Notes to GEOF346 Tidal Dynamics and Sea Level Variations

Transcription

1 Notes to GEOF346 Tidal Dynamics and Sea Level Variations Fall 2017 Helge Drange Geophysical Institute, University of Bergen Mainly off-hour typing so give a word regarding errors, inconsistencies, etc... Also give a word in case of specific requests (derivations, unclear issues), and I will try to update the notes accordingly. The note will be regularly updated so keep printing at a minimum... The notation (closely) follows that used in the text book Sea-Level Science by D. Pugh and P. Woodworth, Cambridge, 2014 ( Note: Symbols/expressions in red are, to the best of the author s knowledge, corrections to typos in the text book. For simplicity, the notation sin 2 a (471) = 2 sin a cos a shows that the explicitly numbered identity or relationship, in this case equation (471), has been used. Norwegian word/phrases are given in quotation marks in footnotes. A (far from) complete Norwegian dictionary is provided in appendix K....and the index section is only slowly being updated. October 20,

2 Contents I Equilibrium theory and derivations thereof 6 1 Equilibrium theory, direct method Geometry of the Earth-Moon system Centre of mass of the Earth-Moon system Newton s law of universal gravitation Gravitational forces and accelerations in the Earth-Moon system Earth s movement around the Earth-Moon centre of mass Net gravity on the Earth caused by the Moon Introducing the zenith angle and simplifying Radial and tangential components of the tidal acceleration Geometric approach Using the definition of the dot and cross products The radial component can be ignored; the tangential component is important Surface elevation caused by the Moon Determining the integration constant for the Equilibrium Tide surface elevation The combined surface elevation caused by the Moon and the Sun Amplitudes Relative contribution from the Sun and the Solar planets Introducing latitude, hour and declination angles The three leading lunar tidal components The Equilibrium Tide caused by the Moon Properties of the three leading lunar components Declination or nodal component, ζ Diurnal component, ζ Semi-diurnal component, ζ Sidereal and synodic periods The Moon s sidereal and synodic monthly periods The synodic lunar day The Earth s sidereal daily period Relationship between the Earth s sidereal day and the Moon s synodic day Key periods and frequencies for the Moon-Earth-Sun system Decomposing the solar and lunar tides into a series of simple harmonic constituents Geometry of the Earth-Sun system Angular speed of the Earth and the Sun Introducing d s Expressing ζ 1 in terms of simple harmonics Expressing ζ 2 in terms of simple harmonics Moon s leading tidal constituents Doodson s system for labelling tidal constituents

3 6 The spring/neap tide Derivation Amplitude and range Timing of maximum amplitude Age of the tide A note Tidal forcing in the momentum equation 40 8 Laplace s Tidal Equations 41 9 The tidal force expressed in terms of the tidal potential The potential of a conservative force Defining the tidal potential Gradient of the factor q/q Gradient of the factor R/R Resulting tidal potential Tidal potential expressed in terms of the zenith angle Radial and tangential tidal forces per unit mass Introducing the tidal potential in the primitive momentum equations Including the effect of elliptic orbits The Sun-Earth system Kepler s second law Kepler s first law Kepler s third law Mean anomaly E Real vs mean length of the radius vector, and Kepler s equation R s vs R s Kepler s equation Solving Kepler s equation R s /R s expressed by means of E Introducing the ecliptic longitude and the longitude of perihelion Right Ascension Equation of time Equation of center Longitudinal differences Equation of time, final expression Equation of time, graphical representation II Tidal related wave dynamics The shallow water equations Starting point and configuration The continuity equation The momentum equations Mainly based on R. Fitzpatrick (): An Introduction to Celestial Mechanics, M. Hendershott (): Lecture 1: Introduction to ocean tides, and Murray & Dermott (1999), Solar Syatem Dynamics, Chap. 2. 3

4 12.4 The final set of equations Gravity waves One-dimensional gravity waves Typical wavelengths for tidal waves One-dimensional gravity waves in a closed channel One-dimensional gravity waves in a semi-closed channel Sverdrup waves and other related waves Derivation Case ω = Case ω 2 = f 2 + c 2 0 kh The resulting wave motion Approximate length of the tidal axes Case f = Case k 2 f 2 /(gd) (short waves) Case k 2 f 2 /(gd) (long waves) Interaction between tidal ellipses and internal oscillations III Appendix 79 A Some key parameters of the Earth, Moon, Sun system 79 A.1 Mass A.2 Length and distance B Spherical coordinates 80 B.1 Two commonly used spherical coordinate systems B.2 Volume and surface elements in spherical coordinates B.2.1 Spherical volume elements B.2.2 Spherical surface elements B.2.3 Relationship with the Earth-Moon system B.2.4 Gradient operator in spherical coordinates C The three lunar tidal components 83 D Elliptic geometry 2 85 D.1 Cartesian coordinates D.2 Semilatus rectum D.3 The y -value of an ellipse relative to the reference circle D.4 Polar coordinates E Binominal theorem for rational exponents 89 F Some trigonometric identities 90 G Miscellaneous notes 94 G.1 Lunar longitude Mainly based on the excellent treatise by M. Capderou (2014): Handbook of Satellite Orbits. From Kepler to GPS, ISBN , DOI / , Springer International Publishing Switzerland. 4

5 G.2 High and low water times and heights G.2.1 High water times and heights G.2.2 Low water times and heights G.2.3 Temporal average H Simplified, harmonic analysis 97 H.1 Outlining of the method H.2 Extraction of the M 2 tide H.3 Note H.4 Note H.5 Extraction of the S 2 tide H.6 Extraction of the N 2 tide H.7 Extraction of the K 2 tide H.8 Extraction of the O 1 tide H.9 Extraction of the K 1 tide H.10 Note H.11 Note I Some wave characteristics 104 I.1 Properties of waves in one spatial dimension I.1.1 Spatial variation I.1.2 Spatial-temporal variations I.1.3 Wave number I.1.4 Other definitions I.2 Two-dimensional waves I.3 Complex notation I.4 Dispersion relationship I.4.1 Non-dispersive waves I.4.2 Dispersive waves I.5 Group speed I.5.1 Derivation J Resources 108 K Norwegian dictionary 109 L Acknowledgement 113 M Index 114 5

6 Part I Equilibrium theory and derivations thereof 1 Equilibrium theory, direct method A key objective in ocean tidal dynamics and analysis is to derive an analytical form of the gravitational attraction on the global ocean caused by the presence of the Moon and the Sun. Once this force is found, it can be added as a forcing term to the momentum equations which, together with the continuity equation, model the full 2- and 3-dimensional flow of the global ocean tide. The derivation of the Equilibrium Tide and subsequent analysis are addressed in Part I. A description of the basic tidal wave characteristics is also provided (Part II). Modelling of the full 2- and 3-dimensional dynamics of the ocean tide is beyond the scope of the presented lecture notes (but might be added at a later stage). In the following sections, the gravitational force is derived based on what one may phrase the direct method. An alternative approach commonly used in tidal dynamics derives the gravitational force from the tidal potential. The latter approach, which to a large extent follows the derivation of the direct method, is presented in Sec Geometry of the Earth-Moon system The configuration of the Earth-Moon system used for deriving the properties of the tidal equilibrium is displayed in Fig. 1. It follows from the figure that r + q = R (1) or, to explicitly label the Earth-Moon system, with the subscript l for lunar, r + q l = R l (2) 1.2 Centre of mass of the Earth-Moon system The centre of mass of the Earth-Moon system is located along the centre line OM at a distance x R (0 < x < 1) from point O (Fig. 1). We then get that m e x R l = m l (1 x) R l (3) or m e x = (4) m e + m l Here m e and m l are the mass of the Earth and Moon, respectively, see appendix A for numerical values. With mean values of r and R (see appendix A), we get that x 0.73 r (5) 6

7 Figure 1: Illustration of the Earth-Moon system with the Earth to the left and the Moon to the right (figure greatly out of scale). O, P and M are the centre of the Earth, an arbitrary point on Earth s surface and the centres of the Moon, respectively. r is the Earth s radius vector (from point O to P ), R is the position vector from the centre of the Earth to Moon s centre (from O to M), and q is the position vector from an arbitrary point P on Earth s surface to M. The line between O and M is sometimes called the centre line and the angle φ the zenith angle or the centre angle. A similar configuration holds for the Earth-Sun system. implying that the centre of mass of the Earth-Moon system is located about three quarters of Earth s radius from the centre of the Earth. The centre of mass of any two- or multiple-body system is also called the system s barycentre. 1.3 Newton s law of universal gravitation Newton s law of universal gravitation (from 1687) states that an attractive force F is set up between any two point masses, varying proportional with the product of the masses (m 1 and m 2 ) and inversely proportional with the distance R between the masses /3.2/, note R! F = G m 1 m 2 R 2 (6) Here G = N m 2 kg 2 is the gravitational constant. 1.4 Gravitational forces and accelerations in the Earth-Moon system The gravitational force at the Earth s centre because of the presence of the Moon, F l, is where R l /R l F l = G m e m l R 2 l is the unit vector along the Earth-Moon centre line. R l R l (7) According to Newton s second law, F = m a, this force leads to an acceleration of the centre of the Earth according to a O = F l = G m l R l m e Rl 2 (8) R l 7

8 Similarly, the gravitational acceleration at point P caused by the Moon is a P = G m l q 2 l q l q l (9) At point P, there is also a gravitational acceleration g towards the centre of the Earth caused by Earth s mass: g = G m e r r 2 (10) r By inserting the numerical values of G, m e and r (appendix A), one obtains g = 9.8 m s 2, as expected. Furthermore, (10) gives the relationship /3.4/ G = g r2 m e (11) 1.5 Earth s movement around the Earth-Moon centre of mass To maintain the Earth-Moon centre of mass at 0.73 r, the Earth can either rotate around the joint centre of mass as a solid body, or the Earth can adjust it s position around the common centre without rotation. The former would imply that only one side of the Earth would face the Moon at all times or, alternatively, that the Earth would make one full rotation for each full passing of the Moon (approximately every 28 days). This is not the case; the Moon can be viewed from all longitudes during the course of a lunar cycle and the Earth does not experience a lunar (near monthly) season. Therefore, the latter movement takes place as explained in Fig. 2. Another way to illustrate the revolution around the common centre of mass of a two-body system is given by an animation at In this case, the two-body system consists of a major (grey coloured) and a minor (blue) body encircling the former. The red point is fixed to the surface of the main body and points towards left at all times, illustrating the revolving movement around the common centre of mass. The above implies that every point on solid Earth describes a circular motion with radius s = 0.73 r, but with different centres, and with the rotation rate ω governed by the rotation rate of the Moon around the Earth. Therefore, each point on Earth, whether on or below Earth s surface, will experience a centrifugal acceleration (i.e., outward-directed acceleration relative to the rotation of the Earth-Moon system) given by (M & P (2008), eqn. 6.28) a ω = ω 2 s (12) 1.6 Net gravity on the Earth caused by the Moon The distance between the centres of the Earth and the Moon vary slowly during the lunar month, but it can be treated as constant for the purposes considered here. If so, the magnitude of the outward-directed centrifugal acceleration at the centre of the Earth has to exactly balance the magnitude of the Moon s gravitational pull at the Earth s centre. This means that a ω = a O (13) 8

9 Figure 2: Illustration of the movement of the Earth (Earth s circumference in blue) around the Earth-Moon centre of mass (red dot), looking from the north, i.e., looking down onto the Earth-Moon system. (1) Upper panel: The Moon is located to the right of the Earth, in the direction of the grey arrow. The black dots show the Earth s centre and a point on the central line, opposite to the direction of the Moon. The red circle shows the path the Earth s centre would follow if it rotates around the joint centre of mass. (2) Second panel: Some time later, the Moon has moved counter-clockwise relative to the Earth. The Earth s centre (orange dot) is located on the circle around the joint centre of mass. Similarly, the point initially marked with the black dot on the opposite side of the Moon describes the same circular revolution as Earth s centre (the left-most yellow dot, with the path traced out by the movement shown by the red dotted arc). (3) Third panel: Later, the Earth s centre (green dot) continue to follow the red, solid circle. The other point follows a similar circular trajectory (the second green dot and the red dotted arc). (4) Lowermost panel: When the Moon is on the opposite side of the Earth, both of the two initial points are tracing out circles with a common radius s = 0.73 r. The resulting circular movement will continue as the Moon rotates around the Earth. In this way, all points on and within the solid Earth will describe circular trajectories with radius s = 0.73 r. Note that the Earth does not rotate as a solid body around the Earth-Moon centre of mass. Rather, any point on Earth, like the left-most dot on the Earth s surface in all of the panels, have the same orientation with respect to a fixed star throughout the movement. This movement is commonly described as revolution without rotation. 9

10 or ω 2 s = G m l R 2 l (14) At the point P at the surface of the Earth, the gravity caused by the Moon will vary according to the distance to the Moon, with the largest gravitational pull for the points closest to the Moon. This pull is directed along the q-vector. At the same time, the centrifugal acceleration in point P is directed opposite to the R-vector. The net acceleration felt at P can then be expressed as or, by means of (14), a = G m l q 2 a = G m l ( ql q 3 l q l q l ω 2 s R l R l (15) R ) l Rl 3 (16) 1.7 Introducing the zenith angle and simplifying The acceleration (16) can be expressed in terms of r, R and φ. From Fig. 1, the law of cosines gives (dropping the subscript l for the time being) or q 2 = R 2 + r 2 2 R r cos φ (17) q = ( R 2 + r 2 2 R r cos φ ) 1/2 (18) = R (1 2 ( R ) 1/2 rr r2 cos φ + R 2 (19) 1 2 r ) 1/2 R cos φ (20) R (1 r ) R cos φ (21) Here the smallness of r/r (appendix A) has been used in the second last expression, and the binominal theorem for rational exponents (460) has been used in the last equality. Consequently, 1 q 3 = 1 ( R 3 1 r ) 3 R cos φ 1 ( R r ) R cos φ (22) and a = G m [( l R r ) ] R cos φ q R The vector sum (2) and the smallness of r/r, and reintroducing the subscript l, give a = G m l r R 3 l ( 3 cos φ R l r ) R l r (23) (24) Substituting G with Earth s gravitational acceleration g from (11) yields a = g m ( ) 3 ( l r 3 cos φ R l r ) m e R l R l r (25) 10

11 With the mass and distance ratios given in appendix A, it follows that /p. 32/ a 10 7 g (26) The tidal acceleration on Earth caused by the presence of the Moon is therefore very small. By decomposing the tidal acceleration into one component in the direction of r and one component tangential to the surface of the Earth, it follows that the former indeed can be ignored. The latter has no counterpart and it is this component that gives rise to the tides. 1.8 Radial and tangential components of the tidal acceleration Geometric approach The component of the tidal acceleration in the direction of R follows directly from (25) a R = 3g m ( ) 3 l r cos φ (27) m e R l a R can be decomposed in the radial and horizontal directions based on Fig. 3. Figure 3: As Fig. 1, but for decomposing a R (see expression 27) in the outward radial direction (a r r) and in the horizontal direction on Earth, pointing towards the Moon (a h ). It follows from the figure that the horizontal component of a R, in the direction towards the Moon, is a h = 3 g m ( ) 3 l r sin φ cos φ (28) m e R l Since (see 471), expression (28) can alternatively be expressed as sin φ cos φ = 1 sin 2φ (29) 2 a h = 3 2 g m ( ) 3 l r sin 2φ (30) m e R l 11

12 Likewise, the radial component of a R is The latter, together with the radial component of (25), adds up to a r is directed radially outward, in the direction of r in Fig g m ( ) 3 l r cos 2 φ (31) m e R l a r = g m ( ) 3 l r (3 cos 2 φ 1) (32) m e R l Using the definition of the dot and cross products Alternatively, a r can be obtained by taking the dot product of a and the unit vector r/r (see the geometic configuration in the left panel in Fig. 5): The definition of the dot product gives so (33) becomes which is identical to (32). a r = a r r = g m ( ) 3 ( l r 3 cos φ R l r m e R l R l r r r ) r 2 (33) R l r = R l r cos φ and r r = r 2 (34) a r = g m ( ) 3 l r ( 3 cos 2 φ 1 ) (35) m e R l In a similar manner, the cross product of a with the unit vector r/r gives the horizontal component of a: a h = a r = g m ( l r r m e The definition of the cross product gives R l ) 3 ( 3 R l r cos φ R l r ) r r r 2 (36) R l r = R l r sin φ and r r = 0 (37) so a h = 3g m ( ) 3 l r sin φ cos φ (38) m e R l The radial component can be ignored; the tangential component is important Since the gravity on Earth s surface caused by the Earth s mass g is orders of magnitude larger than the gravitational acceleration in the radial direction (see 26), a r can safely be neglected compared to g. Since g has no counterpart to a h, this acceleration component aligned 12

13 tangential to the surface of the Earth cannot be ignored. It is a h that gives rise to the tides on Earth. Note that a h only varies with φ, the zenith or central angle in Fig. 1. φ depends on the latitude and longitude of the position P, as well as the declination angle of the Moon (and the Sun). Introduction of these geometric factors are presented in Sec. 3. The horizontal component of the tidal acceleration or tidal force which is the actual tidal acceleration or force is often named the tractive acceleration or force. 1.9 Surface elevation caused by the Moon The tidal acceleration a h will give rise to changes in the sea level. This again leads to a pressure gradient or, alternatively, a pressure force. The pressure force per unit mass (i.e., acceleration) is, per definition (M & P (2008), eqn. 6.6), Here p is pressure and ρ is density. 1 p (39) ρ For a fluid with approximately uniform (constant) density, the hydrostatic approximation gives p = ρ g ζ (40) Here ζ (m) is the free ocean surface, i.e., the elevation relative to a flat ocean, caused by Earth s gravity and the tidal force. This means that the pressure force per unit mass can be expressed as g ζ (41) It follows then from Newton s second law, F = m a, expressed in the form F/m = a, that where h g h ζ = a h (42) denotes the gradient operator tangential to the Earth s surface. It is convenient to express the differential equation (42) in terms of the spherical coordinate system to the right in Fig. 25. In this case the z axis of the spherical coordinate system is oriented in the direction of the centre line R in Fig. 1. In this system, only the e φ component (see 420) has a contribution (since r = konst. and there is symmetry in the Ψ-direction), so (42) becomes g ζ r φ = a h (43) or, by means of (30), ζ φ + 3 m l r 4 sin 2φ = 0 (44) 2 m e R3 Integration over φ, using sin 2φ dφ = 1 cos 2φ (45) 2 (see 477) gives the Equilibrium Tide s surface elevation where C is an integration constant. ζ = 3 m l r 4 cos 2φ + C (46) 4 m e R3 13

14 1.9.1 Determining the integration constant for the Equilibrium Tide surface elevation Conservation of water volume requires that the surface elevation ζ must vanish when integrated over the sphere. This constraint can be used to determine the integration constant C by integrating (46) over a sphere with constant radius r by means of the zenith-angle coordinate system in the right panel of Fig. 25. The first term on the right-hand-side of (46) is a constant multiplied with the factor cos 2φ. Integration over the sphere of the given term therefore corresponds to solving the double-integral (similarly to 418) 2π π r 2 dψ cos 2φ sin φ dφ (47) Ψ=0 φ=0 The integral involving φ can be solved by using the identity (see 478): cos 2x sin x dx = cos x cos 3x + C (48) 2 6 where C is an integration constant. Expression (47) then becomes [ cos φ r 2 2π 2 ] π cos 3φ = r 2 4π (49) Thus, integration of (46) over a sphere with constant radius r leads to 3 m l r 4 ( 4 m e R 3 r 2 4π ) + 4r 2 πc = 0 (50) 3 Consequently, C = 1 m l r 4 4 m e R 3 (51) The Equilibrium Tide is therefore governed by the expression Or alternatively, since the Equilibrium Tide may be put in the commonly used form ζ = 3 m l r 4 ( 2 m e R 3 cos 2 φ 1 ) 3 ζ = 1 m l r 4 (3 cos 2φ + 1) (52) 4 m e R3 cos 2φ = 2 cos 2 φ 1 (53) (54) 14

15 2 The combined surface elevation caused by the Moon and the Sun Expression (54) is also valid for the Sun. Thus, the total surface tidal elevation ζ, with contributions from the Moon (subscript l) and the Sun (subscript l), becomes ζ = 3 m l r 4 2 m e R 3 l ( cos 2 φ l 1 ) m s r 4 ( 2 m e Rs 3 cos 2 φ s 1 ) 3 [ = 3 ( ) 3 ( 2 r m l r cos 2 φ l 1 ) + m ( ) 3 ( s r cos 2 φ s 1 m e R l 3 m e R s 3) ] (55) 2.1 Amplitudes It follows from the above expression that the lunar and solar tides have maximum amplitudes for φ l = φ s = 0 : m l r 4 m s r 4 m e Rl 3 and m e Rs 3 (56) With the values from appendix A, the numerical values of the above factors are 0.27 m (or m with three decimals) for the lunar tide and 0.12 m (0.123 m) for the solar tide. The ratio between the two contributions are thus (for φ l = φ s = 0 ) /p. 40/ ζ l ζ s = 2.18 and ζ s ζ l = 0.46 (57) The lunar tide is therefore about twice as large as the solar tide. Furthermore, the sum of the lunar and solar tides of 0.39 m is, generally speaking, well below the observed tidal amplitude, globally averaged around 1 m, but with amplitudes exceeding 10 m in some locations. The main reason for this mismatch is the influence of the bathymetry, the detailed geometry of the coasts and the ocean basin eigen-modes on the tidal dynamics. 2.2 Relative contribution from the Sun and the Solar planets The maximum gravitational attraction on the Earth from any planet (subscript i) relative to that of the Moon is, from (56), given by the relationship m l m i R 3 i R 3 l (58) With approximate values of any planet s mass m i and the distance from the Earth s centre R i, the relative magnitudes are as shown in Fig. 4. It follows that the Venus is the third most important planet, after the Moon and the Sun, with a contribution approximately 1/18, 000 of that of the Moon. Thus, the Moon and the Sun are, by far, the most important contributors to the Earth s ocean tide. 15

16 Figure 4: Approximate magnitude of the gravitational attraction on the Earth from the Sun and the Solar planets relative to that of the Moon, and the corresponding rank. Background illustration from System. 16

17 3 Introducing latitude, hour and declination angles Up to now, no specific geographic reference has been given beyond the zenith angle shown in the left panel of Fig. 5. To determine the tide at any point P on the Earth s surface relative the the Moon or the Sun (or other celestial bodies), it is convenient to introduce three angles: The northern latitude φ P of P, the hour angle C P between the meridians going through the sub-lunar point 3 V and P (playing the role of longitude), and the declination angle d of the celestial body (d l for the Moon and d s for the Sun), see the right panel in Fig. 5 for the exact definition of the three angles. 3.1 The three leading lunar tidal components We start by considering the two-body Earth-Moon system as illustrated in Fig. 5. Figure 5: The tidal equilibrium configuration defined by R, r and φ (left) and the new angles φ P, C P of and d l (right). x, y, z denotes the Cartesian coordinates with x, y spanning the Earth s equatorial plane and z pointing northward. The x-axis is aligned along the projection of the centre line (or vector R) onto the equatorial plane. O is the centre of the Earth, M is the Moon, and P is any arbitrary point on the surface of the Earth with latitude φ P measured from the equatorial plane. The sub-lunar point V is where the centre line crosses the surface of the Earth (or the point on the surface of the Earth under the Moon) with the declination angle d l. The hour angle C P is the angle on the equatorial plane between the x-axis and the longitude of point P or, which is the same, the angle between the meridians running through V and P. The two position vectors r and R can be expressed in terms of the angles φ P, C P and d l. The projection of R onto the x- and z-axes give R cos d l and R sin d l, respectively, see left panel of Fig. 6. Thus, R = R (cos d l, 0, sin d l ) (59) 3 fotpunkt under månen 17

18 Likewise, the projection of r onto the equatorial plane gives a vector of length r cos ϕ (right panel of Fig. 6). Decomposing this vector onto the x and y axes gives r = r (cos φ P cos C P, cos φ P sin C P, sin φ P ) (60) Expression (54) states that the surface elevation of the Equilibrium Tide ζ is proportional to the factor cos 2 φ 1/3. The task is therefore to express cos φ in terms of φ P, C P and d l. This can be readily done by means of the dot product r R = rr cos φ = rr (cos φ P cos d l cos C P + sin φ P sin d l ) (61) where the first expression is simply the definition of the dot product, whereas 59 and 60 have been used in the second expression. From the above relationship, cos φ = sin φ P sin d l + cos φ P cos d l cos C P (62) which is the last expression on p. 41 in Pugh & Woodworth (2014). /p. 41/ Figure 6: As Fig. 5, illustrating of the decomposition of R (left panel) and r (right panel) onto the Cartesian x, y, z-system. Thus, cos 2 φ 1 3 = cos2 φ P cos 2 d l cos 2 C P +2 cos φ P cos d l cos C P sin φ P sin d l +sin 2 φ P sin 2 d l 1 3 (63) The identity sin(2 a) = 2 sin a cos a (64) can be applied to the φ P - and d l -factors in the second term on the right hand side of (63), resulting in cos 2 φ 1 3 = cos2 φ P cos 2 d l cos 2 C P sin(2 φ P ) sin(2 d l ) cos C P + sin 2 φ P sin 2 d l 1 3 (65) The right hand side of (65) contains terms with variations on three, main time scales. Keeping the latitude of the position P fixed so φ P = const., the first term on the right hand side of (65) 18

19 varies mainly because of cos 2 C P. This term gives rise of the semi-diurnal lunar tide. The second term on the right hand side of (65) varies mainly because of cos C P, describing the diurnal lunar tide. And finally, the third term vary with sin 2 d l, describing the tidal response to the near biweekly (or fortnightly) variations in the lunar declination. The three time scales can be facilitated by expressing cos 2 φ 1 3 = ζ 0 + ζ 1 + ζ 2 (66) where ζ 0 = 3 2 ( sin 2 φ P 1 ) ( sin 2 d l 1 ) 3 3 (67) ζ 1 = 1 2 sin(2 φ P ) sin(2 d l ) cos C P (68) ζ 2 = 1 2 cos2 φ P cos 2 d l cos(2 C P ) (69) Here, ζ 0 describes the biweekly (fortnightly) variations (through sin 2 d l ), ζ 1 describes the diurnal variations (through cos C P ), and ζ 2 describes the semi-diurnal variations (through cos C P ). The transformation of (65) into (66) (69) is given in appendix C. Expression (66) corresponds to the terms in square brackets in expression (3.10) in Pugh & /3.10/ Woodworth (2014). 3.2 The Equilibrium Tide caused by the Moon By combining (55) with expressions (66) (69), the Equlibrium Tide caused by the Moon, ζ l, can be put in the form (see expression 3.12 in Pugh & Woodworth, 2014): /3.12/ ζ l = r m [ ( l 3 C 0 (t) m e 2 sin2 φ P 1 ) ] + C 1 (t) sin 2φ P + C 2 (t) cos 2 φ P 2 (70) where C 0 (t) = C 1 (t) = C 2 (t) = ( ) 3 ( r 3 R l 2 sin2 d l 1 ) (71) 2 ( ) 3 r 3 R l 4 sin 2d l cos C P (72) ( ) 3 r 3 4 cos2 d l cos 2C P (73) R l A similar expression is valid for the solar Equilibrium Tide. 3.3 Properties of the three leading lunar components From (67) (69), it follows that all components depend on the latitude φ P. Secondly, the identities cos 2 d l = 1 2 (1 + cos 2 d l) (74) 19

20 and sin 2 d l = 1 2 (1 cos 2 d l) (75) show that all contributions depend on cos 2 d l, implying that all contributions have a periodicity of T l /2 14 d (see Sec. 4.1 for a discussion of T l, but it is sufficient here to note that T l = d ). And thirdly, only ζ 1 and ζ 2 depend on the hour angle C P. Other characteristics are listed below Declination or nodal component, ζ 0 ζ 0 = 0 for sin 2 φ P = 1/3, or for φ P = ±35 Since the Moon s declination angle d l varies between , sin 2 d l 1/3 does not vanish Diurnal component, ζ 1 Vary with C P sin 2d l = 0 twice a month, so ζ 1 with the lunar day with period 24 h 50 min vanishes twice a month d l -modulation largest when the Moon is high on sky; 18.5 < d l < 28.5 with a period of 18.6 yr Largest at φ P = ±45, vanishes at equator and the poles Semi-diurnal component, ζ 2 Vary with C P with half of the lunar day with period 12h25min d l -modulation decreases with increasing d l ; min(cos 2 d l ) 0.77 for d l = 28.5 Largest for φ P = 0 (equator), vanishes at the poles 20

21 4 Sidereal and synodic periods The solar system is characterised by cyclic variations, like Earth s diurnal variation in solar irradiance due to Earth s rotation, the close to monthly lunar phase as seen from the Earth, and the annual (seasonal) cycle as observed from the Earth. These periodicities need to be uniquely and accurately determined in order to quantify the detailed lunar and solar contributions to the tides. The various periods can be determined from an observer on the (rotating) Earth, or as seen from a (stationary) fixed star. Seen from the Earth, the periodicity is named the synodic period 4 (from synodus, Greek from coming together ). Seen from a distant star, the period is known as the sidereal period 5 (from sidus, or star in Latin). 4.1 The Moon s sidereal and synodic monthly periods A full rotation of the Moon around the Earth as seen from a distant star has a sidereal period T l = d (this is an observed, given value). The synodic period can be found by considering the Sun-Earth-Moon system seen from a fixed star from above (north), as illustrated in Fig. 7. If the Earth were not rotating around the Sun, the Moon s rotation rate as seen from the Earth would equal ω l T l = 2π (76) In this case the Moon, initially located on the line between the Sun and the Earth (point A in Fig. 7), would be back on the same line after the period T l. However, since the Earth does rotate around the Sun with the rotation rate ω es and a rotation period of T es = d, the Moon will not be back at the initial position seen from the Earth after the sidereal period T l. Actually, the Moon needs to move from point B to point C in Fig. 7 in order to be back at the initial position, i.e., back to the position where the Earth, the Moon and the Sun are all aligned. We are therefore searching for a period T > T l in order for the Moon to reach the position C in Fig. 7. Based on (76), T can be derived from the expression ω l T = 2π + ω es T (77) where the last term corresponds to the angle between A, the centre of the Sun and C or, which is equivalent, the angle between B, the centre of the Earth and C in Fig. 7 (both identified by the red, double-lined arcs). Division by the factor 2π T gives 1 = 1 T l T + 1 (78) T es For T l = d and T es = d one obtains that the lunar synodic period T = d (79) Thus, the monthly lunar period seen from a distant star is d, whereas it is d seen from the Earth. 4 synodisk 5 siderisk periode 21

22 Figure 7: Illustration of the Sun (black circle in the centre), the Earth (solid blue dot, encircling the Sun along the blue circle), and the Moon (solid red dot, encircling the Earth along the red circle), viewed from the north. ω l and T l are the lunar rotation rate and period seen from a distant star, respectively. Similarly, ω es and T es are the Earth s rotation rate and rotation period around the Sun. Point A denotes the initial position of the Moon, in this case exactly on the centre line between the Earth and the Sun. Point B refers to the position of the Moon after the sidereal period T l. Point C denotes the full lunar rotation as observed from the Earth, in this case when the Moon is back on the centre line between the Earth and the Sun. The rotation period for the latter is the synodic period. 22

23 4.2 The synodic lunar day Following the logic from Sec. 4.1, the lunar day T seen from Earth due to the Earth s rotation around it s own axis will be somewhat longer than that seen from a fixed star. The situation is illustrated in Fig. 8. Figure 8: Illustration of the lunar day. The Moon is the filled red dot encircling the Earth along the red circle, viewed from above (north). The Earth is the open blue circle. If the Moon was fixed in space and if it was located above point A on the Earth s surface initially, the Moon would be located in the same position after one full rotation of the Earth ( T e = 1 d ). However, during the period T e, the Moon has moved with the rotation rate ω l. The Earth must therefore do an additional rotation, corresponding to moving point A to position B, before the Moon is again located in zenith relative to the initial point A. One full rotation of the Earth can be expressed as where T e = 1 d. ω e T e = 2π (80) Taking into account the simultaneous rotation of the Moon around the Earth, gives where T is the lunar day as seen from the Earth. ω e T = 2π + ω l T (81) Division by 2π T gives 1 = 1 T e T + 1 (82) T l With T l = d and T e = 1 d, we obtain or T = d (83) T = 24 hr 50 min sec 24 hr 50 min (84) The lunar day observed from the Earth, the synodic lunar day, is therefore about 50 minutes longer than the solar day on the Earth. 23

24 4.3 The Earth s sidereal daily period The daily period of the Earth seen from a fixed star is somewhat shorter than the mean solar day because of the movement of the Earth around the Sun. The situation is illustrated in Fig. 9. Figure 9: Illustration of the Earth s sidereal day. The Earth (blue circle) encircles the Sun (black circle), viewed from above (north). If the Earth did not rotate around the Sun, any point on the Earth s surface (illustrated with a person with the Sun in zenith) would be back at the same position relative to the Sun after 1 mean solar day, T e = 1 d. However, during the time T e, the Earth has moved with the rotation rate ω es around the Sun. Therefore, seen from a fixed star, the Earth has made slightly more than one full rotation before the person in the figure has the Sun in zenith. For this reason, the sidereal day, i.e., when the person is in position A, must be slightly shorter than T e. One full rotation of the Earth can be expressed as where T e = 1 d. ω e T e = 2π (85) Taking into account the simultaneous rotation of the Earth around the Sun, gives ω e T = 2π ω es T (86) where T is the Earth s sidereal day and the minus-sign is introduced since T < T e. Division by 2π T gives 1 = 1 T e T 1 (87) T es With T es = d and T e = 1 d, we obtain T = d (88) or T 23 hr 56 min (89) 24

25 Earth s rotation period observed from a fixed star is therefore about 4 minutes less than the mean solar day. 4.4 Relationship between the Earth s sidereal day and the Moon s synodic day Figure 10: The Earth is the blue disc, rotating with a period T e = d relative to a fixed star, exemplified with the first point of Aries, (viewed from above, i.e., north). During the time T e, the Moon (small red disc) rotates around the Earth with a rotation period T l = d, yielding the mean synodic lunar day of d. In this case we consider rotation periods relative to a fixed star, for instance relative to the first point of Aries 6,, see Fig. 10. One full rotation of the Earth relative to is give by where T e = d. ω e T e = 2π (90) Taking into account the simultaneous rotation of the Moon around the Earth, again relative to a fixed star, leads to the relationship where ω l corresponds to the lunar period T l = d. Division by 2π T Thus gives ω e T = 2π + ω l T (91) 1 = 1 T e T + 1 (92) T l T = d (93) or the lunar day as seen fro the Earth (the synodic lunar day) is d. 6 stjernetegnet Væren, eller vårpunktet 25

26 4.5 Key periods and frequencies for the Moon-Earth-Sun system Based on the above sections, a set of basic astronomical periods and frequencies can be defined to describe the Earth-Moon-Sun system. With msd denoting the mean solar day and msh denoting the mean solar hour, these are: Quantity Symbol Unit Period T msd or msh Frequency f = 1/T 1/msd or 1/msh Angular frequency σ = 360 /T deg/msh Angular frequency ω = 2π/T = σ 2π/360 rad/msh Physical angles C s, C l, s, h, p, N, p rad Table 1: Overview of commonly used quantities in tidal analysis. The physical meaning of the seven physical angles are indicated in Table 2 and will be explicitly defined in the subsequent sections. For historical and practical reasons that will become clear(er) in Sec. 5.3, the various periods and rotation rates are put in the form T i, σ i and ω i, where the subscript i is an integer, with the exception of Earth s sidereal day that is denoted with the subscript s, see Table 2. Period Frequency Angular Corresp. speed ange T f σ ω time cycles per deg rad unit time unit per msh per msh deg / rad Mean synodic solar day msd C s Mean synodic lunar day msd C l Lunar sidereal month msd s Tropical (synodic) year msd h Moon s perigee Jyr p Regression of Moon s nodes Jyr N Perihelion 6 20,942 Jyr p Earth s sidereal day s msd Lunar synodic month msd Table 2: Overview of the central astronomical periods and frequencies. msd is the mean solar day, msh is the mean solar hour and Jyr is the Julian year ( d). The second row gives the index of the different quantities, e.g., the period of the Moon s perigee is T 4 = 8.85 Jyr. With the definitions in Table 2, (86) is equivalent to ω 2 = 2π T ω 3 (94) In expressin (86), 2π/T = ω s, the Earth s sidereal day. Consequently (Pugh & Woodworth, p. 48) ω s = ω 0 + ω 3 or σ s = σ 0 + σ 3 (95) /p. 48/ Likewise from expression (91), see also Pugh & Woodworth, p. 48, /p. 48/ 26

27 ω s = ω 1 + ω 2 or σ s = σ 1 + σ 2 (96) 27

28 5 Decomposing the solar and lunar tides into a series of simple harmonic constituents The total tide can be expressed as a sum of many harmonic contributions. Each of these contributions called tidal component, tidal constituent or harmonic constituent is represented by a simple, harmonic cosine function. A capital letter (sometimes a combination of capital and small letters), plus a numerical subscript, is used to designate each constituent. For instance, the semi-diurnal tidal contribution from the Moon and the Sun are named M 2 and S 2, respectively. Each constituent is typically described by its speed (or frequency), expressed as degrees per mean solar hour (hereafter deg/msh) or radians per mean solar hour (rad/msh). The speed of a constituent is 360 /T, where T (msh) is the period. For S 2, the speed is then 360 /(12 msh) = 30 / msh. 5.1 Geometry of the Earth-Sun system The Earth-Sun system, viewed from the Earth (i.e., a geocentric view), can be presented by means of the declination angle d s, the ecliptic plane angle ɛ s = const = and the longitude angle λ s as shown in Fig. 11. From this figure, the following relationships are obtained H = R sin d s (97) H = d sin ɛ s (98) L = R sin λ s (99) The first two expressions give sin d s = L R sin ɛ s (100) Combined with the third expression, one obtains (Pugh & Woodworth, p. 46) /p. 46/ sin d s = sin λ s sin ɛ s (101) Angular speed of the Earth and the Sun For the geocentric view in Fig. 11, the longitude angle λ s describes the Sun s motion around the Earth with a speed given by the duration of the tropical year 7,8 Similarly, C s 360 deg def = = σ 3 (102) msd msh describes the Earth s rotation around it s own axis 360 deg def = = σ 0 (103) 24 msh msh 7 tropisk år 8 tropical comes from the Greek tropicos meaning turn, denoting the turning of the Sun s seasonal motion as seen from the Earth 28

29 Figure 11: A geocentric view of the Earth-Solar system. R is the solar position vector, pointing from the Earth s centre to the centre of the Sun, with a declination angle d s above Earth s equatorial plane. The Sun s path on sky follows the ecliptic, with ɛ s denoting the (approximately) constant angle between Earth s equatorial plane and the ecliptic ( ɛ s = = 23.5 ). The third angle λ s denotes the Sun s eastward directed longitudinal angle with respect to the vernal equinox (the point where the ascending Sun crosses the equatorial plane). The vernal equinox is also known as the first point of Aries. Note the three right-angled triangles in the figure; one red, one blue and one confined by the line from the Earth s centre towards and the sub-solar point (black circle). For the latter, the angle towards Earth s centre is λ s. 29

30 The angles λ s and C s, expressed in degrees from can thus be written as where t is time in msh. λ s = σ 3 t and C s = σ 0 t (104) For the following analysis, we want to express λ s and C s with a shared frequency. Since σ s = σ 0 + σ 3, see expression (95), the two angels can be written, in degrees, as λ s = σ 3 t and C s = (σ s σ 3 ) t (105) Alternatively, λ s and C s can be expressed in terms of radians: λ s = ω 3 t and C s = (ω s ω 3 ) t (106) Introducing d s Both ζ 1 (68) and ζ 2 (69) include a harmonic term with argument 2 d s. This term can be expressed by means of sin d s ; sin 2 d s (471) = 2 sin d s cos d s (461) = 2 sin d s 1 sin 2 d s (107) The declination angle d s < 23.5, so sin 2 d s is a rather small factor. The first order appromation of the binominal theorem (460) can then be applied to (107) ( sin 2 d s 2 sin d s 1 1 ) 2 sin2 d s (108) Expressing ζ 1 in terms of simple harmonics With the above definitions and simplification, (68) can be expressed by means of φ P, d s, ɛ s, λ s and t: 1 ζ 1 = 2 sin(2φ P ) sin 2 d s cos C s ( (108) = sin(2φ P ) sin d s 1 1 ) 2 sin2 d s cos C s ( (101) = sin(2φ P ) sin ɛ s sin λ s 1 ) 2 sin2 ɛ s sin 3 λ s cos C s [( (475) = sin(2φ P ) sin ɛ s 1 3 ) 8 sin2 ɛ s sin λ s + 1 ] 8 sin2 ɛ s sin 3λ s cos C s [( (106) = sin(2φ P ) sin ɛ s 1 3 ) 8 sin2 ɛ s sin ω 3 t + 1 ] 8 sin2 ɛ s sin 3ω 3 t cos(ω s ω 3 )t (109) Recall that the numbers in parentheses point to the expressions used. 30

31 For ɛ s = 23.5, the 1/8-term is small (about 2 percent contribution), so (109) can be expressed as ( ζ 1 = sin(2φ P ) sin ɛ s 1 3 ) 8 sin2 ɛ s sin ω 3 t cos(ω s ω 3 )t = ζ 1 sin ω 3 t cos(ω s ω 3 )t (110) where ζ 1 includes the time-independent contribution to ζ 1. Expression (110) states that the temporal variation of ζ 1 is governed by the basic speed ω s ω 3 = 15 deg/msh, modulated by the (slow) speed ω 3 = deg/msh. The identity ( ) ( ) sin a sin b (468) 1 1 = 2 sin (a b) cos (a + b) 2 2 can be used to split (110) into two simple harmonics. With (111) ω 3 = 1 2 (a b) and ω s ω 3 = 1 (a + b) (112) 2 we obtain a = ω s and b = ω s 2 ω 3. Thus, Expression (113) can alternatively be put in the form ζ 1 = ζ 1 2 [sin ω st sin(ω s 2 ω 3 )t] (113) ζ 1 = ζ 1 2 [sin((ω s ω 3 ) + ω 3 )t sin((ω s ω 3 ) ω 3 )t] (114) The Sun s diurnal contribution has therefore speeds ω s and ω s 2 ω 3 (from 113), symmetrically distributed about the basic speed ω s ω 3 = ω 0 (expression 114). In summary, ζ 1 gives rise to two tidal constituents, originating from the Sun s half-yearly variation of the declination angle. The two constituents are called the solar declination constituents: K 1 Declinational diurnal constituent (will also get a contribution from the Moon, see Sec. 5.2); it s speed is σ s = deg/msh and the period is msh. P 1 Principal (main) solar declinational diurnal constituent; it s speed is σ s 2 σ 3 = deg/msh and period is msh Expressing ζ 2 in terms of simple harmonics ζ 2 can rewritten as ζ 2 = (461) = (101) = (474) = 1 2 cos2 φ P cos 2 d s cos 2C s 1 2 cos2 φ P (1 sin 2 d s ) cos 2C s 1 2 cos2 φ P (1 sin 2 ɛ s sin 2 λ s ) cos 2C s [ 1 2 cos2 φ P 1 1 ] 2 sin2 ɛ s (1 cos 2λ s ) cos 2C s (115) 31

32 If (115) is split in parts with and without λ s -dependency, one obtains ζ 2 = 1 [( 2 cos2 φ P 1 1 ) 2 sin2 ɛ s cos 2C s + 1 ] 2 sin2 ɛ s cos 2λ s cos 2C s = 1 [( 2 cos2 φ P 1 1 ) 2 sin2 ɛ s cos(2 (ω s ω 3 )t) + 1 ] 2 sin2 ɛ s cos 2 ω 3 t cos(2 (ω s ω 3 )t) (116) For the last equality, the angular speeds from Sec have been introduced. Similarly to ζ 1, the cos 2 ω 3 t cos(2 (ω s ω 3 )t) factor in the last term of (116) consists of a temporal variation with basic speed 2 (ω s ω 3 ), slowly modulated by variation with speed 2 ω 3. These two temporal variations can be split into two simple harmonics by means of the identity ( ) ( ) cos a + cos b (469) 1 1 = 2 cos (a + b) cos (a b) (117) 2 2 With the choice one obtains a = 2 ω s 2 ω 3 = 1 2 (a b) and 2(ω s ω 3 ) = 1 (a + b) (118) 2 and b = 2 (ω s 2 ω 3 ). Consequently, ζ 2 = 1 2 cos2 φ P [( sin2 ɛ s ) cos(2 (ω s ω 3 )t) sin2 ɛ s cos 2 ω s t + 1 ] 4 sin2 ɛ s cos(2 (ω s 2 ω 3 )t) (119) In summary, ζ 2 gives rise to three semi-diurnal tidal constituents, originating from Earth s daily rotation: S 2 K 2 is the principal (main) solar semi-diurnal tide with speed 2 (ω s ω 3 ) = 2 ω 0 = 30 deg/msh and a period of 12 msh. is the semi-diurnal declination tide from the Sun and Moon (the latter contribution is given in Sec. 5.2), often called the lunisolar semi-diurnal constituent. It s peed is 2 ω s = deg/msh and the period is msh. The third constituent is the second semi-diurnal solar declination tide. It s speed is 2 (ω s 2 ω 3 ) = deg/msh and the period is msh. Since ɛ s = const = 23 27, sin 2 ɛ s = 0.16, so the 2 ω s and 2 (ω s 2 ω 3 ) constituents in (119) are small compared to S 2. But since K 2 gets a similar contribution from the Moon, this constituent is larger than the third constituent. One may therefore neglect the third component, but not K Moon s leading tidal constituents Moon s declination varies from to in 18.6 years, but it can be considered constant for one month. Moon s tidal constituents follow then those of the Sun derived above. 32

33 A full rotation of the Moon around the Earth as seen from a distant star gives the Moon s sidereal period T l = msd (see Sec. 4.1). The Moon s angular speed is then 360 deg def = = σ 2 (120) msd msh The relative hour angle follows that of the Sun by substituting σ 2 for σ 3 in (105) σ s σ 2 = deg msh (121) where σ s = deg/msd is the sidereal rotation speed of the Earth. Furthermore, the daily lunar period is 24 h 50 min, and the half-daily lunar period is 12 h 25 min (see 93). Duplicating the results from the Sun, one obtains the values as listed in Table 3. Angular speed Period Tidal constituent Symbol (deg/msh) (msh) σ s 2 σ O 1 Principal diurnal lunar tide σ s K 1 Daily declination tide 2(σ s σ 2 ) M 2 Principal semi-diurnal lunar tide 2(σ s 2 σ 2 ) Semi-diurnal lunar declination tide 2 σ s K 2 Semi-diurnal lunar declination tide Table 3: Overview of the major lunar tidal constituents, excluding variations in the Moon s orbit around the Earth. 5.3 Doodson s system for labelling tidal constituents To facilitate a tabular overview of the (many) tidal constituents, Doodson invented a 6-digit system characterising the speed of the constituents. The speed of the individual contributions σ n can be described by means of 6 basic frequencies σ 1...σ 6, /p. 4.1/ where i a...f are integers. σ n = i a σ 1 + i b σ i f σ 6 (122) The frequencies σ 1...σ 6 in the above expression are as listed in table 2 (and are shown below), with the two alternative expressions for σ 1 from (95) and (96): σ 1 = σ s σ 2 = σ 0 σ 2 + σ 3 σ 2 σ 3 σ 4...σ 6 σ 1 = deg/msh, σ s = deg/msh σ 2 = deg/msh σ 3 = deg/msh Frequencies related to the geometry of the planetary motions (not included here) Table 4 summarises the Doodson coefficients for four of the leading tidal constituents. As an example, the speed of the S 2 tide, 2(σ s σ 3 ), can be expressed as 2 σ σ 2 2 σ σ σ σ 6 = 2 (σ s σ 2 ) + 2 σ 2 2 σ 3 = 2 (σ s σ 3 ) (123) 33

34 yielding the Doodson coefficients i a = 2, i b = 2, i c = 2, i d,e,f = 0, or , where the period mark in the number sequence facilitates reading. To circumvent negative coefficients, it is common practice to add 5 to the coefficients i b...i f, see Table 4. Table 4: Overview of the major solar and lunar tides expressed in terms of the Doodson coefficients. Tide Period Speed Symbol Doodson Non-negative coefficients coefficients (msh) (deg/msh) i a i b i c.i d i e i f (add 5 to i b...i f ) M (σ s σ 2 ) S (σ s σ 3 ) K σ s O σ s 2 σ

35 6 The spring/neap tide In regions where the M 2 and S 2 tides dominate, the combined M 2 + S 2 tide generates a prominent fortnightly signal, known as the spring tide 9 when the tidal amplitude is particularly large and the neap tide 10 when the tidal amplitude is small, see Fig. 12. The spring tide occurs when the three-body Earth-Moon-Sun system is closely aligned along a line (often referred to syzygy), implying new or full Moon seen from the Earth. The neap tide occurs when the Moon and the Sun are approximately at right angles to each other seen from the Earth (at first-quarter and third-quarter Moon). Figure 12: Computed M 2 (top panel) and the combined M 2 + S 2 tides for Bergen, Norway, based on harmonic analysis of observed sea level acquired from Derivation The sea surface elevation Z(t) governed by the combined M 2 + S 2 tide can be expressed as Z(t) = Z 0 + H M2 cos(2 σ 1 t g M2 ) + H S2 cos(2 σ 0 t g S2 ) (124) Here Z 0 is the sea surface elevation in the absence of tides, and H M2, 2 σ 1 and g M2 are the amplitude, frequency and (constant) phase of the M 2 tide. Similarly, H S2, 2 σ 0 and g S2 are the corresponding quantities for the S 2 tide. 9 springflo, springfjære 10 nippflo, nippfjære 35

36 The S 2 argument can be expressed similarly with the leading M 2 argument 2 σ 1 t g M2 : Z(t) = Z 0 + H M2 cos(2 σ 1 t g M2 ) + H S2 cos(2 σ 1 t g M2 θ) (125) Consequently, or 2 σ 0 t g S2 = 2 σ 1 t g M2 θ (126) θ(t) = 2(σ 1 σ 0 )t + g S2 g M2 (127) /p. 75/ As will be shown in expression (129) below, the temporally varying part of θ, the angular frequency 2(σ 1 σ 0 ), gives the period of the fortnightly, or spring/neap, tidal signal. The difference σ 0 σ 1 = deg/msh (values from table 2), yields the spring/neap period T = msd (128) Note that the period of the spring/neap tide is not a basic tidal constituent in itself, but occurs as an interference between the two tidal constituents M 2 and S 2 with closely matching periods and comparable amplitudes. The combined M 2 +S 2 tide is prominent in many places, and thus an important part of the total tidal variation. Expansion of the second cosine-term in (125) by means of (463) gives Z(t) = Z 0 + H M2 cos(2 σ 1 t g M2 ) +H S2 {cos(2 σ 1 t g M2 ) cos θ + sin(2 σ 1 t g M2 ) sin θ} = Z 0 + (H M2 + H S2 cos θ) cos(2 σ 1 t g M2 ) + H S2 sin θ sin(2 σ 1 t g M2 ) (129) The above expression consists of slow amplitude modulations given by the factors H M2 +H S2 cos θ and H S2 sin θ (both with the period msd, originating from the speed difference σ 0 σ 1 ), and semi-diurnal variations given by the argument 2 σ 1 t g M2. The slow amplitude modulation can conveniently be expressed by means of a time dependent amplitude α and argument β : H M2 + H S2 cos θ = α cos β (130) H S2 sin θ = α sin β (131) β can be determined upon dividing (131) by (130), yielding ( ) tan β H S2 sin θ = H M2 + H S2 cos θ, so H β S2 sin θ = arctan H M2 + H S2 cos θ (132) Likewise, α can be determined by squaring and adding (130) and (131): α = (H 2 M 2 + H 2 S 2 + 2H M2 H S2 cos θ) 1 2 (133) Z(t) can now be cast in the compact form (by means of the identity 463): /4.8/ Z(t) = Z 0 + α cos(2 σ 1 t g M2 β ) (134) 36

37 6.1.1 Amplitude and range From expression (133), the largest amplitude of the M 2 + S 2 tide occurs when cos θ = 1 : α max = H M2 + H S2 (135) In this case, the (spring) tidal range varies between Z 0 (H M2 + H S2 ) and Z 0 + (H M2 + H S2 ). Similarly, the smallest amplitude occurs when cos θ = 1 : α min = H M2 H S2 (136) Thus, the neap tidal range varies between Z 0 (H M2 H S2 ) and Z 0 + (H M2 H S2 ). With H M2 2.2 H S2 (see Sec. 2.1), this implies that the spring and neap tides have amplitudes α max 3.2 H S2 and α min 1.2 H S 2, respectively, with a ratio α max α min 2.5 (137) Timing of maximum amplitude The largest amplitude α max of the spring tide occurs when θ = 0 or any 2π multiples of θ (see 133). From expression (127), maximum amplitude in the case of θ = 0 occurs at /4.9/ t max = g S 2 g M2 2(σ 0 σ 1 ) (138) Age of the tide The arguments of the M 2 tide from (124) and the combined M 2 + S 2 tide from (134) are, respectively, 2 σ 1 t g M2 and 2 σ 1 t g M2 β (139) A time-varying time lag = (t) between the M 2 and M 2 + S 2 tides can thus be determined from the expression 2σ 1 (t + ) g M2 = 2 σ 1 t g M2 β (140) resulting in (t) = β (t) 2 σ 1 (141) The time lag between the two tides is largest when β reaches its maximum or minimum value, i.e., when dβ dt = 0 (142) with β defined by (132). With the substitution x(t) = H S2 sin θ(t) H M2 + H S2 cos θ(t) 37 (143)

38 the use of the derivative of arctangent (expression 486) and by applying the chain rule, we obtain dβ dt = d 1 dx (arctan x) = 2x dt 1 + x2 dt = 0 (144) or x dx dt = 0 (145) In the above expression, x = 0 is the trivial solution, satisfied for θ = 0 (and any 2π multiples of θ). Maximising β, for a general θ, is therefore given by dx/dt = 0. Vanishing derivative of (143) with respect to t leads to the relationship cos θ = H S 2 H M2 (146) From the equilibrium theory, see Sec. 2.1, H M2 = m and H S2 = m. Thus, cos θ = 0.46, or θ = ±2.05 rad (147) H M2, and H S2 and θ inserted into (132) gives tan β ±0.52 or β ±48 rad (or ± 27 deg) (148) Largest time lag is obtained for β = ±27 deg, yielding maximum time lag from (141) of /p. 76/ max = 27 deg = 0.94 msh = 57 min (149) deg/msh Thus, the combined M 2 + S 2 tide lags/leads the M 2 tide with slightly less than 1 hour at most. This time difference the maximum lag/lead between the new and the full Moon and the maximum spring tidal range is called the age of the tide 11. The time lag between the M 2 and M 2 + S 2 tides, derived from expression (141), is plotted in Fig. 13. During the lunar synodic period T synodic = msd (Sec. 4.1), the time lag maxima occurs at the fraction ±2.05 rad/(2 π) of the period T synodic, or at 2.05 rad 2 π T synodic 9.6 msd and 2 π 2.05 rad 2 π T synodic 19.9 msd (150) Based on observed tidal variations, the age of the tide can be much larger than approximately 1 hr, in many places several days. The difference between the theory presented here and the observed variations can be caused by the influence of the bathymetry, the coastal geometry, and the friction on the tidal waves propagating into and leaving a given location A note For any two waves with closely matching speed and amplitude, the beating frequency equals the speed difference between the two waves. This follows from the derivation of the group speed as outlined in appendix I.5, where the sum of two comparable waves (see expression 559) leads to a modified wave with a slowly changing amplitude and speed (expression 566). 11 tidevannets alder 38

39 Figure 13: Illustration of the age of the tide, from expression (141), through a full lunar synodic period of msd. Maximum time lag of slightly less than 1 hr, called the age of the tide, occurs about 9.6 and 19.9 msd after full Moon. 39

40 7 Tidal forcing in the momentum equation The full 3-dimensional tidal flow, taking into account bathymetry and coastlines, can be described by the primitive equations. The 3-dimensional momentum equation can be expressed in the form (M & P (2008), Chap. 6) Du Dt + 1 p + fẑ u = g ẑ + F (151) ρ In (151), u is the 3-dimensional velocity field, Du/Dt is the total derivative of u, p is pressure, ρ is density, f is the Coriolis parameter, ẑ is the radial, outward-directed unit vector on a sphere, and F is friction. The tidal forcing from the Moon and the Sun at the surface of the Earth can be added to (151) by introducing the pressure force (41) caused by the tidal elevation ζ (55): where ζ = 3 2 Du Dt + 1 p + fẑ u = g ẑ g ζ + F (152) ρ r 4 [ ml m e Rl 3 ( cos 2 φ l 1 ) + m ( s 3 Rs 3 cos 2 φ s 1 )] 3 (153) In the expression for ζ, cos 2 φ 1/3 is written in terms of the latitude φ P and the hour angle C P of a position P on Earth s surface, as well as the declination angle d of the celestial body (see Sec. 3 and expression 65): cos 2 φ 1 3 = cos2 φ P cos 2 d cos 2 C P sin(2 φ P ) sin(2 d) cos C P + sin 2 φ P sin 2 d 1 3 (154) The hour ( longitude ) angle C P needs to be referenced to a fixed geographic point. A common reference point is the point of the vernal equinox 12 (also known as the first point of Aries), see Fig. 11. The angles φ P, C s and δ can be computed in Fortran, C and Python, for any time, from e.g. the Naval Observatory Vector Astrometry Software (NOVAS, see Kaplan et al., 2011, The momentum equation (152), with expressions (153) and (154), together with the continuity equation, models the full 3-dimensional dynamics of the tide. The set of equations, incorporating realistic bathymetry, can only be solved numerically. 12 vårjevndøgn 40

41 8 Laplace s Tidal Equations When tidal forcing is introduced to the (quasi-)linearised version of the shallow water equations, the obtained equations are known as Laplace s Tidal Equations (LTE). Tidal flow is then described as the flow of a barotropic fluid, forced by the tidal pull from the Moon and the Sun. The phrase shallow water equations reflects that the wavelength of the resulting motion is large compared to the thickness of the fluid. The horizontal components of the momentum equation and the continuity equation can then be expressed as u fv t = g (ζ + ζ) x (155) v t + fu = g (ζ + ζ) y (156) ζ t + x (uh) + (vh) y = 0 (157) In the above equations, g ζ is the tidal forcing with ζ obtained from (153) and (154), ζ is the surface elevation, and h is the ocean depth. The horizontal momentum equations are linear, but inclusion of a friction term will typically turn the equations non-linear. Likewise, the divergence terms in the continuity equation are nonlinear because of the product uh and vh. Solution of LTE requires discretisation and subsequent numerical solution. 41

42 9 The tidal force expressed in terms of the tidal potential The direct method described in Sec. 1 can be conveniently carried out by introducing the tidal potential as described below. 9.1 The potential of a conservative force The gravitational force is an example of a conservative force, i.e., a force with the property that the work done in moving a particle from position a to position b is independent of the path taken (think of changes in the potential energy when an object in Earth s gravitational field is moved from one position to a new position; the path taken is irrelevant, only the height difference between the start and end positions matter). It then follows that the gravitational force F can be expressed as the negative gradient of a potential (a scalar) Ω F = Ω (158) Once Ω is found, the resulting force can be readily obtained by taking the gradient of Ω. This simplification is a main motivation of introducing the potential, and it is commonly used in problems involving gravitation and electromagnetism. 9.2 Defining the tidal potential From (16), we have that the tidal acceleration can be expressed as a = G m l ( q q 3 R R 3 ) (159) Let F t denote the tidal force per unit mass (or acceleration). It follows then from the above expression that ( q F t = G m l q 3 R ) R 3 (160) The task is therefore to determine the scalar function Ω, the tidal potential, satisfying F t = Ω (161) Gradient of the factor q/q 3 We start by searching for a scalar with the property that the gradient of the scalar equals q/q 3. It follows from Fig. (1) that q = R r (162) In the above expression, R is a constant (fixed) vector whereas r varies spatially as it can oriented at any location on Earth s surface. In a Cartesian coordinate system with time-invariant unit vectors e x, e y and e z, we may express r = x e x + y e y + z e z (163) 42

43 and Therefore, R = R x e x + R y e y + R z e z (164) q = (R x x) e x + (R y y) e y + (R z z) e z (165) The magnitude of q becomes so q = q = R r = [ (R x x) 2 + (R y y) 2 + (R z z) 2] 1/2 q 3 = [ (R x x) 2 + (R y y) 2 + (R z z) 2] 3/2 (166) (167) Furthermore, In the above expression x ( ) 1 q ( ) 1 = q x ( ) 1 e x + q y ( ) 1 e y + q z ( ) = 1 x [(R x x) 2 + (R y y) 2 + (R z z) 2 ] 1/2 = 1 2 ( ) 1 e z (168) q 2(R x x) [(R x x) 2 + (R y y) 2 + (R z z) 2 ] 3/2 (169) Similar results are obtained for the y- and z-derivatives in (168). Consequently, ( ) 1 = (R x x) e x + (R y y) e y + (R z z) e z q q 3 = q q 3 (170) The q/q 3 -term in (160) can therefore be expressed as (1/q). This is a classical result from both electrodynamics and analysis involving gravity Gradient of the factor R/R 3 The second term on the right-hand side of (160) can also be turned into the gradient of a scalar. For this we use that r R = x R x + y R y + z R z (171) Consequently, (r R) = x (r R) e x + y (r R) e y + z (r R) e x = R x e x + R y e y + R z e z = R (172) The R/R 3 -term in (160) is therefore equivalent to (r R). 43

44 9.2.3 Resulting tidal potential By inserting (170) and (172) into (160), we obtain ( F t = Gm l 1 q + r R ) R 3 + C where C is a constant. (173) C can be determined by noting that F t vanishes at Earth s centre, see the discussion in Sec When r = 0, q equals R, and we get from (173) that C = 1/R. Therefore ( F t = Gm l 1 q + r R R ) = Ω (174) R where the tidal potential Ω was introduced in (161). The tidal potential can therefore be put in the form ( Ω = Gm l 1 q + r R R ) (175) R 9.3 Tidal potential expressed in terms of the zenith angle Next step is to express (175) by means of the zenith angle φ (see Fig. 1). For the 1/q-term the procedure follows that of Sec. 1.7, starting with the law of cosines (17), [ q 2 = R 2 + r 2 2 R r cos φ = R r ( r )] R R 2 cos φ (176) Consequently, [ q = R 1 + r ( r )] 1/2 R R 2 cos φ (177) The factor r/r has been introduced in the above expressions since it is a small number (appendix A), allowing for series expansion. The quantity 1/q can then be expended by applying the binomial theorem (460). We neglect terms proportional to and smaller than (r/r) 3, giving 1 q = 1 [ 1 + r ( r )] 1/2 R R R 2 cos φ = 1 [ 1 1 r ( r ) R 2 R R 2 cos φ + ( 1 2 )( 3 2 ) ( r ) 2 ( r ) ] 2 2 R R 2 cos φ [ 1 1 ( r ) 2 r + R 2 R R cos φ + 3 ( r ) 2 cos φ] 2 2 R = 1 [ 1 + r R R cos φ + 1 ( r ) 2 (3 cos 2 φ 1 )] (178) 2 R For the r R-term, the scalar product gives r R = r R cos φ so r R R 3 = r cos φ (179) R2 44

45 By combining (175), (178) and (179), one obtains the tidal potential expressed in terms of the zenith angle φ, Ω = 3 2 Gm r 2 ( l R 3 cos 2 φ 1 ) (180) 3 /3.8/ Radial and tangential tidal forces per unit mass Once the tidal potential Ω is determined, the radial and tangential tidal forces per unit mass can be determined from (161). For the gradient operator, we use the spherical (zenith) coordinate system shown in the right panel of Fig. 25. In this system, the gradient operator is given by (420). The radial component F t,r of the tidal force per unit mass is then F t,r = Ω r = Gm r ( l 3 cos 2 R 3 φ 1 ) (181) or, by introducing the gravitational acceleration g from (11), /3.9/ F t,r = g m ( l r ) 3 (3 cos 2 φ 1 ) (182) m e R F t,r is directed outward, i.e., in the e r -direction. Similarly, the tangential (horizontal) component F t,h of the tidal force per unit mass is F t,h = 1 Ω r r = 3Gm r l cos φ sin φ (183) R3 or, with sin 2φ = 2 sin φ cos φ and by introducing the gravitational acceleration g from (11), F t,h = 3 2 g m ( l r ) 3 sin 2φ (184) m e R Below /3.9/ F t,h is oriented in the e φ direction, i.e., towards the centre line connecting the centres of the Earth and the Moon (see Fig. 1). F t,r and F t,h are identical to the tidal acceleration in (32) and (30), demonstrating the consistency of the direct method presented in Sec. 1 and the method of tidal potential of this section. 45

46 10 Introducing the tidal potential in the primitive momentum equations The tidal forcing of the ocean can be introduced by adding the tidal forcing per unit mass (160) to the 3-dimensional momentum equation (151). The tidal forcing on the Earth s surface Ω t caused by the Moon (Ω l ) and the Sun (Ω s ) is Ω t = Ω l + Ω s (185) Ω l and Ω s are both on the from of (180), so Ω t = 3 [ ( 2 g r4 ml m e Rl 3 cos 2 φ l 1 ) + m ( s 3 Rs 3 cos 2 φ s 1 )] 3 (186) where G = g r 2 /m e from (11) has been used. The resulting momentum equation becomes Du Dt + 1 ρ p + fẑ u = g ẑ (Ω t) + F (187) Alternatively, the gravity term can be expressed as a gradient of a potential where Ω g = g z. It is therefore common to put (187) in the form g ẑ = Ω g (188) Du Dt + 1 ρ p + fẑ u = (Ω g + Ω t ) + F (189) Note that Ω t is a function of position and time whereas Ω g can be considered time-independent. Equation (187) or (189), together with the continuity equation, is used to simulate the full 3- dimensional flow of the ocean tide. The linearised shallow water equivalent of the above equations is given by (155) and (156). 46

47 11 Including the effect of elliptic orbits For the theory presented, the distances between the Sun, the Earth and the Moon have been kept constant, implying circular orbits. This assumption is correct to lowest order, but smaller but still non-negligible contributions to the tide-generating forces emerge as a result of the elliptic orbit of the Earth around the Sun, and the elliptic orbit of the Moon around the Earth. Kepler s laws are key to both understand and to quantify these contributions. We first consider the Sun-Earth system. J. Kepler (in reference to his own work): If anyone thinks that the obscurity of this presentation arises from the perplexity of my mind,... I urge any such person to read the Conics of Apollonius. He will see that there are some matters which no mind, however gifted, can present in such a way as to be understood in a cursory reading. There is need of meditation, and a close thinking through of what is said The Sun-Earth system 13 The orbit of the Earth about the Sun is an ellipse according to Kepler s first law. The configuration is shown in Fig. 14. Newton s law of universal gravitation states that there is a gravitational force F e on the Earth from the Sun given by (see Section 1.4) F e = GM sm e R 2 s R s R s (190) Newton s second law relates F e (acting on the Earth with mass m e ) and acceleration Combined, expressions (190) and (191) give F e = m e d 2 R s dt 2 (191) d 2 R s dt 2 = GM s R 3 s R s (192) Expressed in polar coordinates (with the position of the Sun as origin), gives the following unit vectors in the directions given by the vector R s (denoted e R ) and in the direction of the angle K (hereafter e K ; with the property e K e s ) e R = (cos K, sin K) (193) e K = ( sin K, cos K) (194) Consequently, R s = R s e R (195) 13 Mainly based on R. Fitzpatrick (): An Introduction to Celestial Mechanics, M. Hendershott (): Lecture 1: Introduction to ocean tides, and Murray & Dermott (1999), Solar Syatem Dynamics, Chap

48 y m e R s Perihelion a ae m s K p x Υ Figure 14: Illustration of the Sun-Earth system. The Earth of mass m e orbits the Sun with mass m s (and assumed to be fixed in space). R s is the radius vector from the Sun to the Earth; K is the true anomaly; a is the semi-major axis; e is the eccentricity; and p is the longitude of the perihelion, or the angle between (the point of reference for the Earth-Sun system) and the positive x -axis. At present, p

49 Noting that both e R and e K change in time and denoting the individual time derivatives with a dot, we get ė R = K ( sin K, cos K) = K e K (196) ė K = K (cos K, sin K) = K e R (197) Consequently, v = dr s dt = Ṙs e R + R s ė R (198) = Ṙs e R + R s K e K (199) Furthermore, a = dv dt = d2 R s dt 2 = R s e R + Ṙs ė R + (Ṙs K + R s K) ek + R s K ė K (200) or, by means of (196) and (197), a = ( R s R s K 2 ) e R + (R s K + 2 Ṙ s K) e K (201) The equation of motion of the Earth, under the gravitational influence of the Sun (expression 190), is therefore given by a = ( R s R s K 2 ) e R + (R s K + 2 Ṙ s K) e K = GM s R 2 s e R (202) From the right-most equality, the radial component of the equation of motion becomes R s R s K 2 = GM s R 2 s (203) and the tangential component is R s K + 2 Ṙ s K = 0 (204) 11.2 Kepler s second law The tangential component (204) is equivalent to implying that the quantity is independent of time. d(r 2 s K) dt = 0 (205) R 2 s K h (206) The above quantity has both a geometrical and physical interpretation. The geometrical interpretation follows from Fig. 15. Per definition, the length l of the outlined (red coloured) arc is given by l = KR s. For small K, or for small changes in K, the area A of the outlined sector is given by the half the base ( R s ) times its height ( l ), i.e., A = 1 2 KR2 s (207) 49

50 Figure 15: Illustration of the Sun-Earth system. The Earth of mass m e orbits the Sun with mass m s (and assumed to be fixed in space). R s is the vector from the Sun to the Earth; K is the true anomaly; a is the semi-major axis; and e is the eccentricity. Temporal changes of A are thus given by da dt = 1 d(krs) 2 = 1 dh 2 dt 2 dt = 0 (208) Therefore, A is constant in time. Thus, Kepler s second law) states that the radius vector from /Fig. 3.9/ the Sun to the Earth (or to any planet) sweeps out equal areas in equal times Kepler s first law The radial component (203) is non-linear in R s, but it can be turned into a linear equation by introducing u = 1 R s or R s = 1 u (209) Temporal derivatives of R s can now be expressed in terms of u = u(k(t)) by repeated use of the chain rule R s = 1 u 2 u = 1 du dk du u 2 = h (210) dk dt dk where h is given by (206). In a similar manner, Inserting into (203) leads to or R s = h d2 u dk 2 dk dt = u2 (h ) 2 d2 u dk 2 (211) (h ) 2 u 2 d2 u dk 2 (h ) 2 u 3 = u 2 GM s (212) d 2 u dk 2 + u = GM s (h ) 2 (213) The homogeneous solution to the above equation is proportional to cos K. Including the constant right-hand side of (213) to the solution gives, with u = 1/R s : 1 R s = A cos K + GM s (h ) 2 (214) where A is a constant of integration. A second integration constant is embedded in the argument of the cosine-function, so that the argument K can be adjusted by a constant phase shift according to the chosen coordinate system. 50

51 Expression (214) is, as expected, the parametric equation of an ellipse, see appendix D. By direct comparison with (457), equation (214) can be written in terms of the ellipse s semi-major axis a and the eccentricity e : A e = a(1 e 2 (215) ) and GM s (h ) 2 = 1 a(1 e 2 ) (216) From expression (214), Kepler s first law states that planets move in ellipses with the Sun at one focus Kepler s third law During one orbital period T K, the area A swept out by R s is From expression (208), integrating over one period, Therefore, A = πab (217) A = h T K 2 (218) T 2 K = 4π 2 a2 b 2 (h ) 2 = 4π2 a4 (1 e 2 ) (h ) 2 (219) where use has been made of expression (434) in the last equality. Furthermore, from (216), (h ) 2 = a(1 e 2 ) GM s (220) By combining (219) and (220), we obtain) /3.14/ a T K = 2π 3 (221) GM s The corresponding angle frequency ω K = 2π/T K becomes ω K = GMs a 3 (222) Note that T K, and therefore ω K, is only dependent of the elliptic parameter a and the environmental parameters G and M s. Kepler s third law, from expression (221), states that the square of the orbital period of a planet is proportional to the cube of its semi-major axis. 51

52 11.5 Mean anomaly E 0 Although the true anomaly K does not change linearly in time for e 0, it is convenient to consider an angle that is 2 π -periodic and that changes linearly in time. This parameter named the mean anomaly and denoted E 0 is defined by with ω K coming from expression (222). E 0 = ω K t (223) From The mean anomaly does not measure an angle between physical objects. It is simply a convenient, uniform measure of how far around its orbit a body has progressed since perihelion. The mean anomaly is one of three angular parameters (known historically as anomalies ) that define a position along an orbit, the other two being the eccentric anomaly and the true anomaly Real vs mean length of the radius vector, and Kepler s equation It is mathematically convenient to introduce the eccentric anomaly E (see Fig. 16), in stead of the true anomaly K, in expression (214). The task is therefore to relate the elliptic radius R s with that of the reference circle R s, for brevity a, with respect to E. We start with deriving an expression relating R s and a, followed by Kepler s equation giving an equation for E R s vs R s Geometric interpretation of the left panel in Fig. 16 gives F A = 0A 0F = a cos E ae = a(cos E e) (224) and F A = R s cos K (225) The combination of the above expressions gives or, alternatively, R s = cos K = a(cos E e) cos K (226) a(cos E e) R s (227) The equation for an ellipse in polar coordinates is given by expression (456) or R s = a(1 e2 ) 1 + e cos K (228) R s (1 + e cos K) = a(1 e 2 ) (229) 52

53 Figure 16: Illustration of the Sun-Earth system. The Earth of mass m e orbits the Sun with mass m s (and assumed to be fixed in space). R s is the vector from the Sun to the Earth; K is the true anomaly; a is the semi-major axis; and e is the eccentricity. Insertion of (227) yields or, after solving for R s, R s ( 1 + e ) a(cos E e) = a(1 e 2 ) (230) R s R s = a(1 e cos E) (231) In stead of the circle radius a, we may use R s the mean and constant radius of the principal circle leading to R s = R s (1 e cos E) (232) The non-circularity of the elliptic orbit is thus given by R s 1 = R s 1 e cos E (233) For small values of the eccentricity e, the right-hand side of (233) can be series expanded by means of the binominal theorem, see appendix (E). We obtain, to lowest order R s R s = 1 + e cos E (234) Next, we need to obtain an equation for E following the orbiting motion of the body M in Fig Kepler s equation The area swept out by the orbiting body M can be interpreted from the right panel of Fig

54 Firstly, the area A E defined by the eccentric anomaly and the reference circle is given by A E = π a 2 E 2π = E a2 2 where E is in radians. Any y -value on an ellipse equals the constant factor (235) b/a = 1 e 2 (236) of the corresponding y -value on the reference circle (see appendix D.3). The area of the tricoloured region A col in Fig. 16 therefore equals A col = A E 1 e2 = E a2 1 e 2 2 (237) The sector area swept out by K is given by area A col minus the pink triangle in Fig. 16, A pink. The latter equals half of the baseline c = ae times the height a sin E 1 e 2 : A pink = a 2 e sin E 1 e 2 2 (238) The sought-after sector area A K swept out by K is therefore given by 1 e A K = A col A pink = a 2 2 (E e sin E) (239) 2 The area A K is proportional to time t, the latter set to zero at perihelion 14, i.e., the point of least distance between the Sun and the Earth or, which is equivalent, when M crosses the positive x -axis in Fig. 16. Therefore, A K = C t (240) where C is a constant to be determined. Let T K and ω K denote the period and frequency of the elliptic motion, respectively. When E = 2π, A K equals the area of the ellipse, A K = πab = πa 2 1 e 2 (241) where the relationship (435) has been used in the last equality. Consequently, C T K = πa 2 1 e 2 (242) so C = πa 2 1 e 2 T K (243) By combining expressions (239), (240) and (243), we get Kepler s equation E e sin E = ω K t (244) yielding a relationship between the eccentric anomaly E and the time t since perihelion. Introducing the mean anomaly from expression (223) gives E 0 = E e sin E (245) 14 The corresponding point for the Earth-Moon system is called perigee. 54

55 Solving Kepler s equation Kepler s equation (245) is transcendental and cannot be solved directly. But the equation can be solved by iteration. As an example, equation (245) can be casted in the form E i+1 = E 0 + e sin E i, i = 0, 1,... (246) Since the magnitude of the E i derivative of the right-hand side is less than unity, max e cos E i = e < 1, the given iterative scheme converge (the Fixed Point Theorem). For the first approximations, this give E 1 = E 0 + e sin E 0 (247) E 2 = E 0 + e sin E 1 = E 0 + e sin(e 0 + e sin E 0 ) (248) The last term on the right-hand side of (248) can be expanded by means of the sum and difference identity (462), leading to E 2 = E 0 + e [sin E 0 cos(e sin E 0 ) + cos E 0 sin(e sin E 0 )] (249) For small arguments x, the following series expansions are valid sin x = x + O(x 3 ) (250) cos x = 1 x2 2 + O(x4 ) (251) Thus, E 2 E 0 + e [sin E 0 + e sin E 0 cos E 0 ] = E 0 + e sin E 0 + O(e 2 ) (252) For an example of how to determine E, see e.g R s /R s expressed by means of E 0 Similar to the procedure outlined in Section , the ratio R s /R s, from (234), can be expanded as follows: R s R s = 1 + e sin E (253) 1 + e[cos(e 0 + e sin E 0 )] (254) = 1 + e[cos E 0 cos(e sin E 0 ) sin E 0 sin(e sin E 0 )] (255) = 1 + e cos E 0 + O(e 2 ) (256) Thus, to first order in e, R s R s = 1 + e cos E 0 (257) 55

56 Introducing the ecliptic longitude and the longitude of perihelion The ratio between the length of the actual and mean radius vector in equation (234) can be expressed in terms of the Sun s geocentric mean ecliptic longitude h, increasing by per mean solar hour, and the longitude of the solar perigee (called perihelion) p0. From Fig. 17, the celestial longitude λs, is λs = p0 + K (258) Figure 17: Left panel is as Fig. 14, but including the principal circle; the eccentric anomaly E (the projection of the Earth, me, onto the principal circle in the y -direction); and the celestial longitude (or true longitude) λs, running from to the actual position of the Earth. In the right panel, an imaginary body (green dot) describes a circular orbit with constant rotation rate ωk as defined on page 54. By defining the mean ecliptic longitude relative to according to h = p0 + E0 (259) E0 = h p0 (260) it follows that Thus, from (257), /3.15/ Rs = 1 + e cos(h p0 ) Rs (261) Furthermore, by combining (276) and (259), we have the relationship λs = h + K E0 (262) In the above expression and elsewhere, the true anomaly K can be written in terms of E, and subsequently E0, by combining the two equivalent expressions (228) and (233): e cos K = 1 e2 1 e cos E 56 (263)

57 The smallness of e leads to, by means of the binominal theorem (460) applied to (263), a series expansion in e : 1 + e cos K = 1 e2 1 e cos E = (1 e2 )(1 + e cos E + e 2 cos 2 E + O(e 3 )) (264) Consequently, to first order in e, = 1 + e cos E + e 2 cos 2 E e 2 + O(e 3 ) (265) cos K = cos E + e (cos 2 E 1) = cos E + e sin 2 E (266) Upon substituting expression (247) for E and following the procedure outlined in Section , we get cos K = cos(e 0 + e sin E 0 ) e sin 2 (E 0 + e sin E 0 ) (267) = cos E 0 cos(e sin E 0 ) sin E 0 sin(e sin E 0 ) (268) e [sin E 0 cos(e sin E 0 ) + cos E 0 sin(e sin E 0 )] 2 (269) = cos E 0 e sin 2 E 0 e [sin E 0 + e sin E 0 cos E 0 ] 2 + O(e 2 ) (270) = cos E 0 2e sin 2 E 0 + O(e 2 ) (271) Similarly, by applying the procedure outlined in Section , cos(e 0 + 2e sin E 0 ) = cos E 0 cos(2e sin E 0 ) sin E 0 sin(2e sin E 0 ) (272) = cos E 0 sin E 0 2e sin E 0 + O(e 2 ) (273) = cos E 0 2e sin 2 E 0 + O(e 2 ) (274) Since the two expressions (267) and (272) are identical to first order in e, K = E 0 + 2e sin E 0 (275) Thus, fom expression (262), or, by using (260), λ s = h + 2e sin E 0 (276) λ s = h + 2e sin(h p ) (277) /3.16/ 11.7 Right Ascension The cotangent rule (485) can be applied to the spherical triangle SS in Fig. 18. This gives cos A s cos ɛ s = sin A s cot λ s (278) where we have used that the angle at S is a right-angle. The above equation is equivalent to sin A s cos λ s = sin λ s cos A s cos ɛ s (279) For cos ɛ s, the half-angle identity (476) can be used: cos ɛ s = 1 tan2 (ɛ s /2) 1 + tan 2 (ɛ s /2) (280) 57

58 P z O λ s S δ s Υ ε s A s S Figure 18: The celestial sphere with the Vernal Equinox ( ), the Sun ( S ), the equatorial plane O S, the ecliptic O S, the celestial north pole P, and the meridian P SS. In the spherical triangle SS, the arc S is the celestial longitude λ s, the arc S is the Right Ascension A s, and the arc SS is the declination δ s. The angle at is the obliquity ɛ s and the angle at S is a right-angle. Caption and drawing is based on Fig. 7.7 in Capderou (2004). 58

59 Thus, Rearranging gives ( sin A s cos λ s 1 + tan 2 ɛ ) ( s = sin λ s cos A s 1 tan 2 ɛ ) s 2 2 (281) sin A s cos λ s cos A s sin λ s = (sin A s cos λ s + sin λ s cos A s ) tan 2 ɛ s 2 (282) or, by means of the sum and difference identity (462), ( sin(a s λ s ) = tan 2 ɛ ) s sin(a s + λ s ) (283) 2 Since ɛ s = 0.41 is relatively small, A s λ s. Series expansion to lowest order gives, from expression (482): ( A s = λ s tan 2 ɛ ) s sin(2λ s ) (284) 2 /3.17/ 11.8 Equation of time The time given on a sundial does approximately, but not exactly, match the time on the clock. The two time frames would be identical if Earth s orbit was circular and if Earth s equatorial plane was identical to the plane swept out by Earth s orbit. In this case Earth s motion could be described with a constant, circular rotation rate. Actually, the clock time is defined in this way: That any given time interval represents the same (circular, ecliptic) motion of the Earth around the Sun. From a geocentric perspective, the latter represents the motion of the mean Sun, where mean represents averaging over a year, i.e., averaging over one full orbit. The motion of the true Sun takes into account the elliptic path (Fig. 14), leading to alternating phases of acceleration and deceleration during the orbit (Section 11.2). Furthermore, the tilt between the celestial longitude λ s and the celestial equator A s (Fig. 18), implies that different distances are travelled during the same time interval. The difference between the Right Ascension of the mean Sun (clock time) and the Right Ascension of the true Sun (sundial time), at any time t, is known as the equation of time Equation of center The effect of the Earth s acceleration/deceleration along it s elliptical orbit relative to a circular orbit with constant rotation rate is called the equation of center, E C. Based on this, E C is the difference between the true anomaly K and the mean anomaly E 0, see Fig. 17: Thus, by means of (275) and to the first order in e, E C = K E 0 (285) E C = 2e sin E 0 (286) E C = ω K t has, per construction, a period of one year, with t = 0 at perihelion. E C is positive for the first half of the year, implying that the actual position of Earth is ahead of an imaginary body moving at constant speed along the reference circle. E C has an amplitude of E C,amp = 2e = rad (287) 59

60 Longitudinal differences The difference in length E R between the Right Ascension A s and the celestial longitude λ s, see Fig. 18, is E R = A s λ s = tan 2 ɛ s 2 sin(2λ s) (288) where use has been made of expression (284) in the last equality. Realtive to perihelion, the argument For small eccentricity, from (276), where (259) has been used in the last equality. Thus, λ s h = E 0 + p (289) E R = tan 2 ɛ s 2 sin 2(E 0 + p ) (290) Equation of time, final expression Based on the above, the expression for the equation of time E T, can be put in the form E T = E C + E R = 2e sin E 0 tan 2 ɛ s 2 sin 2(E 0 + p ) (291) with E 0 = ω K t from (223) and p = 77 = rad Equation of time, graphical representation If we let t denote the day D of the year and let t = 0 correspond to January 1, the passage at perigee on January 3 occurs at D = 3. With these choices, and using that a full year has 365 days (for simplicity), we get [ ] 2π E C = 2e sin (D 3) (292) 365 E C is in radians, but we want to express it in suitable time units. This can be done by transferring the amplitude 2 e from radians to days, and thereafter to minutes, by using the relationship 2 π corresponds to 1 D = 1440 min (293) With e = , we get E C [min] = [ ] [ ] 2π 2π 2 π sin (D 3) = 7.44 sin (D 3) where [min] denotes that E C is given in minutes. Similarly, the argument 2 E 0 in E R becomes (294) 2 E 0 = 2 2π (D 3) (295) 365 The second argument in E R, p, is a constant phase relative to perigee, see Fig. 17. p = 77 or, in radians, p = 77 2 π (296)

61 Thus, E R = tan 2 ɛ [ ( s D 3 2 sin 4 π )] 360 With ɛ s = = 23.45, we obtain [ ( D 3 E R [min] = 9.87 sin 4 π )] 360 [ ] 4 π = 9.87 sin (D 81.07) 365 (297) (298) In summary, [ ] [ ] 2π 4 π E T [min] = 7.44 sin (D 3) 9.87 sin (D 81.07) (299) with E C in minutes and D in days with D = 1, 2,..., 365, with D = 1 for January 1. The resulting variation throughout the year is shown in Fig. 19. /Fig. 3.10/ Figure 19: The equation of time E T (black line), with it s two contributions E C (red) and E R (blue). Positive values means that the time according to a sundial is ahead of the mean time (i.e., ahead of the clock time). Filled circles show the extrema of E T, with maximum value at 14 Feb with E T = min and minimum value at 1 Nov with E T = min. Open circles show the four zero-crossings at 16 Apr, 14 Jun, 1 Sep, and 25 Dec. The stated dates are approximate; they may vary by one to two days depending on the actual year. On longer time scales, e and ɛ s vary slowly, implying changes in E T. 61

62 Figure 20: A sequence showing the true anomaly (blue line/dot), the true anomaly vertically projected onto the reference circle (red dot and red circle, respectively), and the mean anomaly (green line/dot) on the reference circle, during a full period T. The panels run from the upper, left panel and down, column-by-column, with increments of T /7. Thus, the upper, left panel is at time t = 0, the middle, left panel is at t = T /7, etc. The eccentricity is set to 0.4, so the mismatch between the true and mean anomaly is greatly exaggerated compared to the Earth-Sun system. An animation of the above sequence is available from 62

63 Part II Tidal related wave dynamics 12 The shallow water equations A wave is a physical process that transport information such as surface elevation or energy in time and space, without or with little advection of mass associated with the transport. This is in contrast to the geostrophic balance and ageostrophic flow in the atmosphere and ocean that is always associated with advection of mass. Accordingly, a wave signal is propagated without or with little influence of the Earth s rotation, although the propagation velocity can be large. In contrast, advection of mass will always be influenced by the Coriolis effect, linearly increasing with the speed of the fluid. A general overview of definitions and key properties of waves is given in the appendix Starting point and configuration The basis for the shallow water equations is the standard form of the horizontal momentum equation u t + u u x + v u y fv = 1 p ρ x + F x (300) and the continuity equation with u = (u, v). v t + u v x + v v y + fu = 1 ρ p y + F y (301) ρ + (ρ u) = 0 (302) t We consider a homogeneous (ρ = ρ 0 ), frictionless (F H = 0) and barotropic ( u/ z = v/ z = 0) fluid with free surface ζ(x, y, t) as illustrated in Fig. 21. The equations (300) (302) can then be expressed as u t + u u x + v u y fv = 1 p (303) ρ 0 x v t + u v x + v v y + fu = 1 p ρ 0 y (304) u x + v y + w z = 0 (305) 12.2 The continuity equation In the che continuity equation (305), u and v are independent of z (due to the assumption a barotropic fluid), but u/ x 0 and v/ y 0, so that we can have divergent flow. 63

64 Figure 21: Illustration of a homogeneous væ fluid with free surface ζ(x, y, t) and general bathymetry b(x, y). h(x, y, t) describes the total depth of the fluid. z ref = D is a reference level which describes the mean surface height. We consider the homogeneous fluid shown in Fig. 21 and integrates (305) from the bottom z = b(x, y) to the free surface z = b(x, y) + h(x, y, t). This gives ( u x + v ) b+h dz + w b+h b = 0 (306) y b In the above expression, w(z = b + h) is the motion of the free surface. This can be expressed using the total derivative (describing how the surface changes with the motion) w(z = b + h) = Dz dt = D (b + h) (307) b+h dt Similarily, = (b + h) + u t x (b + h) + v (b + h) (308) y = h t + u x (b + h) + v (b + h) (309) y w(z = b) = db dt = u b x + v b y (310) Expressions (309) and (310) inserted in (306) gives ( u x + v ) (b + hb) + h y t + u x (b + h) + v b (b + h) u y x v b y = 0 (311) or h t + x (h u) + (h v) = 0 (312) y which is the continuity equation expressed by the local change of the fluid thickness h and divergence of hu H. 64

65 Alternatively, since the time derivative in (312) can be is expressed by ζ h(x, y, t) + b(x, y) = D + ζ(x, y, t) (313) ζ t + x (hu) + (hv) = 0 (314) y 12.3 The momentum equations The horizontal momentum equations is given by (303) and (304). The pressure p(x, y, z, t) is given by the hydrostatic equation p z = gρ 0 (315) Pressure fluctuations due to changes to the surface height can be expressed as (316) can be written or Similarly, for an arbitrarily depth z 1 (see Fig. 21), where z = D z 1. p b+h D = gρ 0(b + h D) (316) p s p(d) = gρ 0 ζ (317) p(d) = p s + gρ 0 ζ(x, y, t) (318) p(z 1 ) = gρ 0 z + p s + gρ 0 ζ(x, y, t) (319) Over time, there are (very) small spatial variations in the surface pressure p s. We can therefore ignore the contribution from p s. From the expressions (318) and (319) it then follows that p(d) x = p(z 1) x = gρ 0 ζ x (320) The same relationship holds for / y. It is therefore only the surface displacement ζ which gives rise to the pressure force in a homogeneous fluid. For this reason p = gρ 0 ζ is called the dynamic pressure The final set of equations The above gives the shallow water equations expressed in terms of u, v, h and ζ u t + u u x + v u ζ fv = g y x (321) v t + u v x + v v ζ + fu = g y y (322) ζ t + x (hu) + (hv) y = 0 (323) 65

66 For flat bottom we have that h(x, y, t) = D + ζ(x, y, t) (324) Note that the phrase shallow water equations do not imply that the waves are found in shallow waters. Rather, the phrase implies that the wavelength of the waves are much longer than the thickness of the fluid. 66

67 13 Gravity waves 13.1 One-dimensional gravity waves Gravity waves occur at the boundary between the ocean and atmosphere, or more generally between two (or more) fluids with different densities. In the following we consider surface waves. The force acting on these waves is gravity, hence the name (surface) gravity waves. If the waves describe small fluctuations in the wave variables, the wave equation (321) (323) can be linearized. This means that any product of wave variables, such as the advection terms in (321) and (322), can be neglected. Furthermore, we start by neglecting the effect of Earth s rotation, implying that the local derivative term ( / t) is much larger than the Coriolis-term (the effect of the Earth s rotation is discussed in the following section). With f = 0 and by considering a one-dimensional wave movement in the x -direction (v = 0 and / y = 0), equation (321) and (323) can be written as /5.3/ u t ζ t = g ζ x = D u x (325) (326) The velocity component u can be eliminated from the above expressions by considering (325)/ x and (326) t, giving the classical wave equation 2 ζ t 2 gd 2 ζ x 2 = 0 (327) The wave equation (327) can be solved by seeking solutions of the form (see appendix I) ζ = Re {ζ 0 exp[i(kx ωt)]} (328) where Re denotes the real part, ζ 0 is the amplitude, k is the wave number in the x-direction and ω is the angular frequency. Insertion of (328) into (327) leads to the dispersion relation ω 2 + gd k 2 = 0 (329) Since the wave s phase velocity c is given by c = ω/k (see appendix I), the phase velocity of the gravity wave c is given by c = c 0 = ± gd (330) The phase velocity c 0 is independent of the wave number k, so the gravity wave is non-dispersive. Gravity waves in a homogeneous, smooth and barotropid fluid propagate therefore with a phase velocity proportional to gd, and are independent of the wave wave number or wavelength. /p. 100/ Typical wavelengths for tidal waves Since ω = 2π/T and k = 2π/λ, it follows that c = ω k = λ T = gd (331) 67

68 where the last equality is given by (330). Consequently, λ = T gd (332) For the diurnal and semi-diurnal tides, the periods are T 1 24 hr and T 2 12 hr, respectively. The resulting wavelengths for some ocean depths D is given in table 5. Depth Period, diurnal tide Period, semi-diurnal tide D (m) T 1 24 hr T 2 12 hr 20 λ λ Table 5: Typical wavelengths λ 1 (km) and λ 2 (km) for diurnal and semi-durnal tides with approximate periods T 1 and T 2, and for three ocean depths D (m). It follows from the table that the tidal waves have very long wavelengths. Only at shallow water, at depths less than 20 m for diurnal tides and less than 50 m for semi-diurnal tides, the wavelengths are less than 1000 km. The tidal wavelengths are therefore, in general, much longer than the depth of the ocean, in accordance with the phrase shallow water equations as described by the end of Sec One-dimensional gravity waves in a closed channel An externally forced ocean basin or ocean channel will lead to standing waves with wavelengths determined by the geometry of the basin/channel. The resulting standing waves are the eigenmodes of the system, also called seiches. For a closed channel with length L (m) and with boundaries at x = 0 and x = L, kinematic boundary conditions impose no flow across the boundary: u x=0 = u x=l = 0. From equation (325), this results in the two boundary conditions: ζ x = ζ x=0 x = 0 (333) x=l If we assume that the surface elevation ζ can be expressed by means of trigonometric waves travelling in both the positive and negative x -direction, the surface elevation can be written as ζ = A exp[i(ωt kx)] + B exp[i(ωt + kx)] (334) where A and B are the amplitudes for waves travelling in the positive and negative x -directions, respectively. Thus, ζ = ika exp[i(ωt kx)] + ikb exp[i(ωt + kx)] (335) x which, by means of the boundary condition at x = 0 in expression (333) yields A = B. Therefore, ζ = A exp[i(ωt kx)] + exp[i(ωt + kx)] = A{exp( ikx) + exp(ikx)} exp(iωt) = 2A cos(kx) exp(iωt) (336) 68

69 where the identity exp(±iϕ) = cos ϕ ± i sin ϕ (337) is used in the last equality. At the other end of the cannel, at x = L, we get that ζ x = 2kA sin(kx) exp(iωt) x=l = 0 (338) x=l which is satisfied for kl = nπ, n = 1, 2,... (339) From expression (331), it follows that k = ω = 2π gd T gd (340) where ω = 2π/T is used in the last equality. By combining (339) and (340), the following eigenmode, natural or seiche periods are obtained: /5.6 & 7.8/ T n = 2L n, n = 1, 2, 3,... (341) gd The longest period T 1 occurs for n = 1, with the following periods equal to 1/2, 1/3, 1/4,... of T 1. T 1 = 2L/ gd is known as Merian s formula. Since T 1 = 2L/ gd and c = gd, we get that T 1 = 2L c (342) In addition, by using the definitions λ = 2π/k and c = ω/k, we have that From (342) and (343), it follows that T 1 = 2π ω = 2π ck = 2π c λ 2π = λ c (343) λ = 2L (344) Thus, for T 1, the wavelength is twice the length of the channel, implying that we have two equal waves, travelling in opposite directions, yielding a standing wave pattern One-dimensional gravity waves in a semi-closed channel If the channel is open in one end, in our case at x = L, the infinite reservoir of water outside the channel dictates the surface elevation at the opening of the channel. In this case expression (336) at x = L becomes ζ x=l = 2A cos(kx) exp(iωt) x=l = A exp(iω t) (345) where A and ω are the amplitude and the frequency of the externally forced, large-scale ocean tide. Consequently, 2A = A exp(iω t) (346) cos(kl) exp(iωt) 69

70 and from equation (336), ζ = A exp(iω t) cos(kx) cos(kl) (347) The channel response is largest, leading to resonance, when cos(kl) 0, or for kl = n π, n = 1, 3, 5,... (348) 2 Expression (340) and (348) give /7.9/ T n = 4L n, n = 1, 3, 5,... (349) gd Any combinations of D and L yielding periods around 12 or 24 hours will lead to resonance with the semi-diurnal or diurnal tides, giving rise to particularly high tides at the end (the head) of the bay or channel. Should, for instance, D = 40 m and L = 220 km, T hour, a period that is close to that of the semi-diurnal tides, resulting in particularly high tides at the head of the bay or channel. 70

71 14 Sverdrup waves and other related waves Also here we consider long surface waves, implying waves with wavelengths much longer than the thickness of the fluid. The effect of Earth s rotation is now considered by introducing the Coriolis-terms in the momentum equations. Variations in the Earth s rotation are, however, neglected, so f = const. Moreover, it is assumed that the waves has infinite horizontal extent in the two horizontal dimensions. The resulting waves are often called the Sverdrup waves. The basic equations follows from (321) (323) ζ t + D u ζ fv = g t x (350) v + fu t ) = ζ g y (351) = 0 (352) ( u x + v y 14.1 Derivation If we assume a solution on the form u, v, ζ exp[i (kx + ly ωt)] (353) where k = 2π/λ x and l = 2π/λ y are the wave numbers in the x - and y -directions (and λ x,y are the corresponding wavelengths, see Sec. I), respectively, the following algebraic relations are obtained iωu fv = igkζ (354) iωv + fu = iglζ (355) iωζ + D(ku + lv) = 0 (356) The equations (354) (356) constitute a set of three equations with three unknowns, and can be expressed in matrix form as iω f igk f iω igl u v = 0 (357) idk idl iω ζ Non-trivial solutions exist when the equation system s determinant vanishes. This gives the following relationship ω[ω 2 f 2 gd(k 2 + l 2 )] = 0 (358) or, by introducing the horizontal wave number k 2 h = k2 + l 2, ω[ω 2 f 2 gdk 2 h] = 0 (359) If the phase speed of the gravity waves c 2 0 = gd is introduced, one obtains ω[ω 2 f 2 c 2 0k 2 h] = 0 (360) 71

72 The expressions (359) and (360) are the dispersion relationship of the Sverdrup waves. It is generally different physical mechanisms involved for the different solutions of the dispersion relationship. It is therefore convenient to discuss the different solutions separately. Since the expression in square brackets in (359) and (360) includes the wave number k h, the Sverdrup waves are dispersive. Waves with different wave numbers (or wavelengths) will therefore propagate at different speeds Case ω = 0 The trivial solution ω = 0 from (359) is consistent with / t = 0. From the governing equations (350) and (351), we see that this solution is the geostrophic force balance. The stationary (i.e., time-invariant) solution of the shallow water equations is thus geostrophic balance Case ω 2 = f 2 + c 2 0 k 2 h For ω f for all possible wave solutions. ω 2 = f 2 + c 2 0 k 2 h (361) Furthermore, the dispersion relationship is symmetrical with respect to the x- and y-directions. This means that neither the x - or y -direction has a special significance for wave field. We can therefore orient the coordinate system such that the x -axis is aligned with the direction of the wave propagation. In this case, k h = k and l = 0. The resulting dispersion relationship becomes ω 2 = f 2 + c 2 0 k 2 (362) The phase speed c is given by c = ω/k. Thus, c = ω k = 1 k ( f 2 + c 2 0 k 2) 1/2 (363) ( f 2 1/2 = c 0 k 2 c 2 + 1) (364) 0 ( = c ) 1/2 k 2 L 2 (365) ρ where L ρ = c 0 /f is the Rossby deformation radius. The above equation shows that the phase spped c increases with decreasing wave number or increasing wavelength. The group velocity of the wave is given by c g = ω/ k. Differentiation of the dispersion relationship (362) gives 2ω ω = 2kc 2 0 k (366) or c g = k ω c2 0 = c2 0 c (367) Accordingly, the product c c g equals the constant c 2 0. The group speed therefore decreases with increasing wavelengths. 72

73 All possible wave solutions given by (362), expressed in terms of the wave number k and frequency ω, are illustrated in Fig. 22. For small wave number k (long waves), the wave properties Figure 22: Graphic representation of the dispersion relation given by (362). In addition, the dispersion relation for gravity waves (Sec. 13.1) and inertial oscillations (Sec. 14.6) are shown. L ρ is the Rossby deformation radius, IWR is the inertial wave regime where the Earth s rotation is important, and GWR is the gravity wave regime where the effect of Earth s rotation is of small or vanishing importance. follows those of the near inertia waves (see Sec. 14.6) and the effect of the Earth s rotation f is important. For large wave numbers (short waves), the effect of Earth s rotation has little effect. In the latter case, the wave regime is associated with gravitational waves (Sec. 13.1) The resulting wave motion For l = 0, it follows from (354) and (355) that iωu fv = igkζ 0 (368) and From (369), which inserted in (368) gives Since, from expression (362), iωv + fu = 0 (369) u = i ω f v (370) ω 2 f 2 v = igkζ f (371) ω 2 f 2 = gdk 2 (372) 73

74 one gets that v = i f kd ζ (373) so u from (370) becomes u = ω kd ζ (374) Since ω f, it follows from (373) and (374) that u v, or that the wave has a net propagation in the x -direction. Only the real part of the solution has a physical meaning. If we assume that the amplitude ζ 0 is real, it follows that ζ = Re{ζ 0 exp[i(kx ωt)]} = ζ 0 cos(kx ωt) (375) (375) inserted in (374) and (373) gives u = ω ζ 0 kd cos(kx ωt) (376) v = f ζ 0 sin(kx ωt) (377) kd Thus, the Earth s rotation is not felt in the x -direction, while the y -direction is influenced by f. For a fixed point in space, for example at x = 0, the following temporal change takes place, assuming f > 0 (northern hemisphere) and by using ζ 0 = ζ 0 /(kd) > 0 : ωt = 0 : u = ωζ 0 > 0, v = 0 (378) ωt = π/2 : u = 0, v = fζ 0 < 0 (379) ωt = π : u = ωζ 0 < 0, v = 0 (380) ωt = 3π/2 : u = 0, v = fζ 0 > 0 (381) The speed components span out ellipses as shown in the upper panel of Fig. 23 with a resultant movement directed clockwise in the Northern Hemisphere. Moreover, the ellipse s major axis is directed in the x -direction, i.e., in the direction of the wave propagation. The ratio between the semi-major and semi-minor axes is given by ω / f. A particle s position is found by integrating (376) and (377) with respect to time t. gives This x sin( ωt) (382) y cos( ωt) (383) The resulting motion is directed with the clock (anti-cyclonic rotation) in the northern hemisphere, see bottom panel of Fig. 23. Due to variations in water depth, the coastline geometry and non-linear wave-wave interactions, the horizontal tidal movement can be both cyclonic and anti-cyclonic. But for an infinite ocean with flat bottom, which are the underlying assumptions for the Sverdrup waves, the tidal ellipses describe anti-cyclonic movement. The spatial characteristics at a given time, for example at t = 0, yields the relationships shown in Table (6). The corresponding movement in the xy -plane is shown in the lower part of Fig. (24). In this figure, the velocity-component in the direction of the wave propagation, u, is shown with red 74

75 v u ωt = π u = -ω ζ 0, v = 0 ωt = 0 u = ω ζ 0, v = 0 ωt = π/2 u = 0, v = -f ζ 0 y ωt = 0 x ωt = π/2 Figure 23: Illustration of the temporal change of the u - and v -components (top panel) and the corresponding x - and y -positions (lower panel) in the northern hemisphere. The resulting motion is directed with the clock (anti-cyclonic rotation). Argument x u y v ζ kx = 0 0 ωζ 0 fζ 0/k 0 ζ 0 kx = π/2 ωζ 0/k 0 0 fζ 0 0 kx = π 0 -ωζ 0 fζ 0/k 0 ζ 0 kx = 3π/2 ωζ 0/k 0 0 fζ 0 0 Table 6: Overview of the magnitude and sign of x, y, u, v and ζ for four values of the argument kx. 75

76 horizontal arrows. It follows that there is convergence in the left half and divergence in the right half of the figure. Therefore, the surface is lifted in the convergent zone and lowered where divergence takes place (red vertical arrows in the figure). Since the wave form is fixed, the wave propagates in the positive x -direction. It is therefore the fluid oscillation in the x -direction, resulting in convergence and divergence in the fluid, that drives the wave forward. Figure 24: Illustration of the surface elevation ζ and the u -component of fluid particles in the vertical xz -plane (upper part of the figure), and the particle movement in the horizontal xy -plane (ellipses in the lower part of the figure), representing northern hemisphere Sverdrup waves. Convergence and Divergence denote respectively convergent and divergent flow in the x -direction Approximate length of the tidal axes If the horizontal tidal component in the direction of the propagation of the tidal wave is on the form u = u 0 sin( ωt) (384) the length of the, for instance, semi-major tidal axis x can be obtained by integrating (384) over one quarter of a tidal period T, from t = 0 to t = T/4. Thus, with u = dx/dt, T/4 t=0 dx dt dt = T/4 t=0 u 0 sin( ωt) dt (385) Since ω = 2π/T, giving x t=t/4 x t=0 = x = u 0 ω x = u 0 T 2π [cos(π/2) cos(0)] (386) (387) For the M 2 tide, T M2 = hr. In this case, x M2 ( s) u 0 (388) 76

77 For moderately to very strong M 2 tidal currents, u 0 ranges from 0.1 m/s to 1 m/s, respectively, and the corresponding lengths of the semi-major axes are x M2 0.7 km and x M2 7 km (389) The particle trajectories describing the elliptic path are orders of magnitude smaller than the typical wavelength of several 100 km to a few 1000 km for the tidal waves, see Sec It is, however, the co-ordinated movement of the small-scale tidal ellipses that generates the large-scale progression of the tidal wave. Put differently, the progression of the fast, large-scale tidal wave seen as variations in the surface elevation ζ is not associated with advection of mass, but it is a consequence of the coordinated horizontal, smaller-scale, elliptic trajectories of water packages throughout the water column. The latter movements do represent advection of mass. Passive particles released in the ocean, or e.g. buoyant neutral fish eggs or larvae, will therefore, in the absence of other processes and forces, span out an elliptic trajectory during a tidal period, i.e., approximately every 12 hr or 24 hr depending on the tidal regime Case f = 0 In the case of no rotation, f = 0, ω = ± gd k and the phase speed c = ω/k = ± gd. This is the solution of gravity waves Case k 2 f 2 /(gd) (short waves) The dispersion relationship (361) can be put in the form ( ) f 2 ω = ± gd gd + k2 (390) For k 2 f 2 /(gd), implying waves with short wavelengths, ω ± gd k and the phase speed c = ω/k ± gd. Also this solution regime represents the gravity waves. The reason for this is that waves with wavelengths much shorter than the deformation radius L ρ are only weakly influenced by the rotation of the Earth Case k 2 f 2 /(gd) (long waves) In the case of (very) long wavelengths, e.g. for k 2 f 2 /(gd), ω ±f (see 390). This case represents large-scale, horizontal oscillations with the frequency of the Coriolis parameter f. These are the inertial oscillations or waves. For ω = f and since k is small (i.e., k 0 ), expression (354) gives Consequently, u = i v (391) u = Re{iv 0 exp( ift)} = Re{iv 0 [cos( ft) + i sin( ft)]} = v 0 sin( ft) (392) 77

78 and v = Re{v 0 exp( ift)} = Re{v 0 [cos( ft) + i sin( ft)]} = v 0 cos( ft) (393) On the northern hemisphere with f > 0, expression (392) and (393) describe a circular, horizontal motion with radius v 0 and rotation directed with the clock (or anti-cyclonic rotation). These are the inertial oscillations in the ocean Interaction between tidal ellipses and internal oscillations From Sections and 14.6, it follows that tidal ellipses and inertial motion describe similar, rotational movement. Thus, if the period of any of the (major) tidal constituents are similar to the period of the inertial oscillations, strong interactions (resonance) are expected. This is indeed the case at specific geographic latitudes φ, known as critical latitudes. /p. 126/ If we assume that the temporal evolution of the horizontal velocity component u of a tidal wave can be written as u exp( i ωt) (394) the local acceleration term in the momentum equation, u/ t, scales as Similarly, the momentum equation s Coriolis-term f ẑ u scales as ω u (395) f u (396) Here f = 2Ω sin φ is the Coriolis parameter, Ω is the Earth s rotation vector ( Ω = 2π/(24 msh) ), and ẑ is the outward directed unit vector. For the M 2 tide, the speed follows from Tables 2 and 4: ω = 2 (ω s ω 2 ) = 2 ω 1 (397) with ω 1 = rad/msh. Consequently, ω = f occurs for /p. 126/ sin φ = ω 2 Ω, or for φ = ±75 (398) Thus, strong interaction between the M 2 tide and inertial oscillations are expected (and indeed observed) at 75 N, like in the Barents Sea, and at 75 S in the Southern Ocean s Weddel and Ross Seas. Also two times the M 2 tidal period may interact with the inertial oscillations. This gives rise to the critical latitudes φ = ±29 (399) For the K 1 tide with frequency ω = ω 1 +ω 2 = rad/msh (see Tables 2 and 4), the critical latitudes are φ = ±30 (400) /p. 126/ 78

79 Part III Appendix A Some key parameters of the Earth, Moon, Sun system A.1 Mass The mass of the Earth m e, Moon m l and Sun m s are m e = kg (401) m l = kg (402) m s = kg (403) The following mass ratios are then obtained m e 80, m l m e , m s m l m e (404) m s m e (405) (406) A.2 Length and distance Earth s mean radius is r = m (407) Often a, in stead of r, is used to denote the Earth s mean radius. The mean distance between the Earth and the Moon, and the Earth and the Sun, are R l, mean = m (408) R s, mean = m (409) The following distance ratios are obtained a 0.017, R l, mean a , R s, mean R l, mean r R s, mean r 60 (410) (411) (412) 79

80 B Spherical coordinates B.1 Two commonly used spherical coordinate systems Spherical coordinates are, by construction, convenient for any type of spherical problems. Fig. 25 illustrates two commonly used variants of spherical coordinates. Figure 25: Illustration of two spherical coordinate systems. The system to the left is described by Ψ (angle in longitudinal direction, 0 Ψ 2π), ϕ (angle in the latitudinal direction, π/2 ϕ π/2) and r (length in the radial direction, r 0). This coordinate system is commonly used in oceanography and meteorology because of the correspondence with the commonly used Earth s latitude and longitude positions. The system to the right is described by Ψ (azimuth angle, 0 Ψ 2π), φ (zenith angle, 0 φ π) and r (length in the radial direction, r 0). This coordinate system is commonly used in problems involving gravitational or electric potential as will become clear shortly. Note that the set of coordinates in the order (Ψ, ϕ, r) for the system to the left in Fig. 25, and (r, φ, Ψ) for the system to the right, are both right-hand coordinate systems. Also note that the actual naming of the various angles may vary depending on the problem; the azimuth angle Ψ is, as an example, commonly labelled λ in meteorology and oceanography. B.2 Volume and surface elements in spherical coordinates B.2.1 Spherical volume elements The volume element formed by perturbing each of the three coordinates, denoted by δ in the following, can be obtained as follows: For the system to the left in Fig. 25, small changes in r form a line segment of length δr in the direction of r; small changes in ϕ form an upward directed arc of length r δϕ; and small changes in Ψ form an arc in the xy-plane of length r cos ϕ δψ. The resulting volume element is 80

81 the product of the three length contributions δv Ψϕr = r 2 cos ϕ δr δϕ δψ (413) Correspondingly, for the system to the right in the figure, small changes in r form a line segment of length δr in the direction of r, small changes in φ form an downward directed arc of length r δφ, and small changes in Ψ form an arc in the xy-plane of length r sin φ δψ. The resulting volume element is δv rφψ = r 2 sin φ δr δφ δψ (414) B.2.2 Spherical surface elements If r is kept constant at r = a, the surface area S of the sphere spanned out by the two pairs of angles (Ψ, ϕ) and (φ, Ψ) is given the volume elements above when changes in r are ignored δs Ψϕ = a 2 cos ϕ δϕ δψ (415) Integration over the full sphere, i.e., 2π δs φψ = a 2 sin φ δφ δψ (416) Ψ=0 2π Ψ=0 π/2 ϕ= π/2 π φ=0 δs Ψϕ dϕ dψ (417) δs φψ dφ dψ (418) result in the surface area of a sphere with radius a, namely 4πa 2, as expected. B.2.3 Relationship with the Earth-Moon system Note that in the system to the right in Fig. 25, the zenith angle φ is always non-negative. For r = const and 0 Ψ < 2π, the resulting geometry forms a cone with angle 0 φ π centred around the positive z-axis. A cone is also spanned out for r = const, ϕ = const and 0 Ψ 2π in the system to the left in the figure. In the latter case, however, the latitudinal angle π/2 ϕ π/2 takes both signs. It is the non-negative zenith-angle φ in the coordinate system to the right in Fig. 25, similar to the zenith angle φ in tidal theory when the centre line in Fig. 1 is aligned with the z-axis in Fig. 25, that makes the (φ, Ψ)-system particularly suited for problems involving gravitational potential (or problems involving electric potentials), see e.g. Sec B.2.4 Gradient operator in spherical coordinates The gradient operator expressed in terms of the two spherical coordinate systems in Fig. 25 can be expressed in terms of the two system s scale factors. 81

82 For the system to the left in Fig. 25, the scale factors in the Ψ, ϕ, r-directions are r cos ϕ, r and 1, respectively. The resulting gradient operator is therefore ( 1 = r cos ϕ Ψ, 1 r ϕ, ) = 1 r r cos ϕ Ψ e Ψ + 1 r ϕ e ϕ + r e r (419) In the above expression, e Ψ, e ϕ and e r are the unit vectors in the Ψ, ϕ, r-directions, respectively. Similarly, for the right system in Fig. 25, the scale factors in the r, φ, Ψ-directions are 1, r and r sin φ, respectively. The resulting gradient operator becomes ( = r, 1 ) r φ, 1 = r sin φ Ψ r e r + 1 r φ e φ + 1 r sin φ Ψ e Ψ (420) 82

83 C The three lunar tidal components To show the identity between the right hand sides of (65) and (66), one can first note that ζ 1 common to both expressions and can therefor be ignored in the following. Expansion and reordering of ζ 0 gives ζ 0 = 3 ( sin 2 φ 1 ) ( sin 2 d 1 ) is = 3 2 sin2 φ sin 2 d 1 2 sin2 φ 1 2 sin2 d = sin 2 φ sin 2 d sin2 φ sin 2 d 1 2 sin2 φ 1 2 sin2 d = sin 2 φ sin 2 d sin2 φ (sin 2 d 1) 1 2 sin2 d = sin 2 φ sin 2 d 1 2 sin2 φ cos 2 d 1 2 sin2 d (421) In the second last line, the identity sin 2 a + cos 2 a = 1 (422) has been used. Furthermore ζ 2 = 1 2 cos2 φ cos 2 d cos(2c P ) = cos 2 φ cos 2 d 1 2 cos2 φ cos 2 d (423) Here the identity cos 2a = cos 2 a sin 2 a = 2 cos 2 a 1 (424) has been used (with the help of identity (422) in the last equality). With ζ 0 from (421) and ζ 2 from (423), one obtains ζ 0 + ζ 2 = sin 2 φ sin 2 d 1 2 sin2 φ cos 2 d 1 2 sin2 d cos 2 φ cos 2 d cos 2 C P 1 2 cos2 φ cos 2 d (425) The second and last term on the right hand side can be combined into 1 2 cos2 d (426) so ζ 0 + ζ 2 = sin 2 φ sin 2 d 1 2 cos2 d 1 2 sin2 d cos2 φ cos 2 d cos 2 C P (427) Comparison between the right hand sides of (65) and (427), and remembering that ζ 1 has already been taken care of, shows that the two last terms on the right hand side of (427) have 83

84 their counterparts on the right hand side of (65). For the remaining terms on the right hand side of (427), one obtains sin 2 φ sin 2 d 1 2 cos2 d 1 2 sin2 d = sin2 φ sin 2 d = sin2 φ sin 2 d 1 3 (428) which is identical to the remaining terms on the right hand side of (65). 84

85 D Elliptic geometry 15 An ellipse is the locus of points M in the plane such as the sum of the distances MF and MF to two fixed points F and F, called the foci, is constant. A basic configuration is shown in Figure 26. From the definition, an ellipsis can be formed by fixing a non-stretching string of length l at two sticks (the foci points), a distance d < l apart. A third moveable stick will then form the ellipse if the string runs on the outer side of the latter, while the stick is continuously moved with the string fully stretched. Principal circle y Ellipse b M R a E K F c = ae F x Figure 26: Illustration of an ellipse. a and b are the semi-major and semi-minor axes, respectively; F and F are the foci points (of which a point with mass is located in foci point F whereas F is an empty focus (i.e., no mass)); M is a point on the ellipse; R and K are the distance F M and the angle angle between the x -axis and the line segment R, respectively; c is the distance between the centre of the ellipse and the foci points; and e is the eccentricity of the ellipse satisfying c = a e, with 0 e < 1. The angles K and E are called the true anomaly and the eccentric anomaly, respectively. Note that the eccentric anomaly defines the angle between the horizontal axis and a point on the principal circle given by a vertical line running through M. As seen from Fig. 26, the eccentricity e is defined by the relationship c = a e, 0 e < 1 (429) 15 Mainly based on the excellent treatise by M. Capderou (2014): Handbook of Satellite Orbits. From Kepler to GPS, ISBN , DOI / , Springer International Publishing Switzerland. 85

86 Let R = F M and R = F M (430) In the case K = 0, it follows that Therefore, R = a c and R = a + c (431) R + R = 2 a (432) or that the sum of R + R is constant, consistent with the definition. That the constant equals 2 a follows directly from Fig. 26 in the case K = 0, but is otherwise not intuitively given. Another property is obtained when K = π/2. In this case (from 432), and Pythagoras gives In the last equality, the definition (438) is being used. Thus, D.1 Cartesian coordinates R = R = a (433) b 2 = a 2 c 2 = a 2 (1 e 2 ) (434) b a = 1 e 2 (435) In Cartesian coordinates, M is located at the point (x, y) and F is located at (c, 0). Pythagoras gives R 2 = (x c) 2 + y 2 = (x ae) 2 + y 2 (436) Likewise, Therefore Furthermore and by means of (432), we obtain R 2 = (x + c) 2 + y 2 = (x + ae) 2 + y 2 (437) R 2 R 2 = 4 cx = 4 ae x (438) R 2 R 2 = (R R)(R + R) (439) R R = 2 ex (440) Finally, by combining (432) and (440), we get the lengths of R and R expressed in Cartesian coordinates R = a ex (441) R = a + ex (442) The standard Cartesian formula for an ellipse is obtained by combining (436) and (441): giving or, by means of (434), R 2 = (x + ae) 2 + y 2 = (a ex) 2 (443) x 2 a 2 + y 2 a 2 (1 e 2 ) = 1 (444) x 2 a 2 + y2 b 2 = 1 (445) 86

87 D.2 Semilatus rectum The distance p from the x -axis to any point y on the ellipsis is called the ellipse s semilatus rectum, see Fig. 27. y (ae,b 2 /a) b p a ae (ae,0) x (ae,-b 2 /a) Figure 27: Illustration of the semilatus rectum p of an ellipse, the latter with semi-major and semi-minor axes a and b, respectively, and eccentricity e. The semilatus rectum can be derived from expression (445), with x = ae : y 2 b 2 = 1 e2 (446) But from expression (435) therefore 1 e 2 = b2 a 2 (447) y = ± b2 a (448) and the semilatus rectum is therefore p = b 2 /a ( a > b ). Furthermore, the ratio b/a given by (435) implies that p can be expressed as p = a(1 e 2 ) (449) 87

88 D.3 The y -value of an ellipse relative to the reference circle On a reference circle, the positive y value at x = ae is given by Pythagoras y circle = a 1 e 2 (450) On an ellipse, the corresponding y -value is given by expression (446) y ellipse = b 1 e 2 (451) Therefore, for any x -value, the ratio of the y -value of the ellipse and the reference circle is constant and is given by y ellipse y circle = b a = 1 e 2 (452) where the last equality comes from expression (435). D.4 Polar coordinates We first define a Cartesian coordinate system centred on the foci point F. The x -coordinate of F, labelled X, is X = x c = x ea (453) The length R, from (441), can now be expressed as From Fig. 26, R = a ex = a e(x + ea) = a(1 e 2 ) ex (454) X = R cos K (455) Inserting into (454) and rearranging yields the equation for an ellipse in polar coordinates R(K) = a(1 e2 ) 1 + e cos K (456) The inverse of R can be expressed with one term proportional to cos K term: 1 R(K) = e a(1 e 2 ) cos K + 1 a(1 e 2 ) plus a constant (457) Thus, expressions (456) and (457) are the polar counterparts to the more commonly known cartesian expression given by (445). 88

89 E Binominal theorem for rational exponents For all rational 16 numbers r, the following identity exists (1 + x) r = 1 + r x + r r 1 2 x2 + r r 1 2 r 2 3 x (458) For a derivation, see e.g. For x 1, the linear and quadric approximation of (458) becomes (1 + x) r = 1 + r x (459) and (1 + x) r = 1 + r x + r(r 1) x 2 (460) 2 16 A rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero. 89

90 F Some trigonometric identities Pythagorean identity sin 2 a + cos 2 a = 1 (461) Sum and difference identities sin(a ± b) = sin a cos b ± cos a sin b (462) cos(a ± b) = cos a cos b sin a sin b (463) Product identities 2 sin a sin b = cos(a b) cos(a + b) (464) 2 cos a cos b = cos(a b) + cos(a + b) (465) sin 2 a = 1 (1 cos 2a) (466) 2 cos 2 a = 1 (1 + cos 2a) (467) 2 Product sum identities ( a ± b sin a ± sin b = 2 sin 2 ( a + b cos a + cos b = 2 cos 2 ( a + b cos a cos b = 2 sin 2 ) cos ) cos ) sin ( ) a b 2 ( ) a b 2 ( ) a b 2 (468) (469) (470) Double angle identities sin 2a = 2 sin a cos a (471) cos 2a = cos 2 a sin 2 a (472) cos 2a = 2 cos 2 a 1 (473) = 1 2 sin 2 a (474) Triple angle identity sin 3 a = 1 (3 sin a sin 3a) (475) 4 90

91 Half-angle identities cos a = 1 tan2 (a/2) 1 + tan 2 (a/2) (476) Integrals sin 2x dx = 1 2 cos 2x + C (477) cos 2x sin x dx = cos x cos 3x + C (478) 2 6 cos 2 x dx = x sin 2x + + C (479) 2 4 sin 2 x dx = x sin 2x + C (480) 2 4 cos 2x sin x cos x dx = + C (481) 4 Lowest order approximations sin x x for x 1 (482) cos x 1 for x 1 (483) Law of cosines For any triangle with sides a, b and c, with the angle between a and b being C, the following relationship holds: c 2 = a 2 + b 2 2 ab cos C (484) Cotangent formula On a spherical triangle, let the four elements side angle side angle lie adjacent to each other. If the elements are referred to as outer side (OS) inner angle (IA) inner side (IS) outer angle (OA) then the formula cos(is) cos(ia) = sin(is) cot(os) sin(ia) cot(oa) (485) is valid. An example of the location of the four spherical triangle elements is illustrated in Fig

92 OA OS IS IA Figure 28: Illustration of one out of three ways to locate the four adjacent elements outer side (OS) inner angle (IA) inner side (IS) outer angle (OA) on a spherical triangle (outlined in magenta). This ordering is consistent with the cotangent formula written in the form (485). 92

93 Derivative of arctan x d arctan x d x = x 2 (486) 93

94 G Miscellaneous notes G.1 Lunar longitude The following expressions are from Pugh & Woodworth (2014), page 63, with corrections to the text made in red. Startinng point: C 2 (t) = [ ( a R l ) ] cos2 d l [1 e cos(s p)] 3 cos 2 C 1 (487) [ ( ) ] 3 a 3 Ψ = 4 cos2 d l (488) R l /p. 63/ [1 e cos(s p)] e cos(s p) (489) cos 2 C 1 = cos 2[ω 0 t + h s π 2 e sin(s p)] (490) δ ω 0 t + h s π (491) Leading to the simplified version of cos 2 C 1 : cos 2 C 1 = cos 2 [δ 2 e sin(s p)] = cos[2 δ 4 e sin(s p)] = cos 2 δ cos([4 e sin(s p)] + sin 2 δ sin([4 e sin(s p)] cos 2 δ + 4 e sin 2 δ sin(s p) (492) where the smallness of e is used in the last equality, see (482) and (483). highlighted number 4 is mistakenly written as 2. In the book, the G.2 High and low water times and heights G.2.1 High water times and heights We consider the height T (t) of the major M 2 tide and another minor tidal constituent: /p. 368/ T (t) = H M2 cos ωt + A cos nt + B sin nt (493) The latter minor constituent can be cast into the form of a single trignometric function by introducing the amplitude R and the angle θ : The above equations relate A, B and R, θ : A = R cos θ (494) B = R sin θ (495) R 2 = A 2 + B 2, and tan θ = B A (496) Thus, by means of expression (463), equation (493) can also be written as T (t) = H M2 cos ωt + R cos(nt θ) (497) 94

95 Maximum value of (493) occurs for a vanishing temporal derivative: T (t) t = ωh M2 sin ωt na sin nt + nb cos nt = 0 (498) If t = 0 denotes the time for maximum M 2, we can use for small t : cos ωt cos nt 1, sin ωt ωt, and sin nt nt (499) The maximum amplitude H max of the combined tide occurs when the time derivative in (498) vanishes: ω 2 t max H M2 n 2 t max A + n B = 0 (500) As long as M 2 is the leading tidal constituent, R/H M2 1, giving t max = B n H M2 ω 2 (501) With t max inserted into (493), and by using cos ωt max (ωt max) 2, cos nt max 1, sin nt max nt max (502) we obtain H max = H M2 + A ( n ω ) 2 B 2 H M2 (503) G.2.2 Low water times and heights Likewise, if we let t = 0 occur at minimum M 2, expression (493) reads T (t) = H M2 cos ωt + A cos nt + B sin nt (504) With the same simplifications as above, we obtain t min = B n H M2 ω 2 (505) With t min inserted into (504), and by using cos ωt min (ωt min) 2, cos nt min 1, sin nt min nt min (506) we get H min = H M2 + A 1 2 ( n ω ) 2 B 2 H M2 (507) G.2.3 Temporal average The temporal averaged value or any quantity Ψ, denoted Ψ, averaged over many tidal periods m T t, are given by Ψ = 1 m Tt Ψ dt (508) m T t 0 95

96 The temporal average of H max from (503) with A, B given by (495), gives the following for the first term H max = 1 m Tt H max dt = H m Tt max dt = H max (509) m T t t=0 m T t t=0 The second terms gives no contribution due to the symmetric cosine -contribution around zero: A = 1 m Tt A dt = 1 m T t t=0 m 2π m 2π θ=0 R cos θ dθ = 0 (510) For the third term, the sin 2 -dependency gives a contribution due to it s non-negative values: B 2 = 1 m 2π [ ] m 2π R 2 sin 2 θ dθ (480) = R2 θ sin 2θ = R2 (511) m 2π θ=0 m 2π 2 4 θ=0 2 Consequently, from (503), H max = H max ( n ω ) 2 R 2 Likewise, the temporal average of H min, from (507), becomes H min = H min 1 4 ( n ω ) 2 R 2 H M2 (512) H M2 (513) Thus, the Mean High Water (MHW) and Mean Low Water (MLW) are given by /C.1/ MHW = H max (514) MLW = H min (515) The Mean Tide Level (MTL) is the mean of MHW and MLW, so MTL = 0 (516) Finally, the Mean Tidal Range (MTR) is the difference between MHW and MLW, MTR = MHW MLW = 2 H max + 1 ( n ) 2 R 2 2 ω H M2 (517) 96

97 H Simplified, harmonic analysis For description of the method discussed below, see Section 6 in the note by Bjørn Gjevik, available athttp://folk.uio.no/bjorng/tides_unis.pdf. Any integral can be transferred to sums. Thus, the integrals A i = 2 t m tm 0 η(t) cos ω i t dt (518) B i = 2 tm η(t) sin ω i t dt (519) t m 0 can be transferred to sums, and the latter can be computed by numerical methods/software. H.1 Outlining of the method The transformation can be done with the following choices/substitutions: Length of timestep (in our case 1 hr) dt = t (520) Timing t of the M + 1 observations t = 0, t,..., M t (521) The total length t m of the time series t m = M t, so 1 t m = 1 M t (522) The frequency ω i and the associated tidal period T i of the tidal constituent i In this case, the above integrals can be expressed as sums: A i = 2 M B i = 2 M M+1 k=1 M+1 k=1 ω i = 2π T i (523) [ ] 2π η k cos (k 1) t T i [ ] 2π η k sin (k 1) t T i (524) (525) Once A i and B i are numerically computed according to the above formulas, the amplitude H i and the phase κ i can be determined from the expressions H i = A 2 i + B2 i (526) and tan κ i = B i A i, so κ i = arctan B i A i (527) 97

98 Note that the procedure is only valid when the length of the analysed portion of the sea level time series is a multiple of the period t m = m T i. For accurate extraction of a given tidal constituent, it is important to sample over many tidal periods, so m 1. As a rule of thumb, we should sample over a time interval that includes, at least, one spring period ( days). The reason for this is that a shorter time interval will miss the prominent interaction between the M 2 and S 2 tides. Thus, for semi-diurnal tidal constituents with period of around 12 hr, m should be larger than 30. H.2 Extraction of the M 2 tide For M 2, the above considerations are met, as an example, with the following choices: Firstly, from the tidal analysis, we have that T M2 = hr (528) With hourly sea level data, t = 1 hr. Since 50 times the M 2 period is an integer, we can chose m = 50 (i.e., 50 times the M 2 period), since this leads to an integer number of time steps: M = = 621 (529) Thus, 621 hourly time steps can be used. H.3 Note 1 An integer number of time steps are required for carrying out the above given sums. Therefore, as an example, m = 30 times the M 2 period would not work properly since = time steps. For the M 2 tide, m is therefore chose to ensure that m T M2 is an integer. H.4 Note 2 Given the total length of the observed time series used for the analysis, we may chose any multiple of m. In general, the accuracy of our model increases with increasing m. For this reason, we may chose m = 200 and M = H.5 Extraction of the S 2 tide Likewise, for S 2, we have that With t = 1 hr and 60 times the S 2 T S2 = hr (530) period (so m = 60 ), we get M = = 720 (531) (i.e., 720 hourly time steps). 98

99 H.6 Extraction of the N 2 tide For N 2, With t = 1 hr and 50 times the N 2 T N2 = hr (532) period (so m = 50 ), we get M = = 720 (533) (i.e., 720 hourly time steps). H.7 Extraction of the K 2 tide For K 2, T K2 = hr (534) With t = 1 hr and 100 times the K 2 period (so m = 100 ), we get M = = 1197 (535) (i.e., 1197 hourly time steps). H.8 Extraction of the O 1 tide For O 1, With t = 1 hr and 100 times the O 1 T O1 = hr (536) period (so m = 100 ), we get M = = 2582 (537) (i.e., 2582 hourly time steps). H.9 Extraction of the K 1 tide For K 1, T K1 = hr (538) With t = 1 hr and 100 times the K 1 period (so m = 100 ), we get M = = 2393 (539) (i.e., 2393 hourly time steps). 99

100 H.10 Note 3 Care needs to be taken when computing κ i from expression (527) due to the multiple-valued tan κ i -function for 0 < κ i < 2 π. For this reason, programming languages like fortran, matlab and python have a second form of the arctan function with typical syntax: atan2(bi,ai) where B i and A i follows from tan κ i = B i /A i as shown in the right panel in Fig. 29. The atan2 function is constructed to return a unique arctan -function value with magnitude between π/2 and π/2, often called the principal values, in expressions like (527). In this way, atan2(bi, Ai) takes into account the multiple-valued tan κ i -function for κ i > π/2. Figure 29: Left: Graph of y = arctan κ i for π/2 < κ i < 2 π, with the function values 0 < y < 2 π coloured in red. Right: Illustration of tan κ i = B i /A i on a unit circle, with the signs of the quantities A i, B i in the four quadrants (with B i > 0 in the upper and B i < 0 in the lower quadrants, and A i > 0 in the right and A i < 0 in the left quadrants, respectively). The construction of the atan2 function follows from Fig. 29: arctan κ i, with function values between 0 and 2 π, is depicted by the red lines. Note that for any value of κ i, positive or negative and irrespective of its numerical value, there are two values of arctan κ i. But arctan κ i can be made unique by noting that for π/2 < arctan κ i < 3 π/2, the principal value is obtained by subtracting the value π from arctan κ i. Similarly, for 3 π/2 < arctan κ i < 2 π, the principal value is obtained by subtracting the value 2 π from the function value. An algorithm for the above can be based on the sign of A i and B i from tan κ i = B i /A i, see the right panel in Fig. 29. For 0 < arctan κ i < π/2 : B i > 0 and A i > 0. No changes to arctan(b i /A i ). 100

101 For π/2 < arctan κ i < π : B i > 0 and A i < 0. Subtract π to arctan(b i /A i ). For π < arctan κ i < 3 π/2 : B i < 0 and A i < 0. Subtract π to arctan(b i /A i ). For 3 π/2 < arctan κ i < 2 π : B i < 0 and A i > 0. Subtract 2 π to arctan(b i /A i ). Thus, κ i can be computed from expression (527), with coding similar to kappai = atan(bi/ai) if (Ai < 0.) kappai = kappai - pi if ((Ai > 0.) & (Bi < 0.)) kappai = kappai - 2.*pi Or, alternatively, one can use the pre-defined function kappai = atan2(bi,ai) where the actual syntax depends on the actual programming language used. An internet search like atan2 matlab will provide the details of the implementation and use of the atan2-function in matlab. H.11 Note 4 From the above, it follows that different lengths (or number of time steps) of the observed time series is used to extract the different tidal constituents. This is fine, but we need to ensure that the observed time series is sufficiently long so that all tidal constituents of interest are covered by the length of the observed time series. For the plotting, it is convenient to use a time interval covering a few fortnightly tides. For detailed inspection of the tidal constituents and the observed time series, 10 or so tidal periods may be plotted. There are several software packages for tidal analysis. Make e.g. a search matlab tidal analysis or python tidal analysis on the net. 101

102 Figure 30: Observed (black line) and modelled (red line) sea surface hight (SSH, in m) at the Norwegian coastal cities Bergen (top panel), Tromsø (mid panel) and Ny-A lesund (bottom panel) for the first 300 days of The modelled sea surface height is based on the algorithm described in the text, and includes the sum of the M2, S2, N2, K2, O1, and K1 constituents. The difference between the observed and modelled sea surface height can be attributed to an imperfect (simplified) model and the effect of weather. Data from Kartverket, 102

103 Figure 31: As Fig. 31, but showing the residuals, i.e., the difference between observed and modelled sea level height. 103