## 1. Introduction

Atmospheric waves provide important contributions to the energy, momentum, and tracer distributions within planetary atmospheres. Observations of atmospheric waves can serve as a probe of the atmospheric states due to the sensitivity of the waves to the thermal forcing and wind structures that generate, maintain, damp, and modify these waves. For example, measurement of thermal tides in the Martian atmosphere provides indirect insight into the global distribution of aerosol radiative heating that complements the direct measurement of dust and/or water ice cloud optical depths. To properly exploit this information, forward modeling of the waves is necessary using numerical modeling of the atmospheric state. Most often, the sensitivity of the model to variations in aerosol heating or other processes, such as boundary layer mixing, can be used to link the observed wave behavior to the dynamical behavior of the atmosphere. In regions of more complex interaction between the local and global circulation, such as at the Gale Crater landing site of the *Curiosity* rover, models are essential in order to unravel the relative roles of local, regional, and global atmospheric circulation. For example, models have been used to explain aspects of the observed daily variation of surface pressure in terms of global-scale thermal tides, regional-scale flows, and flows over the varied topography of the Gale Crater region (Rafkin et al. 2016; Richardson and Newman 2018).

Implicit in the use of numerical models is the assumption that the dynamical core representation of wave propagation is accurate. If it is accurate, the model provides a direct linkage between the forcing physics and the observable state, and hence the model can be used to extract information about the atmosphere from the wave response. However, it is well known that numerical models do not provide perfect emulation of real atmospheres. This is due to the techniques required to discretize the fluid dynamical solutions both spatially and temporally. In the specific example of the thermal tide on Mars, which we use as our exemplar in this paper, it is known that aspects of the structure of the observed tide are better emulated by some models than others. Thermal tides are ubiquitous in the Martian atmosphere. Various Mars landers and rovers detected similar and repeatable daily surface pressure variations at different geological locations, suggesting that these temporal pressure variations were not localized events, but thermal tides (Leovy 1981; Schofield et al. 1997; Lewis et al. 1999; Guzewich et al. 2016; Banfield et al. 2020). The Oxford Mars General Circulation Model (GCM) was able to simulate all of the daily structures in the Mars Pathfinder data (Lewis et al. 1999), while the Mars Weather Research and Forecasting (WRF) Model (Richardson et al. 2007; Toigo et al. 2012) has struggled to capture some of the higher-frequency, daily repeatable structures (specifically, the daily repeatable surface pressure transgression around 2000 local time) (Guzewich et al. 2016; Fonseca et al. 2018; Richardson and Newman 2018; Newman et al. 2017). For MarsWRF, the cause was ultimately attributed to the dynamical core by testing several different physics parameterization schemes and dust distributions within the model and finding that no combination produced an improved match to observations. Significantly, we have found similar behavior in the Model for Prediction Across Scales (MPAS) model (Skamarock et al. 2012), whose Martian adaptation is described by Lian and Richardson (2022).

## 2. Processes within the WRF and MPAS dynamical core

While most GCMs solve the incompressible, hydrostatic primitive equations, the WRF and MPAS dynamical cores have the capability to solve fully compressible, nonhydrostatic equations. This is useful when investigating regional motions in which the horizontal scale and vertical scale become comparable, or when investigating acoustic wave generation. However, acoustic waves produced by the compressibility of an atmosphere are usually unwanted in atmospheric models because they often lead to numerical instabilities. This is due to violation of the Courant–Friedrichs–Lewy (CFL) criterion: i.e., fluid motions associated with acoustic waves travel too fast to be resolved by model time step on the grid scale.

To mitigate the issues associated with acoustic wave modes, both WRF and MPAS utilize a divergence damping scheme that is intended to suppress the acoustic waves, without affecting gravity waves (Skamarock and Klemp 1992; Klemp et al. 2018). Skamarock and Klemp (1992) demonstrated that, when applied to wind fields in a domain where the effect of planetary rotation can be ignored, the divergence damping had little impact on the propagation properties of gravity waves in a Boussinesq fluid (i.e., divergence damping had little impact on gravity wave phase and amplitude). Gassmann and Herzog (2007) analyzed the divergence damping method in Skamarock and Klemp (1992) using fully compressible equations. They found that the damping must be applied to both horizontal and vertical momentum equations in order to avoid any impact on the gravity wave phases. Klemp et al. (2018) suggested that the proper formula of divergence damping in fully compressible equations required a divergence term different from that in Gassmann and Herzog (2007) and Skamarock and Klemp (1992). With the modified divergence terms, they showed that the divergence damping in the horizontal momentum equations was sufficient to damp acoustic wave modes with negligible effect on gravity waves, while avoiding the complexity of dependence on vertical wavenumber.

The concept of divergence damping, as noted by Skamarock and Klemp (1992), was originally implemented to damp internal and inertial gravity waves in hydrostatic primitive equations. However, none of the aforementioned studies on divergence damping explored the impact of divergence damping on thermal tides, which have horizontal scales much larger than the depth of the atmosphere, and hence are generally considered to be hydrostatic in nature. In the remainder of this paper, we explore the impact on thermal tides and demonstrate that without proper tuning, divergence damping can significantly modify atmospheric thermal tides.

## 3. Linear analysis of divergence damping on waves

To evaluate the impact of divergence damping on thermal tides, we perform linear wave analysis that only includes a single source of atmospheric tides and a single wave dissipation mechanism via divergence damping as detailed below. This avoids the complication associated with various sources of wave generation and dissipation mechanisms in a 3D GCM.

### a. Laplace’s tidal equations

*X*is expressed as

*X*=

*X*+

_{o}*X*′, where

*X*is the mean background state. The perturbation

_{o}*X*′ can be described in wave form:

*ωt*+

*sλ*) are, respectively, the amplitude and phase of thermal tides;

*θ*and

*λ*are, respectively, the latitude and longitude in radians;

*ω*is the wave frequency;

*s*is the zonal wavenumber (positive value for westward wave propagation);

*ρ*is atmospheric density;

*w*is vertical velocity;

*p*is pressure;

*g*is acceleration due to gravity;

*c*and

_{p}*c*=

_{υ}*c*−

_{p}*R*are heat capacities for constant pressure and volume, respectively;

*R*is the specific gas constant;

*a*is the radius of the planet; Ω is the rotation rate of the planet;

*J*is the external heating source that drives the thermal tides; and the divergence in spherical coordinate is expressed as (following Skamarock and Klemp 1992)

*χ*here since it was widely used by previous analysis on divergence damping (e.g., Sadourny 1975; Skamarock and Klemp 1992; Gassmann and Herzog 2007; Whitehead et al. 2011). Further,

*α*in Eq. (4) is the divergence damping coefficient, formulated in WRF and MPAS as

_{d}*α*is a dimensionless coefficient (typically 0.1 in WRF and MPAS, which is suitable for lower-atmosphere applications on Earth),

_{h}*L*is the dissipation length scale (e.g., mean horizontal grid spacing Δ

_{d}*x*), and Δ

*t*is the Runge–Kutta sub–time step in both GCMs. It can be seen that, besides changing

*α*, modifying

_{h}*L*and/or Δ

_{d}*t*can both change

*α*. Here we choose to change

_{d}*α*only while keeping

_{h}*L*= 220 km and Δ

_{d}*t*= 40 s (typical values for global Mars simulations) for a more controlled investigation on the effect of divergence damping. The Runge–Kutta sub–time step Δ

*t*= 40 s (corresponding to a model time step of 120 s) is significantly shorter than the typical value used for Earth simulations with comparable grid resolution. The reason is that Mars has a very thin atmosphere that responds to solar radiation rapidly and very steep terrain variations that leads to strong vertical motions, which prone to the violation of vertical CFL. Finally, the linear differential operator

*F*in Eq. (4) is defined as

*μ*= sin

*θ*.

*ω*and longitudinal wavenumber

*s*(e.g., Chapman and Lindzen 1970) assuming

*n*represents multiple possible solutions (Hough modes; Hough 1897) for the following equations:

*h*is termed the

_{n}*equivalent depth*, which results from the separation of variables and can be determined by eigenvalues

The LTE has two boundary conditions, Θ* _{n}* = 0 at both

*w*′ = 0 at the lower boundary

*z*= 0; second, there is no downward-propagating wave at the top boundary

*z*=

*z*

_{top}. The latter boundary condition implies that, for a homogenous second-order ordinary differential equation,

*c*

_{1},

*c*

_{2}, and

*c*

_{3}are arbitrary coefficients as functions of

*z*, only the solution representing the upward-propagating wave will be retained, i.e., the solution

*k*> 0. This is also called the

_{z}*radiation boundary condition*(see appendix for details).

*T*is the near surface temperature anomaly at the subsolar point and

_{s}*k*is the decaying factor that is correlated to eddy diffusivity,

_{d}*κ*, in the PBL as

_{e}*T*≈ 40 K (roughly the maximum amplitude of diurnal temperature variations measured by multiple Mars landers). The eddy diffusivity on Mars is small due to the thin atmosphere (i.e., roughly 100 times lower surface pressure than on Earth). Assuming

_{s}*κ*= 0.1 m

_{e}^{2}s

^{−1}, a typical value in the Martian surface layer (Martínez et al. 2009), and considering the solutions at

*λ*= 0 such that

*e*-folding scale for

*G*as

_{n}*k*Δ

_{d}*z*≈ 1. For diurnal, semidiurnal and terdiurnal tides of interest,

*k*is estimated to be 0.03–0.05 m

_{d}

^{−}^{1}, corresponding to Δ

*z*≈ 33–20 m. This depth scale is similar to the finest vertical resolution used in the PBL scheme employed by MarsWRF and MarsMPAS.

### b. Model parameters

We solve LTE and VSE using the physical parameters for Mars shown in Table 1.

Physical parameters used by the linear wave analysis. Note that *T _{s}* = 300 K is chosen for demonstration purposes. The results are relatively insensitive to the typical surface temperature at the subsolar point near the equator over a Martian year.

For the present analysis, we consider the diurnal tide, semidiurnal tide, terdiurnal tide, and quadiurnal tide. These are the frequencies of the dominant modes seen in the daily cycles of near-surface pressure observed by Mars landers and rovers (e.g., *InSight* lander and *Curiosity* rover). The mode of thermal tide is defined by the combination of scaled wave frequency *ν* and zonal wavenumber *s* (Table 2).

Parameters for various modes of the thermal tides. DW1 stands for diurnal tide, propagating westward with a zonal wavenumber of 1. Similar notations apply to SW2, TW3, and QW4.

### c. Results

The solutions to the LTE and VSE shown in the following are obtained from the associated Legendre polynomials (ALP) for the LTE (Wang et al. 2016) and the second-order central differential scheme for VSE. There are 90 grid points evenly spaced from *G _{n}* vanishes.

Figure 1 demonstrates the Hough functions (i.e., the eigenvectors Θ* _{n}*, where

*n*= 0, 1, 2, 3, …) for the first three symmetric Hough modes of the DW1, SW2, and TW3 tides. The order of the Hough modes is sorted by the magnitude of the eigenvalues from the largest to the smallest. The Hough functions depict the latitudinal structures of perturbations to physical quantities (e.g., pressure, temperature, or vertical velocity) associated with the different types of thermal tides, as shown by the gravest Hough modes of the diurnal and semidiurnal tides in the MarsWRF simulations (Guzewich et al. 2016). For each mode (each eigenvalue), an equivalent depth

*h*can be obtained (Table 3).

_{n}Equivalent depth *h _{n}* (m) corresponding to different Hough modes for DW1, SW2, and TW3 tides. The subscripts (

*X*,

*Y*) mean the tide

*X*and its dominant symmetric Hough modes

*Y*.

The divergence damping impacts all three types of tides and various modes associated with each tide. Figure 2 shows the effect of divergence damping (*α _{h}* = 0.1) on the first three symmetric modes of the DW1 tide. The effect of divergence damping, measured by the ratio of amplitudes between pressure perturbations with or without it activated (

*r*

_{div}is greater than 1 for all Hough modes at lower altitudes in Fig. 2. This appears to be counterintuitive because divergence damping shall suppress the growth of wave amplitude vertically. However, a nonzero divergence damping coefficient

*α*affects both wave amplitude and phase [see the dispersion relations given by Eqs. (A33) and (A34) in appendix], and the solution of the surface pressure perturbation (i.e., the initial wave amplitude at the lower boundary) can be larger than that of

_{d}*α*= 0 for some wave modes (characterized by

_{d}*ω*and

*h*) under the lower boundary condition

_{n}*w*′ = 0 at

*z*= 0.

Given proper boundary conditions, the divergence damping would show familiar behavior of other wave damping mechanisms (e.g., viscous damping), which suppress the growth of wave amplitudes with increasing altitude. For instance, the ratio *r*_{div} would be smaller than 1 away from the surface and decrease with increasing altitude if the lower boundary condition were *p*′ = constant at *z* = 0 (Fig. 3). With this lower boundary condition, *r*_{div} would also decrease with increasing order of Hough modes because of the decreasing equivalent depth (decreasing vertical wavelength). This behavior is again similar to that of viscous damping, which exhibits a stronger damping effect on shorter vertical wavelengths (e.g., Lian and Yelle 2019; Vadas and Fritts 2005).

Figure 4a shows the effect of the divergence damping (*α _{h}* = 0.1) as a function of tide frequency at the lowest model level.

*r*

_{div}becomes smaller when the frequency of tides increases. For example,

*r*

_{div}for QW4 is more than a factor of 3 smaller than that for DW1. This behavior is consistent with the MarsWRF and MarsMPAS model results, which show that simulations using the typical divergence damping coefficient

*α*= 0.1 overly suppress the high-frequency oscillations associated with daily pressure variations, essentially making DW1 and SW2 the only recognizable thermal tides in these models (see section 4b).

_{h}The strength of the divergence damping is controlled by the nondimensional damping coefficient *α _{h}* in both WRF and MPAS. Figure 4b shows

*r*

_{div}as a function of

*α*at the lowest model level for the QW4 tide (4, 4). The QW4 tide experiences very little damping for

_{h}*α*ranging from 10

_{h}

^{−}^{4}to 10

^{−}^{3}, but it shows slight amplification when

*α*∼ 10

_{h}

^{−}^{2}and becomes noticeably damped when

*α*> 0.1.

_{h}## 4. Effect of divergence damping in WRF/MPAS

The linear wave analysis above shows that divergence damping can impact the amplitude and phase of pressure perturbations associated with thermal tides. Similar impacts exist for wind fields and other physical quantities via polarization relationships derived from the perturbation equations [see Eqs. (A13)–(A16) in appendix]. Here we examine the effect of divergence damping on the diurnal pressure cycles under Martian conditions in WRF/MPAS simulations. Only the daily surface pressure cycles are analyzed because the vertical profiles of thermal tides can be affected by thermal structures, and damped by various mechanisms such as eddy viscosity and other numerical filters such as Smagorinsky viscosity in WRF/MPAS, in addition to divergence damping. These factors make it hard to directly compare the vertical structures between the idealized linear wave model and 3D GCM.

### a. Observations

Multiple Mars landers and spacecraft have detected atmospheric tides via measurements of daily surface pressure variations (e.g., Zurek and Leovy 1981; Haberle et al. 2014; Guzewich et al. 2016; Newman et al. 2017) and daily atmospheric thermal variations (e.g., Conrath 1975; Lee et al. 2009). These variations, with various frequencies, persist throughout the Martian year. Using *Curiosity* rover’s Rover Environmental Monitoring Station (REMS) and *InSight* lander measurements from Planetary Data System (PDS) data over a Martian year as examples, the near-surface pressure shows wavelike variations over a Martian day, with the largest-amplitude wave being the diurnal tide (Fig. 5). Other higher-frequency oscillations are superposed on top of the diurnal tide.

The detailed properties of these diurnal variations can be obtained via spectral analysis. The surface pressure perturbation can be represented by superposition of wave solutions *A _{σ}* is the amplitude of a wave with frequency per sol

*σ*,

*t*is the local time, and

*δ*is the wave phase (Guzewich et al. 2014). For simplicity, we ignore the migrating and nonmigrating nature of tides and their phases. Performing Fourier transformation (FFT) on the measured pressure perturbation

_{σ}

^{−}^{1}.) The largest wave amplitude of 12 Pa corresponds to the diurnal tide (

*σ*= 1 sol

^{−1}), followed by several distinct peaks correspond to higher-order thermal tides such as the semidiurnal tide (

*σ*= 2 sol

^{−1}), terdiurnal tide (

*σ*= 3 sol

^{−1}), quadiurnal tide (

*σ*= 4 sol

^{−1}), etc.

### b. Model predictions

We run the Mars GCM for 10 sols starting from *L _{s}* = 120° as a direct comparison to the observations. The choice of this particular

*L*is somewhat random but the

_{s}*InSight*measurements near this

*L*showed relatively larger amplitudes of high-order oscillations with smaller error bars over 10 sols compared to other

_{s}*L*. The simulations are very similar whether performed using MarsWRF or MarsMPAS, but the specific results shown in this paper use MarsMPAS, which includes a uniform horizontal mesh with roughly 240 km spacing (equivalent to roughly 4° MarsWRF spacing) and a 45-layer terrain-following vertical coordinate with the top of the uppermost layer at roughly 120 km. Higher horizontal grid resolution may introduce some topographic effect to the modeled thermal tides but the effect is not large enough to invalidate our comparisons. The suite of available physics parameterizations is standard between MarsWRF and MarsMPAS (Richardson et al. 2007; Lian and Richardson 2022), and we use the K-distribution method (KDM) radiative transfer (RT) scheme, YSU PBL scheme, surface/subsurface scheme, and a simple microphysics model of the CO

_{s}_{2}condensation–sublimation cycle. To represent aerosol forcing in the Martian atmosphere, we use a fully interactive two-moment dust scheme that generates aerosol radiative properties employed in the KDM RT scheme (Lee et al. 2018). Both the cases with and without divergence damping are compared against observations. Since the purpose of the GCM simulations is to illustrate the impact of divergence damping in otherwise identical models, and not to provide and optimal fit to surface pressure observations, the model distribution of aerosol heating has not been specifically tuned to maximize the best fit match, and for simplicity water ice cloud opacity is not treated.

Figure 6 shows the model-predicted daily pressure cycles compared to the *InSight* measurements. Both the modeled and observed pressure variations show periodic oscillations with comparable magnitude over a sol. Without divergence damping (*α _{h}* = 0), the GCM is able to capture the distinct peaks and valleys in the observed pressure curves during various time of the sol, i.e., the peaks and valleys at midnight, in the early morning and later afternoon/early evening. The exact timing of the predicted pressure perturbation shows some difference with those of the observations, such as the peak near 0700 LT and the valley near 1700 LT. With the divergence damping (

*α*= 0.1), the modeled pressure curve becomes overly smoothed and only exhibits diurnal variation with a peak near 0800 LT and a valley near 1700 LT. Moreover, divergence damping reduces the amplitudes and changes the phases of the thermal tides compared to the case without divergence damping.

_{h}The FFT analysis of the modeled pressure variations suggest that divergence damping can significantly impact the higher-order thermal tides. Figure 7 shows the amplitude–frequency relation for the model-predicted and the observed pressure perturbations. Without divergence damping, the GCM is able to predict almost all dominant modes of the observed thermal tides with comparable amplitudes. However, only diurnal and semidiurnal tides are recognizable once we switch on divergence damping. The amplitudes of higher-order modes become an order of magnitude smaller than those in the case without divergence damping. Additional case studies (not shown in the figure) with various strength of the divergence damping suggest that a damping coefficient of *α _{h}* ∼ 0.001 is able to capture most of the high-order thermal tides without impacting numerical stability (particularly during the dusty southern summer on Mars), compared to

*α*= 0.

_{h}## 5. Conclusions

We performed wave analysis to show how divergence damping can impact the daily variations of atmospheric thermal tides. A linear wave study suggests that divergence damping can affect the wave amplitudes and phases in the entire atmosphere. The specific impact, e.g., either damping or amplifying the wave amplitude near the surface, depends on the wave modes and the boundary conditions. Consistent with this linear wave analysis, spectral analysis of GCM-predicted diurnal pressure perturbations shows that strong divergence damping can suppress the thermal tides with order higher than the diurnal and semidiurnal tides. Thus, the strength of the divergence damping must remain reasonably low to properly represent the observed pressure cycles in numerical models.

We emphasize that the study presented here is to demonstrate the impact of divergence damping on physical quantities in general. The exact behavior of the divergence damping in a 3D GCM may be complicated. For example, MPAS implements a rigid lid approximation; therefore, the reflection of upward-propagating waves near the top of the atmosphere is permitted. Wave-absorbing layers (e.g., Rayleigh damping of vertical velocity, horizontal velocities, and temperature; see Klemp et al. 2008) are introduced to suppress the numerical instabilities introduced by this wave reflection, but these layers also introduce complexity to the linear wave model analysis and change the behavior of the solutions. Further, a more rigorous linear wave analysis should include both the eddy viscosity and various other wave damping mechanisms used in the GCM.

Other mechanisms affecting our linear wave analysis include the excitation sources of the atmospheric tides. We assume that all solar radiation is absorbed by the ground, which in turn exchanges heat with the atmosphere via diffusive mixing. This approach is overly simplified because the radiative heating and cooling of the Martian atmosphere can greatly impact the thermal structures of the atmosphere, e.g., dust aerosols, water ice and the major component of the atmosphere CO_{2} are all radiatively active in both solar and infrared (IR) wavelengths. These atmospheric sources of tidal excitations, like various wave damping mechanisms, need to be considered in the linear wave model in order to establish a better comparison to Mars GCMs.

Numerical diffusion in WRF and MPAS is well-designed to suppress numerical noise and instabilities for climate simulations in general. However, the parameters controlling the numerical diffusion (such as divergence damping, off-centering in the vertically implicit time step, external mode filter and other viscous dissipation) need to be assessed thoroughly for specific planetary atmosphere applications. For instance, prior work on Titan showed that excessive horizontal diffusion reduced the magnitude of the stratospheric superrotation on Titan (Newman et al. 2011). Likewise, we speculate that the default divergence damping coefficient excessively damps short-period thermal tides on Mars. Thankfully, the damping is cleanly “broken out” in the code and its effects are readily tested. While implicit damping is also unavoidable in the numerical solvers of differential equations due to truncation errors (Lauritzen et al. 2011), this damping does not seem to have a deleterious effect. We recommend that other Mars GCMs, especially if they are unable to match the full spectrum of waves in Martian pressure data, should also be examined in terms of detailed numerical damping and dissipation mechanisms in their dynamical cores.

## Acknowledgments.

This work is supported by NASA Solar System Works (SSW) Grant NNH18ZDA001N-SSW. The submission of manuscript has no conflict of interest with this SSW grant.

## Data availability statement.

The Mars *InSight* lander and REMS pressure data used in this study are publicly available from PDS nodes: https://atmos.nmsu.edu/data_and_services/atmospheres_data/INSIGHT/insight.html, https://atmos.nmsu.edu/data_and_services/atmospheres_data/MARS/curiosity/rems.html. The official WRF and MPAS GCMs are available via https://github.com/wrf-model/WRF and https://github.com/MPAS-Dev/MPAS-Model. The codes solving LTE/VSE andperforming FFT analysis of both observed and simulated Martian thermal tides in this study can be obtained from Aeolis Research public repository at https://github.com/AeolisResearch/divergence_damping. The Mars version of WRF or MPAS GCMs and other codes used in this study are available from the corresponding author, Yuan Lian, upon reasonable request.

## APPENDIX

### Tidal Equations and Analytic Solutions

#### a. Thermal tide equations

*ρ*is the density;

*u*,

*υ*, and

*w*are the zonal, meridional, and vertical velocities, respectively;

*p*is the pressure;

*a*is the radius of the planet;

*g*is the gravity;

*θ*and

*λ*are the latitude and longitude in radians, respectively;

*f*= 2Ωsin

*θ*is the Coriolis parameter;

*R*is the gas constant;

*c*) and constant volume (

_{p}*c*);

_{υ}*H*is the pressure scale height (same as density scale height in an isothermal atmosphere);

*α*is the divergence damping coefficient, formulated in MPAS as

_{d}*α*is a dimensionless coefficient (typically 0.1 in WRF and MPAS);

_{h}*L*is the dissipation length scale, which is typically the smallest distance between adjacent grid cells (

_{d}*L*≈ 220 km for 4° grid resolution); Δ

_{d}*t*is the Runge–Kutta split time step (typically three sub–time steps for MarsMPAS; therefore, Δ

*t*= 40 s for model time step of 120 s);

*J*is the source of thermal tides (which will be explained later). The total derivative

*α*are applied to Eqs. (A2) and (A3) in a way similar to that in Skamarock and Klemp (1992). The equation of state for an ideal atmosphere is

_{d}*p*=

*ρ*RT, and the divergence

*X*as a sum of mean and perturbation parts as

*X*=

*X*+

_{o}*X*′, recognizing

*u*= 0,

_{o}*υ*= 0,

_{o}*w*= 0, and

_{o}*O*

^{2}(

*X*′), we can obtain the set of perturbation equations as

*ω*is the wave frequency that depicts phase directions by positive value (westward moving) or negative value (eastward moving), and

*s*is the zonal wavenumber. A few useful properties of the wave perturbations are

*u*′ and

*υ*′. Similarly, Eqs. (A11) and (A12) can be used to eliminate

*T*′. After some mathematical manipulations, we obtain a set of wave equations in terms of

*w*′,

*ρ*′,

*χ*′:

*F*is defined as

*μ*= sin

*θ*. Equations (A13)–(A16) can be further reduced to a single equation by eliminating

*w*′ and

*ρ*′:

*P*′,

*χ*′, and

*J*as sum of all possible solutions (denoted by subscription

*n*):

*e*

^{i}^{(}

^{ωt}^{+}

^{sλ}^{)}is implied in

*L*,

_{n}*M*, and

_{n}*G*since it does not affect the solutions to Eq. (A18). Applying separation of variables to Eq. (A18), we have

_{n}*z*and

*θ*, respectively. Therefore, both sides must be equal to a constant, e.g.,

*h*is called the

_{n}*equivalent depth*. This leads to two equations:

*α*= 0, Eqs. (A20) and (A21) simply become the classical Laplace’s tidal equation and the vertical structure equation seen in many literatures.

_{d}*M*must be eliminated. This can be achieved by reducing Eqs. (A13)–(A15) to

_{n}*M*to Eq. (A21), we can obtain the vertical structure equation as

_{n}*for each pair of normalized wave frequency*

_{n}*ν*and zonal wavenumber

*s*, which define the modes of thermal tides such as diurnal tide (

*ν*= 0.5,

*s*=1), semidiurnal tide (

*ν*= 1,

*s*= 2), and terdiurnal tide (

*ν*= 1.5,

*s*= 3). The equivalent depth thus can be obtained as

Subsequently, the vertical structure equation Eq. (A24) can be solved using the finite difference method using the equivalent depth in Eq. (A25).

*α*= 0 and

_{d}*ϵ*= 0, Eq. (A26) describes undamped waves without external sources.

_{n}#### b. Boundary conditions

*w*′ and

*P*′:

*w*′ = 0 at

*z*= 0 and the separation of variables, we obtain the lower boundary condition in terms of

*L*:

_{n}*y*and

_{n}*ϵ*as

_{n}*T*is the surface temperature anomaly at the subsolar point,

_{s}*k*is the decaying factor that is correlated to eddy diffusivity

_{d}*κ*in the PBL as

_{e}*G*decays exponentially when altitude increases. At altitude high enough,

_{n}*G*≈ 0, and Eq. (A26) becomes

_{n}*A*is a constant and

*k*is the vertical wavenumber, we can solve Eq. (A30) analytically using the second-order polynomial:

_{z}*k*represent the vertical wavenumber and damping rate, respectively:

_{z}*a*and

*b*,

*a*=

*b*,

*a*>

*b*and

*a*<

*b*. Using the typical values for

*α*= 0.1,

_{h}*h*∼ 516 m,

_{n}*ω*= 7.27 × 10

^{−}^{5}

*s*

^{−}^{1},

*L*= 220 km, Δ

_{d}*t*=40 s,

*g*= 3.727 m s

^{−2},

*H*∼ 15 km, |

*η*| = 3.5, and

*γ*= 1.4, the dominant terms within the square root in

*k*for diurnal tide mode (1, 1) can be estimated as

_{z}*k*.

_{z}To obtain meaningful wave solutions, *η* = −3.5), this means *h _{n}* <

*H*. For undamped waves, the wave energy flux (e.g., the heat flux

*ρ*〈

_{o}*w*′

*T*′〉, where 〈

*w*′

*T*′〉 =

*w*′

*T*′

^{*}and

*T*′

^{*}is the conjugate of

*T*′) in the region where

*G*≈ 0 remains constant.

_{n}*z*= ∞, i.e., imag(

*k*) > 0]; therefore, the solution to Eq. (A30) becomes

_{z}*y*with respect to

_{n}*z*, we have

#### c. Impact of divergence damping definitions

*χ*′ with

*χ*′ to

*ϵ*. The dispersion relation for Eq. (A41) is given by

_{n}*h*∼ 516 m,

_{n}*ω*= 7.27 × 10

^{−}^{5}s

^{−}^{1},

*L*= 220 km, Δ

_{d}*t*= 40 s,

*g*= 3.727 m s

^{−2},

*H*∼ 15 km, |

*η*| = 3.5, and

*γ*= 1.4 again, we calculate damping rates [i.e., the imaginary part of

*k*in Eqs. (A31) and (A42)] for a range of

_{z}*α*. Figure A1 shows the ratio of damping rates between the case with the original definition of divergence damping

_{h}*χ*′ and the case with the revised definition of divergence damping

*α*< 0.6.

_{h}*α*. Using Eq. (A42) as an example, the dispersion relation can be rewritten as

_{h}*n*= 1, 2, 3, … is the tide mode such as diurnal, semidiurnal, and terdiurnal tides. It is apparent that

*α*needs to be smaller for higher-order tides to avoid excessive damping in MarsWRF and MarsMPAS.

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