Thanks to the RISE (Rotation and Interior Structure Experiment) radio science experiment, part of NASA's InSight mission (Interior Exploration using Seismic Investigations, Geodesy, and Heat Transport), Mars' rotation and spatial orientation can now be determined with high precision. In particular, variations in the Length-of-Day (LOD) have been measured with an accuracy of 0.002 milliseconds (equivalent to 2 milliarcseconds (marcsec)), and nutations have been resolved to within a few marcsec (Le Maistre et al., 2023). The nutation data confirm that Mars' core is at least partially liquid.

The nutation of Mars is mainly induced by the Sun, since its moons Phobos and Deimos are very small, and their tidal effects remain weak (Baland et al., 2020). Consequently, the dominant nutation frequencies are the annual term and its sub-harmonics (semi-annual, ter-annual, quarter-annual, etc.). The amplitudes of the principal nutation components for the rigid-Mars nutation and the non-rigid contributions are shown in Fig. 1.

The annual term corresponds to one Martian year (about 1.88 Earthyears, i.e., 687 d). Their amplitudes reach more than 500 marcsec in the prograde frequency band (positive periods in Fig. 1) and more than 130 marcsec in the retrograde frequency band (negative periods in Fig. 1). These nutations are particularly valuable because they provide constraints on the interior structure of Mars, especially the size and state of its core (Dehant et al., 2000, 2009, 2011, 2012; Le Maistre et al., 2023). In addition, a nutation of about 5 marcsec amplitude with a period of 826 d has been identified, arising from a combination of terms involving both the orbital period of Phobos and the orbital motion of Mars. Note that the InSight nutation observations of Mars have not revealed the presence of a free mode, unlike for Earth (free FCN – Free Core Nutation at 430 d).

Seasonal variations in Mars' length of day (LOD) are substantial because a significant fraction of the atmosphere – about one quarter – participates in the sublimation and condensation cycle at the polar ice caps. Remarkably, these variations are comparable in magnitude to those on Earth, despite Mars having a much thinner atmosphere.

The internal structure of the Martian mantle has been inferred from seismic data (Banerdt et al., 2020; Drilleau et al., 2021, 2024; Stähler et al., 2021; Khan et al., 2021; Samuel et al., 2023) and spacecraft-derived gravity field measurements (e.g., Smith et al., 1999; Konopliv et al., 2016). These observations provide evidence for subsurface mass anomalies, associated viscous relaxation, and dynamic topography – i.e., topography at the core-mantle boundary caused by flow and pressure from mantle convection, which reflects and sustains the mantle's internal structure (Wieczorek, 2015). Recently, Charalambous et al. (2025) presented evidence for kilometre-scale heterogeneities throughout Mars' mantle using seismic data from InSight. Kiefer et al. (1996) used long-wavelength topographic and gravity data to infer the mantle's thermal structure, revealing the presence of deep-seated anomalies. Their results highlight dominant contributions from spherical harmonic degrees 2 and 3. As evident from any global image of Mars, and particularly from altimetry data, the northern hemisphere is lower than the southern hemisphere – a characteristic known as the Martian dichotomy (Zuber et al., 2000). This dichotomy is quantified by the degree-1 coefficient of Mars' surface topography. This large-scale hemispheric asymmetry is clearly visible on Mars, which exhibits relatively young northern lowlands and heavily cratered, older southern highlands. Recently, Goossens and Sabaka (2025) examined the degree-1 gravity associated with dichotomy, also clearly mentioned in Wieczorek et al. (2019), although they noted that the current resolution is insufficient for a robust determination of the corresponding large-scale density variations. Another distinctive feature of Mars is the presence of the vast Tharsis volcanic province, which has undergone prolonged volcanic activity over an extended period (Wieczorek, 2015). Zuber and Smith (1997) attempted to “remove” Tharsis from the Martian gravitational field in order to isolate its specific contributions. In doing so, they identified Tharsis as contributing primarily to degree-2, orders 0 and 2, as well as to degree-3, orders 0 and 3 – although the latter components were found to be relatively small. Defraigne et al. (2001) proposed the presence of a mantle upwelling plume beneath Tharsis to explain the observed geoid anomaly. However, due to uncertainties in the observations, it remains impossible to discriminate between different mantle models that incorporate such a plume and its effects on core–mantle boundary (CMB) topography. In this study, we assume a solid viscous mantle overlying the liquid core and disregard the possible existence of a basal liquid layer in the mantle, as suggested by Khan et al. (2023). Seismic data indicate that the Martian core is unusually large (Stähler et al., 2021), a result corroborated by radioscience measurements (Le Maistre et al., 2023). With a core of this size, mantle pressures are insufficient to trigger a ringwoodite–bridgmanite phase transition in the lower mantle. Consequently, the predicted CMB topography, ranging from 1 km to 5 km, is greater than in cases with a post-spinel phase transition. The impact of deviation from hydrostatic equilibrium at degree 2–order 0 of that level on the FCN was already highlighted in Defraigne et al. (2001). These results have motivated us to investigate the effects of CMB topography, expressed in spherical harmonics – particularly for degree-1, degree-2, and degree-3 – on the FCN, nutations, and LOD variations. Adding further interest to the liquid–solid boundary, Khan et al. (2023) and Samuel et al. (2023) recently showed that InSight data require the presence of a fully molten, ∼150 km-thick silicate layer overlying the liquid iron core. While this explains seismic observations, it might complicate the determination of the FCN from the geodetic data. This issue is still under study.

For comparison, in the case of Earth, nutation observations as well as variations in the length of day (LOD) provide valuable constraints on the deep interior, and in particular on coupling mechanisms at the core–mantle boundary (CMB). For nutations, following an approach similar to that of Koot et al. (2010), we recently re-evaluated the coupling constants at both the CMB and the inner core boundary. The revised values allow for additional coupling mechanisms beyond viscous and electromagnetic coupling. The incorporation of updated atmospheric and oceanic corrections to the nutation data has led to the determination of a larger CMB coupling constant than that reported by Koot et al., thereby leaving more room for non-hydrostatic or topographic coupling (Cheng et al., 2026). In addition, the electromagnetic coupling strength estimated by Mathews et al. (2002; see also Buffet et al., 2002) is likely overestimated, since the skin depth of magnetic diffusion at diurnal timescales is smaller than the thickness of any plausible electrically conducting layer at the base of the mantle and the lower mantle conductivity might even be smaller than that of the core. Although turbulence could enhance the effective viscosity, it is unlikely to increase it by several orders of magnitude (Shih et al., 2023). Consequently, topographic coupling remains a viable mechanism to account for the observed nutation. However, as shown for the Earth (Puica et al., 2023; Dehant et al., 2025), the pressure torque acting on the CMB is proportional to the product of topography coefficients εlkεl′k′ when the topography is expressed in spherical harmonics and normalized by the core radius. It is therefore of second order in the topography, which additionally decreases with increasing degree of the spherical harmonics (Kaula's law, see Puica et al., 2023). The pressure torque can therefore be expected to be much smaller than the pressure torque associated with the hydrostatic flattening of the core (ε20hydrostatic), in particular for high spherical harmonic degrees l, as the topography amplitude decreases with increasing degree. Nevertheless, the expression of the torque involves functions of inertial wave frequencies, which can lead to resonance effects when the tidal forcing frequency approaches the inertial wave frequency. These inertial waves, governed by the restoring effect of the Coriolis force, have periods of the order of weekly, diurnal, or sub-diurnal cycles and experience limited damping. As a result, they are strong candidates for influencing nutations and short-period variations in LOD, provided a resonance effect occurs. This consideration has motivated us to analyse these cases through a spherical harmonic expansion of the topography up to degree 24, beyond which the topography amplitude becomes too small to affect observations, even near resonance.

In our papers published in 2023 (Puica et al.) and in 2025 (Dehant et al.), we reviewed the state-of-the-art and discussed the literature available at that time. Since then, in a recent paper, Monville et al. (2025) presented topography-coupling computations using local models of the CMB that incorporate rotation, buoyancy, and magnetic effects in a nonlinear regime. Due to the presence of both a magnetic field and stratification in the core, similarly as in the work of Seuren et al. (2026), their models include not only inertial waves but also Magneto-Archimedean-Coriolis (MAC) waves and Rossby waves. They compute the drag force exerted by core flows on the solid mantle and conclude that only large horizontal length scales contribute significantly to the CMB pressure torque. In our study, we focus exclusively on inertial waves (shorter timescale), as Mars currently lacks an active magnetic field. In addition, Guervilly and Dormy (2025) conclude that core convection in the absence of a magnetic field is dominated by the inertial scale, which is hundred times larger than the viscous scale. Note that these studies neglect turbulence. In parallel, Oliver et al. (2025) show that turbulence in Earth's core can generate large topographic torques on the mantle – an effect not considered in the present work. Similarly, Rekier et al. (2025) investigated the form-drag effect, which may provide a viable explanation for the observed CMB coupling constant (Koot et al., 2010; Cheng et al., 2026).

To set the stage, we examine here the variations in Mars' orientation and rotation that could potentially be induced by the pressure torque acting on the topography at the core–mantle boundary (CMB) (see Fig. 2).

In the presence of a rapidly rotating liquid core, inertial waves may develop within the core if suitably excited. These waves are oscillatory motions of fluid primarily controlled by rotation, rather than by buoyancy or elasticity. They propagate along characteristic directions inclined with respect to the rotation axis. For specific geometrical configurations, inertial waves can form global resonant modes within the core. At particular latitudes – known as critical latitudes – the wave characteristics become tangent to the boundary, leading to locally amplified velocities and the generation of internal shear layers that propagate into the fluid interior. If present, such motions could influence the planet's orientation and rotational dynamics. Topography at the CMB may provide a mechanism for exciting these waves. In this paper, we compute the pressure effects exerted on the topography using an analytical approach to determine the torque applied to the CMB. The general expression for this torque is presented in Sect. . We examine the impact on nutations in Sect.  and on LOD variations in Sect. . More precisely, in Sect. , we derive the analytical expression for the dynamic pressure amplitude, expanded in spherical harmonics, by considering the CMB flow boundary conditions (Sect. ). The corresponding torque, also expressed analytically, depends on the topography coefficients developed in spherical harmonics (Sect. ). The nutation amplitudes are then derived from the coupling constant associated with this torque (Sect. ). We compare our results with those obtained for Earth in a recent study (Dehant et al., 2025) in the appendix of this paper. In Sect. , we analyse the LOD case, which is more straightforward, as the analytical expression for pressure takes a simpler form. We also compare our findings with those for Earth, as presented in a recent study by Puica et al. (2023) (see Appendix). Finally, in Sect. , we present our conclusions.

In the context of modelling Mars' orientation and rotational dynamics, the Liouville equations, which describe the conservation of angular momentum, form the theoretical foundation for quantifying how external torques – such as those from the atmosphere or solar tides – and internal torques between core and mantle affect the planet's spin axis orientation – nutations – and length-of-day variations (Wu and Wahr, 1997; Puica et al., 2023; Dehant et al., 2025). The presence of a pressure torque at the bumpy core-mantle boundary (CMB), Γtopo, introduces an additional term in the Liouville equations – beyond the pressure contribution due to the core's hydrostatic flattening (ε20hydrostatic). To quantify this torque, the CMB radius is represented using a spherical harmonic expansion: 1rCMB=R[1+ε00+(ε20)hydrostaticP20cos⁡θ+∑n=1∞∑m=-nnεnmYnm(θ,λ)] where εnm are the normalized coefficients of the CMB topography expansion, R is the mean CMB radius, and Ynm(θ,λ) are the fully normalized spherical harmonic functions, defined in terms of the fully normalized associated Legendre functions Pnm by 2Ynmθ,λ=Pnmcos⁡θeimλ with θ,λ representing colatitude and longitude. Following Wu and Wahr (1997), we express the normalized dynamic pressure Φ as: 3Φ=∑l=1lmaxalkPlkσ2Ylkθ,λ where σ is the frequency in a mantle-fixed reference frame, and the coefficients alk are determined by using the boundary conditions that relate the Poincaré motion (the main flow for nutations) or the global core faster or slower rotation with respect to the mantle (the main flow for LOD tidal effects) to the inertial waves. In our previous work for the Earth (Puica et al., 2023, for LOD; Dehant et al., 2025, for nutations), we employed a fully analytical approach, in contrast to Wu and Wahr (1997), who used a semi-analytical method.

For the nutation case, we consider the x- and y-components of the torque Γtopo (Γ1topo and Γ2topo, respectively) on the diurnal timescale. For Length-of-Day (LOD) variations, we focus on the z-component of the torque (Γ3topo). The equatorial torque components can be expressed as (Dehant et al., 2025) 4Γ1topo=(-iAfΩ2)158π×12∑l=1∞∑k=-ll(-1)kl-kl+k+1εl-k-1+l-k+1l+kεl-k+1Plkσ2alk5Γ2topo=(AfΩ2)158π×12∑l=1∞∑k=-ll(-1)k-l-kl+k+1εl-k-1+l-k+1l+kεl-k+1Plkσ2alk or in a complex form as 6Γ̃topo(σ)=Γ1topo+iΓ2topo=(-iAfΩ2)158π×∑l=1∞∑k=-ll(-1)kl-kl+k+1εl-k-1×Plkσ2alk The polar component can be expressed as (Puica et al., 2023) 7Γ3topo=(iAfΩ2)158π×12∑l=1∞∑k=-ll(-1)kkεl-kPlkσ2alk In these expressions, Af is the mean core moment of inertia and Ω is the planet mean rotation frequency. The factor 158π is coming from the definition of Wu and Wahr of the torque, involving ρfΩ2R5 where ρf is the core density.

In Eq. (), the coefficients alk are substituted by using boundary conditions that relate the Poincaré motion (ωf×r) – where r denotes the position vector of a fluid particle and ωf represents the vectorial wobble of the core relative to the mantle (Sasao et al., 1980) – to the inertial waves induced by the bumpy core-mantle boundary (Greenspan, 1969). In Eq. (), the coefficients alk are substituted using the boundary conditions corresponding to the global core rotation relative to the mantle, with ωf3=-ω3, where ω3 is the third (axial) component of the mantle's rotation. These boundary conditions are described in detail in the Appendices A and B of Wu and Wahr (1997). Since the torque components (Eqs.  and ) are proportional to either εl-k-1alk or εl-kalk, and the coefficients alk themselves are proportional to the topography, the torque components exhibit a quadratic dependence on the topography.

The effects of CMB topography on the tidal variations in Mars' LOD can be calculated using a similar approach to that described above (Puica et al., 2023, for the Earth). For LOD, and analogously to the boundary conditions for nutations (Eqs.  and ), we have a system of equations of the form: 22ALODalk=fεlk where each equation for a particular order l and degree k involves a bracket …lk on the LHS and a function of εlk on the RHS. The solutions yield expressions for the coefficients alk given in Eq. (28) of Puica et al. (2023). The zeros of the determinant of the matrix ALOD, which correspond to the zeros of the …lk brackets, indicate resonance effects.

We then substitute the solution of Eq. () for alk into the expression for the torque, which is in turn substituted into the expression of the tidal variations in LOD, ΔLOD(σ). Finally, the LOD variations can be expressed as a function of the axial component of the mantle wobble, m3: 23ΔLOD(σ)=-iΓ3AmσΩ2=AfAm158π×∑l=1lmax∑k=-ll(-1)kk2εlkεl-kPlkσ2(4-σ2)4σ…lkm3 where the product εlkεl-k clearly shows the quadratic dependence of the LOD on the topography. Resonances occur only when the denominator term …lk=0. Figure 5 shows the determinant of the matrix ALOD. As in Fig. 4 (nutation case), the presence of resonances in LOD is indicated when the curve crosses the zero line on the x axis. The closer these crossing are with respect to the tidal periods involved in LOD (and referenced with the symbols given in the legend of the figure), the higher the resonance would be.

As shown in Fig. 5 there are no resonances close to the Martian tidal frequencies of 668.6, 334.3, 222.9, 167.15, 133.7, and 111.4 Martiand (shown with symbols explained in the legend). The nearest periods to a resonance (shown when the curve is crossing the 0-line on the x axis) are the 1/6 year (or 111.4 d) and 1/4 year (or 167.15 d) tides, which are approximately 7.9 and 4.4 d away from an inertial wave resonance (at 103.5 and 162.7 d), respectively. These distances are too large to produce significant resonant effects, particularly given that the associated spherical harmonic degrees (15 and 22) correspond to extremely small topographic coefficients (ε15-1 and ε22-2). In contrast, for Earth, several tidal periods shorter than one month led to resonances near the 13.63 and 27.6 d tides (see Puica et al., 2023).

Following our previous studies on topographic effects on Earth's LOD and nutations – where potential resonance effects with inertial waves were demonstrated – we applied our analytical framework to planet Mars. The existence of core modes within Mars is plausible, as its core has been shown to be liquid. In the absence of a significant magnetic field, these modes are essentially inertial waves, with the Coriolis force acting as the primary restoring mechanism.

In our model, we considered primary a possible degree-1 topography related to Mars' hemispheric dichotomy, as well as low-degree topographic features potentially associated with the dynamic loading from the Tharsis region at the CMB. We also considered higher degree effects through a Kaula-law extrapolation of the topography.

Our analysis of Martian nutations reveals a potential resonance at 168.60 Martiand, which lies close to the 1/4 year prograde nutation period at 167.15 Martiand – differing by just over one day. This nutation exhibits an amplitude of 18.4 marcsec, which is significant at the level of radioscience observations. However, despite this proximity, the actual topography contribution to nutation remains negligible, as the resonance involves a high degree (degree-18) and therefore small topographic amplitude.

Unlike Earth, where LOD variations can be significantly influenced by low-degree CMB topography, Mars shows no comparable resonant effects near its tidal periods. As a result, we find no evidence of substantial inertial wave resonance contributing to LOD variations on Mars. Generally speaking, in our analytical case, we found a quadratic dependence on the topography amplitude, both for Earth and Mars. As a result, the effects of CMB coupling are generally small, unless resonances occur.

Our study does not consider the effect of turbulence at the CMB. Using numerical simulations of hydrodynamic convection in a rotating spherical shell with boundary topography, Oliver et al. (2025) recently showed that turbulent flow in the core of the Earth can lead to a pressure torque on that mantle of sufficient magnitude to explain LOD variations. These results indicate a linear dependence of the topographic torque on the CMB topography. However, it should be noted that the torque acting on the CMB is proportional to the square of the flow speed. The flow speed considered for Earth is associated with the geodynamo and core convection, processes that are absent in Mars. Similarly, Rekier et al. (2025) investigated the form-drag effect, which may provide a viable explanation for the observed CMB coupling mechanisms.