\def \half{\frac{1}{2}}

\def\Stij {\widetilde{S}_{ij} }
\def\Stjj {\widetilde{S}_{jj} }

\def\ftil{\widetilde{f}}
\def\htil{\widetilde{h}}
\def\util{\widetilde{u}}
\def\Etil{\widetilde{E}}
\def\Ttil{\widetilde{T}}

\def\etal{{\it et al.}}

\def\jour#1 #2 { {\sc #1} #2}
\def\aiaaj#1{{\it AIAA J.}, {\bf #1}}
\def\aiaap#1{{\it AIAA Paper No.~#1}}
\def\jatmo#1{{\it J.~Atmos.\ Sci.}, {\bf #1}}
\def\jcp#1{{\it J.~Comput.~Phys.}, {\bf #1}}
\def\jfm#1{{\it J.~Fluid Mech.}, {\bf #1}}
\def\pof#1{{\it Phys.~Fluids}, {\bf #1}}
\def\pofa#1{{\it Phys.~Fluids A}, {\bf #1}}

\def\qsgs {\tau_{kk}}
\def\ret {Re_\tau}
\def\prt { {\it Pr}_{\rm T} }

\def\delhat{\widehat{\Delta}}
\def\delbar{\overline{\Delta}}
\def\gbar{ \overling{G} }
\def\ghat{ \widehat{G} }
\def\gbt { \widehat{\gbar} }

\def\pb { \overline{p} }
\def\pbh { \widehat{\pb} }
\def\pbhbh { \widehat{\overline{\pbh}} }
\def\pbb { \overline{\pb} }
\def\rhob { \overline{\rho}\, }
\def\rhobh { \widehat{\rhob}\, }

\def\Tt{\widetilde{T}}
\def\Ttt{\widetilde{\Tt}}
\def\Ttb{\breve{\Tt}}
\def\Ttbtb { \breve{\widetilde{\Ttb}} }

\def\ut{\widetilde{u}}
\def\uti { {\widetilde u_i} }
\def\utj { {\widetilde u_j} }
\def\utk { {\widetilde u_k} }
\def\utq { {\widetilde u_q} }
\def\utti{ {\widetilde{\widetilde{u}}_i} }
\def\uttj{ {\widetilde{\widetilde{u}}_j} }
\def\uttk{ {\widetilde{\widetilde{u}}_k} }
\def\uttq{ {\widetilde{\widetilde{u}}_q} }
\def\utb { \breve{\widetilde{u}}}
\def\utbi{ {\breve{\widetilde{u}}_i} }
\def\utbj{ {\breve{\widetilde{u}}_j} }
\def\utbk{ {\breve{\widetilde{u}}_k} }
\def\utbq{ {\breve{\widetilde{u}}_q} }
\def\utbtbi{ {\breve{\widetilde{\breve{\widetilde{u}}}}_i} }
\def\utbtbj{ {\breve{\widetilde{\breve{\widetilde{u}}}}_j} }
\def\utbtbk{ {\breve{\widetilde{\breve{\widetilde{u}}}}_k} }
\def\utbtbq{ {\breve{\widetilde{\breve{\widetilde{u}}}}_q} }

\def\Sij {S_{ij} }
\def\Skk {S_{kk} }
\def\Sjj {S_{jj} }
\def\St { \widetilde{S} }
\def\Stij { \widetilde{S}_{ij} }
\def\Stkk { \widetilde{S}_{kk} }
\def\Stpij { \widetilde{S}^a_{ij} }
\def\Stb { \breve{\St} }
\def\Stbij { \Stb_{ij} }
\def\Stbkk { \Stb_{kk} }
\def\Stt { \widetilde{\St} }
\def\Sttij { \Stt_{ij} }
\def\Sttkk { \Stt_{kk} }
\def\Stbpij{ \breve{\widetilde{S}}^{a}_{ij} }
\def\Stbtb { \breve{\widetilde{\Stb}} }
\def\Stbtbij { \Stbtb_{ij} }
\def\Stbtbkk { \Stbtb_{kk} }

\def\Akk {A_{kk}}
\def\Bkk {B_{kk}}
\def\Lkk {{\cal L}_{kk}}
\def\Mmm {M_{mm}}
\def\Nmm {N_{mm}}
\def\Amm {A_{mm}}
\def\Bmm {B_{mm}}
\def\Lnn {{\cal L}_{nn}}
\def\Mnn {M_{nn}}
\def\Nnn {N_{nn}}
\def\Akk {A_{kk}}
\def\Bkk {B_{kk}}
\def\Lkk {{\cal L}_{kk}}
\def\Mkk {M_{kk}}
\def\Nkk {N_{kk}}
\def\Aij {A_{ij}}
\def\Akk {A_{kk}}
\def\Ajk {A_{jk}}
\def\Bij {B_{ij}}
\def\Bkk {B_{kk}}
\def\Bjk {B_{jk}}
\def\Lij {{\cal L}_{ij}}
\def\Mij {M_{ij}}
\def\Nij {N_{ij}}
\def\Lmn {{\cal L}_{mn}}
\def\Mmn {M_{mn}}
\def\Nmn {N_{mn}}
\def\Llk {{\cal L}_{lk}}
\def\Mlk {M_{lk}}
\def\Nlk {N_{lk}}
\def\Lpq {{\cal L}_{pq}}
\def\Mpq {M_{pq}}
\def\Npq {N_{pq}}
\def\Lrs {{\cal L}_{rs}}
\def\Mrs {M_{rs}}
\def\Nrs {N_{rs}}
\def\Tij {T_{ij}}
\def\eij {e_{ij}}
\def\aij {\alpha_{ij}}
\def\amn {\alpha_{mn}}
\def\akk {\alpha_{kk}}
\def\ajk {\alpha_{jk}}
\def\bij {\beta_{ij}}
\def\bmn {\beta_{mn}}
\def\tij {\tau_{ij}}
\def\tkk {\tau_{kk}}

\def\Ki {{\cal K}_{i}}
\def\Kj {{\cal K}_{j}}
\def\Km {{\cal K}_{m}}

\newcount\ndots
\def\drwln#1#2{\raise 2.5pt\vbox{\hrule width #1pt height #2pt}}
\def\spc#1{\hskip #1pt}
\def\solid{\drwln{24}{.5}\ }
\def\bdash{\hbox{\drwln{4}{.5}\spc{2}}}
\def\dashed{\hbox {\drwln{4}{.5}\spc{2}
 \drwln{4}{.5}\spc{2}\drwln{4}{.5}}\nobreak\ }
\def\ldash{\hbox {\drwln{8}{.5}\spc{2}
 \drwln{8}{.5}\spc{2}\drwln{8}{.5}}\nobreak\ }
\def\bdot{\hbox{\drwln{1}{.5}\spc{2}}}
\def\dotted{\hbox{\leaders\bdot\hskip 24pt}\nobreak\ }
\def\chndash{\hbox {\drwln{8.5}{.5}\spc{2}
 \drwln{3}{.5}\spc{2}\drwln{8.5}{.5}}\nobreak\ }
\def\chndot{\hbox {\drwln{9.5}{.5}\spc{2}
 \drwln{1}{.5}\spc{2}\drwln{9.5}{.5}}\nobreak\ }
\def\chndotdot{\hbox {\drwln{8}{.5}\spc{2}\drwln{1}{.5}\spc{2}
 \drwln{1}{.5}\spc{2}\drwln{8}{.5}}\nobreak\ }
\def\chndotdotdot{\hbox {\drwln{8}{.5}\spc{2}\drwln{1}{.5}\spc{2}
 \drwln{1}{.5}\spc{2}\drwln{1}{.5}\spc{2}
 \drwln{8}{.5}}\nobreak\ }
\def\chnddd{\hbox {\drwln{8}{.5}\spc{2}\drwln{1}{.5}\spc{2}
 \drwln{1}{.5}\spc{2}\drwln{1}{.5}\spc{2}
 \drwln{8}{.5}}\nobreak\ }
\def\circle {$\circ$\nobreak\ }
\def\square {${\vcenter{\hrule height 0.4pt width 6pt
 \hbox{\vrule width 0.4pt height 6pt \kern 5pt
 \vrule width 0.4pt height 6pt}
 \hrule height 0.4pt width 6pt}}$\nobreak\ }
\def\trian {$\triangle$\nobreak\ }
\def\filtr {$\triangle$\nobreak\ }
\def\plus {$+$\nobreak\ }
\def\cross {$\times$\nobreak\ }
\def\diam {$\diamond$\nobreak\ }
\def\filsqr {${\vcenter{\hrule height 2pt
 \hbox{\vrule width 2.2pt height 0.2pt \kern 0.1pt
 \vrule width 2.2pt}
 \hrule height 2.2pt}}$\nobreak\ }
\def\lcross {\hbox{\solid\hspace{-18pt}\cross\hspace{10pt}}}
\def\ltrian {\hbox{\solid\hspace{-18pt}\trian\hspace{10pt}}}
\def\lsquare {\hbox{\solid\hspace{-18pt}\square\hspace{10pt}}}

