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Prepared by Physical Science Services Branch Scientific Services Division

Educational Pamphlet #4 April 1981

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&1.5. DIPARTMENT Of COMMERCE National Oceanic and Atmolpheric Adminiltration National Survey Office of Program Development and Management Illustrations 1 - Map showing principal ocean currents or the world.

Fig. 2 - Example of a typical reversing current showing time vs. velocity ror one day.

3 - Effect of nontidal current on reversing tidal current.

4 - Typical current curves showing daily and mixed types of reversing currents. 5 - Example or a typical rotary current. . 6 - Typical computation of reversing currents.

1 - Sample page of daily predictions for the Narrows, New York Harbor, N.Y.

Pig. 8 - Sample page of Table 2 showing places referred to the Narrows.

9 - Sample page from a tidal current chart. ,

U. S. DEPARTMENT OF COMMERCE Natianal Oceanic and Atmospheric Administration National Ocean Survey Rockville, Md.

TIDAL CURRmfTS

Man in his never ending search for knowledge and in his travels on the has long known about the ocean's and currents. Although he may not have fully understood the reasons for the movements of these , he was very much aware of their benefits and dangers to him. Today the study of the and current is still a major problem facing the oceanographer due to the increased needs of the scientist, engineer, military, and the general public. In presenting a study of the ocean's tides and currents, the usual practice is to consider each separately. In this paper the primary emphasis is on tidal currents with only a brief reference to the tide. For a more complete discussion on the tide, the reader is referred to various C&GS manuals as well as the paper "Significant Aspects of The Tide." Basically the current in the can be broken down into two parts, namely tidal and nontidal. Tidal currents are the horizontal movements of the that accompany the rising and falling of the tide. The horizontal movement of the tidal current and the vertical movement of the tide are intimately related parts of the same phenomenon brought about by the tide- producing of sun and moon. Tidal currents, like the tides, are therefore periodic.

It ie the periodicity or the tidal current that chierly distin- suishes it rrom other kinds of currents in the sea, which are known by the general name of nontidal currents. These latter currents are brought about by causes that are independent or the tides, such as winds, rreshwater runoff, and dirrerences in density and . Currents or this class do not eXhibit the periodicity or tidal currents. The permanent currents in the ge~eral such as the Gulr also raIl into this grouping.

Tidal and nontidal currents occur together in the open sea and in ~nahore tidal waters, the actual cur~ent experienced at any point being the resultant of the two classes of currents. In some places tid~l currents predominate and in others nontidal currents are stronger. Tidal currents generally attain consid- erable velocity in narrow entrances to bays, in constricted parts of , and in passages from one to another. Along the and farther offshore tidal currents 1

3 are generally of moderate velocity; and in the open sea, calcu- lation based on the theory of wave motion, gives a tidal current of less than one-tenth of a knot.

The Tide-Producing Forces The gravitational forces of the various celestial bodies, prin- cipally the sun and the moon upon the rotating earth, result in the periodic changes in the sea. These forces are reflected in the vertical rise and fall of the sea called the tide and in its horizontal motion called the tidal current. Due to its nearness to the earth, the moon is the predominent tide-producing body. The intensity with which the sun (or moon) attracts a particle of matter on the earth varies inversely as the square of the dis- tance. For the solid earth as a whole the distance 1s obviously to be measured from the center of the earth, since that is the center of mass of the whole body. But the waters of the earth, which may be considered as lying on the surface of the earth, are on the one side of the earth nearer to the heavenly bodies and on the other side farther away than the center of the earth. The attraction of sun or moon for the waters of the ocean is thus different in intensity .froM the attraction for the solid earth as a whole, and these differences of attraction give rise to the forces that cause the ocean waters to MOVe relative to the solid earth and bring about the tides and the tidal currents. These forces are called the tide-producing forces. The mathematical development of these forces shows that the tide- producing of a heavenly body varies directly as its mass and inversely as the cube of its distance from the earth. The sun has a mass about 21,000,000 times as great as that of the moon; but it is 389 times as far away from the earth. Its tide- producing force is therefore to that of the moon as 21,000,000 is to (389)3, or somewhat less than one-half.

When the relative motions or the earth, moon, and sun are intro- duced into the equations or the tide-producing rorces, it is round that the tide-producing rorces or both sun and moon group themselves into classes: (a) Those having a period or approxi- mately one haIr a day, known as the semidiurnal rorces; (b) those having a period or approximately I day, 'known as diurnal rorces; (c) those having a period or haIr a month or more, known as long-period rorces. The distribution of the tidal forces over the earth takes place in a regular manner, varying with the latitude. But the response of the various seas to these forces is very profoundly modified by terrestrial features. As a result we find the tides and the currents as they actually occur differing at various places but apparently with no regard to latitude. Other ce1estial bodies 4

CAn be discounted because their size or distance from earth produces a pull that is almost nonexistent. Variations in Velocity ot the Tidal Current The velocity o~ the tidal current at any place varies ~rom day to day. In part this variation arises ~rom changes in meteoro- logical conditions, but in much larger part it is o~ a periodic character due to changes in the position o~ the moon relative to earth and sun. In its movements the tidal current clearly reveals the presence o~ three variations, each related to a partIcular movement o~ the moon. The most noticeable variation, as a rule, is that related to the moon's phase. At the times of new and full moon, the speed of the current is stronger than usual and is called " current." Whp~ the moon is in its first and third quarters, the current flow is less than usual; hence the speed is less than the average. The current at such times is called "neap- curren t . "

It is to be noted, however, that at most places there is a lag o~ a day or two between the occurrence o~ spring or neap cur- rents'and the corresponding phases o~ moon; that is, spring currents do not occur on the days o~ ~ull and new moon, but a day or two later. Likewise neap currents ~ollow the moon's ~irst and third quarters a~ter an interval o~ a day or two. This lag in the response o~ the current expressed in hours or days is known as the "age o~ phase ineq~lity" or "phase age" and is generally ascribed to the er~ects o~ ~riction. The second variation in the velocity or the current is related to the moon's varying distance rrom the earth. In its movement around the earth the moon describes an ellipse in a period of approximately 21~ days. The earth is at one of the roci of this elliptical brbit. Hence, during this period, the moon is at one time nearest the earth and at another time farthest away. When it is nearest the earth, or in perigee, the speed or the current is stronger than on the average, the currents being known as "perigean currents.ft When the moon is :farthest rrom the earth, the speed of the current is weaker than usual. These latter currents are called "apogea[!. currents." In the response to the moon's change in position from perigee to apogee, it is found that, like the response in the case of spring and neap ti-des, .there is a lag in the occurrence of perigean and apogean currents. The stronger currents do not occur on the day when the moon is in perigee but a day or two later. Likewise, weak currents do not occur on the day of the moon's apogee but a day or two later. This interval varies somewhat from place to place, and in some regions it may have a s negative value. This lag is known as the "age of parallax inequality" or "parallax age." The third periodic variation in the speed of the tidal current is that associated with the moon's changing declination. Since the moon moves in an orbit inclined to the plane of the equator, its declination is constantly changing during the month. When it is on or close to the equator, the daily strengths of current do not differ much; in other words, at such times morning and afternoon currents resemble each other. As the declination increases, differences between morning and afternoon currents become pronounced and at the times of the moon's maximum semi- monthly declination these differences are most marked. But like the response to changes in the moon'.s phase and parallax, there is a lag in the response to the change in geclination, this lag beinf known as the "age of diurnal inequality" or "diurnal age. Like the phase and parallax ages, the diurnal age varies from place to place, being generally about 1 day, but in some places it may have a negative value.

When the moon is on or close to the equator and the difference between morning and afternoon currents small, the currents are known as "equa tori al curren ts . " A t the time s ,of the moon's maximum semimonthly declination, when the differences between morning and afternoon currents are at a maximum, the currents are called "tropic currents," since the moon is then near one of the tropics. There are other variations in the speed of the currents, but the three discussed above are the most prominent. These three varia- tions are exhibited by the current the world over but not every- where to the same degree. In many regions the variation from neaps to springs is the principal variation; in certain regions it is the variation from apogee to perigee that is the principal variation, and in other regions it is the variation from equa- torial to tropic currents that is predominant.

The month of the moon's phases the synodic month) is approxi- mately 2~ days in length; the month of the moon's distance (the anomaJ.istic month) is approximately 27~ days in length; the month of the moon's d~~lination (the tropic month) is approximately 27~ days in length. It follows, therefore, that considerable variation in the speed of the current occurs during a year due to the changing relations of the three varia- tions to each other.

Reversing Tidal Currents In the entrance to a or in a and, in general, where a restricted width occurs, the tidal current is or the reversing or rectilinear type; that is, the rlood current runs in one direction ror a period or about 6 hours and the ebb current ror 6 a like period in the opposite direction. The .1'lood current sets inland or upstream and the ebb current sets seawa~d or downstream. The change .1'rom .1'lood to ebb gives rise to a period 0.1' slack water during which the speed of the current is zero. An example 0.1' this type 0.1' current is given in Figure 2 which shows the speed and direction 0.1' the current on a typical day.

Current speeds are given in knots, which is the unit generally used in measuring tidal currents, and represents a speed of 1 nautical mile per hour. Since a nautical mile has a length of 6,080 feet, knots may be converted into statute miles per hour by multiplying by 1.15, or into feet per second by multiply- ing by 1.69.

3r- Strength of Rood '" , ""

t#t 20 ~

2 ~ Strength of ebb

3L I . . . . . I . . . . . I . . , . . I . . . . . I 0 6 12 18 24 Hours Fl.. 2 - Example of a typical reversl". current showll'. time ¥s. velocity for one day.

The curve depicting a reversing current resembles a tide curve. The maximum. velocity of the flood current called the strength of flood corresponds to the time of high water in the tide curve, while maximum velocity of the ebb called the strength or ebb corresponds to the low water. Since the moon in its appar- ent movement around the earth crosses a given meridian on the earth on the average of 50 minutes later each day, the strength of current at most places likewise comes 50 minutes later each day. The current day, ~ike the tidal day and the lunar day, has an average length or 24 hours and 50 minutes.

The current curve shown in Figure 2 represents the current near the sur£ace in the axis o£ a . From observation and also £rom theory it is known that the tidal current extenqs £rom the gur£ace to the bottom. In general it may be said that the 1 velocity of the tidal current decreases from the surface to the bottom, the velocity near the bottom being about two thirds that at the surface. However, the effects of wind and fresh-water flow may bring about considerable variation in the vertical velocity distribution.

The current in a channel is also characterized by a variation in the horizontal distribution of velocity. In a rectangular channel of uniform cross-section, the velocity is greatest in the center of the channel, and decreases uniformly to both sides Combining both the vertical and horizontal variations, it may be said that the average velocity of the current in a section of a regular channel is about three-quarters that of the central surface velocity. Where the current is undisturbed by wind or fresh-water flow, the flood and ebb velocities, and the durations of flood and ebb are approximately equal. In this case, too, the character- istics of the current from the surface to the bottom are much the same. That is, the strengths of the flood and ebb currents, and also the slacks, occur at about the same time from top to bottom. 11', however, nontidal currents are present, the charac- teristics of the tidal flow are modified considerably. The effect of nontidal currents on tidal currents may be derived rrom general considerations. In Figure 3 a purely tidal current is represented by the curve, referred to the line AB as the line of zero velocity. The strengths of the flood and ebb are equal, as are also the dura- tions of flood and ebb. In this case slack water occurs regu- larly 3 hours and 6 minutes (one-quarter of the current cycle of 12 hours and 25 minutes) after the times of flood-and ebb strengths. If now a nontidal current is introduced which sets in the ebb direction with a velocity represented by the line CD, the strength of ebb will obviously be increased by an amount equal to CD and the flood strength will be decreased by the same amount. The current conditions may now be represented by draw- ing, as the new line of zero velocities, the line EF parallel to AB, and distant from it the length or CD. Figure 3 now shows that the nontidal current not only increases the ebb strength while decreasing the flood strength, but also changes the times of slack water. Slack before flood now comes later, while slack before ebb comes earlier. Hence the duration of ebb is increased while the duration of flood is decreased. If the velocity of the nontidal current exceeds that of the tidal current at time of strength, the tidal current in the opposite direction will be completely masked and the resultant current will set at all times in the direction of the nontidal currents. Thus, if in Figure 3 the line OP represents the 8 velocity of the nontidal current, the new axis for measuring the velocity of the combined current at any time will be the line GH and the current will be flowing at times in the ebb direction. There will be no slack waters; but at ~eriod 6 hours 12 minutes apart. there will occur minimum and maximum velocities represented, respectively, by the lines RS and TU.

f"1g.3 . Effect of nootidal current on reversing current.

Insofar as the errect of the nontidal current on the direction of the tidal current is concerned, it is only necessary to remark that the resultant current will set in a direction which at any time is the resultant of the tidal and nontidal currents at that time. This resultant direction and also the resultant velocity may be. determined either graphically by the parallelogram of velocities or by the usual trigonometric computations. Types o~ Reversing Currents Since tides and tidal currents are merely different aspects of the tidal movement of the waters, the former being the vertical movement and the latter the horizontal movement, it is to be expected that tidal currents would show different types, cor- responding to the different types of tide. Observations prove this to be the case. Reversing currents may be readily classed under three types:semidaily, daily, and mixed. The semidaily type is one in which 2 flood strengths and 2 ebb strengths occur in a tidal day with but little inequality between morning and afternoon currents. Figure 2 represents this type of current which is common alor!g the East Coast of the United States. The daily type of tidal current is characterized by I flood and I ebb in a day. The upper diagram ot Figure 4, which represents the typical current in the entrance to Mobile Bay, Ala., exempli- fies .this type ot current. The mixed type of tidal current exhibits 2 and 2 ebbs in a day with considerable 1nequal~ ity in their speed between the torenoon and afternoon cycles. The lower diagram of Figure 4, which represents the typical cur- rent in Rich's Passage, Puget , Washington, illustrates this type of current. 9

In general, it may be said that with reversing currents a given type of current accompanies a like type of tide; that is, semi- daily currents occur with semidaily tides, mixed currents with mixed tides, and daily currents with daily tides. But as noted in considering the variations in strength of current, the varia- tions in the current that involve semidaily components vill approximate corresponding changes in the range of the tide, while in those involving daily components the variation in the current is about half that in the tide. Hence the diurnal inequality in the current at any place is generally less than in the tide at that place.

Oh f!~- 12h ISh Oh 24h Knots 2

0

2

2

0

3 Fig. 4 . Typical current curves showing daily and mixed types of reversing currents.

Relation or Time or Current to Time or Tide In simple wave motion the times of slack water and strength or current bear a constant and simple relation to the times or high and low water. The two principal types of tidal motion are progressive wave and stationary (standing) wave movements. A progressive wave is one whose crest advances horizontally; the times of high and low water progress from one end to the other. The time of slack water comes midway between high and low water

~ 10

and the time of strength of current at time of high and low water. A stationary wave is one in which the water surface oscillates vertically between fixed points called nodes without progression. The points of maximum vertical rise and fall are called antinodes. At the nodes there is no vertical motion and maximum horizontal motion (strength of current). At the anti- nodes the water has no horizontal motion (slack) and maximum vertical motion (high and low water).

Dura tion of Slack In the change of direction of flow from flood to ebb, and vice versa, the reversing tidal current goes through a period of slack water or zero velocity. Obviously, this period of slack is but momentary, and graphically it is represented by the instant when the current curve cuts the zero line o~ velocities. For a brief period each side o~ slack water, however, the cur- rent is very weak, and in ordinary usage "slack water" denotes not only the instant o~ zero velocity but also the period o~ weak current. The question is there~ore frequently raised, how long does slack water last? To give slack water in its ordinary usage a definite meaning, we may define it to be the period during which the velocity of the current is less than one-tenth or a knot. Velocities less than one-tenth of a knot may generally be disregarded for practical purposes, and such velocities are, moreover, difficult to measure either with float or with current meter. For any given current it is now a simple matter to determine the duration of slack water, the current curve rurnishing a ready means for this determination. In general, regarding the current curve as approximately a sine or cosino curve, the duration o.f slack water is a .function o.f the strength o.f current--the stronger the current the less ~he duration o.f slack-~and .from the equation o.f the sine curve we may easily compute the duration o.f slack water tor currents of various strengths. For the normal flood or ebb cycle of 6h 12.0- we may write the equation o.f the current curve y = A sin 0.483lt, in which A is the velocity of the current in knots at time of strength, 0.4831 the angular velocity in degrees per minute, and t is the time in minutes from the instant o.f zero velocity. Setting y. 0.1 and solving .for t (this value o.f t giving half the duration o.f slack) we get ror the duration ot slack the fol- lowing values: Por a current with a strength of 1 knot, slack water is 24 minutes; tor currents of 2 knots strength, 12 minutes; 3 knots, 8 minutes" '4 knots, 6 minutes; 5 knots, 5 minutes; 6 knots, 4 minutes' 8 knots, 3 minutes 10 knots, 2 1/3 minutes. For the daily type of current with a given strength, the duration o.f slack is obviously twice that of a semidaily current with like strength. 11

Velocity or Current and Progression or Tide

In the tidal movement of the water it is necessary to distinguish clearly between the speed of the current and the progression or rate of advance of the tide. In the former case reference is made to the actual speed of a moving par~icle, while in the latter case the reference is to the rate of advance of the tide phase or the velocity of propagation of wave motion, which generally is many times greater than the velocity of the current. It is to be noted that there is no necessary relationship between the velocity or the tidal current at any place and the rate of advance or the tide at that place. In other words, if the rate or advance or the tide is known we cannot rrom that alone inter the speed of the current, nor vice versa. The rate of advance of the tide in any given body or water depends on the type of tidal movement. In a progressive wave the tide moves approxi- mately in accordance with the formula r =..[ir , in which r is the rate of advance or the tide, , the acceleration of gravity, and 1 the depth of the waterway. In stationary wave movement, we have seen that the time of high water and low water occur at very nearly the same time over a considerable area. Thus, there is no progression of the wave present. When a combination of progressive and stationary wave movements occur, the relation between tide and current becomes quite complicated. The velocity of the current, or the actual speed with which the particles ot water are moving past any fixed point, depends on the volume ot water that must pass the given point and the cross section ot the channel at that point. The velocity ot the cur- rent is thus independent ot the rate ot advance ot the tide.

Rotary Tidal Currents Within the channel or a bay or river, the current is compelled to l~ollow the direction or the channel, upstream on the rlood and downstream on the ebb. Out in the open sea, however, this restriction no longer exists, the current having complete rreedom so far as direction is concerned. Orfshore, therefore, tidal currents are generally not or the reversing type. Instead or flowing in the same general direction during the entire period of the rlood and in the opposite direction during the ebb, the tidal currents offshore change direction continually. Such cur- rents are therefore called rotary currents. An example of this type of current is shown in Figure 5, which represents the speed and direction of the current at the beginning of each hour of the ~orenoon for a typical day at Nantucket Lightship, stationed ofr the coast of Massachusetts. The current is seen to have changed its direction at each hourl~ observation, the rotation being in the direction of movement of the hands of a clock, or rrom north to south by way of east, then to north again by way to west. In a period of a little more 12. than 12 hours it is seen that the current has shifted in direc- tion completely round the compass. It will be noted that the tips of the arrows, representing the speeds and directions of the current at the. beginning of each hour, define a somewhat irregular ellipse. If a number of obser- vations are averaged, eliminating accidental errors and temporary meteorological disturbances, the regularity of the curve is con- siderably increased. The average period of the cycle is found to be l2h 25M. In other words, the current day for the rotary current, like the tidal day, is 24h Sam in length. A characteristic feature of the rotary current is the absence of slack water. Although the current generally varies from. hour to hour, this variation from. greatest current to least current and back again to greatest current does not give rise to a period of slack water.

1 I I . . Scale. . of. Knots. . . I ., 0.0 0.4 0.8 1.2

Fil. 5 . Exampleof a typical rotary current. 13

W~en the speed of the rotary tidal current is least, it is known as the minimum current, and when it is greatest it is known as the maximum current. The minimum and maximum speeds of the rotary current are thus related to each other in the same way as slack and strength of the rectilinear current, a minimum speed following a maximum speed by an interval of about 3 hours and being followed in turn by another maximum after a further inter- val of 3 hours. With regard to the current curve, or current ellipse as it may be called, which represents the rotary tidal current at any place, the basic features are the relation of the major and minor axis which determine the ellipticity of the curve, the direction of rotation, and the direction of the major axis. If the major and minor axis are nearly equal the ellipse will be nearly cir- cular; if they differ greatly the ellipse will be flattened. In the northern hemisphere the direction of rotation of the rotary current is, as a rule, with the hands of a clock, while in the southern hemisphere it is counter clockwise. But local hydro- graphic features may bring about a reversal of this general rule. Rotary tidal currents are subject to the periodic variations found in tides and reversing currents. These variations are related to the changes in the phase, .parallax, and declination of the moon. At times of full and new moon the velocity of the rotary current is greater than the average, while at the times of the moon's first and third quarters the velocities are less than the average. Likewise when the moon is in perigee, stronger currents occur, while when the moon is in apogee the currents are weaker. In general it may be taken that the percentage of increase or decrease in the velocity of the current in response to changes in phase and parallax is the same as tM like increase or decrease in the local range of the tide. In response to changes in the declination of the Moon the rotary current eXhibits diurnal inequality like the tide and reversing current. This manifests itself as a difference between Morning and afternoon current ellipses. When the moon is on the equator the two current ellipses of a day are 1rIUch alike; but when the moon .is near its maximum semiMonthly declination the two current ellipses exhibit differences, principally in velocity. Like tides and reversing currents, rotary tidal currents may be grouped under the three types of semidaily, daily, and mixed. The semidaily type of rotary current is one which exhibits two full cycles within a tidal day, morning and afternoon currents differing but little. The daily type is one in which but one cycle occurs in a day, and the mixed type is one Which eXhibits two cycles within a day but with considerable difrerences between morning and a.fternoon currents. In addition to the periodic variations to which rotary tidal currents are subject, they also eXhibit rluctuations arising from the errects or non- tidal currents. Hydraulic Current

In addition to reversl.ng and rotary currents, there is a third type called a hydraulic current. The term applies to a current in a or tidal river that is caused by a difference in head of water at the two entrances. When this difference in head results from tidal action that causes the water at one end to be alternately higher or lower than at the other, the tr.ovement is periodic and may be treated as a reversing type of curr6nt. The currents through the East River which connects New York Har- bor wi th Long I sl and Sound are an example o~ hydraul i c curren ts . When there is tidal action at each entrance to a strait, dif- ~erence in the head will result partly from any difference in the range of tide and partly from any differences in the times of the high and low waters. Theoretically, the current speed will vary as the square root of the difference in head, will be a maximum when this di~ference is greatest, and will be zero or slack water when the difference is zero. Actually there will be a lag of several minutes in response of the currents to this difference in head due to inertia and friction in the waterway itself.

Harmonic Constants The reversing tidal current, like the tide, may be regarded as the resultant of a number of simple harmonic movements, each of the .form ., =A ~(at+e) ; hence, reversing tidal currents may be analyzed in a manner analogous to that used in tides and the harmonic current constants derived. These constants permit the characteristics of the currents to be determined in the same manner as the tidal harmonic constants, and they may also be used in the prediction of the times o.f slack and the times and velocities of the strength of current. It can easily be shown that in inland tidal waters. like rivers and bays. the amplitudes of the various current components are related to each other. not as the amplitudes of the correspond- ing tidal components. but as these latter multiplied by their respective speeds; that is. in any given harbor. if we denote the various components of the current by primes and of the tide by double primes, we have ... . ~ . ~" .K' .0, .- - "". . ~'. - ~"'. k VI'. 0' , m2.';'2.l~2. I' l'-".2m2- 2';'2."2l~2. IAl.O. 1 where the gmall italic letters represent, respectively, the angular speed or the corresponding components. This shows at once that the diurnal inequality in the currents should be approx- imately half that in the tide.

RotQry currents may likewise be analyzed harmonically, but in this case it is necessary to resolve the hourly velocity and direction o~ the current into two components, one in the north 15 and south direction and the other in the east and west direction. Each set of hourly tabulations is then treated independently and analyzed in the usual manner. When the two sets of harmonic constants have been derived the like-named constants of the north and south and east and west directions may be combined into a single resultant, which will be an ellipse, either graphically or by means of the formula H 2. 2 H 2. tan 26 = ~12~~~~~1~ ~22~ ~~z HI ~ 2~I+H22~ 2~2 which may be derived by writing each harmonic constant in the .form .=H1 ~(8-~1) .for the north and south component and v=H2~(8-~2) .for the east and west component.

Reduction Methods and Results

The purpose in studying tidal currents is to determine their speed and direction through direct observations ot the speed and direction ot the current in a tidal body ot water and to MAke such data available to the general public. Two types or reduction--harmonic and nonharmonic--are used in s.tudying tidal currents. A harmonic analysis proceeds through several compli- cated mathematical procedures to determine harmonic constants which in turn are used in the prediction or the daily current at that place. The mathematics involved is beyond the scope of this paper but is explained in detail in other C&GS manuals. "Harmonic Con- stants" has been de.fined in a previous section. A nonharmonic analysis determines the average speed and direction o~ the cur- rent and is a relatively simple reduction. A sample computation is shown in Fig. 6. Here the observed times of slack water and strengths o.f .flood and ebb are compared to the predicted times of their respective phases at a nearby reference station. A reference station is a place for which daily p!gedictions are given in the annual tidal current tables and the type of current is similar in times and velocities to the secondary stations being reduced. Thus in selecting a reference station to which to refer any series of tidal current observations, its characteristics must be studied and compared with the secondary stations tor which new data arc being reduced. It no suitable station is available in the survey area, one must be developed in order that the sta- tions can adequately be reduced. Likewise the observed speeds and directions of the current at a secondary station are summed and averaged for flood and ebb. The velocity of the observed tidal current may be reduced to a mean value by the application of a ractor derived rrom the cur- rent velocity at the rererence station. In using such a station, 16 'I ~-:: ~ ~! ' , I I ~II' .1 ~ . . ...' ~ _I t. I .1 ~w,toii~ ;O' .. "t:1 1 ..' I I -. . -I ~ 9 IA .. 0 0 0 ~',,- ~f\. ...; I ~...:.; i I 'II fIf" ] 'I 1: 1 j 1 ~'"' -, R- 1'1 ~~ ! ~.~] ~ I~-' ~~-. '11 L I .) ~," ¥ 1 1..1 '" '" .. :: I- -, . ~., . "". "':°l",'lt° I~ .. I~~- \i~"-'~ .}I . I a c -;.~~~i ~ ~ 2 M ~ ,:::1 ""a1 ". . . ~ .. ~i.I~~; i~I~1 ""'.~ ..- ~ tit , . ~ "'IA! :I ... ! i! I ~'" .. ~ c ... '; ...: . ; aCt ! '; . J ...... > ~ ~ ,~ ~ )..; ~~~~ ',1 u " I '1 . ~, I c I'll ~~ o~ -- i = ... 1 i 1 ~ i~ I i : u. ~ JJ ,-- t ~. ~I~ . ~ ~ '1 ... o! r ~:... .-) ... 0-' ... - z' ' , 0 I~ ~i c !1 0 &0- c' , ~ I ~ a. ... ~ ..~ Z8 ~ ~ .. ~Il-"'l 0- ~'11 , ~ I~ .. : ..r i :t S . .i 'c , u I~ .. ~ u ~1- g Q -;~ >. ~ ... 10 = ~ .- .u II ij io: . II . I I u~ }I " :1 . ~i fj:;r ~ 1 u. :I. ... iI ii~l- ;I~ ~ . ~ 1..1.I~~ ~ '""1:;;r;:r~ci ~ 0.. 0. ~iiI- - - ~I ;: .c .. ~ ~ .. . ~I". 0 --~ u 11~' 1; ! -QJ - ! .-"""of j ! . ZI,,> i K !~ -~ J if I) I ~ rcI u I

~ 11 the factor is obtained by dividing the best determined average speed at the reference station by the average speed .for the period covered by the observations.

Publishing the Results Two different types of publications are issued by the Coast and Geodetic Survey to include results of the tidal current surveys. Tidal Current Tables are published annually for both the Atlantic and Pacific of North America. The other is a series of Tidal Current Charts for various tidal on both Atlantic and Pacific Coasts. Each of these publications is based on observations obtained over Mm1Y years of current surveys. The Tidal Current Tables have been published annually since 1890. They first appeared as part of the Tide Tables and consisted of brief directions ror obtaining the times of the current for a few locations. Later daily predictions of slack water were given. By 1923, the tables had so expanded that they were issued as a separate publication called "Current Tables, Atlantic Coast" and "Current Tables, Pacific Coast." A few years later, the daily predictions were extended to include the times and speeds of maximum current. Toda¥, each Table is divided into several tables. Table 1 (Fig. 7) giV9S the predicted times of slack water and the predicted times and velocities of the maximum current--f10od and ebb--for each day of the year for a number of places along the Atlantic and Pacific Coasts. Table 2 (Fig. 8) furnishes data which will enable the user to determine the approximate time of slack water and the time and velocity of maximum current at numerous stati~ns on both the Atlantic and Pacific coasts. Given for each station are time differences of slack water and maximum current and flood and ebb velocity ratios with respect to some stations for which daily predictions are also given. Finally the'direction and average speed of maximum flood and ebb are also included in this section The Tidal Cur'rent Tables also contain data on rotary stations, Stream, wind driven currents, etc., all of which may be of interest to the mariner.

Each T~dal Current Chart (Fig. 9) gives a comprehensive view of the hourly speed and direction of the current throughout each of the tidal estuaries tor which a chart has been published. Each chart also provides a means tor determining the speed and direc- tion of the current at various places covered by each chart.

Pig. 7 - Sample page of predictions for the Narrows, New York Harbor, N.Y. Fig. 8 - Sample page of Table 2 showing several places referred to the Narrows. Fig. 9 Sample page from a Tidal Current Chart. 18 '"I "--.OMS.NIW y~. "&.I~. ".Y.. I... 55 F-'LO;)D.01'. '40. "Uf I-fiB. 01'. 1.0. "UI

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Fi8. 7 - Sample pa.. of daily predictions for the Narrows. N- York H8rbor. N.Y 19

TAIlE 2.-CUlaENT DlffEftNCES AND 0"*- CONSTANTS 161

~I~M CI.8fNTS ~ T\Mf DlF. velOCITY FEIfNCES IATlOS Flood Ebb PlACE

- I cwr..."~- 1 "-- IIood 1 -~ ~1""'1'8C' I veioc-A_" I Dirwc'(truel A_. ~ L~. ~ .-Ioc.------~-- i", 1. M. I.. M. : ., c..c. i .,. LOla Ia.AlD, So.th Cout-COfttl..ed ! I. I. .. TIE '.i2 ~ I fl.. ..rldl_. ?~.. I 2250 ShlnnecockInlet -40 81 72 89! -020 -0 40 1.5 / 10 &SO I.' lorD

2255 Fi..e I. , 0.5 IIIi. S. of ~k Beech1 40 38 '7a 18; +0 15 0 00 1.' 1.2! 80 2.4 245 2260 Jones InJet 40 S8i '7a S4 -100 -05& 1.8 1.& &51 &.ll215

2265 Long Beech, Inside, between b..ldges--- 40 Sf V3 40 1 -0 10 +0 10 0.& 0.& T& 0.&1 ITS

la7P Eest RockeweyIn let 40 ~5 15 4~ -1 25 -11& 1.1 1 1.1 40I 1.1 811 2275 AmbroseChennel Lightshlp--- - ,--- ~ 271 V3 .9 s..c table &.

1280 Sandy ~ App. Lighted Horn aooy M- 40 27 'S 55 s..ctable &. I ~IU lAY

2a&5 RockawayInlet-- 40 34 2290 Be en , east of- 40 35

2SOO1895 BeechCane..s'e Channel(IIIldchannel, (b..ldge>' off PI...) 40 38 40 35 2305 Grass Hessock Channel. 403'7

lEI YOlK HARloa EITRAICE I

2310 I ~ose Chennel ent..ence 40 SO 7S 58 231~ Mbr-ose Channe/., SE. of West Benk Lt- 40 sa 7.. Ol 2320 Coney Is/end Lt., 1.6 .lles SSW. of 40 S~ 7.01 2325 AmO.-oseChannel, north end 40 ~. 7. ~ 2330 Coney Island, 0.2 .lle west of 40 S5 7.. 01 233~ Ft. Lafayette, channel .-st of 40 S6 7.. 01 2340 THE NARROWS,.Idchennel 40 S7 7.. OSI lEI YOlKHARI~, U"er I.,

2345 Tompki"sville ~=== 38,

8.1602350 8.155 Red Bey Ridge I«Iok Channel Channe 1 39, 40' I Robbins Light, east of. 39: 23&5 Red 1t)Ok, 1 1111Ie west of- 41 ; 2310 Statue of Liberty, east of- 42

IUDSOI RlfEI, Nidchaftne'-

23'75 The Bei"t...y, northwest of 43'4 ~ I +1 SO +1 S5 O. 1.'- I 15 1. I 195 2.S

238C Desb..osses S~ t 43 1 74 01 +1 S8 +1 40 0.9 1.a 10 1.8!-- a.s 2385 Chelsea Docks 45'4 Ol +130 +1 40 1.0 1.0 aD 1.7 185 2.0 2390 For~y-s8Cond Street 46'4 00 +1 35 +1 4D 1.0 1.1 30 1. 2.3 2395 Nlnety-slJC1'h S'trHt 48 '7a 59 +1 40 +1 50 1.0 1.1 30 1.' -- ! 2.3

2400 Grants Tomb, 123~ S'tr..~ 49 '7a 58 +1.8 +1 85 0.9 1.1 25, 1.4 2.3 2tOS George N8shl~~ton Brldge 51 V3 57 +1.5 +2 00 0.9 1.1 20 1 1.& 200 2.2 2410 Spuyten Duyvl' 53 73 5& +2 00 +2 10 0.9 1.1 ao 1.& -- 1.1 2.15 Rlv...d8 I e 54 V3 55 +2 05 +2 20 0.8 1.0 111 1.4 200 2.0 2420 Dobbs F y 01 V3 53 +. 25 +2 40 0.8 0.9 10 1.a - 1.'

.425 Ta ytown 05 f3 83 +2 40 I +2 55 0.4 0.8 0 1.1 -- 1.1 2430 Osslnlng 10 V3 84 +2 58! +a 10 0.5 0.' aao 0.9 --- 1.S 2435 HBverstrew 12 f3 57 +a 05 I +a 15 0.5 0.7 aI& 0.8 -- 1.3

2440 ~sk 111 17 '7a 57 +3 20 I +a a5 0.5 0.1 0 0.8 ~.2 2445 Bee.. ~nteln Brldge 19 f3 59 +a 25 +a 40 0.5 0.4 0 0.8 - 1.1 2450 HI9hlend Fells 22 '7a 58 +a 35 +a eo 0.1 0.4 & 1.0 188 I 1.2 2455 West A)int, off Duck Island-' U '7a 87 +3 401 +3 55 0.5 0.6 101 1.0 - 1.1

ICur..ent is ..ote..y, 1u..ning clockwise. Mini~ cur..ent of 0.9 knot sets SW. abou~ ~ime of "Slack. flood beglnsw at The Na..rows. Mlnl.um cu ent of 0.5 knot sets ME. about 1 hour before wSlack, ebb begIns" et The ows. i_xl- flood, .{Io 500: _xi- ebb, f(1' 5&-. 'Flood begins, -2. ~5.: maxiMumflood, -QA 058: ebb begins, ~ 05-: -'XlnuM ebb, -1. 50-. -The values for the HudsonRiver ...e for ~he su..er months, when the f..esh-wat.. is a III'nl_.. . FiC. 8 . Sample pa.e of Table 2 showin. places referred to the Narrows.

~ ~Q- TIDAL CURRENT CHAI UPPER CHESAPEAKE BAY - The arrows show the dlT«tlon. and th. fig- tM s~ s~ In knob o( tM ~t at time Indicat" at bottom o( chart. This chart Is design.d (or use with tM AheM_". predicted tlma and l1elocitla o(th. CU1'rent for Baltimore Harbor Approach. Th. doil" predictions are gi~n In th. Tidal Current Tabla. Atlantic Coast of North America published annual/" b" the Envlronm.ntal Sden" s...111ca Admlnistnrtlon. Cod-ctand G.od.tIcSUnIeI/. NOTE: Speeds shown are lor time 01 spring currents. To tktermine tM sp.-t ror a "'- particular time. these speeds must be adjusted bll use 01 the table .. FGdors rorCorrecting ~.. '*- onPo.-l. .~ ...OJ- \. NAtmCAL IOUS . ~~1 :. 5~ 3 2 1 0 "

,.

!

'"\ 2l

References Following is a partial list of references for additional infor mati on on tidal currents: Carson, Rachel. 'iThe Sea Around Us. II New York, Oxford Univer- sity Press, 1951. A famous best seller and very readable

Coker, Robert E. "This Great and Wide Sea," New York, Harper 1962. Readable college text. Defant, Albert. "Ebb and Flow; The Tides or Earth, Air and Water." Ann Arbor, University or Michigan Press, 1960. Deals mainly with ocean tides; very fine.

MaI'lller, H. A. "The Sea." New York, D. Appleton and Co., 1930. Very readable.

Sverdrup, H. U., Johnson, M. W., and Fleming, R. H. "The , Their Physics, Chemistry and General Biology.'f Englewood Clit't's, N.J. Prentice-Hall, 1962. Technical.

Coast and Geodetic Survey, ESSA. "Tidal Current Tables .ft Washington, U.S. Government Printing Office. Two volumes published annually and sold by the U.S. Coast and Geodetic Survey. u.s. Naval Oceanographic Ot't'ice. "American Practical Navigator." Part six, , R.O. Pub. No.9. Washington, Government Printing Ot't'ice, 1958.

u.s. Naval Oceanographic Office. "A Suuunary of Tides and Cur- rents," Informal Manuscript Report No. 0-21-65. UnpublishedWashington, Manuscript.u.s. Naval Oceanographic , Office, .. 1965.

Arx, William S. "An Introduction to . Reading, Mass., Addison-Wesley, 1962.