dcmip_initial_conditions_test_1_2_3_v5.f90

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MODULE dcmip_initial_conditions_test_1_2_3

  !=======================================================================
  !
  !  Functions for setting up initial conditions for the dynamical core tests:
  !
  !  11 - Deformational Advection Test
  !  12 - Hadley Cell Advection Test
  !  13 - Orography Advection Test
  !  20 - Impact of orography on a steady-state at rest
  !  21 and 22 - Non-Hydrostatic Mountain Waves Over A Schaer-Type Mountain without and with vertical wind shear
  !  31 - Non-Hydrostatic Gravity Waves
  !
  !  Given a point specified by: 
  !     lon longitude (radians) 
  !     lat latitude (radians) 
  !     p/z pressure/height
  !  the functions will return:
  ! u   zonal wind (m s^-1)
  ! v   meridional wind (m s^-1)
  ! w   vertical velocity (m s^-1)
  ! t   temperature (K)
  ! phis    surface geopotential (m^2 s^-2)
  ! ps  surface pressure (Pa)
  ! rho density (kj m^-3)
  ! q   specific humidity (kg/kg)
  ! qi  tracers (kg/kg)
  !     p       pressure if height based (Pa)
  !
  !
  !  Authors: James Kent, Paul Ullrich, Christiane Jablonowski 
  !     (University of Michigan, dcmip@ucar.edu)
  !          version 4
  !          July/8/2012
  !
  ! Change log: (v3, June/8/2012, v4 July/8/2012, v5 July/20/2012)
  !
  ! v2: bug fixes in the tracer initialization for height-based models
  ! v3: test 3-1: the density is now initialized with the unperturbed background temperature (not the perturbed temperature)
  ! v3: constants converted to double precision
  ! v4: modified tracers in test 1-1, now with cutoff altitudes. Outside of the vertical domain all tracers are set to 0
  ! v4: modified cos-term in vertical velocity (now cos(2 pi t/tau)) in test 1-1, now completing one full up and down cycle
  ! v4: added subroutine test1_advection_orography for test 1-3
  ! v4: added subroutine test2_steady_state_mountain for test 2-0
  ! v4: modified parameter list for tests 2-1 and 2-2 (routine test2_schaer_mountain): addition of parameters hybrid_eta, hyam, hybm
  !     if the logical flag hybrid_eta is true then the pressure in pressure-based model with hybrid sigma-p (eta) coordinates is 
  !     computed internally. In that case the hybrid coefficients hyam and hybm need to be supplied via the parameter list, 
  !     otherwise they are not used.
  ! v5: Change in test 11 - change to u and w, cutoff altitudes (introduced in v4) are removed again
  ! v5: Change in test 12 - velocities multiplies by rho0/rho, different w0 and vertical location of the initial tracer
  !
  !
  !=======================================================================

  IMPLICIT NONE

!-----------------------------------------------------------------------
!     Physical Parameters
!-----------------------------------------------------------------------

    real(8), parameter ::   a   = 6371220.0d0,  &   ! Earth's Radius (m)
                rd  = 287.0d0,  &   ! Ideal gas const dry air (J kg^-1 K^1)
                g   = 9.80616d0,    &   ! Gravity (m s^2)
                cp  = 1004.5d0, &   ! Specific heat capacity (J kg^-1 K^1)
                pi  = 4.d0*atan(1.d0)       ! pi

!-----------------------------------------------------------------------
!     Additional constants
!-----------------------------------------------------------------------

    real(8), parameter ::   p0  = 100000.d0     ! reference pressure (Pa)


CONTAINS

!==========================================================================================
! TEST CASE 11 - PURE ADVECTION - 3D DEFORMATIONAL FLOW
!==========================================================================================

! The 3D deformational flow test is based on the deformational flow test of Nair and Lauritzen (JCP 2010), 
! with a prescribed vertical wind velocity which makes the test truly 3D. An unscaled planet (with scale parameter
! X = 1) is selected. 

SUBROUTINE test1_advection_deformation (lon,lat,p,z,zcoords,u,v,w,t,phis,ps,rho,q,q1,q2,q3,q4)

IMPLICIT NONE
!-----------------------------------------------------------------------
!     input/output params parameters at given location
!-----------------------------------------------------------------------

    real(8), intent(in)  :: lon, &      ! Longitude (radians)
                lat, &      ! Latitude (radians)
                z       ! Height (m)

    real(8), intent(inout) :: p     ! Pressure  (Pa)                

    integer,  intent(in) :: zcoords     ! 0 or 1 see below

    real(8), intent(out) :: u, &        ! Zonal wind (m s^-1)
                v, &        ! Meridional wind (m s^-1)
                w, &        ! Vertical Velocity (m s^-1)
                t, &        ! Temperature (K)
                phis, &     ! Surface Geopotential (m^2 s^-2)
                ps, &       ! Surface Pressure (Pa)
                rho, &      ! density (kg m^-3)
                q, &        ! Specific Humidity (kg/kg)
                q1, &       ! Tracer q1 (kg/kg)
                q2, &       ! Tracer q2 (kg/kg)
                q3, &       ! Tracer q3 (kg/kg)
                q4      ! Tracer q4 (kg/kg)

    ! if zcoords = 1, then we use z and output p
    ! if zcoords = 0, then we use p 

!-----------------------------------------------------------------------
!     test case parameters
!----------------------------------------------------------------------- 
    real(8), parameter ::   tau     = 12.d0 * 86400.d0, &   ! period of motion 12 days
                    u0      = (2.d0*pi*a)/tau,  &   ! 2 pi a / 12 days
                    k0  = (10.d0*a)/tau,    &   ! Velocity Magnitude
                    omega0  = (23000.d0*pi)/tau,    &   ! Velocity Magnitude
                                t0      = 300.d0,       &   ! temperature
                                h       = rd * t0 / g,      &   ! scale height
                                rr      = 1.d0/2.d0,        &   ! horizontal half width divided by 'a'
                                zz      = 1000.d0,      &   ! vertical half width
                                z0      = 5000.d0,      &   ! center point in z
                                lambda0 = 5.d0*pi/6.d0,     &   ! center point in longitudes
                                lambda1 = 7.d0*pi/6.d0,     &   ! center point in longitudes
                                phi0    = 0.d0,         &   ! center point in latitudes
                                phi1    = 0.d0                      
                            
      real(8) :: height                         ! The height of the model levels
      real(8) :: ptop                           ! Model top in p
      real(8) :: sin_tmp, cos_tmp, sin_tmp2, cos_tmp2           ! Calculate great circle distances
      real(8) :: d1, d2, r, r2, d3, d4                  ! For tracer calculations 
      real(8) :: s, bs                          ! Shape function, and parameter
      real(8) :: lonp                           ! Translational longitude, depends on time
      real(8) :: ud                         ! Divergent part of u
      real(8) :: time                           ! Initially set to zero seconds, needs
                                    ! to be modified when used in dycore

!-----------------------------------------------------------------------
!    HEIGHT AND PRESSURE
!-----------------------------------------------------------------------
    
    ! Height and pressure are aligned (p = p0 exp(-z/H))

    if (zcoords .eq. 1) then

        height = z
        p = p0 * exp(-z/h)

    else

        height = h * log(p0/p)

    endif

    ! Model top in p

    ptop    = p0*exp(-12000.d0/h)   

!-----------------------------------------------------------------------
!    THE VELOCITIES ARE TIME DEPENDENT AND THEREFORE MUST BE UPDATED
!    IN THE DYNAMICAL CORE
!-----------------------------------------------------------------------

    ! These are initial conditions hence time = 0

    time = 0.d0

    ! Translational longitude = longitude when time = 0

    lonp = lon - 2.d0*pi*time/tau

    ! Shape function
!********
! change in version 5: shape function 
!********
    bs = 0.2
    s = 1.0 + exp( (ptop-p0)/(bs*ptop) ) - exp( (p-p0)/(bs*ptop)) - exp( (ptop-p)/(bs*ptop))

    ! Zonal Velocity
!********
! change in version 5: ud 
!********

    ud = (omega0*a)/(bs*ptop) * cos(lonp) * (cos(lat)**2.0) * cos(2.0*pi*time/tau) * &
        ( - exp( (p-p0)/(bs*ptop)) + exp( (ptop-p)/(bs*ptop))  )

    u = k0*sin(lonp)*sin(lonp)*sin(2.d0*lat)*cos(pi*time/tau) + u0*cos(lat) + ud

    ! Meridional Velocity

    v = k0*sin(2.d0*lonp)*cos(lat)*cos(pi*time/tau)

    ! Vertical Velocity - can be changed to vertical pressure velocity by 
    ! omega = -(g*p)/(Rd*T0)*w
        !
!********
! change in version 4: cos(2.0*pi*time/tau) is now used instead of cos(pi*time/tau)
!********
!********
! change in version 5: shape function in w
!********
    w = -((rd*t0)/(g*p))*omega0*sin(lonp)*cos(lat)*cos(2.0*pi*time/tau)*s

!-----------------------------------------------------------------------
!    TEMPERATURE IS CONSTANT 300 K
!-----------------------------------------------------------------------

    t = t0

!-----------------------------------------------------------------------
!    PHIS (surface geopotential) 
!-----------------------------------------------------------------------
      
    phis = 0.d0

!-----------------------------------------------------------------------
!    PS (surface pressure)
!-----------------------------------------------------------------------

    ps = p0

!-----------------------------------------------------------------------
!    RHO (density)
!-----------------------------------------------------------------------

    rho = p/(rd*t)

!-----------------------------------------------------------------------
!     initialize Q, set to zero 
!-----------------------------------------------------------------------

    q = 0.d0

!-----------------------------------------------------------------------
!     initialize tracers
!-----------------------------------------------------------------------

    ! Tracer 1 - Cosine Bells

        ! To calculate great circle distance

    sin_tmp = sin(lat) * sin(phi0)
    cos_tmp = cos(lat) * cos(phi0)
    sin_tmp2 = sin(lat) * sin(phi1)
    cos_tmp2 = cos(lat) * cos(phi1)

        ! great circle distance without 'a'
    
    r  = acos(sin_tmp + cos_tmp*cos(lon-lambda0)) 
    r2  = acos(sin_tmp2 + cos_tmp2*cos(lon-lambda1)) 
    d1 = min( 1.d0, (r/rr)**2 + ((height-z0)/zz)**2 )
    d2 = min( 1.d0, (r2/rr)**2 + ((height-z0)/zz)**2 )
    
    q1 = 0.5d0 * (1.d0 + cos(pi*d1)) + 0.5d0 * (1.d0 + cos(pi*d2))

    ! Tracer 2 - Correlated Cosine Bells

    q2 = 0.9d0 - 0.8d0*q1**2

    ! Tracer 3 - Slotted Ellipse

        ! Make the ellipse

    if (d1 .le. rr) then
        q3 = 1.d0
    elseif (d2 .le. rr) then
        q3 = 1.d0
    else
        q3 = 0.1d0
    endif

        ! Put in the slot
    
    if (height .gt. z0 .and. abs(lat) .lt. 0.125d0) then 

        q3 = 0.1d0      

    endif

    ! Tracer 4: q4 is chosen so that, in combination with the other three tracer
        !           fields with weight (3/10), the sum is equal to one

    q4 = 1.d0 - 0.3d0*(q1+q2+q3)

!************   
! change in version 4: added cutoff altitudes, tracers are set to zero outside this region
! prevents tracers from being trapped near the bottom and top of the domain
!************
        ! use a cutoff altitude for
    ! tracer 2 3 and 4
    ! Set them to zero outside `buffer zone'
!************   
! change in version 5: change from v4 reversed, no cutoff altitudes due to continuity equation being satisfied
!       commented out below
!************

    !if (height .gt. (z0+1.25*ZZ) .or. height .lt. (z0-1.25*ZZ)) then 

    !   q2 = 0.0
    !   q3 = 0.0
    !   q4 = 0.0

    !endif




END SUBROUTINE test1_advection_deformation





!==========================================================================================
! TEST CASE 12 - PURE ADVECTION - 3D HADLEY-LIKE FLOW
!==========================================================================================

SUBROUTINE test1_advection_hadley (lon,lat,p,z,zcoords,u,v,w,t,phis,ps,rho,q,q1)

IMPLICIT NONE
!-----------------------------------------------------------------------
!     input/output params parameters at given location
!-----------------------------------------------------------------------

    real(8), intent(in)  :: lon, &      ! Longitude (radians)
                lat, &      ! Latitude (radians)
                z       ! Height (m)

    real(8), intent(inout) :: p     ! Pressure  (Pa)
                
    integer,  intent(in) :: zcoords     ! 0 or 1 see below

    real(8), intent(out) :: u, &        ! Zonal wind (m s^-1)
                v, &        ! Meridional wind (m s^-1)
                w, &        ! Vertical Velocity (m s^-1)
                t, &        ! Temperature (K)
                phis, &     ! Surface Geopotential (m^2 s^-2)
                ps, &       ! Surface Pressure (Pa)
                rho, &      ! density (kg m^-3)
                q, &        ! Specific Humidity (kg/kg)
                q1      ! Tracer q1 (kg/kg)

    ! if zcoords = 1, then we use z and output p
    ! if zcoords = 0, then we use p

!-----------------------------------------------------------------------
!     test case parameters
!----------------------------------------------------------------------- 
    real(8), parameter ::   tau     = 1.d0 * 86400.d0,  &   ! period of motion 1 day (in s)
                    u0      = 40.d0,        &   ! Zonal velocity magnitude (m/s)
                    w0  = 0.15d0,       &   ! Vertical velocity magnitude (m/s), changed in v5
                                t0      = 300.d0,       &   ! temperature (K)
                                h       = rd * t0 / g,      &   ! scale height
                                k       = 5.d0,         &   ! number of Hadley-like cells
                                z1      = 2000.d0,      &   ! position of lower tracer bound (m), changed in v5       
                                z2      = 5000.d0,      &   ! position of upper tracer bound (m), changed in v5       
                                z0      = 0.5d0*(z1+z2),    &   ! midpoint (m)       
                                ztop    = 12000.d0          ! model top (m)       
                            
      real(8) :: rho0                           ! reference density at z=0 m
      real(8) :: height                         ! Model level heights
      real(8) :: time                           ! Initially set to zero seconds, needs
                                    ! to be modified when used in dycore

!-----------------------------------------------------------------------
!    HEIGHT AND PRESSURE
!-----------------------------------------------------------------------

    ! Height and pressure are aligned (p = p0 exp(-z/H))

    if (zcoords .eq. 1) then

        height = z
        p = p0 * exp(-z/h)

    else

        height = h * log(p0/p)

    endif

!-----------------------------------------------------------------------
!    TEMPERATURE IS CONSTANT 300 K
!-----------------------------------------------------------------------

    t = t0

!-----------------------------------------------------------------------
!    PHIS (surface geopotential)
!-----------------------------------------------------------------------
      
    phis = 0.d0

!-----------------------------------------------------------------------
!    PS (surface pressure)
!-----------------------------------------------------------------------

    ps = p0

!-----------------------------------------------------------------------
!    RHO (density)
!-----------------------------------------------------------------------

    rho = p/(rd*t)
    rho0 = p0/(rd*t)

!-----------------------------------------------------------------------
!    THE VELOCITIES ARE TIME DEPENDENT AND THEREFORE MUST BE UPDATED
!    IN THE DYNAMICAL CORE
!-----------------------------------------------------------------------

    time = 0.d0

    ! Zonal Velocity

    u = u0*cos(lat)

    ! Meridional Velocity

!************   
! change in version 5: multiply v and w by rho0/rho 
!************

    v = -(rho0/rho) * (a*w0*pi)/(k*ztop) *cos(lat)*sin(k*lat)*cos(pi*height/ztop)*cos(pi*time/tau)

    ! Vertical Velocity - can be changed to vertical pressure velocity by 
    ! omega = -g*rho*w

    w = (rho0/rho) *(w0/k)*(-2.d0*sin(k*lat)*sin(lat) + k*cos(lat)*cos(k*lat)) &
        *sin(pi*height/ztop)*cos(pi*time/tau)


!-----------------------------------------------------------------------
!     initialize Q, set to zero 
!-----------------------------------------------------------------------

    q = 0.d0

!-----------------------------------------------------------------------
!     initialize tracers
!-----------------------------------------------------------------------

    ! Tracer 1 - Layer

    if (height .lt. z2 .and. height .gt. z1) then
    
        q1 = 0.5d0 * (1.d0 + cos( 2.d0*pi*(height-z0)/(z2-z1) ) )

    else

        q1 = 0.d0

    endif

END SUBROUTINE test1_advection_hadley



!==========================================================================================
! TEST CASE 13 - HORIZONTAL ADVECTION OF THIN CLOUD-LIKE TRACERS IN THE PRESENCE OF OROGRAPHY
!==========================================================================================

SUBROUTINE test1_advection_orography (lon,lat,p,z,zcoords,cfv,hybrid_eta,hyam,hybm,gc,u,v,w,t,phis,ps,rho,q,q1,q2,q3,q4)

IMPLICIT NONE
!-----------------------------------------------------------------------
!     input/output params parameters at given location
!-----------------------------------------------------------------------

    real(8), intent(in)  :: lon, &      ! Longitude (radians)
                lat, &      ! Latitude (radians)
                z, &        ! Height (m)
                hyam, &     ! A coefficient for hybrid-eta coordinate, at model level midpoint
                hybm, &     ! B coefficient for hybrid-eta coordinate, at model level midpoint
                gc      ! bar{z} for Gal-Chen coordinate 

    logical, intent(in)  :: hybrid_eta      ! flag to indicate whether the hybrid sigma-p (eta) coordinate is used
                                                ! if set to .true., then the pressure will be computed via the 
                                                !    hybrid coefficients hyam and hybm, they need to be initialized
                                                ! if set to .false. (for pressure-based models): the pressure is already pre-computed
                                                !    and is an input value for this routine 
                                                ! for height-based models: pressure will always be computed based on the height and
                                                !    hybrid_eta is not used

    ! Note that we only use hyam and hybm for the hybrid-eta coordiantes, and we only use
    ! gc for the Gal-Chen coordinates. If not required then they become dummy variables

    real(8), intent(inout) :: p     ! Pressure  (Pa)
                
    integer,  intent(in) :: zcoords     ! 0 or 1 see below
    integer,  intent(in) :: cfv         ! 0, 1 or 2 see below

    real(8), intent(out) :: u, &        ! Zonal wind (m s^-1)
                v, &        ! Meridional wind (m s^-1)
                w, &        ! Vertical Velocity (m s^-1)
                t, &        ! Temperature (K)
                phis, &     ! Surface Geopotential (m^2 s^-2)
                ps, &       ! Surface Pressure (Pa)
                rho, &      ! density (kg m^-3)
                q, &        ! Specific Humidity (kg/kg)
                q1, &       ! Tracer q1 (kg/kg)
                q2, &       ! Tracer q2 (kg/kg)
                q3, &       ! Tracer q3 (kg/kg)
                q4      ! Tracer q4 (kg/kg)

    ! if zcoords = 1, then we use z and output p
    ! if zcoords = 0, then we use p

    ! if cfv = 0 we assume that our horizontal velocities are not coordinate following
    ! if cfv = 1 then our velocities follow hybrid eta coordinates and we need to specify w
    ! if cfv = 2 then our velocities follow Gal-Chen coordinates and we need to specify w

    ! In hybrid-eta coords: p = hyam p0 + hybm ps
    ! In Gal-Chen coords: z = zs + (gc/ztop)*(ztop - zs)

        ! if other orography-following coordinates are used, the w wind needs to be newly derived for them

!-----------------------------------------------------------------------
!     test case parameters
!----------------------------------------------------------------------- 
    real(8), parameter ::   tau     = 12.d0 * 86400.d0, &   ! period of motion 12 days (s)
                    u0      = 2.d0*pi*a/tau,    &   ! Velocity Magnitude (m/s)
                                t0      = 300.d0,       &   ! temperature (K)
                                h       = rd * t0 / g,      &   ! scale height (m)
                                alpha   = pi/6.d0,      &   ! rotation angle (radians), 30 degrees   
                                lambdam = 3.d0*pi/2.d0,     &   ! mountain longitude center point (radians)   
                                phim    = 0.d0,         &   ! mountain latitude center point (radians)    
                                h0      = 2000.d0,      &   ! peak height of the mountain range (m)
                                rm      = 3.d0*pi/4.d0,     &   ! mountain radius (radians)
                                zetam   = pi/16.d0,     &   ! mountain oscillation half-width (radians) 
                                lambdap = pi/2.d0,      &   ! cloud-like tracer longitude center point (radians) 
                                phip    = 0.d0,         &   ! cloud-like tracer latitude center point (radians)    
                                rp      = pi/4.d0,      &   ! cloud-like tracer radius (radians)    
                                zp1     = 3050.d0,      &   ! midpoint of first (lowermost) tracer (m)           
                                zp2     = 5050.d0,      &   ! midpoint of second tracer (m)          
                                zp3     = 8200.d0,      &   ! midpoint of third (topmost) tracer (m)    
                                dzp1    = 1000.d0,      &   ! thickness of first (lowermost) tracer (m)           
                                dzp2    = 1000.d0,      &   ! thickness of second tracer (m)
                                dzp3    = 400.d0,       &   ! thickness of third (topmost) tracer (m)       
                                ztop    = 12000.d0          ! model top (m)         
                            
      real(8) :: height                         ! Model level heights (m)
      real(8) :: r                          ! Great circle distance (radians)
      real(8) :: rz                         ! height differences 
      real(8) :: zs                         ! Surface elevation (m)



!-----------------------------------------------------------------------
!    PHIS (surface geopotential)
!-----------------------------------------------------------------------
      
    r = acos( sin(phim)*sin(lat) + cos(phim)*cos(lat)*cos(lon - lambdam) )

    if (r .lt. rm) then

        zs = (h0/2.d0)*(1.d0+cos(pi*r/rm))*cos(pi*r/zetam)**2.d0

    else

        zs = 0.d0

    endif

    phis = g*zs

!-----------------------------------------------------------------------
!    PS (surface pressure)
!-----------------------------------------------------------------------

    ps = p0 * exp(-zs/h)


!-----------------------------------------------------------------------
!    HEIGHT AND PRESSURE
!-----------------------------------------------------------------------

    ! Height and pressure are aligned (p = p0 exp(-z/H))

    if (zcoords .eq. 1) then

        height = z
        p = p0 * exp(-z/h)

    else

                if (hybrid_eta) p = hyam*p0 + hybm*ps   ! compute the pressure based on the surface pressure and hybrid coefficients
        height = h * log(p0/p)

    endif

!-----------------------------------------------------------------------
!    THE VELOCITIES ARE TIME INDEPENDENT 
!-----------------------------------------------------------------------

    ! Zonal Velocity

    u = u0*(cos(lat)*cos(alpha)+sin(lat)*cos(lon)*sin(alpha))

    ! Meridional Velocity

    v = -u0*(sin(lon)*sin(alpha))

!-----------------------------------------------------------------------
!    TEMPERATURE IS CONSTANT 300 K
!-----------------------------------------------------------------------

    t = t0

!-----------------------------------------------------------------------
!    RHO (density)
!-----------------------------------------------------------------------

    rho = p/(rd*t)

!-----------------------------------------------------------------------
!     initialize Q, set to zero 
!-----------------------------------------------------------------------

    q = 0.d0

!-----------------------------------------------------------------------
!    VERTICAL VELOCITY IS TIME INDEPENDENT 
!-----------------------------------------------------------------------

    ! Vertical Velocity - can be changed to vertical pressure velocity by 
    ! omega = -(g*p)/(Rd*T0)*w

    ! NOTE that if orography-following coordinates are used then the vertical 
    ! velocity needs to be translated into the new coordinate system due to
    ! the variation of the height along coordinate surfaces
    ! See section 1.3 and the appendix of the test case document

    if (cfv .eq. 0) then

        ! if the horizontal velocities do not follow the vertical coordinate

        w = 0.d0

    elseif (cfv .eq. 1) then

        ! if the horizontal velocities follow hybrid eta coordinates then
        ! the perceived vertical velocity is

        call test1_advection_orograph_hybrid_eta_velocity(w)

    elseif (cfv .eq. 2) then

        ! if the horizontal velocities follow Gal Chen coordinates then
        ! the perceived vertical velocity is    

        call test1_advection_orograph_gal_chen_velocity(w)  

!        else
!               compute your own vertical velocity if other orography-following 
!               vertical coordinate is used
!
    endif



!-----------------------------------------------------------------------
!     initialize tracers
!-----------------------------------------------------------------------

    ! Tracer 1 - Cloud Layer
      
    r = acos( sin(phip)*sin(lat) + cos(phip)*cos(lat)*cos(lon - lambdap) )

    rz = abs(height - zp1)

    if (rz .lt. 0.5d0*dzp1 .and. r .lt. rp) then

        q1 = 0.25d0*(1.d0+cos(2.d0*pi*rz/dzp1))*(1.d0+cos(pi*r/rp))

    else

        q1 = 0.d0

    endif

    rz = abs(height - zp2)

    if (rz .lt. 0.5d0*dzp2 .and. r .lt. rp) then

        q2 = 0.25d0*(1.d0+cos(2.d0*pi*rz/dzp2))*(1.d0+cos(pi*r/rp))

    else

        q2 = 0.d0

    endif

    rz = abs(height - zp3)

    if (rz .lt. 0.5d0*dzp3 .and. r .lt. rp) then

        q3 = 1.d0

    else

        q3 = 0.d0

    endif

    q4 = q1 + q2 + q3


    CONTAINS

    !-----------------------------------------------------------------------
    !    SUBROUTINE TO CALCULATE THE PERCEIVED VERTICAL VELOCITY 
    !           UNDER HYBRID-ETA COORDINATES
    !-----------------------------------------------------------------------

    SUBROUTINE test1_advection_orograph_hybrid_eta_velocity(w)
    IMPLICIT NONE
    real(8), intent(out) :: w

    real(8) ::  press, &        ! hyam *p0 + hybm *ps
            r, &            ! Great Circle Distance
            dzsdx, &        ! Part of surface height derivative
            dzsdlambda, &       ! Derivative of zs w.r.t lambda
            dzsdphi, &      ! Derivative of zs w.r.t phi
            dzdlambda, &        ! Derivative of z w.r.t lambda
            dzdphi, &       ! Derivative of z w.r.t phi
            dpsdlambda, &       ! Derivative of ps w.r.t lambda
            dpsdphi         ! Derivative of ps w.r.t phi

        ! Calculate pressure and great circle distance to mountain center

        press = hyam*p0 + hybm*ps

        r = acos( sin(phim)*sin(lat) + cos(phim)*cos(lat)*cos(lon - lambdam) )

        ! Derivatives of surface height

        if (r .lt. rm) then
            dzsdx = -h0*pi/(2.d0*rm)*sin(pi*r/rm)*cos(pi*r/zetam)**2 - &
                (h0*pi/zetam)*(1.d0+cos(pi*r/rm))*cos(pi*r/zetam)*sin(pi*r/zetam)
        else
            dzsdx = 0.d0
        endif

        ! Prevent division by zero

        if (1.d0-cos(r)**2 .gt. 0.d0) then
            dzsdlambda = dzsdx * (cos(phim)*cos(lat)*sin(lon-lambdam)) &
                    /sqrt(1.d0-cos(r)**2)
            dzsdphi    = dzsdx * (-sin(phim)*cos(lat) + cos(phim)*sin(lat)*cos(lon-lambdam)) & 
                    /sqrt(1.d0-cos(r)**2)
        else
            dzsdlambda = 0.d0
            dzsdphi    = 0.d0
        endif

        ! Derivatives of surface pressure

        dpsdlambda = -(g*p0/(rd*t0))*exp(-g*zs/(rd*t0))*dzsdlambda
        dpsdphi    = -(g*p0/(rd*t0))*exp(-g*zs/(rd*t0))*dzsdphi

        ! Derivatives of coordinate height

        dzdlambda = -(rd*t0/(g*press))*hybm*dpsdlambda
        dzdphi    = -(rd*t0/(g*press))*hybm*dpsdphi

        ! Prevent division by zero

        if (abs(lat) .lt. pi/2.d0) then
            w = - (u/(a*cos(lat)))*dzdlambda - (v/a)*dzdphi
        else
            w = 0.d0
        endif

    END SUBROUTINE test1_advection_orograph_hybrid_eta_velocity



    !-----------------------------------------------------------------------
    !    SUBROUTINE TO CALCULATE THE PERCEIVED VERTICAL VELOCITY 
    !           UNDER GAL-CHEN COORDINATES
    !-----------------------------------------------------------------------

    SUBROUTINE test1_advection_orograph_gal_chen_velocity(w)
    IMPLICIT NONE
    real(8), intent(out) :: w

    real(8) ::  r, &            ! Great Circle Distance
            dzsdx, &        ! Part of surface height derivative
            dzsdlambda, &       ! Derivative of zs w.r.t lambda
            dzsdphi, &      ! Derivative of zs w.r.t phi
            dzdlambda, &        ! Derivative of z w.r.t lambda
            dzdphi          ! Derivative of z w.r.t phi

        ! Calculate great circle distance to mountain center

        r = acos( sin(phim)*sin(lat) + cos(phim)*cos(lat)*cos(lon - lambdam) )

        ! Derivatives of surface height

        if (r .lt. rm) then
            dzsdx = -h0*pi/(2.d0*rm)*sin(pi*r/rm)*cos(pi*r/zetam)**2 - &
                (h0*pi/zetam)*(1.d0+cos(pi*r/rm))*cos(pi*r/zetam)*sin(pi*r/zetam)
        else
            dzsdx = 0.d0
        endif

        ! Prevent division by zero

        if (1.d0-cos(r)**2 .gt. 0.d0) then
            dzsdlambda = dzsdx * (cos(phim)*cos(lat)*sin(lon-lambdam)) &
                    /sqrt(1.d0-cos(r)**2)
            dzsdphi    = dzsdx * (-sin(phim)*cos(lat) + cos(phim)*sin(lat)*cos(lon-lambdam)) & 
                    /sqrt(1.d0-cos(r)**2)
        else
            dzsdlambda = 0.d0
            dzsdphi    = 0.d0
        endif

        ! Derivatives of coordinate height

        dzdlambda = (1.d0-gc/ztop)*dzsdlambda
        dzdphi    = (1.d0-gc/ztop)*dzsdphi

        ! Prevent division by zero

        if (abs(lat) .lt. pi/2.d0) then
            w = - (u/(a*cos(lat)))*dzdlambda - (v/a)*dzdphi
        else
            w = 0.d0
        endif

    END SUBROUTINE test1_advection_orograph_gal_chen_velocity

END SUBROUTINE test1_advection_orography




!==========================================================================================
! TEST CASE 2X - IMPACT OF OROGRAPHY ON A NON-ROTATING PLANET
!==========================================================================================
! The tests in section 2-x examine the impact of 3D Schaer-like circular mountain profiles on an 
! atmosphere at rest (2-0), and on flow fields with wind shear (2-1) and without vertical wind shear (2-2).  
! A non-rotating planet is used for all configurations. Test 2-0 is conducted on an unscaled regular-size 
! planet and primarily examines the accuracy of the pressure gradient calculation in a steady-state 
! hydrostatically-balanced atmosphere at rest. This test is especially appealing for models with 
! orography-following vertical coordinates. It increases the complexity of test 1-3, that investigated 
! the impact of the same Schaer-type orographic profile on the accuracy of purely-horizontal passive 
! tracer advection.
!
! Tests 2-1 and 2-2 increase the complexity even further since non-zero flow fields are now prescribed 
! with and without vertical wind shear. In order to trigger non-hydrostatic responses the two tests are 
! conducted on a reduced-size planet with reduction factor $X=500$ which makes the horizontal and 
! vertical grid spacing comparable. This test clearly discriminates between non-hydrostatic and hydrostatic 
! models since the expected response is in the non-hydrostatic regime. Therefore, the flow response is 
! captured differently by hydrostatic models.




!=========================================================================
! Test 2-0:  Steady-State Atmosphere at Rest in the Presence of Orography
!=========================================================================
SUBROUTINE test2_steady_state_mountain (lon,lat,p,z,zcoords,hybrid_eta,hyam,hybm,u,v,w,t,phis,ps,rho,q)

IMPLICIT NONE
!-----------------------------------------------------------------------
!     input/output params parameters at given location
!-----------------------------------------------------------------------

    real(8), intent(in)  :: lon, &      ! Longitude (radians)
                lat, &      ! Latitude (radians)
                z, &        ! Height (m)
                hyam, &     ! A coefficient for hybrid-eta coordinate, at model level midpoint
                hybm        ! B coefficient for hybrid-eta coordinate, at model level midpoint

    logical, intent(in)  :: hybrid_eta      ! flag to indicate whether the hybrid sigma-p (eta) coordinate is used
                                                ! if set to .true., then the pressure will be computed via the 
                                                !    hybrid coefficients hyam and hybm, they need to be initialized
                                                ! if set to .false. (for pressure-based models): the pressure is already pre-computed
                                                !    and is an input value for this routine 
                                                ! for height-based models: pressure will always be computed based on the height and
                                                !    hybrid_eta is not used

    real(8), intent(inout) :: p     ! Pressure  (Pa)
                
    integer,  intent(in) :: zcoords     ! 0 or 1 see below

    real(8), intent(out) :: u, &        ! Zonal wind (m s^-1)
                v, &        ! Meridional wind (m s^-1)
                w, &        ! Vertical Velocity (m s^-1)
                t, &        ! Temperature (K)
                phis, &     ! Surface Geopotential (m^2 s^-2)
                ps, &       ! Surface Pressure (Pa)
                rho, &      ! density (kg m^-3)
                q       ! Specific Humidity (kg/kg)

    ! if zcoords = 1, then we use z and output p
    ! if zcoords = 0, then we compute or use p
        !
    ! In hybrid-eta coords: p = hyam p0 + hybm ps
        !
        ! The grid-point based initial data are computed in this routine. 

!-----------------------------------------------------------------------
!     test case parameters
!----------------------------------------------------------------------- 
    real(8), parameter ::   T0      = 300.d0,       &   ! temperature (K)
                                gamma   = 0.0065d0,             &       ! temperature lapse rate (K/m)      
                                lambdam = 3.d0*pi/2.d0,     &   ! mountain longitude center point (radians)   
                                phim    = 0.d0,         &   ! mountain latitude center point (radians)    
                                h0      = 2000.d0,      &   ! peak height of the mountain range (m)
                                rm      = 3.d0*pi/4.d0,     &   ! mountain radius (radians)
                                zetam   = pi/16.d0,     &   ! mountain oscillation half-width (radians) 
                                ztop    = 12000.d0          ! model top (m)         
                            
      real(8) :: height                         ! Model level heights (m)
      real(8) :: r                          ! Great circle distance (radians)
      real(8) :: zs                         ! Surface elevation (m)
      real(8) :: exponent                                               ! exponent: g/(Rd * gamma)
      real(8) :: exponent_rev                                           ! reversed exponent


!-----------------------------------------------------------------------
!    compute exponents 
!-----------------------------------------------------------------------
     exponent     = g/(rd*gamma)
     exponent_rev = 1.d0/exponent

!-----------------------------------------------------------------------
!    PHIS (surface geopotential)
!-----------------------------------------------------------------------
      
    r = acos( sin(phim)*sin(lat) + cos(phim)*cos(lat)*cos(lon - lambdam) )

    if (r .lt. rm) then

        zs = (h0/2.d0)*(1.d0+cos(pi*r/rm))*cos(pi*r/zetam)**2.d0   ! mountain height

    else

        zs = 0.d0

    endif

    phis = g*zs

!-----------------------------------------------------------------------
!    PS (surface pressure)
!-----------------------------------------------------------------------

    ps = p0 * (1.d0 - gamma/t0*zs)**exponent


!-----------------------------------------------------------------------
!    HEIGHT AND PRESSURE
!-----------------------------------------------------------------------

    ! Height and pressure are aligned (p = p0 * (1.d0 - gamma/T0*z)**exponent)

    if (zcoords .eq. 1) then

        height = z
        p = p0 * (1.d0 - gamma/t0*z)**exponent

    else

                if (hybrid_eta) p = hyam*p0 + hybm*ps              ! compute the pressure based on the surface pressure and hybrid coefficients
        height = t0/gamma * (1.d0 - (p/p0)**exponent_rev)  ! compute the height at this pressure

    endif

!-----------------------------------------------------------------------
!    THE VELOCITIES ARE ZERO (STATE AT REST)
!-----------------------------------------------------------------------

    ! Zonal Velocity

    u = 0.d0

    ! Meridional Velocity

    v = 0.d0

        ! Vertical Velocity

        w = 0.d0

!-----------------------------------------------------------------------
!    TEMPERATURE WITH CONSTANT LAPSE RATE
!-----------------------------------------------------------------------

    t = t0 - gamma*height

!-----------------------------------------------------------------------
!    RHO (density)
!-----------------------------------------------------------------------

    rho = p/(rd*t)

!-----------------------------------------------------------------------
!     initialize Q, set to zero 
!-----------------------------------------------------------------------

    q = 0.d0

END SUBROUTINE test2_steady_state_mountain



!=====================================================================================
! Tests 2-1 and 2-2:  Non-hydrostatic Mountain Waves over a Schaer-type Mountain
!=====================================================================================

SUBROUTINE test2_schaer_mountain (lon,lat,p,z,zcoords,hybrid_eta,hyam,hybm,shear,u,v,w,t,phis,ps,rho,q)

IMPLICIT NONE
!-----------------------------------------------------------------------
!     input/output params parameters at given location
!-----------------------------------------------------------------------

    real(8), intent(in)  :: lon, &      ! Longitude (radians)
                lat, &      ! Latitude (radians)
                z,   &      ! Height (m)
                hyam, &     ! A coefficient for hybrid-eta coordinate, at model level midpoint
                hybm        ! B coefficient for hybrid-eta coordinate, at model level midpoint

    logical, intent(in)  :: hybrid_eta      ! flag to indicate whether the hybrid sigma-p (eta) coordinate is used
                                                ! if set to .true., then the pressure will be computed via the 
                                                !    hybrid coefficients hyam and hybm, they need to be initialized
                                                ! if set to .false. (for pressure-based models): the pressure is already pre-computed
                                                !    and is an input value for this routine 
                                                ! for height-based models: pressure will always be computed based on the height and
                                                !    hybrid_eta is not used

    real(8), intent(inout) :: p     ! Pressure  (Pa)
                

    integer,  intent(in) :: zcoords, &  ! 0 or 1 see below
                shear       ! 0 or 1 see below

    real(8), intent(out) :: u, &        ! Zonal wind (m s^-1)
                v, &        ! Meridional wind (m s^-1)
                w, &        ! Vertical Velocity (m s^-1)
                t, &        ! Temperature (K)
                phis, &     ! Surface Geopotential (m^2 s^-2)
                ps, &       ! Surface Pressure (Pa)
                rho, &      ! density (kg m^-3)
                q       ! Specific Humidity (kg/kg)

    ! if zcoords = 1, then we use z and output p
    ! if zcoords = 0, then we either compute or use p

    ! if shear = 1, then we use shear flow
    ! if shear = 0, then we use constant u

!-----------------------------------------------------------------------
!     test case parameters
!----------------------------------------------------------------------- 
    real(8), parameter ::   X       = 500.d0,       &   ! Reduced Earth reduction factor
                    om      = 0.d0,         &   ! Rotation Rate of Earth
                                as      = a/x,          &   ! New Radius of small Earth     
                    ueq     = 20.d0,        &   ! Reference Velocity 
                                teq     = 300.d0,       &   ! Temperature at Equator    
                    peq     = 100000.d0,        &   ! Reference PS at Equator
                                ztop    = 30000.d0,     &   ! Model Top       
                lambdac = pi/4.d0,      &   ! Lon of Schar Mountain Center
                phic    = 0.d0,         &   ! Lat of Schar Mountain Center
                h0      = 250.d0,       &   ! Height of Mountain
                d       = 5000.d0,      &   ! Mountain Half-Width
                xi      = 4000.d0,      &   ! Mountain Wavelength
                cs      = 0.00025d0             ! Wind Shear (shear=1)
                            
      real(8) :: height                         ! Model level heights
      real(8) :: sin_tmp, cos_tmp                   ! Calculation of great circle distance
      real(8) :: r                          ! Great circle distance
      real(8) :: zs                         ! Surface height
      real(8) :: c                          ! Shear

!-----------------------------------------------------------------------
!    PHIS (surface geopotential)
!-----------------------------------------------------------------------

    sin_tmp = sin(lat) * sin(phic)
    cos_tmp = cos(lat) * cos(phic)
    
    ! great circle distance with 'a/X'  

    r  = as * acos(sin_tmp + cos_tmp*cos(lon-lambdac))     
    zs   = h0 * exp(-(r**2)/(d**2))*(cos(pi*r/xi)**2)
    phis = g*zs

!-----------------------------------------------------------------------
!    SHEAR FLOW OR CONSTANT FLOW
!-----------------------------------------------------------------------

    if (shear .eq. 1) then

        c = cs

    else

        c = 0.d0

    endif

!-----------------------------------------------------------------------
!    TEMPERATURE 
!-----------------------------------------------------------------------

    t = teq *(1.d0 - (c*ueq*ueq/(g))*(sin(lat)**2) )

!-----------------------------------------------------------------------
!    PS (surface pressure)
!-----------------------------------------------------------------------

    ps = peq*exp( -(ueq*ueq/(2.d0*rd*teq))*(sin(lat)**2) - phis/(rd*t)    )

!-----------------------------------------------------------------------
!    HEIGHT AND PRESSURE 
!-----------------------------------------------------------------------

    if (zcoords .eq. 1) then

        height = z
        p = peq*exp( -(ueq*ueq/(2.d0*rd*teq))*(sin(lat)**2) - g*height/(rd*t)    )

    else
                if (hybrid_eta) p = hyam*p0 + hybm*ps ! compute the pressure based on the surface pressure and hybrid coefficients
        height = (rd*t/(g))*log(peq/p) - (t*ueq*ueq/(2.d0*teq*g))*(sin(lat)**2)

    endif

!-----------------------------------------------------------------------
!    THE VELOCITIES
!-----------------------------------------------------------------------

    ! Zonal Velocity

    u = ueq * cos(lat) * sqrt( (2.d0*teq/(t))*c*height + t/(teq) )

    ! Meridional Velocity

    v = 0.d0

    ! Vertical Velocity = Vertical Pressure Velocity = 0

    w = 0.d0

!-----------------------------------------------------------------------
!    RHO (density)
!-----------------------------------------------------------------------

    rho = p/(rd*t)

!-----------------------------------------------------------------------
!     initialize Q, set to zero 
!-----------------------------------------------------------------------

    q = 0.d0

END SUBROUTINE test2_schaer_mountain










!==========================================================================================
! TEST CASE 3 - GRAVITY WAVES
!==========================================================================================

! The non-hydrostatic gravity wave test examines the response of models to short time-scale wavemotion
! triggered by a localized perturbation. The formulation presented in this document is new,
! but is based on previous approaches by Skamarock et al. (JAS 1994), Tomita and Satoh (FDR 2004), and
! Jablonowski et al. (NCAR Tech Report 2008) 


!==========
! Test 3-1
!==========
SUBROUTINE test3_gravity_wave (lon,lat,p,z,zcoords,u,v,w,t,phis,ps,rho,q)

IMPLICIT NONE
!-----------------------------------------------------------------------
!     input/output params parameters at given location
!-----------------------------------------------------------------------

    real(8), intent(in)  :: lon, &      ! Longitude (radians)
                lat, &      ! Latitude (radians)
                z       ! Height (m)

    real(8), intent(inout) :: p     ! Pressure  (Pa)
                

    integer,  intent(in) :: zcoords     ! 0 or 1 see below

    real(8), intent(out) :: u, &        ! Zonal wind (m s^-1)
                v, &        ! Meridional wind (m s^-1)
                w, &        ! Vertical Velocity (m s^-1)
                t, &        ! Temperature (K)
                phis, &     ! Surface Geopotential (m^2 s^-2)
                ps, &       ! Surface Pressure (Pa)
                rho, &      ! density (kg m^-3)
                q       ! Specific Humidity (kg/kg)

    ! if zcoords = 1, then we use z and output z
    ! if zcoords = 0, then we use p

!-----------------------------------------------------------------------
!     test case parameters
!----------------------------------------------------------------------- 
    real(8), parameter ::   X       = 125.d0,       &   ! Reduced Earth reduction factor
                    om      = 0.d0,         &   ! Rotation Rate of Earth
                                as      = a/x,          &   ! New Radius of small Earth     
                    u0      = 20.d0,        &   ! Reference Velocity 
                                teq     = 300.d0,       &   ! Temperature at Equator    
                    peq     = 100000.d0,        &   ! Reference PS at Equator
                                ztop    = 10000.d0,     &   ! Model Top       
                lambdac = 2.d0*pi/3.d0,     &   ! Lon of Pert Center
                d       = 5000.d0,      &   ! Width for Pert
                phic    = 0.d0,         &   ! Lat of Pert Center
                delta_theta = 1.d0,         &   ! Max Amplitude of Pert
                lz      = 20000.d0,         &   ! Vertical Wavelength of Pert
                n       = 0.01d0,       &   ! Brunt-Vaisala frequency
                n2      = n*n,          &   ! Brunt-Vaisala frequency Squared
                bigg    = (g*g)/(n2*cp)         ! Constant
                            
      real(8) :: height                         ! Model level height
      real(8) :: sin_tmp, cos_tmp                   ! Calculation of great circle distance
      real(8) :: r, s                           ! Shape of perturbation
      real(8) :: TS                             ! Surface temperature
      real(8) :: t_mean, t_pert                     ! Mean and pert parts of temperature
      real(8) :: theta_pert                     ! Pot-temp perturbation

!-----------------------------------------------------------------------
!    THE VELOCITIES
!-----------------------------------------------------------------------

    ! Zonal Velocity 

    u = u0 * cos(lat)

    ! Meridional Velocity

    v = 0.d0

    ! Vertical Velocity = Vertical Pressure Velocity = 0

    w = 0.d0

!-----------------------------------------------------------------------
!    PHIS (surface geopotential)
!-----------------------------------------------------------------------

    phis = 0.d0

!-----------------------------------------------------------------------
!    SURFACE TEMPERATURE 
!-----------------------------------------------------------------------

    ts = bigg + (teq-bigg)*exp( -(u0*n2/(4.d0*g*g))*(u0+2.d0*om*as)*(cos(2.d0*lat)-1.d0)    ) 

!-----------------------------------------------------------------------
!    PS (surface pressure)
!-----------------------------------------------------------------------

    ps = peq*exp( (u0/(4.0*bigg*rd))*(u0+2.0*om*as)*(cos(2.0*lat)-1.0)  ) &
        * (ts/teq)**(cp/rd)

!-----------------------------------------------------------------------
!    HEIGHT AND PRESSURE AND MEAN TEMPERATURE
!-----------------------------------------------------------------------

    if (zcoords .eq. 1) then

        height = z
        p = ps*( (bigg/ts)*exp(-n2*height/g)+1.d0 - (bigg/ts)  )**(cp/rd)

    else

        height = (-g/n2)*log( (ts/bigg)*( (p/ps)**(rd/cp) - 1.d0  ) + 1.d0 )

    endif

    t_mean = bigg*(1.d0 - exp(n2*height/g))+ ts*exp(n2*height/g)

!-----------------------------------------------------------------------
!    rho (density), unperturbed using the background temperature t_mean
!-----------------------------------------------------------------------

!***********
!       change in version 3: density is now initialized with unperturbed background temperature,
!                            temperature perturbation is added afterwards
!***********
    rho = p/(rd*t_mean)

!-----------------------------------------------------------------------
!    POTENTIAL TEMPERATURE PERTURBATION, 
!    here: converted to temperature and added to the temperature field
!    models with a prognostic potential temperature field can utilize
!    the potential temperature perturbation theta_pert directly and add it
!    to the background theta field (not included here)
!-----------------------------------------------------------------------

    sin_tmp = sin(lat) * sin(phic)
    cos_tmp = cos(lat) * cos(phic)

        ! great circle distance with 'a/X' 

    r  = as * acos(sin_tmp + cos_tmp*cos(lon-lambdac)) 

    s = (d**2)/(d**2 + r**2)

    theta_pert = delta_theta*s*sin(2.d0*pi*height/lz)

    t_pert = theta_pert*(p/p0)**(rd/cp)

    t = t_mean + t_pert

!-----------------------------------------------------------------------
!     initialize Q, set to zero 
!-----------------------------------------------------------------------

    q = 0.d0

END SUBROUTINE test3_gravity_wave


END MODULE dcmip_initial_conditions_test_1_2_3