# Do ocean tidal signals influence recovery of solar quiet variations?

**Part of the following topical collections:**

## Abstract

*Z*, is affected the most such that at some locations the tidal signals explain the majority of the observed daily variation. Further, horizontal tidal magnetic fields at the sea floor are larger in amplitude and exhibit different spatial structures compared to signals estimated at the sea level. We propose a scheme aimed at correcting data for the ocean tidal signals and show that such correction suppresses the tidal signals in the observed field variations.

## Keywords

Sq variations Tidal magnetic fields EM induction## Abbreviations

- Sq
solar quiet

- EM
electromagnetic

- SH
spherical harmonics

- 3-D
three-dimensional

- S3D
a scheme for a source determination incorporating the 3-D ocean effect

- X3DG
a code to compute the EM field in a spherical model of the Earth with a 3-D conductivity distribution

- TPXO8-atlas
global model of ocean tides

## Introduction

Electromagnetic (EM) induction sounding with natural sources contributes to our knowledge of the composition, temperature, and presence of fluids in the Earth’s interior. Among other methods, daily ionospheric solar quiet (Sq) variations can be used to probe the electrical conductivity of the upper mantle (e.g., Schmucker 1999b; Olsen 1998; Koch and Kuvshinov 2013, 2015). Sq variations originate from an Sq electric current system flowing in a thin ionospheric E-layer. This current system is driven by atmospheric tides of predominantly thermal origins in the ambient magnetic field of the Earth and is active on the sunlit side at midlatitudes (Campbell 1989; Yamazaki and Maute 2017).

The Sq source morphology varies daily, seasonally, and yearly depending on short- and long-term solar magnetic activity and the Earth’s orbital position. Therefore, in order to mitigate these various effects, it is reasonable to work with single-day magnetic recordings when analyzing Sq variations. In addition, bearing in mind the complex geometry of the Sq source, one should preferably choose geomagnetically quiet days during or close to the time around equinoxes (March–April, September–October) so that the source will have a relatively simple, symmetric double vortex structure. Moreover, by choosing equinoctial months, we can further avoid possible magnetic field daily variations caused by the magneto-tail current during solstice months (Lühr et al. 2017).

Assuming that Sq variations are periodic, one can represent them via a sum of time harmonics with periods of 24, 12, 8, 6, etc., h (Matsushita and Maeda 1965). At the same time, diurnal and semidiurnal lunar and solar oceanic tides induce magnetic signals with periods close or equal to 12 and 24 h (Egbert and Erofeeva 2002). In short time series, e.g., in single-day data, signals with close periods become inseparable (see “Appendix” for more details). Consequently, ocean tidal signals inevitably affect Sq variations at periods of 12 and 24 h if the length of the time series is as short as one day. If not accounted for, the effect of ocean tidal signals in Sq variations can be mistakenly attributed to being part of the Sq signals, for instance, those induced by the mantle conductivity anomalies. This prompted us to assess the effect of the ocean tidal signals in daily variations and implement a correction scheme for them.

## Governing equations

## Modeling of Sq magnetic signals

### Frequency domain modeling

*r*is the distance from the Earth’s center, \(a = 6371.2\) km for Earth’s mean radius, and \(\nabla _H \mathrm {\Psi }\) is the tangential gradient of the stream function

*n*and

*m*are, respectively, the degree and order of the spherical harmonic (SH) \(Y_{n}^{m} = P_n^{\left| m\right| } (\cos \vartheta ) e^{\text {i}m\varphi }\) with \(P_n^{\left| m\right| }\) given by the Schmidt quasi-normalized associated Legendre polynomials. Note that the stream function, \(\mathrm {\Psi }\), enables a condensed representation of the Sq current system (Sq source); for example, the contours and sign of the stream function specify the direction of \({\mathbf {J}}_{\text {H}}\).

*n*,

*m*combinations describing the Sq source are adopted from Schmucker’s paper (Schmucker 1999a). Following his method (see Appendix B in Schmucker 1999a), the double sum in Eq. (4) is given by

*p*are time harmonics with characteristic Sq variation periods \(T = \frac{24}{p}\). The amplitudes of Sq variations subside toward higher

*p*values, and the time harmonics \(p>6\) corresponding to variations with periods shorter than 4 h no longer contribute to the total signal. By setting \(K = 4\) and \(L=1\), which control the number of, respectively, local-time terms (\(m = p\)) and general terms (\(m \ne p\)) per time harmonic (Schmucker 1999a), we obtain 12 SH combinations for each

*p*, except the \(p=1, n=0, m=0\) combination, which is forbidden since \(\nabla \cdot {\mathbf {B}} = 0\) should be satisfied everywhere (Sabaka et al. 2010). Hence, Eq. (5) gives a total of 71 SH functions (11 for \(p=1\) and \(5\times 12\) for \(p = 2,\ldots ,6\)) which are listed in Table 1.

Subsets of spherical harmonic terms used to describe the Sq source

p = 1 | p = 2 | p = 3 | p = 4 | p = 5 | p = 6 | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

n | m | n | m | n | m | n | m | n | m | n | m |

– | – | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 5 |

1 | 0 | 2 | 1 | 3 | 2 | 4 | 3 | 5 | 4 | 6 | 5 |

2 | 0 | 3 | 1 | 4 | 2 | 5 | 3 | 6 | 4 | 7 | 5 |

3 | 0 | 4 | 1 | 5 | 2 | 6 | 3 | 7 | 4 | 8 | 5 |

1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 6 | 6 |

2 | 1 | 3 | 2 | 4 | 3 | 5 | 4 | 6 | 5 | 7 | 6 |

3 | 1 | 4 | 2 | 5 | 3 | 6 | 4 | 7 | 5 | 8 | 6 |

4 | 1 | 5 | 2 | 6 | 3 | 7 | 4 | 8 | 5 | 9 | 6 |

2 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 7 | 7 |

3 | 2 | 4 | 3 | 5 | 4 | 6 | 5 | 7 | 6 | 8 | 7 |

4 | 2 | 5 | 3 | 6 | 4 | 7 | 5 | 8 | 6 | 9 | 7 |

5 | 2 | 6 | 3 | 7 | 4 | 8 | 5 | 9 | 6 | 10 | 7 |

*p*). The unit magnetic fields are calculated on a regular, global grid at an observation level

*r*, which can be at the ground, \(r = a\), at the sea floor, \(r < a\), or above the ground, \(r > a\), e.g., at a satellite altitude.

### Time domain modeling

## Modeling of tidal magnetic signals

### Frequency domain modeling

Tidal constituents used in this study

Constituent | Name | Period |
---|---|---|

| ||

Lunisolar | K2 | 11 h 58 min |

Principal solar | S2 | 12 h |

Principal lunar | M2 | 12 h 25 min |

Elliptical to M2 | N2 | 12 h 39 min |

| ||

Lunisolar | K1 | 23 h 56 min |

Principal solar | P1 | 24 h 4 min |

Principal lunar | O1 | 25 h 49 min |

Elliptical to O1 | Q1 | 26 h 52 min |

*k*represents the

*k*th tidal constituent, \(\omega _k\) is the corresponding angular frequency, \(\sigma _s\) is the depth-averaged seawater conductivity, \({\mathbf {u}}\) are the depth-integrated seawater transport, and \({\mathbf {B}}^{\text {main}}\) is Earth’s main magnetic field. Note that, in contrast to the Sq extraneous current that flows above the Earth’s surface, the tidal extraneous current is confined to the oceans. We used the TPXO8-atlas global ocean tide model (Egbert and Erofeeva 2002) for the seawater transport \({\mathbf {u}}\), the IGRF-12 model (Thébault et al. 2015) for \({\mathbf {B}}^{\text {main}}\), and a climatological-derived model for \(\sigma _s\) (Grayver et al. 2016). We calculate the magnetic fields \({\mathbf {B}}^{\text {Tides}}\) due to \({\mathbf {j}}^{\text {ext}}\) by solving the system of Eq. (1).

### Time domain modeling

Amplitude and phase modulating factors \(f_k\) and \(u_k\) in year 2011

Constituent | \(f_k\) | \(u_k\) (in rad) |
---|---|---|

| ||

K2 | 0.9802 | 0.3097 |

S2 | 1 | 0 |

M2 | 1.0049 | 0.0368 |

N2 | 1.0049 | 0.0368 |

| ||

K1 | 1.0001 | 0.1547 |

P1 | 1.0015 | 0.0110 |

O1 | 0.9947 | − 0.1920 |

Q1 | 0.9940 | − 0.1884 |

**O**regon State University

**T**idal

**P**rediction

**S**oftware (OTPS2) (Egbert and Erofeeva 2002). As an example, Table 3 lists \(f_k\) and \(u_k\) values for the year 2011. Equation (14) is adapted from Lord Kelvin’s tide prediction equation originally designed for the calculation of tidal heights (NOAA 2016). Note that OTPS2 uses this equation for prediction of both tidal elevations and tidal transports. A summary of the modeling steps described in this section is shown in Fig. 4.

## Results

In what follows, all results are presented by using \({\mathrm {X}} = -{\mathrm {B}}_{\vartheta }\), \({\mathrm {Y}} = {\mathrm {B}}_{\varphi }\), and \({\mathrm {Z}} = -{\mathrm {B}}_r\) conventions.

### Comparison of tidal and Sq magnetic fields

By using the methodology described in sections “Modeling of Sq magnetic signals” and “Modeling of tidal magnetic signals”, we modeled Sq and tidal magnetic fields in the time domain on the geomagnetically quiet, almost equinoctial day of March 16, 2011. Following ISGI (2016), a day is considered quiet if \(\text {mean}(aa)<13\) nT and \(\sum \text {p} <6\) were fulfilled over a 48-h interval (*CC*48 day). In this manner, we can ensure that the chosen day has no significant disturbances 12 h before and after the day of interest. Here, \(\text {p}\) should not be mistaken with the time harmonic *p*, and it denotes a weight assigned to an individual *aa* value according to ISGI (2016). For the chosen day, \(\text {mean}(aa) = 3\) nT and \(\sum \text {p} = 0\). Further, to minimize the influence from auroral and equatorial electrojets, only data from 73 midlatitude observatories between \(\pm\, 6^{\circ }\) and \(\pm\, 60^{\circ }\) geomagnetic latitudes were used.

*Z*component at the surface. For instance, the

*Z*component of the tidal signals at CZT at a period of 12 h is more than 2 times larger than the Sq signal. In fact, the tidal field at CZT on this day accounts for 70% of the sum (Sq+Tides). This indicates that the tidal magnetic signals at some locations significantly contribute to the observed daily variations.

Results for geomagnetic observatory CZT (Crozet Archipelago; South Indian Ocean)

24 h | 12 h | |||
---|---|---|---|---|

Tides | Sq | Tides | Sq | |

| 0.9757 | 15.6691 | 1.6755 | 8.6970 |

| 0.8408 | 11.5317 | 1.5266 | 11.6272 |

| 0.7490 | 0.7692 | 3.0297 | 1.2725 |

Results for a sea floor observation site T12-2 (Philippine Sea)

24 h | 12 h | |||
---|---|---|---|---|

Tides | Sq | Tides | Sq | |

| 0.8448 | 7.2619 | 1.3640 | 2.6494 |

| 1.1960 | 7.7596 | 2.3921 | 9.6571 |

| 1.1238 | 8.3727 | 2.7311 | 6.9216 |

*d*is the depth of the ocean, and \(\mathbf{E}_H\) is the horizontal electric field. The latter equation tells us that—along with the vertical magnetic field—Sq and tidal horizontal electric fields are continuous across the air–sea interface. Note that Eq. (15) is valid at periods longer than several hours.

Since the tidal magnetic signals at the period of 12 h are dominated by the semidiurnal M2 tide, which is known to generate the largest magnetic fields among all tidal constituents (Kuvshinov 2008), it is no surprise that the tidal signals at a period of 12 h, both at the sea level and sea floor (cf. bottom plots of Figs. 6, 7, 8, 9, 10), have larger amplitudes than those at the period of 24 h (cf. bottom plots of Figs. 11, 12, 13, 14, 15). Further, the Sq signals at the 24-h period have larger amplitudes than those at the 12-h period.

Figures 6, 7, 8, 9, 11, 12, 13, 14 also show that sea floor horizontal tidal magnetic fields exhibit a considerably different spatial pattern than those at the sea level. Overall, tidal signals at the sea floor are 2–3 times stronger than those at the sea level. The situation is opposite with the Sq signals; specifically, they are 2–3 times weaker at the sea floor due to the 3-D nature of the problem, namely the heterogeneous conductance of the continents and oceans. Maximum amplitudes of Sq signals are a few times larger than those of tidal signals both at the surface and sea floor. However, due to different global patterns of Sq and tidal signals, tidal signals may become comparable in amplitude and even exceed Sq signals at specific locations. This effect is more pronounced at the period of 12 h than that at 24 h for all three components, with the *Z* component affected most (cf. Figs. 10, 15).

The results presented in Figs. 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 refer to a specific day—March 16, 2011. However, the results of our modeling show that tidal signals vary substantially over the year since individual constituents modulate total signals depending on the phase (cf. Eq. 14). This behavior is shown in Fig. 16 for the geomagnetic observatory VAL (Ireland) and the sea floor site T12-2. Clearly, the peak amplitudes reach up to 5 nT and go as low as 1 nT.

### The correction scheme and its validation

To validate the correction scheme, we performed an experiment using 1 y of data at a ground observatory and sea floor site. The yearly time series are sufficiently long to enable the separation of the effects from Sq and tides in the spectral domain. This can be observed in the power spectral density (PSD) plots of yearly magnetic fields at VAL (Fig. 17, left) and T12-2 (Fig. 17, right). Peaks at precisely 12 and 24 h correspond to the dominant periods of Sq variations, whereas most prominent 12- and 24-h side peaks are induced by, respectively, the M2 and O1 tidal constituents. One can observe that the 24-h side peak is better seen in the sea floor results.

Figure 17 shows data before and after correction for the tidal magnetic signals. At the ground-based site, the 12-h side peak due to the M2 tide is significantly reduced in its amplitude for *Y* and *Z*. The remaining, unaccounted semidiurnal variations in *X* and diurnal variations are probably attributed to atmospheric lunar tidal variations with the same periods or imprecisions in the source definition and conductivity model. At the sea floor site, the tidal signals (both M2 and O1) are well suppressed in all components. Moreover, the corrected signal is reduced at a period of 12 h in the *Y* and *Z* components, which is most likely associated with the correction for the S2 tide. For all three components, small reductions in amplitude are also evident at the periods corresponding to the N2 and Q1 tides.

## Conclusions

In this paper, we assessed the effect of tidal and Sq magnetic signals in daily variations at the surface and sea floor. As expected, sea floor horizontal tidal magnetic fields differ substantially from those at the surface both in amplitude and in spatial structure. In comparison with the surface modeling results, sea floor tidal signals are 2–3 times stronger, while sea floor Sq signals are 2–3 times weaker. Even though Sq signals are a few times larger than the tidal signals for most regions, owing to their substantially different global patterns, tidal signals may exceed Sq signals at many locations. This, in particular, is plausible for the vertical magnetic field component *Z* of the daily variations.

We compared observed and modeled daily variations at several coastal and island observatories and concluded that at some locations the tidal signals could in fact explain the majority of the observed daily variations. We showed that the effect of tidal magnetic signals at numerous locations might be sufficiently strong relative to Sq variations, and neglecting tidal signals might lead to a misinterpretation of data when analyzing daily variations and assuming that these variations are of purely Sq origin. Furthermore, in the context of internal studies, where smaller internally induced Sq signals are to be delineated, the contribution of tidal magnetic signals becomes a significant deteriorating factor.

We proposed a numerical scheme to correct for the effect of tidal magnetic signals. Specifically, we proposed the subtraction of the modeled tidal magnetic fields from observations. Modeling was performed by using realistic tidal transports and conductivity models of the Earth. We validated the correction scheme by using 1-y long time series of magnetic fields at one ground-based and one sea floor site where the tidal signals appear to be substantial. We showed that such correction enables efficient suppression of the tidal signals in the observations.

## Notes

### Authors’ contributions

MG estimated the Sq source and analyzed the data. MG and AG performed the modeling of the tidal magnetic signals. AK provided the three-dimensional modeling code X3DG. AG and AK created the concept of the study. MG drafted the manuscript. All authors read and approved the final manuscript.

### Acknowledgements

The authors thank the British Geological Survey for collecting and distributing high-quality data for INTERMAGNET stations and the staff at the Pacific21 data center for making the data from the Stagnant Slab Project open. We further thank Takao Koyama and two anonymous reviewers for their helpful comments. This study was supported by the ETH Grant No. ETH-3215-2.

### Competing interests

The authors declare that they have no competing interests.

### Ethics approval and consent to participate

Not applicable.

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## References

- Alekseev D, Kuvshinov A, Palshin N (2015) Compilation of 3D global conductivity model of the Earth for space weather applications. Earth Planets Space 67(1):1–11. https://doi.org/10.1186/s40623-015-0272-5CrossRefGoogle Scholar
- Baba K, Utada H, Goto T-N, Kasaya T, Shimizu H, Tada N (2010) Electrical conductivity imaging of the Philippine Sea upper mantle using seafloor magnetotelluric data. Phys Earth Planet Inter 183(1):44–62CrossRefGoogle Scholar
- Campbell WH (1989) An introduction to quiet daily geomagnetic fields. Pure Appl Geophys 131(3):315–331CrossRefGoogle Scholar
- Egbert GD, Erofeeva SY (2002) Efficient inverse modeling of barotropic ocean tides. J Atmos Ocean Technol 19(2):183–204CrossRefGoogle Scholar
- Finlay CC, Olsen N, Kotsiaros S, Gillet N, Tøffner-Clausen L (2016) Recent geomagnetic secular variation from Swarm. **Earth Planets Space 68(1):1–18. https://doi.org/10.1186/s40623-016-0486-1CrossRefGoogle Scholar
- Grayver AV, Schnepf NR, Kuvshinov AV, Sabaka TJ, Manoj C, Olsen N (2016) Satellite tidal magnetic signals constrain oceanic lithosphere-asthenosphere boundary. Sci Adv 2(9):1600798CrossRefGoogle Scholar
- ISGI. http://isgi.unistra.fr/indices_aa.php. Accessed 8 Dec 2016
- Koch S, Kuvshinov A (2013) Global 3-D EM inversion of Sq variations based on simultaneous source and conductivity determination: concept validation and resolution studies. Geophys J Int 195(1):98–116CrossRefGoogle Scholar
- Koch S, Kuvshinov A (2015) 3-D EM inversion of ground based geomagnetic Sq data. Results from the analysis of Australian array (AWAGS) data. Geophys J Int 200(3):1284–1296CrossRefGoogle Scholar
- Kuvshinov AV (2008) 3-D global induction in the oceans and solid Earth: recent progress in modeling magnetic and electric fields from sources of magnetospheric, ionospheric and oceanic origin. Surv Geophys 29(2):139–186CrossRefGoogle Scholar
- Kuvshinov A, Semenov A (2012) Global 3-D imaging of mantle electrical conductivity based on inversion of observatory C-responses—I. An approach and its verification. Geophys J Int 189(3):1335–1352CrossRefGoogle Scholar
- Kuvshinov A, Olsen N (2005) Modelling the ocean effect of geomagnetic storms at ground and satellite altitude. In: Earth observation with CHAMP. Springer, Berlin, pp 353–358Google Scholar
- Lühr H, Xiong C, Olsen N, Le G (2017) Near-earth magnetic field effects of large-scale magnetospheric currents. Space Sci Rev 206:521–545CrossRefGoogle Scholar
- Matsushita S, Maeda H (1965) On the geomagnetic solar quiet daily variation field during the IGY. J Geophys Res 70(11):2535–2558CrossRefGoogle Scholar
- Maus S, Kuvshinov A (2004) Ocean tidal signals in observatory and satellite magnetic measurements. Geophys Res Lett 31(15):15313CrossRefGoogle Scholar
- NOAA. https://tidesandcurrents.noaa.gov/. Accessed 26 Oct 2016
- Olsen N (1998) The electrical conductivity of the mantle beneath Europe derived from C-responses from 3 to 720 hr. Geophys J Int 133(2):298–308CrossRefGoogle Scholar
- Olsen N, Glassmeier K-H, Jia X (2010) Separation of the magnetic field into external and internal parts. Space Sci Rev 152(1–4):135–157CrossRefGoogle Scholar
- Parker BB (2007) Tidal analysis and prediction. NOAA Special Publication NOS CO-OPS 3Google Scholar
- Püthe C, Kuvshinov A, Khan A, Olsen N (2015) A new model of Earth’s radial conductivity structure derived from over 10 yr of satellite and observatory magnetic data. Geophys J Int 203(3):1864–1872CrossRefGoogle Scholar
- Sabaka TJ, Hulot G, Olsen N (2010) Mathematical properties relevant to geomagnetic field modeling. In: Freeden W, Nashed MZ, Sonar T (eds) Handbook of geomathematics. Springer, Berlin, pp 503–538CrossRefGoogle Scholar
- Schmucker U (1984) Sources of the geomagnetic field. In: Landolt-Börnstein. Numerical data and functional relationships in science and technology. New Series. Group V: Geophysics and space research, vol 2: geophysics of the solid earth, the moon and the planets. Subvolume b. Springer, Berlin, pp 31–99Google Scholar
- Schmucker U (1999a) A spherical harmonic analysis of solar daily variations in the years 1964–1965: response estimates and source fields for global induction—I. Methods. Geophys J Int 136(2):439–454CrossRefGoogle Scholar
- Schmucker U (1999b) A spherical harmonic analysis of solar daily variations in the years 1964–1965: response estimates and source fields for global induction—II. Results. Geophys J Int 136(2):455–476CrossRefGoogle Scholar
- Thébault E, Finlay CC, Beggan CD, Alken P, Aubert J, Barrois O, Bertrand F, Bondar T, Boness A, Brocco L et al (2015) International geomagnetic reference field: the 12th generation. Earth Planets Space 67(1):1–19. https://doi.org/10.1186/s40623-015-0228-9CrossRefGoogle Scholar
- Yamazaki Y, Maute A (2017) Sq and EEJ—a review on the daily variation of the geomagnetic field caused by ionospheric dynamo currents. Space Sci Rev 206:299–405CrossRefGoogle Scholar

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