Atmospheric contribution to the dissipation of the gravitational tide of Phobos on Mars


  • Roberto Rondanelli,

    1. Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA
    2. On leave from Department of Geophysics, University of Chile, Santiago, Chile.
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  • Vid Thayalan,

    1. Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA
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  • Richard S. Lindzen,

    1. Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA
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  • Maria T. Zuber

    1. Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA
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[1] Here, we investigate the possibility of a significant atmospheric contribution to the tidal dissipation of the Phobos-Mars system. We apply the classical tidal theory and we find that most of the gravitational forcing is projected onto the first symmetric Hough mode which has an equivalent depth of about 57 km and is significantly trapped in the vertical. Therefore, no significant dissipation occurs through the vertical propagation of energy and subsequent breaking of the tidal wave as the wave amplifies with height. Alternatively, from the energy stored in the first trapped mode we estimate that the time scale required for the dissipative mechanisms to account for the total dissipation of the tides is of order 102 s. This time scale is unrealistically short, since it would contradict observations of propagating thermal tides in Mars atmosphere. Therefore we conclude that the dissipation of the tidal potential that explains the observed acceleration of Phobos most likely occurs within the solid planet.

1. Introduction

[2] The tidal exchange of angular momentum between Phobos, the inner-most natural satellite of Mars, and Mars produces an acceleration in the orbital velocity of Phobos and an increase in the rotation rate of Mars. The most recent estimate for this acceleration comes from the observation of the shadow projected on Mars by the transit of Phobos, as measured by the Mars Orbiter Laser Altimeter instrument on the Mars Global Surveyor spacecraft [Bills et al., 2005]. This observed acceleration of 6.631 × 10−9/year is consistent with an energy dissipation of 3.34 MW. Assuming that all the dissipation occurs in the solid interior of Mars, modeled as a homogeneous Maxwell viscoelastic solid, the effective viscosity of the planet was inferred to be 8.7 × 1014 Pa s [Bills et al., 2005]. The implicit assumption in this calculation is that the tidal dissipation occurs exclusively in the solid planet. In this note we deal with the possibility that significant dissipation occurs in the planetary atmosphere by investigating the response of the Martian atmosphere to the gravitational tidal forcing by Phobos.

2. Summary of the Theory

2.1. Classical Tidal Theory

[3] Here, we briefly summarize the assumptions and main results of the application of classical tidal theory to a thin atmospheric shell as assumed for Mars. Detailed derivations and discussions can be found in work by Chapman and Lindzen [1970]. The tidal fields are assumed to be small perturbations about a basic state, so for instance the total density field ρtot can be written as,

equation image

where ρ0 is the basic state density which depends only on z the height above the surface, and ρ is the perturbation. The coordinates are the colatitude θ measured from the axis of rotation, the azimuth ϕ, the height z from the surface of the planet and the time t. The basic state is in hydrostatic balance and can be considered at rest with respect to the phase speed of the tide (c ∼ 240 m/s > U ∼ 10 m/s, where c is the dimensional phase speed of the tide and U is a typical scale for the horizontal velocity in Mars' atmosphere). The inviscid momentum equations, the adiabatic thermodynamic equation and the ideal gas law are linearized around this basic state.

[4] Since we are interested in the steady state response to a forcing with a known period and magnitude, the resultant tidal fields are assumed to be periodic in time, having the same frequency σ as the forcing (σ = 2 ωL), where ωL = ω − ωM is the relative rotation of the moon with respect to the planet, ω is the rotation rate of the planet and ωM is the rotation rate of the satellite around the planet (Table 1). Here, we are interested in the response to the dominant semidiurnal component of the forcing having a wave number s = 2. Under these assumptions the equations governing the atmospheric response are separable into an equation that contains only vertical dependence and an equation that contains only latitudinal dependence (time and azimuthal dependence are given by the forcing) [Chapman and Lindzen, 1970]. The latitudinal dependence of the perturbation fields is governed by the Laplace Tidal equation,

equation image

where μ = cosθ, f = σ/2ω, g is the gravitational acceleration on Mars, and a is the radius of Mars. Θns are the solutions of the equation, known as Hough functions and hns are the equivalent depths which represent the eigenvalues of the problem, the subscript n is used to identify the eigenfunctions and the eigenvalues. The solutions Θns and the corresponding hns are found by writing Θns as an infinite series of (normalized) associated Legendre polynomials. When the series expansion is substituted into equation 2, the problem reduces to finding the solution of an infinite set of linear equations for the coefficients (for a detailed discussion, see Chapman and Lindzen [1970]). The series expansions are believed to be a complete asymptotic representation of the solutions and have an optimal truncation which depends on n [Ioannou and Lindzen, 1993].

Table 1. Parameters for the Calculation of the Forcing
ωRotation rate of the planet7.08 × 10−5 rad/s
ωMRotation rate of the satellite2.27 × 10−4 rad/s
ωLRelative rotation rate−1.57 × 10−4 rad/s
σFrequency of the forcing−3.14 × 10−4 rad/s
MsMass of Phobos1.08 × 1016 kg
DDistance between Phobos and Mars9.38 × 103 km
aRadius of Mars3.40 × 106 m
εInclination of the orbit of Phobos

[5] The vertical structure of the modes is found through the solution of the equation

equation image

where Q2 = (N2H2/ghn − 1/4) is the index of refraction squared, N is the Brunt-Väisälä frequency, and H is the scale height of the basic state (H = RT0/g, where R is the gas constant for the Mars atmosphere and T0 is the basic state temperature). The functions yn are related to the rest of the tidal fields through the three dimensional divergence perturbation χn as yn = χnex/2, where x = ∫zdz/H is the log-pressure vertical coordinate. Assuming a perfectly spherical planet, the lower boundary condition is wtot = w = 0 and can be written as,

equation image

where γ is the ratio of the heat capacity at constant pressure and the heat capacity at constant volume and i is the complex imaginary unit. Ωn refers to the component of the gravitational forcing projected onto the Hough function of order n. At the upper boundary, we impose the radiation condition, which for propagating modes is equivalent to choosing the solution that has an upward flux of energy [Wilkes, 1949]. Once the gravitational forcing and the eigenvalues are known, equation 3 can be numerically integrated following a procedure similar to the one described by Lindzen [1990, p. 297]. In sum, any given perturbation field, for instance the density, can be written as,

equation image

[6] Since the forcing is symmetric around the equator of Mars, only symmetric eigenfunctions are considered (n = 2, 4,…). For the case of the northerly meridional velocity u and westerly zonal velocity v it is customary to write the fields in terms of associated functions Uns and Vns which contain all the θ-dependence of the fields.

[7] For simplicity, we have assumed a perfectly spherical Mars, without including any energy dissipation in the form of frictional drag with the surface topography, despite the degree 2,2 equatorial features that includes the Tharsis bulge [Smith et al., 1999].

2.2. Tidal Potential

[8] The gravitational tidal potential exerted by Phobos at an arbitrary point a over the surface of the planet, measured from the center of Mars can be written as

equation image

where G is the gravitational constant, Ms the mass of the satellite, and D is the position vector of the satellite in a coordinate system centered at the center of mass of the planet (D = ∣D∣). The first two terms in the series expansion 6 are a constant potential with no consequence in the forcing, and a term which gives an homogeneous forcing equivalent to the acceleration experienced by the center of mass of the planet. For the purpose of specifying the tidal forcing we will be concerned only with the term of order equation image3. Then, the semidiurnal component of the gravitational potential due to satellite of mass Ms has the form,

equation image

where equation image(μ) is the normalized associated Legendre polynomial of degree (2, 2) and ε is the orbital inclination of the satellite. Despite Phobos being relatively close to the surface of Mars compared to the Moon, the smaller mass of Phobos dictates that the gravitational potential of the Moon over the Earth is still about two orders of magnitude larger than the gravitational potential of Phobos over Mars.

3. Results

3.1. Characteristics of the Response

[9] As expected from the relatively large value of f, the expansion of the Θns are dominated by the contribution of the corresponding equation image, that is, only a small correction from the spherical harmonics appears as a result of the rotation of the planet relative to the rotation of the moon. As a consequence, since the gravitational forcing can be expressed in terms of equation image most of the excitation goes into the first symmetric Hough mode–Θ22. From Table 2 we see that the contribution of each of the higher modes to the expansion of the gravitational potential is reduced by about two orders of magnitude relative to the preceding mode.

Table 2. Coefficients of the Expansion of the First Four Hough Modes in Terms of Normalized Associated Legendre Polynomialsa
hn [km]57.702814.92796.825523.91756
equation image53.378915.55347.358914.2823
4 κH/hn0.16040.62031.35662.3636
  • a

    Also shown are the equivalent depths corresponding to each mode (exact and approximated for large f) and the parameter 4 κH/hn, indicative of the vertical propagation of each mode in an isothermal atmosphere (T = 200 K). The data is for Mars and Phobos with f = −2.2152 and s = 2.


[10] One can get a rough idea of the vertical propagation of the modes by considering an isothermal version of equation 3. In that case the quantity equation image becomes simply equation image, where κ = (γ − 1)/γ. Therefore, when 4 κH/hn is less than one the corresponding mode propagates in the vertical, whereas when it is more than one the mode is vertically trapped. From Table 2 we see that at least the first two modes, which receive the bulk of the gravitational forcing, are vertically trapped. The consequence for the problem at hand is that the evaluation of the tidal dissipation cannot be done directly by calculating the upward energy flux equation image, where p and w are the pressure and vertical velocity tidal fields, respectively and the bar refers to average over one wavelength. We know that for vertically propagating modes the tidal fields grow with height as ρ0equation image and therefore it is expected that at some height the waves will become unstable and will break, dissipating the upward energy flux through the generation of turbulence. Such an explicit calculation can still be done for the higher modes which propagate in the vertical, that is n ≥ 6; however, as we will see, we find very modest contributions through this mechanism.

[11] A more realistic approach is to consider a varying vertical profile of temperature and evaluate numerically equation 3. We use as a reference the vertical profiles taken during the descent of the Viking landers [Seiff and Kirk, 1977]. It is evident from the Viking observations that these profiles show not only the background state of the atmosphere but also superimposed wave activity. We construct two profiles V1 and V2 (see Figure 1a) by smoothing the original Viking soundings and taking only the mean vertical lapse rate over five or six layers of about 40 km in height, roughly resembling the observed profiles. We use 1000 vertical levels over a vertical domain of 200 km using the numerical equivalents of the boundary conditions. Note that the magnitude of the tidal fields is not particularly sensitive to the characteristics of the basic temperature profile.

Figure 1.

(a) Basic temperature profiles for the two idealized soundings, Viking 1 (solid line) and Viking 2 (dashed line) (Profiles are idealizations from the Viking soundings [Seiff and Kirk, 1977]). (b) Q2, index of refraction squared for the first Hough mode. (c) Magnitude of the vertical component of the zonal velocity perturbation ∣v2(z)∣ for the first Hough mode. (d) Magnitude of the vertical component of pressure perturbation ∣p2(z)∣ for the first Hough mode.

[12] We present results for the first mode (n = 2). We see in Figure 1b that the refractive index Q2 remains negative over the entire atmosphere. As a consequence the magnitude of the velocity and temperature tidal fields grow only slowly with height, the slow growth is due to the competing effect of the decrease in density with height which amplifies the response and the decaying nature of the solution. Consequently these tidal fields remain within the same order of magnitude throughout the atmosphere (Figure 1c). Our model does not consider explicitly any dissipative mechanism that in the real atmosphere will likely damp any slow growth with height. The behavior of the pressure perturbation shows a strong decay in amplitude with height, which illustrates the fact that the tidal forcing is concentrated near the ground and, for this particular mode, the response does not propagate significantly with height (Figure 1d).

[13] To illustrate the relative (and small) importance of the higher order terms in the expansions of the tidal response Figures 2 and 3 show the latitudinal variation of the magnitude of the surface perturbation of u and v respectively. It can be seen that the magnitude of the sum of the first four modes is about the same order of magnitude (∼10−5 m/s) and it has almost the same latitudinal distribution as the first Hough mode.

Figure 2.

Magnitude of the meridional velocity tidal fields at the surface (∣un(0)Uns(θ)∣) for the first four Hough modes, along with the sum of these first four modes.

Figure 3.

Same as Figure 2 but for the zonal velocity.

[14] For the calculations in which we explicitly included the temperature profiles, we find that the first propagating mode is n = 10 (fifth symmetric mode). As anticipated, the vertical energy flux for this mode is very small and it only contributes a dissipation of the order of 10−11 W. In the next section we look for an alternative, namely that the dissipation occurs through an unknown process acting on the response to the first mode.

3.2. Energy Calculations

[15] Here, instead of explicitly calculating the dissipation in the atmosphere as the vertical flux of energy of the wave, we explore the consequences of assuming that the known value of the dissipation required to explain the orbital acceleration of Phobos occurs through the dissipation of the tidal energy exclusively in the atmosphere. To this end we calculate the total energy of the tide per unit volume E, which can be expressed as the sum of a kinetic and a available potential contribution [see, e.g., Gill, 1982],

equation image

[16] As we deduce from the previous section, the energy is concentrated in the gravest mode, n = 2 and the higher order contributions are at least two orders of magnitude smaller than for n = 2. For the purpose of estimating an order of magnitude for the total tidal energy, it suffices to consider only the contribution of the first mode. We look at the quantity

equation image

where the integral is taken over the volume of the atmosphere, V. When dE/dt is 3.34 × 106W, the dissipation consistent with the acceleration of Phobos orbit, then τdiss is the time scale required to account for such a dissipation within the atmosphere. The value of τdiss for the first mode ranges from 1.01 × 102 to 2.33 × 102 s, depending on whether the temperature vertical profile used is V2 or V1. The contribution of the higher order modes to the total energy (or to the total dissipation time scale) is negligible.

4. Concluding Remarks

[17] The dissipation time scale obtained in the previous section is extremely short compared to known dissipative processes in the Martian atmosphere such as the radiative damping time constant in the lower atmosphere (∼105 s) [Zurek et al., 1992]. As discussed by Chapman and Lindzen [1970], a local condition for neglecting the dissipation in the formulation of the classical tidal theory is that τdissequation image. For the Phobos-Mars system equation image = 3.18 × 103 s and therefore our assumed source of dissipation would not only be important in calculating the tidal response but would be a dominant term in such a way as to effectively damp the tidal response. We can be confident that dissipation time scales as short as that required to explain the tidal dissipation resorting exclusively to the Martian atmosphere are not realistic. Solar diurnal and semidiurnal tides have an even larger forcing period, yet still they have been observed in the surface pressure record, and also the wave structure of the Viking landing profiles has been attributed to vertically propagating thermal tides. No such observations would be possible if the τdiss were as small as required to match the Phobos orbital acceleration. Therefore our calculations give confidence that the source of the tidal friction most likely resides in the planetary interior.


[18] Luis Rademacher provided useful suggestions to perform the numerical calculations. The comments of two anonymous reviewers helped to improve the presentation of the article. VT was supported by a Linden Fellowship through the Earth Systems Initiative at MIT. MTZ acknowledges support from the NASA Planetary Geology and Geophysics Program.