## ABSTRACT

The annual spawning migration of Pacific Coast salmon up the Columbia and Amur rivers has been analysed as an experiment on the maximum range of an un-refuelled vehicle, since the fish do not eat during this migration. The fuel (primarily fat) available to real fish is compared to that computed for an equivalent rigid vehicle with the following performance specifications which may be regarded as optimum ones for marine vehicles.

A choice for the speed of ascent against the average river current corresponding to the least fuel consumption, without seeking systematically either low-or high-velocity water.

A drag coefficient corresponding to a turbulent boundary layer with no separation.

An overall efficiency of 24 % for converting the heat of combustion of the fuel to directed kinetic energy in the wake. Under these assumptions, the vehicles usually require slightly more fuel than the corresponding fish which they are intended to approximate ; in the case of blueback salmon, the vehicle requires about five times as much. One, therefore, concludes that the fish have a performance superior to that of the best engineered rigid vehicles, as defined and operated under the above three specifications. Suggested explanations for the apparently superior performance of the fish are:

a deliberate seeking, by the fish, of low-velocity water close to the bottom or shoreline ;

the ability of the fish to maintain a laminar rather than turbulent boundary layer;

the ability of the fish to extract energy from the turbulent velocity fluctuations of the river.

Analysis of salmon counts over the Bonneville and Rock Island dams on the Columbia River shows that the average velocity of the fish upstream is influenced by the velocity of the water downstream in the following peculiar fashion, depending upon whether one considers water-velocity variations through the course of a year, or variations at a fixed season of the year, from one year to the next. The general effect of the water velocity as it varies through a year, is to slow the fish down when the water velocity is large (at the June flood peak) and to permit faster ascent when the water velocity is less, in both preceding and following months. But at a *fixed* season of the year, prior to the flood peak in June, relatively high-velocity water speeds the fish up. At a fixed season after the flood peak, relatively high-velocity water slows them down. These relationships are in agreement with other biological observations and with an analysis of navigation for minimum fuel consumption.

Data and methods are given permitting a calculation of the average water velocity in the Columbia River, from the mouth to the headwaters, for any season of the year.

## INTRODUCTION

It has been known for a long time that fish and aquatic animals such as porpoises apparently swim with little effort. Attempts to observe or estimate indirectly their energy requirements have frequently, but not always, led to one of two conclusions: (1) these creatures have a much smaller drag than could be achieved with similar, manmade vehicles ; (2) the power output per gram of muscle (ergs/sec.-g.) is considerably larger than might reasonably be expected from physiological experiments on man, or other warm-blooded animals (Gray, 1948, 1949).

There is, therefore, considerable uncertainty at present in the available information to answer quantitatively the question: how well do fish swim? It is the purpose of this paper to examine a natural phenomenon in order to answer this question. This phenomenon is the annual spawning migration of salmon up the Columbia and Amur rivers. In these migrations the following facts can be accepted as well established. Fish weighing from 3 to 40 pounds go distances as great as 1000 miles upstream to spawn and die. It is known that the fish do not eat after leaving the sea, and the weight and chemical composition (as fat, protein and water) both of the fish and the ‘delivered payload’ of reproductive products, at various stages of the migration are reasonably well established. Data on the velocity of the river throughout the year are also available. Fig. 1 shows the elevation of the Columbia River system versus river mile from the mouth, the major tributaries and places where salmon were known to have regularly appeared in the past (Bryant & Parkhurst, 1950; Parkhurst, 1950 *a*, *b*). Fig. 1 can be used to locate by river mile (R.M.) places mentioned in the text.

Such a migration constitutes, from an engineering standpoint, an experiment on the maximum range of an unrefuelled vehicle. From the above data one can determine, at least in principle, the orders of magnitude of the energy consumption and drag which must obtain for the fish to reach its destination.

In this paper we shall compare the performance of the fishes as measured by fuel (fat) consumption with the fuel requirement of hypothetical rigid vehicles of the same length and total surface area, traversing the same reaches of the river on the same time schedule as the salmon migration. To these vehicles we shall assign the best reasonable engineering specifications for large efficiency and low drag presently known. If the computed fuel requirements of our rigid vehicles are larger than the observed fuel available to real fish, one must suppose that the fishes do indeed have abilities superior to those of our ‘best vehicle’, or else seek other possible explanations.

The reason for examining the performance of migrating salmon by making this comparison with rigid vehicles is that we do not know how to calculate the propulsion or drag of non-rigid vehicles. Except for the excellent work of Sir Geoffrey Taylor (1952) on swimming snakes, there seems to be neither a theory nor appreciable experimental data for making such a calculation.

*ρ*is the water density,

*V*

_{w}is the velocity of the water downstream relative to the bottom of the river,

*V*

_{F}is the velocity of the vehicle (cf. fish) upstream relative to the bottom.

*C*

_{f}is the skin friction coefficient. Then it is well known that the drag force

*F*(in dynes, c.g.s. units) on the vehicle is at least as large as the following expression (Goldstein, 1938). For well-polished, streamlined objects the total drag force is in fact only slightly greater than that given by this expression, provided the ‘appropriate’ value of

*C*

_{f}is used. Equation (1) expresses the drag on a flat plate of total (wetted) area S, dragged edgewise through the water.

*C*

_{f}are possible. For a laminar boundary layer For a turbulent boundary layer

*R*is the Reynolds number = length of vehicle × (

*V*

_{F}+

*V*

_{W})/kinematic viscosity. For the length of our vehicle we shall take the length of the corresponding fish, measured to the narrowest point of the tail.

Generally speaking, for large Reynolds numbers (greater than 10^{6}), C_{f (turb)} is appropriate, and C_{f(lam)} for smaller values. The precise value of *R* for transition from laminar to turbulent flow in the boundary layer depends on surface roughness, the degree of turbulence in the water, and the details of the shape of the specific vehicle being considered.

In Fig. 2 we have plotted these two values of *C*_{f}, and also some experimental values of the total drag coefficient for dead fish and fish models, referred to the total surface area. In computing the fuel consumption of our vehicles, we shall use C_{f (turb)}, and it will be seen that this choice ascribes to our vehicles drag forces about one-half to one-third those actually observed for dead fish, but still considerably larger than the very least value using C_{f(lam)} that might be achieved by highly polished, carefully contoured vehicles in smooth water. We have also marked in Fig. 2 the range of Reynolds numbers for the fish considered in this study, which is also roughly the range in *R for* transition from a laminar to a turbulent boundary layer.

*P*(ergs/sec.) expended in moving the vehicle with a speed

*V*

_{F}+

*V*

_{W}relative to the water is at least as large as The corresponding energy

*E*in ergs (lower limit) expended by the vehicle in traversing a distance

*X*(cm.) relative to the bottom of the river, with velocity

*V*

_{F}(cm./ sec.) is

*E = PX/V*

_{F}, which we can write in the form This assumes for the moment that

*V*

_{F}and

*V*

_{w}do not vary over the interval of time

*X/V*

_{F}, nor explicitly with the position in the river.

*E*above to a fuel requirement, expressed as grams of fat, which is the major fuel of real fish.

*W*grams of fat will yield on combustion with oxygen

*WHJ*ergs, where

*H*is the heat of combustion

*(H =*9 ·3 kcal./g. of fat) and

*J*the mechanical equivalent of heat

*(J*= 4 ·18 × 10

^{10}ergs/kcal.). We postulate an efficiency

*N*

_{2}for converting chemical energy (the heat of combustion) to mechanical energy in the ‘engine’ of our vehicle. We also postulate an efficiency

*N*

_{1}for converting mechanical energy of the ‘engine’ to directed kinetic energy in the water.

*N*

_{2}corresponds to the efficiency of a propeller (analogous to the tail of the fish) in converting the rotational energy of an engine to translational energy in the wake. In practice

*N*

_{1}⩽ 0 ·3 for human beings or a good internal combustion engine, and

*N*

_{2}⩽0 ·8 for the very best propellers. We shall assign these optimum values to our vehicle, giving an overall efficiency of 24 % from heat of combustion to directed kinetic energy. So we have to traverse the distance

*X*We can use equation (4) to give the fuel requirements terms of grams of fat per 100 g. of loaded vehicle of initial mass

*M*, at the mouth of the river. At the same time we can take into account the fact that

*V*

_{F}and

*V*

_{w}may vary as our vehicle progresses up the river, by dividing the length of the river into reaches denoted by the subscript

*i*(100 miles in length for the case of the Columbia) and evaluate

*V*

_{Fi},

*V*

_{wi}for each particular reach at the particular time of year, or schedule date for the vehicle’s passage. Adding up the fuel requirement for each reach, from the mouth to spawning grounds, we have for (100

*W/M)*, the fuel requirement per 100 g. of initially loaded vehicle of mass

*M*, It is the quantity computed from the right-hand side of equation (5) which is to be compared with the corresponding fuel consumed by real fish, from changes in their weight and chemical composition.

We have written equation (5) in the form given in order to show the various sources of uncertainty. The factor *C*_{f}*/N*_{1}*N*_{2} we have chosen to be as small as we may reasonably assume. The factor shows the most economical way in which our vehicle should be operated.

Let us suppose for the moment that in a given reach of length *X*_{i,} the water velocity *V*_{Wi} is fixed, and not subject to the vehicle’s choice. What choice of the operating speed *V*_{Fi} uses the least fuel to cover the distance *X*_{i}? This is at . This neglects the small dependence of *C*_{f} on velocity via the Reynolds number. However, this minimum is a rather broad one, so that in the range 0 ·3 *V*_{Wi}*< V*_{Fi}*< V*_{Wi} the fuel consumption is within 15 % of the minimum value. This gives our vehicle considerable latitude in its choice of operating speed *V*_{Fi}, subject only to the proviso that the reach must be covered in a time interval less than or equal to that observed for real fish. The vehicle must, in other words, stay on schedule with real fish.

If we now allow the vehicle some choice for *V*_{Wi} in the reach of length *X*_{it} then in the neighbourhood (0 ·3 *V*_{Wi}*< V*_{Fi}*< V*_{wi}*)* of the optimum *V*_{Fi} = 0 · 5 *V*_{Wi} the fuel requirement varies approximately as the square of *V*_{Wi}. This means, subject to the same proviso as before, that there is considerable fuel saving to be obtained by operating the vehicle in low-velocity water, such as might be found close to the bottom, or the shoreline.

This completes the discussion on the general computation of fuel requirement for a vehicle progressing with velocity *V*_{F} against a counter flow *V*_{w}. In the next two sections we consider the determination of numerical values for *V*_{w} and *V*_{F}.

## THE VELOCITY OF THE WATER

*D*, in cu.ft./sec. equals the velocity

*V*

_{w}(ft./sec.) times the cross-sectional area

*A*(ft.

^{2}) at right angles to the flow. In terms of

*V*

_{w}Note that

*V*

_{w}is the average velocity over the entire cross-section.

The U.S. Geological Survey and the Canadian Water Resources Board have kindly provided the author with unpublished data giving the discharges, areas and hence average velocities at the various metering stations on the Columbia (Fig. 3).

The published record (USGS Water Supply Papers, U.S. Army Engineers Report on the Columbia, 1948) gives the discharge versus date at the metering stations, plus a table of the elevation of the river surface above sea level at the station, as a function of discharge. This is the ‘rating curve’. Daily records of this type go back as far as 1876 at The Dalles, 200 river miles from the mouth. This published record plus the previously unpublished data of Fig. 3 enables the average velocity at a metering station to be determined on any date for which the discharge was known.

Since we shall be using the discharge averaged in a number of ways, we shall need a terminology to specify precisely what kind of an average is used. The discharge itself is normally measured once a day; in the course of a month it may vary quite appreciably. Let us specify by ‘instantaneous’ discharge such a daily measurement. The *mean* discharge will always refer to a mean of instantaneous discharges over a specified number of consecutive days; i.e. mean monthly, mean annual, or between two specified dates. The mean discharge averaged for the same calendar interval for a number of years will be referred to as the average mean discharge. Or we may take the median mean discharge, over a number of years, as a measure of central tendency. One can also define the average instantaneous, or median instantaneous, as the average, for median of discharge on the same day of the year for a number of years.

To obtain velocities at points on the river, other than metering stations, we must determine: (i) how to estimate the discharge for points on the river other than metering stations ; (2) how to determine an area *A* appropriate to the particular discharge at any given point. Given values of *D* and *A* appropriate to each other, we then have a velocity measurement by equation (6).

In order to show how the first question can be answered, we have plotted in Fig. 4, for several metering stations, the average mean monthly discharges divided by the average mean annual discharge at the station. It will be seen that the entire river heaves up and down *en masse* in the course of a normal flood year. The average flood comes about one month later, and more abruptly in the Canadian headwaters than on the lower 700 miles from the mouth, in the United States. It can also be shown by plotting the average mean for a particular month against river mile, that the discharge is a smooth and slowly varying function of distance from the mouth, except for discontinuities at substantial tributaries (see Fig. 5). Instantaneous and mean monthly discharges thus plotted are also very strongly correlated and in the same way, though one can detect the passage downstream of a specific flood crest by careful analysis of measurements very closely spaced in time.

The net consequence is that, given the discharge at a given time at one metering station, the simultaneous discharge for points as much as 400 miles away can be determined with a fair degree of accuracy. In practice we have used on our subsequent figures the discharge at Trinidad (R.M. 455) as our ‘control’ discharge, since this metering station is fairly in the middle of the course traversed by the salmon migration on the Columbia.

Let us now answer the second question: how to determine the cross-sectional area appropriate to a given discharge. We first point out that this area is, at a given point on the river, determined by underwater and above-water ground contours of elevation above sea level, plus the elevation of the water surface above sea level. This latter elevation naturally varies, as given by the rating curve, with the discharge. Now there are two extreme conditions for which the water-surface elevation is well documented : (*a*) for flood conditions of large discharge, since this condition determines what places are likely to be safe; (*b*) low-water conditions of small discharge, which are also fairly well documented, since this condition determines the navigability of the river. Hence, if we can find a reasonable method of interpolation between these two extremes (using the discharge itself as independent or interpolation variable, as obtained under question 1) we can determine the area at any point where the contour maps are available.

Such an interpolation method is suggested by Fig. 3, which gives the discharge versus area for a number of metering stations. It will be seen that, at least in the first 700 miles from the mouth, the assumption that area is a linear function of discharge is a fair approximation (upper left plot of Fig. 3). It is also at sites in this 700-mile stretch that we had contour maps and water-surface elevations requiring this interpolation. Such sites were obtained from the U.S. Coast and Geodetic survey maps (1954, river mile o to 140) plus maps of dam sites, bridges and navigation projects in the Army Engineers 1948 report. The net result of this discussion is that we can obtain velocity measurements at quite a number of points, in addition to the velocities obtainable directly from the metering stations.

Additional, independent velocity measurements can be inferred from the times and places passed by diarists (Symons, 1882; Freeman, 1921) who have descended the Columbia in small boats. These have been corrected for the labours of the oarsmen; a correction from surface to average velocity (Marks, 1941) has been made, and, when plotted in Fig. 6, an estimated correction from the discharge at the date of observation to the discharge appropriate to Fig. 6.

We thus have velocity measurements from three sources of data: (1) directly from the metering stations ; (2) indirectly from contour-depth maps, and interpolated discharge and water elevation data; (3) the corrected surface velocity measurements from the diaries. All of these velocities are plotted in Fig. 6 for discharges corresponding to the months of least (January, February) and largest (June, July) average mean monthly discharge indicated in Figs. 4 and 5, and the average mean annual discharge (approximately April and August). The velocity corresponding to the instantaneous peak discharge of 1894 is also given, since this flood has given a well-documented high water-surface elevation, and also represents an upper limit for velocity in most cases.

All of these measurements seem to be fairly consistent. In Fig. 6 arithmetic averages for 100-mile reaches are shown dotted, and these numerical values are used in computing the fuel requirement from equation (5).

It will be noted that there seems to be a systematic discrepancy between the velocities inferred from Freeman’s diary and the velocities from the Canadian water metering stations (R.M. > 800). This, we suspect, is due to selection of low-velocity sites for metering. Water-metering points are preferably chosen at a point where the water surface is reasonably calm and level, i.e. above a ‘control’ such as rapids. The Canadian reaches are commonly described as the more swift and turbulent portion of the river; actually its swiftest portions were not traversed by Freeman, who had to portage or come down on lines.

The remaining reaches of Fig. 6 seem not to show any systematic discrepancies between the different types of data. We feel that the combined data of Fig. 6 gives a fair picture of the average velocities and fluctuations in velocity encountered by our vehicle, and the fish, in their ascent.

Fig. 6 can be used to check the order of magnitude of fuel requirement quoted, in this paper by using a single velocity averaged for the entire length of a salmon run. Fig. 6 can also be used for a rough examination of cases other than those considered in this paper. It should be noted, however, that fuel consumption of our vehicles varies (in the neighbourhood of the optimum ) as the square of the water velocity, so that such an averaging procedure as described above will slightly underestimate the fuel requirement as given by a more exact calculation.

For completeness one should describe how the velocity varies in a given cross-section (Corbett and others, 1943). The velocity at a metering site is determined by means of a small propeller-type meter, hung at various points in the cross-section; summation of velocities thus measured over the area gives the discharge. Considering velocity measurements on a typical vertical line, velocities ‘on the bottom’ (actually about 6 in. off the bottom) are about one-half the average velocity on the vertical. At a distance equal to 20% of the depth from the bottom, the velocity has arisen to the average value, and increases only slightly in the remaining 80 % of the depth to the surface. This velocity profile is quite characteristic of flow in a broad and shallow channel with turbulent momentum transfer (Prandtl-Tietjens, 1934), which would appear to be a reasonable characterization of a river. In summary, except close to the bottom or shoreline, the actual velocity at any point in the river is of the order of the average over the entire section. Exceptions can occur where the channel is markedly curved or divergent (Goldstein, 1938), giving regions of back flow, but even in such cases the average velocity is still given by the discharge over the cross-sectional area. In our calculations we shall assume that, when in progress, our vehicles encounter the average flow, without seeking systematically either low-or high-velocity water. Whether real fish do likewise remains to be discussed.

## THE SCHEDULE OF ASCENT FOR THE FISH

April to October is the season over which the spawning migrations occur. Over this interval the discharge varies by a factor of about 10, and water velocities by a factor of 2 or 3 (Figs. 4, 6). Hence, the fuel consumption of our hypothetical vehicles will be markedly affected by the season of their ascent, or schedule for arrival at the successive reaches of the river. We should pick these schedules to correspond with the schedules for real fish.

Such schedules can be read off from Fig. 7. In this figure the data points are as follows. At the mouth of the river, Bonneville and Rock Island, the horizontal lines with vertical bar represent major peaks in the weekly catch or count data on Columbia River salmon of Rich’s analysis (Rich, 1942). In these data there are three principal peaks corresponding to the spring, summer and fall runs of chinook salmon

The vertical bar represents the maxima of these peaks, the end-points the interval over which the count or catch fell to one-half its maximum value. The bar end horizontal lines in the neighbourhood of Kettle Falls correspond to reports as to when ‘large numbers’ of salmon appeared in the past (Bryant &, 1950). There is also a report at the headwaters (Golden, R.M. 1100) on the regular annual appearance of large, 40-lb. chinooks.

For determining a schedule of ascent, we have drawn straight lines by eye for the three principal salmon runs, and from these lines the date for the mid-point of each 100-mile reach was used to determine *V*_{w} for that reach, for the pinpose of computing fuel requirements according to equation (5). The spring run of salmon is quite clear cut in this procedure. The distinction between summer and fall runs is not always as sharp as one might wish, but the straight lines certainly represent reasonable approximations to the schedule of actual ascent of fish. We have also drawn a line (data not given) for bluebacks ascending to Arrow Lakes (R.M. 800).

There is also, on Fig. 7, data points as given by Pentegoff, Mentoff & Kumaeff (1928) for the various stages of the run of chum salmon up the Amur River, the schedule for this migration is also approximated by a straight line.

The following points concerning Fig. 7, in relation to the water velocities of Fig. 6, should be noted. Suppose we computed fish velocities from these data (the slope of an ascent line). This turns out to be about 1 ft./sec. for the Columbia River salmon, independent of the season of the year. This, of course, assumes a uniform rate of progress night and day—from biological observations, a most unfish-like behaviour. If we compare this velocity with the water velocities of Fig. 6, this fish velocity for Columbia River salmon is seen to be much less than the hypothetical optimum . Now the water velocity varies by a factor of 2 to 3 over the migration season (April to October), without appearing, from this very approximate data, to have any appreciable effect on the average velocity of the fish. This seems quite surprising, and we shall examine in a subsequent section with more accurate data the relationship between the observed velocity of water (as measured by discharge) and the rate of progress of fish upstream (see, for example, Fig. 9).

The data on chum salmon migrating up the Amur indicate a somewhat different relation between water velocity and implied fish velocity than that for the Columbia. Pentegoff’s data (in which we have found some small errors) gives 69 km./day (= 2 ·6 ft./sec.) for the average water velocity over the entire distance of salmon run, and 50 km./day (=1 ·9 ft./sec.) for the rate of progress of the fish. For these fish, in contrast to the Columbia River salmon, the average rate of progress is somewhat greater, instead of less, than the optimum .

The water velocity on the Amur, it will be noted, is considerably less than for the corresponding length of the Columbia River. Pentegoff’s water velocities were apparently taken from the diary of a boat trip down the Amur in 1897. What correction was applied for the year (1921) of Pentegoff’s observation is not known.

Fig. 7 is primarily intended to give a schedule for ascent of our hypothetical vehicle, rather than allow any firm conclusion about fish velocities, though it does, of course, give lower limits on their average values. In computing fuel requirements with equation (5), for vehicles corresponding to Columbia River salmon, we shall suppose they progress at the optimum with intervals of rest (actually about two-thirds the transit time) sufficient to stay on schedule. For vehicles corresponding to chum salmon on the Amur we shall use just the numerical values quoted above for *V*_{F} and *V*_{w}. This is the simplest assumption for least energy consumption we can make concerning them.

## CALCULATION OF REQUIRED *VS*. AVAILABLE FUEL

The dimensions and other specifications of our vehicles are given in Table 1, labelled by the name of the equivalent fish which they are intended to approximate. The fuel requirements computed from these specifications by equation (5) are listed in table 3. However, two additional fuel increments have been included in Table 3. The computations were carried out for an average flood year (as in Fig. 4), but in some years the flood is much greater than the average. We have estimated, from the measured discharges and velocities associated with a large flood, that an increment of 25 % in the computed fuel for an average year would be sufficient to cover five-sixths of all floods. Hence, a 25 % flood reserve has been added to the fuel requirement computed from equation (5) for an average year.

Secondly, it was pointed out that, at least for the case of the Columbia River, about two-thirds of our vehicle’s transit time would be spent resting, presumably in still water. Hence it seemed reasonable to make an estimate of the fuel consumed during this resting. This fuel consumption we have estimated from the CO_{2} generation and O_{2} consumption of known weight of (quiet) salmon, given in the report of Fish & Hannevan (1948). It amounts to 0·7 g. of fat per 100 g. of fish, for the 700-mile migration from the mouth of the river to Kettle Falls, and corresponding amounts for other distances. This small increment is included in the total fuel requirement given in Table 3.

Calculation of the fuel actually consumed by real fish requires a knowledge of the weight and chemical composition (fat, protein, water) of the fish on entering the mouth of the river from the sea, and on arrival at the spawning grounds. These latter data are given prior to spawning, since a good deal of time and energy is expended in this activity. A summary of these data is given in Table 2; concerning it we make the following comments.

Data for chum salmon in the Amur are taken directly from Pentegoff’s paper, who gives detailed analyses and weights for the various stages of migration. For the Columbia River salmon we have taken average figures from a variety of sources (Shostrom, Clough & Clark 1924; Greene, 1919, 1921)-These average figures indicate that the majority of the fuel consumed is obtained from fat in the muscle tissue, which muscle tissue we have taken as two-thirds the body weight at the mouth of the river. Since the percentage of fat in this tissue fluctuates so widely, according to Shostrom’s data, we have taken the fat portion of muscle tissue as 18 ± 10 % and carried the effect of the ± 10 % fluctuation throughout the calculation. We have assumed the entire body weight shrinks during the migration to 80 % of its value at the mouth, the body tissue being partly consumed and partly going into the reproductive products. The net protein which disappears is rated at 0·44 the fuel value of fat. The residual part of the body weight which is neither muscle tissue nor reproductive products, we have assigned a fuel value two-thirds the value for muscle, at the river mouth, and one-third at the spawning grounds. These figures are estimates based on chemical analyses of body components given by Pentegoff and Greene. We feel these figures do not underestimate the net fuel which might be contributed by this residual.

On comparing the fuel required by our vehicle with the fuel available to real fish, in Table 3, we can make the following observations. The first three entries in column 3 of Table 3 show a decreasing fuel requirement with later season of ascent, a simple consequence of the successively lower water velocities encountered.

Other factors being equal, smaller vehicles require more fuel per 100 g. of body weight than large ones. Both seasonal and size factors conspire to make the vehicle similar to bluebacks require relatively the most fuel, since these fish both are small and ascend (see Fig. 7) during the peak of the flood. The combined advantage of large size and avoiding flood water is shown by comparing big chinooks to the headwaters of the Columbia (1200 R.M.) with spring chinooks to Kettle Falls (700 R.M.).

Except for chum salmon ascending the Amur, and fall chinook ascending the Columbia, the fish have out-performed (used less fuel than) the best designed vehicle we could reasonably imagine. However, the discrepancy between the performance of our vehicles and real fish is not large except in the case of blueback salmon. These, we note, are the smallest fish considered.

## PERFORMANCE DIFFERENCES OF VEHICLES *VS*. FISH

We shall now examine more closely the assumptions and approximations we have made concerning our vehicle, relative to the behaviour and properties of real fish. We shall divide the following discussion into two parts : (1) Evidence on the behaviour of fish permitting the computed fuel consumption of our vehicle to be decreased— i.e. making the performance of our vehicle more nearly that of real fish. (2) Evidence which leads the other way, giving an even better performance by fish over vehicle than the above comparison already indicates.

The first point concerns the average velocity of the water in which fish swim. Do they, as we have supposed for our vehicles, oppose the average flow, or do they systematically seek out low-velocity water? Correspondence and conversation with biologists familiar with these fish give the impression that the fish (and especially bluebacks) do seek out low-velocity water near the bottom and edges of the river. As mentioned in our discussion of the distribution of the velocities in a cross-sectional area, within 6 in. of the bottom the velocity is one-half the average. Moving so close to the bottom would cut fuel consumption to roughly that computed for novelocity selection. Except for the ‘blueback vehicle’, this would reduce the fuel consumption of our vehicles to well below the fuel available to real fish.

An exception to this preference for low-velocity water should be made for the neighbourhood of dams and ladders. Here, judging by the description of fish behaviour as given in the Bonneville Reports (U.S. Army Engineers—annual publication) the fish seem to be attracted by high-velocity water. Presumably this is a special situation to which the fish react in a special way, since only by seeking out moving water can a passage over the dam be found.

The second point concerns the use of the turbulent, as opposed to a laminar boundary-layer friction coefficient. Is it possible, despite the observations on dead fish, for a live and actively swimming fish to achieve a laminar flow, with no separation, over its entire body surface? The observations of Gray (1933, 1936) together with the theory of how drag arises, suggest that it might indeed be possible. Gray clearly showed that a swimming fish sends a travelling wave down its body, and that the propulsive force is applied over the entire body surface, not only with the tail. Now this means that the fish is putting momentum into the fluid over his entire surface, which surface is precisely the site where the loss of momentum by the fluid occurs, which loss is responsible for the drag. Hence, this swimming mechanism (travelling waves) returns to the boundary layer just the momentum which the fluid is losing, which loss would (for a rigid vehicle) ultimately result in a laminar to turbulent transition or separation, or both. It is well known that either replacing the momentum loss of the boundary layer by blowing, mixing, boundary motion parallel to the flow, or by removing the low momentum fluid (suction) will prevent both separation and (in some cases) a laminar-turbulent transition (Goldstein, 1938). This may be precisely what the travelling waves down the fish’s body (boundary motion at right angles to the flow) accomplishes. The gills may also play a role in this process, since they also add momentum to the boundary layer (Breder, 1926).

The third point concerns the intake of energy, by the vehicle (or fish) from the river current. We specified that our vehicle, like the fish, should take in no fuel en route. But it does not necessarily follow that the energy intake (from the river flow) must also be zero. There is, of course, considerable energy in the turbulent velocity fluctuations in the river current, and if the fish could take momentary advantage of these, they would provide a source of energy for propulsion. A precedent for such utilization of turbulent energy for propulsion is given by the Betz-Knoller, or Katzmayr effect (Kuchemann & Weber, 1953). This effect, in the simplest terms, is similar to ‘tacking’ by a sail-boat, but the direction of the current changes (the turbulent fluctuation) rather than the direction of the vehicle, as in the case of a sail-boat. Whether fish can, in fact, utilize this effect is not known.

We should also like to point out that from a strictly engineering standpoint, there is no objection to utilizing the energy of the river current, whether steady or turbulent, to reverse the direction of the chemical reaction which normally provides propulsion. For man-operated vehicles, anchored to the bottom, recharging a storage battery from the propeller driven as a turbine would be the simplest example.

The biological equivalent of ‘recharging the battery’ would be the reversal of the chemical reaction associated with muscle contraction, with work done on the muscle. It is known that such work can store up appreciable chemical energy for a *single* muscle contraction, as experiments on frog and human muscle have amply demonstrated (Hill, 1960). The extent to which fish can utilize such a process to extend their effective fuel supply is, of course, quite conjectural. Such a process could also be interpreted as dynamical ‘damping’ of boundary-layer fluctuations, which has been demonstrated as a means of boundary-layer control (Kramer, 1960).

The above discussion summarizes the arguments which might tend to make our vehicles more fish-like and at the same time consume less fuel. There are, however, cases where we may be compelled to conclude the opposite—more fuel consumption. As a possible example, Fig. 1 shows that salmon, and notably bluebacks, ascend 800 miles to Red Fish Lake against currents which in flood times are probably swifter than those of the Columbia. A second example is given by Greene (1919), who found salmon with 10 % muscle fat on the Snake River, 700 miles from the sea. Presumably these fish have out-performed those on the Columbia by a considerable margin, since they had covered 700 miles and had (presumably) over one-half their fuel still remaining. Still another example would be (for our vehicles) ascent in an exceptionally large flood. However, this example illustrates a significant difference between our vehicles and real fish. Large floods are commonly believed, with good reason, to render passage easier for the fish, but simply add to the fuel requirement of our vehicles (see discussion of Figs. 12 and 13).

From an engineering viewpoint, the ability of the salmon to ascend such great distances with so little fuel consumption presents a near paradox. If we accept our basic assumptions as generally typical and, further, for the bluebacks, allow that they reach the spawning grounds by selecting out the lowest velocity waters and are characterized by a laminar-flow drag coefficient, the performance specifications of the fish (as represented by our vehicles) are still impressive. The tail of the fish is the best propeller, the body chemistry of the fish is the best engine, and the flow characteristics of his body are better than any observed from dead fish. Moreover, the fish shows great competence, both in time and place, in his selection of optimum water velocity under constantly changing environmental conditions from ocean to spawning ground. The maintenance of such top performance specifications under such varied conditions is an astounding engineering feat.

## THE RELATION OF DISCHARGE TO TRANSIT TIME

In this section we wish to examine more closely a question briefly referred to earlier. Is there any observed relation between the time it takes fish to traverse a given reach of river and the velocity of the water in that reach? It should be noted that in the present discussion the words ‘fish velocity’ mean the distance between dams divided by the time interval of transit between them. This velocity is not the ‘fish velocity’ used in computing the fuel requirement according to equation (5), for the Columbia River.

Let us imagine that the Columbia River actually consists of two rivers, a river of water flowing down, with discharge measured in cu.ft./sec., and a river of fish flowing up, whose ‘discharge’ is measured in numbers of fish per day.

Fig. 8 gives a typical plot of ‘fish discharge’ at the Bonneville and Rock Island dams (or metering stations) taken from the U.S. Army Engineers Bonneville Report for 1951. It will be seen that, generally speaking, a peak at the lower dam is followed by a similar peak at the upper dam a few weeks later. If we identify these peaks with the same fish, we then have a measure of the average fish velocity defined as (distance *BV* to *RT)I(T*_{BI}*−T*_{B}*)* when *T*_{R1} and *T*_{B} are the dates of the peaks at Rock Island and Bonneville, and hence *T*_{RI}*− T*_{B} the transit time between them. This property of the fish river should be contrasted with that of the water river, Fig. 4, for which the discharge increases and decreases almost simultaneously at the various metering stations.

Such an identification of fish is clearest and simplest for the case of the bluebacks, which pass up the river each year in a single, well-defined ‘shock wave’. We have chosen to use the median of the blueback count as a measure of the date of passage of fish over the dam, and have plotted in Fig. 9(*a*), *(b)* these median dates, *T*_{B}, *T*_{RI}. The difference *T*_{RI}*− T*_{B}, or interval of transit between dams, is given in Fig. 9(*c*). The abscissa is the mean discharge at Trinidad for the month of July. The data at Bonneville cover the years 1938−54; at Rock Island 1933−54 (Fish & Hannevan, 1948; U.S. Army Engineers Bonneville Reports).

It should be especially noted that the mean discharge has been taken in all these plots for the month of July for each year; actually also the month of transit in most cases, and has *not* been taken on a date given by *T*_{B}, *T*_{RI} or the ‘mean median’ date of transit for each year. The reason we have not made any of the above ‘obvious’ choices is that we are interested in the relation of discharge (of river) to transit time and arrival date of fish. Were we to use instantaneous discharges on dates different in each year, we would find our conclusions distorted by the fact that the river discharge itself is systematically declining in July; i.e. random dates in July would have a strong *negative regression* of date on discharge *at the same date*.

Examination of Fig. 9 will show that, in fact, the arrival date at Bonneville is uncorrelated with discharge (Fig. 9(*a*)), while at Rock Island the larger the discharge for July the later the arrival date (Fig. 9(*b*)). Note that this latter regression is precisely the opposite effect to be expected from the behaviour of the river alone, for which the earlier the date in July the larger the discharge. The transit time, or the difference between Rock Island and Bonneville arrival dates, also increases with increasing July discharge (Fig. 9(*c*)).

Hence, we can state, unambiguously, that for bluebacks a large discharge does affect both arrival date at Rock Island, and the transit time between dams, making them respectively later and larger. All this seems quite reasonable—a large discharge implies a large water velocity, which slows down the fish between Bonneville and Rock Island but not appreciably below Bonneville. The arrival date at Bonneville seems unaffected by discharge. Below Bonneville the water velocities are in general smaller than above —there is, in fact, a small tide in the river below Bonneville.

In the case of the chinook salmon, examination of the data of Fig. 8 shows a rough correspondence between the numbers of fish per day at Bonneville and at Rock Island a few weeks later, though the correspondence is by no means so clear-cut as in the case of bluebacks.

We have endeavoured to analyse the chinook counts in order to obtain their transit time between the dams as follows. A tracing was made of the count at one dam and superimposed on the other. Attention was focused on, say, a 30-day interval at one dam, and the tracing was slid till a sufficiently significant match was found with an equal interval on the other plot. The median dates, *T*_{B} and *T*_{RI}, for the fluctuations in the two plots were then recorded and plotted against each other (Fig. 10). This process was carried out for the years 1938−54 for which counts exist for both dams, and amounts to graphical computation of the interval of maximum lag correlation. This process is admittedly somewhat subjective, but it does provide a measure of the transit time between dams. As can be seen from Fig. 8, the correlation between fish counts in the spring at the two dams is much clearer than in the fall. Also, the fraction of the fish over the lower dam (Bonneville) which also passes over the upper one (Rock Island) varies considerably throughout the year.

The data points on Fig. 10 were then divided into class intervals of (marked on the diagonal line), containing approximately equal numbers of data points. For each of these classes the transit time *T*_{RI}*− T*_{B} was then plotted against the mean discharge at Trinidad for the time interval of the class for that year (same interval for each year). Two such plots of transit time against discharge are given in Figs. 1 and 13. From such plots, for each interval marked on Fig. 10, there can be obtained, with fair accuracy, the median mean discharge and the dispersion of mean discharge for the class interval in , fixed for all years covered by the data. One can also obtain the median transit time *T*_{RI}*− T*_{B} and the dispersion of transit time for the particular (fixed) class interval of . These medians and dispersions of both mean discharge and transit time (marked by arrows on Figs. 12, 13) are plotted in superposition in Fig. 11. It is at once apparent that they all fluctuate together. The median mean discharge, a simple property of the river alone, rises and falls in the normal flood pattern throughout a year (as in Fig. 4) ; the dispersion of the mean discharge is largest when the median mean discharge is largest in June and July. Precisely the same relation obtains for the median of the transit time, and its dispersion, both of which increase and decrease together as the year progresses.

Thus we can say that as a ‘statistical’ year progresses, for ‘average instantaneous’ discharge changing with time in this year, the general effect of a large (June) discharge (with large dispersion) on the fish is to slow them down (increased median transit time) and to increase the dispersion of the transit time.

The situation may be quite different when we examine the relation between mean discharge for a fixed class interval or season of the year, and the corresponding transit time, as in Figs. 9, 12 and 13. In Fig. 12 it will be seen that a mean discharge larger than the average mean discharge is associated with a decreased transit time (negative correlation), while in Fig. 13 precisely the opposite situation obtains. Here a larger mean discharge than ‘average’ gives a larger than average transit time (positive correlation). Fig. 13 gives a result similar to that for the bluebacks (Fig. 9).

Examination of the successive plots, from April to September (using the data of Fig. 10), gives the impression that the correlation ellipses tend to ‘roll over’ from before to after the flood, in the sense that the correlations are negative before the flood peak in June and positive thereafter, as Figs. 12 and 13 illustrate. Numerical values of the correlation coefficients are given in Table 4. While the significance levels are not high by most biological standards, we believe that the effect is genuine. Analysis of counts on silver salmon and steelhead trout gave similar results to those obtained for the bluebacks and chinooks (Osborne, 1959).

The negative correlation of transit time with mean discharge at a fixed season before the flood maximum, is perhaps one way of expressing a fact concerning salmon already well known; i.e. a tendency for fish to stop and wait for high water before continuing a migration ascent. In some cases the high water drowns out insurmountable falls and steep rapids. The negative correlation states that the larger the discharge the less the waiting, or smaller the transit time.

The positive (negative) correlations of transit time with mean discharge at a fixed season after (before) the flood peak, imply a small fuel saving which is deserving of mention. Thus, if a navigator is encountering a head wind which he knows will get stronger, then it saves fuel to speed up a little to reach his destination, whereas if he is encountering a head wind which it is known will get weaker, it saves fuel to slow down. Such behaviour is in agreement with the observed plus and minus correlations for fish in the Columbia. The actual fuel saved is relatively small. We have estimated the effect of this correlation to reduce fuel requirement by about 10% below that for the constant velocity schedules (straight lines) of Fig. 7, for the case of the very largest floods, such as appeared in 1894 and 1948. Ascent of our vehicles in these floods would require about 50% more fuel than for an average year; i.e. about twice the allowed ‘flood reserve’.

## POSSIBLE FUTURE EXPERIMENTS

In the preceding discussion we have compared the fuel consumption of a hypothetical fish-like vehicle with the actual fuel available to real fish. The two figures agreed in order of magnitude, though the comparison indicated that the performance of the fish is in all probability superior to that of our best-designed vehicle.

Let us make the comparison in a slightly different way. Suppose one were to equate the actual fuel consumed by real fish with the theoretical fuel requirement of equation (5) and solve this equality for the undetermined coefficient *C*_{f}*/N*_{1}*N*_{2.}This would determine the effective *C*_{f}*/N*_{1}*N*_{2}which the fish, considered as a rigid vehicle, must achieve in order to perform as well as he does.

The numerical values so obtained would be (except for bluebacks) of the order that we assumed initially; i.e. *C*_{f}*/N*_{1}*N*_{2}{0 ·005/(0 ·3) (0 ·8)} = 0 ·02.

There are, however, other ways of determining this coefficient. Let us imagine a fish swimming in a closed and recirculated water tunnel, with measured values of O_{2} consumption and CO_{2} generation. This would determine the energy consumption, and from the dimension and velocity of fish and the water, one could also determine, just as above, the value of the effective *C*_{f}*/N*_{1}*N*_{2}. Wohlschlage (1954) has performed experiments of just this type, and we have analysed his data, obtaining values of *C*_{f}*/N*_{1}*N*_{2} approximately 100 times greater than those obtained as above for salmon. It should be noted, however, that Wohlschlage’s fish were small (∼ 200 g.), swimming slowly (1 m./min.) in water only a few degrees centigrade above freezing. Thus, the results of Wohlschlage are not strictly comparable with experiments on salmon; we mention them to show that the experiment is practicable, and give the order of magnitude of the conclusion.

We also believe that extended experiments of the above type would permit separate measurements of *C*_{f} and the product *N*_{1}*N*_{2}. Let us suppose, in the above experiment, the kinetic energy content of the momentum-free wake of the fish were also measured (Townsend, 1956). This would determine *N*_{1}*N*_{2}, or the fraction of total energy consumption (from O_{2}, CO_{2} measurements) going into kinetic energy. Hence, *C*_{f} could also be determined. For a 20-lb. chinook swimming at 8 ft/sec. we estimate the velocity fluctuation in the wake immediately behind the fish at 5 −10% the swimming speed. Thus, this experiment seems practicable, though perhaps not easy. One requirement is that the fish swim steadily under controlled conditions, a situation which can be realized experimentally using lights and electric potentials (Paulik & De Lacy, 1958).

## ACKNOWLEDGMENT

It is a pleasure to acknowledge the abundant advice and co-operation of the Staff of the Fish and Wildlife Service. Particular thanks are due to Mr Ralph P. Silliman, who provided the author with the biological background he so badly lacked on this problem, and Mr John Clark of the Bureau of Commercial Fisheries who subjected the manuscript to a most critical and comprehensive review. The author is also indebted to Wendell E. Smith of the State of Washington Fisheries Research Center for enlightening correspondence.

Unpublished data on river flow were provided by the Water Resources Board of the U.S. Geological Survey, Washington, DC. and by the Water Resources Division, Department of Northern Affairs and National Resources, British Columbia, Canada, to whom thanks are also extended.

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*USFWLS, Spec. Sci. Rep. Fisheries*

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*USFWLS, Spec. Sci. Rep. Fisheries*

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*Surface Water Supply of U.S*. Section 12, upper Columbia river basin. Section 14, lower Columbia river basin. Published annually

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