SHARP and FRANCIS: ENERGETICS MODEL FOR YELLOWFIN TUNA POPULATION 



where Mg is the efficiency of the muscle tissue 

 when converting chemical energy to mechanical 

 work; S is the respiration due to activity in 

 mg 02/h; and^ is the acceleration due to gravity 

 (981 cm s'2). The propulsion efficiency is as- 

 sumed to be 0.90 (Lighthill 1970) and is included 

 in the resulting muscle efficiency figure. 



For our purposes we assume M^ to be 0.18. 

 Therefore from Equation (1) 



(Power) (3,600 s/h) 



S = 



(0.18) (143 X 103 g cm) (981 cm/s^) 



mg Oa/h. 



(lA) 

 From the hydrodynamics theory (Streeter 1962) 



Power =-g-Ay3Crf^^ 



where p = the density of seawater (1.025 g/cm^) 

 A= 0.4(Z)2 from Bainbridge (1961) (cm^) 

 V'= is derived from Magnuson's empirical 

 relationships between / and species 

 velocity V (cm/s) 

 Cd = the coefficient of total drag of the fish, 

 which is derived from an empirical re- 

 lation including the results of studies 

 by Pyatetskiy (1971). 



We can therefore rewrite the equation so that 

 respiration due to swimming is equal to 



pAV^Cd 

 ' 2 (7,017.66) 



2.59 X 10-5 (/)2 (y)3 Crfmg Oz/h. 



(2) 



We now have an Equation (2) of three elements 

 for which we have solutions for two (V and Cd) 

 as functions of the third (I) given below. 



V Determination 



From Magnuson (1970), the relation for the 

 minimum velocity (Vioo) for sustained hydro- 

 static equilibrium by tunas is given as 



100 





1/2 



(3) 



where Cj^ = the coefficient of lift for the pectoral 

 fins 

 Af^ = the total lifting area of the pectoral 

 fins (cm2), log Af^ = -1.2154 + 

 1.87 log J 



Ci = the coefficient of lift of the keel 



A^ = the lifting area of the keel (cm^), 

 logA* = -2.7033 + 2.26 log /(cm2) 



Lf = the total weight of the fish in sea- 

 water (dynes). (L, values are ob- 

 tained by multiplying Mf values 

 by appropriate constants as pro- 

 vided by Magnuson (1973) by 

 species and weight class.) 



Mf= mass of the fish = 1.858 x lO'^ 

 (/)3.02i (grams). 



Determination of the Coefficient of 

 Total Drag C^ 



The relation between the total drag coefficient 



iCd) and the Reynolds number (Re) for Atlantic 



bonito, Sarda sarda , reported by Pyatetskiy (1971) 



is taken to be representative in form for scombri- 



/ V 

 form fishes. Re = , where v is the kinematic 



V 



viscosity of seawater or 0.01 cm^/s; / is the fish 

 fork length in centimeters; and V is the fish 

 velocity in centimeters per second. 



An analytical expression was derived for esti- 

 mating the Cd values in the following manner: 

 R. Gooding (Gooding et al. 1973) of the National 

 Marine Fisheries Service Honolulu Laboratory, 

 Honolulu, Hawaii reported respiration rates for 

 unfed K. pelamis from 32 to 36 cm fork length, 

 swimming at or near minimum velocities (Vioo)- 

 From these data it was possible to calculate Cd 

 given the observed respiration rate (St<,tai) was 

 431.5 mg 02/kg h and / = 35 cm. The minimum 

 velocity (Vioo) = 59.1 cm/s and Re = 2.07 x 10^ 

 at this velocity. 



For skipjack tuna of Z = 35 cm, W^et = 200.5 

 g, so that 



Sm = 60.0 mg Oa/h 

 Stotai - S^ = S, = 371.5 mg Oa/h. 



From Equation (2) it is now possible to deter- 

 mine that 



Cd- 



371.5 



2.59 X 10-5 (35)2 (59.1)3 



= 0.057. 



This value of Cd was related to the values 

 graphically displayed by Pyatetskiy (1971) and 

 what was assumed to be a good approximation 



41 



