# Weight decomposition of $\mathfrak{sl}_{d}({\mathbb{R}})$ with respect to
the adjoint representation of $\mathfrak{so}(p,q)$

Jiyoung Han  School of Mathematics, Korea Institute for Advanced Study
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this concise article, we compute the weight decomposition of
$\mathfrak{sl}_{d}({\mathbb{R}})$ with respect to the adjoint representation
of $\mathfrak{so}(p,q)$, where $d=p+q$ and demonstrate in detail that
$\mathfrak{sl}_{d}({\mathbb{R}})$ comprises two irreducible
$\mathfrak{so}(p,q)$-invariant subspaces. This can be employed to establish
the well-known fact that the identity component of ${\mathrm{SO}}(p,q)$ is a
maximal connected subgroup of $\mathrm{SL}_{d}({\mathbb{R}})$.

###### Contents

  1. 1 Introduction
  2. 2 Restricted Root Space Decomposition of $\mathfrak{so}(p,q)$
  3. 3 Weight Decomposition of $\mathfrak{sl}_{d}({\mathbb{R}})$
  4. 4 Proof of Main Theorem

##  1\. Introduction

Let $\mathfrak{g}$ be a Lie algebra and consider a Lie algebra representation
$(\Phi,V)$ of $\mathfrak{g}$. A subspace $W\subseteq V$ is called
_$\mathfrak{g}$ -invariant_ if $\Phi(X)W\subseteq W$ for all
$X\in\mathfrak{g}$, and we say that a $\mathfrak{g}$-invariant subspace $W$ is
_$\mathfrak{g}$ -irreducible_ if there is no proper nontrivial subspace
$W^{\prime}\subset W$ which is $\mathfrak{g}$-invariant. When $\mathfrak{g}$
is a semisimple real Lie algebra, it possesses a restricted root system, thus
exhibiting the property that any representation $V$ for $\mathfrak{g}$
decomposes into $\mathfrak{g}$-irreducible sub-representations.

In this article, our focus lies on the adjoint representation for the
semisimple Lie algebra $\mathfrak{so}(p,q)$ on
$\mathfrak{sl}_{d}({\mathbb{R}})$, where $d=p+q$. The theorem below is likely
familiar to experts; however, given the callenge in locating references, we
will present a detailed proof.

######  Theorem 1.1.

Let $d=p+q\geq 2$ with $p,q\geq 0$. The subalgebra
$\mathfrak{sl}_{d}({\mathbb{R}})$ consists of two
$\mathfrak{so}(p,q)$-irreducible subspaces.

As a corollay, we can re-obtain the well-known result regarding the maximality
of ${\mathrm{SO}}(p,q)$ on $\mathrm{SL}_{d}({\mathbb{R}})$.

######  Corollary 1.2.

The group ${\mathrm{SO}}(p,q)^{\circ}$ is maximal among connected subgroups of
$\mathrm{SL}_{d}({\mathbb{R}})$, where $d=p+q\geq 2$

The paper is organized as follows. In Section 2, we revisit the restricted
root system of $\mathfrak{so}(p,q)$. Based on this root system, we obtain all
of weights and corresponding weight vectors of the compliment of
$\mathfrak{so}(p,q)$ in $\mathfrak{sl}_{d}({\mathbb{R}})$ in Section 3. For
readers seeking a quick overview, we display root and weight vectors for a
concrete example, $\mathfrak{so}(4,2)$, in Appendix. Finally, we present the
proof of Theorem 1.1 in Section 4.

### Acknowledgements

I would like to be thankful to Seonhee Lim and Keivan Mallahi-Karai for their
encouragement throughout this project. The author acknowledges support from a
KIAS Individual Grant MG088401 at Korea Institute for Advanced Study.

##  2\. Restricted Root Space Decomposition of $\mathfrak{so}(p,q)$

Let us briefly recall the restricted root system for $\mathfrak{so}(p,q)$, as
introduced in [1, Page 372-373]. Denote elements of
$\mathfrak{sl}_{d}({\mathbb{R}})$ by two by two block matrix

$X=\left(\begin{array}[]{cc}A&B\\\ C&D\end{array}\right)$

with $A\in\mathrm{Mat}_{p}({\mathbb{R}})$,
$B\in\mathrm{Mat}_{p,q}({\mathbb{R}})$,
$C\in\mathrm{Mat}_{q,p}({\mathbb{R}})$, and
$D\in\mathrm{Mat}_{q}({\mathbb{R}})$, where $p,q\geq 0$ and $p+q=d$. For the
sake of convenience, we will take basis of $\mathrm{Mat}_{d}({\mathbb{R}})$ as
$A_{i,j}$, $B_{i,j}$, $C_{i,j}$ and $D_{i,j}$ as follows: if we let
$\\{E_{i,j}\\}_{1\leq i,j\leq d}$ be a standard basis of
$\mathrm{Mat}_{d}({\mathbb{R}})$,

$\begin{array}[]{ll}A_{i,j}=E_{p+1-i,p+1-j},&B_{i,j}=E_{p+1-i,p+j}\;\text{and}\\\
C_{i,j}=E_{p+i,p+1-j},&D_{i,j}=E_{p+i,p+j}.\end{array}$

The Lie algebra $\mathfrak{so}(p,q)$ consists of matrices $X$ in
$\mathfrak{sl}_{d}({\mathbb{R}})$ for which $A$, $D$ are skew-symmetric and
$B^{t}=C$, so that $\dim\mathfrak{so}(p,q)=\frac{1}{2}(p(p-1)+q(q-1))+pq$.
There is a natural involution $\theta$ in $\mathfrak{so}(p,q)$ defined as the
matrix transpose, leading the Cartan decomposition
$\mathfrak{so}(p,q)=\mathfrak{k}\oplus\mathfrak{p}$, where $\mathfrak{k}$
denotes the set of $X\in\mathfrak{so}(p,q)$ for which $B$ (and consequently
$D$) is a zero matrix, and $\mathfrak{p}$ represents the set of those $X$
whose $A$\- and $D$-components are zero matrices. Then one can take a maximal
abelian subalgebra $\mathfrak{a}$ of $\mathfrak{p}$ whose elements are of the
form

(2.1) $F=\left(\begin{array}[]{c|c}&\begin{array}[]{ccc}&&\\\ &&a_{q}\\\
&\reflectbox{$\ddots$}&\\\ a_{1}&&\end{array}\\\
\hline\cr\begin{array}[]{cccc}&&&a_{1}\\\ &&\reflectbox{$\ddots$}&\\\
&a_{q}&&\end{array}&\end{array}\right)=\sum_{i=1}^{q}a_{i}(B_{i,i}+C_{i,i}).$

Here, withought loss of generality, we assume that $p\geq q$. Denote by

$\\{f_{i}:i=1,\ldots,q\\}$

the dual basis of $\mathfrak{a}$ given by
$f_{i}(B_{j,j}+C_{j,j})=\delta_{ij}$. We say that a nonzero
$\lambda\in\mathfrak{a}^{*}$ is a restricted root for
$(\mathfrak{g},\mathfrak{a})$ if there is a nonzero $X\in\mathfrak{g}$ such
that for all $F\in\mathfrak{a}$,

$[F,X]=\lambda(F)X.$

######  Proposition 2.1.

The restricted roots of $(\mathfrak{so}(p,q),\mathfrak{a})$ are $\pm f_{i}$
for $1\leq i\leq q$ (if $p\neq q$), $\pm f_{i}\pm f_{j}$, and $\pm f_{i}\mp
f_{j}$ for $1\leq i<j\leq q$.

| # of this type |  $\dim$ of each root space |
---|---|---|---
$\pm f_{i}$ | 2q | p-q | 2q(p-q)
$\pm f_{i}\pm f_{j}$ | q(q-1) | 1 | q(q-1)
$\pm f_{i}\mp f_{j}$ | q(q-1) | 1 | q(q-1)
0 | 1 | $\frac{1}{2}(p-q)(p-q-1)+q$ | $\frac{1}{2}(p-q)(p-q-1)+q$
|  |  | $\frac{1}{2}(p(p-1)+q(q-1))+pq=\dim\mathfrak{so}(p,q)$

(Here, notice that $0$ is not strictly a restricted root; however, we include
it for convenience.)

For each restricted root (including $0$), a basis of the root space
$\mathfrak{so}(p,q)_{\lambda}$ can be taken as follows. See also Appendix for
the matrix form in case of $\mathfrak{so}(4,2)$.

  1. i)

$\lambda=\pm f_{i}$ ($1\leq i\leq q$).

For $\lambda=f_{i}$, $\mathfrak{so}(p,q)_{\lambda}$ is generated by

$H(\lambda)_{\ell}=A_{q+\ell,i}-B_{q+\ell,i}-A_{i,q+\ell}-C_{i,q+\ell},\;1\leq\ell\leq
p-q,$

and for $\lambda=-f_{i}$,

$\mathfrak{so}(p,q)_{\lambda}=\left\langle
H(\lambda)_{\ell}=A_{q+\ell,i}+B_{q+\ell,i}-A_{i,q+\ell}+C_{i,q+\ell}:1\leq\ell\leq
p-q\right\rangle.$

  2. ii)

$\lambda=\pm f_{i}\pm f_{j}$ ($1\leq i<j\leq q$).

For $\lambda=f_{i}+f_{j}$,

$\mathfrak{so}(p,q)_{\lambda}={\mathbb{R}}.\left(-A_{i,j}+B_{i,j}+A_{j,i}-B_{j,i}-C_{i,j}+D_{i,j}+C_{j,i}-D_{j,i}\right),$

and for $\lambda=-f_{i}-f_{j}$,

$\mathfrak{so}(p,q)_{\lambda}={\mathbb{R}}.\left(-A_{i,j}-B_{i,j}+A_{j,i}+B_{j,i}+C_{i,j}+D_{i,j}-C_{j,i}-D_{j,i}\right).$

  3. iii)

$\lambda=\pm f_{i}\mp f_{j}$ ($1\leq i<j\leq q$).

For $\lambda=f_{i}-f_{j}$,

$\mathfrak{so}(p,q)_{\lambda}={\mathbb{R}}.\left(-A_{i,j}-B_{i,j}+A_{j,i}-B_{j,i}-C_{i,j}-D_{i,j}-C_{j,i}+D_{j,i}\right),$

and for $\lambda=-f_{i}+f_{j}$,

$\mathfrak{so}(p,q)_{\lambda}={\mathbb{R}}.\left(-A_{i,j}+B_{i,j}+A_{j,i}+B_{j,i}+C_{i,j}-D_{i,j}+C_{j,i}+D_{j,i}\right).$

  4. iv)

$\lambda=0$.

Following the notation of [1],
$\mathfrak{so}(p,q)_{0}=\mathfrak{m}\oplus\mathfrak{a}$, where

$\mathfrak{m}=\sum_{1\leq i<j\leq
p-q}{\mathbb{R}}.(A_{q+i,-q+j}-A_{q+j,-q+i})$

is the centralizer of $\mathfrak{a}$ in $\mathfrak{k}$, which is isomorphic to
$\mathfrak{so}(p-q)$ in our case.

##  3\. Weight Decomposition of $\mathfrak{sl}_{d}({\mathbb{R}})$

Define

(3.1) $\mathfrak{s}=\left\\{X=\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right):\begin{array}[]{c}\mathrm{tr}(A)+\mathrm{tr}(B)=0,\\\
A=A^{t},\;D=D^{t}\\\ B=-C^{t}.\end{array}\right\\}.$

It is not hard to check that
$\mathfrak{sl}_{d}({\mathbb{R}})=\mathfrak{so}(p,q)\oplus\mathfrak{s}$, and
$\mathfrak{s}$ is invariant under the adjoint action of $\mathfrak{so}(p,q)$.

Following the approach in Section 2, we will present the weight decomposition
of $\mathfrak{s}$ for the adjoint representation of $\mathfrak{so}(p,q)$
below, providing a comprehensive overview of the weight decomposition of
$\mathfrak{sl}_{d}({\mathbb{R}})$.

######  Proposition 3.1.

The weights of $\mathfrak{s}$ for the adjoint representation of
$\mathfrak{so}(p,q)$ are $\pm f_{i}$ (if $p\neq q$) and $\pm 2f_{i}$ for
$1\leq i\leq q$, $\pm f_{i}\pm f_{j}$ and $\pm f_{i}\mp f_{j}$ for $1\leq
i<j\leq q$ and $0$.

| # of this type |  $\dim$ of each root space |
---|---|---|---
$\pm f_{i}$ | 2q | p-q | 2q(p-q)
$\pm 2f_{i}$ | 2q | 1 | 2q
$\pm f_{i}\pm f_{j}$ | q(q-1) | 1 | q(q-1)
$\pm f_{i}\mp f_{j}$ | q(q-1) | 1 | q(q-1)
0 | 1 | $\frac{1}{2}(p-q)(p-q+1)+q-1$ | $\frac{1}{2}(p-q)(p-q+1)+q-1$
|  |  | $\frac{1}{2}(p(p+1)+q(q+1))+pq-1=\dim\mathfrak{s}$

For each weight $\omega$, a basis of the weight space $\mathfrak{s}_{\omega}$
can be taken as follows (see Appendix for the matrix form in case of
$\mathfrak{so}(4,2)$).

  1. i)

$\omega=\pm f_{i}$ ($1\leq i\leq q$).

For $\omega=f_{i}$, $\mathfrak{s}_{\omega}$ is generated by

$S(\omega)_{\ell}=A_{q+\ell,i}-B_{q+\ell,i}+A_{i,q+\ell}+C_{i,q+\ell},\;1\leq\ell\leq
p-q,$

and for $\omega=-f_{i}$,

$\mathfrak{s}_{\omega}=\left\langle
S(\omega)_{\ell}=A_{q+\ell,i}+B_{q+\ell,i}+A_{i,q+\ell}-C_{i,q+\ell}:1\leq\ell\leq
p-q\right\rangle.$

  2. ii)

$\omega=\pm 2f_{i}$ ($1\leq i\leq q$).

For $\omega=2f_{i}$,

$\mathfrak{s}_{\omega}={\mathbb{R}}.\left(A_{i,i}-B_{i,i}+C_{i,i}-D_{i,i}\right),$

and for $\omega=-2f_{i}$,

$\mathfrak{s}_{\omega}={\mathbb{R}}.\left(A_{i,i}+B_{i,i}-C_{i,i}-D_{i,i}\right).$

  3. iii)

$\omega=\pm f_{i}\pm f_{j}$ ($1\leq i<j\leq q$).

For $\omega=f_{i}+f_{j}$,

$\mathfrak{s}_{\omega}={\mathbb{R}}.\left(-A_{i,j}+B_{i,j}-A_{j,i}+B_{j,i}-C_{i,j}+D_{i,j}-C_{j,i}+D_{j,i}\right),$

and for $\lambda=-f_{i}-f_{j}$,

$\mathfrak{s}_{\omega}={\mathbb{R}}.\left(A_{i,j}+B_{i,j}+A_{j,i}+B_{j,i}-C_{i,j}-D_{i,j}-C_{j,i}-D_{j,i}\right),$

  4. iv)

$\lambda=\pm f_{i}\mp f_{j}$ ($1\leq i<j\leq q$).

For $\lambda=f_{i}-f_{j}$,

$\mathfrak{s}_{\omega}={\mathbb{R}}.\left(A_{i,j}+B_{i,j}+A_{j,i}-B_{j,i}+C_{i,j}+D_{i,j}-C_{j,i}+D_{j,i}\right).$

and for $\lambda=-f_{i}+f_{j}$,

$\mathfrak{s}_{\omega}={\mathbb{R}}.\left(A_{i,j}-B_{i,j}+A_{j,i}+B_{j,i}-C_{i,j}+D_{i,j}+C_{j,i}+D_{j,i}\right).$

  5. v)

$\omega=0$.

The weight space $\mathfrak{s}_{0}$ is composed of the first
$(p-q)\times(p-q)$ symmetric matrices and diagonal matrices. One can take a
basis of $\mathfrak{s}_{0}$ as follows.

$\begin{gathered}A_{q+i,q+j}+A_{q+j,q+i}\;(1\leq i<j\leq p-q),\\\
2A_{q+i,q+i}+(A_{1,1}+D_{1,1})\;(1\leq i\leq p-q),\;\text{and}\\\
(A_{i,i}+D_{i,i})-(A_{i+1,i+1}+D_{i+1,i+1})\;(1\leq i\leq q-1).\end{gathered}$

###### Proof.

i) For $F\in\mathfrak{a}$ as in (2.1), by putting $\ell^{\prime}=\ell+q$,

$\begin{split}&{\mathrm{ad}}(F)(S(f_{i})_{\ell})=[F,S(f_{i})_{\ell}]\\\
&=\left(\sum_{k=1}^{q}a_{k}(B_{k,k}+C_{k,k})\right)\left(A_{q+\ell,i}-B_{q+\ell,i}+A_{i,q+\ell}+C_{i,q+\ell}\right)\\\
&\hskip
57.81621pt-\left(A_{q+\ell,i}-B_{q+\ell,i}+A_{i,q+\ell}+C_{i,q+\ell}\right)\left(\sum_{k=1}^{q}a_{k}(B_{k,k}+C_{k,k})\right)\\\
&=\sum_{k=1}^{q}a_{k}\left[(E_{p+k,p+1-k}+E_{p+1-k,p+k})(E_{p+1-\ell^{\prime},p+1-i}-E_{p+1-\ell^{\prime},p+i}+E_{p+1-i,p+1-\ell^{\prime}}+E_{p+i,p+1-\ell^{\prime}})\right.\\\
&\hskip
46.97505pt\left.-(E_{p+1-\ell^{\prime},p+1-i}-E_{p+1-\ell^{\prime},p+i}+E_{p+1-i,p+1-\ell^{\prime}}+E_{p+i,p+1-\ell^{\prime}})(E_{p+k,p+1-k}+E_{p+1-k,p+k})\right]\\\
&=a_{i}(E_{p+1-\ell^{\prime},p+1-i}-E_{p+1-\ell^{\prime},p+i}+E_{p+1-i,p+1-\ell^{\prime}}+E_{p+i,p+1-\ell^{\prime}})\\\
&=f_{i}(F)\left(A_{q+\ell,i}-B_{q+\ell,i}+A_{i,q+\ell}+C_{i,q+\ell}\right)=f_{i}(F)S(f_{i})_{\ell}.\end{split}$

The remaining cases can be verified in a similar manner. Given that the total
number of linearly independent weight vectors is equal to $\dim\mathfrak{s}$,
we can thus derive the weight decomposition of $\mathfrak{s}$. ∎

##  4\. Proof of Main Theorem

Theorem 1.1 follows directly from the proposition below.

######  Proposition 4.1.

The subspace $\mathfrak{s}$ defined in (3.1) is
$\mathfrak{so}(p,q)$-irreducible.

###### Proof.

It is sufficient to show that any weight vectors of $\mathfrak{s}$ provided in
Proposition 3.1 is mapped to another weight vector by the adjoint action of
$\mathfrak{so}(p,q)$. Let $[H(\lambda)]$ and $[S(\omega)]$ are the class of
root vectors for $\lambda$ and weight vectors for $\omega$, respectively, if
the root or weight space is one-dimensional, and $H(\lambda)_{\ell}$ and
$S(\omega)_{\ell}$ are as in Proposition 2.1 and Proposition 3.1,
respectively. The following, for instance, demonstrates the transitivity of
the set of all nonzero weights.

  * •

$[S(2f_{1})]\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{H(-f_{1}\mp f_{j})}{H(f_{1}\mp
f_{j})}[S(f_{1}\pm f_{j})]\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{H(-f_{1}\mp f_{j})}{H(f_{1}\pm
f_{j})}[S(\pm 2f_{j})]$;

  * •

$[S(\mp f_{i}\pm f_{j})]\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{H(\pm f_{j}+f_{1})}{H(\mp
f_{i}-f_{1})}[S(f_{1}\pm f_{j})]\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{H(\pm f_{i}-f_{1})}{H(\mp
f_{i}+f_{1})}[S(\pm f_{i}\pm f_{j})]\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{H(-f_{1}\mp f_{i})}{H(f_{1}\pm
f_{i})}[S(-f_{1}\pm f_{j})]$;

  * •

$[S(\pm 2f_{i})]\ext@arrow 9999{\arrowfill@\leftarrow\relbar\rightarrow}{H(\mp
f_{i})_{\ell}}{H(\pm f_{i})_{\ell}}[S(\pm f_{i})_{\ell}]$

It can be verified that each mapping sends a nonzero weight vectors to another
nonzero weight vector. As for the connection to weight vectors of zero weight,
we have the following observaton:

  * •

$[S(-f_{1})_{j}]\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{H(f_{1})_{i}}{H(-f_{1})_{i}}[A_{q+i,q+j}+A_{q+j,q+i}]$

  * •

$[S(-f_{1})_{i}]\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{H(f_{1})_{i}}{H(-f_{1})_{i}}[2A_{q+i,q+i}+(A_{1,1}+D_{1,1})]$

  * •

$[S(-f_{i}-f_{i+1})]\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{H(f_{i}+f_{i+1})}{H(-f_{i}-f_{i+1})}[A_{i,i}+D_{i,i}-(A_{i+1,i+1}+D_{i+1,i+1})]$.

∎

## Appendix

We now illustrate the matrix form of root and weight vectors discussed in
Theorem 2.1 and Theorem 3.1, respectively, in the case of
$\mathfrak{so}(4,2)$.

We have that $\mathfrak{so}(4,2)_{0}=\mathfrak{a}\oplus\mathfrak{m}$, where

$\mathfrak{a}=\left\\{{\small\left(\begin{array}[]{cccc|cc}&&&&&\\\ &&&&&\\\
&&&&&a_{2}\\\ &&&&a_{1}&\\\ \hline\cr&&&a_{1}&&\\\
&&a_{2}&&&\end{array}\right)}:a_{1},\;a_{2}\in{\mathbb{R}}\right\\}\quad\text{and}\quad\mathfrak{m}=\left\\{{\small\left(\begin{array}[]{cccc|cc}&x&&&&\\\
-x&&&&&\\\ &&&&&\\\ &&&&&\\\ \hline\cr&&&&&\\\
&&&&&\end{array}\right)}:x\in{\mathbb{R}}\right\\}.$

The restricted roots for $(\mathfrak{so}(p,q),\mathfrak{a})$ and corresponding
root spaces are as follows.

root | $f_{1}$ | $f_{2}$
---|---|---
$\begin{array}[]{c}\text{root}\\\ \text{vectors}\end{array}$ | $\left(\begin{array}[]{cccc|cc}&&&x_{2}&-x_{2}&\\\ &&&x_{1}&-x_{1}&\\\ &&&&&\\\ -x_{2}&-x_{1}&&&&\\\ \hline\cr-x_{2}&-x_{1}&&&&\\\ &&&&&\end{array}\right)$ | $\left(\begin{array}[]{cccc|cc}&&x_{2}&&&-x_{2}\\\ &&x_{1}&&&-x_{1}\\\ -x_{2}&-x_{1}&&&&\\\ &&&&&\\\ \hline\cr&&&&&\\\ -x_{2}&-x_{1}&&&&\end{array}\right)$
| $-f_{1}$ | $-f_{2}$
| $\left(\begin{array}[]{cccc|cc}&&&x_{2}&x_{2}&\\\ &&&x_{1}&x_{1}&\\\ &&&&&\\\ -x_{2}&-x_{1}&&&&\\\ \hline\cr x_{2}&x_{1}&&&&\\\ &&&&&\end{array}\right)$ | $\left(\begin{array}[]{cccc|cc}&&x_{2}&&&x_{2}\\\ &&x_{1}&&&x_{1}\\\ -x_{2}&-x_{1}&&&&\\\ &&&&&\\\ \hline\cr&&&&&\\\ x_{2}&x_{1}&&&&\end{array}\right)$
| $f_{1}+f_{2}$ | $-f_{1}-f_{2}$
| $\left(\begin{array}[]{cccc|cc}&&&&&\\\ &&&&&\\\ &&&x&-x&\\\ &&-x&&&x\\\ \hline\cr&&-x&&&x\\\ &&&x&-x&\end{array}\right)$ | $\left(\begin{array}[]{cccc|cc}&&&&&\\\ &&&&&\\\ &&&x&x&\\\ &&-x&&&-x\\\ \hline\cr&&x&&&x\\\ &&&-x&-x&\end{array}\right)$
| $f_{1}-f_{2}$ | $-f_{1}+f_{2}$
| $\left(\begin{array}[]{cccc|cc}&&&&&\\\ &&&&&\\\ &&&x&-x&\\\ &&-x&&&-x\\\ \hline\cr&&-x&&&-x\\\ &&&-x&x&\end{array}\right)$ | $\left(\begin{array}[]{cccc|cc}&&&&&\\\ &&&&&\\\ &&&x&x&\\\ &&-x&&&x\\\ \hline\cr&&x&&&-x\\\ &&&x&x&\end{array}\right)$
Table 1. The restricted root system of $\mathfrak{so}(4,2)$

And the following table exhibits weight spaces of nonzero weights in
$\mathfrak{s}$ for $(\mathfrak{so}(4,2),\mathfrak{a})$.

weight | $f_{1}$ | $f_{2}$ | $-f_{1}$
---|---|---|---
$\begin{array}[]{c}\text{weight}\\\ \text{vectors}\end{array}$ | $\left(\begin{array}[]{cccc|cc}&&&x_{2}&-x_{2}&\\\ &&&x_{1}&-x_{1}&\\\ &&&&&\\\ x_{2}&x_{1}&&&&\\\ \hline\cr x_{2}&x_{1}&&&&\\\ &&&&&\end{array}\right)$ | $\left(\begin{array}[]{cccc|cc}&&x_{2}&&&-x_{2}\\\ &&x_{1}&&&-x_{1}\\\ x_{2}&x_{1}&&&&\\\ &&&&&\\\ \hline\cr&&&&&\\\ x_{2}&x_{1}&&&&\end{array}\right)$ | $\left(\begin{array}[]{cccc|cc}&&&x_{2}&x_{2}&\\\ &&&x_{1}&x_{1}&\\\ &&&&&\\\ x_{2}&x_{1}&&&&\\\ \hline\cr-x_{2}&-x_{1}&&&&\\\ &&&&&\end{array}\right)$
| $-f_{2}$ | $2f_{1}$ | $2f_{2}$
| $\left(\begin{array}[]{cccc|cc}&&x_{2}&&&x_{2}\\\ &&x_{1}&&&x_{1}\\\ x_{2}&x_{1}&&&&\\\ &&&&&\\\ \hline\cr&&&&&\\\ -x_{2}&-x_{1}&&&&\end{array}\right)$ | $\left(\begin{array}[]{cccc|cc}&&&&&\\\ &&&&&\\\ &&&&&\\\ &&&x&-x&\\\ \hline\cr&&&x&-x&\\\ &&&&&\end{array}\right)$ | $\left(\begin{array}[]{cccc|cc}&&&&&\\\ &&&&&\\\ &&x&&&-x\\\ &&&&&\\\ \hline\cr&&&&&\\\ &&x&&&-x\end{array}\right)$
| $-2f_{1}$ | $-2f_{2}$ | $f_{1}+f_{2}$
| $\left(\begin{array}[]{cccc|cc}&&&&&\\\ &&&&&\\\ &&&&&\\\ &&&x&x&\\\ \hline\cr&&&-x&-x&\\\ &&&&&\end{array}\right)$ | $\left(\begin{array}[]{cccc|cc}&&&&&\\\ &&&&&\\\ &&x&&&x\\\ &&&&&\\\ \hline\cr&&&&&\\\ &&-x&&&-x\end{array}\right)$ | $\left(\begin{array}[]{cccc|cc}&&&&&\\\ &&&&&\\\ &&&-x&x&\\\ &&-x&&&x\\\ \hline\cr&&-x&&&x\\\ &&&-x&x&\end{array}\right)$
| $-f_{1}-f_{2}$ | $f_{1}-f_{2}$ | $-f_{1}+f_{2}$
| $\left(\begin{array}[]{cccc|cc}&&&&&\\\ &&&&&\\\ &&&x&x&\\\ &&x&&&x\\\ \hline\cr&&-x&&&-x\\\ &&&-x&-x&\end{array}\right)$ | $\left(\begin{array}[]{cccc|cc}&&&&&\\\ &&&&&\\\ &&&x&-x&\\\ &&x&&&x\\\ \hline\cr&&x&&&x\\\ &&&-x&x&\end{array}\right)$ | $\left(\begin{array}[]{cccc|cc}&&&&&\\\ &&&&&\\\ &&&x&x&\\\ &&x&&&x\\\ \hline\cr&&-x&&&x\\\ &&&x&x&\end{array}\right)$
Table 2. The weight decomposition of $\mathfrak{s}$ for the adjoint action of
$\mathfrak{so}(4,2)$

The weight space $\mathfrak{s}_{0}$ of zero weight is generated by

${\small\left(\begin{array}[]{cccc|cc}&x_{1}&&&&\\\ x_{1}&&&&&\\\ &&&&&\\\
&&&&&\\\ \hline\cr&&&&&\\\
&&&&&\end{array}\right),\left(\begin{array}[]{cccc|cc}2x_{2}&&&&&\\\
&2x_{3}&&&&\\\ &&&&&\\\ &&&-x_{2}-x_{3}&&\\\ \hline\cr&&&&-x_{2}-x_{3}&\\\
&&&&&\end{array}\right)},\;\text{and}{\small\left(\begin{array}[]{cccc|cc}&&&&&\\\
&&&&&\\\ &&-x_{4}&&&\\\ &&&x_{4}&&\\\ \hline\cr&&&&x_{4}&\\\
&&&&&-x_{4}\end{array}\right).}$

## References

  * [1] Anthony W. Knapp, _Lie groups beyond an introduction_ , Second edition. Progr. Math., 140 Birkhäuser Boston, Inc., Boston, MA, 2002. xviii+812 pp. ISBN:0-8176-4259-5


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bimodal size distribution of aerosol nucleation particles

###### Abstract

We present a new implementation of the hydrological cycle scheme into a
general circulation model of the Martian atmosphere. The model includes a
semi-Lagrangian transport scheme for water vapor and ice, and accounts for
microphysics of phase transitions between them. The hydrological scheme
includes processes of saturation, nucleation, particle growth, sublimation and
sedimentation under the assumption of a variable size distribution. The scheme
has been implemented into the Max Planck Institute Martian general circulation
model (MPI–MGCM) and tested assuming mono- and bimodal log-normal
distributions of ice condensation nuclei. We present a comparison of the
simulated annual variations, horizontal and vertical distributions of water
vapor and ice clouds with the available observations from instruments onboard
Mars orbiters. The accounting for bi-modality of aerosol particle distribution
improves the simulations of the annual hydrological cycle, including predicted
ice clouds mass, opacity, number density, particle radii. The increased number
density and lower nucleation rates brings the simulated cloud opacities closer
to observations. Simulations show a weak effect of the excess of small aerosol
particles on the simulated water vapor distributions.

JGR-Planets

Moscow Institute of Physics and Technology, Moscow, Russia Space Research
Institute, Moscow, Russia Max Planck Institute for Solar System Research,
Göttingen, Germany National Institute of Information and Communications
Technology, Koganei, Japan Department of Geophysics, Tohoku University,
Sendai, Japan

D. S<EMAIL_ADDRESS>
A new microphysical scheme for water cycle on Mars implemented into a general
circulation model.

Accounting for bi-modality of aerosol particle distributions improves
simulations of water ice in the model compared to observations.

The fine fraction of atmospheric aerosols weakly affects simulations of water
vapor in the model.

##  1 Introduction

Water in its different phases is a very important element of the current
Martian climate, being a sensitive marker of meteorology in the atmosphere. It
affects the Martian climate mostly through radiative effects of water ice
clouds and scavenging dust from the atmosphere. Water was first detected in
the Martian atmosphere more than a half century ago (Spinrad et al., 1963).
The next generation of studies broadly utilized data from orbiting and landing
spacecraft, e.g., from Mars Atmospheric Water Detector (MAWD) onboard the
Viking Orbiter (Jakosky and Farmer, 1982). To date, the main sources of
information about the water distribution in the Martian atmosphere are the
Thermal Emission Spectrometer (TES) onboard Mars Global Surveyor (MGS) (Smith
et al., 2001; Smith, 2004), the Mars Climate Sounder (MCS) and the Compact
Reconnaissance Imaging Spectrometer for Mars (CRISM) onboard Mars
Reconnaissance Orbiter (MRO) (Smith et al., 2009), the LIDAR instrument
onboard the Phoenix Lander (Whiteway et al., 2009) and the Planetary Fourier
Spectrometer (PFS), the Visible and Infrared Mineralogical Mapping
Spectrometer (OMEGA) and the Spectroscopy for Investigation of Characteristics
of the Atmosphere of Mars (SPICAM) instruments onboard Mars Express (Fedorova
et al., 2006; Fouchet et al., 2007; Melchiorri et al., 2007; Tschimmel et al.,
2008; Sindoni et al., 2011; Maltagliati et al., 2011; Trokhimovskiy et al.,
2015).

The history of the water cycle modeling starts from the work of Davies (1981)
who developed the model to test the hypothesis that the observed seasonal and
latitudinal distribution of water on Mars is controlled by the sublimation and
condensation of surface ice deposits in the polar regions, and the meridional
transport of water vapor. Then, James (1990) who used a 1D model to show the
role of water ice clouds in the water migration from north to south. The first
comprehensive microphysical model of clouds was developed by Michelangeli et
al. (1993) following the earlier attempts undertaken after measurements of
water vapor vertical profiles (Kulikov and Rykhletskii, 1984). Colaprete et
al. (1999) used microphysical models and Haberle et al. (1999) employed a
Martian general circulation model (MGCM) to reproduce observations provided by
Mars Pathfinder. Richardson and Wilson (2002) and Richardson et al. (2002)
used the Geophysical Fluid Dynamics Laboratory (GFDL) MGCM to simulate the
annual water cycle on Mars and compared it with the Viking MAWD data. Although
the simulated climate was overly wet, these studies revealed the key
mechanisms of the water transport. A more sophisticated model, which included
transport, phase transitions and microphysical processes, has been developed
by Montmessin et al. (2004). Later a microphysical model for Mars dust and ice
clouds has been applied in combination with a model of the planetary boundary
layer (PBL) for interpretation of measurements by the LIDAR instrument on the
Phoenix Mars lander (Daerden et al., 2010). Observations of temperature
inversions in the atmosphere of Mars (Hinson and Wilson, 2004) have motivated
modelers to include effects of radiatively active water ice clouds (RAC) in
MGCMs (Wilson et al., 2007, 2008; Haberle et al., 2011; Madeleine et al.,
2012; Pottier et al., 2017). These studies have demonstrated that accounting
for RAC helped to reduce global temperature biases between simulations and
observations at northern spring and summer (see also the work of Urata and
Toon, 2013). More MGCMs that include water cycle have been developed to date:
DRAMATIC (Dynamics, RAdiation, MAterial Transport and their mutual
InteraCtions) MGCM (Kuroda, 2017), NASA Ames GCM (Kahre et al., 2017), GEM-
Mars (The Global Environmental Multiscale model for Mars) GCM (Neary and
Daerden, 2018), the Laboratoire de Météorologie Dynamique (LMD) MGCM
(Montmessin et al., 2004; Navarro et al., 2014) and the Oxford University MGCM
(Steele et al., 2014). The cloud scheme described by Montmessin et al. (2004)
was implemented at least in the latter two models, while the Oxford MGCM also
used data assimilation scheme to nudge the simulated temperature to available
observations.

To successfully reproduce water cloud formations in the atmosphere of Mars,
microphysical models require a correct parameterizationprediction of the size
distribution of aerosol particles, which serve as cloud condensation nuclei
(CCN). Several observations have provided the evidence that this distribution
is bimodal (Montmessin et al., 2006; Määttänen et al., 2013; Fedorova et al.,
2014). The distribution is called bimodal if its density function has two
peaks, or modes. Montmessin et al. (2002) implemented such distribution into
their one-dimensional model, prescribing two peaks with constant effective
radii and variance for fine and large modes with a fixed ratio between them.
They indicated that this assumption improved the simulations. For instance, it
resulted in decrease of the effective radii of ice particles condensing on the
CCN. In this study, we focus on the effects of the bimodal dust distribution
on the global hydrological cycle.

We present a new implementation of the hydrological scheme based on the
Montmessin et al. (2002, 2004) and Navarro et al. (2014) approaches into the
Max Planck Institute (MPI) MGCM (also known as MAOAM – Martian Atmosphere
Observation and Modeling). This model with the employed physical
parameterizations have been described in detail in the works of Hartogh et al.
(2005, 2007) and Medvedev and Hartogh (2007). The most recent applications of
this MGCM along with the current setup are presented in the works of Medvedev
et al. (2013, 2015, 2016) and Yiğit et al. (2015). The earlier version of the
hydrological scheme has been described in the work of (Shaposhnikov et al.,
2016).

The structure of this paper is as follows. In section 2, we outline the newly
developed hydrological scheme, discuss its main parameters and their
importance. Section 3 presents the simulations with the bimodal dust
distribution and comparison with observations. In section 4, we use different
scenarios (with mono-modal and bimodal dust particle distribution) to
demonstrate the sensitivity of the model to the bi-modality. Conclusions are
summarized in section 5.

##  2 Martian General Circulation Model

###  2.1 Transport

The MPI-MGCM employs a spectral dynamical core to solve the primitive
equations of hydrodynamics on a sphere. The physics and tendencies are
calculated on a 3D grid, and then are transformed into spectral coefficients
at every time step ($\Delta t=$ 20 seconds). In the vertical, the grid is
defined in the hybrid $\eta$-coordinate (Simmons and Burridge, 1981)
discretized into 50 levels, terrain-following near the surface and pressure
based near the top, which was located at $\sim$100 km in our simulations. The
horizontal grid is based on the Gauss-Kruger map projection with 32 and 64
bins in latitude and longitude, respectively. This discretization corresponds
to a T21 triangular spectral truncation, which is a typical resolution of
currently employed MGCMs, with a few exceptions for high-resolution
experiments (Kuroda et al., 2015, 2016; Pottier et al., 2017). Finite spatial
resolution can be a source of numerically-induced features in simulations,
which we discuss in the text.

The spectral dynamical core is not well suited for simulation of the tracer
transport. Instead, we adopted the advection based on a semi-Lagrangian
explicit monotone second-order hybrid scheme and on the time splitting method
in three spatial directions (Mingalev et al., 2010). A thorough examination of
performed runs has confirmed that this scheme maintains a high order of
conservation of water masses and solution accuracy appropriate for general
circulation modeling (Shaposhnikov et al., 2016). In addition to advection,
transport includes diffusion and mixing associated with subgrid-scale
processes. The importance of vertical eddy mixing for modeling the water cycle
was emphasized by Richardson and Wilson (2002). In our simulations, we use the
Crank-Nicolson implicit method with the Richardson number-based diffusion
coefficients (Becker and Burkhardt, 2007) to solve the vertical diffusion
equation.

###  2.2 Bimodal dust distribution

We employ a predetermined dust scenario that represents a seasonal evolution
of the zonally averaged aerosol optical depth $\tau$ in the thermal IR based
on MGS-TES and MEX-PFS measurements with the global dust storms removed (the
so-called “MGCM dust scenario” (Medvedev et al., 2011)). Thus, dust is not
transported in the model. We use four bins to represent the cloud condensation
nuclei size distribution: 0.03, 0.1, 0.3, 1 $\mathrm{\mu}$m. For the ice
particle size distribution, a two-moment scheme is used at every bin, for
keeping separately track of the ice mass and number of particles (Rodin,
2002). Note that the particles were assumed to be spheres, although it may be
a poor approximation for smallest nuclei. The size of ice particles determines
some of their other microphysical properties and the sedimentation rate. The
latter is accounted for through corrections to the advective vertical velocity
for every particle bin according to the formula obtained by Korablev et al.
(1993) with the Cunningham correction. Then, the CCN number density in each
bin can be calculated from, for instance, the bimodal log-normal dust
distribution (Fedorova et al., 2014), as shown in Figure 1.

Figure 1: The operation scheme for calculating the bimodal dust distribution.
$N$ is the number density of large/small particles, $\tau$ is the dust optical
depth in a vertical column (“MGCM dust scenario”), $H$ is the altitude of the
grid cell, $r_{\textrm{eff}}$ is the effective radius, $v_{\textrm{eff}}$ is
the effective variance, $\gamma$ is the population ratio, $n(r)$ and
$n(r)_{*}$ are the bimodal and mono-modal size distribution functions,
correspondingly.

Following Montmessin et al. (2002), the mono-modal log-normal density function
$n(r)_{*}$ can be defined as

$n(r)_{*}=\frac{1}{\sqrt{2\pi\sigma_{*}^{2}}r}\exp{\left(-\frac{(\ln{r}-\mu_{*})^{2}}{2\sigma_{*}^{2}}\right)},$
(1)

where the index ∗ indicates the fine or large modes, respectively, and the
parameters $\sigma_{*}$ and $\mu_{*}$ are related to the effective radius
$r_{\textrm{eff,*}}$ and variance $v_{\textrm{eff,*}}$:

$r_{\textrm{eff,*}}=\exp{\left(\mu_{*}+\frac{5}{2}\sigma_{*}^{2}\right)},\quad
v_{\textrm{eff,*}}=\exp{\left(\sigma_{*}^{2}\right)}-1.$ (2)

Then, the procedure of number density calculation for each bin includes the
following steps:

  1. 1.

Number density of large particles and the dust mass mixing ratio in the cell
vary according to the annual dust scenario. Following the work of Montmessin
et al. (2004), the number density $N_{\textrm{large}}$ of large particles can
be calculated from the equation

$N_{*}=N_{0}\frac{\tau}{\tau_{\textrm{ref}}}e^{-\frac{H}{H_{0}}},$ (3)

where $H$ is the altitude of the cell; $N_{0}$ = 15 cm-3,
$\tau_{\textrm{ref}}$ = 0.1, $H_{0}$ = 10 km are the adopted in this work
parameters. The dust mass mixing ratio $Q$ in the cell is produced by the
“MGCM dust scenario” and varyvaries in time and space.

  2. 2.

The effective radius for large particles, $r_{\textrm{eff,large}}$, is
calculated following (Maltagliati et al., 2011):

$r_{\textrm{eff,*}}=\left(\frac{Q}{4/3\pi\rho_{\textrm{ice}}\rho_{\textrm{air}}N_{*}}\right)^{\frac{1}{3}}\left(1+v_{\textrm{eff,*}}\right),$
(4)

where $v_{\textrm{eff,large}}$ = 0.35, $\rho_{\textrm{ice}}$ and
$\rho_{\textrm{air}}$ are values of density of ice and air, respectively.

  3. 3.

Using constant values for the effective variance and radius of small particles
$r_{\textrm{eff,small}}$ = 0.05 $\mathrm{\mu}$m and $v_{\textrm{eff,small}}$ =
0.1, the probability density function of the bimodal distribution $n(r)$ is
obtained as a sum of $\gamma*n(r)_{\textrm{small}}$ and
$n(r)_{\textrm{large}}$. Although observations indicate that the population
ratio $\gamma$ may vary in time and space (see Figure 2b), we adopt the
constant $\gamma=10^{3}$ in the bimodal case for all altitudes. The importance
of this assumption can be a subject of future studies.

  4. 4.

Having the CCN sizes and number density of all particles, it is
straightforward to calculate the number of particles in each bin and use it in
the microphysical equations. For that, one has to specify the number density
of small particles $N_{\textrm{small}}$ (which will be close to the number
density of all particles in the case of large population ratio) with the
equation (3) and parameters $N_{0}$ = 3$\times$104 cm-3, $\tau_{\textrm{ref}}$
= 0.1, $H_{0}$ = 6 km. All the mentioned parameters were selected according to
the work of Fedorova et al. (2014).

Figure 2: Evolution of the probability density function of the bimodal aerosol
particles distribution with a) altitude and b) population ratio $\gamma$ in
the MPI-MGCM averaged over latitude and longitude around $L_{s}=120^{\circ}$.
Different curves correspond to a) different altitudes and b) different
$\gamma$ (see the legend).

We use a bimodal distribution that includes much smaller particles than
suggested by Montmessin et al. (2002). The effective radius of the fine mode
is only 0.03 $\mathrm{\mu}$m and the number density ratio up to 103-104
compared to $\sim$0.2 $\mathrm{\mu}$m and 25, correspondingly. As it was
described above, the assumed distribution has a variable effective radius for
the large mode in the addoptedadopted fixed dust scenario. This allows for
taking into account a variation of the distribution of large particles and
their size decrease with height. Figure 2a demonstrates that the second peak
corresponding to the large mode shifts to the left with increasing altitude.
As a result, above 40–60 km (depending on latitude and season), two modes
merge and the distribution appears as a mono-modal, with small particles only.
Some studies suggested that the source of such particles can be a
micrometeoric smoke (Bardeen et al., 2008), andwhile others assumed that the
corresponding particle lifting can be caused by electrical forces (Kok and
Renno, 2006). Other studies indicate that eddy diffusion and dynamic effects
are more plausible sources of the fine mode (Fedorova et al., 2014).

Note that to date, the dust bi-modality was reliably observed only in some
seasons (Montmessin et al., 2006; Määttänen et al., 2013; Fedorova et al.,
2014). Therefore, the assumption of bi-modality throughout the entire Martian
year still must be validated with measurements. The simulations presented here
demonstrate the sensitivity of the water cycle in the model to this input
parameterbi-modality, and the noticeable improvements it produces.

###  2.3 Nucleation

The water vapor saturation pressure in a given spatial grid cell provides
information about the amount of the gaseous water held by the air parcel. In
the model, the saturation pressure $p_{\textrm{sat}}$ is expressed as a
function of temperature $T$ with a modification of the August-Roche-Magnus
formula (Curry and Webster, 1998):

$p_{\textrm{sat}}=611\;\textrm{Pa}\;\exp\left(22.5\left(1-\frac{273.16\;\textrm{K}}{T}\right)\right).$
(5)

Ice clouds are formed whenever water vapor nucleates on CCN. For
quantification of water ice condensation on dust nuclei, a heterogeneous
nucleation rate is evaluated according to Jacobson (2005):

$J_{\textrm{het}}=4\pi r_{c}^{2}\beta^{2}Z_{\textrm{n(het)}}\Delta
t\exp{\left(\frac{-\Delta G^{*}_{\textrm{het}}}{k_{b}T}\right)},$ (6)

where $J_{\textrm{het}}$ is the heterogeneous nucleation rate (number of
embryos, cm-2 s-1) of new particles on a surface, $r_{c}$ is the critical
radius, $\beta$ is the number of gas molecules striking the substrate surface
per second, $\Delta t$ = 10-13 s is the characteristic time that a gas
molecule spends on the surface before bouncing off, $Z_{\textrm{n(het)}}$ and
$\Delta G^{*}_{\textrm{het}}$ are the Zeldovich nonequilibrium factor and the
critical change in Gibbs free energy, respectively (adjusted for
heterogeneity), $k_{b}$ is the Boltzmann constant, $T$ is the temperature.

The number of active CCN also depends on the parameter $m_{h}=\cos{\theta}$,
where $\theta$ is the “contact angle” (for the liquid phase, it is the angle
of the interface between the droplet and its nucleus). We use the value
$m_{h}=0.96$ in calculations, which is consistent with the typical range
between 0.93 and 0.97 for Martian dust (Trainer et al., 2009). There are
indications that the value of this parameter significantly affects the
distribution of water in simulations (Navarro et al., 2014). Although the
laboratory studies indicate that the contact parameter is temperature
dependent (Iraci et al., 2010), our numerical experiments with the constant
contact angle produce results that better fit the observations compared to the
previously used parameterizations like those described by Määttänen and
Douspis (2014). Since these parameterizations differ for particles of
different type, shape and size, we select the simplest one with a fixed
contact angle.

###  2.4 Particle Growth and Sublimation

Once water vapor is condensed, the ice particles are prone to growth
determined by various factors. The rate of particle growth $\frac{dr}{dt}$
depends on the current water vapor saturation ratio $S$, the saturation
pressure ratio over a curved surface $S_{eq}$, the molecular $F_{D}$ and the
heat $F_{H}$ diffusion resistances (Montmessin et al., 2002):

$\frac{dr}{dt}=\frac{S-S_{eq}}{r(F_{D}+F_{H})}.$ (7)

The resistances in (7) reflect the contribution of the molecular and the heat
diffusion in the condensation process:

$F_{D}=\frac{\rho_{ice}RT}{D^{\prime}_{i}M_{H_{2}O}p_{sat}},$ (8)
$F_{H}=\frac{L_{ice}\rho_{ice}}{K_{air}T}\left(\frac{L_{ice}M_{H_{2}O}}{RT}-1\right),$
(9)

where $R$ is the universal gas constant, $M_{H_{2}O}$ is the water molar mass,
$L_{ice}$ and $K_{air}$ are the temperature dependent latent heat of ice and
thermal conductivity of air, respectively, parameterized in accordance with
Murphy and Koop (2005) and Gori and Corasaniti (2004). Here $D^{\prime}_{i}$
is the molecular diffusion coefficient corrected for collision geometry and
sticking probability (Jacobson, 2005, Eqns. 16.17–16.19):

$D^{\prime}_{i}=D\left[1+Kn_{i}\left(\frac{1.33+0.71Kn_{i}^{-1}}{1+Kn_{i}^{-1}}+\frac{4(1-\alpha_{i})}{3\alpha_{i}}\right)\right]^{-1},$
(10)

where $D$ is the molecular diffusion coefficient of water vapor in the Martian
atmosphere, $Kn_{i}$ corresponds to the Knudsen number of the condensing vapor
with respect to particles of size $i$ and $\alpha_{i}$ is the mass
accommodation (sticking) coefficient of vapor obtained by Kong et al. (2014).

Ultimately, the hydrological cycle in the Martian atmosphere is critically
dependent on the water sublimating from the surface. The model accounts for
this with the turbulent flux at the bottom of the atmosphere $E_{w}$
(Montmessin et al., 2004):

$E_{w}=\rho_{\textrm{air}}C_{d}u_{*}(q_{vg}-q_{va}),$ (11)

where the friction velocity $u_{*}$ depends on zonal and meridional wind
velocities at the lower layer of the atmosphere, the variable $q_{vg}$
describes the water vapor partial pressure at this level, and $q_{va}$ is the
saturation mixing ratio at the midpoint of the bottom layer. The saturation
ratios are directly determined from the local pressure. The drag coefficient
$C_{d}$ is set to $0.005$. In the model, the northern polar cap is the only
continuous source of water in the atmosphere, specified as an infinite
reservoir of the surface ice northward of 80∘N. This latitude corresponds to
the middle of the model grid closest to the extent of the observed cap. The
shape of the constant ice cap is a disc, while condensation and sublimation
can modify temporarily the surface ice in other regions over the course of
simulations. The MPI-MGCM uses the observed surface thermal inertia and albedo
everywhere, except at the ice caps, where the albedo is set to 0.3 (Hartogh et
al., 2005; Medvedev and Hartogh, 2007).

##  3 Simulations

The model runs have been initialized with the distribution of water vapor in
the atmosphere obtained in our earlier experiments (Shaposhnikov et al.,
2016). As a reminder, the previous simulations started with a linear
latitudinal gradient of water vapor (from 0 in the south to 200 ppm in the
north) and no ice outside the caps was prescribed anywhere in the atmosphere
at the beginning. Then, runs have been performed for several Martian years.
The results to be presented here are based on daily averaged quantities, if
not stated otherwise, and represent the second model year of simulations,
counting from the restart. In this paper we explore processes, whose
timescales are much shorter than one Martian year and do not require long-term
stability of the simulated hydrological cycle. The issues of seasonal cycle
repeatability are beyond the scope of this paper.Water production over the
shown year differs from that of the previous model year within a few percent
and alternate signaling that the model seems to achieve a quasi-stable state.
Details of the hydrological cycle repeatability and interannual variations are
beyond the scope of the paper, here we only prove that the model meets general
requirements of water mass conservation, numerical stability and accuracy.

###  3.1 Annual Variations

Figure 3 presents a comparison of the simulated water vapor column density and
the one observed with SPICAM. The SPICAM data are an average over 5 Martian
years (MY27–31). The model reproduces both the seasonal asymmetry of the
hydrological cycle and the total amount of water vapor in the atmosphere. The
discrepancy between the observed and simulated quantities does not exceed
$\sim$10 pr.$\mathrm{\mu}$m, and varies with seasons. The maximum amount of
vapor occurs near the north pole between $L_{s}=90^{\circ}$ and 140∘. The
model also reproduces the specific southward migration afterwards. The excess
of simulated water vapor column at $L_{s}=170^{\circ}-180^{\circ}$ in 0∘N –
45∘N can be explained by an overly strong vertical eddy diffusion, which
transports water vapor over the subliming polar cap. Another deficiency of the
model results compared to SPICAM is the lack of vapor after $L_{s}\sim$270∘.
Near the south pole, this can be related to the restriction imposed on the
surface temperature. Southward of 85∘S, CO2 ice is assumed to be present
permanently on the surface, thus preventing temperature increase. It is also
seen that the modeled cycle is somewhat delayed compared to the observations.
The simulated scarcity of water in middle latitudes near $L_{s}\sim$300∘ could
be explained either by inadequate meridional eddy transport, or insufficient
evaporation of the south polar cap in the “mean” dust scenario lacking the
seasonal dust storm. Despite the mentioned discrepancies, the simulated water
cycle generally reproduces the observations well (the root mean square error
is 4.3 pr.$\mathrm{\mu}$m). The simulated water vapor column density in the
summer season at the north pole with the maximum near 50-70 pr.$\mathrm{\mu}$m
agrees with the observations and other GCMs. The simulated vapor is also
absent in the north and south polar regions during winter. The model exhibits
a good repeatability of the vapor cycle, as can be judged by the match between
the end and beginning of the model year.

Further comparison of the simulated water vapor column with SPICAM data for
the particular Martian years is shown in Figure 4. The upper panel displays
the comparison at 70∘N, where the amount of water vapor is strongly affected
by sublimation from the north polar cap. It illustrates the time lag of the
simulated vapor in the atmosphere with respect to the observations. The main
reason of this is presumably the delayed ice cap sublimation caused by low
surface temperatures. Nevertheless, the simulated total vapor column is in a
good agreement within the observed year-to-year variability.

The middle panel shows a reasonably good agreement of the simulations and
observations at the equator, with observed interannual variations being much
smaller than in midlatitudes. The best agreement between the MGCM and
observations occurs in MY28 and the worst in MY29. Regarding the latter, it is
important to mention that the observed values were most scattered also during
the observational period in MY29. Note that, near $L_{s}=100^{\circ}$,
observations for MY27 and MY30 show a rapid growth of water vapor up to 15
pr.$\mathrm{\mu}$m, which is not reproduced in the model, and for which there
are no observational data in other years. This may be linked to the transient
baroclinic wave activity (Barnes et al., 1993; Kuroda et al., 2007).

The lower panel corresponds to 55∘S, i.e., to the region influenced by the
south ice polar cap. Figure 4c shows some lack of water vapor in the
simulations during $L_{s}=270^{\circ}-360^{\circ}$. However, this simulation
is very similar to the observations in MY28, when the Martian atmosphere was
affected by the global dust storm. As it was mentioned in section 2.2, we do
not include dust storms in the MGCM dust scenario, and the apparent agreement
between the model and observations during $L_{s}=270^{\circ}-360^{\circ}$ on
55∘S is accidental. The lack of vapor at this latitude can indicate weaker
sublimation from the south pole cap. Finally, the regular spikes in the
simulated vapor between $L_{s}=220^{\circ}$ and 260∘ are caused by the finite
size of model grid cells. This phenomenon is the result of a staged
disappearance of seasonal frost, which is imposed by the sparse model
gridding. Sublimation can be fast enough to completely remove the surface
frost from a grid cell before its southward neighbor is exposed to sunlight.
Other models with a comparable spatial resolution demonstrate similar effects
(see, for example, Figure 1b of Lefèvre et al. (2008)).

Because SPICAM measurements of water ice are not as plentiful as of vapor, we
compare the simulated annual cycle of ice with that based on the TES data.
Figure 5 (left column) presents the seasonal variations of zonally averaged
water vapor and ice from the TES data (MY24–26). The simulated quantities are
shown on the right panels. It is seen that the water vapor retrieved from the
TES observations is close to that measured by SPICAM (Korablev et al., 2006).
The bottom panels present the column opacity of water ice clouds at 12 micron.
To calculate the cloud opacity $\tau_{\textrm{abs}}$ of ice particles from the
MGCM output, we use the equation from the works of Montmessin et al. (2004)
and Warren et al. (2006):

$\tau_{\textrm{abs}}=\frac{3Q_{\textrm{abs}}(r,\lambda)M_{c}}{4\rho_{\textrm{ice}}r},$
(12)

where the absorption efficiency $Q_{\textrm{abs}}(r,\lambda)$ depends on ice
particle sizes $r$ and the wavelength $\lambda$, $M_{c}$ is the integrated
cloud mass predicted by the model in kg m-2 and $\rho_{\textrm{ice}}$ is the
ice density. The model opacity is compared with the quantities obtained from
the TES retrievals (Smith, 2002). It is seen that the model demonstrates a
good agreement with the observations. The simulations show no ice clouds over
the north polar cap during the aphelion season and predict a dense north polar
hood during the perihelion season, which could not be measured by TES. The
model reproduces the aphelion equatorial cloud belt, yet slightly displaced
northwards compared to observations.

###  3.2 Longitudinal Variations

The seasonal and spatial coverage of SPICAM data allows for comparing both
zonally averaged water cycle and longitude-latitude maps of water vapor column
density with MGCM results. The examples of such maps, where SPICAM data are
averaged over 5 Martian years, are presented in Figure 6. Leaving aside
quantitative differences mentioned in Section 3.1, one may notice the similar
variations in longitudes between $L_{s}=90^{\circ}$ and 150∘. These structures
are present, although less apparent, at other seasons as well. For instance,
on the first panel (averaged over the season $L_{s}=0^{\circ}$ – 30∘), the
structures are located near the equator between 90∘W and 140∘W (see also
Figure 7) and northward between 0∘E and 50∘E. The structures are clearly
visible in Figure 6d. It is evident that their extremes occur between 40∘N and
70∘N. These peaks are very robust and stable for more than one hundred Martian
sols of simulations.

Figure 7: Diurnally averaged water vapor column density in precipitable
microns (log-scale) in orthographic projection over the season $L_{s}\sim
120^{\circ}$ (northern hemisphere summer). Arrows show average horizontal wind
velocities in m s-1.

These quasi-symmetric (with respect to the north pole) peaks are most likely
associated with stationary planetary waves (Richardson, 1999; Banfield et al.,
2004; Haberle et al., 2017). Planetary waves in the atmosphere are caused by
inhomogeneity of the topography and surface characteristics (albedo and
thermal inertia), whose longitudinal structure on Mars is dominated by zonal
wavenumbers 2 and 3. The same longitudinal harmonics are clearly seen in
Figure 6. The numerical experiments show that the peaks start to form near the
season $L_{s}\sim 90^{\circ}$ and disappear later (around $L_{s}\sim
150^{\circ}$). The main peak is located near 140∘W, in a good accordance with
the observations. It should be recalled that the major highland (Olympus Mons
and Tharsis Montes) is located close to these longitudes. The other two wave
peaks are less stable, and migrate between 0∘ and 130∘E.

###  3.3 Vertical Distribution

In the solar occultation mode, SPICAM IR (1.38 $\mathrm{\mu}$m-band) channel
has conducted measurements of 82 vertical water profiles during MY28 (Betsis
et al., 2017; Fedorova et al., 2018). The eclipses were recorded from
$L_{s}\sim 255^{\circ}$ to $L_{s}\sim 300^{\circ}$, 49 of them in the northern
hemisphere and 33 in the southern one. According to the solar occultation
technique, each profile corresponds to specific time and location. We have
chosen the interval between $L_{s}=255^{\circ}$ and $L_{s}=267^{\circ}$ for
comparison to exclude the period with the major dust storm of MY28. The
results for 6 profiles from that period are shown in Figure 8 (see also Table
1).

Figures 8a and 8d present a comparison of the observed and simulated water
vapor density. The model output was averaged over one hour around the
corresponding local times (given in Table 1). Blue lines are for the same as
SPICAM data longitudes, while yellow lines show the model results
longitudinally averaged in addition. Because of the limited amount of
available measurements, we cannot verify whether the variations predicted by
the model near the surface are realistic. Above 60–70 km, the uncertainty of
retrievals becomes too large. In the middle atmosphere between 40–60 km, the
simulated profiles are in a good agreement with the SPICAM retrievals. We
admit that the selection of a few SPICAM profiles may be not representative,
but a more comprehensive comparison can be a subject of further studies.
Figures 8b and 8e present the simulated water ice particle number density for
the sum over all ice particle bins. It can be seen that the ice particle
number density is systematically underestimated by the model. As the result,
the simulated particle radii are too large compared to observations for some
orbits and altitudes (Figures 8c and 8f). The reason for this discrepancy is
not fully understood. It can be because of the insufficiently accurate
parametrization of the bimodal distribution in this season, which lead to the
insufficient number of CCN in the model, the slow nucleation rate, or too
rapid condensation due to the parameters chosen in the equations (6)–(7) or
the microphysical time step (will be explained in the section 4). Despite of
that, the comparison is generally favorable, especially for vapor where the
Pearson correlation coefficient between the observational and model data does
not fall below 0.91. Particularly interesting is that the model tends to
reproduce the shape of the profiles of water vapor and ice for particular
local times. Disagreements in particle sizes occur mostly for cells with a
small amount of water and have very little effect on the whole cycle.

Finally, to complete our comparison with observations, we focus on the ice
mixing ratio retrievals. For that, we use 410 assembled limb-viewing
observations from CRISM (Guzewich et al., 2014) to verify the compliance of
the MGCM simulations with the experiment. To retrieve the CRISM total column
optical depth $\tau(\lambda)$, the following equation was used by Smith et al.
(2013):

$\tau(\lambda)=\sum\limits_{i=1}^{\textrm{layers}}n_{i}Q_{\textrm{ext}}(\lambda)\Delta
p_{i}/p_{\textrm{surf}},$ (13)

where the sum was taken over all atmospheric levels, $n_{i}$ is the ice mixing
ratio, $Q_{\textrm{ext}}$ is the extinction coefficient, $\Delta p_{i}$ is the
difference in atmospheric pressure across the model layer, and
$p_{\textrm{surf}}$ is the surface pressure. Thus, the retrieved mixing ratios
are normalized to the pressure. Combining (13) and (12), we obtain the method
for calculating mixing ratios from the model output.

Figure 9 compares the ice mixing ratios normalized to pressure from
simulations and CRISM observations. The data are averaged over longitude and
season $L_{s}\sim 40^{\circ}-110^{\circ}$. The top and middle panels show the
results from CRISM and simulations, respectively. It is seen that the major
part of water ice is located above the equator at $\sim$ 20–40 km. This so-
called aphelion cloud belt (ACB) regularly occurs over the mentioned season.
During the northern spring, the north polar cap starts to sublime. Then, winds
transport the vapor to the equatorial regions, where air is still not too
warm. This leads to nucleation on the existing dust particles, and formation
of the clouds. It is seen that the MGCM reproduces this cycle well despite a
few deficiencies, which we focus at below.

Firstly, the model produces slightly thinner ice clouds, especially in the
center of the belt. Secondly, the model clouds are $\sim$5 km lower than the
CRISM data show. Possible reasons for these discrepancies could be smaller
simulated vertical velocities and some excess of large ice particles, as was
mentioned in the previous sections. The vertical transport in low latitudes is
weak at this season, whereas the large size of ice particles leads to an
enhanced sedimentation (and cloud lowering) as well as to a reduced
transparency. Nevertheless, the shape and location of the clouds agree well
with the observations. It is also seen that most of the ice is spread between
45∘S and 60∘N, which is supported by the observations. The belt has a halo in
the center and two thinner tails. In addition, it has some slope in the
latitude-altitude plane: ice particles are located higher near the north pole,
then the cloud layer altitude drops towards the equator and rises again closer
to the south pole. This smile-like shape, which results from a combination of
temperature distribution and sedimentation, is successfully reproduced by the
model. Some northward offset of the halo center can be caused by the location
of the main source of sublimation in the season. The model reproduces lower
atmospheric polar clouds above south pole, which were not observed by CRISM.

##  4 Comparison With the Mono-modal Distributions

Having compared the simulations versus the available observations and verified
the ability of the model to reproduce the hydrological cycle, we turn to a
comparison of different model scenarios: three with the mono-modal and one
with the bimodal dust particle size distributions. Under the first mono-modal
scenario M1, the number of large particles is the same as in the bimodal one,
but there are no small particles at all. Thus, the total number of all
particles in M1 is less than in the bimodal case. The rationale for this
scenario is based on the fact that the available observations constrain the
number of large particles, while the number of small particles is less known,
and one can hypothesize a total lack of them. The second mono-modal scenario
M2 includes the same total number of particles as the bimodal one, but also
without small particles. To obtain it from the bimodal case, we simply set
$\gamma=0$. We do not consider the mono-modal scenario with only small
particles, because it has no observational justification. The third mono-modal
scenario M3 is the same as M1, but the microphysical code was turned on only
every tenth model time step (every 200 seconds), that is, with a tenfold
increase of the microphysical time step. We postpone the consideration of this
scenario until later. The simulations under these mono-modal scenarios have
been performed similarly to the bimodal one described in the previous section.

The comparison of the annual water cycle was done in a similar manner as in
Figure 5. For that, we plotted the seasonal dependence of the zonally averaged
water vapor and ice in Figure 10. It is seen that the water vapor
distributions are similar for all scenarios. This occurs because the vapor is
strongly controlled by temperature. Therefore, changing the modality of dust
scenario has only a slight integral effect on the vapor. However, the
distribution in Figure 10e corresponding to the M2 scenario looks more
diffused than the bimodal one (Figure 10a). The distribution for the M3 mono-
modal scenario (Figure 10g) is very similar to that for the M2 run, and the M1
mono-modal scenario produced somewhat smaller amount of vapor (Figure 10a).
These discrepancies can be explained by water mass exchanges between ice and
vapor. The larger amount of dust particles facilitates the exchange, because
it leads to more ice particles. This can explain the difference between the M1
and bimodal cases. However, the simulations also reveal a significant effect
of the microphysical time step. As it was mentioned above, M1 and M3 runs are
absolutely identical except that in the M3 case, the microphysics was ran only
every tenth dynamical time step. We next consider the effect of the
microphysical time step on the whole water cycle.

Figure 11: Evolution of (a) the number of water ice particles for the CCN size
bin of 0.1 $\mathrm{\mu}$m and (b) water vapor saturation ratio in an explicit
time-stepping microphysical scheme in an isolated grid cell (equator, altitude
$\sim$ 30 km, $L_{s}$ $\sim$ 100∘). Advection, diffusion and any water mass
exchanges with other cells have been disabled. The initial ice mass and number
of ice particles were set to zero. Solid color lines denote the results for
different microphysical time steps from 1 to 200 seconds (see the legend). The
blue line corresponds to the results with disabled ice particles growth (only
nucleation was permitted). In the other experiments, ice particles can
nucleate and grow.

In order to illustrate processes occurring when the microphysical time step is
long, we consider a grid cell with disabled water mass exchanges with the
surrounding cells, i.e., with disabled advection, sedimentation and diffusion.
The initial amount of ice mass and number of ice particles were set to zero
and the results of calculations are presented in Figure 11. The resulting
water supersaturation led to fast nucleation and particle growth (purple
line). We also ran the code without particle growth, only with nucleation
(blue line). It is seen that the microphysical time steps longer than one
second yield an instantaneous nucleation. The particle growth on the second
step produces a sharp reduction of the water vapor mass, decrease in the water
vapor saturation ratio (Figure 11b) and, eventually, leads to a complete
cessation of nucleation. Using smaller microphysical time steps can slow down
this process, thus reproducing it more realistically. The formation of a
smaller number of ice particles decelerates the decrease of the water vapor
mass, which, in turn, slows down the nucleation. With time, the number of ice
particles tends to a constant value, because the nucleation stops and mass
exchange with other grid cells was disabled. Therefore, longer microphysical
time steps tend to increase the amount of simulated ice particles (Figure
11a). During a single step, the amount of formed ice particles becomes as
large as if the nucleation was not slowed down by the particle growth during
this time. This happens because we calculate the nucleation and particle
growth consequently. The same approach is used in other MGCMs (e.g. Navarro et
al., 2014) and apparently produces the same effect (i.e., dependence of the
results on the microphysical time step).

The right column of Figure 10 presents the comparison of the water ice cloud
opacity in the atmospheric column during one Martian year. All the shown
values are diurnally and zonally averaged. The M1 mono-modal dust scenario
produces very thin ice clouds (Figure 10d), which are caused by an
insufficient amount of CCN in the atmosphere. Figures 10d and 10f show that a
larger number of CCN leads to a greater amount of ice mass in the atmosphere,
thus increasing the column opacity of ice. The amount of large CCN in the M2
scenario is greater than it was observed in the Martian atmosphere, and the
simulated cloud opacities are two times greater than in the measurements
(Figure 10b). It is seen that the tenfold increase of the microphysical time
step in M3 also approximately doubled the simulated column ice opacity,
although the latter is still below the observed quantities (Figure 5c). The M3
simulation illustrates that application of longer time steps can bring model
results closer to observations, however this process is caused by purely
numerical aspects and has no physical ground. On the contrary, the bimodal
dust scenario produces better agreement with the measurements (Figure 10b),
although the utilized microphysical time step (20 seconds) should probably be
decreased in future applications. The distributions of opaque clouds over
winter poles are similar in the simulations despite the mentioned differences
in the magnitudes. Also interesting is that the greater number of large
particles (M2 scenario) produces the aphelion cloud belt not symmetric with
respect to the seasons.

Figure 12 presents the vertical profiles of (averaged over a Martian year)
water ice characteristics simulated using the M1 mono-modal and bimodal dust
particle size scenarios. Note that identical CCN sizes are marked by the same
colors. As it was described in section 2.2, the ice mass and number density
increase in every CCN bin separately (the so-called two-moment scheme). One
can see that the amount of ice mass in the M1 mono-modal dust scenario (Figure
12b) is much smaller than in the bimodal one (Figure 12a) for all bins except
in the bin with the radius 0.3 $\mathrm{\mu}$m. This bin becomes the main
source of ice mass in the M1 mono-modal scenario. This is caused by the choice
of parameters of the dust distribution. For the same reason CCN with an
average size 0.03 $\mathrm{\mu}$m become the main source of ice mass for the
bimodal scenario. This is not surprising because these CCN correspond to the
first peak of the bimodal distribution (see, e.g., Figure 2). The amount of
ice mass in the bimodal scenario is greater than in the M1 case, because of
the number of ice nuclei. As was discussed above, the number of small dust
particles is approximately 103–104 times greater than the number of large
particles. Such abundance of condensation nuclei allows for catching much more
vapor in the bimodal simulation and, as a result, to produce heavier clouds.
In addition, the larger number of ice particles results in a decrease of their
mean radius, which reduces sedimentation and leaves more ice airborne.

It is easy to see from Figures 12c and 12d that the number density of ice
particles corresponding to the small CCN in the M1 mono-modal experiment is
approximately an order of magnitude smaller than in the bimodal. The effect of
the bimodal distribution is best illustrated by the last two panels, where the
ice particle effective radii averaged over a Martian year are shown for the
both simulations. Because of the small amount of dust particles covered by ice
in the M1 mono-modal scenario, the resulted average radii reach 10–15
$\mathrm{\mu}$m in the middle atmosphere, which absolutely disagrees with the
observations (Fedorova et al., 2014; Guzewich et al., 2014). Otherwise, the
bimodal scenario produces much smaller ice particles for the CCN with small
cores in a good agreement with the measurements, as it was discussed in
section 3.3. Some local excesses of the radii are compensated by averaging. It
should be mentioned that, even in the bimodal scenario, we obtain some surplus
of ice radius sizes, especially near the surface, where it could be caused by
the insufficiently rigorous sublimation/condensation scheme. Hence, a further
development of the model should address this limitation. Based on the
foregoing, we can state with certainty that the use of the bimodal
distribution with an added peak of small particles increases the number of
nucleation particles in the atmosphere contributing to the growth of the ice
mass, increases the concentration of ice particles and reduces their radii,
which, in turn, improves the simulated opacity of the clouds compared to
observations.

Finally, after considering the effects of the bi-modality of the dust
distributions on the annual cycle, we turn our attention to the details of the
cycle. For that, we consider again the CRISM observations. They are presented
in Figure 9 along with the latitude-altitude distributions of ice clouds
simulated with bi- and M1 mono-modal scenarios. First, it is seen that there
are less clouds over the south pole in the mono-modal scenario. It is an
interesting result, which points at the insufficient amount of CCN in the M1
mono-modal scenario above the south pole during the seasons $L_{s}\sim
40^{\circ}-110^{\circ}$. Potentially, some clouds nucleated on small particles
can occur there, which can be proved or refuted by future Mars missions.
Second, note that the lower panel is plotted with a different scale. The
displayed ice mixing ratio for the aphelion cloud belt in the M1 mono-modal
scenario is approximately three times smaller than that in the bimodal one.
The reasons for that are described in the previous paragraph. Additionally,
huge sizes of the particles lead to a strong sedimentation, which confines the
clouds at levels closer to the surface. This effect is illustrated by Figure
9c showing that the clouds are located lower than in Figure 9b. Generally, the
shape of the clouds simulated with the mono-modal scenario, their sizes and
the specific meridional slope of the cloud belt are close to those in the
bimodal experiment.

##  5 Summary and Conclusions

We presented a new implementation of the hydrological cycle scheme in the
Martian atmosphere into the Max Planck Institute Martian general circulation
model (MPI-MGCM, also known as MAOAM). The hydrological model includes a semi-
Lagrangian scheme for describing transport of water vapor and ice particles,
and a microphysical model for calculating phase transitions between them. The
microphysical scheme generally follows the approach of Montmessin et al.
(2002, 2004) and Navarro et al. (2014), but utilizes different microphysical
parameterizations, which are described in section 2. Simulations with MPI-MGCM
have been performed using a predetermined distribution of dust that
represented a seasonal and spatial evolution of atmospheric aerosol based on
the Thermal Emission Spectrometer onboard Mars Global Surveyor (MGS-TES) and
the Planetary Fourier Spectrometer onboard Mars Express (MEX-PFS) measurements
with the global dust storms removed. The number of particles (serving as
nuclei) of particular size (subdivided into bins) was calculated using the
bimodal log-normal dust distribution (the distribution is called bimodal if
its density function has two peaks). Unlike in the previous works (e.g.,
Montmessin et al., 2002), we employed a distribution that included much
smaller particles and the ratio of number density peaks up to 103–104, which
are apparently supported by observations for certain seasons (Fedorova et al.,
2014). The hydrological scheme also includes saturation, nucleation, particle
growth, sublimation and sedimentation depending on the average radius of
particles and the surface microphysics.

The simulated annual variations, horizontal and vertical distributions of
water vapor and ice clouds have been compared with SPICAM (the Spectroscopy
for Investigation of Characteristics of the Atmosphere of Mars onboard Mars
Express), MGS-TES and CRISM (the Compact Reconnaissance Imaging Spectrometer
for Mars onboard Mars Reconnaissance Orbiter) observations. The simulated
general amount of vapor in the summer season at the north pole with the
maximum near 50–70 precipitable microns is close to the SPICAM observations.
The simulated vapor is absent in cold seasons during north and south winters,
thus indicating that the conversion between vapor and ice as well as their
transport are captured realistically. We showed that the model is able to
replicate the horizontal distribution of water vapor with the quasi-symmetric
with respect to the north pole peaks of concentration between
$L_{s}=90^{\circ}$ and 150∘. These structures are most likely associated with
stationary planetary waves (Richardson, 1999; Banfield et al., 2004; Haberle
et al., 2017).

The comparison of the model results with the SPICAM profiles demonstrated a
good agreement for water vapor in the middle atmosphere (40–60 km), and a
systematic underestimation of the simulated ice density. However, the latter
disagreement occurred mostly for cells with a small amount of water and, thus,
affected the whole cycle very little. The model also predicted an overly dense
north polar hood during the perihelion season, which largely could not have
been measured by TES, and prevented cloud formation right above the north
polar cap during the aphelion season, consistently with the observations. The
simulated aphelion cloud belt (water ice clouds above the equator during the
seasons $L_{s}\sim 40^{\circ}-110^{\circ}$) is in a good agreement with the
CRISM observations.

Simulations for mono-modal and bimodal particle size distributions
demonstrated that the latter scenario most strongly affects the modeled ice
clouds mass, opacity, number density and particle radii bringing them closer
to observations. The simulations showed much weaker effect of the bi-modality
(excess of small aerosol particles) on water vapor distributions. The use of
the second peak of small CCN in the bimodal distribution increases the number
of particles nucleated in the atmosphere, which contributes to the growth of
ice mass, increases the concentration of ice particles and reduces their
radii. This, in turn, improves the simulated opacity of the clouds compared to
observations.

More generally, our results highlight the importance of the dust size
distribution with the peak of small particles for modeling water ice in the
atmosphere of Mars. We must cautiously state that this result may be a model-
dependent and can also be affected by the hypotheses and parameterizations
made in our model (e.g, the lack of interactions between dust and water ice).
Also, it is reasonable to expect that these distributions throughout all
seasons and locations are not perfectly bimodal, but have more complex shapes.
More measurements and MGCM simulations that self-consistently account for dust
transport (e.g., Navarro et al., 2014; Kahre et al., 2017) can further clarify
this and shed the light on the modality of the dust size distribution.

###### Acknowledgements.

The data supporting the MPI-MGCM simulations can be found at
https://mars.mipt.ru, https://zenodo.org/record/1045331 (Shaposhnikov et al.,
2017) or obtained from D. Shaposhnikov (shaposhnikov@phystech.edu). We express
our deep gratitude to Alexander Trokhimovskiy, Daria Betsis, Chris Mockel and
Scott Guzewich for assistance with the observational data and helpful
discussions. This work was performed at the Laboratory of Applied Infrared
Spectroscopy of Moscow Institute of Physics and Technology and at Max Planck
Institute for Solar System Research. The work was partially supported by the
Russian Science Foundation grant 100027.07.32.RSF27 and German Science
Foundation (DFG) grant ME2752/3-1.

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Table 1: Characteristics of the SPICAM orbits Orbit | Longitude (E) | Latitude (N) | Local time (h) | $L_{s}$
---|---|---|---|---
4407A3 | 54.19 | -25.75 | 5.14 | 254.94
4409A2 | 215.82 | -28.63 | 5.03 | 255.30
4421A1 | 168.45 | 41.52 | 7.55 | 257.41
4428A1 | 207.70 | 46.97 | 7.92 | 258.65
4435A1 | 247.16 | 51.49 | 8.31 | 259.89
4461A2 | 109.05 | -60.03 | 2.34 | 264.53

Figure 3: Annual water cycle on Mars from the SPICAM data (Trokhimovskiy et
al., 2015) and simulated with the MPI-MGCM. The top panels show water vapor
column density a) observed with SPICAM averaged over 5 Martian years and
zonally, and b) simulated with the MGCM averaged daily and zonally. The lower
panel presents the difference between the SPICAM data and the simulations.

Figure 4: Annual water cycle measured by SPICAM (markers) (Trokhimovskiy et
al., 2015) and simulated with the MGCM (red lines, averaged daily and zonally)
at three characteristic latitudes (70∘N, 0∘ and 55∘S). Color and shape of
markers indicate the Martian year of observations (see the annotations)

Figure 5: Comparison of the simulated annual water cycle averaged daily and
zonally with TES observations (Smith et al., 2001) averaged zonally and over 3
Martian years (MY24–26). The upper panels are for the precipitable water vapor
from TES (left) and from the model (right). The bottom panels are the same,
but for the column opacity of water ice particles at the wavelength 12 $\mu$m.

Figure 6: Horizontal distributions of water vapor column density (in
preciptable microns) observed with SPICAM (upper parts of each panel) and
simulated with the MGCM (daily averaged). In each pair of plots, the water
vapor measured by SPICAM is averaged over 5 Martian years.

Figure 8: Vertical profiles of water vapor number density in cm-3 (a, d),
water ice density in cm-3 (b, e) and effective ice radii in $\mathrm{\mu}$m
(c, f) from the SPICAM data (MY28, green markers) (Betsis et al., 2017;
Fedorova et al., 2018) and MGCM simulations (blue and yellow lines) for
several orbits given in Table 1. Each orbit corresponds to particular time and
location (see the Table 1). Red errorbars denote the standard deviation of the
observations, blue lines indicate hourly averaged simulated quantities on the
same as the SPICAM data coordinates and time, yellow dots and errorbars show
longitudinally averaged simulations at the local time of observations and the
standard errors of the mean of the MGCM results, respectively.

Figure 9: Comparison of the observed by CRISM (a) (Smith et al., 2013;
Guzewich et al., 2014) and simulated ice mixing ratios normalized to pressure
(see the eq. (13)) with the bimodal (b) and M1 mono-modal (c) dust size
distribution scenarios from the surface up to 60 km averaged zonally and over
the season $L_{s}\sim 40^{\circ}-110^{\circ}$.

Figure 10: Comparison of the annual water cycle using the bimodal (a, b), M1
mono-modal (c, d), M2 mono-modal (e, f) and M3 mono-modal (g, h) CCN scenarios
(see the section 4). The left panels illustrate the water vapor in
precipitable $\mathrm{\mu}$m. The right ones show the column opacity of water
ice particles at the wavelength 12 $\mathrm{\mu}$m. The data are daily and
zonally averaged.

Figure 12: Vertical profiles of the water ice mass mixing ratio (in mg kg-1,
upper row), number density (in cm-3, central row) and ice particle radii (in
$\mathrm{\mu}$m, lower row) simulated with the bimodal (a, c, e) and M1 mono-
modal (b, d, f) dust scenarios. All profiles are based on the output averaged
over latitude, longitude and seasons. Solid color lines denote the results for
4 different bins: with CCN radii 0.03, 0.1, 0.3 and 1 $\mathrm{\mu}$m (see
also the legend for notations).