Main

Tides are universal phenomena and often play essential roles in planetary and galactic systems wherever gradients in gravitational attraction are important1,2,3. As the Earth’s sole natural satellite, the Moon and its gravitational interaction with Earth have attracted extensive research and curiosity over several hundred years4. Periodic lunar tidal effects have been observed in the Earth’s crust, oceans, near-ground geomagnetic field, atmosphere and ionosphere5,6,7,8,9,10,11,12,13,14,15,16. These lunar tides mainly have semidiurnal (and semimonthly) periods17,18,19,20,21 and are of fundamental importance to conditions on our planet. For example, the Earth’s crustal tide can trigger seismic22,23,24 and volcanic activity25, and the ocean tide can influence the flow of heat from equatorial to polar regions26 and the evolution of primordial terrestrial species from aquatic life27. Atmospheric tides have a global impact on rainfall28, and ionospheric tides can affect radio transmission and low-Earth-orbit satellite altitude9,29. If we follow the four states of matter, they progress from solid, liquid and gas to plasma, which is the dominant component of the Earth–Moon space environment. In the past, lunar tides were mainly found to affect the first three states: solid Earth tides, liquid ocean tides and neutral gas-dominated atmospheric tides. Whether lunar tides can influence the plasma-dominated regions, which are much more extensive in space, has not yet been explored.

The Earth’s plasmasphere is the most ideal and representative plasma-dominated place in which to study the existence of lunar tides in plasma, because its basic properties have been studied extensively and there are massive amounts of observational data. The plasmasphere is a collisionless magnetized plasma region that extends along geomagnetic field lines from the upper ionosphere (its source region), filling a roughly torus-shaped volume in near-Earth space. It is composed of cold (1–2 eV) and dense (102–104 cm−3) plasma with quasi-equal numbers of electrons and ions (~80% H+, ~10–20% He+ and ~5–10% O+), and it plays a key role in particle exchange and storage within the magnetosphere30,31. Given its cold, dense plasma properties, the plasmasphere can be regarded as a ‘plasma ocean’, and the plasmapause represents the ‘surface’ of this ocean because of the dramatic change of plasma properties across this boundary. The location of the plasmapause is thus an important parameter in the study of plasma dynamics within the inner magnetosphere32,33,34,35.

Previous studies have shown that the plasmapause location reflects the dynamics of the entire cold plasma, which can strongly affect energetic particle distributions in the inner magnetosphere by significantly influencing the excitation and propagation of electromagnetic waves, and subsequently impact radiation belt and ring current dynamics36,37,38. Hence, to explore the existence of a lunar tidal effect in this cold plasma ocean, it is natural to examine whether these ‘sea surface’ variations may be associated with the lunar cycle. Figure 1a,b shows two typical satellite images of the plasmasphere obtained by the Extreme Ultraviolet (EUV) imager on board NASA’s Imager for Magnetosphere-to-Aurora Global Exploration (IMAGE) spacecraft in polar perspective and the EUV camera aboard China’s Chang’e-3 lunar lander in lunar perspective, respectively, with corresponding plasmapause outlines indicated by white curves. The findings reported in this article demonstrate a significant lunar tidal effect in the plasmapause location, providing a causal link by which the Moon exerts an influence on magnetospheric dynamics. This expands our understanding of Earth–Moon interactions in a direction that has not been previously considered, and provides important clues for future investigations in broader regions and two-body celestial systems, including other planetary systems in our solar system and beyond.

Fig. 1: Overview of the plasma lunar tide observation in Earth–Moon space.
figure 1

a, A plasmaspheric image taken by the IMAGE EUV imager at 14:14 UT on 26 June 2000 viewed above the Earth’s North Pole. b, A plasmaspheric image captured by the Chang’e-3 (CE-3) EUV camera at 13:01 UT on 21 April 2014 viewed from the Moon. The EUV emission intensity (in Rayleigh; 1 Rayleigh = 106 photons cm−2 s−1 sr−1) is indicated by the colour bars at the right of the panels. A sharp drop in the emission intensity represents the plasmapause as denoted by the white curves. The white arrows indicate the direction of the Sun. The black circles indicate the apparent size of the Earth. c, Variations of ∆LPP versus MLT at LP = 0 (full Moon). d), Variations of ∆LPP versus MLT at LP = 6 (third quarter Moon). e, Variations of ∆LPP versus MLT at LP = 12 (new Moon). f, Variations of ∆LPP versus MLT at LP = 18 (first quarter Moon). ∆LPP is plasmapause perturbation data in units of RE (1 RE = 6,371 km). The high tide peaks are marked by the red dashed lines. The vertical error bars in panels c–f denote the 95% confidence intervals.

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A large database of plasmapause positions35 determined by 50,778 plasmapause crossings from the Time History of Events and Macroscale Interactions during Substorms (THEMIS) mission and other missions is used in this study (the largest such database compiled to date, detailed in the Methods, and examples of THEMIS and Cluster plasmapause crossings are shown in Extended Data Fig. 1). The survey period is from November 1977 to December 2015, covering almost four solar cycles (21, 22, 23 and 24). This provides a unique opportunity to study lunar influences on plasmapause position by eliminating external factors, such as effects from solar activity. In this investigation, 35,982 (~71% of the total) in situ plasmapause crossings under low-activity geomagnetic conditions (Dst > −50 nT, AE < 300 nT and Kp ≤ 3, where Dst is the disturbance-storm-time index, AE is the aurora electrojet index and Kp is the 3-hour averaged geomagnetic activity index, respectively.) were selected to reduce the possible effects of varying solar wind and geomagnetic activity and minimize statistical uncertainties. To extract the lunar tide signal, a database of plasmapause perturbations categorized by lunar phase (LP) was compiled and used for the following investigations (Methods and Extended Data Figs. 2 and 3).

Figures 1 and 2 show conclusive evidence of a lunar tidal effect on the plasmapause location. For convenience in this study, LP is defined as the magnetic local time (MLT) of lunar position (Extended Data Fig. 4). Figure 1c–f shows the perturbations in plasmapause position as a function of MLT for LP of 0 h (full Moon), 6 h (third quarter Moon), 12 h (new Moon) and 18 h (first quarter Moon), respectively. These panels reveal that the high tide peaks (marked by red dashed lines) of the perturbations progress regularly with LP, and the MLT (longitude) of the high tide bulges is ~6 h (90°) ahead of the LP. This is further demonstrated below in detail. The plasmapause perturbation data (∆LPP) was binned in 2 h windows in both MLT and LP, and the resulting two-dimensional distribution of average values in each bin is shown in Fig. 2a. Figure 2b shows the two-dimensional discrete Fourier transform39 amplitude of this dataset, which unambiguously illustrates the diurnal and monthly (lunar cycle) variations. Figure 2c depicts the distribution of ∆LPP binned in 6 h windows in both MLT and LP, more clearly showing the existence of lunar tides in the plasmasphere, hereafter referred to as lunar tidal wave in the plasmapause (LTWP). Since there is only one high tide and one low tide for each MLT or LP, the period of LTWP is diurnal and monthly, contrasting with most of the previously observed lunar tides in other regions of the Earth system, which have predominantly semidiurnal and semimonthly periods40. It is also clearly shown in Fig. 2c that the MLT of the high tide is about 6 h ahead of the LP (longitude difference ~90°)—that is, they have a well-defined linear relationship that is expressed by MLT of the high tide = (0.956 ± 0.051) × LP + 6.336 ± 1.29. Given the clear synchronization with LP, it is logical to define a local time coordinate with respect to lunar (instead of solar) position, which we label lunar local time (LLT; with 12 h corresponding to lunar zenith, as illustrated in Fig. 2e). Figure 2d depicts LLT variations of the normalized perturbations in plasmapause position for each LP. As shown in Fig. 2c, the amplitude of ∆LPP is ±0.13 Earth radii (RE). These three panels clearly show the features of the lunar tide in the plasmasphere: the high tide (maximum of about 0.12RE) occurs at LLT = 18 h, while the low tide (minimum of about −0.14RE) occurs at LLT = 6 h, as illustrated in Fig. 2e.

Fig. 2: The characteristics of LTWP.
figure 2

a, The variation of ∆LPP binned in 2 h windows in both MLT and LP, where ∆LPP is plasmapause perturbation data in units of RE. b, The two-dimensional discrete Fourier transform amplitude of ∆LPP in a. The two red spots represent frequencies with the highest amplitude in MLT and LP, respectively. The black and white dashed lines indicate frequencies equal to 0 and 1/24 for reference, respectively. c, The variation of ∆LPP binned in 6 h windows in both MLT and LP. d, Variations of the normalization of c as a function of LLT at different LPs. The LLT is defined as (12 − LP + MLT) mod 24 with LLT = 12 corresponding to lunar noon. Detailed definitions of MLT, LP and LLT can also be found in the Methods and Extended Data Fig. 4. The grey shaded band represents the normalized 95% confidence interval error, which is defined here as the 95% confidence interval error divided by the difference between the maximum and the minimum of ∆LPP. e, Illustration of the LWTP in the LLT frame in the Earth–Moon space. The background plasmapause is shown by the blue circle. The modulation of the plasmapause by lunar tide is shown by the red dashed circle; left for low tide and right for high tide.

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To rule out the possibility of data anomalies and ensure the reliability of the results, the database is divided into two subdatasets in three different ways. In all cases, similar tidal signals are observed (Methods and Extended Data Figs. 5–7). We have made a video showing tidal changes at the plasmapause related to LP (Supplementary Video 1). It is worth noting that because the plasmapause location has also been found to be modulated by solar rotation (for example, corotating interaction regions) with a period of about 27 days41, one could reasonably suspect that the signal we are observing is in fact modulated by the Sun. However, our use of low geomagnetic activity data with a long-term average and the strong correlation of the high tide of LTWP with the LP rules out the possibility of the solar rotation effect for this signal.

The essential question now is how does such a lunar tide with diurnal and monthly periods occur in the plasmasphere? The motion of the cold plasma in the plasmasphere is primarily subject to E × B drift (where E is electric field and B is magnetic field) and thus the electric field is essential in determining the position of the plasmapause. The electric field in the inner magnetosphere is composed of the steady corotation electric field (Ecoro) that is determined by Earth’s magnetic moment and rotation, and the varying magnetospheric convection electric field (Econv) that is controlled by solar wind and geomagnetic activity. Figure 3a shows the perturbations (∆Er) of the radial component of the electric field Er (positive Er towards the Earth) measured by the Van Allen Probes (Methods) binned in 2 h windows in both MLT and LP, with the corresponding amplitude of its two-dimensional discrete Fourier transform shown in Fig. 3b. Both diurnal and monthly variations are clearly apparent in these plots, similar to the case for ∆LPP. Figure 3c shows ∆Er binned in 6 h windows in both MLT and LP, which more clearly shows the existence of a similar tidal disturbance in the electric field. Figure 3d shows variations of Er with LLT, which varies from 0.57 to 0.69 mV m−1. The maximum (minimum) Er occurs at LLT = 6 (18) h, which is opposite to LTWP (blue line in Fig. 3d). There is a strong negative correlation between the mean values of LTWP and Er for each sector with a correlation coefficient of −0.94. By subtracting the baseline of the Er (0.63 mV m−1, which agrees well with the averaged corotation electric field of 0.61 mV m−1 over the range of L values (McIlwain L, corresponding to the equatorial radius of a drift shell of a charged particle)42 from 3 to 6) in Fig. 3d, ΔEr is obtained and fitted by the function ΔEr = (0.0155 ± 0.0022) × cos((0.2618 ± 0.0002) × LLT) + (0.0578 ± 0.0012) × sin((0.2618 ± 0.0002) × LLT) with a correlation coefficient of 0.99.

Fig. 3: Independent electric field observation and cold plasma convection modelling.
figure 3

a, The perturbations (∆Er) of the radial component of the electric field Er (positive Er towards the Earth in this paper) measured by the Van Allen Probes binned in 2 h windows in both MLT and LP. b, The two-dimensional discrete Fourier transform amplitude of ∆Er in a. The black and white dashed lines indicate frequencies equal to 0 and 1/24 for reference, respectively. c, The perturbations of ∆Er binned in 6 h windows in both MLT and LP. d, Variations of Er (black) and ∆LPP (blue) with LLT, where ∆LPP is plasmapause perturbation data in units of RE. The vertical error bars in panel d denote the 95% confidence intervals. e,f, Perturbations of the plasmapause derived from the two plasmapause formation models, zero parallel force (ZPF) surface (e) and LCE (f).

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Subsequently, models for plasmapause formation are considered to evaluate whether such an electric field perturbation is consistent with and can reproduce the observed LTWP. One model is based on the kinetic theory of plasmapause formation (that is, via the physical process of interchange motion, driven unstable at the location where the Roche limit surface or ZPF surface, which becomes tangential to the magnetospheric convection-corotation streamlines)43,44,45,46. The simulated ∆LPP in Fig. 3e demonstrates that this model can successfully reproduce this tidal phenomenon (Methods). The correlation coefficient between observed and simulated ∆LPP is about 0.70, and the ∆LPP (approximately −0.15–0.14RE) obtained from the model agrees well with the observations (−0.14–0.12RE). Another model considers that the plasmapause coincides with the last closed equipotential (LCE) of the magnetospheric convection electric field, and thus it is called the LCE model47. This model is suggested to work under quiet or quasi-static geomagnetic conditions30. In our study, we apply this model with the limitation of Kp = 3. It is found that LTWP can be qualitatively reproduced with a 38% downscaling of observed ∆Er (Fig. 3f and Methods).

Therefore, both observations and model simulations suggest that the perturbations of plasmapause position result from the disturbed radial electric field. Understanding the causal link between the LP and the perturbed electric field is a subject of ongoing research. One possibility is that the neutral winds in the ionosphere, which are modulated by LP, can generate an electric potential difference that can be mapped along field lines threading the plasmasphere and perturb the radial electric field that modulates the plasmapause position. Preliminary model calculations support this hypothesis and will be further developed.

Another possible origin of the observed lunar tidal signal might be the Moon’s gravitational field perturbation on the ZPF surface, since the position and shape of the virtual ZPF depend on the field-aligned component of the Earth’s gravitational field. However, including the Moon’s gravitational effect in the ZPF theory just introduces a very small extra perturbation of ~0.01% on the plasmapause position (Methods), and thus is negligible.

Our discovery of this plasma tidal effect with distinct characteristics may indicate a fundamental interaction mechanism in the Earth–Moon system that has not been previously considered. Furthermore, reflected by this plasma tide, the plasma flow in the entire Earth–Moon space exists as a persistent background variation of the magnetosphere and can modulate Earth–Moon space dynamics continuously, although the observed perturbations caused by the lunar tide are often relatively small compared to those arising from solar activity. Since this plasma tide effect appears to be predictably fundamental, it may be expected not only in the plasmasphere, but over a much wider set of phenomena. For example, plasmapause surface waves can alter energy transport from the magnetosphere to the upper atmosphere48, while whistler mode chorus and hiss waves, and also electromagnetic ion cyclotron waves near the plasmapause contribute significantly to electron acceleration and loss in the Van Allen radiation belt 49,50. We suspect that the observed plasma tide may subtly affect the distribution of energetic radiation belt particles, which are a well-known hazard to space-based infrastructure and human activities in space. It is therefore worthwhile to look for evidence of this effect in future studies—for example, by checking for correlations of variations in the distribution of ‘zebra stripes’51 with LP. These have been suggested to be formed by a weak electric field independent of corotation52; however, the observed electric field modulated by this lunar plasma tide may contribute. As for how the LP adjusts the radial electric field, we suspect that the Moon may modulate the radial electric field by affecting the upper atmosphere.

Whether the magnetic field or plasma lunar tides seen in space are related to the Earth’s crustal and oceanic tide53 is also a question worthy of discussion. The configuration and structure of Earth’s cold plasma in relation to the magnetosphere is not unique to the Earth, and similar structures have been found at other planets54,55,56 and astrophysical objects57. Here, we can summarize that the three fundamental elements necessary for this plasma tidal signal are the existence of a two-body celestial system, plasma and a magnetic field. Because the planetary environments in the stellar system that meet all these conditions are very common, this plasma tide may be observed universally throughout the cosmos. Therefore, the finding of this lunar tidal effect in the plasmasphere not only extends our knowledge of the Earth–Moon system, but also open new perspectives for further studies of tidal interactions in other planetary and larger-scale systems.

Methods

Plasmapause database

The compilation of the plasmapause database is detailed in ref. 35. Here we briefly introduce the data sources and plasmapause determination methods. According to different satellite observation datasets, a variety of plasmapause determination methods have been developed. For the THEMIS and Polar satellites, the spacecraft potential (that is, the electric potential of the spacecraft body relative to the ambient plasma) and the electron thermal velocity are used to calculate the electron density. A detailed introduction of this method is given in ref. 58 and ref. 59. The estimated error (a factor of 2) of this method is smaller than the typical density changes across the plasmapause, thus this method has been widely used to determine plasmapause positions60,61. An example of the satellite potential and corresponding electron density measured by THEMIS is shown in Extended Data Fig. 1a,b. For the plasma wave instruments of various satellite missions (for example, International Sun–Earth Explorer-1, Plasma Wave Instrument of Dynamics Explorer 1, Plasma Wave and Sounder of Akebono, Plasma Wave Experiment of Combined Release and Radiation Effects Satellite, Plasma Wave Instrument of Polar, Radio Plasma Imager of IMAGE, Waves of High Frequency and Sounder for Probing of Density by Relaxation of Cluster, and Electric and Magnetic Field Instrument Suite and Integrated Science of Van Allen Probes), the electron density (ne (cm−3)) is given by ne = (\(f^{\,2}\)UHR − \(f^{\,2}\)ce)/8,9802, where fUHR is the upper hybrid resonance frequency, fce = eB/me is the electron cyclotron frequency, e is the electron charge magnitude, B is the strength of the magnetic field and me is the electron mass62. Since some satellites are not equipped with a magnetometer, B is obtained by the Tsyganenko 2007 external magnetic field model combined with the International Geomagnetic Reference Field internal magnetic field model63,64. In terms of deducing electron densities by the plasma wave instruments, this method has been proven to be very successful in many studies62,65,66,67,68. Extended Data Figure 1c,d illustrates a corresponding example of the spectrogram of the plasma waves measured by Cluster-4 and the electron density deduced from the upper hybrid resonance frequency from 02:00 universal time (UT) to 03:30 UT on 25 January 2002. The criterion for identifying the plasmapause adopted in previous studies and used in this investigation is that the electron density changes by a factor of five or more within a short distance of ~0.5RE Based on this criterion, the black vertical dashed lines mark the plasmapause position.

Plasmapause perturbations at different LP

To extract the lunar tide signal in the plasmapause positions, the averaged background profile of the plasmapause is constructed with the following steps. The MLT is divided into 241 bins with intervals of 0.1 h, and the average position of the plasmapause in each bin is calculated to obtain the averaged background profile of the plasmapause. This profile is subtracted from the plasmapause positions. Through this process, the dawn–dusk asymmetry and geomagnetic activity-induced variations of the plasmapause are almost eliminated. A database of plasmapause perturbations as a function of LP is formed and used for the following investigations.

Statistical uncertainty considerations and data processing

Although the database contains 35,982 plasmapause crossings, we need to divide these events into several bins in MLT and LP phase, to explore the possibility of a lunar tide signal (for example, we use 12 bins each—that is, MLT = 2:2:24 h and LP = 2:2:24 h, respectively). This reduces the level of confidence we can assert for the mean values of plasmapause location for each bin, due to the lower number of counts.

Extended Data Figure 2a shows the distribution of perturbations in plasmapause radial distance in the MLT–LP frame, ranging from −0.3 to 0.3RE. Extended Data Figure 2b shows the corresponding 95% confidence interval (ranging from 0.05 to 0.2RE, and higher on the dusk side due to plasmapause variations associated with evolving plasmaspheric plume structures, as shown in Fig. 1a,b). At first glance, Extended Data Fig. 2a appears to be dominated by random fluctuations in the range of ±0.3RE; however, a low-frequency underlying signal is also discernible. Given that these fluctuations are of the same order as the confidence interval, we apply spatial smoothing or low-pass filtering to enhance the underlying signal.

It is important to note that the smoothing window size within a period of an oscillatory signal does not change its periodic nature (although it can reduce its amplitude). Extended Data Figure 3a shows a simple example in which a sine function with added noise is smoothed using different smoothing windows. The function is y = sin(x) + rand(−1, 1), where rand(−1, 1) represents a random number between −1 and 1. Its period is 2π and the data within a period (0 ≤ x ≤ 2π) is shown in the figure by a black line. The red, green, blue and yellow lines represent the results of smoothing with a window size equal to 1/4π, 1/2π, 3/4π and π, respectively. It is shown that these smoothed results can more clearly display the changing trend of the data without changing the period of the data, but the amplitude of the data will decrease with the increasing size of the smoothing window.

Extended Data Fig. 3b–e shows the distribution of perturbations of plasmapause position in the MLT–LP frame with smoothing window equal to 3, 6, 9 and 12 h, respectively. We obtain a clear lunar tidal signal, which has diurnal (and monthly) periodicities, and this lunar signal becomes clearer as the smoothing window is increased. Therefore, to obtain clear results (accurate period and amplitude), we chose 6 h (a quarter of a period) for the smoothing window in this study.

Descriptions of MLT, LP and LLT

The MLT, LP and LLT variables all refer to spatial, azimuthal locations and are defined in Extended Data Fig. 4. The MLT is usually used to define azimuthal locations in the Sun–Earth reference frame. When looking from the North Pole, the MLT increases anticlockwise from 0 at midnight to 6 at dawn, 12 at noon, 18 at dusk and 24 at midnight. For convenience in this study, the LP is defined as the MLT of the lunar position with LP values of 0/24, 6, 12 and 18 corresponding to the full Moon, third quarter Moon, new Moon and first quarter Moon, respectively. The LLT is based on the MLT defined with respect to the Moon (instead of the Sun). LLT is always equal to 0 along the magnetic meridian intersecting the far side of the Earth–Moon line, and is always equal to 12 along the magnetic meridian intersecting the near side of the Earth–Moon line. For an azimuthal location P at MLT in Earth’s frame, when the Moon is at LP, the corresponding LLT is defined as (12 − LP + MLT) mod 24.

Van Allen Probes observations of radial electric field

In this study, the electric field data measured by the Van Allen Probes satellites between L values of 3–6 from January 2013 to May 2019 were used. The sensitivity of the Electric Fields and Waves instrument on board the Van Allen Probes is 0.1 mV m−1 or 10% of the amplitude69,70,71. Note that for Van Allen Probes spacecraft, the electric field component (Ex) along the spin axis is estimated using the assumption that E · B = 0 or the parallel electric field is zero under the conditions of |By/Bx| <4 and |Bz/Bx| <4. To obtain electric field data near the magnetic equator, we selected the electric field data within the magnetic latitude range between −15° and 15° and take its radial component to obtain observations of variations in Er with LLT. Here, the size of the smoothing window is 6 h; for example, LLT = 6 stands for 3 < LLT < 9. The method used to determine ∆Er is similar to the method of obtaining the ∆LPP—that is, the MLT is divided into 240 bins with intervals of 0.1 h, and the average radial component of the observed magnetospheric electric field (Er) in each bin is calculated to obtain the averaged background profile of the Er, and the perturbations (∆Er) of the Er equal to Er minus its interpolated background profile.

Including the disturbed electric field in plasmapause formation models

Since the motion of plasma in the plasmasphere is mainly controlled by the E × B drift, the configuration of the plasmasphere is primarily determined by the electric field, assuming a steady magnetic field. Accordingly, the electric field is assumed to be electrostatic (curl-free) and contains two parts. The first is Ecoro, calculated from the differentiation of the corotation potential72 ΦC = −ΩEμME/(4𝜋LRE), where ΩE is the rotation speed of the Earth, μ is the magnetic permeability, ME is the magnetic moment of the Earth’s dipole and L denotes the L value. The second is Econv, which is calculated with the Volland–Stern potential model73,74. Plasmapause formation models are adopted to verify whether the LTWP is generated by the lunar-tide-inducing electric field perturbations.

One model is based on the ZPF theory30. In this theory, the plasmapause position is expressed by LPP = (2GM/3Ω2RE3)1/3 = (2GM/3RE3)1/3Ω−2/3, where M is the mass of the Earth, G is the universal gravitational constant, RE is the Earth’s radius and Ω is the cold plasma angular velocity. The ZPF theory involves consideration of gravitation on cold plasma along magnetic field lines in the frame rotating with angular speed Ω. The above result is calculated for the case of the Earth in isolation. We therefore check whether the addition of lunar gravitation in this formulation can directly provide a perturbation to LPP that would explain the observed variations with LP. This results in a cubic equation for LPP, given by LPP3 − (2GM/3Ω2RE3)(1 − (MmLPP2/MLm2)cosφ) = 0 + O((LPP/Lm)3), where Mm is the mass of the Moon, Lm is the Earth–Moon distance and φ is the LP. The modification is in the scale of MmLPP2/MLm2 or 0.01% for LPP = 6.0 and Lm = 60.0, and is thus negligible. This indicates that variations in LPP must be due to perturbations in Ω in the ZPF theory.

For example, perturbations in Ω will result in changes in LPP that are given to first order by ΔLPP = −2/3(2GM/3RE3)1/3Ω−5/3ΔΩ = −2/3LPP(ΔΩ/Ω). Assuming a steady axisymmetric magnetic field strength near the plasmapause and ideal magnetohydrodynamics, we have ΔΩ/Ω = ΔEr/Er. Finally, we get ΔLPP = −2/3LPP(ΔEr/Er), where Er is the radial component of electric field and is the sum of Ecoro and Econv. Applying the observed ΔEr in Fig. 3d, the modelled perturbations of the plasmapause positions are shown in Fig. 3e.

Another model is based on LCE theory, which is suggested to be used under quiet or quasi-static geomagnetic conditions30, an electric field model consisting of curl-free Ecoro and Econv (Kp = 3) components and perturbation of the electric field. Here the observed tidal disturbance electric field ΔEr, which is a function of LP, is superimposed onto the convection-corotation electric field by changing Ecoro. We introduced an adjustable factor (f) and a linear scaling function w(L) for low L values (3 < L < 3.2) to define a model for the corotation potential:

$${\varPhi}_{{{{\mathrm{C1}}}}}\left( L \right) = {\varPhi}_{{{\mathrm{C}}}}\left( L \right) \times \left[ {1 - w\left( L \right) + w\left( L \right)/\left( {1 + f{\Delta}E_r/\overline E _r} \right)} \right]$$

where

$$w\left( L \right)\left\{ \begin{array}{l}0{{{\mathrm{,}}}}\quad \quad \quad \quad \quad \quad \quad L \le L_0\\ \left( {L - L_0} \right)/\left( {L_1 - L_0} \right){{{\mathrm{,}}}}\,L_0 < L < L_1\\ 1{{{\mathrm{,}}}}\quad \quad \quad \quad \quad \quad \quad L > L_1\end{array} \right.$$

and \(\overline E _r\) is the mean of Er, and the values of L0 and L1 are 3.0 and 3.2, respectively. Scanning the value of f in increments of 0.05, we find that the modelled ∆LPP best matches the observed ∆LPP when f = 0.3. The modelled perturbations of the plasmapause positions are shown in Fig. 3f. Under such conditions, when the electric potential distribution is deviated to get the Er distribution, the mean value of ∆Er is calculated to be 38% of the observed one between L values of 3 and 6.

The reasons for superimposing 38% ∆Er (as opposed to 100%) under different LPs onto the convection-corotation electric field might be as follows. (1) The amplitude of Er is of the same order as the Van Allen Probes Electric Fields and Waves instrument measurement error, hence the amplitude might be not well determined. (2) The radial electric field measured by the Van Allen Probes satellites will in general overestimate the value at the equator, assuming equipotential field lines. The electric field data used in this study are selected for the region of −15° < magnetic latitude < 15°. The error introduced by the magnetic latitude effect can be estimated based on the dipole magnetic field model. The maximum and average errors of off-equator observations are ~7% and ~2%, respectively. (3) The LCE model we have used here is a simplified and idealized model, which may qualitatively but not quantitatively describe the tidal phenomenon.

Validation of the observed tidal signal

To eliminate any interference and confirm the tidal signal in subsets of the data, we divided the dataset into two nearly equal subdatasets using different three methods: (1) by year, 1977–2006 (51%) versus 2007–2015 (49%); (2) by solar radio flux at 10.7 cm (F10.7), F10.7 ≤ 103 (50%) versus F10.7 > 103 (50%); and (3) by season, January–June (48%) versus July–December (52%). The 6 h smoothed results are each depicted in Extended Data Fig. 5, showing that the tidal signal can be distinguished in all the six subsets. Extended Data Figure 6 shows the non-smoothed results binned in 2 h windows in both MLT and LP, with corresponding two-dimensional discrete Fourier transform39 amplitudes shown in Extended Data Fig. 7, confirming that the diurnal and monthly variations are dominant. These results not only eliminate the possibility of data anomalies, but also increase the credibility of our results.

Visualizations of the lunar tide in the plasmasphere

Supplementary Video 1 shows tidal changes at the plasmapause using the same observational data as in Fig. 2a (here the smoothing window is 12 h for a clearer display). In panel a, the dashed line represents the background plasmapause position from all the 50,778 crossings and the solid line shows the perturbed plasmapause locations for different LPs, while the red cross represents the centre of the high tide for different LPs. Panel b shows the perturbations of plasmapause positions as a function of MLT for different LPs. This video shows that the ‘high tide’ peak of the perturbations moves regularly with the LPs. Note that the plasmapause perturbations are multiplied by 40 here.