ৻ቯߩ૏⋧ㅦᐲߢㅴⴕߔࠆ㘑ᵄ਄ߩ᳇ᵹߦ㧟ᰴర⋥ធᢙ୯⸘▚ࠍㆡ↪ߔࠆߎߣߦ
ࠃࠅ㧘᳇ᵹߩੂᵹ․ᕈࠍ⺞ߴߚ‫ޕ‬⒳㘃ߩᵄ㦂㧔 c / u*0, 4, 8, 10, 12, 16 and 20㧕ߢ㧘ߘࠇߙࠇߦߟ޿ߡੂᵹ߇⛔⸘⊛ߦቯᏱ⁁ᘒߦߥࠆ߹ߢ⸘▚ߒߚ‫⚿ߩߘޕ‬ᨐ㧘᳇ᵹ෸߮ੂᵹߩ᜼േߦᒝ޿ᵄ㦂ଐሽᕈ߇⷗ࠄࠇߚ‫ߚ߹ޕ‬㧘ᵄߦࠃࠅࠁ߇߹ߐࠇߚᵹࠇߪ㧘⥃⇇㜞ߐㄭறߢᄢ߈ߊᄌൻߒߚ‫⚿ߩߘޕ‬ᨐ㧘ᵄߩ⊒㆐₸ߣᒝ޿㑐ଥ߇޽ࠆᵄߦࠃࠆㆇേ㊂ࡈ࡜࠶ࠢࠬ(wave-induced momentum flux)ߪ⥃⇇㜞ߐࠃࠅૐ޿႐ᚲߢߪᱜ㧘㜞޿႐ᚲߢߪ⽶ߣߥࠆߎߣ߇ࠊ߆ߞߚ‫ޕ‬ᴾࠠ࡯ࡢ࡯࠼㘑ᵄ㧘ੂᵹ㧘ᄢ᳇࡮ᶏᵗ⋧੕૞↪㧘Ⴚ⇇ጀ㧘㧟ᰴర⋥ធᢙ୯⸘▚ ✜   ⸒ᵹᚑಽߪ㧘᳇஥ߩᐔဋ㘑ㅦಽᏓߩ⛽ᜬߦᔅⷐߣߐࠇߡ޿

ᴾ

ࠆ߽ߩߩ㧘ߘߩ㘑ᵄߩ⊒㆐߳ߩ⋥ធ⊛ߥᓇ㗀ߪήⷞߐࠇ
㘑ᵄߦࠃࠆᄢ᳇࡮ᶏᵗ㑆ߩㆇേ㊂࡮ᾲ࡮‛⾰ߩ੤឵ߪ㧘ߡ޿ࠆ‫ޕ‬

ᄢ᳇஥ߩᾲᏪૐ᳇࿶ߩ⊒㆐߿㧘ᶏᵗ஥ߩ㜞ầ߿㜞ᵄߩ⊒ߎࠇߦኻߒ㧘Belcher and Hunt (1998)߿ Townsend (1972)㆐ࠍࠦࡦ࠻ࡠ࡯࡞ߔࠆ‫߼ߚߩߘޕ‬㧘ߎࠇࠄߩ⃻⽎ߩᱜ⏕ߪ㧘ੂᵹਛߢߪᵹ૕ߪੂᵹ᷵ߦࠃࠅ㜞ㅦߢ⥃⇇㜞ߐࠍㅢߥ੍᷹ߦߪ㘑ᵄߣ᳇ᵹߣߩ⋧੕૞↪ࠍචಽᛠីߔࠆᔅⷐ

ㆊߔࠆߚ߼㧘⥃⇇㜞ߐઃㄭߩᵹࠇ႐ߩ᭴ㅧߪ㧘᳇ᵹ߆ࠄ߇޽ࠆ‫ޕ‬

㘑ᵄ߳ߩㆇേ㊂ャㅍߦ߅޿ߡ㊀ⷐߢߪߥ޿ߣㅀߴߡ޿ࠆ‫ޕ‬᳇ᵹߣ㘑ᵄߩ⋧੕૞↪ߦߟ޿ߡߪ㧘Jeffreys (1925)㧘

৻ᣇ㧘Sullivan et al. (2000)ߪ㧘㘑ᵄ߇ᒻࠍᄌ߃ߥ޿ߢ㧘Phillips (1957)㧘Miles (1957)╬ߩℂ⺰⊛⎇ⓥࠍߪߓ߼㧘ᵴ৻ቯㅦᐲߢㅴ߻ㅴⴕᵄ਄ߩ᳇ᵹࠍ 3 ᰴరߩ⋥ធᢙ୯⸘▚⊒ߦ⎇ⓥߐࠇߡ߈ߚ‫ޕ‬Jeffreys ߪ㧘᳇ᵹ߇㘑ᵄߩ㘑ਅ஥ߢ(DNS)ߦࠃࠅ⺞ߴ㧘᳇ᵹ߇ੂᵹߩ႐วߢ߽⥃⇇㜞ߐ߇ᐔဋ೸㔌ߔࠆߎߣߦࠃࠅ࿶ജߩಽᏓ߇ᵄ㗡ߩ਄ᵹ஥ߣਅᵹ஥

㘑ㅦಽᏓߦᓇ㗀ࠍਈ߃ࠆߎߣࠍ␜ߒߚ‫ࠄ߇ߥߒ߆ߒޕ‬㧘ߢ㕖ኻ⒓ߣߥࠅ㧘ߘߩ⚿ᨐ㧘࿶ജᛶ᛫߇Ⴧടߔࠆߣ޿߁

ㅴⴕᵄ߇ሽ࿷ߔࠆߣ߈ߩ᳇ᵹߩ࡟ࠗࡁ࡞࠭ᔕജ߿ੂࠇᒝ

઒⺑ࠍឭ᩺ߒߚ‫ߚ߹ޕ‬㧘Phillips ߪੂᵹߩ࿶ജᡓੂߣ᳓㕙ᐲ╬ߩੂᵹ⛔⸘㊂ߩⓨ㑆ಽᏓߩ⹦⚦ߥ⸃ᨆߪߥߊ㧘ㅴⴕᵄߩ౒㡆ߦࠃࠅ᳇஥߆ࠄᶧ஥߳ࠛࡀ࡞ࠡ࡯߇ャㅍߐࠇࠆᵄ߇᳇ᵹߩੂᵹ᭴ㅧߦਈ߃ࠆᓇ㗀ߦ㑐ߔࠆℂ⸃ߪਇචಽ

ߣߔࠆℂ⺰ࠍឭ᩺ߒߚ‫৻ޕ‬ᣇ㧘Miles ߪߖࠎᢿᵹߩ✢ᒻ቟ߢ޽ࠆ‫ߚ߹ޕ‬㧘ㅴⴕᵄߣหߓㅦᐲߢ⒖േߔࠆᐳᮡ♽ߢߪ㧘ቯᕈ⸃ᨆߦࠃࠅ㧘᳇ᵹߦࠃࠆ㘑ᵄߩ⊒㆐₸ࠍ᳞߼ߚ‫ߎޕ‬ᐔဋ㘑ㅦߩᵹ✢ߪ⥃⇇㜞ߐߢ cat’s eye ߣ๭߫ࠇࠆ㐽ᦛ✢

ߩℂ⺰ߪ㧘⊒㆐₸߇⥃⇇㜞ߐ z c 㧔ᐔဋ㘑ㅦ߇㘑ᵄߩ૏⋧ࠍឬߊ߇㧘᳇ᵹ߆ࠄ㘑ᵄ߳ߩㆇേ㊂ャㅍߦߘࠇ߇ਈ߃ࠆ

ㅦᐲ c ߣหߓߦߥࠆ㜞ߐ㧕ߦ߅ߌࠆᐔဋ㘑ㅦಽᏓߩᦛ₸ߦᲧ଀ߔࠆߎߣࠍ੍᷹ߔࠆ‫ޕ‬ᓥߞߡ㧘⊒㆐₸ߪ㧘㘑ᵄߩ૏ᓇ㗀߽ਇ᣿ߢ޽ࠆ‫ޕ‬ߘߎߢ㧘ᧄ⎇ⓥߢߪ㧘㘑ᵄ߇ᒻࠍᄌ߃ߥ޿ߢ৻ቯ

⋧ㅦᐲ c ߣ᳇ᵹߩ⇇㕙៺ᡂㅦᐲ u* ߩᲧߢ޽ࠆᵄ㦂ㅦᐲߢㅴ߻ㅴⴕᵄ਄ߩ᳇ᵹ႐ࠍ㧘3ᰴరߩDNSߦࠃࠅ( c / u* )ߦଐሽߔࠆߎߣߦߥࠆ‫ߩߎߒ߆ߒޕ‬ℂ⺰ߢߪ㧘ੂ᳞߼㧘߹ߕ㧘㘑ᵄ߇᳇ᵹߩᐔဋ㘑ㅦ႐ߦਈ߃ࠆᓇ㗀ࠍ⺞ߴࠆ‫ޕ‬ᰴߦ㧘⥃⇇㜞ߐㄭறߩ᳇ᵹߩ᭴ㅧ㧘෸߮᳇ᵹ߆ࠄ㘑ᵄ߳ߩࠛࡀ࡞ࠡ࡯ャㅍ߳ߩᓇ㗀ࠍ⺞ߴࠆ‫ޕ‬ᦨᓟߦ㧘ㅴⴕᵄ਄ߩ᳇ᵹߩੂᵹ᭴ㅧࠍ⺞ߴ㧘㘑ᵄ߇ੂᵹ႐ߦਈ߃ࠆᓇ㗀ࠍ⼏⺰ߔࠆ‫ޕ‬ᴾ ᰴర⋥ធᢙ୯⸘▚ᴾᧄ⎇ⓥߢ⠨߃ࠆᵹࠇߪ㧘Fig.1 ߦ␜ߔࠃ߁ߥ㧘ਅ㕙߇ᤨ

Fig. 1 Computational domain.

㑆⊛ߦᄌᒻߒߥ޿㧞ᰴర⊛ߥㅴⴕᵄ㕙㧘਄㕙߇᳓ᐔߥ៺
ᡂߥߒߩႺ⇇ߢ޽ࠆ㗔ၞౝߩ᳇ᵹߢ޽ࠆ‫ޕ‬᳓ᐔᣇะߪ๟

v bot

0

(7)

ᦼႺ⇇᧦ઙࠍ઒ቯߔࠆ‫ޕ‬ㅴⴕᵄߣหߓㅦᐲߢ⒖േߔࠆᐳ

w bot

wo

(8)

ᮡ♽ࠍ↪޿ߡ⸘▚ߔࠆߚ߼㧘ㅴⴕᵄߩᒻ⁁ߪᤨ㑆⊛ߦᄌൻߒߥ޿‫ߚ߹ޕ‬ᵹࠇߪ㧘৻ቯߩ᳓ᐔ࿶ജ൨㈩ߦࠃߞߡ㚟ߎߎߢ㧘 u o ෸߮ w o ߪ᳓㕙ᵄߩㆇേߦࠃࠆㅦᐲᚑಽߢ޽

േߐࠇࠆߣߒ㧘ቢోߦ⊒㆐ߒߚቯᏱੂᵹߦኻߔࠆ⸘▚ࠍࠅ㧘એਅߩᑼߢ⴫ߐࠇࠆ‫ޕ‬ⴕ߁‫ޕ‬ၮ␆ᣇ⒟ᑼߪ㧘એਅߦ␜ߔ㕖࿶❗ᕈᵹ૕ߦኻߔࠆ
࠽ࡧࠖࠛ࡮ࠬ࠻࡯ࠢࠬᣇ⒟ᑼߣㅪ⛯ߩᑼߢ޽ࠆ‫ޕ‬
wu i
wu i
u j
wt
wx j

w 2ui
1 wp
Q

U wx iwx j 2(1)uo

akc cos(kx)^1  2ak cos(kx)`

(9)

wo

akcsin ( kx)^1  2ak cos(kx)`(10)ᢙ୯⸘▚ߦ߅޿ߡߪ㧘ᐳᮡᄌ឵ߒߚၮ␆ᣇ⒟ᑼࠍᒝ଻
ሽᒻߢᦠ߈㧘ࠦࡠࠤ࡯࠻ᩰሶࠍ↪޿ߡ㧘ⓨ㑆ᓸಽߪ 2 ᰴ♖ᐲਛᔃᏅಽ㧘ᤨ㑆ᓸಽߪ 2 ᰴ♖ᐲߩ Adams-Bashforth ᴺ

wu i
wx i

(2)

0

ߎߎߢ㧘ᷝሼ i ( 1, 2, 3) ߪߘࠇߙࠇᵹࠇᣇะ x㧘ࠬࡄࡦᣇะ y㧘㋦⋥ᣇะ z ࠍ⴫ߔ‫ޕ‬z ߪ㧘ᐔဋ᳓㕙ࠍේὐ(z=0)ߦߣ

ߢᏅಽൻߔࠆ‫ޕ‬ᣇ⒟ᑼߪ fractional step ᴺߦࠃࠅᤨ㑆⊒ዷߐߖ㧘ㅪ⛯ߩᑼ(2)ߪ㧘એਅߩ࿶ജߦኻߔࠆࡐࠕ࠰ࡦᣇ⒟ᑼࠍ⸃ߊߎߣߦࠃࠅḩ⿷ߐߖࠆ‫ޕ‬ࠆ‫ߚ߹ޕ‬㧘 p ߪ࿶ജ㧘 U ෸߮Q ߪߘࠇߙࠇ㧘ⓨ᳇ߩኒᐲw2 p(u,v, w) ߪ ߘ ࠇ ߙ ࠇ2෸ ߮ േ ☼ ᕈ ଥ ᢙ 㧘 (u1,u 2 ,u 3 )x, y, z ᣇะߩᵹㅦߢ޽ࠆ‫ޕ‬⸘▚ߔࠆᐳᮡ♽ߪ㧘Benjamin (1959)߇ឭ᩺ߒߚએਅߩ
ᄌ឵ᑼ(3)ߦࠃࠅ ( x, y, z ) ᐳᮡࠍᵄ㕙ߦᴪߞߚ৻⥸ᐳᮡ♽([ ,K , ] ) ߳ᄌ឵ߒߚ߽ߩߢ޽ࠆ‫ޕ‬
ª[ º
« »
«K »
«¬] »¼ª x  ia exp^ k ( z  ix)`º
«
»
y
«
»
«¬ z  a exp^ k ( z  ix)`»¼

w[

ᴺࠍ↪޿ߚ‫ޕ‬ᓐࠄߩᣇᴺߢߪ㧘ᑼ(11)ࠍ߹ߕ㧘એਅߩࠃ߁
ߦᄌᒻߔࠆ‫ޕ‬(3)

2S / O 㧕ߢ޽ࠆ‫ߣߩߎޕ‬߈ࡗࠦࡆࠕࡦ J ߪ㧘w ([ ,K , ] )w ( x, y , z )ªw2 p w2 p w2 p º

« 2 
»
wK 2 w] 2 ¼» m¬« w[
1 § wU wV wW · ª§ 1 · w 2 p º¨¸  «¨1  ¸»'t ¨© w[ wK w] ¸¹ «¬© J ¹ wK 2 »¼

(4)

0 ߣ߅ߊߎߣߦࠃࠅ㧘ᵄ

ᒻ z bot ߪએਅߩࠃ߁ߦㄭૃߐࠇࠆ‫ޕ‬z bot | a cos(kx)  a 2 k cos 2 (kx)(5)ߎߎߢ㧘ᷝሼ bot ߪ᳇ᶧ⇇㕙ߢߩ୯ࠍ␜ߔ‫ߚ߹ޕ‬㧘ߎߩᵄᒻߦኻᔕߒߡ㧘᳇ᶧ⇇㕙ߢߩᵹㅦߪએਅߩ୯ߣߥࠆ‫ޕ‬u bot

c  u o  (12)m 1ߎߎߢᷝሼ m ߪ෻ᓳ࿁ᢙࠍ⴫ߒߡ޿ࠆ‫ޕ‬m ࿁⋡ߩ෻ᓳᤨߦ㧘m  1 ࿁⋡ߩ෻ᓳߢᓧࠄࠇߚ࿶ജ p ࠍᑼ(12)ߩฝㄝߦ

1  2ak exp^ k ( z  ix)`ߣߥࠆ‫߅ߥޕ‬㧘ᄌ឵ᑼ(3)ߢ ]1 § wU wV wW ·¨¸  (11)'t ¨© w[ wK w] ¸¹ᚑಽߢ޽ࠆ‫ޕ‬ᑼ(11)ߩ⸃ᴺߦߪ Sullivan ࠄ߇ឭ᩺ߒߚ෻ᓳ

ㇱࠍߣࠆ‫ߚ߹ޕ‬㧘 a ߣ k ߪߘࠇߙࠇ㧘ㅴⴕᵄߩᝄ᏷෸߮

J

1 w2 p w2 p
J wK 2 w] 2ߎߎߢ㧘 (U , V , W ) ߪ ([ ,K , ] ) ᣇะᵹㅦߩ৻⥸ᐳᮡ෻ᄌߎߎߢ㧘i ߪ⯯ᢙන૏ߢ޽ࠅ㧘‛ℂᄌᢙߣߒߡߪᑼ(3)ߩታ
ᵄᢙ㧔 O ࠍᵄ㐳ߣߒߡᵄᢙ k(6)↪޿㧘⋥ធᴺ (Shumann and Sweet, 1988; Nagaosa, 1999)ߦ
7
ࠃࠅᑼ(12)ࠍ⸃ߊ‫ߩߡోߡߒߘޕ‬ὐߦ߅޿ߡ |   u | 10߇ḩ⿷ߐࠇࠆ߹ߢ෻ᓳ⸘▚ࠍⴕߞߚ‫ޕ‬

u* h / Q )ߪ 150 ߣߒߚ‫ߚ߹ޕ‬એਅߢߪ㧘ᷝሼ  ࠍ↪޿ߡ㧘ㅦᐲࠍ⇇㕙៺ᡂㅦᐲ u* 㧘㐳ߐࠍQ / u* ߦࠃࠅήᰴరൻߒߚ୯ࠍ⴫ߔ‫▚⸘ޕ‬㗔ၞߪᵹࠇᣇะ㧘ࠬࡄࡦᣇะ㧘㋦⋥ᣇะࠍߘࠇߙࠇ L x 8h 㧘 L y Sh 㧘
L z h ߣߒߚ‫ߚ߹ޕ‬㧘ᩰሶὐᢙߪ㧘ߎࠇࠄਃᣇะߦ 180࡟ࠗࡁ࡞࠭ᢙ( ReW

 100  128 ߣߒ㧘ᩰሶ㑆㓒ߪ㧘᳓ᐔᣇะߪ╬㑆㓒( 'x 

6.7 㧘 'y 

4.7 )㧘㋦⋥ᣇะߪ਄ਅႺ⇇ߢᦨዊ( 'z 0.2 )㧘ᵹ〝ਛᔃߢᦨ ᄢ( 'z O ( 2S / k)ᵄߩᵄ㐳ߪ,

߼㧘ᵄᒻ൨㈩ ak

1.9 )ߣߒߚ‫ޕ‬ㅴⴕ4h / 3 ߣߥࠆࠃ߁ߦ k ࠍቯ

0.1 ߣߒߚ‫ޕ‬ᵄ㦂 c / u* ߪ㧘0, 4, 8, 10, 12,

16 ෸߮ 20 ߩ 7 ࠤ࡯ࠬߦߟ޿ߡ⸘▚ߒ㧘ੂᵹ⛔⸘㊂߇ቯᏱߦߥࠆ߹ߢᤨ㑆⊒ዷߐߖߚ‫߅ߥޕ‬㧘⸘▚♖ᐲࠍ⏕⹺ߔࠆߚ߼㧘 ReW150 ߢߩ㐿᳓〝ੂᵹߩ⸘▚ࠍታᣉߒ㧘ᐔဋᵹㅦ߿࡟ࠗࡁ࡞࠭ᔕജ╬ߩಽᏓ߇‫ ߫߃଀ޔ‬Kasagi et al.

(1992)ߩᐔⴕᐔ᧼㑆ੂᵹߩ⚿ᨐߣ z   70 ߢ৻⥌ߔࠆߎߣࠍ⏕⹺ߒߚ‫ޕ‬ ㅴⴕᵄ߇ᐔဋᵹㅦ႐ߦਈ߃ࠆᓇ㗀 ⛔⸘㊂ߩቯ⟵
2 ᰴర⊛ߥᵄ㕙਄ߩ᳇ᵹߢߪ㧘ᵹࠇ (x) ᣇะߦ㧘⇇㕙ߩᵄᒻߣหߓᵄᢙߩᄌൻ߇ᐔဋᵹߦ↢ߓࠆ‫ޕ‬એਅߩ⸃ᨆߢߪ‛ℂ㊂ q ࠍ㧘ᤨ㑆 t 㧘ࠬࡄࡦᣇะᐳᮡ y 㧘෸߮㧘ㅴⴕᵄ೉ౝߩ⇣ߥࠆᵄߩห৻૏⋧ I ( 0 d I d 2S )ߦ߅ߌࠆ୯ߢᐔဋߒߚ૏⋧ᐔဋᚑಽ q (I , z ) ߣ㧘ੂᵹᚑಽ q c ߣߦಽߌࠆ(Hussain and Reynolds, 1970)‫ޕ‬q ( x, y, z , t ) q (I , z )  q c( x, y, z , t )q (I , z )
N 1

u

¦³
n 0

c / u*
c / u*1
NL y 'T
§
q¨¨ x
©

§

O¨ n 
©

·I ·¸, y, z , t ¸¸ dydt
2S ¹¹

ⴕᵄߩᢙߢ޽ࠅ㧘ᧄ⎇ⓥߢߪ N(14)

6 ߢ޽ࠆ‫ߚ߹ޕ‬㧘૏⋧q ( z ) ( x, y, t ᐔဋ)ߣ㧘ᧄ⎇ⓥߢ․ᓽ⊛ߥ⇇㕙ᵄᒻߦࠃ~ (I , z ) ߣߦಽߌࠆߎߣ߇ߢ߈ࠆ‫ޕ‬ࠆᵄ⁁ᚑಽ qq ( z )  q~ (I , z )1
L x L y 'T
12S³

0 , (b) c / u * 4 , (c) c / u *12 , and (e) c / u * 20 .

8 , (d)஥ߦ૏⟎ߔࠆ‫ޕ‬ㅴⴕᵄ㧔 c / u* ! 0 㧕ߩ႐วߪ㧘ᵹ✢ߪ㧘⥃⇇㜞ߐઃㄭߢ cat’s eyeߣ๭߫ࠇࠆ㐽ᦛ✢ࠍឬߊᐔ ဋ ᚑ ಽ q (I , z ) ߪ ߐ ࠄ ߦ 㧘 ࠕ ࡦ ࠨ ࡦ ࡉ ࡞ ᐔ ဋ

q ( z)

p for (a)and the wave-correlated pressure ~(13)ߎߎߢ㧘'T ߪᐔဋߔࠆᤨ㑆㑆㓒㧘N ߪ⸘▚㗔ၞౝߩㅴq (I , z )Fig. 2 The phase averaged streamlines (white lines)(15)ߚ߼㧘ߘߩ਄ਅߢᵹࠇߩะ߈߇⇣ߥࠆ‫߅ߥޕ‬㧘c / u*

4 ߢߪ㧘 cat’s eyeߪߜࠂ߁ߤᵄߩ⼱ߩ਄ߦ૏⟎ߒ㧘ߘߩ⌀਄ߩᵹࠇߪ㧘㕒ᱛᵄ⁁ო㕙ߩᵄ㗡਄ߩᵹࠇ㧔 Fig. 2(a)ෳᾖ㧕ߣ㘃ૃߢ޽ࠆ‫ߚ߹ޕ‬㧘 c / u *8

෸߮ 12ߢߪ㧘 cat’s eyeߪ㧘ߘߩਛᔃߩ਄ᵹ஥ߣਅᵹ
஥ߢ㕖ኻ⒓ߥᒻ⁁ࠍߒߡ޿ࠆ‫ޕ‬ᵄ㦂߇ߐࠄߦᄢ߈ߊ
ߥࠆߣ㧔 c / u *20㧕㧘⥃⇇㜞ߐ߇⸘▚㗔ၞߩ਄┵ߦ૏⟎ߔࠆߚ߼㧘cat’s eyeߪ Fig.2(e)㧔 0  kz  3㧕ߦߪq( x, y, z , t ) dxdydt

2S⴫␜ߐࠇߡ޿ߥ޿‫߅ߥޕ‬㧘ߎߩᤨߩᵹ✢෸߮ p̃ ߪᵄ(16)³ q (I , z) dI
0㗡ߩ਄ᵹ஥ߣਅᵹ஥ߢ߶߷ኻ⒓ߢ޽ࠆ‫ޕ‬ᴾ ᐔဋᵹㅦߩ㋦⋥ಽᏓᴾ ᵹ✢෸߮࿶ജߩⓨ㑆ಽᏓೋ߼ߦ㧘ᵹㅦ u , w ߩ૏⋧ᐔဋᚑಽ u , w ߦኻᔕߔFig.ߦᧄ⸘▚ߢ↪޿ߚోߡߩᵄ㦂 c / u* 㧘෸߮㕒ᱛ᳓ᐔო㕙਄ߩ᳇ᵹߦ߅ߌࠆ㧘㕒ᱛ♽ߦኻߔࠆᐔဋ

ࠆᵹ✢㧘෸߮࿶ജߩᵄ⁁ᚑಽ p̃ ߩ╬㜞✢ࠍ Fig.2ߦ␜ᵹㅦ ( u ! c ) / u* ߩಽᏓࠍᐔဋ᳓㕙߆ࠄߩ〒㔌 z ߔ‫ޕ‬૏⋧ᐔဋᚑಽ෸߮ᵄ⁁ᚑಽߪ㧘3.1▵ߢㅀߴߚࠃ

ߩ㑐ᢙߢ␜ߔ‫ޕ‬ਅ㕙Ⴚ⇇߇ᵄ⁁ო㕙ߩ႐วߪ㧘ᧄ⎇߁ߦᧄ᧪ 0 d I d 2S ߢቯ⟵ߐࠇߡ޿ࠆ߇㧘Fig.2ߢߪ㧘ⓥߢ↪޿ߚ akᵹࠇߩ᭽⋧ࠍࠊ߆ࠅ߿ߔߊߔࠆߚ߼㧘 0 d I d 2 S ߩz   3.2 ߢߪᵄߩሽ࿷ߦࠃࠅᐔဋࠍߣࠆߎߣ߇࿎㔍࿑ߣหߓ߽ߩࠍ 2 S d I d 4 S ߩ▸࿐ߦ⛯ߌߡᦠ߈‫ޔ‬ੑߢ޽ࠆ‫ ߡߞ߇ߚߒޕ‬z   3.2 ߩ㗔ၞߪ⴫␜ߒߡ޿ߥ޿‫ޕ‬๟ᦼಽ ( 0 d I d 4S ) ߩ࿑ߣߒߡ޽ࠆ‫ޕ‬㕒ᱛᵄ⁁ო㕙㧔 c / u*0㧕ߩ႐ว㧘࿶ജಽᏓߪᵄ㗡ߩ㘑਄஥ߣ㘑ਅ஥ߢ㕖ኻ⒓ߢ޽ࠅ㧘 p̃ ߩᦨዊ୯ߪᵄ㗡ߩ߿߿㘑ਅ

0.1 ߩᤨ㧘ᵄ㗡ߢ z 

3.2 ߩߚ߼㧘߹ߕ㧘㕒ᱛ᳓ᐔო㕙਄ߩᵹࠇߢߪ㧘ო㕙ㄭறߩ☼ᕈᐩጀߢߪ u / u *u / u*z  ߦ㧘 z  ! 30 ߩኻᢙ㗔ၞߢߪ(1 / 0.4) ln( z  )  5.5 ߦẋㄭߒ㧘ㆊ෰ߩ㐿᳓〝ੂᵹߩ&05ߣ߽ࠃߊ৻⥌ߔࠆ(Nagaosa, 1999)‫  ޕ‬ߎࠇߦኻߒ㕒ᱛᵄ⁁ო㕙 c / u *0 ߢߪࡃ࠶

ࡈࠔ࡯ጀઃㄭ 5  z   30 ߢᐔဋᵹㅦ൨㈩߇߿߿ዊߐߊ
ߥࠆ‫⚿ߩߘޕ‬ᨐ㧘ኻᢙ㗔ၞ z  ! 30 ߢߪᐔဋᵹㅦ
ߪ߿߿ዊߐߊߥࠆ‫ߒ߆ߒޕ‬㧘ᐔဋᵹㅦ൨㈩ߦߪᱴࠎߤ㆑
޿߇ߥ޿‫ޕ‬ㅴⴕᵄ਄ ߩ ᵹ ࠇ (c / u* ! 0) ߢ ߪᵄ 㦂 c / u* ࠍ Ⴧടߐߖࠆߣ, ࡃ࠶ࡈࠔ࡯ጀ㧔 z   30 ߹ߚߪ੹࿁
ߩ᧦ઙ⸳ቯਅߢ kz  1 㧕ߢߩᐔဋᵹㅦ൨㈩ߪ৻ᣤ
Ⴧട㧔 c / u*

4 㧕ߒߚᓟᷫዋߒ c / u *Ⴧടߔࠆ c / u *8 , ౣ߮

10, 12, 16 ෸߮ 20 ߣ޿߁ⶄ㔀ߥ᜼േࠍ␜ߔ‫ߒ߆ߒޕ‬㧘ኻᢙ㗔ၞ z  ! 30 ߢߩᐔဋᵹㅦ
൨㈩ߩᄌൻߪዊߐ޿‫ޕ‬৻ ᣇ  ኻ ᢙ 㗔 ၞ ߢ ߩ ᐔ ဋ ᵹ ㅦ ߪ  ᵄ 㦂 ߇

c / u*

4 ߢ৻ᣤ߿߿Ⴧടߒߚᓟ , ᷫዋߒ , c / u *8ߩᤨߦᦨዊߣߥࠆ‫ ޕ‬c / u* ߇ߐࠄߦᄢ߈ߊߥࠆߣౣ߮
Ⴧടߒ㧘ᦨᄢᵄ㦂ߢ޽ࠆ c / u*

20 ߢᦨᄢߣߥࠆ‫ޕ‬㕒ᱛ᳓ᐔო㕙ߩ႐วߣߩᐔဋᵹㅦߩᏅߪ㧘ᵄߩ⊒㆐ࡄ࡜ࡔ࡯࠲ E 㧔ᓟㅀ▵ࠍෳᾖ㧕ߣ⋧㑐߇⦟ߊ㧘

E ! 0 ߩᤨ㧔ߔߥࠊߜ㧘᳇ᵹ߆ࠄ㘑ᵄߦࠛࡀ࡞ࠡ࡯
߇ᵈ౉ߐࠇࠆᤨ㧕ߪ㧘㕒ᱛ᳓ᐔო㕙ߩ႐วࠃࠅ߽ᐔ

Fig. 3  The vertical distributions of the ensemble
averaged velocities. Bold solid line denotes
the flow over a flat surface. ٤, c / u * 0 ;+, c / u * 4 ; ٨ , c / u * 8 ; ٌ ,c / u* 10 ; ً , c / u * 12 ; ‫ غ‬,
c / u * 16 ; ‫ع‬, c / u * 20 .

ဋᵹㅦ߇ᷫዋߒ㧔ߚߛߒ c / u*

4 ߩᤨએᄖ㧕㧘E  0ߩᤨߪჇടߔࠆ‫߅ߥޕ‬㧘 c / u*

4 ߩᤨߪ㧘᳇ᵹ߆ࠄߢ û 㧘 ŵ ߩਔ⠪߇ᭂዊ୯ࠍᜬߜ (Fig.4(a), (b))㧘ߐ㘑ᵄ߳ࠛࡀ࡞ࠡ࡯߇ャㅍߐࠇࠆߚ߼㧘ო㕙ㄭறࠄߦ㧘૏⋧ T u~ , T w̃ ߇ᕆỗߦᷫዋߔࠆ (Fig.4(c), (d))‫ޕ‬(z   10) ߢߪᐔဋᵹㅦ߇ᷫዋߔࠆ‫ߒ߆ߒޕ‬㧘ㅴⴕᵄߎࠇߪ㧘 z | z c ߢᐔဋᵹߦᒝ޿ࠪࠕ࡯߇↢ߓߡ޿ࠆߩ⼱ߩㇱಽߦ↢ߓࠆᓴⅣᵹ㧔cat's eye㧕ߩߚ߼㧘ࡃ࠶ߎߣࠍ␜ߔ‫ߚ߹ޕ‬㧘ߎߩ⥃⇇㜞ߐߢߩᵹㅦߩᕆỗߥࡈࠔ࡯ጀߦᒝ޿ࠪࠕ࡯߇↢ߓ㧘ߘߎߢߩᐔဋᵹㅦ൨

ᄌൻߪ㧘 Hristov et al. (2003)ߩ᷹ⷰ⚿ᨐߣቯᕈ⊛ߦ৻㈩߇㕖Ᏹߦᄢ߈ߊߥࠆ‫⚿ߩߘޕ‬ᨐ㧘ኻᢙ㗔ၞߩᐔဋ⥌ߔࠆ‫ޕ‬ᵹㅦߪ⧯ᐓჇടߔࠆߣ⠨߃ࠄࠇࠆ‫ޕ‬ᴾ ᵄߦ⿠࿃ߔࠆㆇേ㊂ࡈ࡜࠶ࠢࠬ

 ⥃⇇㜞ߐ߇ᵹㅦߩⓨ㑆ಽᏓߦਈ߃ࠆᓇ㗀ᰴߦ㧘ੂᵹ႐ߢߩᵹㅦߩᵄ⁁ᚑಽ u~ , w̃ ߣ⥃⇇㜞ߐߣߩ㑐ଥࠍ⺞ߴࠆ‫ᧄޕ‬ᢙ୯⸘▚ߩ⚿ᨐߢߪ㧘 u~ ෸߮ w̃ ߦ߅޿ߡ㧘ㅴⴕᵄߩᵄ㐳ߦኻᔕߔࠆᵄᢙ k ߩᚑಽ߇ථ⿧ߒߡ
޿ࠆߚ߼㧘ߚߣ߃߫ u~ ߪ, z ߛߌߩ㑐ᢙ û (z ) ߣ㧘 x ᣇะߦㅴⴕᵄߣหߓ๟ᦼࠍᜬߟᱜᒏ㑐ᢙߩⓍߣߒߡ㧘એਅߩࠃ߁ߦㄭૃߢ߈ࠆ‫ޕ‬>1u~ (I , z ) | uˆ ( z ) exp(ikx)  uˆ * ( z ) exp(ikx)
2| uˆ | ( z ) cos(kx  T u~ ( z ))

@

(17)

ߘ ࠇ ߙ ࠇ û ߩ ᝄ ᏷ ߣ ૏ ⋧ ( S d T u~ d S ) ߢ ޽ ࠆ ‫ޕ‬

0 ߣߥࠆ㜞ߐ z ߢߪ㧘 u~ ߪਅ㕙ߩㅴⴕᵄߣห૏

⋧ߢ޽ࠅ㧘 T ũ ! 0 ( 0) ߪ㧘૏⋧ߩㅴߺ㧔ㆃࠇ㧕ࠍ⴫ߔ‫ޕ‬

Fig.4ߦ㧘⒳‫ߩޘ‬ᵄ㦂 c / u* ߦኻߔࠆ u~ ෸߮ w̃ ߩᝄ᏷ߣ૏⋧ߩಽᏓࠍ㧘z / z c ߩ㑐ᢙߢ␜ߔ‫ ߒߛߚޕ‬z c ߪ㧘

 u ! (z c )ᦼ⊛ߥᄌൻࠍߔࠆ‫߫߃଀ޕ‬㧘ᑼ(13), (15)ࠍㆇേᣇ⒟ᑼߦઍ
౉ߔࠆߣ㧘છᗧߩ㜞ߐ z ߢߩㆇേ㊂ࡈ࡜࠶ࠢࠬߪ
~ !   uc wc !  1 w  u ! 1  z   u~ w 
Re wzh(18)ߣߥࠆ‫ߢߎߎޕ‬㧘Ꮐㄝ╙৻㗄ߪᵄߩሽ࿷ߦ⿠࿃ߔࠆㆇേ
㊂ࡈ࡜࠶ࠢࠬߢ޽ࠅ㧘ᵄ㕙ߢߩㆇേ㊂੤឵ߣኒធߥ㑐ଥߎߎߢ㧘 û * ߪ û ߩ౒ᓎⶄ⚛ᢙߢ޽ࠅ㧘 | uˆ | ෸߮ T u~ ߪ㧘T u~ ( z )㧞ᰴర⊛ߥㅴⴕᵄߩሽ࿷ߦࠃࠅ㧘᳇ᵹߪਥᵹᣇะߦ๟

0 ߣߥࠆ㜞ߐߢ޽ࠆ‫⇇⥃ޕ‬㜞ߐ z c ߩㄭற߇޽ࠆ‫߅ߥޕ‬㧘Miles ߩḰጀᵹࡕ࠺࡞ߢߪ㧘   ũw̃ ! ߪ㧘z  z c ߢ㜞ߐߦࠃࠄߕ৻ቯ㧘 z ! z c ߢ   ũw̃ ! 0 ߢ޽ࠅ㧘☼ᕈߣੂᵹᚑಽࠍήⷞߒߡ޿ࠆߚ߼ zz c ߢਇㅪ⛯ߣߥࠆ‫ޕ‬ ⒳‫ߩޘ‬ᵄ㦂ߦ߅ߌࠆ |   ũw̃ !|ߩಽᏓࠍ Fig.5 ߦ␜ߔ‫ޕ‬~ ߪᄢ߈ߊᄌൻߒ㧘ߘࠇߦ઻޿
⥃⇇㜞ߐߦ߅޿ߡ u~ ߣ w  ũw̃ ! ߽ᄌൻߔࠆ‫ߢߎߎޕ‬㧘ᑼ(17)ߩቯ⟵ߦࠃࠅ㧘~ !  1 | uˆ || wˆ | cos(T ~  T ~ )  u~w
u
w
2

(19)

~ . (a), | uˆ | ; (b), | wˆ | ; (c), T ~ ; (d), T ~ .
Fig. 4 Amplitude and phase of u~ and w
uw
The symbols are same as figure 3.

ߢ ޽ ࠆ ߆ ࠄ 㧘 'T {| T u~  T w~ | S / 2 ߣ ߥ ࠆ 㜞 ߐ ߢߣ⴫ߖ㧘৻⥸ߦ E ߪ㧘㘑ᵄߩ⊒㆐₸ࠍ⴫ߔઍ⴫⊛ߥࡄ࡜  ũw̃ ! 0 ߣߥࠆ‫' ߼ߚߩߘޕ‬T ߇ zࡔ࡯࠲ߣߐࠇߡ޿ࠆ‫ߢߎߎޕ‬㧘V 㧘 U a , U w ߪ㧘㘑ᵄߩz c ࠍႺߦ S / 2એ਄߆ࠄ S / 2 એਅߦᄌൻߔࠆߣ㧔Fig.4 ෳᾖ㧕㧘z  z c ߢ๟ᵄᢙ㧘ⓨ᳇෸߮᳓ߩኒᐲߢ޽ࠆ‫ޕ‬ߪ   ũw̃ ! ߪᱜߢ޽ࠆ߇⥃⇇㜞ߐㄭறߢᕆỗߦᷫዋߒ㧘ᧄ⎇ⓥߢߩ DNS ߆ࠄ᳞߼ߚ E ࠍ㧘Mastenbroek (1996)

z ! z c ߢߪ⽶ߩ୯ࠍขࠆ‫ߚ߹ޕ‬㧘ᵄ㦂߇ᄢ߈ߊߥࠆߦߟ

ߦࠃࠆ㧞ᰴߩੂᵹࠢࡠ࡯ࠫࡖ࡯ࡕ࠺࡞ࠍ↪޿ߚ⚿ᨐ෸߮ࠇߡ⥃⇇㜞ߐߢߩᄌൻ߇Ṗࠄ߆ߦߥߞߡ޿ࠆ‫ߪࠇߎޕ‬㧘

Cohen and Belcher (1999)ߩ RDT (Rapid Distortion Theory)c ࠍḩߚߔ⥃⇇㜞ߐߩߩ⚿ᨐߣᲧセߒߚ߽ߩ߇ Fig.6 ߢ޽ࠆ‫⚿ߩࠇߕ޿ޕ‬ᨐ߽㧘૏⟎߇ᤨ㑆⊛ߦᄌേߒ㧘㋦⋥ᣇะߦ᏷ᐢߊಽᏓߔࠆߚ߼

ᵄ㦂߇ᄢ߈ߊߥࠆ߶ߤ E ߇ᷫዋߒߡ޿ߊ௑ะߪ৻⥌ߒߡߛߣ⠨߃ࠄࠇࠆ‫ޕ‬ᴾ

޿ࠆ߇㧘DNS ߢߪ㧘ੂᵹࡕ࠺࡞߿ RDT ߣᲧߴ㧘ૐᵄ㦂 ᵄߩ⊒㆐₸ߥߞߡ޿ࠆ‫ߩߎޕ‬㆑޿ߪ㧘࡟ࠗࡁ࡞࠭ᢙ㧘෸߮㧘ᵄ㕙ߢᵄ㦂߇ᄢ߈޿႐ว㧘 u ( x , y , z c , t )(c / u*  8) ߢ E ߇ᄢ߈ߊ㧘㜞ᵄ㦂 (c / u* ! 12) ߢዊߐߊ᳇ᵹ߆ࠄ㘑ᵄ߳ਈ߃ࠄࠇࠆන૏ᤨ㑆޽ߚࠅߩࠛࡀ࡞ࠡ
࡯ E ߪᵄ㕙ߦ߆߆ࠆ࿶ജ p bot ߣߖࠎᢿജ W bot ߦࠃߞߡߩ☻ᐲߩ㆑޿ߦࠃࠆߣ⠨߃ࠄࠇࠆ‫ޕ‬

Meirink and Makin (2000)ߪੂᵹࡕ࠺࡞ࠍ↪޿ߡ㧘ૐᵄ㦂એਅߩࠃ߁ߦ⴫ߐࠇࠆ(Mastenboek et al., 1996)‫ޕ‬(c / u*  5) ߢߩ E ߩ࡟ࠗࡁ࡞࠭ᢙ ReW ( 10 4 ) ଐሽᕈࠍdzdE
  p bot ( wo  bot u o ) !
dx
dt
dz bot  W bot (u o wo ) !
dx⺞ߴ㧘 E ߪ㧘 ReW  800 ߢߪ ReW ߩჇടߣ౒ߦჇടߒ㧘

E

(20)

ReW | 800 ߢᦨᄢߣߥࠆ߇㧘ߘߩᓟߪᷫዋߦォߓ㧘
ReW ! 10 4 ߢߪ㧘 ReW ߇ 75 એਅߩᤨߣห╬ߩዊߐߥ৻ቯ
୯ߦẋㄭߔࠆߎߣࠍ੍᷹ߒߚ‫ޕ‬Fig.6 ߩੂᵹࡕ࠺࡞෸߮

ߎߎߢ E ߪ㘑ᵄߩࠛࡀ࡞ࠡ࡯ߢ޽ࠅ㧘ㅦᐲ෸߮ᵄᒻ߇ᑼ

RDT ߪ㕖☼ᕈᵹ૕ࠍ઒ቯߒߡ޿ࠆߚ߼㧘 ReW ! 10 4 ߩ႐(5) 㧙 (10) ╬ ߢ ਈ ߃ ࠄ ࠇ ࠆ ࠬ ࠻ ࡯ ࠢ ࠬ ᵄ ߩ ႐ ววߣห᭽㧘 E ߇ዊߐ޿୯ߣߥߞߡ޿ࠆߣ⠨߃ࠄࠇࠆ‫ޕ‬(1 / 2) U w a 2 kc 2 ߣ ㄭ ૃ ߢ ߈ ࠆ ‫ ޕ‬㘑 ᵄ ߩ ⊒ ㆐ ₸E / EV ߪ㘑ᵄߩ⊒㆐ࡄ࡜ࡔ࡯࠲ E ࠍ↪޿ߡ

E

J
J

E
EV

E

U a § u* ·¨ ¸
U w ¨© c ¸¹߹ߚ㧘Fig.6ߩੂᵹࡕ࠺࡞෸߮ RDTߢߪ㧘ᵄᢙ kߢήᰴరൻߒߚ⴫㕙☻ᐲࠍ kz 0

10 5 ߆ࠄ kz 0

10 3 ߦჇടߐߖࠆߣ㧘 E ߪૐᵄᢙߢჇടߒ㧘㜞ᵄᢙߢᷫዋ

2

(21)

ߒߡ޿ࠆ‫⴫ࠆߌ߅ߦ▚⸘ᧄޕ‬㕙☻ᐲߪ㧘ᐔမߢṖࠄ

Fig. 5  The vertical distributions of the moduli of
ensemble averaged wave-induced momentum
fluxes. ٤ , ٌ , ‫ غ‬and ٖ denote positive~ ! 0) , and ٨ , ً , ‫ ع‬and ٕ
flux (  u~w~  0)
denotenegative
flux (  u~w
forc / u* 8, 10, 12 and 16.
߆ߥო㕙਄ߩ⴫㕙☻ᐲ z 0

0.11Q / u* ߣ╬ߒ޿ߣߔࠆߣ (Townsend, 1976)㧘 k

Su* /(100Q ) ߢ޽ࠆ߆ࠄ kz 0Fig. 6 The growth rate parameter E . ٨ denote our
DNS results; ‫ع‬, the second order closure model
for kz 0 10 3 ; ‫ غ‬, the second order closure
model for kz 0 10 5 ; ً , RDT for
kz 0 10 3 and ٌ, RDT for kz 0 10 5 .

3

3.5 u 10 ߣߥࠆ‫ߩߎޕ‬୯ߪ㧘࿑ਛߩੂᵹ

ࡕ࠺࡞෸߮ RDTߦᲧߴߡᄢ߈ߊ㧘ߘߩ⚿ᨐ㧘 E ߪૐᵄ㦂ߢߐࠄߦᄢ߈ߥ୯㧘㜞ᵄ㦂ߢߐࠄߦዊߐߥ
୯ߣߥࠆߣ⠨߃ࠄࠇࠆ‫ޕ‬ᴾ ㅴⴕᵄ߇ੂᵹ႐ߦਈ߃ࠆᓇ㗀
ᴾ ੂᵹ⛔⸘㊂ ᴾ
ᰴߦ㧘ㅴⴕߔࠆ㘑ᵄ߇᳇ᵹਛߩੂᵹ᭴ㅧߦਈ߃ࠆᓇ
㗀ࠍ⺞ߴࠆ‫ߕ߹ޕ‬㧘⒳‫ߩޘ‬ᵄ㦂ߦ߅ߌࠆੂࠇᒝᐲ

 uc2 ! 㧘 v c2 ! 㧘 wc2 ! ߣ࡟ࠗࡁ࡞࠭ᔕജ  ucwc!
ߩ ㋦ ⋥ ಽ Ꮣ ࠍ Fig.7 ߦ ␜ ߔ ‫ ߪ ࠄ ࠇ ߎ ޕ‬㧘 ߶ ߷kz  2 (z   64) ߩ㗔ၞߢ㗼⪺ߥᵄ㦂ଐሽᕈࠍ␜ߒߡ
2޿ࠆ‫  ޕ‬u c ! ߪ㧘ࡃ࠶ࡈࠔ࡯ጀߢߩᐔဋᵹㅦ൨㈩㧔Fig.3㧕ߦኻᔕߒߚᵄ㦂ଐሽᕈࠍᜬߜ㧘ᵹㅦ൨㈩ߩᄢ
2߈޿ᵄ㦂ߢᄢ߈޿୯ࠍߣߞߡ޿ࠆ‫৻ޕ‬ᣇ㧘  v c ! ߪ㧘ᐔဋᵹ߇⥃⇇㜞ߐઃㄭߢᄢ߈ߊᄌൻߒߡ޿ࠆᵄ㦂c / u*

8, 10, 12 ߢჇടߒߡ޿ࠆ‫ߚ߹ޕ‬㧘  wc2 ! ߪ v c2 ! ߣห᭽ߩ᜼േࠍ␜ߔ‫ޕ‬ᵄ㕙਄ߢߩੂࠇᒝᐲߩⓨ㑆ಽᏓࠍ⺞ߴࠆߚ߼㧘ੂࠇᒝᐲߩᵄ⁁ᚑಽ

㧘㧘ߩಽᏓࠍ Fig.8㧙10 ߦ␜ߔ‫ޕ‬

Fig. 7  The vertical distributions of the turbulent
Reynolds stresses. The symbols are same as
figure 3.

ߎࠇࠄ߽ᵄ㦂ଐሽᕈߣหᤨߦ㧘⥃⇇㜞ߐߢߩᵹࠇߩ㗼⪺ߢߪ㧘ߥᄌൻࠍ␜ߒߡ޿ࠆ‫ ߫߃଀ޕ‬c / u*10㧕‫ޕ‬

8 ߩ႐ว㧘 ߪ⥃⇇㜞ߐ㧔ᵄ㕙߆ࠄߩ㜞ߐࠍ hc ߣߒߡ㧘 khc | 0.5 㧕ߢᕆỗߦᄌൻߒߡ޿ࠆ㧔Fig.8 (c)㧕‫ߡߒߘޕ‬㧘z  z bot  hc ߢߪ㧘ߩᦨᄢ୯ߪᵄ㗡ߩ㘑਄஥㧘ᦨዊ୯ߪᵄ㗡ߩ㘑ਅ஥ߦ૏⟎ߒ㧘 z ! z bot  hc ߢߪㅒߦ㧘ᦨᄢ୯߇ᵄ㗡ߩ㘑ਅ஥㧘ᦨዊ୯߇ᵄ㗡ߩ㘑਄஥ߦ૏⟎ߒߡ޿ࠆ‫৻ޕ‬ᣇ㧘 ෸߮ߦᲧߴ㧘⥃⇇㜞ߐߦ߅ߌࠆᄌൻߪዊߐ޿㧔Fig.9,

Sullivan ࠄߪ㧘ᓐࠄ߇⸘▚ߒߚዊᝄ᏷㧔ਥߣߒߡak

0.1 㧕ߩᵄ㕙਄ߩ᳇ᵹߢߪ  v c2 ! ߦᄌൻ߇⷗ࠄࠇߥ

߆ߞߚߚ߼㧘ㆊ෰ߩ㕒ᱛᵄ⁁ო㕙਄ߩ᳇ᵹ⸘▚(Cherukat

et al, 1998; De Angelis et al., 1997)ߦ߅޿ߡ

߇㘑਄஥ߢ

Fig.

8 

Streamwise

componentsofFig. 9 Spanwise components of the wave-correlatedthewave-correlated Reynolds normal stresses/ u*2for (a) c / u *(c) c / u *0 , (b) c / u*

8 and (d) c / u*

Reynolds normal stressesc / u*

0 , (b) c / u*and (d) c / u* 20 .

4,

20 . Solid

/ u*

2

for (a)4 , (c) c / u *8line in (c) is the critical height.
Ⴧട㧘㘑ਅ஥ߢᷫዋߒߚේ࿃ߪ㧘ᵄߩᝄ᏷߇ᄢ߈߆ߞߚ

ᵄߩ႐วߦߪ㧘ᵄ㗡ߩ㘑਄஥ߣ㘑ਅ஥ߩߎߣߦࠃࠆᵹࠇߩ೸㔌࡮ౣઃ⌕ߩߚ߼ߛߣផ⺰ߒߚ‫ᧄޕ‬߈޿‫ߚ߹ޕ‬㧘c / u*⸘▚㧔 akeyeߩㄭறߢ

0.1 㧕ߢߪ㧘㕒ᱛᵄ⁁ო㕙( c / u*೸㔌ߪߥ޿߇㧘0 )ߩ႐ว㧘ߪㆊ෰ߩ⎇ⓥ(Cherukat et al, 1998; DeAngelis et al., 1997)ߣห᭽ߦ㧘ᵄ㗡ߩ㘑਄஥ߢᄢ߈ߊ㘑ਅ஥ߢዊߐߥ୯ߦߥߞߡ޿ࠆ㧔Fig.9 (a)㧕‫ߒ߆ߒޕ‬㧘ᑼ(15)2ߦࠃࠅ v c ߣߪหߓ I ଐሽᕈ߇޽ࠆߎߣࠍ⠨ᘦߔࠆߣ㧘22⚿ዪ㧘ᵄ㗡ߩ㘑਄஥ߩᄢ߈ߥ v c ߣ㧘㘑ਅ஥ߩዊߐߥ v cߩᏅ߇ᄢ

4 ߩ႐ว㧘ᵄߩ⼱ߦ૏⟎ߔࠆ cat’sߪⶄ㔀ߥಽᏓߣߥࠆ߇㧘ᵄ㕙߆ࠄ㔌ࠇࠆߦߟࠇߡ㧘ઁߩᵄ㦂ߣห᭽㧘ᵄߩᵄ㐳ߣหߓᵄ㐳ᚑಽ߇ථ⿧ߔࠆ‫ޕ‬ᴾ ੂᵹࠛࡀ࡞ࠡ࡯↢ᚑߣᢔㅺ ᴾ
Fig.12 ߦ c / u*(0,4,8,20) ߢߩએਅߩࠃ߁ߦቯ⟵ߐวߣ߶߷หߓ୯ߣߥࠆߎߣ߇ࠊ߆ࠆ(Fig.㧣)‫৻ޕ‬ᣇ㧘ㅴⴕࠇࠆήᰴరൻߒߚੂᵹࠛࡀ࡞ࠡ࡯ߩ↢ᚑ₸ Pk ࠍ␜ߔ‫ޕ‬wu i
Pk u c i u c j
(22)
wx  jᵄ㧔 c / u* ! 0 㧕ߩ႐ว㧘 c / u*↢ᚑ₸ߩಽᏓߪᵄ㦂ߦࠃߞߡᄢ߈ߊᄌൻߒߡ޿ࠆߎߣ߇

ߣ߇㧘 x ᣇะߦ৻ᵄ㐳ಽߩᐔဋࠍߣࠆߣᛂߜᶖߒ޽߁‫ޕ‬
2ߒߚ߇ߞߡ㕒ᱛᵄ⁁ო㕙ߩ  v c ! ߪ㧘㕒ᱛ᳓ᐔო㕙ߩ႐

4 ߩᤨߦߪߩಽᏓߪ8ߢࠊ߆ࠆ‫ޕ‬ᵄ㦂߇ 4 ߩᤨߪ Pk ߪᵹㅦ႐ߩ௑ะߣห᭽ߢ㧘cat'sߪಽᏓ߇਄ᵹ஥ߦߕࠇ㧘⼱㑆ߢᦨᄢ㧘ᵄ㗡ߢᦨዊߣߥࠅ㧘eye ߩᒻ⁁ࠍߒߚႺ⇇਄ߩᵹࠇߣ㘃ૃߒߚಽᏓࠍ␜ߒߡ

c / u*

0 ߩᤨߣ޽߹ࠅᄌൻ߇ߥ޿‫ߒ߆ߒޕ‬㧘c / u*2ᐔဋࠍߣߞߚ  v c ! ߪჇടߒߡ޿ࠆ㧔Fig.㧣㧕‫ߦࠄߐޕ‬޿ࠆ‫ޕ‬ᰴߦ c / u*ᵄ㦂߇ᄢ߈ߊߥࠆߣ㧔 c / u*ߩᦨᄢ୯ߣᦨዊ

⇣ߥࠆ‫ޕ‬z  z c ߢߪ㧘㜞ߊߥࠆߦߟࠇߡ Pk ߩᦨᄢ୯ߩ૏

0 ෸߮ 4 ߩᤨߣౣ߮ૃ

⟎߇㘑਄஥ߦ⒖േߒ㧘 z ! z c ߢߪㅒߦ㘑ਅ஥߳⒖േߒߡ

20 㧕㧘୯ߩᵄ㗡ߦኻߔࠆ૏⟎ߪ c / u*2

ߚ㈩⟎ߣߥߞߡ߅ࠅ㧘  v c ! ߩ୯߽㧘 c / u*

0 ߩ႐ว

8 ߢߪಽᏓߩ௑ะ߇⥃⇇㜞ߐߩ਄ਅߢ

޿ߊ‫ߦࠄߐޕ‬ᵄ㦂߇ᄢ߈ߊߥࠆߣᦨᄢ୯ߪࠢ࡟ࠬ࠻ߩ㘑਄஥ߦ૏⟎ߒ㧘⽶ߩ Pk ߇࠻࡜ࡈߩ㘑ਅ஥ߦ⷗ࠄࠇࠆ‫ޕ‬ߣ߶߷หߓߣߥࠆ‫ޕ‬ߩᄢ߈ߐߪ㧘৻⥸ߦ㧘 ߦᲧߴߡዊߐ޿㧔Fig.10㧕‫ޕ‬ᐔⴕᐔ᧼㑆ੂᵹߢߩੂᵹࠛࡀ࡞ࠡ࡯ߩ↢ᚑ₸ߦߪ࡟ߪ⼱㑆ߢᦨᄢ㧘

2ࠗࡁ࡞࠭ᔕജߦࠃࠆነਈ  u cwcwu  / wz  ߩߺߢ޽ࠆ߇㧘ᵄ㗡ߢᦨዊߣߥࠅ㧘  wc ! ߇Ⴧടߒߡ޿ࠆ㧔Fig.㧣㧕‫ޕ‬ᰴరᵄ㕙਄ߩੂᵹߢߪᐔဋ㋦⋥ᣇะᵹㅦ w ߩሽ࿷ߦࠃࠅ

߹ߚ c / u *

8 ߩ႐ว

ߣห᭽ߦ㧘
2

ᦨᓟߦ㧘⒳‫ߩޘ‬ᵄ㦂ߦ߅ߌࠆ࡟ࠗࡁ࡞࠭ᔕജⓨ㑆ಽᏓࠍ Fig.11ߦ␜ߔ‫ޕ‬ߩߪ㧘ᵄ㕙ߩߏߊㄭறߢ

u cu cwu  / wx  㧘 Pk 33  wcwcww / wz  㧘u cwcwu  / wz  㧘Pk 31 ucwcww / wx ߆ࠄ᭴ᚑ㧠ᚑಽ Pk11

Pk13

ߪ㧘Ᏹߦᵄ㗡ߩ㘑਄஥ߢᱜ㧘㘑ਅ஥ߢ⽶ߩ୯ࠍขࠆ

ߐࠇߡ޿ࠆ‫ߩߡోޕ‬ᵄ㦂ߦ߅޿ߡᐔဋ㋦⋥ᵹㅦ߆ࠄߩነ

ߎߣ߇ࠊ߆ࠆ‫ߒߛߚޕ‬㧘㕒ᱛᵄ⁁ო㕙ߦᲧߴߡㅴⴕ

ਈ Pk 33 ෸߮ Pk 31 ߪዊߐߊ㧘ᵹࠇᣇะᐔဋᵹㅦ߆ࠄߩ࡟ࠗ

Fig. 10 Vertical components of the wave-correlated2/ u*for (a)Reynolds normal stressesc / u * 0 , (b) c / u* 4 , (c) c / u * 8and (d) c / u* 20 .

Fig.

ࡁ࡞࠭ᔕജߦࠃߞߡ↢ߓࠆ Pk13 ߩነਈ߇ᄢ߈޿‫ߩߎޕ‬ᄢ

cat’s eye ߦࠃߞߡ↢ߓࠆᐔဋᵹߩࠪࠚࠕ߇ࡃ࡯ࠬ࠹

߈ߥ Pk13 ߆ࠄߩነਈߪࠢ࡟ࠬ࠻ߩ㘑਄஥ߦ߅޿ߡᐔဋᵹ

ࠖࡦࠣ⃻⽎ࠍଦㅴߔࠆ௛߈߇޽ࠆߣ⠨߃ࠄࠇࠆ‫ߐޕ‬ߩࠬࡄࡦᣇะ᷵ᐲ߆ࠄੂᵹ᷵߳ߩࠛࡀ࡞ࠡ࡯ャㅍ߇ථ⿧ࠄߦᵄ㦂߇ᄢ߈ߊߥࠆߦߟࠇߡ㧘cat’s eyeߪࡃ࠶ࡈࠔ

ߒߡ޿ࠆߎߣࠍᗧ๧ߒߡ޿ࠆ‫ߚ߹ޕ‬㧘ᵄ㦂߇ᄢ߈ߊߥࠆ

࡯ጀ߆ࠄ㆙ߑ߆ࠅ㧘ߘߩ⚿ᨐࡃ࠶ࡈࠔ࡯ጀߢߩᢔㅺ

ߦߟࠇߡ㧘 Pk11 ߩነਈߪჇടߒ㧘ߘߩ⚿ᨐ㧘࠻࡜ࡈߩ㘑₸ߩᄌേߪዊߐߊߥࠆ‫ޕ‬ᴾᴾ

਄஥ߢ⽶ߩ↢ᚑ₸߇↢ߓࠆ‫ޕ‬ੂᵹࠛࡀ࡞ࠡ࡯ߩ↢ᚑ₸ߦട߃ߡ㧘☼ᕈᢔㅺ₸ߩ⹦
⚦ߥℂ⸃ߪㅴⴕᵄ਄ߩ᳇ᵹߩࡕ࠺࡝ࡦࠣߦߪ㊀ⷐߥⷐ⚛
ߢ޽ࠆ‫ޕ‬c / u*(0,4,8,20) ߢߩએਅߩࠃ߁ߦቯ⟵ߒߚή

Hk

Wave-correlated Reynolds stress2/ u*
for (a) c / u * 0 , (b)c / u* 4 , (c) c / u * 8 and(d)c / u * 20 .

 ⚿   ⺆ᴾᧄ⎇ⓥߢߪ㧘3 ᰴర⋥ធᢙ୯⸘▚ߦࠃࠅ৻ቯㅦᐲߢ⒖േߔࠆㅴⴕᵄ਄ߩ᳇ᵹࠍ⸘▚ߒ㧘ㅴⴕᵄߦࠃࠆ᳇ᵹߩᐔ

ᰴరൻߐࠇߚ☼ᕈᢔㅺ₸ࠍ Fig.13 ߦ␜ߔ‫ޕ‬

wu c i wu ciwx  j wx  j

11 ဋᵹ႐෸߮ੂᵹ႐߳ߩᓇ㗀ࠍ⼏⺰ߒߚ‫ޕ‬ਥߥ⚿⺰ࠍᰴߦ

(23)

␜ߔ‫ޕ‬

(1) ኻᢙ㗔ၞߦ߅ߌࠆᐔဋᵹㅦߩᄢ߈ߐߪ㧘ᵄߩ⊒㆐₸ోᵄ㦂ߦ߅޿ߡ㧘ᵄ㕙ߢߩߖࠎᢿᔕജ߇ᒝ޿㗔ၞߣ⦟޿⋧㑐ࠍᜬߜ㧔 c / u*

4 ߩ႐วએᄖ㧕㧘⊒㆐₸߇ᱜߢᢔㅺ₸ߩ⛘ኻ୯ | H k | ߇㜞ߊߥߞߡ޿ࠆ‫ߚ߹ޕ‬㧘ᵄߩᤨߪ㕒ᱛ᳓ᐔო㕙਄ߢߩᵹࠇߦᲧߴᐔဋᵹㅦ߇ᷫዋߒ㧘㦂߇ 0 ෸߮ 4 ߢߪ㧘ߖࠎᢿജ߇ᦨᄢߣߥࠆ㗔ၞߪߘߩ⽶ߩᤨߪㅒߦჇടߔࠆ‫ޕ‬ઁߩᵄ㦂ߦᲧߴߡࠢ࡟ࠬ࠻ߦㄭߊߦ૏⟎ߒߡ޿ࠆߚ~ ߪ⥃⇇㜞ߐ z ߩㄭறߢᕆ
(2) ᵹㅦߩᵄ⁁ᚑಽ u~ ෸߮ wc

߼㧘 | H k | ߩᦨᄢ୯߽ࠢ࡟ࠬ࠻ߩㄭறߦ૏⟎ߒ㧘ᄢ߈~ ߩ⥃⇇㜞ߐߦ߅ߌࠆᄌൻߦ
ỗߦᄌൻߔࠆ‫ ߩߎޕ‬u~ ෸߮ wߥ୯ࠍขࠆ‫ޕ‬ᐔဋᵹߦኻߒߡ⥃⇇㜞ߐߩᓇ㗀߇㗼⪺ࠃࠅ㧘  ũw̃ ! ߪ z  z c ߢᱜߩ୯ࠍᜬߜ㧘⥃⇇㜞ߐㄭறߦ⷗ࠄࠇࠆ c / u*ߩᄢ߈ߥᄌേ߇⷗ࠄࠇࠆ‫ߩߎޕ‬ᄢ߈ߥᄌേߪ㧘ࡃ࠶ߢᕆỗߦᷫዋߒ㧘 z ! z c ߢߪ⽶ߩ୯ߦߥࠆ‫ޕ‬(3) ోߡߩᣇะߩੂࠇᒝᐲ෸߮࡟ࠗࡁ࡞࠭ᔕജߩⓨࡈࠔ࡯ጀਛߩᵹࠇ߇ cat’s eye ߦࠃߞߡᒝ޿ࠪࠚࠕࠍ

㑆ಽᏓߪᵄ㦂ߦᒝߊଐሽߒߡ޿ࠆ‫ߚ߹ޕ‬㧘ੂࠇᒝᐲᜬߟߎߣߦࠃࠆ߽ߩߛߣ⠨߃ࠄࠇࠆ‫࠶ࡃߜࠊߥߔޕ‬ߪ⥃⇇㜞ߐߢᕆỗߦᄌൻߔࠆ‫ޕ‬ᵄ㕙ㄭறߩ࡟ࠗࡁ࡞

ࡈࠔ࡯ጀਛߢߪ㧘☼ᕈᢔㅺ₸߳ᄢ߈ߊነਈߔࠆੂᵹ

࠭ᔕജߩᵄ⁁ᚑಽ᷵ߩࡃ࡯ࠬ࠹ ࠖࡦࠣ⃻⽎߇ 㗫❥ߦ⊒↢ߒ ߡ߅ࠅ㧘୯ࠍߣࠆ‫ޕ‬8 ߢߪ㧘⥃⇇㜞ߐߩㄭறߢᢔㅺ₸ߪ㧘㘑਄஥ߢᱜ㧘㘑ਅ஥ߢ⽶ߩ

Fig. 12  The phase-averaged production for (a)c / u*

0 , (b) c / u*
20 .

4 , (c) c / u *8 and(d) c / u *Fig. 13  The phase-averaged dissipation for (a)c / u*

0 , (b) c / u*
20 .

4 , (c) c / u *8 and(d) c / u *⻢ ㄉ

wave-induced air flow, Nature, Vol. 422, pp. 55-58.
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mechanics of an organized wave in turbulent shear flow,࠲࡯ߩࠬ࡯ࡄ࡯ࠦࡦࡇࡘ࡯࠲ NEC-SX6ࠍ↪޿ߡታᣉ

J. Fluid Mech., Vol, 41, pp. 241-258.

ߒߚ‫ߚ߹ޕ‬㧘ᧄ⎇ⓥߪᢥㇱ⑼ቇ⋭⑼ቇ⎇ⓥ⾌⵬ഥ㊄࡮Jeffreys, H. (1925): On the formation of water waves by⺖㗴⇟ภ16000599ߩេഥࠍฃߌߡⴕࠊࠇߚ‫ޕ‬

wind, Proc. R. Soc. Lond. A, Vol. 107, pp. 189-206.
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numerical simulation of passive scalar field in aturbulent channel flow, Trans. ASME, J. Heat Transfer,

Belcher, S. E. and Hunt, J. C. R. (1998): Turbulent flow

Vol. 114, pp. 598-606.

over a wavy boundary, Ann. Rev. Fluid Mech., Vol. 30,

Mastenbroek, C. (1996): Wind-wave interaction, PhD

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Direct numerical simulation of turbulent airflow over traveling wind waves
Naoto KIHARA*, Hiromasa UEDA, and Hideshi HANAZAKI*** Graduate School of Science, Kyoto University** Graduate School of Engineering, Kyoto University
Synopsis
Characteristics of the airflow over a train of waves moving at a constant speed are investigatedusing a direct numerical simulation. We simulate the flow for various wave ages ( c / u* 0, 4, 8, 10, 12,
16 and 20), where c is the phase speed of the wind wave and u* is the friction velocity on the wave
surface. The results show that both the mean flow profiles and the turbulent statistics depend strongly on
the wave age. For example, the mean-flow speed in the log layer generally decreases at small wave ages,
i. e., when the wave growth rate is positive. Vertical profiles of the flow change significantly at thecritical height z c , where the mean flow speed and the wave speed c agree. Then, the wave-induced
vertical flux of the horizontal momentum is positive below z c , decreases rapidly near z c , and becomes
negative above z c .
Keywords: Wind-wave, turbulent flow, air-sea interaction, boundary layer, direct numerical simulation