H{j) = {f^j%)-'/jlU'^jlr\ (90) 



1=1 



between the inertial frequency a),„ = 2 sin(latitude) cpd 

 and the buoyancy frequency «U)»a),„, and zero other- 

 wise, where 



1 

 Similarly for the spectrum of 5C/C 

 Fk ic (t^: ;;«) = < (5C/C)2 ) G{w)H{j) 



= {(5C/C)l)(n/nofGUmj). (91) 



For a' not too close to n(z), the dispersion relations 



k„=JTjB-\il\u.'^-oj\J'\ ky(z)=JT,B-\i(z)/na (92) 



for the horizontal and vertical components of wave num- 

 ber. The spectrum in k„,j space is accordingly 



■P'»c/c(V:') = -f"tc/c('^>;) 



dm 

 ~dk'„ 





for 0<k„<ky=JTrB'^n{z)/no. Equations (90)-(93) are 

 essentially WKB approximations, and they fail near the 

 boundaries and the turning depths (GM72). 



Coherence scales can be estimated from the Fourier 

 transforms: 



Jjj m 



cosil?_:if(ik 



-'",„ Jo 



dk„F{k„ , w) 





da cosikff cosaX) 





dk^ F{k„ , ii})jQ(k„x) 



= j" ' rfco G{w)J2 H(i)Ja[k„x) = 1 - 877-'j^(a),./no) [log(n/a.u) -l]\x\/B, 



X! H(j) coskyz = 1 - (iTJ^ - 1) {n/na)\z\/B , 

 which suggest the coherence scales 



^''"8jJlog(n/c.J-il ' ''"tt;,-! ' 



(94) 



Setting a),„ = 7. 3x10"' sec"' (30° latitude), wo = 5.3xlO"' 

 sec"', B = lkm, gives the values in Table II. (The 

 near-bottom value of Ly is meaningless. ) The assump- 

 tion of spherical symmetry (so popular to scattering 

 theoriests) is useless to oceanographic application. 



VII. FLUCTUATIONS IN MODEL OCEAN 



Armed with the specific ocean model described in 

 Sees. IV-VI, we can now proceed with the evaluations 

 of the general expressions (66) and (75) for the quanti- 

 ties <IA'I^> and <X^>. Let us first look at < IXI^>. 



Substitution of the spectrum (91) into (66) yields 

 {\xY)-{{^ClC)\){f)^BRF^kR), (95) 



i=^i(i?) = -Z -f 5 \ drsec^enM -y ( "7 — ^r] 



\l/2 



TABLE n. Typical values for u,„=7.3xl0"' sec"' (30° lattitude). 

 no=5.2xlO"' sec"', andB=l km. The near-bottom value of 

 ly is meaningless. 



= T-^l( rfxsec=0«Vi(A) 



and 



-^'<^'=K^7I^2(^^^ 



2 (A^ + 1)'/^ 



i72log7X2TTW2 



1)"^+1 



(A2+1)1/2_1 



with 



A=(w/aiiJtane 



(96) 



(97) 



(98) 



i="i(fl) as here defined is a dimensionless number of or- 

 der one when R is of the order of R, the range of a loop 

 {R=\Bti€^'^ = 2Q. 8 km). It is for this reason that the 

 factor R has been explicitly separated out in Eq. (95). 



The quantity { j"' > represents the average of _;"' 

 weighted by the internal wave spectrum H{f). We have 



<;"')=(L;"'(/+/.)"')/E (/-/.)- 



= 0. 730, 0. 647, 0. 519, 0.435, 0. 379, 0. 340 '99) 



for j^ = 0, 1, . . . , 5. An approximate expression is 

 log(4/, + l)/(7rj,-l). 



For axial rays, S = A = 0, /i(0) = l, and n = n is a con- 

 stant. Hence F-^KR) becomes simply proportional to R; 

 we have 



i^i(fl)^3^^ 1 = 1.436 |Hf^(i?) 

 IT a;,„no R R 



(100) 



For upward rays turning near the surface, the major 

 contribution to the integral defining F^^R) comes from 

 the ray apex (Fig. 1); here the equation of the ray is, 

 approximately, 



z(x)^S-(l/2(R)(r-3-)S 



and 



224 



