# Caspian Sea Tidal Modelling Using Coastal Tide Gauge Data

Authors: Mahmoud Pirooznia; Seyyed Rouhollah Emadi; Mehdi Najafi Alamdari
Journal: Journal of Geological Research
 (2016)

---

## Abstract

The purpose of this paper is to model tidal conditions in the Caspian Sea using data from coastal tide gauges of Anzali, Noshahr, and Neka Ports. Harmonic analysis method was used to identify and examine 40 tidal components. The results illustrate that the annual (Sa) and semiannual solar (Ssa) components on all of the ports listed have the highest range in comparison with the other components which are, respectively, 16 cm, 18 cm, and 15 cm for annual components and 2.8 cm, 5.4 cm, and 3.7 cm for semiannual components.


---

## Body

## 1. Introduction

The gravitational attraction between Earth, moon, and sun has a well-known influence on Earth’s ocean levels [1].Tidal analysis and prediction are the primary steps in the studies of each hydrodynamic and coastal management matter. It is dealing with the analysis of the sea level heights observations using physical and geophysical approaches advanced from experiences or physical reasoning and based on spectral analysis [2, 3].In general, for the modelling of tides, there are three main ways: first is using hydrodynamic models (Laplace differential equations (LTE)), second is empirical modelling, and third is combination of hydrodynamic models and empirical models [4].Knowing the causes of water level fluctuations of the seas has been one of the major challenges in all scientific fields and attracts the attention of many researches. Oceanic effects, aerologic effects, tides, climate change, and vertical movement of Earth’s shell can be noted as five factors that contribute to the impacts of climate change on the water level fluctuations [5].The prediction is used in science and engineering to confirm the sympathetic of a given phenomenon by stating what its behavior has been till now and will be at a given time and then verifying that it is so [3], although without considering the physical effects of the large events (e.g., tsunami, storms, and seabed earthquake) the prediction of water levels can be achieved [6].Not only the tides currently are exactly identified in the most global ocean, but also we have learned and computed new features of tidal dynamics [7].Tidal effects due to their significant impacts on sea water level are of a great importance and researchers have always been looking forward to modelling them [8].However, numerous undesirable natural events may have an effect on the predictions, but still there are determinable parameters which are essential to handle them exactly; for example, astronomical arguments and nodal corrections are singing the important parts in dealing with tidal analysis and prediction [9].In this study, the effectiveness of tidal components by the use of coastal tide gauge’s observations is surveyed. Harmonic analysis method was used to determine tidal amplitude and phase of 40 components.(1) Study Area and Relevant Data. Caspian Sea is surrounded by five countries, including Iran, Russia, Azerbaijan, Turkmenistan, and Kazakhstan (Figure 1). The Caspian Sea is the largest remaining section of the old Tethys Sea breakdown that was spread from the Arctic to Indian Ocean through the first to third geological period. In third geological period, appearance of Caucasus and Asian mountains leads to this big Sea dividing, so rise of European continent and construction of Iranian plateau were the main reasons of creation of Caspian Sea. Having the length of about 1030 km and the width of 435 to 196 km, the Caspian Sea locates between the 47°57′ an 36°33′ circuits and 46°43′ and 54°53′ hour circles [10].Figure 1
Caspian Sea location.


## 2. Tides

Tides are the regular ebbing and flowing movement of the sea happening as the result of attractions of celestial near-earth bodies such as the sun and the moon. Tidal acceleration of celestial objects such as the moon at one point is the difference between the gravity acceleration of that celestial body and the mass centre of the Earth at that point. Viscoelastic earth changes due to tidal forces are roughly one-third of the surface international water. The vector field of these forces can be replaced with a scalar field named potential of tides [11].In each point of Earth, this potential can be computed by [12](1)
U
R
e
,
λ
,
ϕ
=
G
M
R
∑
i
=
2
∞
R
e
R
i
P
i
cos
⁡
θ
,where G is the universal constant of gravity, M is the mass of absorbing body (moon or sun), R
e is the average radius of Earth, R is the geocentric distance, and θ is the geological distance of a point with R
e
,
λ
,
ϕ coordinates [1]. Main course of this relation is achieved when i
=
2. However, in some cases, i
=
3 is also used for the moon. Thus, the main indicator term of tidal potential is described as follows:(2)
U
2
R
e
,
λ
,
ϕ
=
3
G
M
R
e
2
4
R
3
cos
⁡
2
θ
+
1
3
.And regarding spherical trigonometry relations [4],(3)
U
2
R
e
,
λ
,
ϕ
=
∑
i
=
0
2
U
2
m
R
e
,
λ
,
θ
3
G
M
R
e
2
4
R
3
cos
2
⁡
ϕ
cos
2
⁡
δ
cos
⁡
2
H
+
sin
⁡
2
ϕ
sin
⁡
2
δ
cos
⁡
H
+
1
-
3
s
i
n
2
ϕ
1
3
-
s
i
n
2
δ
.It can be seen clearly from this relation that the first parameter of the relation relates to the half-daily effects and the second and third parameters, respectively, show the effects of daily and long period of tides. This relationship suggests that the tidal potential is a function of the absorbent body (moon and sun). It should be noted that, except for the gravitational forces mentioned above, other factors may be effective in producing and intensifying the tides [13].In general, the sea water level fluctuations can be obtained as the result of the interaction of following dynamic processes: tides, changes in atmospheric pressure, the dynamic effects of ocean circulation, wind effects, the effects of temperature, water salinity changes, the effects of river discharge into the oceans, and melting of polar ices.


## 3. Tidal Harmonic Analysis

In this study, the Fourier harmonic analysis method is used to determine the effect of tidal components. As tidal components frequency, water level in each moment and time was given, and the amplitude and phase of needed component were determined [14]:(4)
u
φ
,
λ
,
t
=
M
S
L
ϕ
,
λ
+
∑
i
=
1
n
a
i
Φ
,
λ
cos
⁡
ω
i
t
+
b
i
φ
,
λ
sin
⁡
ω
i
t
,where u
(
φ
,
λ
,
t
) is water level in t moment obtained by tide gauge, MSL(
ϕ
,
λ
) is mean sea level, ω
i
=
2
π
f
i is angular frequency derived from tidal components frequency, t is the observation time, and a
i and b
i are the Fourier coefficients we need to determine. Considering (4) and proration, the amplitude and the phase are calculated as(5)
u
φ
,
λ
,
t
1
u
φ
,
λ
,
t
2
⋮
u
φ
,
λ
,
t
m
m
×
1
︸
L
m
×
1
=
1
cos
⁡
ω
1
t
1
sin
⁡
ω
1
t
1
⋯
cos
⁡
ω
n
t
1
sin
⁡
ω
n
t
1
⋮
⋮
⋮
⋯
⋮
⋮
1
cos
⁡
ω
1
t
m
sin
⁡
ω
1
t
m
⋯
cos
⁡
ω
n
t
m
sin
⁡
ω
n
t
m
m
×
2
n
+
1
︸
A
m
×
2
n
+
1
a
1
b
1
⋮
a
n
b
n
M
S
L
2
n
+
1
×
1
.Ifm was observed value from tide gauge and n was the tidal component, then(6)
A
m
×
2
n
+
1
X
2
n
+
1
×
1
=
L
m
×
1
.A is Vandermonde (pattern) matrices [15]. As the result,(7)
X
^
=
A
T
A
-
1
A
T
L
.And the amplitude and phase of tidal components are determined from the following equation [16, 17]:Amplitude:(8)
A
i
=
a
i
2
+
b
i
2
.
Phase:(9)
Φ
i
=
t
a
n
-
1
b
i
a
i
o
r
Φ
i
=
2
t
a
n
-
1
b
i
A
i
+
a
i
.


## 4. Tide Modelling

The following equation is used to analyze and predict the tide [18, 19]:(10)
h
t
=
Z
0
+
∑
f
n
t
H
n
cos
⁡
δ
n
t
-
g
n
+
V
n
t
0
+
u
n
t
.Andn
=
1,2
,
…
,
N tidal components.In the above equation,Z
0 is the average sea level, N is the number of tidal components, δ
n is angular frequency or speed (degree per hour), V
n
(
t
0
) is astronomical argument, f
n
(
t
) is nodal factor, u
n
(
t
) is nodal phase, H
n is the component amplitude, and g
n is the phase lag.In above relations, nodal correctionsf
n
t
,
u
n
t for each tidal component and astronomical argument (
V
n
(
t
0
)
) must be calculated in order to determine the revised amplitude and phase. In this step, V
n
(
t
0
) is computed as(11)
V
n
t
0
=
i
b
λ
s
t
+
i
c
λ
h
t
+
i
d
λ
p
t
+
i
e
λ
N
t
+
i
f
λ
P
′
t
+
Φ
n
,where i
b
,
i
c
,
i
d
,
i
e
,
i
f are the constituent’s Doodson numbers in which solar coefficients are preferred to lunar coefficients and the desired time is the time data starting in zero hour UT (Universal Time) and remaining terms are as follows:(i)
λ
s
(
t
) or simply “s” is the Mean Longitude of Moon.
(ii)
λ
h
(
t
) or simply “h” is the Mean Longitude of Sun.
(iii)
λ
p
(
t
) or simply “P” is the Longitude of Lunar Perigee.
(iv)
λ
N
(
t
) or simply “N” is the Longitude of Lunar Ascending Node.
(v)
λ
P
′
(
t
) or simply “P
′” or “P
1” is the Longitude of Perihelion.
(vi)
Φ
n is stationary phase (a multiple of 90°).Stationary phase exists in most of Doodson numbers’ tables. Algebraic formula to determine the geographic Astronomical Longitude (the Celestial Sphere)λ
s
t, λ
h
t, λ
P
t, λ
N
t, λ
P
′
t or in some references in form of (s
,
h
,
P
,
N
,
P
′) has been provided [20, 21]. In this study, Task-2000 method has been used.Task-2000 is based on zero hour epochs UT, January 1900. This formula is accurate at least from 1800 to 2100 [22]:(12)
λ
s
t
=
277.0247
+
129.38481
·
I
Y
+
13.17639
·
D
L
,
λ
h
t
=
280.1895
-
0.23872
·
I
Y
+
0.98565
·
D
L
,
λ
P
t
=
334.3853
+
40.66490
·
I
Y
+
0.11140
·
D
L
,
λ
N
t
=
259.1568
-
19.32818
·
I
Y
-
0.05295
·
D
L
,
λ
P
′
t
=
281.2209
+
0.017192
·
I
Y
,where(13)
I
Y
=
y
e
a
r
-
1900
,
D
L
=
I
L
+
I
D
A
Y
-
1
.In this formula, IL is the number of leap years from 1900 onward that is calculated by the following equation:(14)
I
L
=
I
Y
-
1
4
.IDAY is the number of days from January 1st of that year.


## 5. Forming the Observations Time Series of Coastal Tide Gauge

Features of tide gauge stations which are formed from their time series data were in Table1 and geographical position of tide gauge stations could be seen in Figure 2.Table 1
The location of Caspian Sea coastal tide gauge stations.
Station name
Latitude
Longitude
Time spam
Anzali
37.4780°
49.4623°
21/3/2005–20/3/2014
Noshahr
36.6584°
51.5047°
21/3/2006–21/3/2014
Neka
36.8502°
53.3656°
1/1/2000–31/8/2012Figure 2
Display studied stations.The analysis of time series Anzali tide gauge station and modelling time series by (10) could be seen in Figure 3. Residuals are small and acceptable. More amplitude to frequency of 40 tidal components of Anzali tidal gauge station could be seen (Figure 4). The values of time series analysis in Anzali station as well as the values of time series modelling analysis are shown in Table 2. It is shown that the modelling was done well. The same process was continued for stations Noshahr and Neka (Figures 5–8). Also, results of time series analysis and results of time series modelling in stations Noshahr and Neka are shown in Tables 3-4.Table 2
The amplitude and phase of 40 tidal components for Anzali tide gauge station, obtained from observations and modelling.
Constituent
Amp_obs (m)
Phase_obs (deg)
Amp_mod (m)
Phase_mod (deg)
K2
0.003015
295.7626
0.00208
286.43
L2
0.001672
271.2841
0.00171
271.46
M2
0.008682
269.2956
0.0084
269.245
N2
0.003698
158.6862
0.00374
159.22
Ma2
0.003133
296.7903
0.00314
296.79
R2
0.017878
73.22793
0.01812
78.22
S2
0.012064
67.14119
0.01206
67.141
T2
0.005672
144.9542
0.0065
142.954
J1
0.001707
121.5101
0.0017
121.8201
K1
0.022738
11.43623
0.0233
11.3
M1
0.007332
67.24207
0.0073
68.45
O1
0.006945
165.1889
0.00693
165.1923
P1
0.00704
217.9986
0.007245
218.43
Q1
0.008204
232.6201
0.008325
232.56
S1
0.008428
83.31833
0.008267
84.673
M3
0.0072
93.20983
0.00687
93.65
S3
0.002717
119.0538
0.00281
119.487
M4
0.005868
233.6994
0.00589
233.996
S4
0.006028
85.39538
0.006518
85.7
M5
0.015845
21.3593
0.01522
21.487
M6
0.004693
239.5097
0.00467
240.43
S6
0.004883
193.316
0.0049
193.76
M8
0.008275
265.0774
0.00865
265.9
S8
0.003187
159.8985
0.00376
158.34
Mf
0.006459
109.5878
0.006217
107.789
Mm
0.007524
134.4543
0.00776
136.89
Msf
0.000792
77.13807
0.000734
76.54
Oo1
0.006236
355.2012
0.006236
355.34
Ssa
0.028563
55.42608
0.02862
53.56
Ms4
0.070979
15.67987
0.0704
15.456
Mn4
0.018667
186.2534
0.01889
186.789
Mk3
0.02533
282.7036
0.0255
282.408
Sa
0.163234
212.3174
0.163198
212.456
Mo3
0.007534
219.1039
0.007423
218.56
No3
0.026181
8.642835
0.026789
9.43
2N2
0.006548
46.04756
0.006437
47.13
So3
0.028834
137.9187
0.02889
136.67
Sk3
0.006522
44.34284
0.00657
44.389
S01
0.001092
68.53536
0.001034
68.576
Mk4
0.022139
331.8166
0.0234
331.8834Table 3
The amplitude and phase of 40 tidal components for Noshahr tide gauge station, obtained from observations and modelling.
Constituent
Amp_obs (m)
Phase_obs (deg)
Amp_mod (m)
Phase_mod (deg)
K2
0.001265
205.3896
0.001276
204.9
L2
0.000134
104.9257
0.000144
103.89
M2
0.010247
275.8484
0.01028
276.896
N2
0.002429
110.3829
0.002534
110.789
Ma2
0.002005
73.94501
0.002007
74.289
R2
0.00083
248.9937
0.00076
249.3397
S2
0.001758
10.64485
0.001777
10.5689
T2
0.00104
252.2587
0.001067
252.678
J1
0.000575
38.258
0.000589
38.378
K1
0.003133
206.5065
0.003145
206.45
M1
0.000804
197.7024
0.000809
197.745
O1
0.001833
74.5144
0.001845
74.78
P1
0.001664
171.5155
0.001643
172.504
Q1
0.000887
255.0884
0.00076
255.56
S1
0.002105
352.2459
0.00167
352.214
M3
0.000044
284.4275
0.000245
284.545
S3
0.000222
10.67813
0.00678
10.63413
M4
0.000773
269.5824
0.00989
269.6724
S4
0.000389
334
0.000567
333.89
M5
0.000592
308.7407
0.000345
308.437
M6
0.000816
107.9719
0.00424
107.901
S6
0.001758
297.3552
0.00987
296.9552
M8
0.001783
300.9042
0.00345
300.405
S8
0.035218
128
0.03789
128.2362
Mf
0.005655
63.45231
0.00679
64.82
Mm
0.008585
157.1591
0.00567
156.2271
Msf
0.000698
171.7511
0.000778
171.842
Oo1
0.001602
126.1866
0.00189
125.731
Ssa
0.054341
86.44686
0.05345
86.461
Ms4
0.000389
142.9945
0.00927
143.18
Mn4
0.000375
12.73982
0.00678
13.7
Mk3
0.000346
67.02355
0.00345
66.425
Sa
0.188683
208.5804
0.18907
208.2541
Mo3
0.000228
52.28991
0.00789
51.75
No3
0.000178
216.8859
0.000845
217.323
2N2
0.000519
37.64077
0.000612
37.772
So3
0.000414
105.4683
0.000432
105.94
Sk3
0.000306
334.2423
0.000351
334.58
S01
0.000225
195.3219
0.000376
194.218
Mk4
0.000424
74.62624
0.000387
74.947Table 4
The amplitude and phase of 40 tidal components for Neka tide gauge station, obtained from observations and modelling.
Constituent
Amp_obs (m)
Phase_obs (deg)
Amp_mod (m)
Phase_mod (deg)
K2
0.000609
248.292
0.00071
248.35
L2
0.000094
210.8618
0.000078
211.23
M2
0.003386
211.6024
0.00421
212.316
N2
0.000451
33.09812
0.000418
33.789
Ma2
0.000203
2.616894
0.000206
3.269
R2
0.00027
266.0452
0.000261
266.169
S2
0.001491
353.4808
0.00152
353.724
T2
0.000322
84.26617
0.000341
83.972
J1
0.0002
222.6129
0.000237
222.7034
K1
0.003984
232.586
0.004112
232.213
M1
0.000075
18.18968
0.000087
18.226
O1
0.001979
22.82131
0.002141
24.718
P1
0.000709
221.6676
0.000801
237.18
Q1
0.000071
199.8688
0.000076
200.1073
S1
0.001333
265.0889
0.001343
265.623
M3
0.000014
238.5899
0.000017
238.3041
S3
0.000127
71.53864
0.000143
70.946
M4
0.000041
106.7546
0.000052
106.9928
S4
0.00007
53.35676
0.000083
53.468
M5
0.000036
289.5994
0.000038
289.783
M6
0.00004
274.9504
0.000047
275.7935
S6
0.000042
59.32671
0.00004
58.87
M8
0.000071
0.493216
0.000082
0.78465
S8
0.000007
343.1489
0.000007
343.1376
Mf
0.003968
358.5273
0.003965
357.5186
Mm
0.001392
37.48796
0.001323
38.418
Msf
0.002203
332.5874
0.002212
333.64
Oo1
0.000043
172.098
0.000043
172.487
Ssa
0.037471
90.08432
0.0375
91.508
Ms4
0.000045
225.9218
0.00005
226.5
Mn4
0.000063
309.734
0.00006
310.47
Mk3
0.000063
168.7787
0.00007
169.681
Sa
0.151559
218.163
0.1518
217.2045
Mo3
0.000028
42.31016
0.000027
43.401
No3
0.000055
308.456
0.000054
307.731
2N2
0.000032
294.6541
0.000034
295.391
So3
0.00005
94.33536
0.00006
95.591
Sk3
0.000114
73.86118
0.000145
74.584
S01
0.000171
47.30391
0.000182
47.835
Mk4
0.000058
116.3563
0.000072
116.2482Figure 3
Time series along with the tidal modelling and residual of Anzali tide gauge station.Figure 4
Amplitude to frequency of 40 tidal components of Anzali tidal gauge station.Figure 5
Time series along with the tidal modelling and residual of Noshahr tide gauge station.Figure 6
Amplitude to frequency of 40 tidal components of Noshahr tidal gauge station.Figure 7
Time series along with the tidal modelling and residual of Neka tide gauge station.Figure 8
Amplitude to frequency of 40 tidal components of Neka tidal gauge station.


## 6. Conclusion

Tidal analysis of tide gauge stations illustrates the absence of efficiency in tidal components except for annual and semiannual components. However, unlike other stations, in Anzali station, MS4 components were effective.The results also indicate that the annual (Sa) and semiannual solar (Ssa) components on all of the ports listed have the highest range in comparison with the other components which are, respectively, 16 cm, 18 cm, and 15 cm for annual components and 2.8 cm, 5.4 cm, and 3.7 cm for semiannual components.By comparing the results with other studies [23, 24], the results of tidal analysis of this study are confirmed. The effect of the tides in the Caspian Sea is very small and there are only annual and monthly variations.In [23], spectral analysis on tide gauge observation of Caspian Sea was done where the annual and semiannual component effect were observed. So it is corresponding with tidal analysis of this study.Ignoring the modulation of solar perihelion and nodal modulation use can be mentioned as the weak points which were effective in tidal harmonic analysis; 18.6 time series must exist for resolving all frequencies. As for the other defects of this study, we can point out that there is no easy way to indicate the significance of amplitude and phase together with lack of appropriate solution for coastal areas that affects the shape of tidal waves.In this study, the method of modelling and predicting of sea level is explained. With modelling sea level, the trend of sea level in future will be predicted and it helps coastal management. Also, in areas where effect of tide is high, by having sea level prediction with modelling, tide can be used to assess other factors affecting sea level variations like tsunami, vertical movement of the crust, and so forth.The main problem with Caspian Sea tide gauge stations data is improper collecting and compiling them. Moreover, the sampling intervals in Anzali and Noshahr tide gauges are daily and each three hours, respectively, that can affect the tidal modelling and will decrease the accuracy of the computation.



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*Source: 6416917-2016-07-26.xml*