ABSTRACT
The kelp rockfish (Sebastes atrovirens) hovers at a range of angles
to horizontal using slow movements of its pectoral and caudal fins. It
adjusts its buoyancy using a swimbladder, which increases the volume of
the fish without increasing its mass and thus reduces its net density. I
found that the center of buoyancy, the point at which the force due to the
swimbladder acts, is anterior and ventral to the center of gravity. This
creates rotational instability. I formed a model to examine the effects of
the buoyant force, the force due to gravity, and the forces of the pectoral
and caudal fins on the rotational and vertical translational equilibria of
the fish. 1 found that the net torque is much more sensitive to changes in
the angle of the fish than is the net vertical force.
INTRODUCTION
The teleost swimbladder is a specialized organ forming a gas-filled
space within the body cavity of a fish. Although a fish can control the
volume of its swimbladder by secreting or resorbing gasses, this is a slow
process which occurs over several hours. The swimbladder is inherently
instable due to the inverse relationship between pressure and the volume
of à gas in an enciosed space. If a fish in equilibrium moves down even
slightly, the ambient pressure increases and its swimbladder is
compressed. As a result, the fish becomes denser and begins to sink
faster. Conversely, if the same fish is moved above its usual depth, the
reduction in pressure causes the gasses in the swimbladder to expand.
sending the fish towards the surface of the water. Fish caught at depth
and brought rapidly to the surface have trouble swimming normally and
often are seen with their swimbladder expanded into the mouth cavity,
Extensive studies have been carried out on the physiology of the
swimbladder, the effects of pressure changes, and other functions such as
respiration, pressure sensitivity, and sound production. For a detailed
treatment of the swimbladder, see Gee (1983) in Eish Biomechanics and
Alexander (1966).
The center of buoyancy of a fish is the point at which the buoyant
force, primarily the upwards force of the swimbladder, acts. Alexander
(1965) postulated that the conter of buoyancy lay anterior to the center
of gravity. This conclusion was arrived at by noting the orientation of a
freshly killed fish in a beaker of water. He observed that the fish in
question lay ventral side up with its tail on the bottom and its head
slightly raised. In a later work, Alexander (1983) describes the technique
he used to determine the center of buoyancy of a living Nautilus. His
central assumption was that the center of buoyancy is directly above the
animal's center of mass in its natural orientation. Based on observations
of dead fish, this cannot be true for teleost fish with swimbladders.
This raises the question of how to deter mine the center of
buoyancy and the effects of this unstable force on the ability of a fish to
hover. The fish supplements the buoyant effect of a swimbladder with
forces created by its pectoral fins and caudal fin. These three forces and
the force due to the weight of the fish are the principal forces acting on a
fish as it hovers. An examination of their interactions and the net effect
on the torque and vertical translation of a hovering fish help in the
interpretation of actual behaviors.
MATERIALS and METHODS
The blue rockfish (Sebastes mystinus) and kelp rockfish (Sebastes
atrovirens) were caught at depths of 5 to 20 feet in the vicinity of
Hopkins Marine Station.
Injections
The reaction of a fish to the inability to use a fin was examined by
selectively anesthetizing fins. A quinaldine solution consisting of three
drops quinaldine (202 isopropyl) in 900 ml sea water was used as a
general anesthetic. The fish were immersed in this solution for one to one
and one half minutes or until they were easy to handle. I held the fish
firmly, leaving a pectoral muscle exposed, and injected the fish with
approximately 0.3 cc dibucaine (200 microM) subder mally with a fine
needle. I returned the fish to a large, deep (30 cm) tank of sea water. The
fish were observed for fifteen to twenty minutes. I recorded the fish's
orientation, pectoral and caudal fin use, swimming or resting behaviors,
and recovery times. As a control, I also observed the effects of just
quinaldine. After completing observations, I allowed each fish a minimum
of 24 hours to recover between injections.
Center of Buoyancy
To find the center of buoyancy, a kelp rockfish was immersed in a
tank of saltwater, the density of which had been adjusted to that of a fish
without a swimbladder. When the fish was tethered to a line and
sub merged, the only force present was the buoyancy of the swimbladder
and the fish rotated until the center of buoyancy lay above the pivot
point. This line was marked by a string held vertical by a float. By
tethering the fish at five different points (the base of the caudal fin, the
insertion of the anal fin, anterior to the pelvic fins, at the insertion of the
dorsal fin, and at the break between spiny rays and soft rays of the
dorsal fin) I marked five lines on the fish's skin, the point of their
intersection being the center of buoyancy. I assume that the center of
buoyancy lies at one-half of the lateral thickness of the fish. I then
measured the distance to the center of buoyancy from the anterior lip
and from the dorsal margin of the fish.
The density of one individual S atrovirens was calculated from the
weight in air divided by the volume without the swimbladder. The
rockfish was killed in a bucket of quinaldined sea water. Before weighing.
1 dried the fish, removing excess water from the gills and the mouth
cavity. The volume of a fish without swimbladder was determined by
displacement. The dimensions of the fish, the weight in air, the weight in
water with and without a swimbladder, and the area of the pectoral fins
were measured. The volume of the swimbladder was deter mined by
emptying the swimbladder using a syringe.
1 found the center of gravity by a method analogous to that used
for the center of buoyancy. The fish, partially frozen, was suspended
using a thin metal rod as a pivot. The fish rotated about the pivot until
the center of gravity lay directly beneath. A plumb line marked the line
along which the center of gravity lay. I used the same five holes as aboye,
1 marked the five lines with a razor and the point of intersection
indicated the center of gravity. I measured the distance from this point to
the anterior lip and the dorsal margin of the fish. (see Figure 1)
Mathematical Analysis
To hover in equilibrium, the net translational forces and the net
torques acting on a fish must equal zero. The three translational axes are
vertical, transverse, and along the long axis of the fish, each passing
through the center of gravity. The three types of torques are pitch, roll.
and yaw. I chose to examine the pitch of the long axis of the fish about
the transverse axis through the center of gravity, the roll of the vertical
axis about the long axis through the center of gravity, and the vertical
translation.
I calculated the vertical and rotational equilibria for the fish for a range
of angles &, between the long axis of the fish and horizontal. The fish
was oriented such that at &am -+90 degrees, it was vertical with its head
pointing down.
For purposes of calculating the vertical translational forces acting
on the fish, I assumed that the force from the caudal fin (Fcf) is always
perpendicular to the long axis of the body. I then calculated the vertical
component (Fcf.y) of this force from
Fcf,y - Fef (cos &).
The forces due to gravity (Fwt) and to the swimbladder acting at the
center of buoyancy (Fch) are always vertical, parallel, and acting in
opposite directions to each other. I resolved the pectoral fin force (Fpf)
into a net vertical force (Fpf.y) by examining the perpendicular and
parallel components of (Fpf) relative to 0, the angle of the plane of the
fin to the long axis of the fish. (see Appendix 1 and Figure 2)
Fpf,y -Fpf lcos 0 cos & -sin O sin & l.
The net vertical translational force as shown in Figure 1 is:
Fnet - Fcb -Fwt + Fpf,y + Fcf.y
In calculating the net torque about the center of gravity, I assumed
that the caudal fin force is always perpendicular to the radius (Rcf)
between the point at which the force acted and the center of gravity. The
pectoral fin force is always perpendicular to the plane of the pectoral fin.
The force due to the swimbladder is vertical so the effect of buoyancy
varied with the angle &a to the horizontal. For a complete derivation and
force diagrams, see Appendix 2 and Figure 4. The net torque on the fish
Tnet -Tef-Tcb-Tpf.L
Cpf.
lalso examined the roll of the fish about its long axis. In this case,
& is the angle between the dorsal-ventral axis of the fish and vertical.
The following equations are evaluated for angles from 490 to-90 degrees,
The fish is oriented to an upright position at 0 degrees and lying on its
left side at +90 degrees.
In this case, I assumed that the tail fin has no effect on roll or
vertical movement. The equations are set up to simulate the conditions of
the pectoral fin injections, i.e. that only the right pectoral fin was being
used. Only the vertical component of the pectoral fin force contributed to
the roll torque.
The net torque is:
Tnet -Ech Epf
The net translational force is:
Fnet - Fcb -Fwt + Fpf.y
For force diagrams and derivations, see Appendix 3 and Figure 5.
RESULTS
Injections
Initial trials showed the effects of the general anesthetic.
quinaldine. After returning the anesthetized fish to fresh sea water,
they were initially belly-up with their head near the surface of the water.
The pectoral fins were extended sideways. Approximately thirty seconds
after entering the tank the fish sank to the bottom and came to rest
belly-up. The first movement was a slow rocking from side to side. After
fifteen to thirty seconds, the fish began swimming movements using both
the pectoral fins and the caudal fin, bringing it upright on the bottom. The
fish came to rest leaning on one pectoral fin and then shifted to lean on
the other fin. This tipping continued for about two and one half minutes
The fish then began to swim around the tank, slowly but with a normal
orientation. The fish alternated swimming with periods of resting on the
bottom. The amount of leaning decreased with time. The fish appeared to
have fully recovered from the anesthetic by fourteen and one half
minutes in the fresh sea water.
After injecting the left pectoral muscle area with dibucaine, their
initial behavior was the same as above. The first tail movements began
about one and three-quarters to two minutes after entering the tank. The
fish attempted to come to vertical, but could not, instead hovering
horizontally on their left side. The left pectoral fin stuck out from the
body at appproximately 90 degrees. The fish began to swim while still in
their sideways orientation after about two and one half minutes. They
swam in a circle about the axis of the left fin using mostly the right
pectoral fin. The fish increased the velocity of right fin rowing motions
but remained on their left side. This continued for one and one quarter
minutes when the fish settled on the bottom in an upright position,
leaning to the left and resting on the left pectoral fin. The fish stayed
there for several minutes, occasionally beating their right pectoral fin or
moving their tail sinusoidally. With time, their orientation became
increasingly vertical, but at eight minutes the left pectoral fin had not
regained its full range of motion. At nine and one half minutes or after a
long period of inactivity, I poked the fish and forced them to swim.
Swimming was now upright, but sluggish. Full recovery occured from
thirteen to eighteen minutes after exposure to quinaldine.
Center of Buoyancy
The density of the salt solution, and thus the density of the fish
without its swimbladder was 1.0792 g/cm3.
The experimental fish's total length was 27.5 cm and had a weight
in air of 302.85 g. The thickest cross section, measured at the insertion of
the pectoral fins, was 4.0 cm while the doral-ventral depth was 7.6 cm.
The weight of the fish in water was 1.2 g and its weight in water after I
deflated the swimbladder was 15.4 g. The pectoral fins were 2.2 cm wide
at the base, had an area of 26.1 cm2. The pectoral fins were slightly more
than 7/ cm long, and the dorsal edge at the insertion was 7.5 cm from the
anterior lip of the fish and 5.2 cm from the dorsal margin. The center of
buoyancy (CB) ws 9.5 cm from the anterior lip of the fish and 4.8 cm from
the dorsal margin. The center of gravity (CG) was 10.4 cm from the
anterior lip and 4.0 cm from the dorsal margin of the fish. (see Figure 1)
From the above data, I calculated the constants needed to solve the
torque and translational force equations. The downward force due to the
weight of the fish was 15107.4 dynes and the buoyant force was 14292.4
dynes. Because Fwt is greater than Feb, there is a net negative force and
the fish would slowly sink if it did not compensate using its pectoral and
caudal fins. The radius (rcb) from the center of buoyancy to the center of
gravity was 1.2 cm and the angle J between this radius and the long
axis of the fish was 38.42 degrees. The radius (rpf) from the center of
gravity and the center at which the pectoral force acted was 3.7 cm and
the angle ß between this radius and the long axis of the fish was
41.63 degrees. The radius (ref) from the center of gravity to the center at
which the caudal fin forces acted was 12.5 cm. This radius fell along the
same line as the long axis of the fish.
From a videotape of a hovering kelp rockfish, I calculated the
angular velocity of the pectoral fins during adduction as approximately
pi/2 radians per second. The angle  of the plane of the fin to the long
axis of the fish was about 60 degrees. The calculated force of the pectoral
fin (Fpf) is 644 dynes.
Mathematical Analysis
The only force which was not calculated above was the force due to
the caudal fin. I examined two cases. In Case 1, I set the net torques in
equilibrium and solved for Fcf. This value was used in the vertical
translation equation to examine the net vertical force on the fish as a
function of pitch or roll. In Case 2, I reversed the process, setting the
translational forces equal to zero, solving for Fcf, and examining the net
torque at a particular angle &am to horizontal. The results are presented in
Figures 6-9.
PITCHING
Case 1: Rotational Equilibrium Examine Net Translation
The intersection of the force line with zero force in Figure 6
indicates a point of equilibrium. At an angle of slightly more than twenty
degrees, the net vertical force and the net torque are both zero. At this
angle, the fish is oriented with its head slightly tilted down. The small
change in the net vertical force with the change in angle at this point
indicates a relatively stable equilibrium. The translational force at +10
& «130 degrees range from 418.65 to-287.02 dynes, a fairly small
force. Thus, the fish can hover in rotational equilibrium with a small
amount of vertical movement over a wide range of angles.
Case 2: Translational Equilibrium examine Net Torque
The torques due to Fpf and Fef are constant. The effect of the
caudal fin is to turn the fish in a positive sense, sending the head of the
fish down. The torque caused by the pectoral fins rotates the fish in the
opposite direction, but does not balance the torque due to the tail.
Although the net torque reaches equilibrium in Figure 7 at the same
angle that the vertical forces do in Figure 8, it is a relatively unstable
equilibrium. The rapid rate of change of the net torque with the change
of angle to horizontal at this point demonstrates that the torque is very
sensitive to changes in orientation. The slightest deviation from this angle
will cause the fish to rotate. At +10 « & «430, torque ranges from¬
5313.77 to +4142.82 dynes cm and rapily increases at greater angles. If
the fish maintains translational equilibrium, there is only one angle at
which it will not pitch.
ROLLING
Case 1: Rotational Equilibrium. Examine Net Translation
The vertical force due to Fcb and Fwt is constant and negative. The
fish must compensate with its right pectoral fin or sink. In Figure 8, the
angle at which the fish is in rotational and vertical equilibrium is slightly
greater than -10 degrees. At this angle, the fish is leaning slighly to its
right. The vertical component of the right pectoral fin force is sufficient to
offset the net negative force of the fish and the fish does not roll. The rate
of change of the vertical force with change in the angle is relatively small.
The net translational force between -10 and 0 degrees ranges from 162.6
and -815 dynes. Although the fish can only use one pectoral fin, it is still
able to maintain a state of rolling equilibrium over a range of angles with
à small amount of vertical movement. The vertical force of the pectoral
fin must be a minimum of +815 dynes to keep itself from sinking
rapidly. Very little additional force from the pectoral fin is needed for a
wide range of angles (see Figure 10). Between -40 or +40 degrees and 0
degrees there is an increase in the Fpf of only 249 dynes.
Case 2 : Translational Equilibrium Examine Net Torque
The torques due to Feb and Fpf either act in the same direction or
oppose each other, depending on whether the fish is leaning to its left or
right. There is one point of net equilibrium in Figure 9 for the rolling fish.
At slightly less than -10 degrees, the fish is stable. The rate of change of
the torque with angle to vertical is large enough that further rotation in
either direction will cause him to roll away from equilibrium. The net
torque between -10 and 0 changes by 1960 dyne cm. The Fpf needed to
return to equilibrium rapidly gets very large. For a 10 degree roll, over
1000 dynes are needed to return to a stable state. The torque is again
much more sensitive to changes in the angle to vertical.
DISCUSSION
These data suggest that it is advantageous for the kelp rockfish fish
to be in rotational equilibrium and allow itself a slight vertical movement.
From an examination of pitching and rolling rotations relative to the
vertical stability of the fish, it is seen that the rate of change of the
torque with the angle is much greater than the rate of change of the
vertical force with angle. The fish can tolerate a range of angles with a
small amount of vertical movement. In contrast, the torque is much more
sensitive to altering the angle.
These findings correlate well with observations made at Monterey
Bay Aquarium. The kelp and blue rockfish in the large kelp tank hovered
using only their tail and pectoral fins. Each individual seemed to have a
preferred angle &a to the horizontal which they maintained within a few
degrees. At the same time, they would slowly sink a distance from a few
inches to approximately a foot. The fish then used a few tail flicks to
return to their previous depth and orientation. This cycle was repeated
several times during the period that I made my observations. These fish
undergo vertical migrations as a part of their daily and life cycle to feed
reproduce, and hide from predators. The most dramatic changes in
relative pressure occur within the first 33 feet below the surface. This is
also the range of depths at which the effect of waves, tides, and surge is
at a maximum. Nearshore fish live in an incredibly unstable environment
yet are able to hover in a state approaching equilibrium. The fish could
increase their stability by slightly altering their horizontal orientation
while in rotational equilibrium. The fish can thus compensate for changes
in their environment by altering the angle at which they are in rotational
equilibrium and undergoing small vertical movements.
The high degree of rolling instability when the fish is lying on its
left side and the large pectoral fin force needed to return the fish to
vertical correlate well with my results from anesthetizing the left
pectoral fin. The mathematical model de monstrates that it is difficult for
a fish to use just its right pectoral fin and swim normally. An increase in
the frequency of fin strokes causes the fish to swim on its left side along
à constant horizontal plane. In the wild, a fish with an injured pectoral fin
would be at severe disadvantage. It would have no control over its
orientation and could become easy prey. This illustrates the importance
to a fish of having control over the unstable system created by the
swimbladder.
The development of a technique for locating the center of
buoyancy opens up a new approach for studying fish mechanics. I only
examined three out of the six conditions necessary for total equilibrium,
For a comprehensive study of the effects of the buoyant force and the fin
forces, the model l presented can be expanded to include the other three
axes. It would also be interesting to look for physical differences between
fish of the same species which hover in different orientations.
APPENDICES
APPENDIX 1:
The four forces I examined for vertical translational equilibrium
were those due to the weight acting at the center of gravity, the
buoyancy acting at the center of buoyancy, the pectoral fins and the tail
fins (see Figure 3). The Feb and Feg are vertical and oppose each other.
Fef was assumed to always perpendicular to the radius between the
caudal fin and the center of buoyancy. It therefore varled as the angle to
horizontal changed. The Fpf is perpendicular to the plane of the pectoral
fin. To find the net vertical force of the pectoral fins, I first simplified the
Fpf into its perpendicular and parallel components (see Figure 2). I then
found the vertical components of these two forces. The sum of Fpf.L.y and
Fpf. v is the net vertical force of the pectoral fin.
APPENNIX 2
TEANSAON EBeu ong oxis)
CENER O GE
Fut Mito
-5.1 (48 6/82)
50 dyes
CENEP OE BOVANE
Fee Vo8
2
-14. 2 an (l.Ollo n )281
75
Fs - 292.1 dynes
CuML EN
Fetv - Fef Cos ap;
PEHAS
Pf. 12 Sn
Fo  Fof e
FpE n,VFuSin&
Fflv- Fpfsa
Héy - Fes Lese cosa-Sae sind.
Ner Foeck
net  Kg-ut tggy t Fet,V
net - ke-Fut t fot Lsse cos-Sinesnalt fig Cose
APPENDIX 2:
The three forces which contribute to the net torque are Fcb, Fpf,
and Fcf. The torque caused by Feb changes with the angle to horizontal.
The torque caused by Fef is constant because Fef is perpendicular to Rcf.
The pectoral fins are more complex. I once again broke the Fpf into its
perpendicular and parallel components. I found the torque due to these
two forces and used this as the net torque caused by the pectoral fin.
APRENbIX 2
RoATONAL EoBen (orgas)
CENER OF Biotey
Tg  Fg sie os (tß
CUSAL EN
Te  ke se Sn )
le  Ft set
PECOEALENS
o Fl pe s8
Lo, - F  s0058
P  Sn 8
Cot Fpé pfame 28
NET TORQUE
tet tet -e p
et Fela - Re lee cas amp; B
Felpt Lasoas t Sneönd
APPENDIX 3:
The equations for rolling torque and for vertical translational force
were much simpler. They only involved two or three forces (see Figure
5). The caudal fin force was assumed not to contribute in this case. The
right pectoral fin force was simplified by assumung that only the
per pendicular component contributes to rolling motion. The parallel
component is perpendicular to the plane of the paper. The assumptions
asserted in the previous two appendices are also valid for this case.
APPENbIX
VERTCAL AXS
JANSLMIONA EQUILBEN
Ph Peo
Foh v Fpf os &a;
NE FORCE
oet - Fog - Fit t Fp V
Foet - Re - ft t Fptd
COATIONAL EGLBEIN Crol)
Rhtt PEo
Lef P  Sino
CEMEP O BUOAN
La Re Sn
NET TORQUE
Toet - et tot
Tet - Rasdand t otle Sn
FIGURE LEGENDS
FIGURE 1: Diagram of a S atrovirens marked with locations of key
anatomical points. All abbreviations are explained in the text.
FIGURE 2: Force diagram of the pectoral fin forces. Fpf is broken into
perpendicular and parallel components. These components are used to
determine net vertical force. Explanations in the text.
FIGURE 3: Force diagram of the vertical translational forces acting on
the fish. (a) is the angle of the fish from horizontal. Explanations in
text.
FIGURE 4: A side view of the pitching rotational forces on the fish. The
long axis is rotating about the transverse axis through the center of
gravity.
FIGURE 5: A cross sectional view of the fish looking along the long axis
from posterior to anterior end. The torques and vertical forces are
both shown. Positive torque is oriented counterclockwise.
FIGURE 6: Graph of the net translational force versus the angle (a)
from horizontal. The net pitching torque has been set to equal zero.
FIGURE 7: Graph of the net pitching torque versus the angle from
horizontal. The net vertical force has been set to equal zero.
FIGURE 8: Graph of the net vertical force versus the angle from
vertical. The net rolling torque has been set to equal zero. The fish can
only use its right pectoral fin.
FIGURE 9: Graph of the net rolling torque versus the angle from
vertical. The net vertical force has been set to equal zero. The fish can
only use its right pectoral fin.
FIGURE 10: Graph of the right pectoral force needed for equilibrium
versus angle from vertical. The crosses represent Case 1 - net rolling
torque set to zero. The squares represent Case 2 - net vertical force set
to zero.
PF


8
CE
CF
C- center dt guty
(6- Cecter ot buogr
sodius fow
C n C
P- peoal tin
ius ow
P h
e - Coudal in
Ce ladius Gow
ce h C
Eleute 1: PSCAL HERSURENENS
51
f
Bf
24
Fof
— — — — — — -   — — — — — - -
2 - —- - - - - --
PF
LV

pe,1
HEAS
Plane of
Pectoral n
Flouse 2: Pecioe FN Foeces
(Side vier)
HAL
Fish axis
honzonta
Fegy
Fr

E
tish axis
e
4----zoa
— — ——— — — —
C
c8
1626
Fec 4
e e
FGuRé 3: Veercan TnsToN o
(Side view)
— —— — — —
HEAD
Fofl
Fo 4
T
EGUKE 4:
PF
CE
Feg
-—— — —— ——
e
T6
8
RATION FoRCES
(Gide Vied)
Ta
E
fsh
X5
horizoma
6841
Fe

C
T4
C8
Fit
Vertical
Gute 5: RoLLN Foes
(aoss sechonal vien
Pfy
FFlright
ish axis
NEUTEAU
o0
o0
NET TRANSLATIONAL FORCE (dynes)
(Thousands)
No
00
A
FIGURE 6
25
—
o0
—
NET PITCHING TORQUE (dyne x cm)
(Thousands)
N
4
0N



N
FIGURE 7
2
EO
2
—
2
a
a
2
0
0
o0
NET TRANSLATIONAL FORCE (dynes)
(Thousands)
oNo



FiGuRE 8
28
—
8 Z
o
a
E
7

NET ROLLING TORQUE (dyne cm)
(Thousands)
oS


FlGUgE
N

2
.
—
2 2
O

.
8
D
8 -
+
8
RGHT RECTORAL FORCE (duys)
(Thousards)
oI
Mo

5-
Fibuke 10
8
3