This document discusses improving the representation of harmonic tides in models. It finds that the accuracy of tides in recent models has deteriorated by up to 40% compared to older models. It also discusses tidal constituents like M2, S2, and K2, and how they are represented using parameters like amplitude and phase. Tables show tidal frequencies from older sources and more accurate values recommended by the International Hydrographic Organization tidal committee to improve long model runs. The conclusion recommends using these committee frequency values or recalculating phases each time step from orbit equations to improve tidal representation.

1. Improvement of the harmonic tides in
Simona
Werner Kramer
07-06-2018 1

2. Representation of the 18.6-year nodal cycle
in DCSMv6 – Firmijn Zijl
Last year it was found
that the accuracy with
which the tide is
represented in the
models
DCSMv6 and DCSMv6-
ZUNOv4 has
deteriorated
significantly over the
years; at important
locations with -40%
(Zijl, 2015).

3. Restults for CSM8 cadzand
25-07-2016 3

4. Tides in Simona
U.S. Coast and Geodetic Survey, Paul
Schureman, (1928)

5. Moon gravity force on Earth
𝒂 = −
𝐺 𝑀moon
𝑟2
𝒓
Definition: The tidal force is an apparent force that stretches a body towards
the center of mass of another body due to a gradient (difference in strength)
in gravitational field from the other body

6. Moon gravity force on Earth
apparent force in the non-inertial frame moving with earth
force is not pulling the water (normal component)
the tangential components pushes water along the surface

7. M1, M2 and mixed tides
Earth is making a full rotation each 23:56
plane of the orbit of moon is make an angle with the equator

8. S2 and M2 tides
gradient in gravitational field of the sun is weaker

9. earth & moon orbits
https://www.youtube.com/watch?v=jWCBhV
feAQU

10. angles equator, ecliptic and plane of moon’s
orbit
5°
23.5°

11. orbit parameters

12. orbit parameters typical time scales
𝜏 1 lunar day= 1.035050 solar days (24h)
𝑠 27.321 solar days
ℎ 365.242 solar days
𝑝 8.85 Julian years
𝑁 18.61 Julian years
𝑝𝑙 20942 Julian years

13. Tidal constituents
𝑦 = 𝑓𝐻 cos(𝑉 + 𝑢 − 𝜅)
𝑦 = 𝑓𝐻 cos(𝑉𝑡=0 + 𝜔𝑡 + 𝑢 − 𝜅)
nodal
precession
109 M2 M2 28.98410420 BZZZZZZ 2 0 0 0 0 0 0 y
128 S2 S2 29.99999994 BBXZZZZ 2 2 -2 0 0 0 0 z
131 K2 K2 30.08213720 BBZZZZZ 2 2 0 0 0 0 0 y
𝜏 𝑠 ℎ 𝑝 𝑁 𝑝𝑙 𝑓

14. CSM8 harmonic boundaries
25-07-2016 14
BOUNDAries
HARMonic
GEneral
# Rvw set Csm8 , set konstanten ( 01-01-1900 , GMT )
# Gebaseerd op gecalibreerde rvw (oc76 + handm)
'M2 ','S2 ','N2 ','K2 ',
'O1 ','K1 ','Q1 ','P1 ',
'NU2 ','L2 ‘
CONSTAnts
S : P901, TID= .00000000E+00, AZEro= .0000000E+00
AMPL= 1.81594 .62490 .36832 .16089 .06360
.06010 .01966 .01819 .06382 .05695
PHASE= 1.96559 2.70036 1.73841 2.47748 5.72382
1.24285 4.81509 1.05558 1.45477 1.75424
S : P902, TID= .00000000E+00, AZEro= .0000000E+00
AMPL= 1.57991 .53022 .32657 .13741 .06480
.06194 .02034 .01849 .05709 .05045
PHASE= 1.98618 2.69086 1.71814 2.47395 5.74498
1.30748 4.82661 1.14444 1.48267 1.83034
# TIHARM = 360

15. Simona harmonic tides
REFDATE: calculate 𝑉0,𝑓 and 𝑢
TSTART -> TSTOP: each step y = f𝑀 cos 𝑉0 + 𝜔𝑡 + 𝑢
TIHARM recalculate 𝑓 and 𝑢
REFDATE = 1996-01-01
REFDATE = 2015-01-01 0
9993600 10519200
525600

16. Radial speeds in WAQUA vs Schureman
25-07-2016 16
M2 8.43113E-03 28.9840899 28.9841042 -0.1250771
MSP2 8.44308E-03 29.0251710 NA
MKS2 8.45503E-03 29.0662521 NA
M2(KS)2 8.47892E-03 29.1483798 NA
2SN(MK)2 8.54440E-03 29.3734835 NA
LABDA2 8.56829E-03 29.4556113 29.4556253 -0.1229112
2MN2 8.58949E-03 29.5284915 NA
L2 8.58949E-03 29.5284915 29.5284789 0.11054092
L2A 8.58949E-03 29.5284915 NA
L2B 8.59219E-03 29.5377734 NA
2SK2 8.70275E-03 29.9178507 NA
T2 8.71470E-03 29.9589318 29.9589333 -0.0132949
S2 8.72665E-03 30.0000129 30.0000000 0.11270683

17. 25-07-2016 17
Restults with IHO Tidal committee
frequencies

18. Restults with IHO Tidal committee
frequencies
25-07-2016 18

19. Conclusion
• Specified radial frequencies were not precise
• Only problem for long runs or large TSTART
• Use REFDATE (and TIHARM!)
• Solution1a: use values provided by IHO tidal
committee, single precision
• Solution1b: use values provided by IHO tidal
committee, double precision
• Solution2: calculate phases (V) every time step
from orbit equations
25-07-2016 19