*U.S. Dept. of Commerce / NOAA / OAR / PMEL / Publications*

Copyright ©1996 by Blackwell Science Ltd. Further electronic distribution is not allowed.

We believe that the increased patchiness in the late larval and early juvenile
stages resulted from an increasing ability of these animals to swim and respond
to their environment (Olla
*et al.*, 1996, see pp. 167-178 in this supplement). With time, diffusion
tends to equalize the concentration of larvae over space. Larvae could balance
this effect by actively swimming toward the center of the patch. To evaluate
this possibility, we considered an idealized analytical model of a larval patch.
In polar coordinates, the change in concentration (C) of larvae with time (*t*)
in an axisymmetric patch of constant radius is given by the advection-diffusion
equation

(4)

where *r* is the distance of the larvae from the center of the patch,
*U* is the assumed constant larval swimming speed (toward the center
of the patch), and *K*_{H} the horizontal eddy diffusivity. Equation
4 allows a bounded steady state solution:

(5)

where *C*_{0} is the concentration at the patch center. For
a wide range of oceanic scales and in the absence of horizontal shear, experiments
show that the eddy diffusivity coefficient can be approximated by 2 × 10^{-3} R^{4/3} cm^{2}
s^{-1}, where *R* is the radius of the patch (Okubo,
1980). From eqn (5), the radial swimming speed (*U*) required to
maintain a portion of larvae within
a circle of radius *R* is

*U* = -2 × 10^{-3} ln(1 - ) *R*^{1/3}
(6)

and for = 0.99,

*U* = 9.2 × 10^{-3} *R*^{1/3}.

The greatest change of patch radius with swimming speed occurs for *U* < 0.7 cm s^{-1}
(Fig. 6). For a patch with radius comparable
to observed eddies in Shelikof sea valley (~15 km), a minimum radial swimming
speed of 1.0 cm s^{-1} would be required to maintain a patch
against diffusion. (Since it is unlikely that larvae could swim exactly to patch
center, this estimate only forms a minimum requirement.) Given that larvae swim
one to two body lengths per second, swimming ability can be important in maintaining
a patch once the larvae reach a length of 10-20 mm. This mechanism most
likely explains the observed increase in **P** for older larvae
(Fig. 3).

**Figure 6.** The relationship between larval swimming speed and
the radius of a patch they are capable of maintaining in the presence of horizontal
diffusion. See text for details of the model. Dashed line represents the approximate
radius of eddies in Shelikof Strait and the resulting minimum radial swimming
speed to maintain the larval patch against diffusion.

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