ISSN 00380946, Solar System Research, 2016, Vol. 50, No. 2, pp. 90–101. © Pleiades Publishing, Inc., 2016. Original Russian Text © D.S. Shaposhnikov, A.V. Rodin, A.S. Medvedev, 2016, published in Astronomicheskii Vestnik, 2016, Vol. 50, No. 2, pp. 100–111.

The Water Cycle in the General Circulation Model of the Martian Atmosphere D. S. Shaposhnikova, b, A. V. Rodina, b, and A. S. Medvedevc a

Space Research Institute, Russian Academy of Sciences, Moscow, Russia Moscow Institute of Physics and Technology (State University), Moscow, Russia cMax Planck Institute for Solar System Research, Göttingen, Germany email: [email protected]

b

Received July 15, 2015

Abstract—Within the numerical generalcirculation model of the Martian atmosphere MAOAM (Martian Atmosphere: Observation and Modeling), we have developed the water cycle block, which is an essential component of modern general circulation models of the Martian atmosphere. The MAOAM model has a spectral dynamic core and successfully predicts the temperature regime on Mars through the use of physical parameterizations typical of both terrestrial and Martian models. We have achieved stable computation for three Martian years, while maintaining a conservative advection scheme taking into account the water–ice phase transitions, water exchange between the atmosphere and surface, and corrections for the vertical veloc ities of ice particles due to sedimentation. The studies show a strong dependence of the amount of water that is actively involved in the water cycle on the initial data, model temperatures, and the mechanism of water exchange between the atmosphere and the surface. The general pattern and seasonal asymmetry of the water cycle depends on the size of ice particles, the albedo, and the thermal inertia of the planet’s surface. One of the modeling tasks, which results from a comparison of the model data with those of the TES experiment on board Mars Global Surveyor, is the increase in the total mass of water vapor in the model in the aphelion sea son and decrease in the mass of water ice clouds at the poles. The surface evaporation scheme, which takes into account the turbulent rise of water vapor, on the one hand, leads to the most complete evaporation of ice from the surface in the summer season in the northern hemisphere and, on the other hand, supersaturates the atmosphere with ice due to the vigorous evaporation, which leads to worse consistency between the amount of the precipitated atmospheric ice and the experimental data. The full evaporation of ice from the surface increases the model sensitivity to the size of the polar cap; therefore, the increase in the latter leads to better results. The use of a more accurate dust scenario changes the model temperatures, which also strongly affects the water cycle. Keyword: Mars, water cycle, numerical modeling, atmosphere, climate, general circulation model, MAOAM, advection, ice sedimentation, water phase transformation, surface water exchange DOI: 10.1134/S0038094616020039

INTRODUCTION

the most likely destination for the first manned mis sion to another planet. However, the main thing is that Mars is so far the only planet that holds promise in terms of human development (Koroteev, 2006; Shee han, 1996).

Studying the planets and small bodies of the Solar System is of paramount importance for understanding its origin and development. But above all, it provides the key to finding the likely paths of the future evolu tion of our planet and understanding how to keep the Earth habitable for the future generations. Mars is the fourth planet from the Sun in the Solar System and the closest one to the Earth among the outer planets. At present, Mars is the most interesting and the most explored planet of the Solar System after the Earth. The climate conditions on Mars are, although unsuitable for life, the most similar to those on the Earth. Presumably, in the past, the Martian cli mate could have been warmer and wetter; there was liquid water on its surface, and it even rained. Mars is

The Martian climate is mainly determined by the processes occurring in its atmosphere, such as the movement of air masses, convective mixing, radiative transfer, and the transfer of impurities. There is no complete solution to the problem of the experimental measurement of, e.g., the full velocity field in the atmosphere of the Earth and other planets because this problem is strongly nonlinear. Therefore, the unknown parameters are derived from those obtained in experiments by building climate models of general or global atmospheric circulation (GCMs). The 90

THE WATER CYCLE IN THE GENERAL CIRCULATION MODEL

majority of the wellknown models are based on numerical solution of the equations of hydrodynamics. One of the first attempts to numerically describe the atmosphere of Mars was made by Leovy and Mintz (1969), who successfully adapted the GCM developed at the University of California (Los Angeles) to the Martian conditions. Since then, GCMs have become popular. Currently, there are several models of suffi cient complexity, which were developed in the United States, France, Britain, Japan, Canada, and Germany. They are used to investigate a wide range of processes and phenomena in the Martian atmosphere and to interpret observational data. The appearance of mass measurements of water vapor led to attempts to simu late the water cycle in the Martian GCMs. However, this has been achieved to a satisfactory extent only in the Laboratoire de Météorologie Dynamique Mars General Circulation Model (LMD MGCM) (Mont messin et al. 2004; Navarro et al. 2014) and the joint software product of the LMD and Oxford University (Forget et al. 1999; Bottger et al. 2003; Montmessin and Forget, 2003). Although the amount of water on Mars is negligible compared with the Earth and its radiation effects in the energy balance are also negligi ble, water vapor is a very sensitive tracer and indicator of transfer processes. Hence, a successful simulation of the annual water cycle is also a measure of our understanding of the ongoing processes in the Martian atmosphere and of the accuracy of the numerical gen eralcirculation models. This work represents a new and independent development of the water cycle block, which is incorporated into the MAOAM model (Martian Atmosphere: Observation and Modeling), developed at the Max Planck Institute for Solar Sys tem Research (Germany) by A.S. Medvedev et al. (Hartogh et al., 2005; 2007; Medvedev and Hartogh, 2007; Medvedev et al., 2011). Furthermore, the MAOAM model is used to make a full computational cycle. The MAOAM model has a spectral dynamic core and successfully predicts the circulation and tem perature on Mars through the use of physical parame terizations, which are typical both of terrestrial models (vertical turbulent diffusion, surface physics, and grav itational waves) and Martian ones (heating in the CO2 bands in the near IR range, dust effects, parameteriza tion of radiation in the СО2 band at 15 μm, which takes into account the local thermodynamic equilib rium (Kutepov et al., 1998)). The model uses the accurate topography of Mars from the MOLA laser altimeter measurements, and data on the thermal inertia of the surface from the TES instrument. The MAOAM GCM includes the atmosphere from the surface to the middle thermosphere (~150 km). The good numerical properties of the dynamic core and the account for the parameterization of the gravita SOLAR SYSTEM RESEARCH

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tional waves on a subgrid scale (Medvedev and Klaas sen, 2000; Yigit et al., 2009) allow the model to simu late well the heating of the atmosphere at the poles (Medvedev and Hartogh, 2007) and in the mesosphere (Medvedev et al., 2015). To achieve this goal, we needed to develop the aerosol block and the surface physics block. To build these blocks, we first developed the scheme of the threedimensional advection (transfer) of passive impurities by atmospheric flows in such a way that it did not loose conservatism when taking into account the phase changes of water–ice. In addition, it was necessary to take into account the water exchange between the atmosphere and the surface and the effect of the deposited ice on such physical parameters as the albedo and the thermal inertia of the planet’s surface. During the implementation of the objectives, we achieved stable computation for several Martian years, while maintaining the stability and conservatism of the scheme. The modeling results were obtained by select ing the model parameters and using the spectral numerical scheme that has a lower numerical viscosity and, therefore, ensures a more intense meridional cir culation, compared with the previous grid schemes. COMPUTATIONAL GRID When developing a threedimensional advection scheme, we should first describe the grid used for the computation. The model architecture is such that heightwise it is most convenient to use a nonstationary pressure grid parameterized via the hybrid η coordi nates (Simmons and Burridge, 1981; Simmons and Chen, 1991) and the surface pressure ps as follows: p ( η ; p s ) = a ( η ) + b ( η )p s ,

(1)

p ( η = 0; p s ) = 0 and p ( η = 1; p s ) = p s ,

(2)

a(η) where  = η(1 – η), p00 is a constant, and b(η) = η2. p 00 It is evident from the equations that η takes values from 0 to 1 and the grid is not uniform. This parame terization allows one to track the unevenness of the surface relief and reduce the movement of impurities between the cells in the vertical direction. In the horizontal plane, we use the Gauss–Kruger map projection. The cell size is the same within the same latitude but different for different latitudes. Lon gitudinally, the cells have different sizes, which is spec ified by the distribution of Gaussian latitudes.

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HYBRID ADVECTION SCHEME

ICE SEDIMENTATION

To calculate the threedimensional advection, we use the secondorder explicit monotonic hybrid scheme developed at the Polar Geophysical Institute (PGI) RAS (Mingalev et al., 2010) and the method of splitting into spatial directions. The idea of the method is that the threedimensional transfer equa tion can be decomposed into three unidimensional ones for each of the spatial directions, which can be solved one by one. The alternation of the order in which the equations are solved helps minimize the numerical errors of this approximation.

The sedimentation (or precipitation) is the deposi tion of dispersed phase particles in a liquid or gas under the influence of a gravitational field or a centrif ugal force.

The PGI scheme has substantial advantages such as a higher order of conservatism and stability and the retaining of sharp gradients of the solution, which is very important for the atmospheric parameters such as the density and the mass concentration of the sub stance. The water vapor and water ice masses are trans ferred separately. When they are transferred along the parallels, the cell width in the direction of the transfer (longitude) hx and the areas of the faces between the contiguous cells are the same, which simplifies the equations. When they are transferred vertically, we need to take into account the different height of the cells by selecting a different grid spacing. In the case of the meridional transfer, we should: (1) Take into account the different height of the cells in the direction of the transfer (latitude) hy by selecting a different grid spacing. (2) Take into account the different areas of the faces between the contiguous cells by dividing by the different cell width by longitude hx. (3) Seam together the grid boundaries at the poles in a periodic manner so as to retain the conservatism of the solution. In the case of a vertical transfer, the boundary con ditions do not allow impurities to flow away through the boundaries in contrast to the horizontal transfer, whereby the periodicity of the boundaries ensures the absence of features at the junctions. Note that in the case of the transfer along the meridians, it is enough to seam the grid with its mirror image (on the other side of the planet) and apply, and after correcting for the different area of the faces between the contiguous cells, the same procedure as for the transfer along the parallels. In addition, for the vertical velocities of ice parti cles, we use the Stokes formula to calculate the correc tion due to their sedimentation (precipitation).

In the model for water ice particles, we calculated the sedimentation by using the Stokes formula with the Cunningham correction. As already mentioned, the vertical cells represent the different pressure levels. dp Then, the derivative  acts as velocity. We now calcu dτ late the correction to this derivative by the formula from (Korablev et al., 1992): 2

2 R ice ⎛ dp λ  = 2 ρ ice g   1 + A + ⎞ , dτ 9 R ice⎠ ν ⎝

(3)

where A+ is the Cunningham correction; ρice is the ice density; g is the acceleration of gravity; Rice is the radius of ice particles (we consider them to be spheri cal; 4 μm), ν is the kinematic viscosity; and λ is the average length of the free path in gas. The Cunningham correction is calculated as (Bur lakov and Rodin, 2011): ⎧ R ice ⎫ A+ = 1.246 + 0.42exp ⎨ –0.87   ⎬. λ ⎭ ⎩

(4)

We calculate the kinematic viscosity by the Suther land formula (Fuks, 1955; Piskunov, 2010) and substi tute the quantities for Mars: –6

1.579 × 10 Pа s  T , ν =  ρa 258 К 1 +  T

(5)

where ρa is the atmospheric density. The average length of the free path 〈λ〉 and atmo spheric energy is calculated by the formulas known 3ν p from general physics 〈λ〉 =  and ρa =  . 〈 u〉 RT dp Thus, we now have the necessary correction to  , dτ which should be added to the numerical scheme of vertical advection. WATER PHASE TRANSFORMATIONS The water phase transformations are calculated as follows. First, we use the Clapeyron–Clausius equa tion to calculate the temperature (T) dependence of the pressure of saturated vapors psat in the computa SOLAR SYSTEM RESEARCH

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tional grid cell (Curry and Webster, 1999) and substi tute the quantities for Mars: 273.16 K psat = 611 Pа exp 22.5 ⎛ 1 – ⎞ . ⎝ T ⎠

(6)

Then we divide the result by the atmospheric gas pressure p and multiply by the ratio between the molar masses of water M H2O and atmospheric gas Mair to con vert pressure into the maximum relative mass concen tration of water rsat, which has the dimension of the amount of water mass in the atmospheric air mass (mainly CO2):

M H2 O p sat r sat =   . M air p

(7)

Then, we multiply it by the difference between the pressures at the lower and upper cell boundaries Δp and divide by the acceleration of gravity g to obtain from the mass concentration the mass of precipitated water per unit area ρsat: Δp ρ sat = r sat . g

(8)

The multiplication by the cell area yields the max imum mass of water vapor msat, which may be con tained in a cell: msat = rsatSϕ. (9) At the first step, all the available ice evaporates; at the second step, the excess of water (above msat) trans forms back into ice. We do not take into account the effect of water condensation and evaporation on the atmospheric air temperature because tests have shown that it is small (within 1%), but may lead to an increase in computational errors. SURFACE WATER EXCHANGE The next step after making the water vapor phase transitions scheme is to describe the water exchange between the atmosphere and the surface. Originally, we planned to use the following set of simple rules. If, after the evaporation of all the ice, the resulting mass of water in the subsurface cell exceeds that of supersaturation msat, the excess water trans forms into the surface ice and is stored on the surface. Otherwise, all the available ice evaporates from the surface to the saturation level. This scheme proved to be not fully conservative, as shown by the further experiments, due to a program ming error. In addition, the rate of evaporation from the surface proved to be insufficient. The modeling results showed that, apart from the temperatures and, hence, the saturation level in the bottom cell, we needed to take into account the surface temperature SOLAR SYSTEM RESEARCH

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and the horizontal wind velocities, which ensure the turbulent rise of water vapor. As a result, we decided to use the flow chart from (Flasar and Goody, 1976; Montmessin et al., 2004): Ew = ρCdu*(rvg – rva). (10) Here, Ew is the flow of water vapor from the sur face; ρ is the atmospheric density (calculated from the Clapeyron equation); Cd is the pressure drag coeffi 2

2

cient (for plane 0.005); u* = u + ν is the average horizontal velocity; rvg is the mass concentration of the supersaturated vapor on the surface; and rva is the mass concentration of the supersaturated vapor in the sub surface layer. Multiplying Ew by the time step and the bottom cell’s area, we derive the mass of water vapor that needs to be evaporated during one step. For the ice flow onto the surface, we calculate, instead of using the instantaneous storage, the mass of the sedimented ice Fsed, which depends on the dimen sionless sedimentation rate η:

F sed = r i η,

(11)

dp τ, r is the mass concentration of water where η =   i dτ Δp dp ice;  is the sedimentation rate (3) at the center of dτ the bottom cell; τ is the time step; and Δp is the height of the bottom cell in pressure units. At the north pole (north of 75°), instead of using large values of the concentration of surface ice, we allowed the ice to evaporate, even at negative concen trations. Thus, we modeled the north polar cap on Mars, which is the main reservoir of water on this planet. In addition, we assumed that ice is present on the surface if its temperature drops below 230 K to the north of 65° in the season with Ls = 90° to Ls = 120°. Navarro et al. (2014) showed that this assumption is justified and consistent with experimental data. MAOAM MODEL ADAPTION We modified the model’s core code to ensure better consistency with the experimental data. First, we added the impact of the precipitated water ice on the albedo and the thermal inertia of the planet’s surface. If the ice on the surface was above 5 μm, the surface albedo was set at 0.3 and the thermal inertia, at 600 J m–2 K–1 s–1/2. These parameters showed the best agreement with the experiment. Second, the surface temperature of the south polar cap to the south of 85° south latitude was set equal to the CO2 evaporation temperature. This modification of the code was necessary to take into account the per manent CO2 polar cap at the south pole.

SHAPOSHNIKOV et al.

TES ice cloud opasity at 825 cm–1

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Fig. 1. Comparison of the TES experiment (Lefevre et al., 2008) (left) and the MAOAM results (right). The upper line in the plots shows the average seasonal distribution of water vapor in precipitable micrometers; the lower line shows that of water ice particles. The vertical axis plots the latitude; the horizontal one, time in Ls.

Furthermore, it should be noted that, we originally used a static constantopacity dust distribution, which depends on the season, latitude, and altitude (Conrath, 1975), as the model dust scenario. Supple menting the dust scenario with the dependence of opacity on season and latitude on the basis of the TES data averaged over several years largely improved the agreement with experiment. RESULTS To compare the calculated results with experimen tal data, we had to make calculations for several Mar tian years. We used the model version with the T21 tri angular spectral resolution (64 × 32 cells in longitude and latitude) for a period of three Martian years. Since we used a PGI Accelerator for compiling and per formed the MPICH parallelization of the code into eight streams, we were able to solve this problem on a PC with Intel® Core™ i72600 CPU @ 3.40GHz × 8.

It took about five days to make calculations for one Martian year. The initial data was zero ice in the atmosphere and on the surface. The presence of water vapor was set at all the atmospheric levels with a linear gradient from 0 in the south to 200 ppm in the north. A comparison of the program results for the second calculated year with the TES experimental data is shown in Fig. 1. The results for the third year showed no qualitative differences from those for the second, which leads us to conclude that starting from the sec ond year, the model reaches a sufficiently steady regime and the results can be compared with the experiment. The plots show that the results are gener ally consistent. There are visible seasonal effects and an evident asymmetry of the distribution of water vapor: there are more clouds in the summer season in the northern hemisphere than in the summer season in the southern hemisphere. The plots show the presence of ice clouds in the equatorial zone and their asymme SOLAR SYSTEM RESEARCH

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THE WATER CYCLE IN THE GENERAL CIRCULATION MODEL TES T average in K. Ls 270

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Fig. 2. Comparison of the TES experiment (Leovy and Mintz, 1969) (left) and the MAOAM results (right). The upper line in the plots shows the longitude averaged temperature in K in the season Ls = 270°; the lower line, shows the velocities in the eastward direction in m/s. The vertical axis plots the height in the pressure units (mbar); the horizontal axis plots the latitude in degrees.

try depending on the season. Unfortunately, the total amount of water vapor in the aphelion is somewhat underestimated. Today, it is assumed that the typical values of precipitated water in the aphelion in the northern hemisphere are in the range of 40–70 μm (Trokhimovskiy et al., 2014), but we have only 30– 50 μm, which can generally be regarded as satisfactory, as compared with the LMD MGCM results. The pos sible reasons for the discrepancy with the experiment is the lack of modeling time, insufficiently accurate temperatures in the model (Fig. 2), the absence of dis persion of ice particles by size, etc. Note that to transform the mass of the precipitated ice per unit area Mi into opacity σi, we need to use the special formula (Warren et al., 2006): 3Q exp M ice σ t =  . 4ρR ice

(12)

Here, ρice is the ice density; Rice is the radius of ice particle; and Qext is the extinction coefficient (~1 for the wavelength 12 μm). The latter is strongly depen dent on the wavelength at which the atmosphere is studied as well as the radius of ice particles (Irvine and Pollack, 1968). This is why the absolute values of ice opacity in the atmosphere may vary by an order of SOLAR SYSTEM RESEARCH

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magnitude; given the presence of several tracers of dif ferent size, they may change the distribution pattern. It is clearly evident in Fig. 2 that although the aver age velocities in the eastward direction in the season Ls = 270° are consistent with the experimental data, the model temperatures are not always consistent with the experiment. Since the microphysics and sedimen tation formulas are exponentially dependent on tem perature (see formulas (6)–(9)), even a small change in temperature leads to substantial changes in the water cycle. Figures 3–6 show the results of the MAOAM mod eling under different assumptions. The modeling in column 1 ignored the effect of surface ice on the albedo and the temperature inertia of the surface; in columns 2–4, they were taken into account. In columns 1 and 2, we used instant evapora tion and stocking of ice onto the surface; in 3 and 4, we used the formula taking into account the turbulent vapor rise. Moreover, in columns 3 and 4, we corrected the surface temperature by changing the emittance parameter for better agreement with the TES experi ment. In columns 1–3, we located the water ice polar cap to the north of 80° north latitude and used the constantopacity dust scenario. The latter column dif fers from the previous ones in terms of an increased water ice polar cap (to 75° N) and an approximation

SHAPOSHNIKOV et al. 1 MAOAM H2O gas

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Fig. 3. The results of the MAOAM modeling. The different columns show the modeling results under different assumption (see the text). Panel A shows the precipitated water vapor in µm; B shows the precipitated atmospheric ice in opacity units; and C shows the surface water ice in mm. The horizontal axis in all the panels plots the season in Ls; the vertical axis plots the latitude.

of the presence of ice on the surface to the north of 65° north latitude at surface temperatures below 230 K in the seasons from Ls = 90° to Ls = 120° and the use of the dust transparency dependence on season and lati tude from the TES data.

Evidently, the best agreement of the precipitated water vapors with the experiment is achieved in the latter case. On the one hand, procedure (11) of surface evaporation leads to the fuller evaporation of ice from the surface during the summer season in the northern SOLAR SYSTEM RESEARCH

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THE WATER CYCLE IN THE GENERAL CIRCULATION MODEL 3 MAOAM H2O gas

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Fig. 4. The results of the MAOAM modeling. The different columns show the modeling results under different assumptions (see the text). Panel A shows the precipitated water vapor in µm; B shows the precipitated atmospheric ice in opacity units; and C shows the surface water ice in mm. The horizontal axis in all the panels plots the season in Ls; the vertical axis plots the latitude.

hemisphere (Fig. 4 C3); on the other hand, it super saturates the atmosphere with ice due to vigorous evap oration, which leads to a worse agreement of the atmo sphere ice with the experiment (Figs. 4 B3 and 4 B4). SOLAR SYSTEM RESEARCH

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The total evaporation of ice from the surface increases the model sensitivity to the size of the polar cap; there fore, increasing this size yields better results (Figs. 4 A3 and 4 A4). The use of a more accurate dust scenario

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SHAPOSHNIKOV et al. 1 MAOAM T on 0.5 mbar

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Fig. 5. The results of the MAOAM modeling. The different columns show the modeling results under different assumptions (see the text). Panel D shows the atmospheric temperature at a height of 0.5 mbar in K; E shows the surface temperature in K; and F shows the surface carbon dioxide ice in mm. The horizontal axis in all the panels plots the season in Ls; the vertical axis plots the latitude.

changed the model temperature, and this change also had a substantial effect on the water cycle (Figs. 6 D3 and 6 D4). The excessive ice content in the model (Fig. 1) may be due to the use of only one size of ice particle. The results

obtained by Navarro et al. (2014) are similar in the case of using weak dispersion by size. Adding several tracers to the model should seriously reduce the amount of ice in the atmosphere at the poles because of the strong effect of the ice particle size on the sedimentation rate. SOLAR SYSTEM RESEARCH

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THE WATER CYCLE IN THE GENERAL CIRCULATION MODEL 3 MAOAM T on 0.5 mbar

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Fig. 6. The results of the MAOAM modeling. The different columns show the modeling results under different assumption (see the text). Panel D shows the atmospheric temperature at a height of 0.5 mbar in K; E shows the surface temperature in K; and F shows the surface carbon dioxide ice in mm. The horizontal axis in all the panels plots the season in Ls; the vertical axis plots the latitude.

Figure 7 shows the change in the average surface density of water, depending on the season in Ls. For clarity, all the plots are shifted to the origin. It is evi dent that, first, the advection scheme has full conser SOLAR SYSTEM RESEARCH

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vatism with changes in the total mass of water within the machine accuracy and, second, the model has not yet reached the final stationary cycle and the amount of water in the atmosphere continues to grow.

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SHAPOSHNIKOV et al. Water delta, kg/m2 0.015 0.012

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Fig. 7. Change in the average surface density of water in kg/m2 (vertical axis) depending on the season in Ls (hori zontal axis). All the plots are shifted to the origin. The label indicates as follows: Vapor for water vapor in the atmosphere; Ice for ice in the atmosphere, Total amount for the total amount of water, and Surface ice for ice on the surface.

CONCLUSIONS In this work we developed the water cycle block for the Martian atmosphere, which was incorporated and tested within the numerical generalcirculation model of the Martian atmosphere MAOAM. We were able to achieve stable computation for three Martian years, while maintaining a conservative advection scheme taking into account the water–ice phase transitions, water exchange between the atmosphere and the sur face, and corrections to the vertical velocity of ice par ticles due to sedimentation. The studies show a strong dependence of the amount of water that is actively involved in the water cycle on the initial data, model temperatures, and the mechanism of water exchange between the atmo sphere and the surface. The general pattern and sea sonal asymmetry of the water cycle depends on the size of ice particles, the albedo, and the thermal inertia of the planet’s surface. The best consistency of the calculated results with the TES experiment was achieved under the following assumptions: (1) The amount of the initial water vapor is defined with a latitudinal gradient from 0 to 200 ppm from the south to the north; there is no ice in the atmosphere and on the surface. (2) The surface albedo is set at 0.3; the thermal inertia, at 600 J m–2 K–1 s–1/2, given the presence of a surface ice cover thicker than 5 μm. (3) We use the dust scenario with opacity that depends on season and latitude (from the TES data).

(4) The waterice polar cap has a size of 75° north latitude; ice is present on the surface between 65° north latitude and 75° north latitude at surface tem peratures below 230 K in the season from Ls = 90° to Ls = 120°. (5) The evaporation of ice from the surface follows the scheme taking into account the turbulent rise of water vapor; the ice flow from the surface depends on the horizontal wind velocities. (6) Sedimentation is calculated for ice particles with a size of 4 μm. Although the simulation of the annual cycle is, in general, satisfactory (compared with the single foreign analogue), one of the subsequent modeling tasks, which results from the comparison of the model with the TES experiment, is to increase the total mass of the water vapor in the model in the aphelion season and decrease in the mass of water ice clouds at the poles. The surface evaporation scheme, which takes into account the turbulent rise of water vapor, on the one hand, leads to the most complete evaporation of ice from the surface in the summer season in the northern hemisphere and, on the other hand, supersaturates the atmosphere with ice due to vigorous evaporation, which leads to worse consistency between the amount of the precipitated atmospheric ice and the experi mental data. The full evaporation of ice from the sur face increases the model’s sensitivity to the size of the polar cap; therefore, the increase in the latter leads to better results. The use of a more accurate dust scenario changes the model temperatures, which also strongly affects the water cycle. ACKNOWLEDGMENTS This work was performed at the Laboratory of HighResolution Infrared Spectroscopy of Planetary Atmospheres, Moscow Institute of Physics and Tech nology, and supported by the Ministry of Science and Education of the Russian Federation, project no. 11.G34.31.0074. REFERENCES Bottger, H.M., Lewis, S.R., Read, P.L., and Forget, F., GCM simulations of the Martian water cycle, Proc. 1st Int. Workshop on Mars Atmosphere Modelling and Observations, Granada, 2003. Burlakov, A.V. and Rodin, A.V., 1D microphysical model of H2O condensation clouds in the Martian atmosphere, Sovr. Probl. Distants. Zondir. Zemli Kosmosa, 2011, vol. 8, no. 4, pp. 159–168. Conrath, B.J., Thermal structure of the Martian atmo sphere during the dissipation of dust storm, Icarus, 1975, vol. 24, pp. 34–46. Curry, J.A. and Webster, P.J., Thermodynamics of Atmo spheres and Oceans, Acad. Press, 1999. SOLAR SYSTEM RESEARCH

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THE WATER CYCLE IN THE GENERAL CIRCULATION MODEL Flasar, F.M. and Goody, R.M., Diurnal behaviour of water on Mars, Planet. Space Sci., 1976, vol. 24, pp. 161–181. Forget, F., Hourdin, F., Fournier, R., Hourdin, C., Tala grand, O., Collins, M., Lewis, S.R., Read, P.L., and Huot, J.P., Improved general circulation models of the Martian atmosphere from the surface to above 80 km, J. Geophys. Res., 1999, vol. 104, pp. 24–155. Fuks, N.A., Mekhanika aerozolei (Aerosols Mechanics), Moscow: USSR Acad. Sci., 1955. Hartogh, P., Medvedev, A.S., Kuroda, T., Saito, R., Villan ueva, G., Feofilov, A.G., Kutepov, A.A., and Berger, U., Description and climatology of a new gen eral circulation model of the Martian atmosphere, J. Geophys. Res., 2005, vol. 110, p. 15. Hartogh, P., Medvedev, A.S., and Jarchow, C., Middle atmosphere polar warmings on Mars: simulations and study on the validation with submillimeter observa tions, Planet. Space Sci., 2007, vol. 55, pp. 1103–1112. Irvine, W.M. and Pollack, J.B., Infrared optical properties of water and ice spheres, Icarus, 1968, vol. 8, nos. 1–3, pp. 324–360. Korablev, O.I., Krasnopolsky, V.A., and Rodin, A.V., Verti cal structure of Martian dust measured by solar infrared occultations from the Phobos spacecraft, Icarus, 1992, vol. 102, pp. 76–87. Koroteev, A.S., Pilotiruemaya ekspeditsiya na Mars (Manned Mission to the Mars), Moscow: K.E. Tsiolk ovsky Russian Cosmonautics Academy, 2006. Kutepov, A.A., Gusev, O.A., and Ogibalov, V.P., Solution of the nonLTE problem for molecular gas in planetary atmospheres: superiority of accelerated lambda itera tion, J. Quant. Spectrosc. Radiat. Transf., 1998, vol. 60, pp. 199–220. Lefevre, F., Bertaux, J.L., Clancy, R.T., Encrenaz, T., Fast, K., Forget, F., Lebonnois, S., Montmessin, F., and Perrier, S., Heterogeneous chemistry in the atmo sphere of Mars, Nature, 2008, vol. 454, pp. 971–975. Leovy, C.B. and Mintz, Y., Numerical simulation of the atmospheric circulation and climate of Mars, J. Atmos. Sci., 1969, vol. 26, pp. 1167–1190. Medvedev, A.S. and Klaassen, G.P., Parameterization of gravity wave momentum deposition based on nonlinear wave interactions: basic formulation and sensitivity tests, J. Atmos. Sol.Terr. Phys., 2000, vol. 62, pp. 1015– 1033. Medvedev, A.S. and Hartogh, P., Winter polar warmings and the meridional transport on mars simulated with a general circulation model, Icarus, 2007, vol. 186, pp. 97–110. Medvedev, A.S., Yigit, E., Hartogh, P., and Becker, E., Influence of gravity waves on the Martian atmosphere: general circulation modeling, J. Geophys. Res., 2011, vol. 116, p. 14. SOLAR SYSTEM RESEARCH

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Medvedev, A.S., GonzalezGalindo, F., Yigit, E., Feofilov, A.G., Forget, F., and Hartogh, P., Cooling of the Martian thermosphere by CO2 radiation and gravity waves: an intercomparison study with two general cir culation models, J. Geophys. Res. Planets, 2015, vol. 120, pp. 913–927. Mingalev, V.S., Mingalev, I.V., Mingalev, O.V., Oparin, A.M., and Orlov, K.G., The way to generalize monotonous hybrid scheme of the 2nd order for gas dynamics equations for irregular spatial grid, Zh. Vychisl. Mat. Mat. Fiz., 2010, vol. 50, no. 5, pp. 923– 936. Montmessin, F. and Forget, F., Waterice clouds in the LMDs Martian general circulation model, Proc. 1st Int. Workshop on Mars Atmosphere Modelling and Observa tions, Granada, 2003. Montmessin, F., Forget, F., Rannou, P., Cabane, M., and Haberle, R.M., Origin and role of water ice clouds in the Martian water cycle as inferred from a general cir culation model, J. Geophys. Res., 2004, vol. 109, p. 109. Navarro, T., Madeleine, J.B., Forget, F., Spiga, A., Mil lour, E., Montmessin, F., and Maattanen, A., Global climate modeling of the Martian water cycle with improved microphysics and radiatively active water ice clouds, J. Geophys. Res., 2014, vol. 119, no. 7, pp. 1479–1495. Piskunov, V.N., Dinamika aerozolei (Aerosols Dynamics), Moscow: Fizmatlit, 2010. Sheehan, W., The Planet Mars: a History of Observation and Discovery, Tucson: Univ. Arizona Press, 1996. Simmons, A.J. and Burridge, D.M., An energy and angu larmomentum conserving vertical finitedifference scheme and hybrid vertical coordinates, Mon. Wea. Rev., 1981, vol. 109, pp. 758–766. Simmons, A.J. and Chen, J., The calculation of geopoten tial and the pressure gradient in the ECMWF atmo spheric model: influence on the simulation of the polar atmosphere and on temperature analyses, Q. J. R. Met. Soc., 1991, vol. 117, pp. 29–58. Trokhimovskiy, A., Fedorova, A., Korablev, O., Mont messin, F., Bertaux, J.L., Rodin, A., and Smith, M.D., Mars’ water vapor mapping by the Spicam IR spec trometer: five Martian years of observations, Icarus, 2014, vol. 251, pp. 50–64. Warren, S.G., Brandt, R.E., and Grenfell, T.C., Visible and nearultraviolet absorption spectrum of ice from trans mission of solar radiation into snow, Appl. Opt., 2006, vol. 45, no. 21, pp. 5320–5334. Yigit, E., Medvedev, A.S., Aylward, A.D., Hartogh, P., and Harris, M.J., Modeling the effects of gravity wave momentum deposition on the general circulation above the turbopause, J. Geophys. Res., 2009, vol. 114, p. 14.

Translated by A. Kobkova