### 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 ρ_{tot} can be written as,

where ρ_{0} 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 ω_{L}), where ω_{L} = ω − ω_{M} is the relative rotation of the moon with respect to the planet, ω is the rotation rate of the planet and ω_{M} 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,

where μ = cosθ, *f* = σ/2ω, *g* is the gravitational acceleration on Mars, and *a* is the radius of Mars. Θ_{n}^{s} are the solutions of the equation, known as Hough functions and *h*_{n}^{s} 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 Θ_{n}^{s} and the corresponding *h*_{n}^{s} are found by writing Θ_{n}^{s} 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 Forcingω | Rotation rate of the planet | 7.08 × 10^{−5} rad/s |

ω_{M} | Rotation rate of the satellite | 2.27 × 10^{−4} rad/s |

ω_{L} | Relative rotation rate | −1.57 × 10^{−4} rad/s |

σ | Frequency of the forcing | −3.14 × 10^{−4} rad/s |

*M*_{s} | Mass of Phobos | 1.08 × 10^{16} kg |

*D* | Distance between Phobos and Mars | 9.38 × 10^{3} km |

*a* | Radius of Mars | 3.40 × 10^{6} m |

ε | Inclination of the orbit of Phobos | 1° |

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

where *Q*^{2} = (*N*^{2}*H*^{2}/*gh*_{n} − 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* = *RT*_{0}/*g*, where *R* is the gas constant for the Mars atmosphere and *T*_{0} is the basic state temperature). The functions *y*_{n} are related to the rest of the tidal fields through the three dimensional divergence perturbation χ_{n} as *y*_{n} = χ_{n}*e*^{x/2}, where *x* = ∫^{z}*dz*/*H* is the *log-pressure* vertical coordinate. Assuming a perfectly spherical planet, the lower boundary condition is *w*_{tot} = *w* = 0 and can be written as,

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. Ω_{n} 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,

[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 *U*_{n}^{s} and *V*_{n}^{s} 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

where *G* is the gravitational constant, *M*_{s} 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 ^{3}. Then, the semidiurnal component of the gravitational potential due to satellite of mass *M*_{s} has the form,

where (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.