1 Introduction

Although modern atmospheric science has its origins in attempts to understand the Earth's atmosphere, thanks to satellite data and computer models, it has been extended to encompass extraterrestrial atmospheres, the most similar being on Mars. So far, the comparisons have focused on the similarities and differences of various dynamical mechanisms (see, e.g., the review by Leovy [2001]) and the Martian point of comparison has enriched our understanding of the Earth. Significant differences between terrestrial and Martian atmospheres include the strong control of Martian atmospheric temperature by dust, the larger role of topography, the stronger diurnal and annual cycles, and the larger role of atmospheric tides. Significant similarities include jets, zonal circulation patterns, and the existence of fairly similar Hadley cells.

These deterministic mechanisms are pertinent at the very largest spatial scales, but what about the rest? On both Mars and Earth, typical Reynolds numbers are greater than 1011 (Table 1), so that from planetary down to dissipation scales (centimetric and millimetric, respectively, Table 1) the flow is turbulent so that we would expect high‐level (statistical) turbulent laws to be obeyed. As reviewed in Lovejoy and Schertzer [2013], these laws are of the form: , where ΔI is a fluctuation in the parameter I, is the vector separating two points, and ϕ a turbulent flux (or a power of a turbulent flux). For example, we recover the Kolmogorov law if I is a velocity component, is the distance between two points, ϕ = ε1/3 where ε is the energy flux to smaller scales, H is a scaling exponent, and the fluctuation ΔI ( ) is taken as the difference in I between two points separated by vector (in cases where H < 0, other definitions of fluctuations are needed). For atmospheric applications, the main limitations of the classical laws are their assumption that the turbulent flux is fairly homogeneous (constant or quasi‐Gaussian) and that the turbulence is statistically isotropic: they take to be the usual vector norm of the vector displacement . However, since the 1980s the development of multiplicative cascades to account for intermittency have allowed the fluxes to be wildly variable (multifractal), and the development of generalized scale invariance has allowed the turbulent laws to be extended beyond isotropy to strongly stratified flows. The resulting anisotropic cascade picture has been shown to be highly accurate for the Earth's atmosphere, numerical models of the atmosphere, and atmospheric reanalyses with significant deviations only occurring at scales larger than about 5000 km (see the extensive review by Lovejoy and Schertzer [2013]).

Table 1. Mars Versus Earth, Turbulent Characteristics Near Their Surfaces Earth Mars Densitya (ρ) (surface, kg/m3) 1.24 (10°C) 7 × 10−3 (−20°C) Dynamic viscosityb (η), (kg m−1s−1) 1.8 × 10−5 (10°C) 1.3 × 10−5 (−20°C) Kinematic viscosity (v = η/ρ), m2/s 1.5 × 10−5 1.9 × 10−3 L w c 2 × 107 1.07 × 107 ΔUd 27 70 Re = L w ΔU/v 4 × 1013 4 × 1011 εe (W/kg = m2/s3) 10−3 4 × 10−2 L diss f (m) 1.4 × 10−3 2.0 × 10−2

(A note on terminology: In this paper we use the term “stratification” to refer to vertical stratification, which is apparently scaling with different exponents in the horizontal and vertical direction implying that the stratification becomes more and more extreme at large scales. However, we find that there is an exactly (mathematically) analogous scaling anisotropy in the EW versus NS direction, and this is important in this paper. Also, in the following, we use the word scaling as in the physics literature, to refer to a physical quantity that varies in a power law way as a function of space and/or time scale. The corresponding exponents are “scale invariant” and the notion of scale can be considerably generalized from the usual Euclidean notion so as to take into account anisotropy.)

The advantage of these high‐level laws is that although they describe the statistical properties over wide ranges of space‐time scales, they are quite simple, and they provide the theoretical basis of stochastic modeling and stochastic forecasting [e.g., Lovejoy et al., 2015]. Theoretically, the reason that the turbulent laws are obeyed is because the dynamic equations are formally scale invariant down to the dissipation scale. We therefore expect them to hold on Mars as well as Earth, although the relevant fluxes and exponents could be different. Lovejoy et al. [2014] took the first step in making a statistical Earth‐Mars comparison by showing that with the exception of the diurnal and annual cycles and harmonics that the second‐order temporal statistics (the spectra) for wind and temperature of the two planets were virtually the same, as long as the Martian time scales were rescaled by a factor of about 5, itself theoretically predicted from turbulence theory and from the differences in the overall solar forcing and atmospheric thicknesses. Lovejoy et al. [2014] used both in situ (Viking Lander) data and MACDA (Mars Analysis Correction Data Assimilation) reanalyzes [Montabone et al., 2014]; the aim of this paper is to extend this statistical analysis to moments other than the second, to the pressure and both horizontal wind components, to provide analyses in space and in time, and finally to directly analyze turbulent fluxes and hence cascades structures. The methods are the same as those that were applied to terrestrial reanalyses in Lovejoy and Schertzer [2011]. Beyond establishing striking quantitative and qualitative similarities between Martian and terrestrial atmospheric variability, this paper underlines the importance of going beyond deterministic mechanistic descriptions to consider the simple higher level statistical laws needed for an understanding—and improved modeling—over a much wider range of space‐time scales.

Let us briefly comment on turbulent fluxes which are fluxes in Fourier, not real space (i.e., across surfaces in Fourier space i.e. across scales). It seems that the energy flux is important in both for determining the horizontal velocity dynamics, whereas the buoyancy force variance flux controls the vertical structure of the horizontal wind. These are anisotropic generalizations of the classical Kolmogorov and Bolgiano‐Obukhov laws. But most of the exponents observed in the atmosphere have yet to be explained and the corresponding fluxes identified. Fortunately, this is not necessary for analyzing the data and for estimating their intermittency characteristics.

This paper is organized as follows. In section 2, we describe the MACDA reanalysis data set used for the statistical analysis for the Martian Atmosphere. In section 3, we compare the Fourier‐space spectral scaling of atmospheric fields on Earth and Mars. In section 4, we compare real‐space statistics of atmospheric fields on Earth and Mars. In section 5, we discuss and conclude.