subroutine AuxFinalize
    !
    ! モジュール内部の変数の割り付け解除を行います. 
    !
    ! Deallocate variables in this module. 
    !

    ! 宣言文 ; Declaration statements
    !
    implicit none

    ! 実行文 ; Executable statement
    !

    if ( .not. auxiliary_inited ) return

    ! 割り付け解除
    ! Deallocation
    !
    if ( allocated( z_TempInpolM ) ) deallocate( z_TempInpolM )
    if ( allocated( z_TempInpol  ) ) deallocate( z_TempInpol  )

    auxiliary_inited = .false.
  end subroutine AuxFinalize

  subroutine AuxVars( xy_Ps,      xyz_Temp, xyr_Temp, xyz_Press,  xyr_Press, xy_SurfHeight, xyz_Height, xyr_Height, xyz_Exner,  xyr_Exner )
    !
    ! 温度の半整数σレベルの補間, 気圧と高度の算出, 
    ! エクスナー関数の計算を行います. 
    !
    ! Interpolate temperature on half sigma level, 
    ! and calculate pressure and height, 
    ! and calculate exner function. 

    ! モジュール引用 ; USE statements
    !

    ! 座標データ設定
    ! Axes data settings
    !
    use axesset, only: z_Sigma, r_Sigma, z_DelSigma, r_DelSigma
                              ! $ \Delta \sigma $ (半整数). 
                              ! $ \Delta \sigma $ (Half)

    ! 物理定数設定
    ! Physical constants settings
    !
    use constants, only: Grav, GasRDry, CpDry
                              ! $ C_p $ [J kg-1 K-1]. 
                              ! 乾燥大気の定圧比熱. 
                              ! Specific heat of air at constant pressure

    ! 時刻管理
    ! Time control
    !
    use timeset, only: TimesetClockStart, TimesetClockStop

    ! 宣言文 ; Declaration statements
    !
    implicit none

    real(DP), intent(in):: xy_Ps       (0:imax-1, 1:jmax)
                              ! $ p_s (t-\Delta t) $ . 地表面気圧. Surface pressure
    real(DP), intent(in):: xyz_Temp    (0:imax-1, 1:jmax, 1:kmax)
                              ! $ T $ .     温度. Temperature
    real(DP), intent(out), optional:: xyr_Temp   (0:imax-1, 1:jmax, 0:kmax)
                              ! $ \hat{T} $ . 温度 (半整数レベル). 
                              ! Temperature (half level)
    real(DP), intent(out), optional:: xyz_Press  (0:imax-1, 1:jmax, 1:kmax)
                              ! $ p $ . 気圧 (整数レベル). 
                              ! Air pressure (full level)
    real(DP), intent(out), optional:: xyr_Press  (0:imax-1, 1:jmax, 0:kmax)
                              ! $ \hat{p} $ . 気圧 (半整数レベル). 
                              ! Air pressure (half level)
    real(DP), intent(in ), optional:: xy_SurfHeight(0:imax-1,1:jmax)
                              ! $ z_s $ . 地表面高度. 
                              ! Surface height. 
    real(DP), intent(out), optional:: xyz_Height (0:imax-1, 1:jmax, 1:kmax)
                              ! 高度 (整数レベル). 
                              ! Height (full level)
    real(DP), intent(out), optional:: xyr_Height (0:imax-1, 1:jmax, 0:kmax)
                              ! 高度 (半整数レベル). 
                              ! Height (half level)
    real(DP), intent(out), optional:: xyz_Exner (0:imax-1, 1:jmax, 1:kmax)
                              ! Exner 関数 (整数レベル). 
                              ! Exner function (full level)
    real(DP), intent(out), optional:: xyr_Exner (0:imax-1, 1:jmax, 0:kmax)
                              ! Exner 関数 (半整数レベル). 
                              ! Exner function (half level)


    ! 作業変数
    ! Work variables
    !
    real(DP):: xyz_PressWork  (0:imax-1, 1:jmax, 1:kmax)
                              ! $ p $ . 気圧 (整数レベル). 
                              ! Air pressure (full level)
    real(DP):: xyr_PressWork  (0:imax-1, 1:jmax, 0:kmax)
                              ! $ \hat{p} $ . 気圧 (半整数レベル). 
                              ! Air pressure (half level)
    real(DP):: xyr_TempWork   (0:imax-1, 1:jmax, 0:kmax)
                              ! $ \hat{T} $ . 温度 (半整数レベル). 
                              ! Temperature (half level)

    integer:: k               ! 鉛直方向に回る DO ループ用作業変数
                              ! Work variables for DO loop in vertical direction

    ! 実行文 ; Executable statement
    !

    ! 計算時間計測開始
    ! Start measurement of computation time
    !
    call TimesetClockStart( module_name )

    ! 初期化
    ! Initialization
    !
    if ( .not. auxiliary_inited ) call AuxInit

    ! 温度の補間
    ! Interpolate temperature
    !
    if ( present( xyr_Temp ) ) then
      call AuxVarsInternalRoutine( xy_Ps,      xyz_Temp, xyr_Temp = xyr_Temp )
    end if

    ! 気圧の計算
    ! Calculate pressure
    !
    if ( present( xyz_Press ) ) then
      call AuxVarsInternalRoutine( xy_Ps,      xyz_Temp, xyz_Press = xyz_Press )
    end if

    if ( present( xyr_Press ) ) then
      call AuxVarsInternalRoutine( xy_Ps,      xyz_Temp, xyr_Press = xyr_Press )
    end if

    ! 高度の計算
    ! Calculate height
    !
    if ( present( xyz_Height ) .and. present( xy_SurfHeight ) ) then
      xyz_Height(:,:,1) = xy_SurfHeight(:,:) + GasRDry / Grav * xyz_Temp(:,:,1) * ( 1. - z_Sigma(1) )

      if ( present( xyr_Temp ) ) then
        xyr_TempWork = xyr_Temp
      else
        call AuxVarsInternalRoutine( xy_Ps,      xyz_Temp, xyr_Temp = xyr_TempWork )
      end if

      do k = 2, kmax
        xyz_Height(:,:,k) = xyz_Height(:,:,k-1) + GasRDry / Grav * xyr_TempWork(:,:,k-1) * r_DelSigma(k-1) / r_Sigma(k-1)
      end do
    end if

    if ( present( xyr_Height ) .and. present( xy_SurfHeight ) ) then
      xyr_Height(:,:,0) = xy_SurfHeight
      do k = 1, kmax
        xyr_Height(:,:,k) = xyr_Height(:,:,k-1) + GasRDry / Grav * xyz_Temp(:,:,k) * z_DelSigma(k) / z_Sigma(k)
      end do
    end if

    ! エクスナー関数の計算
    ! Calculate exner function
    !
    if ( present( xyz_Exner ) ) then
      if ( present( xyz_Press ) ) then
        xyz_PressWork = xyz_Press
      else
        call AuxVarsInternalRoutine( xy_Ps,      xyz_Temp, xyz_Press = xyz_PressWork )
      end if

      xyz_Exner = ( xyz_PressWork / RefPress ) ** ( GasRDry / CpDry )
    end if


    if ( present( xyr_Exner ) ) then
      if ( present( xyr_Press ) ) then
        xyr_PressWork = xyr_Press
      else
        call AuxVarsInternalRoutine( xy_Ps,      xyz_Temp, xyr_Press = xyr_PressWork )
      end if

      xyr_Exner = ( xyr_PressWork / RefPress ) ** ( GasRDry / CpDry )
    end if

    ! 計算時間計測一時停止
    ! Pause measurement of computation time
    !
    call TimesetClockStop( module_name )

  end subroutine AuxVars

  subroutine AuxInit
    !
    ! auxiliary モジュールの初期化を行います. 
    ! NAMELIST#auxiliary_nml の読み込みはこの手続きで行われます. 
    !
    ! "auxiliary" module is initialized. 
    ! "NAMELIST#auxiliary_nml" is loaded in this procedure. 
    !

    ! モジュール引用 ; USE statements
    !

    ! 座標データ設定
    ! Axes data settings
    !
    use axesset, only: z_Sigma, r_Sigma
                              ! $ \sigma $ レベル (半整数). 
                              ! Half $ \sigma $ level

    ! NAMELIST ファイル入力に関するユーティリティ
    ! Utilities for NAMELIST file input
    !
    use namelist_util, only: namelist_filename, NmlutilMsg

    ! ファイル入出力補助
    ! File I/O support
    !
    use dc_iounit, only: FileOpen

    ! 種別型パラメタ
    ! Kind type parameter
    !
    use dc_types, only: STDOUT ! 標準出力の装置番号. Unit number of standard output

    ! 文字列操作
    ! Character handling
    !
    use dc_string, only: StoA

    ! 宣言文 ; Declaration statements
    !
    implicit none

    integer:: unit_nml        ! NAMELIST ファイルオープン用装置番号. 
                              ! Unit number for NAMELIST file open
    integer:: iostat_nml      ! NAMELIST 読み込み時の IOSTAT. 
                              ! IOSTAT of NAMELIST read

    integer:: k               ! 鉛直方向に回る DO ループ用作業変数
                              ! Work variables for DO loop in vertical direction

    ! NAMELIST 変数群
    ! NAMELIST group name
    !
    namelist /auxiliary_nml/ RefPress
          !
          ! デフォルト値については初期化手続 "auxiliary#AuxInit" 
          ! のソースコードを参照のこと. 
          !
          ! Refer to source codes in the initialization procedure
          ! "auxiliary#AuxInit" for the default values. 
          !

    ! 実行文 ; Executable statement
    !

    if ( auxiliary_inited ) return
    call InitCheck

    ! デフォルト値の設定
    ! Default values settings
    !
    RefPress        =   1.0e+6

    ! NAMELIST の読み込み
    ! NAMELIST is input
    !
    if ( trim(namelist_filename) /= '' ) then
      call FileOpen( unit_nml, namelist_filename, mode = 'r' ) ! (in)

      rewind( unit_nml )
      read( unit_nml, nml = auxiliary_nml, iostat = iostat_nml )   ! (out)
      close( unit_nml )

      call NmlutilMsg( iostat_nml, module_name ) ! (in)
    end if

    !  補間用の係数の計算
    !  Calculate coefficient for interpolation
    !
    allocate( z_TempInpolM(1:kmax) )
    allocate( z_TempInpol(1:kmax) )

    z_TempInpolM(1) = 0.
    z_TempInpol (1) = 0.

    do k = 2, kmax
      z_TempInpolM(k) = ( log( r_Sigma( k-1 ) ) - log( z_Sigma( k ) ) ) / ( log( z_Sigma( k-1 ) ) - log( z_Sigma( k ) ) )
      z_TempInpol(k) = ( log( z_Sigma( k-1 ) ) - log( r_Sigma( k-1 ) ) ) / ( log( z_Sigma( k-1 ) ) - log( z_Sigma( k )   ) )
    end  do

    ! 印字 ; Print
    !
    call MessageNotify( 'M', module_name, '----- Initialization Messages -----' )
    call MessageNotify( 'M', module_name, '  z_TempInpolM = %*r', r = real( z_TempInpolM ), n = (/ kmax /) )
    call MessageNotify( 'M', module_name, '  z_TempInpol  = %*r', r = real( z_TempInpol  ), n = (/ kmax /) )
    call MessageNotify( 'M', module_name, '  RefPress     = %f', d = (/ RefPress /) )
    call MessageNotify( 'M', module_name, '-- version = %c', c1 = trim(version) )

    auxiliary_inited = .true.
  end subroutine AuxInit

  subroutine AuxVarsInternalRoutine( xy_Ps,      xyz_Temp, xyr_Temp, xyz_Press,  xyr_Press )
    !
    ! 温度の半整数σレベルの補間, 気圧の算出, 
    ! エクスナー関数の計算を行います. 
    !
    ! Interpolate temperature on half sigma level, 
    ! and calculate pressure, 
    ! and calculate exner function. 

    ! モジュール引用 ; USE statements
    !

    ! 座標データ設定
    ! Axes data settings
    !
    use axesset, only: z_Sigma, r_Sigma
                              ! $ \sigma $ レベル (半整数). 
                              ! Half $ \sigma $ level

    ! 宣言文 ; Declaration statements
    !
    implicit none

    real(DP), intent(in):: xy_Ps       (0:imax-1, 1:jmax)
                              ! $ p_s (t-\Delta t) $ . 地表面気圧. Surface pressure
    real(DP), intent(in):: xyz_Temp    (0:imax-1, 1:jmax, 1:kmax)
                              ! $ T $ .     温度. Temperature
    real(DP), intent(out), optional:: xyr_Temp   (0:imax-1, 1:jmax, 0:kmax)
                              ! $ \hat{T} $ . 温度 (半整数レベル). 
                              ! Temperature (half level)
    real(DP), intent(out), optional:: xyz_Press  (0:imax-1, 1:jmax, 1:kmax)
                              ! $ p $ . 気圧 (整数レベル). 
                              ! Air pressure (full level)
    real(DP), intent(out), optional:: xyr_Press  (0:imax-1, 1:jmax, 0:kmax)
                              ! $ \hat{p} $ . 気圧 (半整数レベル). 
                              ! Air pressure (half level)


    ! 作業変数
    ! Work variables
    !
    integer:: k               ! 鉛直方向に回る DO ループ用作業変数
                              ! Work variables for DO loop in vertical direction

    ! 実行文 ; Executable statement
    !


    ! 初期化
    ! Initialization
    !
    if ( .not. auxiliary_inited ) call AuxInit

    ! 温度の補間
    ! Interpolate temperature
    !
    if ( present( xyr_Temp ) ) then
      do k = 2, kmax
        xyr_Temp(:,:,k-1) =   z_TempInpolM(k) * xyz_Temp(:,:,k-1) + z_TempInpol (k) * xyz_Temp(:,:,k)
      end do

      xyr_Temp(:,:,kmax) = xyz_Temp(:,:,kmax)
      xyr_Temp(:,:,0)    = xyz_Temp(:,:,1)

    end if

    ! 気圧の計算
    ! Calculate pressure
    !
    if ( present( xyz_Press ) ) then
      do k = 1, kmax
        xyz_Press(:,:,k) = xy_Ps * z_Sigma(k)
      end do
    end if

    if ( present( xyr_Press ) ) then
      do k = 0, kmax
        xyr_Press(:,:,k) = xy_Ps * r_Sigma(k)
      end do
    end if


  end subroutine AuxVarsInternalRoutine

  subroutine InitCheck
    !
    ! 依存モジュールの初期化チェック
    !
    ! Check initialization of dependency modules

    ! モジュール引用 ; USE statements
    !

    ! NAMELIST ファイル入力に関するユーティリティ
    ! Utilities for NAMELIST file input
    !
    use namelist_util, only: namelist_util_inited

    ! 格子点設定
    ! Grid points settings
    !
    use gridset, only: gridset_inited

    ! 物理定数設定
    ! Physical constants settings
    !
    use constants, only: constants_inited

    ! 座標データ設定
    ! Axes data settings
    !
    use axesset, only: axesset_inited

    ! 実行文 ; Executable statement
    !

    if ( .not. namelist_util_inited ) call MessageNotify( 'E', module_name, '"namelist_util" module is not initialized.' )

    if ( .not. gridset_inited ) call MessageNotify( 'E', module_name, '"gridset" module is not initialized.' )

    if ( .not. constants_inited ) call MessageNotify( 'E', module_name, '"constants" module is not initialized.' )

    if ( .not. axesset_inited ) call MessageNotify( 'E', module_name, '"axesset" module is not initialized.' )

  end subroutine InitCheck